Under the Spell of the Gauge Principle (Advanced Series in Mathematical Physics, Vol 19)

UNDER THE SPELL OF THE

GAUGE PRINCIPLE

ADVANCED SERIES IN MATH~MATICAL PHYSICS

Editors-in4 harge H Araki (RIMS, Kyoto) V G Kac (Ml7) D H Phong (Columbia University) S-T Yau (Harvard University)

Associate Editors L Alvarez-Game (CERA!) J P Bourgu~gnon ( fcoie Po/~echni~ue, Palaiseau) T Eguchi (University of Tokyo) B Julia (CNRS, Paris) F Wilczek (lnstitute for Advanced Study, Princeton)

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Mathematical Aspects of String Theory edited by S-T Yau Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras by V G Kac and A K Raina Kac-Moody and Virasoro Algebras: A Reprint Volume for Physicists edited by P Goddard and D Olive Harmonic Mappings, Twistors and 0-Models edited by P Gauduchon Geometric Phases in Physics edited by A Shapere and F Witczek Infinite Dimensional Lie Algebras and Groups edited by V Kac Introduction to String Field Theory by W Siege/ Braid Group, Knot Theory and Statistical Mechanics edited by C N Yang and M L Ge Yang-Baxter Equations in Integrable Systems edited by M Jimbo New Developments in the Theory of Knots edited by T Kohno Soliton Equations and Hamiltonian Systems by L A Dickey Form Factors in Completely Integrable Models of Quantum Field Theory by F A Smirnov ~ o n - P e ~ u ~ a t i v e Quantum Field Theory - Mathema~cal Aspects and Applications Selected Papers of Jurg Friihlich Infinite Analysis - Proceedings of the RIMS Research Project 1991 edited by A Tsucbiya, T Eguchi and M Jimbo Braid Group, Knot Theory and Statistical Mechanics (11) edited by C N Yang and M L Ge Exactly Solvable Models and Strongly Correlated Electrons edited by V Korepin and F N L Ebler State of Matter edited by M Aizenman and H Araki

Advanced Series in Mathematical Physics voi. 19

UNDER THE SPELL OF THE

GAUGE PRINCIPLE

G. ' t H o oft i ~ s t ~ t ~ t e for Theoretical Physics

University of Utrecht The ~ e t h e r l a ~ ~ s

World Scientific New Jersey London ffong Kong

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The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to reproduce the selected reprints found in this volume: The American Physical Society (Phys. Rev. 0); Springer-Verlag (Trends in EZemen- fury Particle 7'Fyory); Uppsala University, Faculty of Science (Reporfs of the CELSIUWNNE Lectures).

While every effoa has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.

UNDER THE SPELL OF THE GAUGE PRINCIPLE

Copyright 0 1994 by World Scientific Publishing Co. Re. ttd.

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To my mother

To the memory of my father

PREFACE

Our view of the world that we live in has changed. And it continues to change, as we learn more. What became abundantly clear in the course of the twentieth century is that the terminology most suited to describe the Laws of Physics is no longer that of everyday life. Instead, we have learned to create a new language, the language of mathematics, to describe elementary particles and their constituents. To facilitate this work we often introduce familiar words from the ordinary world to describe some law, principle or quantity, such as “energy”, “force”, “matter”. But then these are given a meaning that is much more accurately defined than in our social lives. Sometimes the relation with the conventional meaning is remote, such as in “potential energy” and “action”. And sometimes they mean something altogether different, for instance “color”, or “charm”.

The notion of “gauge invariance” also originated from a concept in daily life: instruments can be “gauged” by comparing them with other instruments, but the way they work is independent from the way they are gauged. The notion has been redefined with mathematical precision for theories in physics. Gauge symmetries, both “global” (i.e. space-time independent) and “local” (space-time dependent) became key issues in elementary particle physics. This book contains a selection of my work of over twenty years of research. The guiding principles in this work were symmetry and elegance of the magnificent edifice that we call our universe. And the most important symmetry was gauge symmetry.

Tremendous successes have been achieved in the recent past by the application of advanced mathematics to physics. However, some of the papers published in recent physics journals seem to be doing nothing but preparing and extending fragile mathematical constructions without the slightest indication as to how these structures should be used to build a theory of physics. I think it is important always to keep in mind that the mathematics we use in trying to understand our world is just a tool, a language sharp and precise enough to help us where ordinary words fail. Mathematics can never replace a theory. And, although I love mathematics, none of my papers were intended to improve our understanding of pure mathematics.

vii

What I have always sought to do is quite the opposite. Using mathematics as a tool I have tried to identify the most urgent problems, the most baffling questions that axe standing in our way towards a better understanding of our physical world. And then the art is to select out of those the ones that are worth being further pursued by a theoretical physicist. More often than not I end up immersed in mathematical equations. The next question is then always: “HOW will these equations help me answer the question I started off with? How do I interpret my mathematical expressions?’) In this light one has to read and understand the articles reproduced in this book.

This book is a collection of what I consider to be the more salient chapters of my work. It is by no means meant to be complete. There axe several important subjects in quantum field theory on which I have also made investigations, in particular monopoles and statistics, instanton solutions, lattice theory) classical and quantum gravity, and fundamental issues in quantum mechanics. All these subjects were either too technical or too incomplete to be included in a review book such as this, but that does not mean that I would consider them to be less important. I also could have included more work that I did with my cO-authors, Martinus Veltman, Bernard de Wit, Peter Hasenfratz, Tevian Dray, Stanley Deser, Roman Jackiw, Karl Isler, Stilyan Kalitzin and others. In any case, if this book was supposed to be my “collected works” I sincerely hope it to be incomplete because I plan to continue my investigations.

It is impossible to produce important contributions to science without the insight, advice, help and support of numerous colleagues and friends. Among them, of come, the ccmuthors I just mentioned. Before Veltman became my teacher I learned a lot from N. G. van Kampen, who attempted to afflict me with his passion for precise arguments, discontent with half explanations, and total dedication to theoretical physics. Later I benefitted from so many inspiring discussions with equal-minded colleagues that I find it impossible to print out all their names. Often the importance of a discussion only became manifest much later at a time that I had forgotten who it was who told me.

I thank them all.

... Vlll

CONTENTS

PREFACE vii

Chapter 1. INTRODUCTION 1

Chapter 2. RENORMALIZATION OF GAUGE THEORIES 11 Introductions ..................................................................... 12 [2.1] “Gauge field theory”, in P m . of the Adriatic Summer Meeting on

Particle Physics, eds. M. Martinis, S. Pallua, N. Zovko, Rovinj,

[2.2] with M. Veltman, “DIAGRAMMAR”, CERN Rep. 73-9 (1973), reprinted in “Particle interactions at very high energies”, NATO Adv.

“Gauge theories with unified weak, electromagnetic and strong interactions”, E.P.S. Znt. Conf. on High Energy Physics, Palermo, Sicily, June 1978 ........................................................ 174

Yugoslavia, Sep.-Oct. 1973, pp. 321-332 ................................ 16

Study Znst. Series B, vol. 4b, pp. 177-322 ............................... 28 [2.3]

Chapter 3. THE RENORMALIZATION GROUP 205 Introduction .................................................................... 206 [3.1] The Renormalization Group in Quantum Field Theory, Eight

Graduate School Lectures, Doorwerth, Jan. 1988, unpublished ....... 208

Chapter 4. EXTENDED OBJECTS 251 Introductions ................................................................... 252 [4.1] “Extended objects in gauge field theories”, in Particles and Fields,

eds. D. H. Bod and A. N. Kamal, Plenum, New York, 1978, pp. 165-198 .............................................................. 254 “Magnetic monopoles in unified gauge theories”, Nucl. Phys. B79 (1974) 276-284 .................................................... 288

[4.2]

ix

Chapter 5. INSTANTONS Introductions [5.1]

[5.2]

[5.3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D14 (1976) 3432-3450 “HOW instantons solve the U(1) problem”, Phys. Rep. 142

“Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking”, In Recent Developments in Gauge Theories, Cargkse 1979, eds. G. ’t Hooft et aL, New York, 1990, Plenum, Lecture 111, reprinted in Dynamical Gauge Symmetry Breaking, A Collection of Reprints, eds. A. Farhi, and R. Jackiw, World Scientific, Singapore, 1982, pp. 345-367 . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . , . . . . . . . , . .

(1986) 357-387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . .

Chapter 6. PLANAR DIAGRAMS Introductions (6.11

. , . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . “Planar diagram field theories”, in Progress in Gauge Field Theorp, NATO Adv. Study Inst. Series, eds. G . ’t Hoot% et al., Plenum, 1984, pp. 271-335 “A two-dimensional model for mesons”, Nucl. Phys. B75 (1974) 461470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . Epilogue to the TwGDimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ..... ............ ........... .... , ........ ... [6.2]

Chapter 7. QUARK CONFINEMENT Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.11

(7.21

“Confinement and topology in non-Abelian gauge theories”, Acta Phys. Awtr . Suppl. 22 (1980) 531-586 “The confinement phenomenon in quantum field theory”, 1981 Cargtse Summer School Lecture Notes on Fundamental Intemctions, eds. M. Uvy and J.-L. Basdevant, NATO Adv. Study Inst. Series B: Phys., vol. 85, pp. 639-671 “Can we make sense out of “Quantum Chromodynamics”?” in The Whys of Subnuclear Physics, ed. A. Zichichi, Plenum, New York, pp. 943-971 .......................................................

.. . .. . . . . . . . . . .. . . . . . . .. .

...................... ...... [7.3)

Chapter 8. QUANTUM GRAVITY AND BLACK HOLES htroduct ions i8.11 “Quantum gravity”, in %rids in Elementary Particle Theory,

eds. H. Rollnik and K. Dietz, Springer-Verlag, 1975, pp. 92-113 (8.21 “Classical N-particle cosmology in 2 + 1 dimensions”, Class.

Quantum Gmv. 10 (1993) S79-S91 . . . . .. . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . [8.3] “On the quantum structure of a black hole”, Nucl. Phys. B256

(8.41 with T. Dray, “The gravitational shock wave of a massless particle”,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . .

. . .. .

(1985) 727-736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuc~. Phys. B253 (1985) 173-188 ..,..... . ............................

297 298

302

321

352

375 376

378

443 453

455 456

458

514

547

577 578

584

606

619

629

X

[8.5] “S-matrix theory for black holes”, Lectures given at the NATO Adv. Summer Inst. on New Symmetry Principles an Quantum Field Theory, eds. J. hohlich et al., Cargbse, 1992, Plenum, New York, pp. 275-294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Chapter 9. EPILOGUE 665 19.11

666 “Can the ultimate laws of nature be found?”, Celsius-Limb Lecture, Feb. 1992, Uppsala University, Sweden, pp. 1-12 . . . . . . . . . . . .

INDEX 678

xi

CHAPTER 1

INTRODUCTION

When I began my graduate work, renormalization was considered by many physicists to be a dirty word. We had to learn about it because, inspite of its ugliness, renormalization seemed to work, at least in one limited area of particle physics, that of quantum electrodynamics. &normalization was thought to be ugly because the procedure was ill-formulated and apparently an ad hoc and imprecise cure for a fundamental shortcoming in the quantum field theories of the day. But for those who analyzed the situation sdliciently carefully the physical reasoning behind renormalization wasn’t that ugly at all. Indeed it seemed to be a quite natural fact that the fundamental interaction constants one puts into a theory do not need to be identical to the chargea and m m one actually measures. If the theory is formulated on a very fine but discretised lattice instead of a continuum of space and time, the rele tion between the input parameters and the measured quantities will be quite direct. But now it 80 happens that if we take the limit where the mesh size approaches to zero, the input parameters needed to reproduce the measured quantities walk off the scale. This seems to be no disaster as long as we can’t measure these input, or “bare”, parameters directly.

Nowadays we know very well that such theories are not infinitely precise from a mathematical point of view, except possibly when the bare coupling strengths run to zero instead of infinity (such as in quantum chromodynamics). This latter, rare, property was not yet known about at the time, so in some sense the critics were right: our theories were mathematically empty. It is here that one needs more than just mathematical skill, namely insight into the requirements for our theories as needed to answer our questions. It would be too much to ask for infinite mathematical precision. If, in lieu of that, one could construct a theory that allows for an infinite crsgmptotic expansion in terms of some parameter that is measured by our experimental friends to be small, what may be achieved is a theory that is more than sufficiently precise to meet all our purposes.

My advisor and colleague, Tini Veltman, had an even more pragmatic view. He had reached the conclusion that Yang-Mills gauge field theories should be used in describing elementary particles by elimination: all the alternatives were uglier, and indeed quite a few aspects of the weak interactions pointed towards a renormalizable Yang-Mills theory for them. In any case, it was his perseverence that kept u8 on this track. Even though there was a moment that he was practically certain to have proven Yang-Mills theories to be nonrenormalizable (he had been studying the pure massive case and was not interested in the Higgs mechanism), he kept pointing towards the facts experimenters had told us about the weak interactions: Yang-Mills theories had to be right!

How could gauge invariance have anything to do with renormalization? To analyze this question it is crucial to understand the physical requirements of a particle theory at short distance scales. Suppose we want a theory that describes all known particles at large distance scales, and essentially nothing else at small distances. If many new particle types would arise at very small distances, that is, if we would have a large series of increasingly massive particles, then we would be stuck with a large, in general infinite number of uncontrollable parameters. Such a theory would have zero predictive power. Thus, as measured at a short distance scale, what we want is a theory containing only very light particles. It is all a question of economy; we are dealing here with in some sense the most efficient theories of nature, carrying the least possible amount of structure at small distances.

Ideally, we would like to have a system in which besides the mass parameters also all coupling parameters have the dimension of a negative power of a length, so that at small distance scales the particles are nearly free. In a renormalizable theory we settle for the next best thing, namely dimensionless couplings. One can then prove a very powerful theorem: the onlv perturbative field theories with massless particles and dimensionless couplings am theories featuring fvndamental particles with spin 0, 1 or 3. F’urthermore, the spin 1 fields have an invisible component: the longitudinal part decouples. All for the better, because the longitudinal parts mry essentially no kinetic terms, so they behave as ghosts, particles that would contribute in a disastrous way to the S matrix if they were real; unitarity and positivity of the energy would be destroyed.

Now suppose we do want the particles at large distance scales to have a finite maas. Massive particles with spin 1 have a larger number of physical degrees of freedom than massless ones. For massless spin 1 particles only two of the four components of their vector field A,($) are independently observable (the two possible helicities of the radiation field); massive particles on the other hand carry three independent degrees of freedom: the values m, = - 1 , O and +l. If we would simply add a tiny mass term to the field equations we would give the bad ghost component of the vector fields at all distance scales the status of a physically observable field. The value of this field component is ill-controlled at short distances. Its wild oscilla tions would effectively produce strong interactions, more precisely, interactions with a negative mass dimension, or in other words: such a theory is nonrenormalizable.

The only mechanism that can protect the ghost components from becoming observable under all circumstances is a complete local gauge invariance. This invariance must be =act at small distance scale, and therefore should also survive at large distance scale. The great discovery of Peter Higgs was that local gauge invariance is not at all incompatible with finite masses for the vector particles at large distance scales. All one needs to do is add physical degrees of freedom to the model that can play the role of the needed transverse field components. If these fields behave as scalar fields at short distances they don’t give rise to any ghost problem there. At large distances they conspire with the vector field to describe the three independent components of a massive spin 1 particle.

The trick used is that although the local gauge invariance has to be exact, the gauge transformation rule can be more general than in a pure Yang-Mills system. It may also involve the scalar field, even if the energetically preferred value of this scalar field is substantially different from zero. Now if we had been dealing with merely a global gauge symmetry this would have been an example of spontaneous symmetry breaking: the vacuum, with its nonvanishing scalar field, is not gauge invariant. It was tempting to refer to the present situation as “spontaneous breakdown of local gauge symmetry”, but this would be somewhat inaccurate. In our theoriea the vacuum does not break local gauge invariance. Any state in Hilbert space that fails to be invariant under local gauge transformations is an unphysical state. Strictly speaking, the vacuum is entirely gauge invariant here. The reason why we can do our calculations in practice as if the symmetry were spontaneously broken is that one can choose the gauge condition such that the scalar field points in a (more or less) fixed direction. In view of the above, we prefer not to use the phrase “spontaneous local symmetry breakdown”, but rather the phrase “Higgs mechanism” to characterize this realization of the theory where the vector particles are massive.

Space and time are continuous. This is how it has to be in all our theories, because it is the only way known to implement the experimentally established fact that we have exact Lorentz invariance. It is also the reason why we must restrict ourselvea to renormaliaable quantum field theories for elementary particles. As a consequence we can consider unlimited scale transformations and study the behavior of our theories at all scales (Chapter 3). This behavior is important and turns out to be highly nontrivial. The fundamental physical parameters such as masses and coupling constants undergo an effective change if we study a theory at a different length and time scale, even the ones that had been introduced as being dimensionless. The reason for this is that the renormalization procedure that relates these constants to physically observable particle properties depends explicity on the mass and length scales used.

It waa proposed by A. Peterman and E. C. G. Stueckelbergl, back in 1953, that the freedom to choose one’s renormalization subtraction points can be seen as an invariance group for a renormalizable theory. They called this group the “renormalization group”. In 1954 Murray Gell-Mann and Francis Low observed that

‘E. C. G. Stueckelberg and A. Peterman, Helu. Phva. Acta 20 (1953) 499.

the optimal choice of the subtraction depends on the energy and length scales at which one studies the system. Consequently it turned out that the most important subgroup of the renormalization group corresponds to the group of scale transformations. Later, Curtis G. Callan and independently Kurt Symanzik derived from this invariance partial differential equations for the amplitudes. The coefficients in these equations depend directly on the subtraction terms for the renormalized interaction parameters.

The subtraction terms depend to some extent on the details of the subtraction scheme used. For gauge field theories Veltman and the author had introduced the so-called dimensional renormalization procedure. It turns out that if the subtraction terms obtained from this procedure are used for the renormalization group equations these equations simplify. Furthermore there is a purely algebraic relation between the dimensional subtraction terms and the original parameters of the theory. This enables us to express the scaling properties of the most general renormalizable theory directly in terms of all interaction constants via an algebraic master equation. In deriving this equation one can make maximal use of gauge invariance. One can extend this master equation in order to derive counter terms for nonrenormalizable theories such as perturbative quantum gravity, but it would be incorrect to relate these terms to the scaling behavior of this theory, because here the canonical dimension of the interaction parameter, Newton’s gravitational constant, does not vanish but is that of an inverse mass-squared.

In the 1960’s and early 1970’s it was thought that all renormalizable field theories scale in such a way that the effective interaction strengths increase at smaller distance scales. Indeed, there existed some theorems that suggest a general law here, which were based on unitarity and positivity for the Feynman propagators. I had difficulties understanding these theorems, but only later realized why. I had made some preliminary investigations concerning scaling behavior, and had taken those theories I understood best: the gauge theories. Here the scaling behavior seemed to be quite different. Now that we have the complete algebra for the scaling behavior of all renormalizable theories we know that non-Abeiian gauge theories are the only renormalizable field theories that may scale in such a way that all interactions at small distances become weak. The reason why they violate the earlier mentioned theorems is that in the renormaliied formulation of the propagators the ghost particles play a fundamental role and the positivity arguments are invalid. This new development was of extreme importance because it enabled us to define theories with strong interactions at large distance scales in terms of a rapidly convergent perturbative formulation at small distances.

The new theory for the strong interactions based on this principle was called “quantum chromodynamics” . According to this theory all hadronic subatomic particles are built from more elementary constituent particles called “quarks”. These quarks are bound together by a non-Abelian gauge force, whose quanta are called “gluons”. The crucial assumption was that quarks and gluons behave nearly as free particles as long as they stay close together, but attract each’other with strong binding forces if they are far apart. This fits with the renormalization group

properties of the system, but the fact that complete separation of the quarks from each other and the gluons is impossible under any circumstance could not be understood from arguments based on perturbative formulations of the theory. Only a nonperturbative approach could possibly explain this.

Quantum chromodynamica is one of the very few renormalizable field theories in four spacetime dimensions that one can hope to understand nonperturbatively. But how could one possibly explain the confinement phenomenon?

Permanent quark confinement must have everything to do with gauge invariance. In his early searches for an explanation the author hit upon a feature in gauge theories that at first sight is entirely unrelated to this problem: the existence of magnetic monopoles. Purely magnetically charged particles had been considered before in quantum field theories, notably by Paul A. M. Dirac in 1934. He had derived that, in natural units, the product of the magnetic charge unit g and the electric charge unit e had to be an integer multiple of 27r. This implies that one can never apply perturbative methods to the interactions of magnetic monopoles because they interact super strongly. What was discovered, also independently by Alexander M. Polyakov2 in Moscow, was that certain extended solutions of the non-Abelian field equations carry a magnetic charge that obeys Dirac’s quantization condition. These solutions behave like classical particles in the perturbative limit and the strong interactions among them are classical (i.e. unquantized) interactions.

These monopoles are interesting in their own right, since they may be a feature of certain grand unified schemes for the fundamental particles and may play a role in cosmological theories. But also they may i n d d provide for a quark confinement mechanism. Quark confinement is now generally considered to be a consequence of superconductivity of the vacuum state with respect to monopoles. It is related to the Higgs mechanism if one exchanges electric with magnetic forces. Magnetic monopoles were a consequence of the topological properties of gauge theories. Are they the only consequences? Algebraic topology turns out to be a very rich subject, and indeed there is much more, as explained in Chapter 4. Magnetic monopoles are stable objects in three dimensions. If one search- for objects stable in one dimension one obtains domain walls. Stability in two dimensions is enjoyed by “vortices”. One may now also ask for topologically stable field structures in four dimensions rather than three. In a spacetime diagram such structures would show up as a special kind of “events”. An example of such a configuration that looked interesting had been considered by Alexander Belavin, Albert Schwartz, Yuri Tyupkin and A. Polyakov.

But what was the phgsicai interpretation of such “events”? Would they change and enrich the theory as much as magnetic monopoles? They certainly would, as seen in Chapter 5. Events of this sort could be calculated to be extremely rare in most theories, except when the interactions are strong, or if the temperature is very high. In every respect they represented some kind of “tunneling transition”. But tunneling from what to where? At first sight the tunneling was merely from some gauge field configuration to a gaugerotated field configuration. What was special about that? One answer is: the side effects. The event goes associated with

2 ~ . M. poiyakov, JETP ktt. ao (1974) 194.

an unusual transition between energy levels in the Dirac sea. The consequence of this transition is that in most of our standard theories some apparent conservation laws are violated in these events. For the weak interaction theories the violated conservation law is that of baryon number and lepton number. In the strong interaction the result is nonconservation of chiral U(1) charge. There are still some hopes that baryon number violation will be seen in future laboratory experiments (although I do not share this optimism), but chiral charge nonconservation in the strong interactions had already been observed for some time, causing confusion and embarassment among theorists who failed to understand what was going on. The new, “topoiogical” events, which we called “instantons”, explain this nonconservation in a satisfactory way. It was finally understood why the strong interactions render the 9 particle so much heavier than the pions.

The mechanism that keeps quarks permanently bound together in quantum chromodynamics would be a lot easier to understand if we had a soluble version of QCD. Ordinary perturbation expansion, which had served us so well in the previous quantum field theories, was useless here because our question concerns the region of strong couplings. Does there not exist a simplified version of QCD that is exactly soluble, or better even, is the coupling constant expansion the only expansion we can perform, or does there exist some other asymptotic expansion? At first sight the coupling constant g seems to be the only variable available to expand in, but that is not true. There are at least two 0the1-s.~ One variable is the dimension D of space-time. In 2 space-time dimensions the theory becomes substantially simpler; however it is not exactly soluble in 2 dimensions, and an expansion in D - 2 does not seem to be very enlightening. A very interesting expansion parameter is 1/N, where N is a parameter if we replace the color gauge group SU(3) by S U ( N ) . At first sight the theory does not seem to simplify at all as N + 00, since there still is an infinite class of highly complicated Feynman graphs.

But it does, in a very special way. If we keep g2N = 0’ fixed we obtain a theory which, again, as expanded in powers of 3’ produces an infinite series of Feynman diagrams, and they are too complicated to sum up, even in the limit. But the simplification that does take place is a topological one: all surviving diagrams in the N + 00 limit are “planar”, which means that they can be drawn on a 2-dimensional plane without any crossings of the lines. The twedimensional structures one obtains this way remind one very much of string theory, such that the strings connect every quark with an antiquark. This is the topic of Chapter 6.

But what is the use of this observation if one still cannot sum the required diagrams? First of all, the series of diagrams can be summed in 2 dimensions. In the exact theory for D = 2, N = 00, one obtains a beautiful spectrum of hadrons consisting of permanently confined quarks. It would be nice if now the expansion with respect to D - 2 could be performed, but that is not easy. There are two other reasons for further studying QCD in 4 dimensions in the N 4 00 limit. One is that one may consider the theory in a lightcone gauge. The stringy nature of the interactions is then fairly apparent and one may hope to obtain a sensible description

3A third involves the topological parameter 8, connected with instantons.

of hadrons this way.4 Another has to do with more rigorous mathematical aspects of the theory. One may expect namely that the diagrammatic expansion of the theory in terms of powers of B2 converges better than that of the full theory. If one counts the total number of diagrams one finds this to diverge only geometrically with the order, whereas this number diverges factorially in the full theory.

Now if any single diagram would only give a contribution to the amplitudes that is strictly bounded by a universal limit, then one would have an expansion that converges within some circle of convergence in the complex 3 plane. But we are not that fortunate. The contribution of a single diagram is not bounded. This is because there were infinities that had to be subtracted. For each individual diagram the renormalization procedure does what it is supposed to do, namely produce a finite, hence useful expression. But when it comes to summing all those diagrams up, the result of these subtractions is a new divergence. Fortunately, the ultraviolet divergent diagrams only form a small subset of all diagrams, and maybe there is a different way to get these under control. Just because the total number of diagrams is much better behaved than in the full theory one can try to perform resummation tricks here. What we were able to prove is that, if some Higgs mechanism provides for masses so that also the infrared divergences are tamed, and if furthermore the coupling constant is small enough, then the resummation procedure named after Bore16 is applicable.

Originally we did this in Euclidean space, but analytic continuation towards Minkmki space does not produce serious new problems. The only drawback is that the proof is only rigorous if the coupling constant 3 is so small that the theory is utterly trivial. From a physical point of view nothing new is added to what we already knew or suspected from ordinary perturbation theory. One does not obtain the hadron spectrum or even a confinement mechanism this way. It is rather from the more formal, mathematical point of view that this result is interesting, because it shows that this limit can be constructed in all mathematical rigor. A procedure to remove the constraint that there should be maases, or, equivalently, that Q should be kept small, is still lacking. Indeed, to solve that problem a better understanding of the vacuum state is needed. Atempts to Borel resum such theories bounce off against the fact that we do not know the vacuum expectation values of an infinite claw of composite operators.

Then, in Chapter 7, we finally attack the confinement problem directly. Some details of this mechanism were first exposed when Kenneth G. Wilsona produced the l/g expansion of a gauge theory on a lattice. A linearly rising electric potential energy between two quarks then emerges naturally. As indicated earlier, confinement has to do with superconductivity of magnetic monopoles. But it is also directly related to gaugeinvariance. It had been argued by V. Gribov7 that the

4There was a recent report of substantial progreaa in lightcone QCD. See S. J. Brodsky and G. P. Lepage, in Perturbatiue Quantum Chmmod~mics, ed. A. H. Miiller (World Scientific, Singapore, 1989), p. 93. '&mile Borel, Lqom sur lea dries divergentes, Paris, Gauthier-Villers, 1901. 6K. G. Wilson, Phys. Rev. D10 (1974) 2445. 7V. N. Gribov, Nucl. Phys. B139 (1978) 1.

usual procedure for fixing the gauge freedom in non-Abelian gauge theories is not unambiguous. His claim that this ambiguity may have something to do with quark confinement was originally greeted with skepticism. But it turned out to be true. If one meticulously fixes the gauge in an unambiguous manner one finds a new phe nomenon, namely a sea of color-magnetic monopoles and antimonopoles. It is these monopoles that may undergo Bose condensation. Whether Bose condensation actually occurs or not is not something one can establish from topological arguments alone, because the ordinary Higgs mechanism would be impossible to exclude, even if one has no fundamental scalar field at hand; composite scalar fields could also be used for the Higgs mechanism. So one has to look at dynamical properties of the theory, such as asymptotic freedom.

What could be established was a precise description of the confinement mechanism in these terms, as well as alternative modes a theory such as QCD could condense into. Since these alternatives could not be ruled out we have not actually given a “proof” that QCD explains quark conhement. But the logical structure of the confinement mode is now so well understood that no basic mystery seems to be left.

And thus elementary particle physics ran out of mysteries. The problem of uniting special relativity with quantum mechanics has been solved. The solution is called “renormalizable quantum field theory”, which in general includes gauge theories. The small distance divergence is in its most essential form only logarithmic, and the solution of that should be looked for by postulating hierarchies of field theories, each hierarchy d i d at its own characteristic distance scale.

But this does not solve the biggest mystery of all: why do these theories work? &om a mathematical point of view they work because they are the most economical constructions in terms of physical degrees of freedom: we postulated the least amount of structure possible at small dietance scales. That postulate is what makes these theories so unique. But why should we postulate optimal economy? And what determines which structures exist at all at small distance scales? How does our chain of hierarchies get started? We all think we know where to look for the answer to such questions: the gravitational force. Gravity adds a new fundamental constant to our physical world, namely Newton’s constant. It is incredibly small, indicating that gravity dominates the natural forces at a tremendously tiny distance scale. At this distance scale, called the Planck scale, everything in physics will have to be reconsidered. It is here namely that all our presently known techniques fall short.

Numerous attempts have been made to attack this problem. Basically all we want is unite geneml relativity with quantum mechanics. Now general relativity even without quantum mechanics is a highly complex theory, and the roles played by the concepts “energy” and “time” in general relativity are on the one hand equally crucial as, and on the other hand fundamentally different from the one8 they play in quantum mechanics. Well-known are the approaches where a new kind of symmetry is introduced, called “supersymmetry” . The resulting “supergravity theory” was potentially interesting because it seemed to be not as divergent as gravity alone. But I never gave this approach by itself much chance of success because it did

not address the most fundamental aspect of the difficulty, which is the fact that Newton’s constant has the dimensionality of a length squared, so that at distance scales shorter than the Planck length the gravitational force, supersymmetric or not, runs out of control.

The same objection, though with a little more hesitation, can be brought against the “superstring” approach to quantizing gravity. The space-time in which the superstring moves is a continuous space-time, and yet we have a distance scale at which a smooth metric becomes meaningless. On the other hand a flat background metric is usually required at ultrashort distances, even in string theories. It is my conviction that a much more drastic approach is inevitable. Space-time cewes to make sense at distances shorter than the Planck length. Here again I reject a purely mathematical attack, particularly when the math is impressive for its stunning complexity, yet too straightforward to be credible. The point here is also that our problem is not only a mathematical one but more essentially physical as well: what is it precisely that we want to know, and what do we know already?

There is a different, more “physical” way to see what goes wrong at small distances. Suppose we want to describe any physical phenomenon that is localized at a distance scale smaller than the Planck scale. According to an established paradigm in quantum mechanics this system will contain momenta that will be spread over more than one Planck unit of momentum, and its energy will be of the same order of magnitude. But then it is easy to see that this energy would be confined to within its Schwarzschild horizon, which will stretch beyond one Planck radius. Hence gravitational collapse must already have occurred: such a system is a black hole, and its size will be larger than a Planck unit. This proves that confining any system to within a Planck unit is impossible.

Indeed, black holes form a natural barrier. Obviously they must be a central theme in any theory with gravity and quantum mechanics that boasts to be complete. This discussion is conspicuously missing in any superstring theory of quantum gravity, which is why I don’t believe these theories can be complete. In my attempts towards a better understanding of this problem I begin with black holes. How can their presence be reconciled with the laws of quantum mechanics? A brilliant discovery had been made before by Stephen Hawkings: quantum field theory gives black holes a behavior fundamentally different from unquantized general relativity: they radiate. Upon studying this system further one stumbles upon more surprises. It is a fascinating subject. One can imagine thought experiments that may ultimately lead to the resolution of our most fundamental questions. This is Chapter 8.

One of the weaknesses a scientist may fall victim to later in his career is that he (or she) may begin to ponder about the deeper significance of his theories in a wider context, the direction they are heading to, the expectations they may hold for the more distant future. What would the most natural, the most satisfactory, the most complete results look like? Could there be an ulttimate theory? Somewhat surprising, perhaps, is that our present insights do indeed suggest that there may

‘S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.

be such a thing as an ultimate description of all physical degrees of freedom and the laws according to which these should evolve. As argued above, space-time itself at the Planck length does not seem to allow for any further subdivision. Could an ultimate physical theory be formulated around that length scale? Our last chapter deals with such questions.

CHAPTER 2

RENORMALIZATION OF GAUGE THEORIES

Introductions ..................................................................... 12

[2.1] “Gauge field theory”, in Pmc. of the Adriatic Summer Meeting on Particle Physics, eds. M. Martinis, S. Pallua, N. Zovko, Elovinj, Yugoslavia, Sep.-Oct. 1973, pp. 321-332 ................................ 16

[2.2] with M. Veltman, “DIAGRAMMAR”, CERN Rep. 73-9 (1973), reprinted in “Particle interactions at very high energies”, NATO Adv. Study Inst. Series B, vol. 4b, pp. 177-322 ............................... 28 “Gauge theories with unified weak, electromagnetic and strong interactions”, E.P.S. Int. Conf. on High Energy Physics, Palermo, Sicily, June 1978 ........................................................ 174

[2.3]

CHAPTER 2

RENORMALIZATION OF GAUGE THEORIES

Introduction to Gauge Field Theory [2.1]

The pioneering paper of Chen Ning Yang and Robert Mills9 in 1954 has inspired many physicists in the late 1960's to construct theories for the weak and the strong interactions. In the following papers it is assumed that the reader is somewhat familiar with this fundamental idea, namely to consider field equations that are invariant under symmetry transformations that vary from point to point in space- time. It is basically just a generalization of the Maxwell equations which indeed have such a symmetry:

Q(x, t ) -t exp ieh(x, t) @(x, t ) ,

A J x , t ) A,(x, t ) - O p W , t ) 9

or a space-time dependent rotation in the complex plane. Yang and Mills replaced these by higher dimensional rotations, which in general are non-Abelian.

It was clear however that the Yang-Mills equations would describe massless spin one particles, just as the Maxwell equations, and that these would interact as if they were electrically charged. And it was evident that such particles do not exist in the real world. This leads us to ask two questions:

1. Can one add a mass term to the Yang-Mills equations, such that they become physically more plausible?

2. Since power counting suggests renormalizability, can one renormalize the quantum version of this theory? As explained in the previous chapter, the answer to both these questions is yes, provided that one invokes what is now known as the Higgs mechanism. Without the Higgs mechanism the power counting argument would fail because the longitudinal components of the vector fields (which decouple in the massless case) have no kinetic part in their propagator and therefore interact nonrenormalizably.

' C . N. Yang and R. L. Mills, Phys. Rev. 96 (1954) 191.

But this had to be proved. The problem Veltman was studying in the late 1960's was that the contributions of the ghost particles, all in the longitudinal sector, did not seem to add up properly to give a unitary theory. The ghost problem itself had been studied by Richard Feynman for the massive case, and Bryce DeWitt, Ludwig Faddeev and Victor Popov in the massless case. It was not immediately realized that the massless theory is not simply the limit of a massive one where the mass is sent to zerolo because the transverse component would continue to contribute in loop diagrams: it can be pair-produced.

The papers by Faddeev and POPOV,~~ mostly in Russian, were not immediately available to us, but a short paper by them in Physics Letters12, in which they summarized their arguments, made their procedure quite clear to me. Basically the message was that the 5'-matrix amplitudes, just like all quantum mechanical amplitudes, are functional integral expressions. Like always in integrals, when one performs a symmetry transformation in the integrand, one has to keep track of the Jacobian factors. These are usually just big determinants. I decided to rewrite these determinants as Gaussian integrals, because then one can more easily read off what the Feynman rules for these are. These Feynman rules are just as if there are some extra complex scalar particles, but because of a sign switch we found that that these scalar particles had to be seen as fermions rather than bosons. And then the trick was that one had to combine the Faddeev-Popov prescription with the Higgs-Kibble mechanism. So we were dealing with two kinds of ghost fields rather than one.

In papers (2.1) and (2.2) we consider this procedure as given. But we did not wish to trust the details of the functional integral approach, because the way it dealt with the renormalization infinities did not seem to be rational. Instead, we consider the perturbative formulation of the theory. We then prove that, order by order in the perturbation expansion and after renormalization, this theory indeed obeys all physically relevant requirements such as unitarity and causality. That it was wise to be this careful soon became clear: there may sometimes be anomalies, and in that case the procedure does not work. A theory with anomalies in the local gauge sector is inconsistent.

The fkst paper is an introduction to the second, illustrating the prescriptions in a simple example. In this example the three gauge bosons B1*293 all get the same mass M, see eq. (2.11). That the masses are all the same is not a consequence of the local gauge symmetry but of a somewhat hidden global symmetry (2.7). This symmetry is explicitly broken by the A2 term, so that higher order corrections may cause a relative mass shift. We first see this mass shift in the fermions. The explicit expression for Peak is:

cbreak = A 2F(Zl$l+ 41x1) + cbreakJnt, to be obtained by substituting (2.10) in the last part of (2.5).

'OH. van Dam and M. Veltman, Nucl. Phys. B2l (1970) 288. "L. D. FBddeev, Theor. Math. Phys. 1 (1969) 3 (in Russian); Theor. Math. Phys. 1 (1969) 1 (English translation). 12L. D. F'addeev and V. N. Popov, Phys. Lett. 26B (1967) 29.

13

This symmetry structure is very special. It is one of the few systems with computable mass splittings for some of the elementary fields. This was the reason for my interest in this model.

Introduction to DIAGRAMMAR [2.2]

Perturbation expansions with respect to small coupling constants are often looked upon as ugly but necessary tools, and repeatedly physicists attempt to avoid them altogether. It cannot be sufficiently emphasized however that perturbation expansions are an absolutely essential ingredient in quantized gauge field theories. Many of our cherished particle theories can only be defined perturbatively. This means that their treatment can only be considered as being mathematically rigorous if we consider all observable quantities as formal power series in terms of a small parameter. This is sometimes referred to as “nonstandard analysis”: we extend the field of numbers to the field of power expansions. The strength of this modification of our mathematics lies in the fact that the expansions need not have a finite radius of convergence (in general they don’t). If we substitute a finite number, say g2 for the coupling constants we can trust the series to be meaningful only as long as the next term is a smaller correction than the previous one. Later we will see that in most cases the series will behave as C N ( g 2 ) N N ! . This then implies that observables cannot be computed more accurately than with margins of the order of e-1/C9a, in many cases more than good enough!

It should be noted that shifts of vacuum expectation values are not difficult to deal with within this philosophy of nonstandard analysis. Most importantly, one can observe that the perturbation expansion for a quantum field theory is equivalent to dividing the amplitudes up in secalled Feynman diagrams. Feynman diagrams are nothing more than book-keeping devices for the various contributions to the amplitudes. They are central in the next paper, “DIAGRAMMAR”, written together with M. Veltman. With this word we intended to indicate that our prescriptions are nothing but the “grammatical rules” for working with diagrams. It also means “diagrams” in Danish.

There are circumstances where one hopes to be able to do better than perturbation expansion. In asymptotically free theories one might be able to replace the perturbation parameter g2 by the much smaller number at an arbitrarily small distance scale, so that our margin should be tightened significantly, perhaps all the way to zero. Thus a theory such as QCD can perhaps be given a completely rigorous mathematical basis. This has never been proven, but it indeed seems to be plausible. Another point is that one expects phenomena that are themselves of the order of e-l/@, as explained in Chapter 5. These cannot be handled with Feynman diagrams.

In “DIAGRAMMAR” we first show how to work with the diagrams, then how transformations in the field variables correspond to combinatorial manipulations of the diagrams, exactly as in expressions for functional integrals. If a series of diagrams is geometrically divergent (for instance if we sum the propagator inser-

tions, then unatady tells us exactly that the analytic sum is to be taken. We show how the Faddeev-Popov procedure is translated in diagrammatic language. Most importantly, it is now seen how and why these tricks continue to work when the theory is renormalized. The infinities cancel if the anomalies cancel.

Then dimensional regularization and renormalization are explained. There are three steps to be taken. First we have to define what it means to have a non-integer number of dimensions. Fortunately this is unambiguous at the level of the diagrams - though not at all so “beyond” the perturbation expansion! Secondly we have to deal with integrations that diverge even at non-rational spece-time dimension near four; they are easily tamed by analytic continuation from the regions where the integrals do converge. In practice one does this by partial integrations. Thirdly we obeerve that precisely at integer dimensions some infinities are not tamed by partial integrations: the logarithmic ones. These show up as powers of l/e, where e = 4 - n, TI is the number of spacetime dimensions. They have to be cancelled out by inserting the proper counterterms into the Lagrangian. One then ends up with finite, physically meaningful expressions.

In Section 11 of “DIAGRAMMAR” a transformation is described that had been introduced before as “Bell-”reiman transformation”, by my c+author M. Veltman. The name is something of a joke. There is no reference to either Bell or Tkeiman. It would have been more appropriate to call them “Veltman transformations”.

The Slavnov-Taylor identities play a crucial role in the renormalization proce dure of gauge theories. They get the attention they deserve in DIAGRAMMAR. The proofs here are as they were first derived. Nowadays a more elegant method exists: the Becchi-Rouet-Stora-Tyupkin quantization procedure. These authors observed that our identities follow from a global symmetry.

Now I had tried such symmetries myself long before DIAGRAMMAR was written, but without success. The crucial, and brilliant, ingredient invented by BRST was that the symmetry generators had to be anticommuting numbers.

Introduction to Gauge Theories with Unifled Weak, Electromagnetic and Strong Interactions [!MI

The last paper in this chapter needs little further introduction. It displays the enormous progress made in just a few years of renormalizable gauge theory. The J / @ particle, the last crucial ingredient needed to render credibility to what was to become the standard model, had just been found. I was a little too optimistic in applying asymptotic freedom to understand the J/$J energy levels, hence my underestimation of the splitting between the ortho and the para levels in Table 1. But these levels, at that time not yet seen, would soon be discovered and everything fell into place. Of course most of the more exotic models speculated about in this paper were ruled out during the years that followed. What remained was to be called the “Standard Model”. The solution to the eta problems, Section 10 was basically correct, but would be understood better in 1975, with the discovery of the instanton effects, see Chapter 5.

CHAPTER 2.1

GAUGE FIELD THEORY*

G. ’t Hooft CERN - Geneva

On leave from the University of Utrecht, Netherlands

1. INTRODUCTION There are several possible approaches to quantum field theory.

One may start with a classical system of fields, interacting throuqh non-linear equations of motion which are subsequently “quantized”. Alternatively, one could take the physically observed particles as a starting point; then define a Hilbert space, local operator fields, and an interaction Hamiltonian. More ambitious, perhaps, is the functional integral approach, which has the advantage of being obviously Lorentz covariant.

All these approaches have one unpleasant and one pleasant feature in common. The unpleasant one is that in deriving the S-matrix for the theory, one encounters infinities of different types. In order to get rid of these, one has to invoke a rather ad hoc “renormalization procedure”, thus changing and undermining the theory half- way. The pleasant feature, on the other hand, is that one always ends up with a simple calculus for the S matrix: the Feynman rules. Few physicists object nowadays to the idea that these Feynman diagrams contain more truth than the underlying formalism, and it seems only rational to abandon the aforementioned principles and use the diagrammatic rules as a starting point.

It is this diagrammatic approach to quantum field theory which we wish to advertise. The short-circuiting has several advantages. Besides the fact that it implies a considerable simplification, in particular in the case of gauge theories, one can simply superimpose the renormalization prescriptions on the Feynman rules. As for unitarity and causality, the situation has now been reversed: we shall have to investigate under which conditions these Feynman rules describe a unitary and causal theory. Within such a scheme many more or less doubtful or complicated theorems from the other approaches can be proved completely rigorously.

Clearly, Feynman diagrams merely describe an asymptotic expansion of a theory for the coupling constants going to zero, and strict ly speaking, nothing is known about the theory with finite coupling constants. But the other approaches are not better in this respect, if it comes to calculations of physical effects. A really rigorous formulation of quantum field theory with finite interactions has not yet been given and we are of the opinion that attempts at such a for-

Proceedings of the Adriatic Summer Meeting on Particle Physics. Edited by M. Martinis et al. (North Holland, 1973).

16

G. 't HOOFT

mulation can only succeed if the perturbation expansion, formulated in the simplest possible way (Feynman diagrams) is well understood.

In these notes, which must be considered as an introduction to the CERN report called "DIAGRAMMAR" 111, we shall outline the basic step8 that have to be taken in order to formulate and understand a gauge field theory. We shall illustrate our arguments with a simple example of a non-Abelian gauge model (which is not realistic physically).

In Section 2 we give a review of the construction process of a model in general, showing the Brout-Englert-Higgs-Kibble phenomenon.

In Section 3 the quantization problem is formulated and the essential steps in the proof of renormalizability are indicated.

2. CONSTRUCTION OF A MODEL In the construction of a model one must try to combine the ex-

perimental observations with the theoretical requirements. There is of course no logical prescription how to do that, and therefore we shall here only consider the theoretical principles 121 and leave it to the reader to alter our little example in such a way as to obtain finally physically more interesting results.

a) First we choose the GAUGE GROUP. This is a group of internal symmetry transformations that depend on space and time 131. In our example this group will be SU(Z)XU(l) (at each space-time point). Let us denote an infinitesimal gauge transformation as eT, which is generated by an infinitesimal function of space-time, Aa(x), a - 1,2,3,0. The condition that tha infinitesimal gauge transformations generate a qroup is

where [T(l), T(2)] 13 again an infinitesimal gauge transformation, generated by

(2.2)

Here the indices between the parentheses denote different choices for A , and fabc are structure constants of the group. In our example,

fabc = cabc for a,b,c # 0 - 0 otherwise.

(Often we shall write gAa instead of Aa, where g is some coupling constant. )

cles: one for each generator T. In our example: By choosing the gauge group we also fix the set of vector parti-

17

GAUGE FIELD THEORY

a B,,(x), a = 1,2,3 and ACl(x) . b) Now we choose the other fields. They all must be representa-

1 z tions of the gauge group. Example: A Bose field with "isospin" (a two-component representation of S U ( 2 ) ) and "charge" 0 (a scalar

1 for U ( 1 ) transformations). A Fermi field Qi with "isospin" 5 and "charge" 1; a Fermi field X with "isospin" 0 and "charge" 1.

rules (2.11, ( 2 . 2 ) . There is one way to satisfy the requirements. In our example the infinitesimal transformations are

c) The GAUGE TRANSFORMATION LAW must satisfy the commutation

B;' = B; - a ha + glEabc A b c B,, ; u 3

$ ' =$- 1 igl 1 T ~ A ~ $ , as1

3 Q'= Q - 1 1 T ~ A ~ Q + ig2Ao$ , z ig1 all

X' = x + i g 2 A O X . (2.31

d) Next we write down a Lagrange density, which we shall call the SYMMETRIC LAGRANGIAN, ginv(x). It is a functior. of the fields and their space-time derivatives at the point x. The corresponding Lagrange equations

6 Y 6 g 6Ai(x) - a apAi(X) = O

where Ai(x) is any of the fields Ba, A $, Q or X will describe to a first approximation the propagation of these fields, the quanta of which are the physical particles. The fact that we have a quantum theory will necessitate "higher order" corrections (the loops in the Feynman graphs).

tions ( 2 . 3 ) . In our example we take

u v' i

This Lagrangian must be invariant under the gauge transforma-

18

G. 't HOOFT

D,,X = a,,x + ig A x . ( 2 . 6 ) 2 u

~ l l terms in the Lagrangian must have dimension 4 or less. Dimension is counted as follows: a derivative has dimension one, a Bose field has dimension one and a Fermi field has dimension 5 . Given that Bo- se propagators behave as -2 and Fermi propagators as i; for Ikl +

this requirement will enable us to estimate the maximal "overall" divergence of an integral in a diagram, if only the external lines are known. For instance, the overall divergence of the diagram in fig. 1 is linear, independent of its internal topology (in fig. 1 there are 11 Fermion and 8 Boson propagators,

3 1 1 k

1 k2

boson ( - -) 1 fermion ( - i;) - - - - ', I' 5' vertices from and ( $ * @ I 2 , respectively, (for in-

stance). ,*' ,,

Fig. 1

and 7 loops for integration, 7.4-11-1-8.2 = 1 . ) are independent of space time. In the example: Parity (PI, Charge conjugation (C) , Time reversal (T) , and, if X 2 + 0, an unexpected SU (2) symmetry:

e) Examine the GLOBAL SYMMETRIES that is, all symmetries that

0; Qi + " E i j 0 3 + A z i E i j $ ; + A 3 i 0 i t (2 .7 )

A i space-time independent.

interaction terms that do not violate these symmetries nor the local The importance of the global symmetries is that all possible

19

GAUGE FIELD THEORY

gauge symmetry, must be present in the Lagrangian (2.5). This is why the term with X 1 must be present.

X J, and @ particles. But note that our Lagrangian does not contain a mass term for the vector particles, simply because no gauge invariant mass term exists. So these spin-one particles are massless and they interact with each other. As a consequence there are huge infrared divergencies of a type that cannot be cured as in quantum electrodynamics. It is expected that the eventual solution of the quantized equations will be drastically different from the classical ones (2.4) and completely governed by these infrared effects (.conjecture: strongly interacting Regge particles), but nobody knows how to solve this problem.?

Nevertheless, the model is interesting, and that is because we can take the variable uo to be negative. If we then assume for a moment that the fields in d. are classical, then the Hamilton density

2 If p o p m and m >O, then the model contains massive Fermions

2

, obtained from , contains a term

The energy is minimal not if 4 4 , but if

Because of gauge invariance, we can always rotate until @i is parallel to the spinor (1,O). Small fluctuations of the various fields near this equilibrium state are now described by massive equations of motion, as we shall see in f.

We can now proceed in two ways. Either we I) first quantize the "symmetric" theory and then construct the new

vacuum, corresponding to this "equilibrium state", or we 11) first define new fields 4 , with the equilibrium value (2.9) sub-

tracted, and then quantize this "asymmetric" theory. We choose possibility no. 11. Therefore: f) Shift those fields which have a vacuum expectation value 141.

In the example

(2.10)

Here the number F satisfies, up to possible quantum corrections, eq.

20

G. 't HOOFT

( 2 . 9 ) , and 2 and Ya a r e t h e new f i e l d v a r i a b l e s (real) . The Lagran- gfan I n terms of t h e new f i e l d s Is t h e NON-SYMMETRIC LAGRANGIAN. I n our example:

where M2 = 1 g:F2 ; Mi = 2A1F2 and

6 = p i + AlF2 .

(2.11)

(2.12)

Tha i n t e r a c t i o n tenn is r a t h e r complicated now and n o t so r e l evan t f o r t h e discussion. In our example It Is

l a int = - i glauYaedc B > ~ + g B, , ( z~ , ,Y~ - Y a z) a u

- 3 9 2 2 B u ( Z 2 + Y z ) - $ g M B 2 2 u

(2.13)

The tenn gbreak arises from t h e A 2 term and breaks t h e g l o b a l SU(2) symmetry ( t h e Fennion masses are a l s o s p l i t ) .

leave8 t h e sp ino r $,) in (2.10) inva r i an t . This is t h e new symmetry group, broken by t h e A 2 tenn.

A 8 w e do not i n s i s t on eq. ( 2 . 9 ) f o r F, our s h i f t is f r e e , and w e g e t a new " f r e e parameter" B i n (2 .11 ) . I n general , we s h a l l re- q u i r e t h a t a l l diagrams where t h e 2 p a r t i c l e vanishes i n t o t h e vacuum cancel ( f i g . 2 ) . I n lowest o rde r t h i 8 corresponds to B-0.

t h a t Note t h a t t h e r e is a subgroup of SU(2)local xsu (2 ) g loba l 1

21

GAUGE FIELD THEORY

g) The obtained Lagrangian does not at all look invariant under the local gauge transformations, in particular the vector-mass term and the i2 term. (Of course, the U(1) invariance here is obvious). But we can rewrite the original gauge transformation law (2.3) in terms of the new fields. We get the NEW GAUGE TRANSFORMATION LAW:

Y,' - Ya + 5 1 glcabc AbYc - 3 glAaZ - MAa , 1

2 ' a 2 + 5 glhaYa.

No change in:

BE' = B~ - a + glEabcA b c B,, . 1 ! P

In the general case,

A; = Ai + %:Aa + g siajAaAj ,

(2.14)

(2.15)

where 1; the derivative a for the vector fields.

Under these gauge transformations the Lagrangian is still inva- - riant. Its nonsymmetric form is merely a consequence of the practi- tally arbitrary, nonsymmetric new coefficlents tt.

Although we have much more bookkeeping to do now, the general principle, the invariance under a gauge transformation, is not changed by the shift. Only the transformation law is a little bit different.

are either coefficients with the dimension of a mass, or

P

Note that the new t coefficients are of zeroth order in gl. h) The model contains PHYSICAL PARTICLES and GHOSTS. Ghosts cor-

respond to those fields that can be turned away independently by gauge transformations. To determine which fields are ghosts it Ls sufficient to consider only the lowest-order parts of the gauge transformation, described by the t coefficients. In our example one can take a,,A,, = 0, and then either turn the Ya fields away or one of the spin-components of the vector fields WE. The first choice is obviously the most convenient one. Compare (2.14) ; we choose ha =

+ perturbation expansion in gl. Our model evidently contains three massive spin-one particles Ba a massless photon Ap (with only two possible helicities), one massive spinless particle 2, and three massive Fermions X and $i (of which the latter form a doublet).

i) It is instructive to consider first the classical field equations. Just as in electrodynamics one must choose a gauge condition in order to remove the gauge freedom. In general, we can write

1 Ya+

lJ,

22

G. 't HOOFT

this condition in the form ca(x) - 0 , (2.16)

where Ca(x) is some function of the fields that must not be gauge invariant. The number of components of Ca must be identical to the nwn- ber of generators in the wou 1 d have

co = ca -

One may impose such

gauge group, here

co E

ca - avAIJ or ya

a gauge condition placing the original Lagrangian '&Inv by

i. p n v - 3 (cap .

four. In subsection h) we

a PAP

aLIBIJ

in an elegant way by re-

(2 .17)

The equations of motion corresponding to this Lagrangian are fulfilled if any small variation of the fields does not change us choose as a small variation an infinitesimal gauge transformation, described by Ab(x) :

d4x. Let

2-3- Ca. b C a . Ab , (2.18)

where - stands formally for the variation of Ca under a gauge trans formatfon, and must be unequal to zero. Hence Ca must be zero. Ap- plying other variations to the Lagrangian, we get back the original equations of motion, now with the gauge condition Ca= 0.

6 Ab 6C"

Ad

In our example it is convenient to choose

where A is a free parameter. Consequently, after a gauge transformation,

more explicitly,

(2.19)

23

(2 .20)

GAUGE FIELD THEORY

The reason why this particular choice for Ca(x) is convenient: is that the bilinear interaction

-M Yaa,,Bt

in zinv, eq. (2.11), is cancelled, so that the B are decoupled. The importance of the gauge parameter A will be made clear in the next section.

and Y propagators IJ

The model of our example becomes more interesting if we assign a U(1) charge not to X but to the field +. We then get something closely resembling the Weinberg model 151. Its gauge structure is a bit more complicated.

3. THE MATHEMATICAL STRUCTURE OF GAUGE FIELD THEORY The models constructed along the lines given in the previous

section are renormalizable (under one additional condition, see 11) in this section). For the proof of this statement several steps must be made. We indicate those here without going into any of the technical details (which can all be found in "DIAGRAMMAR"). We give the steps in an order which is logical in the mathematical sense. This is not the order in which the theorems have been derived, but the order in which a final proof should be given. Historically, the Feynman rules were first found by functional integral methods but the proof was heuristic and not very rigorous. As our first step we shall

1) postulate the Feynman rules. These are always uniquely defined by a Lagrangian. As in the classical case, one must choose a gauge function c : ~ ) (XI , example. The Lagrangian tion, but

x = where describes

for which we shall again take (2.19) In our however is not the one of the previous sec-

a new "ghost particle", called Faddeev-Popov ghost, occurring only in closed loops, and it depends on the choice of ca(x) :

where Oa is a new complex

24

G. 't HOOFT

The 0 particle may not occur among the external lines and because the vertices only connect two @ lines, it only occurs in single loopa. A further prescription is that there must be one more minus sign for eadr 4 loop ("wrong statistics").

free particle and therefore drops out. The mass of the SU(21-0 depend. on the gauge parameter A .

i i ) Reqularize the theory, that is, we must modify the theory slightly in terms of a small parameter, say c p such that all divergent integrals become finite and the physical situation is attained in the limit c+O. The modification must be as gentle as possible and should not destroy gauge invariance so that we can still use all our theorems. The most elegant method 171 is the dimensional regularization procedure, in which E - 4-d, d is the "number of space-time dimensions". What we mean by this can be formulated completely rigorously as long as we confine ourselves to diagrams (but that we have

5 1 2 3 4 already decided). For some theories, however, containing y =y y y y

or the tensor coBy6* the extension to arbitrary dimensions cannot be made, and the method does not work. Indeed, such theories suffer, more often than not, from the so-called "Bell-Jackiw" anomalies 161.

They are not renormalizable.

the S matrix is independent of the choice of Ca(xL. For this we must consider non-local field transformations ("canonical transformations") and prove the Slavnov identities. These laentities are essential and from them the gauge independence of the S matrix follows. It is in the proof of the Slavnov identities where the "group property", eqe. (2.1) and (2.21, enters.

Note that the @ corresponding to the Abelian group U ( 1 ) is a

iii) Now that all Green's functions are finite, we prove that

iv) Find a set of gauge functions C:(x) such that a) the theory is "renormalizable by power counting" for

b) the S matrix is unitary in the space of states with all A < - (compare subsection 2d and fig. 1);

energy*) the "cutting rules" ;

E < X . This can be established by means of c) the gauge functions must stay Lorentz-invariant.

..................................................................... *) Note that the ghosts in our example all have mass AM. ..................................................................... (One could also require renormalizability and complete unitarity but give up Lorentz invariance for a while; this seems not very practi-

25

GAUGE FIELD THEORY

-1.)

approaches its physical value zero.

The most secure preacription is to add "local" counter terms to the Lagrangian that cancel one by one the divergencies in all possible diagrams. In the dimensional procedure they have the form of ordinary terms but with coefficients $ , , etc. It must be proved that only local counter terms are sufficient to remove all divergencies, otherwise the cutting rules, e.g. unitarity are violated. Here we make use of the causality-dispersion relations (also derived from cutting rules).

gauge invariance? We show that the new, "renormalized" Lagrmgian is invariant under new, "renormalized" gauge transformation laws. We must convince ourselves that the gauge group (in our example SU(2)xU(1) is not changed into another one (for example, U(l)xU(l)x xU(1) XU (1) ) . In fact, the new laws are equivalent to the old ones for "renormalized" fields. It then follows that the theory is "multi- plicitatively" renormalizable. We could only prove this if the gauge function Ca (x) has been chosen without bilinear field combinations, such as

Now we must consider the limit where the regulator parameter B

v) Renormalize, according to some well-defined prescription.

0

vi) Evidently, we altered the Lagrangian. Does this not spoil

Ca(x) = a,B: + a(B,) a 2 . vii) Finally show that not only the regularized but also the E-

normalized from the obvious fact that a variation of Ca(x) may have to be accompanied by a slight change of the original variables gl, M, etc. It is this latter remark which makes this point far from easy to deal with, but it follows from the "multiplicative" renormalizability in the case of linear C.

S matrix is independent of the choice of C"(x1, apart

REFERENCES

1 1 I G. 't Hooft and M. Veltman, "DIAGRAMMAR", CERN 73-9 (1973),

121 See also: B.W. Lee, Phys. Rev. D5 (1972) 823; Chapter 2.2 of this book.

B.W. Lee and J. Zinn Justin, Phys. Rev. D5 (1972) 3121, 3137,

3155.

131 C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191.

26

G . 't HOOFT

141 F. Eng le r t and R. Brout, Phys. Rev. Letters 13 ( 1 9 6 4 ) 3 2 1 ;

P.W. Higgs, Phys. Letters 12 ( 1 9 6 4 ) 132; Phys. Rev. Letters 13

( 1 9 6 4 ) 508; Phys. Rev. 145 ( 1 9 6 6 ) 1156;

G . S . Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Letters

13 ( 1 9 6 4 ) 585; T.W.B. Kibble, Phys. Rev. 155 ( 1 9 6 7 ) 6 2 7 .

151 S. Weinberg, Phys. Rev. Letters 19 ( 1 9 6 7 ) 1264.

161 J .S. Bell and R. Jackiw, Nuovo Cimento 60A ( 1 9 6 9 ) 47 .

171 See however a l s o the approach of A. Slavnov, Theor. and Math.

Phys. 13 ( 1 9 7 2 ) 174.

*Reprinted from Proceedings of the Adriatic S m e r Meeting on Particle Physics, Rovinj, Yugoslavia, September 23-October 5, 1973.

*Note added: This paper was w r i t t e n i n 1973 , b e f o r e "asymptotic freedom" l e d t o t h e general acceptance of QCD. The c o n j e c t u r e , which r e f e r s t o gauge t h e o r i e s without Higgs mechanism, is now g e n e r a l l y bel ieved t o be c o r r e c t .

27

CHAPTER 2.2

DIAGRAMMAR

G. 't Hooft and M. Veltman CERN - European Organization for Nuclear Research - Geneva

Reprint of CERN Yellow Report 73-9 : Diagrammar by G . 't Hooft and M. Veltman

with the kind permission of CERN

28

G. 't HOOFT and M. VELTMAN

1 s INTRODUCTION

With the advent o f gauge theo r ies i t became necessary t o reconsider many we l l -es tab l i shed ideas I n quantum f i e l d theor ies . The canonica l formalism, fo rmer l y regarded as t h e most conven- t i o n a l and r i go rous approach, has now been abandoned by many authors. The p a t h - i n t e g r a l concept cannot rep lace t h e canonica l formal ism i n de f in ing a theory, s ince pa th i n t e g r a l s i n f o u r dimensions are meaningless w i thout a d d i t i o n a l and r a t h e r ad hoc renormal iza t ion p resc r ip t i ons .

Whatever approach i s used, t h e r e s u l t i s always t h a t t h e S-matrix i s expressed i n terms of a c e r t a i n se t of Feynman d i a - grams. Few p h y s i c i s t s ob jec t nowadays t o the i dea t h a t diagrams conta in more t r u t h than the under ly ing formalism, and i t seems on ly r a t i o n a l t o take the f i n a l s tep and abandon operator forma- l i s m and pa th i n t e g r a l s as inst ruments o f analys is .

Yet i t would be very shor ts igh ted t o t u r n away completely from these methods. Many u s e f u l r e l a t i o n s have been derived. and many more may be i n t h e fu tu re . What must be done I s t o pu t them on a s o l i d f o o t i n g . The s i t u a t i o n must be reversed: diagrams form the bas is from which every th ing must be der ived. They de f i ne the opera t iona l ru les , and t e l l us when t o worry about Schwinger terms, subtract ions, and whatever o ther mytho log ica l ob jec ts need t o be introduced.

The development of gauge theo r ies owes much t o pa th i n t e g r a l s and i t i s tempt ing t o a t tach more than a h e u r i s t i c va lue t o path i n t e g r a l de r i va t i ons . Although we do no t r e l y on pa th i n t e g r a l s i n t h i s paper, one may t h i n k of expanding t h e exponent of t h e i n t e r a c t i o n Lagrangian i n a Tay lo r ser ies, so t h a t t he a lgebra o f t h e Gaussian i n t e g r a l s becomes exac t l y i d e n t i c a l t o the scheme of manipulat ions w i t h Feynman diagrams. That would leave us w i t h t h e problems of g i v i n g the co r rec t IE p r e s c r i p t i o n i n t h e propagators , and t o f i n d a decent renormal iza t ion scheme.

There i s another aspect t h a t needs emphasis. From the ou tse t t h e canonica l operator formal ism i s no t a pe r tuba t ion theory. w h i l e diagrams c e r t a i n l y are p e r t u r b a t i v e objects . Using d i a - grams as a s t a r t i n g p o i n t seems therefore t o be a c a p i t u l a t i o n i n t h e s t rugg le t o go beyond p e r t u r b a t i o n theory. I t i s un th inkab le t o accept as a f i n a l goa l a p e r t u r b a t i o n theory, and i t i s no t our purpose t o forward such a not ion. On t h e cont ra ry , i t becomes more and more c l e a r t h a t pe r tu rba t i on theory i s a very use fu l dev ice t o d iscover equations and p roper t i es t h a t may ho ld t r u e even if the pe r tu rba t i on expansion f a i l s . There are a l ready seve- r a l examples of t h i s mechanism: on t h e s implest l e v e l t he re i s f o r ins tance t h e treatment o f unstable p a r t i c l e s , wh i l e i f i t

29

DIAGRAMMAR

comes t o unfathomed depths the Callan-Symanzik equation may be quoted. A l l such treatments have i n common t h a t g l o b a l proper- t i e s are es tab l i shed f o r diagrams and then ex t rapo la ted beyond pe r tu rba t i on theory. Global p roper t i es are those t h a t ho ld i n a r b i t r a r y o rder of p e r t u r b a t i o n theory f o r t h e grand t o t a l o f a l l diagrams en te r ing a t any g iven order. I t i s here t h a t very n a t u r a l l y t he concept of t he g l o b a l diagram enter's: i t i s f o r a g iven order o f pe r tu rba t i on theory, f o r a g iven number o f e x t e r n a l l i n e s t h e sum of a l l c o n t r i b u t i n g diagrams. Th is ob ject , very o f ten presented as a blob, an empty c i r c l e , i n the f o l l o w i n g pages, i s supposed t o have a s ign i f i cance beyond pe r tu rba t i on theo- r y . P r a c t i c a l l y all equations o f t h e canonica l formalism can be r e w r i t t e n i n terms of such g l o b a l diagrams, thereby opening up t h e arsena l o f t h e canonica l formal ism f o r t h i s approach.

A f u r t h e r de f ic iency i s r e l a t e d t o t h e d ivergencies o f t he pe r tu rba t i on ser ies. T r a d i t i o n a l l y i t was poss ib le t o make the theory f i n i t e w i t h i n t h e context o f t he canonica l formalism. For i ns tance quantum electrodynamics can be made f i n i t e by means o f P a u l i - V i l l a r s regu la to r f i e l d s , represent ing heavy p a r t i c l e s w i t h wrong m e t r i c o r wrong s t a t i s t i c s . J u d i c i a l choice o f masses and coup l ing constants makes every th ing f i n i t e and gauge i n v a r i a n t and tu rns t h e canonica l formal ism i n t o a reasonably well-behaving machine, f r e e o f ob jec ts such as 6[0), t o name one. Unfor tunate ly t h i s i s no t t he s i t u a t i o n i n the case o f gauge theor ies . There the most s u i t a b l e r e g u l a t o r method, t h e dimensionel r e g u l e r i z e t i o n scheme, i s de f ined exc lus i ve l y f o r diagrams, and up t o now nobody has seen a way t o in t roduce a dimensional canonica l formal ism or pa th i n t e g r a l . The very concept of a f i e l d , and the no t i on o f a H i l b e r t space are too r i g i d t o a l l ow such genera l i za t ions .

The treatment o u t l i n e d i n the fo l l ow ing pages i s no t supposed t o .be complete, but r a t h e r meant as a f i r s t , more o r l ess pedagogical attempt t o implement the above p o i n t o f view f e a t u r i n g g l o b a l diagrams as pr imary ob jects . The most impor tant p roper t i es of t he canonica l as w e l l as pa th i n t e g r a l formal ism are reder ived: u n i t a r i t y , causa l i t y , Faddeev-Popov determinants, e tc . The s t a r - t i n g p o i n t i s always a s e t of Feynemn r u l e s succ inc t l y g iven by means o f a Legranglen. No d e r i v a t i o n o f these r u l e s i s g iven: corresponding t o any Lagrangian [ w i t h very few l i m i t a t i o n s concer- n ing i t s form1 the r u l e s are simply defined.

Subsequently, Green's func t ions , a Hilb81-t space and an S-matrix are def ined i n terms o f diagrams. Next we examine pro- p e r t i e s l i k e u n i t a r i t y and c a u s a l i t y o f the ' r e s u l t i n g theory. The bas i c t o o l f o r t h a t are t he c u t t i n g equations der ived i n the t e x t . The use of these equations r e l a t e s very c l o s e l y t o the c l e s s i c a l work o f Bogoliubov, and Bogoliubov's d e f i n i t i o n o f ceusa l i t y i s seen t o hold. The equations remain t r u e w i t h i n the framework o f the continuous dimension method: renormal iza t ion can the re fo re be

30


t reated B l a Bogoliubov.

O f course, the c u t t i n g equations w i l l t e l l us i n general t ha t thb theory i s not unitary, unless the Lagrangian from which we s ta r ted s a t i s f i e s ce r ta in re la t ions. I n a gauge theory moreover, the S-matrix i e only un i ta ry i n a "physical" H i l b e r t space, which I s a subspace o f the o r i g i n a l H i l b e r t space [ the one tha t wae suggested by the form o f the Lagrangian).

To i l l u s t r a t e i n d e t a i l the complications o f gauge theory we have turned t o good o l d quantum electrodynamics. Even I f t h i s theo- r y lacks some of the complications t h a t may a r i se i n the general case I t turns out t o be s u f f i c i e n t l y structured t o show how every- t h ing works.

The metr ic used throughout the paper i s

The fac to rs 1 i n the f o u r t h components are only there fo r ease o f notat ion, and should not be revereed when tak ing the complex conjugate of a four-vector

I n our Feynman ru les we have e x p l i c i t l y denoted the relevant f ac to rs (Zn14 i , but o f ten omitted the 6 funct ions f o r energy- momentum conservation.

31

D I AG RAMMA R

2 . DEFINITIONS

2.1. D e f i n i t i o n o f the Feynman Rules

The purpose o f t h i s sec t i on i s t o s p e l l ou t t h e p rec i se fo rm o f t he Feynman r u l e s f o r a g iven Lagrangian. I n p r i n c i p l e , t h i s i s very s t ra igh t fo rward : the propagators a re de f ined by t h e quadra t ic p a r t o f t he Lagrangian, and t h e r e s t i s represented by ve r t i ces . As i s we13 known. the propagators are minus t h e i nve rse o f t h e operator found i n the quadra t ic term, f o r example

1 2 -1 1: = ? - + ( a 2 - m l g * (k2 + m2 - i ~ l ,

- i y k + m

k2 + m2 - i e ' 1: - -$(y'a, + m)$ + ( i y k + m 1 - I =

Customari ly, one der ives t h i s us ing commutation r u l e s o f t he f i e l d s , e tc . We w i l l s imply s k i p the d e r i v a t i o n and def ine the propaga- t o r , i n c l u d i n g the IE p r e s c r i p t i o n f o r t h e pole.

S i m i l a r l y , v e r t i c e s a r i se . For instance, i f t h e i n t e r a c t i o n Lagrangian conta ins a term p rov id ing f o r t he i n t e r a c t i o n o f fermions and a sca la r f i e l d one has

I 1

A 9 I n th is . and s i m i l a r cases the re is no d i f f i c u l t y i n d e r i v i n g t h e r u l e s by the usua l canonica l formalism, I f however de r i va t i ves , o r worse non- loca l terms, occur i n L, then compl icat ions ar ise . Again we w i l l s h o r t - c i r c u i t a l l d i f f i c u l t i e s and def ine o u r ve r t i ces , i n c l u d i n g non- loca l v e r t i c e s , d i r e c t l y f rom t h e Lagran- g ian. Furthermore, we w i l l a l l ow sources t h a t can absorb o r emit p a r t i c l e s . They are an Impor tant t o o l i n t h e analys is . I n t h e r e s t o f t h i s sec t ion we w i l l t r y t o d e f i n e p r e c i s e l y t h e Feynman rules f o r t h e genera l case, i n c l u d i n g f a c t o r s II, etc. B a s i c a l l y t he rec ipe i s t he s t ra igh t fo rward genera l i za t i on o f t h e simple cases shown above.

The most genera l Lagrangian t o be discussed here i s

(2.11

The and d m o t e se ts o f complex and r e a l f i e l d s t h a t may be

32


scalar, spinor, vector, tensor, etc., f i e l d s . The index i stands f o r any spinor, Lorentz, isospin, etc., index. V and W are matr ix operators tha t may contain der ivat ives, and whose Four ier transform must have an inverse. Furthermore, these inverses must s a t i s f y the Killen-Lehmann representation, t o be discussed la te r . The In te rac t i on Lagrangian XI ($*,$,41 i s any polynomial i n ce r ta in coupling constants g as w e l l as the f i e l d s . This i n t e r a c t i o n La- grangian i s allowed t o be non-local, 1.e. not only depend on f i e l d s I n the po in t x, but a lso on f i e l d s a t other space t i m e po ints XI, XI', ... . The coe f f i c i en ts i n the polynomial expansion may be funct ions o f x. The e x p l i c i t form o f a general t e r m inl: [x ) i s I

[ d4Xld4x2 8 (x, x1,x2, ... 1 %,I*. . .

(2.21 * $, [x,), ..., $, ( X m L ..., 4, [ X n L ... .

1 m n

The u may contain any number o f d i f f e r e n t i a l operators working on the various f i e l d s .

Roughly speaking propagators are defined t o be minus the inverse o f the Four ier transforms o f V and W, and ver t ices as the Four ier transforms o f the coef f ic ients a i n 1: I'

The act ion S i s defined by

I n 1: we make the replacement

* i k x $I(xl - [ d4k bi(kle ,

4i(x1 = 1 d4k ci(kle i k x ,

ai i (x, xl, x2, ... 1 = 1 z . . .

33

DIAGRAMMAR

i k x + i k (x -x l )+ ik (x-x21+ ... - 0 (k,k1,k2 ,... I.

= I d4k d4kl d4k 2 , . . e , e 1 2 il...

The a c t i o n t imes I takes the form

each term in teg ra ted over the momenta invo lved. The i and w conta in a f a c t o r i k ( o r -1k 1 f o r every d e r i v a t i v e a/ax a c t i n g t o the r i g h t [ l e f t l n i n V andnW, respec t ive ly . The a con tk ln a f a c t o r i k f o r every d e r i v a t i v e a/ax a c t i n g on a f i e l d w i t h argument A:. JlJ

The propagators are defined t o be:

- - - Here w i s re f lec ted , 1.e. - = W . I n the r a r e case o f r e a l fermions the propagator mustw&$ mind& the inverse o f t he a n t i - symmetric p a r t o f W. Furthermore, t he re i s t h e usual i e presc r ip - t i o n f o r the poles o f these propagators. The momentum k i n Eqs. ( 2 . 5 ) i s t he momentum f low i n the d i r e c t i o n of t he arrow.

The d e f i n i t i o n o f t h e v e r t i c e s i s :

...164(k + k + ... 1. (2.6 1 k2 * 1

x a [k, kl, il.. .

34

G. ‘t HOOFT and M. VELTMAN

The summation i s over a l l permutations o f the ind ices and momenta indicated. The momenta are taken t o f l o w inwards. Any f i e l d $* corresponds t o a l i n e w i t h an arrow po in t i ng outwards; a f i e l d $ gives an opposite arrow. The 4 f i e l d s give arrow-less l ines. The fac to r ( - 1 l p i s only o f importance i f several fermion f i e l d occur. A l l fermion f i e l d s are taken t o anticommute w i t h a l l other fermion f i e l d s . There i s a f a c t o r -1 f o r every permutation exchanging two fermion f i e l d s .

The coe f f i c i en ts u w i l l o f t en be constants. Then the sum over permutations r e s u l t s simply i n a factor . I t i s convenient t o include such fac to rs already i n XI; f o r instance

e,cx1 - - g$* ( x I 3$ ( X I 3:6:

gives as vertex simply the constant g.

As indicated, the c o e f f i c i e n t s a may be funct ions of X. corresponding t o some a r b i t r a r y dependence on the momentum k i n (2.6). This momentum i s not associated w i t h any o f the l i n e s of the vertex. I f we have such a k dependence, i .e. the c o e f f i c i e n t a i s non-zero f o r some non-zero value o f k, then t h i s vertex w i l l be ca l led a source. Sources w i l l be ind icated by a cross or other convenient notat ion as the need ar ises.

A diagram i s obtained by connecting ve r t i ces and sources by means o f propagators i n accordance w i th the arrow notations. Any diagram i s provided w i t h a combinatorial f a c t o r t ha t corrects fo r double counting i n case i d e n t i c a l p a r t i c l e s occur. The compu- t a t i o n of these factors i s somewhat cumberaorne; the recipe i s given i n Appendix A.

Further, i f f e n i o n s occur, diagrems are provided w i t h a sign. The r u l e i s as fo l lows:

i l there i s a minus s ign f o r every closed ferrnion loopj

addi t ion o f boson l i n e s have the same sign; i i l diagrams t h a t are r e l a t e d t o each other by the omission o r

35

DIAGRAMMAR

i i i l diagrams r e l a t e d by the exchange of two fermion l ines , in ternal or e x t e r n a l , have a r e l a t i v e m i n u s s ign .

EXAMPLE 2.1.1 E lec t ron -e l ec t ron s c a t t e r i n g i n quantum-electrodynamics. Some diagrams and t h e i r r e l a t i v e s i g n a r e given i n t h e f i g u r e . The f o u r t h and f i f t h diagrams a r e r e a l l y t h e same diegram and should not be counted s e p a r a t e l y :

+l +1 +1

m x -1 -1

T h i s r a i s e s t h e ques t ion o f t h e r ecogn i t ion of topologi - c a l l y i d e n t i c a l diagrams. I n Appendix A some cons ide ra t ions p e r t i n e n t t o t h e topology of quantum electrodynamics are presented .

2.2 A Simple Theorem 3 Some Examples

THEOREM Diagrams a r e i n v a r i a n t f o r t h e replacement

i n t h e Lagrangian (1.11, w h e r e X i s any mat r ix t h a t may i n c l u d e d e r i v a t i v e s b u t m u s t have an inve r se .

T h e theorem i s l t r i v i a l t o prove. Any o r i en ted propagator ob ta ins a f a c t o r X t h a t cance l s a g a i n s t t h e f a c t o r X occurr ing i n t h e v e r t i c e s . I t is l e f t t o t h e r eade r t o g e n e r a l i z e t h i s theorem t o i n c l u d e t ransformat ions of t h e $J'S and 4's.

Some examples i l l u s t r a t i n g our d e f i n i t i o n s a r e i n o rder . 4 EXAMPLE 2.2.1 Charged s c a l a r p a r t i c l e s w i t h J, i n t e r a c t i o n

[2.71 * * * = $?a2 - m21$ + [ a $1' + J + $ J ( i c ) . 2:2:

36


* Only 9 1x1 and'9[x l i n the same po in t x occur, and i n accordance wi th the descr ip t ion given a t the beginhing o f the previous sect ion we have here a l o c a l Lagrangian. The funct ions J [ x I end J* (XI ere source functions.

Propagator: - k

Vertex:

x

1 1 AF 1 - [Zn14i k2 + m2 - ic'

4 (2nl ig ( 8 funct ion f o r energy-momentum

conservation is understood).

Sources :

* Tkp funct ions J [ k l , J [ k l are the Four ier transforms o f J r x l and J [XI

J [ x l = d4ke-ikxJ[kl.

Note t h a t k i s the momentum f lowing from the l i n e I n t o the source.

Some diagrams:

* * J (-p1 1 J [-pt) J [P,] J (P41

pi; + m2 - i e p: + m2 - i c p: + m2 - ic p: + m2 - i c

6 4 (PI + P2 - Pg - P4) x

37

DIAGRAMMAR

* * J ( - p l ) J [ -p21 J [P3) J ( P 4 )

p2 + m2 - i E pz + m - iE p2 + m2 - 1 s p i + m2 - is 2 1 3

1 2 1 1 x 7 g 1 d4k

k2 + m2 - IE [ k + p1 + p212 + m2 - i E '

The f a c t o r 1/2 I n t h e second case I s t h e combina tor la l f a c t o r occur r ing because o f t he i d e n t i c a l p a r t i c l e s I n t h e i n t e r - mediate s ta te .

EXAMPLE 2.2.2 Free r e a l vec tor mesons

2 1 JW2 - a w l - - 4 [auwv v u 2 u* 1 1: E - -

I n terms o f f o u r r e a l f i e l d s 6 : M a , a = 1, .... 4. a

The m a t r i x V i n momentum space I s

-6 (k2 + m 2 1 + k,kB. aB

The vec tor meson propagator becomes:

( 2 . 8 )

2 -1 1 6aB+k k /m

- - i k j - ' ( V l a ~ = - *FaB= [ 2r 41 [ Z n I 4 i k2 + m2 - IE

(2.91


Indeed

68A + k8kX/m 2

( k 2 + m21 - kaksl r6a8 k2 + m2 - 1~

[k2 + m21 - kakA 1

k2 + m2 - i c

k2

m - kakX 2 3 = 6aX. k2 + m2

+ kakX 2

EXAMPLE 2.2.3 Elect ron i n an e x t e r n a l electromagnetic f i e l d A U

.t = -Il(vlJalJ + m l J , + ieA,$y'$ . (2.101

4 Note t h a t 6 = $*y4. Because o f our l i t t l e theorem t h e mat r ix y can be omitted i n g i v i n g r u l e s for t h e diagrams. Then:

k

As a f u r t h e r a p p l i c a t i o n o f our theorem we may subst i tue

J, -t ( -yva + ml$ t o g i v e v

Th is gives t h e equivalent r u l e s :

39

DIAGRAMMAR

2.3 I n t e r n a l consistency

Two p o i n t s need t o r e a l and complex f i e l d s complex f i e l d (a one can

4 = - ( A + 191, fi

be inves t iga ted . F i r s t , t he separat ion i n is r e a l l y a r b i t r a r y , because f o r any always write

where A and 6 are r e a l f i e l d s . The quest ion is whether the r e s u l t 8 w i l l be independent o f t h e representa t ion chosen. Th is indeed is t h e case, and may be best expla ined by cons ider ing an example:

i - 4 * (2nl i J i

4 - ( 2 ~ 1 i J i

The diagram conta in ing two sources i s :

On t h e o ther hand, l e t us w r i t e 41 - ( l / f i ) ( A + 16)

+ & v + h V 6 - b V A + 1: - + i V i j A j 2 1 i j6j 2 1 i j 1 2 1 i j j

* + AiJi - iE,J,I + - (JiAi + iJi6i 1 *

v5

G. 't HOOF1 and M. VELTMAN

Def in ing t h e r e a l f i e l d X,

we have

w i t h

where t h e superscr ip ts s and a denote the symmetrical and t h e ant isymmetr ica l par t , respec t ive ly .

Le t Y now be the i nve rse o f V. Thus

VY - 1 , YV = 1.

W r i t i n g V - Vs + V', Y = Y s + Ya one f i nds , comparing t h e r e f l e c - t e d VY w i t h YV

VSYS + vBYa = 1 ,

v9ya + \jays I y a p + y s p I 0 . The inverse of t he ma t r i x W is therefore

w-1 = [ - ysa i Y ;:] , *

where Y - V - I . The source-source diagram becomes

as before. C l e a r l y t h e separat ion i n t o r e a l and imaginary parts amounts t o separat ion i n t o symmetric and ant isymmetr ic p a r t s of t he propagator.

41

DIAGRAMMAR

The second p o i n t t o be i nves t i ga ted i s t h e quest ion of separat ion o f t h e quadrat ic , 1.e. propagator de f i n ing , p a r t i n e. Thus l e t there be g iven the Lagrangian

* * 1: = 4ivij4i + 4iv;j4j

-1 One can e i t h e r say t h a t one has a propagatorl-(V + V'] 4 - f i e l d , o r a l t e r n a t i v e l y a propagator - ( V ) ' and 8 ver tex :

f o r t he

However, these t w o cases g i ve the same r e s u l t . Sumning over a l l poss ib le i n s e r t i o n s o f t he ver tex V ' one f i n d s :

1 1 1 - - + [ - - 3v' I - - I + V V V

1 1 1 = - v 1 + v' /v v + v" ... = - -

This demonstrates the i n t e r n a l consistency o f our scheme of d e f i n i t i o n s . We leave i t as an exerc ise t o the reader t o v e r i f y t h a t combinational f a c t o r s check i f one makes t h e replacement 4 + [ A + i B ) / & , f o r ins tance f o r t he diagrams:

2.4 D e f i n i t i o n o f the Green's Funct ions

Le t there be g iven a genera l Lagrangian o f t he form (2.11. Corresponding t o any f i e l d we in t roduce source terms [we w i l l need many, bu t w r i t e on ly one f o r each f i e l d )

* * e + l + J $ 1 1 + $IJ, + 4 i K i

According t o t he prev ious r u l e s such terms g i ve r i s e t o the fo l l ow ing v e r t i c e s :

42


- [2n14iK[k l ,

where t h e k dependence i m p l i e s Four ie r t ransformat ion.

Remember now t h a t t he Lagrangian was a polynomia l i n some coup l ing constants. F o r a g iven order i n these coup l ing constants we may consider t h e sum o f a l l diagrams connecting n sources. A l l n sources are t o be taken d i f f e r e n t , because we want t o be able

t o very the momenta independently. Each o f these diagrams, and a f o r t i o r i t h e i r sum, w i l l be of t h e form

Ji [ k 13' (k21 Ki [ k )Gi ... in(kl, ..., k I. n 1 I 2 n " 1

The f u n c t i o n Gi w i l l be c a l l e d the n-po in t Green's f u n c t i o n f o r t h e g iven ex te rna l l i n e conf igura t ion f o r t h e order i n the coup- l i n g constants spec i f i ed , Factors (Zn14i from the source v e r t i c e s are inc luded. The k denote t h e momenta f l o w i n g f rom t h e sources i n t o the diagrams. +he propagators t h a t connect t o the sources are inc luded i n t h i s d e f i n i t i o n .

The f i r s t example o f Sect ion 2.2 shows some diagrams c o n t r i - b u t i n g t o the second-order f o u r - p o i n t Green's func t ions .

2.5 D e f i n i t i o n o f t he S-Matrix s Some Examples

Roughly speaking the S-matrix obta ins f rom t h e Green's func- t i o n s i n two steps: [ i l t h e momenta o f t he e x t e r n a l l i n e are pu t on mess she l l , and [ii) t h e sources are normalized such t h a t they correspond t o emission o r absorpt ion o f one p a r t i c l e . Both these statements are somewhat vague. and we must p rec i se them, b u t they

43

DIAGRAMMAR

* ( a l ( k ) I Kji(klJ:al(kl =

I J d

r e f l e c t t he e s s e n t i a l phys i ca l content o f t he reasoning below.

1 f o r I n t e g e r sp in

kO - f o r h a l f - i n t e g e r sp in . m

Consider t h e diagrams connecting two sources:

The corresponding expression Is

The two-point Green's f u n c t i o n w i l l i n genera l have a po le (Or poss ib l y many s i n g l e po les) a t some value -PI2 o f t h e squared four-momentum k . I f the re i s no po le the re w i l l be no corresponding S-matrix element1 such w i l l be the case I f a p a r t i c l e becomes unstable because of t he In te rac t i ons . A t t he po le the Green's f u n c t i o n w i l l be o f t h e form

2 K ( k l

i j k + M 4 G [k, k ' l = ( 2 ~ 1 i S 4 [ k + k ' l h a t k2 + -M .

The m a t r i x res idue K can be a f u n c t i o n o f t he components k , w i t h the r e s t r i c t i o n thatijk2 = -M2. lJ

F i r s t we w i l l t r e a t t he cur ren ts for emission of a p a r t i c l e , corresponding t o n ominy p a r t i c l e s o f t h e S-matrix. Def ine a new se t o f cur ren ts ,+a', one f o r every non-zero eigenvalue o f K, which are mutua l l y or thogonal and e igenstates o f t h e m a t r i x K(k ) 1

(a)*J(bl = 0 I f a # b , Ji 1

This i s poss ib le on ly if a l l elgenvalues of K are p o s i t i v e . I n

44


the case o f negative eigenvalues nonna l iza t ion i s done w i t h minus the r igh t -hand s ide o f Eq. (2.141.

The sources thus def ined are the proper ly normalized sources f o r emission o f a p a r t i c l e or a n t i p a r t i c l e [ t he l a t t e r fo l lows from consider ing )?[-k1 1. The proper ly normalized sources f o r absorption o f a p a r t i c l e or a n t i p a r t i c l e fo l low by considering Eqs. (2.131 and t2.141 bu t w i t h k replaced by -k i n K.

The above procedure defines the cur ren ts up t o a phase fec- t o r . We must take care t h a t the phase fac to r f o r the emission of a c e r t a i n p a r t i c l e agrees w i t h t h a t f o r absorpt ion o f t h a t same p a r t i c l e . Th is i s f i x e d by r e q u i r i n g t h a t t he two-point Green's f u n c t i o n p rov id d w i t h such sources has p rec i se l y the residue 1 ( o r k o h l f o r k9 = -m2.

the S-matrix, see below1 f o r n ingo ing p a r t i c l e s and m outgoing p a r t i c l e s are def ined by

The mat r i x elements o f t he mat r ix S' (almost, bu t no t exac t ly

n

m

are a l l pos i t i ve . 'mo The energies k , ..., k and p , ..., The minus signl?or the m88enta p% the Green's f u n c t i o n appears because i n the ma t r i x element the momenta o f the outgoing p a r t i - c l es are taken t o be f lowing out, wh i l e i n our Green's func t ions the convention was t h a t a l l momenta f l o w inwards.

EXAMPLE 2.5.1 Scalar p a r t i c l e s . Near the phys i ca l mass po le the two-point Green's funct ion w i l l be such t h a t

(2.161 G(k, k '1 1 4 [2nl icS4(k - k '1 z2(k2 + N21'

.. For any k t he proper ly normalized ex te rna l cur ren t i s J - Z, and the p reac r ip t i on t o f i n d S'-matr ix elements w i t h ex te rna l sca la r p a r t i c l e s i s

45

EXAMPLE 2.5.2 Fermlons as i n OED. Near t h e p h y s i c a l mass po le the two-point Green’s f u n c t i o n w i l l be such t h a t

K [ k )

k2 + M -6 (2.181 2’

1

2 [ i y k + M l

GI (k, k’ 1

(271) 1s4(k - k ’ l 4 = 2

w i t h

Asset o f e igenstates of t h i s m a t r i x -3 prov ided by the s o l u t i o n s u ( k l o f t he D i rac equat ion

* a a

[ l y k + M 1 u = O , u u - 1 , a = l , 2 . (2.201

Because o f t he n o r ~ ~ ) i z a t i o n cond i t i on Eq. (2.141, we must take f o r t he cur ren ts J ( k l

(2.21 1

w i t h t h e D i rac sp lnors normalized t o 1, see Eq. (2.201. Note t h a t t he f a c t o r 1/2M cancels upon m u l t i p l i c a t l o n o f t h i s source w i t h t h e propagator numerator [2.19).

The a n t i p a r t i c l e emission cur ren ts a re obta ined by cons ider ing K [ - k l . The so lu t i ons are t h e a n t i p a r t i c l e sp lnors

i a ( k l , a = 3, 4.

For outgoing a n t i p e r t i c l e s and p a r t i c l e s K I - k l end K[k ) ...

must be considered t o ob ta ig the proper absorp t ion currents . The s o l u t i o n s are t h e spinors u ( k l , a = 3, 4 and c a t k l , a = 1, 2.

46


The phase condi t ion GaKua = 2M d i c t a t e s a m i n u s s ign f o r t h e source f o r emission of an a n t i p a r t i c l e (see Appendix A ) .

Let u s now complete t h e S-matrix d e f i n i t i o n . The p r e s c r i p - t i o n given above results i n z e r o when appl ied t o t h e two-point func t ion , because there w i l l be two f a c t o r s K2 + M2 [one f o r every sou rce ) . T h u s we ev iden t ly do not ob ta in e x a c t l y t h e S-matrix t h a t one has i n s u c h cases. T h i s discrepency i s related t o t h e t rea tment of t h e o v e r - a l l d f u n c t i o n of energy-momentum conserva- t i o n , when pass ing from S-matrix elements t o t r a n s i t i o n p r o b a b i l i - t i es . Anyway, t o g e t t h e S-matrix we m u s t a l s o e l low l ines where p a r t i c l e s go through without any i n t e r e c t i o n , and a s s o c i a t e a f a c t o r 1 w i t h s u c h lines. T h e s e psr t ic les must. of course , have a mass as given by t h e po le of t h e i r propagator .

Matrix e lements of t h e S-matr ix , i nc lud ing p o s s i b l e l i n e s going s t r a i g h t through, w i l l be denoted by graphs w i t h e x t e r n a l l ines t h a t have no t e rmina t ing c ros s . The convention i s : l e f t ere

incoming p a r t i c l e s , r i g h t outgoing. Energy f lows from l e f t t o r i g h t . The d i r e c t i o n of t h e arrows is of course not r e l a t e d t o t h e d i r e c t i o n of t h e energy f low.

We emph8SlZe aga in t h a t t h e above S-matrix elements inc lude diagrams conta in ing i n t e r a c t i o n

diagram shown i s included i n t h e 3-particle-in/3-particle-out S- matrix element.

- f r e e l ines . For i n s t a n c e , t h e

T h e d e f i n i t i o n of t h e S-matrix given above a p p l i e s if there is no gauge symmetry. F o r gauge t h e o r i e s some of t h e propagators correspond t o ”ghost” p a r t i c l e s t h a t are essumed not t o have phy- s i c a l re levance . I n de f in ing t h e S-matr ix , t h e sources m u s t be r e s t r i c t e d t o emit or absorb only phys ica l p a r t i c l e s . Such sources w i l l be c a l l e d p h y s i c a l sources and have t o be d e f i n e d i n t h e p r e c i s e contex t of t h e gauge symmetry. To show i n s u c h ca ses t h a t t h e S-matrix i s u n i t a r y requires t h e n s p e c i a l e f f o r t .

2.6 D e f i n i t i o n of S t

T h e mat r ix elements of t h e mat r ix St a r e d e f i n e d as u s u a l by

* < a l s t [ 0> = 43ISla> , ( 2 . 2 2 )

where t h e complex conjugat ion impl i e s a l s o t h e replacement i e + -1e i n the propagators .

47

D I AG RAMMAR

The matr ix elements of St can also be obtained i n another way. In addi t ion t o the Lagrangian 1: def in ing S, consider t h e conjugated Lagrangian f t . I t i s obtained from 1: by complex conjugation and reversa l of t h e order o f t h e f i e l d s . The latter i s only re levant f o r fermions. This tt may be used t o def ine another S-matrix; l e t 3 denote the matrix obtained i n the usual way from et, however with the opposi te s ign f o r the is i n the propagators and a l so t h e replacement i + -i i n the notor ious f a c t o r s ( h ) 4 i . We claim t h a t t he matr ix elements of 6 are equal t o those o f S+, In formula we ge t

< a l s u , i I + l B >=<als ( l t , - i l l e >. T h e p roof rests mainly on t h e o b s e r v a t i o n t h a t an incoming p a r - t i c l e s o u r c e is o b t a i n e d by c o n s i d e r i n g [ n o t a t i o n of S e c t i o n 2.5, Eq. (2.1311:

and t h e complex c o n j u g a t e of an o u t g o i n g p a r t i c l e s o u r c e by s t u d y o f :

o r , e q u i v a l e n t l y ,

- T h i s is i ndeed w h a t c o r r e s onds t o complex con ju -

= K j i * ? where K g a t i o n &’! t h e p r o p a g a t o r d e f i n i n g p a r t of 1:

Note t h e change of s i g n of the d e r i v a t i v e : w h a t worked t o t h e r i g h t works now t o t h e l e f t , w h i c h i m p l i e s a m i n u s s i g n . Vt i s o b t a i n e d by t r a n s p o s i t i o n and complex c o n j u g a t i o n of V. C l e a r l y complex c o n j u g a t i o n and exchange of incoming and o u t g o i n g s ta tes c o r r e s p o n d s t o t h e use of Vt i n s t e a d of V.

EXAMPLE 2.6.1 Fermion coup led t o complex sca la r f i e l d

5 * I: = - icy3 + m)$ + $*[a2 - mz)g + g i ( 1 + y ,

48


The lowest o rder S-matrix element I s :

According t o Eq. (2.221

5 There are minus s igns because of y4, y except a4 + a4. I n c l u d i n g the s ign change f o r [2,1 1, we ob ta in f o r S:

exchange and 3 -P -3 changes 4

which equals t h e r e s u l t f o r Si found above.

t To summarize, t h e m a t r i x elements o f S can be obta ined e i t h e r d i r e c t l y f r o m t h e i r d e f i n i t i o n (2.221, or by the use of d i f f e ren t Feynmsn ru les . These new r u l e s can be obta ined f rom t h e o l d ones by reve rs ing a l l arrows i n v e r t i c e s and propagators and rep lac ing a l l ve r tex func t i ons and propagators by t h e i r complex conjugate ( f o r t h e propagators t h i s means us ing t h e Hermi t ien conjugate propagators). Also, t h e fac to rs ( Z n I 4 i and the l e terms i n the propagators e re t o be complex conjugated. The i n - and ou t - s ta te source func t ions are de f ined by the usua l procedure, i n v o l v i n g now t h e Hermi t ien conjugate propagators,

L e t us f l n e l l y , f o r t h e seke of c l a r i t y , fo rmula te the d e f l - n i t i o n o f i ngo ing and outgoing s ta tes i n the diagram language f o r bo th t h e o l d and the new r u l e s :

49

DIAGRAMMAR

in )I--c- antiparticle

Y_ particle

I a>

out - antiparticle - particle

2.7 Definition of Transition Probabilities j

Cross-Sections and Lifetimes

The S-matrix elements are the transition amplitudes of the theory. The probability amplitudes are defined by the absolute value squared of these amplitudes. Conservation of probability requires that the S-matrix be unitary

This property will be true only if the diagrams satisfy certain conditions, and we will investigate this in Section 6.

In the usual way, lifetimes and cross-sections can be deduced from the transition probabilities.

Consider the decay o f particle a into particles 1, 2, 3, ...' n. The decay width r [ = inverse lifetime T I is

1 d3P 1 d3Pn 64[P - p1 - P2 * - * -Pnl - = r = / 4 T 2Pl0t2lr1 zpn0 t 2 n ~ 3 2p omr l 3' * " ,

t x<alM 11, 2 , ... , n>.a, 2 , ..., nlNla >, (2.24)

t t where the matrix elements of M [and M 1 are those of S [and S 1 without the energy-momentum conservation 6 function

G. ‘t HOOF1 and M. VELTMAN

The p ..., p are t h e energies o f p a r t i c l e s a, I, 2. ..., n. o l v l d i % ~ r by nns 6.587 x 1022 MeV sec g ives r i n sec-1, p r o v i - ded a l l was computed i n n a t u r a l u n i t s ( h = c - 1) w i t h t h e MeV as t h e on ly u n i t l e f t .

Next, consider the s c a t t e r i n g o f a p a r t i c l e a on a p a r t i c l e B a t r e s t g i v i n g r i s e t o a f i n a l s t a t e w i t h n p a r t i c l e s . The cross-sect ion u f o r t h i s process i s

+ Here pa i s the three-momentum of t he incoming p a r t i c l e a i n t he r e s t system of t he B f f r t i c l e [ t a r g e t p a r t i c l e l . M u l t i p l y i n g by (hc I2 = (1.9732 x 10- MeV cmI2, one ob ta ins u i n cm2.

51

DIAGRAMMAR

3. DIAGRAMS AND FUNCTIONAL INTEGRALS

I n a l l p roo fs we w i l l r e l y on l y on combinatorics o f diagrams. However, the r u l e s and d e f i n i t i o n s may look somewhat ad hoc, and the purpose o f t h i s sec t i on i s t o show a fo rmal equivalence w i t h c e r t a i n i n t e g r a l expressions.

Imagine a wor ld w i t h on ly a f i n i t e number of space-time p o i n t s x , a = 1, ..., N j p = 1, .... 4. Fo r s i m p l i c i t y , we consider’only the case o f r e a l boson f i e l d s . The a c t i o n S i s now

where W i s taken t o be symmetric. Der i va t i ves occur as d i f fe rences . The diagram r u l e s corresponding t o t h i s a c t i o n are as g iven I n Sec t ion 2.

Suppose now t h a t S i s r e a l i f I$ i s r e a l . The f o l l o w i n g theorem holds.

The rules def ined i n Sec t ion 2 f o r connected diagrams are p r e c i s e l y the same as t h e mathematical r u l e s t o ob ta in the f u n c t i o n r def ined by

where t h e r igh t -hand s ide i s understood as a se r ies expansion i n terms o f t h e c o e f f i c i e n t s i n JI. Here C i s an a r b i t r a r y constant no t depending on the sources i n 1: d ing disconnected ones a re obta ined by expanding elr.

The s e t of a l l diagrams i n c l u - I’

Ins tead o f Eq. [3.2) we w i l l use the condensed n o t a t i o n

C3.31 i r isr41 e = c / P 4 e

The theorem i s easy t o prove. F i r s t c a l c u l a t e t h e i n t e g r a l f o r f r e e p a r t i c l e s , us ing f o r t h e a c t i o n the expression

(3.41

w i t h a r b i t r a r y source func t ions J. Def ine

52


r c a n be f o u n d by meking a s h i f t i n t h e i n t e g r a t i o n v e r i a b l e s 0

so t h a t

T h e pr imed f i e l d s are n o t c o u p l e d t o t h e J: We f i n d

b e c a u s e t h e i n t e g r a l o v e r t h e (0’ i s now a s o u r c e i n d e p e n d e n t c o n s t a n t and c a n be a b s o r b e d I n t h e c o n s t a n t C I n E q . [3.31. Note t h e f a c t o r 112 b e c a u s e of i d e n t i c a l s o u r c e s .

0 We see t h a t I’ is n o t h i n g b u t t h e f ree p a r t i c l e p r o p a g a t o r . Even t h e itz p r e s c r i p t i o n c0n be I n t r o d u c e d c o r r e c t l y if w e i n t r o - duce a smooth c u t - o f f f o r t h e i n t e g r a l f o r l a r g e v a l u e s of t h e f i e l d s (0

T h i s makes t h e i n t e g r a l well d e f i n e d even i n d i r e c t i o n s where (0W+=O. The result is t h e u s u a l I& a d d i t i o n t c W.

Let us now c o n s i d e r also i n t e r a c t i o n terms. T h e p e r t u r b a t i o n e x p a n s i o n I s

a i2 a b 1 LI(X ,(01LI(X ,OI . . . = 1 + i 1 LI(X ,$I + 2: icL, r xal

e a a , b

(3.101 or

r .

w h e r e t h e s o u r c e s are i n c l u d e d i n S . From t h e d e f i n i t i o n o f So we have 0

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D IAGRAMMAR

a Consequently we can, replace everywhere the fields 4 [x I by the derivative 6/6Ji[x 1 1

The remaining integral I s precisely the one computed before and is equal to exp Iiro1 with r0 given in Eq. (3.0). Expanding this exponential

(3.14) ir 1 -1 2 2 8

e o = 1 - - i 1 JW-IJ + i2[JW J ) ... , we see that every term in Eq. 13.131 corresponds to a diagram with vertices and propagators as defined in Section 2. Diagrams not coupled to sources are called vacuum renormalizations and must be absorbed in the constant C. Note the combinatorial factors due to the occurrence of identical sources.

The functional integral notation 13.31 for t h e amplitude in the presence of sources is elegant and compact, and it is very tempting to write the amplitudes of relativistic field theories in this way. Many theorists can indeed not resist this temptation. Let us see what is involved. In our definition r3.3) we restricted ourselves to a finite number o f space-time points. In any realistic theory this number is infinite end summations are to be replaced by integrations. So a suitable limiting procedure must be defined, but unfortunately these definitions cannot always be given in a manner free of ambiguities. Generally speaking, difficulties set in about at the same point that difficulties appear in the usual operator formalism, in particular we mention higher order derivatives o r worse non-localities in the Lagrangian. We shall, therefore, in this report not try to formulate such a definition but attach to the result of manipulations with functional integrals a heuristic value. Everything is to be verified explicitly by combinatorics of diagrams. But having verified certain basic algebraic properties we can happily manipulate these "path" integrals. One of the most interesting manipulations used with great advantage in connection with gauge theories is to change of field variables. Both local and non-local canonical transformations turn out to be correctly described by the path integral formulae, as we shall see later.

54


We may f i n a l l y mention t h a t t he var ious s ign prescr ip t ions , i n t h e case o f fermion f i e l d s . cannot be obtained i n a simple way, They can be corrected f o r by hand afterwards, or i n a more sophis- t i c a t e d way an algebra o f anticommuting va r iab les can be i n t r o - duced. Such work can be found i n t he l i t e r a t u r e and i s q u i t e s t ra igh t fo rward .

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4. KALLEN-LEHMANN REPRESENTATION -1 -1

T h e q u a n t i t i e s -V and -W (see Eqs. [2,511 t h a t a r e d e f i - ned a s propagators I n t h e theory a r e more p r e c i s e l y c a l l e d t h e - bare propagators . T h i s i s c o n t r a s t t o t h e two-point Green’s func t ion t h a t when d i v i d e d by - ( 2 ~ ) * 6 4 ( k + k’l i s c a l l e d t h e dressed propagator . Both t h e bare and dressed propagators a r e requi red t o s a t i s f y t h e KallBn-Lehmenn rep resen ta t ion , but i t w i l l t u r n out t h a t t h e dressed propagators s a t i s f y t h i s automa- t i c a l l y i f t h e bare propagators do.

Consider any propagator and decompose it I n t o i n v a r i a n t func t ions . For i n s t ance , f o r vec to r mesons

The KallBn-Lehmann rep resen ta t ion f o r t h e I n v a r i a n t func t ions i s

T h e func t ions p(s ’1 m u s t be r e a l . Fo r bare propagators we w i l l i n s i s t t h a t t h e p ( s ’ 1 a r e a sum of 6 func t ions

f o r bare propagators (4.31 p(s’1 = 1 ai6(s’ - mi) i

2 1‘ w i t h r e a l c o e f f i c i e n t s ai , and r e a l p o s i t i v e m

Now In t roduce t h e Four i e r t ransform of t h e propagators

i k x AFij(xl = I d4ke A ( k l . F i j

(4.41

Corresponding t o t h e decomposition of A ( k l i n i n v a r i a n t func t ions we w i l l have a decomposition involv ing d e r i v a t i v e s . For i n s t ance , f o r vec to r mesons

F

where A and A a r e t h e Four i e r t ransforms of f and f above. T h e staJement gha t any func t ion f ( x l setisfles *he Kalf6n-Lehrnann r ep resen ta t ion I s equiva len t t o t h e s ta tement

56


f ( x i = e [ x o l f + ( x i + e [ - x o l f - [ x i , (4.61

+ - where f [ f 1 i s a p o s i t i v e (negat ive1 energy func t ion

The p roo f is very simple. Using

1 d4keikxekk 0 16(k2 + 9’1 . (4.71

t h e Four ie r representa t ion

(4.81

one der ives immediately t h e r e s u l t . Of course t h i s d e r i v a t i o n is on ly co r rec t if t h e i n t e g r a l i n Eq. (4.21 ex i s t s , and t h i s i s i n genera l t he case. I f no t one must i n t roduce regu la to rs . t o be discussed below. However, we need t h e representa t ion (4.61 no t on l y f o r t he i n v a r i a n t func t ions , bu t also f o r t he complete propagator such as i n Eq. (4.51. Whether t h i s i s t r u e depends on t,he convergence o f t he d i spe rs ion i n t e g r a l s . If t h e func t i ons f and f go f o r x = 0 s u f f i c i e n t l y smoothly over i n t o one another , then t h e exppession (4.61 has no p a r t i c u l a r s i n g u l a r i t y a t x = 0 and one may igno re t h e ac t i on of d e r i v a t i v e s on t h e e f inct ions.-Now f rom the above equal ion (4.z) i t i s c l e a r t h a t f ( x o l = f I - x 1, so i n any case f [01 = f [01. Th is is enough t o t r e a t t h e c8se of one d e r i v a t i v e such as occur r ing i n t h e case o f fermion propagators. Consider now

F o r x = 0 we can do the k i n t e g r a l 0 0

C l e a r l y a f+ - aof- i n xo- 0 on ly i f the d i spe rs ion i n t e g r a l is superconv8rgen t

F o r bare propagators, see Eq. (4.31, t h i s can on ly be t r u e i f some o f t h e coe f f i c i en ts a a re negative, some posot ive. Th is i m p l i e s t h e ex is tence of negat ive m e t r i c p a r t i c l e s , which i n t u r n i

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may lead to unphysical results, such as negative lifetimes or cross-sections, or lack of unitarity depending on how one defines things.

Let us now assume that the superconvergance equation (4.111 holds. Then one obtains Indeed

6(x0)aof+ - 6(xoiaof- = o ,

as well as

6"X01f+ - 6'(xolf- = 0 ,

G. ‘t HOOF1 and M. VELTMAN

5. INTERIM CUT-OFF METHOD: U N I T A R Y REGULATORS

Because of d ivergencies , the d e f i n i t i o n of diagrams given i n Sec t ion 2 may be meaningless. Moreover, t h e propagators may not s a t i s f y t h e KallBn-Lehmann rep resen ta t ion because o f l ack of superconvergence (see Sec t ion 41. To avoid such d i f f i c u l t i e s we in t roduce what we w i l l c a l l u n i t a r y r egu la to r s . T h i s r e g u l a t o r method works f o r any theory i n t h e sense t h a t i t a l lows a proper d e f i n i t i o n of a l l diagrams and i s moreover very s u i t a b l e i n connect ion w i t h t h e proof of u n i t a r y , c a u s a l i t y , etc. I t f a i l s i n the case of Lagrangians i n v a r i a n t under a gauge group, f o r w h i c h we w i l l l a t e r i n t roduce a more s o p h i s t i c a t e d method.

T h e p r e s c r i p t i o n i s exceedingly simple : c o n s t r u c t t h ings so t h a t any propagator i s rep laced by a s u m of propagators . T h e e x t r a propagators correspond t o heavy p a r t i c l e s , and c o e f f i c i e n t s [ t h e ai i n Eq. (4.311 a r e chosen such t h a t the high momentum behaviour i s very good. T h i s i s achieved a s fo l lows . In t roduce i n t h e o r i g i n a l Lagranglen l, Eq. (2.11, sets of r e g u l a t o r f i e l d s

and 0; i n t h e fol lowing way:

(5.11 x The c o e f f i c i e n t s a and b and t h e mass m a t r i c e s Mx and m m u s t

be chosen so as t o assure t h e proper high momentum behaviour , x x x The propagators f o r t h e r e g u l a t i n g f i e l d s $ a r e

-1 x -1 I -a rv + M l id

‘ F i j x RA [5 .21

Because i n t h e i n t e r a c t i o n Lagranglen $ i s everywhere rep laced by + 2 of propagators w i l l occur i n t h e diagrams,

( s i m i l a r l y $* and 91, only t h e fo l lowing combination

1 1 1

x x - - - I , (5.31

Choosing t h e appropr i a t e c o e f f i c i e n t s and mass ma t r i ces a l l diagrams w i l l become f i n i t e . If every e igenvalue of t h e mass ma- t r i x i s made very l a r g e , diagrams t h a t were f i n i t e before t h e i n t r o d u c t i o n of r e g u l a t o r s w i l l converge t o t h e i r unregulated

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value.

EXAMPLE 5 . 1 Charged sca la r p a r t i c l e s w i t h 4 i n t e r a c t i o n 4

(5 .41 1 2

X I I J J + $ l . 1 1

m 4 I k2 + m 2 ' JI propagator: -

(5.61 1 1 1 $ propagator: - -

[Za14i k2 + m2 + M2

The combination occur r ing i n the diagrams is

1 MZ - (2n14i t k 2 + m21 (k2 + m2 + M21

(5 .71

T h i s behaves as k-4 a t h igh momenta. The diagrams shown i n Exem- p l e 2.2.1 are now f i n i t e f o r l a rge bu t f i n i t e M2.


8. CUTTING EQUATIONS

6.1 Prel iminar ies

I n order t o keep the work o f t h i s sect ion transparent, we w i l l suppress indices, der ivat ives, etc. I n pa r t i cu la r , f o r v e r t i - ces we r e t a i n only a factor (2n i4 ig i n momentum space, t ha t i s i g I n coordinate apace. There i s no d i f f i c u l t y whatsoever i n re in t roducing the neceseary de ta i l s .

I t i s assumed t h a t diagrams are su f f i c i en t l y regulated, so tha t no divergencies occur.

The s t a r t i n g po in t i s the decomposition of the propagator I n t o p o s i t i v e and negative energy pa r t s

w i t h x - x, - x,. Here we used the notat ion

I n view of the r e a l i t y o f the spect ra l funct ions p we have

- A& A i l

Consequently

(6 31

As ueual

1 e 1 Tx 1 i f xo ' 0

i f xo < 0 (6 .51 e(x1 = - 7 dT =

2 m -43

O(x1 + 0 r - x ) = 1 . The representation (6.51 can be v e r i f i e d by choosing the in tegra- t i o n contour w i t h a large semi-c i rc le i n the upper ( lower] complex

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p l a n e f o r xo p o s i t i v e ( n e g a t i v e ) .

Cons ide r now a diagram w i t h n ver t ices . Such a diagram re- p r e s e n t s i n c o o r d i n a t e s p a c e an e x p r e s s i o n c o n t a i n i n g many pro- p a g a t o r s depending on arguments xl ' .... x . We w i l l d e n o t e such an e x p r e s s i o n by n

F ( x l , x ..., x I . 2' n

For example, t h e t r i a n g l e diagram r e p r e s e n t s t h e f u n c t i o n :

3 F(x l . x x 2 ' 3 = [ i g l A31A23A,2 ,

(6.6)

Every diagram, when m u l t i p l i e d by t h e a p p r o p r i a t e s o u r c e f u n c t i o n s and i n t e g r a t e d o v e r a l l x c o n t r i b u t e s t o t h e S -ma t r ix . T h e con- t r i b u t i o n t o t h e T-matr ix , d e f i n e d by

S = l + i T ( 6 . 7 1

i s o b t a i n e d by m u l t i p l y i n g by a f a c t o r -1. U n i t a r i t y of t h e S- m a t r i x i m p l i e s an e q u a t i o n f o r t h e imaginary p a r t o f t he so def ined T m a t r i x

(6.81 t T - Tt = I T T .

T h e T-matr ix , or r a t h e r t h e diagrams. are a l s o c o n s t r a i n e d by t h e r equ i r emen t of c a u s a l i t y . As y e t nobody has found a d e f i n i t i o n of c a u s a l i t y t h a t co r re sponds d i r e c t l y t o t h e i n t u i t i v e n o t i o n s i i n s t e a d f o r m u l a t i o n s have been proposed i n v o l v i n g t h e off-mass- s h e l l Green's f u n c t i o n s . We w i l l employ t h e c a u s a l i t y r equ i r emen t i n t h e form proposed by Bogoliubov t h a t has a t l e a s t some i n t u i - t i v e a p p e a l and i s most s u i t a b l e i n connec t ion w i t h a d i a g r e m e - t i c a n a l y s i s . Roughly speak ing Bogo l iubov ' s c o n d i t i o n can be p u t a s f o l l o w s : i f a space - t ime p o i n t x i s i n t h e f u t u r e w i t h res- p a c t t o some o t h e r space - t ime p o i n t x a t h e n t h e diagrams i n v o l -

v o l v e o s i t i v e energy f l o w f rom x t o x on ly .

1

and x can be rewritten i n te?ms of f u n c t i o n s t h a t i n - 2 ving xd 2 1

T h e t r o u b l e w i t h t h i s d e f i n i t i o n i s t h a t space - t ime p o i n t s

62


cannot be accura te ly p inpo in ted w i t h r e l a t i v i s t i c wave packets corresponding t o p a r t i c l e s on mass-shell. Therefore t h i s d e f i n i - t i o n cannot be formulated as an S-matrix cons t ra in t . I t can on ly be used f o r t h e Green's func t ions .

Other d e f i n i t i o n s r e f e r t o the p roper t i es o f t h e f i e l d s . I n p a r t i c u l a r t he re i s t he proposal o f Lehmann, Symanzik and Zimmermann t h a t t he f i e l d s commute ou ts ide the l i g h t cone. De f in ing f i e l d s i n terms o f diagrams, t h i s d e f i n i t i o n can be shown t o reduce t o Bogoliubov's d e f i n i t i o n . The fo rmula t ion o f Bogoliubov causa l i t y i n terms of c u t t i n g r u l e s f o r diagrams w i l l be g iven i n Sect ion 6.4.

6.2 The Largest T i m e Equation

L e t us now consider a f u n c t i o n F [x , ..., x 1 corres- n ponding t o some diagram. We define new & n k o n s F, where one are under l ined. Consider

n o r more o f t h e va r iab les x 1' .*., x

F(xl, xZ, . .*, xis . a * # 2, . . . a Xnl . (6.91 - This f u n c t i o n 13 obtained from the o r i g i n a l f u n c t i o n F by t h e f o l low ing :

1 1

11 1

ill 1

i v 1

V l

A propagator Aki i s unchanged i f n e i t h e r x underl ined. A propagator Aki is rep laced by A i l if x but no t xi i s underl ined. - A propagator Aki i s rep laced by Aki i f x bu t no t xk i s under l ined.

nor x k 1

are

k

1

A propagator A i s replaced by A* i f x and x are under l ined. k i k i k 1

For any under l ined x rep lace one fac to r 1 by -1. Apart f rom tha t , the r u l e s f o r t he v e r t i c e s remain unchanged.

[6.101

Equations (6.11 and (6.41 lead t r i v i a l l y t o an impor tant equation, t he l a rges t t ime equation. Suppose the t ime x i s l a r g e r than any o ther t ime component. Then any functionif i n which x i s not under l ined equals minus the same f u n c t i o n bu t w i t h x now under l ined 1

1

(6.111

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The minus s ign i s a consequence o f p o i n t (v) . I n view o f what f o l l o w s i t i s u s e f u l t o i nven t a diagramnatic representa t ion o f t he newly-defined func t ions :

Any f u n c t i o n F i s represented by a diagram where any ver tex corresponding t o an under- l i n e d v a r i a b l e i s prov ided w i t h a c i r c l e .

EXAMPLE 6.2.1 If F(xl, xzI x31 i s g iven by Eq. (6.6) then

3 + * - F(xl, x x I = ( I g I A31A23A12

2 ’ 3 L A

(6.12)

I f the t ime component of x3 i s l a r g e s t we have, f o r instance:

From such a diagram i t i s impossib le t o see i f a g iven l i n p donngcting a c i r c l e d t o an unc i r c led ve r tex corresponds t o a A or A func t ion . But due t o Eq. (6.21 t he r e s u l t i s t he same anyway. The impor tant f a c t i s t h a t energy always f l ows from the u n c i r c l e d t o the c i r c l e d ver tex, because of t he 8 f u n c t i o n i n Eq. (6.21. O f course the re i s no r e s t r i c t i o n on t h e s ign o f energy f l o w f o r l i n e s connecting two c i r c l e d o r two u n c i r c l e d ve r t i ces .

6.3 Absorpt ive P a r t

To ob ta in the c o n t r i b u t i o n of a diagram t o the S-matrix t he corresponding f u n c t i o n F (x I ..., x 1 must be m u l t i p l i e d w i t h t h e appropr ia te source fu r lc t ions f 8 r t he i ngo ing and out - go ing l i n e s and in teg ra ted over a l l x * For instance, t h e func- t i o n F(xl, . . . I x 1 corresponding t o the diagram: 6

G. 7 HOOF1 and M. VELTMAN

must be mul t ip l i ed by

i p x i p x - i k x - i k x 1 le 2 Se 1 3 e 2 4 9

1' . * * x6 ' and subsequent ly i n t e g r a t e d over x

The r e a t r i c t i o n t h a t Eq. (6.111 only holds if x i s l a r g e r than t h e o t h e r time components makes i t impossible t&obvrite down t h e analogue of Eq. (6.111 i n momentum space.

We w i l l now write down an equat ion w h i c h fo l lows d i r e c t l y from t h e l a r g e s t time equat ion , but holds whatever the time

corresponding t o some diairam. We have order ing of t h e var ious x . Thus cons ider a func t ion Ffx 1' . * * a Xnl

T h e sumnation goes over a l l poss ib l e ways t h e t t h e v a r i a b l e s may be underl ined. For i ne t ance , there i s one term without underl ined v a r i a b l e s , n terms w i t h one underl ined v a r i a b l e , etc. There i s a l s o one term, t h e l a s t , where a l l v a r i a b l e s a r e underl ined. Under c e r t a i n condi t ions , t o be d iscussed i n Sec t ion 7 , t h a t term is r e l a t e d t o t h e first term

* F(xl, x2, ..., x 1 - F(xl, x2, ..., x 1 . (6.141 n n - - -

I' T h e proof of Eq. (6.131 i s t r i v i a l . Let one o f t h e X, say x have t h e l a r g e s t time component. Then on t h e le f t -hand s i d e of Eq. (6.131 any diagram w i t h xi not underl ined cance l s aga ins t t h e term where i n add i t ion xi i s underl ined, by v i r t u e of t h e l e r g e s t time equation.

Now we can mul t ip ly Eq. (6.131 by t h e appropr i a t e source f a c t o r s and i n t e g r a t e over a l l X. We ob ta in a set of func t ions F depending on t h e var ioua e x t e r n a l momenta and f u r t h e r i n t e r n a l momenta [ loop momenta). Here, t h e bar on F denotes t h e Four i e r transform. T h e func t ions I? a r e composed of A*, A and A* func t ions

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DIAGRAMMAR

m

(6.15)

(6.16)

We now observe t h a t I n t h e r e s u l t i n g equat ion many terms w i l l be zero , due.,to c o n f l i c t i n g energy 0 func t ions . l a k e , f o r example, a diagram”1 w i t h one under l ined po in t which I s not connected t o an outgoing l i n e :

+ Because of t h e 8 func t ions i n t h e A- [ s e e Eq. [6.1511, energy I s forced t o flow towards t h e c i r c l e d ve r t ex . S ince energy conserva t ion holds i n t h a t ve r t ex t h i s is impossible and we conclude t h a t t h i s diagram is zero. The same is t rue i f ve r t ex 5 I s c l r - c led in s t ead . F u r t h e r , if t h e momenta p i and p2 r ep resen t lnco- ming p a r t i c l e s , w h i c h Impl ies energy f lowing from t h e o u t s i d e I n t o v e r t i c e s 1 and 6 , then a l s o t h e diagrams w i t h ve r t ex 1 and/or ve r t ex 6 c i r c l e d a r e zero. Also t h e diagram w i t h 2 end 5 c i r c l e d i s zero , even i f now energy may f low i n e i ther d i r e c t i o n between 2 and 5. because a l l o t h e r l ines ending I n 2 or 5 f o r c e energy flow towards these v e r t i c e s . We t h u s come t o t h e fo l lowing resul t .

A diagram conta in ing c i r c l e d v e r t i c e s g i v e s r ise t o a non- zero con t r ibu t ion If t h e c i rc led v e r t i c e s form connected reg ions t h a t conta in one or more outgoing l ines . And a l s o t h e unc i r c l ed v e r t i c e s m u s t form connected reg ions Involving incoming l ines.

T h u s , f o r example,

- ::I I n t h i s and I n t h e fo l lowing diagrams, incoming p a r t i c l e s a r e

e t t h e l e f t , outgoing a t t h e r i g h t (energy f lows i n a t t h e l e f t , out a t t h e r i g h t ) .

66


i s zero because i n ve r tex 4 we have a c o n f l i c t i n g s i t u a t i o n . Examples o f non-zero diagrams are:

Note t h a t an i ngo ing l i n e may be attached t o a c i r c l e d reg ion.

Since t h e c i r c l e d v e r t i c e s form connected reg ions we may drop t h e c i r c l e s and i n d i c a t e the reg ion w i t h the help o f a boundary l i n e : ~ = q

Here i s an example of another diagram:

Note t h a t no spec ia l s i g n i f i c a n c e i s a t tached t o t h e c u t t i n g o f an ex te rna l l i n e .

Taking t h e above i n t o account, Eq. t6.131 now reduces t o

16.17) Here f is,the F o u r i e r t rans form o f t he f u n c t i o n F wi thout under- l i n i n g s , F t h e Four ie r t rans form of t he func t i on F w i t h a l l va r iab les underl ined. The funct ions Fc correspond t o a l l non-zero diagrams conta in ing bo th c i r c l e d and u n c i r c l e d ve r t i ces . They correspond t o a l l poss ib le c u t t i n g s o f t he o r i g i n a l diagram w i t h the p r e s c r i p t i o n t h a t f o r B cu t l i n e t h e propagator f u n c t i o n Afk) must be rep laced by A*[k) w i t h the s ign such t h a t energy i s fo rced

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DIAGRAMMAR

t o f l o w towards t h e shaded region. Equation (6.171 i s Cutkosky's c u t t i n g r u l e .

Remembering t h a t t he T-matr ix i s obtained by m u l t i p l y i n g by -1, we see t h a t Eq. (6.171 i s o f exac t l y the same s t r u c t u r e as t h e u n i t e r i t y equation (6.81. There i s one no tab le d i f f e rence : Eq. (6.171 holds f o r a s i n g l e diagram, wh i l e u n i t a r i t y i s a p roper ty t r u e f o r a t r a n s i t i o n amplitude, t h a t i s f o r t h e sum o f diagrams c o n t r i b u t i n g t o a g iven process.

Equation (6.171 holds f o r any theory described by a Legran- glen, whether i t i s u n i t a r y o r not.+The Feynman r u l e s f o r F! are, however, d i f f e r e n t f rom those f o r T (Sect ion 2.61. Therefore, i f Eq. (6.171 i s t r u l y t o imp ly u n i t a r i t y a number of p roper t i es must hold. Th is w i l l be discussed l a t e r .

6.4 Causa l i t y

Again consider any diagram, t h a t represents a f u n c t i o n F(x.,, and xj. L e t us suppose t h a t t h e t ime component o f x i s l a r g i r than xio. The f o l l o w i n g equat ion holds independently o f t he t ime o rde r ing o f t h e o ther t ime components

x,l. Take any two var iab les , say x

Again terms cancel i n pa i r s . We do no t need the diagrams where x i s under l ined, because we know f o r sure t h a t x i s never the A g e s t t ime. 10

Equation (6.181, when m u l t i p l i e d by the appropr ia te source func t i ons and i n t e g r a t e d over a l l x except x and x , i s t h e s i n - g l e diagram vers ion of Bogoliubov's c a u s a l i t y cond i i ion . H i s n o t a t i o n i s

1

(6.191 Here t h e f i r s t term descr ibes cu t diagrams [ i n c l u d i n g the case o f no c u t a t a l l - - t h e u n i t p a r t o f Sl w i t h x and x no t c i r -

c i r c l e d . S i s t he S-matrix obta ined f rom the conjdgate Feynman r u l e s (1.0. ell v e r t i c e s under l ined) , and w i l l o f t e n be equal t o S . Fur ther g ( x I i s t h e coup l ing constant, made i n t o a f u n c t i o n of space-time.

cled, and-the second term denotes diagrams w i t h i x bu t j no t xi

+

68


S i m i l a r l y we can consider the case where x > x Then we have an equation where now x Separating o.Ff the term w i t h no va r iab le under l ined we may combine equations, w i t h the r e s u l t

i s never t o beiflnder&ed. j

F [ x ~ , x ..., x I = -e(x - x I F(xl, .... xk, ..., xn1- 30 i o i - 2 ' n

-e[xio - x I C 0 F [ x ~ , ..., xk, ..., Xn). - j0 j

The prime ind i ca tes absence o f the term without under l ined va- r i ab les . The index i imp l i es absence o f diagrams w i t h xi under- l ined.

The summations i n Eq. (6.201 s t i l l contain many I d e n t i c a l terms, namely those where n e i t h e r x no r x are underlined. Also these may be taken together t o g i ve i J

i not

F(xl, em., x I c n X

i underl ined j not

t6.211

The f i r s t term on the r ight-hand s ide of Eq. (6.21) i s a se t o f cu t diagrams, w i t h x _and x They represent the product &wi th Jhe r e s t r i c t i o n t h a t x and x are ve r t i ces of S. We can now apply the same equation, w i t h the same po in ts x and x , t o the diagrams f o r S i n t h i s product. Doing t h i s as many dmes as necessary, the r ight-hand s ide o f Eq. (6.211 can be reduced e n t i r e l y t o the sum o f two terms, one containing a funct ion e(x - x 1 m u l t i p l y i n g a func t i on whose Four ie r transforms contaif ig 0 fdF(ctions f o r c i n g energy f l o w from 1 t o j, the other containing the opposite combination. This i s prec i se l y o f the form ind i ca ted i n Sect ion 6.1.

always i n the unshaded region.

j

i

Let us now r e t u r n t o Eq. (6.211. I n t roduc ing fo r 8 the Four ier

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DIAGRAM MAR

r e p r e s e n t a t i o n Eq. (6 .51 , we can see B as a n o t h e r k ind of p ropa - g a t o r c o n n e c t i n g t h e p o i n t s x and x . M u l t i p l y i n g by t h e appro - p r i a t e source f u n c t i o n and i n $ a g r e t d g over a l l xi we o b t a i n t h e f o l l o w i n g d i a g r a m n a t i c e q u a t i o n :

The b l o b s t a n d s f o r any d i ag ram or c o l l e c t i o n of d iag rams . The p o i n t s 1 and 2 i n d i c a t e two a r b i t r a r i l y s e l e c t e d vertices. The "self i n d u c t a n c e " I s t h e c o n t r i b u t i o n due t o t h e 8 f u n c t i o n , and i s o b v i o u s l y n o n - c o v e r i a n t :

O f course, i n t h e diagrams on t h e r i g h t - h a n d s i d e summation o v e r a l l cuts w i t h t h e p o i n t s 1 and 2 i n t h e p o s i t i o n shown is i n t e n d e d .

T h i s i s p e r h a p s t h e r i g h t moment t o summarize t h e Feynman rules f o r t h e c u t diagrams. As an example we t a k e t h e s i m p l e sca la r t h e o r y :

P r o p a g a t o r i n unshadowed r e g i o n :

1 1 - -I (Zn14i k2 + m2 - IE

P r o p a g a t o r i n shadowed r e g i o n :

1 1 - - ( 2 S l 4 I k2 + m2 + IE

Cut l ine :

70


4

4 Vertex i n unshadowed region: (2s) i g .

Vertex i n shadowed reg ion : -(2sr) i g .

F o r a spin-1/2 p s r t i c l e every th ing ob ta ins by m u l t i p l y i n g w i t h t h e f a c t o r - i y k + m:

1 -1yk + m

[2nl;?i k2 + m2 - i E -I L k

(6.251

The most simple a p p l i c a t i o n concerns t h e C8Se o f on l y one propaga to r connecting t w o sources. We w i l l l e t these sources emit and absorb energy, b u t we will no t pu t anyth ing on mass-shell. Indeed, nowhere have mass-shell cond i t ions been used i n the der lve t lons .

Thus consider :

J2tk1 Jk 4 1 i2 (271 iJ l (k l - k k2 + m2 -

w i t h J 1 and J (6.171 reads:

non-zero on ly if k > 0. The u n i t a r i t y equat ion 2 0

The complex conjugat ion does apply t o every th ing except t h e sources J. The second term on the r igh t -hand s ide i s zero, because o f t h e cond i t i on ko > 0. The equation becomes

IJ = 4 1 - (2aI i 1

k2 + m2 - i~ J

k2 + rn2 + I€

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4 Note t h a t t h e v e r t e x i n t h e shadowed r e g i o n g i v e s a f a c t o r -[2n1 I. W i t h

1 a - i e

we see the e q u a t i o n h o l d s t r u e .

A l s o EQ. (6.221 can be v e r i f i e d :

i

We now o b t a i n ( n o t e t h e m i n u s s i g n f o r vertex i n shadowed r e g i o n )

+ - i e 8 ( - k o + po16[(k - p12 + mzl]+ PO p=o .

The f o u r - v e c t o r p has z e r o s p a c e components [see e x p r e s s i o n (6.23)). The po i 8 t e g r a t i o n i s t r i v i a l and g i v e s t h e d e s i r e d re- s u l t .

6 .5 Di spe r s ion R e l a t i o n s

Equa t ions (6.221 a r e no th ing bu t d i s p e r s i o n r e l a t i o n s , v a l i d f o r any diagram. Let r be t h e f o u r t h compoQent of t h e momentum f l o w i n g th rough t h e s e l f - i n d u c t a n c e . Let f [ T I be t h e f u n c t i o n co r re spond ing t o t h e cut diagrams e x c l u d i n g t h e r - l i ne i n t h e second term on t h e r i g h t - h a n d s i d e of Eq. (6.221, w i t h T d i r e c t e d from 1 t o 2. S i m i l a r l y f o r f [ T I . If f and f ’ r e p r e s e n t t h e lef t - hand s i d e and t h e f irst term on t h e r i g h t - h a n d s i d e , r e s p e c t i v e l y , w e have

72


(6.261 O f course. a l l the funcpons f depend on the var ious external momenta. The func t i on f [TI w i l l be zero f o r large p o s i t i v e T, namely as soon as r becomes larger than the t o t a l amount o f energy f lowing i n t o the diagram i n the unshadowed region. S i m i l a r l y f [TI i s zero f o r large negative T.

The dispersion r e l e t i o n s Eq. c6.261 are very important i n connection w i t h renormalization. I f a l l subdivergencies o f a diagram have been removed by su i tab le counter terms. then a l l cut diagrams w i l l be f i n i t e [ i nvo l v ing products of subdiagrams w i t h ce r ta in f i n i t e phase-space i n t e g r a l e l . According t o Eq. (6.261 the i n f i n i t i e s i n the diagram must then a r i se because of non-converging dispersion i n teg ra l s . Sui tab le subtractions. 1.e. counter terms, w i l l make the i n teg ra l s f i n 1 t e . t

It may f i n a l l y be noted t h a t our dispersion r e l a t i o n s are very d i f f e r e n t from those usual ly advertised. We do not disperse i n some external Lorentz invar iant . such as the centre-of-mass energy or momentum transfer.

A

'Note added: Indeed, a rigorous proof of multiplicative renormalizability of many theories, a t the expense of exclusively local counter terms a t a l l orders, can be constructed without too much trouble from these dispersion re lat ions .

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DI AG RAMMAR

7. U N I T A R I T Y

If t h e c u t t i n g equat ion (6.171, diagrammatical ly r ep resen ted as :

t corresponding t o T - Tt - I T T , i s t o imply u n i t a r i t y , t h e fo l lowing m u s t hold:

I 1 The diagrams i n t h e shadowed reg ion m u s t be those t h a t

i l l The A func t ions m u s t be equal t o what i s obta ined when

Re fe r r ing t o our d i scuss ion of t h e mat r ix St, i n Sec t ion 2.6, we note t h a t po in t (1) w i l l be t r u e i f t h e Lagrangian genera t ing the S-matrix i s i t s own conjugate .

occur+in s+, summing over in t e rmed ia t e s t a t e s .

Po in t ( i l l amounts t o t h e fo l lowing . T h e two-point Green's func t ion , on which t h e d e f i n i t i o n of t h e S-matrix sources was based, contained a matr ix K [see Eq. (2.121 and fo l lowing) . I n cons ider ing S S one wil l ' incounter ( p a r t i c l e - o u t o f S, par- t i c l e - i n of S ~ I :

t

S S t 1 Kt I-klJ*(') (klJ:(k)KQm(-k) -- i j j a (7.11

k - i n t h e sum over In te rmedia te s ta tes . The K are from t h e propaga-

have K t m ( - k l = K Q [kl . . A l s o if J (k lK( -k ) - J ( k 1 t h e n K [ - k l J showink t h a t J an! J* a r e t h e appropr i a t e e igen currents of S and St. If u n i t a r i t y i s t o be t rue we r e q u i r e t h a t t h i s sum (7.11 occurr ing i n S'fS equals t h e mat r ix K occu r r ing when c u t t i n g a propagator .

t o r s a t t ached t o t h e sources . Because of 1: = (CIconjugqte we* * * J ,

i m

The proof of t h i s i s t r i v i a l . Suppose K i s diagonal w i t h . The cu r ren t de f in ing eJda t ions [2.13) and d iagonal e lements (2.141 imply t h a t tAe currents a r e o f t h e form

74


There are no cur ren ts corresponding t o z8ro eigenvalues. Obviously

and t h i s remains t r u e i f one prov ides the cur ren ts w i t h phase fac to rs , e tc .

F o r sp in -1 /Z4par t i c les th ings are s l i g h t l y more complicated, because o f y manipulat ions. For instance, one w i l l have

4 Kt[-kly4 = y K(k1 . (7.31

Also t h e normal iza t ion o f t he cur ren ts is d i f f e r e n t . One f i n d s the co r rec t expression when summing up particle-out/particle-in states, but a minus s ign e x t r a f o r antiparticle-out/antlpartlcle- i n s ta tes. Th is f a c t o r is found back i n the p r e s c r i p t i o n -1 f o r every fermion loop. A few examples are perhaps usefu l .

EXAMPLE 7.1

Propagators and v e r t i c e s have been g iven before. The appropr ia te source func t i ons and r e l a t e d equations are g iven i n Appendix A.

There are four two-point Green's functions:

-Iyk + u a ( k l 2 2k0 ; a = 1, 2 , - ia[kl - k k2 + m2 4m

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DIAGRAMMAR

; a - 3 , 4 . - - i a ( k l u ( k l 7 k i k + m a 2k0

k + m 4m - Note t h e minus s ign f o r t he incoming a n t i p a r t i c l e wave-function.

Scalar p a r t i c l e se l f -energy (we w r i t e a l s o 6 f unc t i ons ) :

2 - i y p + m -iy[p - k l + m

p2 + m2 - IE -g ti4[k - k’l I d4P

[p - k12 + m2 - iE’

Note t h e minus s ign f o r t h e closed fermion loop. Cut diagram (remember - [ 2 n 1 4 i f o r ve r tex i n shadowed r e g i o n I :

Decay o f sca la r i n t o two fermions:

2 The superscr ip t u now i n d i c a t e s a n t i p a r t i c l e sp inor . The complex conjugate, bu t w i t h k’ i ns tead of k, i s

The product o f t h e two summed over in te rmed ia te s ta tes i s

76

G. ’t HOOFT and M. VELTMAN

Note t h e m i n u s s ign f o r t h e q-spinor Bum.

S ince po =m we have

and similarly f o r q. The q i n t e g r a t i o n can be performed

-[2n) 2 2 g 641k - k’l 1 d4pBIpolb[p2 + m21e[k0 - pol x

w h i c h i n d e e d equals t h e result f o r t h e cut diagram. T h e minus s ign f o r t h e closed fermion loop appears here as a m i n u s s ign i n f r o n t of t h e e n t i p a r t i c l e sp inor summation, One may wonder what happens i n t h e ca se where an a n t i p a r t i c l e l ine is c u t , but when there is no closed fermion loop. An example is provided by t h e a n t i p a r t i c l e self-energy as compared t o p a r t i c l e se l f - energy :

Somehow there m u s t a l s o b e an e x t r a m i n u s s ign f o r t h e first diagram. Indeed it is there, because t h e first diagram con ta ins an incoming a n t i p a r t i c l e whose wave-function has a m i n u s s ign .

A l l t h i s demonstrates a t i g h t i n t e r p l a y between s t a t i s t i c s [minus s ign f o r fermion loops) and t h e t ransformat ion p r o p e r t i e s under Lorentz t ransormat ions of t h e epinors . The l a t t e r r e q u i r e s the normalizat ion t o energy d iv ided by mass a8 given before , and a l s o r e l a t e s t o t h e m i n u s s ign f o r e n t i p a r t i c l e source summation.

If an i n t e g e r sp in f i e l d is assigned Fermi s t a t i s t i c s i n t h e form of m i n u s s igns f o r l i n e in te rchanges t h e n u n i t a r i t y w i l l be v io l a t ed .

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8. INDEFINITE METRIC

I f the numerator K o f a propagator has a negat ive eigen- va lue a t the po le then u n i t a r i t y cannot ho ld because Eq. (7.21 cannot hold. U n i t a r i t y can be res to red i f we In t roduce the convention t h a t a minus s ign i s t o be attached whenever such a s t a t e appears. I t i s sa id t h a t t h e s t a t e has negat ive norm, and t r a n s i t i o n p r o b a b i l i t i e s as w e l l as cross-sect ions and l i f e t i m e s can now take negat ive values. Th is i s , of course, p h y s i c a l l y uncacceptable, and p a r t i c l e s corresponding t o these s ta tes are c a l l e d ghosts. I n theo r ies w i t h ghosts spec ia l mechanisms must be present t o assure absence o f unphys ica l e f f e c t s . I n gauge theo r ies negat ive m e t r i c ghosts occur simultaneously w i t h c e r t a i n o ther p a r t i c l e s w i t h p o s i t i v e met r ic , i n such a way t h a t t he t r a n s i t i o n p r o b a b i l i t i e s cancel. A l s o t he second type of p a r t i c l e , although completely decent, i s c a l l e d a ghost and has no phys i ca l s ign i f i cance.

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I


9. DRESSED PROPAGATORS

The pe r tu rba t i on se r ies as formulated up t o now w i l l i n genera l be d ivergent i n a c e r t a i n reg ion. Consider the case o f a sca la r p a r t i c l e i n t e r a c t i n ? w i t h i t s e l f and poss ib l y o ther p a r t i c l e s . Le t 64 (k - k ' l r r k 1 denote t h e c o n t r i b u t i o n o f a l l self-energy diagrams t h a t cannot be separated i n t o two pieces by c u t t i n g one l i n e . These diagrams are c a l l e d i r r e d u c i b l e self-energy diagrams. The two-point Green's f u n c t i o n f o r t h i s s c a l a r p a r t i c l e i s o f the form

The f u n c t i o n iF i s c a l l e d t h e dressed propagator. Th is i n cont ras t t o t h e propagator of t he sca la r p a r t i c l e , c a l l e d the bare propagator. If we denote t h i s bare propagator by A F we f i n d

- A,= AF + AFl'AF + A F r A F r A F + a * * r 9 . a

Summing t h i s se r ies

1 - rAF

corresponding t o the diagrams:

+ +

r9.31

+ ...

where t h e hatched b lobs stand f o r t he i r r e d u c i b l e se l f -energy d i agr ams .

The func t ion r i s p ropor t i ona l t o the coup l ing constant o f t h e theory. I t i s c l e a r t h a t t h e pe r tu rba t i on se r ies converges on ly i f r A F < 1. But if AF has a po le f o r a c e r t a i n va lue o f t he four-momentum then t h i s se r ies w i l l c e r t a i n l y no t converge near t h i s pole, unless r happens t o be zero there. And i f we remember t h a t t he d e f i n i t i o n o f t he S-matrix i nvo l ves p r e c i s e l y the behaviour of t he propagators a t t he poles we see t h a t t h i s problem needs discussion.

There are t w o poss ib le so lu t i ons t o t h i s d i f f i c u l t y . The s implest s o l u t i o n i s t o arrange th ings i n such a way t h a t indeed I'

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DIAGRAMMAR

i s ze ro a t t h e pole. T h i s can b e done by in t roduc ing a s u i t a b l e ve r t ex i n t h e Lagrangian. T h i s ve r t ex con ta ins two scalar f i e l d s and equa l s m i n u s t h e va lue of T a t t h e pole . F o r i n s t a n c e , suppose

1 1 AF = -

( Z n I 4 i k2 + m2 - IE (9.41

2 T h e f u n c t i o n l'lk 1 can be expanded a t t h e po in t k2 + m2 = 0

(9 .5 ) 2 r(kl = ro + (k2 + m ITl + r2(k1 .

2 r and r a r e cons t an t s , I' i s of o rde r ( k + m 2 I 2 . In t roduce i n tRe Lagrangian a term t h a t l eads t o t h e ve r t ex : 1 2

Ins t ead of r we w i l l now have a func t ion T':

+

r' w i l l be o f orde r ( k 2 + m 2 1 2 and t h e series (9.2) converges a t t h e pole. A c t u a l l y t h i s reasoning i s c o r r e c t only i n lowest o r d e r because t h e new ve r t ex can a l s o appear i n s i d e t h e hatched blob. Through t h e in t roduc t ion of fur ther h igher o rde r v e r t i c e s of t h e t ype (9 .61 t h e r equ i r ed result can be made accurate t o a r b i t r a r y order .

T h i s procedure, involv ing mass and wave-function renorma- l i z a t i o n corresponding t o ro and r type terms, r e s p e c t i v e l y , embodies c e r t a i n inconveniences. F h s t of a l l , i t may be t h a t I' and r con ta in imaginary p e r t s . T h i s occurs i f t h e p a r t i c l e becomes hns t ab le because of t h e i n t e r a c t i o n . Such t r u l y phys ica l effects a r e p a r t of t h e conten t of t h e theory and t h e above n e u t r a l i z i n g procedure cannot be c a r r i e d through. Furthermore, i n t h e case of gauge t h e o r i e s , t h e freedom i n t h e choice of terms i n t h e Lagrangian i s l imi t ed by gauge Invar iance , and it is not sure t h a t t h e procedure can be c a r r i e d through without gauge inva r i ance v i o l a t i o n .

0

An a l t e r n a t i v e s o l u t i o n is t o use d i r e c t l y t h e sumed express ion (9.3) f o r t h e p ropagators i n the diagrams. These diagrams must then , of course , con ta in no f u r t h e r i n t e r n a l se l f -energy p a r t s , t h a t is they m u s t be ske le ton diagrams. T h e func t ion r occurr ing i n t h e propagator m u s t t h e n be c a l c u l a t e d

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G. 't HOOFT and M. VEkTMAN

w i t h a ce r ta in accuracy i n the coupling constant. For instance, i n lowest order f o r any Green's func t i on the rec ipe i s t o compute t r e e diagrams using bare propagators. I n the next order there are diagrams w i t h one closed loop [no self-energy loops) and bare propagators, and t r e e diagrams w i t h dressed propagators where r i s computed by considering one-closed-loop self-energy diagrams.

I t i s c lear that . th1s rec ipe leads t o a l l kinds of complications, which however do not appear t o be very profound. Mainly, kinematics and the per turbat ion expansion have t o be considered together. With respect t o renormalization the complications are t r i v i a l , because then only the beheviour o f the propagator for very large k2 i s of importance. But then the ser ies expansion Eq. (9.21 i s permitted.

For the r e s t of t h i s sect ion we w i l l consider the impl ice- t i o n s o f the use o f dressed propagators f o r c u t t i n g ru les.

As a f i r s t step we note tha t the dressed propagator s a t i s f i e s the KellBn-Lehmann representation. This fo l lows from the causal i ty r e l a t i o n Eq. 16.211, o r i n p i c t u r e Eq. [6.221 f o r the two-point Green's function, where x and x are taken t o be the in-source and out-source vertices!. repec&vely. The f i r s t term on the r ight-hand side o f these equations i s then zero. Indeed, t h i s gives prec ise ly Eq. (6.11, the deCOmpOSltlDn of the (dressedlpropagator i n t o p o s i t i v e and negative frequency parts. The de r i va t i on of the c u t t i n g re la t ions, which i s besed on t h i s decompoaition goes through unchanged.

However, w i t h respect t o u n i t a r i t y there i s a f u r t h e r sublety. Using dressed propagators the p resc r ip t i on is t o use only diagrams wfthout self-energy inser t ions, t ha t i s skeleton diagrams. Now the A obtained by c u t t i n g a dressed propagator. The dressed propagator l a a series, and c u t t i n g gives the r e s u l t (we use again the exam- p l e o f a scalar p a r t i c l e 1

funct ion corresponding t o a dressed propagator i s evident ly

- * Re bF = i F ( - R e r1AF + po le p a r t

or 1

f2nI411k2 + m2 + ir/12n1 1 I

Re iF = Re 4

[9.71

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where we used

k2 + m2 + i r / (2n14 % Z[k2 + f12 - i a l + 0"k2 + N2I21

(9.91

near t h e pole. Now Re r is obta ined when c u t t i n g the i r r e d u c i b l e self-energy diagrams. I n diagramnatic form we have:

Note t h e occurrence of dressed propagators i n t h e second term on t h e r i g h t .

The s u b t l e t y h in ted a t ebove is t h e fo l l ow ing . The m a t r i x S conta ins ske le ton diagrams w i t h dressed propagators. C u t t i n g these diegrams apparent ly r e s u l t s i n expressions obta ined when c u t t ng se l f -energy diagrams. Indeed, these a r i s e au tomat ica l l y i n S S. I f S conta ins skeleton diagrams o f t h e type: +

t then S S conta ins the type:

t t Even i f S and S conta in no reduc ib le diagrams, the product S S never the less has se l f -energy s t ruc tu res . They correspond t o what i s obta ined by c u t t i n g B dressed propagator.

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10. CANONICAL TRANSFORMATIONS

10.1 I n t r o d u c t i o n

I n t h i s s e c t i o n we s t u d y the behav iour of t h e t h e o r y under f i e l d t r a n s f o r m a t i o n s . F i e l d s by themse lves are n o t ve ry r e l e v a n t q u a n t i t i e s , from t h e p h y s i c a l p o i n t of view. The S-matr ix i s supposed t o d e s c r i b e t h e p h y s i c a l c o n t e n t of t h e t h e o r y , and there i s no d i r e c t r e l a t i o n between S -ma t r ix and f i e l d s . Given t h e Green ' s f u n c t i o n s f i e l d s may be d e f i n e d l t h e Green's f u n c t i o n s , however, can be cons ide red as a ra ther a r b i t r a r y e x t e n s i o n of t h e S-matr ix t o o f f - m a s s - s h e l l v a l u e s o f t h e e x t e r n a l momenta. Within t h e framework of p e r t u r b a t i o n t h e o r y i t i s p o s s i b l e . up t o a p o i n t , t o d e f i n e t h e f i e l d s by c o n s i d e r i n g t h o s e Green's f u n c t i o n s t h a t behave as smoothly as p o s s i b l e when going o f f mass - she l l . For gauge t h e o r i e s t h i s i s s t i l l insuf f ic ien t t o f i x t h e t h e o r y , there being many c h o i c e s of f i e l d s (and Green's f u n c t i o n s ) t h a t g i v e t h e same p h y s i c s CS-matrix) w i t h e q u a l l y smooth behaviour . For g r a v i t a t i o n t h e s i t u a t i o n i s even more b e w i l d e r i n g , up t o t h e p o i n t of f r u s t r a t i o n .

I n t h e s t u d y of f i e l d t r a n s f o r m a t i o n s p a t h i n t e g r a l s have been of g r e a t heuris t ic v a l u e . T h e e s s e n t i a l c h a r a c t e r i s t i c s w i l l be shown i n t h e n e x t s e c t i o n .

10.2 Path Integrals

A few s imple e q u a t i o n s form t h e b a s i s of a l l p a t h i n t e g r a l m a n i p u l a t i o n s , and w i l l be l i s t e d he re .

Let a be a complex number w i t h a p o s i t i v e non-zero imaginary p a r t . Furthermore, z = x + i y is a complex v a r i a b l e . We have

* m m 2 2 i a [ x +y 1 / ldzeiaZ = I dx I dye -OD -0

I n t r o d u c i n g p o l a r c o o r d i n a t e s

[ l o . 11

t10.2)

I n c i d e n t a l l y , r e a l i z i n g t h a t t h e e x p r e s s i o n (10.11 i s a pu re s q u a r e

Let now z be a complex n-component v e c t o r , and A a complex

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n x n mat r ix . The genera l i za t i on o f Eqs. [10.11, [10.21 i s

* i l z , A 2 1 nnin

det [A) ’ \ dzl ... dz e n

C10.41

where

/ dzj - \ dx j dyj , zJ = x j + i y j . (10.51 -0 -00

Equation [10.41 f o l l o w s t r i v i a l l y f rom Eqs. (10.11, 110.21 i n the case where A is diagonal. Next no te t h a t t h e i n t e g r a t i o n measure i s i n v a r i a n t f o r u n i t a r y t rans format ions U. To see t h i s w r i t e

U = A + i B ,

where A and B are real matrices. The f a c t t h a t UUt = 1 imp l i es

- - - - AA + BB 1 BA - AB 0 (10.61

where the w igg le denotes r e f l e c t i o n . I f now zc = uz then x c = Ax - By and y’ = Ay + Bx. That i s , t h e 2n dimensional vec tor xl y transforms as

The determinant o f t h i s 2n x 2n t rans format ion m a t r i x is 1. I n f a c t , t he ma t r i x i s orthogonal, because m u l t i p l i c a t i o n w i t h i t s transpose g ives 1, on account of t he i d e n t i t i e s (10.61.

Because o f t h e invar iance o f bo th t h e i n t e g r a t i o n measure as w e l l as t he complex sca la r product under u n i t a r y t ransformat ions, we conclude t h a t Eq. (10.51 holds f o r any complex ma t r i x A t h a t cen be diagonal ized by means of a u n i t a r y t rans format ion and where a l l d iagonal ma t r i x elements have a p o s i t i v e non-zero imaginary p a r t .

Consider now a path i n t e g r a l i n v o l v i n g a Lagrangian depen- d ing on a se t of r e a l f i e l d s Ai [see Section31

84

(10.71


where the a c t i o n S i s g iven by

S[A1 - 1 d 4 d ( A ) . (10.81

Suppose we want t o use o the r f i e l d s 6 t h a t are r e l a t e d t o the f i e l d s A as fo l lows i

i

AI(x) - B,[xl + f tx, B1 . [10.91 i

The f, are a r b i t r a r y funct ions (apar t f rom t h e f a c t t h a t Eq. (10.9f must be any sapce-time

&)Ai = de t

or

According

@Ai = det

i n v e r t i b l e ) . They may depend on t h e f i e l d s B a t p o i n t i n c l u d i n g the space-time p o i n t x.

t o well-known r u l e s

r 10.101

r 10.11)

The determinant i s simply t h e Jacobian o f t h i s t ransformat ion. I t i s very clumsy t o work w i t h t h i s determinant, espec ia l l y if we r e a l i z e t h a t i t invo lves t h e f i e l d s a t every space-time p o i n t separately. Fo r tuna te l y the re e x i t s a n i c e method t h a t makes th ings easy. L e t I$ be a complex f i e l d . According t o Eq. t10.41 we have

where C i s an i r r e l e v a n t numerical f a c t o r . and

(10.121

(10.131

However, Eq. "J.111 invo lves a determinant, w h i l e Eq. [10.121 i s f o r t h e inverse of a determinant. We must I n v e r t Eq. (10.121.

The expression on the r igh t -hand s ide o f Eq. [30.121 i s a pa th i n t e g r a l of t he type considered i n Sec t ion 3. I t invo lves a complex f i e l d 4 as w e l l as f i e l d s Bi t h a t appear s imply as sour-

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CBS. T h e “ a c t i o n “ is [see Eq. [10.1111

T h e diagrams corresponding t o Eq. [10.121 involve a +-propegator t h a t i s simply 1. There are only v e r t i c e s involv ing two 4 - l i n e s :

-- i j ’

i j c.c-.

4 (2nl iY(k ’ , k , 61 .

The blob con ta ins 8 - f i e l d s , bu t w e have no t i n d i c a t e d t h e m e x p l i c i t l y . The only diagrams t h a t can occur are c losed +-loops involv ing one, two or more Y-vert ices . We w i l l write down t h e f i rs t few:

Zeroth o rde r i n Y : 1 a F i r s t o rde r

The e x p l i c i t l y w r i t t e n f a c t o r 1/2 i s a combina tor ie l f a c t o r . It fo l lows a150 by t r e a t i n g t h e pa th i n t e g r a l a long t h e l i n e s i n d i t h ca ted i n Sec t ion 3. I n f a c t , i t i s e a s i l y e s t a b l i s h e d t h a t i n n o rde r there w i l l be a con t r ibu t ion of n t imes t h e f i r s t - o r d e r l oop w i t h a f a c t o r l / n l S i m i l a r l y f o r t h e two-Y loop. I n t h i s way, it i s seen t h a t t h e whole series adds up t o an exponent ia l , w i t h i n t h e exponent only s i n g l e c losed loops

86

r 10.161


T h i s can be e a s i l y inve r t ed simply by r ep lac ing r by -r. t h a t i s by t h e p r e s c r i p t i o n t h a t every c losed 4- loop m u s t be given a m i n u s s i g n . We so a r r i v e a t t h e equat ion

w i t h the a d d i t i o n a l r u l e t h a t every closed 4-loop m u s t be given a m i n u s s ign . S[+l i s given i n Eq. (10,141. F i n a l l y S"B1 fo l lows by s u b s t i t u t i n g t h e t ransformet ion Eq. (10.9) i n t o t h e a c t i o n f o r ' the A-f ie ld , Eq. [ l0 .81

SOIB) = S[B + f(B11 . [ l o . 181

Summarizing, t h e theory remains unchanged i f a f i e l d t r a n s f o r - mation I s performed, provided closed loops of ghost p a r t i c l e s [ w i t h a m i n u s s ign/ loopl a r e a l s o in t roduced . T h e v e r t i c e s i n t h e ghos t loop are determined by t h e t ransformat ion law.

I n t h e fo l lowing s e c t i o n s we w i l l d e r i v e t h i s same r e su l t , however without t h e use of pa th i n t e g r a l s .

10.3 Diagrams and F i e l d Transformations

I n t h i s s e c t i o n we w i l l cons ider f i e l d t ransformat ions of t h e very s imples t type. T h i s we do i n o rde r t o make t h e mechanism t r a n s p a r e n t .

Consider t h e Lagranglen

1: [ 4 , 4 * . A ) = - 1 4 V 4 + 2 i i j j 2 1

A ( a 2 - m 2 ) A i + eI[$l + Ji$i + HIAi-

(10.191 We assume V t o be symmetric. 1 i s any i n t e r a c t i o n Lagranglen not involv ing the A-f ie lds ; t h e l a g t e r a r e e v i d e n t l y f ree f i e l d s . T h e diagrams corresponding t o t h i s Lagranglen are wal l -def inedl i f necessary use can be made of t h e previously-given r e g u l a r i z a t i o n

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procedure. The +- and A-propagators are:

The v e r t i c e s stemming f rom cI I n v o l v e + - l i n e s only.

by t h e replacement We want t o study another Lagrengian i* obta ined f rom

where a i s any mat r ix . The A remain unchanged.

THEOREM The Green's func t i ons of t he new Legrangian e0 are equal t o those o f t he o r i g i n a l Legrangian. I n p a r t i c u l a r , A remains a f r e e f i e l d , t h a t I s a l l Green's func t i ons i n v o l v i n g an A as e x t e r n a l l eg are zeroI except f o r A-propagators connecting d i r e c - t l y two A-sources.

The proof of t h i s theorem is very simple, and cons is t s mainly of expanding the quadra t ic p e r t of c O . We have (dropping e l l Ind ices1

Here t h e w igg le denotes r e f l e c t i o n .

Rather then t r y i n g t o i n v e r t t h e complete ma t r i x i n t h e quadra t ic p a r t we w i l l t r e a t as the propagator p a r t on l y t h e terms $,V4 and A l a 2 - m21A. The remaining terms are t rea ted as i n t e r a c t i o n terms. We ob ta in the f o l l o w i n g A-t$ v e r t i c e s :

G. 'r HOOFT and M. VELTMAN

They w i l l be c a l l e d "spec ia l ve r t i ces " . I n a d d i t i o n t o these the re w i l l be many v e r t i c e s i n v o l v i n g A- l ines, because o f t h e rep lace- ment (10.211 performed i n t h e i n t e r a c t i o n Lagrangian.

Consider now any ver tex conta in ing an A - f i e l d [exc lud ing the A-sources H I . Because the A - l i ne a r i ses due t o t h e sub_sti- t u t i o n (10.211 the re always e x i s t s a s i m i l a r ve r tex w i t h t h e A - l i n e rep laced by a # - l i ne . But t he A - l i n e can be connected t o t h a t ver tex a l so a f t e r t ransformat ion t o a # - l i n e v i a one o f t h e specie1 ver t i ces . The sum of t h e two p o s s i b i l i t i e s i s zero.

EXAMPLE 10.3.1 I n the transformed Lagrangian c O , Eq. (10.22) we have a new ver tex

4 (2n1 iJa

connecting an A - l i ne w i t h t h e source of t h e 4 - f i e l d . I n add i t ion , we have t h e o r i g i n a l ver tex :

and then a l s o t he diagram i n v o l v i n g one s p e c i a l ver tex. The f a c t o r V i n the ver tex cancels:

against t h e propagator, l eav ing a minus sign. The diagram cancels against t h e prev ious one.

EXAMPLE 10.3.2 F o r t h e two-point Green's func t i on a t t h e zero loop l e v e l we have:

B V va 19 w i t h a 4 , etc. + c - - - * - m - - - 4 +

Only t h e f i r s t term survives: a l l otheas cancel i n p a i r s .

Diagrams w i t h loops i n v o l v i n g A- l ines a l so cancel s ince they i n v o l v e v e r t i c e s a l ready discussed.

The above theorem can e a s i l y be genera l ized t o the case o f more complicated transformations, such as

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+ + + + f ( A l

w i th f any f u n c t i o n o f t h e A f i e l d s .

EXAMPLE 10.3.3 Consider t h e t r a n s f o r m a t i o n

The f o l l o w i n g s p e c i a l v e r t i c e s a r e generated [we l e a v e t h e f a c t o r s [ Z n 1 4 i as understood1 :

0 Va ,0

N:--- \

\ 0 \

0

aVa

Again, by t h e same t r i v i a l mechanism as b e f o r e we example:

0 0

0 J Va 0 Ja -:--- a:-- +

\

\ -V- \

' \

\

have, f o r

= O

The s i t u a t i o n becomes more comp l i ca ted i f we a l l o w t h e f u n c t i o n f a l s o t o depend on t h e + - f i e l d s . even i n a n o n - l o c a l way.

10.4 L o c a l and Non- loca l T rans fo rma t ions

S t a r t i n g aga in f r o m t h e Lagrangian (10.191 we cons ide r now t h e g e n e r a l s u b s t i t u t i o n

I$ -f I$ + f [ A , + I . We w i l l now have s p e c i a l v e r t i c e s i n v o l v i n g t h e u n s p e c i f i e d f u n c t i o n f. The f u n c t i o n f w i l l be rep resen ted as a b l o b w i t h a c e r t a i n number o f A- and + - l i n e s o f which we w i l l i n d i c a t e o n l y t h r e e e x p l i c i t l y . The p ropaga to r m a t r i x V i s always t h e r e as a f a c t o r and i s i n d i c a t e d by a do t . The + - l i n e a t tached a t t h a t p o i n t w i l l be c a l l e d t h e o r i g i n a l @ - l i n e o f t h e s p e c i a l ve r tex .

90


Now t h i s o r i g i n a l $ - l i n e may be connected t o any o f t he o l d v e r t i c e s o f t h e theory o r t o another o r i g i n a l $ - l ine . Again the cance l l a t i on mechanism works.

EXAMPLE 10.4.1

No new fea tu re ar ises i n these cases. However, if an o r i g i n a l $ - l i n e i s connected t o any o f t he o the r Ip-l ines o f 8 spec ia l ver- t ex (except t he f V f ver tex t h a t i s cancel led ou t a l ready as shown above) no cance l l a t i on occurs. F o r example, t he re i s no th ing t h a t cancels t h e f o l l o w i n g cons t ruc t ion :

Thus we ge t a non-zero e x t r a c o n t r i b u t i o n on ly if the o r i g i n a l # - l i n e i s connected t o another # - l i n e o f t he spec ia l ver tex. A l l t h i s means t h a t t h e new Lagrangian conta ins all t h e con t r i bu t i ons o f t he o r i g i n a l Lagrangian p l u s a new k i n d o f diagram where ell t he o r i g i n a l $ - l i nes are connected t o one o f t h e o the r o f t h e spec ia l vertex. Such diagrams conta in a t l e a s t one closad loop o f spec ia l v e r t i c e s w i t h "wings" o f old v e r t i c e s as w e l l as spec ia l ve r t i ces .

- l i n e s

EXAMPLE 10.4.2 Some new diagrams:

Considering these diagrams one immediately no t i ces t h a t t he f a c - t o r s V i n the s e c i a l v e r t i c e s are always m u l t i p l i e d w i t h t h e propagators -V-'. T h i s then g ives as ne t r e s u l t t h e simple propa-

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g a t o r f a c t o r -1 f o r tne "ghost" connect lng t h e f vertices. see Eq. (10.141.There is no momentum dependence i n t h a t propagator . If now t h e s p e c i a l ver t ices a r e l o c a l , i . e . only p o l nomial dependence on t h e momenta and no f a c t o r s s u c h as l / k , t h e n these vertices can be represented by a p o i n t and t h e c losed loop momen- t u m i n t e g r a t i o n becomes t h e i n t e g r a l over a polynomial. W i t h i n t h e r e g u l a r i z a t i o n scheme t o be in t roduced l a t e r such i n t e g r a l s a r e zero, and noth ing s u r v i v e s .

Y

I f , however, t h e f i e l d t r a n s f o r m a t i o n i s non-local these new diagrams s u r v i v e and g i v e an a d d i t i o n a l c o n t r i b u t i o n w i t h r e s p e c t t o what w e had from t h e o r i g i n a l Lagrangian.

How can w e g e t back t o t h e o r i g i n a l Green's f u n c t i o n s ? A t f irst s l g h t i t seems t h a t we m u s t s lmply s u b t r a c t these diagrams, 1.e. i n t r o d u c e new v e r t i c e s i n t h e t ransformed Lagrangian t h a t produce p r e c i s e l y s u c h diagrams, bu t w i t h t h e o p p o s i t e s i g n . T h i s however i g n o r e s t h e p o s s i b i l i t y t h a t tne o r i g i n a l t$-line of a s p e c i a l v e r t e x connects t o any o f tnese new v e r t i c e s . T h e c o r r e c t s o l u t l o n i s q u i t e simple: i n t r o d u c e v e r t i c e s t h a t reproduce t h e c losed loops of t h o s e c losed loops l e v e l t h e e x t r a terms

-e- Now there is a l s o t h e

on ly , without t h e "wings", and g i v e each a m l n u s s i g n . T h u s a t t h e Lagrangian are as d e p i c t e d below:

\

p o s s i b i l i t y t h a t t h e o r i g i n a l $ - l i n e o f a s p e c i a l v e r t e x connects t o these c a u n t e r loops. I n t h i s way c o u n t e r "winged" diagrams arise a u t o m a t i c a l l y and need not t o be in t roduced by hand.

92


EXAMPLE 10.4.3. Lagrangian i n c l u d i n g counter c losed loops:

The f o l l o w i n g c a n c e l l a t i o n occurs i n case o f a

= o

The crosses denote counter c losed loops.

The s o l u t i o n may thus be summarized as fo l lows. S t a r t f rom a Lagrangian .?,($I. Perform t h e t rans format ion g ( x l + @l(x) +

f [ x , $1. Add a ghost Lagranglen fgBost. Th i s gkost Lagrangian must be such t h a t i t g ives r i s e t o wingless' ' c losed loops of f unc t i ons f, w i t n one o f t he ex te rna l l i n e s removed, a t which p o i n t another ( o r t he samel f u n c t i o n f i s attached. The connec- t i o n i s by means o f a propagator - l /12n14i . O f course, every f u n c t i o n a l so c a r r i e s a f a c t o r [2n14i. Fu r the r the re i s a minus s i g n f o r every such c losed loop.

I

ns i nd ica ted , t h e v e r t i c e s i n such loops are t h e f w i t h one l i n e taken o f f . Symtml ica l l y

P i c t o r i a l l y

Th is agrees indeed p r e c i s e l y w i t h the r e s u l t found w i t h t h e he lp o f pa th i n t e g r a l s .

I t may be t h a t t h e reader 1s somewhat wor r ied about combi- n a t o r i a l f ac to rs i n these cance l la t ions . A well-known theorem s ta tes t h a t comblna tor ia l f a c t o r s a re impossib le t o expla in ; everyone must convince himself t h a t the above ghost ioop pres- c r i p t i o n leads exac t l y t o t h e requ i red cance l la t ions . There i s

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r e a l l y no th ing d i f f i c u l t here i we do not want t o suggest t h a t t he re i s . A good gu ide l i ne 1s always g iven by t h e pa th i n t e g r a l formulat ion. Another way 1s t o convince onesel f t h a t , f o r every p o s s i b i l i t y o f spec ia l l i n e s and v e r t i c e s connecting up, t he re i s a s i m i l a r p o s s i b i l i t y a r i s i n g f rom the gnost Lagrangian. That i s , t he prec ise t a c t o r i n f r o n t o f any p o s s i b i l i t y i s no t r e l e - vant, as l ong as i t i s known t h a t i t i s t h e same as found i n t h e counterpart.

10.5 Concerning the Rigour of t h e Der i va t i ons

There i s no th ing mysterious about the prev lous deriVatiOnS -- prov ided we remember t h a t we are working w i t h i n t h e context o f pe r tu rba t i on theory. Thus t h e t ransformat ions must be such t h a t the new Lagrangian i s o f t he type as descr ibed a t t h e beginning. In p a r t i c u l a r t h e quadra t ic p a r t o f t h e new Lagrangien must be such t h a t propagators e x i s t . For instance, a s u b s t i t u t i o n of t h e fo rm 4 + 4 - $ is c l e a r l y i l l e g a l . I n shor t , t h e canonica l t rans - fo rmat ion must be i n v e r t i b l e .

There i s a fu r the r problem connected w i t h t h e IE presc r ip - t i o n . The propagators are no t s t r i c t l y - V - I , i n t h e n o t a t i o n o f t he prev ious sect ion. bu t have been mod i f ied through t h e IE addi- t i o n . The f a c t o r s V appearing i n the spec ia l v e r t i c e s must there- f o r e be mod i f ied a l so such t h a t t h e k e y - r e l a t i o n L-V-lIV = -1 remains s t r i c t l y t rue. I n connection w i t h gauge f i e l d theory t h e func t i ons f themselves conta in o f t e n f a c t o r s V - l , care less hand- l i n g o f such f a c t o r s may lead t o e r ro rs . As we w i l l see the co r rec t i.z p r e s c r i p t i o n f o r Faddeev-Popov ghosts can be e s t a b l i s - hed by p rec i se considerat ion of tnese circumstances.

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11. THE ELECTROMAGNETIC FIELD

11.1 Lorentz Gauge; Bell-Treiman Transformations and Ward I d e n t i t i e s

The theorem about t ransformat ions of f i e l d s proved i n t h e foregoing s e c t i o n a p p l i e s t o any Lagrangian I: and 1s q u i t e gene- r a l . We w i l l now e x p l o i t I t s consequences i n t h e case of t h e e lec t romagnet ic f i e l d Lagrengian, i n o rde r t o d e r i v e i n p a r t i c u - l a r t h e so -ca l l ed "genera l ized Ward i d e n t i t i e s " f o r t h e Green's f u n c t i o n s of t h e theory .

We s t a r t cons ider ing t h e f ree e lec t romagnet ic f i e l d case , s i n c e it con ta ins a l l t h e main f e a t u r e s of t h e problems we want t o s tudy . T h i s i nc ludes t h e case of i n t e r a c t i o n s w i t h o t h e r p a r t i c l e s 1e.g. e l e c t r o n s ) provided these i n t e r a c t i o n s are i n t r o - duced i n a gauge- invar ian t way.

r h e Lagrangian g iv ing r lse t o t h e Maxwell equat ions i s

A s i s well known, i n t r y i n g t o apply t h e canonica l formalism t o t h e quan t i za t ion of t h i s Lagrangien, many d i f f i c u l t i e s are e n - countered. Such d i f f i c u l t i e s a r e e s s e n t i a l l y d u e t o some redundancy o f t h e l ec t romagnet ic f i e l d v a r i a b l e s , meaning t h a t some combination of t h e m i s a c t u a l l y decoupled from t h e o t h e r ones and themselves .

w i t h i n t h e scheme def ined i n Sec t ion2 , t o g e t ruaes f o r diagrams and then a theory from a given Lagrangian, such d i f f i - cu l t i e s mani fes t themselves i n t h e f a c t t h a t t h e matrix V t h a t d e f i n e s t h e propagator has no inve r se i n tne case of t h e Lagrangian r11.11.

i J

To avoid such problems convent iona l ly one adds t o t h e Lagran- g ian 111.11, w i t h some mot iva t ions , a term -1 /213 A 1 2 . Then one ob ta ins l J l J

Now t h e propagator i s well def ined:

1 61JV k v - ~.--. k2 - iE

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DIAGRAMMAR

and we have t n e ver tex :

k - This c e r t a i n l y de f ines a theory, bu t i n f a c t we nave no i dea i f the phys i ca l content I s as t h a t descr ibed by t h e Maxwell equations. Th is is main ly t h e sub jec t we want t o study now. I n do ing t h a t we w i l l encounter many o ther somewnat r e l a t e d problems which must be solved when cons t ruc t i ng a tneory f o r t he e lect romagnet ic f i e l d . The key t o a l l these problems i s prov ided by t h e genera- l i z e d Ward i d e n t i t i e s .

I n o rder t o de r i ve these i d e n t i t i e s we add t o t h e Lagran- g len (11.21 a f r e e r e a l s c a l a r f i e l d 8, coupled t o a source JB. We g e t

1 2 2 x * * ( A , ei = - $ r a A 1’ + J A + B(a - p IB + J ~ B . P V U P

(11.31 We w i l l now per form a Bel l -Tre iman t ransformat ion. Th is i s a canonica l t rans format ion t h a t 1s i n form a l so a gauge t ransformation. Here

A + A - U l J

E a B (11.41 lJ

depending on a parameter E. The replacement f11.41 i n l”, Eq. 111.31, g ives up t o f i r s t order i n t h i s parameter

L - * ( A - E a 8 , BI = .c*#rA, BI + E [ a A ) a a B - Ea BJ . lJ lJ l J v l J v l J l J

(11.51 As stated, t he t ransformat ion (11.41 i s i n form just a gauge t rans format ion

A + A + a h , U l J l J

( I l . E I

w i t h A any f u n c t i o n o f x. Note t h a t t h e f i r s t Lagrangian, Eq. 111.11, apar t from the source term is i n v a r i a n t under such a t ransformet ion. On the contrary , t h e second Lagrangian 111.21, and a f o r t i o r i Eq. l11.31, i s no t gauge i n v a r i a n t because the added t e r m -1/21a,,AU12 is not . P r e c i s e l y the source term and t h i s -1/2(alJAPJ2 term a re respons ib le f o r t h e dif ference, i n Eq. (11.51, between the o r i g i n a l Lagrangian L** [AU, BJ and the

96

0. 't HOOFT and M. VELTMAN

transformed on@.

L e t u s come back t o our local canonical transformation [11.4). The difference between the o r i g i n a l and t h e new Lagranglan gives rise t o the following e x t r a ve r t i ce s :

c(2nI 4 - k,,J,, . Here t h e dotted lines denote the 8 - f i e ld . and t h e sho r t double l ine a t t h e sources denotes t h e gauge f a c t o r a t h e gauke transformation (11.61 i n f r o n t of A.'

appearing i n

(Note: A der iva t ive a . ac t ing on a f l e l d i n an Interact ion term of t h e Lagrangian. gibes i times t h e momentum of tne f i e l d flowing towards the vertex.J

O f course t h e ver tex term ~[Znl k,,J would give no contr l - bution If t h e source of t h e elactromagnehc f l e l d J,, were a "gauge-invariant source". namely i f a,,Jcl = U. I n t he discussion which follows we want. h0WeVeI'. t o allow i n general a l s o "non- gauge-invariant sources". whose four-divergence i s d i f f e r e n t from zero.

4

As a consequence of t he general theorem proved i n t he foregoing sect ion. t h e 8 - f i e ld remains a f r e e f i e l d . j u s t as i n t h e theory described by t h e Lagranglan c". To f i r s t order i n E. considering t h e n-point Green's function w i t h one B source we have :

+

+ . . . . .

- - rK ------ a- - w@--

+ a

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DIAGRAMMAR

I n the f i r s t diagram the 6 - A propagator. One has:

ver tex is fo l lowed by a photon lJ

Apart f rom a s ign t h i s Is p r e c i s e l y the same f a c t o r as occur r lng i n t h e a,,BJ,, ver tex 1eik,,[2nl4iJu1 and we may use the same n o t a t i o n f o r I t :

The r e s u l t l n g Ward i d e n t i t i e s are then:

= +. . . + ro" %

(11.8) This r e s u l t looks perhaps a l i t t l e strange, s ince u s u a l l y one tends t o f o r g e t t he l i n e s t h a t go s t r a i g h t through w i thout i n t e r a c t i o n . A s a mat te r o f f a c t , t h i s i s al lowed on ly i n t h e case o f gauge-Invariant sources, f o r which, s ince then k J = 0 , we have: u u

Equation [11.8) i s an equat ion f o r Green's func t ions . O f course i n d e f i n i n g Green's func t i ons one should no t employ p a r t i c u l a r p roper t i es o f sources, sucn as auJp = 0. As we w i l l see, t h e S-matrlx will be defined us ing gauge- invar iant sources f o r t h e e lect romagnet ic f l e l d , and f o r such cases the r igh t -hand s ide o f Eq. (11.81 Is zero.

We remark t h a t t he War0 i d e n t i t y , Eq. [11.81, is t r i v i a l l y t r u e as I t stands i n t h e case o f no i n t e r a c t i o n t h a t we are considering. I t i s i n f a c t s u f f i c i e n t t o r e a l i z e t h a t here, f o r oxample f o r the four -po in t Green's func t i on , we have:

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DIAGRAMMAR

It is worth w h i l e t o no te t h a t t h e above d e r i v a t i o n goes through a l s o I f e l e c t r o n s , o r any o t h e r p a r t i c l e s , a r e p re sen t , provided t h e photon is coupled t o t h e m i n a gauge- invar ian t way. Let u s cons ider i n some d e t a i l s t h e i n t roduc t ion of t h e e l e c t r o n s i n t o t h e theory. The Lagrangian t o start rrom 1s

where L”(A,,, B) I s given i n tq. 111.31. The new p i eces o f - t h i s Lagrangian involv ing t h e e l e c t r o n f i e l d , t h e source terms .ley + $Je excepted, are i n v a r i a n t under t h e t ransformat ion 111.41 i f a l s o t h e e l e c t r o n f l e l d I s t ransformed

- - + l E e B c1 - 2 JI + $e $[1 + iEeB1 + O ~ E 1 . Then, up t o f i r s t o rder i n E , we have

The e x t r a v e r t i c e s of t h e transformed Lagrangian a r e t h e n t h e same a s t h o s e of t h e f r e e f i e l d case, given i n (11.71, t o g e t h e r w i t h t h e fo l lowing ones involv ing tne e l e c t r o n sources :

-(2nI4ljeLq + k l l e e ,

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DIAGRAM M A R

Therefore, f o r Green's func t i ons i n v o l v i n g no e x t e r n a l e l e c t r o n l i n e s , t h e Ward i d e n t i t i e s a re exac t l y t he same as i n Eq. (11.81, even i f i n t h i s case the bubbles conta in any number of closed e l e c t r o n loops. When e x t e r n a l e l e c t r o n l i n e s are present, t h e Ward i d e n t i t i e s rece ive a d d i t i o n a l con t r i bu t i ons o f t he form:

11.2 Lorentz Gauge: S-Matrix and U n i t a r i t y

L e t us now i n v e s t i g a t e t h e S-matrix, keeping I n mind f o r s i m p l i c i t y t h e f r e e e lect romagnet ic f i e l d Lagrangian 1:. Eq. 111.2'1. The m a t r i x K [see Sections 2 ana 71 i s here

From the c u t t i n g equation, Sec t ion 7, we de r i ve t h e f a c t t h a t u n i t a r i t y would ho ld i f t h e sources were normalized according t o Eq. (7 .21 . nowever, t he complex conjugate o f a fou r -vec to r i s defined t o have an a d a i t i o n a l minus s ign i n i t s f o u r t h component (see the I n t r o d u c t i o n ) , so we are forced t o a t t r l b u t e t o the fou r th component o f h vec tor p a r t i c l e e negat ive m e t r i c (Sect ion 81. The sources Jtae, a = 1, ..., 4 are chosen such t h a t

U

(12.91

= - 1 i f a - 4 .

On the o ther hand, i t I s w e l l known t h a t due t o gauge inva- r i a n c e we do have a p o s i t i v e m e t r i c theory, bu t w i t h on ly two photon p o l a r i z a t i o n s . i n the system where k we l a b e l t he sources as f o l l o w s

= (0. 0, K , l k o l , lJ

100


‘I) - 11, 0, 0, OJ 5 d2) = ( 0 , 1, 0, 01 I

J l 3 ] - to, 0, 1, 01 ,

J[41 = (0, u, 0, 11 .

u

IJ

lJ

we now p o s t u l e t e t h a t on ly t h e f i r s t two o f these sources emi t phys i ca l photons. I n terms of a non-covariant ob jec t z - (0, 0, -K, IK 1 obtained f rom k DY space-ref lect ion, we have

lJ 0 v

whereas

Considering c e r t a i n se ts of cu t diagrams we can epply the Ward i d e n t i t i e s t o t h e l e f t - and t h e r igh t -hand side, t o see t h a t t h e terms p r o p o r t i o n e l t o k end kv, respec t i ve l y cancel emong themselves. For ins tance, a#! t he l e f t -hand s ide o f a c u t diagram one can apply t h e Ward i d e n t i t y :

As be fore t h e double l i n e i n d l c a t s s a f a c t o r ik,,. Note t h e t a l l p o l e r i z a t i o n s can occur a t t h e in te rmea ia te s ta tes , 1.8. t h e ou t - s ta tes f o r S end t h e i n - s t a t e s f o r St, but t h e o ther l i n e s ere phys ica l .

I n our case the r ign t -hand s ide o f t h i s Ward i d e n t i t y i s zero because t h e sources on t h e r igh t -hand s ide absorb energy only, end the re fo re :

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DIAGRAMMAR

-1 So, due t o t h e Ward i d e n t i t i e s , one may r e p l a c e t h e f a c t o r KPv i n t h e In te rmedia te s t a t e s , found from t h e c u t t i n g equat ion , by

Jh13Jh11 + Jh21J$’3, w h i c h imp l i e s u n i t a r i t y i n a H i l b e r t space w i t h only t r a n s v e r s e photons.

We m u s t make a s l i g h t d i s t i n c t i o n between phys ica l sources and gauge- invar ian t sources . The combination

= k J(3’ + ,J t4’ = (0, 0, ko, IKJ JIJ 0 P P

i s gauge i n v a r i a n t because a,J, = 0, b u t on mass s h e l l t h i s sour - ce i s p ropor t iona l t o kP and it g i v e s no c o n t r i b u t i o n due t o t h e Ward I d e n t i t i e s , so d e s p i t e i t s gauge inva r i ance , I t i s unphysi- c a l , i n t h e sense t h a t i t emits nothing a t a l l , not even ghos t s .

11.3 Other Gauges: T h e Faddeev-Popov Ghost

t iefore going on, l e t u s come back f o r a moment and t r y t o i n t e r p r e t t h e Ward i d e n t i t y w e have proved. S t a r t i n g from a gauge- invar ian t Lagrangian, E q . 111.1), w e added t h e term

i n o rde r t o d e f i n e a propagator . T h e Ward i d e n t i t i e s fo l low by performing a Bell-Treiman t r ans fo rma t ion , 1.e. a canonica l t ransformat ion t h a t i s a l s o a gauge t ransformat ion . As we a l r eady noted, only t h e above term (and t h e source term1 g i v e s rise t o a c o n t r i b u t i o n , namely ~(a , f i , l [a2Bl . T h i s is t h e on ly coupl ing of t h e B - f i e l d formal ly appearing i n t h e t ransformed Lagrangian. Now t h e f ac t t h a t B remains a f r e e f i e l d , 1 .e . g ives ze ro when p a r t of a Green’s func t ion , i m p l i e s t h a t aPAP i s a l s o free. T h i s i s a c t u a l l y t h e conten t o t t h e Ward i d e n t i t y . The re - f o r e we conclude t h a t t h e a d d i t i o n of t h e term -1 /2ta A l 2 does not change t h e physics of t h e theory . P P

T h i s d e f i n e s our s t a r t i n g po in t . I f , due t o a gauge inva- rlance, a Lagrangian Is s i n g u l a r , 1.e. t h e p ropagators do not ex i s t , t h e n a “good” Lagrangian can be obta ined by adding a term

102


2 -4; , wherc C behaves under gauge transformations as C + C + + t A . Here t is any field-independent quantity that may contain derivatives. The argument given above, snowing that the addition of the term -1/2[a,,A,,12 does not modify the physical content of the theory, can equally well De applied here. C will appear to-be a free field, as can be seen by performing a Bell-Treiman transformation. However, for this simple recipe to be correct, as in the case explicitly considered C = apAU, one needs t, defined by the gauge transformetion C + C + tA, not to depend on any fields.

The difficulty with Yang-Mills fields is that the gauge transformations are more complicated, so much so that no simple C with the required properties exists. [Actually there exists a choice of C for Yang-r.iills fields that is acceptable from this point of view., namely C = aA3, a + =. See R.L. Arnowitt and S.I. Fickler, Phys. Rev. 12/, 1821 (1962). I For this reason we will now study quantum electrodynamics using a gauge function C that has more complicated properties under gauge transformations.

We take for understood the theory corresponding to the Lagrangian

Ihe sources will be taken to be gauge invariant and are included in xinv. Later we will consider also non-gauge-invariant sources.

2 Let us now suppose that we would like to have a A,, + XA instead of a,,A,, for the function C. We can go from t!e above’ Lagrengian with C = +,Au to the case of C = of a non-local Bell-lreiman transtormation

+ X A ~ by means lJ

A,, + A,, + xa -2 ay[AvJ2 ,

or, more explicitly,

x’JAtlx’) , [11.111

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DIAGRAMMAR

Here we have a s i t u a t i o n as described i n Subsection 10.5. For the subsequent manipulat ions t o be t r u e we must i n f a c t supply a - i e t o the denominator. Note t h a t

1: 1s unchanged under t h i s somewhat strange gauge t ransformat f%, and

I n view o f t h e s t r u c t u r e o f our canonica l t rans format ion we may expect t h a t a ghost Lagrangian must be added.

Per forming t h e t rans tormat ion (11.11) one gets t o f i r s t order i n E t he f o l l o w i n g spec ia l ver tex [see Sect ion I D ) :

Here, as usual, t h e dot i n d i c a t e s t h e fac to r V whlch i s minus the inverse photon propagator, and t h e Short dodble l i n e a fac- t o r I+,. The dot ted l i n e represents t h e func t i on A(x - x’l, which i s j u s t l i k e a sca la r massless p a r t i c l e propagator. The theorem proved I n Sect lon IU shows t h a t t h e theory described ny the transformed LagrangIan remains unchanged i f we a l so prov ide f o r ghost loops, const ructed by connecting t h e o r i g i n a l photon l i n e s t o one o f t h e o the r photon l i n e s o f t h e spec ia l ver tex. F o r example, we have:

i

Here we have cancel led t h e photon propegetors against t h e dbts, so t h a t t he new ver tex :

104


appears which can be formed by i n t r o d u c i n g a massless complex f i e l d (I i n t e r a c t i n g w i t h the photon v i e t h e i n t e r a c t i o n Lagrangian e = 2A$ A,,a,,$. A l l t he c losed loops const ructed t h i s way must ha6e a minus s i g n I n f r o n t .

- We come t o t h e conclus ion t h a t t h e Lagrangian

w i t h the p r e s c r i p t i o n t h a t every c losed 9 loop ge ts a minus sign, reproduces t h e same Green’s func t i ons as oefore, when we had C = auAP and no ghost p a r t i c l e s . The (I p a r t i c l e i s c a l l e d t h e Faddeev-Popov ghost. rhe 10 p r e s c r i p t i o n f o r i t s propagator is t he usual one. I t 1s perhaps noteworthy t n a t t h e Faddeev-Popov ghost i s not t h e gnost w i t h propagator -1 o f Sec t ion 10. Ihe F-P ghost i s t he I n t e r n a l s t r u c t u r e o f t h e t rans format ion func t ion , see t ransformat ion [ I I . l I J .

EXAMPLE 11.3.1 The Feynman r u l e s from t h e Lagrangian (11.121 are:

1 k - W-v (Zn14i k2 - i c ’

*----a

6x Q P

(-1 f o r every c losed loop1 ,

1 - 4

[2n1 i k2 - IE

+ 6 6 1 , a6 BY

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DIAGRAMMAR

Let u s cons ider photon-photon s c a t t e r i n g i n t h e zero loop approximation. T h e fo l lowing diagrams:

x + X . M + X a) b) C ) d )

m u s t g i v e zero. T h i s i s indeed t h e case

4 1 1 4 4 2 a1 -t (2n1 2x6 [ - k X l 7 2 (2nl 2Xk 6 = 412nl i A ' aB [2n1 1 K a ~6

4 2 C J + 4(2nJ i h 6 b , a6 BY

Note t h a t I n t h e con t r ibu t ions from diagrams [ e l , IbJ , [ c ) , many terms Vanish, because our e x t e r n a l sources a r e t a k e n t o be gauge i n v a r i a n t .

O f course, t o g e t a con t r ibu t ion from t h e Faddeev-Popov ghos t , one has t o cons ider examples o f diagrams conta in ing loops. Indeed, one notes f o r i n s t ance :

Actua l ly t o prove t h i s c a n c e l l a t i o n we need a gauge- invar ian t r e g u l a r i z a t i o n method, w h i c h w l l l be provided i n t h e fo l lowing .

We want t o understand now whether a gene ra l p r e s c r i p t i o n can be given w h i c h , I n a gene ra l gauge def ined by t h e func t ion C ,

106


a l lows us t o w r i t e down immediately the Lagrangian f o r t he ghost p a r t i c l e 4 .

2 I n t h e case considered o f C = alJA,, + X A v , i t i s easy t o

see t h a t t he ghost Lagrangian i n 111.121 comes out by t h e f o l l o - wing p resc r ip t i on .

Take the f u n c t i o n C. under a gauge t rans format ion one has t o f i r s t o rder i n A

C + C + d A

w i t h & some operator, which i n tne ac tua l case depends on t h e f i e l d s A . then the ghost Lagrangian i s simply

!J

Here

f rom which 111.12) fo l lows.

The g e n e r a l i t y of t h i s p r e s c r i p t i o n can be understood by l ook ing a t our manipulat ions. The non- loca l t rans format ion 111.711 was taken such t h a t a A + a A + XA2. I f we consider a gauge t rans format ion lJ l J l J lJ

2 + a ~ + a h , alJAIJ ll lJ

then we must so lve the equation

2 The f a c t t h a t t h e propagator l / k appeared I n our loops i s

thus simply r e l a t e d t o the behaviour a A + a A + a2A. Secondly, t h e spec ia l v e r t i c e s are P l J l J l J

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where VUv I s I n t h e usua l way r e l a t e d t o t h e photon propagator. Then t h e v e r t l c e s appearing i n the ghost loops a r i s e f rom t h e p r e s c r i p t i o n : remove one f a c t o r A,, f rom t h e expression AA2 and rep lace i t by a A. D iagranmet ica l l y : U

U

H f a c t o r o f 2 prov ides f o r t h e two poss ib le ways o f removing an A U - f i e l d f rom AAE. Th is is indeed p r e c i s e l y what one ob ta ins t o f i r s t order I n A by submi t t i ng XA2 t o a gauge t ransformat ion, see Eu. 111.131. U

The beauty o f t h i s r e c i p e t o f i n d the ghost Lagrangian is t h a t t he re i s no reference t o the f u n c t i o n C t h a t we s t a r t e d from. To cons t ruc t t h e ghost Lagranglan one needs on ly the C s c t u a l l y requi red. A lso we need on ly i n f i n i t e s i m a l gauge t rans - format ions.

EXAMPLE 11.3.2 The above p r e s c r i p t i o n works a l so i n t h e case of C = i3 A t h a t we s t a r t e d from. I n f a c t we have

V l J

a~ + a P A U + a ~ , 2 U U

meaning t h a t t h e ghost Lagrangian should be

w i t h no coup l ing however o f t h e + - f i e l d t o t h e e lect romagnet ic f i e l d . Therefore, as we a l ready know, no ghost loop I s t o be added I n t h i s case.

Even if t h e above manipulat lons are q u i t e s o l i d one may have some doubts concerning t h e f i n a l p r e s c r i p t i o n f o r t h e ghost La- grangian. To e s t a b l i s h correctness o f t he theory, as we d i d i n t h e case of C = aUH,,, Ward i d e n t i t i e s are needed. I n t h e genera l case such i d e n t i t i e s are more complicated than those g i ven before and we w i l l c a l l them Slavnov-Taylor I d e n t i t i e s .

11.4 The Slavnov-Taylor I d e n t i t i e s

In t h i s sec t i on we w i l l a e r i v e t h e Slavnov-Taylor i d e n t i t i e s , us ing on ly l o c a l canonica l t ransformat ions. One can always keep i n mind t h e e x p l i c i t case o f C = a H + AA2, bu t we now s t a r t ,,u U

1 08


adapting a general notat ion a n t i c l p a t m g the general case.

Le t there be given a funct ion C, t h a t behaves under a gene-

P l J U r a l gauge transformation A + A + a h as fo l lows

c + c + rm + ~ J A + or**) . (11.14)

Here we s p l i t up the e a r l i e r factor 4 - Ifi + iJ wizh the p a r t It depending on the f i e l d A . For C - a H + XA and g = 2M a

WE have A - a IJ l J l J IJJ

v ll'

Our s t a r t i n g point w i l l be the Lagrangian

w i t h the - l / l oop p resc r ip t i on f o r ghost loops. As before. we add t o t h i s Lagrangian B piece descr ib ing a f ree f i e l d B

1 2 2 2 - ata - p IE + J ~ B

end perform the l o c a l Hell-Treiman transformetion

A,, + Au + tuO

Working t o f i r s t order i n the f i e l a B

r11.1ti1 Here we used the transformation

I -

a + a + 10 + ;la + ore2) . Again we d i s t i ngu ish the field-independent part 0 and the f i e l d - dependent B . I n the actual case o f i = XA a we have

l J l J

i~ - zxa Ba d = D I P P J

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T h e vertices i n v o l v i n g t h e 8-f ie la i n t h e t r a n s f o r m e d L a g r a n g i a n (11.16) are ( t h e o v e r - a l l f a c t o r ( Z V I 4 i is a l w a y s l e f t u n d e r s t o o d ) :

-i from -cia, - ----7

The ver tex c o r r e s p o n d i ? g t o 9 &I$ i s n o t drawn slnce I n - t h e actual case [C = +,A,, + XAE) d is zero. I h e v e r t i c e s CmB a n d CRB a p p e a r w i t h a m i n u s s i g n i n t h e L a g r a n g i a n : t h i s s i g n i s n o t i n c l u d e d i n t h e v e r t e x d e f i n i t i o n a n d t h u s m u s t b e p r o v i d e d s e p a r a t e l y . The s t a t e m e n t t h a t B r e m a i n s a i n t h e 8 - v e r t i c e s :

I n t h e s e c o n d term a t t h e

f ree f i e l d becomes t n e n t o first order

r i g h t - h a n o s i d e we e x h i b i t e d e x p l i c i t l y t h e w h o l e l o o p t o w h i c h t h e 8 - l i n e i s a t t a c h e d , t o g e t h e r w i t h t h e a s s o c i a t e d m i n u s s i g n . T h e b l o b s are b u i l t u p f r o m d i a g r a m s c o n t a i n i n g p r o p a g a t o r s a n d v e r t i c e s as g i v e n i n t h e p r e v i o u s sec t ion . T h e $ p a r t i c l e s go a r o u n d i n l o o p s o n l y , and s u c h l o o p s are i n c l u d e d i n t h e b l o b s .

I n t h e a b o v e e q u a t i o n t h e r e 1s o n e c r u c i a l p o i n t . C o n s i d e r t h e v e r y s l m p l e s t case, n o i n t e r a c t i o n and o n l y o n e p h o t o n s o u r c e . s ince C = alrA,, + XAE, one h a s t o z e r o t h o r d e r i n X :

110


T h e first diagram [ s i m i l a r l y t o t h e second one) con ta ins t h e B propag3tor po le , a phOt?n propagator po le , t h e ve r t ex f a c t o r [Zn)4im and tce f a c t o r t . The las t diagram has only thg B po le and a f a c t o r t,, = a,. ThN equat ion cac be t rue only i f rn i s ze ro p r e c i s e l y on t h e photon pole . We haa m = a' + -k2, and we m u s t theref o r e always take

[ 2 r I 4 i i = (ZnI4l[a2 + i E ) + - [2n)4i1k2 - I€ ) .

As w i l l be seen, t h i s m i s m i n u s t he i n v e r s e of t h e ghos t propaga- t o r . I n t h e gene ra l case , cons ider ing lowest o rde r i d e n t i t i e s , one m u s t check on t h e 1 i n t h e p ropagators i n t h e manner des- c r ibed here.

We w i l l perform some manipulat ions on Eq. [11.171. Consider t h e f i rs t diagram on t h e r ight-hand s i d e . Using:

6i = = - cc- C -av c

t h e c o n t r i b u t i o n of t h i s aiagram becomes:

w i t h a new ve r t ex : P --

Trea t ing t h i s f i rs t ve r t ex a s a photon source we see t h a t we can i t e r a t e , g e t t i n g :

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ZJ means sum over a l l t h e photon sources. The l a s t diagram a r i s e s because t h e f i r s t ver tex i s t r e a t e d as a source. Some new v e r t i c e s enter , wnose meaning i s unambigoualy defined. F o r example, we have:

4 - which corre.spond p r e c i s e l y t o a ver tex 12n1 i e o r t o an l n t e r a c - t i o n term $'*a,Ba 4 . I n p a r t i c u l a r , s ince t h i s term i s i n v a r i a n t under t h e In terckange B -t $, we have:

/ \ \

3 - - -4 1 ---*, This equa l i t y , t r i v i a l i n t h e case o f quantum electrodynamics, becomes much more t r i c k y i n t h e case o f non-Abelian gauge symme- t r i e s . Then a s i m i l a r i d e n t i t y f o l l o w s f rom t h e group s t ruc tu re , and i s c l o s e l y r e l a t e d t o the Jacob1 i d e n t i t y f o r t he s t r u c t u r e constants o f t he group.

The r igh t -hand s ide of Eq. [11.181 can now be i n s e r t e d i n p lace o f t he f i r s t diagram on the r igh t -hand s ide o f t he o r i g i n a l EQ. [11.171. We note t h a t some cance l l a t i ons occur. The second diagram on t h e r igh t -hand s i d e o f con f i gu ra t i on where t h e ghost has

Eq. [11.171 conta ins a l so t h e no f u r t h e r i n t e r a c t i o n :

Th is cancels p r e c i s e l y t h e l a s t diagram of t he i t e r a t e d equation, a f t e r i n s e r t i o n i n Eq. 111.171.

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This process can be repeated i n d e f i n i t e l y and one f i n a l l y ar r ives a t the equation:

The ghost l i n e going through the diagram may have zero, one, two, etc., ver t ices of t h e type:

If one notes now tha t the mass of the 8 - f i e l d i s an absolutely f r e e parameter i n our discus?ion. we can a lso g ive now the B- f i e l d the propagator - [ Z r 1 4 i m ' l . I n t h i s way we can simply reduce the f i r s t diagram on the r ight-hand t o :

side o f Eq. [11.191

and a lso incluae i n I t the l a s t diagram o f t h a t :

*--Q -m-' c = -

Eq. [11.191. Observing

m we get the i d e n t i t y :

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Note: Ward i d e n t i t y ~ 1 1 . d 1 . One has indeed:

I n t h e case of C = a,,A,, t h i s equat ion co inc ides w i t h t h e - k - K - t - 3 .

CL 1- i k 1 = 4 -

4 2 lJ = [ 2 ~ 1 i J g

1271 i k &

so t h a t t he f i r s t diagram of Eq. (11.201 can a l so be drawn as:

Furthermore, i n t h i s case o f C = a A rhe ghost does no t i n t e r a c t and then : l J l J

Equat ion [11.201 can be genera l ized as t 'o l lows. Suppose t h a t a source is coupled t o severa l e lect romagnet ic f i e l d s , f o r i ns tance

JA2 o r JA4 P P

o r even more genera l t o a l o c a l bu t otherwise a r b i t r a r y f u n c t i o n R [ A ) o f t he A . Example:

P

4 lJ lJ

J[aPAlJ + AAZ + KH , ... I . Ihe on ly way t h l s comes i n t o the above d e r i v a t i o n i s through t h e behaviour o f t h i s whole term under gauge t ransformat ions. Suppose one has i n the Lagrangian [11.151 a l so the coup l ing

J R [ A 1 . 1.1

Furthermore, under a gauge t ransformat ion


+ A + a h , AIJ lJ IJ

one has

i s t h e p a r t indgpenQe?t of t h e A and conta ins the A depen- IJ dent pa r t s . Both r and p may contaffn de r i va t i ves .

Now, per forming the Bel l -Tre iman t rans format ion w i t h the B- f i e l d one ob ta ins the v e r t i c e s :

f rom ~ r t l ,

f rom J&I , -- -x

where the double l i n e denotes a c o l l e c t i o n o f p o s s i b i l i t i e s , inClUQing a t l e a s t one photon l i n e . The whole d e r i v a t i o n can be c a r r i e d through unchanged, and g iven many sources Ja coupled t o many f i e l d func t i ons Ra we ge t :

L

where r and p under g8uge t r h s f o r m a t i o n s

are de f ined by t h e behaviour o f t h e func t i ons Ra

The above i d e n t i t i e s are t h e Slavnov-Taylor i d e n t i t i e s .

11.5 Equivalence o f Gauges

From the precedlng sec t ions we ob ta in t h e f o l l o w i n g pres- c r i p t i o n for handl lng a theory w i t h gauge invar iance. F i r s t choose a non-gauge-invariant f u n c t i o n C and add -1/2C2 t o t h e Lagrangian. Next cons ider the p r o p e r t i e s o f C under i n f i n i t e s i m a l gauge transformations, f o r ins tance

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A ghost p a r t must a l so be added t o t h e Lagrangian.

w i t h t h e p r e s c r i p t i o n of p r o v i d i n g a f a c t o r -1 f o r every c losed 0 loop. C i e a r l y t h e choice of C i s m i l i t e d by t h e f a c t t h a t t h e operator m, d e f i n i n g t h e ghost propagator, must have an inverse. Moreover C must together w l t h 1: t o rs , bu t t h i s i s automatic if t""breaks the gauge invar iance.

def ine non-s ingular A propaga-

Suppose we had taken a s l l g h t l y d i f f e r e n t C

Now under a gauge t rans formet ion

C' -c C' + [k + i ) A + ~ [ r + ;)A ,

where r and p are, respec t i ve l y . t h e f ie ld- independent and f i e l d - dependent p a r t s r e s u l t i n g From a gauge t rans format lpn o f R. For example, if R = A2 then R + R + 2A a A and we nave r = 0 and p = 2AIJau* IJ U P

The ghost Lagrangian must be changed accord ing ly . and we ge t

* 1' = J: - 1 [ c + ERI' + 0 [ii + i + EF. +

(11.221 i n v 2

We now prove t h a t t o f i r s t order i n E t he s-matr ix generated by x' equals t h e s -mat r ix generated by 1. l h i s is c l e a r l y s u f f i c i e n t t o have equivalence o f any two gauges t h a t can be connected by a se r ies of i n f i n l t e s i m a l steps.

Le t us compare the Green's func t i ons o f &* w l t h those o f e. The d i f ference is given by Green's funct ions con ta in ing one E-vertex. From Eq. [11.221 these v e r t i c e s are:

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G. 't HOOFT ond M. VELTMAN

The d i f ference between an f d and Green's func t i on w i t h the same external legs i s then:

We exhib i ted e x p l i c i t l y the minus sign associated w i t h the ghost loops. upenlng up the top vertex t h i s d i f ference i s :

I f now a l l the o r i g i n a l sources are gauge invar ient , namely a J,, = 0. we see tha t t h i s d i f ference i s zero as a consequence of !he slavnov-Taylor i d e n t i t i e s , Eq. 111.21). The diagrams i n Eq. (11.211, where the ghost l i n e i s attached t o such curre2ts. g ive ?o contr ibut io?, since f o r these currents we have r,, - a,, land rvJv = 01 and p - 0. As the S-matrix i s defined on the besis of gauge-invariant sources, we see t h a t the S-matrix i s i nva r ian t under a change o f gauge as given above.

11.6 Inc lus ion o f Electrons: Wave-Function Renormalizetion

The preceding discussion has been ca r r i ed out i n such 8 way tha t the i nc lus ion of e lectrons changes p r a c t i c a l l y nothing. The main d i f ference 1s t ha t we now must introduce sources t h a t e m i t or absorb electrons and such sources are not gauge invar iant . This complicates somewhat the discussion o f the equivalence o f gauges.

Thus conslder e lect ron 8ource terms

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under gauge t r a n s f o r m a t i o n s ( A -P A + a A 1 P P U

Let u s c o n s i d e r t h e d i f f e r e n c e of t h e Green's f u n c t i o n s of two d i f f e r e n t gauges i n t h e p r e s e n c e of an electron source . For sim- p l i c i t y we w i l l on ly draw one o f them e x p l i c i t l y . The Slavnov- Tay lo r i d e n t i t y is:

J.

I h e f i r s t t h r e e terms a r e p r e c i s e l y t h o s e found i n c o n s i d e r i n g t h e d i f fe rence of t h e Green's f u n c t i o n s , or more p r e c i s e l y af o b j e c t s o b t a i n e d when u n f o l d i n g a v e r t e x i n t h e d i f f e r e n c e o f t h e Green 's f u n c t i o n (see p r e c e d i n g s e c t i o n l .

Fo ld ing back t h e C and R s o u r c e t o o b t a i n a g a i n t h e t rue d i f f e r e n c e of t h e Green's f u n c t i o n and u s i n g t h e Slavnov-Taylor i d e n t i t y we g e t :

I n g e n e r a l , such a t y p e of dlagram w i l l have no p o l e a s one goes t o t h e e l e c t r o n mass - she l l . Thus p a s s i n g t o t h e S-matr ix t h e c o n t r i b u t i o n of most diagrams will d i s a p p e a r . B u t i n c l u d e d i n t h e above se t are also dlagrams o f t h e t y p e :

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w h i c h do have a pole. However, we m u s t not f o r g e t t h a t t h e S- matr ix d e f i n i t i o n I s based on the two-point func t ion . T h e sources m u s t be s u c h thet t h e r e s idue of the two-point f u n c t i o n Inc luding t h e sources i s one.

Consider t h e two-point func t ion f o r t h e e l e c t r o n :

A l s o t h i s func t ion w i l l change, and f o r any of t h e two sources we w i l l have p r e c i s e l y t h e same change a s above:

I n accordance w i t h our d e f i n i t i o n of t h e S-matrix we m u s t r e d e f i n e our sources s u c h t h a t the r e s i d u e of the two-point func t ion remains one8 t h a t I s

where

6 - value a t t h e po ie o f -& It i s seen t h e t inc luding t h e r e d e f i n i t i o n of t h e sources t h e S-matrix I s unchanged under 5 change of gauge.

I n convent ional language. t h e above shows t h a t t h e e l e c t r o n wave-function renormal iza t ion i s gauge dependent, b u t t h a t i s of no consequence f o r t h e S-matrix.

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12. COPlBINATORIAL METHODS

There ere essentially three levels of sophistication with which one can do combinatorics. On the first level one simply uses identities of Vertices that are a consequence o f gauge invariance. Example: electrons interacting with photons. The

- - + A + a a

P P JI + $ + lee$ , JI + tp - leB$ ,

pert of the Lagrangian containing electrons is

Of course this LagrangIan remains invariant under th-se ransformations, but we want to understand this in terms o f diagrams. Une has, not cancelling anything, as extra terms

- i e [Znl 4 11-iyp + m l ,

4 ie[2n1 i ( i y q + ml ,

How does the cancellation manifest itself? In the Lagrangian one must write

120


t o see i t . Here we take the f i r s t vertex and w r i t e

Then one sees e x p l i c i t l y how the three ve r t i ces together g ive zero

4 -1el2nI i{iyq + l y k + m - i y q - m - i y k ) - 0

and t h i s remains t rue i f we replace m everywhere by m - i c . Only then can one say t h a t the ve r t i ces contain fac to rs which are inverse propagators: now we know f o r sure tha t the fo l l ow ing Green's funct ion i d e n t i t y holds [ i n lowest order]:

where the short double l i n e ind icates the inver ted propagator inc lud ing i c . Exp lo i t i ng t h i s f a c t we obtain:

which i s a Ward i d e n t i t y given before.

The above makes c lear how tne very f i r s t l e v e l combinatorics is t h e bu i l d ing block f o r the second-level combinatorics. This second-level combinatorics uses the f a c t t h a t a13 terms cancel when one has gauge Invariance. But t h i s f a c t must be v e r i f i e d e x p l i c i t l y by means o f f i r s t - l e v e l combinatorics t o ascertain tha t the i E p resc r ip t i on i s conslstent w i t h the gauge invariance. I t is j u s t by such a type o f reasoning t n a t t h e ic i n the ghost propagator is f ixed, as shown before.

The t h i r d l e v e l o f comblnatorics is t h a t when one uses non- l o c a l canonlcal transformations. These can be used t o der ive the Slavnov-Taylor i d e n t i t i e s d i r e c t l y , as was done by Slevnov. One must be very carefu l about the IE prescr ip t ion: our procedure whereby t h i s i d e n t i t y was derived by means of f i r s t - and second- l e v e l combinatorics shows t h a t the usual -I€ presc r ip t i on f o r the ghost propagator i s the correct one.

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13. REGULARIZATION AND RENORMALIZATION

13.1 General Remarks

As noted before, the regulation scheme introduced in the beginning is not gauge invariant. We need a better scheme that can also be used in case of gauge theories, Abelian or non- Hbelian. An elegant method is the dimension regularization scheme which we Will discuss now.

A good regularization scheme must be such that for certain values of some parameters [the masses A in the unitary cut-off methodl the theory is finite and well defined. The physical theory obtains in a certain limit (masses oecoming very big), and one requires that quantities that were already finite before regularization was introduced remain unchanged in this limit.

In order to obtain a finite physical theory it will be necessary to introduce counter-terms in the Lagrangian. H theory is said to be renormalitable if by addition of a finite number Of kinds of counterterms a finite physical theory results. This physical theory must of course not only be finite, but also unitary, causal, etc. With regard to a regularization and a renormallza- tion procedure for B gauge theory, some problems arise which are peculiar to this kind o f theory. As demonstrated before, the Ward identities (or more generally the Slavnov-Taylor identitiesl must hold, because they play a crucial role in proving unitarlty. Since regularization implies the introduction of new rules for diagrams. and because new vertices, corresponding to the renormalization countertens are added, it becomes problematic as to whether our final renormalized theory eatisfies Ward identities. A s far as the regularization procedure is concerned, a first step is to provide a scheme in which the Ward identities are satisfied for any value of the regularizetion parameters.

This goal is obtained in the dlmensional regularization scheme. It must be stressed, however, that this does not guarantee that the counterterms satisfy Ward identitles and indeed, in general, one has considereble difficulty in proving ward Identities for the renormallzed theory. The point is this: in the unrenormalized theory there exists Ward identities, and an invariant cut- off procedure guarantees that the countertens satisfy certain relations. From these reletions one can derive new Ward identities satisfied by the renormalized theory. However, these Ward Identi- ties turn out to be different: one may speak of renormalized Ward identities. That is, one can prove that the renormalized theory has a certain symmetry structure [giving rise to certain Ward identitiesl, but one has to show also that this symmetry is the same as that of the unrenormallzed theory. Let us remark once

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more t h a t Ward i d e n t i t i e s are needed f o r t h e renormalized theory because they are needed i n p rov ing u n i t a r i t y . Indeed, Ward i d e n t i - t i e s have no th ing t o do w i t h r e n o r m a l i z a o i l i t y bu t every th ing t o do w i t h u n i t a r i t y .

Another problem t o be considered is t h e f o l l o w i n g one. I f one admits t h e p o s s i b i l i t y t h a t t h e renormalized s y m e t r y i s d i f f e r e n t f rom the unrenormalized one, then f o r m a l l y t h e f o l l o - wing can happen. I f one c a r r i e s tnrough a renormal iza t ion program, one must f i r s t make a cholce o f gauge i n t h e unrenormalized theo- r y . Perhaps then the symmetry o f t he renormalized Lagrangian depends on t h e i n i t i a l choice o f gauge. Given a gauge theory one must show e x p l i c i t l y t h a t t h i s i s no t the case, and t h a t t h e var ious renormalized Lagrangians belonging t o d i f f e r e n t gauge choices i n t h e unrenormalized theory are r e l a t e d by a change o f gauge w i t h respect t o t h e renormalized symmetry. I n quantum electrodynamics t h l s problem i s s ta ted as t h e gauge independence o f t h e renormalized theory.

13.2 Dimensional Regu lar iza t ion Method: One-Loop Diagrams

I n t h e dimensional r e g u l a r i z a t i o n scheme a parameter n i s in t roduced t h a t i n some sense can be v i s u a l i z e d as t h e dimension o f space time. For n 4 a f i n i t e theory r e s u l t s 1 the phys i ca l theory ob ta ins i n the l i m i t n = 4.

As a f i r s t step, we de f i ne the procedure f o r one-loop diagrams. The example we w i l l t r e a t makes c l e a r e x p l i c i t l y t h a t t h e dimensional method I n no way depends on t h e use o f Feynman parameters. Ac tua l l y , s ince f o r two or more c losed loops u l t r a - v i o l e t d ivergencies may a l so be t rans fe r red f rom t h e momentum i n t e g r a t i o n s t o the Feynman parameter i n t e g r a t i o n s t h e use of Feynman paremeters i n connection w i t h the dimensional regu la - r i z a t i o n scheme must be avoided. o r a t l e a s t be done very j u d i c i o u s l y . The procedure for m u l t i l o o p diegrams w i l l be defined i n subsequent subsections.

Consider a se l f -energy diagram w i t h two sca la r in te rmed ia te p a r t i c l e s i n n dimensions:

I n t h e in tegrand the loop momentum p i s an n component vector .

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This expression makes sense I n one-, two-, and three-dimensional space1 i n f o u r dimensions t h e i n t e g r a l i s l o g a r i t h m i c a l l y d i v e r - gent. To evaluate t h i s i n t e g r a l we can go t o t h e k rest - f rame tk = 0, 0, 0, iu1. Next we can in t roduce p o l a r coord inates i n t h e remaining space dimensions

2 m W 2n n n

-OD 0 0 0 0

n-2 In I dPo w dw dol I de2 s i n El2 dEly s i n e3 .,.

1 x 2

1-Po + w2 + m2 - i ~ J [ - ( p ~ + p12 + u2 + M2 - i c ]

Here w is t h e l eng th of t h e subspace. The in tegrand has

and one can i n t e g r a t e 'n-2'

(13.31

(13.2) vec to r p i n t h e n-1 dimensional no dependence on the angles 8 us ing 1' '." - 11

These and Other u s e f u l formulae w i l l be g iven i n Appendix 8 . The r e s u l t is

. 113.41 1 1

(-p2 0 + u2 + m2j [ - (po + p12 + w2 + F I ~ I

Th is I n t e g r a l makes sense a l so f o r non-integer, I n f a c t also f o r complex n. We can use t h i s equat ion t o define I n i n t h e reg ion where t h e i n t e g r a l ex i s t s . And ou ts iae t h a t reg ion we de f ine I n as t h e a n a l y t i c con t inuat ion i n n o f t h i s expression.

Apparent ly t h i s expression becomes meaningless for n 5 1. f o r then the w i n t e g r a l d iverges near w - 0. or f o r ns 4 due t o t h e u l t r e v i o l e r behaviour. Tne lower l i m i t divergence i s no t very serious and main ly a consequence o f our procedure. A c t u a l l y f o r n + 1 not on l y does t h e i n t e g r a l diverge, b u t a l s o the r

124


f unc t i on i n t h e denomlnator, so t h a t 11 f rom Eq. t13.41 1s an undetermined form -/-. L e t US f i r s t f i x n t o be i n t h e reg ion where the above expression ex l s t s , for i ns tance 1.5 < n < 1.75. Next we per form a p a r t i a l i n t e g r a t i o n w i t h respect t o w2

n-2 1 2 2 [n-31/2 1 dw 2 2 d 2 (n-11/2 2 1w I I -

2 " - I d @ dw w = - dw [ w 1 2

. For n i n the g iven domain the surface terms are zero. Using zrlz) = I'Iz + 11 and repeat ing t h i s opera t ion A t imes we o b t a i n

We have de r i ved t h i s equat ion f o r 1 < n < 4. However i t 1s meaningful a l so f o r 1 - 2X < n < 4. The u l t r a v i o l e t behavlour is unchanged, b u t t h e divergence near w = 0 i s seen t o cancel eg ianst t he po le o f t h e l' func t ion , Note t h a t

tnus f o r z -+ 0 t h i s behaves as l / z .

The l a s t equation I s a n a l y t l c i n n f o r 1 - 2A < n < 4. Since i t coinc ides w i t h t h e o r i g i n a l I n for 1 < n < 4 i t must be equal t o the a n a l y t i c COntinUetiOn of I n ou ts ide 1 < n < 4. C lear ly , we now have an e x p l i c i t sxpressron f o r I n f o r a r b i t r a r i l y smal l values of n ( t a k i n g X s u f f i c i e n t l y la rge) . It I s equa l ly obvious t h a t i f t h e o r i g i n a l expression which we s t a r t e d from had been convergent, then i t s va lue would have been equal t o In f o r n 4 4. Moreover, c u t t i n g equations can be der ived us ing on ly t ime and energy components, and as long as t h e l a s t g iven expression for I n ex i s t s , and po and w i n t e g r a t i o n s can be exchanged C1.e. f o r n < 41 these c u t t i n g equations can be esta- b l i shed.

cont inuat ion . Fo r reg ion 1 < n < 4.

L e t us now see What happens f o r n >_ 4. Again we w i l l use the method o f p a r t i a l i n t e g r a t i o n s t o per form t h e a n a l y t i c

s i m p l i c i t y we se t X - 0. F i r s t f i x n i n t h e Next we i n s e r t 9. dw

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Next we per form p a r t i a l i n t e g r a t i o n w i t h respect t o po and . Again i n t h e g iven domain the sur face terms are zero. Fu r the r

x E . . . ) =

-2m 2 2p0p + 2p2 - 2M

x {...I . I n s e r t i n g t h i s I n the r igh t -hand s ide of Eq. (13.41 g ives

In=-I -n + 6 -1; 2 n 113.6)

o r

w i t h

The i n t e g r a l I; is convergent f o r 1 < n < 5. Now t n e I given n

126


before and t h i s expression are equal f o r 1 < n < 41 s ince t h e l a s t expression I s a n a l y t i c f o r n e 5 w i t h a simple po le a t n = 4, i t must be equal t o the a n a l y t i c con t inua t ion o f I . n

The above procedure may be repeated i nde f in i t e l y . . One f i n d s t h a t In i s o f t h e form

113.81

where t h e func t i on f 1s well-behaved f o r a r b i t r a r i l y l a rge n, The I’ f u n c t i o n shows simple po les a t n = 4, 6, 8, ... .

We see now why t h e l i m i t n + 4 cannot be taken: t he re i s a po le f o r n = 4. I t I s very tempt ing t o say t h a t one must i n t r o - duce a counterterrn equal t o minus t h e po le and i t s residue. But i f u n i t a r i t y i s t o be maintained t h i s counterterm may no t have an imaginary par t , 1.e. i t must be a polynomia l i n u, rn and m. Thus we must f i n d t h e form of t he res idue o f t h e pole. I t w i l l t u r n out t o be of t h e requ i red form. l o show t h a t i t i s o f t h e polynomia l form i n t h e genera l case o f many loops necess i ta tes use o f t h e c u t t i n g equations. For t h e one-loop case a t hand we w i l l s imply compute In suing Feynman parameters. One has

1 1

[p2 + 2pkx + K 2 x + m 2 x + rn 2 (1 - XII 2 ’ In = dx I drip 0

S h i f t i n g i n t e g r a t i o n va r iab les (p’ = p + kx l , making t h e Wick r o t a t i o n , and in t roduc ing n-dimensional p o l a r coordinates, one computes

I n t h i s way we have e x p l i c i t l y In i n t h e fo rm o f Eq. (13.81. F o r n = 4, 6, 8, ..., t h e in tegrand i s a simple polynomial, f o r n = 4 t h e i n t e g r a l g ives s imply 1. Thus t h e po le term i s

in2 2 PPLInl = -- 1 4 - n ’

where PP stands f o r “po le p a r t ” . Using the equat ion

(13.91

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one may compute

2 + k x i 1 - XI] + C.

nere C is E constant r e l a t e d t0 t h e n dependence Other than I n the exponent o f t he denominator, COntaining f o r ins tance I n n. f rom vn/2. I n genera l C i s a po lynomia l j u s t as the po le p a r t o f I n . Since we could have taken as our s t a r t i n g p o i n t an I n m u l t i p l i e d by bn-4, where b I s any constant, we see t h a t C I s undetermined. Th is I s t h e a r b i t r a r i n e s s t h a t always occurs i n connection w i t h renormal izat ion.

We now must do some work t h a t w i l l f a c i l i t a t e the t reatment o f t he m u l t i l o o p case. L e t us consider the c u t t i n g equation. One has :

+ 2 2 = f 1k J = - I Z n I / dnp6[-po)a(p2 + m z l x #

I n the r e s t frame

OD

x dpo6L-po ls [ -p~ + w2 + m 2 J8(po + p16(-2p0p + M 2 2 2 - m - p I . -W

113.10) 2 The func t ion f - ( k I can be obta ined by changing t h e s igns o f t he arguments o f the 8 func t ions .

I n coord inate space one has

The Four ie r t ransform o f t h i s statement I s

128

G. 't HOOFT ond M. VELTMAN

OD

f - t k - T I . 1 2=i + -

T can be considered as an n vector w i t h a l l components zero except the energy component. A8 long as the w i n t e g r a l is s u f f i - c i e n t l y convergent Li.e. n < 41 one may exchange f r e e l y the w end T in tegrat ion. Doing t h e po and T In tegrat ions one obtains o f courae the o l d r e s u l t f o r 1,. which is not very i n te res t i ng . Let us therefore leave the o I n teg ra t i on i n f r o n t o f the w i n t e g r a l and compute the po i n teg ra l . One f i nds f o r the po i n t e g r a l i n f+

where

/(u2 - m2 - m212 - 4m 2 2 M K'

The 0 funct ion expresses the f a c t t h a t p must be p o s i t i v e and furthermore tha t I C ~ must be p o s i t i v e I n order for the w i n teg ra t i on t o g ive a non-zero r e s u l t . For f- one obtalns the same. except f o r the change e I p - M - m l + 0(-p - M - m l .

Also t h e w i n teg ra t i on can be done, and I s o f course independent of X ( t h i s fo l lows as ueual by considering the i n t e g r a l f o r 1 < n < 4 and doing the necessary p a r t i a l in tegrat ions) . F o r X - 0 one f i n d s

The complete funct ion f ( k l = In obtains as given above. We must study

where

1 29

DIAGRAMMAR

4T'

Since

where u i s p o s i t i v e and f i n i t e a t th resho ld T + m +

i n t e g r a l i s not w e l l de f ined f o r n 5 1. This i s how a t w = U, p rev ious l y found i n Eq. (13.41, mani fests But t h i s i s again no problem, and r e a l l y due t o the

(13.121

m, t h i s the divergence i t s e l f here. f a c t t h a t

our d e r i v a t i o n i s co r rec t on ly f o r n > 1. We w i l l come back t o t h a t below. And the re 1s no t r o u b l e i n cons t ruc t i ng +he a n a l y t i c con t inua t ion t o smal ler values o f n. l h i s can be done by performing p a r t i a l i n t e g r a t i o n s w i t h respect t o t h e f a c t o r IT - PI - mJ i n Eq. (13.121.

For n 4, however. t he I n t e g r a l d iverges f o r l a r g e values o f 1.1. Th is can be handled as fo l lows. The expression 113.121 f o r g L l ~ l i s no th ing bu t a d ispers ion r e l a t i o n , and we may per - form a sub t rac t i on

AS before, by i n s e r t i n g d.r/dT i t may oe shown t h a t gIO1 has a po le a t n = 4. The remainder of g', however. i s p e r f e c t l y we l l - behaved f o r n < 5. Again we see t h a t t he po le terms ( i n n = 41 have the proper polynomlal behaviour: they are l i k e subt rac t ions I n a d i spe rs ion r e l a t i o n .

We must now c lea r t h e behaviour o f t he example the f u n c t i o n

I n t h e complex T plane a c u t a long the r e a l

up a f i n a l po in t , namely the quest lon o f i n t e g r a l near threshold. Consider as an

we have a po le a t T = 0 and f o r non-integer a x i s f rom T = 1 t o T = m. M u l t i p l y t h i s

130


-1 func t ion w i t h [T - l~ - i s ) around t h e po in t T = u. One ob ta ins

and i n t e g r a t e over a small circle

2 n i f l l ~ I . On t h e o t h e r hand t h e contour may be en larged; we.get

(* + c o n t r i b u t i o n of t h e 1 dT L T - lJ - 1 E f ( u 1 = 2ni 4

T o r i g i n ,

where C i s as i n t h e fo l lowing diagram:

T h e c l r c l e a t i n f i n i t y may be ignored provided a < 4. S ince now t h e in tegrand has a qu i t e s i n g u l a r behaviour a t T - 1. t h i s po in t m u s t be t r e a t e d c a r e f u l l y . The contour may be d iv ided i n t o a con t r lbu t lon of a smal l c i r c l e w i t h r a d l u s E around t h i s po in t , end t h e rest.

I n cons ider ing t h e i n t e g r a l over t h e c i r c l e , T may be s e t t o one except i n t h e f a c t o r (T - 11". Moreover, we may in t roduce t h e change o f v a r i a b l e T = T - 1. Writ ing T = cei6:

r

On t h e o t h e r hand, tne con t r ibu t ion of t h e two contour l ines from t h e c i r c l e t o some po in t b above and below t h e cut con ta ins a p a r t

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The i n t e g r a n d is t h e jump across t h e cut. T h i s c a n be i n t e g r a t e d t o g ive

T o g e t h e r w i t h t h e c o n t r i b u t i o n from t h e c i rc le

- 1 ba+l Ie-ira - e l n u ] a + 1

T h i s i s i n d e p e n d e n t of E , and t h e l i m i t E + 0 c a n b e t a k e n . Note t h a t t h e result i s -2ni i n t h e l i m i t a + -1. as s h o u l d b e f o r a c l o c k w i s e c o n t o u r .

I n t h e e x p r e s s i o n fo r f * i p ) ( a n d f - t v ) ) w e h a v e n o t b o t h e r e d a b o u t t h e p r e c i s e b e h a v i o u r a t t h e s t a r t of t h e cut. T h i s is i n p r i n c i p l e a c c o u n t e d f o r by t h e 8 f u n c t i o n . T h i s 11 f u n c t i o n gives o n l y t h e c o n t r i b u t i o n a l o n g t h e c u t 1 t h e small c i rc le has b e e n i g n o r e d . T h i s i s a l l o w e d o n l y i f n > 1 ( c o r r e s p o n d i n g t o a > -11. O t h e r w i s e o n e m u s t c a r e f u l l y s p e c i f y w h a t h a p p e n s a t t h r e s h o l d i n f + , f - , a n d i n t h e s u b s e q u e n t i n t e g r a l s over T.

2 To see i n d e t a i l how t h i s g o e s c o n s i d e r a f u n c t i o n o f T h a v i n g a c u t from T~ = t o T~ = a, b u t o t h e r w i s e a n a l y t i c l a n d g o i n g s u f f i c i e n t l y f a s t t o z e r o a t i n f i n i t y l . I n t h e T p l a n e t h e f u n c t i o n has c u t s f r o m -a t o -00 a n d +a t o +a, The d i s p e r s i o n r e l a t i o n l e a d s i n t h e T p l a n e t o t h e f o l l o w i n g c o n t o u r :

C o n s i d e r t h e r i g h t - h a n d s i d e c o n t o u r . T h e c u t s tarts a t t h e p o i n t T = a. We may wri te

2 w h e r e C& s t a n d s f o r t h e c i r c l e a t a w i t h r a d i u s 6. and P ~ T I i s t h e jump o v e r t h e c u t , 1.e.

132

I G. ‘t HOOF1 and M. VELTMAN

1

+t?’J = l i m ftr’ + 16’1 - f[r2 - 16’) . 6 ’+o i i

T h i s i s t h e p rec i se equivalent of our f’. It I s e s s e n t i a l t o first take t h e l i m i t 6’ - 0 before t h e l i m i t 6 = 0.

L e t u s now return t o the question of subtract ions. It i s now possible t o t u r n t he reasoning around. We know t h a t f o r In and i t s Fourier transforms t h e following p rope r t i e s hold:

11 I n l k l has poles f o r n = 4, 6 , ..., and tne < [ X I have no pole f o r n = 4.

111 I n [ X ) = OfxoJf~Lxl + 81-XoJf,lXl f o r < 4. 1111 I ~ ~ X I , f f i t x l are Lorentz-invariant.

+ The funct ions f n and fi are non-singular f o r n = 4: they a r e t h e cut diagrams and these are not aivergent f o r n = 4. Concer- ning point 1111, the der ivat ion of the cu t t i ng equation requires n < 4. O f couree It i s possible t o def ine things by ana ly t i c continuation, but our dispers ion-l ike r e l a t i o n s as exhlbited above w i l l hold only f o r n < 4. From (11) and l i i i J it follows immediately t h a t t h e pole i n n = 4 of In tx l can a t most be a 6 function. The precise reasoning i e as follows. Applying a Lorentz tranaformation t o [ i l l and i n s i s t i n g on Lorentz invariance we f l n d

for x 1 0 . 0

x t o .

Consider nDW+the ana ly t i c continuation of t i l l t o l a rge r n values. If f and f - have no pole f o r n = 4, t h e n t he only way t h a t In tx l can have a pole f o r n = 4 is i n t h e point xo Remember now t h a t we nave shown t h a t t h e Fourier transform of

0, 2 = 0 .

I n 1s O f t h e form

- x f i n i t e function f o r n < 5 n - 4

1.9. t h e Fourier transform of In o x l s t s f o r n < 5. Since t h e Fourier transform of the pole pa r t i s a function which is non- zero only f o r xo = x = 0, t h e only p o s s i b i l i t y i s t h a t t h i s pole pa r t i s a polynomial i n t h e external momenta [and t h e varlous masses i n t h e problem), 1.9. i n coordinate space a 6 function and de r iva t lves of 6 functions.

T h u s we know now t h a t t h e res idue of I n a t the pole 1s a polynomial. Oifferent ia t ing I n w i t h respect t o t h e external momentum k. i n t h e region n < 4 where we can use a well-defined

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representat lon, see Eqs. (13.4) and (13.51, we see t h a t t h i s d e r i v a t i v e is f i n i t e f o r n = 4. I t f o l l o w s t h a t t h e po le p a r t I s independent o f k, which i s indeed what i s found before, Eq. (13.91. Also, t h i s i s a l l we need t o know f o r renormal iza t ion purposes.

13.3 M u l t i l o o p Diagrams

We must now extend the method t o diagrams conta in ing a r b i - t r a r i l y many loops. Th is i s q u i t e s t ra igh t fo rward . F i r s t note t h a t e x t e r n a l momenta land sources) span a t most a four-d imensional space. We s p l i t n-dimensional space i n t o t h i s four-d imensional space and the r e s t

The & are t h e components o f the p i i n four-dimensional space. The in tegrand w i l l depend on the sca la r products o f t he w i t h themselves ant t h e e x t e r n a l vectors, and furthermore on the sca la r products of t he p i w i t h themselves. Again, the i n t e g r a t i o n over those angles t h a t do no t appear i n the in tegrand may be performed. Th is i s done as fo l lows. Consider t h e I n t e g r a l

The argument o f t h i s I n t e g r a l i s al ready i n t e g r a t e d over p i + l .., p k i t he re fo re t h e i n t e g r a l depends on ly i n the sca la r product o f P i w i t h PI ... P i - 1 . These vec tors .span an i-1 dimenslonal space, and we may w r i t e

(13.14)

Now the in tegrand w i l l no longer depeEd on t h e d i r e c t i o n o f F, and In t roduc ing p o l a r coord inates I n P space

Tne oi represent the lengths o f dimensional space the vec tors

00

0 1 4-4-i dwi

the vec tors

(13.151

- Pi. I n t roduce i n k-

134


q 1 =

0 b v i ou 9 1 y

* q2 =

2

0

w

0

but note

113.161

I13.17)

wi th wk running from -OD to +w. we have there fore comparing E q s . (13.141 and (13.151 w i t h (13.16) and (13.171

n-4-k 0(w,I *

In- k-41/2 dn-4Pk ’’

where we used r(1/21 = G. Finally note

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Since wl, wz, etc., ere p o s i t i v e we can w r i t e

B 1 w k l - B tde t q l . We a r r i v e thus a t t h e equat ion

(13.181 n-4-k x 1 dkql ... d q [de t q) B(det q l . k k

Next we have

Th is equat ion can be a r r i v e d e t by ted ious work. Jus t w r i t e out t he determinants w i t h the he lp of t h e E symbol w i t h k ind ices . Note t h a t f o r a = 0 t he equat ion fo l lows t r i v i a l l y beceuse o f

E E = k: il...l 1 ... I k 1 k

A lso f o r k = 1 the equat ion i s t r i v i e l .

Now the in tegrand is a func t lon of t h e s c a l a r products a i J = ( q l , qdJ. L e t us consider t h e opera t ion on such e func t i on o f t he operetor

a aq

de t - . One has

136

k 2 tdet q l det

w i t h these equations

G. 't HOOF1 and M. VELTMAN 1

one can construct the ana ly t i c cont inuat ion of the above equation t o small values o f n

X n-4-k= 1 I dql" *dqk (n-4) (17-51.. . (n-3-k) I dq l...dq [det q) k

D 1 $ dq l...dq kdet q l ln-4J...ln-3-kl k

2 e

Performing t h i s operation X times gives

x I dkql ... dkqKO[det q) [det q J n-4-k+2A [det - 7 i a i J

(13.191

Rather than worrying whether t h i s de r i va t i on i s correct o r not we simply take t h i s equation as the d e f i n i t i o n of the k-loop diagrams f o r non-integer n. I t 1s not very d i f f i c u l t t o v e r i f y t ha t f o r f i n i t e diagrams the result coincides w i t h the above equation f o r n = 4 lsee below). I n doing such work i t i s o f ten advantageous t o go back t o the special coordinate system w i t h which we started1 the expression i n terms o f the ql i s mainly useful f o r invariance considerations. For instance s h i f t s o f i n t e g r a t i o n var iables such as

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l e a v e bo th dkql and dkq2 a s well as d e t q unchanged as long as t h e i n t e g r a l s converge lwhich t h e y do f o r s u f f i c i e n t l y small n ) .

Equat ion (13.1YI i s much t o o compl l ca t ed f o r p r a c t i c a l work. and we w i l l i n s t e a d use t h e n o t a t i o n

A l l t h e above work I s t o show how t h i n g s can be d e f i n e d for n o n - i n t e g e r n.

Ihe f i n a l s t e p c o n s l s t s of showing t h a t t h e a l g e b r a i c p r o p e r t i e s n e c e s s a r y f o r dlagrammatlc a n a l y s i s hold f o r t h l s i n t h e same way a s t h e y do f o r i n t e g e r n. We have seen a l r e a d y t h a t t h i s i s so f o r sh i f t s of i n t e g r a t i o n v a r i a b l e s . Furthermore, c l e a r l y

(a lways assuming t h a t n i s chosen such t h a t t h e I n t e g r a l s conver- g e l .

N e x t there i s t h e f o l l o w i n g problem. It seems t h a t t h e d e f i - n i t i o n depends on t h e number o f l oops . Then i n c o n s i d e r i n g c u t t i n g e q u a t i o n s one h a s r e l a t l o n s i n v o l v i n g a l l k inds of p o s s i b i l i t i e s on both s i d e s of t h e cu t , and i t may seem hard t o unde r s t and what happens. I h e answer t o t h i s i s t h a t t h e d e f i n i t i o n I n v o l v e s f o r k l oops a k-dimensional q-space, bu t n o t h i n g p r e v e n t s u s from u s i n g a l a r g e r space . T h u s i n any e q u a t i o n one can use f o r k simply the l a r g e s t number of c l o s e d loops t h a t o c c u r i n any one term. And t h e r e i s no d i f f i c u l t y i n d e c r e a s i n g t h e k used if t h e number of l oops is s m a l l e r . t n a n t h i s k : Simply perform t h e Inte- g r a t i o n s o v e r t h o s e d i r e c t i o n s t h a t od n o t a p p e a r I n t h e i n t e - grand.

Let u s now c o n s i d e r t h e l i m i t n -t 4 i n t h e c a s e where the o r i g i n a l i n t e g r a l e x i s t s . T h i s means t h a t there are no d i f f i c u l - t i e s w i t h t h e u l t r a v i o l e t Dehaviour. Using Eq. (13.19) w i t h X s u f f i c i e n t l y l a r g e w e may first d o t h e q1 i n t e g r a t i o n . Using c o o r d i n a t e s as d e s c r i b e d i n t h e beg inn ing one a r r i v e s a t an i n t e g r a l of t h e t y p e as encoun te red i n t h e one-loop c a s e

138


Keeping n i n the neighbourhood o f f o u r i t is poss ib le t o perform par t181 i n t e g r a t i o n w i t h respect t o w1 on l y

1

r[T t n - 4 1 + 11 00 I dwlo!-3 [ - $]f(u;) 1 o

E 2 Next we w r i t e wT-3 = w 1 . w 1 - wl(l + E I n w 1 + O ( E 11, w i t h E = n - 4. Since w 1 . d q = 1/2dw2, we f i n d

00

1 1 do: I n o1 [- $ ] f [ w 2 I + r[? t n - 41 + 11

We see t h a t t h e i n t e g r a l reduces t o what one would have w i t h q1 - 0, p lus a f i n i t e amount p r o p o r t i o n a l t o n - 4. Since q1 was the "unphysicaln p a r t o f t he momentum p i we see t h a t we recover I n t h i s way t h e expresslon t h a t we s t a r t e d from. Moreover, if t h e o r i g i n a l expression was f i n i t e , then t h e r e s u l t f o r values of n c lose t o f o u r d i f f e r s by f i n i t e terms p r o p o r t i o n a l t o n - 4.

I f t h e i n t e g r a l does no t e x i s t s f o r n - 4 , then we are In te res ted i n cons t ruc t i ng an a n a l y t i c con t inua t ion t o l a r g e r n values. This can be done as fo l lows. Se lec t a number o f c losed loops i n t h e diagram. C a l l t h e loop-momenta associated w i t h these loops p i ... ps [S loops se lec ted l . Next i n s e r t

a ( p I 1 = - fa,

s o n j -1 l=l d ( P l ) j

which is symbolic f o r

and per form p a r t i a l i n teg ra t i ons . The r e s u l t is t he o r i g l n a l l n t e g r e l I p l u s en i n t e g r a l I' t h a t is b e t t e r convergent w i t h

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respect t o th8 I O O P S selected. So lv ing t h e Equation Wi th respect t o I we g e t

1 I=-I’ sn - X

i n analogy w i t h Eq. (13.61, obtained i n ( - A ] i s the number obta ined by count ing p i ... ps i n t h e lntegrand. That i s t he t h i s method: the re i s a d i r e c t r e l a t i o n

t h e one-loop case. Here t h e powers o f t h e momenta very n i c e t h i n g about between power count ing

and t h e l o c a t i o n of t he po les i n t h e complex n plane.

Here we f i n d a po le f o r n = X/s. If 4s. - X = 2 , 1, 0. we have quadrat ic , l i nea r , o r l oga r i t hm ic d ivergencies w i t h respect t o t h e p i ... ps in teg ra t i ons .

I t i s c l e a r t h a t we now can have poles f o r n = X/s, w i t h X and s In tegers . Diagrams w i t n many c losed loops g i v e many po les i n t h e complex n plane. lhe f i r s t po le would be a t n - 4 - 2/s, 4 - l /s , 4 for quadrat ic , etc.. divergent integrals, t h e next I s one [genera l l y however two) u n i t s l/s f u r t h e r i n the d i r e c t i o n of inc reas ing n. e tc .

13.4 The Algebra o f n-Dimensional I n t e g r a l s

I t i s now very impor tant t o know how the p rev ious l y d iscussed combinatorics surv ives all t he d e f i n i t i o n s . I n do ing vec tor a lgebra one manipulates vec tors and Kronecker 6 symbols accord ing t o t h e r u l e s

2 P P = p ,

V l J = t i ,

‘pv’va pa

a n . 61JV

The on ly p lace where the dimension comes i n is i n t h e t r a c e o f t he 6 symbol. Th is 6 symbol a l so appears n a t u r a l l y when performing i n t e g r a l s ~ f o r instance [see Appendix B l

140


{I’[ a - :)k,,kv + F b - 1 - :)6yvIm - k21 1 . J

The i n d i c e s u, v are supposedly contracted w i t h i n d i c e s p , v o f e x t e r n a l q u a n t i t i e s [such q u a n t i t i e s a re zero i f t h e va lue o f t h e index i s l a r g e r than f o u r ) or o the r i n t e r n a l quan t i t i es . I n do ing combinatorics one w i l l c e r t a i n l y meet i d e n t i t i e s o f t he form

2 6,,vPpPv - P ’

Now note

All these i n t e g r a l s a re computed accord lng t o t h e p rev ious l y g iven rec ipes. The i n d i c e s p, v simply spec i f y t w o e x t r a dimensions. t h e two-dimensional space spanned by the ob jec ts w i t h which p and v speci fy cont rac t ions .

C l e a r l y t h e two equations are cons is ten t on ly i f we use the r u l e 6 6 - n.

PV uv A l l t h i s can a l so be rephrased as fol lows. I f we take t h e

r u l e &,,,, - n, then the a lgebra of t he i n t e g r a l s I s t he same as t h a t o f t h e integrands.

The s i t u a t i o n i s somewhat more complicated i f the re are fermions and y matr ices. Now y matr ices never occur i n f i n a l answers; on ly t races occur. The only re levan t r u l e s a re

E Y P . YV1 = 26,,v 4 ( 4 = u n i t ma t r i x ) ,

The numbers 2 and 4 a re no t d i r e c t l y r e l a t e d t o t h e d imens iona l i t y o f space-time, and they p l a y no r o l e i n comblna tor ia l r e l a t i o n s . Tne 6 must of course be t rea ted as i n d i c a t e d above.

uv

141

DIAGRAMMAR

Some considerat ions are i n o rder now t o e s t a b l i s h t h a t our regu la r i zed diegrams s a t i s f y Ward i d e n t i t i e s f o r any va lue of t he parameter n. The combina tor ia l p roo f o f Ward i d e n t i t i e s i n - volves [IJ vec tor a lgebra and [ill s h i f t i n g o f i n t e g r a t i o n var iab les . We have a l ready shown i n subsect ion 13.3 t h a t s h i f t i n g o f i n t e g r a t i o n va r iab les is a c t u a l l y al lowed because o f t h e invar iance o f de t q i n Eq. (13.191. I t is a lso easy t o see now tha t , i n t h e sense de f lned above, t he vec tor a lgebra goes through unchanged f o r any n.

O i f f i c u l t i e s a r i s e as soon as i n t h e Ward i d e n t i t i e s t h s r e appear q u a n t i t i e s t h a t have t n e des i red p r o p e r t i e s on ly i n four- dimensional space, l l k e y5 o r t h e completely antisymmetric tensor tzUvpo. Then the scheme breaks down, and the re are v i o l a - t i o n s o f t he Ward I d e n t i t i e s p r o p o r t i o n a l t o n - 4. I f the re are i n f i n i t i e s (1 ,s. poles f o r n = 41 the re may be f i n i t e v i o l a t i o n s o f t he Ward i d e n t i t i e s i n the l i m i t n = 4. Th is i s what happens i n t h e case o f t he famous Bel l -Jack iw-Aaler anomalies.

13.5 Renormalization

Ever s i n c e t h e invent ion o f r e l a t i v i s t i c quantum electm- dynamics, work has been devoted t o the problem o f renormaliza- t i o n . Mainly, t h e r e i s t h e l i n e of Bogoliubov-Parasiuk-Hepp- Zimmerman and t h e l i n e of Stueckelberg-Petermann-Bogoliubov- Epstein-Glaser . Of course, both t rea tments have a l o t i n comvon; the BPHZ method however seems a t a disadvantage i n t h e sense t h a t t h e r e seem t o be unnecessary complicat ions. Unfortunately these complicat ions are such a s t o i n h i b i t g r e a t l y t h e t reatment o f gauge t h e o r i e s .

Tne SPBEG method, on t h e o the r hand, can be taken over unchanged, and a l so accomodates n i c e l y w i t h t h e dimensional r e g u l a r i z a t i o n scheme. The fundamental i ng red ien ts are u n i t a r i t y and c a u s a l i t y p r e c i s e l y i n the form of t he c u t t i n g equations. The on ly compl icat ion t h a t remains i n the case o f gauge theo r ies is t he problem of dresses/bare propagators discussed i n Sect ion 9. It seems u n l i k e l y t h a t t h i s i s a fundamental d l f f i c u l t y .

We w i l l sketch i n a rough way how renormal iza t ion proceeds i n t h e sPBEG method. For more d e t a i l s we r e f e r t o the works of these authors.

I n the fo rego ing a d e f l n i t i o n f o r diagrams a l s o f o r non- i n t e g e r n [ = number o f dimensions) has been given. Tn is d e f i n i t i o n i s such t h a t :

5 I1 c u t t i n g equations ho ld f o r all n: iil Ward I d e n t i t i e s ho ld f o r a l l n, prov ided t h e E tensor and y

do no t p lay a r o l e i n the comDinatorics o f these Ward iden-

142


t i t i e s ;

t h e complex n plane. iii) divergences manifest themselves as po les for n = 4 in

Ihe problem I s now t o add counterterms t o t h e Lagranglan such t h a t t h e po les cancel out. The e s s e n t i a l p o i n t I s t h a t t h i s ha5 t o be done order by o rder I n pe r tu rba t i on theory, and s ince t h i s i s p r e c i s e l y the o r i g i n o f a l o t of t r o u b l e i n connection w i t h gauge theo r ies we w i l l focus a t t e n t i o n on t h a t po in t . ’

cons ider one-loop self-energy diagrams I n quantum e l e c t r o - magnetics. They are:

The Feynman r u l e s are:

1 -1yp + m

1 ~ ~ 1 ~ 1 p2 + m2 - IE

- - P v - 1 ti lJV .-----. EW41 k2 - f e ’

They f o l l o w f rom the Lagrangian

The f i r s t diagram g ives

2 Iy = - e 1 dnP tr { y ” [ - ~ y [ p + k) + m1yv(-iyp + m ) )

(p2 + m2 - IEI [ [ p + k12 + m2 - IE] n

u v l l v We must work out the t r a c e us ing on ly {y y = 461Jv. The technique I s t o reduce a t r a c e o f X matr ices t o t races o f X - 2 metr lces. For f o u r matr ices

= 26,,,,4 and t r ( y y I =

a v B v - ... = - t r [ y y y y 1 + tr (y ~ a v B y y y J - -tr [y a u v B y y y I +i B61JQ6vB

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o r

u a v 6 tr (y y y y J = 46tra6v6 - 46 6 + 46 6 trv a0 1.4 av

This g i v e s

Using Feynman parameters and the equations of Appendix B t h i s integral can be computed giving

2 n/2 r ( 9 ) 2 x t l - X J (ktrkv - k 2 6 u v l

2-n/2 [rn2 + k2x11 - XI] I dx

4 i e fl In - - r(21

The q u a d r a t i c divergence ( p o l e a t n - 21 nas cance l led ou t through (1 - n / 2 1 r ( l - n / 2 I = r [ 2 - n /2 l .

The second diagram g i v e s

From t h e a n t i c o m u t a t i o n r u l e s , and t h e r u l e 6 = n tr’

F u r t h e r

y”ypy” = -y’y’yp + ~ y p = [ Z - n l y p

( i f n = 4 t h i s reduces t o t h e well-known Chisholm r u l e . Such r u l e s a r e n o t v a l i d i n n-dimenslonal space]. l h e i n t e g r a l may now be worked o u t

Both I: and 1: nave a p o l e a t n = 4. The res idues a r e e a s i l y

144


computed

2 tkpkv - k &,,,,I a PP~I:J - in2e2 - n - 4

8 1

PP[I:l - 21n2e2 n-4 [4md- l y k l . If we introduce i n the Lagrangian (remember t h a t ver t ices and terms i n the Lagrangian d i f f e r by a f a c t o r 12n14i terms

the counter-

z then the two-point functions up t o order e poles f o r n = 4, and one can take the l i m i t n - 4. To have a l l one-loop diagrams f i n i t e a c o u n t e r t e n t o cancel vertex divergen- c ies must be introduced.

w i l l be f r e e of

Performing a similar calculation (simplified very much from the beginning i f only the pole part is needed, see Appendix B ) for the vertex diagram leads t o the counterterm

3 2 i e $YvJIAv

8n (n - 41

The remarkable f a c t i s t h a t the counterterms are gauge inva r ian t by themselves. Tnis would not have been so if the coef f ic ients of h a $ and -iAvWV$ had been d i f f e ren t . I t w i l l become c lea r tha t t h i s is special t o t h i s gaugei i n the case o f a gauge where the ghost loops are non-zero the r e s u l t i s d i f f e ren t , and the counterterms are i n genera not gauge inva r ian t by themselves.

There are now two separate questions t o be discussed. I n - c luding the counterterms, a l l r e s u l t s up t o a ce r ta in order are f i n i t e . This order 1s such t h a t one has one closed loop but no "countervertex", o r a t r e e diagram inc lud ing a t most one countervertex. An example i s electron-electron scat ter ing t o order e4 I n amplitude. Some examples of con t r i bu t i ng diagrams are:

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DIAGRAMMAR

The c ros ses denote counter terms. T h e first and t h i r d diagrams (and t h e second and fou r th1 t o g e t k r are f i n i t e .

The ques t ion is now w h e t h e r it i s p o s s i b l e t o make t h e theory f i n i t e up t o t h e next o rde r i n e2 by in t roduc ing f u r t h e r counterterms of t h e type shown above. N e x t one may ask how t o understand i n more d e t a i l t h e gauge s t r u c t u r e of t h e renormalized theory , 1.e. t h e Lagrangian inc lud ing t h e counter terms. Moth ques t ions w i l l be i n v e s t i g a t e d now.

1 3 . 6 Overlapping Divergences

The problem is t h e fol lowing. Can t h e renormal iza t ion procedure be carried through o rde r by order . F i r s t we m u s t s t a t e more p r e c i s e l y what we mean, because i n t h e Lagranglan we have counter terms o f orde r e2 and e3. To t h i s purpose we in t roduce a parameter ns and a l l counterterrns found from t h e a n a l y s i s of one-loop diagrams g e t t h i s parameter a s c o e f f i c i e n t . T h u s we have now

If now i n a diagram t h e number of c losed loops is i t n a n we may a s s o c i a t e w i t h each diagram a f a c t o r Li. The S matr ix 1s f i n i t e up t o f i rs t o rde r i n [ L 4 n l , i n tne l i m i t n = 1, L = 1. T n u s a t most one closed loop no 0 ver t ex , or no closed loop one n ver t ex .

I t may perhaps be noted t h a t if t h e number of ingoing and outgoing l ines i s given, t h e n spec i fy ing L + q i s equ iva len t t o spec l fy ing a c e r t a i n o rde r i n e.

Let u s now cons ider diagrams of orde r [L + q l Z , f o r i n s t a n c e photon se l f -energy diagrams. There are diagrams of o rde r L2, i . e . two closed loops , of o rde r Ln and of o rde r 11’:

146


b Lz:- +

a

t12 : - I t i s now necessary t o i n t roduce a c l a s s i f i c a t i o n . We d i v i d e

a l l dlagrams i n t o two se ts :

2 2 i l t he se t o f a l l diagrams of order L , Ln o r n t h a t can be disconnected by removing one propagaotr i here [a, e, f, and i l r

l i l t h e r e s t , c a l l e d the over lapplng diagrams.

s e t (11 are t h e non-overlapping dlagrams. No new divergences occur, as can be v e r i f i e d r e a d i l y . I n f a c t , on bo th Bides o f t he propagator i n quest ion one f inds p r e c i s e l y what has been made prev ious ly :

a + e + f + l = (u+ - l X l - 0 - + - I The over lapping diagrams may conta in new divergencies, and we must t r y t o prove t h a t these new d ivergencies behave as l o c a l counterterms. I n p r i n c i p l e , t he proo f i s very simple and based on t h e use o f c u t t i n g equations. The f i r s t observat ion i s t h a t a l l cu t diagrams (always t a k i n g together diagrams o f a g iven order i n [L + n) on each s ide o f t h e cu t1 are f i n i t e , t h a t i s t he re i s no po le f o r n = 4. The p roo f o f t h i s i s easy1 these cu t diagrams are o f t h e s t r u c t u r e o f a product o f diagrams o f lower order I n (L + nl, which are supposedly f i n i t e , and an i n t e - g r a t i o n over in te rmed ia te s ta tes . Since f o r g iven energy t h e a v a i l a b l e phase space i s f i n i t e the r e s u l t f o l l ows . Th is assumes t h a t t h e diagrams have no- in tegrab le s i n g u l s r i t l e s i n phase space; t he l a t t e r would correspond t o i n f i n i t e t r a n s i t i o n p r o b a b i l l t i e s i n lower order. which we take no t t o e x i s t .

Now l e t us number the v e r t i c e s t o wnich the ex te rna l l i n e s are connected. Here the re are on ly two such ve r t i ces , and we c a l l them 1 and 2. According t o our c u t t i n g r e l a t i o n s we have, i n t e - g r a t i n g over a l l x except over x and x 1 2

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w i t h x = x2 - XI. Here d t h e unshadowed region, and A- a l l cu t diagrams w i t h x2 i n the unshadowed region. Th is i s of p r e c i s e l y the same s t r u c t u r e as discussed before1 we know t h a t i f A* are f r e e o f po les f o r n = 4, then f ( x 1 can on ly have a p o l e p a r t t h a t i s a 6 f u n c t i o n or d e r i - v a t i v e o f a 6 func t ion . I n o the r words, t he p o l e p a r t I n n = 4 must be a polynomial I n t h e ex te rna l momentum. F o r a r b i t r a r y diagrams t h i s must be t r u e f o r any combination o f "external ' v e r t i c e s ( 0 v e r t i c e s t h a t have a t l e a s t one ex te rna l l i n e ] , and thus f o r any o f t he e x t e r n a l momenta. Due t o the f a c t t h a t power count ing an0 t h e ex is tence o f po les f o r n = 4 a re i n a one-one r e l a t i o n s h i p i t can e a s i l y be es tab l i shed t h a t t n e polynomials are a t most o f a c e r t a i n degree; here, photon se l f energies, a t most of degree 2. To do a l l t h i s p roper l y i t I s necessary t o d i s t i n g u i s h between o v e r - a l l divergences and subdivergences [ t h e former disappear if any of t he propagators i s opened), and show t h a t t he re are no subdivergences i f a l l t he lower order counterterms a re inc luded. Next i t must be shown t h a t a f t e r a c e r t a i n number o f d i f f e r e n t i a t i o n s w i t h respect t o the e x t e r n a l momenta the o v e r - a l l divergence disappears. A l l t h i s i s t r i v i a l : opening up a propagator i n a self-energy diagram I s equ iva len t t o consl - de r ing diagrams w i t h four e x t e r n a l l l n e s o f lower order i n [L + 9). thus a l ready f i n i t e . Over -a l l divergences correspond t o o v e r - a l l power count ing ( p a r t i a l d i f f e r e n t i a t i o n w i t h respect t o a l l loop momentel. and d i f f e r e n t l a t i o n w i t h respect t o any e x t e r n a l momentum lowers t h i s power by one.

+ conta ins a l l cu t d i a g r a m w i t h x l i n

13.7 The Order o f the Poles

Renormalizetion may now be performed order by o rder i n 1L + + rll. One s t a r t s w i t h dlagrems w i t h L1 (one c losed loop l end f inds the necessary counterterms. They ge t a f a c t o r r) i n t he Lagrangian. Next one considers diagrams o f o rder L2 [ two c losed loops) o r [Lr)J1 (one closed loop, one counter ve r tex ] o r nz [ t r e e dlagrems, two counter Vert iCeSl. Tne t o t a l can be mede f i n i t e by adding f u r t h e r counterterms. Such terms ge t a f a c t o r n2 i n t he Lagrangian. I n t h i s way one can go on and ob ta in a counter Lagrangian i n the form of a power se r ies i n . F o r r) = 1 the complete theory i s f i n i t e t t he re are no po les f o r n = 41.

We have a l ready seen t h a t t h e terms of o rder rj conta in simple po les only. We now want t o i n d i c a t e t h a t t he terms of order n conte in po les of t h e fo rm I n - 41-2, and furthermore, t h e coe f f i c i en t o f such a p o l e i s determined by t h e order r) terms.

F i r s t of a l l , i t i s t r i v i a l t o see t h a t t he re are no qua- d r a t i c pole3 a t o rder q2 i f t he re are no po les a t o rder rj. If t he re are no poles a t order n then the re are no subdivergences i n two-loop diagrams. But o v e r - a l l divergences of diagrams o f

2

148


2 order L are simple poles ( t h i s fo l lows from p a r t i e l in tegrat ion) , 1.e. there are no quadratic poles.

To see t n a t there are i n general quadratic poles consider the 5um o f phqton self-energy diagrams [ c l and ( d ) . The sub- i n t e g r a l i n [ c ) may be done and gives rise t o an expression of the form

where p i s the momentum going I n t o the sub-loop. This equation i s symbolic i n 80 fsr as t h 8 t we have ind i cs ted only t h e momentum dependence w i t h respect t o power counting. The above expression can then be obtained from dimensional arguments. The sum of d ia- grams [ c l and Id) i s o f the form

S i m p l i f i c a t i o n can go too farJ we w r i t e p - (p2 + m2)1/2. Next

and the i n t e g r a l becomes

Since

r ( 4 - n l , r [ i - $ ] = 2-n 2 r[2 - $1, r i 3 - n l = - 3 - n

we f i n d f o r t he Qouble po le

---m ----m _- (4 1 n)' 2m2 1 1 2 2 2 2 2 2

3 - n 4 - n 4 - n 2 - n 4 - n 4 - n

I f we w r i t e

2 2 (m2~n-3 = m2 + m (n-41 I n m ,

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DIAGRAMMAR

2 n/2-1 = m2 + m2 n - 4 2 2 (m l ,

2 we d iscover t h a t t he I n m t e r m has no po le

8 2 - n 2 (4 - n l

The res idue o f t he po le f o r n - 4 i s zero. This i s as should be because a term I n m2 i s non-local, and no t admiss ib le as counterterm. Remember t h a t m2 stands f o r I n v a r i a n t s made up from ex te rna l momenta, masses, tec.

The above argument i s very genera l and i n fact based only on loop and power count ing. I t can be extended t o a r b i t r a r y o rde r , and a p r e c i s e r e l a t i o n between the c o e f f i c i e n t s of t h e var ious h igher o rde r poles can be found. ( G . ' t Hooft , Nucl. Phys. 861 (1973) 455, see Chapter 3 of t h i s book.)

13.8 Order-by-Order Renormalization

Consider the f o l l o w i n g two-loop diagram:

---u-u Suppose the se l f -energy bubble is a f u n c t i o n o f n and the momentum k o f t h e form:

-0- f [ k l = - 2 1 2 2 2 fl[k 1 + f 2 [ k I + (n-41f3[k l . 2 We have omi t ted a fac to r k 6 - kpkv. Th8 two-loop diagram gives

VV

I f we simply throw away the po le p a r t s the r e s u l t is f o r n = 4

f; + 2 f f 1 3 '

150


However, t h i s r e s u l t i s wrong because i t v i o l a t e s u n i t a r i t y . Suppose now we do order-by-order subt rac t ion . Then one has a f t e r t h e treatment o f one c losed loop:

2 The phys i ca l r e s u l t i s t he l i m i t n = 4 abd i s equal t o f 2 [ k I . C u t t i n g two-loop diagrams should the re fo re no t i n v o l v e f3. Next i nc lude t h e proper counterterms and consider:

- o - O - + + + e +-

L = f2 + O [ n - 41 . Indeed one ob ta ins t h e c o r r e c t r e s u l t cons is ten t w i t h u n i t a r i t y . Th is demonstrates t h a t order-by-order reno rma l i za t i on i s not equ iva len t t o throwing away po les and t h e i r res idues o f t he u n r e n o n e l l z e d S-matrix. I n a way t h a t i s a p i t y , because the l a s t method I s so much easier , But t he re I s no escape, one must do th ings s tep by step, t h a t i s order by order i n the parameter 11.

13.9 Renormalization and Slavnov-Taylor I d e n t i t i e s

I t has been shown t h a t t he one-loop counterterms are gauge i n v a r i a n t by themwelves, I n the case o f Lorentz gauge quantum electrodynamics. The quest ion i s whether we can understand t h i s , and t o what ex ten t t h i s is genera l phenomenon.

As a f i r s t step we note t h a t combine tor ia l p roo fs can be read backwards, a t l e a s t if they are o f t he l o c a l v a r i e t y . I n o ther words, Ward i d e n t i t i e s cen be t rans la ted backwards i n t o a symmetry If the Lagrangian. Therefore, i f the Ward i d e n t i t i e s of t he Legranglen I n c l u d i n g counterterms are I d e n t i c a l t o those w i thout counterterms, then bo th Lagranglans s a t i s f y the same gauge invar iance. Let us consider t h e S-T i d e n t i t i e s o f t he unrenormalized theory i n c l u d i n g e lec t ron sources:

We have drawn one photon and olie e lec t ron source e x p l i c i t l y , leav lng o the r sources understood.

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DIAGRAMMAR

Since these i d e n t i t i e s a re t r u e f o r any va lue o f t h e dimen- s i o n a l parameter n, they are i n p a r t i c u l a r t r u e f o r the po le pa r t s .

Consider now one-loop diagrams. The po le p a r t s i n t h e above i d e n t i t y are:

11 t h e po le p a r t s found i n the Green's func t i ons themselvest ii) t h e po le p a r t a r i s i n g from a c losed loop, i n v o l v i n g t h e

e lect ron-ghost ver tex shown i n the l a s t diagram.

Now t h i s l a s t ver tex i s defined by t h e behaviour o f t h e Lagrangian under gauge transformations, b u t - i t i s no t present I n t h e Lagrangian i t s e l f . I n 1: one on ly f i n d s J$J if we i n t roduce counterterms i n t h e Lagrangian t h a t make the Green's func t i ons f i n i t e [ i n c l u d i n g Green's func t ions w i t h i ngo ing and outgoing ghost l i n e s ) then the above S-T i d e n t i t y can remain t r u e on ly if we throw away the p o l e p a r t of t h e type [ill. That i s , f o r t h e renormalized Lagrangian, t he S-T i d e n t i t i e s take the form:

Here the double cross stands f o r minus the diagram such as:

C l e a r l y we can understand t h i s i d e n t i t y as

p o l e p a r t o f any

an S-T i d e n t i t y i f we rede f ine the behaviour of the term SJ, under gauge t ransformat ions. I f we say t h a t under a renormalized gauge t rans format ion J) t ransforms as

Z n - 4 $ ' J , + ieh$ + i- Jw

where 2 is minus the res idue of t he po le p a r t of t h e type [ii), then t h e above i d e n t i t y i s again p r e c i s e l y an S-T i d e n t i t y . We conclude t h a t t he Lagrangian i n c l u d i n g counterterms i s i n v a r i a n t f o r renormalized gauge transformations.

To make the statement more p rec i se we must add on a f a c t o r Q t o t h e po le term i n t h e renormalized gauge t ransformat ion. Since we allow is on ly t r u e up t o f i r s t order i n n. I t i s un fo r tuna te l y somewhat

as a f i r s t s tep on ly one c losed loop t he statement

152


complicated t o extend t h i s work t o a r b i t r a r y o rder i n n, b u t we w i l l do i t i n t h e next sect ion.

I n quantum electrodynamics th ings are never r e a l l y complicated, because i n t h e usua l gauges [Lorentz, Landau, etc.1 there i s no ghost ver tex, ergo the re are no po le p a r t s o f t he type ( i l l . Then t h e renormalized Lagranglan i s i n v a r i a n t w i t h respect t o gauge t ransformat ions t h a t transform an e l e c t r o n source coup l ing as before, 1.0.

JI + J I + ieAJI

w i t h the same e as i n t h e unrenormalized Lagranglan. The r e s u l t i s

i s i n v a r i a n t under t h e t rans format ion [ A ) ,

I s i n v a r i a n t under t h e t rans format ion [ A ) , 'unren

unren + 'counter 1:

t he re fo re

'counter alone i s I nva r ian t .

I n t h e genera l case t h i s i s n o t t rue .

13.10 Higher-Order Counterterrns

Doing th ings order by order we suspect [end have shown t h i s t o be t r u e up t o f i r s t o rder i n 01 t h a t Lren is of t h e form

+ '

2

' ren a 'unren counter '

- nz1 + n x2 + ... , counter 1:

, etc., con ta in f a c t o r s l / [ n - 41. For q = 1 t h e n i t e . The renormallzed Lagranglen '(,I i s i n v a r i a n t

under gauge t rans format ions of t he form

A + A + (to + t Q + t2q2 + ... ) A , P I J 1

where the t and t' con ta in f a c t o r s l / [ n - 41.

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DIAGRAMMAR

Let us suppose t h i s l a t t e r s t a t e m e n t t o be true up t o o r d e r k i n n. C o n s i d e r now:

1 1 d i ag rams c o n t a i n i n g o n l y vertices o f t h e un renorma l i zed Legrangian w i t h k + 1 c l o s e d loops, such d i ag rams e x h i b i t p o l e s up t o d e g r e e k + 1 and s a t i s f y t h e un renorma l i zed S-T i d e n t i t i e s ,

o f c l o s e d loop:

ii) d iag rams c o n t a i n i n g c o u n t e r t e r m s ( c o r r e s p o n d i n g t o v e r t i c e s . . . L k , to ... t k , t b . . . ti) b u t w i t h a t least one

i i i l d i ag rams o f o r d e r q k + l [ c o n t a i n i n g t h u s no c l o s e d l o o p l .

Example f o r two c l o s e d l o o p photon s e l f - e n e r g y diagrams:

S e t (1) = d iag rams [ a ) , ( b l , [ c ) . Set [ i l l - d iag rams [ d l t o ( h l . S e t [ i i i l = diagram ( 1 1 p l u s a new d iag ram o f t h e form t h a t is needed t o remove t h e d i v e r g e n c i e s of t h e o v e r l a p p i n g d i ag rams . Now (1) + [ i l l + I i i i ) g i v e a f i n i t e t h e o r y as n + 4 and q + 1. Tha t i s s imply how t h e y were d e f i n e d . Next ( 1 1 + [ i i l s a t i s f y Ward i d e n t i t i e s [ t h e t h e o r y was assumed t o be i n v a - r i a n t up t o o r d e r qk). T h e r e f o r e t h e r e s i d u e s o f t h e p o l e of set [ i l l 1 s a t i s f y t h e S-T i d e n t i t i e s . Because t h e d i ag rams [ill) are t ree d iag rams t h e dimension n a p p e a r s nowhere e x c e p t i n t h e p o l e f a c t o r s l / l n - 41. T h e r e f o r e t h e d i ag rams [ i i i l s a t i s f y t h e S-T i d e n t i t i e s , which means i n v a r i a n c e o f t h e r enorma l i zed Lagrangian under t h e r enorma l i zed gauge t r a n s f o r m a t i o n c o n s i d e r i n g terms of o r d e r n k - I on ly . T h i s comple t e s t h e proof by i n d u c t i o n .

Acknowledgements

The a u t h o r s are d e e p l y i n d e b t e d t o O r . R . B a r b i e r i , who helped i n w r i t i n g a f i rs t approx ima t ion t o t h i s r e p o r t . One of t h e a u t h o r s (M.V.1 w i s h e s t o e x p r e s s h i s g r a t i t u d e f o r t h e hosp i - t a l i t y and p l e a s a n t a tmosphere encoun te red a t t h e S c u o l a Normale a t P i s a ; t h e l e c t u r e s g i v e n there have been a f i r s t tes t of t h e p r e s e n t mater ia l .

F i n a l l y , w e would l i k e t o t h a n k t h e CERN Document Reproduc- t i o n S e r v i c e f o r t h e i r e x c e l l e n t work on t h i s r e p o r t ( i n p a r t i c u - l a r . Mme Pau le t t e E s t i e r f o r t h e speedy and accurate t y p i n g o f t h e t e x t ) , and h e E . Ponssen o f t h e CERN MSC Drawing Office f o r h e r e f f i c i e n t c o - o p e r a t i o n .


APPENDIX A

FEYNMAN RULES

The most f requen t l y encountered Feynman r u l e s w i l l be summa- r i z e d here. A lso combine tor ia l f a c t o r s w i l l be discussed. The ex te rna l sources t o be employed f o r S-matrix d e f i n i t i o n must be normalized such t h a t they emi t o r absorb one p a r t i c l e . The norma- l i z a t i o n formulae f o l l o w f rom the propagators1 the " ingo ing and outgoing l i n e wave-functions" i n d i c a t e d below are t h e product o f those sources, t h e associated propagator and t h e associated mass- s h e l l f a c t o r k2 + m2.

Spin-0 p a r t i c l e s

Propagator:

1 1 - IZa14i k2 + m2 - is

I n shadowed reg ion :

1 1 - - 12a14i k2+ m2+ I s '

Cut propagator:

Wave-function:

1 .

- [A. 11

IA.21

IA.31

(A.41

Spin-1/2 p a r t i c l e s

Propagator: - I A . 5 1 1 -iyk+m

k I Z d 4 i k2 + m2 - ic c


1 -I_

1 2 ~ 1 ~ 1

- i y k + m 2 k2+ m + IE '

IA.61

155

DIAGRAMMAR

Cut propagator:

1 Ingoing particle wave-function : u a ( k l q I a = 1, 2

1. Ingoing antiparticle wave-function: - i i a ( k l q 8 a = 3, 4

J Outgoing particle wave-function:

Outgoing antiparticle wave-function:

i i a ( k l q : a = 1, 2

u a ( k l q : a = 3, 4 ( A . 9 1

Note the minus sign f o r the incoming antiparticle. The momentum k is directed inwards for incoming particles and outwards for outgoing particles. AS usual i = u*y4.

8re solutions of the Dirac equation [note that = +

kO

(iy’k + tnlua(k1 0 , a - 1, 2 , ’ f-iy’k + mlue(kl - 0 , B = 3, 4 .

IJ 5 1 2 3 4

= y y y y In the 4 x 4 representation, with y

-1

0

( A . 1 0 1

( A . 1 1 1

1 80

’ 1

these solutions are given in the fallowing table:

156


L

m - X

2k0

f u1 [ k l

1

0

kg - m + ko

kl + i k 2

m + ko

The arrows denote spin up/down assignments i n t h e k r e s t frame. Normalization:

Spin eummations:

[A . 121

I n connection w i t h p a r i t y P, charge conjugation C, and time- r e v e r s a l T, t h e fol lowing matr ices and transformation proper t ies a re o f relevance.

The mat r ice y4 is t h e transformation mat r ix connected w i t h space r e f l e c t i o n

157

DIAGRAMMAR

0 3 1 , 2 , 4 a + uat-Tt, kol = -y u [k, kol , p a r t i c l e

a + 4 a + u ( - k , ko) = -y u (k, k 1 , a n t i p a r t i c l e a = 3. 4 .

p = 1 . 2 , 3 Y Y Y

y’ l J ’ 4 . (A.141

T h e m a t r i x C t r a n s f o r m s a n i n c o m i n g p a r t i c l e i n t o an i n c o r n i n g a n t i p a r t i c l e , etc.

a - 4 . B - 1 u a [ k l - -C;’[kl

a = 3 , 8 - 2

a - 1 , 8 - 4

a - 2 , B - 3 ;‘[kI = uB[k1C-’

-a [ - - t r a n s p o s e ) -1 a c y c 5 - y

tA.151

-1 5 -5 c Y C ’ Y #

2 4 - -1 where C = Y Y , C = C = -C.

I n c o n n e c t i o n w i t h t i m e - r e v e r s a l we h a v e t h e m a t r i x D

-4-8 + = Dy u ( k , kol a = l , B = 2

T h i s c h a n g e s s p i n , d i r e c t i o n of three-momentum a n d f u r t h e r m o r e e x c h a n g e s i n - a n d o u t - s t a t e s ;


' 0 0 0 1'

0 0 - 1 0

0 1 0 0 ' D = C =

.-1 0 0 0 ,

' 0 1 0 0'

- 1 0 0 0 (A.171 0 0 0 1 '

, 0 0 -1 0,

6 + k kv/mz 1 UV - [Zn14i k2+ m2 - IE


2 6 + k k / m 1 lJV l l v --

' + m + i c 2 (2nl I k r-- Cut propagator:

P a r t i c l e wave-functions:

* e (k1 w i t h e (k1e (kl = 1 and k e = 0 . ll u l l l J l J

There are on ly th ree wave-functions, because the propagator mat r ix has one eigenvalue zero. I n t h e k r e s t system the var ious assignments are

Ingoing Out going

Spin-z component zero: e - (0, 0, 1, 01 e = [ O D 0, 1. 01 . lJ lJ

For outgoing p a r t i c l e s we must take these expressions t o have

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DIAGRAMMAR

co r rec t phases. Indeed, the res idue o f t he two-point spin-up/ spin-up ampli tude i s equal t o one.

Combinator ia l f a c t o r s

These are best expla ined by cons ider ing a few examples. L e t t h e i n t e r a c t i o n Lagrangian f o r a sca la r f i e l d be

The v e r t i c e s are :

The lowest o rder se l f -energy diagram is:

Draw two p o i n t s x1 and x ver tex :

and draw i n each of these p o i n t s the a 2

Now count i n how many ways t h e l i n e s can be connected w i t h the same t o p o l o g i c a l r e s u l t . Ex te rna l l i n e 1 can be at tached i n s i x , a f t e r t h a t l i n e 2 i n th ree ways. A f te r t h a t t he re are two ways t o connect t h e remaining l i n e s such t h a t t he des i red diagram r e s u l t s . Thus the re are a l toge the r 6 x 3 x 2 combinations. Now d i v i d e by the permuta t iona l f a c t o r s o f t he v e r t i c e s (here 3! f o r each a ve r tex ) . F i n a l l y d i v i d e by the number o f permutations o f t h e p o i n t s x t h a t have i d e n t i c a l ve r t i ces . Here 2: The t o t a l r e s u l t i s

6 x 3 ~ 2 - 1 - - 3: 3: 2: 2

As another example consider the diagram:

160


There are th ree x po in ts :

w w w XI x 2 x3

L i n e 1: s i x ways. L ine 2: four ways. Then we have f o r instance:

XI x3

There are 6 x 3 x 2 ways t o connect t h e r e s t such as t o g e t t he des i red topology. We must d i v i d e by ve r tex f a c t o r s [3: 3: 4:) and by 2: [permutat ion o f t h e i d e n t i c a l ve r tex p o i n t s XI and x2). The r e s u l t i s

6 x 4 ~ 6 ~ 3 ~ 2 I - 1 3: 31 4: 2 : 2 .

F i n a l example: two i d e n t i c a l sources connected by a sce' lar l i n e :

1 J-J Factor : 2: . For two non-dent ica l sources t h e f a c t o r i s 1:

J1-J2 Factor : 1 . Topology o f quantum electrodynamics

We now show t h a t t h e v e r t i c e s o f any diagram of quantum electrodynamics can be numbered i n a unique way. L e t t h e ex ter - n a l momenta be k ... knm

Step 1

S t e r t a t t h e e l e c t r o n l i n e w i t h the lowest momentum index. Fo l low t h e arrow o f t h e l i n e . Number the v e r t i c e s consecut ive ly .

Step 2

Go back t o the f i r s t ver tex [o r lowest numbered ver tex of which t h e photon l i n e was n o t exp lo i ted) . Fo l low t h e photon l i n e . We a r r i v e then e i t h e r a t a ver tex t h a t i s al ready numbered, a t an e x t e r n a l photon l i n e , or a t another e l e c t r o n l i n e . I n t h e f i r s t

161

DIAGRAMMAR

two casesr take the next ver tex a long the e l e c t r o n l i n e o f s tep 1 and r e s t a r t a t s tep 1. When a r r i v i n g a t a new ver tex, number again consecut ive ly f o l l o w i n g the e l e c t r o n l i n e along t h e arrow. When h i t t i n g the end, o r an a l ready numbered ve r tex r go back against t he arrow and number a l l t he v e r t i c e s on the e l e c t r o n l i n e before the ver tex t h a t was the entrance p o i n t o f t h a t l i n e . A f t e r t h a t r e s t a r t 2.

Example :

2 3 I 1

11 lo 12 -

162


APPENDIX B

SOME USEFUL FORMULAE

w i t h 0 s 81 s n s except 0 <_ 0 1 <_ 2n. If f ( x 1 depends only on

one may perform t h e i n t e g r a t i o n over ang le s us ing

(8.21

leading t o

Keeping t h e p r e s c r i p t i o n s and d e f i n i t i o n s o f Sec t ion 13 i n mind, t h e fo l lowing equat ions hold f o r a r b i t r a r y n

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2 a [p2 + 2kp + m 1 I dnP

(m2 - k21] ,

X 1

rI a1

n/2

2 a-n/2 - P$VPA i n

[p2 + 2kp + m 2 I a (m2 - k 1 I dnP

n/2

2 2 a-n/2 ill - I - k p l x

2 P P

[p2 + 2kp + m21a I m - k 1 r ( a 1 I dnP

The above equations conta in i n d i c e s p , v, A . These ind i ces are understood t o be contracted w i t h a r b i t r a r y n-vectors ql, 92, etc. I n computing the i n t e g r a l s one f i r s t i n t e g r a t e s over the p a r t o f

IB .11 t o (8.4). Af te r t h a t t h e expressions are meaningful a l s o f o r non- in teger n. Note t h a t f o rma l l y Eqs. IB.61 t o IB.101 may be obtained from Eq. (8.51 by d i f f e r e n t i a t i o n w i t h respect t o k, o r by us ing p2 = [p2 + 2pk + m21 - 2pk - m2,

n-space or thogonal t o the Vectors k, 91, 92, etC., us ing Eqs.

To show t h a t i n t e g r a l s over polynomials g i v e zero w i t h i n the dimensional r e g u l a r i z a t i o n scheme i s very simple. Consider, f o r example,

where a i s some i n t e g e r g rea te r than o r equal t o zero. According t o Eq. [13.191 i n t he case of on ly one loop, we have

164


By partial integrations (see Subsection 13.21 we get now

which gives zero for X > a.

A nice example, suggested by 8. Lautrup is the following. Consider the following integral

k I = I d4k* ,

IJ (k + m 1

which gives zero, because of symmetric integration, if one regu- larizes, for example, as follows

1 2 21

I -* d4kky[ 1

1! [k2 + m212 (k2 + A 1

It ia also zero in the dimensional cut-off scheme according to Eq. (B.61.

Let us now shift the Integration variable, forgetting about regulators

k + P Iy = 1 d4k 2 2 ' [[k + p12 + m I

Expanding the denominators, we get

2 which by symmetric integrat ion (k + 0, kVkv +. +Vvk ) gives V

2 d4k + O(p 1 f 0 . 2

IJ [k2+ m2I3

Using dimensional regularization, which means k + 0 but k k +

-t [6 /n1k2, we get from Eq. tB.121 u l J v lJV

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DIAGRAMMAR

2 + o t p 1 .

From Eqs. (B.51 and (B.71

n/2 2 n/2-3 4 3 - 9 = ir (m I I dnk I

1

(k2 + m 2 I 3 r(31

n/2 2 n/2-2 2 z r(31 ' = i n (m 1 k2

2 3 dnk [k2 m

Then

1

I n t h e l i m i t n + 4, remembering t h a t

1 n z-t-n (-11 r [ z i = - - n: z + n '

t he c o e f f i c i e n t o f t he p,, term tu rns out t o be exac t l y zero. O f course, f r o m Eqs. lE.51 t o (8.7). I i n Eq. [8.111 g ives zero t o any order i n p. lJ

I n computing po le p a r t s i t Is very advantageous t o develop denorninetors. Take Eq. [B .51 . Fo r a = 2 we f i n d the po le p e r t

2 PP[(E.51, a = 2 1 = n-4 - 2 i n

zo a (8 .13)

This 20 is a bas i c fac to r . Every l o g e r i t h m i c a l l y d ivergent i n t e g r a l has t h i s fac to r , and f u r t h e r vectors, 6 func t ions . etc.

'aPB - 1 PP I dnP 2 3 - zo 4 6ciB

[p2 + m I tB.14)

166


The f o u r on the r igh t -hand s ide f o l l o w s f rom symmetry considera- t i o n s i t h e c o e f f i c i e n t fo l lows because m u l t i p l i c a t i o n w i t h 6ap g ives t h e prev ious i n t e g r a l . We leave i t t o the reader t o f i n d the genera l equation.

Fo r o ther than l o g a r i t h m i c a l l y d ivergent i n t e g r a l s t h e deno- mina tor must be developed. F o r ins tance

'a 2 2 -kaZo pp dnP + 2pk + m )

(P (8.16)

where we used Eq. (B.141 together w i t h

1 2 2 = 11' 4 - 4 $ + .'.-.I} . (P + 2pk + m I P

L inea r l y , quadra t i ca l l y , etc.. d ivergent i n t e g r a l s t h a t have no dependence on masses o r e x t e r n a l momenta can be p u t equal t o zero.

The r e s u l t Eq. (8.161 coinc ides w i t h w h a t can be deduced f r o m Eq. (8.6).

167

DIAGRAMMAR

APPENDIX C

DEFINITION OF THE FIELOS FOR DRESSED PARTICLES

A l s o the ma t r i x elements of f i e l d s and products o f f i e l d s (such as encountered i n cur ren ts ) can be de f ined i n terms of diagrams. I t i s then poss ib le t o der ive, o r r a t h e r v e r i f y , t he equations o f motion fo r t he f i e l d s . Th is prov ides f o r t he l i n k between diagrams and t h e canonica l operator formalism.

Since th ings tend t o be t e c h n i c a l l y complicated we w i l l l i m i t ourse lves t o a simple case, namely th ree r e a l sca la r f i e l d s i n t e r a c t i n g i n the most simple way. The Lagrangian i s taken t o be

1 2 2 1 2 2 - mBIB + - 1 2 2 C r a - m C I C + 2 1 = A ( a - mA)A + - B(a 2

+ gABC + JAA + J B + JcC . ( C . 1 1 B

b b FA' 'FB and The bare propagators w i l l be denoted by t h e symbols A

t h e dressed prnpagators by AFA, +B and aFc. For ex amp 1 e

1 1 I-

[ 2 n 1 4 i k2 + m i - i~ '

- 1 1 *FA - 4 2 2

(2nl i Z i [ k 2 + MA) - rlA(k 1 - i c

The po le p a r t of the dressed propagator p lays an impor tant r o l e and w i l l be denoted by AF

The r e s u l t (C.2) has been obta ined as f o l l o w s [see Sect ion 9, i n p a r t i c u l a r Eq. (9.311. The f u n c t i o n rA[k2) i s the sum o f a l l i r r e d u c i b l e self-energy diagrams f o r t he A - f i e ld . The dressed propagator i s of the form (k2 + m i - r A ) - ' # Th i s expression w i l l have a po le for some value of k2, say f o r k2= -MA. Then we can expand rA around the p o i n t k2 = -MZ

2

168


2 2 TA[k I - 6 m i + (k2 + NAIFA + rlA(k21 ,

(C.41 2 2 2 6mA = m - MA , Z i = l - F A , A

where I' leads t A A Eq. (C.21.

is of order (k2 + M i l 2 . I n s e r t i o n o f t h i s expression

Next t o the propagators we def ine external l i n e factors N (k21, etc. They are the r a t i o o f the dressed propagators and t a e i r pole par ts

2 'FA NA(k I - 'A'FA

tc.51

2 I n the l i m i t k2 = -MA t h i s i s prec ise ly the f a c t o r occurring i n external l i n e s when passing from Green's func t i on t o S-matrix. F i n a l l y we have the important A+ and A- funct ions

f 1 1 2 AA = - - 8[fk 16(k2 + MA] . nni3 Z; 0

Consider now any Green's funct ion i nvo l v ing a t l eas t one A - f i e l d source. For a l l except t h i s one source we fo l low the procedure as used i n obta in ing the S-matrix, t h a t is a l l dressed propagators and associated sources are replaced by fac to rs N and the mass-shell l i m i t is taken. For the s ingled out A- f ie ld source we replace the dressed propagator and source by N~(k21, but do not take the l i m i t k2 = -M2. The Four ier transform w i t h respect t o k o f the funct ion so obtained i s defined t o be the matr ix element ( f o r a given order i n the coupling constant w i t h the appropriate i n - and out-states1 o f an operator denoted by

(The no ta t i on used here should not be confused w i t h notat ions of the type used i n Section 9.1 I t i s , roughly speaking, obtained f r o m the S-matrix by tak ing o f f one external A- l ine and rep lac ing t h a t l i n e by the f a c t o r ~A(k21. Diagramnatical ly:

169

2 f NA(k )

w i t h t h e no ta t i on : (C. 71 - , *--+ , - A- , 8-, C-l ine.

I t i s t o be noted t h a t t h e propagators used are completely dressed operators, and the re fo re self-energy I n s e r t i o n s are no t t o be contained i n Eq. [C.71. I n p a r t i c u l a r t he re are no con t r i bu t i ons of t he type:

(C.81

2 However, t h e f a c t o r NA[k ) imp l i es r e a l l y t h e i n s e r t i o n of i r r e - duc ib le se l f -energy par ts . Working ou t Eq. ( C . 5 1 we see (compare Eq. (C.41)

1 + (I. ( k 2 1 - i(2rr1 4 2 (mA - MA) 2 - A *

Diagrammatical ly:

4 2 6 = (271 i [ m A - MA] + (2n1 i [ k 2 + MAIFA . Remember t h a t T A s t a r t s w i t h a B- and a C- l ine , and t h a t NA i s attached t o a 8-C ver tex (see Eq. (C.711. We see t h a t t he r igh t -hand s ide o f Eq. (C.7) cons is ts of all skeleton diagrams s t a r t i n g w i t h an amputated 8-C ver tex, apar t f r o m the 6-correct ion.

We now de f ine the product o f t h i s ob jec t and the ma t r i x St. I t i s obtained by connecting diagrams of 6S/6A t o diagrams of St

170


+ by meang o f A func t ions :

t Th is i s a c o l l e c t i o n of cu t diagrams, w i t h S corresponding t o the p a r t i n the shadowed reg ion . Th is d e f i n i t i o n of t h e product i s t h e same as t h a t encountered i n the expression S h .

I t tu rns out t h a t t h e d i f f e r e n t i a t i o n symbol 6/6 i n W 6 A has more than fo rmal meaning. Wi th t h e he lp o f t he c u t t i n g equations i t is easy t o show t h a t

The second term has t h e p o i n t x t o t h e r i g h t o f t he c u t t i n g l i n e . Th is can be expressed f o r m a l l y by w r i t i n g 6'(StS)/GA[xl = 0, which i s what i s expected if u n i t a r i t y holds, StS = 1. One may speak o f genera l ized u n i t a r i t y , because the A - l i n e i s o f f mass- s h e l l .

The A - f i e l d cu r ren t j A ( x ) i s de f ined by

[C. 121

By v i r t u e o f Eq. (C.111 i t f o l l o w s t h a t j A [ x l i s Hermit ian.

equat ion o f motion To de f i ne t h e ma t r i x elements o f t h e f i e l d A cons ider t h e

C. 131

This i s not d i r e c t l y t he equat ion o f motion t h a t one would w r i t e down g iven the Lagrangian (C.11, because we have the mass fli [de f ined by t h e l o c a t i o n of t h e po le o f t h e dressed propagator) i ns tead o f m i .

eouat ion The equat ion o f motion (C.131 can be r e w r i t t e n as an i n t e g r a l

171

DIAGRAMMAR

The re ta rded A f u n c t i o n i s

1 1 i k x 1 2 ARA[x) - - -

k2 + MA - iEko 4 2 d4ke (2nl i ZA

Th is f u n c t i o n i s zero unless x > 0. I n f a c t 0

(C.151

I n passing, we note t h e i d e n t i t i e s

* AR - A+ - -AF ,

(C.171

'F AR + A- - Equation (C.141 defines t h e A - f i e l d i n terms o f diagrams. I t s a t i e f i e s the weak o r asymptot ic d e f i n i t i o n

l i m < alA(x1 IB, = l i m < alAin(xl 1s >. x +-w x +-w 0 0

The f i e l d A i n ( x l i s a f r e e f i e l d s a t i s f y i n g t h e equat ion o f motion [C.131 w i t h j - 0.

We w i l l w r i t e Eq. (C.141 i n terms o f diagrams, and t o t h a t purpose we must i n t roduce the "ordered product". We w r i t e

1 1 1 t - Aln(x1 = - A [xfStS - - :A zA ZA i n ZA i n ( X I S :S +

(C.18) + ZA I d4XcAi[X - X 4 1 - 6s' 6Atx'I '

+ The double do ts imp ly t h a t A t n 1s n o t t o be connected by a A - l i n e t+O S t , Below t h i s w i l l be shown d iagrammat ica l ly . The f u n c t i o n AA has been de f ined before. Due t o t h e presence o f t h i s A+ o n l y t h e mass-shel l va lue o f t h e f a c t o r N i s requ i red in 6St/6A, and

172


t h i s ia 1 1 2 ~ . Using Eq. (C.181 we can r e w r i t e the i n t e g r a l equation o f motion (C.14) i n the form

r

tc. 191

Keeping i n mind Eq. (C.171 as w e l l a8 Eq. (C.51 we see t h a t A(x)/ZA can be p i c tu red as fol lows:

+

This then i s the diagrammatic expression f o r the ma t r i x elements o f t h e f i e l d A, S im i la r expressions can be der ived f o r the f i e l d s B and C. A l l k inds o f r e l a t i o n s from canonical f i e l d theory can be der ived using these expressions. Fo r instance

2 2 FA(a - j A [ X ) - B ( X I C ( X ) - ~ A [x l - 2

2 A ") A(x I

6m

'A'B'C zA zA

fC.21) w i t h FA and 6m from Eq. (C.41.

173

CHAPTER 2.3

GAUGE THEORIES WITH UNIFIED WEAK, ELECTROMAGNETIC, A N D STRONG INTERACTIONS*

G . ‘T HOOFT University of Utrecht

1. Introduction

Only half a decade ago, quantum field theory was considered as just one of the many different approaches to particle physics, and there were many reasons not t o take it too seriously. In the first place the only possible “elementary” particles were spin zero bosons, spin 4 fermions, and photons. All other particles, in particular the p , the N * , and a possible intermediate vector boson, had to be composite. To make such particles we need strong couplings, and that would lead us immediately outside the region where renormalized perturbation series make sense. And if we wanted to mimic the observed weak interactions using scalar fields, then we would need an improbable type of conspiracy between the coupling constants to get the V-A structure’). Finally, it seemed to be impossible to reproduce the observed simple behaviour of certain inclu- sive electron-scattering cross sections under scaling of the momenta involved, in terms of any of the existing renormalizable theories2). N o wonder that people looked for different tools, like current algebra’s, bootstrap theories and other nonperturbative approaches.

Theories with a non-Abelian, local gauge invariance, were known3), and even considered interesting and suggestive as possible theories for weak

*Ftapporteur’s talk given at the E.P.S. Iiiternational Conference on High Energy Physics, Palermo, Sicily, 23-28 June 1975. 0 European Physical Society

174

interactions4i5), but they made a very slow start in particle physics, because it seemed that they did not solve very much since unitarity and/or renormalizability were not understood and it remained impossible to do better than lowest order calculations.

When finally the Feynman rules for gauge theories were settled6) and the renormalization procedure in the presence of spontaneous symmetry breakdown underst~od’-’~), it was immediately realized that there might exist a simple Gauge Model for all particles and all interactions in the world. The first who would find the Model would obtain a theory for all particles, and immortality. Thus the Great Model Rush began29~35~36*44-53~56~5a).

First, one looks a t the leptons. The observed ones can easily be arranged in a symmetry pattern consistent with experiment6): SU(2) x U(1). But if we assume that other leptons exist which are so heavy that they have not yet been observed then there are many other possibilities. To settle the matter we have to look at the hadrons.

The observed hadron spectrum is so complicated with its octuplets, nonets and decuplets that it would have been a miracle if they would fit in a simple gauge theory like the leptons. They don’t. To reproduce the nice SU(3) x SU(3) structure one is forced to take the “quarks”, the building blocks of the hadrons, as elementary fields. The existing hadrons are then all assumed to be composite. To bind those quarks together we need strong forces and here we are, back at our starting point. What have we won?

We have won quite a lot, because the tools we can use, renormalized gauge theories, are much more powerful than the old renormalizable theories. Not only do we have indications that they exhaust all possible renormalizable interactionsz0) but there is also a completely new property: the behaviour of some of these theories under scaling of all coordinates and momentaz1). If you look at such a system of particles through a microscope, then what you see is a similar system of particles, but their interactions have reduced. The theory is “asymptotically free”z1-z3). The old theories always show a messy, strongly interacting soup when you look through the mic ro~cope~~) . If you assume that any theory should be defined by giving its behaviour at small distances, then the old theories would be very ill-defined, contrary to the gauge theories.

But when it comes to model building, then it is still awkward that the forces between the quarks are strong, because that makes gauge theories not very predictive and there are countless possibilities. I have seen theories with 3, 4, 6, 9, 12, 18 and more quarks. How should we choose among all these

175

different group structures? Before answering the question let us first make up the balance. What we are certain of is:

1) Gauge theories are renormalizable, if firstly the local symmetry is broken spontaneously, and secondly the Adler-Bell-Jackiw anomalies are arranged to cancel.

2) Global symmetries may be broken explicitly, so we can always get rid of G 01 ds tone b osons .

3) We can make asymptotically free theories for strong interactions. Then the following statements are not absolute but have been learnt from general experience:

4) The Higgs mechanism is an expensive luxury: each time we introduce a Higgs field we have to accept many new free parameters in the system.

5 ) There are always more free parameters than there are masses in the theory, so we can never obtain a reliable mass relation for the elementary constituent particles. Of course, masses of composite systems are not free but can be calculated.

6) To break large groups like SU(4) or SU(3) x SU(3) by means of the Higgs mechanism is hopelessly complicated. Theories with a small gauge group like SU(2) x U ( l ) or large but unbroken groups are in a much better shape.

Of course, these are practical arguments, that distinguish useful from useless theories. But do they also distinguish good from false theories? Personally I tend to believe this. I find it very difficult to believe that nature would have created as many Higgs fields as are necessary to break the big symmetry groups. I t is more natural to suppose that just one or two Higgs fields are present and some remaining local symmetry groups are not broken a t all.

2. Strong Interaction Theory

a. Towards permanent color binding

There is a general consensus on the idea that gauge vector particles corresponding to the color group SU(3)c can provide for the necessary binding force between quarks, which transform as triplet representations of this group. The states with lowest energy are all singlets. This theory explains the observed selection rules and the SU(6) properties of the hadrons. But now there are essentially two possibilities.

176

The first possibility is that SU(3)' is broken by the Higgs mechanism, so that the masses of all colored objects are large, but finite. The $J particles can be incorporated in this scheme: they may be the first colored objects, as you heard in the sessions on color theories. Besides the disadvantages of such theories I mentioned before, it is also difficult to arrange suppression of higher order contributions to KO - K O mixing and KL -+ p+p- decay25) and the theoriee are not asymptotically free.

The alternative possibility is that SU(3)' is not broken at all. All colored objects like the quarks and the color vector bosons have strictly infinite r n a e ~ e s ~ ~ ~ ~ ~ ) . I suspect that this situation can be obtained from the former one through a phase transition. Let me explain this.

In the Higgs broken color theories there exist "solitons", objects closely related to the magnetic monopole solutions2") in the Georgi-Glashow Now let me continuously vary the parameter p2 in the Higgs potential

from negative to positive values (Fig. 1).

p2< 0

Fig. 1. The Hi= potential before and after the phase transition.

We keep A fixed. Now the vacuum expectation value F H ~ ~ . of the Higgs field 4 is first roughly proportional to 1p1 and so are the vector boson mass M v and the soliton mass,

4 ~ M v M I " - , !12

as indicated on the left-hand side of Fig. 2.

177

Fig. 2. The phase transition.

M H ~ ~ ~ ~ = mass of unbroken Higgs field F H ~ ~ ~ ~ = vac. exp. value of Higgs field

M , = mass of soliton Fs = vac. exp. value of soliton field

g = color (electric) coupling constant j = soliton coupling constant ( g j = 2an)

What I assume is that when p2 becomes positive, it is the soliton’s turn to develop a non-zero vacuum expectation value. Since i t carries color-magnetic charges, the vacuum will behave like a superconductor for color-magnetic charges. What does that mean? Remember that in ordinary electric supercon- ductors magnetic charges are confined by magnetic vortex lines; as described by Nielsen and Olesen3’). We now have the opposite: it is the color charges that are confined by “electric” flux tubes. So we think that after the phase transition all color non-singlets will be tied together by “strings” into groups that are color singlets. In this phase the Higgs scalars play no physical role whatsoever and we may disregard them now.

The great shortcoming of this theory is that it is intuitive and as yet no mathematical framework exists. But there are various reasons to take i t seriously.

Theoretically: i) we see it happen if we replace continuous space-time by a sufficiently coarse

latticez7): the action becomes that of the Nambu string. ii) We see it happen in the only soluble asymptotically free gauge theories:

Schwinger's mode131) and, even better, in the SU(o0) gauge theory in two space-time dimension^^^).

Experimentally: a) the flux lines would behave like the dual string and thus explain the straight

b) This is the simplest and probably the only asymptotically free theory that

c) The theory is closely related to the rather successful MIT bag

In principle there exists an intermediate possibility: we probably have several phase-transitions when we go from unbroken SU(3) to for instance SU(2), U ( l ) and finally complete breaking. We will not consider the possibility that we are in SU(2) or U(1), but we must remember that from the low lying states it is difficult to deduce in what phase we really are.

If we are in the unbroken phase then SU(3) color must commute with weak and electromagnetic SU(2) x U(1), and with SU(3) flavor'). Consequently, we are obliged to introduce charm35), or even more new quarks perhaps36). Now let us consider the 11, particles.

Regge-trajectories%).

explains Bjorken behaviourz2).

b. Charmonium

The most celebrated theory for $J is that it is a bound state of a charmed quark and its a n t i p a r t i ~ l e ~ ~ ) . Charmed quarks are assumed to be rather heavy. The size of the bound state wave function will therefore be small and we can look a t the thing through a microscope (i.e. apply a scale transformation). Then we see rather small couplings, so we may use perturbation expansion to describe the system. The mathematics is do-able here! At first approximation the gluon gauge field behaves exactly like Maxwell fields, even the SU(3) structure constants can be absorbed in the coupling constant. We can calculate the

*)i.e. the familiar broken symmetry group that transforms p , n and X into each other. The word "flavor" has been proposed by Gell-Mann to denote both isospin and strangeness.

179

Table.

3.7

3.6

3.5

3.4

3.3

3.2

3.1

3.0

ORTHO 11

hadrons

T T T T J pc

2:' 1:' 0" 1-- 0-+ 1 +-

180

Tsble. SU(4) predictions

pew Meeona

charm J P = 0 - JP = 1' mass (GeV)

+ + + o + o

1 2

0 0 0 3.1

0 - 0 - -1 2

3 2+ 5

2 2 t t 2 + t 3.7 + +

2 + + o 2 t + o + + o

0 2.5 + o

1 + o

0

annihilation rate and level splitting8 exactly as in positronium. The annihilation rate of the positronium vector state is

For charmonium the formula would be')

where a, = g1/3+ (the subscript s standing for "strong").

It fits well in the renormalization group theory if a, is around 1/3 at 2 GeV. That it is raised to the sixth power explains the stability of 4. Note that this

*)The quark flavora in this paper are called p , n, p' and A. More modem is the nomenclature u, d , c a d u

181

is the rate with which the two quarks annihilate. It is independent (to first approximation) of the details of the hadronic final state3g).

Now this gauge theory for strong interactions gives precise predictions on the other charmonium states (Table 1) and the charmed hadrons (Table 2). They can be found in some nice papers by Appelquist, De Rcjula, Politzer, Glashow and others3?).

It is very tempting to assume that this new quark is really the charmed one as predicted by weak interaction theories (see Sect. 3), but it could of course be an “unpredicted” quark.

If the predictions from this theory come out to be roughly correct then that would be a great success both for the asymptotically free gauge theory for strong interactions, and for the renormalization theories that predicted charm.

3. The Weak Interactions

a. SU(2) x U ( l ) theories

A simple SU(2) x U ( l ) pattern seems to be compatible with all experimental data on pure leptonic and semileptonic processes (neutral currents and charm), but with the pure hadronic weak interactions we still have a problem: there should be AZ = 3/2 and AZ = 1/2 transitions with similar strength and we only see A1 = 1/2. Also, these interactions seem to be somewhat stronger than other weak interaction processes. The traditional way to try to solve the problem is the possibility that A1 = 1/2 is “dynamically enhanced”. Seen through our “microscope” a t momenta of the order of the weak boson mass, the AZ = 1/2 and AI = 3/2 parts of the weak interaction Hamiltonian may be equal in strength, but when we scale towards only 1 GeV, then AZ = 1/2 may be enhanced through its renormalization group equations. This mechanism really works, but can only give a factor between 6 and at most 14 in the amplitudes40). But there are many uncertainties, since we do not know exactly from where to where we should scale, and what the importance is of the higher order corrections. I t has been argued that a similar mechanism might depress leptonic decay modes of charmed particles, thus giving them a better camouflage that prevents their detection41) and the mechanism could influence parity and isospin violation4’).

An interesting alternative explanation of the AZ = 1/2 rule has recently been given by De Ri?jula, Georgi and G l a s h ~ w ~ ~ ) . In usual SU(2) x U ( l ) theories only the left-handed parts of the spinors may be SU(2) doublets, all

1 82

right-handed parts are singlets. These authors however take the right-handed parts of p' and n to form a doublet. This adds to the hadronic current

P'Yp(1 - 75)n I ( 5 ) which is only observable in purely hadronic events or charm decays. Note that there is no Cabibbo rotation. We get among others a term,

cos 0, F ' 7 p ( 1 + 7 5 ) A E Y p ( 1 - 75)d (6) in the effective Hamiltonian. Note the absence of Cabibbo suppression by factors sin 8, and the AZ = 1/2 nature of this term.

Fig. 3. a) The new strangeness changing processes, b) diagram for the decay A'- 4 2?r, c) conventional picture of K- 4 27r.

The predictions on neutrino production of charmed particles, and the charm decay rates are radically changed by this theory. We can have

vp + p-Cf+, (7) etc. with no sin2 8, suppression, and pseudo AS = 2 processes can take place like

UP + p - p i ( + DO p - v - K t (8)

because of Do-Do mixing (as in K&o mixing). Experts however are skeptical about the idea. The charmed quark in this current has to violate Zweig's rule and the charmed quark loop must couple to gauge gluons before the current can contribute.

183

The idea is very recent and I have not yet seen any detailed calculations. We will soon know what the contribution of such diagrams can be. Furthermore, the K L - KS mass difference, when naively calculated in this model seems to be too large compared with experiment.

b. Other and larger groups

SU(2) x U ( l ) theories do not really unify weak interactions and electromagnetism. The U( 1) group may be considered as the fundamental electromagnetic group, and its photon is merely somewhat mixed with the neutral component of the weak SU(2) gauge field. True unification occurs only then when we have a single compact group.

1. Heavy lepfons The first example was originally invented to avoid neutral currents: the

Georgi-Glashow O(3) modelz9). Now we have a large class of based on SU(2) x U ( l ) , 0 ( 3 ) , 0 ( 4 ) , O(4) x U(1), SU(3), SU(3) x U(1), SU(3) x SU(3) etc., all predicting new leptons. An extensive discussion of these models is given by Albright, Jarlskog and Tjia53). Table 3 gives the predicted leptons in these schemes. Albright, Jarlskog and Wolfenstein also analysed the possibilities to detect such objects by neutrino productions4).

Of course, we can always extend the lepton spectrum in other ways, for instance by adding new representation^^^), so that we get more members in theseries (,”), (:), ( L ) , e t c .

2. The Pali-Salam Model56) Theoreticians are eagerly awaiting the discovery of the first heavy lepton.

But that may take quite a while and in the meantime we search for more guidelines to disclose the symmetry structure of our world. One such guideline is that eventually we do not expect that baryons and leptons are essentially different. That is, they might belong to just one big multiplet. This, and other ideas of symmetry and simplicity led Pati and Salam to formulate their most recent “completely unified model”. Leptons and quarks from just one representation of SU(4) x SU(4), of which color SU(3) and weak SU(2) x U(l ) are subgroups. If, as argued before, color is unbroken then we are free to mix the photon (also unbroken) with colored bosons, so there is no physical difference between the charge assignments

184

w3. Heavy leptons

Gauge group Muonic leptons Refs.

sup) x U(1) cc- UP Weinberg, Salam' ~ 4 ' 8 ' ' ) ~

UP Bjorken and Llewellyn

MO BCg and Zee50) Smith no. s5') cc-

Bjorken and Llewellyn

Prentki and Z ~ m i n o ~ ~ ) " M+ Smith no. 551) Mo P-

IA- u,, M+ Prentki and Z u m i n ~ ~ ~ )

Bjorken and Llewellyn Smith no. 451)

fi- Mo M+ Mi

Georgi and GlashowZ9) O(3) N SU(2) P- up M+ Bjorken and Llewellyn

Mo Smith no. 651)

"9

Mo O(4) = SU(2) x SU(2) cc- M+ Pais45)

M+ UP

Mo cc- Pais'6) ~ ~ ~~

cc- UP

M- Schechter and Singer") SU(3) x U(1)

Aft Albright, Jarlskog, T j i 8 ) Mo

M- MfIo

185

Table 3. (Cont inued)

Gauge group Leptons

~~

Refs.

U(3) left right

Eo eS M o M +

UP uc Salam

MQI EO) Pati '') p- M" M + E - Eo e+ and

E- Eol p- Mol

left right

M" pt Eo E+

Ve PP S a l m

EO' MO' Pati 4') e- E" E+ M - M o p+ and

M - MoI E - Eo'

Y e , Yp

SU(3) x SU(3) e- P+

LO

Acluman, Weinberg4')

-113 -1/3 -1/3 - 1 - 1 0 0 -1 213 213 213 0 1 1

because the photon may freely mix with components of the unbroken color vectors fields*).

*)If the integer-charge assignment is adopted however, then leptons couple directly to color gluom, and this would make interpretation of the experimentally observed ratio R = a(hadrons)/a(p+p-) in e t e - annihilation more complicated. So R must be computed from the non-integer charges. This 16plet yields R = 10/3.

186

4. The Higgs Scalars

The ugly ducklings of all Unified Theories are the Higgs scalars. They usually bring along with them as many free parameters as there are masses in the theory or more. This makes the theories so flexible that tests become very difficult.

These scalars are needed for pure mathematical reasons: otherwise we cannot do perturbation expansions, and we have no other procedure a t hand to do accurate calculations. But do we need them phy~ically?~')

Various attempts have been made to answer this question negatively. First of all they are ugly, and physics must be clean. But that is purely emotional. Then: we do not observe scalars experimentally. But: there is so much that we do not observe: quarks, I.V.B.'s, etc.

Linde and Veltman raised the point that the scalars do something funny with gravity: their vacuum-expectation value gives the vacuum a very large energy-density. That should renormalize the cosmological constant. (Otherwise our universe would be as curved as the surface of an orange5"). On the other hand the net cosmological cosntant is very many orders of magnitude smaller than this. How will we ever be able to explain this miraculous cancellation? In this respect it is interesting to note an observation made by Z ~ m i n o ~ ~ ) ) : in supersymmetric models there is no cosmological constant renormalization, even a t the one-loop level.

Ross and Veltman then suggest that perhaps one should choose the scalars in such a way that the vacuum-energy density vanishes")). In Weinberg's model that means that one should add an isospin 3/2 Higgs field. Such a Higgs field could reduce neutral current interactions, and that would be welcome to explain several experiments.

Personally I think that one has to consider the renormalization group to see what kind of scalars are possible. If we scale to small distanceslo6) then the theory has nearly massless particles. Only a nearly exact symmetry principle can explain why their masses are so small at that scale. For fermions we have chiral symmetry for this. Scalar particles can be forced to be massless if they are the Goldstone bosons of some global symmetry. They then have the quantum numbers of the generators of that symmetry group. Since all global symmetry groups must commute with local gauge groups, it is difficult to get light scalar particles that are not gauge singlets.

According to this argument it is impossible to have scalar Higgs particles, except when they are strongly interacting, since in that case we cannot scale very far because of non-asymptotic-freedom.

187

A notable exception may be constructions using supersymmetries (see Sect. 5) .

In connection with Higgs’ scalars I want to clear up a generally believed misconception. I t is not true that theories with a Higgs phenomenon in general cannot be asymptotically free. For many simple Higgs theories one can obtain asymptotic freedom, provided that certain very special relations are satisfied, between coupling constants that would otherwise be arbitrary62). These theories do not have a stable fixed point at the origin but I cannot think of any physical reason to require that. Examples of such theories are supersymmetric theories, with possible supersymmetry breaking masses. Quark theories of this nature do not exist.

Fig. 4. Example of an unstable ultraviolet fixed point at the origin of parameter space. The arrows indicate the change of the coupling constants as momenta increase. The solid line is the collection of asymptotically free theories.

For weak interactions however I do not think that asymptotic freedom would be a good criterion, because any change of the theory beyond, say 10000 GeV would alter the relations between coupling constants completely.

5. Supersymmetric Models

Symmetry arguments can be deceptive. In the nineteenth century physicists argued that the Moon must be inhabited by animals, plants and people. This was based on theological and symmetry arguments (earth-moon-symmetry), similar to the ones we use today.

188

Keeping this warning in mind, let us now consider the supersymmetric modelseg). For supersymmetry we have a special session. But an interesting attempt by F ~ a y e t ~ ~ ) must be mentioned here, who constructs a Weinberg- like model with supersymmetry. There, fermions and bosons sit in the same multiplet. Thus the photon joins the electreneutrino. But which massless boeon should join the muon-neutrino? This problem has not been solved, to my knowledge.

Even if supersymmetry would be ruled out by the overwhelming experimental evidence that fermion and boson masses are not the same, I think we do not have to drop the idea. Suppose that quantum field theory begins at the Planck length; we cannot go further than the Planck length as long as we do not know how to quantize General Relativity. And suppose that Quantum Field Theory is supersymmetric but the supersymmetry is very slightly broken (it is a global symmetry, 80 we may do that) and the breaking is described by a coefficient of the order of Suppose further that the representations happen to be such that there are no supersymmetric mass terms. Then at the mass scale of 1 GeV we would have all dimensionless couplings completely supersymmetric, but mas8 terms arise that break supersymmetry, because l(GeV)a is in our natural units defined by the Planck length. I would like to call this relaxed supersymmetry and it is conceivable that many interesting models of this nature could be built106).

6. PC-Breaking

Theories without scalars are automatically PC invariant. To describe PC- breaking we can either introduce elementary scalar fields with PC = -1 and put PGodd terms in the Lagrangian, or believe that all scalars are in principle composite. Then PC-breaking must be spontaneous as described by T.D. Leeea). Usually however the scalars are not specified, but only the currents66).

De RGula, Georgi and Glashow observe that their chiral current can easily be modified to incorporate PGbreaking, a scheme already proposed by Mohapatra and Pati6') in 1972:

A J = ~ ' 7 ~ ( l - 75)(n cos g5 + i A sin g5),

g 5 < 1. (9)

7. Quantum Gravity

Quantum gravity is still not understood, but an interesting formal interpretation is given by Christodouloum). He gives a completely new definition of

189

time, in terms of the distance in "superspace" between two three-dimensional geometries. But his formalism is not yet in a shape that enables one to give interesting physical predictions, and he has not yet considered the problem of infinities.

Renormalizable interactions between gravity and matter have not yet been found69*70) but Berends and Gastmans find suggestive cancellations in the gravitational corrections to anomalous magnetic moments of leptons71).

Numerous authors tried to put terms like &R2 or/and &RPv RP" in the Lagrangian, and thus (re)obtain renormalizability. It is about as clever as jumping to the moon through a telescope. Personally, I am convinced that if you want a finite theory of gravity, you have to put new physics in72). Very interesting in that respect are the attempts by Scherk and Schwarz to start with dual models.

8. Dual Models

Though originally designed as models for strong interactions, the dual models have become interesting for other types of interactions as well. They are the only field theoretical scheme that start from an infinite mass spectrum, forming Regge-trajectories with slope a'. In the limit a' + 0 they can mimic not only many renormalizable theories but also gravitation (always in combination with matter fields). If the tachyon-problem and the 26-dimension problem can be overcome then one might end up with a big renormalizable theory that unifies e ~ e r y t h i n g ~ ~ ) .

Scherk and Schwarz point out that for a' # 0 those models are equally or better convergent than renorma1izal.de theories.

9. Two-Dimensional Field Theories

Field theories in one-space and onetime dimension are valuable playgrounds for testing certain mathematical theorems that are supposed to hold also in four dimensions. There is no time now to discuss all the interesting developments of the past years74) but I do want to mention just three things.

In a recent beautiful paper S. Coleman75) explains the complete equivalency between two seemingly different structures: the massive Thirring model on the one hand, and the SineeGordon model on the other. The solitons (extended particle solutions) in one theory correspond to the fermions of the other. Thus we get one of the very few theories that can be expanded both at small and at large values of the coupling constant.']

*)Note added: this result would later be challenged, the equivalence of these two models is not complete.

190

Solitons, and their quantization procedure have been studied in two and four dimensions by many g r o ~ p s ~ " ~ ~ ~ ~ ' ~ ) . Even in theories with weak coupling constants, solitons interact strongly (they have very large cross sections!) so they could make interesting candidates for an alternative strong-interaction theory. N o convincing soliton theory for strong interactions does yet exist, but several magnetic-monopole quark structures have been c o n ~ i d e r e d ~ ~ * ~ ~ ~ ~ " ) .

Thirdly we mention the gauge theories in two dimensions with gauge group U(N) or SU(N) . They are exactly soluble for either N = 1, mfermion = 0, or for: N + 00, mfermion arbitrary, in which case the 1/N expansion is possible. These theories exhibit most clearly the quark confinement effect, when color is ~ n b r o k e n ~ l t ~ ~ ) . Quark confinement is almost trivial in two space- time dimensions because the Coulomb potential looks like Fig. 5 .

V

Fig. 5. The Coulomb potential iu one space-like dimension.

QED in two dimensions (Schwinger's model) is not a good confinement theory because the electrons manage to screen electric charges completely, so the dressed electron is free, except for one bound state. At N + 00, probably a t all N > 1, we get an infinity of bound states, "mesons" that interact with strength proportional to 1/N. And they are on a nearly straight trajectory (actually a series of daughters because there is no angular momentum).

10. The Two Eta Problems

The latter one-space onetime dimensional model has lots of physically interesting properties. One is, that we can test ideas from current algebra. As in its four-dimensional analogue, we have in the limit where the quark masses mp, m, + 0 an exact SU(2) x SU(2) symmetry, and the pion mass goes to zero in the s e n ~ e ~ ~ ~ ' ~ ~ )

191

m: -, C(mp + m,) . (10)

C=gJ; ;= l /&. (11)

The proportionality constant contains the RRgge slope. In two dimensions we have

So if m, is small, then mp and m, must be very small, in the order of 10 MeV. Now we can consider two old problems associated with the eta- meson. The first is that our theory seems to have U(2) x U(2) symmetry, not only SU(2) x SU(2). Thus, there should also be an isospin-zero particle, 11, degenerate with 80, whose mass should vanish. Experimentally however, mi > m:. Also we do not understand why the ?ro - q splitting goes according to TO = p p - fin; 0 = p p + nn despite of possible mass differences of p and n. This problem reappears every now and then in the literature. Fritzsch, Gell- Mann and Leutwyler”) gave in 1972 a beautiful and simple solution: there is an Adler-Bell-Jackiw anomaly associated with the chiral U( 1) subgroups1):

a, j,” = 2mj5 + C C , , , , ~ ~ T~(G,,G,~). (12)

The symmetry is broken and the eta-mass is raised. But then came the big confusing counter argument: we can redefine the axial current so that it is conserved, by writing

) 2 3: = j[ - 2Ccp,,p Tr A, &Ap - apA, + :g[Aa, Ap] , (13)

so there should be a massless eta after all. Considerable effort has been made the past years to show that this counter argument is wrong. There are four counter counter arguments, all based on the fact that 3: is not gauge-invariant:

i) 3; is not gauge invariant, therefore the massless eta carries color, so it will be confined and thus removed from the physical spectrum8’).

ii) The gluon field that occurs explicitly in the “corrected” equation for the axial current is a long-range field. The Goldstone theorem does not apply when long-range interactions are present.

iii) In the Lorentz-gauge there is no explicit long-range Coulomb force. But then there are negative metric states and this eta may be a ghost.. It will be cancelled by other ghosts with wrong metricm).

iv) But the simplest argument is that we can calculate the eta mass exactly in two dimensions and see what happens. In two dimensions there is an anomaly only in the U(l) fieldlo7):

(

8, jt = 2mj6 + CrPuF,,, . (14)

192

It is exactly this anomaly that raises the mass of the eta (it can be identified with Schwinger’s photon), despite of the fact that we can find a new axial current:

3,” = j,” - 2C~,uAu 9 (15)

which is conserved+).

The second eta problem is the decay

It breaks G-parity and thus isospin. If we assume this decay to be electromagnetic then current algebra shows us that it should be suppressed by factors of rn:/rni, which it does not seem to bem). I think there is a very simple explanation in terms of the present theory: the proton- and neutron- quark masses are free parameters in our theory, not “determined” by electromagnetism. Their difference follows from known hadron mass splitting8 and tends to unmix Fp and fin, that is, mix T O and q. We get

with the correct order of magnitudea6). The essential difficulty with the current algebra argument was that all SU(2) breaking effects were assumed to be electromagnetic. That would be beautiful, we could do current algebra by replacing 6, by 6, + iqA, everywherea7).

In a gauge theory this is wrong. As I emphasized in the beginning, mass differences cannot be explained by electromagnetism alone, they are arbitrary parameters and must be fixed by experiment. We’ll have to live with that. Only for composite systems mass differences can be calculated. This suggests of course that we must make a theory with composite quarks. We leave that for a next generation of physicists.

Note that the breaking of SU(2) x SU(2) is governed by proton- and neutron quark masses alone. They are of order 10-20 MeV and differ by 5 MeV or so. So the breaking term of SU(2) x SU(2) also breaks SU(2) rather badly.

*)This argument is not quite correct, because in two apace-time dimensions the Goldstone realisation of a continuow aymmetry is impossiblea4). However, if we let first N = 00, then m = 0 then we do get nevertheless the Goldstone mode. This ir possible because the meson- menon interactions decrease like 1/N. Note that in any CMC there arc no parity doublets for large N.

193

That is why you may not use PCAC and isospin together to get factors of mf/mi in the decay q -+ 3 ~ ' ) .

11. Misellaneous

a. Perturbation expansion

The renormalizability of the perturbation theory for gauge fields being well settled, there are still new developments. The dimensional renormalization procedure is the solution to all existence and uniqueness problems for the necessary gauge-invariant counterterms.

But if one wishes to circumvent the continuation to non-integer number of dimensions then the combinatorics is very hard. The Abelian Higgs-Kibble model can now be treated completely to all orders within the Zimmermann normal product formalism"") if there are no massless particles. Slavnov identities can be satisfied to all orders in this procedure also in the non-Abelian Higgs-Kibble model if the group is semi-simple (no invariant U(l) group) and if no massless particles are present. Unitarity and gauge invariance have explicitly been proven this way in a particular SU(2) model.

Massless particles are very complicated this way, but advances have been made by Lowenstein and Becchig9) in certain examples of massless Yang-Mills fields, and Clark and Rouetgo) for the Georgi-Glashow model.

It is noted by B. de Witg1) that there is a technical restriction on the allowable form of the gauge-fixing term, relevant for supersymmetric models. The gauge-fixing term cannot have a non-vanishing vacuum expectation value in lowest order, otherwise contradictions arise. Of course, the Slavnov identities make the vacuum expectation value of this term vanish automatically in the usual formulation.

b. The background field method

The algebra in gauge theories is often quite involved. For certain calculations it would be of great help if gauge-invariance could be maintained throughout the calculation. On the other hand we must choose a gauge condition, which by definition spoils gauge-invariance right from the beginning. The trick is now to use the so-called background gaugeg2): the fields are split into a c-

number, called background field, and a q-number, called quantum field. Only

*)As yet we have no satisfactory explanation for a possible A I = 3 component in 1 -+ 3n.

194

the quantum part must be fixed by a gauge condition, whereas gauge-invariance for the c-numbers can be maintained. The method was very successful in the case of g r a ~ i t y ~ ’ * ~ ~ ) and has also been applied to calculate anomalous dimensions of Wilson operator^^^^^^). The method can be generalized for higher order irreducible graphsIo5).

c. Two-loop-beta

The Callan-Symanzik beta function has now been calculated up to two loops for several gauge theoriesg5tg6). We have

P(g) = Ag3 + Bg5 + O(g7) . Now A can have either sign, and can also be very close to zero. In general, B will not vanish (it can be either positive or negative). We can then get either an UV or an IR stable fixed point close to zero. In certain supersymmetric models, A vanishes. The question is whether perhaps p(g) vanishes identically for such a model. The answer appears to be: no, because for these models B has now been calculated also and it is non-zerog6).*)

d. The infrared problem

Massless Yang-Mills theories are very infrared divergent. As explained, we expect extremely complicated effects to occur, like flux tube formation and color confinement. A general argument is presented by Patrascioiug7) and Swiecag8) that shows that if we have a local gauge-invariance and if we can have isolated regions in space (-particles) with non-vanishing total charge, then there must exist massless photons coupled to that charge. This is only proven for Abelian gauge symmetries, but if it would also hold for non-Abelian invariance, then the absence of massless colored photons must imply the absence of any colored particles (= color confinement).

e. Symmetry restoration at high temperatures

Just like a superconductor that becomes normal when the temperature is raised above a certain critical value, so can the vacuum of the Weinberg model become “normal” a t a certain temperature”). The critical temperature is typically of the order of

Mw e

kT- -

*)Note added later: this is now known to be a mistake; p vanishes in all orders in these models.

195

This assumes that Hagedorn’s limit on high temperatures”’) is invalid. Indeed it is invalid in the present quark theories, but the specific heat of the vacuum is very high because there are so many color components of fields. Observe that for SU(N) theories Hagedorn might be correct in the limit N - r o o .

f. Symmetry restoration at high external fields

If we consider extremely strong magnetic fields then also the symmetry properties of the vacuum might change”’). One can speculate on restoration of color symmetry, e - p symmetry, parity or CP restoration, and vanishing of Cabibbo’s angle. In still larger fields formation of magnetic monopoles2’) would make the vacuum unstable, as in strong electric fields.

g. Symmetry restoration at high densities

A very high fermion density means that $$ and I&$ have a vacuum expectation valuelo2). This also can have a symmetry restoring effect.

T.D. Lee, Margulies and Wicklo3) argue that chiral SU(2) x SU(2) might be restored a t very high nuclear densities. Thus the mass of one nucleon would go to zero and perhaps very heavy stable nuclei could be formed. The most recent calculations show a very remarkable phase transition at no more than twice the normal nuclear density. Although the result is of course model dependent, this work seems to predict stable large nuclei with binding energy of 150 MeV/nucleon.

At still higher densities we can speculate on more transition points. Again we think that the quark picture is more suitable than Hagedorn’s picture”*).

12. Conclusions

a. Unifying everything

What I hoped to have made clear at this conference is that gauge fields are likely to describe all fundamental interactions including, in a sense, gravity. This is a breakthrough in particle physics and deserves to be called: “unification of all interactions”.

196

b. Unifying nothing

But when we consider our present theory of strong interactions, the unbroken color version, then we see that it is unlikely to be really unified with weak and electromagnetic interactions, unless we go at ridiculously high energies, because the gauge coupling constants probably still differ considerably, and SU(3)m'0' commutes with SU(2) x U(1). Also if we look at weak and electromagnetic interactions, we see that true unification has not yet been reached. At small distances strong interactions become weak, weak interactions become strong and electromagnetic ones stay electromagnetic, but no unification yet.

Perhaps our knowledge of the particle spectrum is still far too incomplete to enable us to unify their interactions.

I have given air to my own feeling that we are going in the wrong direction by choosing larger and larger gauge groups.

Perhaps we can use the confinement mechanism again to build quarks and leptons from still more elementary building blocks (chirps, growls, etc.).

Instead of "unifying" all particles and forces, it is much more important to unify knowledge.

Acknowledgement

I would like to thank D.A. Rms and M. Veltman for many interesting discussions.

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53. C.H. Albright, C. Jarlskog and M.O. Tjia, Nucl. Phys. B86 535 (1975). 54. C.H. Albright and C. Jarlskog, Nucl. Phys. B84 467 (1975). C.H. Albright and

C. Jarlskog and L. Wolfenstein, Nucl. Phys. B84 493 (1975). 55. For a review and 108 more references, see: M.L. Per1 and P. Rapidis, SLAC-

PUB-1496 preprint (Oct. 1974). 56. J.C. Pati and A. Salam, Phys. Rev. D10 275 (1974); D11 703 (1975); D11

1137 (1975); Phys. Rev. Lett. 31 661 (1973), preprint IC/74/87 Trieste (Aug. 1974). J.C. Pati, “Particles, forces and the new mesons”, Maryland tech. rep.

57. Many attempts have been made to formulate “dynamical symmetry breakdown”, i.e. without the simple Higgs scalar. See for instance: T.C. Cheng and E. Eichten, SLAGPUB-1340 preprint (Nov. 1973); E.J. Eichten and F.L. Fein- berg, Phys. Rev. D10 3254 (1974). F. Englert, J.M. Frkre and P. Nicoletopoulos, Bruxelles preprint (1975); T. Goldman and P. Vinciarelli, Phys. Rev. D10 3431 (1974).

Llewellyn Smith, Nucl. Phys. B56 325 (1973).

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75-069 (1975).

200

58. A.D. Linde, Pis’ma Zh. Eksp. Teor. Fiz. 19 (1974) 320 ( J E T P Lett. 19 (1974) 183). J. Dreitlein, Phys. Rev. Lett. 33 1243 (1974). M. Veltman, Phys. Rev. Lett. 34 777 (1975).

59. B. Zumino, Nucl. Phys. B89 535 (1975). 60. D.A. Ross and M. Veltman, Utrecht preprint (1 April 1975). 61. See also: H. Georgi, H.R. Quinn and S. Weinberg, Harvard preprint (1974). 62. M. Suzuki, Nucl. Phys. B83 269 (1974). E. Ma, Phys. Rev. D11 322 (1975).

63. See Abdus Salam’s talk on Supersymmetries at this conference. 64. P. Fayet and J. Iliopoulos, Phys. Lett. S1B 475 (1974). P. Fayet, Nucl. Phvs.

B78 14 (1974); preprint PTENS 74/7 (Nov. 1974, Ecole Normale SupCrieure, Paris), Nucl. Phys. B O O 104 (1975).

N.P. Chang, fhys. Rev. D10 2706 (1974).

65. T.D. Lee, Phys. Rev. D8 1226 (1973); Phys. Rep. Qc (Z) . 66. See A. Pais and J. Primack, Phys. Rev. D8 3063 (1973) and references therein. 67. R.N. Mohapatra and J.C. Pati, Phys. Rev. D11 566 (1975). 68. D. Christodoulou, Proceedings of the Academy of Athens, 47 (Jan. 1972); CERN

69. G. ’t Hooft and M. Veltman, Ann. Inst. If. Poinmrd 20 69 (1974). 70. S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32 245 (1974), Phys. Rev.

D10 401 (1974), Lett. Nuovo Cimento 11 218 (1974), f h y s . Rev. D10 411 (1974). S. Deser, H.S. Tsao and P. van Nieuwenhuizen, Phys. Lett. SOB 491 (1974), Pbys. Rev. D10 3337 (1974).

71. F.A. Berends and R. Gastmans, Leuven preprint (Oct. 1974). M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 3203 (1975).

72. See also: G. ’t Hooft, in Trends in Elementary Particle Theory, Proceedings of the International Summer Institute on Theoretical Physics in Bonn 1974, ed. by H. Rollnik and K. Dietz, Springer-Verlag p. 92.

73. J. Scherk and J.H. Schwarz, Nucl. Phys. B81 118 (1974); Phys. Lett. 67B 463 (1975).

74. D.J. Gross and A. Neveu, Phys. Rev. D10 3235 (1974). R.G. Root, Phys. Rev. D11 831 (1975). L.F. Li and J.F. Willemsen, f h y s . Rev. D10 4087 (1974). J. Kogut and D.K. Sinclair, Phys. Rev. D10 4181 (1974). R.F. Dashen, S.K. Ma and R. Rajaraman, Phys. Rev. D11 1499 (1975). S.K. Ma and R. Rajaraman, Phys. Rev. D11 1701 (1975).

75. S. Coleman, Harvard preprint (1974). B. Schroer, Berlin preprint FUB HEP 5 (April 1975). J. Friihlich, Phys. Rev. Lett. 34 833 (1975).

76. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10 4114, 4130, 4138 (1974). P. Vinciarelli, 1993-CERN preprint TH (March 1975).

preprint TH 1894 (June 1974).

20 1

M.B. Halpern, Berkeley preprint (April 1975). K. Cahill, Ecole Polytechnique A preprint 205-0375 (March 1975, Paris); Phys. Lett. 53B 174 (1974). C.G. Callan and D.J. Gross, Princeton preprint (1975). L.D. Faddeev, Steklov Math. Inst. preprint, Leningrad 1975.

77. S. Mandelstam, Berkeley preprints (Nov. 1974; Feb. 1975); Phys. Rev. D11 3026 (1975).

78. A.P. Balachandran, H. Rupertsberger and, J. Schechter, Syracuse preprint rep. no. SU-4205-41 (1974). G. Parisi, Phys. Rev. D11 970 (1975). A. Jevicki and P. Senjanovic, Phys. Rev. D11 860 (1975). See also interesting ideas by L.D. Fad- deev, preprint MPI-PAE/Pth 16, Max-Planck Institute, Miinchen, June 1974.

79. This follows immediately if we assume that a term

c(pp + an)

C I O

in the sy,mmetric Lagrangian for a sigma model of the pion, with c and c' of the same order of magnitude. See ref. 107.

80. H. Fritzsch, M. Cell-Mann and H. Leutwyler, Phys. Lett. 47B 365 (1973); W. Bardeen, Stanford report (1974, unpublished).

81. J.S. Bell and R. Jackiw, Nuovo Cimento 60A 47 (1969). 82. J. Kogut and L. Susskind, Phys. Rev. D9 3501 (1974); D10 3468 (1974); D11

83. S. Weinberg, Phys. Rev. D11 3583 (1975). 84. S. Coleman, Commun. Math. Phys. 31 259 (1973). 85. D. Sutherland, Phys. Lett. 23 384 (1966). J.S. Bell and D. Sutherland, Nucl.

86. See for instance ref. 107, and W. Hudnall and J. Schlechter, Phys. Rev. D9

87. M. Veltman and J. Yellin, Phys. Rev. 154 1469 (1967). 88. C. Becchi, A. Rouet and R. Stora, Phys. Lett. 62B 344 (1974); lectures given

at the International School of Elementary Particle Physics, Basko Polje, Yu- goslavia Sept. 1974, and a t the Marseille Colloquium on Recent Progress in Lagrangian Field Theory and Applications, June 1974; preprint 75, p. 723 (Mar- seille, April 1975).

89. J. H. Lowenstein and W. Zimmermann, preprints MPI-PAE/PTh 5 and 6/75 (Miinchen, New York, March 1975); Commun. Math. Phys. 44 73 (1975). C. Becchi, to be published.

in the quark Lagrangian corresponds to a term

3594 (1975).

Phys. B4 315 (1968).

2111 (1974). I. Bars and M.B. Halpern, Phys. Rev. D11 956 (1975).

90. T. Clark and A. Rouet, to be published. 91. B. de Wit, Phys. Rev. D 1 2 1843 (1975). 92. B.S. DeWitt, Phys. Rev. 162 1195; 1239 (1967). J. Honerkamp, Nucl. Phys. B18

269 (1972); J. Honerkamp, Proceedings Marseille Conf., 19-23 June 1972. G. 't Hooft, Nucl. Phys. B62 444 (1973). G. 't Hooft, lectures given at the XIIth Winter School of Theoretical Physics in Karpacz, Poland, (Feb. 1975). M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phvs. Rev. D 1 2 3203 (1975).

202

93. S. Sarkar, Nucl. Phys. B82 447 (1974); B83 108 (1974). S. Sarkar and H. Strubbe, Nucl. Phys. BOO 45 (1975).

94. H. Kluberg-Stern and J.B. Zuber, preprint DPh-T/75/28 (CEN-Saclay, March 1975); Phys. Rev. D12 467, 482 (1975).

95. D.R.T. Jones, Nucl. Phys. B75 531 (1974). A.A. Belavin and A.A. Migdal, Gorky State University, Gorky, USSR, preprint (Jan. 1974). W.E. Caswell, Phys. Rev. Lett. 33 244 (1974).

96. D.R.T. Jones, Nucl. Phys. B87 127 (1975). S. Ferrara and B. Zumino, Nucl. Phys. B70 413 (1974). A. Salam and J. Strathdee, Phys. Lett. 61B 353 (1974).

97. A. Patrascioiu, Inst. Adv. Study, Princeton preprint COO 2220-45 (April 1975). 98. J.A. Swieca, NYU/TR4/75 preprint (1975, New York). 99. C.W. Bernard, Phys. Rev. DD 3312 (1974). L. Dolan and R. Jackiw, Phys. Rev.

DO 3320 (1974). S. Weinberg, Phys. Rev. DO 3357 (1974). B.J. Harrington and A. Yildiz, Phys. Rev. D11 779 (1975).

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102. R.J. Harrington and A. Yildiz, Phys. Rev. D11 1705 (1975). A.D. Linde, Lebe-

103. T.D. Lee and G.C. Wick, Phys. Rev. DO 2291 (1974). T.D. Lee and M. Mar-

104. J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 1353 (1975). 105. G. ’t Hooft, “The background field method in gauge field theories”, lectures

given a t the XIIth Winter School of Theoretical Physics in Karpacz, Feb. 1975, and at the Symposium on Interactions of Fundamental Particles, Copenhagen, August 1975.

106. The general idea to use the Renormalization Group to relate the theory a t our mass scales of order 1 GeV to theories at very small distances (Planck length), also occurs in: H. Georgi, H.R. Quinn and S. Weinberg, Harvard preprint (1974), and M.B. Halpern and W. Siegel, Berkeley preprint (Feb. 1975).

107. G. ’t Hooft, lectures given a t the International School of Subnuclear Physics “Ettore Majorana”, Erice 1975.

(Oct. 1974).

dev Phys. Inst. preprint no. 25 (1975).

gulies, Phys. Rev. D11 1591 (1975).

203

CHAPTER 3 THE RENORMALIZATION GROUP

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . [3.1]

206

208 The Renormalization Group in Quantum Field Theony, Eight G d u a t e School Lectum, Doomrth, Jan. 1988, unpublished . . . . . . .

205

CHAPTER 3

THE RENORMALIZATION GROUP

Introduction to the Renormalization Group in Quantum Field Theory [3.1]

My first calculations of renormalization coefficients for gauge theories date from 1970. But at that time the delicate gauge dependence of the counter terms were not yet completely understood and it was hard to avoid making err01-s.'~ By 1972 however I learned about a procedure that would simplify things considerably. While at CERN I met J. Honerkamp14 who explained to me the secalled background field method, that had been applied by B. DeWitt in his early gravity calculations. The basic idea was due to Feynman. It simply amounts to writing all fields A* (which may include the gauge fields as well as possible scalars) as

where the "classical" fields A*)'' are required to obey the classical equations of motion, in the possible presence of sources, whereas Aav'J" are now the "quantized variables", so that they serve as the integration variables in a functional integral.

My first reaction was that this could only be a book-keeping device that would neither change the physics nor the kind of calculations one will have to do. But then Honerkamp told me about the trick: the gauge condition needs to be imposed only on the quantum fields, while it may depend explicitly on the classical fields. While fixing the gauge degrees of freedom for the quantum field, one may keep the expressions gauge invariant with respect to background gauge transformations. If we now renormalize the one-loop diagrams the counterterms needed will automatically be gauge invariant.

13The sign of the p-function for gauge theories was clear to me, already then, but at that time I was unable to convince my advisor of the significance of this observation. 14J. Honerkamp, Nucl. Phys. B48 (1972) 169.

206

The method enables one to produce a complete algebra for all one-loop p functions for all renormalizable theories in four space-time dimensions. We also use it in perturbative quantum gravity, and Chapter 8.1 gives another introductory text to this method. It does not work so nicely for diagrams with more loops because the overlapping divergences require counterterms also containing the quantum fields, so that the advantages disappear.

The paper of this chapter is an unpublished set of graduate lecture notes, which had to be polished a bit.

207

CHAPTER 3.1

THE RENORMALIZATION GROUP IN QUANTUM FIELD THEORY Eight Graduate School Lectures

Doorwerth, 25-29 January 1988

C. ' t Hooft

Inst i tute f o r Theoretical Physics

Princetonplein 5, P.O. Box 80.006

3508 TA UTRECHT, The Netherlands

1 Introduction

The values one should assign to the f r ee parameters of a quantum

f ie ld theory, such as masses ( m i ) and coupling constants of various

types ( h i , g,), depend on the renormalization procedure adopted.

Although the physical properties of these theories should not be scheme

dependent, the calculational procedures indeed do, in a r a the r

non-trivial way, depend on the subtraction scheme.

I t w a s proposed by Stueckelberg and Peterman'" in 1953 tha t one

should require f o r any decent renormalization procedure tha t its re su l t s

be invariant under the group of non-linear transformations among these

f r e e parameters, and they noted tha t t h i s requirement would imply

consistency conditions upon the renormalization constants. They called

t h i s group the "Renormalization group". One f inds t h a t modern

renormalizable f ie ld theories meet these requirements by construction,

and most of the requirements are rather t r i v i a l , with one exception.

Renormalization is always associated with a regularization

procedure, which needs some s o r t of cut-off parameter A . One subgroup

of the renormalization group coincides exactly with a redefinit ion of A:

h + K h , (1.1)

and one f inds tha t t h i s subgroup corresponds with the group of scale

transformations: we a r e comparing a theory with i tself with all masses

and all momenta scaled by a common factor K. J u s t because the renormalization procedure is ra the r delicate one

208

finds that even those coupling parameters A i that one would expect to be

scale-invariant since they are dimensionless, actually transform

non-trivially. And so it turned out that the renormalization group has

an important application in quantum field theory. Stueckelberg and

Peterman had been thinking about the general diffeomorfism group in the

multi-dimensional parameter space ; but all that remains nowadays is the

simple, one- dimensional group of scale transformations. Nevertheless,

the name "Renormalization group" stuck.

The renormalization group becomes particularly non-trivial when the

scale transformations that we consider are large. We get extra

information on the behavior of amplitudes a t extremely large or extremely s m a l l external momenta that would otherwise be difficult to

obtain. But this not only holds for quantum field theory. The same or

similar procedures can be applied to statist ical models. Here one also

uses the words "renormalization groupoorz1 although the rationale for

that is somewhat more obscure: a model is scale transformed and then

compared with the original. But it is sometimes observed that in a

rigorous treatment the scale transformation can only be performed

one-way, from smaller scales to larger scale theories, so that the group

is actually a semi-groupt3'! In these lectures we briefly discuss renormalization. For the basic

imput of our theory we do not need many details, such as the

combinatoric proofs of its c o n s i ~ t e n c y ' ~ ~ , the uniqueness of the results

as consequences of requirements such as causality and unitarity , and the independence of physically measurable quantities from freely

chosen gauge fixing parameters . A l l we want to know is the dependence

of "counterterms" on cut-off parameters A, and the relations between

"infinities" that follow.

t5,Bl

171

We then discuss a simple theory with one coupling constant and one

m a s s parameter such as and quantum electrodynamics. We note that there are three different kinds of theories: asymptotically free,

infrared free, and scale-invariant.

A44

Some calculations are done in the next chapters and then we discuss

the more general theories, with an arbitrary number of masses and

coupling constants.

Also for the more general case we want to compute the

renormalization group coefficients, but now we need a more powerful

209

method. We f ind that the various coefficients can be put i n a simple

algebra, as a consequence of various internal symmetries of our models.

One then needs to do only a few calculations to obtain the most general

s e t s of renormalization group coefficients, up to one loop.

Two-loop calculations a re much too lengthy to discuss here in any

detail , but we show that it is important to know the two-loop

coefficients if we want to give a rigorous definition of cer ta in

(asymptotically f r ee ) quantum field theories. We touch upon the point

raised in the beginning: maybe we can define models rigorously, but

proofs that allow us to sum the perturbative expansion without being

troubled by its possibly very divergent nature a re s t i l l lacking.

Quantum chromodynamics cannot be understood without the

renormalization group. Many attempts have been made t o improve our

formulation of t h i s theory, fo r instance by "resumming" its perturbation

expansion. Most of t h i s subject goes beyond the scope of these lectures

but we briefly indicate some sources of the bad behavior of the

perturbative axpansion.

2. Renormalization

Even though a mathematically completely rigorous description of quantum

f i e l d theory in 3+1 dimensions is not known, it is generally thought

that many different quantum field systems exist: models that combine the

requirements of quantum mechanics on the one hand, and special

re la t ivi ty on the other. In these lectures we will explain why, and

which diff icul t ies we encounter.

A central point in quantum field theory is that although we do not

know how t o construct rigorous models with f in i t e , non-trivial coupling

s t rengths , there a re many perturbative models. These a re a l l defined by

postulating the (renormalized: see la ter) coupling s t rengths , h i , to be

infinitesimal, and everything we want to know about the system is

expressed as perturbative ser ies in h i :

fthj} = f, + f , i X i *+ f ,jjhihj + . . . , (2.1)

in which a l l coefficients fNij (depending in some definite way On the

masses m j ) a re uniquely computable, but nothing is said about the

convergence or divergence of the ser ies . Indeed, quite generally one

. .

210

[ B I expects divergence of the form

IfNlj .. I 3 CodvN! , (2.2)

so t h a t f o r f i n i t e h i expressions (2.11, as they stand, are not

meaningful.

We w i l l not t r y to prove eq. (2.2) in these lectures. Here only we

note tha t , if t rue , it would imply that (2.1) may define f{A,) within a

t iny "error bar": it makes a lot of sense in perturbative f ie ld theory

t o cut off the se r i e s (2.1) as soon as the N + l s t term becomes larger

than the Nth. This happens when

at which point

and if t h i s is a good measure fo r the uncertainty in f then indeed

such models may be quite acceptable physically as soon as the coupling

s t rengths h i s t ay reasonably s m a l l .

A standard way to construct a perturbative quantum field

theoretical model is to start off with a Lagrange density 2 and then

construct functional integrals fo r the wanted amplitudes. One then

discovers that the "bare" coupling constants as they occur in the

Lagrangian are not the appropriate parameters t o use in the expansion

series (2.11. The coefficients f would then be infinite.

ASi

N i j . .

We w i l l now explain what t h i s really means. Imagine that we

introduced some s o r t of cut-off in our theory. This could be done in a

number of ways. For one, we could simply postulate that no states In

Hilbert space a re considered in which any particle has spacelike

momentum k with Ikl>ba,, , f o r some large number ha, . In t h i s case

we require uni tar i ty only to hold rigorously fo r S matrix elements

out(kl,k2 ,... Ip1,p2 ,... ) i n fo r which the total energy, kl0+kzo+ ... =

pl0+pzO+ ... does not exceed 2Amar , because only then we a re guaranteed

tha t there a re no intermediate states IQ) containing any of the

forbidden particles i n the uni tar i ty condition

- - - - - - #We use St i r l ing 's formula: N! 3 .

21 1

( 2 . 5 )

This would make the theory f in i t e but no longer Lorentz-invariant. I t

would be uni tary up till energies 2 k a x . A better way t o make the theory f in i t e is t o add "unphysical"

particles. These a re particles that contribute to the uni tar i ty relation

( 2 . 5 ) intermediate s t a t e s 10) that may have "negative metric": the

sign of IQ)(QI is wrong, as if

(The normalization of a one-particle state is determined by the value,

and s ign, of the residue of the pole in the propagator).

Loop integrals can then be made to converge so f a s t that the

finite-momentum cut-off ha, of the previous paragraph can safely go

to inf ini ty such tha t the l i m i t exists, which now is Lorentz invariant

by construction. I f we choose the masses of all these unphysical

particles t o exceed a cer ta in number A,, then we also have uni tar i ty

up till energies hpv or 2hPV because energy conservation would

forbid intermediate s t a t e s I Q) containing these "sick" particles. The

subscript PV stands f o r Pauli and Villars who first proposed t h i s

regularization me t hod' *' . A thi rd way t o make amplitudes f in i t e is much more subtle and w i l l

be used very often here. We imagine changing the theory a l i t t l e b i t by

postulating that the number of space-time dimensions, n , deviates

s l ight ly from 4:

I el

n = 4-E , c infinitesimal. (2.71

I t is not possible t o give a mathematically rigorous definition of what

t h i s means unless we formulate everything in perturbation expansions (or

if we abandon translation invariance, see ref 'lo] 1. In Feynman diagrams n only occurs in the closed loop integrals

over space-time momenta k :

Sdnk + Sd4-"k , (2 .8 )

and f o r large enough n such integrals may be well-defined, if the

212

integrand contains only explicit external momenta p i that span a d < n

dimensional subgroup of Minkowski space :

f ( n - d ) / 2 m

0

n-d-1 2 r( ( n - d ) / 2 ) Jddkl Idr 2 r f{k,,pi;r } . (2.91

The subtle point is that the integral (2.9) may diverge just about

as badly as the original integral with n integer, but this time there is a unique way to define its "finite part" a s long a s n is non-integer. In

most cases this f inite part can simply be defined as the analytic continuation of the integral from n values where it converges, but

sometimes it is also infra-red divergent because of poles in the

integrand at finite or zero values of kl and/or r . In the latter

case one can still define the finite part by splitting these divergent

pieces apart , or by repeated partial integrations.

What one now observes is that these "dimensionally regularized"

expressions, though finite at generic n , may still produce poles a t

n + 4 . They are of the form 1/c (for one-loop diagrams), or 1/c2 , l/c3 , ... at higher loops. The divergent term l/c plays a role very similar to logbax or logApV in the other regularization schemes'.

Again, at sufficiently small E unitarity is only very slightly violated,

t h i s time because the phase space integrals (which always converge,

unless there is some lnfra-red divergence which must be handled

separately) w i l l approach the physical values.

Coming back to the coefficients f of eq. (2.11, one now observes that, quite generally, they diverge, either as ELpv m or as

1/& + m . So a t first sight it seems that these theories are

inconsistent.

N i j . .

However, the coefficients A , that we started off with are not

directly physically observable. Throughout, we w i l l adopt the philosophy (renormalization Ansatz) that one is allowed to choose these arbitrary

numbers to vary with A,,,dx or Apv or E : for instance, in the

'In those schemes one may also encounter higher divergences going as a power of A (linearly or quadratically). These have been eliminated in the dimensional scheme. One might say that our "finite part" procedure eliminated them.

213

dimensional procedure we substi tute into the ser ies (2.1):

(2.10)

where now X R i a re defined to be t ruly f ini te constants, independent of

E (or A if another regulator w a s chosen). The coefficients a‘:’ a re I Vl of second o r third order in hR and the higher order coefficients a

A R . of successively higher order in

A t first s ight it seems odd that Asi a re both inf ini te and

infinitesimal. We have to keep in mind that eq. (2.10) only makes sense

in the l i m i t A + 0 f irs t , then E + 0 . Indeed, the renormalization

group w i l l enable us to replace (2.10) by an expression in which the

l i m i t s may be reversed.

R

Since both hR and As a r e to be considered infinitesimal the

perturbation expansion in hR is equivalent to the one in As . But it

is important to note that since the a‘y’ a re higher order in h the

expansions a re term by term different.

There is lots of arbi t rar iness in (2.10). A l l we want is that any

f in i t e value fo r hRi produces some f ini te values fo r the physically

measured quantit ies f { h B j ) in the l i m i t E + 0. The se r i e s of

functions fNij has poles at E=O and s o does the ser ies (2.10). We

want to define the se r i e s (2.10) in such a way tha t , when substi tuted

into (2.1) we get new expansions in hRi that a r e all f r ee from

divergences at E=O . If t h i s is possible the theory is called

renormalizable.

To find out whether a theory is renormalizable is in principle very

simple:

( i l Write down all interaction terms in the Lagrangian tha t

agree with the chosen symmetries of your model.

( i i ) Expect al l possible terms to be required in the ser ies

(2.11) fo r the bare coupling constants, as long as these

( a ) have the correct symmetry transformation rules under

whatever symmetry that w a s imposed on the model, and

( b ) have the correct dimension.

(.i.fi) I f , following the above rules , a cer ta in subset of

couplings does not get renormalized once they are put equal

to zero then, and only then, we can leave such couplings out.

214

Requirement (l ib) follows from the assumption that the infinite par ts of

a[ r 1 cannot be chosen freely (two different choices cannot possibly both

yield a f in i te f 1, so they should follow from the s t ructure of the

theory, but then they should not depend on our choice of units f o r

lengths and masses.

In general, if s p a c e - t h e has 4 dimensions, only a f in i te number of

coupling constants with positive or vanishing dimension can be

constructed. But then, if we choose all coupling constants of positive

or vanishing dimensions then s tep (iif) allows us to leave all the

others out. Only a f ini te number of coupling constants a r e le f t . Eq.

(2.10) then only contains a f ini te number of possible terms at each

order. They should suff ice to render everything f ini te . They do. One

condition is tha t there are no "anomalies", but th i s is just the obvious

(and very important) requirement that also our regulator procedure,

whichever we chose, obeys the chosen symmetries.

The student should be cautioned that the above is not a proof of

the renormalizability of field theories, but must be seen as a summary

of the resul t of such a proof, which can be found in the

l i terature[ 4-71 .

3. theory To i l lustrate our methods we could either take the simplest theory which

is jus t a single self-interacting scalar, called theory, or we

could take more complicated cases containing fermions and/or gauge

fields. The disadvantage of the pure scalar theory is tha t there is no

field renormalization at one loop, which makes it a l t t l e too simple. We

w i l l take t h i s simplest case anyway, but we'll do as if there is a f ie ld

renormalization.

The Lagrangian is

Here J(x) is a space-time dependent source insertion which we put in

there f o r technical convenience. Later it w i l l be put equal t o zero. The

sub- or superscript B stands for "bare" or unrenormalized. These

quantit ies are not directly observed. We write

(3.2)

215

where hR , mR and $R are t ruly f ini te . The normalization of J is

not yet f ixed.

Physically observable features of the model are reveiled by

computing vacuum expectation values (OIAx) 10) , (OIJ(xl)J(x2) 10) , etc. The first of these is zero. The second gives us the "physical"

propagator. In diagrams Fig. 1.

X x + x+x + x- x + .

Fig. 1

The f i r s t diagram f o r the self energy insertion gives only a momentum-

independent contribution. It can be removed by a m a s s renormalization.

The second diagram has a non-trivial momentum dependence but is of too

high order in he f o r our considerations. In theories with fermions or

gauge particles we have also diagrams of the kind given in Fig. 2

We'll now pretend that there is a diagram of t h i s kind in our example.

In 4-E dimensions the integral would typically give (the k2 in the

numerator has been put there just f o r dimensional reasons; in more

elaborated theories the expressions are more complicated) :

- - k 2 2 2 2 2

r2(p) = (2n)e-4f(XB/2)~d'-Ek ( (k+p) + m - k ) ( k + m -ic)

- ( k - x p I 2 - 1

2 2 = ( 2 n ) E - 4 i ( h B / 2 ) ~ d ~ ~ d 4 - E k

0 ( k 2 + m 2 + x ( 1 - x ) p -ic)

216

(3.5)

Here, 7 is Euler's constant: I'(c/2) = WE: - 1 + O ( c ) . The

insertion of the integral over the variable x is the famous "Feynman trick". Again, in our scalar theory the diagram of Fig. 2 actually does

not occur. We have only the "seagull" diagram in Fig 1, which gives the

p independent expression

For pedagogical reasons we w i l l pretend, temporarily, as if we are dealing with the more complicated expression (3.5) in our scalar field theory.

There are two things in eq. (3.5) that are of importance to us: the

fact that there emerges a l /c term and the fact that we get a logarithmic function of p2 . First look at the 1/c term.

Let us substitute eqs. (3.2-4) into our Lagrangian. We get

+ higher orders. (3.7)

In the counter t e r m s the distiction between B and R w a s neglected

because that can be absorbed in the higher order corrections. The source

term was not included because we are free to choose its normalization. We now add the correction terms in (3.7) to the diagrammatic rules

treating them as if they were new interactions.

We see that extra terms should be added to the self energy

expression of the form

- (2c"'/c)(p2+m2) - 2b111m/c . (3.8)

217

So if we choose

(3.9)

b[” = - A B / 2 idx (2-2x+3x2) mB , (3.10) (4nI2 0

then the divergences of the form 1/c2 in eq. (3.5) a re cancelled fo r

all momenta p . If we work with (3.6) we must choose

(3.11)

Notice that the source is now coupled to #R , not @ . We discuss

the choice fo r a[’’ la ter .

Now consider the logarithm in (3.5). I t seemed to come together

with the 1/c term, and, as we w i l l see, it is indeed closely related.

The important point is t ha t , in dimensionally regularized expressions,

the quantit ies inside the logarithm a re not dimensionless! Clearly th i s

is rather absurd, because if we would change our units of m a s s and

momentum these expressions would change. However, such changes can

easily be absorbed in finite renormalization terms in the Lagrangian, so

indeed they w i l l not be physically observable. In the other

regularization schemes (momentum cut-off or Pauli-Villars) one always

gets terms such as

(3.12)

replacing the logarithm of (3.5). One may now understand why we compared

1/c with logh . The renormalization procedure in these older schemes

is j u s t as in the dimensional scheme: we put counter terms proportional

to Alogh (and higher powers of t h i s to remove the higher order

divergences) in the Lagrangian.

In an “apparently scale invariant“ theory, that is, if m=O , one

would at f i r s t sight expect (3.5) to scale like p 2 . But th i s is not

so, because of the logarithm. I t is unbounded above and below, so at a

cer ta in value of p2 the self energy (3.5) w i l l vanish, whereas for

218

very large or very small p2 the logarithm w i l l dominate. But because

the logh terms can be removed (or altered) by renormalizations, there

is also the freedom to choose at w i l l the value of p 2 where the

self-energy vanishes. This is how one may first observe that a scale

transformation of a theory maps the theory into i tself only if th i s

transformation is accompanied by (finite) renormalizations of the kind

(3 .3 ,4) .

Also h needs renormalization. This is seen when we consider the

one-loop corrections to the scattering diagrams (Fig. 3)

= x+x+x a b

Fig. 3

C

Consider one of the terms in Fig. 3, €or instance b . It is given by the

integral

(3.13)

Here p is the momentum through the horizontal channel in Fig. 3.

The divergence at c=O can be removed, as before, by plugging in eq.

(3.21, which would give in addition to (3.13):

(3.14)

Therefore we choose

21 9

a [ * ’ = 3h2/16r2 , (3.15)

where the factor 3 came from the fac t tha t we want to cancel the

divergences of the total of the three diagrams in Fig. 3. Again, the

logarithm determines the scaling behavior of the theory when the m a s s

effects become unimportant.

4. analysis of the scaling behaviour of dimensionally renormalized

theories

When E is an i r ra t ional number dimensionally regularized amplitudes can

be uniquely defined. With this we mean that the mathematical procedure

to extract the “f ini te par t” of an integral in any Feynman diagram is

unambiguous. The deeper reason for this is that the integrands, buil t

out of propagators and vertices are allways rational expressions in the

momenta to be integrated over. Integrals over pure polynomials and pure

powers such as

can all be postulated to be zero. Convergent integrals are postulated to

have their usual values. This way one succeeds in defining uinambiguous

subtractions f o r all power-like divergences. Only the logarithmically

divergent integral ,

r d x / x , 0

(4.2)

cannot be made f in i te because infrared divergences (x + 0) a r e entangled

with u l t r a v i o l e t divergences (x + LO). If E is i r ra t ional there a r e no

logarithmic divergences (for half integer, third integer E logarithmic

divergences show up at two, three loops).

Consequently, all relations between diagrams tha t can be proved via

par t ia l integration, redefinitions of integration variables and

combinatorics of diagrams, will hold automatically, and we need not

worry about omitted boundary terms in integrals. This includes all Ward

and Slavnov-Taylor identities, but it also includes s c a l i n g relations

between diagrams.

Thus, at E * 0 all “naive“ scaling relations obtained by counting

dimensions hold exactly.

220

What a re the dimensions of the various parameters at E f 0 7 In a

functional integral we exponentiate the action S , so its dimension

must be chosen to be zero. We have

S = Id4-'* !!(XI , (4.3)

so the Lagrange density J? has dimension E-4 . Let us write

J? = p4-" !e0 , (4.4)

where p is an arbi t rary quantity with the dimension of a m a s s (unit of

mass) and J?, is a dimensionless object.

-&(&$I2 , carries the dimensionless

unit 4 , therefore the dimensions must match. So we find the dimension

of 4 . We w r i t e

The kinetic term in J? , being

9B = Y2 90 * (4.5)

Similarly,

(4.6) B B E B m = pmo , hB = p ho . Thus, although A is scale-invariant in four dimensions, hB in

4-E dimensions is not. T h i s should be kept in mind when one uses eq.

(3.2).

From now on we should insist that AR is kept dimensionless, but

then eq. (3.2) can only be written down if we have some quantity p to

our disposal. I.( w i l l be chosen to be some arbi t rary mass; in fact it

w i l l be related to that value of p2 for which the self-energy or the vertex-correction vanishes. Thus we now write

I t is important to know that the series (4.7) are constructed in

such a way that finite expansion coefficients are obtained for all

choices of the (infinitesimal) quantity AR . Therefore it should do no

harm if we substitute in (4.7)

R' hR = A + el& + e2e2 + ... ,

221

R R' m = m + flc + f2E2 + ... ,

but t h i s does give an entirely different ser ies :

111 lll I - V l ' V + AR" + a[Vl',-V ) , hB = p E [ c a E

V = l v = 1 (4.9)

lll lll

(4.10)

and s i m i l a r expressions f o r the bare fields +B , where the new

are functions of AR' and m , quite coefficients a and b r V 1 as functions of AR and different from the original a

m ; also and m donot coincide with A and m .

B I-Vl' v R" + b[Vl'E-V ) m = p [ C b E + m

v = 1 V = 1

R' IVl' and b [ v l '

IVl

R AR" R" R R

Clearly eqs (4.9) and (4.10) a re a more general choice f o r the bare

parameters than (4.7). The positive powers of E a re not necessary. In

(4.7) the coefficients fo r the positive powers were chosen t o vanish.

They can allways be adjusted t o zero by subst i tut ing (4.8) with

judicious choices for ev and fv . Since p is essentially our unit of mass, a l l logarithms obtained in

our integrations will have entr ies made dimensionless with powers of p :

log((p2+m2)/p2) . This is why p w i l l determine the order of magnitude of

the momenta at which our loop corrections w i l l (nearly) vanish. A t those

momenta perturbation expansion w i l l converge relatively rapidly (at

least the f i r s t few terms). If we want to understand how the system

behaves at momenta very f a r away from these we have to make the

t ransi t ion to other p values. Therefore, it is of importance to consider

scale transformations :

p = p ' ( l +u ) , u infinitesimal. (4.11)

This tu rns expressions (4.7) into

V = l m

(4.13)

Notice that eq. (4.12) is of the form (4.91, not (4.7). In order to

R IVl R m = p'( l+u)[mR + c b (A ,mR)c-') . v = 1

222

bring it into the form (4.7) we have to make a transformation (4.8). Let

us t ry

X R = (1-crc) SiR . (4.14)

This removes the positive powers of E from (4.12):

(4.15)

['I A stands for aa["/8h , but also the expression for the bare where a

m a s s changes :

(4.16)

Note that we had to write the coefficients as functions of the newly

chosen iR , which is now kept fixed a s E += 0 .

The series (4.15) and (4.16) and a similar one for the fields 4 which we won't need for the moment, are now to substitute the old eqs

14.7). We see that the zeroth coefficients are

(4.17)

I t is essential that, with (4.141, we are free to keep either hR

or K R fixed as E += 0. They are both finite. When p is chosen to be

the unit of mass X R is fixed, and when p' is chosen we'll decide to

keep hR' fixed, which implies that X R is fixed. We wish to compare

the two descriptions in the l i m i t x R =

A R . The conclusion is that when we make the (infinitesimal) transition

from p to p' with (4.111, we have to replace hR and rn by:

R -R 111 rnR' = rnR + urn - oh b .

c=O , so in (4.17) we may put

R

(4.18)

223

R R R a I I I R R R u ( h , m ) = A - b ( A , m l - r n . ahR

(4.19)

(4.20)

The function p(AR,mR) describes the anomalous dimension of AR ; the

function u also has a canonical part . From the above it follows that

these functions can be computed from the counter terms needed in

dimensional renormalization. A t the one-loop level it is the only

counter term; at two and more loops we also have the coefficients

, which govern higher poles E , E , a

etc. One w i l l f i r s t have to substract these higher poles before being

able to determine the coefficients of the lowest ones. S t i l l , u and p are uniquely prescribed as perturbation expansions in

[21,[31, ... bt21,131,. . . -2 -3

AR.

In theory we have in lowest order in A (see (3.13):

P(A) = .I1’ = 3A2/16n2 ;

and from the coefficient b“’ in (3.111,

(4.21)

(4.22) 2 u ( A , m ) = -m + Am/32n ,

We have not yet written down the complete ser ies tha t replaces

(4.7) a f t e r the replacement p + p’ . at E f 0 we substi tute

R AR = hR’ + pu - &A u ;

mR = mR’ + uu

We get from (4.7):

(4.23)

(4.24)

131 dour 01 di f fe rs by a factor 2m from Symanzik’s a notation usually he puts u=l, which we are here not allowed to do.

. But in his

224

and

m

V = 1

3 -V IVJ R' R' R' I V + l l + m = p'[mR' + C E [b (A ,m 1 + cbfVJ - uh b

+ pu b[I1 + au b':' I ) . (4.25)

Note that, the way they are defined, all coefficients a , b , hR

and m are dimensionless. Obviously, the functional dependence of

a and b["' of hR' and m should be independent of p' . As

argued before, the series (4.7) is the only choice of counter terms that

should lead to a finite set of amplitudes in the l i m i t c 0 . Therefore, all "correction terms" proportional to u should cancel. Those

terms proportional to Q that are of zeroth order in c were made to

cancel by construction, but for the higher order ones this observation

R

R'

-1

leads to important identities between the coefficients a ["I and bEVJ .

R 2 Since for u > l the coefficients a ["I are at least of order (A )

the equations (4.26) completely determine the higher order poles once

we know the lowest order ones.

In ref[12 'we stated that usually the coefficients a ["I are

independent of the masses m , and the b[" are proportional to m.

This follows from simple power counting arguments when the divergent

parts of integrals are considered a s functions of the momenta and the

masses m . However in some more complicated models not only the masses

and coupling constants are renormalized. and not only are the fields # multiplicatively renormalized, but one may also find renormalizations of

the vacuum expectation values of the fields, so that one should

substitute

# + b + C , (4.27)

where C contains poles in c . For instance, if there is a #3 term next to the #4 term in the Lagrangian one might be inclined to adjust C so as to remove any terms in the Lagrangian that are linear in # . But in this case the renormalized #3 and tP4 couplings are defined in

225

a more delicate way and then the above power counting argument breaks

down, because a condition on C such a s

<#J> = 0 , (4.28)

introduces logarithmic dependence on masses in the theory.

Consider now the formal sum

m I V l R R hB = hR + c a ( A , m )c+ ;

U=l m

mB = mR + 1 b[V'(hR,mR)&-" . v= 1

Putting

(4.29)

(4.30)

We f ind that (4.26) also holds f o r u=O , and (4.26) can be rewrit ten as

I " R R ~

am + a(h , m + E h = 0 ;

I " R R ~

am + a ( h , m ) - R + 1 m = 0 .

(4.32)

(4.33)

Equations of t h i s so r t a r e solved by considering t ra jector ies i n R the space spanned by the parameters hR and m .

dhR(t) - - - f j ( A R , m R ) - E h R ; (4.34)

- - (4.35)

d t

d t

Then

B d B 8 h B ( t ) = -ch ; m (tl = - m . Z

Let's take the case (as in most two-parameter theories)

R 6 = B, (hRI2 ; CL = -1 + Alh .

Then the dependence between hB and A R is

B CAR A = h R - c/B2 '

but we also require that

(4.36)

(4.37)

(4.38)

hB = hR + 0 ( h R l 2 , (4.39)

226

so

(4.40)

This expression has been computed without special conditions on the

order of the l i m i t s hR -. 0 and c -. 0 . So it replaces the ser ies

(3.2) or (4.7) when c S AR . Note tha t in that case hR becomes

negative unless Bz<O . In the l i m i t &=O we have

B A R C C = - c / B Z ; h = R c - Bzh

2 R hB = - c / B Z + c 2 / ( B 2 h ) + ... , (4.41)

so it starts out with a value independent of hR . Actually, theories with negative values f o r the coupling constant

a re in trouble: the Hamiltonian w i l l not be bounded from below. But

because AB is so tiny these diseases (havivg to do with tunnelling out

of the unstable vacuum) are not noticed at any f ini te order in

perturbation expansion. Also we must s t ress that (4.40) is only valid as

long as hR and hB s tay small, because only then the perturbative

expansions have some validity. So if B 2 > 0 the negative value of (4.41)

cannot be trusted because we had to integrate the differential euations

across the pole, where it cannot be correct. Now we could have taken the

contour in t space to be complex, avoiding the pole, but this would

require us to consider complex scale transformations and complex values

of the coupling constant. where the Hamiltonian is also not

well-behaved. Thus, eq. (4.41) has a formal validity only when both E

and hR are infinitesimal, in which case their relative s izes a re not

restricted.

We do see tha t the sign of Bz is very important. If Bz<O, so

tha t all bare coupling constants at c=O are positive, we cal l our theory

asymptot ica l ly free. Let us consider this case for a moment, and assume

tha t fi depends only on hR , not on m (which is the case in most

theories), such that

R

with g(0) = B, < 0 . One then proves from (4.32) , R

(4.42)

and (3.39):

1 h - &/m

hB = hR exp(-[ dh 1 (4.43)

227

Let us take

- B(A) = B2 + B3A + B4A2 + ... ,

so tha t

E / E = e/B2 - eB3A/B,2 + ... ,

(4.44)

(4.45)

then t h i s t u rns (4.43) into

which f o r s m a l l E becomes

2 A B = -&/B2 +* loge +e2(-- - % log XR +0(1)) + O ( E 3 loge) (4.47)

B~~ BZ2AR B2

The f i r s t two terms of t h i s a r e independent of AR ; these a re

completely determined by the mathematics of the system: only with a AB

that approaches zero th i s way when ~4 the theory w i l l give f in i t e

amplitudes. But the E t e r m is arbi t rary. The higher order terms,

E loge , etc . a r e therefore not important. They could be absorbed by

2

3

AR . We also note that eqs. (4.34) and (4.35) a re is the s a m e equations

f o r AR and mR as eqs (4.19) tha t describe the response of AR and

m to an infinitesimal scale transformation p + p’=p(l+a) , when c=O

. For t h i s reason the l i m i t E + 0 in some respects corresponds t o a

l i m i t towards sho r t distance scales, j u s t like the l i m i t A -+ m f o r a

more conventional regulator method.

R

228

The renormalization group equation

W e have not yet discussed the renormalization of the fields # ( X I as

it follows from eq. (4.7). I t should be treated in exactly the same way

as the mass renormalizations. Thus, in addition to eq. (4.33) we have

Here,

The differentiation a/a# is lacking of course because the field + was outside the brackets in (4.7).

Related to these equations is the change in the definition of the

when the unit of m a s s p is changed. A transition renormalized field

(4.11) is associated with transformations (4.19) and: +R

The combined transformations (4.111, (4.19) and (5.3) should leave

the entire description of physical features invariant

(renormalization group invariance) :

where TR = ($R(pl)6R(pz) This can be rewritten as

However, remember that

... $R(pH)) is an "amputated" Green function.

R rn , and #R in (5.31, were defined to be

dimensionless. But it is tempting to redefine them such that they have

their canonical dimensions:

pmR = mren (physical m a s s ) ;

p#R = 6'"" (physical field) . (5.61

We then only keep the "anomalous parts" (coming from one loop and

higher) of OL and y :

229

R w N (5.7) g j p ( a + r n ) = a ; r+l 7 .

This is the renormalization group equation as i t follows from the

rules of dimensional renormalization. The renormalization group

equations used by Callan and Symanzik[13' a r e equivalent but they

look a l i t t l e bi t different. For Symanzik, p is the mass parameter

(hence h i s 01 is identical to one) and he does not have our a term, in

stead of which h i s equations have a non-vanishing r igh t hand side.

1111

-

For our purposes we f ind the " renormalization group f low-equations"

or Gell-mann Low equations"41, (4.111, (4.19) and (5.3) more

informative: if

then

(5.10)

N

(as stated before, in most cases f3 is independent of rnren and a is

linear in mren).

Three important cases must be distinguished:

(i) theories in which f3 (for at least one of its coupling

constants) s t ays positive. This happens in most models; in

all 4 dimensional theories that donot contain non-Abelian

gauge f ie lds (such as A+4 , QED, and Yukawa theories). In

t h i s case the solution of (5.10) must show a strong, possibly

divergent, effective coupling constant at high energies.

Suppose

@(A) = bh2

Then we would get

(5.11)

(5.12)

230

(5.13)

This is the so-called Landau singularity. The student is

advized to study this singularity in the "chain of bubble

approximation" for scalar field theories. The chain of

bubbles represents faithfully what happens in an N

component scalar field theory in the l i m i t N + OD , AN

fixed . (if) Theories in which a l l /3 coefficients s ta r t out negative.

This may happen in non-Abellan gauge theories only (it is

guaranteed only in a pure non-Abelian gauge theory). Then,

according to (5.22) and (5.13) A(p) goes to zero

logarithmically. This means that at high energy the

perturbative expansion becomes better and better! We s a w that

in this case also the bare coupling constants formally tend

to zero. There is a general consensus that theories of this

type must have a rigorous mathematical footing (see

introduction) but this could be proven only in very special cases[1s1. The phenomenon is called Asymptot ic f reedom and

was perhaps the most compelling reason for assuming a non-Abelian theory such as QCD €or describing strong

interactions. There is no Landau ghost in the f a r

ultraviolet. There is one in the infrared (at very low values of p 1, but one may well imagine that this region should be

properly described by integrating out the field equations

defined at high p values, so that the Landau ghost may be

just some sor t of physical bound state.

(ill) In some very special cases one might find that P(A) = 0 at

all orders in perturbation theory. An example is N=4

supersymmetric Yang-Mills theory and superstrings in 2

dimensions Such a theory must be s t r ic t ly scale-invariant.

Quite generally one then also finds that such a theory must

also be conformally invariant.

One must be careful to observe that the form taken by the

renormalization group equations and the details of the expressions for

the a and /3 functions depend a lot on details of the regularization

scheme. In Pauli-Villars logh plays the role of E and much of the

231

rest goes the same way, except that many higher order coefficients w i l l

look different. A simple way to see this is as follows.

Suppose that two schemes choose a different convention for the

finite part of the subtraction constants. Consequently,

R R’ h = h + f ( h R ’ ) ,

where f(h) = O(h2) , a given function. Now

2 A R = f3(hR) = b2(hRI2 + b3(hRI3 + ... ; ac a R’ R’ R’ 2 R’ 3 + - A = p ’ ( h 1 = bz’(h 1 + b3’lh 1 ... . ac

Assume for simplicity that

(5.14)

(5.15)

f(A) = ch2 . R’ I t follows that (write h = h ):

2 aa hR = b2(h2+2ch3+c2h4) + b3(h3+3ch4+ . . . I + b4A4 + ... =

a 2 = (l+Zch)& = (1+2ch)(b2’A + b3’A3 + b4’h4 + ...) ,

(5.16)

(5.17)

or

(5.18) 2 b2’ = bz ; b3’ = b3 ; b4’ = bl+cb,+c b2 ; ... .

Observe that b2 and b3 are “universal“: they are scheme independent; bz

because it is the lowest order term, depending on the leading divergence

at one loop, which is the same for all regulators; b3 is universal

because c happens to cancel out. This two-loop coefficient for the

beta function ceases to be universal if we have more than one coupling

constant“”. The student is suggested to find a plausible substitution

(5.14) in that case such that the two-loop coefficients change.

Having the freedom to redefine our coupling constants a s suggested

in (5.14) we could decide to choose c such that b,’=O . I t is not

hard to convince oneself that if b2+0 a substitution of the form

(5.14) with f ( h ) = O ( h 3 ) can be found in such a way that

b , ’ = O , n = 4 , 5 , . . . m. (5.19)

This f(h) is unique apart from scale transformations which obviously

232

leave the functional form of p ( h ) invariant.

The words “min ima l subtraction” have already been reserved for the

dimensional renormalization counter terms (2.10) or (4.7) (which are

unique in contrast with the more arbitrary choice (4 .9)) . We might call

the choice that gives (5.191, implying the exact 6 function

p ( h ’ ) = b2(k’l2 + b3(h’I3 (no further terms) , (5.201

an “ultra-minimal’’ subtraction.

A word of caution is of order. The original beta function f3(hR)

is sometimes believed to have a fixed point:

R so that if h = ho the theory becomes scale-invariant. Of course this

fixed point is totally unrelated to a possible fixed point of the

truncated series (5.20). A t such a point hR = A. the realation (5.14)

w i l l tend to show singulariries. But we do learn that the question

whether or not f3 has a fixed point may depend on how k is defined

when it becomes large. I t is argued by many researchers that the

question whether the theory is scale lnvariant should not depend on such definitions, and that therefore the existence of zeros of the f3

function should be scheme independent. But we s t ress that it is not known how to give any rigorous definition of 4 dimensional field

theories beyond the perturbation expansion. Indeed there is reason to

hope that a definition can be given precisely in the case that there is

not such a zero, when the theory is asymptotically free @ ( h ) < O ) . In

that case condition (5.20) determines h uniquely apart from scale

transformations.

6 The higher order terms in the many parameter case

The procedure of chapter 4 and the equations (4.26) can be generalized

to an arbitrary number of parameters hk which may be found to have dimension

in n=4-& dimensions. Some of these may be masses (for which P k

vanishes), others are three- or four-particle coupling constants (which

233

have dk=O ) . We could even include fields +, @, or gauge f ixing

parameters a . As before, we write m

X,B = pDk[hkR + c a : ’ ~ { ~ R } c - u ] , (6.2) v = l

where in the brackets { } a re those objects on which these quantit ies

a re allowed to depend. The renormalization of physically observable

coupling s t rengths A should not depend on gauge fixing parameters a

o r field s t rengths 4, 1,9, so these form a se t by

Consider the transformation

p = p’(l+(r) .

The equivalent of (4 .121 , ( 4 . 1 3 ) is

R A: = ( p ’ I D k ( l + d ( k ) o ) hk + &p(k)hk (r

[ R

m + c (a:ul+p(k)a:v+l l ( r )&-u] .

V = l

Let us w r i t e

themselves.

+ p ( ) ( ) a y ( r +

(6.3)

(6.4)

where the functions P k must be chosen such tha t the first term in

(6.4) is j u s t hk . The t3 t e r m should remove the positive powers of

E in (6.4).

R’

Eq. (6.4) t u rns into

I11 r 1 1 R ’ hkB = (p’IDk(I+d(k)U‘) hk + P k c + p(k)ak 0‘ - ak, t p ( t ) A t +

t

[ U + l I

[ R’

m

+ c + p ( k ) a k [U+l I 0‘ + c &a:f:q - a k , t p ( l , h t R ’ ( r ) E - ” ] *

V = l t t

(6.6)

I U l Here, a k , t stands f o r aaLul/ahtR . Thus we f ind how to express P k in terms of the counter terms ak :

(6.7) R I 1 1 @ , { A R } = -d(k)hE + c a::: p(t)ht - p ( k ) a k #

t I U l R’ and since ( l+d(k) (r )ak { h } in (6.6) should be the same functions as

a:”{h) in (6.21, we have the identities fo r ak : I U l

234

replacing our previous eqs. (4.26). In most cases the coefficients

a k f V 1 are polynomials of very low degrees in the masses and other

dlmensionful parameters, which simplifies the use of equs. ( 6 . 8 ) .

7. An algebra for the one-loop counter terms

The first terms of the a, 6 and 7 coefficients a re by far the most

important. They determine the leading scaling behaviour of the theory,

and whether it is asymptotically free or not. The counter terms needed

to make the one-loop Feynman diagrams f ini te a re particularly simple to

compute, although, if you start out doing first the complete one-loop

calculation it can still be quite cumbersome. So it w i l l be of help tha t

a universal "master formula" can be written down tha t gives us all

necessary one loop counter terms once and f o r all.

A l l 4 dimensional renorrnalizable field theories can be written in

the form

where summation over repeated indices a, p, v , i, j , is implied. @ i

a r e a bunch of scalar f ields and #i are (possibly chlral) fermions.

G a re the gauge fields, PV

(7.2) C' = a A' - a A' + P bc A ~ A ~ b c , pv P V V P

where pbc is any combination of coupling constants and s t ructure

functions tha t sa t i s f ies the Jacobi identity f o r some compact gauge Lie

group. The function V{@) is any quartic polynomial in the f ie lds @

such as +n1'@~/2 + Ad4/4! in a simple scalar f ield theory leaving open

the question whether or not spontaneous symmetry breakdown occurs. The

derivatives D a r e covariant derivatives, P

where again possible coupling constants were absorbed into the

235

coefficients T and U . These must sa t i s fy the usual commutation rules

(and the U may be spl i t into a lef t handed and a r ight handed par t ) :

(7.4)

The functions S and P must be l inear in + , f o r instance, S=m++g$

in a Yukawa theory. And of course the ent i re Lagrangian must be

invariant under the local gauge group.

We can now add to th i s Lagrangian a counter term A 2 that can be

chosen such tha t it absorbes all (one loop) infinit ies. The way to

construct A 2 is as follows.

We are interested in the complete s e t of one-loop diagrams. For instance we could inser t (gauge-invariant1 source terms in the

Lagrangian and consider the dependence of the vacuum-vacuum transit ion

amplitude on the strengths of these sources, expanded to one-loop order.

A practical way to compute this amplitude is to s h i f t the f ie ld

variables :

(7.5)

where the "classical" fields a re c numbers satisfying the classical

equations of motion (with the source terms in) , and the "quantum" fields

a re integrated over. Let us denote the bosonic quantum fields together

as {pi) . The Lagrangian is now expanded in these:

Now the fac t that the classical f ields sa t i s fy the equations of

motion implies that , when integrated over, the linear term g1

vanishes. The quadratic term generates propagators for the quantum

f ie lds and vertices with two quantum fields and other background fields.

The higher order terms neglected in (7.5) generate vertices tha t cannot

contribute to one-loop graphs, so we ignore them.

We rewrite the last term of (7.61 as

(7.7)

Here, N and M depend on the classical background fields. We w i l l see

236

to it that W does not depend on the background fields.

In general, Lagrangian (7.7) is a gauge theory and should be

quantized and renormalized according to the usual rules. A gauge-fixing

term w i l l be needed, and ghost f iels. As a gauge fixing term one could

for instance take

with

(7.9)

The reason why this choice is so clever is that we fixed the gauge, but

still kept invariance under the transformation

Consequently the required counter terms

invariance. Since we w i l l only need their

allowed counter terms are gauge invariant

have been the

unobservable) may

We put (7.8)

In our Lagrangian

case, since field

well depend on the gauge

(7.10)

w i l l also exhibit this gauge

dependence on A;' , the only

ones. This would normally not

renormalizations (which are

fixing.

and the associated ghost (also Invariant under (7.10))

(7.7). In general one can choose things such that

flu = -gpvg . (7.11) ij ij

Now M w i l l in general contain a mass term, but we w i l l not include it

in the propagator. Rather, we keep M as a vertex, realizing that from

a certain order on (at sufficiently high powers of M 1 we w i l l have

only convergent diagrams, whereas we are really only interested in

divergent renormalization counter terms. In other words, if there is a

mass m we expand the propagator,

l/(k2+m2-f&) = l/(k2-ic) - m2/(k2-ic)2 + m4/(k2-i~)3 - ... (7.12)

treating the correction terms as vertex insertions.

237

Consider now the one loop diagram of Fig. 4.

Fig. 4

The amplitude is

1 (6jk6jl+6j16jk)p Sd"k 4(2nI4i (k2-ie)((k+p)2-ic) '

If n = 4-c t h i s is

which has a pole at c=O:

(7.13)

(7.14)

(7.15)

A counter t e r m in A!! is needed:

(7.16)

The complete A!! containing a l l possible combinations of M and # can only have a relatively s m a l l number of terms. This is because simple

power counting arguments tell us that they have to be polynomials of

dimension 4: Now W is of dimension 0, N of dimension 1 and M of

dimension 2. If W were non-trivial we could have an inf ini ty of

possible terms but now we must have

2 A!! = -(1/8n C ) MijMji/4 .

+ a4N N a N + + a,(N N + N 1 . (7.17)

Now it is easy t o see that we can put some restr ic t ions on M and

P V P V P V P P

N :

M i j = M . . J l ; Nyj = -Ny i , (7.18)

by absorbing the symmetric par t of a fl into M . However, one can do

much more by observing a new "gauge" symmetry tha t is apparent i n our

system. Write

P

238

with

(7.19)

X = M - N r u c - ' . P

We then have invariance under

(7.20)

(7.21)

X' = X + [A,XI . Here, A is infinitesimal.

Note tha t the dimension of t h i s gauge group is as large as there

are f ie lds in the system, so it could be larger than the original gauge

symmetry, and tha t the symmetry is exact also af t e r a gauge fixing term

is added (This was made possible by the fact that the background fields

transform non-trivially; the gauge fixing term is allowed to depend on

the background f ie lds not in the same way as from the quantum f ie lds) .

A consequence is that AP must have the same symmetry. The only

combination of terms from (7.17) invariant under (7.21) is

AP = (1/8n2e)Tr(aX2+bY Y ) , PW PW

where

Y = a ~ - a ~ + N N - N N . PV P V U P P V V P

From (7.13) we f ind that

a = a. = -1/4 .

(7.22)

(7.23)

(7.24)

Similarly"71 one can f ind, by computing a few other diagrams,

b = (ao-as)/2 = a,/2 = -1/24 . (7.251

The excercise can be extended to include f e r m i ~ n s " ~ ' . The

Lagrangian we then start off with is

239

plus higher orders in the quantum f ie lds which a r e irrelevant for one

loop calculations.

By making much use of our previous resul t , eq. (7.221, and taking

the fermion minus sign into account, one f inal ly obtains

+ H2/8 + H H /48) , (7.27) PV PV

where

We see that t h i s expression contains a few more simple numerical

coefficients t ha t had to be fixed by explicitly computing some diagrams.

The r e su l t obtained here, essentially algebraically, is not yet in

the form we would like to see it, because the background f ie lds a re

still hidden in the expressions X, Y , F , H, ... . W e now have to

subst i tute V, S, P from (7.1) and the s t ructure constants i n here. The

f ie lds 'pi in (7.7) are in fact a combination of @i and Ae q'. This

is a dull and lengthy process. But the resul t is worth while. One

f inal ly obtains' *" :

P

(7.29) 2 (871 & ) A 2 =

Ca Gb [ - ( l l / 1 2 ) c b + (1/24)Gb + (l/6)Gb1 - AV - $ ( A S + f r , A P ) J I , PJ P

i n which

(7.30)

(7.31)

(7.32)

(7.33)

240

V i = aV/a#i ; S,{ = aS/aai , etc.

And writing

* w = S+iP ; w = s-iP ,

(7.34)

(7.35)

one gets *

AW = (1/4)W,iW,iW +(1/4)WW:iW,i + W,iW'w,i +

+(3/2)G W +(3/2)W$ + W,i#jTr(S,iS,j + P, ,Pl j ) . (7.36)

One may wonder why there are no field renormalization terms such as c(aF#I2 . The answer is that these can be rewritten a s

-c#(a2#) (7.37)

which, via the equation of motion of the background fields. can be

expressed in the other terms. I t also can be noted that an infinitesimal

renormalization of the field # ,

# + # ' = # + S # (7.38)

adds to the Lagrangian terms proportional to the equation of motion:

!t+ Z + S $ X , (7.39)

where X vanishes because of the equation of motion. So field

renormalizations, whlch by the way need not be gauge-invariant, are not

seen if we use our background field method.

eqs (7.291 - (7.36) are our master formula for all one-loop counter

terms. I t is very instructive to play with a few examples to see what

kind of renormalization group flow equations they produce.

241

8 . Asymptotic freedom

Let us look at the one-loop resul ts , eqs (7.29)-(7.36) more closely. How

should we use them?

The term AV is easiest to understand. Take the case

There is only one f ie ld , so i=j=l ;

v,, = a2v/a+’ = 4~4‘ + m2 ,

and the first term in (7.33) is

(8.2)

2 and including the factor 1/8n E we get (note that A(m2)=2mAm 1:

AA = (3/16n2~)A2 , A ( m ) = Arn/32n2c ; (8.4)

compare eqs. (3.15) and (3.11). The constant $m4 in (8.3) is immaterial.

If the f ie ld has N components, and V=A(52)2/8 , (a convenient

choice of normalization) then

AV = A2(N/4 + 2)($f2/4 ; (8.6)

Ah = (N+8)A2/16n2~ , (8.7)

which reproduces (8.4) fo r N=l since we had a factor 3 difference in

our definition of A . 2 The next t e r m in (7.33) contains T . This represents the gauge

coupling, 1.e. which representation 4 is of our gauge group. Suppose

that in (7.3)

which would imply that 4 is in the adjoint representation of the gauge

group, and

242

Then

contributing t o AV :

iaj Ta = gc . ij

so that we get a negative counter t e r m

AA = -12Ag2/8n2c .

The th i rd term in (7.33) gives again a positive contrubution:

(8.9)

(8.10)

(8.11)

(8.12)

I t is easy to see now tha t i f we insis t on keeping the Hamilton density

positive in a purely scalar theory most @ functions w i l l allways be

positive (at least one of them), because of the definite sign of the

f i r s t term in (7.29). But we also see that t h i s is no longer obviously

the case if gauge interactions a re included. In practice however one can

prove from all equations (7.29)-(7.36) (with much pain) that i f we have

scalars and only vectors (no spinors) then the middle term of (7.29)

alone is not able to render the system asymptotically f r ee (all p

negative). There a re too many terms with the wrong sign.

But now consider the pure gauge theory. W e then only have:

A!! = -(11/12)CIabGa G b /8n2c , fJv P

(8.14)

and surely the C a s i m i r operator Clab is positive (or zero in the

Abelian case). Mostly we diagonalize it:

clab = C,(a)aab , (8.15)

with ClrO . Let u s simply w r i t e C,(a) = C, . Since the coupling

constant g was absorbed into the s t ructure constants f we have tha t

C, is proportional to g . For an SU(2) gauge theory it is 2

(8.16) 2 c, = 2g .

243

(8.14) renormalizes the fields A . (One could have chosen to

w r i t e (7.29) differently so as to avoid field renormalizations. but th i s

notation came out to be the easiest .) Let us rewrite (absorbing

temporarily the 1/8n2c into the coefficient Cl

it

but then

(8.17)

(8.18)

(8.191

(8.20) Ag = -(11/61CIg/8n2c ,

which is -(11/3)g3/8n2e in SU(2).

Thus, the beta function for g is negative. Note tha t both

fermions and bosons w i l l tend to turn p in the other direction:

Fields in the adjoint representation have

c2,3 = cl * (8.22)

but in the elementary representation of SU(N) we have

C2,3 = C1/2N , (8.23)

(note that in (8.22) and (8.23) we consider chiral fermions. Massive,

complex fermions have twice these values; furthermore, scalar f ie lds

were taken to be real . If they a re complex we also need a factor two).

If we have 11 massive fermions in the fundamental representation (for

SU12)) or 164 (for SU(3) then p w i l l switch sign.

Thus, QED, which has Cl=O , can never be asymptotically free. A

purely scalar theory also can’t , but also a scalar-fermion theory with

Yukawa couplings cannot be asymptotically free. This is because in

(7.32) onlu the terms with U W have a negative sign. 2

In gauge theories with fermions and scalars the si tuation is

complicated. If one imposes the conditions that all beta functions come

244

out negative one usually will f ind that all coupling constants should

keep fixed ratios with each other under scale transformations:

because only then they won't run away in different directions.

I t is an interestlng game to t r y to construct such models. There a re plenty of them. The rat ios hi often turn out to be rather ugly

algebraical numbers except when one invokes symmetries such as supersymmetry. Supersymmetric Yang-Mills is asymptotically f ree (M2) or

scale-invariant (N=4) . In practice one finds tha t the fermions must

couple suff ic ient ly strongly to the gauge particles which happens

typically if they a r e in the adjoint representation or higher. So in QCD

it w i l l be hard to add scalars while keeping asymptotic freedom because

the quarks a r e all in the fundamental representation. Perhaps th i s is the reason why fundamental scalars that couple to the strong gauge group

cannot exist .

Constructing weakly interacting systems while insist ing on (8.24) , though an interesting game, is probably physically irrelevant because

then the coupling constants run so slowly tha t the Landau ghost is

extremely far away in the ultraviolet, where, certainly if it is beyond the point where gravitational forces a re expected, none of our theories

can be believed anyway.

Similarly to our equations (7.29)-(7.36) one can devise expressions

tha t generate the two-loop contributions to the beta functions. But even

in th i s compact algebraic notation the resul t is very lengthy' " I .

9. Perturbation expansion at high orders and t h e renormalization group

The renormalization group gives us a good idea about the force l a w

between quarks in QCD. The strength of the coupling constant increases

with distance. So we get f o r the inter-quark force:

g ( r l 2 F ( r ) = - 2 ' r

with

245

and since f3cO we have a force law in which the strength of the force

decreases not as quickly with the distance as . A t s m a l l distance

scales quarks interact quite weakly, and indeed t h i s f a c t , which w a s

strongly indicated by experimental observations in the late ~ O ' S , has

been a major reason to believe that a non-Abelian gauge theory must be

responsible f o r the strong interactions.

l/r2

But a lso fo r theoretical reasons an asymptotically f r ee gauge

theory (with fermions and possibly also scalars) is the only likely

theory f o r any strongly interacting se t of particles. One would believe

that at least at very t iny distance scales the theory should be

precisely defined. According to our own philosophy th i s is only the case

if the coupling constant tends to zero. Now in any weakly interacting

theory the coupling constants a r e s m a l l enough to make the perturbation

expansion useful even if absolute formal convergence is lacking. So they

need not become any smaller at small distance scales. But for strongly

interacting theories t h i s is necessary.

But even in asymptotically f ree theories the coupling constant

tends to zero very slowly. Is this convergence f a s t enough to make the

theory well-defined?

The answer to th i s question is not known. A s should be evident from

J. S m i t ' s lectures at th i s graduate school it is often taken f o r granted

that i f , i n a latt ice gauge theory, the bare coupling constant is

postulated to tend to zero according to

as a 3 0 , where A and B a re determined by the renormalization

group, then a sensible l i m i t is obtained, independent of the details of

the latt ice.

There is no proof of this , apar t from numerical evidence, which

however at best can give only indications of how the system w i l l tend t o

behave. Our problem is t ha t , even at s m a l l distances, all features of

our system a r e defined as perturbation ser ies in g2 , which diverge as

C"g2"n! , as far as we know. The e r ro r , behaving a s exp(-1/Cg2)

decreases rapidly a s the scale becomes smaller. Does t h i s uncertainty

decrease fast enough to become irrelevant in the l i m i t ? Or could the

e r ro r amplify as one integrates the equations back to large distances?

246

The question tu rns out to be not easy, particularly if we a re dealing

with models that are not exactly solvable.

One way t o attack t h i s problem is to search the source of the

offending factors n! i n perturbation expansions. When one sums over

large s e t s of Feynman diagrams one discovers that the number of diagrams

with n loops increases like n! . Only in special cases they donot.

One case is the so-called spherical model. This is (4')' theory with N

components, where we let N tend t o infinity, such that A N = x is

kept fixed. This model is exactly solvable. The 6 function can be given

in closed form, and indeed the limiting theory is well-defined. I t is

asymptotically f r ee if xC0 , but in t h i s case the negative sign of A

does no harm. After all, A i tself tends to zero. The amount of action

needed f o r a cer ta in field configuration to tunnel into a mode tha t has

negative energy is found to increase indefinitely with N , so tha t such

a tunnelling is inhibited in the l i m i t .

N

A less t r i v i a l model is planar f ie ld theory. Here, we have fields

with two indices i and j which both can take N values. A typical

example is SU(N) gauge theory. If one now allows N t o tend to

inf ini ty , again keeping A N or g'N fixed, one recovers all planar

Feynman diagrams. There are infinitely many of these, and nobody knows

how t o sum these analytically. But their number now only grows

geometrically with N ( that is, without the N! 1. But there is another source f o r factors n! This can be understood

in detail using the renormalization group, but here we give only an

heurist ic argument.

Consider a self-energy insertion diagram with one loop. I t shows

the following typical k dependence:

r(k2) Y Ak210gk2 , (9.4)

as we have seen, and we know that the log is a crucial feature following

from the renormalization group.

Now consider n of these diagrams connected in a se r i e s to the

propagator. This diagram then behaves as

P(k2) = (k2-ic)-l An(logk2)" . (9.5)

Suppose tha t , i n t u rn , t h i s diagram occurs inside a loop diagram, which

247

we have to integrate:

and assume that the integral converges by power counting. If it didn't

we make it converge by using subtractions.

Thus, at large k , the function F(k) tends to k-4 . Assuming no

infrared divergence the integral behaves as

1

Substituting k2 + es , we get m

Sds 0

This is how n! appears that

I n! .

can ruin perturbation expansion.

(9.8)

I t should be clear that the logarithmic behavior in eqs. (9.4)-

(9.6) can be understood by studying scaling via the renormalization

group. One then f inds tha t the coefficients in f ront of the n! can be

obtained directly from the 6 function coefficients.

248

References

1.

2. 3. 4. 5 6

7 8 9

10 11 12 13

14 15

16 17 18

E.C.G. Stueckelberg and A. Petennan, Helv. Phys. Acta 26 (1953) 499; N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959) see for instance: M.N. Barber, Phys. Repts 29C (1977) 1 N.G. van Kampen, private communication G. 't Hooft, Nucl. Phys. B33 (1971) 173; ibid. B35 (1971) 167; M. Veltman, Physica 29 (1963) 186 G. 't Hooft and M. Veltman, DIAGRAMMAR, CERN report 73/9 (1973), repr. in "Particle interactions at Very High Energies", NATO Adv. Study fnst. Series, Sect B, Vol. 4b, p. 177. G. 't Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318 W. Pauli and F. Villars, Rev. Mod. Phys. 21 (1949) 434 G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566; J.F. Ashmore, Nouvo Cimento Letters 4 (1972) 289 G.L. Eyink, 1987 Ph.D. Thesis, Bruxelles C.G. Callan, Phys. Rev. D2 (1970) 1541 G. 't Hooft, Nucl. Phys. B61 (1973) 455 K. Symanzik, Comm. Math. Phys. 18 (1970) 227; ibid. 23 (1971) 49; 45 (1975) 79 M. Gell-Mann and F. Low, Phys. Rev. 95 (1954) 1300 For instance, G. 't Hooft, Comm. Math. Phys. 86 (1982) 449; ibid. 88 (1983) 1 R. van Damme, Nucl. Phys. B227 (1983) 317 G. 't Hooft, Nucl. Phys. B62 (1973) 444 See ref. 16 and: G.'t Hooft, Nucl. Phys. B254 (1985) 11

249

CHAPTER 4

EXTENDED OBJECTS

Introductiom

[4.1]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

“Extended objects in gauge field theories”, in Particles and Fields, eds. D. H. Bod and A. N. Kamal, Plenum, New York, 1978, pp. 165-198 .............................................................. 254 “Magnetic monopoles in unified gauge theories”, Nucl. Phys. B79 (1974) 276-284 288

[4.2] .... .... . ... . .. .. .. ... . . .. . ... . .. .. .. .. .. . . . . . . . .. . .

25 1

CHAPTER 4

EXTENDED OBJECTS

Introduction to Extended Objects in Gauge Field Theories [4.1]

Up till now we maintained that quantum field theories (in four dimensions) can onlg be understood perturbatively. What we really meant however was that it is the infinite series of successive quantum corrections that must be addressed term by term. Even though the renormalization group allows us to partially resum this expansion, the manipulations are still permissible only when the coupling constant(s) is (are) sufficiently small.

Extended objects form another topic that suggests the possibility to transcend beyond the perturbation expansion. We get particles with masses inversely proportional to the coupling constants, and we get “instantons” that will produce tunneling effects with amplitudes depending exponentially on the inverse coupling constant. Indeed, phenomena described in this chapter may perhaps be called “nonperturbative”, but we always have to remember that what we compute will require an infinite series of multiloop corrections, as usual. So, again, we must always assume the coupling constants to be sufficiently small.

The first paper in this chapter is an introduction to kinks, vortices, monopoles and instantons. This is a logical order. In its Section 4 it is explained how in certain gauge theories 2-dimensional objects (vortices) become unstable and how this leads to the existence of %dimensional things (monopoles). This is also the way I discovered them. Alexander Polyakov had followed a slightly different route. He had considered “hedgehog” configurations of scalar fields and asked whether one could construct particles this way. Following the arguments that I explain in Section 2, he found that gauge fields had to be added. That these things actually carry magnetic charge had been suggested to him by Lev Okun.

Note that this paper was written before the third generation of quarks and leptons had been firmly established. In a conservative “Standard Model” at that time therefore instantons gave rise to processes with A B = AL = 2 rather than 3.

252

Introduction to Magnetic Monopoles in Unified Gauge Theories [4.2]

The next paper is the one in which the discovery of magnetic monopoles as valid solutions in gauge theories was published. It gave rise to a renewed interest in monopoles by theoreticians, experimentalists and cosmologists alike. Experimental detection has never been confirmed, but, as I had pointed out in the Palermo Confer- ence, Chapter 2.3, monopoles play a significant role in theory anyway. Most crucial is their part in the quark confinement mechanism, see in particular Chapter 7.

One later observation should be added to this paper. The coefficient C(p) waa computed by numerical variation techniques, and found to be close to one when p is small. It waa an important result of Prasad and Sommerfield15 that the equations simplify essentially when p = 0. The second order field equations then become the square of a first order equation (duality). This equation can be solved exactly and one then finds that C(0) = 1.

“M. K. P r d and C. M. Sommerfield, Php. Rev. Lett. 96 (1975) 760.

253

CHAPTER 4.1

EXTENDED OBJECTS IN GAUGE FIELD THEORIES

G. 't Hooft University of Utrecht

The Netherlands

1. INTRODUCTION AND PROTOTYPES I N 1+ 1 DIMENSIONS

With the exception of the e l ec t ron , the neutrino, and some o the r elementary p a r t i c l e s , a l l ob jec t s i n our physical world are known t o be extended over some region i n space. Thus, the subject "extended objects" is too large t o be covered i n j u s t four lectures . However, most of these extended ob jec t s may be considered as bound s t a t e s of more elementary const i tuents . I f we exclude a l l those, then a very i n t e r e s t i n g small s e t of pecul iar objects remains: objects t ha t cannot be considered as j u s t a bunch of p a r t i c l e s , but aa some smeared, but a l s o more o r less loca l i zed , configuration of f i e l d s . equations t o allow such smeared lumps of energy ("solitons") t o be s t ab le .

These f i e l d s m u s t obey very s p e c i a l types of f i e l d

These may be two a l t e r n a t i v e reasons f o r alump of field-energy t o be s t a b l e against decay:

i ) A conservation l a w might t e l l us t h a t the t o t a l mass-energy of the decay products would be l a rge r than our ob jec t itself. easy t o prove t h a t t h i s cannot happen i n a l i n e a r ( i .e . non in t e r - act ing) theory: I f e l e c t r i c charge is given by

It is

Q - i e j[O*aoO - (ao4*)dl d3x , (1.1)

and the energy by

Particles and Fields, Edited by D. H. Bod and A. N. Kamal (Plenum 1978).

254

G. 't HOOFT

(1.3)

whereas a bunch of charged p a r t i c l e s a t rest, wi th a t o t a l charge equal t o Q , could have a t o t a l mass-energy n o t exceeding

+ Consequently, a f i e l d conf igu ra t ion wi th a$ # 0 would not be s t a b l e by charge conserva t ion alone. I n i n t e r a c t i n g f i e l d theori-es however, one may cons t ruc t many i n t e r e s t i n g s t a b l e lumps of f i e l d s simply because t h e above arguments do n o t apply when t h e r e a r e i n t e r a c t i o n s . and w i l l no t be t r e a t e d i n t h i s series of t a lks .

These have been considered by T.D. Lee and coworkers'

i i ) Topological s t a b i l i t y . In these l e c t u r e s we w i l l only consider t opo log ica l ly s t a b l e s t r u c t u r e s . The s ta tement t h a t our ob jec t s are r e a l l y f i e l d conf igura t ions i m p l i e s t h a t w e cons ider f i e l d equat ions without bo ther ing about t h e f a c t t h a t small oscil- l a t i o n s about t he s o l u t i o n s ought t o b e quant ized.

To exp la in topo log ica l s t a b i l i t y w e f i r s t t u r n our a t t e n t i o n t o models i n one space - one t i m e dimension2. s c a l a r f i e l d (I s a t i s f y i n g

Consider a simple

t o which corresponds a Lagrangian:

The two cases w e cons ider are

a> V(4) = (0'- F2)' (see Fig: 1) .

(1.6)

(1.8)

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EXTENDED OBJECTS IN GAUGE THEORIES

F

Fig. 1. Potential of the form X ( $ 2 - F 2 ) 2 / 4 !

I t i s customary then t o write 4 = F + 4’ ,

(1.9) 1 2 , 2 - x $ 1 4

v = T r n $ + - $ - 4 1 3 + 41 with

(1.10) m2 = - X F ~ ; g - X F . V($) = A(1- COB a) (“Sine-Gordon” model- see Fig. 2 ) . (1.11)

1 3

b) F

Here w e write

SO

ant

that

1 x 2 41 v = - m 2 4 2 - - 4” + ...

(1.12)

(1.13)

In both cases m stands for the mass of the physical par t i c l e , A is the usual coupling constant-expansion parameter.

Fig. 2 . Potential i n “Sine-Gordon” model.

0. 't HOOFT

The energy of a f i e l d configuration is

H = k d 3 x , (1.14)

The examples I gave here have the property t h a t there is more than one way t o make a zero-energy state' ("vacuum"):

a ) $ - n F , n - 2 1 ,

b) 4 = nF , n = ..., -2, -1, 0, 1, 2 , ... (1.15)

The world i n these models is assumed t o be a one-dimensional l ine. We envisage the s i t u a t i o n t h a t a t one s i d e of the l i ne Q, has one value f o r n, a t the o ther s i d e a d i f f e r e n t value (see Fig. 3). We can have a t r a n s i t i o n region c lose t o the o r i g i n which c a r r i e s energy :

4 C n~ , n in t ege r ; ax+ f 0 . (1.16)

Assme t h a t the s i t u a t i o n is s t a t iona ry :

at4 * 0 . (1.17)

We wish a f i n i t e t o t a l energy; 4 m u a t approach one i f i ts poss ib le vacuum values a t x+*. we f ind a d i s c r e t e set of allowable boundary conditions.

m u s t approach zero a t x + 200. So Thus

I n a s t a t iona ry so lu t ion of t he equation (1.6) a l s o the energy is an extremum. The lowest-energy configuration under the given boundary conditions a l s o s a t i s f i e s t h i s equation, It is obviously s t a b l e . I n our examples the so lu t ions are e a s i l y given:

'-w2 ) X

(1.18)

Fig. 3. A static form for $ with d i f f e r e n t boundary condi t ions a t x + fa.

257


t h e r e f o r e

Case a):

1 2 $(x ) = F tanh-mx ,

Case b) :

mx $(x) = -i;- 2F a r c t g e . Note t h a t , as x + 0 3 ,

1 F tanh 1"" + F ( l - 2e-mx)

2F a r c t g e

,

e-mx) , + F( I - - mx IT IT -

(1.19)

(1.20)

41.21)

(1.22)

(1.23)

(1.24)

(1.25)

thus t h e vacuum value is reached exponen t i a l ly , wi th t h e mass of the phys ica l p a r t i c l e i n the exponent ( see Fig. 3). The t o t a l energy of t hese ob jec t s ("sol i tons") i s

(1.26)

Case a ) :

E - 2 m 3 ~ ~ , Case b ) :

E = h 3 1 x . They behave as p a r t i c l e s i n every sense ; f o r i n s t ance one can show t h a t t h e r e l a t i o n s between energy, momentum and v e l o c i t y are a s ind ica t ed by r e l a t i v i t y theory.

W e d i s t i n g u i s h two types of s t a b i l i t y requirements:

1) Topological s t a b i l i t y . Since the boundary condi t ions i n our case form a d i s c r e t e set t h e r e .can be no continuous t r a n s i t i o n

258

G. 't HOOFT

towards t h e vacuum boundary condi t ion (which is the one f o r a bunch of ordinary elementary p a r t i c l e s ) .

2) S t a b i l i t y aga ins t s ca l ing . Let us r e t u r n f o r a moment t o n space l ike dimensions ( in s t ead of one). In the s t a t i o n a r y case w e a l w a y s have

H = S + V , s = -(a , 1: x V = IV($)dnx . (1.27)

The energy must be s t a t i o n a r y under any i n f i n i t e s i m a l v a r i a t i o n . L e t us try one s p e c i a l va r i a t ion :

then

S + An" S ; V + h"v ,

W e must have

na (S + V) = ( n - 2 ) S + nV - 0 .

(1.28)

(1.29)

(1.30)

But S and V are both pos i t i ve . i n the case n - 1. Since the decomposition (1.27) is poss ib l e i n a l l scalar f i e l d theo r i e s , s c a l a r t heo r i e s have only s t a t i o n a r y s o l i t o n s i n one space-dimension. I n two o r more dimensions i t would always be e n e r g e t i c a l l y favorable f o r a system t o s h r i n k u n t i l i t becomes p o i n t l i k e . That i s not an extended ob jec t by d e f i n i t i o n .

That is only compatible wi th (1.30)

F ina l ly , we emphasize t h a t d e s c r i p t i o n of p a r t i c l e s i n terms of quant ized f i e l d s i n most cases is only poss ib l e i f t he coupling is not t o o s t rong:

( i n 1+1 dimensions X has t h e dimension of a mass-squared). Therefore , t he mass of s o l i t o n s , being p ropor t iona l t o m3/X, is always much g r e a t e r than t h a t of t he o r i g i n a l p a r t i c l e s i n the theory.

259


2. SOLITONS I N 2 SPACE DIMENSIONS AND STRINGS I N 3 SPACE DIMENSIONS

There is another reason why s c a l a r t h e o r i e s have no t o p o l o g i c a l s o l i t o n s i n 2 o r more space dimensions. We can only have topologi- c a l s t a b i l i t y i f t he boundary condi t ion d i f f e r s from: I$+c n a t a n t a t 1x1 +-. assme t h a t V($) has a whole continuum of minima. w e could take n f i e l d components (n= number of space- l ike dimen- s ions ) and have V($) - minimal i f mapping

Imagine t h a t w e have s e v e r a l f i e l d components 8 and For in s t ance

= F. We could th ink of

as a boundary condi t ion. This would imply topo log ica l s t a b i l i t y . But then

and

diverges a t l a r g e x unless n = 1. This i n f r a r e d divergence is c lose ly r e l a t e d t o the ex i s t ence of massless Goldstone bosons i n such a theory.

The only way known a t present t o re-obtain a topo log ica l ly s t a b l e s o l i t o n i n more than one space l ike dimension i s t o add a gauge f i e l d :

The space-components of the vec tor f i e l d A can be arranged such t h a t (D$)' * 0 more r ap id ly than 1 /x2 , and t h i s way w e can approach a phys ica l vacuum so r ap id ly t h a t t h e t o t a l energy converges. (Note a l s o t h a t by adding a gauge f i e l d the massless Goldatone boson d isappears , so t he theory has b e t t e r i n f r a r e d convergence. There may s t i l l be a massless vec tor boson but t h a t is usual ly less harmful, a s we w i l l see.)

The s imples t theory wi th a s o l i t o n i n 2 space dimensions is scalar quantum-electrodynamics i n the Higgs mode4 9 :

260

G. 't HOOFT

Here q is the unit of charge. The 3 + 1 dimensional analogue of t h i s model is w e l l known i n physics: conductor. JI is the s implif ied f i e l d ("order parameter") t h a t e s s e n t i a l l y descr ibes two-electron bound states with t o t a l sp in zero (hence i t is a s c a l a r boson f i e l d ) . vector po ten t i a l . q = 2e.

i t describes the super-

A,, is the ordinary

Inside the superconductor the photon behaves as a massive particle and electromagnetic f i e l d s are of sho r t range. Long- range magnetic f i e l d s are forbidden because i n order t o c rea t e them we need p o t e n t i a l differences (according t o Maxwell's laws) and those donot occur in s ide a superconductor. But when placed i n a s t rong magnetic f i e l d a superconductor w i l l f ind an exci ted state i n which i t can allow magnetic f lux t o penetrate. w i l l go through the superconductor i n narrow tubes. The f lux i n such a tube turns out t o be quantized. I f w e consider a plane orthogonal t o these tubes then we get equations f o r extended pa r t i c l e - l i ke objects i n t h i s plane. They are the 2 dimensional s o l i t o n s t o be considered now. The f lux tubes ("vortices", "s t r ings") , i n the th ree dimensional world are derived from them by adding a t h i r d coordinate ( p a r a l l e l t o the axie) on which the f i e l d s do not depend.

This f lux

The vacuun is described by

144 - F (2.6)

W e construct our s o l i t o n just as i n the previous sec t ion ; w e choose t o approach a vacuum rapidly at Is] + -, but i n such a way tha t continuity requires a t r a n s i t i o n region a t x -t 0. Write

x1 - r COB 8 ,

xp - r s i n 8 , i e then we choose as r -t =, JI + Fe .

Continuity requires t h a t JI produces a zero somewhere near the o r i g i n of x-space. energy t o produce such a zero: s t a b l e lump of energy near the o r ig in of x-space. L e t us look a t the lowest-energy configuration. Assume it is time-independent and A. = 0.

Since the vacuum value of lJll is F, i t costs w e w i l l obtain a topologically

As a gauge condition we choose:

and, by symmetry arguments :

26 1


One can e a s i l y check t h a t t he above is a se l f - cons i s t en t ansa tz f o r a s o l u t i o n of t he f i e l d equat ions.

To ob ta in the equat ion f o r x and A i t is e a s i e s t t o reexpress the Lagrangian i n terms of x and A:

m

/ & d 2 z = I 2nrdr(- i r2($)2 - L(a)2 - ( i + q A r ) 2 2 x 2 d r

0 m

- X ( ~ 2 - F 2 ) 2 ] - nr2A21 0 2

Assuming I+(-) I -C 0 w e may ignore the boundary term. boundary condi t ions a t r + 0 w e m u s t r equ i r e

As fo r

(2.10)

A t r + w t h e t h i r d and fou r th terms i n (2.9) converge only i f

(2.11)

Since our system is assumed t o be s t a t i o n a r y ( a / a t = 01, and the time-components of A,, vanish, the energy w i l l be j u s t minus the Lagrangian.

m

E = I 6 ( r ) d r , 0

( 2 . 1 2 )

(2 .13 )

We m u s t f i n d the minimum of t h i s energy under t h e above boundary condi t ions. The coupled d i f f e r e n t i a l equat ions ,

262

G. 't HOOFT

(2.14)

are no t exac t ly so luble . The s o l u t i o n is sketched i n Fig. 4. As i n t he 1+ 1 dimensional case, w e expect t h a t t h e f i e l d s a t l a rge d i s t ance from the o r i g i n dev ia t e only exponent ia l ly from the vacam va lues , with the masses of t he two massive p a r t i c l e s (vec tor boson and Higgs p a r t i c l e ) i n the exponents. behaved, t opo log ica l ly s t a b l e s o l i t o n , p a r t i c l e - l i k e i n 2 + 1 dimen- s i o n s , s t r i n g - o r vortex-l ike i n 3 + 1 dimensions.

Thus w e ob ta in a w e l l

What w i l l t h e mass of t h i s par t ic le be? Without so lv ing t h e equat ions e x p l i c i t l y , we can make some s c a l i n g arguments. Put

(2.15) 2 2 2 2 F = mH/2A = %/2q ,

where mH and % are, r e spec t ive ly t h e Higgs and the vec tor boson masses, and

then

(2.17)

Fig. 4. Sketch of the s o l u t i o n s t o Eq. (2.14).

263


Note t h e coupling cons tan t downstairs . Of course, i n 2 + 1 dimen- s i o n s , q2 has t h e dimension of a mass, hence 4 , not %. dimensions, t he energy of a s t r i n g is propor t iona l t o i ts length , and (2.17) g ives the energy pe r u n i t of length , which is i n u n i t s of mass-squared.

I n 3 + 1

I n the beginning of t h i s s e c t i o n w e a n t i c i p a t e d t h a t t h i s vortex would be a magnetic f l u x tube. L e t us now v e r i f y t h a t by computing t h e magnetic f i e Id.

A t s p a t i a l i n f i n i t y t h e vec tor p o t e n t i a l is found t o be , from (2.111,

Ai -F - E x / q r 2 . (2.18) il

Therefore

i Aidx -t 2nlq . f (2.19)

So, indeed, w e have a magnetic f i e l d going along the vor tex wi th t o t a l f l u x 2n/q, t h i s i n s p i t e of t h e f a c t t h a t t he photon had become massive which would imply t h a t e lec t romagnet ic f i e l d s are only short-range. of t h a t f lux , t he re fo re magnetic f l u x i n a superconductor is quant ized, by u n i t s

There is no way of producing a broken f r a c t i o n

2nlq n l e . (2.20)

There is c l e a r l y also a phys ica l reason f o r our vortex o r s t r i n g t o be s t a b l e : t h e magnetic f l u x i t conta ins cannot spread ou t i n the superconductor and i t is exac t ly conserved.

3. THE BOUNDARY CONDITION - HOMOTOPY CLASSES

The above w a s a pedes t r i an way t o ob ta in the vor tex i n a superconductor. W e w i l l now reformula te t h e boundary condi t ion more p rec i se ly i n order t o be ab le t o produce more complicated ob jec t s later.

Remember w e chose our boundary condi t ion t o be J, + eieF a t r -F Q). W e could have chosen

every choice of n would y i e l d a p a r t i c u l a r conf igu ra t ion wi th lowest energy, dependent on n. I f t h e r e would have been any way

264

G. 't HOOFT

t o make a continuous t r a n s i t i o n from one n t o another, without v io l a t ing l J l l - F a t large dis tance from the o r ig in , then t h a t would cause an i n s t a b i l i t y f o r whatever s ta te had the higher energy. But there is no way t o make t h i s continuous t r ans i t i on . We say t h a t the d i f f e r e n t choices of n are homotopically d i f f e r e n t 2 . The boundary has the topological shape of a c i r c l e . (In N dimensions i t is the SN-1 sphere.) The vectors

a l so form a circle. c i r c l e are ca l l ed "homotopic" i f one can continuously be transformed i n t o the other. A l l c lasses of mappings t h a t are homotopic t o each other form a "homotopy class". For the mappings of a c i r c l e onto a c i r c l e , o r any S classes are l a b e r i a by an i n t ege r n running from -OD t o m.

Two continuous mappings of a circle onto a

sphere onto an SN-1 sphere, these homotopy

I n order t o formulate the boundary condition i n the general case w e introduce two types of vacuum: a supervacuum is a region of space o r space-time where a l l vector f i e l d s vanish and a l l scalar f i e l d s have t h e i r vacuum value: i n a preassigned d i r ec t ion i n isospace, f o r instance the z- direct ion:

- F and are pointing

Now i n a gauge theory, l i k e electromagnetism, we are allowed t o make gauge ro t a t ions ; f o r instance i n the model of the previous se c t ion :

After t h i s , i n general space-time-dependent, gauge r o t a t i o n w e obtain non-vanishing vector-potentials and the s c a l a r f i e l d s w i l l point i n a r b i t r a r y d i r e c t i o n in isospace. have a vacum. This w e w i l l c a l l a normal vacum, but no longer s upervacum.

Physically, w e s t i l l

Any vacuum configuration is defined by specifying the gauge r o t a t i o n Q(x, t ) t h a t produces i t from the supervacum. Sometimes, i f t he re is no complete spontaneous symmetry breaking, Q is only defined up t o a space-time independent constant rotat ion. We w i l l ignore t h a t f o r a moment.

265


Returning t o t h e model of t h e previous s e c t i o n , based on the gauge group U(1), t h e gauge r o t a t i o n s Q are determined by a po in t on the u n i t c i rc le , and so t he gene ra l vacuum is determined by an angle t h a t may vary as a func t ion of space-time. Now, i n f i n i t y i n two dimensions is cha rac t e r i zed by an angle 8 ( the d i r e c t i o n i n which w e go t o i n f i n i t y ) , so the boundary condi t ion is defined by a mapping

of t h e u n i t circle onto an angle 8. As w e saw befo re , t h i s mapping m u s t be i n one of a d i s c r e t e set of homotopy c l a s s e s , l abe led by - m < n < + - :

Because n m u s t s t a y i n t e g e r t he re cannot be continuous t r a n s i t i o n s from a state containing n s o l i t o n s of the same type i n t o a state containing fewer s o l i t o n s . Phys ica l ly t h i s is again the s ta tement t h a t magnetic f l u x going through a two-dimensional su r f ace is conserved.

Now we a r e i n a p o s i t i o n t h a t w e can e a s i l y gene ra l i ze our model i n t o a similar model bu t wi th a non-Abelian gauge f i e l d i n s t e a d of electromagnetism: the non-Abelian superconductor. We assume t h a t t h e gauge r o t a t i o n s (3 .1 ) and (3 .2 ) a r e rep laced by real, or thogonal , 3 x 3 ma t r ix r o t a t i o n Q(x) , w i th determinant one. They form t h e non-Abelian group S O ( 3 ) . W e assume, t h a t , a s i n the U(1) superconductor, a Higgs f i e l d o r set of Higgs f i e l d s with i n t e g e r i sosp in break t h e symmetry spontaneously and completely. Does t h i s model allow f o r s t a b l e f lux tubes a s i ts Abelian counter- p a r t does? Do w e have s o l i t o n s i n two-dimensional space?

To analyse t h i s ques t ion w e merely have t o i n v e s t i g a t e the homotopy c l a s ses of t he mappings of SO(3) matrices i n the pe r iod ic space of angles 8. It i s w e l l known t h a t the group SO(3) i s l o c a l l y isomorphic wi th SU(2), the group of 2 x 2 un i t a ry mat r ices with determinant one. The correspondence is made by i d e n t i f y i n g >vec tors wi th traceless, 2 x 2 t enso r s . An SU(2) matr ix ac t ing on the t enso r corresponds t o an SO(3) r o t a t i o n of t h e 3-vector.

The SU(2) matrices are easier t o handle. They can be decomposed as

( 3 . 3 )

a+ * a* - i f apt. 11- 1 0

266

G. 't HOOFT

The requirements Gilt = 1 and d e t $2 = 1 correspond to:

ao,ag are real and

1 3 2 a g = l .

R=O ( 3 . 4 )

Clearly, the SU(2) matrices form an S3 sphere (the surface of a sphere embedded i n 4 dimensions).

The SO(3) matrices do not form an S, sphere, however. The reason is t h a t an SU(2) r o t a t i o n

on spinors leaves 2 x 2 tensors invariant . Consequently, two oppo- s i t e points on the S3-sphere correspond t o the came SO(3) matrix (see Fig. 5). It is t h i s feature t h a t will allow a non-tr ivial homotopy c l a s s t o emerge.

The homotopy classes here are e n t i r e l y d i f f e r e n t from those i n the U( 1 ) model: any closed contour drawn on an S3 sphere can be deformed continuously t o a dot (see Fig. 6a), so the mappings of SU(2) onto the u n i t c i r c l e are characterized by only one homotopy c l a s s , namely t h a t of the supervacuum. But i f we a r e only concerned with SO(3) matrices one may construct one d i f f e r e n t homotopy c l a s s : a l l contours t h a t s t a r t on one point of the S, sphere and close by ending up a t the point opposite t o A (see Fig. 6b). These a r e the only two homotopy classes i n t h i s case, t h i s i n contrast with the s i t u a t i o n i n the ordinary superconductor where we had an i n f i n i t y of homotopy classes .

What is the consequence for our S0(3)-superconductor? The exis tence of one non-tr ivial homotopy c l a s s implies t ha t w e can

Fig. 5 . The opposite points on SJ-sphere are i d e n t i f i e d .

267


...........

............. ......... .... .........

F i g 6 . Two (and the only two) homotopy c l a s s e s of closed paths fo r SO(3). Paths i n (b) cannot be deformed i n t o the paths i n (a) .

construct a non-tr ivial boundary condition f o r a s t a b l e s o l i t o n i n 2 dimensions. So our model does a d m i t a s t a b l e f lux tube. However, now there is no longer an addi t ive f lux conservation l a w : i f two o f these s o l i t o n s come together then the boundary of the t o t a l sys tem is given by a contour i n SO(3) space t h a t goes through the S3 sphere twice. That contour is i n the t r i v i a l homotopy c l a s s , together with the boundary of a supervacuum. So the p a i r of s o l i t o n s is not topologically s t a b l e which means they

s t ab le . Pa i r s of vo r t i ce s , regardless t h e i r o r i en ta t ion , may ann ih i l a t e each other , forming a shower of ordinary pa r t i c l e s .

may annihilate each other. Thus, only single f l u x tubes are

268

G. 't HOOFT

4 . STRINGS AND MONOPOLES

Next, le t UB d i scuss a model very similar t o t h e prev ious one. Again our gauge group is S0(3), bu t now we assume only one Higgs t r i p l e t , breaking i t down t o U(1). This is t h e so-ca l led Georgi- Glashow model6 tntroduced f i v e yea r s ago as a poss ib l e candidate f o r t h e weak i n t e r a c t i o n s . It conta ins a massless U ( 1 ) photon, a massive charged "intermediate vec tor boson" , and a n e u t r a l rnassiwe Higgs p a r t i c l e . and charm, a t t h e p r i c e of having t o in t roduce heavy l ep tons , t h i s model was doomed t o become obsole te a8 a real is t ic d e s c r i p t i o n of observed weak in t e rac t ions . However, being one of t h e very few t r u l y unifying models i t has aome p e c u l i a r f e a t u r e s which w e now wish t o focus on.

Designed t o avoid the necess i ty of n e u t r a l cu r ren t s

To create a s i t u a t i o n exac t ly as i n the previous model, w e cons ider an ordinary superconductor' i n a world whose high-energy behavior is descr ibed by t h e Georgi-Glashow model. parameter then provides f o r breaking of t h e r e s i d u a l U ( 1 ) symmetry.

The order

According t o the previous s e c t i o n t h i s superconductor d i f f e r s from t h e ord inary superconductor by t h e f a c t t h a t two f l u x tubes poin t ing i n the same d i r e c t i o n may a n n i h i l a t e each o t h e r , because the gauge group i s S0(3), no t U ( 1 ) . But phys ica l ly t h i s looks very odd, s i n c e one would expec t magnetic f lux t o be conserved. The only poss ib l e explana t ion of t he breaking up of (double) flux tubes is t h a t a p a i r of magnetic monopoles is formed. They ca r ry magnetic charge +4n/e.

This way one i s n a t u r a l l y l e d t o accept t he i d e a t h a t magnetic monopoles occur i n the Georgi-Glashow model. They are not confined t o the superconductor, which we had only introduced f o r pedagogical reasons.

L e t us now focus on the near vacuum surrounding t h i s magnetic monopole. At one s i d e , say the p o s i t i v e z -d i rec t ion , we assume a double magnetic flux tube t o emerge. Around t h i s f l ux tube w e do not have a supervacuum b u t a gauge r o t a t i o n of t h a t , which is i n ~ ~ ( 2 1 nota t ion :

where Q, is the angle around the z-axis. A t the o the r s i d e i s no magnetic f lux coming i n , so

n(z + -,$I = indep. ($1 . (4.2)

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According t o homotopy theory no d i s c o n t i n u i t y i n 62 is needed elsewhere (except a t the l o c a t i o n of t h e monopole, where no 62 i s needed because i t is not a vacuum conf igura t ion) . Indeed w e may cons t ruc t 51 everywhere a s follows:

The Higgs f i e l d i n t h i s model i s vacum takes t h e form

o i + s i n ieI, 4 . (4.3)

an i sovec to r , which i n a super-

In the surroundings of a s i n g l e monopole i t is

(4.4)

where i t is understood t h a t the 62 of Eq. (4.3) has f i r s t been w r i t t e n i n SO(3) no ta t ion .

It is c l e a r t h a t t he boundary condi t ion (4.5) f o r the Higgs f i e l d leaves a s t a b l e s o l i t o n a t the center . I t i s a l s o clear from the previous d iscuss ion t h a t t h i s s o l i t o n w i l l c a r ry a mag- n e t i c charge equal t o 4nle.

We sk ip f u r t h e r d i scuss ion of t he monopole i n these notes because w e have l i t t l e t o add t o the e x i s t i n g l i t e r a t u r e 6 ’ e * g . J u s t no te t h a t , as w a s t h e case f o r o the r s o l i t o n s , i t s mass is inverse ly p ropor t iona l t o the coupling parameter:

5. INSTANTONS

5.1. Construct ion

We have seen extended o b j e c t s i n one, two and t h r e e dimensions. None of the models of t h e previous s e c t i o n s , i n which these o b j e c t s occur are of d i r e c t i n t e r e s t i n high energy physics . The models

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G. 't HOOFT

of i n t e r e s t are two d i f f e r e n t types of gauge theo r i e s t h a t are most ex tens ive ly s t u d i e d these days: the SU(2) X U ( 1 ) gauge theory f o r t he weak and e lec t romagnet ic i n t e r a c t i o n s , "spontaneously broken" t o U ( 1 ) by an i sodouble t Higgs f i e l d ' O a l l , and a pure gauge theory based on SU(3) f o r t h e s t rong in t e rac t ions12 .

The weak i n t e r a c t i o n gauge group has e s s e n t i a l l y a four component Higgs f i e l d because the Higgs doublet is complex. Therefore w e expect a topologica l ly s t a b l e ob jec t no t i n t h r e e but i n four dimensions. world is t opo log ica l ly an S, sphere. l e t

Why? The boundary of a four dimensional Also t he complex Higgs doub-

form an S, sphere. again charac te r ized by non- t r iv l a l homotopy classes, labe led by an i n t e g e r n t h a t can run from -m t o *. Thus w e can cons t ruc t boundary condi t ions f o r a four dimensional space under which configu- r a t i o n s wi th n s o l i t o n - l i k e ob jec t s are s t a b l e .

The mappings of S, spheres on S, spheres a r e

In c o n t r a s t t o t h e cases i n lower dimensions, a l s o pure gauge theo r i e s allow f o r topologica l ly s t a b l e s o l i t o n - l i k e ob jec t s in four dimensions. This is most e a s i l y understood when we consider t he s imples t non-Abelian group SU(2). Remember t h a t one way t o de f ine a vacuum a t+the boundary of a region is t o spec i fy the gauge r o t a t i o n s n(x) t h a t produce t h i s vacuum out of the supervacuum. The gauge r o t a t i o n s 51 themselves form an S3 sphere (see s e c t i o n 31, so here too we have the n o n - t r i v i a l homotopy classes a t the boundary. I n f a c t , the S, sphere of t h e gauge r o t a t i o n s 51 themselves, is the same as the S, sphere of the complex doublets

so one can say t h a t the 4 dimensional s o l i t o n s i n the pure gauge theory are e s s e n t i a l l y the same as those i n the theo r i e s broken by Higgs isodoub le ts .

I n order t o d i s t i n g u i s h our 4-dimensional o b j e c t s from the 3-dimensional s o l i t o n s and the s t r i n g s and s u r f a c e s , we g ive them a new name: "Instantons". The name emphasizes t h a t i f one of t he four dimensions is t i m e , then these ob jec t s occupy only a s h o r t t ime- in te rva l ( i n s t an t ) .

We w i l l now argue t h a t t h e r e is an important quan t i ty t h a t is s e n s i t i v e t o the presence of ins tan tons :

27 1


where Ga is the usual covariant c u r l of the gauge vector f i e l d : vv pt"

GE is gauge inva r i an t , and i t is a pure divergence:

However, as one can ve r i fy e a s i l y , $ is not gauge invariant . us see what the consequence is of t ha t .

Let

Consider a region of space-time enclosed by a l a rge S3 sphere. It is i r r e l e v a n t as y e t whether the space-time is Euclidean o r Minkowskian; w e are only concerned about topological s t a b i l i t y and not y e t i n t e re s t ed i n any equation of motion f o r the f i e l d s , not even the space-time metric. Suppose w e have a supervacuum a t the surrounding sphere :

A a - O , P therefore

\ = o . (5.4)

Because of (5.4) a t the boundary, and because of (5.3), w e f ind by means of Gauss' l a w :

regardless of what values the f i e l d s take in s ide the sphere.

Now w e take a non-tr ivial homotopy class of gauge ro t a t ions t o g e t a new vacuum boundary condition:

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G. ‘t HOOFT

I f w e write the vector f i e l d in an SU(2) notat ion,

then the transformation l a w is

We f ind t h a t t he vacuum a t the boundary has

w i t h

U 1a2.3 s

(5 7)

(5.10)

Now although the f i e l d (5.9) is physical ly equivalent t o the vacuum (for instance, G i V a t the boundary vanishes), w e nevertheless g e t a non-vanishing value f o r K,,. This w a s possible because K,, is not gauge-invariant :

(5.11) 4

where 2 2 2

1x1 = x4 + xi . P V

I f we now apply Gauss’ l a w w e f ind t h a t i n s ide the S, sphere G cannot vanish everywhere because

(5.12)

boundary

Merely because the vacuum at our boundary i e topological ly d i s t i n c t from the supervacum we g e t a non-tr ivial phyeical f i e l d configura- t i o n inside.

273


Note t h a t t h e r e s u l t (5.12) ho lds t r u e a l s o i f t h e SU(2) group considered would be embedded i n a l a r g e r gauge group. This impl ies t h a t i n s t an tons remain topo log ica l ly s t a b l e even i f the gauge group is enlarged. Remember t h e Abelian f l u x tubes t h a t could become uns tab le i f the U ( 1 ) group were p a r t of a l a r g e r group; such a th ing does n o t happen f o r i n s t an tons .

Now le t ua i n s e r t t h e f i e l d equat ions . It is easiest t o con- s i d e r f i r s t Euclidean apace, where t h e boundary choice. (5.6) has n i c e r o t a t i o n a l p rope r t i e s . Is t h e r e a s o l u t i o n t o the Eucl idean f i e l d equat ions t h a t approaches (5.9) a t x 2 + and s t a y s r e g u l a r a t the o r i g i n ? We t r y

(5.13)

The f i e l d equat ion , DVGpv= 0, now reads

(5.14) 2 - ( r 2 f ) = 4r f - 1 2 ( r 2 f 1 2 + 8 ( r 2 f 1 3 . d L (d log r )2

This equat ion is sca l e - inva r i an t , which makes i t a s easy t o s o l v e as t h e one-dimensional s o l i t o n , whose equat ion is t r a n s l a t i o n - i n v a r i a n t . 1 3 Besides the t r i v i a l s o l u t i o n , f = l / r 2 , w e can have

, X a r b i t r a r y , f ( r ) - 1 1 r + A

which has t h e requi red p r o p e r t i e s .

The t o t a l a c t i o n f o r t h e s o l u t i o n is

(5.15)

(5.16)

Without a c t u a l l y computing S i t is easy t o d e r i v e an i n e q u a l i t y f o r it. I n Eucl idean space Ga and ca a l l have r e a l components only. Therefore VV P V

(5.17)

(5.18)

because of (5 .12) , and w e can choose e i t h e r s ign . For our s o l u t i o n ,

(5.19)

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G. 't HOOFT

and w e f i n d

(5.20) 2 2 S = - 8 T /g . a "a The i n e q u a l i t y is s a t u r a t e d , because GPv= %Vfor our solu_tion.

Indeed, any s o l u t i o n of t he ( f i r s t o rder ) equat ion GPv = G-,,,, automafica&ly s a t i s f i e s t he second order equat ion D G because D,,G p o t e n t i a l ('d. 2).

= 0 = 0 f o r any gauge f i e l d der ived from a vector lJ lJv

5.2. Euclidean F ie ld Theory and Tunneling

A s o l u t i o n of the classical f i e l d equat ion i n Euclidean space What we r e a l l y would may seem t o be of l i t t l e phys ica l re levance.

l i k e t o cons ider are ins t an ton events i n Minkowski space. Conei- de r again the S, boundary of a compact reg ion i n Minkowski space. The mapping Q(x) a t t h e boundary may be deformed a b i t , as long as we s t a y i n t h e same homotopy class. Thus, the in s t an ton can be seen t o c y r e s p o n d t o a t r a n s i t i o n from a supervacuum a t t i - - (where Q(x) - 1) t o a gauge r o t a t e d vacuum a t t - +=. can have

There w e

(5.21)

A t in te rmedia te t i m e s t h e r e must be a reg ion where G,," # 0 , because of Eq. (5.12). Because w e have a vacuum a t the beginning and a t t h e end t h i s f i e l d conf igura t ion cannot be a s o l u t i o n of t he classical equat ions. Rather , i t should be considered as a quantum-mechanical tunnel ing t r a n s i t i o n . How do we. compute the amplitude of such a tunnel ing t r a n s i t i o n ? It is i n s t r u c t i v e t o compare t h i s problem wi th the motion of a s i n g l e p a r t i c l e through a one-dimensional p o t e n t i a l w e l l :

2

H = + V(x) , (5.22)

i n a reg ion where V(x) >> E , assuming t h a t Planck 's cons tan t is small. To first approximation

(5.23)

The amplitude f o r a tunnel ing process is propor t iona l t o e' with

275


... S - - L i " n

a

4 2m(V- E) dx . X

(5.24)

On the o t h e r hand, i f a t r a j e c t o r y is i n an e n e r g e t i c a l l y allowed reg ion , E > V, then the wave func t ion o s c i l l a t e s . The number of o s c i l l a t i o n s i s given by

X. b 1 pdx = I 42m(E- V) dx - - R *. 2niI 2nn 2 m

X a

R I s r e l a t e d t o t h e a c t i o n i n t e g r a l :

I f we normalize the energy t o zero then

R - Ldt = S . I

(5.25)

(5.26)

This i s t h e t o t a l a c t i o n of a motion from xa t o Xb wi th given energy. Note t h a t the tunnel ing process i s given by (5.24) which is the same express ion except t h a t t h e s i g n of V - E is f l ipped . The s i g n of t h e p o t e n t i a l i n :

d2x 3V m d P = a x ' (5.27)

is a l s o f l i pped i f w e r ep lace t by it. I f t h e r e i s no c l a s s i c a l l y allowed movement a t zero energy from xa t o xb a t t h e r e i s an allowed t r a n s i t i o n f o r imaginary t i m e s . We f i n d t h a t the t o t a l a c t i o n 3 f o r t h i s imaginary t r a n s i t i o n determines the tunnel ing amplitude". t o rep lac ing Minkowski space by Euclidean space.

t i m e s then

I n quantum f i e l d theory t h a t corresponds

The phys ica l tunnel ing amplitude from t h e supervacuum t o a gauge r o t a t e d vacuum i n a d i f f e r e n t homotopy c l a s s i s governed t o f i r s t approximation by t h e t o c a l a c t i o n 3 of a classical s o l u t i o n i n Eucl idean space. So, w e w i l l f i n d amplitudes propor t ion t o

The ine t an ton h a s , a s w e s a w , 5 = - 8 n Z / g 2 .

-8n2/g2 . e (5.28)

The supe rpos i t i on of t h e supervacuum and t h e gauge r o t a t e d vacua

276

G. ‘t HOOFT

g ives rise t o d i f f e r e n t p o s s i b i l i t i e s f o r the t r u e ground s t a t e of H i lbe r t space, l abe led by an a r b i t r a r y angle 0. phenomenon i e t o be found i n Refs. 15 and 16.

Discussion of this

5.3. Symmetry Breaking through Ins t an tons

The above holds f o r a gauge theory without fermions. Ins- tantone t h e r e rear range t h e ground state of t he theory. not show up as s p e c i a l events”.

They do

A new phenomenon occurs i f fermions a r e coupled t o the theory. Consider some gauge-invariant fermion cu r ren t J,,. s t rong background gauge f i e l d Ga p o l a r i z a t i o n :

I f t h e r e is a then t h e r e w i l l be a vacuum ,,v ’

That e f f e c t is computed i n a t r i a n g u l a r Feynman diagram (see Fig.7). One of t h e e x t e r n a l l egs denotes the space-time po in t x where J,,(x) is measured. The o t h e r two l egs are the two gauge photon e x t e r n a l l i n e e .

Fig. 7 . Triangular anomaly diagram.

277


Let us assume, without y e t spec i fy ing J , t h a t i t i s conserved according t o the Noether theorem. Then our diagram, which w e could c a l l

should s a t i s f y

(5.30)

(5.31)

Gauge invar iance f u r t h e r d i c t a t e s t h a t only t r ansve r se photons a re coupled :

karuaL3 - O (5.32)

(5.33)

Now the t r i a n g l e diagram, when computed, shows an i n f i n i t y t h a t must be subt rac ted . I t can happen t h a t t h e r e a r e only two sub- t r a c t i o n cons tan ts t o be chosen. Eqs. (5 .31 ) - (5 .33 ) form t h r e e condi t ions f o r these two cons tan ts . They can be incompatible. I n t h a t case w e have t o keep gauge invar iance i n order t o keep renormal izabi l i ty . Consequently (5.31) is The r e s u l t can be w r i t t e n a s

( 5 . 3 4 )

Here N is an i n t e g e r determined by the d e t a i l s which were l e f t ou t i n my discussion. This i s c a l l e d the Adler-Bell-Jackiw-anomaly.

Now consider Eq. ( 5 .12 )20 . We f ind t h a t an i n t e g e r number of charge unitsQ belonging t o the cu r ren t J are consumed by the ins tan ton: u

AQ - /d4x auJU = 2N . (5.35)

I n the s t rong- in t e rac t ion gauge theory, a cu r ren t J with t h i s property is the c h i r a l charge ( t o t a l number of quarks wi th h e l i c i t y + minus quarks wi th h e l i c i t y -). Since the r i g h t hand s i d e of (5.34) is a s i n g l e t under f l a v o r SU(3) i t is t he n i n t h a x i a l vec tor cur ren t which is nonconserved through t h e ins tan ton . This probably expla ins why t h e r e is no SU(3) s i n g l e t (or SU(2) s i n g l e t ) pseudo- s c a l a r p a r t i c l e as l i g h t as t he pion2’ s 2 2 .

278

G. 't HOOFT

Since the vec to r c u r r e n t s remain conserved, t he i n s t a n t o n appears t o f l i p the h e l l c i t y of one of each type of fermion involved (see Fig. 8 ) . be wri t ten i n terms of an e f f e c t i v e i n t e r a c t i o n Lagrangian. o r i g i n a l c h i r a l symmetry w a s

This symmetry breaking e f f e c t o f t h e i n s t a n t o n can The

" (N) l e f t U ( N ) r i g h t ( 5 . 3 6 )

It i s broken i n t o

s' (N) l e f t SU(N) r igh t '(''vector * ( 5 . 3 7 )

An e f f e c t i v e Lagrangian t h a t does t h i s breaking is

( 5 . 3 8 )

Actua l ly t h e e f f e c t i v e i n t e r a c t i o n is more complicated when c o l o r i n d i c e s are inc luded22 , and C s t i l l con ta ins powers of g. In the exponent, 0 i s an a r b i t r a r y angle a s soc ia t ed wi th t h e symmetry breaking.

Fig. 8. n

The i n s t a n t o n appears t o f l i p t h e h e l i c i t y type of fermion involved.

of one of each

279


Now le t us turn t o the instantons i n the weak and electromag- n e t i c guage theory SU(2) x U(1), with leptons, and non-srrange, s t range and charmed quarks which have color indices. This theory is anomaly-free, according t o its commercials, a s f a r as such is needed fo r renormalizabili ty. Indeed, i n t r i a n g l e diagrams where a l l three ex te rna l l i n e s are gauge photons of whatever type, the anomalies (must) cancel. But i f one of the ex te rna l l i n e s is one of the famil iar conserved currents l i k e baryon or lepton current ( t o which no known photons are coupled) then the re a re anomalies. We f ind , by in se r t ing the anomaly equation (5.34) with the correct values f o r N , t h a t one instanton gives

AQE - 1

AQM = 1

AQN = 3

AQc - 3

(electron-number nonconservation) ,

(muon-number nonconeervation) , (non-strange quark nonconservation) , (strangelcharm quark nonconservation) .

In t o t a l , two baryon and two lepton un i t s a r e consumed by the instanton (strange and non-strange may mix through the Cabibbo angle). Thus w e can g e t the decays

P N -t

NN +

P P -t

The order

+- +- e v or p v e , p - -

v v e p ’

e+p+ , e tc .

of magnitude of these decays is given by

(5.39)

With a Weinberg angle s in2% = .35, and assuming t h a t the weak intermediate vector boson determines the s c a l e , then we ge t l i f e t imes of the order of (give o r take many orders of magnitude)

T = 1 o Z z 5 s e c , (5.40)

corresponding t o one deuteron degay i n 10’ 3 7 universes.

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G. 't HOOFT

6 . SOME REFLECTIONS ON CLASSICAL SOLUTIONS AND THE QUARK CONFINEMENT PROBLEM

Clear ly , t h e s tudy of non- t r iv i a l classical f i e l d configura- t i o n s g ives us a welcome extens ion of our knowledge and understanding of f i e l d theory beyond the usua l pe r tu rba t ion expansion. One non-perturbative problem is p a r t i c u l a r l y i n t r i g u i n g : ment i n QCD. There has been wide-spread specu la t ion t h a t t h e new. classical s o l u t i o n s are somehow respons ib le f o r t h i s phenomenon. Some authors be l i eve t h a t some plasma of i n s t an tons o r ins tan ton- l i k e ob jec t s does t h e t r i c k . We w i l l not go t h a t f a r . We w i l l however e x h i b i t some s imple models wi th i n t e r e s t i n g p rope r t i e s . They w i l l c l e a r l y show t h a t a "phase t r ans i t i on ' ' towards permanent confinement is not a t a l l an absurd i d e a (it is a c t u a l l y much harder t o m a k e models with ' 'nearly confinement" of quarks).

quark confine-

Our first model is based on an SU(3) gauge theory i n 3 + 1 dimensions. The gauge group he re is n e i t h e r "color" "f lavor" , so l e t us ca l l i t "horror" (or : " t e r ro r " ) , f o r reasons t h a t w i l l become clear later ( t h e model does not desc r ibe the observat ions on quarks very w e l l ) .

L e t us assume t h a t t he SU(3) symmetry is spontaneously com- p l e t e l y broken by a convent ional Higgs f i e l d . The Higgs f i e l d m u s t be a %on-exotic" r ep resen ta t ion of ho r ro r (mathematically: a r ep resen ta t ion of SU(3) /2 (3 ) ) , e.g. an o c t e t o r decuple t repre- s e n t a t i o n .

I n the following we w i l l show t h a t t h i s model admits, l i k e the U ( 1 ) , and the SU(2) analogues, s t r i ng -vor t i ce s , bu t t hese a r e again d i f f e r e n t . t h a t they b ind t o the s t r i n g s i n the combination t h a t we a c t u a l l y see i n hadrons2 3 .

Afte r t h a t w e w i l l in t roduce quarks and show

To see the topologica l p rope r t i e s o f v o r t i c e s i n t h i s model w e cons ider , as we d id i n s e c t i o n 3, a two dimensional space, enclosed by a boundary which has t h e topology of a circle. vacuum a t t h i s boundary is s p e c i f i e d by a gauge r o t a t i o n fl(e), so the number of t dpo log ica l ly s t a b l e s t r u c t u r e s is given by t h e number of homotopy classes of t h e mapping e+f l (e ) . The group SU(3 h s an invari subgroup Z(3) given by t h e t h r e e elements I, eini931 and e' "V5I. Our phys ica l f i e l d s have been requi red t o be nva r i an t under Z(3). So, t he mappings wi th fl(2T)-

the t r ivial one: Q(2n) - fl(0). See Fig. 9. We conclude t h a t t h e r e are s t a b l e vo r t i ce s . Opposi te ly o r i en ted v o r t i c e s are d i f - f e r en t : t h e i r boundaries are i n d i f f e r e n t homotopy classes. But two v o r t i c e s o r i en ted i n t h e same d i r e c t i o n are in the same class as one vor tex i n the o t h e r d i r e c t i o n (see Fig. lo), because

The

e'2ni/ 4 N O ) are acceptable . They form two homotopy classes bes ides

281

EXTENOED OBJECTS IN GAUGE THEORIES

........ ................ ......... .................... ............... ....................... ................... .......................... ....................... ............................. ............ ........................... ........... ........................... ........ ............... ......................... .................... ......................... ............................ .

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............... ................. ........ ............ ..................... ........ (a) ( b ) (C)

Fig. 9 . Three homotopy c las ses of the mapping 8 + Q ( 8 ) .

Fig. 10. Decay of two paral le l vort ices into a s ing le vortex i n opposite d irect ion.

282

G. 't HOOFT

e 4.rri/3 = e-2ni/3, so two p a r a l l e l v o r t i c e s will decay i n t o an e n e r g e t i c a l l y more favorable s i n g l e vortex i n t h e o the r d i r ec t ion . Note t h a t th ree-s t r ing connections are poss ib le .

Quarks are now introduced as the end-points of s t r i n g s . They could be compared wi th magnetic monopoles i n s i d e a superconductor. Note t h a t quarks d i f f e r from ant iquarks and because of t h e topolo- g i c a l p rope r t i e s of these s t r i n g s they only occur i n t h e form of "hadronic" bound s t a t e s (see Fig. 11).

We could go i n t o lengthy d e t a i l s about a c t u a l cons t ruc t ion of quark monopoles i n the model, f o r i n s t ance by embedding S U ( 3 ) / Z ( 3 ) i n a l a r g e gauge group t h a t does not conta in Z(3) i t s e l f . these monopoles themselves are allowed as classical so lu t ions .

Then

But t he model is not very v i a b l e from a phys ica l po in t of v i e w because:

(i) The "horror" degrees of freedom are not known t o e x i s t ; where a r e the "horror"-vector bosons etc.?

(ii) The Dirac monopoles w i l l have monopole charges of t he order of 2n/gh0rror. So, i f quarks should behave as approximately f r e e par tons i n s i d e hadrons, ghor or should be so l a rge t h a t no pe r tu rba t ion expansion is l i k e l y t o make sense, and

(ffl) unl ike monopoles (see s e c t i o n 4), quarks are l i g h t , r e l a t i v i s - t i c , p a r t i c l e s .

( i v ) It i s n ' t Q.C.D.

One th ing on the o t h e r hand is very clear in t h i s model: because the s t r i n g s are topologica l ly s t a b l e quarks are abso lu te ly and permanently confined.

Fig. 11. Allowed quark and an t iqua rk bound states.

283


The next model w e cons ider is e n t i r e l y d i f f e r e n t from the

We claim previous one. It is pure SU(N) co lo r gauge theory (quantum- chromodynamics) b u t i n 2 space - one t i m e dimension. t h a t t h e r e is an e l egan t way t o formulate the quark confinement mechanism he re , although the dynamics is very complicated. For pedagogical. reasons l e t us f i r s t add a set of Higgs f i e l d s t o t h e sys tem, which must be a r ep resen ta t ion of SU(N)/Z(N) as i n the previous model. There is a complete spontaneous symmetry br'eak- down and obviously no quark confinement. L e t us a l s o leave out the quarks now. Since w e have only 2 space dimensions t h e model has no f lux tubes bu t i n s t e a d s o l i t o n p a r t i c l e s . The boundary of t h i s space can be i n one of N homotopy classes, so i f w e have more than N / 2 s o l i t o n s they may decay: an t i - so l i t ons is only conserved modulo N. important s t e p z 4 : one s o l i t o n (o r creates one an t i - so l i t on ) a t G,t) . enough t i m e now t o formulate a p r e c i s e d e f i n i t i o n of $ ( t , t ) . s u f f i c e s t o state t h a t in p a r t i c u l a r t h e Green's func t ions

t h e number of s o l i t o n s minus We now make a very

de f ine an opera tor f i e l d $ G , t ) t h a t des t roys

It There is no t

and

can be def ined accu ra t e ly i n terms of ord inary pe r tu rba t ion theory.25 We can r equ i r e t h a t $ @ , t ) produces o r des t roys o n 4 "bare" so l i - t p s : the vacuum (not supervacuum) remains a t a l l x ' with Ix'- X I 1 E b u t a l a r g e f i e l d conf igura t ion is produced where I P ' - t l < E ; then E -+ 0. axioms. We could assume t h a t they may be more o r less reproduced by an e f f e c t i v e Lagrangian:

-+

Then (6.1) and ( 6 . 2 ) s a t i s f y t h e Wightman

Here M is t he ca l cu lab le mass of t h e s o l i t o n . There w i l l be many in t e rac t ion ' t e rms ; the one w e wrote down is necessary t o make (6.2) d i f f e r e n t from zero. Observe a Z(N) symmetry t h a t leaves (6.11, (6.2) and ( 6 . 3 ) i n v a r i a n t :

Now what would happen i f we vary t h e Higgs p o t e n t i a l such tha t i ts second d e r i v a t i v e a t t h e - o r i g i n tu rns from negat ive t o

284

G. 't HOOFT

pos i t i ve? The f i e l d 4 remains w e l l defined. goes t o zero i f t h e Higgs mass goes t o zero. It i s the re fo re n a t u r a l t o assume t h a t a f t e r t h e t r a n s i t i o n M2 i n (6.3) has changed s ign.

The s o l i t o n mass M2

Our e f f e c t i v e Lagrangian now could be

We imagine t h a t then our Z ( N ) symmetry (6.4) w i l l be spontaneously broken! the dynamics, bu t if t h i s assumption is c o r r e c t then the o r i g i n a l model confines its quarks! Why? There a r e now N d i f f e r e n t vacum states f o r t h e f i e l d 4:

We stress t h a t we do not y e t understand the d e t a i l s of

<p2 = ... 2lriIN - e

I f t h e r e are two d i f f e r e n t vacuum conf igura t ions i n one plane then w e ob ta in "Bloch walls" sepa ra t ing them. These Bloch walls can be considered t o be c losed s t r i n g s . It is no t hard t o see t h a t i f we now add t h e quarks t o the o r i g i n a l model, then these w i l l form end- p o i n t s of t h e s t r i n g s ("Dirac monopoles"). Since the s t r i n g s ca r ry energy p e r un i t of length these quarks w i l l be permanently confined by a l inear p o t e n t i a l .

W e l i k e to look at the t r a n s i t i o n from t h e SU(N) gauge theory t o t he Z ( N ) scalar theory as a dual t ransformation. The s o l i t o n s i n one theory are the elementary p a r t i c l e s of t h e o t h e r . W e no te a p e c u l i a r antagonism: i f t h e S U ( N ) symmetry is spontaneously broken, t he Z ( N ) symmetry is i n t a c t and v i c e versa . Af t e r t he dua l t ransformat ion from t h e unbroken S U ( N ) theory t o the broken Z ( N ) theory, quark confinement i n t h e lat ter is obvious (once w e accept t h e symmetry breaking antagonism). 26 What about 3+1 dimensional quantmchromodynamics? Our f i n a l specu la t ion is t h a t the "horror" model discussed i n the beginning of t h i s s e c t i o n is i n a similar way t h e dua l transform of quantmchromodynamics. The dua l t r ans - formation i n 3'+1 dimensions is much more complicated than i n 2 + 1 dimensions, b u t aga in , one can show t h a t c e r t a i n d u a l Green's func t ions corresponding t o the dua l gauge f i e l d may be constructed. The s u b j e c t is obscured by t h e f a c t t h a t w e are no t a b l e t o cons- t r u c t Lagrangians from t h e Green's func t ions , and t h a t the ho r ro r coupling cons tan t is i nve r se ly p ropor t iona l t o the co lo r coupling cons tan t so t h a t i t is impossible t o compare t h e two per tu rba t ion series. However, i f t h e s e dua l t ransformat ions can be given a more sound foot ing then the quark confinement phenomenon w i l l no longer look as myster ious as i t does now.

285


REFERENCES

1) T.D. Lee and G.C. Wick, Phys. Rev. D9, 2291 (1974); T.D. Lee and M. Margul ies , Phys. R e v , = , 1591 (1975).

2) S . Coleman, i n New Phenomena i n Subnuclear Phys ics , I n t e r n a - t i o n a l School of Subnuclear Phys ics " E t t o r e Ma jorana-'I, Erice 1975, P a r t A, ed. A. Z i c h i c h i (Plenum P r e s s , New York, 1977).

"State" h e r e does n o t mean "state i n H i l b e r t space" b u t r a t h e r a s t a t i o n a r y c lass ical f i e l d c o n f i g u r a t i o n .

3)

4) H.B. Nie lsen and P. Olesen, Nucl. Phys. E, 45 (1973); H.B. Nie lsen i n Proceedings of t h e AdraticSummer Meeting on P a r t i c l e Physics , Rovinj , Yugoslavia 1973, ed. M. M a r t i n i s e t a l . (North Holland, Amsterdam, 1974).

5) B. Zumino, i n Renormalizat ion and I n v a r i a n c e i n Quantum F i e l d Theory, NATO Adv. Summer I n s t . , C a p r i 1973, ed . E.R.Caianiel lo (Plenum P r e s s , New York, 1974) .

H. Georgi and S.L. Glashow, Phys. Rev. L e t t . 32, 438 (1974). 6)

7) For s i m p l i c i t y our o r d e r parameter is chosen t o c a r r y a s i n g l e charge q=e, so that s i n g l e f l u x t u b e s are quant ized by u n i t s a l e .

8) G. I t Hooft , Nucl. Phys. E, 276 (1974); Nucl. Phys. m, 538 (1976); A. Polyakov, JETP L e t t . - 20, 194 (1974).

9) B. J u l i a and A. Zee, Phys. Rev. z, 2227 (1975).

10) S. Weinberg, Phys. Rev. L e t t . 2, 1264 (1967).

11) G. ' t Hooft , Nucl. Phys. E, 167 (1971).

12) J. Kogut and L. Susskind, Phys. Rev. =, 3501 (1974); K.G. Wilson, Phys. Rev. - D10, 2445 (1974).

13) A.A. Be lav in e t a l . , Phys. L e t t . E, 85 (1975).

14) S. Coleman, i n The Why's of Subnuclear Phys ics , Erice l e c t u r e n o t e s , Erice 1977.

15) R. Jackiw and C. Rebbi, Phys. Rev. L e t t . 37, 172 (1976).

16) C . Callan, R. Dashen and D. Gross, Phys. L e t t . =, 334 (1976).

1 7 ) It i s to b e remarked however t h a t t h e i n s t a n t o n e f f e c t s v i o l a t e P a r i t y v i o l a - p a r i t y c o n s e r v a t i o n i f 0 is n o t a m u l t i p l e of 71.

t i n g e v e n t s could then b e searched f o r .

286

G. 't HOOFT

18) S.L. Adler, Phya. Rev. 177, 2426 (1969).

19) J.S. Bell and R. Jackiw, I1 Nuovo Cimento g, 47 (1969).

20) The f a c t o r i should be added i n (5.12) s ince w e are now dea l ing wi th a Minkowski metric where time components of vec to r s are taken t o be imaginary.

21) H. Fr i t z sch , M. Gell-Mann and H. Leutwyler, Phys. L e t t . *, 365 (1973).

22) G. ' t Hooft, Phys. Rev. L e t t . - B37, 8 (1976), and G. ' t Hooft, Phys. Rev. m, 3432, (1976).

23) F. Engler t , Lectures given a t t h e C a r g h e Summer School, J u l y 1977.

24) S. Mandelstam, Phys. Rev. z, 3026 (1975).

25) G. 't Hooft, t o be published.

26) Compare t h e dua l t ransformation i n t h e 2 dimensional I s i n g model: t h e ordered phase is transformed i n t o t h e disordered phase and v i c e versa.

287

CHAPTER 4.2

MAGNETIC MONOPOLES IN UNIFIED GAUGE THEORIES

G. 't HOOFT CERN. Geneva

Received 31 May 1974

Abstract: I t is shown that in all those gauge theories in which the electromagnetic group U(1) is taken to be a subgroup of a larger group with a compact covering group, like SU(2) or SU(3), genuine magnetic monopoles can be created as regular solutions of the field equations. Their mass is calculable and of order 137 Mw, whereMw is a typical vector boson mass.

1. Introduction

The present investigation is inspired by the work of Nielsen et al. [ 11, who found that quantized magnetic flux lines, in a superconductor, behave very much like the Nambu string [2]. Their solution consists of a kernel in the form of a thin tube which contains most of the flux lines and the energy; all physical fields decrease exponentially outside this kernel. Outside the kernel we do have a transverse vector potential A , but there it is rotation-free: if we put the kernel along the z axis, then

A(x) a (v, -x, 0)/(x2 + Y 2 ) - (1.1)

A(x) can be obtained by means of a gauge transformation n(cp) from the vacuum. Here cp is the angle about the z axis:

SZ(0) = SZ(2n) = 1 . It is obvious that such a string cannot break since we cannot have an end point:

it is impossible to replace a rotation over 2n continuously by S2(cp) + 1. Or: magnetic monopoles do not occur in the system. Also it is easy to see that these strings are oriented: two strings with opposite direction can annihilate; if they have the same direction they may only join to form an even tighter string.

Now, let 11s suppose that the electromagnetism in the superconductor is in fact described by a unified gauge theory, in which the electromagnetic group U(l) is a subgroup of, say, SO(3). In such a non-Abelian theory one can only imagine non- oriented strings, because a rotation over 4n can be continuously shifted towards a fixed SZ. What happened with our original strings? The answer is simple: in an SO(3) gauge theory magnetic monopoles with twice the flux quantum (i.e., the

Nuclear Physics B79 (1974) 276-284. North-Holland Publishing Company

288

G. 't Hooft, Magnetic monopoles

Schwinger [3,4] value), occur. Two of the original strings, oriented in the same direction, can now annihilate by formation of a monopole pair [ 5 ] .

From now on we shall dispose of the original superconductor with its quantized flux lines. We consider free monopoles in the physical vacuum. That these monopoles are possible, as regular solutions of the field equations, can be understood in the following way. Imagine a sphere, with a magnetic flux @ entering at one spot (see fig. 1). Immediately around that spot, on the contour C, in fig. 1, we must have a magnetic potential field A , with $(A dx) = @. It can be obtained from the vacuum by applying a gauge transformation A :

A = V A . (1.2)

$ + $ enih , (1 -3)

This A is multivalued. Now we require that all fields, which transform according to

to remain single valued, so must be an integer times 2n: we then have a complete gauge rotation along the contour in fig. 1.

In an Abelian gauge theory we must necessarily have some other spot on the sphere where the flux lines come out, because the rotation over 2kn cannot continuously change into a constant while we lower the contour C over the sphere. In a non-Abelian theory with compact covering group, however, for instance the group 0(3), a rotation over 4n may be shifted towards a constant, without singularity: we may have a vacuum all around the sphere. In other theories, even rotations over 2n

Fig. 1. The contour C on the sphere around the monopole. We deplace it from CO to C1, etc., until It shrinks at the bottom of the sphere. We require that there be no singularity at that point.

289


may be shifted towards a constant. This is why a magnetic monopole with twice or sometimes once the flux quantum is allowed in a non-Abelian theory, if the electromagnetic group U(l) is a subgroup of a gauge group with compact covering group. There is no singularity anywhere in the sphere, nor is there the need for a Dirac string.

This is how we were led to consider solutions of the following type to the classical field equations in a non-Abelian Higgs-Kibble system: a small kernel occurs in the origin of three dimensional space. Outside that kernel a non-vanishing vector potential exists (and other non-physical fields) which can be obtained from the vacuum* by means of a gauge transformation 52(8,cp). At one side of the sphere (cos8 -+ 1) we have a rotation over 4n, which goes to unity at the other side of the sphere (cos 8 + -1). For such a rotation one can, for instance, take the following SU(2) matrix:

Now consider one rotation of the angle cp over 277. At 8 = 0, this 52 rotates over 4n (the spinor rotates over 2n). At 8 = R , this 52 is a constant. One easily checks that

ant = 1 . (1.5)

In the usual gauge theories one normally chooses the gauge in which the Higgs field is a vector in a fixed direction, say, along the positive z axis, in isospin space. Now, however, we take as a gauge condition that the Higgs field is 52(8,cp) times this vector. As we shall see in sect. 2, this leads to a new boundary condition at infinity, to which corresponds a non-trivial solution of the field equations: a stable particle is sitting at the origin. It will be shown to be a magnetic monopole. If we want to be conservative and only permit the normal boundary condition at infinity, with Higgs fields pointing in the z direction, then still monopole-antimonopole pairs, arbitrarily far apart, are legitimate solutions of the field quations.

2. The model

We must have a model with a compact covering group. That, unfortunately, ex- cludes the popular SU(2) X U(l) model of Weinberg and Salam [ 6 ] . There are two classes of possibilities.

(i) In models of the type described by Georgi and Glashow [7], based on S0(3), we can construct monopoles with a mass of the order of 137 Mw, where Mw is the

* As we shall see this vacuum will still contain a radial magnetic field. This is because the incoming field in fig. 1 will be spread over the whole sphere.

290


mass of the familiar intermediate vector boson. In the Georgi-Glashow model, Mw < 53 GeVlc2.

of processes with energies around hundreds of GeV, but may need extension to a larger gauge group at still-higher energies. Weinberg [8] proposed SU(3) X SU(3) which would then be compact. Then the monopole mass would be 137 times the mass of one of the superheavy vector bosons.

plest one. We take as our Lagrangian:

(ii) The Weinberg-Salam model can still be a good phenomenological description

We choose the first possibility for our sample calculations, because it is the sim-

(2.1) P=- 'Ga p 1 2 2 4 pu p v -iDpQaDpQa -21.1 Qa -$X(Q2I2

where

G& = a, w," - a& t e egbCw,b W; ,

DpQa = a p Q a +eeabcWiQc (2.2)

FV; and Q, are a triplet of vector fields and scalar fields, respectively.

vacuum expectation value [6,7,9] : We choose the parameter p2 to be negative so that the field Q gets a non-zero

(Qa)2 = F2 , p2 = -$AF2 . (2.3)

Mw,;2 = eF Y (2.4)

M ~ = J x F . (2.5)

Two components of the vector field will acquire a mass:

whereas the third component describes the surviving Abelian electromagnetic interactions. The Higgs particle has a mass:

We are interested in a solution where the Higgs field is not rotated everywhere towards the positive z direction. If we apply the transformation !2 of eq. (1 S ) to the isospin-one vector F(O,O, 1) we get

F(sin 8 cos cp, sin 8 sin cp, cos 8). (2 -6) We shall take this isovector as our boundary condition for the Higgs field at space- l i e infinity. As one can easily verify, it implies that the Higgs field must have at least one zero. This zero we take as the origin of our coordinate system.

We now ask for a solution of the field equations that is time-independent and spherically symmetric, apart from the obvious angle dependence. Introducing the vector

(2.7) 2 2 r* = (XYYYZ) Y r a = r ,

29 1

G. 't Hoof& Magnetic monopoles

we can write

where epab is the usual E symbol if p = 1,2,3, and E4& = 0. In terms of these variables the Lagrangian becomes

m

L = JL'd3x =4n r2 d r [ -r2 (Fr -4rW dW - 6W2 - 2er2W3

0

t iXF2r2Q2 - iXr4Q4 - :W4] , (2.9)

where the constant has been added to give the vacuum a vanishing action integral. The field equations are obtained by requiring L to be stationary under small variations of the functions W(r) and Q(r). The energy of the system is then given by

E = - L , (2.10) since our system is stationary.

r + OQ. From the preceding arguments we already know that we must insist on Before calculating this energy, let us concentrate on the boundary condition at

Q(r) + F / r . (2.1 1)

The field W must behave smoothly, as some negative power of r :

W ( r ) + ar-" . (2.12)

From (2.9) we find the Lagrange equation

= r2 [4r !!.!! t 12W t 6 e r 2 W 2 t 2e2r4W3 +2er2Q2 t2e2r4WQ2] . (2.13) dr So, substituting (2.1 1) and (2.12),

t 2eF2r2 t 2e2aF2r4-n . (2.14)

The only solution is

n = 2 , a = - l / e (2.1 5)

So, far from the origin, the fields are

292

G. 't Hmft, Magnetic monopoles

W;(x, r ) + - Epabr&r2 , Qa(X9 t ) + Fra/r . (2.16)

Now most of these fields are not physical. To find the physically observable fields, in particular the electromagnetic ones, FPv , we must first give a gauge invariant definition, which will yield the usual definition in the gauge where the Higgs field lies along the z direction everywhere. We propose:

(2.17)

because, if after a gauge rotation, Q, = IQ I(O,O, 1) everywhere within some region, then we have there

F~~ = a,w,3 - a,$ ,

as one can easily check. (Observe that the definition (2.1 7) satisfies the usual Maxwell equations, except where QQ = 0; this is one other way of understanding the possibility of monopoles in this theory.) From (2.16), we get (see the definitions (2.2)):

(2.18)

(2.19)

(2.20)

Again, the E symbol has been defined to be zero as soon as one of its indices has the value 4 . So, there is a radial magnetic field

B~ = ra/er3 , (2.2 1)

with a total flux

4 n / e .

Hence, our solution is a magnetic monopole, as we expected. It satisfies Schwinger's condition

eg- 1 (2.22)

(in units where h = 1). In sect. 4 , however, we show that in certain cases only Dirac's condition

1 e g = z n , n integer,

is satisfied.

293


3. The m a s of the monopole

Let us introduce dimensionless parameters:

w = W / F 2 e , q = Q / F 2 e ,

x = e F r , 0 = Ale2 = M$M$ . (3.1)

From (2.9) and (2.10) we find that the energy E of the system is the minima! value of

] (3.2) t x q dq - t f q 2 t 2x2wq2 t x4w2q2 - iDx2q2 + ipx444 t $ . d x

The quantity between the brackets is dimensionless and the extremum can be found by inserting trial functions and adjusting their parameters.

We found that the mass of the monopole (which is equal to the energy E since the monopole is at rest) is

Mm =-MwC(P) , (3.3) 4n e2

where C(0) is nearly independent of the parameter 0. It varies from 1.1 for to 1.44 for 0 = 10 *.

Only in the Georgi-Glashow model (for which we did this calculation) is the parameter Mw in eq. (3 .3) really the mass of the conventional intermediate vector bostm. In other models it will in general be the mass of that boson which corresponds to the gauge transformations of the compact covering group: some of the superheavies in Weinberg's SU(3) X SU(3) for instance.

= 0.1

4. Conclusions

The relation between charge quantization and the possible existence of magnetic monopoles has been speculated on for a long time [ 101 and it has been observed that the gauge theories with compact gauge groups provide for the necessary charge quantization [ l l ] . On the other hand, solutions of thefield equations with ab- normally rotated boundary conditions for the Higgs fields have also been considered before [ 1,121. Nevertheless, it had escaped to our notion until now that magnetic monopoles occur among the solutions in those theories, and that their properties are predictable and calculable.

* These values may be slightly too high, as a consequence of our approximation procedure.

294

G. ’t Hooft, Magnetic monopoles

Our way of formulating the theory of magnetic monopoles avoids the introduction of Dirac’s string [3], We expect no fundamental problems in calculating quantum corrections to the solution although they might be complicated to carry out.

in that model the mass is so high that that might explain the negative experimental evidence so far. If Weinberg’s SU(2) X U(l) model wins the race for the presently observed weak interactions, then we shall have to wait for its extension to a compact gauge model, and the predicted monopole mass will be again much higher. Finally, one important observation. In the Georgi-Glashow model, one may introduce isospin $ representations of the group SU(2) describing particles with charges * ie. In that case our monopoles do not obey Schwinger’s condition, but only Dirac’s condition

The prediction is the most striking for the Georgi-Glashow model, although even

q g = i ’ where q is the charge quantum and g the magnetic pole quantum, in spite of the fact that we have a completely quantized theory. Evidently, Schwinger’s arguments do not hold for this theory [ 131. We do have, in our model

A q g = 1 ,

where Aq is the charge-difference between members of a multiplet, but this is certainly not a general phenomenon. In Weinberg’s SU(3) X SU(3) the monopole quantum is the Dirac one and in models where the leptons form an SU(3) X SU(3) octet [ 141 the monopole quantum is three times the Dirac value (note the possibility of fractionally charged quarks in that case).

We thank H. Strubbe for help with a computer calculation of the coefficient C(@, and B. Zumino and D. Gross for interesting discussions.

References

[ 11 H.B. Nielsen and P. Olesen, Niels Bohr Institute preprint, Copenhagen,(May 1973); B. Zumino, Lectures given at the 1973 Nato Summer Institute in Capri, CERN preprint TH. 1779 (1973).

Breach, New York, 1970) p. 269; L. Susskind, Nuovo Cimento 69A (1970) 457; J.L. Gervais and B. Sakita, Phys. Rev. Letters 30 (1973) 716.

[2] Y. Nambu, Proc. Int. Conf. on symmetries and quark models, Detroit, 1969 (Gordon and

[3] P.A.M. Dirac, Proc. Roy. SOC. A133 (1934) 60; Phys. Rev. 74 (1948) 817. [4] J. Schwinger, Phys. Rev. 144 (1966) 1087. [ 5 ] G. Parisi, Columbia University preprint CO-2271-29. [6] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264. 171 M. Georgi and S.L. Glashow, Phys. Rev. Letters 28 (1972) 1494. [8] S. Weinberg, Phys. Rev. D5 (1972) 1962.

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[9] F. Englert and R. Brout, Phys. Rev. Letters 13 (1964) 321; P.W. Higgs, Phys. Letters 12 (1964) 132; Phys. Rev. Letters 13 (1964) 508;Phys. Rev. 145 (1966) 1156; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Letters 13 (1964) 585.

[ 101 D.M. Stevens, Magnetic monopoles: an updated bibliography, Virginia Poly. Inst. and State University preprint VPI-EPP-73-5 (October 1973).

[ 11 1 C.N. Yang, Phys. Rev. D1 (1970) 2360. I121 A. Neveu and R. Dashen, Private communication. [ 13) B. Zumino, Strong and weak interactions, 1966 Int. School of Physics, Erice, ed.

[14] A. Salam and J.C. Pati, University of Maryland preprint (November 1972). A. Zichichi (Acad. Press, New York and London) p. 71 1.

296

CHAPTER 6

INSTANTONS

Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5.1]

[5.2]

[5.3]

298

302 "Computation of the quantum effects due to a four-dimensional pseudoparticle", Phys. Rev. D14 (1976) 3432-3450 . . . . . . . . . . . . . . . . . . "How instantons solve the U(1) problem", Phys. Rep. 142

"Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking", In Recent Developments in Gauge Theories, Cargbe 1979, eds. G. 't Hooft et aZ., New York, 1990, Plenum, Lectutv 111, reprinted in Dynamical Gauge S p m e t r y B d n g , A Collection of Reprints, eds. A. Farhi, and R. Jackiw, World Scientific, Singapore, 1982, pp. 345-367 .. . . . . . . . . . . . . . -. . . . . . . . . . . . . . . .

(1986) 357-387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . 321

352

297

CHAPTER 5

INSTANTONS

Introduction to Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle [ 5.11

An introduction to instantons was already given in the first paper of the previous chapter. They are topologically stable twists in four dimensions. The classical solution in Euclidean space describes tunneling phenomena. In contrast with all other amplitudes in quantum field theory the instanton related amplitudes are exponentially damped when the coupling constant is small.

Now this confronted me with a problem. As explained earlier, quantum field theory cannot be trusted with infinite accuracy, because the loop expansion has a built-in deficiency: the divergence at very high orders, irrespective of the value of the coupling parameters. The margin of error in any amplitude due to this divergence in general cannot be avoided (with the possible exception of asymptotically free theories), and it is of the same order of magnitude as the instanton effects. Therefore, are the instanton effects real?

Fortunately, the instanton effects violate a symmetry, so that they can clearly be distinguished from ordinary perturbative contributions which in the corresponding channels vanish strictly. The dynamical origin of the instanton effects due to tunneling seems to be straightforward. And there seems to be no obstacle against calculating them. It was not hard to estimate what the results of such calculations would be.

But are these exponentially damped phenomena really completely self- consistent? The true coupling constant of the theory must be the renormalized one. Is that also the parameter in the exponent? Are instanton effects really finite? This was my motivation for performing some very detailed calculations. I wanted to know exactly how the nuts and bolts fit together here, and what the possible surprises would be. I decided to calculate the prefactor in front of the exponent.

298

The surprise was that everything works out fine. There were quite a few hurdles to be taken. The prefactor can really be seen as a oneloop quantum correction in the exponent itself. To compute it we needed the renormalized determinants of infinite dimensional matrices M, and several tricks were needed to obtain these. We first had to remove the zero modes, and for that we had to understand what their physical intepretation is. There are zero modes corresponding to translations and scale transformations, but also three due to global gauge rotations. The latter can only be understood in relation with the gauge fixing procedure. The zero modes, eight in total (if the gauge group is SU(2)) , give rise to an extra factor g-’ in front of the exponent.

The original paper contained a few minor errors. Now all errors act multiplic& tively on the final result, so as far as the final amplitude is concerned none of the errors was minor. But of course the real importance of our calculation was to see if it could be done at all, and if the answer would come out finite. The answer we found to this is of course not affected by small errors. Anyway, the version reproduced here has been reedited to remove all errors known to us. We owe some of these refinements to meticulous calculations by Ore and Hasenfratz. l6

The word “instanton” does not occur in the original publication. I had proposed this name in a previous paper, but the editors of Phys. Rev. Lett. and Phys. Rev. opposed the use of self-invented phrases. Their alternative (“Euclidean Gauge Field Pseudoparticle, EGFP for short”) was so ugly that the name instanton caught on quickly.

Introduction to How Instantons Solve the U(1) Problem [6.2]

Because of their ‘honperturbative’’ nature, instantons give rise to a number of con- trove=. A very important recent issue is the question whether the symmetry breaking effects associated with instantons continue to be exponentially damped when collision processes at very high energies are considered. A number of authors have presented calculations suggesting sharply increasing amplitudes, and at energies above a few tens of TeV’s the exponential could be completely cancelled so that the amplitudes could even hit the unitarity barrier. It is true that at such energies tunneling is formally no longer necessary. The transition could take place via so-called sphdemns. These are metastable field configurations resembling magnetic monopoles, but such that they can decay with equal probabilities into two channels with different winding numbers, hence different B and L.

But that the amplitudes should become strong is a highly dubious claim. Pre- cisely because Feynman diagrams can no longer be trusted (they diverge) c d c u b tions based on diagrammatic techniques may well be incorrect. My own attempt at constructing a physically reasonable scenario for the efficiency of sphaleron production at high energies suggests that exponential suppression factors, comparable to

16F. R. Ore, Phys. Rev. Dl8 (1977) 2577; A. Hasenfrats and P. Hasenfratz, Nucl. Phys. B193 (1981) 210.

299

the one at low energies, exp (-87r2/g2), continue to be present at all energies. Only in high temperature plasmas such as the one that must have existed in the very early universe, the conditions for efficient sphaleron production seem to be realized.

So much for instantons in weak interaction theories. In the strong interactions they also occur, and here the exponential suppression is not important. The symme try violated by instantons is chiral symmetry. This observation finally provided for a satisfactory answer to a famous problem in QCD: the U( 1) problem. According to its Lagrangian namely, QCD possesses a nearly perfect global (chiral) symmetry of the form U ( 2 ) @ U ( 2 ) , only broken by the light quark masses into the diagonal U ( 2 ) subgroup of this group. This group has eight independent generators. PhenomenG logically however, there is only a global SU(2) @ SU(2) @ U(1) chiral symmetry, a group with seven generators, spontaneously broken into SU(2) @ U(1). The three broken symmetry generators give rise to three Goldstone bosons, 7r+ , 7ro and 7r-. This means that one U(1) subgroup of U(2) @ U ( 2 ) is missing. The corresponding Goldstone boson would be one with the quantum numbers of the q particle. The mystery is that for this the q particle seems to be too heavy (550 MeV). Why is it so much heavier than the pions (140 MeV)? Remember that we have to compare the squares of the masses rather than the masses themselves. A spurious interaction, notably absent in the QCD Lagrangian, must be responsible for this explicit chiral symmetry breaking.

In my 1975 Palermo paper (Chapter 2.3) I already indicated what a number of authors had observed: chiral U(1) is explicitly broken by an anomaly. The only remaining problem is: how does this anomaly raise the mass of the q particle? The answer is that the extra interaction responsible for this is precisely the one effec- tuated by the instanton. Indeed, the effective interaction vertex due to instantons has precisely the quantum numbers of an q mass term.

But things are not quite this simple. In a number of papers, Rodney Crewther criticised this point of view. He claimed that the theory, including all its instantons, still obeys what he called “anomalous Ward identities”, and that these identities prohibit an q mass term.

These arguments seemed to be formal mathematics to me, not reflecting the true physical nature of the instanton-induced interactions. Then, in 1984, a pupil of Crewther’s, G. A. Christos, published a review paper in Physics R e p o 7 - t ~ ~ ~ clarifying Crewther’s arguments against the instanton solution of the U(1) problem. This paper was written in a style that compelled me to react. The “anomalous Ward identities” do not refer to amplitudes relating to physically observable quantities, and do not have a proper large distance limit. The easiest way to see what happens is to write down effective Lagrangians that exhibit all symmetries and selection rules (including the spurious Ward identities) of the expected instanton interactions. This was published in another issue of Physics Reports, of which we include a reprint.

17G. A. Christos, Phys. Rep. 116 (1984) 251.

300

Introduction to Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking [5.3]

The next paper is only indirectly related to instantons. It shows how topological winding numbers can be used to deduce information about the spectrum of light bound states in theories such as QCD. The paper, one of a series of lectures presented in a summer school in Cargbe in 1979, speculates on how the Standard Model may be extended beyond the magic threshold of 1 TeV. At present most particle physicists adhere to the theory that beyond that energy range particles form supersymmetric multiplets. This could be a natural way to protect the Higgs scalar from becoming too heavy. At the moment this is written there is however still no single direct evidence for supersymmetry, and certainly in 1979 there wasn’t.

If there is no supersymmetry then there must be strong interactions in the TeV range. The symmetry pattern of these interactions must then be such that the presently observed particle spectrum is the spectrum of low-lying bound states. How do we compute this spectrum in a given theory? Take QCD as an example. At low energies we have a chird Q model. Chird symmetry is spontaneously broken, but in other versions of this dynamicd system c h i d symmetry may be realized in a different way. We could have massless nucleons instead of massless pions. How many massleas bound states would be protected by the symmetries? What are the rules for more complicated variations of the theme “QCD”? This is the question the paper addresses. And an important theorem is discussed: all triangle anomalies for all fermionic currents (in particular the ones to which no gauge fields are coupled) must be the same for the bound state spectrum as they are for the original “bare” constituent particles. This leads to index theorems constraining the possibilities for the bound state spectrum.

The theorems were not powerful enough to constrain the spectrum completely, so I tried to deduce more theorems. They are discussed in Section 3.12.

The latter however are mere conjectures. It turns out that they cannot be completely true because the resulting spectrum would have fractional occupation numbers which is clearly absurd. My own conclusion was therefore that the various chiral symmetries must be spontaneously broken 80 that instead of massless fermions we get massless scalars. This would kill the bound state theories for quarks and leptons, and, disappointingly, such was my conclusion. Many authors since then argued that the conjectures of Section 3.12 are too stringent. But even the anomaly matching conditions alone (which surely must be true) are so restrictive that no really attractive bound state model for quarks and leptons could be produced. Does this imply that we have to accept the supersymmetry scenario? Or is Nature simply too complicated for us to guess how things go from what we know at present? Only the h t u r e will tell how stupid we are now.

301

P H Y S I C A L R E V I E W D 0 Amcrican Physical Society

CHAPTER 5.1 1 5 D E C E M B E R 1 9 7 6

Computation of the quantum effects due to a four-dimensional pseudoparticle*

G. ’t Hooft’ Physics hbomrorier, Harvard Universily, Cambridge, Masrachuserrs 02138

(Received 28 June 1976)

A detailed quanlitativc calculation is carried out of the tunneling process dcwribed by the Belavin-Polyakov- Schwan-Tyupkin field confi~umtion. A certain chiral symmetry is violated as a consequence of the Adler- Bell-Jackiw anomaiy. ?he collective motions or the pseudoparticle and all contributions from single loops of scalar, spinor. and vector fields are takcn into acwunt. The result is an elTcctivc interaction Lagrangian for the rpinors

I. INTRODUCTION

When one attempts to construct a realistic gauge theory for the observed weak, electromagnetic, and strong interactions, one is often confronted with the difficulty that most simple models have too much symmetry. In Nature, many symmetries are slightly broken, which leads to, for instance, the lepton massea, the quark masses, and C P violation, These symmetry violations, either explicit o r spontaneous, have to be introduced artlflcially in the existing models.

There 1s one occasion where explicit symmetry violation is a necessary consequence of the laws of relativistic quantum theory: the Adler-Bell-Jackiw anomaly. The theory we consider i s an SU(2) gauge theory with an arbitrary set of scalar fields and a number, N’, of massless fermions. The apparent chiral symmetry of the form UW’) X U(”, is actually broken down to SUyU/) X SU(N/) X U(1). This paper is devoted to a detailed computation of this effect.

The most essential ingredient in our theory Is the localized classical solution of the field equations in Euclidean space-time, of the type found by Belavin st nl.’

Although the main objective of this paper 1s the computation of the resulting effective symmetry- breaking Lagrangian in a weak-interaction theory, we present the calculations in such a way that they can also be used for possible color gauge theories of stong interactions based on the same classical field configurations. For such theories our intermediate expressions (12.5) and (12.8) will be applicable. Our final results a re (15.1) together with the convergence factor (15.8).

Our general philosophy has been sketched in Ref. 2. We a r e dealing with amplitudes that depend on the coupling constant g in the following way:

The coefficient a, involves one-loop quantum cor-

302

rections, and it determines the scale of the amplitude. Clearly, then, to understand the main features of such an amplitude, complete understanding of all one-loop quantum effects is desired. For instance, if one changes from one renormalization subtraction procedure to another, so that 2-2 + O(g‘), then this leads to a change in (1.1) by an overall multiplicative constant. Thus, the renormalization subtraction point p may enter a s a dimensional parameter in front of our expressions. This is just one of the reasons to suggest that our results will also have interesting applications in strong- interaction color gauge theories.

The underlying classical solutions only exist in Euclidean space, but they give rise to a particular symmetry- breaking amplitude that can easily be continued analytically to Minkowski space. We interpret this amplitude as the result of a certaln tunneling effect from one vacuum to a gauge-rotated vacuum. We recall that, Indeed, tunneling through a barr ier can sometimes be described by means of a classical solution of the equation of motion in the imaginary time direction.’Ps

vacuum amplitude in the presence of external sources, thus obtaining full Green’s functions. Of course, we must limit ourselves to gauge- invariant sources only, but that wlll be no problem. It turns out to be trivial to amputate the obtained Green’s function and get the effective vertex.

We first give in Sec. II the functional integral expression for the amplitude, first in a conventional Feynman gauge: C,= 8,A,. Later, we go over to the so-called background gauge: C,= D,Ar. This i s actually only correct up to an overall factor, as wi l l be explained in Sec. XI. It is just for pedagogical reasons that we ignore this complication for a moment. It is in this gauge that the quantum excitations take a simple form: “Spin- orbit” couplings commute with the operator La

We compute in Euclidean space the vacuum- to-

The various calculational steps a r e the following.

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E TO A , .

= - t L,,L,,, where

so that we can look at eigenstates of L'. (In other gauges only total angular momentum j= t+ spin t isospin is conserved, not La.)

described by an eigenvalue equation, In Sec. III we consider the quantum fluctuations

nJ,-.W, (1.3)

in order to compute det9n. Now this looks like an ordinary scattering problem (in 4 + 1 dimensions), and, indeed, we show that the product of all non- zero eigenvalues E, in some large box, can be expressed in terms of the phase shift q(k) a s a function of the wave number k.

In Sec. IV we show that Eq. (1.3) i s essentially the same for scalars, spinors, and vectors, from which we derive the important result that the product of all nonzero eigenvalues is the same for scalar fields a s for each component of the spinor and vector fields. So, we turn to the (much easier) scalar case first.

But even for scalars Eq. (1.3) has no simple solutions In terms of well-known elementary functions. We decide not to compute d e m by 801v- ing ( l . Z ) , but we compute instead

det[(l +ra)X(l tw')]

The factors 1 t x a drop out if we divlde by the same determinant coming from the vacuum (h, the case An'= 0). The equation

(1.4)

is a simple hypergeometric equation that can be solved under the given boundary condition (Sec. V) .

Now we must find the product of all eigenvalues A, but that diverges badly even if we divide by the values they take in the vacuum. We must find a gauge-invariant regulator, and the regulator determinant must be calculable. Dimensional regularization is not applicable here, but we can use background Pauli-Villars regulators. They give messy equations unless they have a space- time-dependent mass:

The rules a r e formulated in Sec. V. In Sec. VI we compute the product of the eigen-

values using this regulator. In Sec. VII we make the transition to regulators with fixed mass by observing that a change toward fixed regulator mass

303

must correspond to a local, space-time-dependent counterterm in the Lagrangian. The effect of this counterterm i s computed.

Using the result of Sec. IV we now find also the contributions of all nonvanishing eigenvalues for the vectors and spinors. But there a re also vanishing eigenvalues. They are listed in Sec. VIII. For the vector fields, we have eight zero eigenvectors in addition to the ones computed via the theorem of Sec. IV. They are to be interpreted a s translations (Sec. W , dilatation (Sec. X), and isospin rotations (Sec. XI). The last need special care and can only be interpreted correctly when different gauge choices a re compared. This leads to the factor mentioned in the beginning.

In Sec. XI1 we combine the results so far obtained and add the fermions. This intermediate result may be useful to strong-interaction theo- rtes. In Sec. Xm we reexpress the result in terms of the dimensionally renormalized coupling constant gD, a s opposed to the previous coupling constant which was renormalized in a Pauli-Villars manner. In Sec. XIV the external sources for the fermions are considered and the amputation operation for the Green's function is performed. We obtain the desired effective Lagrangian, but there is still one divergence. So far, we only had massless partlcles, and as a consequence of that there i s still a scale parameter p over which we must integrate. Asymptotic freedom gives a natural CUtDff for this integral in the ultraviolet direction, but there is still an infrared divergence. In weak-interaction theories the Higgs field is expected to provide for the infrared cutoff. Sec- tion XV shows how to compute this cutoff.

q, Ti which are used many times throughout these calculations.

The Appendix lists the properties of the symbols

11. FORMULATION OF THE PROBLEM

Let a field theory in four space-time dimensions

e = - t G:YC:Y - D, *'Dub - %,D,$ t ?,& 9, , be given by the Lagrangian

(2.1)

Giv=O,,Rv- B,AO,tgr,,A~A,A: . (2.2)

where the gauge group is SU(2):

The SU2) indices will be called isospin indices. The scalars*, taken to be complex, may contain several multiplets of arbitrary isospin:

(2.3)

The spinors (I itre taken to be isospin-b doublets.

C . ' t H O O F T

The total number of doublets i s N'. Mass terms and interaction terms between scalars and spinors a r e irrelevant for the time being.

We inserted a source term in a gauge-invariant way, with respect to which we will expand, in order to obtain Green's functions. The indices s , t = 1,. . . , N', called flavor indices, label the different isospin multiplets. Isospin and Dirac indices have been suppressed. d,, must be diagonal in the isospin indices but may contain Dirac y matrices.

The system (2.1) seems to have a chiral UyU') X UYJ') global symmetry, butactuallyhasanAdler- Bell-Jackiw anomaly4 associated with the chiral U(1) current, breaking the symmetry down to SU(N') X SU(") X U(1). The aim of this paper is to find that part of the amplitude that violates the chiral U(l) conservation.

tude is The functional integral expression for the ampli-

w = out (0 I @I.

=JaADdl,WD@

xexp {I [e- t CI2(A) + B y (@)I d x } , (2.4)

to be expanded with respect to d. Here C,(A) is a gauge-fixing term, and are the corresponding ghost terms.

A s is argued in Ref. 2, the U(1)-breaking part of this amplitude comes from that region of superspace where the A field approaches the solutions described in Ref. 1:

where 2, and p a r e five free parameters associated with translatlon invariance and scale invariance. The coefficients 9 a r e studied in the Appendix. Conjugate to (2.5) we have its mirror image, described by the coefficients 5 (see Appendix).

Now these solutions form a local extremum of our functional integrand, and therefore it makes sense to consider separately that contribution to Win (2.4) that is obtained through a new perturbation expansion around these new solutions, taking the integrand there to be approximately Gaussian. The fields 8 , @,, and $ all remain infinitesimal so that their mutual interactions may be neglected in the first approximation. Of course, we must also integrate over the values of zll and p. This will be done by means of the collective- coordinate formalism.5 One writes

and those values of A'" that correspond to translations or dilatations a r e replaced by collective co-

304

ordinates.

e(A") - t (D, A:")'t t (D,AY)' - gA: '"c,,C::'A','"

The integrand in (2.4) now becomes

- D , 8 * D u 8 - G U D u + t @+- f C,'+ efbu +W4'",8 ,$) ' , (2.7)

where

S"= IC(Aa')d%

= - 8nZ/$ , (2.8)

and the "covariant derivative" D, only contains the background field A:', for instance:

D , , q '"= B,,At q U + g c a k A ~ E ' A ~ q U , (2.9)

etc. We abbreviate the integral over (2.7) by

sCL- k q ' m A ~ q u + i j ~ , $ - o*ml.o- +*ma&, (2.10)

where the last term describes the Faddeev-Popov ghost, Thus, expression (2.4) is (ignoring temporarily the collective coordinates, and certain factors f rom the Gaussian integration; see Sec. u()

W = exp(- 84/gz)(detJn,)~L'Zdet~,(det;m,)-'

x det mr,, . (2.11)

The determinants will be computed by diagonalization:

J b $ = E i $ 8 (2.12)

after which we multiply all eigenvalues E. Since there are infinitely many very large eigenvalues, this infinite product diverges very badly. There are two procedures that will make it converge:

(I) The vacuum- to-vacuum amplitude in the absence of sources must be normalized to 1, so that the vacuum state has norm 1. This implies that W must be divided by the same expression with A" = 0.

(ii) We must regularize and renormalize. The dimensional procedure is not available here because the four-dimensionality of the classical solution is crucial. We will use the so-called background Paull-Villars regulators (Secs. IV and V).

Taking a closer look a t the eigenvalue equations (2.12) as they follow from (2.7), we notice that the background field in there gives rise to couplings between spin, isospin, and (the four-dimensional equivalent of) orbital angular momentum, through the coefficients qaWv in (2.5). Now these couplings

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E T O A

simplify enormously if we go over to a new gauge that explicitly depends on the background fielde

G(Aq") = 0, AtqU

8,A~uU+g<,0A~C1A,6u". (2.13)

Thus the third and eighth terms in (2.7) cancel.

tion, to be dlscussed In See. XI: The gauge for the vacuum-to-vacuum amplitude in the absence of sources, used for normalization, in the region A, - 0, is usually invorlant under global lsospin rotations, but the classical solution (2.5) and the gauge (2.19) are not, Associated with this will be three spurious zero eigenvalues of Sn, that cannot be dlrectly associated with global isospin rotations. The question is resolved in Sec. XI by careful comparison of the gauge C, with C, and some intermediate choices of gauge.

There will be five other zero eigenvalues of X A that of course must not be inserted in the product of eigenvalues directly, since they would render expression (2.11) infinite. They exactly correspond to the infinitesimal translations and dilatations of the clnesical solution, and, (LB

discussed before, must be replaced by the corresponding collective coordinates (Seca. M and X).

The matrices 9R are now (ignoring temporarily the fermion source)

This choice of gauge will lead to one complica-

3nAAt"-- 2gt,,G:,01A,0eu,

rn&=-DS+. In order to substitute the ClaSslcal solution (2.5)

with z = 0 md p = 1 (generalization to other z and p will be straightforward), we introduce the apace- time operators

(2.15) 8

-G= - t { q , " x ' g , with

[L;,L:I*fo,,&L;, (2.16)

L'EL,'. L,P= - t (x ,e, - ~ ~ 8 , ) ~ .

They represent rotations in the huo invariant SU(2) subgmupa of the rotation group SO(4).

Isoepin rotatione will be generated by the operators T' tor the scalars, T.=f7. for the spinors,

305

(2.17)

where 1.1 = (x - 2)'. This clearly displays the isospin-orbit coupling.

The vector and splnor fields also have a spin- isospin coupling. Far the spinors we define the spin operators

(2.181

satisfying

and

(2.20)

For the vector fields we define

SYP-~LAF, S;Ar=-tfq,,,A:V, (2.21)

sIp=s:= t . For the scalar fields s, = &= 0. Thus right- and left- handed spinore are (3,O) and (0, f ) representations of 50(4), and vectors a r e (i,t) representations. Scalars of course are (0,O) representations. h terms of the operators S and T, the spin-

isospin couplings turn out to be universal for all particles. Substituting the classlcal value for G$ in (2.14) we find

(2.22)

with 317=311, or - X t or 311, or 32,. Observe the abEenC8 of spin-orblt and Isoepin-

orbit couplings that contain xu or 8 /8x , explicitly. It all goes via the orbital angular momentum operator L, and thnt implies that La commutes with 9R. This would not be so In other gauges. Further, 9R commutes wlth 3, = el+ 8, + and & and 8,. Eigen- vectors of 311 can thus be characterized by the quantum numbers

G . ' t H O O F T

s, and s1 (both either 0 or t) , t (total isospin, arbitrary for the sca la rs , a for the spinors, 1 for the vector and the ghost),

l = O , i , l , ... , j , = I - s, - t , I - sl - t + 1,. . . , I +s, + t , a s long a s j , 2 0 ,

j , 3 = - j ,,... , + j , ,

sl==- S1). . . , + s a t

1 ; = - 1 , ... , + 1 .

(2.23)

For normalization we need the corresponding operator 9R for the case that the background field is zero:

(2.24)

111. DETERMINANTS AND PHASE SHIFTS

The eigenvalue equation (2.12) with I a s in (2.22) differs in no essential way from an ordinary Schrlldinger scattering problem. In this section we show the relation between the corresponding scattering matrix and the desired determinant.

Temporarily, we put the system in a large spherical box with radius R. At the edge we have some boundary condition: either 9(R) = 0, or W(R) = O (or a linear combination thereof). Here rIr stands for any of the scalar, spinor, or vector fields. In the case W(R) = 0 the vacuum operator 3lt, has a zero eigenvalue corresponding to Q =constant, and also the lowest eigenvalue of Sn may go to zero more rapidly than 1/R2 when R - -. Such eigenvalues have to be considered separately (negative eigenvalues can be proved not to exist).

We here consider a l l other eigenvalues of N. They approach the ones of "2, if R - -, We wish to compute the product

The scattering matrix S(k) =ez'"(b' will be defined by comparing the solution of

Xly = kZO (3.2)

with

Xo#,= k2rIro (3.3)

both with boundary condition 9- CP' at r = 0. Let

) r ~ r , ( ~ ) - ~ ~ - 3 / Z ( ~ - I b ( ? + o > + &b(r+o)

= 2Cr-3'a cosk(r+a) for large Y (3.4)

30

The level distance Ak = k(n + 1) - k(n) is in both cases, asymptotically for large R,

(3.8)

We find that

provided that the integral converges at both ends. At k-0 the integral (3.9) converges provided

that the interaction potential decreases faster than l/P a s r - m ; at k- - the integral converges if the interaction potential i s less singular than l/P a s r - 0. The latter condition is satisfied if we compare 311 and Sn, at the same values for the quantum number I ; the f i rs t condition is satisfied if (L,+ T)' for the interacting matrix is set equal to La for the vacuum matrix. If we consider the combined effect of all values for Lz and (4+ T ) a both for the vacuum and for the interacting case then we can split the integral (3.9) somewhere in the middle, and combine the k-- parts, so that we get convergence everywhere.

6


An easier way to get convergence is to regularize:

(3.10)

Regulators will be introduced anyhow, SO we will not encounter difficulties due to non-convergence of the integral in (3.9).

IV. ELIMINATION OF THE SPIN DEPENDENCE

In Eq. (2.22) the operators T * L , and T - S , do not commute. Only in the case that

l j l - Z I = s + t

(as defined in 2.23) do they simultaneously diagonalize. If

I j , - l l c s t t

jl+o, I+O, s = + , t # ~ ,

and

then we have a set of coupled differential equations for two dependent variables.

solving this set of equations analytically, but here we can make use of a unique property of the equation

;mQ=E+, (4.1)

In any other case there would be no hope of

which enables us to diagonalize it completely. If sI = 0 the equation could describe a left-handed fermion wlth isospin t:

(4.2) y5J,=+J,.

V - r - D l l (4.3)

But then we can define, if E # 0 ,

with

3n4J' = EJ,' , Y5$' = - $'

Now (I' has s; = i, and hence we found a solution for the set of coupled equations with s:=& from a solution of the simpler equation with sI = 0. The operater y* Din Eq. (4.3) does not commute with La, so if J, has a given set of quantum numbers ZJ,, t then Q 1s a superposition of a statewithl' = 1 t i and one with Z ' = 1 - i. Now I does commute with L', so if we project out the state withZ'=l++ or I' - I - 4 thenwe get a new solution in both cases. Thus one solutionwith s, = 0 and quantum numbers

307

Z,j,,# generates two solutions with s;=$, l ' = l i $ , j;=jl, t ' = f . In terms of the operators L , S, and T the new solutions to the coupled equations can be expressed in terms of the S= 0 solutions a s follows:

L:ts:=L,, l ' = l * f ,

T '= T I J' = J , (4.5)

It is easy to check explicitly that if 0 satisfies (2.22) with S, = 0, then the two wave functions Q' both satisfy (2.22) when the operators L , S , T are replaced by the primed ones.

Asymptotically, for large r , W = (8/87)*, and hence the phase shift q(k) is the same for the primed case as for the original case. Consequently, the integral over the phase shifts a s it occurs in (3.9) is the same for spinor and vector fields (with sI= i) a s it is for scalar fields (with s,=O).

The above procedure becomes more delicate if E = 0. Indeed, although scalar fields can easily be seen to have no zero-eigenvalue modes, spinor and vector fields do have them. In conclusion, the nonzero eigenvalues for the vector and spinor modes a re the same a s for the scalar modes, but the zero eigenvectors a re different.

In the following sections we compute the universal value for the product. Note that also the regularized expressions (3.10) are equivalent because the q(k) match for all k. The regulator of Eq. (3.10) corresponds to new fields with Lagranglans

c = - & ( D , B , ) ~ - ~ M ~ B , ~ - ~ B ; E , , , G : , B C , (4.7)

C = i [ - ( y -D)z - Mz] x for spinors , (4.8)

for vectors,

and

a=- (o,,[)*o,~-&'[*~ for s c a a r s . (4.9)

Within the background field procedure it is obvious that such regulator fields make the one-loop amplitudes finite. Later (Sec. X U ) we wfll make the link with the more conventional dimensiona1 regulators.

V. A NEW EIGENVALUE EQUATION AND NEW REGULATORS

As stated in the Introduction, the solutions to the equations 3R-9 = EP even in the scalar case cannot be expressed in terms of simple elementary functions. But eventually we only need det3lZ/ demo, and this can be obtained in another way.

G . ' I H O O F ?

We write

V=!(1+*3)fSL(l+r'),

v o = ~ ( l + ~ ) a u e ( l + ~ ) ,

and, formally,

det@ZhRJ = det(V/VJ . The equation

(5.1) V* = x*

corresponds to the expression

This Is a hypergeometrlc equation. The physical region is 1/(1+ Ra) < X 6 1. In the Hilbert space of square-integrable wave functions the spectrum is now discrete, which implies that we can safely take the limit R-m. The solutions for rb a r e just polynomials:

* ( x ) = a&,%", Y.

(5.7)

where n Is defined by

(n + I + j , + 1) (n + I + j ,+ 2) = la+ A . (5.9)

If n = integer Otherwise rb is not square-integrable. So we find the eigenvalues

A,= (n+l+ jl + 1 - t ) (n+l+ j, + 2 + t ) , (5.10)

n=0,1,2 ,..., Po=t( t+ l ) .

0 then the series (5.7) breaks off.

The vacuum case, Vo*=XoU, is solved by the same equation, but with j, = I , t = 0.

The product of these eigenvalues, even when divided by the vacuum values, still badly diverges so we must regularize. The regulators of Sec. N a r e not very attractive here because they spoil the hypergeometric nature of the equations. More convenient here is a set of regulator fields with masses that all depend on space-time in a certain way. They are given by the Lagrangians (4.7)-(4.9) but with Ma replaced by

(5.11)

308

We choose Ma here so large that anywhere near the origin the regulator i s heavy. Far from the origin the classical solution is expected to be close enough to the real vacuum, so that there the details of the regulators a r e irrelevant.

Of course the regulator procedure affects the definition of the subtracted coupling constant. In Sec. VII we link the regulator (5.11) with the more acceptable one of Sec. IV, and In Sec. XIII we make the link with the dimensional regulator.

The eigenvalues of the regulator are

A:= ( n + l + j , + 1 - I ) ( f l + l + j , + 2 + f)+nn'. (5.12)

The regulators M, with i = 1,. . . ,R a r e aa usual of alternating metric e , = i 1. Consequently, detsR i s replaced by

This converges rapidly if

$ el=- 1 ,

and

$ e , l n M , = - lnM=finite.

Let i = 0 denote the physical field, then

e o = l , Mo=O, e,=O, etc. $

(5.13)

(5.14)

(5.15)

(5.16)

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E T O A . . .

VL THE REGULARIZED PRODUCT OF THE NONVANISHING EIGENVALUES

We now consider the logarithm of the regularized product of the nonvanishing eigenvalues, for a scalar field with total isospin t :

with

XYI = (n+I + j l t 1 - t ) (n+Z + j , + 2 + t ) + M t (6.2)

(we imply that e, = 1 and Mo = 0). The summation goes over the values of all quantum numbers. Now for given n , l , j , the degeneracy is @ j , + l ) (2ja+ 1) = ( Z j , + 1) (21 + 1). The values of 1 , j , , and n are restricted by

(6.3) us2 +jl - t 2 0 ,

~ = j , - 1 + 1 ) 0 , 1 5 2 t , n=O.

[Later we will divide n(f) by the vacuum value lTo(t), which Is obtained by the same formulas as above and the following, but with 1 replaced by zero, and the degeneracy will be (2t+ 1) (21 + l)'.] We go over to the variables u and T a s given

by (6.3) and s with

s =n+l +jl + f , s rt + u+ 4. (6.4) We find that

The summation over u and r gives

X ln[S'+M,' - ( t+ t ) ' ] .

(6.6)

The vacuum value lI,(t) is obtained from (6.6) by

309

replacing t with zero and adding an additional multiplicity 21 + 1, thus

x ln(sa+M,l- t ) . (6.7)

Now we interchange the summation over s and i , letting first s go from t + 3 to A and taking A- in the end. We get

with

AM'($)= $;, (s3- sga)ln(sa+M,*- $a). (6.0) a

Let us first consider the regulator contribution. Then M i s large. We may c6nsider the logarithm 88 a slowly varying function, and approximate the summation by means of the Euler-Maclaurin formula,

- By"(%) * * .] 1; , (6.10)

and we obtain

An(@) = indep(9) + $'(- f M a - A'ln A- A WL

- 4 A - M- 4 - i lnM)

++ 94(2 w+ 1)+0 (-$) + o( i ) .

(6.11)

The f i rs t term stands for an array of expressions, all independent of $, and is not needed because It cancels out in Eq. (6.8). For Ao(@) the ser ies (6.10) will not converge at

x = p 80 it cannot be used. After some purely algebraic manipulations we find

(6.12)

G ' t H O O F T

Now we insert (6.11) and (6.12) into (6.8):

s(2t + 1 - s) (s - t - i) Ins

+ t ( l + 1)[4 $ S Ins - 2A21nA - 2AInA- i l n A + A 2 + i I n M - %t( t+ 1)- i

We made use ofC:e,=O, C:e ,M,2=0, C:e,lnMiZ= - InM. The limit A - m exists. Defining

R = !!?( $ s Ins - 4 A' InA - $ A InA - InA+ f AZ

= 0.248 154 471 , (6.14)

we find that , 2 t t 1 2 t + l 1 In[II(t)/IIo(t)] = 7 I(*+ 1)( ;In M + 4 R - k t ( t + 1) - 2) + s ( Z + 1 - .)(a - 1 -

*- 1

R is related to the Riemann zeta function b ( z ) as follows:

R = h - t ' ( - l )

y = 0.511215664 9 is Euler's constant,

and

= 0.931 548 254 315 844.

VII. THE FIXED MASS REGULATOR

Equation (6.15) gives the regularized product of all nonvanishing eigenvalues of Jn. But the regulator used was a very unsatisfactory one, from a physical point of view, because the regulator mass p depends on space- time:

This p must be interpreted as the subtraction point of the coupling constant g. Now g does not occur in II(t)&(#), but it does occur in the expression for the total action f o r the classical solution, and as we emphasized in the Introduction, any change in the subtraction procedure is important. The problem here i s that we wish to make a space- time-dependent change in the subtraction point, from p to a fixed po. We solve that in the following way.

The effect of a change in the regulator mass can

31 0

be absorbed by a counterterm in the Lagrangian, and hence is local in space-time. So we expect that, if we make a space-time-dependent change in the regulator mass , then this change can be absorbed by a space- time-dependent counterterm. Moreover, since our regulators are both gauge- invariant, this counterterm is gauge invariant. For space- time-independent regulators, this counterterm can be computed by totally conventional methods:

(7.2)

From locality we deduce that the same formula must also be true for space-time-dependent regulator mass p(x) , simply because no other gauge- invariant, local expressions of the same dimensionality exist. Inserting the classical value for G," I

(7.3)

and expression (7.1) for p , we get

as"'= J A S 8%

- l6 12=* t(f + 1) (2t+ 1) 3 2 ~ ~ x 9

= $t ( f+l ) (Zt+l ) :In "+ & 2M

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E T O A , . .

In the expression

$91,

with ll(t)/II,(t) a s computed in (6.15), we must correct 9’ with the above W’, in order to get the corresponding expression with g subtracted with

l( t + 1) (21 + 1) I . In(n(t)/n,(fll= 3 [*l+ + 4R + g s(2t

31 1

fixed mass regulators, a s defined in (4.9). The regulator masses M, in there must be such that

e , h M , = - i n N , . (7.5)

Expression (7.4) must be added to (6.15). Thus we get

+ 1- s) (s - t - 4) Lns- 4 1 ( t + 1) - $3
We note that the coefficient of the regulator term in (6.15) has the correct value. It matches the coefficlent of (7.2) that has been computed independently. The regulator in this expression, (7.6), is the same a s the one used in (%lo), and so we can use the result of Sec. IV to do the spinor and vector fields.
In Sec. N we proved that the nonzero eigenvalues for vector and spinor fields a re the same a s for scalar flelds, but we must take some multl- plicity factors into account. Equation (7.6) holds for one complex scalar multiplet with lsospin f . Fields wlth integer isospin may be real and then we have to multiply by 1. The vector field has four components but is real, and hence i ts value for In(n/n,,) is twice expression (7.6), with t = 1. The complex Faddeev-Popov ghost has Fermi statistics and contributes with one unit, but opposite sign. Thus, altogether, the vector field contribute6 just like one complex scalar with t = 1.

For fermions we must compute d e t q , but the theorem of Sec. IV applies to at. The fermlona have four Mrac components. So, altogether, fermions contribute just like two complex scalars, but the slgn in In(fI/nd is opposite because of fermi etatletlcs.

The above summarizes in words the complete contribution of all nonzero eigenstates to the functional determinants. But the spinor and vector flelds have a few more modes, with E m 0, and also the regulators have corresponding new modes, with E = p:.

VIII. THE ZERO EIGENSTATES

First we conslder the vector fields. We have sI = i, I - 1. Careful study of the operator a, Eq. (2.22), enables us to list the square-intergra- ble zero elgenstates a s follows:

(i) jl‘t, l z 0 : Q=(l+f)’,

(1,=s,=f), (8.1) multiplicity = (2 j , + 1) (2 j l + 1) = 4.

(ii) j , = o , I = + : * = r ( ~ + r ~ ) ’ * .

There a re two possibilities for the other quantum numbers:

(a) j , = 0, multiplicity = 1 , (b) j , = 1 , multiplicity = 3.

(82 )

(8.3)

This completes the set of zero eigenstates. We interpret these a s follows. States (1) have jl=j, = i t that is, the quantum numbers of an infinitesimal translation. The translations a re considered in Sec. M. State (iia) is the only singlet. It will correspond to the infinitesimal dilatation, Sec. X. State (ilb) is just an anomaly. It will be discussed in Sec. XI. It is indirectly connected with infinitesimal global Isospin rotations.

Spinors have similar sets of elgenstates, but their interpretation will be totally different. If t = 1, then the eigenstates a r e essentially the same a s the vector ones, but their multiplicity is half of that because s1 = 0. In this paper we limit ourselves to t = $ . Then there Is just one zero eigenstate:

jl =o, I = j , = 0, Q = (1 +r2)-9‘a. (8.4)

Its multiplicity is of course N’ If there a re N’ flavors. It leads to an ”-fold zero in the amplitude (note that in (2.11) the amplitude ia proportional to det‘JR, and thus is proportional to the product of the eigenvalues of “t,; if we have Nf zero elgenvalues then W has an N’-fold zero]. But this zero will be removed if we switch on the fermlon source $in the Lagrangian (2.1). In Sec. X N we will construct the resulting Nf -point Green’s function.

In strong-interaction theories the fermion mass will also remove this zero. The zero eigenstates must also be included in the regulator contributions. From Sec. Vn on, our regulator mass is fixed and is essentially equal to pw Every zero eigenvector of the operator 3ll, Eq. (2.22), wil l be accompanied by a factor pCa for the regulator (a zero eigenvector of Jrr, i s accompanied by a factor IllJ-‘).

C . ' t H O O F T

IX. COLLECTIVE COORDINATES: I . TRANSLATIONS

Clearly, zero eigenvalues make no sense if they would be included in the products carefully computed in the previous sections. In the case of the vector fields, which we will now discuss, they would render the functional integral infinite because they a re in the denominator. It merely means that the integration in those directions is not Gaussian.

angular dependence and index dependence can be read off from the quantum numbers. Written in full, the mode corresponds to the quantum field fluctuation (with arbitrarily chosen norm):

Let us f i rs t consider the four modes (8.1). The

A tq"(v) = 2qauv(l +rz)-z , u= 1 , . . . , 4 . (9.1)

This can be seen to be the s p a c e - t h e derivative of the classical solution up to a gauge transformation:

(9.2)

(9.3)

with

A"(v) = - qevA.8(l + x *)-I . The gauge transformation is there because our gauge-fixing term depends on the background field.

If we want to replace the varlable DAaU in this particular zero- mode direction by the collective variables &', then we must insert the corresponding Jacobian factor'

(9.4)

where AEq"(u) is the solution (9.1). The factor 2/g comes from the factor g/2 in (9.2). This way the re- nult ia independent of the normalization of AEq"(u) of (9.1). The factors 2r-'/' arise from the fact that we compare this integral with Gaussian integrals of the form JdAexp(-hA'), and in these Gaussian integrals the factors fi that go with each eigenvalue had been suppressed previously. We could also have drag& along all factors 6 at each of the eigenvalues of the matrices VI, and then we would have noticed that the factors fi going with the corresponding modes of the regulators, which are still Gaunnian, would have been left over. In (9.4) we just indude these factors from the beginning.

The norm of the solution (9.1) is

$ A tau( u)A z '"(X)d ' X = 2n2QA , (9.5)

31

so, together with the regulator, these four modes yield the factor

( :)4 ( $)'(2z2)'d42 = 2'n'po4g-'d'2 I (9.6)

The integral over the collective coordinates Z"

will yield the total volume of space-time, if no massless fermions a r e present. If there are massless fermions, then we must include the sources 3, which break the translation invariance. In that case the z integration is rather like the integration over the location of an interaction vertex in a Feynman diagram in coordinate conflguration, as we will see in Sec. XIV.

X. COLLECTIVE COORDINATES: 2. DILATATIONS

From the quantum numbers of the zero eigenstate (8.2) we deduce its angular and index dependence:

A ~ q " = q ~ L y ~ Y ( 1 + ~ * ) ~ 2 . (10.1)

This is a pure infinitesimal dilatation of the classical solution:

(10.2)

Thus, going from the integration variable Aau in this direction to the collective variable p, we need a Jacobian factor:

The norm of the solution (10.1) is

(A ~q")2d4x = tra . (10.4)

Thus, from this mode we obtain the factor

a t p = 1. Our system is not scale invariant because of the nontrlvial renormalization-group behavior. The complete p dependence for p # 1 will be deduced from simple dimensional arguments (including renormalization group) in Sec. W.

XI. GLOBAL GAUGE ROTATIONS AND THE GAUGE CONDITION

Discussion of the legitimacy of the background gauge-fixlng term h a s been deliberately post poned to this section, because we wanted to derive first the existence of the three anomalous zero eigenstates (8.3). They have the explicit form

2

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E T O A .

(arbitrary normalization)

$:(b)=2~~.rrSwrxA(l+X2)-'. (11.1)

$ t ( b ) = 4 , $ " ( ~ ) , (11.2)

They a r e a pure gauge artifact

$'(b)=tl..rv~&"XA(1 +Yl)-' , (11.3)

but $'(b) is not square-Integrable. What is going on? Note that $"(a), slnce they a re x dependent, do not generate a closed algebra of gauge rotations. They may net be replaced by a collectlve coordinate Zor global isospin rotations. To analyze this sltuation we first go back to a background- independent gauge-fixing term,

C;(%) = a8 ,,( AY'+A ",") - ae, A :u', (11.4)

where a is a free parameter. In this gauge we know exactly how to handle all zero eigenmodea: There a re five for translations and dilatations and also three for global isospin rotations because global isospin is still an Invariance in this gauge. To understand the latter we put the system in a large spherical box with volume V and assume that all (vector and ghost) fields vanish on the boundary. Let

A:@,%) = Cb (11.5)

generate an infinitesimal global isospin rotation. Then there i s a zero eigenmode:

4TU(b) = W ; ( b )

= 2q,,,b~eeu~Y(1 +%*)-I . (11.6)

The subscript 1 is to remlnd us that this is a solution i n the gauge C,. Similarly a s in the foregoing two sectlons, we can replace the integral over aA by an integral over the collective coordinates dA(b) by inserting the corresponding Jacobian factor

with

(11.7)

(11.8)

which diverges as the volume V of space-time goes to infinity. The integral over the gauge rotation is just the volume of the group and yields

(11.9)

where the factor f' comes from our normalizatlon of A, in (11.6). Thus, in this gauge, the zero

313

elgenmode yields the factor

(11.10)

Now we will study different gauges, and for that we need to change the boundary condition (we know from See. Ill that that will not affect the finite eigenvalues, but the zero eigenmodes change) into a gauge-invariant one: The ghosts 0 and gauge generators A must satisfy

(11.11)

where R is the radius of the box, and the vector fields

(11.12) a

A,(R)=O, =A, , (R)=O,

where A, is the vector component parallel to the boundary.

Now observe the following. The gauge term C,, Eq. (11.4), does not fix the gauge completely which can be seen in hvo ways: (a) Global isospin rotations a re still an invariance; (b) one component of the gauge term C, is identleally zero:

f q(x)d4x = 0. (11.13)

This leaves us the possibility of adding a constant to C,, which is orthogonal to it, with which we fix the remaining global gauge

c(%) = aB,A :'"

wlth a , ~ free parameters and +~,,(b,y)=DUOsb, as in (11.6). The integral over group space is now replaced by a Gaussian integral. The Gaussian volume is corrected for by the Faddeev-Popov ghost,

c:= cwpe,, D,,+w

- K ~ a b ( ~ : u ( b , y ) D , C ( ~ ) d 4 ] . b

(11.15)

so that the combined contrlbution of vector fields and ghosts is now independent of a and K : The zero eigenvalues a re replaced by

&-' = Klw and

@Q*=Ka l .

(11.16)

(Remember that the ghost, with the new boundary

G . ' I H O O F T

( ~

condition, has now an eigenstate @ = constant.) Thus, instead of (ll.lO), this gauge gives

( X f a ) 3 ( A ~ a o r ) - 3 ~ * = ( x / V ) 3 / 2 . (1 1.17)

Conclusion: If the redundant eigenmodes are fixed by an additional component in the gauge-fixing term, then a correction factor is needed: Equa- tion (11.10) divided by (11.17):

!$ ( (11.18)

Here V is the volume of the spherical box. The background gauge C, =D,Azq" has a problem

simllar to the gauge C,. Both the ghost (under the new boundary condition) and the vector field have one eigenvalue that vanishes like 1 /V as V- - (not l/i?', a s the other eigenvalues). Let $:(b) be the three ghost eigenstates and $&(b)=D,$:(b) be the vector ones. Let A, be the ghost eigenvalue

@$",(a) = - h,JI:(b) . (11.19)

It is easy to see that

(11.20)

From (11.1) and (11.3) we see that this is O(l/V). It i s safer to have a gauge condition that fixes this gauge degree of freedom a s V--,

C,(%) = aD,AO,qU(x)

- KC $ : ( b , % ) J $ ~ " ( b , ~ ~ ~ ~ '3

(1 1.2 1)

although the result, (11.25), will turn out to remain the same even if K = 0, Q = 1. The ghost Lagran- gian is

314

) d ~ ,

and

(1 1.24)

where no summation over b is implied. In this gauge we find the contribution from the lowest eigenmodes:

( ~ ~ ) 3 ( x p I o r ) - 3 / 2 = ~ , 3 / 2 . (11.25)

Using

(11.26)

we find

A, = 4na/V. (11.27)

Thus, together with the correction factor (11.18), we find the correct contribution for the three eigenmodes (9.3), together with that of their regulator:

XII. ASSEMBLING THE VECTOR, SCALAR, AND SPINOR TERMS

The eight zero-eigenvalue modes for the vector field give the factors (9.6), (10.5), and (11.28). Multiplying these gives

2'' x8 g-' PO' d' zd p (12.1)

for p = 1 (later we will find the p dependence).

modes both for the scalar and for the vector fields a re essentially contained in formula (7.6). A s we saw before, the vector fields, combined with the Faddeev-Popov ghost, together count as two real, or one complex, scalar with t = 1.

Let there be Na( t ) scalar multiplets for each isospin t , where each complex scalar multiplet counts a s one, and each real scalar multiplet counts a8 one-half. Then from (7.6) we obtah the total contribution from vector and scalar nonzero modes:

The contributions from the nonvanishing eigen-

(12.2)

(12.3)


and P I

& ( f ) =C(1)[2R- t w+t s(2t+ 1- s) (s- 1 - 4) Ins .+ 1)- 4. The numerical values for C ( t ) and a(t) are listed in Table I. Combining (12.1), (12.2), and the classical action (2.8) gives the total amplitude in the absence of fermions:

(12.4)

&zdp &a 2 ' 'noP 7 ew{- ~+ln(pg)[e-)~~a(t)C(tj] - a ( l ) - E N ' ( W t ) ) . (12.5)
Note that the coefficient multiplying lnp, coincides with the usual Callan-Symanzik p coefficient for $ ( p d In such a way that (12.5)becomesindependent of the subtraction point p, if we choose d ( p J to obey the Gell-Mann- Low equation. We now also insert the p dependence if p # 1 by straightforward dimensional analysis.
The interpretation of (12.5) Is best given in the language of path integrals: If 10) is the vacuum, and la) is the gauge-rotated vacuum, then (12.5) i s the total contribution to (6 10) from all paths in Euclidean space that have apseudoparticle located at z within d'z, having a scale between p and p+dp. The fermions can be introduced in two ways:

(1) If they have a mass m << l/p then only the lowest eigenvalue will depend critically on m.

(11) If they are rigorously massless then the lowest eigenvalue depends critically on the external source 8. In this paper we iimit ourselves only to fermions with isosph f -t. In case (I) the contribution of the lowest modes wi l l simply be

(12.6)

The nonvanishing eigenmodes are again obtained . from (7.6), which represents the eigenvalues of X,'. Now we wish to compute detSn, and we take into account that the Dirac field has four components. Thus the fermion nonvanishing eigenmodes wil l give

em[$ ~ ( 4 ) bcco+ ~ f a ( i ) ] . (12.7)

Together with (12.8) we find the total fermion factor that multiplies (12.5),

p"'m"exp[- fNfln(p,p) + ~ f a ( + ) l , (12.8)

where we again inserted the factors p as they follow from dimensional arguments. Note that the well-known Callan-Symanzik /3 coefficient for go( pJ again matches the term in front of lnpw

Equation8 (12.5) and (12.8) could be used a s a starting point for a strong-interaction color theory

31 5

where Euclidean pseudoparticles form a p lasma like statistical ensemble.' For that, one also needsto extend from SU(2) to SU(3). We will not do that in this paper.

must be replaced by the eigenvalue of the lowest mode a s it is perturbed by the source insertion.

In Sec. XW we consider case (11). In that case m

XIII. DIMENSIONAL RENORMALIZATION

The regulators used in Sees. VII-W are what we call fixed mass Pauli-Villars regulators and they only make sense in the background-field formalism. They a r e given by the Lagrangians (4.7)-(4.9). In this section we wish to switch to another regulator scheme which is much more widely used in gauge theories: the dimensional method.' Let us emphasize again that if one switches to another regulator, then that affects the definition of g(p) and that influences our calculation by an overall constant. We know that in the dimensional procedure the limit of large cutoff is replaced by a limit n - 4, where n is the number of space-time dimensions, roughly in the following way:

I ~ A - + finite. (13.1) 4 - n

In (12.5) and (12.7) the regulator mass p, plays the role of the cutoff A. Clearly then, the finite part in (13.1) will be relevant. In this section we derive that finite part, in ordinary perturbation

TABLE I. Numerical values of the coefficients C(t) and qt) a8 they occur in the text.

0 0 0

1 %-rln 2 -Lr =0.145 873 f 6 I2 1 4 ~ f t + f l n ~ - y = 0.443 307

+ 10 20R+4In3 -;In 2 -j(9=0.853182

R=$(lnZnty)*-i;?~~=0.2487&4471033784 1 -

theory. It correaponds to a finite counterterm in the Lagrangian. It is easy to compute this finite counterterm when, again, one makes use of the background fields. There are some diagrams to be computed and the rest is algebra. This algebra is identical to the algebra devised in Ref. 9. Symmetry arguments restrict the possible form of the finite counterterm in just two independent terms, X z and Y,,Y,,,, in the language of Ref. 9. The first of these is obtained by comparing the integral

(13.2) 1 & / d " k i F G T in the limits po - co and n -+ 4. For definite- nes8, we specify the theory at n # 4 dimensions: All trivial factors (2.)' must also be replaced by (2x)", which leads to the factor (2r)-" in (13.2).

The integral (13.2) is in this limit

(13.3)

where y is again Euler's constant. So here

(13.4) 1 1 1 2 2 Inpo - 4-n - - y + -1114s.

(13.8)

with 11 1 3 3

A = - a ( l ) + -(ln4x - y) + - = 7.05399103 (13.9)

and 1

A(#) = -a(l) + -(ln 4 r - r)C(#). 12 Numerically,

A(0) = 0 , A(!.) = 0.30869069, A(1) = 1.09457662,

A(-) = 2.48135610 2

( 13.10) 3

2 Similarly for the fermion factor

pNfmNf exp [- f N' (hp+ A)- N'B] (13.11)

with 1 1 B = - 2 4 - ) + -(ln4r - 7) 2 3

= 0.35952290. (13.12)

316

The coefficient in front of Y,,Y,,, is obtained in the same way by comparing the integral

in the two limits, where the signs ei are defined an in Eqs. (5.15) and (5 .16 ) , replacing M by PO. This time we get

So we see that for both the X z terms and the Y,yYpv terms the substitution (13.4) is to be made, However, we have to remember that in dimensional regularization the number of the fields Aq,U is n rather than 4. For these fields one therefore must replace

1 1 1 Inpo -, - 4 - n - ~ y + - I n 4 n - l . 2 (13.7)

In conclusion, (12.5) is to be replaced by

If we define the subtracted coupling constant as in Ref' lo,

with a, depending only on g, but not on n or p, then we can make the following replacements in (13.8) and u3.11):

g,(d - E m , (13.14) 1

lnp+ - 4 - n - WW). Here the superscript B stands for the dimensional procedure which defines g$. We see that the expression in t e r m s of g E ( p ) differs slightly from the one in terms of g(po).


XIV. THE PERMION SOURCE AND THE GREEN3 FUNCfION

We now consider the fermion zero eigenmode (Q.4), and assume that the fermlon mas8 (12.6) vanishes. In that case the source Sat in (2.1) must be taken into account, since the lowest eigenvalue wlll now mainly be determined by this source. W e determine the lowest eigenvalues E(i ) , i m 1,. . . ,h” of the operator

%=- hD,b#,+Q., (14. I) by perturbatlon theory, taking 8 as the small perturbation. The method is the standard one (the author thanks S. Coleman for an enlightening discuseion on thls point). The unperturbed, degenerate elgenmodes a re (taking for simplielty P‘1)

4f(t)nC(l+la)-st%a6,t, (14.2)

a = 1,2, s , t = 1,. . . ,”. The coefficients u’ contatn besldes the isospin index a a Dirac index. They satisfy

+‘**.+Pa&O,

or

(14.3)

y,u -- u. (14.4)

The coefficient C 18 determined by normalizing #,

C‘.u*u~crx(l + x y

=#/a uu*u=i. (14.5) Let

4, ( m) 1% I NO)

= $-j cr41+~’)-3~:4,(x)u~, (14.8)

then E ( i ) are the N f elgenvalues of H. We wish to compute v E(i)=deW

(14.7)

For large 9, this amplitude has exactly the space-tlme structure of an N’-potnt Green’s function, where each source point is connected to the origin by two fermion lines. The integral over the collective coordinate E [whlch is at the origin in Eq. (14.1)] will correspond to the integration

317

in coordinate configuration over the vertex variable.

Equation (14.7) should really be considered as our final result for the space-time dependence of the fermion Green’s function. But it would be enlight8ning If we could represent it in terms of an effective Lagrangian.

flrst be written in the form We found that the effective Lagrangian can best

(14.8)

where w is some fixed Dirac spinor wlth isospin 4. Owing to Fermi statistics, the varioue terms of the determlnant in (14.1) wlll arise with the appropriate minus signs 60 that we may limit ourselves to sources a,, that are dtagonal ln s and t:

&,=aWJ,,. (14.9)

Here d may still contain Dirac matrices.

from the effective lnteractlon (14.8) would be In the presence of thts source, the amplitude

(14.10)

where the minus sign comes from the Ferml statis-tics and the S,(%) are the Dirac propagators for massless fermions in coordinate configuration,

(14.11)

Comparing this with (14.71, at large 2, we find that we must require (leaving aside temporarily the other contributions to the overall constant)

(14.12)

where we may sum over the isospin index a but not over the Dirac components. Now from (14.3) one can derive

(14.13)

so we must require the w, to be such that

c W,GQ = 4(l+ Y,) . a

(14.14)

Thue, the w, a r e some parity reflection Of un. There is clearly no gauge-invarlant solution to

(14,14), 80 our effective Lagrangian (14.8) is apparently not gauge invariant. But note that we only wish to reproduce the amplitude (14.7) for gauge-invariant currents a,,. Thus, any gauge rotation of (14.8) does the same job. We get a gauge-invariant So‘f if we average over the whole

G . * t H O O F T

group of gauge rotations:

(14.15)

where the brackets ( ) denote the average for all gauge rotations of w. We can then derive

(w,q=t%.,(1+YJ,

and, for instance,

(14.16)

(a @P) (ail,)) = & ( 2 q 6: - %; )t"'

x ~ ;"L (1+YS)J l f l~PZ(1+Y5)$r l .

(14.17)

The Lagrangian (14.15) only acts on the left- handed spinors. The parity- reflected Euclidean pseudoparticles will give a similar contribution

XV. CONVERGENCE OF T

The entire expression that we now have for the effecti

&*ff(z)&z =~10+SNf,,6+2Nfg-8& p-5*SN~dpew - all2{ + A - 5

318

acting on the right-handed spinors only. So in total we get So" of (14.15) plus its Hermitian conjugate.

Note that we obtain products of fermion fields, such as (14.17), that violate only chiral U(1) invariance. They have the chiral-symmetry properties of the determinant of an N f X N f matrix in flavor space and are therefore still invariant underchiral SU(N') X SU(N'). The symmetry violation is associated with an arbitrary phase factor c * ' ~ in front of the effective Lagrangians. If other mass terms or interaction terms occur in the Lagrangianthat also violate chiral U( l ) , then they may have a phase factor different from these. We then find that our effective Lagrangian may violate P invariance, whereas C invariance is maintained. Thus we find that not only U(1) invariance but also PC invariance can be violated by our effect.

HE p INTEGRATION

ve Lagrangian is

+ WPP) b- ; 7 "(t )C(t ) - : N f ]

I : N a ( t ) A ( t ) - N f B ] ( f r 8.1 (T,w)(Z#,))+H.c., (15.1)

with

<waGB)=i 6,(1+y,), etc.,

and the numbers A,A(t) ,B,C(t ) a s defined before. The p dependence has been changed because the

effective Lagrangian (14.8) is not dimensionless. We see that this integral converges as p- 0

(except when there a re very many scalars). But there Is an infrared divergence a s p- -. In an unbroken color gauge theory for strong interactions this is just one of the various infrared dis- asters of the theory to which we have no answer. But in a weak-interaction theory it 1s expected that the Higgs field provides for the cutoff. Let there be a Higgs field wlth lsospin q and vacuum expectation value F. Let its contribution to the original Lagrangian be

sH= - D,+*D,O - v(o). (15.2)

Formally, no classical solution exlsts now, because the Higgs Lagrangian tends to add to the total action of the pseudoparticle a contribution proportional to F*, but this can always be re-

duced by scaling to smaller distances, until the action reaches the usual value 8na/gl when the field configuration is singular.

On the other hand, it is clear that the quantum corrections, a s can be seen in (15.1), act in the opposite way. There must be a region of values for p where the quantum effects compete with the effects due to the Higgs fields.

slightly the philosophy of Sec. rl. In Euclidean space it is not compulsory to conslder only those classical fields for which the action is stationary. We will now look at approximate solutions of the classical equations, so that the total action is only a slowly varying function of one collective parameter, p.

We simply postulate the gauge field A to have the same configuration a s before, with certain value for p , and now choose the Higgs field configuration in such a way that the total actlon is extreme. Only those infinitesimal variations that a r e pure scale transformations donot leave the action totally invariant, but nevertheless the parameter p gets the full

To handle this situation rigorously we alter


treatment a s a collective variable.

values for p will be those where the quantum effects and the Hlggs contribution a r e equally important. Since the quantum effects a r e small we expect that there

As will be verified explicitly, the dominant

P<< VM,, (15.3)

which implies that the Higgs particle may be considered a s approximately massless. Let us scale toward

p = l , l F 1 - 1 , MH*'"XF2<<l.

The equation for this field will be approximately

Dab=O,

a'(?-- Q) = Fa. (15.4)

The solution to that i s a zero-eigenvalue mode of the famlllar operator (2.22):

j , = O , l = j , = q , (15.5)

The contribution to the classical action is

Sn = j [- DuWDu* - V(b)]d'x. (15.6)

The first term is (observing that XuAE1=o)

- aU(PDub)d% = - pea,* d ' x

= - 4l?qP. (15.7)

The second term in (15.6) is of order IF', where A is a small coupling constant. If we scale back to arbitrary p, then the Higgs field factor in the total expression is

expS"=exp[- 4a'qF'd- O(xF'p')]. (15.8)

We see that the second term in the exponent may be neglected at first approximation.

Thus (15.8) multiplies the integrand in (15.1) and the p integration is now completely convergent. The integration over p yields a factor

t (4l?qFP)*-'"l)X'-Cr(~~'+ c - 2) , (15.9)

where

(15.10)

ACKNOWLEDGMENT

The author wishes to thank S. Coleman, R. Jacklw, C. Rebbi, and all other theorists a t

319

Harvard for their hospitality, encouragement, and discussions during the completion of this work.

APPENDIX: PROPERTIES OF THE q SYMBOLS

The group SO(4) i s locally equivalent to SO(3) x SO(3). The antisymmetric tensors A yy in SO(4) having six components form a 3 + 3 representation of SO(3) X SO(3). The self-dual tensors

A UY = tfUu.6A.6 (A l l

transform as 3-vectors of one SO(3) group. We now define the q symbols, in a way very similar to the Dirac y matrices:

is a covariant mapping of SO(3) vectors on self- dual SO(4) tensors. A convenient representation is

L , ~ = L , if ~ , ~ = 1 , 2 , 3

(A31 va4u=- %"* %u'= %u , 'I.44'0.

q... = ( - i ) 6 ~ 4 * 0 9 0 u y . (A4)

The symbols qa, will then do the same with vectors of the other SO(3) group and tensors Bw that

Let us also define

G . * I H O O F T

'Work supported In pa r t by the Natlonal Sclence Foun- dation under Grant No. MPS 76-20427.

ton leave from the Unlverslty of Utrecht. IA. A. BBlavln el d.. Phys. Lett. E. 85 (1975). 'G. 't Hooft. Phys. Rev. Lett. 3. 8 (1978); R. Jackiw

and C . Rebbl, ibfd. 37, 172 (1976); Phys. Rev. D U , 517 (1978); C. Callan, R. Dashen. and D. Cross. Phys. Lett. e. 334 (1076). See also F. R. Ore. Jr.. Phys. Rev. D (to be publlshed). %I. Jscklw and 9. Coleman (pr lvate communlcatlon). 'J. 8. Bell and R. Jacklw. Nuovo Clmento E, 47 (1969); 'J. L. Germla and B. Saklta, Phye. Ftev. 0 2 . 2943

9. L. Adler. Phys. Rev. 111. 2426 (1969).

(1976); E. Tomboulla, ibdd. 2, 1678 (1976).

320

'J. Honerkamp. Nucl. Phye. E, 289 (197% J. Honer- kamp, In Proceedings of the Colloquium on Renoma- llzation of Yag-Mills Fields and Applications to Par- ticle Physics, 1972. edlted by C. P. Korthals-Altes (C.N.R.S., Mareellle, France. 19721.

'A. M. Polyakov, Phys. Lett. KB, 82 (1975). 'Go. 't Hooft and M. Veltman, Nucl. Phye. E, 189 (1972); C. G . Bolllnl and J. J. Glamblagl. phys. Lett. - 40B, 586 (1972); J. F. Ashmore, Lett. Nuovo Clmento 4, 289 (1972); G. 't Hooft and M. Veltman. Report No. CERN 73-9, 1973 (unpublished).

' G . 't Hooft. Nucl. Phys. s, 444.(1973). %. ' t Hooft. Nucl. Phye. F&, 455 (1973).

CHAPTER 5.2

HOW INSTANTONS SOLVE THE U(l) PROBLEM

C . 't HOOIT

Institute for Theorerical Physics, Princetonplein 5, P. 0. Box 80.006, 3508 T A Utrecht, The Netherlands

NORTH-HOLLAND - AMSTERDAM

321

Editorial Note

A review article should not lead to any major controversy even if i t often implies that the author takes sides when conflicting results or approaches exist. Yet, when it covers a delicate and topical question, it may happen that some important controversy escapes the editor’s attention. The very important and difficult “U(1) problem” was reviewed by G.A. Christos in a Physics Reports article published in 1984.

The present article by G. ’t Hooft is meant to be a critical supplement to this former review. The editor is thankful to the present author to thus help to clarify this difficult question for the readers of Physics Reports.

M. Jacob

322

PHYSICS REPORTS (Review Section of Physics Letters) 142, No. 6 (1986) 327-387. North-Holland, Amsterdam

HOW INSTANTONS SOLVE THE U(1) PROBLEM

G. 't HOOFT lnrfihrfe for Theoretical PhyJics, Princeronplein S. P.O. Box M.lN$ 3508 T A Ufrechf. The Nefherlands

Received I8 April 1986

Conunu:

1. Introduction 2. A simple model 362 tities 381 3. QCD 361 9. Conclusion 382 4. Symmetries and currents 370 Appendix A. The sign of A& 382 5. Solution of the U(1) dilemma 383 6. Fictitmur symmetry 375 References 306 7. The "exactly conserved chiral charge" in a canonically

360 8. Diagrammatic interpretation of the anomalous Ward iden-

373 Appendix E. ?he integration over cha instanton size p

quanthcd theory 377

0 Elsevier Science Publishers BY. (North-Holland Physics Publishing Division)

323

C. $1 HooJt. How insfanfons solve fhe U ( I ) problem

1. Introduction

In addition to the usual hadronic symmetries the Lagrangian of the prevalent theory of the strong interactions, quantum chromodynamics (QCD) shows a chiral U( 1) symmetry which is not realized, or at least badly broken, in the real world 111. Now although it was soon established that the corresponding current conservation law is formally violated by quantum effects due to the Adler- Bell-Jackiw anomaly [2] it was for some time a mystery how effective U(l) violating interactions could take place to realize this violation, in particular because a less trivial variant of chiral U(l) symmetry still seemed to exist. Indeed, all perturbative calculations showed a persistence of the U(l) invariance.

With the discovery of instantons [3], and the form the Adler-Bell-Jackiw anomaly takes in these nonperturbative field configurations, this so-called U(l) problem was resolved [4,5]. It was now clear how entire units of axial U(l) charge could appear or disappear into the vacuum without the need of (nearly) massless Goldstone bosons. In a world without instantons the q and q’ particles would play the role of Goldstone bosons. Now the instantons provide them with an anomalous contribution to their masses.

In the view of most theorists the above arguments neatly explain why the q particle is considerably heavier than the pions (one must compare m: with mf), and q’ much heavier than the kaons.

Not everyone shares this opinion. In particular Crewther [6] argues that Ward identities can be written down whose solutions would still require either massless Goldstone bosons or gauge field configurations with fractional winding numbers, whereas experimental evidence denies the first and index theorems in QCD disfavor the second. If he were right then QCD would seem to be in serious trouble.

In a recent review article this dissident point of view was defended [7]. The way in which it refers to the present author’s work calls for a reaction. At first sight the disagreement seems to be very deep. For instance the sign of the axial charge violation by the instanton is disputed; there are serious disagreements on the form of the effective interaction due to instantons, and the Crewther school insists that chiral U(l) is only spontaneously broken whereas we prefer to call this breaking an explicit one. Now as it turns out after closer study and discussions [8], much of the disagreement (but not all) can be traced back to linguistics and definitions. The aim of this paper is to demonstrate that using quite reasonable definitions of what a “symmetry” is supposed to mean for a theory, the “standard view” is absolutely correct; chiral U(l) is explicitly broken by instantons, and the sign of AQ, is as given by the anomaly equation; the effective Lagrangian due to. instantons can be chosen to be local and polynomial in the mesonic fields, and the q and q’ acquire masses due to instantons with integer winding numbers.

In order to make it clear to the reader what we are talking about we first consider a simple model (section 2) which we claim to be the relevant effective theory for the mesons in QCD (even though it is dismissed by ref. [7]). The model shows what the symmetry structure of the vacuum is and how the q particles obtain their masses. It also exhibits a curious periodicity structure with respect to the instanton 0 angle, which was also noted in [7] but could not be cast in an easy language because of their refusal to consider models of this sort.

Now does the model of section 2 reflect the symmetry properties of QCD properly? Since refs. [6,7,8] express doubt in this respect we show in section 3 how it reflects the exactly defined operators and Green’s functions of the exact theory. The calculation of the q-mass is redone, but now in terms of QCD parameters. We clearly do not pretend to “solve” QCD, so certain assumptions have to be

324

G. ‘ I Hoof!, How inslclnfons solvr rhr U(I) problrm

made. The most important of these, quite consistent with all we know about QCD and the real world, is

( q q ) = F # 0. (1.1)

Now refs. [6,7] claim that in that case one needs gauge field configurations with fractional winding numbers. Our model calculations will clearly show that this is not the case.

How can it be then if our calculations appear to be theoretically sound and in close agreement with experiment, that they seem to contradict so-called “anomalous Ward identities”? To answer this question we were faced with unraveling some problems of communication. The current-algebraic methods of refs. [6,7] were maiuly developed before the rise of gauge theories but are subsequently applied to gauge-noninvariant sectors of Hilbert space. Their language is quite different from the one used in many papers on gauge theories [4] and due to incorrect “translations” several results of the present author were misquoted in 171. After making the necessary corrections we try to analyze, in our own language, where the problem lies.

There are two classes of identities that one can write down for Green functions. One is the class of identities that follow from exactly preserved global or local symmetries. Local symmetries must always be exact symmetries. From those symmetries we get Ward identities (91 (in the Abelian case), or more generally Ward-Slavnov-Taylor [lo] identities, which also follow from the (exact) Becchi- Rouet-Stora [ll] global invariance.

On the other hand we have identities which follow from applying field transformations which may have the form of gauge transformations but which do not leave the Lagrangian (or, more precisely, the entire theory) invariant. These transformations are sometimes called Bell-Treiman transformations [12] in the literature, but “Veltman transformations” would be more appropriate [ 131. The identities one gets reflect to some extent the dynamics of the theory and form a subclass of its Dyson-Schwinger equations. Thus when refs. [6,7] perform chiral rotations of the fermionic fields, which do not leave 0 invariant, they are performing a Veltman transformation, and their so-called “anomalous Ward identities” fall in this second class of equations among Green functions.

It is in the second derivative with respect to 0 that refs. [6,7] claim to get contradiction with our model calculations [8]: the second derivative of an insertion of the form

iOFF, (1.2)

in the Lagrangian vanishes, whereas the simple “effective field theory” requires an insertion of the form

err det( qLqR) t h.c. , (1.3)

of which the second derivative produces the q mass. So they claim that in the real theory the q mass cannot be. explained that simply.

In section 5 we explain why, in the modem formalism this problem does not arise at all. (1.2) should not be confused with ordinary Lagrange insertions and after resummation correctly reproduces (1.3). The canonical methods are not allowed if one tries first to quantize the gauge fields A,, and only afterwards the fermion fields. The fermions have to be quantized and integrated out first. The phenomenon of “variable numbers of canonical fermionic variables” resolves the dilemma. We conclude, that the q mass is what it should be and there is no U(l) problem.

325

G. ‘f HooJf, How inslonrons solve the U( I ) problem

There is a further linguistic disagreement on whether the U(1) breaking of QCD should be called an explicit or a spontaneous symmetry breaking. In the effective Lagrangian model the symmetry is clearly broken explicitly. Several authors however refer to the 0 angle as a property of the vacuum [5 ] in QCD. Of course as long as one agrees on the physical effects one is free to use whatever terminology seems appropriate. We merely point out that all physical consequences of the instanton effects in QCD (in particular the absence of a physical Goldstone boson) coincide with the ones of an explicit symmetry breaking. Our 0 angle is as much a constant of Nature as any other physical parameter, to be compared for instance with the electron mass term which breaks the electron’s chiral invariance.

Only if one adds nonphysical sectors to Hilbert space one may obtain an alternative description of the 0 angle as a parameter induced by boundary effects producing a spontaneous symmetry breakdown. However, any explicit symmetry breaking can be turned into “spontaneous” symmetry breaking by artificially enlarging the Hilbert space. One gets no physically observable Goldstone boson however, so, the most convenient place to draw the dividing line between spontaneous and explicit global symmetry breaking is between the presence or absence of a Goldstone boson.

We explain this situation in section 6, where we show that any nonphysical symmetry can be forced upon a theory this way. In section 7 we show how the spurious U(1) symmetry that is used as a starting point in [7], actually belongs to this class. Section 8 shows that if one takes into account the enlarged Hilbert space with the variable 0 angles then the effective theory of section 2 neatly obeys the so-called anomalous Ward identities. The effective model shows the vacuum structure so clearly that all problems with “fractional winding numbers” are removed.

The decay amplitude q-3n posed problems similar to that of the q mass. As explained correctly in [7] this problem is resolved as soon as the U(1) breaking is understood so no further discussion of this decay is necessary. It fits well with theory.

Appendix A is a comment concerning the sign of axial charge violation under various boundary conditions.

Appendix B discusses the instanton-induced amplitude. The contribution from “small” instantons can be computed precisely but the infrared cutoff is uncertain. (Rough) Estimates of the amplitude in a simple color SU(2) theory give quite large values, which confirms that instantons may affect the symmetry structure of QCD sufficiently strongly such as to explain the known features of hadrons.

2. A simple model

Before really touching upon some of the more subtle aspects of the “U(1) problem” we first construct a simple “effective Lagrangian” model. Whether or not this model truly reflects the symmetry properties of QCD (which we do claim to be the case) is left to be discussed in the following sections. For simplicity the model of this section will be discussed only in the tree- approximation.

To be explicit we take the number of quark flavors to be two. Generalization towards any numbers of flavors (two and three are the most relevant numbers to be compared with the situation in the real world) will be completely straightforward at all stages in this section;. So we start with the “unbroken model” having global invariance of the form

‘See however the remark following eq. (2.24).

326

G. ‘ I Hwff, How insrancorn solve h e U( 1 ) problem

U(t),@U(L), , with L = 2 , (2.1)

where the subscripts L and R refer to left and right, respectively. We consider complex meson fields with the quantum numbers of the quark-antiquark composite operator iRiqLi . They transform under (2.1) as:

4,; = u;4&’ I (2.2)

to be written simply as

4’ = U L 4 U R ’ . (2.3)

Since we have no hermiticity condition on 4, there are eight physical particles a, q, n, and a, (a=1,2,3):

4 = i (a + is) + 4 (a + in ) * ‘I , (2.4)

where 7’s’*3 are the Pauli matrices. We take as our Lagrangian:

2’= -Tr dp&’p4t - V(4).

A potential V, invariant under (2.1) is

VO(4)=-p2Tr&$’t i ( A l -A,)(Tr&$’)’+ ~A,Tr(&$’)’ 2

= -k (a2 +q2 + a2 t a2)+ 1 A (a2 + q2 t a’ + a2)2 t - 4 ((aa +qn)’+ (a A a)’), 2 8 2

(2.7)

( a ) = f , a = f t s , (2.8)

Assuming, as usual

we get, by taking the extremum of (2.7),

f ’ = 2p2/A, ;

(2.10) A A 2 v, = (fs t t(s2 + q2 t a2 + d))’ t -2 ((fa + sa t qa)’ + (a A a)’),

from which we read off

m : = A l f 2 = 2 p 2 , m : = O , m f = A , f ’ , m $ = O . (2.11)

There are four Goldstone bosons, as expected from the U(2)@U(2) invariance, broken down to U(2) by (2.8).

327

G. ' I Hwff, How iiuracuons d u e the 4 1 ) probltm

We now consider two less symmetric additional terms in V:

v, = urn t u:;

U, = '' m eix&q, " = f m eix Tr 4 = im eix(u t ir)) ,

and:

v, = u* t u: ;

U, = " K eie det( &qL) " = K eie det I$

= K ei'((u t iq)' - (a t in)2).

(2.12)

(2.13)

Here, m, x , K and 0 are all free parameters. The terms between quotation marks are there just to show the algebraic structure, up to renormalization constants. Notice that V, still has the U(l) invariance

Therefore we are free to rotate

x - + x + o , e+e+20 .

(Here the 2 would be repiaced by L in a t h e o j with L flavors.) Consider first the theory with K = 0;

V = V , + V , .

(2.14)

(2.15)

(2.16)

Then we can choose o = n - x , and

V, = - m u . (2.17)

So x is unphysical. Equation (2.9) is replaced by

f 2 = Z~'IA, t 2 m 1 ~ , f . (2.18)

Consequently, in (2.11) we get

1 m, = m i = mlf.

Note that with the sign choice of (2.17), f must be the positive solution of (2.18).

(2.19)

328

G. ‘ I Hoofr. How insraniom ~olvc rhc U(1) problcm

Now take the other case, namely m = 0;

v = v, t v, . It is now convenient to choose

w = f ( m - e) . So here the angle e is unphysical.

v, = -2K(cr* + Trz - Q 2 - a’) . f = 2p21h, + ~ K M , .

The masses of the light particles become

(2.20)

(2.21)

So the K term contributes directly to the q mass and not to the pion mass. In case of more than two flavors the determinant in (2.13) will contain higher powers of 4, and extra factors f will occur in (2.24). But the mechanism generating a mass for the q’ will not really be different from that of the q.

It is interesting to study the case when both m and K are unequal to zero. We then cannot rotate both x and 0 independently and one of the two angles is physical. Since obviously the model Lagrangian is periodic with period 277 in 0, its physical consequences will be periodic in x with period T , because of the invariance (2.15). In the general case we now also expect a nonvanishing value for

( v ) = g . (2.25)

So we write 9 = g i h, where g is a c-number and h a field. The conditions for f and g are:

It will be convenient however to choose o in (2.15) such that g = 0. According to (2.26) one then must have

msin x + 4 ~ f sin 0 = O f (2.23

(2.W

(2.29)

329

G 'I Hoof(. How imranfons solvc fhr U( I ) problem

apart from an 9-s mixing. The effect of this mixing however goes proportional to

16x2 sin' 0 = m' sin' ,y

and therefore is of higher order both in K and m. Consider now the function

(2.30)

F(w) = rn cos(,y t 0) t 2Kf cos(0 t 20). (2.31)

Then (2.27) requires o to be a solution of

F'(O) = 0 (2.32)

(after which we insert the replacement (2.15)), and (2.29) implies:

fmt = F"(o) > 0 . (2.33)

So, o must be chosen such that (2.31) takes its minimum value. Furthermore, the replacement w--, o t ?r switches the sign of the first but not of the second term in F(o), so, if o is chosen to be the absolute minimum of (2.31), then indeed also

(2.34)

except when there are two minima. In that case there is a phase transition with long range order (due to the massless pion) at the transition point.

This phase transition is at

x = n 1 2 ,

sin 8 = - m / 4 ~ f , c o s e < o ,

or, if a rotation (2.15) is performed:

x=O, sin B = m/4xf, o< 8 < n12.

A further critical point may occur at

x = o e = r12 m=4Kf

(2.35)

(2.36)

(2.37)

330

G. ‘ I Hooft. How i n ~ l M f ~ t ~ JOIVC hc U(1) problcm

where also m, vanishes. Of course these values of the parameters are not believed to be close to the ones describing real mesons. Phase transitions of this sort were indeed also described in ref. [7]. But because no specific model such as ours was considered, the periodicity structure in x and 0 was discussed in a somewhat untransparent way. Although obviously in our model we have a periodicity in 0 with period 2 r , ref. [7] suspected a period 4 ~ . Indeed if o in (2.15) shifts by an amount T, then m eiY- -m eiy and one might end up in an unstable analytic extension of the theory. Clearly the properties of the minimum of the potential F(o), eq. (2.31), can only have period 2~ in 0.

3. QCD

If Quantum Chromodynamics with L quark flavors, were to have an approximate U(L), X U(L); symmetry only broken by quark mass terms, the model of the previous section with V = V, + V, could then conveniently describe the qualitative features of mesons, but with q and n almost degenerate. (In the case L > 2 one merely has to substitute the 2 x 2 matrix 4 by an L X L matrix.) That would be the case if somehow the effects of instantons could be suppressed.

Let us now consider instantons and write in a shorthand notation the functional integral I for a certain mesonic amplitude in QCD. For the ease of the discussion we assume all integrations to be in Euclidean space-time:

AS we will see shortly, it is crucial that the integration over $ is done first and the one over A afterwards.

Since we have no way of solving the theory exactly certain simplifying assumptions must be made. We now claim that the assumptions to be formulated next will in no way interfere with the known symmetry properties of the low-energy theory. Discussion of this claim will be postponed to sections 4-7.

An (anti-)instanton is a field configuration of the A fields with the property

(3.6)

33 1

G. ’t Hoofr, How iwrmrotu solve fhr U(I) problem

where AV is the volume of a space-time region. Outside AV we have essentially [FFvI = O but we cannot have [ A [ = 0 there because then (3.6) would vanish as the integrand is a total derivative.

The assumption we make is that the A integral can be split into an integral over instanton-locations and an integral over perturbative fluctuations around those instantons. We do this as follows. Let US divide space-time into four-dimensional boxes with volumes AV of the order of l(fm)‘. Each box may or may not contain one instanton or one anti-instanton. (There could be more than one instanton or anti-instanton in a single box, but we choose our boxes so small that such multi-instantons in one box become statistically insignificant.) The essential point is that since an instanton in a box AV will do nothing but gauge-rptate any of the fields outside AV, the instanton-numbers in each box are independent variables. Notice that at this point we do not require these twisted field configurations in the boxes to be exact solutions to the classical field configurations. This is why we have no difficulties confining each instanton to be completely inside one box, with only gauge rotations of the vacuum outside.

Let us then write

A = Ai,,%, +- SA (3.7)

where Ainrl is due to the instantons only, then the integral over 6A will essentially commute with the integral over the instantons. The 6A integral is assumed to be responsible for the strong binding between the quarks. The confinement problem is not solved this way but is not relevant here since we decided to concentrate on low-energy phenomena only.

Note that the integration over Aim,, is more than a summation over total winding number v. Rather, if we write

v = u+ - v_ (3.8)

then the integral over A,,,, closely corresponds to integration over the locations of the v+ instantons and the v- anti-instantons. It is important that we restrict ourselves to instantons with compact support (namely, limited to the confines of the box AV in which they belong). A larger instanton, if it occurs, should be represented as a small one in one of the boxes, with in addition a tail that is taken care of by integration over SA. “Very large” instantons are irrelevant because they would be superimposed by small ones. In short, in eq. (3.71, Aim,, is defined to be a smooth field configura!ion that accounts for all winding numbers inside the boxes, and SA is defined to contribute to FF by less than one unit in each box.

Now consider an isolated instanton located within one of our boxes, located at x = x , . What is discussed at length in the literature is the fact that the + integration is now affected by the presence of a zero mode solution of the Dirac equation. If there were no other anti-instantons and no source term J then the fermionic integral, being proportional to the determinant of the operator yp(8,, t igA,), would vanish because of this one zero eigenvalue. If we do add the source t e h J$q5 the integral need not vanish. In ref. [4] it was derived that the instanton exactly acts as if it would contain a source for every fermionic flavor. Thus with one instanton located at x = x , the fermionic integral

has the same e#ed as the integral

332

G. ‘ I Hoof(, How instanlorn solve rhr U( I ) problem

where K may be computed from all one-loop corrections [4]. Indeed it was shown that the zero eigenmodes for all flavors which extend beyond the volume AV conveniently reproduce the fermionic propagators connecting xl with the sources J. The fact that (3.9) does have the same quantum selection properties as (3.10) can also be argued by realizing that a gauge-invariant regulator for the fermions had to be introduced, and instead of the lowest eigenmodes one could have concentrated on the much more localized highest fermionic states. The correctly regularized fermionic integral contains a mismatch by one unit for each flavor between the total number of left handed and right ha_nded fermionic degrees of freedom. Since this happens both for the fermions and the antifennions $ the determinant in (3.10) consists of products of L fermionic and L antifermionic fields.

Since we do require that the SU(L), BSU(L), is kept unharmed by the instantons, the determinant is at first sight the only allowed choice for (3.10) but, actually, if one does not suppress the color and spin indices, one can write down more expressions with the required symmetry properties.

Next consider u+ instantons, located at x = x i . Following a declustering assumption which, at least to the present author’s taste, is quite natural and does not require much discussion, we may assume these to act on the fermionic integrations as

”+ K“+ 1 W D$ es+n det(+L(xi) &(xi)). (3.11)

i- 1

Let us add the 8 dependence and integrate over the instanton locations xi:

(3.12)

The denominator v+! is due to exchange symmetry of the instantons. We now extend our declustering assumption to the anti-instantons as well. This assumption was

vigorously attacked in [6-81. Indeed one might criticize it, for instance by suggesting that “merons” play a more Crucial role [14]. We insist however that the assumption in no way interferes with the symmetry properties of our model. We will see in sections 7 and 8 that the anomalous Ward identities will be exactly satisfied by our model. To avoid confusion let us also stress that our declustering assumptions refer to the QCD part of the metric only, not to the contributions of the fermions which we denote explicitly. So there is no disagreement at all with the findings of ref. [15]. Indeed, our approach here is closely analogous to theirs.

Thus, consider u- anti-instantons. The complete instanton contribution to the functional integral is

(3.13)

The summations are now easy to carry out:

which is precisely the effective interaction V, of eq. (2.13). The remaining integrals over the fermionic

333

G. 7 Hoofl, How inrronfonr solve fhc U ( 1 ) problem

fields J, and the perturbative fields SA may well result (n the effective Lagrangian model of section 2. Notice that, before we interchanged the Ainr, and $, $ integrations, we have made the substitution (3.10). This will be crucial for our later discussions. Once the substitution (3.10) has been made, the (A-field-dependent) extra fermionic degrees of freedom have been taken care of, and only then one is allowed to interchange the A and the J, integrations. This is how (2.13) follows from (3.14).

4. Symmetries and currents

Let us split the generators A, and A , for the U(L), X U(L), transformations into scalar ones, A' and A', and pseudoscalar ones, A: and A:, The infinitesimal transformation rules for the various fields considered thus far are:

In a theory with L flavors the factor 4 in eq. (4.8) must be replaced by 2L. In a classical field theory the currents are most easily obtained by considering transformations (4.1)-(4.8) with space-time dependent A, (x ) . Their effect on the total action can be written as

SS = d4x(- F,A,(x) - JLJ,A,(x)) (4.9)

(here i = 1,. . . ,8). Since according to the equatio'ns of motion 6s = 0 for all choices of A, (x ) , one has

a , J l ( x ) = F,(x) . (4.10)

A Lagrangian which gives invariance under the space-time independent A, must have F, = 0, so that the current J L is conserved.

We are now mainly concerned about the current JF5 associated with A!. The QCD Lagrangian (3.1) produces the current

334

Jp5 = i&,* 9

and, prior to quantization:

G. ’ I Hoo/r, How inrlenlonr m1w the U( I ) problem

(4.11)

a,, J,,, =I 2irn&&. (4.12)

As is well known, however, eq. (4.12) does not survive renormalization. Renormalization cannot be performed in a chirally invariant way and therefore the symmetry cannot be maintained, unless we would be prepared to violate the local color gauge-invariance. But violation of color gauge invariance would cause violation of unitarity, so, in a correctly quantized theory, (4.12) breaks down. A diagrammatic analysis [2] shows that, at least to all orders of the perturbation expansion, one gets

(4.13)

with

e“ = bpy .gF:g . (4.14)

We read off that, if we may ignore the mass term, then in a space-time volume V with u+ instantons and u- anti-instantons

J d‘r d,, J F 5 = -2iL(v+ - v-) . (4.15) V

Here the factor i is an artefact of Euclidean space. Defining the charge Q, in a 3-volume V, by

Q, =[ J,, d3x=i J4d3x= Q, - QL , I “3 “3

each instanton causes a transition*

(4.16)

AQ, = 2 L . (4.17)

This is called the “naive” equation in ref. [7]. Since we were working in a finite space-time volume V the nature of the “vacuum” has not yet entered into the discussion. Remarks on the language used here and in ref. (71 are postponed to appendix A. Now let us write the corresponding equation in our effective Lagrangian model. Here,

J,,, =2iTrW,,4*)4 - 4*3,,41 (4.18)

and

* Apct fmm the dispused sign thext aic also diffetences in sign mmentiona with ref. 171.

335

G. ’ I Hooff. How inrfanrons rolvc fhc U(I) problem

d P ~ P , , = - 2 m ( ~ c o s ~ + u s i n ~ ) + 1 6 ~ ( ~ ~ a - ~ ) c o s 6 + 8 ~ ( ~ Z + ~ 2 - u Z - a * ) s i n 8 . (4.19)

Before comparing this with eq. (4.13) of the QCD theory let us chirally rotate over an angle i x . The mass term in the original Lagrangian then becomes

-$m$ cos ,y - i$rny,S sin ,y , (4.20)

and then (4.13) becomes

iLgz 1 6 ~

dPJP, = 2im$-y5$ cos x - 2m& sin x - 7 F;”F:”. (4.21)

Therefore the first term in (4.19) can neatly be matched with the first terms of (4.21). An issue raised in refs. [7,8] is that there is an apparent discrepancy if we try to identify the last

terms of (4.19) with the last term of (4.21). The last term of (4.21) contains the color fields only and there is absolutely no 8 dependence here. But the last terms of (4.19) do show a crucial 0 dependence. It is essentially

8 K h ( e “ det 4) . (4.22)

Where did the 8 dependence come from? One way of arguing would be that the 8 dependence of (4.22) is obvious. Chiral transformations

are described by eq. (2.15) and any symmetry breaking term in a Lagrangian can obviously not be invariant at the same time. So the 8 dependence of (4.22) is as it has to be. The symmetry breaking in QCD is not visible in its Lagrangian but is due to the 8 dependence of the regularization procedure.

However, although this argument may explain why (4.19) shows a 8 dependence and (4.21) does not, it does not explain why nevertheless these two theories can describe the same system. This is (partly) what the dispute is about. We claim that one can identify in the effective theory

(4.23)

so that, if 8 = 0, one may identify Ffi with the 17 field. At the same time we would also like to put

6 = 4RRqL , (4.24)

but this 8 phase seems to be in disagreement with the canonical quantization procedure if A:, qR, qL and Q were to be considered as independent canonical variables. Another way of formulating this problem is that the right hand side of (4.23) seems to commute with the chiral charge operator Q, while the left hand side does not.

Notice that if we could somehow suppress instantons essentially F p would vanish. The left hand side of (4.23) would vanish also, because K + O . This suggests one simple answer to our problem: equation (4.23) violates axial charge conservation, but that is to be expected in a theory where axial charge is not conserved. Unfortunately some physicists insist in considering the U(l)

336

G. ‘ I Hoof, How inrcpnronr solve the U(1) problrm

violation by instantons as being “spontaneous” rather than explicit and therefore they rejected this simple answer. A rather curious attempt to bypass the problem was described in [7]. They first propose to replace our V, of eq. (2.13) by

V: = K Tr(log(Cpl4’))’ . (4.25)

But this also does not commute with the axial charge operator and furthermore the logarithm is not single-valued so (4.25) makes no sense at all. So then they propose

(4.26)

It is not obvious how this expression should be read such that it does make sense. If it is equivalent to (4.25) then clearly no improvement has been achieved. The problem of a multivalued logarithm has merely been substituted by the problem of an infrared divergent integral in x space. Equation (4.26) is then a clear example of linguistical gymnastics that should be avoided: formally it appears to be chirally invariant, yet it is equivalent to the local term (4.29, which is not.

We conclude in this section that the aforementioned problem is not solved by the logarithmic potentials V : of eqs. (4.29, (4.26). Let us call this problem the “U(1) dilemma”. The correct resolution of the U(l) dilemma will be given in the next sections.

5. Solution of the U(1) dikmma

We must keep in mind how and why an effective Lagrangian is constructed. The word “effective” is meant to imply that such a model is not intended to describe the system in all circumstances. Rather, the model gives a simplified treatment of the system in a given range of energies and momenta. In this case we are interested in energies and momenta lower than, say, 1 GeV.

Now the cQmplete theory contains variables at much higher frequencies. In as far as they play a role at lower energies, we must assume that they have been taken care of in the effective model. Consequently, the simple identification (4.24) is not correct as it stands. It should be read as

6 = ( 4RRqL)lOW lreepu.dcr ’ (5.1)

But what does “low frequency” mean? In a gauge theory the concept “frequency” need not be gauge-invariant. Therefore the splitting between high frequency and low frequency components of the quark field# must depend in general on the gluonic fields A,. This is why the contribution of the high frequency components of the quark fields to the axial current JP5 may depend explicitly on the A fields, a fact that is correctly expressed by the so-called “anomalous commutators” of [6,7]. After integrating out the high frequency modes of the quark fields, but before integrating out the A fields, we have an expression for the axial current which has the following form:

J,,,=2iTrWPCp*) 4 - Cp*~,4) + JLAA). (5.2)

It is &(A) which is responsible for the nontri@al axial charge of the quantity FF in (4.23). Let Q; be the charge corresponding to JLs. How does FF commute with Q;?

337

G. ' 1 H w f t , How inntantom solve the U(1) problem

Rather than FF itself, it is the integral over some space-time volume AV,

\ FF= .f - V i " , AV

(5.3)

that is relevant in (4.13). (We use the short hand notation of eq. (3.4).) Let us take AV so small that

J F F = O or tl AV

(5.4)

(i) If FF = 0 then we are not interested in its quantum numbers. (ii) Whenever the right hand side of (5.3) is tl we have an amplitude in which k2L units of axial

(iii) The higher values of the right hand side are negligible. The creation or-annihilation of axial charges occurs because of the extra high frequency modes of

or IGL and JIR that make the functional integral non-invariant. Let us call their contribution 2.

charge are created.

JL, If vf = 1 then

where the subscripts under the multiplication symbols denote the numbers of variables to be integrated over. Then if

we have for all integrals over the anticommuting fields:

(5.7)

so that

Z . (5.8) Z+e-Zi"L

In this discussion we only include the high frequency components of the JI fields. We see two things: the effective interaction 2 due to an instanton trapsforms exactly as our insertion U, of eq. (2.13), and-secondly that, in a simplified picture where FF takes integer values only (eq. (5.3)), the quantity FF, after inregration over the high frequency fermionic modes, transforms with a factor

e + 2 i u L , (5.9)

338

G. ’ I Hwff. How instomons solve the U(1) problem

so that there is no longer any conflict. with (4.23). The transformation rules (4.1-4.8) hold for the effective fields. The terms containing A: tell us how the various fields commute with Q,:

(5.10)

(5.11) (5.12)

(5.13)

6. Fictitious symmetry

The chiral U(1) symmetry breaking in QCD is an explicit one because the functional measure Il D$ fails to be chirally invariant when regularized in a gauge-invariant way [21]. This neatly explains why no massless Goldstone bosons are associated with this symmetry. Yet in several treatizes the words “spontaneous symmetry breaking” are used. How can this be?

Any broken global symmetry can formally be considered as a “spontaneously” broken one by a procedure consisting of two steps.

(i) Enlarge the physically accessible Hilbert space by adding all those Hilbert spaces of systems that would be obtained by applying the phoney symmetry transformation:

Q ’ = @ x S (6.1)

where @ is the original Hilbert space and S the space of physical constants describing symmetry breaking.

(ii) Define the symmetry operator(s) as acting both in S and in 18. We then obtain transformations in 4’ that obviously leave the Hamiltonian H invariant. This procedure allows one to write down Ward identities for theories with symmetries broken explicitly by one or more terms in the Lagran- gian. Since such identities were excessively used and advocated by Veltman in his early work on gauge theories with mass-insertions, we propose to refer to the above transformations as Veltman transformations [12].

Consider for example quantum electrodynamics with electron mass term

-m*&.,lbk - m6Rd$, ’ (6.2)

Then S is the space of complex numbers m. In this theory then, m is promoted to be an operator rather than a c-number. The chiral transformation

For the factor el’ see scction 7

339

G. ‘t Hooft . How instantons solve the U ( J ) problem

L , ‘

t _ _ _ _ _

is obviously an invariance of this theory. If in the “physical world”

( m ) = m = r e a I

then one could argue that the symmetry (6.3) is “spontaneously broken”. The canonical charge operator Q associated with (6.3) is now

,,’ ,I R

t l Lt\. __-- -- ” v.

which commutes with (6.2). Thus, Q is exactly conserved. But, since m is not a dynamical field, the new term cannot be written as an integral over 3-space, unless we enlarge the Hilbert space once again.

Let us now consider a Feynman diagram in which the mass term (6.2) occurs perturbatively as a two-prong vertex. Let there be a diagram with v + insertions of the last term in (6.2), going with m, and v- of the first term, going with m*. We have

Only by brute force one could produce a current of which the fourth component would give a charge satisfying (6.6):

j,, = J,,, + K,,

Fig. 1. Propagating ekctron (solid line) with mass terms. The propagators are expanded in m. yielding artificial panicles (sehhru, dotted lines) that any away two units of axial charge. but no energy-momentum. Total chiral charge Q, is conserved. Here Q,(f,) = Q,(t2)= 1.

340

G. ’1 Hoojr. How insranrons solve rhc U( I ) probleni

Clearly, K, is not locally observtble. There is a nonobservable “Goldstone ghost” (the pole of D’). It goes without saying that Q, although exactly conserved, and J , , are not very useful for canonical formalism. Yet the current Ji5,rym and the charge Qk as used in ref. [7] are precisely of this form. This will be explained in the next section.

A neat way to implement the symmetry (6.3) is to treat the parameter m as a field: the “schizon”, or “spurion”. as those auxiliary objects are sometimes called to describe explicit symmetry breaking, such as isospin breaking by electromagnetism. The schizon field has a nonvanishing vacuum expectation value (6.4). Diagrammatically, a propagating electron could be represented by a diagram (fig. 1). Defining Q = 2 2 for the schizons we see that Q is absolutely conserved. Of course Q is also “spontaneously broken”.

7. The “exactly conserved chiral charge” in a canonically quantized theory

The fictitious symmetry described in the previous section can be mimicked in a gauge theory in a way that looks very real. Consider instead of (4.11), the current

J,,.,,, = 4 5 + K,

Then, in the limit m+O, one has

ap J,S.aym = 0 . (7.2)

The corresponding charge,

(7.3)

generates “exact” chiral transformations. How does this operator act in Hilbert space?

Conceptually the most transparent way is to first choose the temporal gauge: To answer this question we must formulate the canonical quantization of the gluon field carefully.

A,=O, (7.4)

which leaves us formally the set of all states (A@), Jl(x), G(x)) at a given time t, where Jl and $ should be seen as Grassmann numbers. Let us call the Hilbert space spanned by all these states the “huge” Hilbert space.

Then (7.4) leaves us invariance under all time-independent gauge transformations R = O(x), so that the Hamiltonian in this space is invariant under a group G composed of gauge transformations R(x) that may vary from point to point. This generates an invariance at each x, according to Noether’s theorem. Writing

341

G. ‘ I hoof^. How insronronr solve rhe U( I ) problem

RIA, 4, 6) = I A ” J I ” ~ ” )

where the subscript f j indicates how the fields are gauge-transformed, we have

[ H , R ] = O .

We can impose the gauge conditions of the second type:

nip) =IT) I

for ail infinitesimal R, acting nontrivially only in a finite region of 3-space:

AC(x) = A(x) t D,A(x) ,

(7.5)

A infinitesimal, and with compact support.

“huge” one). States [ Y ) satisfying (7.7) are said to be in the “large” Hilbert space (which is not as large as the

Finally, we consider all 0 with nontrivial winding number v

R,lT) = eiB’lY) . (7.8)

These states IT) are said to constitute the small, or physical Hilbert space at given 8. Now notice that JpS,rym does not commute with 0:

(7.9) iLgz

[ J p 5 . r y m ’ 01 = Dv’ ~puap‘mp

Therefore, J,,,sy, cannot be considered to be an operator for states in the ‘‘large’’ Hilbert space. Acting on a state satisfying (7.7) it produces a state not satisfying (7.7).

Now the charge Q5,sym of eq. (7.3) does commute with all R with v 7 0, but not with the others:

[Qs.sym* 41 =2iLvR, . (7.10)

Therefore, QJ,rym does act as an operator in the large Hilbert space, but not in the physical Hilbert space, because it mixes different 0 values. We can write

(7.11)

We see that in every respect Q5,sym behaves as 6 of the previous section, and JlrS,sym as j .

possible to construct physical states in which 8 would depend on space-time: A Goldstone boson would emerge in the theory if, besides the states satisfying (7.8), it could be

7 e = e(x, . (7.12)

Here it is obvious that (7.12) would be in contradiction with (7.7) and (7.8): If we would compare

342

G. ‘I Hooff, How insfuntons solve fhe U(J) problem

different 0“ but with the same u, such that the support of a(’) would be in a region near d’), and that of d2) near x ( * ) , then the combination

0l“’~‘’’-’

would have winding number zero. So the second gauge constraint would exclude any states for which f I (x ( ’ ) )# fJ(x( ’ ) ) . This is an important contrast with systems such as a ferromagnet, where local fluctuations are allowed, which, because of their large correlation lengths, correspond to massless excitations. Because of the similarity between (7.11) and (6.5) we can consider eie as a “schizon” field just as

the electron mass term. Since 0 cannot have any space-time dependence this schizon field cannot carry away any energy or momentum, just as m in the previous section.

Although Q5,1ym does not act in the “physical” Hilbert space, it is possible to write Ward identities [6,7] due to its formal conservation,

a,K, = -- g2i F i Y q Y . 321~’

One can take

op = KAO) >

and assume

[Qs,sym* K1 = O

while putting D,+O.

Now (7.17) is not obvious. Substituting (7.15) gives

(7.15)

(7.16)

(7.17)

(7.18)

On the other hand we showed in section 5 that F& has nontrivial chiral transformation properties. This however corresponds to

343

G. ‘ I Hooff, How iwlanlow so/w chc U ( / ) problem

Indeed,

where the right hand side follows from the substitution (4.23) and the commutation rule (5.12). C is a constant and f the

Equation (7.19) is a fundamental starting point of the discussions in refs. [6-81. The difference between (7.19) and (7.20) must apparently be made up by the contribution of KO to Q,,,,,. Now KO is not a physically observable field. Assigning to it the conventional commutation rules to be deduced from its composition in terms of color gauge fields is only allowed if one works in the “huge” Hilbert space including the gauge noninvariant states.

All we have to do to incorporate the fictitious symmetry generated by Q,,,,,,, into our model of effective fields described in section 2, is to add a schizon field, enlarging the Hilbert space. Let us call the schizon field

expectation value.

s = ei’ . (7.22)

Our new identification is

- 128di FF = 7 Im(S det 4) ,

Lg

S

(7.23)

- - j K o f O

-- I K o - O

WE. 2. Effective instanton action and its Q, symmetry properties. (a) Due lo fermionic zero modes 2L units of Q, are absorbed at lhc aite of the instanton. In the same time the charge generated by KO is not mnscrvcd. (b) In the effective theory the fennions are replaced by he +&Id. and KO by a sehizon S. The 4 carry 2 units of Q, each, and S has -2L units. S has a nonvanishing vacuum expeaation vslue.

344

C. ' I HOO~I, How inrm~ons solve (he U(1) problem

and if we postulate, in addition to (5.10)-(5.13) for QJ,sym also (see 7.11):

[Qs.sym, SI = - 2 ~ 9 (7.24)

then with (5.12) we find that F@ commutes with QJ,sym. Substituting e" det 4 by S det I$ in (2.13) we see that indeed our effective field theory obeys the fictitious symmetry generated by QS,,,,,,. It must therefore also obey the so-called anomalous Ward identities. See fig. 2.

8. Diagrammatic Interpretation of the anomalous Ward identities

The conclusion of the previous sections is that the model of section 2, with the substitution

cis+ s (8.1)

obeys all anomalous Ward identities. It also exhibits in a very transparent way how the symmetries are now spontaneously broken. There are two vacuum expectation values:

Both break Q,,,,, conservation. We can now draw Feynman diagrams in the Wigner representation by explicitly adding the vacuum bubbles due to (8.2) and (8.3). See fig. 3. By summing over the bubble insertions (geometric series which are trivial to sum), one reobtains the Goldstone representation of the particles. The a bubbles tend to make the pions and eta massless, but the terms with K

(and the quark masses m) contribute linearly to m2 for the various mesons. These diagrams clearly visualize where the masses come from and how the QS,,ym charges are absorbed into the vacuum,

In ref. [7] an apparent problem was raised by their equations (4.27) and (4.28): they suggest the

V,: 0 t-- . 9p @$ ua: 77;-

A n A 2iP.L *. ,,# K

Fig. 3. Fcynmrn rules in the Wiper mode. Thc u blob, render the pion and the q massless. But the S blob give n mass to q. HCI we drew explicitly che 1' propagator (L = 3). Its mass comes from the i n s d o n at the right. Note that mi. is linear in (S) and in (a): m:. rf.

345

G. ‘ I Hoofl. How insronrons solve the U ( 1 ) problem

need for field configurations with fractional winding number u, which would correspond with the breaking up of our S field into components with smaller Q,,sym charges. In our diagrams we clearly see that there is no such need. If we have a Green function with an operator that creates only two chiral charges,

xop = 2

then the sigma field can absorb these two, or add 2(L - 1) more and have them absorbed by S. The vacuum simply isn’t an eigenstate of Q, (nor Q5.,ym) as it was assumed.

9. Conclusion

The disagreements between the approach of the Crewther’s school to the U(l) problem, using anomalous Ward identities, and the more standard beliefs are not as wide as they appear. Their anomalous Ward identities, if applied with appropriate care, are perfectly valid for a simple effective field theory that clearly exhibits the most likely vacuum structure of QCD. It is important however to realize that the relevant exactly conserved chiral charge Q,,,ym is not physically observable, something which explains the need for the introduction of a spurion field S in the effective theory. It appears that the consequences of working with an unphysical symmetry were underestimated in ref. (71. Some of the difficulties signalled in [7] were due to the too strong assumption that the vacuum is an eigenstate of Q,. That such assumptions are unnecessary and probably wrong would have been realized if they had taken the effective theory more seriously.

The fact that the effective theory of section 2 displays the correct symmetry properties does not have to mean that it is accurate. Indeed it could be that instantons tend to split into “merons” [16], a dynamical property that might be a factor in the spontaneous chiral symmetry breaking mechanism [14]. But these aspects do not affect the symmetry transformation properties of the fields under con- sideration. More fields, describing higher resonances, could have to be added. The baryonic degrees of freedom are most likely to be considered as extended solutions of the effective field equations (skyrmions). That these pkyrmions I171 indeed possess the relevant baryonic quantum numbers was discovered by Witten [lS].

The author thanks R.J. Crewther for his patience in extensive discussions, even though no complete agreement was reached.

Appendix A. The sign of AQ,

In ref. [7] the present author’s work was claimed to b e i n error a t various places. Although some minor technical corrections on the computed coefficients in the quantum corrections due t o instantons were found (in the original publication see ref. [20]) and even some insignificant inaccuracies in the notation of a sign might occur, we stress here that none of those claims of ref. [7] were justified. In particular there are no fundamental discrepancies i n the sign of AQs.

Let us here ignore the masses of the quarks. As formulated i n section 4, an instanton in a finite space-time volume V causes a transition with

346

G. ' I Hooff. How irufcmfons solve fhe U( I probkm

where Q, is the gauge-invariant axial charge. Since QS,,,,,,,, as defined in section 7, is now strictly conserved one obviously has

AQs.,ym = 0 (A21

which is of course only defined in the ''large'' Hilbert space comprising all 0 worlds. Instead of (Al-A2), we read in ref. [7]:

A Q , = O ; AQs,lym=-2L. ('43)

These are not the properties of a closed space-time volume such as we described, but represent the features of a Green's function where the asymptotic states are 0-vacua. Since QS,,ym contains explicitly the operator d l d 0 in the ''large'' Hilbert space (cf. eq. (7.11)), the @-vacuum is not invariant under Q,,,ym. This is why (A3) is not in conflict with Ql,sym conservation. But we also see that (Al-A2) and (A3) hold under different boundary conditions (the reason why AQs = O for Green's functions in a &vacuum is correctly explained in ref. (71).

Appendix B. The integration over the instanton size p

To get even a rough estimate of the size of our instanton's contribution to an amplitude requires lengthy calculations. The most essential ingredients for such a calculation are provided in our previous paper (5.11 (see refs. I191 for the original publication, and the corrections given in [20]). Let us take for simplicity SU(2) as our color gauge group. An instanton with given size p then gives rise to an effective interaction of the form

where N f is the number of fermions in the doublet representation, p is the size of the instanton, p is an arbitrary mass unit enabling us to obtain a renormalization group invariant expression, and N " ( t ) is the number of scalar field representations with color t. The parameter C was computed in the previous paper, and it also depends on the scalars and fermions present:

where A, A(t), and B are numbers computed in [5.1].

347

G. 'f H o o f f , f fow inr/antom solve thc U(f) problem

W = exp( -8?r2/g2)(det det det W8,,

and the determinants can be computed by diagonalization:

It is clear that (B3) is highly divergent unless we formulate very precisely an appropriate subtraction procedure. A convenient method is to apply first a variety of Pauli-Villars regularization to the fields Aq" and $ and then add correction terms to be obtained by comparing this regularization scheme to for instance dimensional regularization.

It is then found that (B3)is not the complete answer: there are zero eigenvalues of W A , n+ and a&,, which have to be considered separately. The corresponding eigenmodes must be replaced by collective coordinates including an appropriate Jacobian for this transformation. Since (B3) must be compared with the vacuum transition (in absence of instantons) the collective coordinate integration is to be divided by a norm factor determined by a Gaussian integral. The result is the effective Lagrangian (for the case that the color gauge group is SU(2)):

where N' is the number of fermions in the doublet representation, p is a scalar parameter for the instanton, p is an arbitrary mass unit enabling us to obtain a renormalization group invariant expression, and N'(f ) is the number of scalar field representations with color f .

Defining coefficients a(f) as in table 1, we have now:

A = - a ( l ) + Y ( l o g 4 n - y ) + f =7.053W103 A(!) = -a(t) + A(log4a - y)C(f)

A( 112) = 0.308 690 69 A(1) = 1.09457662

A(3/2) = 2.481 356 10 B = -24112) + f(log4n - y ) = 0.35952290.

348

G. ’I Hoo11. How imlanlotu solve the U( I ) problem

Table 1

I C(l) 4) 0 0 0 I /2 1 2R - 1 log2 - 17/72 1 4 8R t I log2 - 1619 312 10 20Rt410g3- j log2-265136

1 1 = I J R = (log2nt 7 ) t 1 =0.2487544?’/. 1

Finally, the spinors o are normalized by

z @.k = 1(1 -t- Y,) (B7)

and required to be smeared in color space, such that for instance

(@.q3) = f4Jl -t- YA (B8) and, in the case L = N‘ = 2:

(B9) 1 (i (&o)(Lj$,)) = 24 (2SB,:6? - 8 p ~ ) & ’ ’ 4 ; v + Y5)$%?(1 + n)dP

8-1

where s and t are flavor indices and a,, pi color indices.

be natural to choose The p integral may seem to diverge in most interesting cases (N‘> 1). Note however that it would

and substitute g-’ by the running value g( p)-*. At large p one might take g a p and thus improve the convergence. Of course the infrared end of the integral is quite uncertain because in our perturbative procedure the effects of confinement etc. have not been taken into account. This inhibits a precise evaluation of the amplitude. A rough estimate (for a color SU(2) theory) is obtained if we take at large p

g2( l /p )+ 16.rrp2u (B11)

where (I is the string constant. Then quarks with color charge 112 at a distance p from each other feel a force

Our integral becomes, in the case N‘ = 2,

S“‘ = 216.rr10 eA-”2, I - exp - 8 r 2 / g 2 ( l / p ) .

349

G. 'I Hoof/, How insfontons solve /he U ( I ) problem

From (B11) we get

p dp 2 g d g / l 6 m

and using z = l /g2 the integral in (B13) is

so that

where Ql is the Lagrangian (B9). The result is uncomfortably large, but then the approximations used here (eq. B l l )

could at best only be expected to yield the order of magnitude of the expected interaction, which is clearly a strong one. Note that our coupling constant is subtraction scheme dependent. Since we did not yet include two loop order effects the subtraction scheme used strongly effects the final result. A factor 47r difference in the definition of p in (Bl) reduces the coefficient in front of the effective action (B16) by a factor ( 4 ~ ) ~ = 26n3. This is just to illustrate how sensitively the amplitude obtained depends upon the assumptions.

References

[l] S.L. Glubow, in: H a h md their inleraaionc. Ericc 1967. 4. A. Zichi (Acad. Rar, New York. 1968) p. 83, S.L. Glubow, R. J d w and S.4. Shci. Phys. Rev. 187 (1969) 1916; M. O e l l - M i ~ , in: h. Tbid TOpial Cod. on Plrtidc Physia, Honolulu 1969. edr. W.A. S i md S.F. Tuan (Western Periodicals, La Angela. 1970) p. 1, and b: Ekmcnlary Partidc Physics. Wadming 1972. ed. P. Urbrn (Springer-Verlag, 1972); Actr Phyrifl AuslrircP Suppl. IX (1972) 733;

Phys. Lea. 47B (1973) 36.5.

I.S. &u and R. J&w, Nwvo Cim. 6oA (1969) 4?; S.L. Adler and W.A. Budecn, Phys. Rev. 182 (1969) 1517.

H. Fflczacb d M. GeU-MIIUI, ROC. XVI Intern. Conf. on H.E.P.. chiago 1972, cdr. J.D. Jackson lad A. RO~CIIS, Vol. 2, p. 135, md

121 S.L. Adkr, Php. Rev. 177 (1969) 2426;

131 A.A. BcIaVio, A.M. Pdyllrov, AS. Schwm md Yu.S. Tyupkio, Phys. Lett. S9B (1975) 85. [4] G. 'I Hooh, Php. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432;

R. Jackiw md C. Rebbi, Php. Rev. Lett. 37 (1976) 172; C.G. Cdhn Jr., R.F. Darhen and D.J. O m , Phys. Lett. 63B (1976) 334; Phys. Rev. D17 (1978) 2717.

IS] S. Cokmm. in: The Whyx of Subnudur Physics. Ericc 1977, ed. A. zifhichi (Pknum Prar. New York. 1979) p. 805. 161 R. Crcwther, Php. Lett. 70B (1W) 349; Riv. Nuovo Cim. 2 (1979) 63;

R. Crcwther, in: F a md Rape*s of Gauge Theories, schladming 1978, ed. P. Urban (Springer-Verlag, 1978); A m Phys. Aualriaca suppl. XIX (1978) 47.

171 G.A. chrirtoa, Phya. Repom 116 (1984) 251. 181 R. Crewther, private communiation. 191 J.C. Wud. Phya. Rev. 78 (1993) 1824;

Y. Takahashi. Nuovo cim. 6 (1957) 370.

J.C. Taylor, Nud. Phys. B33 (1971) 436. [la] A. Shmov, '(heor. .ad k t h . Phyr. 10 (1972) 153 (in Russian). transl. Them. md Math. Phys. 10. p. 99;

350

G. ' I Hoof l , How i ~ ~ f a ~ ~ o n i solve the U ( f ) problem

[ l l ) C. Becfhi, A. Rouet and R. Ston, Comm. Math. Phys. 42 (1975) lD, AM. Phys. (N.Y.) 98 (1976) 281. [lZl M. Velunm, Nuel. Php. B7 (I=) 637; Nuel. Php. 921 (1970) 288. 1131 Then u no paper by &I1 or Treiman on thh plrrieulu subject. See ref. [12]. [14] A.R. ulitnitrky. The diwnte c h i spllncny b r d i o g in OCD II a manifatation of IIIC Rubalrov-Clllan effcn. Novaibint prepiat 1986. 1151 C. Let ud W. Budecn, Nud. Flip. B153 (1979) 210. 1161 C.G. CAlm JI., R.F. Duhen d D.J. Gron. Phys. Len. 669 (1977) 375; Php. Rev. D17 (1978) 2717; Phys. Lett. 789 (1978) Wn. 1171 T.H.R. Skyme, Proc. Roy. Soe. A260 (1%1) 127. [l8] E. Wittea, Nucl. Phya. B223 (1983) 422,433. 1191 0. 't Hwft , Phys. Rev. D14 (1976) 3432. (201 F.R. On, Php. Rev. D16 (1977) ZSV,

0. 'I H d , Phys. Rev. D18 (1978) 2199; A. Hlrcnfratz a d P. H u e n h t z . Nud. pbp. Bl!B (1981) 210.

[21) K. Fujiklwr, Flip. Rev. Lett. 42 (1979) 1195; Php. Rev. DZ1 (19210) 2848.

351

CHAPTER 5.3

NATURALNESS, CHIRAL SYMMETRY, AND SPONTANEOUS CHIRAL SYMMETRY BREAKING

G . 't Hooft Institute for Theoretical Physics

Utrecht, T h e Netherlands

ABSTRACT

A properly called "naturalness" is imposed on gauge theories. It is an order-ofmagnitude restriction that must hold at all energy scales IJ. To construct models with complete naturalness for elementary particles one needs more types of confining gauge theories besides quantum chromodynamics. We propose a search program for models with improved naturalness and concentrate on the possibility that presently elementary fermions can be considered as composite. Chiral symmetry must then be responsible for the masslessness of these fermions. Thus we search for QCD- like models where chiral symmetry is not or only partly broken spontaneously. They are restricted by index relations that often cannot be satisfied by other than unphysical fractional indices. This difficulty made the author's own search unsuccessful so far. As a by-product we find yet another reason why in ordinary QCD chiral syrmnetry must be broken spontaneously.

1111. INTRODUCTION

The concept of causality requires that macroscopic phenomena follow from microscopic equations. Thus the properties of liquids and solids follow from the microscopic properties of molecules and atoms. One may either consider these microscopic properties to have been chosen at random by Nature, or attempt to deduce these from.even more fundamental equations at still smaller length and time scales. In either case, it is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a

Dynamical Symmetry Breaking, A Collection of Reprints, Edited by A. Farhi and R. Jackiw (World Scientific. 1982).

352

G. 't HOOFT

philosophy which we would l i k e t o apply to the unif ied gauge theories: the e f fec t ive in te rac t ions a t a large length scale, corresponding t o a low energy scale u l , should follow from the properties a t a much smaller length scale , o r higher energy scale ~ 2 , without the requirement tha t various d i f fe ren t parameters a t the energy sca le 1.12 match with an accuracy of the order of u , /p2 . That would be unnatural. On the other hand, i f a t the energy scale ~2 some parameters would be very small, say

then t h i s may s t i l l be na tura l , provided tha t t h i s property would not be s p o i l t by any higher order e f fec ts . We now conjecture tha t the following d o p a should be followed: - a t any energy sca le p , a physical parameter or set of physical parmeters a i ( p ) i s allowed t o be very small only i f the replacement ui(1.1) - o would increase the synaDetry of the system. - In what follows t h i s is what w e mean by naturalness. It i s c lear ly a weaker requirement than tha t of P. Diracl) who i n s i s t s on having no small numbers a t a l l . It i s what one expects i f a t any m a s s sca le p > p o some ununderatood theory with s t rong in te rac t ions determines a spectrum of pa r t i c l e s with various good o r bad symmetry properties. I f a t p - yo cer ta in parameters come out t o be small, say 10'5, then tha t cannot be an accident; i t must be the consequence of a near synmetry.

For instance, a t a mass sca le

p - 50 GeV,

the e lectron mass me is IO? This is a small parameter. It is acceptable because me = o would imply an addi t ional ch i r a l aymetry corresponding to separate conservation of l e f t handed and r igh t handed electron-like leptons. This guarantees tha t a l l renormalizations of I112 nd I113 we compare naturalness f o r quantum electrodynamics and +t theory.

a r e proportional t o me i t s e l f . In sects .

Gauge coupling constants and other (sets of) in te rac t ion constants may be small because put t ing them equal t o zero would turn the gauge bosona o r o ther pa r t i c l e s i n t o f r e e pa r t i c l e s so t h a t they are separately conserved.

I f within a set of small parameters one i s several orders of ma~nitude smaller than another then the smallest must s a t i s f y our "dogma" separately. Ad w e w i l l see, naturalness w i l l put the severest r e s t r i c t ion on the occurrence of scalar pa r t i c l e s i n renormalizable theories. In f a c t we conjecture tha t t h i s is the reason why l i g h t , weakly in te rac t ing scalar pa r t i c l e s are not seen.

353

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

It is our aim to use naturalness as a new guideline to construct models of elementary particles (sect. 1114). In practice naturalness will be lost beyond a certain mass scale po, to be referred to as "Naturalness Breakdown Mass Scale" (NBMS). This simply means that unknown particles with masses beyond that scale are ignored in our model. The NBMS is only defined as an order of magnitude and can be obtained for each renormalizable field theory. For present "unified theories", including the existing grand unified schemes, it is only about 1000 GeV. In sect. 5 we attempt to construct realistic models with an NBMS some orders of magnitude higher.

One parameter in our world is unnatural, according to our definition, already at a very low mass scale (voh. is the cosmological constant. Putting it equal to zero does not seem to increase the symnetry. Apparently gravitational effects do not obey naturalness in our formulation. We have nothing to say about this fundamental problem, accept to suggest that onty gravitational effects violate naturalness. Quantum gravity is not understood anyhow so we exclude it from our naturalness requirements.

eV). This

On the other hand it is quite remarkable that all other elementary particle interactions have a high degree of naturalness. No unnatural parameters occur in that energy range where our popular field theories could be checked experimentally. We consider this as important evidence in favor of the general hypothesis of naturalness. Pursuing naturalness beyond 1000 GeV will require theories that are immensely complex compared with some of the grand unified schemes.

A remarkable attempt towards a natural theory was made by Dimopoulos and Susskind z), These authors employ various kinds of confining gauge forces to obtain scalar bound states which may substitute the Higgs fields in the conventional schemes. In their model the observed fermions are still considered to be elementary.

Most likely a complete model of this kind has to be constructed step by step. One starts with the experimentally accessibleaspects of the Glashow-Weinberg-Salam-Ward model. This model is natural if one restricts oneself to mass-energy scales below 1000 GeV. Beyond 1000 GeV one has to assume, as Dimopoulos and Susskind do, that the Higgs field is actually a fermion-antifermion composite field. Coupling this field to quarks and leptons in order to produce their mass, requires new scalar fields that cause naturalness to break down at 30 TeV or so. Dimopoulos and Susskind speculate further on how to remedy this. To supplement such ideas, we toyed with the idea that (some of) the presently "elementary" fermions may turn out to be bound states of an odd number of fermions when considered beyond 30 TeV. The binding mechanism would be similar

354

G. 't HOOFT

to the one that keeps quarks inside the proton. However, the proton is not particularly light compared with the characteristic mass scale of quantum chrmodynamics (00). Clearly our idea is only viable if something prevented our "baryons" from obtaining a mass (eventually a small mass may be due t o some secondary perturbation).

The proton ows its mass to spontaneous breakdown of chiral symmetry, or so it seems according to a simple, fairly successful model of the mesonic and baryonic states in QCD: the Gell-Mann-Llvy sigma model3). Is it possible then that in some variant of QCD chiral symmetry is not spontaneously broken, or only partly, so that at least some chiral syutnetry remains in the spectrum of fermionic bound states? In this article we will see that in general in SU(N) binding theories this is not allowed to happen, i.e. chiral symmetry must be broken spontaneously.

1112. NATURALNESS IN QUANTUM ELECTRODYNAMICS

Quantum Electrodynamics as a renormalizable model of electrons (and muons if desired) and photons is an example of a "natural" field theory. The parameters a, m, (and m,,) may be small independently. In particular The relevant symmetry here is chiral symmetry, for the electron and the muon separately. We need not be concerned about the Adler-Bell-Jackiw anomaly here because the photon field being Abelian cannot acquire non-trivial topological winding numbers4).

(and m,,) are very small at large p .

There is a value of p where Quantum Electrodynamics ceases to be useful, even as a model. The model is not asymptotically free, so there is an energy scale where all interactions become strong:

(1112)

where Nf is the number of light fermions. If some world would be described by such a theory at low energies, then a replacement of the theory would be necessary at or below energies of order uo.

4 1113. 4 -THEORY

A renormalizable scalar field theory is described by the Lagrangian

(1113) 2 1 4 P - -lo,,+) - Am242 - A+ . the interactions become strong at

u 5 m exp(l6a 2 /3X) , but is it still natural there?

(1114)

355


because X = o would correspond to a non-interacting theory with total number of $ particles conserved. But is small m allowed? If we put m = o in the Lagrangian (1113) then the symmetry is not enhanced*). However we can take both m and X to be small, becauseif X = m - o we have invariance under

There are two parameters, X and m. Of these, X may be small

(1115)

This would be an approximate symmetry of a new underlying theory at energies of order po. Let the symmetry be broken by effects described by a dimensionless parameter E . Both the mass term and the interaction term in the effective Lagrangian (1113) result from these symmetry breaking effects. Both are expected to be of order E . Substituting the correct powers of p0 to account for' the dimensions of these parameters we have

Theref ore,

(1116)

(1117)

This value is much lower than eq. (1124). We now turn the argument around: if any "natural" underlying theory is to describe a scalar particle whose effectiue Lagrangian at low energies will be eq. (III3), then its energy scale cannot be given by (1114) but at best by (1117). We say that naturalness breaks down beyond m/A. It must be stressed that these are orders of magnitude. For instance one might prefer to consider X/n2 rather than A to be the relevant parameter. po 'then has to be multiplied by T . Furthermore, X could be much smaller than E because A = o separately also enhances the synnnetry. Therefore, apart from factors n, eq. (1117) indicates a maximum value for po.

Another way of looking at the problem of naturalness is by comparing field theory with statistical physics. The parameter m h would correspond to (T-Tc)/T in a statistical ensemble. Why would the temperature T chosen by Nature to describe the elementary particles be so close to a critical temperature Tc? If Tc then T may not be close t o Tc just by accident.

1114. NATURALNESS IN THE WEINBERG-SAW-GIM MODEL

o

The difficulties with the unnatural mass parameters only occur in theories with scalar fields. The only fundamental scalar

*) Conformal symmetry is violated at the quantum level.

G. 't HOOF1

field that occurs in the presently fashionable models is the Higgs f'eld in the extended Weinberg-Salam model. The Higgs mass-squared,

agrangian. It is small at energy scales v >>mw Is there an approximate symmetry if % + 07 With some we might consider a Goldstone-type symmetry:

1, is up to a coefficient a fundamental parameter in the

stretch of imagination

+(XI + +(XI + const. (1118)

However we also had the local gauge transformations:

$(XI + n(x) $(XI . (1119)

The transformations QII8) and QII9) only form a closed group if we also have invariance under

But then it becomes possible to transform + away completely. The Higgs field would then become an unphysical field and that is not what we want. Alternatively, we could have that (1118) is an approximate synmetry only, and it is broken by all interactions that have to do with the symmetry (1119) which are the weak gauge field interactions. Their strength is g2/4n - 0(1/137). So .at best we can have that the symmetry is broken by 6(1/137) effects. Therefore

4 Also the A $ symmetry. Therefore

term in the Higgs field interactions breaks this

Now

(11111)

(11112)

where FH is the vacuum expectation value of the Higgs field, known to be')

FH - ( 2 G f i ) - ' / * - 174 GeV . We now read off that

(11113)

*) Some numerical values given during the lecture were incorrect. I here give corrected values.

357


This means t h a t a t energy scales much beyond FH our model becomes more and more unnatural . Actual ly , f a c t o r s of n have been omitted. In p r a c t i c e one f a c t o r of 5 o r 10 is s t i l l not t o t a l l y unacceptable. Notice t h a t the a c t u a l value of dropped ou t , except t h a t ..

(1111s)

Values f o r % of j u s t a few G e V a r e unnatural .

1115. EXTENDING NATURALNESS

Equation (11114) t e l l s us t h a t a t energy s c a l e s much beyond 174 G e V the s tandard model becomes unnatural . As long as the Higgs f i e l d H remains a fundamental scalar nothing much can be done about t h a t . We the re fo re conclude, with Dimopoulos and Susskind2) t h a t the "observed" Higgs f i e l d must be composite. A non- t r iv i a l s t r o n g l y i n t e r a c t i n g f i e l d theory must be ope ra t ive a t 1000 GeV o r so. An obvious and indeed l i k e l y p o s s i b i l i t y i s t h a t the Higgs f i e l d H can be w r i t t e n as

H = ZGJl , (11116)

where'Z is a renormalizat ion f a c t o r and J, i s a new quark-like o b j e c t , a fermion with a new color- l ike i n t e r a c t i o n 2 ) . We w i l l r e f e r t o t h e ob jec t as mete-quark having meta-color. The theory w i l l have a l l f e a t u r e s of QCD so t h a t we can copy the nomenclature of QCD with the p r e f i x "meta-". The Higgs f i e l d i s a meta-meson.

It is now tempting t o assume t h a t t he meta-quarks transform the same way under weak SU(2) x U ( 1 ) as ordinary quarks. Take a doublet with left-handed components forming one gauge doublet and r i g h t handed components forming two gauge s i n g l e t s . The nkta- quarks a r e massless. Suppose t h a t the meta-chiral symmetry i s broken spontaneously j u s t as i n ordinary QCD. What would happen?

What happens is i n ordinary QCD w e l l descr ibed by the Gell- Mann-Levy sigma model. The l i g h t e s t mesons form a q u a r t e t of r e a l f i e l d s , 0 transforming as a i j '

21ef t 2 r i g h t

r ep resen ta t ion of

S U ( 2 p f t 0 SU(2)right.

Since the weak i n t e r a c t i o n only dea l s with SU(2)left t h i s q u a r t e t can a l s o be considered a s one complex doublet r ep resen ta t ion of weak Su(2). In ordinary QCD we have

358

a a 1J

4 i j b s i j + i .r. .n , and

<a >vacuum = & f n * 9 1 MeV . The complex doublet i s

4 i -- then

and

.I [A] x 64 MeV . "i 'vacuum

(11118)

(11119)

(11120)

We conclude t h a t i f w e t r a n s p l a n t t h i s theory t o the TeV range then w e g e t a s c a l a r doublet f i e l d with a non-vanishing vacuum expectat ion value f o r f r e e . A l l w e have t o do now is t o match the numbers. If w e scale a l l QCD masses by a s c a l i n g f a c t o r K then we match

F~ = 174 ~ e v = K 64 MeV i

K = 2700 . (11121)

Now the mesonic s e c t o r of QCD i s usua l ly assumed t o be reproduced i n the I/N expansion 5 , where N i s the number of co lo r s ( i n QCD we have N - 3). The 4-meson coupling constant goes l i k e 1/N. Then one would expect

f T a f i . (11122)

Therefore

K = 2700 4 , (11123)

i f the metacolor group is SU(N).

Thus w e ob ta in a model t h a t reproduces the W-mass and p r e d i c t s t he Higgs mass. The Higgs is the meta-sigma p a r t i c l e . The ordinary sigma is a wide resonance a t about 700 MeV3), so t h a t we p r e d i c t

(11124)

and i t w i l l be extremely d i f f i c u l t t o d e t e c t among o the r s t rongly i n t e r a c t i n g ob jec t s .

359


1116. WHAT NEXT?

The model of the previous s e c t i o n i s t o our mind near ly inevi- t a b l e , but t he re are problems. These have t o do with the observed fermion masses. A l l leptons and quarks owe t h e i r masses t o an i n t e r a c t i o n term of the form

sJlHJ, , (11125)

where g is a coupling constant , J, i s the l ep ton or quark and H i s the Higgs f i e l d . With (11116) t h i s becomes a four-fermion i n t e r a c t i o n , a fundamental i n t e r a c t i o n i n the new theory. Because i t is non-renormalizable f u r t h e r s t r u c t u r e is needed. In r e f . 2 t he obvious choice i s made: a new "meta-weak in t e rac t ion ' ' gauge theory e n t e r s with new super-heavy in t e rmed ia t e vec to r bosons. But s i n c e H is a s c a l a r t h i s boson must be i n the crossed channe1,a r a t h e r awkward s i t u a t i o n . (See op t ion a i n Figure 1.) A simpler theory i s t h a t a new s c a l a r p a r t i c l e i s exchanged i n the d i r e c t channel. (See opt ion b i n Figure I . )

me ta-quarks

mete-quarks mete-quarks Figure 1.

360

G. 't HOOFT

Notice tha t i n both cases new scalar f i e l d s are needed because i n case a) something must cause the "spontaneous breakdown" o f the new gauge symmetries. Therefore choice b) i s simpler. We removed a Higgs sca l a r and we get a s ca l a r back. Does naturalness improve? The answer i s y e s . The coupling constant g i n the in t e rac t ion GI1251 s a t i s f ies

(11126)

Here g 1 sca l a r s mass, and Z i s from (IIK16) and i s of order and g2 a r e the couplings a t the new ver t ices , M, i s the new

2 - f . m 4 (K m,,l2 (1800 G e V ) 2 '

(11127)

Suppose tha t the heaviest lepton o r quark i s about 10 GeV. For that fermion the coupling constant g is

"f g * p - 1/20 . We get

Naturalness breaks do& a t

an improvement of about a fac tor 50 compared w i t h the s i t ua t ion in sect. 1114. Presumably we a r e again allowed t o multiply by factors l i k e 5 or 10, before ge t t ing in to rea l trouble.

Before speculating on how to go on from here t o improve naturalness s t i l l fur ther we must assure ourselves tha t a l l other a l l eys a re bl ind ones. An int r iguing poss ib i l i t y i s tha t the presently observed fermions a re composite. We would get option c> Figure 2.

361


or y Fig. 2

The do t t ed l i n e could be an ordinary weak i n t e r a c t i o n W or photon, t h a t breaks an i n t e r n a l symmetry i n t h e binding f o r c e f o r t he new components. The new binding fo rce could e i t h e r act a t t h e 1 TeV or a t the 10-100 TeV range. It could e i t h e r be an extension o f meta- co lo r or be a (color)" or paracolor fo rce . Is such an idea v i ab le?

Clear ly , compared with the energy scale on which t h e binding forces take place, the composite fermions must be near ly massless. Again, t h i s cannot be an accident . The c h i r a l symmetry responsible f o r t h i s must be present i n the underlying-theory. Apparently then, t h e underlying theory w i l l possess a c h i r a l syaunetry which i s not (or not completely)spontaneou8ly broken, bu t r e f l e c t e d i n the bound state spectrum i n the Wigner mode: some massless c h i r a l ob jec t s and p a r i t y doubled massive fermions. This p o s s i b i l i t y i s most c l e a r l y descr ibed by the o-model as a model f o r the lowest bound states occurr ing i n ordinary quantum chromodynamics.

1117. THE U MODEL

The fermion system i n quantum chromodynamics shows an axial synunetry. To i l l umina te our problem l e t us consider the case of two f lavors . The l o c a l c o l o r group i s SU(3),. The s u b s c r i p t c he re s tands for co lo r . The f l a v o r symmetry group i s SU(Z)L l e f t and r i g h t and the group elements must be chosen t o be space- t i m e independent. W e s p l i t t he fermion f i e l d s J, i n t o l e f t and r i g h t components:

SU(Z), @ U(1) where the subsc r ip t s L and R stands f o r

J, I ( l + Y 5 ) J , L + 4(1T5)JIR (11128)

J,L transforms as a 3 (11129)

(11130)

fB zL 8 l R 0 2s

and $, transforms as a 3c 8 I L @ 2, 0 Is where the ind ices r e f e r t o the var ious groups. 1 stands f o r t h e Lorentzgroup S0(3,1), l o c a l l y equivalent t o SL(2,c) which has two

C

362

G. 't HOOFT

d i f f e r e n t complex doublet representat ions 21 and 24 (corresponding t o the transformation law f o r the neutr ino and an t ineu t r ino , r e spec t ive ly ) . The f i e l d s $L and $8 have the same charge under U(I) , whereas a x i a l U(1) group (under which they would have opposite charges) is absent because of i n s t an ton e f f e c t s 4 ) .

The e f f e c t of the co lo r gauge f i e l d s is t o bind these fermions i n t o mesons and baryons a l l of which must be co lo r s i n g l e t s . It would be n i ce i f one could descr ibe these hadronic f i e l d s a s representat ions of SU(2)L 8 SU(2)R @ U ( I ) and the Lorentz group, and then c a s t t h e i r mutual i n t e r a c t i o n s i n the form of an e f f ec t ive Lagrangian, i nva r i an t under the f l a v o r symmetry group. In the case a t hand t h i s is possible and the r e s u l t i n g construct ion is a successful d one-time popular model f o r pions and nucleons: t he u model3Y. We have a nucleon doublet

(11131 )

where

N transforms a s a I @ 2 L @ l R 9 2 , ( I I I32a)

and NR transforms as a l c @ l L @2R 8 2g . (III32b)

Further w e have a q u a r t e t of real scalar f i e l d s (a ,n) which t r a n e f o m a s a 1

L C - +

O zL 8 21 @ it. The 7.agrangian is C

(11133)

Here V must be a r o t a t i o n a l l y i n v a r i a n t function.

Usually V i s chosen such t h a t i t s absolute+minimum is away from the o r ig in . Let V be minimal a t a - v and n - 0 . Here v is j u s t a c-umber. To ob ta in the physical p a r t i c l e spectrum we wr i t e

0 - v + 8 (11134)

+ i n t e r a c t i o n terms . (11135)

Clearly, i n t h i s case the nucleons acquire a mass term m = gOv and the s p a r t i c l e has a mass m: - 4v2V"(v2), whereas t t e pion remains s t r i c t l y massless. The e n t i r e mass of the pion must be due t o e f f e c t s t h a t e x p l i c i t l y break SU(2)L x SU(2)R , such as a small

363


mass term m 51) for the quarks (11128). We say that in this case the flavor group sU(2)~ @ sU(2)~ is spontaneously broken into the isospin group SU(2).

9

Another possibility however, apparently not realised in ordinary quantum chromodynamics, would be that SU(2)L @SU(2)R is not spontaneously broken. We would read off from the Lagrangian (11133) that the nusleons N would form a massless doublet and that the four fields (a,n) could be heavy. The dynamics of other confining gauge theories could differ sufficiently from ordinary QCD so that, rather than a spontaneous symmetry breakdown, massless "baryons" develop. The principle question we will concentrate on is why do these massless baryons form the representation (III32), and how does this generalize to other systems. We would let future generations worry about the question where exactly the absolute minimum of the effective potential V will appear.

1118. INDICES

We now consider any color group Gc. The fundamental fermions in our system must be non-trivial representation of Gc and we assume "confinement" to occur: all physical particles are bound states that are singlets under Gc. Assume that the fermions are all massless (later mass terms can be considered as a perturbation). We will have automatically some global symmetry which we call the flavor group GF. (We only consider exact flavor symmetries, not spoilt by instanton effects.) Assume that GF is not spontaneously broken. Which and how many representations of GF will occur in the massless fermion spectrum of the baryonic bound statgs? We must formulate the problem more precisely. The massless nucleons in (11133) being bound states, may have many massive excitations. However, massive Fermion fields cannot transform-as a 21: under Lorentz transformations; they must go as a 21: @ 2 1 . That is because a mass term being a Lorentz invariant proiuct of two fields at one point only links 21: representations with 21: representations. Consider a given representation T of +. Let p be the number of field multiplets transforming as r 821: and q be the number of field multiplets r @21. Mass terms that link the 21: with 21: fields are completely invariant and in general to be expected in the effective Lagrangian. But the absolute value of

1 1 - p - q (11136)

is the minimal number of surviving massless chiral field multiplets. We will call 11 the index corresponding to the . By definition this index must be a (positive or integer. In the sigma model it is postulated that

364

G. 't HOOFT

(11137)

index (IL @ 2R) - -1 index (r)- o for all other representations r.

This tells us that if chiral synrmetry is not broken spontaneously one massless nucleon doublet emerges. We wish to find out what massless fermionic bound states will come out in more general theories. Our problem is: how does (11137) generalize?

1119. ABSENCE OF MASSLESS BOUND STATES WITH SPIN 312 OR HIGHER

In the foregoing we only considered spin o and spin 112 bound states. Is it not possible that fundamentally massless bound states develop with higher spin? I believe to have strong arguments that this is indeed not possible. Let us consider the case of spin 312. Massive spin 312 fermionr are described by a Lagrangian of the form

(11138)

Just like spin-one particles, this has a gauge-invariance if m + 0 :

v ) ) + J,)) + aP. l (X) s (11139)

where n(x) is arbitrary. Indeed, massless spin 312 particles only occur in locally supersymmetric field theories. The field q(x) is fundamental 1 y unobservable.

Now in our model JI would be shorthand for some composite field: J, would be'observables. If m = o we would be forced to add a gauge fixing term that would turn rl into an unacceptable ghost particle*).

-+ $$$. However: then all components of this, including 0,

We believe, therefore, that unitarity and locality forbid the occurrence of massless bound states with spin 312. The case for higher spin will not be any better. And so we concentrate on a bound state spectrum of spin 112 particles only.

*) Note added: during the lectures it was suggested by one attendant to consider only gauge-invariant fields as Y,,,, =aRJlv - awJ,,. However, such fields must satisfy constraints:a [a&pw]=o. Composite field will never automatically satisfy such constraints.

365


11110. SPECTATOR GAUGE FIELDS AND -FERMIONS

So far, our model consisted of a strong interaction color gauge theory with gauge group Gc, coupled to chiral fermions in various representations r of Gc but of course in such a way that the anomalies cancel. The fermions are all massless and form multiplets of a global symmetry group, called GF. For QCD this would be the flavor group. In the fermion symmetries besides metacolor.

metacolor theory GF would include all other

In order to study the mathematical problem raised above we will add another gauge connection field that turns GF into a local symmetry group. The associated coupling constants may all be arbitrarily small, so that the dynamics of the strong color gauge interactions is not much affected. In particular the massless bound state spectrum should not change. One may either think of this new gauge field as a completely quantized field or simply as an artificial background field with possibly non-trivial topology. We will study the behavior of our system in the presence of this "spectator gauge field". As stated, its gauge group is GF.

Note however, that some flavor transformations could be associated with anomalies. There are two types of anomalies:

i) those associated with Gc x GF, only occurring where the color field has a winding number. Only U ( 1 ) invariant subgroups of GF contribute here. They simply correspond to small explicit violations of the + syrmnetry. From now on we will take as GF only the anomaly-free part. Thus, for QCD with N flavors, GF is not U(N) x U(N) but

GF - SU(N) 8 SU(N) @ U ( 1 ) . ii) those associated with GF alone. They only occur if the spectator gauge field is quantized. To remedy these we simply add "spectator fermions" coupled to GF alone. Again, since these interactions are weak they should not influence the bound state spectrum.

Here, the spectator gauge fields and fermions are introduced as mathematical tools only. It just happens to be that they really do occur in Nature, for instance the weak and electromagnetic SU(2) x U ( 1 ) gauge fields coupled to quarks in QCD. The leptons then play the role of spectator fermions.

11111. ANOMALY CANCELLATION FOR THE BOUND STATE SPECTRUM

Let us now resume the particle content of our theory. At small distances we have a gauge group Gc 8 GF with chiral fernions in several representations of this group. Those fermions which are

366

G. t HOGFT

trivial under Gc are only coupled weakly and are called "spectator fermions". All anomalies cancel, by construction.

At low energies, much lower than the mass scale where color binding occurs, we see only the GF gauge group with its gauge fields. Coupled to these gauge fields are the massless bound states, forming new rhpresentations r of Gp, with either left- or right handed chirality. The numbers of left minus right handed f e d o n fields in the representations r are given by the as yet unknown indices Il(r). And finally we have the spectator fermions which are unchanged.

We now expect these very 1ight.objects to be described by a new local field theory, that is, a theory local with respect to the large distance scale that we now use. The central theme of our reasoning is now that this new theory must again be anomaly free. We simply cannot allow the contradictions that would arise if this were not so. Nature must arrange its new particle spectrum in such a way that unitarity is obeyed, and because of the large distance scale used the effective interactions are either vanishingly small or renormalizable. The requirement of anomaly cancellation in the new particle spectrum gives us equations for the indices a(r) , as we will see.

The reason why these equations are sometimes difficult or impogsible to solve is that the new representations r must be different from the old ones; if Gc = SU(N) then r must also be faithful representations of Gp/Z(N). For instance in QCD we only allow for octet or decuplet representations of (SU(3))flavor, whereas the original quarks were triplets,

However, the anomaly cancellation requirement, restrictive as it may be, does not fix the values of E(r) completely. We must look for additional limitations.

11112 @PELQUIST-CARAZZONE DECOUPLING AND N-INDEPENDENCE

A further limitation is found by the following argument. Suppose we add a mass term for one of the colored fermions.

Clearly this links one of the left handed fermions with one of the right handed ones and thus reduces the flavor group + into G' C GF. Now let us gradually vary m from o to infinity. A famous tffeorern 5 ) tells us that in the limit m + - all effects due to this massive quark disappear. All bound states containing this quark should also disappear which they can only do by becoming very heavy. And they can only become heavy if they form representations r' of Gb with total index &'(r') = 0 . Each representation r of % forms

367


an a r r ay of represen' tations r ' of G k . Therefore

(11140)

Apparently t h i s expression must vanish.

Thus w e found another requirement f o r the indices E(r). The ind ices w i l l be nearly bu t not q u i t e uniquely determined now. Calculat ions show t h a t t h i s second requirement makes our indices t ( r ) p r a c t i c a l l y independent of the dimensions n i of GF. For instance, i f Gc = SU(3) and i f we have l e f t - and righthanded quarks forming t r i p l e t s and s e x t e t s then ..

where n1,2 r e f e r t o the t r i p l e t s and n t o the s e x t e t s . Gc i s anomaly-free i f 3 , 4

n1 - n2 + 7(n3- n4) = o . (11142)

Here we have three independent numbers n i . If we w r i t e the representat ions r a s Young tableaus then E(r) could s t i l l depend e x p l i c i t l y on n i .

However, suppose t h a t someone would s t a r t a s approximation of Bethe-Salpeter type t o discover the zero mass bound s t a t e spectrum. He vould s tudy diagrams such a s Fig. 3

Fig. 3

The r e s u l t i n g indices E(r) would follow from tdpological p rope r t i e s of the in t e rac t ions represented by the blobs. It is unl ikely t h a t t h i s topology would be seriousZy e f f ec t ed by d e t a i l s such a s the contributions o f diagrams containing add i t iona l closed fermion loops. However, t h a t is the only way i n which e x p l i c i t n-dependence enters. I t i s therefore n a t u r a l t o assume a ( r ) toben-independent. This l a t t e r assumption f i x e s E(r) completely. What i s the r e s u l t of these ca l cu la t ions?

368

6. 't HOOFT 11113. CALCULATIONS

Let C be any (reducible or irreducible) gauge group. Let chiral fermions in a representation r be coupled to the gauge fields by the covariant derivative

a where Aa are the gauge fields and X (r) a set of matrices depending on the lJ representation r. Let the left-handed fermions be in the representations rt and the right-handed ones in rR. Then the anomalies cancel If

(11144)

b The object dabc(r) = Tr{Xa(r), X (r)) lc(r) can be computed for any r. In table 1 we give some examples. The fundamental representation ro is represented by a Young tableau: 0 . Let it have n components. We take the case that Tr X(ro) = 0 . Write

Tr I(ro) = n , Tr X(r) = o , Tr Xa(r) Xb(r) - C ( r ) Tr Aa(ro) A b (ro)

dabc (r) - K(r) dabc(ro) .

Tr I(r) - N(r) ,

We read off C and K from table 1.

s

(11145)

Now 11144 must hold both in the high energy region and in the low energy region. The contribution of the spectator fermions in both regions is the same. Thus we get for the bound states

(roR)) dabc(r) - n babC(roL) - dabc

C (11146)

where a,b,c are indices of Gp and ro is the fundamental representation of %, We have the factor nc written explicitly, being the number of color components.

Let us now consider the case G, = SU(3); Gp = SUL(n) @ SUR(n) @U(l). We have n "quarks" in the fundamental representations. The representations r of the bound states must be in +/Z(3). They are assumed to be built from three quarks, but we are free to choose their chirality. The expected representations

369

CHIRAL WMMETRY AND CHIRAL WMMETRY BREAKING

1

1

n+ 2

n2-3

K(r)

1

- 1

n-?: 4

(n+3) (e6) 2

370

G. 't HOOFT

are given in table 2 , where also their indices are defined. Because of left-right symmetry these numbers change sign under interchange of left ++ right.

Table 2

representation index rep re sent a tion index

For the time being we assume no other representations. In eq. 11146 we may either choose a, b and c all t o be SU(n)L indices, or choose a and b to be SU(n)L indices and c the U(1) index. We get two independent equations:

1 4(n*3)(d6)kIf-1 ln(n*7)t2,+(n2-9)!t3 - 3, if n > 2 , 2 f

and 2 1 l(n+2)(n+3)k

f f -1 ln(h+3)k2?+(n -3)k - 1 , if n > I . If 3

(11147)

The Appelquist-Carazzone decoupling requirement, eq. ( I I I40) , gives us in addition two other equations:

a , , - a2+ + i3 - 0 9

1,- - E2- + !L3 = o , both if n > 2 . (11148)

For n > 2 the general solution is

37 1


P I + = = 11 , e2+ = e2- - 3 e - 3 ,

113 - 21-5.

1

(11149) I

Here II i s s t i l l a r b i t r a r y . Clearly t h i s r e s u l t i s unacceptable. tle cannot allow any of the ind ices L t o be non-integer. Only f o r t he case n - 2 (QCD with j u s t two f l avor s ) there i s another so lu t ion . In t h a t case 112- and 113 descr ibe the same rep resen ta t ion , and a , - an empty r ep resen ta t ion . We get

(11150)

According t o the o-model, I l l + - t2+ - 0 ; k = 1 . The a-model i s therefore a c o r r e c t so lu t ion t o our equations.

I n the previous s e c t i o n w e promised t o determine t h e indices completely. This i s done by imposing n-independence for the more general case including a l s o o the r co lo r r ep resen ta t ions such a s s e x t e t s besides t r i p l e t s . The r e s u l t i n g equations are not very i l luminat ing, with r a t h e r ugly c o e f f i c i e n t s . One f inds t h a t i n general no s o l u t i o n exists except when one assumes t h a t a l l mixed r ep resen ta t ions have vanishing indices . With mixed r ep resen ta t ions we mean a product of two o r mre non-tr ivial representat ions of two o r more non-fielian inva r i an t subgroups of GF. I f now w e assume n-independence t h i s must a l s o hold i f the number of s e x t e t s is zero. So L2+ and k2- must vanish. t1e ge t

a,+ - a , - - 1 / 9 ,

e3 = -119 . If a l l quarks were sextets, not t r i p l e t s , w e would ge t

e l + = a , - - 2 / 9 , -219 . a 3 -

(I115 1 )

(11152)

In the case Gc SU(5) the indices were a l s o found. See t ab le 3.

372

G. 't HOOFT

R l + =

5+ = -

Table 3 indices f o r Gc Su(5) I 2 5

I 2 5

= 1/25 a3 w1 = 1/25 5-

This c l e a r l y suggest3 a general tendency f o r SU(N) co lo r groups to produce indices fl/N o r 0 .

11114. CONCLUSIONS

Our r e s u l t t h a t the ind ices w e searched for are f r a c t i o n a l i s c l e a r l y absurd. 1Je nevertheless pursued t h i s ca l cu la t ion i n order to exh ib i t the general philosophy of t h i s approach and t o f ind out what a possible cure might be. Our s t a r t i n g po in t was t h a t c h i r a l syrmuetry is not broken spontangpuely. Most l i k e l y t h i s is untenable, a s s eve ra l authors have argued synnnetry i n QCD leads t o t roub le i n p a r t i c u l a r i f the number of f l avor s i s more than two. A daring conjecture i s then t h a t i n QCD the s t range quark, being r a t h e r l i g h t , is responsible f o r the spontaneous breakdown of c h i r a l symmetry. An i n t e r e s t i n g p o s s i b i l i t y is t h a t i n some generalized vers ions of QCD c h i r a l synrmetry is broken only p a r t l y , leaving a few massless c h i r a l bound states. Indeed t h e r e a r e examples of models where our philosophy would then give in t ege r i nd ices , but s ince w e must drop the requirement of n-dependence our r e s u l t was not unique and it was always ugly. No such model seems to reproduce anything reeembling the observed quark-lepton spectrum.

. We f ind t h a t e x p l i c i t c h i r a l

F ina l ly the re i s the remote p o s s i b i l i t y t h a t the paradoxes associated with higher sp in massless bound states can be resolved. Perhaps the A(1236) plays a more s u b t l e r o l e i n the a-model than assumed so f a r (we took i t t o be a p a r i t y doublet) .

We conclude t h a t w e are unable t o cons t ruc t a bound state theory f o r the p re sen t ly fundamental fermions along the l i n e s

373


sugges ted above.

We thank R. van Damme f o r a c a l c u l a t i o n y i e l d i n g t h e i n d i c e s i n t h e c a s e G = SU(5).

C

REFERENCES

1 . P.A.M. Dirac, Nature 139 (1937) 323, Proc . Roy. SOC. - A165 (1938) 199, and i n : C u r r e n t Trends i n t h e Theory o f F i e l d s , ( T a l l a h a s s e e 1978) AIP Conf. Proc . No 48, P a r t i c l e s and F i e l d s S u b s e r i e s No 15, e d . by Lannut i and Williams, p . 169.

2. S. Dimopoulos and L. Susskind , Nucl. Phys. (1979) 237. 3. M. Gell-Nann and M. Lgvy, Nuovo C i m . (1960) 705.

B.W. Lee, C h i r a l Dynamics, Gordon and Breach, New York, London, P a r i s 1972.

3432. S . Coleman, "The Uses o f I n s t a n t o n s " , Erice L e c t u r e s 1977. R. Jackiw and C. Rebbi , Phys. Rev. L e t t . 37 (1976) 172. C. C a l l a n , R. Dashen and D. Gross, Phys. L e t t . - 63B (1976) 334.

4 . G. ' t Hooft , Phys. Rev. L e t t . 37 (1976) 8 ; Phys. Rev. - D 1 4 (1976)

5 . G. ' t Hoof t , Nucl. Phys. B72 (1974) 461. 6. T. A p p e l q u i s t and J. Carazzone, Phys. Rev. D11 (1975) 2856. 7 . A. Casher , C h i r a l Symmetry Breaking i n Q u a r k o n f i n i n g T h e o r i e s ,

T e l Aviv p r e p r i n t TAUP 734179 (1979) .

374

CHAPTER 6

PLANAR DIAGRAMS

Introductions ................................................................... 376

16.11 “Planar diagram field theories”. in Progress in Gauge Field Theory. NATO Adv . Study I w t . Series. eds . G . ’t Hooft et al., Plenum. 1984. pp . 271-335 ............................................. 378 “A two-dimensional model for mesons”. Nucl . Phys . B75 (1974) 461470 .................................................................. 443 Epilogue to the Two-Dimensional Model .............................. 453

[6.2]

375

CHAPTER 6

PLANAR DIAGRAMS

Introduction to Planar Diagram Field Theories [6.1]

Consider an SU(N) gauge theory, possibly enlarged with scalars and/or fermions in the elementary or the adjoint representation. Take the limit g -+ 0, N + 00,

g2N = ij2 fixed. In this limit only those Feynman diagrams survive that can be drawn on a plane without any crossings of the lines. We call these “planar diagrams”. Planar diagrams resemble the world sheets of strings, and 80 one is led to hope that perhaps the stringlike behavior of the gluon field confining quarks can be understood in this limit. Unfortunately, planar diagrams form a set that is still too large to be summed analytically by any known method.

However, there is another important aspect to the planar diagrams. The perturbative expansion with respect to i j may converge much better than in theories with finite N. The reason for this is that the total number of planar diagrams is much smaller than the total of all diagrams. The only complication is that each individual diagram may give huge amplitudes at very high orders, because of the need for renormalization subtractions. The renormalized diagrams are finite, but huge.

The bulk of the first paper in this chapter is devoted to the question of whether perturbation expansion defines a theory with infinite accuracy in the infinite N limit. The answer is affirmative, but we could prove this only if the coupling constant is sufficiently small and all masses sufficiently large, a physically uninteresting caw. So here we have a rigorously defined theory. Our construction is very technical. A casual reader may well be interested only in the first part where the planarity of the diagrams is proved, or possibly the less technical Section 16, where Bore1 summability is deduced. In Appendix A a curious model is discussed which is on the one hand an asymptotically free gauge theory and on the other hand there are a sufiicient number of scalar fields to render all particles massive. To achieve this it was compulsory to also have fermions present.

376

Appendix B is the well-known “spherical model”, a scalar N-component field in the N --.) 00 limit. This model is extremely instructive. In this limit one may allow the coupling constant to be negative (it has to become negative at high energies!); at sufficiently large N the vacuum will become sufficiently stable. The model shows that if the original renormalized maas parameter vanishes there will still be a “spontaneously generated‘’ physical m a a for the particle. There is a (negative) lower limit for the allowed values of the renormalized mass, at which the bound state mass tends to zero. The dotted region in Figure 21 is an unphysical solution for m and M for the one given (negative) value of mR.

Introduction to A Tw-Dimensional Model for Mesons [6.2]

Since our original motivation for studying the N + 00 limit was our hope to find solvable versions of QCD and to understand quark confinement we now present one successful attempt. In one-space, one-time dimension this limit can indeed be solved. This is because there a gauge choice can be made such that the gauge field self-interactions disappear. The surviving planar diagrams are then the so- called rainbow diagrams, see Figure 2. They obey simple closed Dyson-Schwinger equations which can be solved. The propagator can be found in closed form, the mesonic bound states can be expressed as BetheSalpeter like integral equations, which can easily be solved numerically.

At the end of this paper we add an epilogue.

377

CHAPTER 6.1

PLANAR DIAGRAM FIELD THEORIES

Gerard 't Hooft Institute for Theoretical Physics

Princetonplein, 5 P.O. Box 80.006 3508 TA Utrecht, The Netherlands

ABSTRACT

In this compilation of lectures field theories are considered which consist of N component fields qi interacting with N x N component matrix fields Ai' with internal (local or global) symmetry group SIJ(N) or SO(N). The double expansion in 1/N and 2 = Ng2 can be formulated in terms of Feynman diagrams with a structure. If the mass is sufficiently large and g sufficiently small then the(extreme1y non-trivial) expansion in g2 at lowest order in 1/N is Borel summable. Exact limits on the behavior of the Borel integrand for the z2 expansion are derived.

-!lanar)ty

1. INTRODUCTION

In spite of conbiderable efforts it is still not known how to compute physical quantities reliably and accurately in any four- dimensional field theory with strong interactions, It seems quite likely that if any strong interaction field theory exists in which accurate calculations can be done, then that must be an asymptotically free non-Abelian gauge theory. In such theories the small-distance structure is completely described by solutions of the renormalization group equations1 ; and there are reasons to believe that the continuum theory can be uniquely defined as a limit of a lattice gauge theory2, when the size of the meshes of the lattice tends to zero, together with che coupling constant, in a way prescribed by this renormalization group3. Indeed, one can prove using this formalism4 that this limit exists up to any finite order in the perturbation expansion for small coupling.

However, this result has not been extended beyond pertur-

Progress in Gauge Field Theory, NATO Adv. Srudy Inst. Series Edited by G 't Hoot3 er al. (Plenum, 1984).

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G. 't HOQFT

bation expansion. It is important to realize that this might imply that theories such as "quantum chromodynamics" (QCD) are not based on solid mathematics, and indeed, it could be that physical numbers such as the ratio between the proton mass and the string constant do not follow unambiguously from QCD alone. In view of the qualitative successes of the recent Monte-Carlo computation techniques5 the idea that hadronic properties could be shaped by forces other than QCD alone seems to be far-fetched, but it would be extremely important if this happened to be the case. More likely, we may simply have to improve our mathematics to show that QCD is indeed an unambiguous theory. Either way, it will be important to extend our understanding of the suaunability aspects of higher order perturbation theory as well as we can. The following constitutes just such an attempt.

There are two categories of divergences when one attempts to sum or resum perturbation expansion for a field theory in four space-time dimensions. One is simpJy the divergence due to the increasingly large numbers of Feynman diagrams to consider at higher orders. They grow roughly as n! at order g2n. This is a kind of divergence that already occurs if the functional integral is replaced by some ordinary finite-dimensional integral of similar type:

Here the diagrams themselves are bounded by but the numbers K(n) of diagrams of order n pansion is only asymptotic:

I(g2) + 1 K(n) Cn g2n ; n n K(n) + a n!

geometric expressions are such that the ex-

where a is determined by one of the stationary points of the action S, called "instantons".

However in four-dimensional field theories the diagrams themselves are not geometrically bounded. In some theories it has been shown6 that diagrams of the nth order that required k ultraviolet subtractions (with essentially k 5 n) can be bounded at best by

379


bn k! g2n . (1 .3 )

But since the total number of such diagrams grow at most as n!/k! we still get bounds of the form ( 1 . 2 ) for the total amplitude, however with a replaced by a different coefficient. This is a different kind of divergence sometimes referred to as ultraviolet 11renormalons'14 7.

If a field theory is asymptotically free the corresponding coefficient b is negative and one might hope that the ultraviolet renormalons are relatively harmless. But in massless theories a similar kind of divergence will then be difficult to cope with: the infrared renormalone, which, as the word suggests, are due to a build-up of infrared divergences at very high orders: individual diagrams may still be convergentbuttheir sum diverges againwithn! The theories we will study more closely are governed by planar diagrams only. Their numbers grow only geometrically8 so that divergences due to instantons are absent. These diagrams are akin to but more complicated than Bethe-Salpeter ladder diagrams and by trying to sum them we intend to learn much about the renormalon divergences.

Infinite color quantum chromodynamics is of course the most interesting example of a planar field theory but unfortunately our analysis cannot yet be carried out completely there. We do get bounds on the behavior of its Bore1 functions however (sect. 17) .

Examples of large N field theories that we can handle our way are given in sect. 3 and appendix A.

2. FEY" RULES FOR ARBITRARY N

In order to show that the set of planar Feynman diagrams becomes dominant at large N values we first formulate a generic theory at arbitrary N, with a coupling constant g as expansion parameter in the usual sense.

Let us express the fields as a finite number K of N-component vector? I@(x) (ail ,... ,K; i = l , . .., N), and a small number D of N xN matrices AbiJ (x) , where b=I ,. . . ,D and i,j=l,. . . ,N. Usually we will take 9 to be complex and AiJ to be Hermitean, in which case thes-try group will be U(N) or SU(N) . The case that Jli are real and AiJ real and symmetric can easily be included after a few changes, in which case the symmetry group would be O(N) or SO(N) , but the complex case seems to be more interesting from a physical point of view. In quantum chromodynamics K would be proportional to the number of flavors and D is the number of space-time dimensions plus two for the (non-Hermitean) ghost field.

In general then the Lagrangian has the form

380

G. 't HOOFT

i(A,$,$*) = - 1 Jlra(Mtb + gM1 abc A c + ig2M;bCdACAd)jlb a,...

a b a b 1 a b c a b c 1 2 a b c d a b c d !& A A + ~ g R 1 A A A + T; g R2 A A A A ) , (2 .1 )

with

Aa.j 1 * (Aaji)* . Here the usual matrix multiplication rule with respect to the indices i,j, ... is implied, and Tr stands for trace with respect to these indices. The objects M and R carry no indices i,j but only "flavor" indices a,b,... . Furthermore M o , ~ and R o , ~ may contain the derivatives of $(XI or A(x). So the case that J, are fermions is included: then a,b,... may include spinor indices. The coupling constant g has been put in (2.1) in such a way that it is a handy expansion parameter.

venient to split the fields AaiJ into complex fields for i > j and real fields for i = j. One can then denote an upper index by an incoming arrow and a lower index by an outgoing arrow. The propagator is then denoted by a double line. In fig. 1 , the A propagator stands for an AiJ propagator to the right if i > j ; an Aji propagator to the left if i < j and a real propagator if i = j.

In order to keep track of fhe indices i,j , . . . it is con-

It is crucial now that the coefficients M and P in the Lagrangian respect the U(N) (or O ( N ) ) symmetry: they carry no indices i,j, ... . Hence the vertices in the Feynman graphs only depend on these indices v ia Kronecker deltas. We indicate such a Kronecker delta in a vertex by connecting the corresponding index lines. Since we have a unitary invariance group these Kronecker deltas only connect upper indices with lower indices, therefore the index lines carry an or-Lentation which is preserved at the vertices. This is where the unitary case differs from the real orthogonal groups: the restriction to real fields with O(N) symmetry corresponds to dropping the arrows in Fig. 1: the index lines then carry no orientation.

As for the rest the Feynman rules for computing a diagram are as usual. For instance, fernionic and ghost loops are associated with extra minus signs.

In some theories (such as SU(N) gauge theories) we have an

Tr Aa = o . extra constraint:

(2.3)

381


i i

3 3

. b Pab(k) = fRoI,k (N x N matrix propagator)

- I i i a -b Pia(k) = <Mo )ba (N-vector propagatof)

C

a

Fig. 1: Feynman Rules at arbitrary N. The accolades { cyclic

1 stand for symmetrization with respect to the indices a,b,...

382

G. 't HOOFT

In that case an extra projection operator is required in the propagator:

The second term in (2.4) corresponds to an extra piece in the propagator, as given in Fig. 2. We will see later that such terms' are relatively unimportant as N + Q).

For defining amplitudes it is often useful to consider source terms that preserve the (global) symmetry:

(2.5) pource = J ab (x)$*'(x)$~(x) + lJtb(x) Tr Aa(x)Ab(x) . Q

The corresponding notation in Feynman graphs is shown in Fig. 3.

3. THE N + Q D LIMIT AND PLANARITY

As usual, amplitudes and Green's functions are obtained by adding all possible (planar and non-planar) diagrams with their appropriate combinatorial factors. Note that, apart from the optional correction term in ( 2 . 4 ) , the number N does not occur in Fig. 1. But, of course, the number N for the amplitudes, and that is when index-loop gives rise to a factor

i l a i I N . i

will enter into expressions an index-line closes. Such an

(3.1)

We are now in a position that we can classify the diagrams (with only gauge invariant sources as given by eq. (2.5)) according to their order in g and in N . Let there be given a connected diagram. First we consider the two-dimensional structure obtained by considering all closed index loops as the edges of little (simply connected) surface elements. All N xN matrix-propagators connecr these surface elements into a bigger surface, whereas the N-vector-propagators form a natural boundary to the total surface. In the complex case the total surface is an oriented one; in the real case there is no orientation. In both cases the total surface may be multiply connected, containing "worm holes". For convenience we limit ourselves to the complex (oriented) case, and we close the surface by attaching extra surface elements to all N-vector-loops.

Let that surface have F faces (surface elements), P lines (propagators) and V vertices. We haye F - L + I + Pt, where L is the number of N-vector-loops*) and I the number of index-loops; and we write (footnote: see next page)

383


Fig. 2: Extra term in the propagator i f T r A i s to be projected out.

Jab JI

b

a

(N-vector source)

a b (matrix source)

Fig. 3: Invariant source insertions.

384

G. 't HOOFT

where V source inser t ions) . The diagram is now associated with a fac tor

i s t h e number of n-point ver t ices . (Vq is the number of B

v3+2v4 I-Pt r = 8 N

Here Pt is the number of t i m e s t h e second t e r m of ( 2 . 4 ) has been inser ted t o obtain t race less propagators. By drawing a dot a t each end of each propagator w e f ind t h a t t h e t o t a l number of dots is

2~ = l n V n , n

and eq. (3.2) can be wr i t ten a s

2P-2V F-L- 2Pt r - 8 N

( 3 . 3 )

( 3 . 4 )

Now we apply a well-known theorem of Euler:

F - P + V = 2 - 2 H , (3.5)

where H counts the number of "wormholes" i n t h e surface and i s therefore always pos i t ive (a sphere has H = 0 , a torus H = 1 , etc.) . And so,

4v3+vq 2-2H-L-2Pt r = (g2N) N ( 3 . 6 )

Suppose we take t h e l i m i t

N + a0 , g + o , g2N = g2 (fixed) . ( 3 . 7 )

I f there are N-vector-sources then there must be a t l e a s t one N- vector-loop:

L > I .

The leading diagrams i n t h i s l i m i t have H = 0 , Pt = o and L = 1. They have one overa l l mul t ip l ica t ive fac tor N , and they a r e a l l planar: an open plane with t h e N-vector l i n e a t i t s edge (Fig. 4a).

*) "N-vector" here stands f o r N-component vector i n U(N) space, so the N-vector-loops a r e the quark loops i n quantum-chromodynamics.

385


Fig. 4: Elements of the c l a s s of leading diagrams i n the N - P -

l i m i t . a) I f vector sources are present. b) In the absence of vector sources (e .g . pure gauge theory).

386

G. 't HOOFT

If there are only matrix sources then L 30. The leading diagrams all have the topology of a sphere and carry an overall factor N2 (see Fig. 4b). We read off from eq. (3.6) that next to leading graphs are down by a factor I/N for each additional N-vector-loop (- quark loop in quantum chromodynamics) and a factor 1/N2 for each "wormhole". Also the difference between U(N) and SU(N) theories disappears as l/N2. It will be clear that this result depends only on the field variables being N-vectors and N x N matrices gnd the Lagrangian containing only single inner products or traces (not the pmduats of inner products and/or traces). Diagrams with L - 1 and H - o are the easiest to visualize. In the sequel we discuss convergence aspects of the summation of those diagrams in all orders of I. Our main examples are I ) U(N) (or SU(N)) gauge theories with fermions in the N represen-

tat ion; 2) purely Lorentz scalar fields, both in N and in N x N represen-

tations of U(N). That theory will be called T ~ X $ I ~ , or - T r Q 4 , if A is given the unusual sign. Both SU(N) gauge theory and -T~XI#~ are asymptotically free9. The latter has the advantage that one may add a mass term, so that it is also infrared convergent. However the fact that X has the wrong sign implies that that theory only exists in the N + - limit, not for finite N, A model that combines all "good" features of the previous model is:

representation, N fermions in the elementary representation and a fermion in the adjoint representation. A global SU(N) symmetry then survives. All vector, spinor and scalar particles are massive, and it is asymptotically free if x2@ * 1;

3) an SU(N) Higgs theory with N Higgs fields in the elementary

(3 .8)

where h is a Yukawa coupling constant and X the Higgs self coupling. The reason for mentioning this made1 is that it is asymptotically free in the ultraviolet, and it is convergent in the infrared, so that our methods will enable us to construct it rigorously in the N -)QI limit (provided that masses are chosen sufficiently large and the coupling constant sufficiently small), and positivity of the Hamiltonian is guaranteed also for finite N so that there is every reason for hope that the theory makes sense also at finite N, contrary to the - X T ~ ~ I ~ theory. This model is described in appendix A.

4. THE SKELETON EXPANSION

From now on we consider diagrams of the type pictured in figure 4a (H - 0 , L - 1 ) . They all have the same N dependence, so once we restricted ourselves to these planar diagrams only we may drop the indices i,j, ... and replace the double-line pro agators by single lines. Often we will forget the tilde (-) on gq because

387


the factor N is always understood. Only the (few) indices a,b,.. . of eq, (2.1) , as far as they do not refer to the SU(N) group(a), are kept. The details of this surviving index structure are not important for what follows, as long as the Feynman rules (Fig. 1) are of the general renormalizable type,

Our first concern will be the isolation of the ultraviolet divergent parts of the diagrams. For t is we use an ancient

graph, planar or not, but for the planar case it is particularly useful.

devicelo called "skeleton-expansion" * 5 . It can be applied to 'any

Consider a graph with at least five external lines. A one- particle irreducible subgraph is a subset of more than one vertices with the internal lines that connect these vertices, that is such that if one of the internal lines is cut through then the subgraph still remains connected. We now draw boxes around all one-particle irreducible subgraphs that have four of fewer external lines. In general one may get boxes that are partially overlapping. A box is m d m a Z if it is not entirely contained inside a larger box.

Theorem: All maximal boxes are not-overlapping. This means that two different maximal boxes have no vertex in common.

Proof: If two maximal boxes A and B would overlap then at least one vertex XI would be both in A and B. There must be a vertex xp in A but not in B, otherwise A would not be maximal. Similarly there is an x3 in B but not in A. Now A was irreducible, so that at least two lines connect xi with xp. These are external lines of B but not of A U B. Now B may not have more than 4 external lines. So not more than two external lines of A U B are also external lines of B. The others may be external lines of A. But there can also be not more than two of those. So A U B has not more than four external lines and is also irreducible since A and B are, and they have a vertex in common. So we should draw a box around A U B. But then neither A nor B would be maximal, contrary to our assumption. No planarity was needed in this proof.

The skeleton graph of the diagram is now defined by replacing all maximal boxes by single "dressed" vertices. Any diagram can now be decomposed into its "skeleton" and the "meat", which is the collection of all vertex and self-energy insertions at every two-, three- and four-leg irreducible subgraph. In particular the self-

*) The method described here differs from Bjorken and Drell" in that we do not distinguish fermiona from bosons, so that also subgraphs with four external l'ines are contracted.

388

G. 't HOOFT

energy insertions build up the so-called dressed propagator. We call the dressed three- and fourvertices and propagators the "basic Green functions" of the theory. They contain all ultraviolet divergences of the theory. The rest of the diagram, the "skeleton" built out of these basic Green functions is entirely void of ultraviolet divergencies because there are no further (sub)graphs with four of fewer external lines, which could be divergent.

The skeleton expansion is an important tool that will enable us to construct in a rigorous way the planar field theory. For, under fairly mild assumptions concerning the behavior of the basic Green functions we are able to prove that, given these basic Green functions, the sum of all skeleton graphs contributing to a certain amplitude i n Euotidean apaoe is abeolutely convergent (not only Bore1 sumPable). This proof is produced in the next 6 sections. Clearly, this leaves us to construct the basic Green functions themselves. A recursive procedure for doing just that will be given in sects. 12-15. Indeed we will see that our original assumptions concerning these Green functions can be verified provided the masses are big and the coupling constants small, with one exception: in the scalar -XTr+4 theory the skeleton expansion always converges even if the bare (minimally subtracted) mass vanishes! (sect. 18)

5. TYTE IV PLANAR F E Y " RULES

We wish to prove the theorem mentioned in the previous section: given certain bounds for the basic Green functions, then the sum of all skeleton graphs containing these basic Green functions inside their "boxes" converges in the absolute. In fact we want a little more than that. In sects. 13-15 we will also require bounds on the total sum. Those in turn will give us the basic Green functions. We have to anticipate what bounds those will satisfy. In general one will find that the basic Green functions will behave much like the bare propagators and vertices, with deviations that are not worse than small powers of ratios of the various momenta. Note that all our amplitudes are Euclidean.

First we must know how the dressed propagators behave at high and low momenta. The following bounds are required:

Here P (k) is the propagator. From now on we use the absolute value symbol for momenta to mean: lpl - h-'. Then the field renormalization factor Z(k) is approximately:

ab

389


where u is a coefficient that can be computed from perturbation expansion. The mass term in (5 .1 ) is not crucial for our procedure but m in (5.2) can of course not be removed easily.

To write down the bounds on the three- and four-point Green functions in Euclidean space we introduce a convenient notation to indicate which external momenta are large and which are small.

For any planar Green function we label not the external momenta but the spaces in between two external lines by indices 1,2,3,. .. which have a cyclic ordering. An external line has momentum

pi,i+l dEf pi -pi+l We have automatically momentum conservation,

(5.3)

(5.4)

and the pi are defined up t o an overall translation,

pi + pi+q , all i . (5.5)

A channel (any in which possibly a resonance can occur) is gives by a pair of indices, and the momentum through the channel is given by

So we can look at the pi as dots in Euclidean momentum space, and the distance between any pair of dots is the momentum through some channel. If we write

So the brackets are around wmenta.that form close clusters.

(5.7)

Our bounds for the three- and four-point functions are now defined in table 1 .

390

G. ‘t HOOFT

TabZe 1

Bounds for the 3- and 4-point dressedGreenfunctions. Zij stands for Z(pi-p*). All other exceptional mmentum configurations can be obtained by cyclic rotations and reflections of these. Ki are coefficients close to one.

Here a and @ are small positive coefficients. g(x) is a slowly varying running coupling constant. For the time being all we need is some g with

y x Ki g(x) 6 8 for all x , (5.9)

where we also assume that possible summation over indices a,b,,.. is included in the K coefficients. Clearly the bare vertices would satisfy the bounds with a = B = 0 . Having positive a and B allows us to have any of the typical logarithmic expressions coming from the radiative corrections in these dressed Green functions. Indeed we will see later (sect, 13) that those logarithms will never surpass our power-laws.

Table 1 has been carefully designed such that it can be re- obtained in constructing the basic Green functions as we will see in sects. 12-14. irst we notice that the field renormalization

bounds for the propagator (5.1). The power-laws of Table 1 can be conveniently expressed in terms of a revised set of Feynman rules. These are given in Pig. 5. We call them type IV Feynman rules after a fourth attempt to refonnulate our bounds (types I, I1 and

factors Z(pi-pj)- F cancel against corresponding factors in our

391


1 - (dressed elementary (k2+rn2) propagator)

1 +3a g[mX( Ikl I Ikp I , Ikg I) I (dressed 3-

kl /L k2 vertex )

- 1 (composite propagator) 2

lkI2*

e k 1kl-O (external line) 1

Fig. 5: Type IV Feynman Rules. lkl stands for .

392

G. 't HOOFT

I11 occur in refs. 1 1 , 12 and are not neede here). The trick is

an exchange of two or more of the original particles in the diagram we started off with.

to introduce a new kind of propagator, - s , that represents

The procedure adapted in these lectures deviates from earlier workll in particular by the introduction of the last two vertices in Fig. 5. Notice that they decrease whenever two of the three external momenta become large.

It is now a simple exercise to check that indeed any diagram built from basic Green functions that satisfy the bounds of Table 1 can also be bounded by corresponding diagram(s) built from type IV Feynman rules. The four-vertex is simply considered as a sum of two contributions both made by connecting two three-point vertices with a type 2 propagator, and the factors lkl-" from the propagators in Fig. 5 are considered parts of the vertex functions (the mass term of the propagator may be left out; it is needed at a later stage). Type 2 propagators will also be referred to as "composite propagators" .

Elementary power counting now tells us that the superficial degree of convergence, 2, of any (sublgraph with El external single linea and E2 external composite lines is given by

2 = (1-a)El + (2-B)Ez - 4 . (5.10)

Since we consider only skeleton graphs, all our graphs and subgraphs have

El + 2E2 > 5 . (5.1 1)

Thus, 2 is guaranteed to be positive if we restrict our coefficients by

o < a < 1 1 5 ;

(5.12)

(Infrared convergence would merely require a < 1; B < 2, and is therefore guaranteed also.) So we know that with (5.12) all graphs and subgraphs are ultraviolet and infrared convergent. The theorem we now wish to prove is: the sum of all convergent type IV diagrams contributing to any given amplitude with 5 (or more) external lines converges in Euclidean space. It is bounded by the sum of all type IV tree graphs (graphs without closed loops) multiplied with a fixed finite coefficient.

A further restriction on the coefficients a and B will be necessary (eq. (8.15)).

393


6 . NUMBER OF TYPE IV DIAGRAMS

The total number G(E,L) of connected or irreducible planar diagrams with E external lines and L closed loops in any finite set of Feynman rules, is bounded by a power law (in contrast with the non-planar diagrams that contribute for instance to the Lth order term in the expansion such as ( 1 . 1 ) for a simple functional integral) :

for some C1 and Cg.

In some cases C1 and C2 can be computed exactly and even closed expressions for G(E,L) exist'. These mathematical exercises are beautiful but rather complicated and give us much more than we really need. In order to make these lectures reasonably self- sustained we will here derive a crude but simple derivation of ineq. (6.1) yielding C coefficients that can be much improved on, with a little more effort.

Let us ignore the distinction between the two types of propagators and just count the total number G(E,L) of connected planar o 3 diagrams with a given configuration of E external lines and L closed loops. We have (see Fig. 6)

G(E+l ,L) = G(E+2,L-I) + 1 G(n+l ,Li)G(E+I-n,L-LI) . ( 6 . 2 ) n,L1

G(E,L) = o if E < 2 or if L < o ;

G(2,o) = 1 . ( 6 . 3 )

Fig. 6: Eq. (6.2)

n

E-n

We wish to solve, or at least find bounds for, G(E,L) from (6 .2 j with boundary condition (6.3). A good guess is to try

394

which is compatible with (6 .3 ) if

coc: > 1 . Using the inequality

1 4 k 1 2 nil n (k-n)

we find that the r.h.8. of (6 .2 ) will be bounded by E+2 L-1 2 E+2 L 16CoC1 C, COCl c2

( E + I ) ~ L~ ( E - I ) ~ ( L + I ) ~ ’ +

which is smaller than E+1 L

cocl c2

E~(L+I)‘ ’

G. ‘t HOOFT

(6 .4)

(6 .5)

(6 .6)

(6 .7 )

if

This is not incompatible with (6.5) although the best “solution“ to these two inequalities is a set of uncomfortably large values for C , , C , and C , . But we proved that they are finite.

The exact solution to eq. (6 .2) is

2L (2E-2) ! (2E+3L-4) ! G(E’L) L! (E-I)! (E-2)! (2E+2L-2)! ’

which we wiXl not derive here. Using

(A+B)! < 2A+B-I ’ A!B!

we find that in (6 . I ) ,

C1 < 16 ; Cp < 16 .

(6 .10)

(6.1 I )

(6.12)

395


For fixed E, in the limit of large L,

Cq + 27/2 . (6 .13)

Similar expressions can be found for the set of irreducible diagram. Since they are a subset of the connected diagrams we expect C coefficients equal to or smaller than the ones of eqs. (6 .12) and (6 .13) . Limiting oneself to only convergent skeleton graphs will reduce these coefficients even further.

We have for the number of vertices V

V - E + 2 L - 2 , (6 .14)

and the number of propagators P:

P - V + L - I . (6 . 15)

So, if different kinds of vertices and propagators are counted separately then the number of diagrams is multiplied with

(6.16)

where C and C are some fixed coefficients. This does not alter our result quafitatively . Also if there are elementary 4-vertices then these can be considered as pairs of 3-vertices connected by a new kind of propagators, as we in fact did. So also in that case the numbers of diagrams are bounded by expressions in the form of eq. (6 .1) .

V

7. THE W E S T FACETS

We now wish to show that every planar type IV diagram with L loops is bounded by a coefficient CL times a (set of) type IV tree graph(s), with the same momentum values at the E external lines. This will be done by complete induction. We will choose a closed loop somewhere in the diagram and bound it by a tree insertion. Now even in a planar diagram some closed loops can become quite large (i.e. have many vertices) and it will not be easy to write down general bounds for those. Can we always find a "small" loop somewhere?

We call the elementary loops of a planar diagram facets. Now Euler's theorem for planar graphs is:

Take an irriducible diagram. Write

396

G. 't HOOFT

L = 1 F n , n

(7.2)

where F are the number of facets with exactly n vertices (or. "tr corners 1. Let I t

P - P i + P e , (7.3)

where Pi is the number of internal propagators and Pe is the number of propagators at the edges of the diagram. Then, by putting a dot at every edge of each facet and counting the number of dote we get

1 n F~ - 2pi + pe . n

(7.4)

For the numbers V of n-point vertices we have similarly n

C n v n = 2 p + E , (7.5) n

but in ourcase we only consider +point vertices (compare eqs. (6.14) and (6.15)):

3 v = 2P + E . (7.6)

Combining eqs. (7.1) - (7.6) we find 1 (n-6)Fn = 2E - Pe - 6 . n

(7 .7 )

This equation tells us that if a diagram has

L 3 2 E - 8 (7.8)

then either it is a "seagull graph" (Pe 4 1 ) which we usually are not interested in, or there must be at least one subloop with 6 or fewer external lines:

Fn > o for some n 9 6 . (7.9)

So diagrams with given E and large enough L must always contain facets that are either hexagons or even smaller.

In fact we can go further: theorem: if a planar graph (with only 3-vertices) and all its irreducible subgraphs have 2E - Pe 2 6 then the entire graph obeys

397

(7.10)


This simple theorem together with eq. (7 .7) t e l l s us t h a t any diagram with a number of loops L exceeding the bound of (7.10) must have a t l e a s t one elementary f ace t with 5 of fewer l i n e s attached t o i t . Although we could do without i t , it is a convenient theorem and now w e devote the rest of t h i s s ec t ion t o i t s proof ( i t could be skipped a t f i r s t reading).

F i r s t we remark t h a t i f w e have the theorem proven for a l l irreducible graphs up t o a c e r t a i n order, then i t must a l s o hold f o r reducible graphs up t o the same order. This i s because i f w e connect two graphs with one l i n e w e ge t a graph 3 with

I f L1, E l and L 2 , E 2 s a t i s f y (7.10) then so do L 3 and E3 (remember t h a t E and L are integers and the smallest graph with L > o h a s E 6; propagators t h a t form two edges of a diagram are counted twice i n P ) . e

Forthe i r reducible graphs w e prove (7 .10) by a r a the r unusual induction procedure f o r planar graphs. We consider the ou te r r i m of an i r reducible graph and a l l the ( i n general not i r reducible) graphs in s ide i t (see Fig. 7 ) . Let t h e e n t i r e graph have E ex- t e rna l l i n e s and Pe propagators a t i t s s ides . The subgraphs i inside the r i m have e i external l i n e s and Pei propagators a t t h e i r s ides . We count:

p e = ~ + l e i , i

and the number of loops L of the e n t i r e diagram is

( 7 . 1 2 )

Now each f ace t between the subgraphs and the r i m must have a t least 6 propagators as supposed, therefore

(7.14)

b u t i f some of the subgraphs a r e s ing le propagators we need t o be more precise

Pe + 1 P e i + 2 1 ei 2 6 1 (ei-l) + 6 + 2N2 , i

(7.15)

398

G. ‘t HOOFT

I

Fig. 7: Proving the theorem of sect. 7. The number E counts the external lines of the entire graph. Pe the number of sides and ei and P,i do the same for the subgraphs 1 and2. N is the number of single propagators.

where Np is the number of single propagators, each contributes with e = 2 in eq. (7.131, and have Pei (7.12) , and

I Pei < 2ei - 6 ,

of which = 0 . Now we use

(7.16)

as required, whereas C(2e-6) + 2N2 = o for the single propagators, to arrive at

E > J e i + 6 . 1

(7 .17)

From the assumption that all subraphs already satisfy (7.10) we get, writing L1 = F L and El = 9 . : l i 1

399


(7.18)

and from (7.13)

L <L1 + El . (7.19)

' With (7.17) which reads E > El + 6 we now see t h a t (7.10) again holds f o r the e n t i r e diagram. The quadratic expression i n (7.10) i s the sharpest t h a t can be derived from (7.17)-(7.19), and indeed large diagrams t h a t saturate the inequal i ty can be found, by joining hexagons i n t o c i r c u l a r pat terns .

Our conclusion is t h a t i f we wish t o use an induction procedure to express a bound f o r diagrams with type I V Feynman rules and L loops i n terms of one with a smaller number L' loops we can t r y t o do t h a t by replacing successively t r i a n g l e s , qudrangles and/or pentagons by type I V t r e e in se r t ions , u n t i l the bound (7.10) is reached. I n p a r t i c u l a r i f E = 5 t h i s leads us t o a t r e e diagram. The next three sect ions show how t h i s procedure works i n d e t a i l .

8. TRIANGLES

Consider a ( l a rge ) diagram with type I V Feynman ru l e s . We had already decreed t h a t it and a l l i t s subgraphs a r e u l t r a v i o l e t and infrared convergent (divergent subgraphs had been absorbed into the v e r t i c e s and propagators before) . With eqs. (5.10)-(5.12) t h i s mean8 t h a t each subgraph has

so, in p a r t i c u l a r , there a r e no self-energy blobs. F i r s t w e use the inequal i ty of Fig. 8 t o replace composite propagators by ordinary dressed propagators one by one u n t i l ineq. (8.1) forbids any f u r t h e r such replacements. The inequal i ty i s r ead i ly proven: we w r i t e f o r the!propagators with i ts ve r t i ce s : .

Fig. 8. A composite propagator is smaller than an elementary one.

400

G. 't HOOFT

where k is the momentum through t h e propagator, Ipl I 2 I k! . Ip21 > l k l , and a i = a f o r a dressed propagator, and a i - B-1 f o r a composite propagator. A t the l e f t hand s ide y = B-1, and a t the r i g h t hand s ide y = a. Clearly we have

a and B being both c lose t o zero due t o (5 .12) .

Now consider all elementary t r i a n g l e loops i n our diagram. Under what conditions can we replace them by type IV 3-vertices (Fig. 9 ) ? Due t o (8.1) there can be a t most one elementary (dressed external l i n e , the o the r s are composite: E2 - 2 o r 3;

Fig. 9: Removal of elementary t r i a n g l e f ace t s .

El - 1 or 0 . We write

a1 = a o r 8 -1 (8.4)

t o cover both cases. Now l e t us replace t h e ve r t ex functions by bounde t h a t depend only on the momenta of the i n t e r n a l l i nes :

with Y - 1+3a ; R ( ~ ) I 23a*,

401

PLANAR DIAGRAM FtELD THEORIES

or y = 8+2a ; R(y) = 1 . (8.6)

The integral over the loop momentum is then bounded by 8 terms all of the form

where for a moment we ignored the mass term. It can b e added easily later. We have convergence for all integrals:

z = 2 1 6 - 4 = 1-2p-al > 0 . (8.8)

and

61 > I +a -1(2a+~) - 1(1+2a+al) = l(l-2a-al-8) ; (8.9)

The integral (8.7) can be done using Feynman multiplicators:

(8.10)

Now i f lpl 3 141 > lp+ql (the other cases can be obtained by permutation) then lql > 4 Ipl , so our integrals are bounded by

(8.11)

where C is the sum of integrals of the type

which can be further bounded (replacing ~ 1 x 2 by 1~1x2) by

(8.13)

if all integrals converge, of course. All entries in the r

402

G. 't HOOFT

functions must be positive. In particular, we must have

2 - 62 - 63 = 6 1 - lz > o . (8.14)

Now with (8.8) and (8.9) this corresponds to the condition:

B > 2 a , (8.15)

this is the extra restriction on the coefficients a and t3 to be combined with (5.12), and which we already alluded to in the end of sect. 5 . A good choice may be

u - 0.1 , 6 - 0.3 . (8.16)

We conclude that we proved the bound of Fig. 9, if a and B have values such as (8.16), and the number C in Fig. 9 is bounded by the sum of eight finite numbers in the form of eq. (8.13).

9. QUADRANGLES

We continue removing triangular facets from our diagram, replacing them by single 3-vertices, following the prescriptions of the previous sections. We get fewer and fewer loops, at the cost of at most a factor C for each loop. Either we end up with a tree diagram, in which case our argument is completed, or we may end up with a diagram that can still be arbitrarily large but only contains larger facets. According to sect. 7 there must be quadrangles and/or pentagons among these.

Before concentrating on the quadrangles we must realize that there still may be larger subgraphs with only three external lines. In that case we consider those first: a minimal triangular subgraph is a triangular subgraph that contains no further triangular subgraphs. If our .diagram contains triangular subgraphs then we first consider a minimal triangular subgraph and attack quadrangles (latdr pentagons) in these. Otherwise we consider the quadrangles inside the entire diagram.

Let us again replace as many composite propagators ( 0 0 ) 1 by single dressed propagators (-) as allowed by ineq. (8.1) for each subgraph. Then one can argue that as a result we must get at least one quadrangle yomewhere whose own propa ators are all of the elementary type ( -1, not composite ( 0 ' a). This is because facets with composite propagators now must be adjacent to 4-leg subgraphs (elementary facets or more complicated), and then these in turn must have facet(s) with elementary propagators. Also (although we will not really need this) one may argue that there will be quadrangles with not more than one external composite propagator, the others elementary (the one exception is the case when one of the adjacent quadranglular subgraphs has itself only

403


b C

+

Fig. 10. Inequality for quadrangles. a, b and c may each be 1 or2.

pentagons, but that case will be treated in the next section). As a result of these arguments, of all inequalities of the type given in Fig. 10 we only need to check the case that only one external propagator is composite, a = b = c * 1 .

But in fact they hold quite generally, also in the other cases. This is essentially because of the careful construction of the effective Feynman rules of type IV in Fig. 5.

Rather than presenting the complete proof of the inequalities of Fig. 10 (5 different configurations) we will just present a simple algorithm that the reader can use t o prove and understand these inequalities himself. In general we have integrals of the form

We could write this as a diagram in Fig. 1 1 , where the 6i at the propagators now indicate their respective powers. The vertices are here ordinary point-vertices, not the type IV rules. Now write

with

(9.3)

Inserted in a diagram, this is the inequality pictured in Fig. 1 1 .

404

G. ‘t HOOFT

a 6

We use it for instance when p1-p2 is the largest momentum of all channels, and if

W l < 261 ; w p < 262 ; wi < 2 , (9.4)

where Z is the degree of convergence of the diagram: 2 = 2C6i - 4. We continue making such insertione, everytime reducing the diagram% t o a convenient momentum dependent factor times a less convergent diagram. Finally we may have

2 < 261 , z < 262 , (9.5)

for two of its propagators. Then we use the inequality pictured in Fig. 12:

-26. d4k n (k-pi) I

i

Notice that in (9.5) we ineq. (9.4). This C can

have a strictly unequal sign, contrary to be computed using Feynman multiplicators,

405


Fig. 12. Inequality holding if 6 1 , ~ are strictly larger than 42 (ineqs. (9.41, (9.5)).

much like in the previous section. We get

(9 .7 )

We see that ineqs. (9.2) and (9.6) have essentially the same effect: if two propagators have a power larger than a certain coefficient they allow us to obtain as a factor a corresponding "propagator" for the momentum in that particular channel. This is how proving the ineqs. of Fig. 10 can be reduced to purely algebraic manipulations. We discovered that ineq. (8.15) is again crucial. We notice that if the internal propagators of the quadrangle were elementary ones then the superficial degree of convergence of any of the other subgraphs of our diagram may change slightly, since in eq. (5.10) 6 > 2a, but the left hand side of eq. (5.11) remains unchanged, because

AE1 = -2AE2 , ( 9 . 8 )

so our condition that all subgraphs be convergent remains fulfilled after the substitution of the inequalities of Fig. 10.

However, if one of the internal lines of the quadrangle had 2 been a composite one (-), then a subgraph would become more

divergent, because we are unable to continue our scheme with something like a three-particle composite propagator ( *). A crucial point of our argument is that we will never really need such a thing, if we attack the quadrangle subgraphs in the right order.

10. PENTAGONS. CONVERGENCE OF THE SKELETON EXPANSION

As stated before, the order in which we reduce our diagram

406

G. 't HOOFT

into a tree diagram is:

1 ) remove triangular facets; 2) remove triangular subgraphs if any. By complete induction we

prove this to be possible. Take a minimal triangular subgraph and go to 3;

3) remove quadrangular facets as far as possible. If any cannat be removed because of a crucial composite propagator in them, then

4 ) remove qudrangular subgraphs, After that we only have to 5) remove the pentagons. 6) If we happened to be dealing with a subgraph by branching at

point 2 or 3, then by now that will have become a tree graph, because of the theorem in sect. 7. Go back to 1.

We still must verify point 5 . If indeed our whole diagram contains pentagons then we can replace all propagators by elementary ones. But if we had branched at steps 2 or 3 then the subgraphs we are dealing with may still have composite external propagator(s). In that case it is easy to verify that there will be enough pentagons buried inside our subgraphs that do not need composite external lines. In that case we apply directly the inequality of Fig. 13. The procedure for proving Fig. 13 is

Fig. 13. Inequality for pentagons.

exactly as described for the quadrangles in the previous section. Again the degree of divergence of any of the adjacent subgraphs has not changed significantly. This now completes our proof by induction that any planar skeleton diagram with 5 external lines is equal to CL times a diagram with type IV Feynman rules, where C is limited to fixed bounds. Since also the number of diagrams is an exponential function of L we see that for this set of graphs perturbation expansion in g has a finite radius of convergence. The proof given here is slightly more elegant than in Ref. 1 1 , and also leads to tree graph expressions that are more useful for our manipulations.

407


If the diagram has 6 or more external lines then still a number of facets may be left, limited by ineq. (7.10), all having 6 or more propagators. If we wish we can still continue our procedure for thes.e but that would be rather pointless: having a limited number of loops the diagram is finite anyhow. The difficulty would not so much be that no inequalities for hexagons etc,. could be written down; they certainly exist, but our problem would be that the corresponding number C would not obviously be bounded by one universal constant. This is why our procedure would not work for nonplanar theories where ineq. (7.10) does not hold. In the non-planar case however similar theorems as ours have been derived6.

1 1 . BASIC GREEN mTNCTIONS

The conclusion of the previous section is that if we know the "basic Green functions", with which we mean the two- three- and four-point functions, and if these fall within the bounds given in Table 1, then all other Green functions are uniquely determined by a convergent sum. Clearly we take the value for the bound g2 for the coupling constant (ineq. (5.9)) as determined by the inverse product of the coefficient C2 found in sect. 6 and the maximum of the coefficients in the ineqs. pictured in Figs. 9, 10 and 13, times acombinatorial factor.

Now we wish not only to verify whether these bounds are indeed satisfied, but also we would like to have a convergent calculational scheme to obtain these basic Green functions. One way of doing this would be to use the Dyson-Schwinger equations. After all, the reason why those equations are usually unsoluble is that they contain all higher Green functions for which some rather unsatisfactory cut-off would be needed. Now here we are able to re-express these higher Green functions in terms of the basic ones and thus obtain a closed set of equations.

These Dyson-Schwinger equations however contain the bare coupling constants and therefore require subtractions. It is then hard to derive bounds for the results which depend on the difference between two (or more) divergent quantities. We decided to do these subtractions in a different way, such that only the finite, renormalized basic Green functions enter in our equations, not the bare coupling constants, in a way not unlike the old "bootstrap" models. Our equations, to be called "difference equations" will be solved iteratively and we will show that our iteration procedure converges. So we start with some Ansatz for the basic Green functions and derive from that an improved set of values using the difference equations. Actually this yill be done in various steps. We start with assuming some function g(x) for the f loat ing coupting conetrmt, where x is the momentum in the maximal channel (see Table 1):

G. 't HOOFT

( 1 1 . 1 )

and a aet of functions g(i)(x) with

g(x) - max Ig(;)(X)l . def i

(11.2)

Here g i)(x) is the set of independent numbers that determined the basic f, reen function at their "sylmnetry point":

Ipi-p.I = x for all i,j . (11.3) 3

The index i in ghi) then simply counts all configurations in (11.3). With "in ependent" we mean that in some gauge theories we assume that the various Ward-Slavnov-Taylor' basic Green functions are fulfilled. This is not a very crucial point of our argument so we will skip any further discussion of these Ward or Slavnov-Taylor identities.

identities among the

If the values of the basic n-point Green functions (n = 3 or 4 ) at their symmetry points are Ai(x), then the relation between A. and g is: 1 i

, ( I I .4)

where K'

(5.1) renormalizable couplings.) Our first Ansatz for gi(x) is a set of functions that is bounded by (11.21, with g(x) decreasing asymptotically to zero for large x as dictated by the lowest order term(s) of the renormalization group equations. We will find better equations for gi(x) as we go along. In any case we will require

are coefficients of order one, and Z(x) is defined in " and (5.2). (We ignore for a moment the case of super-

(11.5)

for some finite coefficient iT. Our first Ansat2 for the basic Green functions away from the

symmetry points will be even more crude. All we know now is that they must satisfy the bounds of Table 1 . In general one may start with choosing (11.4) to hold even away from the symmetry points, and

x - max Ip.-p.1 . 1 3 i ,j

( I 1.6)


After a few iterations we will get values still obeying the bounds of Table 1 , and with Uncertainties also given by Table 1 but K; replaced by coefficients 6Ki. Thus we start with

6Kio) = K. 1 . (11.7)

We will spiral towards improved AnsLttze for the basic Green functions in two movements:

i) the "small spiral" is the use of difference equations to obtain improved values at exceptional momenta, given the values g i ( x l a t the symmetry points. These difference equations will be given in the next section.

ii) The "second spiral" is the use of a variant of the Gell-Mann- Low equation to obtain improved functions g;(x) from previous Ansstze for g;(x), making use of the convergent "small spiral" at every step. What is also needed at every step here is a set of integration constants determining the boundary condition of this Gell-Mann-Low equation. It must be ensured that these are always such that g(x) in ineq. (11.2) remains bounded:

g(x) < go 3

where go is limited by the coefficients Ki and the various coefficients C from sects. 6, 8, 9 and 10, as in ineq. (5.9).

12. DIFFERENCE EQUATIONS FOR BASIC GREEN FUNCTIONS

The Feynman rules of our set of theories must follow from a Lagrangian, as usual. For brevity we ignore the Lorentz indices and such, because those details are not of much concern to us. Let the dressed propagator be

P(P> = -GZ1(p) , (12.1)

and let the corresponding zeroth order, bare expressions be indicated by adding a superscript 0. In massive theories:

so that

P(p+k) - P(k) = P(p+k)Gq,(plk)k,,P(p) . (12.4)

This gives us the "Feynman rule" for the difference of two dressed

410

G. 't HOOFT

Fig. 14. Feynman rule for the difference of two dressed propagators. The 3-vertex at the right is the function G,,,(p Ik).

2lJ F ig . 15. Difference equation (12.6) for G

41 1


Fig. 16.Some arbitrarily chosen terms in the skeleton expansion for G2VUX'

propagatore, depicted in Fig. 14, (Note that, in this section only, p and k denote external line momenta, not external loop momenta.)

Go We have also this Feynman rule for bare propagators. There follows directly from the Lagrangian:

2 P

GiP - 2p,, - k,, .

Continuing this way we define

G2,,(plk+q) - G2P(plk) G2,,,,(plklq)q,, , with

G i p u -6 VV -

In Feynman graphs this is sketched in Fig. 15. once more we get

G2,,,,(plklq+r) - G2,,,,(plklq) G2,,vX(plklq

(12.5)

(12.6)

(12.7)

Differentiating

r)rX . (12.8)

Of course GzP,,~ can be computed formally in perturbation expansion. The rules for computing the new Green functions G G,,,,, GPYl are easy to establish. Let p1 be one of the external Y;op momenta as defined in eq. (5.3). For a Green function G(p1) we have

(12.9)

where fi(qi) are bare vertex and/or propagator functions adjacent to the external facet labeled by 1. The remainder F is independent of p1. We write

412

G. 't HOOFT

which is just the rule for taking the difference of two products. We find the difference of two dressed Green functions G in terms of the difference of bare functions f. Therefore the "Feynman rules" for the diagrams at the right hand sides of Figs. 14, 15 and the 1.h.s. of Fig. 16 consist of the usual combinatorial rules with new bare vertices given by the eqs. (12.5) and (12.7). These bare vertices occur only at the edge of the diagram.

We see that the power counting rules for divergences in GzU,,~ are just as in 5-point functions in gauge theories. Since the global degree of divergence is negative we can expand in skeleton graphs. See Fig. 16, in which the blobs represent ordinary dressed propagators and dressed vertices or dressed functions G,, and G,,,,.

Notice that one might also need G3,,(pl,p2lk) defined by

G3(P1 ,p2+k) - G3 (P1 rP2) G3p (P1 rP2 Ik) ok p ( 1 2 . 1 1 )

In short, the skeleton expansion expresses G 2 p V ~ but also G 3 ~ v etc. in terms of the few basic functions Gp,,, GpVv, G3,, and the basic Green functions G2,3,1+.Also the function GkU, defined similarly, can thus be expressed. The corresponding Feynman rules should be clear and straightforward.

We conclude that the basic Green functions can in turn be expressed in terms of skeleton expansions, and, up to overall constants, these equations, if convergent, determine the Green functions completely. Notice that we never refer to the bare Lagrangian of the theory, so, perhaps surprisingly, these sets of equations are the same for all field theories. The difference between different field theories only comes about by choosing the integration constants differently .

Planarity however was crucial for this chapter, because only planar diagrams have well defined "edges": the new vertices only occur at the edge of a diagram.

13. FINDING THE BASIC GREEN FUNCTIONS AT EXCEPTIONAL MOMENTA (THE "SMALL SPIRAL")

In this section we regard the basic Green functions at their synmetry points as given, and use the difference equations of

41 3


Sect. 12 to express the values at exceptional momenta in terms of these. If pi-pj is the momentum flowing through the planarchannel ij, then in our difference equations we decide to keep

p = max Ip;-p. I 3 i,j

(13.1)

fixed. So the left hand side of our difference equations will show two Green functions with the same value for u, one of which may be exceptional and the other at its symmetry point, and therefore known. (Weuse the concept of "exceptional momenta" as in ref, 14.)

Now the right hand side of these difference equations show a skeleton expansion of diagrams which of course again contain basic Green functions, also at exceptional momenta. But these only come in combinationsofhigher order, and the effect of exceptional mu- menta is relatively small, so at this point one might already suspect that when these equations are used recursively to determine the exceptional basic Green functions then this recursion might converge. This will indeed be the case under certain conditions as we will show in sect. 15.

Our iterative procedure must be such that after every step the bounds of Table I again be satisfied. This will be our guide to define the procedure. First we take the 4-point functions, and consider all cases of Table 1 separately.

The right hand side of our difference equations (Fig. 16) contains a skeleton expansion to which we apply the theorem mentioned in the end of sect. 5 and proven in sects. 5-10: the skeleton expansion for any 5-point Green function converges and is bound by free diagrams constructedwith type IV Feynman rules, Since thq 5-point functions in Fig. 16 are irreducible, the internal lines in the resulting tree graph will always be composite propagators, as in the r.h.s. of Fig. 13. So we simply apply the type IV Feynman rules for 5 tree graphs to obtain bounds on the 5-point function in various exceptional regions of momentum space. Table 2 lists the results.

The power of g2(A3) in the table applies where we consider the function Gqu. The other functions G3pv and G2,,vh have one and zero powers of g ( A 3 ) , respectively. In front of all this comes a - - power- series of the form

m 1 c"g$ - cg;(l-cs~)-l n= 1

which converges provided that

(13 .2)

414

G. ‘t HOOFT

Table 2 Bounds for the irreducible 5-point function at some exceptional momentum values.

(13.3) -4 go - yx Ig(l l ) l < c

We now write an equation such as (12.8) as follows:

E4{(((12)1 312 413) G~,{((523)2 413) +

-t (Pl-PS),, G4p{(((12)1 5312 413) Y (13.4)

where r = p1-p5. In this and following expressions the tilde (-) indicates which quantities are being replaced by new ones in the iteration procedure.

If the Ansatz holds for G4{ ((523)2 413) then the new exceptional function will obey

n

cg; Choosing - , = Y - (13.6)

1-cg;

and considering that to a good apiroximation (since Ip1-p5l << I p4-p 5 I :

245 z41 s (13.7)

41 5


we find

(13.8)

What is now needed is a bound €or the last term in (13.8). Let

x12 = lp121/m> 1

(remember that Ipl stands for 6-1, and

(13.9)

(13.10)

where u is defined in (5.2). When

x12 > exp ( -u /2a) = xo (13.1 I )

this f is an increasing function, so that if

xgm 6 A1 < A2 (13.12)

then

The range 1 < x <xo is compact, so there exists a finite number L such that

(13.14)

as soon aa

x12 <x52 - (13.15)

So we find that after one iteration given by (13.4), the new K2 coefficient satisfies

K: < y + K ~ L . Similarly we derive

(13.16)

41 6

G. 't HOOFT

K q < y + L , (13.17)

when the difference equation is used to express E{ ( ( 1 2 ) 1 3412) in terms of G{ (523412). Also we use

Z;c{ ((1211 (341213) 'G;c{(52(34)2)3} +

+ (p1-P5), G4,,{ ((12)1 5(34)2)3) 9 (13.18)

to find that after one step

K: < y + K;L < y + YL + ~2 ; (13.19)

and for the three-point function

K1 <K; + L < y + 2L. (13.20)

The remaining coefficients K4-6 must be computed in a slightly different way. Consider Kq. We replace p1 by p5 now in such a way that

and work with induction. Write

Inspecting Tables 1 and 2 we find now

( 1 3.23)

Applyink the same technique we compute the fifth exceptional configuration of Table 1 . We separate p1 from p3 until A1 -.) A2. Then we separate alternatively p2 from p4 and p1 from p3 keeping A1 m A 2 . This makes the rate of convergence slightly slower:

(13.24)

41 7


where

(13.26)

We find

The sum can certainly be bounded:

1 < L' Q 2L(1-2a-fl)-1 . ( 1 3.28)

Therefore

(13.29)

Thus all coefficients Ki have bounds that will be obeyed everywhere in the "small spiral" induction procedure. Note that these coefficients would b ow up if a,B + 0 . In particular in (13.11) we need a > 0. Only if go + o we can let a,B + 0. It will be clear from the above arguments that our bounds are only very crude. Our present aim was to establish their existence and not to find optimalbounds.

2 K6 g m a x ( 1 , y L ' ) .

!!

In sect. 15 we show that the "small spiral" of iterations for the exceptional Green functions, given the non-exceptional ones in (13.27), actually converges geometrically.

14. NON-EXCEPTIONAL MOMENTA (THE 1 * ~ ~ ~ ~ ~ ~ SPIRALII)

In order to formulate the complete recursion procedure for determinihg the basic Green functions we need relations that link these Green functions at different symmetry points. Again the difference equations are used:

Here pi and pji) are external Loop momenta. They are non- exceptional. JWe use a shorthand notation for (14.1). Writing

418

G. 't HOOFT

Similarly we have

(14.2)

(14.3)

These are just discrete versions of the renormalization group equations. The right hand side of (14.21, [not (14.3)Il is to be expanded in a skeleton expansion which contains all basic Green functions at all p , also away from their symmetry points. There we insert the values obtained after a previous iteration. It is our aim to derive from eq. (14.2) a Gell-Mann-Low equation' of the form

(14.4) N + lg(p)l Pi(d ,

where B(') are the first k coefficients of the B function, and they must coincide with the perturbatively computed B coefficients. Often (depending on the dimension of the coupling constant) only odd powers occur so that k - N-2 is odd. The rest function p must satisfy

Ipi(dl Q, (14.5)

for some constant QN <a. This inequality must hold in the sense that Ig(p)IN p ( p ) must be a convergent expansion in the functions g(v'), with

P ' z m (14.6)

(so that p' may be smaller than p), in such a way that the absolute value of each diagram contributes to % and their total sum remains finite.

Now clearly eq. (14.2) is a difference equation, not a differential equation such as (14.4). Up till now differential equations were avoided because of infrared divergences. Just for ease of notation we have put (14.4) in differential firm because the mathematical convergence questions that we are to consider now are insensitive to this simplification.

Consider the skeleton expansion of Gti) in (14.2). At each of the four external particle lines a factor g(pi) occurs with pi 2 p,

419


so it may seem easy to prove (14.4) from (14.2) with N = 3 or 4. However, we find it more convenient* to have an equation of the form (14.4) with N 6 7, qyd our problem is that the internal vertices of the We will return to this question later in this section.

G,$ might have momenta which are le4S thanu.

In proving the difference equation variant of (14.4) from (14.2) we have to make the transition from GI, to g2 and Gg to g, and this involves the coefficients Z(p), associated to the functions G2, by equations of the form

G2(u) = - u2Z-’(u) s

G3(u) = ~z’~’~(~)g3(~) ; (14.7)

G~(P) Z-2(u>gc(u)

where gg, g4 are just various components of the coupling constant gi. In the following expressions we suppress these indices i when we are primarily interested in the dependence on u (= lpl at the symmetry point). Now from (14.2) and (14.3) we find not first order but third order differential equations for Gp, basically of the form

( 1 4 . 8 )

where G2 XxA is just a shorthand notation for the combination of expandabie functions G~,A,,,, obtained after taking differences three times. Write

then

(14.9)

(14.10)

* Closer analysis shows that actually N = 3 or 4 is sufficient to prove unique solubility. Only if we wish an exact, non-perturbative definition of the free parameters we need the higher N values. Note that not only QN but also go may deteriorate as N increases.

420

G. ‘t HOOFT

Here A and B are free integration constants; A is usually determined by Lorentz invariance and B by the mass, fixed to be equal to m. In lowest order:

A = m~p(m) ; B - lm2Ui(m) . (14.12)

This strange-looking form of the integration constants is an artifact coming from our substitution of difference equations by differential equations. Using difference equations we can impose Lorentz invariance by spmetrization in momentum space, so that only one (for each particle) integration constant is left: the mass term. We choose at all stages iUp(m) = Z(m) - 1 .

A convenient way to implement eq. (14.12) is to formally define U2(p) = 2 if o 5 p 5 m, and replace the lower bound of the integral in (14.11) by zero. Then after symmetrization: A - B = 0 .

Equation (14.11) has a linearly convergent integral, whereas (14.10) is logarithmic. Together they determine the next iterative approximation to G . In fact we have

( 1 4 . 1 3 )

and in f({g)), Z occurs only indirectly. So the iteration converges fastest if we replace (14.10) by

where the tilde denotes the new function Up(p).

One can however also use (14.9) with Up replaced by 52. We find

&! 2’’ = - 1 dr(I-T)uGp,AXX(tp) . ( 1 4 . 1 5 ) ap

m/u As stated before, the @( f) terms have been removed by symmetrization.

This equation allows us to remove the Z factors from the functions 6 3 . 4 and arrive at first order renormalization group integrodifferential equations for gi(p).

For the 3-point functions we must write

421


(14.16)

(14.17)

(14.18)

A potential difficulty in writing down the renormalization group equation even for N = 4 is the convolutions in (14.15) and (14.18) which contain Green functions at lower v values, and so they depend on g(p') with p' < 1-1. So a further trick is needed to derive (14 .4 ) . This is accomplished by realizing that the integrals in (14.15) and (14.18) converge linearly in u . Suppose we require at every iteration step (see eq. 11.5) :

and (14.19)

for some B < m , go < =. Then it is easy to show that if u 1 5 p , then

if

(14.20)

So with C large enough and go small enough we can make & as small as we like. Inequality (14 .20 ) is proven by differentiating with u . -This enables us to replace g(T1.l) by g(U) in (14.15) and (14.18) while the factor T-& does no harm to our integrals. So we find bounds for @ ??l and @ E3 in terms of a power series of g(p). We must terminate the series as soon as the factors T-' accumulate to give T-~. This implies that N must be kept finite, otherwise go + 0 .

au au

The same inequality (14.16) is used to go from N = 4 to N = 7 in these equations. If in a skeleton diagram a vertex is not associated with any external line, then it may be proportional to a factor g(p') with replaced by g(p) at the cost of a factor (p/ul)'. At most three of these extra factors are needed. If the three corresponding vertices are chosen not to be too far away from one of the external vertices of the diagram (which we can always arrange), then this just corresponds to inserting an extra factor

< p . But using (14.16) we see that it may be

422

G. 't HOOFT

(Cr at an external vertex. We now note that such factors still leave our integrals convergent. In the ultraviolet of course the diagrams converge even better than they already did, and in the infrared our degree of convergence was at least I-a (or 2-8) as can easily be read off from eq. (5.10): adding the external'propagators to any diagram one demands

Z + (2+2u)E1 + 28E2 < 4(Ei+E2-1) , (14.22)

Thus infrared convergence requires

1 -a -T& > o ( 14.23)

where T 5 5 is the number of times our inequality (14.20) was applied.

From the above considerations we conclude that an equation of the form (14.4) can be written down for any finite N, such that % in inequality (14.5) remains finite. We do expect of course that QN might increase rapidly with N, but then we only want the equation for N 2 7. We are now in a position to formulate completely our recursive definition of the Green functions G2, C3, G4 of the theory:

1) We start with a given set of trial functions G2(u), G3(u), G4(u) for the basic Green functions at their symmetry points. They determine our initial choice for the floating coupling constants gi(v) and the functions Z . ( p ) . We require their asymptotic behaviour to satisfy (5 .2 f , (5.9) and (11.5) (= eq. (14.19)).

2) We also start with an Ansatz for the exceptional Green functions that must obey the bounds of Table 1 .

3) Usa the difference equations of sect. 13 to improve the exceptional Green functions (the new values are indicated by a tilde (-))'. These will again obey Table 1 as was shown in sect. 13. Repeat theprocedure. It will converge towards fixed values for the exceptional Green functions (as we will argue in sect. 15). This we call the "small spiral".

4) With these values for the exceptional Green functions we are now able to compute the right hand side of the renormalization group equation for G2, or rather Z-', from (14.151, using (14.20):

where l(p2 is again bounded. Here Yijk are the one-loopy coefficients This gives us inproved propagators. See sect. (15.b).

423


5) Now we can compute the right hand side of eq. (14.4). Before integrating eq. (14.4) it is advisable to apply Ward identities (if we were dealing with a gauge theory) in order to reduce the number of independent degrees of freedom at each p. As is well known, in gauge theories one can determine all subtraction constants this way except those corresponding to the usual free coupling constants and gauge fixing parameter^'^. So the number of unknown funotions gi(u) need not exceed the number of "independent" coupling constants of the theory*.

6) Eq. (14.4) is now integrated, giving improved expressions for gi(P). Now go back to 2. This is the "second spiral", which will be seen to converge towards fixed values of g;(p).

The question of convergence of these two spirals is now discussed in the following section.

15. CONVERGENCE OF THE PROCEDURE

a) Except7:onaZ Mmenta

In sect. 13 a procedure is outlined to obtain the Green functions at exceptional momenta, if the Green functions at the symmetry point are given. That procedure is recursive because eqs. (13.4), (l3.18), (13.22) and (13.25) determine the Green functions G2,3,4 in terms of the symmetry ones, and GI,,,, G3,,,,, G2*,,1. But the latter still contain the previous ansatz for G2,3,4. Fortunately it is easy to show that any error 6G2,3,4 will reduce in size, so that here the recursive procedure converges:

Let us indicate the bounds discussed in sects. 5 and 13 as

and assume that a fikst trial Gil) has an error

(15.1)

(15.2)

with some &(') 5 2.

(15. I). Furthermore they were one order higher in g2. So we have Now GI,,,, G3,,,,, G~,,vA also satisfy inequalities of the form

* We put "independent" between quotation marks because our requirement of asymptotic freedom usually gives further relations m n g various running coupling constants, see appendix A

424

G. 't HOOFT

(5.3)

when the function GqP itself converges like

1 cngn ¶

and BI,,, is the bound for Gq,, itself, as given by Table 2.

error 6Gn in the Green functions. But there is a factor in front, The procedure of sect. 13 can be applied unaltered to the

(15.4)

This gives €or the newly obtained exceptional Green functions an error

I SG:)l < L ( ~ ) B (15.5)

and e(2) < c(') if we reduce the maximally allowed value for g, as given by (5.9), somewhat mre:

+ 0.6527 gmax . (15.6) ~max We stress that the above argument is only valid as long as the Green functions at their symmetry points were kept fixed and are determined by g(v ) , bounded by (15.6).

b ) The Z Factors

(15.7) (15.8)

where the 0(g2) terms are again bounded by a coefficient times g2(u). These equations must be solved iteratively, because the right hand side of (15.7) contains skeleton expansions that again contain Z(u), hidden in the function l ( u 1 ) . It is not hard to convince oneself that such iterations converge. A change

425

PLANAR DIAGRAM FIELD THEORlES

(15.9)

yields a change in the function 1 ( P I ) bounded by

1621 ,< &"'g; 1 , so that

(15.10)

(15.11)

with c ( * ) < c ( l ) if go is small enough.

c ) The coupZing constants

We now consider the integro-differential equation (14 .4 ) . The solution is constructed iteratively by solving

where the tilde denotes the next "improved"function gi(p). Our first Ansatz will be a solution of (15.12) with piig(v),vl replaced by zero. This certainly exists because the B coefficients are determined by perturbation expansion and therefore finite. The integration constants must be chosen such that for all u > m we have

where gmX is the previously determined maximally allowed value of g(p) and K is again a constant smaller than 1 to be determined later. In practice this requirement implies asymptotic freedom3:

( 1 5 . 1 4 )

(It is constructive to consider also complex solutions.)

If we now substitute this g(p) in the right hand side of eq. (15.12) we may find a correction:

g(v) g(ll) = g(v) + 6g(v) , (15.15)

for which we may require

ISg(v)l E gmax for all p . (15.16)

426

G. 't HOOFT

We must start with:

E+K < 1 (15.17)

Will a recursive application of eq. (15.12) converge to a solution? Let the first Ansatz produce a change (15.16). The next correction is then, up to higher orders in 6g, given by

(where Mij is determined by differentiation of (15.12) with respect to gi(p).

Our argument that l p l < came from adding the absolute values of all diagrams contributing to p , possibly after application of (14.20) several times. Replacing (14.20) then by

To estimate Sf(p) we must find a limit for the change in p .

which indeed is true if g satisfies (15.18), or

(15.20)

as can be derived from (14.20) and (l4.21), we find that we can write

16Pl < EC' , (15.21)

with C' slightly larger than C, and

ISf(v)l 2 &(N+l)C' g(vIN . (15.22)

Now asymptotically,

M.. (11) -+ M!. /log v , (15.23) 1 3 1 J

0 where Mi. is determined by one-loop perturbation theory. If there is only dne coupling constant it is the number 312. In the more general case we now assume it to be diagonalized:

M?. = M(i)6.. , (15.24) 1 J 1J

with one eigenvalue equal to 312. (Our arguments can easily be ex-

in which case the standard triangle form must be used.) The asymp- tended to the special situation when Mij 0 cannot be diagonalized,

427


totic form of the solution to

P

where p(i) are integration constants. If M(i) < 3 - I then we choose v(i) = QD. If M(i) >T - I we set p(i) * m. Then in both cases we get

N

(15.26)

where C" is related to C' and the first 8 coefficient. In a compact set of p values where the deviation from (15.25) is appreciable we of course also have an inequality of the form (15.16).

If M(i) - N / 2 - 1 then we simply pick another N value (which raising or lowering it by one unit. needs not be integer here),

We see that we only need to consider N ( 4 . Comparing (15.26) with ( l S . l 6 ) , noting that C" is independent of A, we see that if

then our procedure converges. Since C" stays constant with decreasing bx, we find that a finite gmX will (15.27).

(15.27)

or decreases satisfy

Also we should check whether i ( p ) satisfies the Ansatz (14.19) ,j with unshanged 8 . This however is obvious from the construction of g through eq. (15.18). Notice that the masses are adjusted in every step of the iteration for the Z functions, by choosing A and B in section 14. They are necessary now because we wish to confine the integrals (15.25) at II > m, limiting the solutions g(p) to satisfy Ig(p) I 6 g.

16. BOREL SUMMABILITY~~

The fact that we obtained eq. (14.4) holding for m 6 P <= is OUK central result. For simplicity of the following discussions we ignore the next-to-leading terms of the coefficients B. Let us take the case that the leading ones have = 3. We find that

(16.1)

G. 't HOOF7

(the next-to-leading Bcoefficients give an unimportant correction in the denominator of the form log logu2). The coefficients bi are fixed by the leading renormalization group B coefficients. We must choose C such that

lg2(v)I < g L x if p > m . (16.2)

This is guaranteed if either

Now the requirement of asymptotic freedom is usually so stringent that the constant C is the only free parameter besides the masses and possible dimension I coupling constants (a situation corresponding to the necessity of choosing all p(i) - QI in eq. (15.25)). But still we can do ordinary perturbation expansion, writing

sib) = big + Q(g3) ; (16.4)

(16.5)

where g is now a regular expansion parameter. The 0(g3) terms are fixed by the asymptotic freedom requirement and can be computed perturbatively. We now claim that perturbation expansion in g is indeed Borei-summable:

0

(16.6a)

where the path C must be choosen to lie entirely in the region limited by eqs. (16.3) and (16.5) (see Fig. 17). Since in that region the Green functions G(g2) are approximately given by their perturbative values the integral (16.6b) will converge rapidly along this path, if

R e z > o , (16.7)

429


Im( 1 /g2>

Fig. 17. The integration path C in eq. (16.7).

and F(z) will be bounded by

IF(z)~ < A exp(IzI/gkax) , (16.8)

where hx is the allowed limit for g as derived in the previous sect ions.

However, eqs. (16.7) and (16.8) are not quite sufficient to prove Bore1 summability because we also want to have analyticity of F(z) for an open region around the origin. That this requirement is met c?n be seen as follows. Let us solve a variant of equation ( 1 4 . 4 ) of the following form:

(16.9)

where e(x) is the step function. The coefficients B(') and the functional p are the same as before (constructed the same way via difference equations of sects. 12 and 13) . Clearly we have, if P > A: g(u,A) = g(A,A). And eq. (16.1) now reads

(16.10)

430

G . 't HOOFT

Our point is that this solution exists not only in the region (16.31, but also if

max (b ) Re C < - - log A2 E -R-log A' . (16.11)

gLX

We can now close the contour C (see Fig. 18).

Fig. 18. The contour C of eq. (16.13); the forbidden regions of G(g,Ai) are shaded.

Now compare two different A values: A 1 and A p , and compare the Green functions computed with these two A values, both as a function of g; taking

(16.12)

We take these Green functions at (possibly exceptional) momentum values p, but always such that A >> Ipl. If they are computed directly following our algorithm then a slight A-dependence may still exist, coming from two sources: one is the fact that gi(u,A) depends on A because p(u,{g)) as a functionat of g, may depend on g(u') with p' > A. But clearly, since all integrals involved in the construction of p converge we expect this dependence to go like

or probably

431


(16.13b)

(because linearly convergent equations can often be made quadratically convergent by symmetrization); here c(g) C o if g+o. The second source of A-dependence comes from the application of difference equations to compute Green functions away from the synmetry point (sects. 12,13). Again, this error must behave like ( l6.13), because of convergence of the integrals involved.

The change in the Borel function F(z) can be read off from eq. (16.6b)

(16.14)

C

i f A2 2 A1. Under what conditions does this vanish in the limit A 1 + w ? In our

integral g'2 ranges from -R-log -$ to R. So if Re z =G o then (3 (16.15)

If lg'21 + then c(g) C 0 . This we may use on the far left of the curve c. Therefore ineq. (16.14) can be written as

(Id. 16)

By comparing a series of A values of the form An = 2", one easily sees that (16.16) guarantees convergence as soon as

Re z > - 1 , (16.17)

which is a quite large region of analyticity of F ( z ) . Note that

(16.17) together with (16.8) implies complete Borel sumability of this theory.

= - 1 is the location of the first renormalon singularity4s7. Eq.

17. THE MASSLESS THEORY

In the sections 14-16 the mass m2 had to be non-zero. This was the only way we could obtain the necessary ineq. (16.3a),

us to draw the contour C. What hap ens if we take the + 0 , considering not g(m2) but g(p ! ) at some fixed l~ as

432

G. 't HOOFT

expansion parameter? Our method to construct the entire theory then fails, but for some z values F(z) will still exist. The argument is analogous to the one of the previous section where we dealt with an ultraviolet problem. Now our difficulty is in the infrared. Comparing two different small values for m, we have

C if mp zm1. Now if Re z > o then

Thus

(17.1)

(17.2)

(17.3)

( I 7.4)

Now we have convergence as m2 + o as long as R e z < I , (17.5)

and,indeed, p - 1 is a point where an infrared renormalon singularity is to be expected. Thus, the Bore1 transform F ( z ) of the massless theory is analytic in the region

- 1 < Rez < 1 . (17.6)

This result guarantees that perturbation expansion in g2 diverges not worse than

g2" n!

but is clearly not enough for Bore1 summability.

18. THE - A Tr EaODEL

A special case is the pure scalar planar field theory, with just one coupling constant X and a mass m. If m is sufficiently large and X sufficiently small then our analysis applies, and we find that all planar Green functions are uniquely determined. However, in this special case there is mre: the Green functions can be uniquely determined as long as the masses in a l l channels

433


are non-negative, and X is negat ive . This w a s discovered by comparing wi th t h e much s imple r " sphe r i ca l model" (appendix B) which shows t h e same proper ty .

The argument i s f a i r l y s imple. L e t us f i r s t t ake t h e theory a t m2 va lues which are so l a r g e t h a t every th ing is wel l -def ined. Now decrease t h e "bare mass" m$ cont inuously. The change i n t h e dressed propagator

P(k,m2) = -Gp1(k,m2) (18.1)

is determined by

a G (k,m2) : G'(k,m2) a m z 2 (18.2)

which w e take t o s tar t o u t a t s u f f i c i e n t l y l a r g e R'.

Now t h e Feynman r u l e s f o r

G"(k,m2) f am2 a G'(k,m2) , (18.3)

can e a s i l y be w r i t t e n down, j u s t l i k e those f o r

(18.4)

Since G;' and G l are s u p e r f i c i a l l y convergent w e can a g a i n express them i n terms.of a ske le ton expansion con ta in ing only t h e funct ions G4 and -G> and t h e propagators P. They are a l l posi t ive (remember t h a t G4 s tarfs ou t as - A , with X < o ) , am' aZZ integrals and sununations converge. Only GL has one s u r v i v i n g minus s i g n from d i f f e r e n t i a t i n g one pmpaga to r wi th m . Thus:

G'; > o ; (18.5)

G I < o . (18.6)

I f w e l e t m2 dec rease then c l e a r l y G 2 w i l l s t a y p o s i t i v e and G)2 negat ive. The i r abso lu t e va lues grow however, u n t i l a po in t is reached where e i t h e r t h e sum of a l l diagrams w i l l no longe r converge, o r t h e two-point func t ion G2 becomes zero . As soon as t h i s happens t h e theory w i l l be i l l -de f ined . A tachyonic po le tends t o develop, followed by ca t a s t rophes i n a l l channels . The po in t w e wish t o make i n t h i s s e c t i o n however is t h a t as long as t h i s does n o t happen, indeed a l l summations and i n t e g r a l s converge, so t h a t our i t e r a t i v e procedure t o produce t h e Green func t ions w i l l a l s o converge. For a l l those v a l u e s of A and m2 t h i s theory w i l l be Bore1 summable.

This r e s u l t on ly holds fo r t h e s p e c i a l case cons idered h e r e , namely - X T r $4 theory , because a l l ske le ton diagrams t h a t con-

434

G. 't HOOFT

tribute to some Green function, carry the same sign. They can never interfere destructively. Notice that what also was needed here was convergence of the diagram expansion. Now we know that at finite N the non-planar graphs give a divergent contribution. Thus the "tachyons" will develop already at infinite m2: the theory is fundamentally unstable. Of course we knew this already: X after all has the wrong sign. The instantons that bring about the-decay of our "false vacuum" carry on action S proportional to -N/X which is finite for finite N.

19. OUTLOOK

Apart from the model of sect, 18, the models we are able to construct explicitly now lack any appreciable structure, so they are physically not very interesting. Two (extremely difficult) things should clearly be tried to be done: one is the massless planar theories such as SU(=) QCD. Clearly that theory should show an enormously intricate structure, including several possible phase-transitions. We still believe that more and better understanding of the infrared renormalons that limited analyticity of our bore1 functions in sect. 17 could help us to go beyond those singular points and may possibly "solve" that model (i.e. yield a demonstrably convergent calculational scheme). Secondly one would try to use the same or similar skeleton techniques at finite N (non-planar diagrams). Of course now the skeleton expansion does not converge, but, in Borel-summing the skeleton expansion there should be no renormalons, and all divergences may be due entirely to instantonlike structures. More understanding of resummation techniques for these diagrams by saddle point methods could help us out. If such a program could work then that would enable us to write down SU(3) QCD in a finite (but small) box. QCD in the real world could then perhaps be obtained by gluing boxes together, as in lattice gauge theories.

Another thing yet to be done is to repeat our procedure now in Minkowski apace instead of Euclidean space. Singling out the obvioue singularities in Minkowski space may well be not so difficult, so perhaps this is a more reasonable challenge that we can leave for the interested student.

APPENDIX A. ASYMPTOTICALLY FREE INFRARED CONVERGENT PLANAR HIGGS MODEL

In discussing examples of planar field theories €or which our analysis is applicable we found that pure SU(o0) gauge theory (with possibly a limited number of fermions) is asymptotically free as required, but unbounded in the infrared - so that even the ultraviolet limit cannot be treated exactly (see sects. 14 and 15).

435


-X T r (p4 is asymptotically free aad can be given a mass term ao t ha t infrared convergence is a l s o guaranteed. A t N + Q O t h i s is a f ine planar theory, but at f i n i t e N the vacuum is unstable. The only theory t h a t s u f f e r s from none of these de fec t s is an SU(-) gauge theory i n which a l l bosons ge t a mass due t o the Higgs mechanism. But then a new scalar self-coupling occurs t h a t tends to be not asymptotically f ree . Asymptotic freedom is only secured i f , curiously enough, several kinds of fermionic degrees of freedom are added. The following model is an example ( s imi l a r examples can a l s o be constructed a t f i n i t e N, such as SU(2)).

I n general a renormalizable model can be wr i t t en as

where G i a t he covariant c u r l , .+i is a set of s c a l a r f i e l d s and 6 a set of spinors. V is a q u a r t i c and W a l i n e a r polynomial i n 0. We write

abc b c = a ~ ~ - a ~ ~ + g APAV

a u v v II

C:b = - T r ( I J ~ ~ U ~ U ~ ) - -2Tr(UaUb+Uaub) s s P P . (A. 2)

The most compact way t o w r i t e the complete set of one-loop 8 functions is t o express them i n terms of the one-loop counter- Lagrangian, [ 8 ~ ~ ( 4 - n ) l - ~ A C , where AL has been found t o be1', a f t e r performing t h e necessary f i e l d renormalizations,

+ + V f j i y 3 Vi(T2.#)i + T(+T%b+)2 3

1 1

+ 2 $(62w+WU2)g + 4i(V. +$W.$)Tr(S.S +P.P.) 2 J J 1 j 1 1

- Tr(S2+P2) + Tr[S,P]2 .

436

(A. 3)

G. ' t HOOFT

Here Vi stands for aV/Wi, etc. The scaling behavior of the coupling constants is then detetmined by

(A. 4)

We now choose a model with U(N)l cal x SU(N)global symmetry. Besides the gauge field we have a scafar 4s in the Nlocal X

Nglobal representation, and two kinds of fermions:

. We choose local Nglobal in N i (2)a

x N and JI j JI(1)i in Nlocal local

and

+ h.c. . I a i

(A. 5)

(A .6 )

Writing

c;b = c;b._ 2 p ; Cfb = 322 gab , (A. 7)

we find: "2

-8n2 a .I Az2 I 2 z 4 * (A. 8) ap 2 '

and with expression (A.3) after some algebra, and in the limit N +a:

NX; K2 = Nh2, by substituting (A.5) and (A.6) in the

AG2 9 -$4 + 9 "p2 ; (A. 9)

(A. 10)

2 h g AT .t -2y + 3;2y - - 3 i 4 - 4ci;2y + 4Z4 . 4

These equations (A.8)-(A.10) are ordinary differential equations whose solutions we can study. The signs in all terms are typical for any such models with three coupling constants g, X and h. Only the relative magnitudes of the various terms differ from one model to another. For asymptotic fre-edom we need that the second and last terms of (A.10) and the last of (A.9) are sufficiently large. Usually this implies that the fermions must be in a sufficiently large representation of the gauge group, which explains our choice for the f e d o n i c representations. Our model has an asymptotically free solution if all coupling constants stay in a fixed ratio with respect to each other:

N A N A X = X T 2 ; h - h g , (A. 1 I )

and then, from (A.8)-(A.10) we see:

437


x = -!- ( r n - 5 ) , a (A. 12)

So indeed we have a solut ion with pos i t i ve A .

It now must be shown t h a t i n t h i s model a l l p a r t i c l e s can be made massive v i a the Higgs mechanism. We consider spontaneous breakdown of SU(N)local x SU(N)global i n t o the diagonal SU(N)global subgroup. Take as a mass term

(A. 13)

We can w r i t e V as 2 v = + + k - ~ 2 6 ~ ~ l + const. (A. 14)

Clearly t h i s i s minimal i f

(A. IS)

o r a gauge r o t a t i o n thereof. All vector bosons get an equal mass:

-D*@ D@ * - g2P2A2 ; (A. 16) P

Mi = 2g2F2 , (A. 17)

and of course the s c a l a r s get a mass:

$ = hF2 . (A. 18)

Thus the mass r a t i o i s given by

J-- ,112 = 0.6303 m a . . . . (A. 19)

This is a f ixed number of t h i s theory, but it w i l l be affected by higher order corrections. The fermions can each be given a maas term:

(A. 20)

and the Yukawa force w i l l give a mixing of a d e f i n i t e strength. The model described i n t h i s Appendixiis probably t h e simplest completely convergent planar f i e l d theory with absolutely s t a b l e vacuum. It is unlikely however t h a t it would have a d i r e c t physical s ignif icance.

438

G. 't HOOFT

APPENDIX B. THE N-VECTOR MODEL IN THE N + W LIMIT (SPHERICAL MODEL). SPONTANEOUS MASS GENERATION

When only W-vector fields are present (rather than NxN tensors)then the N - ) w limit is easily obtained analytically. This is the quite illustrative spherical model. Let the bare Lagrangian be

The only diagrams that dominate in the N +a+ limit are the chains of bubbles (Fig. 19).

a b Fig. 19. a) Dominating diagrams for the &point function.

b) Mass renormalization.

Let us remve some factors H~ by defining N

hg = NAB/16n2 . (B. 2)

The diagrams of Fig. 19 are easily summed. Mass and coupling constant need to be renormalized. Dimensional tenoylization is appropriate hete. In terms of the finite constants XR(p) and mR(p), chosenlat some subtraction point p , and the infinitesimal & = 4-n, where n is the number of space-time dimensions, one finds :

and

The sum of all diagrams of type 19a gives an effective propagator of the form

439


where q is the exchanged momentum, y is Euler's constant, and

0

and m is the physical mass in the propagators of Fig. 19a; that is because these should include the renormalizations of the form of Fig. 19b.

- = - + 0 Fig. 20.

Fig. 20 shows how m follows from %:

From ( B . 3 ) we see that the renormalization-group invariant combination is

N

so that inevitably XR < o at large p. Indeed, in this model the vacuum would become unstable as soon as N is made finite. In the limit N +a however everything is still fine.

2 Now in Fig. 21 we plot both 9 and the composite mass M2, determined by the pole of F(q4 , as a function of the physical mass m2. We see that at negative % there are two solutions for m2, but one should be rejected because $ would be negative, an indication for an unstable choice of vacuum. The observation we w'sh to make in this appendix is that in the allowed region for 9 we get an entirely positive 4-point function in Euclidean space LF(q) > 0 ) . If we chose-% to be fixed and vary 1~ (or rather vary XB! then at 9 > o ai?Z X valu s are allowed, at negative 4 only sufficiently small values. At 9 = o we see a

3 2

s 2

440

G. 't HOOFT

I ' I

Fig. 21. Mass ratios at given value for xR(p).

"spontaneous"Ngeneration of a finite value for m . Perturbation expansion in XR would show the "infrared renormalon" difficulty. Apparently here the difficulty solves itself via this spontaneous mass generation.

REFERENCES

1 . E.C.G. Stueckelberg and A. Peterman, Helv. Phys. Acta 26 (1953),

2. K.G. Wilson, Phys. Rev. D10, 2445 (1974). 499; M. Gell-Mann and F. Low, Phys. Rev. 95 (1954) 1300.

K.G. Wilson, in "Recent Developments in Gauge Theories", ed. by G. 't Hooft et al., Plenum Press,New York and London, 1980, p. 363.

3. G. 't Hooft, Marseille Conference on Renormalization of Yang- Mills Fields and Applications to Particle Physics, June 1972, unpublished; H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); D.J. Cross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).

4. G . 't.Hooft, in "The Whys of Subnuclear Physics", Erice 1977, A. Zichichi ed. Plenum Press, New York and London 1979, p. 943.

5 . M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42, 1390

6. C. de Calan and V. Rivasseau, Conrm. Math.*Phys. 82, 69 (1981); (1979).

91, 265 (1983).

441


7) G. Parisi, Phys. .Lett. 76B, 65 (1978) and Phys. Rep. 49, 215

8) J. Koplik, A. Neveu and S. Nussinov, Nucl. Phys. B123, 109 (1979).

(1977). W.T. Tuttle, Can, J. Math. 14, 21 (1962).

9) See ref. 3. 10) J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics

1 1 ) G. 't Hooft, Commun. Math. Phys. 86, 449 (1982). 12) G. 't Hooft, Commun. Math. Phys. 88, 1 (1983). 13) J.C. Ward, Phys. Rev. 78 (1950) 1824.

(IkGraw-Hill, New York, 1964).

Y. Takahashi, Nuovo Cimento 6 (1957) 370. A. Slavnov, Theor. and Math. Phys. 10 (1972) 143 (in Russian). English translation: Theor. and Math. Phys. 10, p. 99. J.C. Taylor, Nucl. Phys. B33 (1971) 436.

14) K. Symanzik, Comm. Math. Physics 18 (1970) 227; 23 (1971) 49; Lett. Nuovo Cim. 6 (1973) 77; "Small-Distance Behaviour in Field Theory", Springer Tracts in Modern Physics vol. 57 (G. HEhler ed., 1971) p. 221.

15) G. 't Hooft, Nucl. Phys. B33 (1971) 173. 16) G. 't Hooft, Phys. Lett. 119B, 369 (1982). 17) G. 't Hooft, unpublished.

R. van Dame, Phys. Lett. IlOB (1982) 239.

442

Nuclear Physics B75 (1974) 461-470. North-Holland Publishing Company

CHAPTER 6.2

A TWO-DIMENSIONAL MODEL FOR MESONS

C. 't HOOFT CERN, Geneva

Received 21 February 1974

Abstract: A recently proposed gauge theory for strong interactions, in which the set of planar diagrams play a dominant role, is considered in one space and one time dimension. In this case, the planar diagrams can be reduced to selfenergy and ladder diagrams, and they can be summed. The gauge field interactions resemble those of the quantized dual string, and the physical mass spectrum consists of a nearly straight "Regge trajectory".

It has widely been speculated that a quantized non-Abelian gauge field without Higgs fields, provides for the force that keeps the quarks inseparably together [ 1-41 . Due to the infra-red instability of the system, the gauge field flux lines should squeeze together to form a structure resembling the quantized dual string.

complicated force in nature. It may therefore be of help that an amusingly simple model exists which exhibits the most remarkable feature of such a theory: the infinite potential well. In the model there is only one space, and one time dimension. "liere is a local gauge group U(&'), of which the parameter N is so large that the perturbation expansion with respect to 1 /N is reasonable.

If all this is true, then the strong interactions will undoubtedly be by far the most

Our Lagrangian is, like in ref. [4] ,

( x ) = -A* ( x ) ; B I r r

The Lorentz indices p,v, can take the two values 0 and 1. It will be convenient to use

443

G. 't Hooft , Wodimensbnut model for mesons

light cone coordinates. For upper indices:

and for lower indices

1 A, =- (A1 f A o ) , etc., 4

where 1 0 PI = P 9 P o = - P .

Our summation convention will then be as follows,

x P p = , x cc P p = X , P , - - x+p+ + x - p - = x + p - + x - p + = x + p - + x - p + *

"he model becomes particularly simple if we imp0 tion:

A - = A + = O .

c+- = +A+ I

In that gauge we have

and

1: = -3Tr (a-A+)2 - q"(ya + M(") + m-A+) q" .

the light-cone gauge c

(4)

ndi-

(7) "here is no ghost in this gauge. If we take x+ as our t h e direction, then we notice that the field A + is not an independent dynamical variable because it has no time derivative in the Lagrangian. But it does provide for a (non-local) Coulomb force between the Fermions.

"he Feynman rules are given in fig. 1 (using the notation of ref. [4]). The algebra for the y matrices is

Y+Y- + Y-Y+ = 2 * (8 .b)

Since the only vertex in the model is proportional to y- and 72 = 0, only that part of the quark propagator that is proportional to y+ will contribute. As a consequence

444

G. 't Hooft, 7bodhenmbmi modei for mesons

Fig. 1. Planar Feynman ruler in the light cone gauge.

we can eliminate the y matrices from our Feynman rules (see the righthand side of rig. I).

We now consider the limit N+ 00; g2N fixed, which corresponds fo taking only the planar diagrams with no Fermion loops (4). They are of the type of fig. 2. AU gauge field lines must be between the Fermion lines and may not cross each other.

do not interact with themselves. We have nothing but ladder diagrams with self- energy insertions for the Fermions. Let us first concentrate on these selfcnergy parts. Let il'(k) stand for the sum of the irreducible selfcnergy parts (after having eliminated the y matrices). The dressed propagator is

They are so much simpler than the diagrams of ref. [4], because the gauge fields

-ik- (9) mZ i 2k+k- - k- r(k) - ic

0 b

P Fig. 2. Large diagram. a and b must have opposite U(N) charge, but need not be each other's antiparticle.

445

G. 't Hooft, I\uodimensional model for mesuns

AL=LaL Fig. 3. Equations for the planar selfenergy blob.

Since the gauge field lines must all be at one side of the Fermion line, we have a simple bootstrap equation (see fig. 3).

Observe that we can shift k, + p+ --* k, , so re) must be independent of p+, and

(1 1) Let us consider the last integral in (1 1). It is ultra-violet divergent, but as it is well known, this is only a consequence of our rather singular gauge condition, eq. (5 ) . Fortunately, the divergence is only logarithmic (we work in two dimensions), and a symmetric ultra-violet cut-off removes the infinity. But then the integral over k, is independent of r. It is

ni 2Ik-tp-1 '

so,

This integral is infra-red divergent. How should we make the infra-red cutsff ? One can think of putting the system in a large but finite box, or turning off the interactions at large distances, or simply drill a hole in momentum space around k = 0. We shall take X < lk_l< 00 as our integration region and postpone the limit h + 0 until it makes sense. We shall not try to justify this procedure here, except for the remark that our final result will be completely independent of A, so even if a more thorough discussion would necessitate a more complicated momentum cut-off, this would in general make no difference for our final result.

We find from (1 2), that

446

G. 't Hooft. Two-ciimensionaI model for mesons

and the dressed propagator is -ik-

Now because of the infra-red divergence, the pole of this propagator is shifted towards k, + = and we conclude that there is no physical single quark state. This will be confirmed by our study of the ladder diagrams, of which the spectrum has no continuum corresponding to a state with two free quarks.

"he ladder diagrams satisfy a Bethe-Salpeter equation, depicted in fig. '4. Let Jl(p, r) stand for an arbitrary blob out of which comes a quark with mass m1 and

Fig. 4. Eq. (15).

momentum p, and an antiquark with mass m2 and momentum r - p . Such a blob satisfies an inhomogeneous bootstrap equation. We are particularly interested in the homogeneous part of this equation, which governs the spectrum of two- particle states:

(15) t k, r) *

+2p+p- tg Ip-l - ie]-yJ'@,, dk+ dk- , - nX

where 4

writing

we have for cp

447

G. 't Hooft , 7bodimensionaI model for mewns

One integral has been separated. This was possible because the Coulomb force is instantaneous. The p + integral is only non-zero if the integration path is between the poles, that is,

sgn (P- - r - ) = -sgn (P-1 * and can easily be performed. Thus, if we take r- > 0, then

The integral in eq. (20) is again infra-red divergent. Using the same cut-off as before, we find

where the principal value integral is defined as

cp(k- t k) dk- cp(k- - k) dk- = 'f t 'J (22) p F k - k t ) dk - ' ( k - t ie)'

Substitutiiig (2 I ) into (20) we find

' (k-- ie)' '

and is always finite.

(2.3) l l i c infra-red cut-off dependence has disappeared ! In fact. we have here the exact forni of the Haniiltonian discussed i n ref. 141. Let us introduce diiiiensionless units:

"Mi,' ntu;., g? 7 i7.J. = .v ;. ('4) - a,.2 =- - - - I , -2r+r- = - p' ;

A g' R'

448

G. ‘t Hmft, livo-dimendoonrrl model for mesons

p is the mass of the two-particle state in units of g/+. Now we have the equation

We were unable to solve this equation analytically. But much can be said, in particular about the spectrum. First, one must settle the boundary condition. At the boundary x = 0 the solutions I&) may behave like x*@1, with

q c o t g n B 1 + a 1 = o , (26) but only in the Hilbert space of functions that vanish at the boundary the Hamil- tonian (the right-hand side of (25)) is Hennitcan:

In particular, the “eigenstate”

in the case a1 = a2, is not orthogonal to the ground state that does satisfy do) =

Also, from (27) it can be shown that the eigenstates @ with ~ ~ ( 0 ) = @(1) = 0 form a complete set. We conclude that this is the correct boundary condition *. A rough approximation for the eigenstates I p k is the following. Ihe integral in (25) gives its main contribution ify is close to x. For a periodic function we have

dl) = 0.

The boundary condition is q(0) = Cp(1) =O. So ifal ,a2 5- 0 then the eigenfunctions can be approximated by

@(x)=sin&nx, & = 1 , 2 ,...( (28)

fi@) 2 = 8% . with eigenvalues

(29)

This is a straight “Regge trajectory”, and there is no continuum in the spectrum! The approximation is valid for large k, so (29) will determine the asymptotic form

Footnote see next page.

449

G. 't Hmft , Tbodimenswnal model for mesvns

of the trajectories whereas deviations from the straight line are expected near the origin as a consequence of the finiteness of the region of integration, and the contribution of the mass terms.

Further, one can easily deduce from (27), that the system has only positive eigenvalues if a l . a2 > -1. For a1 = a2 = -1 there is one eigenstate with eigenvalue zero (9 = 1). Evidently, tachyonic bound states only emerge if one or more of the original quarks were tachyons (see eq. (24)). A zero mass bound state occurs if both quarks have mass zero.

The physical interpretation is clear. "he Coulomb force in a onedimensional world has the form

va Ix,-xzl, which gives rise to an insurmountable potential well. Single quarks have do finite dressed propagators because they cannot be produced. Only colourless states can escape the Coulomb potential and are therefore free of infra-red ambiguities. Our result is completely different from the exact solution of twodimensional massless quantum electrodynamics [S , 2 ] , which should correspond to N = 1 in our case. The perturbation expansion with respect to l /N is then evidently not a good approximation; in two dimensional massless Q.E.D. the spectrum consists of only one massive particle with the quantum numbers of the photon.

In order to check our ideas on the solutions of eq. (25), we devised a computer program that generates accurately the first 40 or so eigenvalues p2 . We used a set of trial functions of the type Axo1(l-x)2-@1 t E(~-xY~x*-P~ t Bg!Ck sin knx. The accuracy is typically of the order of 6 decimal places for the lowest eigenvalues, decreasing to 4 for the 40th eigenvalue, and less beyond the 40th.

A certain W.K.B. approximation that yields the formula 2 Pf,) - n2n t (ff 1 + a*) log n t CS'(al ,ff2), n = 0,1, ...

n+-

was confumed qualitatively (the constant in front of the logarithm could not be checked accurately).

In fig. S we show the mass spectra for mesons built from equal mass quarks. In the case mq = m; = 1 (or u1 = a2 = 0) the straight line is approached rapidly, and the constant in eq. (30) is likely to be exactly an2.

In fa. 6 we give some results for quarks with different masses. The mass difference for the nonets built from two triplets are shown in two cases:

(a) m , = O ; m2 = 0.200 ; m3 = 0.400,

(b) ml 10.80; m2 = 1 .OO ; m3 = 1.20,

in units of g / / n . The higher states seem to spread logarithmically, in accordance with

This will certainly not be the last word on the boundary condition. For a more thorough study we would have to consider the unitarity condition for the interactions proportional to 1/N. That is beyond the aim of this paper.

450

G. ’t Hooft, Wodimensmnal model for mesons

0 U’ In’ 3w’ , w 2 -p2

Fig. 5:“Regge trajectories” for mesons built from a quark-antiquark pair with equal mass, m, varying from 0 to 2.1 1 in units of g/Jw. ’IXe squared mass of the bound states is in unitsg2/n.

n = 4

n = 3

n . 2

n : 1

0 n’ - p’ - nrc’ Fig. 6. Meson nonets built from quark triplets. The picture is to be interpreted just as the previous figure, but in order to get a better display of the mass differences the members of one nonet have been separated vertically, and the nth excited state has been shifted to the left by an amount nn2. In case (a) the masses of the triplet are m l = 0.00;m2 - 0.20;m3= 0.40 and in case (b) ml = 0.80; m2 = 1.00;m3 = 1.20. Again the unit of mass isg/Jn.

45 1

G. 't Hmft, Twodimensional model for mesons

eq. (30). But, contrary to eq. (30), it is rather the average mass, than the average squared mass of the quarks that determines the mass of the lower bound states.

Comparing our model with the real world we find two basic flaws. First there are no transverse motions, and hence there exists nothing like angular momentum, nor particles such as photons. Secondly, at IV = 3 there exist also other colourless states: the baryons, built from three quarks or three antiquarks. In the 1 /N expansion, they do not turn up. To determine their spectra one must use different approximation methods and we expect those calculations to become very tedious and the results difficult to interpret. The unitarity problem for finite N will also be tricky.

Details on our numbers and computer calculations can be obtained from the author or G. Komen, presently at CERN.

"he author wishes to thank A. Neveu for discussions on the two dimensional gauge field model; N.G. van Kampen, M. Nauenberg and H. de Groot, for a contribution on the integral equation, and in particular G. Komen, who wrote the computer programs for the mass spectra.

References

[ l ] M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1006. (21 A. Casher, J . Kogut and L. Susskind, Tel Aviv preprint TAW-373-73 (June 1973). [ 3 ] P. Olesen, A dual quark prison, Copenhagen, preprint NBI-HE-74-1 (1974). [4] C. 't Hooft, CERN preprint TH 1786, Nucl. Phys. B, to be published. (51 J. Schwinger,Phys. Rev. 128 (1962) 2425.

452

Epilogue to the Two-Dimensional Model

Apparently we find quite realistic meson spectra. The quark masses are free parameters, and in the limit m, -t 0 the pions (and eta's!) are massless, which indicates that the chiral symmetry is spontaneously broken. This spontaneous symmetry breakdown may be surprising in view of the fact that there are theorems forbidding such a breakdown in two dimensions. The reuon why these theorems are violated is that the massless mesons do not interact at all if N = 00. One may expect that at finite N infrared divergences due to the maasless pion and eta propagators will eventually restore chiral symmetry nonperturbatively.

The paper gave rise to another controversy. The only gauge choice in which the solutions can be given in elegant and practically closed form is the light cone gauge. However in this gauge the zero modes seem to be treated incorrectly, a problem related to the rather dangerous light cone boundary conditions. All we say about this in the paper is that the pole in the propagator is cut off by a principal value integration prescription. That this is actually correct is not so easy to see. As usual, one has to ask what the physical significance of the danger is. To understand the infrared structure of our model we have to take periodic boundary conditions, and these are only consistent with our procedure if the periodic variable is z-. Now our "gauge condition" is A- = 0, but actually this implies also a constraint: the total magnetic field generated by the closed loop formed by our periodic one- space is constrained to be zero. This is the total magnetic flux caught by our loop. Physically, we expect in the infrared limit that the loop should be so large that this single constraint should become unimportant. The principal value prescription can then be seen to be related to an extra gauge condition for the zero modes. The point here is that besides A- = 0 we could fix the residual global gauge invariance by demanding A + ( k = 0) = 0. And finally, to judge whether the periodic boundary condition in the lightlike direction is legal, one has to check whether masslees modes that survive can move with the speed of light so that a,causal behavior might result. Well, we find that all observable objects, the physical meaons, come out to be massive (as long as rn, # 0), so there is no problem here. The h a l check comes when one notices that the light cone gauge in the left direction gives the same bound states as the light cone gauge in the right direction. Apparently our procedure was gauge independent.

An interesting problem was raised by D. Gross in a private discussion with the author.18 Since we start off with a model with conservation of parity, all bound states should either be parity even or parity odd. The even states should be coupled to the current && and the odd states to the current &76$~2. One expects the bound states to be alternatingly even and odd under parity. In terms of the solutions of the integral equation (25) one derives that the scalar current couples to a solution p(z) uio the integral /' 0 (s 2 - z ) p ( z ) d z , 1-2

"C. G . Callan, N. Coote, and D. J. Gross, Phys. Rev. D1S (1976) 1649.

453

and the pseudoscalar current via the integral

1 J 0 (22 2 + =)v(x)dx, 1-2

where ml and m2 are the masses of the quark and the antiquark inside the meaon. If the masses are equal it is evident that for the even solutions the first integral vanishes and for the odd ones the second integral. But if the masses are unequal this is far from evident. The reason why it is 80 difficult to see is that our light cone gauge condition is not parity invariant.

Yet it is true. To see this, consider the product of the two integrals, and prove that it is equal to

where the operator H is, up to a constant, the right-hand side of Q. (25), and P is defined by

r l 1

This commutator expectation value must vanish for the eigenstates of H, hence one of the two integrals must vanish. All states are either even or odd under parity. This result collfirm8 once again that our procedure is gauge independent.

454

CHAPTER 7

QUARK CONFINEMENT

Introductions ................................................................... 456

I7.11 “Confinement and topology in non-Abeliau gauge theories”, Acta Phys. Austr. Suppl. 22 (1980) 531-586 ......................... 458

[7.2] “The confinement phenomenon in quantum field theory”, 1981 Garghe Summer School Lecture Notes on findamental Intemctions, eds. M. IRvy and J.-L. Baedevant, NATO Adv. Study Inst. Series B: Phya., vol. 85, pp. 634-671 ............................ 514 “Can we make sense out of “Quantum Chromodynamics”?” in The Whys of Subnuclear Physics, ed. A. Zichichi, Plenum, New York, pp. 943-971 ....................................................... 547

[7.3]

455

CHAPTER 7

QUARK CONFINEMENT

Introduction to Confinement and Topology in Non-Abelian Gauge Theorieg [7.1] Introduction to the Confinement Phenomenon in Quantum Field Theory [7.2] Introduction to Can We Make Sense out of Quantum Chromodynamics? [7.3]

How can we understand the behavior of quarks in the theory of Quantum Chre modynamics? Is “ionization” of mesons and baryons into “free quarks” indeed absolutely forbidden? What is the mechanism for this absolute and permanent ver- dict of uconfinementl’? And &ally, how do we calculate the details of the properties of mesons and baryons once the QCD Lagrangian is given?

The answers to these questions came very gradually. Our understanding became more and more complete. Precise calculations are still very diflicult these days, but there seems to be no longer any fundamental obstacle. Confinement is a state of aggregation, comparable to the solid state, the liquid state, the gaseous state or the plasma state of ordinary materials. Other modes a system such as quantum chromodynamica could have condensed into, are various versions of Higgs phasea or the Coulomb phase. Only in the Coulomb phase physical massless vector particles would persist in the physical spectrum, but, as will be demonstrated, these particles would have little left of what was once their non-Abelian character; they could only survive disguised as more or less ordinary photons (an exception to this would be QCD with such a large number of fermions, that the sign of the /3 function is reversed).

I wrote a number of papers on the subject, and a couple of summer school lecture notes. Three summer school lecture notes are reproduced here; for some of the technical details I refer the reader to the original papers. There is 80 much material to be discussed that the three papers reproduced here hardly overlap; they

show different approaches to the confinement problem. In the first two papers the notion of “dual transformations”, or the transition from order parameters to disorder parameters and back, plays a very important role.

It is instructive to consider these transformations in a box with periodic boundary conditions at finite temperature T = l/P. This is done in the first paper, from Section VI onwards. Combining methods from field theory and statistical physics it is discovered there that powerful identities are obtained by rotating this box over 90’ in Euclidean space. One finds distinct possibilities for the system to behave at large distance: the Higgs mode, the Coulomb mode and the confinement mode. The first and the lest are each other’s dual.

The last paper addresses the question of calculability. Its results are modest, if not disappointing. Even if our theory is asymptotically free, replacement of perturbation expansion by some guaranteedly convergent resummation procedure seem to be impossible. Only very modest formal improvement on the most naive perturbative expansions could be obtained. The much more pragmatic approach of Monte Carlo simulations on lattices seems to give much more satisfactory results in practice, but formal proofs that these methods converge to unique spectral data and transition amplitudes when the lattice becomes infinitely dense have never been given. I have nothing to add to this except the conjecture that indeed the Monte Carlo calculations define the theory rigorously seem to be quite plausible.

A small correction must be made in this last paper. As was correctly pointed out to me by Miinster, the instanton singularities and the infrared renormalon singularities in QCD will not show up independently as suggested in Fig. 6, but they merge, due to a divergence of the integration over the instanton size parameters p.

457

Acta Physica Austriaca, Suppl. XXII, 531-586 (1980) 0 by Springer-Verlag 1980

CHAPTER 7.1

CONFINEMENT AND TOPOLOGY IN NON-ABELIAN + GAUGE THEORIES

G. 't HOOFT Institute for Theoretical Physics Princetonplein 5, P.O.Box 80006

3508 TA Utrecht, The Netherlands

ABSTRACT

Pure non-Abelian theories with gauge group SU(N) are considered in 3 and 4 space-time dimensions. In 3

dimensions non-perturbative features invalidate ordinary coupling constant expansions. Disorder operators can be defined in 3 and 4 dimensions. Confinement is first. explained in 3 dimensions as a spontaneous breakdown of a topologically defined 2 (N) global symmetry of the theory. In 4 dimensions confinement can be seen as one of the various possible phases of the system by considering a box with periodic boundary conditions in the "thermodynamic limit". An exact duality equation allows either electric or magnetic flux tubes to be stable, but not both.

Special attention is given to the explicit oc- + Lecture given at XIX.Internationale Universitstswochen ftir Kernphysik,Schladming,February 20 - 2 9 , 1980.

458

currence of instanton configurations with an instanton angle 0 in deriving duality.

I. INTRODUCTION

The first non-Abelian gauge theory that was recognized to describe interactions between elementary particles, to some extent, was the Weinberg-Salam-Ward- GIM model[ll.That model contains not only gauge fields but also a scalar field doublet H. Perturbation expansion was considered not about the point H = 0, but about the "vacuum value"

F H = ( o ) .

Such a theory is usually called a theory with "spontaneous symmetry breakdown" [23. In contrast one mkght consider "unbroken gauge theories" where perturbation expansion is only performed about a symmetric "vacuum". These theories are characterized by the absence of a mass term for the gauge vector bosons in the Lagrangian. The physical consequences of that are quite serious. The propagators now have their poles at k =o and it will often happen that in the diagrams new divergences arise because such poles tend to coincide. These are fundamental infrared divergencies that imply a blow-up of the interactions at large distance scales. Often they make it nearly impossible to understand what the stable particle states are.

2

A particular example of such a system is "Quantum Chromodynamics", an unbroken gauge theory with gauge group S U ( 3 ) , and in addition some fermions in the 3 -

representation of the group, called "quarks". We will

investigate the possibility that these quarks are permanently confined inside bound structures that do not carry gauge quantum numbers. First of all this idea is not as absurd as it may seem. The converse would be equally difficult to understand. Gauge quantum numbers are a priori only defined up to local gauge transformations. The existence of g l o b a l quantum numbers that would correspond to these local ones but would be detectakle experimentally from a distance is not at all a prerequisite. We are nevertheless accustomed to attaching a global significance to local gauge transformation properties because we are familiar with the theories with spontaneous breakdown. The electron and its neutrino, for example, are usually said to form a gauge doublet, to be subjected to local gauge transformations. But actually these words are not properly used. Even the words "spontaneous breakdown" are formally not correct for local gauge theories (which i.s why I put them between quotation marks). The vacuum never breaks local gauge invariance because it itself is gauge invariant. All states in the physical Hilbert space are gauge-invariant. This may be confusing so let me illustrate what I mean by considering the familiar Weinberg et a1 model. The invariant Lagrangian is

HereHlsthe scalar Higgs doublet. The gauge group is SU(2)xU(1), to which correspond ka(Ga ) and A ( F u v )

The subscripts L and R denote left and right handed components of a Dirac field, obtained by the projection operators ; ( I f y,) .

0

u uv lJ

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eR is a singlet;

JI, is a doublet.

D stands for covariant derivative. P

The function r l ( l HI takes its minimum at1 HI=F. Usually one takes

(1.2)

F + h, ) .

h2 and perturbs around that value: H = (

One identifies the components of$Lwith neutrino and electron:

vL 9, = (e 1 . L

However, this model is n o t fundamentally different from a model with "permanent confinement". One could interpret the same physical particles as being all gauge singlets, bound states of the fundamental fields with extremely

a strong confining forces, due to the gauge fields A,, of the group SU(2). We have scalar quarks (the Higgs field H ) and fermionic quarks (theJIL field) both as fundamental doublets. Let us call them q. Then there are "mesons" (qq) and "baryons" (qq). The neutrino is a "meson". Its field is the composite, SU(2l-invariant

H* qL = F vL + negligible higher order terms.

The eL field is a "baryon", created by the SU ( 2 ) - invariant

461

Hi I ) , ~ = FeL + ... I

the eR field remains an SU(2) singlet. Also bound states with angular momentum occur: The neutral intermediate vector boson is the "meson"

* i H D H = - ,F2A(3 ' + total derivative + higher P 2 - P

orders I (1.5)

if h7e split off the total derivative term (which corresponds to a spin-zero Higgs particle). The Wf are obtained from the "baryon"EijHiDPHj and the Higgs particle can also be obtained from H H. Apparently some mesonic and baryonic bound states survive perturbation expansion, most do not (only those containing a Higgs "quark" may survive).

*

Is there no fundamental difference then between a theory with spontaneous breakdown and a theory with confinement? Sometimes there is. In the above example the Higgs field was a faithful representation of SU(2). This is why the above procedure worked. But suppose that all scalar fields present were invariant under the center Z(N) of the gauge group SU(N), but some fermion fields were not. Then there are clearly two possibilites. The gauge symmetry is "broken" if physical objects exist that transform non-trivially under Z such as the fundamental fermions. We call this the Higgs phase. If on the other hand all physical objects are invariant under Z such as the mesons and the baryons, then we have permanent confinement.

N'

N'

Quantum Chromodynamics is such a theory where these distinct possibilities exist. It is unlikely that one will ever prove from first principles that permanent

462

c o n f i n e m e n t t a k e s place, s i m p l y b e c a u s e o n e c a n a l w a y s imag ine t h e Higgs mode t o o c c u r . I f no f u n d a m e n t a l

scalar f i e l d s e x i s t t h e n o n e c o u l d i n t r o d u c e c o m p o s i t e f i e l d s s u c h as

or

and p o s t u l a t e n o n v a n i s h i n g vacuum e x p e c t a t i o n v a l u e s

f o r them:

1 dab8 + F2 dab3 <Hab> = F

I n t h a t case t h e r e would be no c o n f i n e m e n t . Whether or n o t F are e q u a l t o z e r o w i l l c?.epend o n d e t a i l s of t h e dynamics . T h e r e f o r e , dynamic!; n u s t be a n i n q r e d i e n t

OF t h e c o n f i n e m e n t mechanism] n o t o n l y t o p o l o g i c a l . a r g u - men t s . What we w i l l a t t e m p t i n t h i s l e c t u r e i s t o show t h a t t o p o l o g i c a l a r g u m e n t s imply for t h i s t h e o r y t h e

e x i s t e n c e o f p h a s e r e g i o n s , s e p a r a t e d by s h a r p p h a s e t r a n s i t i o n b o u n d a r i e s ( u s u a l l y of f i r s t o rde r ) . One r e g i o n c o r r e s p o n d s t o wha t i s u s u a l l y c a l l e d " s p o n t a n e - o u s breakdown"] and w i l l be r e f e r r e d t o as Higqs p h a s e . Ano the r c o r r e s p o n d s t o a b s o l u t e q i a r k c o n f i n e m e n t . S t i l l a n o t h e r p h a s e e x i s t s which allows f o r l o n n r a n 9 e Coulomb- l ike f o r c e s t o o c c u r . (Coulomb phase.)

1 , 2

I t i s i l l u s t r a t i v e t o c o n s i d e r f j r s t i!ure qauge t h e o r i e s i n 3 s p a c e - t i m e d i m e n s i o n s . These c i r f e r i n two i m p o r t a n t ways f rom t h e i r 4 d i m e n s i o n a l c o u n t e r p a r t s . F i r s t , t h e y a re n o t s c a l e - i n v a r i a n t i n t h e c l a s s i ca l l i m i t . A consequence of t h a t i s t h a t t h e y 20

463

n o t h a v e a c o m p u t a b l e smal l c o u p l i n g c o n s t a n t ex - p a n s i o n . A l r e a d y a t f i n i t e o r d e r s i n t h e c o u p l i n g

c o n s t a n t g phenomena o c c u r t h a t are associated t o a

complex sort of vacuum instability. This is explained i n S e c t s 2 and 3 . S e c o n d l y , t h e t o p o l o g i c a l p r o p e r t i e s

a re d i f f e r e n t . I n t h r e e d i m e n s i o n s a " d i s o r d e r pa-

rameter", b e i n g i3n o p e r a t o r - v a l u e d f i e l d @ (x, t ) c a n

b e d e f i n e d . I f

-+

t h e n t h e r e i s a b s o l u t e c o n f i n e m e n t , as w e e x p l a i n i n S e c t s , 4 and 5 . The r e rns jn ing s e c t i o n s a rc d e v o t e d t o t h e four-dimen:; ior ia l c a ' 0 . They o v e r l a p t o some e x t e n t

l e c t u r e s q ive r . a t ~ a r g S s e [ 3 1 e x c e p t f o r a more e x p l i c i t

c o n s i d e r a t i o n of e f f e c t s d u e t o i n s t a n t o n s and t h e i r a n g l e 8 . Here it i s c o n v e n i e n t t o i n t r o d u c e t h e " p e r i -

odic box" (a c u b i c or r e c t a n g u l a r box w i t h p e r i o d i c or p s e u d o - p e r i o d i c boundary c o n d i t i o n s ) .Again w e have order and d i s o r d e r o p e r a t o r s b u t now t h e y are d e f i n e d n o t on space-time p o i n t s ( x , t ) b u t on l o o p s ( C , t ) . I n S e c t . 7

it i s e x p l a i n e d how t o i n t e r p r e t t h e s e o p e r a t o r s i n

terms o f m a g n e t i c and e l e c t r i c f l u x o p e r a t o r s , and m a g n e t i c f l u x i s d e f i n e d i n t e r m s of t h e boundary con-

d i t i o n s of t h e box . T h e r e are a l s o o t h e r c o n s e r v e d

q u a n t i t i e s ( S e c t . 8 ) , t o be i n t e r p r e t e d as e l ec t r i c f l u x a r d j - n s t a n t o n a n g l e .

I n S e c t . 9 we c o n s i d e r t h e " h o t box" , a box a t

f i n i t e t e m p e r a t u r e T = l / k B , and e x p r e s s t h e :free

e n e r g y F of s u c h a box i n t e r m s of L u n c t i o n a l i n t e g r a l s

w i t h t w i s t e d p e r i o d i c boundary c o n d i t i o n s i n 4 - d i -

m e n s i o n a l E u c l i d e a n s p a c e . I n S e c t . 1 0 w e n o t i c e a n

e x a c t d u a l i t y r e l a t i o n f o t t h e f r e e enercfy of e l e c t r i c

464

and magnetic fluxes. Section 11 shows that if electric confinement is assumed, magnetic confinement is excluded, and vice versa. The Coulomh phase realized for instance in the Georgi-Glashow model[4] is dually symmetric, as explained' in sect. 12.

1I.DIFFICULTIES IN THE PERTURBATION EXPANSION FOR QCD IN 2+1 DIMENSIONS

Consider pure gauge theory in 2+1 dimensions. The Lagrangian is

with

a abc b c G;" = all A: - a v A~ + gf A,,A,, . (2.2)

The functional integrals to be studied are

I D A exp(ijL d2x' dt + source terms). (2.3)

has to be dimensionless, 2

has dimension (mass)' and g has dimension Clearly ( L d x dt

a therefore A,, (mass) . f

Gauge-invariant quantities are ultra-violet convergent if they are regularized in a gauge invariant way (for instance by dimensional regularization). This is because the only possible counter terms would be

( 2 . 4 )

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Which would all be of too high dimension. Conventionally this would imply that all Green's functions in Euclidean space would be well defined perturbatively. The physical theory (in Minkowsky space) would then be obtained by analytic continuation.

problem, even in Euclidean space. Consider namely diagrams of the following type:

However, in our case we do have an infra-red

p-LP k fP

(2.5)

Simple power counting tells us that the small self- energy insertion is proportional to g Ikl , where k is the momentum circulating in the large blob. Because of the two propagators the k integration has an infrared divergent part:

2

/d3k -* (remainder). (k -iE)

(2.6)

Here the divergence is o n l y logarithmic, but it becomes worse if more self-energy insertions occur in the k- propagator. How should one cure such divergences?

vergence let us consider on easier case first: 90 theory without mass term:

To understand the physics of this infra-red di- 3

466

t h i s t i m e i n f o u r space- t ime dimensions: Power c o u n t i n g

t e l l s u s t h a t 4 h a s dimension (mass) 111 and g h a s d i -

r .ension (mass) [ I ] .This t h e o r y i s a l i t t l e b i t more a r t i - f i c a l because a n u l t r a v i o l e t d i v e r g e n c e h a s t o be sub- t r a c t e d by a mass-counter t e r m :

AL = g 2 $ 2 , (2.8)

b u t s t i l l w e c o u l d c o n s i d e r s t a r t i n g p e r t u r b a t i o n t h e o r y w i t h no r e s i d u a l mass. If w e t a k e t h e same diagram ( 2 . 5 )

t h e n now t h e s e l f - e n e r g y i n s e r t i o n behaves as

g2 log / k / I

and t h e k i n t e g r a t i o n i s a g a i n i n f r a - r e d d i v e r g e n t :

la4k log I k l ( r emainder ) . ( k 2 - i c ) 2

(2 .9 )

(2 .10 )

I n this case however t h e c u r e s e e m s obvious : we were

doing p e r t u r b a t i o n expans ion a t a v e r y s i n g u l a r p o i n t . N o problem ar ises i f w e f i r s t i n t r o d u c e a s m a l l m a s s t e r m

L + L - - ' v2 @2 , 2 (2.11)

and t h e n l e t P L S O i n t h e end. How are t h e i n f i n i t i e s

such a s ( 2 . 1 0 ) " r e g u l a r i z e d " i f w e do t h a t ? I f k2 i n ( 2 . 1 0 ) is r e p l a c e d by k2+U2 t h e n t h e 2 l i m i t -CO d o e s n o t e x i s t . However one may a r g u e t h a t

t h i s i n f i n i t y i s n o t v e r y p h y s i c a l . Suppose we sum diagrams o f t h e t y p e :

467

( 2 . 1 2 ) i t h e n t h e k i n t e g r a t i o n i s o f t h e form

( r e m a i n d e r ) . (2 .13) 1 +a,g 2 l o g l k l + p 2 - i s

2 C l e a r l y t h e l i m i t U $0 e x i s t h e r e . I t h a s a s m a l l imaginary p a r t due t o a t a c h y o n i c p o l e i n t h e d r e s s e d p r o p a g a t o r , which c a n n o t be a d m i t t e d i n a real t h e o r y , b u t t h i s is probably due t o t h e f a c t t h a t $ t h e o r y is u n s t a b l e and w e i g n o r e i t . The p o i n t i s t h a t t h e "summed" t h e o r y i s supposed t o be f r e e of i n f r a - r e d d i v e r g e n c e s . I n w r i t i n g down (2.13) we r e p l a c e d a s o m e t i m e s d i -

v e r g e n t sum of b u b l e s by a n a n a l y t i c and c o n v e r g e n t e x p r e s s i o n . T h i s e x p r e s s i o n c o u l d have been o b t a i n e d d i r e c t l y i f w e wrote t h e Dyson-Schwinger e q u a t i o n s f o r t h e s e diagrams :

3

( 2 . 1 4 )

I f eq . ( 2 . 1 4 ) i s used t h e n t h e diagram

..... ....... .. ....... .. ........ ......... ... ...... ....... ..... .:::::::::. a- (2 .15)

y i e l d s ( 2 . 1 3 ) . Now ( 2 , 1 4 ) i s o n l y a t r u n c a t e d Dyson- Schwinger e q u a t i o n . I f w e would u s e t h e complete set

468

of equations we would get the complete amplitude and this may be assumed to be free of divergences. (Also, in a good theory, free of tachyonic poles or cuts). A good theory should be a solution to its Dyson- Schwinger equations with dressed propagators not more singular than l / k . 2

Let us return to pure gauge theory in 2+1 dimensions. We can now roughly estimate how the integral ( 2 . 6 ) has to be cut-off, by replacing the bare propagators by dressed ones. This is not exactly according to Dyson-Schwinger equations but good enough for our purpose. The dressed propagator is

( 2 . 1 6 )

2 Let us assume that f(z) has an expansion l+alz + a2z but is non singular at z + m . (Thlsasymptotic expansion for f is not quite right, as as ws will see later). Our integral is

+...

( 2 . 1 7 )

where R ( k ) is the non-singular remainder. Let us expand R:

R(k) = Ro + R k + R k kL, + ..., ( 2 . 1 8 ) P P !Jv P

and split the integral in two pieces:

( 2 . 1 9 )

469

I

w i t h g2 < < E < < 1 . W e w r i t e ( 2 . 1 9 ) as

2 2 2 I ( g 1 = I , ( E I 9 ) + I2 ( E l 9 1 (2 .20 )

I t is e a s y now t o e s t a b l i s h how I behave for s m a l l E

and g2: I t 2

2 4

d3k 2 ( l + m + - a l g a 2 9 +. . . ) ( Ro+Rpkp+Rpvkpkv.. . ) 4

I k ( > E k PI2

... ) 2 a2 - 4 m ~ ~ ( & + g a , l o g E - - = 1 n E

( 2 . 2 1 ) 2 And i f w e w r i t e k -

I-r - then

= 1 Bn(g21n + 4mg 2 E ~ ~ ( 2 + a, log - E + . . . I 2 n 9 9

( 2 . 2 2 )

Tak ing t h e t w o series t o g e t h e r w e f i n d t h a t , of c o u r s e , t h e E dependence disappears I b u t t h e a s y m p t o t i c g2-ex- p a n s i o n does n o t o n l y c o n t a i n powers o f g2:

470

( 2 . 2 3 )

SO now w e found where t h e i n f r a - r e d i n f i n i t y of t h e i n t e g r a l ( 2 . 6 ) goes: I ( g ) h a s no proper T a y l o r expans ion i r . g . E q . (2 .23) shows terms wi th l o g g . One consequence of t h a t i s t h a t f ( g / i n ( 2 . 1 6 ) w i l l a l so d e v e l o p l o y a r i t h m s f u r t h e r down i n i t s expans ion series. I f w e c o r r e c t t h e p r e v i o u s a n a l y s i s f o r t h i s w e f i n d h i g h e r powers of l o g a r i t m s f u r t h e r down t h e series.

2

2 2 2

Ik I

S i n c e a and Ro a r e known, t h e c o e f f i c i e n t i n 1 f r o n t of t h e loqx-ithm i s w e l l d e t o r r d r e o . On t h e c t h e r hand, t h e c o e f f i c i e n t B1 can o n l y be de te rmined i f f ( z ) i s c o m i e t e l y known, because it must b e i n t e g r a t e d o v e r . I n g e n e r a l , a t any f i x e d power of CJ , o n l y t h e

l e a d i n g power of i o y g2 h a s a w e l l de te rmined c o e f - f i c i e n t . A l l o t h e r c o e f f i c i e n t s c a n o n l y be e v a l u a t e d i f t h e complete n o n - p e r t u r b a t i v e s o l u t i o n of f(z) i s

known. A t h i g h e r o r d e r s 3 1 S G t h r e e - a+ more-pcint Green f u n c t i o n s a r e requirea. Thus, t h e c o e f f i c i e n t s f o r t h e non- lesd ing powers of l o g g 2 c a n never k-e

de termined by p e r t u r b a t i v e m e a n s R lGne.

2

47 1

111. VACUUM STRUCTURE IN TEE PERTURBFTIVE 2+1

DIMENSIONAL THEORY. GAUGE INVARIANCE

Does t h e disease observed i n t h e p r e v i o u s s e c t i o n have a n y t h i n g t o do w i t h confinement? One might

a r g u e t h a t t h e d r e s s e d p r o p a g a t o r does n o t e x i s t i n a t h e o r y w i t h conf inement . However, s i n c e it i s n o t gauge- i n v a r i a n t , it might e x i s t i n c e r t a i n c o n v e n i e n t gauges , and w e do n o t have t o worry a b o u t s p u r i o u s p o l e s or c u t s i n t h i s p r o p a g a t o r because t h e y need n o t c o r r e s p o n d to p h y s i c a l states. I n fact t h e r e are r e a s o n s t o believe

t h a t i n c e r t a i n c o v a r i a n t gauges t h e p r o p a g a t o r i s a c t u a l l y v e r y smooth and convergent .

Confinement i s a p r o p e r t y of t h e vacuum i n t h e f i e l d t h e o r y and , as we w i l l a r g u e now, i t i s t h e vacuum t h a t d e t e r m i n e s o u r unknown c o e f f i c i e n t s . The f i r s t unknown c o e f f i c i e n t w a s B1 i n ( 2 . 2 2 ) . I t i s

Although unknown, i t s dependence on t h e remainder o f

t h e g r a p h , R , i s v e r y s imple : it o n l y depends on R ( k = o ) . T h i s i m p l i e s t h a t once it h a s been de termined

f o r one graph , i t w i l l be known f o r a l l o t h e r g r a p h s as w e l l . L e t u s t a k e t h e d iagram w i t h R = l :

472

It corresponds to the computation of

Of course it is ultra-violet divergent, but that divergence can be assumed to be subtracted in the usual way. We conclude that if we assume some value for

subtr. CO = <A 2 (X) > vacuum

then all amplitudes can be expanded up to

2 2 r(kl...lg ) = I'o(kt-.*) + g r l ( k , * * * )

+ 0(g6) .

(3.3)

(3.4)

0 In particular, r 2 depends on Co. At higher orders one will need

4 subtr . c* = <A (x)> vacuum '

subtr. vacuum c3 = cap am J (3.5)

etc. One may observe that A2 in (3.3) is not gauge-

invariant. Therefore, the actual value of CO should 0 not affect the r2 of gauge-invariant Green functions.

Indeed, gauge-invariant Green functions have Ro= 0, as a consequence of certain Ward-Takahashi identities.

473

But of course, the relevant quantities are

subtr. uv p" vacuum

a = < G G > 1

subtr. vacuum

bl = <$JI>

By power counting it is possible to tell where in the power series these coefficients center. a has dimension 3; b has di.mension 2 ; a2 has dimension 5. And g2 has dimension 1. Therefore, al will enter at order q8; at order g 6 a:id a2 at order cj

One may hope that iterative procedures exist to determine these coefficiznts semi-perturbatively: determine the known coefficients for f(i); extrapolate to find a decent function f o r z +. (Padti?), then substitute this f ! z ) tc! find the next coefficients and repeat. Whatever one does, the procedure will not be as straightforward as the determination of <€I>

from a Higgs Lagrangian. The unknowns this time form an infinite series. They are the vacuum expectation values of all composite operators.

1 1

bl 12 .

It may seem that these vacuum experation values are only needed in the 2+1 dimensional theory, and that the problems discussed here are actually irrelevant for the real world which is in 3+1 dimensions. This, we emphasize, is not true. In 3+1 dimensions the same problem occurs, however not within the ordinary perturbation expansion. This perturbation expansion namely diverges at high orders. If one tries to rearrange the perturbation expansion to obtain

474

convergent e x p r e s s i o n s , one f i n d s t h a t a m p l i t u d e s c a n b e c o n v e n i e n t l y expresaeci by a B o r e 1 formula:

m 2 I’(g2) = B ( z ) ,“Ig dz/g2 .

0

According t o t h e r e n o r m a l i z a t i o n group B(z) h a s dimension p r o p o r t i o n a l t o z . I f t h i s dimension c o i n c i d e s w i t h t h a t o f <G ~v G D v > , etc. s i n g u l a r i t i e s ar ise and t h e i r t r e a t m e n t g o e s a l o n u t h e same l i n e s a s i n t h e p r e v i o u s s e c t i o n . We r e f e r t o r e f s . [5 ,61for f u r t h e r d e t a i l s .

IV. 2+1 DIMENSIONS NON PERTURBATIVELY:

TOPOLOGICAL OPERATORS AND THEIR GREEN FUNCTIONS

I n t h i s s e c t i o n a new f i e l d i s i n t r o d u c e d t h a t

w i l l e n a b l e u s t o g e t a b e t t e r u n d e r s t a n d i n g of t h e p o s s i b l e vacuum s t r u c t u r e s i n 2 i 1 dimensions. To set up our arguments s t e p ty s t e p we begin w i t h adding a Higgs f i e l d [ T ] . T h e gauge group i s SU(N) and t h e gauge symmetry i s snontancoualy and comple te ly broken. The Higgs f i e l d s , H , must be a set of un ique r e p r e s e n - t a t i o n s of S U ( N ) / Z ( N ) such a s t h e o c t e t and d e c u p l e t r e p r e s e n t a t i o n s of SU(3) / Z ( 3 ) . Thus a l l ( v e c t o r and scalar) f i e l d s are i n v a r i a n t under t h e c e n t e r Z ( N )

o f t h e gauge group SU(N) . ( T h i s is t h e subgroup of matrices e ” in” 1, where n i s i n t e q e r . ) Quarks , which are n o t i n v a r i a n t under Z (14) , are n o t y e t i n t r o d u c e d a t t h i s s t a g e .

B e s i d e s t h e mass ive photons and Higgs p a r t i c l e ( s ) t h i s model c o n t a i n s one o t h e r c lass of p a r t i c l e s :

475

ex tended s o l i t o n s o l u t i o n s t h a t are s t a b l e kecause of

a t o p o l o g i c a l c o n s e r v a t i o n law . C o n s i d e r , t h e r e f o r e , a r e g i o n R i n two-dimensional s p a c e su r rounded by

a n o t h e r r e q i o n B where t h e e n e r g y d - . n s i t y i s z e r o

+

(vacuum). I n B , t h e Hicjgs f i e l d H ( x ) s a t i s f i e s

where F i s a f i x e d number. l h e r e must be a qauge r o t a t i o n R ( 2 ) so t h a t

where Ho i s f i x e d . n(2) i s de te rmined u p t o e l e m e n t s of Z ( N ) . We a l s o r e q u i r e a b s e n c e of s i n g u l a r i t i e s so

R (2) i s c o n t i n u o u s . Cons ide r a c l o s e d c o n t o u r C ( 8 ) i n B p a r a m e t r i z e d by a n a n g l e 8 w i t . h 0 2 8

C ( 0 ) = C ( 2 n ) . Cons ide r t h e case t h a t C goes c l o c k w i s e around R. S i n c e 3 i s n o t s imply connec ted w e may have

2Tr and

wi th 0 5 n ( N t n i n t e g e r . Because o f c o n t i n u i t y , n i s

+ For a g e n e r a l i n t r o d u c t i o n t o s o l i t o n s see e . g . Coleman [81. For a n i n t r o d u c t i o n t o t h i s p r o c e d u r e see ' t Hooft [ 9 1 .

476

conserved. I f n f 0 and i f ke r e q u i r e absence of s ingu- larities i n R t hen t h e f i e l d c o n f i g u r a t i o n i n R cannot be t h a t of t h e vacuum o r a gauge r o t a t i o n t h e r e o f , so t h e r e must he som f i n i t e amount of energy i n R. The

f i e l d c o n f i g u r a t i o n w i t h lowest energy E , i n t h e case n = 1, describes a s t ab le s o l i t o n wi th mass M=E. I f

N > 2 t hen s o l i t c n s d i f f e r from a n t i s o l i t o n s (oh ich correspond t o n=N-1) and t h e nunber of s o l i t o n s minus a n t i s o l i t o n s is conserved modulo N.

An a l t e r n a t i v e way to r e p r e s e n t t h e f i e l d s cor responding t o a s o l i t o n c o n f i g u r a t i o n is t o ex tend n(x') t o be also w i t h i n R (which, however, may be p o s s i b l e on ly i f o-e admit a s i n g u l a r i t y f o r n a t some p o i n t x' i n €?) . iJe t hen apply n t o t h e above f i e l d c o n f i g u r a t i o n . T h i s has t h e advantage t h a t everyohere i n B w e keep H=Hol regardless of t h e number n of s o l i t o n s i n R I b u t t h e p r i c e of t h a t is t o allow for a s ingu la r i i -y zo i n R. W e w i l l r e f e r t o t h i s a s t h e "second r e p r e s e n t a t i o n " of t h e s o l i t o n , f o r l a te r use .

fo l lows . L e t lAi(Z),H(Z)> be a s t a t e i n H i l b e r t space which is an e i g e n s t a t e of t h e space components of t h e v e c t o r f i e l d s and the Higgs f ields, w i t h Ai(3) and H(2) as g iven e igenvolues . Then

-1 0

A set of operators @(;) is now d e f i n e d hs

+ [xol where Q gauge r o t a t i o n w i t h t h e p rope r ty t h a t

f o r eve ry closed cu rve C ( e )

w e have t h a t enc loses %o once

( 4 . 5 )

where t h e minus s i g n ho lds f o r c lockwise and t h e + s i g n

477

+ for anticlockwise C. When xo is outside C, then

+ [X,l + + The singularity of R at x=x must be smeared over

an infinitesimal region around xo but we will not consider this "renormalization" problem in this paper. In what sense is #(x) a l o c a l operator? The operator formalism in qauge theories is most conveniently formulated in the gauge Ao(x,t)=O. Then, timc-independent continuous gauge rotations D(") still form an invariance group. For physical states[$> however,

+O

+

+

( 4 . 7 )

where R is any single-valued gauge rotatiqn. So #(;)

would have been trivial were it not that R[xo3has a singularity at xo.

physical states, and the details of Q apart from ( 4 . 5 )

and ( 4 . 6 ) are irrelevant. It is now easy to verify that

+

The operator a(;) loads physical states into

@(a@($) I $ > = O ( $ ) # ( Z ) I $ > I

if I $ > is physical state, because koth the left- and the right-hand sides of ( 4 . 9 ) are completely defined by the singularities alone of the combined gauge rotations. Also, when R(x) is a conventional gauge- invariant field operator composed of fields at 2 then obvious1 y

+

478

but n o t n e c e s s a r i l y [ R ( g ) , @ ( $ ) l = 0 f o r x' # 9 , f o r t = 3 ( 4 . 9 )

Because of ( 4 . 8 ) and ( 4 . 9 ) @ i s c o n s i d e r e d t o b e a local f i e l d o p e r a t o r when it 8cts on p h y s i c a l s t a t e s .

a b s o r b s one t o p o l o q i c a l u n i t , so w e say t h a t @(;I i s t h e a n n i h i l a t i o n ( c r e a t i o n ) o p e r a t o r f o r one "bare" s o l i t o n ( a n t i s o l i t o n ) a t x and @ ( x ) i s t h e c r e a t i o n ( a n n i h i l a t i o n ) o p e r a t o r f o r one " b a r e " s o l i t o n ( a n t i s o l i t o n ) .

From i t s d e f i n i t i o n it must be c lear t h a t @(;I

-b +

L e t u s now i l l u s t r a t e how one can compute Green f u n c t i o n s i n v o l v i n g #(x) by o r d i n a r y s a d d l e - p o i n t t e c h n i q u e s i n a f u n c t i o n a l i n t e g r a l . Let u s c o n s i d e r < T ( @ ( O , t l )

sponding f u n c t i o n a l - i n t e g r a l e x p r e s s i o n :

-I- @ (o,O))>= f ( t l ) by computing t h e c o r r e -

~ D A D H e x p S ( A , H I

IDA-DH e x p s (A,H) ' C a t l ) = ( 4 . 1 0 )

where C i s t h e set of f i e l d c o n f i g u r a t i o n s where t h e f i e l d s make a sudden gauge jw.p a t t = O d e s c r i b e d by

back by a t r a n s f o r m a t i o n Q[ol-C f i e l d s r.iust he

c o n t i n u o u s everywhere else.

Q[ol (see ( 4 . 5 ) and ( 4 . 6 ) ) and a t t=tl t h e y jump

This w a s how f ( t ) f o l l o w s froin t h e d e f i n i t i o n s 1 b u t it i s more e l e g a n t t o t r a n s f o r m a l i t t l e f u r t h e r . By gauge t r a n s f o r m i n g back i n t h e r e g i o n 0 < t < t w e g e t t h a t t h e f i e l d s are c o n t i n u o u s e v e r y w h e r e e x c e p t a t x=O,O < t < tl . So t h e r e i s a Dirac s t r i n g [ lo1 g o i n g from (0,o) t o ( 0 ,t). A t 0 < t < tl we o b t a i n i n t h i s way t h e s o l i t o n i n i t s "second r e p r e s e n t a t i o n " : a non-

1

-b

479

trivial field configuration with a singalarity at the origin. In short:<@(x)@ (O)> is obtained ky integrating over field configurations with a Dirac string in space- time frcn 0 t o X.

+

Let us compute f (t,) for Euclidean time tl = iT, T real. Let the field theory be pure SU(N) without Higgs Scalars. We must find the 1eastnecJative action configuration with the given Dirac strinq. The conditions ( 4 . 5 1 , ( 4 . 6 ) can be realized for an Abelian subqroup of gauge rotations a , so let us take

the Last diagonal element being different from the others because we mist. have dctR = 1 for all 8 . Here 8 Ls t h e angle around the time axis (remernber space-time is here thme diaer.slona1) . The singularity+ at the Dirac string must be t h e one obtained when $2

acts on the vacuum.

CO 1

The transformations ( 4 . 1 1 ) form an Abelian subgroup, so, as an ansatz, the field configuration with this string singularity may be chosen to be an Abelian subset of fields corresponding to this subgroup:

7 l a Xij A,,(x) a = a,,(x) XiJ(N) ;

(4.12)

(4.13) F - - au av - a v ap IJV

Here A; to SU(N). Within t h i s set of f i e l d s w e j u s t have t h e l i n e a r Maxwell equation..;, and t h e Dirac s t r i n g i s t h e one corresponding t o t w o o p p o s i t e l y charged Dirac monopoles, one a t 0 and one s t tl. The i r maqnetic cha rges are +2n/gN.

are the convent ional Gell-Mann matrices extended

The t o t a l a.ction of this c o n f i g u r a t i o n i s

S = 3 1 N(N-l)lFpv F d 3 x = P V

( 4 . 1 4 )

where 2, and Z 2 are t h e se l f - ene rg ie s of t h e monopoles, which d ive rge but can be s u b t r a c t e d , l e a v i n g a spnce- t i m e independent r enorma l i za t ion cons t an t .

&

W e now assume t h a t (4.14) is indeed an a b s o l u t e extremum f o r t h e to ta l a c t i o n of a l l f i e l d c o n f i g u r a t i o n s with t h e g iven s t r i n g s i n g u l a r i t y . We t h i n k t h a t t h i s

assumption is p l a u s i b l e but p r e s e n t no p roof . We t hus o b t a i n a f i r s t approsirtiation t o (4.10) :

tl = i - c , T real . (4.15)

Here A is a f i x e d c o n s t a n t ob ta ined a f t e r s u b t r a c t i o n of t h e i n f i n i t e s e l f - a c t i o n a t t h e sou rces . Note t h e

481

p l u s s i g n i n our exponent due t o t h e a t t rac t ive f o r c e

between t h e monopole-antimonopole p a i r . Computation of

t h e terms of h i g h e r o r d e r i n g T s u f f e r s from t h e

o b s t a c l e s mentioned i n sect. 2 . I n any case, a t l a r g e

T we expec t no convergence. O f c o u r s e , when t h e Higgs

mechanism is t u r n e d on t h e n t h e s o l i t o n a c q u i r e s a

f i n i t e mass, say M , and t h e n a t l a r g e T we e x p e c t

2

t, = i T ,

T = rea l . ( 4 . 1 6 )

j u s t as any o r d i n a r y ( d r u s s e d ) p r o p a g a t o r , However, t h e r e are il.lso ir , ternc: t ims, i n p a r t i c u -

l a r t h e N-so l i ton processCs , d e s c r i b e d e s s e n t i a l l y by

( 4 . 1 7 )

where N i s t h e group parameter . Again w e choose {xk} t o be Eucl idean . The re i s some freedom i n choos ing t h e

D i r a c s t r i n g s , fo r i n s t a n c e w e c a n l e t one s t r i n g l e a v e a t each p o i n t X ~ , . . . , X ~ - ~ and l e t t h e s e a l l assemble a t xN. Because of t h e modulo N c o n s e r v a t i o n l.zv~, :he N-1

q u a n t a coming i n a t x arc e q u i v a l e n t t o one quantum leavi r lg a t x N '

T o f i n d t h e f i e l d conf igura t j -on w i t h l eas t n e g a t i v e a c t i o n we c o u l d t r y a g a i n t.he a n s a t z ( 4 . 1 2 )

b u t t h e n t h e r e s u l t i s t h a t X ~ , . . . , X ~ - ~ r e p e l each

o t h e r and a l l a t t r a c t xfl , an unsymmctric and t h e r e f o r e u n l i k e l y r e s u l t . Ohviously, . s n y f i e l d conf i q u r a t i o n where t h e sicjns i n t h e exponents such as (11.15) are p o s i t i v e , c o r r e s p o n d i n g t o a t t r a c t i o n , w i l l Tive much

larger, therefore dominating, contributions to the amplitude. So let us try to produce such a field configuration now.

Observe that pure permutations of the N spirm components are good elements of S U ( N ) , so by pure gauge rotations we are allowed to move the unequal diagonal element of (4.11) up or down the diagonal. So let us now again choose one Dirac strring leaving at each of

and all entering at x but this time all N- 1 N’ xl,. . . ,x these strings have their unequal diagonal element (see (4.11)) in a different position along the diagonal At xN the combined rotation is again of type (4.11) as one can easily verify, so this is a more symrietric configuration. Since the qzuge transformations we performed so far are ali diagonal elementc of SU(N) they actually form the subgroup[U ( 1 ) IN-’ of SU (N) which is Abelian still. Let us define

k = l,...N . (4.18)

with N

= o . k= 1 ’(k)

so one of thece X matrjces is actu&lly redurdant. Our present ansatz is

(4.19)

without bothering about the invariance

483

( 4 . 2 0 )

( 4 . 2 1 )

w i t h

( 4 . 2 2 )

Notice t h e chanrje of s i g n i r , (4.21) when f i e l d s of

d i f f e r e r l t i n d e s o v e r l a p . We now ch(:ose i1 I?i:-ac monopole c o r r e s p o n d i n g t o t h e k t h subgroup U ( 1 ) a t x is j u s t t h e f i e l d of one monopole a t x (remember t h a t ,

because we s t a y i n t h e Abel ian s u b c l a s s of f i e l d s , l i n e a r

s u p e r p o s i t i o n s a r e a l l o w e d ) . The d i a g o n a l terms i n

( 4 . 2 1 ) t l len o n l y c o n t r i b u t e t o t h e monopole “ s e l f -energy” (more p r e c i s e l y : s e l f - a c t i o n ) and o n l y c o n t r i b u t e t o t h e

a l r e a d y e s t a b l i s h e d Z f a c t o r t h a t must be s u b t r a c t e d .

Only t h e cross t e r m s g i v e n o n - t r i v i a l e f f e c t s :

SO A ( k ) (x) k ,

k

Again w e assume t h a t t h i s r e p r e s e n t s t h e least n e g a t i v e a c t i o n c o n f i g u r a t i o n w i t h o u t p r e s e n t i n g t h e c o m p l e t e p r o o f .

484

A good check o f t h i s equa t ion i s t h a t if N-1 p o i n t s come close and t h e Nth s t a y s f a r away then w e recover ( 4 . 1 4 ) a p a r t from o v e r a l l normal iza t ion , e x a c t l y as one should expec t . So w e conclude:

f (x, , . . . ,x ) = ~ ’ e x p [ - “ 1 ’ + O ( 1 0 c J ( j 2 T ) ) I . N 2g2N k>P. Ixk-xBr

( 4 . 2 4 )

f o r Eucl idean {X,).

The above was maLn1.y to i l l u s t r a t e how t o compute i n lowest-order p e r t u r b a t i o n expansion t h e Green f u n c t i o n s t h a t w e w i l l d i s c u s s i u r t h e r i n s e c t . 5 . Wedid n o t prove t h a t t h e f i e l d conf i a u r a t i o n s w e choose t o expand about ars r e a l l y t h e minima:. ancs (which w e do expec t ) but t h e r e i s no need t o e l a b o r a t e on t h a t p o i n t f u r t h e r here .

V. SPONTPNEOUS SYMMETRY BREAKING AND CONFINEMENT

I n sect .4 w e found t h a t some SU(N) gauge t h e o r i e s i n 2+1 dimensions p o s s e s s a t o p o l o g i c a l quantum number, conserved modulo N , and t h a t Green f u n c t i o n s c o r r e - sponding t o exchange o f t h i s quantum may be computed. Our f i e l d behaves as a l o c a l complex scalar f i e l d ( rea l f o r S u ( 2 ) i n a l l r e s p e c t s . Me expec t t h a t t h e s e Green f u n c t i o n s s a t i s f y a l l t h e usu3al Wightman axioms [ 11 1 except t h a t they are more s i n g u l a r t han u s u a l at small d i s t a n c e s . I n f a c t , t h e Green f u n c t i o n s as c o m - puted i n sect.4 can be e x a c t l y reproduced i n a theo ry with free scalar par t ic les and non-polynomial sou rces

485

u s i n g s u p e r p r o p a g a t o r t e c h n i q u e s [ 1?1 . We l e a v e t h a t as a n e x e r c i s e €or t h e r e a d e r . I n p u r e S U ( N 1 t h e t r o u b l e i s t h a t o n l y t h e quantum c o r r e c t i o n s g i v e s o m e i n t e r e s t - i n g s t r u c t u r e t o t h e s e Green f u n c t i o n s . The terms o f h i g h e r o r d e r s i n c j2 i n ( 4 . 1 5 ) and ( 4 . 2 4 ) c o r r e s p o n d t o quantum l o o p c o r r e c ! t ? o n s . They have no t y e t been c o m -

p u t e d , t o t h e e x t e n t t h a t t h e y c a n b e c o m - p u t e d . However, l e t u s assume t h a t t h e r e s u l t i n g Green f u n c t i o n s c a n be computed and are rough ly g e n e r a t e d by some e f f e c t i v e LacJrimyian w i t h n e c e s s a r i l y s t o n g coup- l i n g , f o r i n s t a n c e :

- - X I (aN + ( @ * I N ) - X2V(@*@) . N!

Here w e have assumed t h a t t h e Higgs mechanism ( e i t h e r by some e x p l i c i t o r Sy some dynamical Higgs f i e l d ) makes a l l f i e l d s mass ive , g e n e r a t i n g a l s o a large

s o l i t o n m a s s M. O f c o u r s e we expect many possible

i n t e r a c t i o n terms b u t o n l y t h e m o s t i m p o r t a n t o n e s are added i n ( 5 . 1 ) : t h e h t e r m p roduces t h e 11-

s o l i t o n i n t e r a c t i o n and i s r e s p o n s i b l e f o r t h e non- v a n i s h i n g r e s u l t (4 .24) f o r t h e e x p r e s s i o n ( 4 . 1 7 ) , t h e N-point f u n c t i o n . The X 2 t e r m i s o f c o u r s e a l s o e x p e c t e d and i s i n c l u d e d i n ( 5 . 1 ) f o r r e a s o n s t h a t w i l l become c lear l a t e r .

1

Observe now a Z ( E J ) g l o b a l s y m e t r y t h a t l e a v e s t h e Lagrang ian ( 5 . 1 ) , and t h e Green f u n c t i o n s con- s i d e r e d i n sect.4 i n v a r i a n t :

486

T h i s is simply t h e symmetry a s s o c i a t e d w i t h t h e t o p o l o g i c a l l y conserved s o l i t o n quantum number. Modulo N conse rva t ion laws correspond t o Z ( N ) g l o b a l symmetries.

As w e s a w , t h e mass t e r m i s e s s e n t i a l l y due t o t h e Higgs mechanism. Roughly, V2 a U: where l-IH is t h e second d e r i v a t i v e of t h e Higgs p o t e n t i a l a t t h s o r i g i n . W e can now cons ide r e i ther swi tch ing o f € t h e Higgs f i e l d , or j u s t changing t h e s i g n of p i so t h a t < H > + c. What happens t o M2 ( t h e s o l i t o n mass)? I f it s t a y s p o s i t i v e then t h e p h y s i c a l s o l i t o n has n o t gone away and t h e t o p o l o g i c a l conse rva t ion l a w r e n a i n s v a l i d . The symmetry of t h e theo ry is as i n t h e Higgs mode and we d e f i n e this node t o be a "dynamical Higgs mode". However, a very good p o s s i b i l i t y is t h a t M2 also swi tches its s i g n , so t h a t <#>+F#O.The t o p o i o g i c a l g l o b a l Z ( N ) symmetry may g e t spontaneously broken. T h i s mode can be recognized d i r e c t l y from t h e Green f u n c t i o n s of sect.4. The c r i t e r i o n is

2

I F I * = , l i m r t t , ) , t, = iT . (5.3) TI-...

J u s t f o r amusement we might n o t e t h a t ( 4 . 1 5 ) indeed seems t o g i v e F f 0 b u t of course t h e auantum c o r r e c t i o n s may n o t be neg lec t ed and we do n o t knc,!. a t p r e s e n t how t o a c t u a l l y lcompute t h e l i m i t (5.3).

Spontaneous breakdown of t h e t o p o l o g i c a l Z ( H )

symmetry i s a new phase t h e system may choose, depending on t h e dynamics. W e may compare a bunch of molecules t h a t chooses t o be i n a gaseous, l i q u i d or s o l i d phase depending on t h e dynamics and on t h e v a l u e s of c e r t a i n i n t e n s i v e parameters .

vacuum w i l l now have a Z (PI) deqeneracy, t h a t i s , an L e t u s s tudy this new phase more c l o s e l y . The

487

N-fold degeneracy. Labeling these vacua by an index from 1 to N we have

( 5 . 4 )

Since the symmetry is discrete there are no Goldstone particles; all physical particles have some finite mass. Again we are able to construct a set of topologically stable objects: the Bloch walls that separate two different vacua. These vortex-like structures are stable because the vacua that surround them are stable (Bloch walls are vortex-like because our model is in 2 + 1 dimensions). The width of tne Bloch wall or vortex is roughly proportional to the inverse of the lowest mass of all physical particles, and therefore finite. The Bloch wall carries a definite amount of energy per unit of length.

We will now show the relevance of the operator

+ k A(C) = Tr P exp 4 ig Ak(x) dx , C (5.5)

for these Bloch walls. Here C is an arbitrary oriented contour in 2-dim. space and P stands for the path ordering of the integral. $j components of the gauge vector field in the matrix notation. A(C) is a non-local gauge-invariant operator that does not commute with @ . Let us explain that. As is well known, if C' is an open contour, then the opera tor

are the space

+ k x2 + + A(C',x,xZ) = P exp 1 ig Ak(j;)dx ,

+ x;c'

transforms under a gauge rotation fl as

Now the operator @(go) was defined by a 3auge transformation Si that is multivalued when followed over a contour that encloses x . So when we close C ' to obtain C, and C encloses

+ O+ xo once,

+ + P(C) = Tr A(C,x, ,xl 1 I (5.7)

then the value of A makes a jump b y a factor ex? (f21~i/N) wnen the operator @(xJ acts, so +

P(C)Q(Zo) = @(~o)P(C)exp(2~in/N) , (5 .a ) where n counts the number of times that C winds around x in a clockwise fashion minus the number of tines -+

+ 0 it winds around xo anticlockwise. E q . (5.8) is

an extension of eq.(4.9) for non-local o?erators A(C). As we will see, (5.8) can be generalized to 3+1 dimensions. Now let us inter?ret (5.8) in a framework where a(;) is diagonalized. Then, as we see, A(C) is an operator that causes a jump by a factor exp(2nin/N) of 4(;)for all 2 inside C. So A(C) causes a switch from one vacuum to another vacuum within C in the case that Z ( N ) is spontaneously broken. In other words A ( C ) creates a "bare" Bloch wall or vortex exactly at the curve C.

Our model does not yet include quarks. Quarks are not invariant under the center Z(N), so they do not admit a direct definition of is to be expected when one considers the physics of the system. The vortices that were locally stable

a(;). This difficulty

489

w i t h o u t q u a r k s may now become l o c a l l y u n s t a b l e due t o v i r t u a l quark-ant iquark pa i r c r e a t i o n . Most a u t h o r s

t h e r e f o r e c o n s i d e r quark confinement t o b e a b a s i c p r o p e r t y of t h e g l u e s u r r o u n d i n g t h e q u a r k s , i n wich q u a r k s must be i n s e r t e d p e r t u r b a t i v e l y . Such a procedure is j u s t i f i e d by t h e e x p e r i m e n t a l e v i d e n c e ; a l l hadrons can b e l a b e l e d a c c o r d i n g t o t h e number and t y p e s of q u a r k s t h e y c o n t a i n ; none of them is sa id

t o b e composed of an u n s p e c i f i a b l e or i n f i n i t e number of q u a r k s . The number of g l u o n s on t h e o t h e r hand can n o t e a s i l y be g i v e n . W e s h o u l d n ’ t s a y it is z e r o f o r most h a d r o n s , because w e need t h e v e r y s o f t g l u o n s t o p r o v i d e t h e b i n d i n g f o r c e .

How t o i n t r o d u c e q u a r k s a t t h e p e r t u r b a t i v e l e v e l is f u r t h e r e x p l a i n e d i n r e f . l131.The outcome is t h a t quarks are t h e end p o i n t of a v o r t e x . The c o n v e n t i o n a l o p e r a t o r

-b

k 2 X

[P e x p ( j i g Pk(G)dx I J I ( X , ) I (5.9) +

creates n o t o n l y a quark p a i r b u t a l so a v o r t e x i n between them. T h i s v o r t e x is t o p o l o g i c a l l y s t a b l e i f < a > = F # 0. I f w e have a c o n f i g u r a t i o n w i t h N

q u a r k s t h e n @ makes a f u l l r o t a t i o n over 271 when it f o l l o w s a c l o s e d c o n t o u r around. T h i s is why a “baryon” c o n s i s t i n g of N quarks i s n o t c o n f i n e d t o a n y t h i n g else. E v i d e n t l y , f o r r e a l baryons N must be 3.

Our c o n c l u s i o n is as follows. I n SU(N) gauge t h e o r i e s where a l l scalar f i e l d s are i n r e p r e s e n t a t i o n s t h a t are i n v a r i a n t under t h e c e n t e r Z ( N ) of S U ( N )

( such as oc te t or d e c u p l e t r e p r e s e n t a t i o n s of S U ( 3 ) ) , t h e r e e x i s t s a n o n - t r i v i a l t o p o l o g i c a l g l o b a l

Z ( N ) invariance. If the Higgs mechanism breaks S U ( N )

completely then the vacuum is Z ( N ) invariant. However, we can also have spontaneous breakdown of Z ( N ) symmetry. If that breakdown is comnlete thenwe can have no Higgs mechanism for S U ( N 1 , because in that mode "colored" objects are germanently and completely confined by the infinitely rising linear gotentials due to the Bloch-wall-vortices. 5Je can also envisage the intermediate modes where a Higgs mechanism breaks S U ( N ) partly, and Z ( N 1 is partly broken. Finally, if neither Higgs' effect, nor spontaneous breakdown of Z ( N ) take place, then there must be massless particles causing complicated long range interactions as we will show more explicitly for the 3+1 dimensional case. That may either corrPsDond to a point where a higher order phase transition occurs, or to a new phase, e.q. tne Coulomb or Georgi-Glashow phase, where an effective Abelian photon field survives at lorgdiatances, see sect. 12.

Eqs. ( 4 . 8 ) and 5 . 8 ) are the basic commutation relations satisfied by our topological fields @.

They suggest a dual relationship between A and @.

Indeed, one could start with a scalar theory exhibiting global Z ( N ) invariance and then define the topological operator A(C) through eq. ( 5 . 8 1 , but it is impossible to see this way that A can be written as the ordered exponent of an integral of a vector potential, and also the gauge grou? S U ( N ) cannot be recovered. As we will exnlain later, the center Z ( N ) is more basic to this all then the complete group S U ( N ) .

A good name for the field 8 is the "disorder parameter" [14] since it does not commute with the other, usual, fields wich have been called order

491

parameters in solid-state physics. The fact that in the quark confinement Fhase the degenerate vacuum states are eigenstates of this disorder narameter shows a close analogy with the superconductor where the vacuum state is an eigenstate of the order parameter.

VI. SU(N) GAUGE THEORIES IN 3 + 1 DIMENSIONS

In the previous sections the construction of a scalar field and the successive formulation of the spontaneous breakdown of the topological Z(N) symmetry were only possible because the model was in 2 space, 1 time dimensions. Also the boundary between different but equivalent vacua can only serve as an vortex in 2+1 dimensions. It would have the topology of a sheet in 3+1 dimensions and therefore not be useful as a vortex of conserved electric flux. So in 3+1 dimensions the formulation of quark confinement must be considerably different from the 2+1 dimensional case. Nevertheless extension of our ideas to 3+1 dimensions is possiole.

We concentrate on longe-range topological phenomena. One topological feature is the instanton, corresponding to a gauge field configuration with non-trivial Pontryagin or Second Chern Class numer. This however has no direct implication for confinement. What is needed for confinement is something with the space-time structure of a string, i.e. a two dimensional manifold in 4 dim. space-time. Instantons are rather event-like, i.e. zero dimensional and can far instance give rise to new types of interactions that violate otherwise apparent symmetries.

492

As we will see, they do Glay a role, though be it a subtle one. A topological structure which is extended in two dimensional sheets exists in gauge theories, as has been first observed by Nielsen, Olesen [15land Zumino C16l.They are crucial. We will exhibit them by comgactifying space-time. For the instanton it had been convenient to compactify space-time to a sphere For our purposes a hypertorus

s4

s1 x s1 s, s1

is more suitable f171.0ne can also consider this to be a four dimensional cubic box with periodic boundary conditions. Inside, space-time is flat. Tne box may be arbitrarily large. To be explicit we put a pure SU(N) gauge theory in the box (no quarks yet). Now in the continuum theory the gauge fields themselves are representations of SU (NI /Z (N) I where 2 (N) is the center of the group SU(N) :

Z(N) = (e2’in”I; n = 0, . . . l ~ - ~ ~ . Tnis is because any gauge transformation of the type (6.1) leaves A (X) invariant. A consequence of this is the existence of another class of topological quantum numbers in this box besides the familiar Pontryagin number. Consider the most general possible periodic boundary condition for A,,(x) a plane {XlIX2) in the 12 direction with fixed values of x3 and

!J

in the box. Take first

x4. One may have

493

Here, a RA s t a n d s s h o r t f o r

a2 are t h e p e r i o d s . 1 '

lJ

The p e r i o d i c i t y c o n d i t i o n s fo r 51 ( X I follow by

c o n s i d e r i n g ( 6 . 2 ) a t t h e c o r n e r s of t h e box: 1,2

where Z is some e lement of Z ( N ) . One may now perform c o n t i n u o u s gauge t r a n s f o r m a t i o n s on

A,,(x) I

( X I 1x2) + R ( x l 1 x 2 ) A ~ ( x l ,x2) r ( 6 . 5 )

where R ( x , ~ x ~ ) such t h a t n 2 ( x 1 ) -+ I or such t h a t R1(x2) -+ I , b u t n o t b o t h , because Z i n ( 6 . 4 ) remains i n v a r i a n t under (6 .5) as one can e a s i l y v e r i f y . W e c a l l t h i s e lement 2 ( 1 , 2 ) because t h e 1 2 p l a n e w a s chosen. By c o n t i n u i t y Z ( l , 2 )

cannot depend on x or x 4 . For each ( p v ) d i r e c t i o n such a Z e lement e x i s t , t o be labeled by i n t e g e r s

( n o n - p e r i o d i c ) can be a r r a n g e d e i t h e r

3

VlJ ' n = -n U V

d e f i n e d modulo N . C l e a r l y t h i s g i v e s

d (d-1) N = N 6

t o p o l o g i c a l classes o f gauge f i e l d c o n f i g u r a t i o n s . Note t h a t t h e s e classes d i s a p p e a r i f a f i e l d i n t h e fundamental r e p r e s e n t a t i o n of S U ( N ) i s added t o t h e system ( t h e s e f i e l d s would make u n a c c e p t a b l e jumps a t t h e boundary) . Indeed, t o u n d e r s t a n d q u a r k confinement i t is n e c e s s a r y t o u n d e r s t a n d p u r e gauge sys tems w i t h o u t q u a r k s f i r s t .

494

As we shall see, the new topological classes will imply the existence of new vacuum parameters besides the well-known instanton[l8] angle 0 The latter still exists in our box, and will be associated with a topological quantum number v , an arbitrary integer.

VII. ORDER AND DISORDER LOOP INTEGRALS

To elucidate the physical significance of the we first concentrate on topological numbers n

gauge field theory in a three dimensional periodic box with time running from -w to m. To be specific we will choose the temporal gauge,

PV

(this is the gauge in wich rotation towards Euclidean space is particularly elegant). Space has the topology

There is a n i n f i n i t e set of homotopy classes (S,I3 . of closed oriented curves C in this space: C may wind any number of times in each of the three principal directions. For each curve C at each time t there is a quantum mechanical operator A(C,t) defined by

called Wilson loop or order parameter. Here P stands for path ordering of the factors A(2,t) exponents are expanded. The ordering is done with respect to the matrix indices. The A(x',t) are also operators in Hilbert space, but for different x' , same t, all A(2,t) commute with each other. ay analogy

when the

495

with ordinary electromagnetism we say that A ( C ) measures magnetic flux t h rough C , and in the same time c r e a t e s an electric flux line a long C. Since A ( C ) is gauge- invariant under purely periodic gauge transformations, our versions of magnetic and electric flux are gauge-invariant. Therefore they are not directly linked to the gauge Tovariant curl Ga ( 2 ) .

P V

There exists a dual analogon of A ( C ) wich will be

3 called B(C) or disorder loop operator f 1 3 1 . C is again a closed oriented curve in (S,) . A simple definition of B ( C ) could be made by postulating its equal-time commutation rules with A(C) :

[ A ( C ) , P - ( C ' ) ] = 0;

[B(C), B ( C ' ) I = 0;

( 7 . 3 ) A ( C ) B ( C ' ) = B ( C ' ) A ( C ) exp 21~in/N ,

where n is the number of times C ' winds around C in a certain direction. Note that n is only well defined if either C or C' is in the trivial homotopy class (that is, can be shrunk to a point by continuous deformations). Therefore, if C' is in a nontrivial class we must choose C to be in a trivial class. Since these commutation rules (7.3) determine B(C) only up to factors that commute with A and B , we could make further requirements, for instance that B ( C ) be a

unitary operator.

An explicit definition of B ( C ) can be given as follows. As in sect. 4 , we go to the temporal gauge, A, = 0. We then must distinguish a "large Hilbert space" H of all field configurations A ( x ) from a "physical Hilbert space a C H. This ?f is defined

496

to be the subspace of H of all gauge invariant states :

wheren is any infinitesimal gauge transformation in 3 dim. space. Often we will also writen for the corresponding rotation in H:

[C' 3 Now consider a pseudo-gauge transformation Q defined to be a genuine gauge transformation at all pointsx $ C',but singular on C'. For any closed path X ( 8 ) with 0 < 6 < 21r twisting n times around C' we require

+

(7.6)

This discontinuity is not felt by the fieldsA(g,t) are invariant under Z ( N ) . They do feel the singularity at C ' however. We define B(C') as

wich

p 1 but with the singularity at C' smoothened; this corresponds to some form of regularization, and implies that the operator differs from an ordinary gauge transformation. Therefore, even for I$> E H we have

For any regular gauge transformation ilwe have an Q'

497

such that

'L Therefore, if I$> E ?f then B(C') I+> E H, and B(C') is gauge-invariant. We say that B(C') maesures electric flux t h r o u g h C' and creates a magnetic flux line d o n g C' .

We now want to find a conserved variety of Non-Abelian gauge-invariant magnetic flux in the 3-direction in the 3 dimensional periodic box. One might be temped to l ook for some curve C enclosing the box in the 1 2 direction so that A(C) maesures the flux through the box. That turns out not to work because such a flux is not guaranteed to be conserved. It is better to consider a curve C' in the 3-direction winding over the torus exactly once:

B ( C ' ) creates one magnetic flux line. But B(C') also changes the number n12 into n12 + 1. This is because

makes a Z(N) jump according to ( 7 . 6 ) . If Q , , 2 ( g ) in ( 6 . 2 )

are still defined to be continuous then Z in ( 6 . 4 )

changes by one unit. Clearly,n12 measures the number of times an operator of the type B(C') has acted, i.e. the number of magnetic flux lines created.n, also conserved by continuity. We simply define

is

n = E ij ijk mk '

498

( 7 . 1 0 )

with % the total magnetic flux in the k-direction. Note thatm corresponds to the usual magnetic flux (apart from a numerical constant) in the Abelian case. Here, m is only defined as an integer modulo N.

-+

+

VIII. NON-ABELIAN GAUGE-INVARIANT ELECTRIC FLUX IN THE BOX

As in the magnetic case, there exists no simple curve C such that the total electric flux through C, measured by B(C), corresponds to a conserved total flux through the box. We consider a curve C winding once over the torus in the 3-direction and consider the electric flux creation operator A(C). But first we must study aomc new conserved quantum numbers.

Let I$> be a state in the before mentioned little Hilbert space % * Then, according to eq. (7.5) ,

I $> is invariant under infinitesimal gauge transformations n. But we also have some non-trivial homotopy classes

of gauge transformations n. These are the pseudoperiodic

ones : n(al,x21x3) = R(0,x2,x3)Z1 I s2Cxl Ia21x3) = R(xl ,GIx3)Z2 I n(xl ,x2,a3) = R(xl ,x2,C) Z3 I

‘1,2,3 (8.1) E center Z ( N ) of S U ( N ) ,

and also those nwhich are periodic but do carry a non-trivial Pontryagin number V , A little problem arises when we try to combine these two topological features. The z,,2,3 can be labeled by three integers k1,2,3between 0 and N :

2.rr ikt /M zt = e (8.2)

499

But how is v defined? The best definition is obtained if we consider a field configuration in a fourdimensional space, obtained by multiplying the box(S,)3 with a line segment:

Now choose a boundary condition: F(t=l)=RA(t=O). Then, if the fields in between are continuous,

is uniquely determined by R . On S4this would be the integer v,Now however, it needs not to be integer anymore because of the twists in the periodic boundary conditions for ( S , ) .We find

where v is integer and ;fi is the magnetization defined in the previous section. Notice that v is only well defined if not modulo N. Taking this warning to heart, we write Q [ g , v ] for any n in the homotopy class [ k , ~ ] .

smoothly under n [ g , v ] ,

the Z ( N ) transformations of eq.(8.1) , but also their boundary conditions do not change. These ncommute therefore with the magnetic flux &. the same equation (8.1) and have the same v , they may act differently on states of the big Hilbert space H,

but since they differ only by regular gauge transformations they act identically on states inH,

and % are given as genuine integers,

-b

Notice that not only do the A (x) transform IJ since they are invariant under

If two 52 satisfy

Ir

def ined i n (7.5). W e may s imul taneous ly d i a g o n a l i z e t h e Hamiltonian H , t h e magnetic f l u x z , and Q [ k , v l :

“ , v I l $ > = e (8.5)

where o($,v) are s t r i c t l y conserved numbers. Now t h e 52 o p e r a t o r s form a group. Defining f o r each 52 t h e number P as i n ( 8 . 4 ) w e have

so

i f 3 is an i n t e g e r . W e f i n d t h a t w must be l i n e a r in k and v : +

where ei are i n t e g e r numbers de f ined modulo N , and 8

is t h e f a m i l i a r i n s t a n t o n ang le , def ined t o l i e between 0 and 27l.

Now l e t u s t u r n back to A ( C ) def ined i n e q . ( 7 . 2 ) . I f C i s t h e curve cons idered i n t h e beginning of t h i s s e c t i o n , A ( C ) is n o t i n v a r i a n t under Q [ z , v l because

( 8 . 1 0 )

501

T h e r e f o r e ,

If ( 8 . 1 2 )

( 8 . 1 3 )

(8 .1 4 )

T h e r e f o r e A ( C ) i n c r e a s e s e3 by one u n i t :

e3 is a good i n d i c a t o r f o r e lec t r ic f l u x i n t h e

3 - d i r e c t i o n , up t o a c o n s t a n t . It is s t r i c t l y conserved. However if w e l e t 0 r u n from 0 t o 2.rr t h e n t u r n s i n t o + 2. It i s t h e r e f o r e p h y s i c a l l y perhaps m o r e a p p r o p t i a t e t o i d e n t i f y

+ e + e + - 2 m (8.16)

as b e i n g t h e t o t a l electric f l u x i n t h e t h r e e d i r e c t i o n s of t h e box.

IXaFREE ENERGY OF A GIVEN FLUX CONFIGURATION

Again w e f o l l o w r e f . [ 3 ] b u t f o r completeness w e add t h e P o n t r y a g i n winding number v.

s ta te (e,m,8) a t t e m p e r a t u r e T = l/kf3:

L e t u s w r i t e down t h e f r e e energy F of a g i v e n + +

502

Here H i s t h e Hamiltonian and *H t h e l i t t l e H i l b e r t space , P a r e p r o j e c t i o n ope ra to r s . P,($) i s simply def ined t o select a given set of n i j =

t h e t h r e e space- l ike i n d i c e s of eq. ( 6 . 6 ) . P e ( e ) P e ( e )

i s def ined by s e l e c t i n g states I $ > w i t h

mk:

- 2ni (z.z)+ + ( G - Z ) + i e v I$> ' (9.2) N $>= e n t L v 1

Therefore Pe

1 - -,

2 ~ i -f + e i + -, - - (k .e)- - ( m . k ) - i e v ","I . (9.3) N N

c e N' k,v

Now e'8H i s t h e e v o l u t i o n ope ra to r i n imaginary t i m e d i r e c t i o n a t i n t e r v a l 8 , expressed by a f u n c t i o n a l i n t e g r a l over a Eucl idean box w i t h sides (a l Ia2 ,a3 ,B) :

W e may f i x t h e gauge f o r 3 choosing

(2 ) f o r i n s t a n c e by ( 2 )

= 0 I

A ( 2 ) 2 ( X I Y I O ) = 0 I

1 (X ,O,O) = 0 . (9.5)

W e a l r e a d y had A 4 ( Z , t ) = 0. Since only s t a t e s i n ;f a r e considered, w e i n s e r t a l s o a p r o j e c t i o n ope ra to r I DSl were I is t h e t r i v i a l homotopy class. R E 1

503

( 1 ( 2 ) "Trace" means that we integrate over all P.

therefore we get periodic boundary conditions in the 4-direction. Insertions of I DQ means that we have

s 1 E I

periodicity up to gauge transformations, in the completely unique gauge

Eq. (9.3) tells us that we have to consider twisted boundary conditions in the 41, 4 2, 43 directions and Fourier transform:

Here W(~,;fi,v,a }is the Euclidean functional integrsll with boundary condition:> f ixsc? by chnosinq n number U . Because of the gauqe choice (9 .61 t h i s

functional integral Rust include integration over the belonging to the Given homoton;7 classes as they determine the boundary conditions such as ( 6 . 2 ) .

The definition of W is completely Euclidean symmetric. In the next chapter I show how to make use of this symmetry with respect to rotation over 90 in Euclidean space.

U

- - mk ; ni4 = ki; a4 = B , snd a Pontryagin ij 'i jk

0

X. DUALITY

The Euclidean symmetry in eq. (9.7) suggests to consider the following SO(4) rotation:

L e t us i n t r o d u c e a n o t a t i o n f o r t h e f i r s t t w o components of a vector:

x IJ = (H,x4) I

% x = ( x 1 , x 2 ) t

x = ( X , , X , ) . .L

(10 .2)

We have, from e q . ( 9 . 7 ) :

Notice t h a t i n t h i s formula t h e t r a n s v e r s e e l ec t r i c and magnet ic f l u x e s are F o u r i e r t r a z s f o r m e d and i n t e r c h a n g e p o s i t i o n s . Notice also t h a t , apart from a s i g n d i f f e r e n c e , t h e r e i s a complete e lectr ic- magnet ic sylnmetry i n t h i s e x p r e s s i o n , i n s p i t e of t h e f a c t t h a t t h e d e f i n i t i o n of F i n t e r m s of W w a s n o t so symmetric. Eq. (10.3) i s a n e x a c t p r o p e r t y of o u r system. N o approximat ion w a s made. W e r e f e r t o it as " dua 1 it y 'I .

X I . LONG-DISTANCE BEHAVIOR COMPATIBLE W I T H DUALITY

Eq. ( 1 0 . 3 ) shows t h a t t h e i n s t a n t o n a n q l e 8

p l a y s no role i n d u a l i t y . I t does however a f f e c t

505

+ the physical iiiterpretation of e as electric flux, see (8.10). From now on we put 6 = 0 for simplicity, and omit it.

Let us now assume that the theory has a mass gap. No massless particles occur. Then asymptotic behavior at large distances will be approached exponentially. Then it is excluded that

-+ + + + F(e,m,a,B) + 0, exponentially as a,B + ,

-+ + for all e and m, wich would clearly contradict (10.3). This means that at least some of the flux configurations must get a large energy content as a, B -+ m. These fl-ux lines apparently cannot spread out and because they were created along curves C it is practically inescapable that they get a total energy wich will be proportional to their length :

+

E = lim F = pa . B +w

(11.1)

However, duality will never enable us to determine whether it is the electric or the magnetic flux lines that behave this way. From the requirement that W in ( 9 . 7 ) is always positive one can deduce the impossibility of a third option, namely that only exotic combinations of electric and magnetic fluxes behave as strings (provided 8 = 0 ) .

For further information we must make the physically quite plausible assumption of factor izab ili ty 'I :

( 1 1 . 2 )

506

Suppose that we have confinement in the electric domain:

where p is the fundamental string constant. Then we can derive from duality the behavior of Fm(G).

First we improve (11.3) by applying statistical mechanics to obtain Fe for large but finite 8 . One obtains:

e -8Fe(e1 te2,0,gt8) + C(a,8) - -

+ - + - n +n n +n = c 1 y1 ' y 2 ti N I (n+-n;-e1)6N(ni-ni-e2). f 2 ntlniln;lnil

(11.4) "1 tn2

Here -8~a1 y1 = ha2a3 e I

y2 = Aala3 e I

-8~a2

1 N-l 2nikx/N tiNW = f j c e k=O

1 if x = 0 (mod N) (1 1.5)

0 if x = other integer number. = {

f The sum i s over all nonnegative integer values of ni (the orientations f are needed i f N 3 ) . The y's are Boltzmann factors associated with each string-like flux tube.

We now insert this, with (11.21, into ( 1 0 . 3 ) putting e3 = m3 = 0. One obtains

-+ 21 y ' cos(2ma?/N) (11.6) a -8Fm ( m, m2 , 0, a , P 1 a e = C'e

507

where C' is a g a i n a c o n s t a n t and

y; = h a l @ e I

y; = h a 2 B e

-pa2a3

-Pa la3

w i t h

+ 2 nml -pa2a3 El ( m l I a ) = ~ ~ ( I - c o s - N I a l e I

and s i m i l a r l y f o r E 2 and E3.

b e no m a g n e t i c c o n f i n e m e n t , b e c a u s e i f w e l e t t h e box become w i d e r t h e e x p o n e n t i a l f a c t o r

One reads o f f f rom eq. ( 1 1 . 8 ) t h z t t h e r e w i l l

c a u s e s a r a p i d decrease of t h e e n e r g y of t h e m a g n e t i c f l u x . Notice t h e o c c u r r e n c e o f t h e s t r i n g c o n s t a n t p

i n t h e r e . Of colirse we c o u l d e q u a l l y w e l l h a v e s t a r t e d

f rom t h e p r e s u m p t i o n t h a t t h e r e w e r e m a g n e t i c c o n f i n e m e n t . One t h e n would c o n c l u d e t h a t t h e r e would be no e l e c t r i c c o n f i n e m e n t , b e c a u s e t h e n t h e e l ec t r i c f l u x would h a v e ar, e n e r g y g i v e n by (11.8) .

XII. THE COULOMB PHASE

(11 .7 )

( 1 1 .8 )

To see wha t m i g h t happen i n t h e a b s e n c e o f a mass qap o n e c o u l d s t u d y t h e ( € i r s t ) Gctorgi-Slashow model [ 4 ] Here SU(2) i s "Droken s p o n t a n e o u s l y " i n t o

U(1) by an isospin one Higgs field. Ordinary perturbation expansion tells us what happens in the infrared limit. There are electrically charged particles: Wf They carry two fundamental electric flux units ("quarks" with isospin f would have the fundamental flux unit qo = f 5 e). There are also magnetically charged particles (monopoles, [19].They also carry two fundamental magnetic flux units:

(the charged vector particles).

1

2.rr 4a g = - = - - .

90 (12.1)

A given electric flux configuration of k flux units would have an energy

2 2

2a2a3 gok al E = (12.2)

At finite 6 however pair creation of Wf so that we should take a statistical average over various values of the flux. Flux is only rigorously defined mociulo 2qo. We have

takes place,

2 W

(12.3)

Similarly, because of pair creation of magnetic monopoles

2 W A n a

(12.4)

k=-m e-a, L a3

These expressions do satisfy duality, eq. (10.3). This is easily verified when one observes that

and

Notice now that this model realizes the dual formula in a symmetric way, contrary to the case that there is a mass gag. This dually symmetric mode will be referred to as the "Coulomb phase" or "Georgi-Glashow phase 'I .

Suppose that Quantum Chromodynamics would be enriched with two free paramaters that would not destroy the basic topological features (for instance the mass of some heavy scalar fields ii7 the adjoint representation). Then we .dould have a phase diagran as in the Figure below.

lllvu \ mode I

Fig. 1

510

Numerical calculations[20] suggest that the phase transition between the two confinement modes is a first order one. Real QCD is represented by one point in this diagram. Where will that point be? If it were in the Coulomb phase there would be long range, strongly interacting Abelian gluons contrary to experiment. In the IIiggs mode quarks would have finite mass and escape easily. It could be still in the Higgs phase but very close to the border line with the confinement mode. If the phase transition were a second order one then that would imply long range correlation effects requiring light physical gluons. Again, they are not observed experimentally. If, wich is more likely, the phase transition is a first order one then even close to the border line not even approximate confinement would take place: quarkswould be produced copiously. There is only one possibility: we are in the confinement mode. Electric tlux lines cannot spread out. Quark confinement is absolute.

'

51 1

REFERENCES

1. S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. A. Salam and J. Ward, Phys. Lett. - 13 (1964) 168.

S.L. Glashow, J. Iliopulos and L. Maiani, Phys. Rev. D2 (1970) 1285.

2. E.S. Akrs and B.W. Lee, Physics Reports no. 1.

S. Coleman in: Laws of Hadronic matter, Erice July 1973, ed. by A . Zichichi, Acad. Press, NY and London.

3 . G. It Hooft, lectures given at the Cargese Summer Institute on "Recent Developments in Gauge Theories", 1979 (Plenum, New York, London) ed. G. 't Hooft et al. Lecture no 11.

4. H. Georgi and S.L. Glashow, Phys. Rev. Lett. - 28

5. G. It Hooft, in "The Whys of Subnuclear Physics", (1972) 1494.

Erice 1977, ed. by A. Zichichi (Plenum, New York and London) p. 943.

6. G. Parisi, lectures given at the 1977 Cargese Summer Institute.

7. P. Higgs, Phys. Rev. 145 (1966) 1156. T.W.B. Kibble, Phys. Rev. 155 (1967) 1554.

8. S. Coleman, in "New Phenomena in Subnuclear Physics", Erice 1975, Part A, ed. by A. Zichichi (Plenum press, New York and London).

9. G. 't Hooft, in "Particles and Fields", Banff 1977, ed. by D.H. Boa1 and A.N. Kamal, (Plenum Press, New York and London) p. 165.

10. P.A.M. Dirac, Proc. Roy. Soc. A133 (1931) 60; Phys. Rev. 2 (1948) 817.

statistics, and all that (Benjamin, New York and Amsterdam, 1964).

11. R.F. Streater and A.S. Wightman, PCT, spin and

12. H. Lehmann and K. Pohlmeyer, Comm. Math. Phys. 20

512

(1971) 101; A. Salam and J. Strathdee, Phys. Rev. E (1970) 3296.

13. G. 't Hooft, Nucl. Phys. B138 (1978) 1. 14. L.P. Kadanoff and H. Ceva, Phys. Rev. B3 (1971)

15. H.B. Nielsen and P. Olesen, Nucl. Phys. - B61 (1973)

16. B. Zumino, in "Renormalization and Invariance in

3918.

45;ibidem B160 (1979) 380.

Quantum Field Theory", ed. E.R. Caianiello, (Plenum Press, New York) p. 367.

17. G. 't Hooft, Nucl. Phys. B153 (1979) 141. 18. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976)

172. C. Callan, R. Dashen and D. Gross, Phys. Lett. 63B (1976) 334 and Phys. Rev. E (1978) 2717.

A.M. Polyakov, JETP Lett. 20 (1974) 194. 19. G. 't Hooft, Nucl. Physics E (1974) 276

20. M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. p20 (1979) 1915.

513

CHAPTER 7.2

THE CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

Gerard 't Hooft Instituut voor Theoretische Fysica der Rijksuniversiteit

De Uithof Utrecht, The Netherlands

0. ABSTRACT

In these written notes of four lectures it is explained how the phenomenon of permanent confinement of certain types of particles inside bound structures can be understood as a consequence of local gauge invariance and the topological properties of gauge field theories.

I. INTRODUCTION

The lagrangian of a quantum field theory describes the evolution of a certain number of degrees of freedom of the system, called f i e l d s as a function of space and time. only straightforward in a small region of space-time and therefore these fields should be interpreted only as the microscopic variables of the theory. Only in the simplest cases these microscopic variables also correspond to actual physical particles' (the macroscopic objects) but very often the connection is less straightforward for two reasons. One reason is that there may be a local gauge invariance. This is a class of transformations that transform a set of fields into another set of fields with the postulate that the new set describes the same physical situation as the old one. only constitute some orthogonal subset of the original set of fields and one of the problems that we will study in these lectures is how to associate this subset to observable objects.

In many cases this evolution is

Therefore the physical fields

However even these observable objects ("transient particles") may in some cases not yet be the macroscopic physical particles. That is one second point : various kinds of Bose-condensation may

1981 CargLse Summer School Lecture Notes on Fundamental Interactions. NATO A&. Study Inst. Series E: Phys.. Vol. 85, Edited by M. U v y and J.-L. Basdevant (Plenum).

514

G. 't HOOFT

take place after which the spectrum of physical particles may look entirely different once again. nomena as we go along.

We will study these condensation phe-

The most challenging application of our theoretical considerations is "quantum chromodynamics" or generalizations thereof. The "microscopic" lagrangian there contains only vector particles ("gluons") and spinors ("quarks") but neither of these are really physical. The spectrum of physical particles always consists of bound states of certain numbers of quarks and/or antiquarks with unspecified numbers of gluons. We will obtain a qualitative understanding of this transition from the microscopic to the macroscopic dynamical variables.

The first models that we will consider may seem to be a far way off from that desired goal but studying them will turn out to be crucial for obtaining a suitable frame and language in order to put the more advanced systems in a proper perspective.

11. SCALAR FIELD THEORY

Let us consider the lagrangian of a complex scalar field theory :

(2.1)

Equivalently we can use real variables :

r We write $* rather than $ not operators, and the lagrangian must be seen in the context of a functional integral as described by B. de Wit in his lectures at this School.

because here the fields are c-numbers,

The lagrangian (2.1) describes a system with two species of Bose- particles both with the same mass m tum mechanical sense. species, both equal to their own antiparticles, or consider $ a8 a particle and +* as its distinct antiparticle. At this level these descriptions are equivalent.

but distinguishable in the quan- One can either consider $1 and $2 as the two

Obviously there is a synnetrgunder exchange of $ and $ , or 1 2 more generally

$** e-iA$* (2.3)

515

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

This is a group of rotations in the complex plane called U(1) and is only global, that is, A must be independent of space-time.

By Noether's theorem the model contains a conserved current,

= +*aU+ - (aU+*)+ j U

(2.4)

with aPjU = 0 (2.5)

Classically (2.5) is only true if e,+* are required to obey the Euler-Lagrange equations generated by (2.1). Quantum mechanically ( 2 . 5 ) follows if we substitute for 4 and +* the corresponding operators + and e t .

Our model is trivial in the sense that, if m2 > 0, the microscopic fields + and + directly correspond to the expected macroscopic physical particles (apart from possible stable bound states) and the symmetry (2.3) is also a symmetry between these particles.

1

111. BOSE CONDENSATION

Bose condensation is a well-known phenomenon in quantum statistical physics. of interest let us first consider an ideal non-relativistic Bose gas. The states are

Just in order to make the connection with our case

where

and 11,12,13 are positive integers. L is the length of a side of the box in which the particles are contaimed. The energy of the states is

(3.3)

where M is the mass of the (non-relativistic) particles. for the thermodynamic free energy F,

We take

(3.4)

516

G. 't HOOFT

where N = zn(ci), 6

T is the temperature in natural units, and v is the chemical potential.

1/T i

We can easily solve (3.4) :

e - m = m ~e 1 2 0(V - k /2M)n .I OD

k n=O k 1 - exp 8 ( ~ - k2/2M> (3.5)

and in the limit of infinite volume :

The particle number density is

(3.7)

One easily notes that the formulas (3.5) - (3.7) explode if the chemical potential v becomes larger than or equal to zero, a typical property of Bose gases.

Since this is a+non-relativistic system it is convenient to introduce a field +(XI in the following way :

where a(g) is an operator that annihilates one particle with k in the usual way.

momentum -+ The hamiltonian is then

(3.9)

where for convenience we included the chemical potential term 80 that this hamiltonian describes the complete system :

- BH e'" = Tr e

It ir obvious here that should not be allowed to be positive.

517


Now however we can take into account the repulsive forces between the particles. expect an extra, poeitive contribution to H. is

When many particles are close together we A simple model for that

(3.11)

As long as But if p is positive then the stabilize the system.

Of course it is difficul't t o find the free energy of this revised system exactly, but an easy approximation, valid for h not too large, is to substitute #I by a c-number (as defined in (3.8) it was an operator), and subsequently minimize H . mately the energy of the lowest eigenstate :

is negative the term is just a small perturbation. term is the only one that can

bte then find approxi-

t - 4 o = P

F E E 0

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

to be contrasted with (3.7), still approximately valid for <O. As p turns from negative to positive a phase transition is said to take place; we suddenly get large values for the fields 0 already in the lowest eigenstate of H. This is called Bose condensation. It takes place whenever the pressure is so high that the chemical potential becomes negative. Also the A term must be sufficiently small.

The model described by our hamiltonian (3.11) resembles somewhat the model of the previous section. Note however that the mass

518

G. ' t HOOFT

M comes in the derivative term and never vanishes. In the relativis- stic field theory M is replaced by 112, and -p by the mass-squared, m2. In that theory Bose-condensation can take place also : we must extend the allowed values for m2 to negative values, clearly a more profound change in the particle properties. But what we get in return is that in the relativistic model, the vacuum itself, without any external pressure, can become a Bose-condensate.

IV. GOLDSTONE PARTICLES

In the previous section we discussed Bose-condensation in a statistical system only in a the details (for instance, Q, and were not canonical variables in the usual sense). relevant to us : the relativistic complex scalar case. We return to the lagrangian (2.1) and now assume m2 to be negative. convenient to rewrite it as

sketchy we,y because we will not need

We will be more precise for the case that is more

It is

2 2 where m = - AF -112 F4 , has been added to the lagrangian.

< 0 automatically, and an irrelevant constant,

The hamiltonian density of the system is

t i X i 2 2 x = n n + ai4 ai4 + #Q, 4 - F (4.2) 1. where n,mi are the canonical momenta associate with o,Q, which

are now operators rather than fields. an extremum for

!i! The +,$ dependent part has

2 ?Q, = F

from which

(4.3)

(4.4)

where w is arbitrary but fixed to the same value everywhere in space.

t Now if there were no n n term inacthen (4.4) would be the exact solution to the Schriidinger equation for the lowest energy state. Every between 0 and 2n would describe a lowest-energy

51 9


eigenstate of H =/Xd x, each with eigenvalue E = 0. The question t 0 is whether the n TI term causes sufficient fluctuations to lift this degeneracy. The answer is not so simple : in 1 space - 1 time dimension, yes; in 2 or more space dimensions, no. So let us limit ourselves to 3 space + 1 time dimensions. energy state) is degenerate and characteriz d by a phase angle W.

But ( 4 . 4 ) is not exactly valid due to the n TI term. We replace it

3

Then the vacuum (= lowest

F by

< O l g l O > = F'eLW ( 4 . 5 )

where 10 > is the vacuum state and U i s the vacuum angle. now on we will consider only the world surrounded by a vacuum with

divergences, subtractions must be mdUe in F and,+, and we could choose these such that F' F.

From

= 0. Further, F' is close to F. In fact, because of ultraviolet

Actually our theory only makes sense if either h is chosen to be rather small and F of order l/&, or if an adequate ultraviolet cutoff has been introduced. The reasons for this are deeper field theoretic arguments connected with the renormalization group that I will not go into. the fluctuations of 6 around F are also relatively small and it makes

Let us assume that h is rather small. Then

sense to split

+ ' F + r l

and the lagrangian becomes

Writing

L

This becomes

1' 02' where " int " stands for higher order terms in n that one particle, ql, obtained a mass

M - F a rl

But its companion became massless. 2

( 4 . 6 )

(4.7)

( 4 . 8 )

Notice now

( 4 . 9 )

520

G. ‘t HOOFT

The occurrence of a.massless particle as soon as the vacuum expectation value of a field 4 is not invariant under t e continuous symmetry (2.3) has been first observed by J. Goldstonele, and it is an exact property of the system, not related to our perturbative approximation (no higher order mass corrections). We conclude that after the phase-transition caused by Bose-condensation, the symmetry (2.3) is 8pOntatteOl48Zy broken (the degeneracy of $1 and $2 is not reproduced in ~ 1 , rl2) and at the same time 8 messless particle appears: the Goldstone particle. microscopic fields $,$* in the lagrangian do not reflect accurately the physical spectrum, but the transition towards the 0 fields was still very simple.

Here we see the first example where the

It is correct to characterize the vacuum by

< OlI$(;)lO > = F # 0

and the vacuum is infinitely degenerate. Characterization of the Hfgg8 mode, next sectionm) will be very different!

V. THE HIGGS MECHANISM

We now switch on electromagnetic interactions*)simply by adding the Maxwell term to the lagrangian and replacing derivatives by covariant derivatives :

where

- a ~ F,,v = allAV v ’ D,,$ = (a,, + iqA,,)$ (5.2)

q is the electric charge of the particle $.

the can

Indeed the previously introduced current j eq (2.41, is now conserved electromagnetic current. now be replaced by

But the’invariance (2.3)

i W $ , $* + -iA(x) $* Q + e + A - l/q a,,A(x) (5 .3)

U

*)It may seem to a su erficial reader that these notes are just repeating the story I ) of the early 70’s. However we are now not primarily interested in perturbative quantization but rather non- perturbative characterization of what happens.

521


which is a local invariance

As already mentioned in the Introduction, the consequence of this local invariance is that only a subspace of all (@,$I*, A 1, namely the gauge-nonequivalent values , correspond to physxal'observables.

Traditionally, one now proceeds by choosing a gauge fixing procedure so that most, if not all, degeneracy is removed. One then performs perturbation expansions in h and q2. can again ask whether or not

At zero A and q2 one

< $ >o = F f 0 ? (5.4)

and since the higher order corrections to < $ > in A and q2 the qualitative distinction whether or not < $ > = 0 remains valid at every order. Goldstone mechanism : after Bose-condensation of charged particles we get the Higgs mechanism 3 ) . However, there is a difficulty with (5.41, because $ is not gauge invariant. Smearing (5.4) over all of space-time may yield zero or not, depending on the gauge chosen. In a trivial gauge

are of higher order 0

And so we get the local varPant on the

(5.5)

we have

always. bative considerations, where it is correct (as good as it can be) but not as an absolute non-perturbative criterion for the Higgs mode. Another criterion that cannot be used is whether or not the vacuum is degenerate. The problem there is that transformation (5.3) yields physically equivalent states, contrary to its global equivalent ( 2 . 3 ) 1 1 to different angles are now one and the same state. The vacuum is never degenerate if the symmetry is local. never "spontaneously broken". in connectionwith gauge theories ? Because, as I will show now, there certainly is such a thing as a Higgs mode and it usually can be described in some or other reasonable perturbation expansion around a Goldstone (= global) field theory.

We therefore propose to use criterion (5.4) only in pertur-

Therefore all those vacuum states corresponding

Local symmetries are Then why is this phrase so often used

Let us return to perturbation theory momentarily. We then write as usual

+(XI = F + d x ) (5.6)

522

G. ‘t HOOFT

(5.7)

A convenient renormalizable gauge is obtained by adding the gauge fixing term

so that

1 1 2 2 - * 1 2 2 1 2 2 c +cc = - +a A l 2 - p A ~ ~,,q D , , ~ - pnnl - pAV2 + int u v (5 .9 )

with

MA - fi qF M - m F rl

(5.10)

It is easily read off from this lagrangian that the vector particle A has a mass MA and ‘1, a mass %. The longitudinal component of the vector field A and ‘12 are h ts, which both cancel against the Faddeev-Popov-keWi t t ghost ‘ssY all having the same mass MA.

So perturbation theory suggests that the Higgs theory behaves in a way very different from the symmetric or Coulomb theory : one of the two scalar fields $I disappears and the vector field obtains a mass so that the photon field is short-range only. distinction that should survive beyond perturbation theory. Thus the criterion that electromagnetic forces become short-range is much more fundamental than either the vacuum value of the scalar field (5.4) or the “degeneracy of the vacuum”. new phenomenon in the Higgs mode contrary to the “unbroken” or Coulomb mode. distinguish a Higgs theory from just any non-gauge theory with massive vector particles.

This i8 a

But, there is yet another

This is important because the above does not yet

VI. VORTEX TUBES

The non-relativistic version of the theory of the previous section is the superconductor : if electrically charged bosons (Cooper’s bound state of an electron pair) Bose-condense then there the electric fields become short-range. Also magnetic fields are repelled completely (Meissner effect). Except when they become too strong. Then, because of magnetic flux conservation, they have to be allowed in. forms narrow flux tubes.

What happens is that a penetrating magnetic field These flux tubes carry a multiple of a

523


precisely defined quantum of magnetic flux. the perturbative Higgs theory can explain this.

A 1

By gauge transformations the "vacuum value" I$ can be changed into

iw(x) < @(XI >o = Fe

ttle amendement to

of the Higgs field

(6.1)

if w(x) is a continuous, differentiable function of space and time, However we can also consider perturbation theory around field configurat ions

where p(x) = F nearly everywhere, but at some points p may be zero. At such points w needs not be well defined and therefore in all the rest of space w could be multivalued. For instance, if we take a closed contour C around a zero of p(x) then following around C could give values that run from 0 to 27 instead of back to zero. The energy of such a field configuration is only finite provided that Di#(x) goes to zero sufficiently rapidly at m.

not go to zero fast enough there must be a supplementary vector potential A;(x).

Since a i # does

The easiest way to find that is by taking a gauge transformation that is regular at 00 but singular at the origin.

In the new gauge 3 . 0 may vanish rapidly at 0 and therefore Ai(x) also. So in the oid gauge

(6.4)

which is the magnetic flux. One can compute the energy of the flux tube by assuming cylindrical symmetry and substituting (6.2) into (5.1). fixed, and minimize8 the hamiltonian derived from (5.1) One typi-

One then varies AV and p with the boundary condition (6.4)

cally finds that the energy of a flux with length ll is 6 )

E - all ( 6 . 4 )

(6.6)

If finally a magnetic field is admitted inside a superconductor it can only come in some multiple of these vortices, never spread

524

G. 't HOOFT

out because of the Meissner effect. as an aspect of the infinite conductivity of the material that is only broken down in sufficiently strong magnetic fields. vortex configurations that we discussed here were first derived in the relativistic theory by Nielsen, Olesen and Zumino6). tence of these macroscopic stable objects can be used as another characterization of the Higgs mechanism. beyond perturbation expansion.

Classically one may considerthis

The stable

The exis-

They should also survive

VII. DIRAC'S MAGNETIC MONOPOLES

magnetic charge B la Diracj5. It is not (yet) a dynamic particle but just a source or sink of magnetic flux, a spectator particle not dynamically involved in the lagrangian of *he theory. A Dirac monopole can be visualized as the end point of an infinitely thin coil carrying a large electric current, The vector potential it is very large close to the coil, because of this electric current :

At this stage it is U ful to introduce the notion of a single

where @is the magnetic flux of the coil and the integral is over any contour going closely around it. therefore 2 becomes large.

Close t o the coil dx is small,

Nevertheless the effect of the coil on its surroundings comes only through the end points, if a gauge transformation exists that removes this large vector potential :

Such g! ge transformations A would be multivalued, but we require that elx in (5.3) remains single-valued. allowed to make are multiples of 2n. mation (7.2) turns single-valued field configurations into single- valued field configurations if

So the jumps that A is Therefore'the gauge transfor-

Q = 2nnfq (7.3)

This is how Dirac found that the total amount of magnetic flux carried by a magnetic monopole must be quantized in units 2n/q where q is the smallest possible electric charge in the universe. This condition must be satisfied whenever we want a rotationally invariant quantized theory with magnetic monopoles and single valued fields. It is illustrative now to see what would happen with such a spectator particle inside a Higgs theory (or superconductor).

525


It is not accidental that the monopole quantum 2s/q coincides with the Nielsen-Olesen-Zumino vortex quantum. This implies that the monopole will be sitting at the end of an integer number of such vortices. Antimonopoles may be sitting at the other ends. Now the energy of such configurations is approximated by eq. ( 6 . 4 ) . proportional to their separation distance. the monopoles inside a superconductor are kept together by an infinite potential well, the potential being simply linearly proportional to their separation. This is the first observation of a confinement feature in quantum field theory, although the confined objects were as yet spectators, not any of the participants of the field equations.

It is And so we notice that

That will come later (sect. XIII).

VIII. THE UNITARY GAUGE

So far we limited ourselves strictly to Abelian gauge theories. We knew what the microscopic 'field variables are, and now we know what particles and vortices survive at macroscopic distance scales. The latter depend critically on what kind, if any, of Bose condensation took place. microscopic physical variables. Formally the space H of these variables is given by

Now in our introductioq we also mentioned the

H = R/G (8.1)

where R is the space of field variables and G is the (local) gauge group. How do we enumerate the variables in H ?

Traditionally one imposes a gauge condition on the fields in R, thus obtaining a subspace in R which could be representative for H. In section V we used the gauge fixing term (5.8). This is good enough if one intends to do perturbation expansion2~5). particles one obtainscanceleach other and can be dealt with. How- ever we claim that if a non-perturbative characterization of the physical variables is required then this is not good enough. Imagine t.hat one tries to solve the Dyson-Schwinger equations of the theory in some nonperturbative way. Whenever a computed S-matrix element shows a pole one can never be sure whether er not this is due to a ghost or whether it is physical. Furthermore as we will see the ghosts will produce their own topological features called "phantom solitons" which are entirely non-physical. Therefore if we want some understanding of the physical variables we must go to a "unitary gauge" (agaugewith no ghosts)

The ghost

Often the axial gauge

A. = 0 (8.2)

is used in order to understand the physical Hilbert space. However,

526

G. 't HOOFT

this leaves invariance with respect to time-independent gauge transformations :

(8.3)

and so there is still a redundancy in our set of variables. not suitable for our purposes.

It is

A completely ghost-free gauge can be formulated if we have a charged scalar field (0 (if no such field is present one may consider building such a field by composing, say, two fermion fields). We do not require the Higgs phenomenon to take place. condensation takes place at large distance scales one can look at the gauge

Regardless what

This fixes the gauge function A locally, point by point in space- time, contrary to gauges such as eq. (5.8) where the condition on A requires solving a second order partial differential equation (the cause of the ghosts).

Within the unitary gauge (8.4) all components of the vector field A,, are entirely observable. The complex scalar field I$ is reduced to a real field p that can only take positive values. would be a convenient description of the space of microscopic physical field variables were it not for one deficiency in the condition (8.4) : the original space of variables R certainly allows the scalar field + to vanish at certain points in space-time. defined by (for any I$ ER)

This

These points,

have the topological structure of a set of closed curves in 3-space, or closed surfaces in 3 + 1 dimensional spaceytime. At these points the condition (8.4) becomes singular : if (0 = pel8 then we must choose

A = -8 (8.6)

but the gradient of 6 is easily seen to explode close to a zero of + and therefore the vector potential 2, transforming as the gradient of A , will grow as the inverse power of the distance to this zero. Thus we find that the string-like structures, defined by (8.5) are separate degrees of freedom, giving a boundary condition on p (p * 0) and a prescribed singular boundary behavior of A,,. This completes our discussion of the microscopic physical degrees of

527


freedom for any Abelian gauge theory : we have observable vector fields % strings *), on which there is a boundary condition for both p and A,,

Note that only in the Higgs theory these physical variables are in the same time the macroscopic physical variables, although of course the macroscopic variables will be "dressed with a cloud of virtual particles" (the string becomes a vortex with finite thick- ness). macroscopic variables are harder t o discuss. It appears that our vortices "Bose-condense" to form long range, non-energetic magnetic field lines : the ordinary magnetic field f .

a truncated scalar field p (p > 0) and all possible closed

In the "unbroken" Abelian theory of electromagnetism the

IX. PHANTOM SOLITONS

The gauge (8.5) is called "unitary gauge" because in that gauge all surviving fields will be physically observable. Their quanta will all contribute in the unitarity relation

(9.1)

However as soon as practical calculationp are considered smoother gauge conditions are required. (8.5) is hard to implement if 4 oscillates wildly at small distances. a transition towards "smoother" gauge conditions not only ghost particles arise but also what we call "phantom solitons" : extended structures which are stable for topological reasons but nevertheless unphysical gauge artif acts 8).

We will now argue that after

Intermediate between the "renormalizable gauge" (5.8) and the unitary gauge we could choose

arg(4) + K a A = 0 (9.2) u u where arg ( 4 ) = Im(log 4 ) and K is an arbitrary gauge parameter. The gauge condition (9.2)is smoother than the unitary gauge because at small distances, by power counting, the second term dominates and we come close to the renormalizable Lorentz gauge. One finds ghosts in this gauge which propagate with a mass

so for small K they become unimportant.

(9.3)

*)That is, strings without ends; they couldrun from m to -.

528

G. ‘t HOOR

Now imagine a Nielsen-Olesen-Zumino vortex tube in the form of What do the field configurations in the a closed curve,

look like ? The gauge(9.2) $8 that particular gauge for !U e (9.2)

W - 1d4x(q”(arg(I$))* + ~4) (9.4)

has an extremum. Let us assume this is a minimum. The system then likes to arrange arg(4) to be zero as much as possible but not with too large vector potentials A .

IJ

Fig 1

In fig 1 we pictured a cross section of the vortex. intersected twice, in opposite directions. A t the intersection points the scalar field I$ makes one complete rotation, again in opposite directions. the optimal gauge (9 .2 ) , keeping Q as much as possible oriented towards the positive real axis. complete “twist” from top to bottom had to be preserved. This twist will cover an entire sheet spanned by the vortex. will be obtained (i.e. (9.4)will be minimallif that sheet has minimal surface. sheet are easy to solve if the sheet is considered to be locally sufficiently flat. We think that structures of this sort will further obscure the physical interpretation of whatever solutions will be found to the Dyson-Schwinger equations in the gauge (9.2) or completely renormalizable gauges. For instance, bubbles made out of these sheets will form a whole Regge-like family of phantom particles.

The plane is

The configuration in the figure is close to

We see that the topology of the

Certainly (9.2)

The equations for the field configurations inside the

Clearly the sheet is a gauge artifact.

529


X. NON-ABELIAN GAUGE THEORY

We now intend to perform the same procedure in non-Abelian gauge theories. that the gauge group is SU(N), with arbitrary-N. field variables are : a matrix vector field l J ( x ) where p is a Lorentz indel! and i, j run from 1 to N; and 2 t ere may be Dirac spinor fields $ i ~ , $ i ~ where F is a flavor index. assumed to play a significant role but may be present too.

For simplicity we restrict ourselves to the case The microscopic

Scalar fields are not

A n element of the gauge group G (here SU(N)) is a space-time dependent unitary matrix ~(x). Any transformation of the form

(10.1)

is postulated to describe the same physical situation as before, but of course gives a different set of values to the microscopic variables. If R is the space of microscopic variables, then RIG is the space of microscopic phys i ca l variables. Again we ask the question how to categorize or enumerate these physical variables.

In perturbation theory it is customary to impose a gauge condition, which implies that we find a subspace of R (the set of fields in R that satisfy the gauge condition) representative for RIG. renormalizable gauge condition is

A

a~ - 0 (10.2) v v

but it is easy to see that this subspace of R does not accurately describe the physical degrees of freedom in R/G (even though it is accurate enough for perturbation theory). infinitesimal perturbation in R

To see this, consider an

(10.3) A + A + & A P L J lJ

where &A,, is entirely localized in a particular region &V of space- time. We may however have

In order to impose the gauge condition (10.2) we now find an infinitesimal gauge transformation n = elgA that restores (10.2). It must

530

G. 't HOOFT

satisfy

(10.5)

but the inverse of the operator a,,D,, is non-local. space of R satisfying the gauge condition we find a perturbation which is spread out all over space-time, well outside 6V. this perturbation outside 6V is unphysical and in perturbation theory we learnt how to deal with this : the theory has "ghosts".

So in our sub-

Clearly

We know claim that beyond perturbation theory these ghosts obscure the physical contents of our theory. look for a "unitary gauge". in two steps. in our case L = U(I) .

Therefore we shall Our strategy is to determine this gauge

Let L v-pne of the largest Abelian subgroups of G,

L = u(1IN-l (10.6)

We call G/L the "non-Abelian part" of the gauge group and our first step will be to fix the "non-Abelian part" of the gauge redundancy. We choose a gauge condition C thatreduces the space R into a subspace that could be called

H1 = R/(G/L) If the gauge group G has N2-1 geperators and L has N-1 generators, then we choose N2-N real components for the gauge condition C, all invariant under L C Gbut not G itself. The second step is the choice of an N-1 component gauge condition that fixes the remaining invariance under L :

(10.7)

H - Hl/L - R/G (10.8)

But this second step is precisely the same as fixing the gauge in an ordinary Abelian gauge theory such as electromagnetism and is therefore much more trivial. To understand the physical contents of the theory one could jdst as well stop after obtaining (10.7) which is expected to describe Abelian charged particles and photons.

A variant on the Lorentz condition that reduces R to R/(G/L) is easy to find :

DOA'~ = o (10.9) P l J

0

lJ there D is the L-covariant derivative, containing the diagonal part of the vector field A only. of A only. Eq. (lo.!) has indeed N -N components.

qh is2the set of off-diagonal elements lJ

This gauge suffers from the ghost problem as much as the ordinary Lorentz gauge, and is therefore not suitable for understanding all physical degrees of freedom.

531


XI. UNITARY GAUGE

A unitary gauge must be picked in a way similar to the Abelian case. We need a field that transforms without derivatives under gauge transformations. field, call it X, transforms as the adjoint representation under G.

We will limit ourselves to the case that this

x + nxn-l (11.1)

Such a field namely can always be found. be

The simplest choice would

(11.2)

which is one of the components of the covariant curl G choice has the disadvantage of not being Lorentz-invariant. may choose a composite field :

This UV' One

(11.3)

This however does not work if G = SU(2) because then X would be proportional to the identity matrix. tor part. We could choose

We need a non-vanishing isovec-

* * ik 2 'k X1' = G ~ v D Gdv (11.4)

but this choice looks rather complicated. Perhaps the most practical choice would be to take ones refuge to an extra scalar field in the theory, giving it a sufficiently high mass value so that the theory is not changed perceptibly at low energies.

Our gauge condition will be that X is diagonal :

(11.5)

where the eigenvalues Ai may be ordered :

532

(11.6)

6. ' t HOOFT

What is the subgroup of the gauge transformations n under which (11.5) is invariant ? If we require

X' = n m - l = x then

fx,nl = 0

therefore, n is also diagonal :

and rince detn 1, we have

N C w i m O ill

(11.7)

(11.8)

(11.9)

Indeed, this is the largest Abelian subgroup L of G. If we write

(11 .lo) iw ~ = e

then the diagonal part Ao of A

1

transforms as lJ lJ

A' + AO - -a lJ lJ glJ

and the off-diagonal part ACh as U

ACh ij + e A i(wi - w j ) ch ij lJ lJ

(11.11)

(11.12)

Apart from the gauge transformations (11.11) and (11.12) all So our physical degrees of our fields are physically observable.

freedom are

- N-1 "massless" photons - 1/2 N(N-1) "massive" charged vector fields - N scalar fields Ai with the restriction : A1 > ,I2> ... > AN.

There is of course another constraint : depending on our choice for

533


X, we have that

satisfies (11.21, (11.31, or local U(1)N-l symmetry to be

(11.4). Of course we still have the removed by either conventional gauge

fixing, or the procedure described in the previous chapters.

X I I . A TOPOLOGICAL OBJECT

So far we assumed that the eigenvalues .xl, ..., ,% coincide nowhere. What if they do, at some set of points in space-time ? At those points, the invariance group is larger and a problem emerges with our enumeration procedure. nal points : a) are pointlike in 3-space, describing particle-like trajectories in space-time, and b) correspond to singularities in the fields 9J and $ . Abelian case, but the fact that these things are particle-like differs from the Abelian case, where the singular points were string-like.

The reason why the singularities in the generic case are pointlike is that the dimensionality of the space of field variables with two coinciding eigenvalues A is three less than the space with non-coinciding eigenvalues. Therefore the dimensionality of the points in space or space-time where two eigenvalues coincide is three less than that of space-time itself. This statement holds for any generic N x N hermitian matrix field X(x).

We now argue that these exception-

The argument for the singularity is similar to the

What is the physical nature of such a particle-like singulari- were ordered, we only need to consi- ty ? Since the eigenvalues

der the case thattwo successive A's coincide :

(12.1)

for certain j. point. Di and D2 may safely be considered to be diagonalized because the other eigenvalues did not coincide. The three fields Ea(x) are small because we are close to the point where they vanish. With respect to that SU(2) subgroup of SU(N) that corresponds to rotations among the jth and j + lst components, the fields form an isovector.

Let us consider a close neighbourhood of such a Prior to the gauge-fixing we may take X to be (12.21, where

534

G. 't HOOFT

One may write the center block as

+ + x AI + E . 0

(12.2)

(12.3)

yhere U

E field: the field E has a hedgehog conf&uration. i.e. diagonalization of X, corresponds to rotating and E away such that E3 is positive ( A > A +l). Thus is our unitary gauge,

are the Payli spin matrices. Close to a zero point of this But gauge fixing

2 j j

(12.4)

By now the reader may recognize this field configuration as the one for a magnetic monopole9). Indeed, fixing the L-gauge as well cannot be done without accepting a string-like singularity connecting zeros of opposite signature : the Dirac string.

terized with respect to the U(1IN subgroup of the extended gauge group U(N) :

The magnetic charges of the monopole ban most easily be charac-

+ 21 m = ( 0 , ..., 0, 5, - - g, 0, ... , 0) (12.5)

where the f 2n/g are at the jth and j+lst position. fundamental electric charge of the elementary representation. then see that m' actually only acts in the subgroup U(l)N/U(l) of SU(N) because the sum of all its charges vanishes. It is constructive to notice a subtile difference between this magnetic charge spectrum and the spectrum of the electrically charged gauge

g is here the We

535


particles A'' : from (11.12) we read off that these have electric charges '

..

if = (0, ..., 0, +g, 0, ..., 0, -g, 0, ..., 0) (12.6)

where 2 g occur at arbitrary, not necessarily adjacent, positions i and j. Again however the sum of the charges vanishes, so that we are really working in the Cartan group IJ(1IN-l, not U(l)N. Magnetic monopoles with

are possible sionality of general they

only if three or more eigenvalues coincide. The dimen- such points is at-least 8 less than space-time so in do not occur at single points. Rather, they should be

considered as bound states of "elementary" monopoles (12.5).

We conclude that we arrive at a picture where an Abelian gauge theory is enriched with magnetic monopoles, but because of the slightly different charge spectrum this picture is in general not "self ddal". monopoles we have here are "dynamical", that is, they will inevitably take part in the dynamics of the system.

Contrary to the case discussed in section VII, the

XIII. THE MACROSCOPIC VARIABLES

We have now arrived at a point where we could sketch a possible strategy for precise calculations for the dynamics of the system :

1) Consider the physical degrees of freedom in the space H1 = R/(G/L). We find N-1 sets of Maxwell fields, electrically charged fields (among which vector fields), and magnetically charged particles. The particular case of interest is now the possibility that magnetically charged particles "Bose-condensepl. If we ever are to understand such a mechanism in detail; the following step is probably necessary :

2 ) Eliminate the electric charges. With as much precision as possible we must compute all light-by-light scattering amplitudes and express them in term of an effective interaction lagrangian for the photon fields :

+ higher orders (13.1)

r(Au) =

536

G . t HOOFT

3) Now perform the "dual transformation". Since we have only Maxwell fields and magnetic charges interacting with them, we could replace 3 by 3 and 3 by -3, then introduce operator fields in the usual way for the monopole particles, which now look like ordinary electrically charged objects.

4) Work out the self-interactions among these magnetic monopoles. Then the question Set up a perturbation theory now in terms of 2T/g.

is :

5) Does, in terms of this perturbation theory, Bose condensation occur among these monopoles ? Is it reasonable to start with

(13.2)

If so, then the vacuum is a magnetic superconductor. formally have a negative mass-squared. tor electric charges are confined. apply qualitatively, after the interchange electric * magnetic.

The monopoles

The descriptions of section VII In this magnetic superconduc-

XIV. THE DIRAC CONDITION IN THE ELECTRIC-MAGNETIC CHARGE SPECTRUM

Fig 2 represents the spectrum of possible charges in the case that the gauge group G is SU(2).

m

- Fig 2

537


Horizontally is plotted the electric charge Q, vertically the magnetic charge 2. Elementary and bound state charges are indicated. The crosses represent a fundamental SU(2) doublet which may or may not have been added to the theory. The pure Maxwell field equations are now invariant under rotations of the figure about the origin. is why one is always free to postulate that the fundamental fields carry no magnetic monopole charge. fig 2 have to be rectangular ? tization condition allows more kinds of lattices and "oblique" lattices indeed may result in a non-Abelian gauge theory.

This

But does the lattice obtained in We will now argue that Dirac's quan-

The Dirac condition for a magnetic charge quantum m and the electric charge quantum q was

qm = 2nn (14.1)

where n is integer. condition for the Lorentz force that the magnetically charged particle exerts on the electrically charged object. But now consider two particles 1 and 2 both with various kinds of magnetic and electric charges.

To be precise this correspopds to a quantization

Then the Lorentz force quantization corresponds to

(14.2)

where the index i refers to the label of the species of photons and 1-112 is an integer relevant for particles 1 and 2, to be referred to as the Dirac quantum of particle (1) with respect to particle (2).

In our specific case of SU(N) broken down to IJ(1)"l we usually require

N

i=l 1 q;=O

N C m . = O

i=l (14.3)

Our charge lattice in this case will be spanned by 2(N-1) basic charges, to be labelled by an index A = 1, ..., 2N-2. Because of the invariance

we may always take

(14.4)

(14.5) m p ) = 0 for A = 1, ..., N-1 1

538

G. 't HOOFT

The gluons will provide us with a basis of electric charges:

for A - 1, ...* N-1 (A) A A+ 1 q i = g6i . - g6i

(The fundamental representation, if it occurs, could have

(14.6)

(14.7)

The magnetic monopoles have the remaining basic charges:

for A N, ...* 2N-2 ,(A) = & &A+l-N - & &A+2-N g i 8 % (14.8) i

It was Witten") who observed that monopoles mAy also carry electric charges. He found

p = - egL (A)

471 mi , for A = N, ..., 2N-2 i (14.9)

where that for any value of 8 the Dirac condition (14.2) is fulfilled.

is the instanton angle of the theory, 0 < 8 < 2n. Notice

It can be seen that this phenomenon, eq. (14.91, follows from the lagrangian

there

and Ga 2 corresponds to PV uv 4 E 3".sa

a

(14.10)

(14.11)

(14.12)

The canonical argument can be found in refs 101, 111, 8) .

In the case of SU(2) the charge lattice indeed becomes tilted now (Pig 3). charge lattice indeed turns back into itself, but the "elementary" monopole labelled by (2) in fig 3 is replaced by the "monopole gluon bound state" labelled (3). It seems that no fundamental distinction

It is remarkable that if 8 runs from 0 to 271 then the

539


m

will be possible between monopoles and monopole-gluon bound states.

XV. OBLIQUE CONFINEMENT

We can now ask which of the objects in the lattice of fig 3 will form a Bose-condensate. If it is a purely electrically charged object, (1) in fig 3, then we get the familiar Higgs theory or electric superconductor. will then be confined by linear potentials, because of the arguments presented in section VIZ. les may conceivably undergo Bose condensation, for instance charge 9(2) in fig 3. charges Bose-condense because if the electric ones condense then the magnetic ones have m2 -+ +-, not negative, and vice versa. case of larger N, Bose condensation can only take place among charges forming anylinear sublattice of the original charge lattice, as long as its members all have vanishing Dirac quanta with respect to each other.

All charges that are not on the horizontal axis

By duality we now expect that also monopo-

One cannot however have both electric and magnetic

In the

Now if 8 is switched on, then point (3) gradually takes the place of (2). (2) and (3) cannot both Bose condense because they have a non-vanishing relative Dirac quantum. So it is either (2) or (3). It is likely that Bose condensation in the (2)-direction is replaced by condensation in the (31-direction at n < 8 < 37r. This would then be a phase transition in 8, possibly of first order, just like the transition between Higgs and confinement.

Various attempts at dynamical calculations however indicate that

G. 't HOOFT

ate a n the confinement mechanism is not strong. could simply be that the monopoles then carry large electric charges and therefore may have larger self-energies contributing positively to their mass-squared Suggestions have been made that ate = n the Higgs mode reappears i2) or a Coulomb mode (no Bose condensation at all).

An explanation

I suggest yet a different condensation mode that could possibly occur in theories withe close ton. If neither (2) nor (3) condense because they carry large (but opposite) electric charges , then perhaps ( 4 ) which is a bound state of these two with much smaller electric charge condenses. This would only be possible if the lattice is oblique (8 # 0) so this mode is referred to as "oblique confinement". A theory with oblique confinement shows some peculiar features. stress that these will not occur in ordinary QCD because there we know that 0 5 0. with "technicolor" as we will show shortly.

We

OUT observations may be relevant for certain models

Returning to the case that our gauge group was SU(2), we first argue that the "quarks" (or "preons") in this oblique confinement mode are not confined in the usual sense. That is, if we attach a flavor quantum number to every type Qf preon, then physical particles transforming as the fundamental representation of the flavor group do occur. The preons are the crosses in fig 4 .

F i g 4 .

541


Since they are not on the line connecting the origin to the condensed object (arrow) they are confined. However, a bound state of a monopole (without flavor quantum numbers) and one preon does occur on that line (@), and hence is physical, and has the same flavor as the preon itself. But a price had to be paid : spin and statistics-properties of the physical object are opposite to that of the original preon! If the preon is a fermion, the liberated object is a boson and vice-versa. This is a consequence of a general rule : if two particles have an odd relative Dirac quantum, then the orbi ta l angular momentum in any bound state of the two is half-odd-integer, a well-known property of the SchrSdinger equation of an electrically charged particle in the field of a magnetic point-s~urcel~). That also the s t a t i s t i c s of the bound state eta an extra fermionic contribution has been shown by Goldhaberl4f. Let me give an outline of the argument.

If we wish to consider the statistics of two identical composite states A and B, both composed of a magnetic monopole M and an electric charge Q, (fig 5 ) then we would like to separate the center-of- mass motions of A and B from the orbital motions of M and Q inside A and B.

Fig 5

542

G. 't HOOFT

The particles Q see two Dirac strings, ending at both M ' s . The M ' s see two Dirac strings ending at both Q's. motion of A can only be split from the orbital motion of M and Q if the magnetic strings run in a direction opposite to that of the electric strings (see fig 51, so that if M hits an electric string, in the same timeQ hits a magnetic string. M and Q inside A and B, but only consider A and B as a whole. A feels both strings at B (and vice versa), in fact, these two are connected in such a way that one string results, running from infinity to infinity. This could be expected because A and B have the same electromagnetic charge combination (they were identical) so their relative Dirac quantum vanishes. Their relative motion (A against B) is as if they only were electrically charged. the familiar Coulomb SchrBdinger equation by removing this complete string by a single gauge transformation :

Now the center-of-mass

Now let us ignore the motion of Then

Therefore we obtain

i 4 ~ ~ +AB + +AB (15.1)

where 4AB is the angle by which A rotates around the B-string.

Now notice that if we interchange A and B this angle is 180°, so that

This t ion

XVI .

is how one can see an extra minus sign appearing in the commuta- properties of the particles A and B.

FERMIONS OUT OF BOSONS AND VICE-VERSA

Some exotic models can be constructed if we use obliaue confinement as a starting point. theory without fernions but with a scalar field 4 in the fundamental representation. Let 8 be close to T , and let us assume that the familiar Higgs mechanism takes place, described by

Consider for instance an SU(3) gauge

(16.1)

(though formally incorrect, as explained in section V, the notation is useful and adequate for our purpose). SU(3) then "breaks spontaneously" into SU(2).. One heavy neutral Higgs particle arises. The original octet of gauge fields splits into one neutral heavy vector particle and one SU(2) complex doublet of heavy vector particles.

543


The SU(2) triplet remains massless, and now we will assume that it produces oblique confinement in the SU(2) sector. of heavy vector bosons will appear as physical particles but disguised as fermions! without invoking anything resembling supersynrmetry.

The SU(2) doublet

Here we have an example of fermionic gauge "bosons"

Another model worth considering is a simplistic weak interaction theory, based upon SU(2)tc x (SU(2) x U(l))ew. for "technicolor" and ew for "electro-weak". are all in the usual representation of (SU(2) x U(l))ew and singlets under SU(2)tC. Now add one fermion multiplet transforming just like all other fermions under (SU(2) x U(l))ew but also as a 2 under tc. Assume oblique confinement. Then this fermion will be liberated, but disguised as a boson. Probably it will have a spin-zero component. Since it is in line with the condensing monopole bound states it may Bose-condense itself. So we obtain a scalar field transforming just as the fermions under (SU(2) x U(l))ew and a non-vanishing vacuum expectation value : a model for the Higgs particle. Indeed, its Bose condensation could well be responsible for the oblique confinement mode in the first place, so here it was not even necessary to consider the monopole-dyon bound state.

There "tc" stands Quarks and fermions

Unfortunately, elegant as it may be, this model seems to suffer from the same shortcomings as the more conventional technicolor ideas : it is hard to reproduce the required Yukawa couplings between this scalar field and the other fermions.

XVII. OTHER CONDENSATION MODES

It will be clear from the previous sections, by looking at the electric-magnetic charge lattice, that even more exotic forms of oblique confinement can be imagined. Just assume condensation of bound states with three or more monopoles. Such a condensation mode would be required for instance if we would wish to liberate the fundamental triplets in an SU(3) gauge theor?. A fundamental exercise tells us the properties ay. In any case, all these different confinement modes will be separated from each other by sharp phase transition boundaries, which should show up in the solutions of the theory when the parameter 9 is varied.

that these triplets do not switch their spin-statistics

In principle each point on the electric-magnetic charge lattice may correspond to a possible phase of the system. There may however be features which cannot easily be understood in terms of our intermediate physical degrees of freedom with Abelian electric and magnetic charges. detail by Bais15). with an isospin-two Higgs field, $ab (a, b = 1, 2, 3 ) .

We have in mind a condensation mode studied in more The simplest example is an SU(2) gauge theory

544

G. 't H O O n

Let us assume

(17.1)

If we were just dealing with a global syrmmetry, we would say that SU(2) is spontaneously broken into a subgroup D (the invariance group of eq. (17.1)). ding to rotations of'spinors over 90" :

D = (k I, k ial, k io2, 2 ia

Now D is the discrete subgroup of SU(2) correspon-

(17.2)

If the aymetry is local then D is not really a global invariance of the vacuum. constructed which are characterized by the following boundary condition at infinity :

3

What we do see is that magnetic vortex tubes can be

(17.3)

where Sl is multivalued. turn8 into itself multiplied with an element D1 of D. are non-commuting.

If we go around the vortex once, then n

Physically this means the following (fig 6) These vortices

Fig 6

If two strings, characterized by different, non cormnuting elements D1 and D2, approach each other at right angles (fig 6a), then cannot pass each other without leaving a connecting string D3 (fig 6b).

they

The element D3 is given by

-1 -1 (17 .4 ) D g = D D D D 1 2 1 2

545


Clearly the non-commuting properties of the original gauge group were crucial for underetanding this phenomenon, so that our Abelian physical variables are not useful here. Indeed, we could ask the question whether the dual of this "Bais mode" exists, with electric strings having similar properties ? As yet, the answer to that question is unknown.

REFERENCES

1.

2.

3.

J. Goldstone, Nuovo Cim. 19 154 (1961) F.S. Abers and B.W. Lee, Phys Reports E, 1 (1973) and references therein.

P.W. Higgs, Phys. Lett. l2, 132 (19691, Phys. Lett. l3, 508 (1964), Phys. Rev. 145, 1156 (1966)

F. Englert, R. Brout, Phys. Rev. Lett. 12, 321 (1964) G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Lett. 3

T.W.B. Kibble, Phys. Rev. 155, 627 (1967) 585 (1964)

4. B . S . DeWitt, Phys. Rev. 162, (1967), 1195, 1239

5. G. 't Hooft and M. Veltman, "DIAGRAMMAR", CERN report 7319

6.

(1973)

H.B. Nielsen, P. Olesen, Nucl. Phys. E, 45 (1973) B. Zumino, in "Renormalization and Invariance in Quantum Field

Theory ed. E.R. Caianiello Plenum Press New York (1974) p. 367

7. P.A.M. Dirac, Proc. Roy. Soc. A133, 60 (1931)

8. G. 't Hooft, Nucl. Phys. B, to be published

9. A.M. Polyakov JETP Lett. 20, 194 (1974). G. 't Hooft, Nucl. Phys. E, 276 (1974) E. Witten, Phys. Lett. - 86B, 283 (1979)

Phys. Rev. 14, 8 m 1 9 4 8 )

10. 11. A. Salam and J. Strathdee, Lett. in Mathematical Physics 2, 505

(1980)

12. C. Callan, private communication (1979)

13.

14. 15. F.A. Bais, private comunication.

R. Jackiw and C. Rebbi, Phys. Rev. Lett. a, 1116 (1976) G. 't Hooft and P. Hasenfratz, Phys. Rev. Lett 2, 1119 (1976) A.F. Goldhaber, Phys. Rev. Lett. 36, 1122 (1976)

546

CHAPTER 7.3

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

G. 't Hooft Institute for Theoretical Physics University of Utrecht, Netherlands

1. INTRODUCTION

"Quantum Chromodynamics" is a pure gauge theory of fermions

and vector bosons that is assumed to describe the observed strong

interactions.

beyond the usual perturbation expansion. Not only must we ex-

plore the mathematics of solving the field equations non-perturbatively; it is more important and more urgent first to find a decent formulation of these equations themselves, in such a way

that it can be shown that the solution is uniquely determined by

these equations. We understand how to renormalize the theory to any finite order in the perturbation expansion, but it is ex-

pected that this expansion will diverge badly, for any value

of the coupling constant. define the theory. But the renormalization procedure is not known

to work beyond the perturbation expansion. A clear example of the possible consequences of such an unsatisfactory situation was the

recent suprising demonstration

ries have parameters 0 , in the form of an angle, that describe certain symmetry breaking phenomena in the theory, but never show up

within the usual perturbation expansion because they occur in the

To get an accurate theory it is mandatory to go

1-5)

Thus the expansion itself does not yet

6-1 0 ) that all non-Abelian gauge theo-

The Whys of Subnuclear Physics, Edited by A. Zichichi (Plenum).

547

G. 't HOOFT

combination

where g is the gauge coupling constant. increase in our understanding of non-perturbative field theory but this understanding is not yet complete and, in principle, more of such surprises could await us.

This discovery marks an

The situation can be compared with the infinity problems in the older theories of weak interactions. solved by the gauge theories for which an acceptable and unique regularization and renormalization scheme was found. sent strong interaction theory, again a "regularization scheme" must be found, this time for "regularizing" the infinities encountered in summing the perturbation expansion.

Those problems were

For our pre-

An interesting attempt t o give a non-perturbative formulation of the (renormalized) theory is the introduction of a space-time lattice in various ways . But, here also, a proof of uniqueness could not be given (will the continuum-limit yield one and only one theory?) and of course the 8 phenomenon mentioned before was not observed in the lattice scheme. So, the lattice theories are still a long way off from answering our fundamental questions.

11-1 3 )

It is more important for us to make use as much as possible of the important pieces of information contained in the coupling constant expansion. Because of asymptotic freedom this ex-

pansion tells us precisely what happens at asymptotically large external momenta and it would be a waste to throw that information away.

1 4 - 1 6 )

Which tools could we use to extend our definitions? One is a

study of the theory at complex values of the coupling constant. That this is possible in some particular cases is explained in Section 4. at certain points in the complex g2 plane and stay regular at others.

We may find that Green's functions must become singular


Suppose we would discover that there are only singularities on the real axis for - 5 g2 < -a (unfortunately the true situation is

much less favourable). Then we could make a mapping:

with the inverse

td=..4QIL(4-u)-2

The whole complex plane is now mapped onto the interior of the unit circle, with the singularities at the edge. the perturbation expansion in terms of U instead of g have convergence everywhere inside the circle*), which implies convergence for all g2 definite improvement. we find in Section 4 is too complicated for this method to work: the origin turns out to be an essential singularity. shall show how this knowledge of the complex structure may be used to give a very slight improvement in the perturbation expansion

(Section 9).

If *we rewrite then we

not too close to the negative real axis. A

Unfortunately, the structure for complex g2

Still, I

The other tool to be used is the Bore1 resummation procedure, to be explained in Section 5. considered whose perturbation expansion terms are defined to be the previous ones divided by (n-l)! where n is the order of the expansion terms. The singularities of these new'functions can be found and the analytic continuation procedure as sketched above can be applied to obtain better convergence. types of singularities: the other due to renormalization phenomena. ly controversia1,they are the only ones that should occur in

A new set of Green's functions" is

There seem to emerge two one due to the instantons in the theory,

The latter are slight-

*) We make use of a well-known theorem for analytic functions that says that the rate of convergence of an expansion around the origin is dictated by the singularity closest to the origin.

549

G. 't HOOFT

quantum electrodynamics, giving that theory of n! type divergence.

The first few terms for the electron 8-2 do not seem to diverge

the way suggested by this singularity. perturbation expansion for QED but in practice the "improvement" seems to be bad.

Formally we can improve the

Our work is not finished. What remains to be done is to prove that all singularities in the Borel variable have been found and to find a prescription how to deal with those singularities that are on the positive real axis. integrals that link the Borel Green's functions with the original Green's functions make sense and a good theory is found (see Sec- tion 9).

Finally, it must be shown that the

The problem of quark confinement can probably be related to certain singularities on the positive real axis, because these singularities arise from infra-red divergences (Section 8 ) .

2. DEFINITION OF THE COUPLING CONSTANT AM) NASS fA3AMETEP.S IN TEWS OF M G E PfOMENTUM LIMITS

This section contains the mathematical definitions of the parameters in the theory, so that the statements in the other sections can be made rigorous and free of unnecessary assumptions. could be skipped at first reading.

It

The dimensional renormalization scheme is a convenient way of defining a perturbation series of off-mass shell Green's functions with some coupling constant g (U) as an expansion parameter. The

subscript D stands for dimensionally renormalized . Each term of the perturbation series is finite. There is an arbitrariness in the choice of the subtraction point U (which has the dimension of a mass). change in gD and

1 7 - 2 0 ) D

The theory is invariant under a simultaneous provided that

550

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS?

Here we show explicitly the minus sign for the first coefficient. This minus sign 'is a unique property of non-Abelian gauge fields and is responsible for "asymptotic freedom" (at increasing we get decreasing g2, see Refs. 14-16).

2 1 - 2 3 ) The coefficients 61 and 62 are known . Since we shall

try to go beyond perturbation expansion we must be aware of two facts: First, the perturbation expansion is expected to diverge for all g2 and is, therefore, at this stage, meaningless as soon as we substitute some finite value for g2. procedure has only been defined in terms of the perturbation expansion (Feynman diagrams). meaning at all as a finite number. The correct interpretation of these series is that they are asymptotic series valid for infinitesimal g only, or, equivalently, valid only for asymptotically large momentum: p2 - O(p2) + W. Thus, gi may not be so good to use as a variable for a study of analytic structures at finite complex values.

Second, the dimensional

Consequently, gi may not have any

We shall now introduce another parameter gi, that may just as well be used instead of gi. ments :

It is defined by the following require-

When + w, then

(2.2a)

(2.2b)

55 1

G. 't HOOF1

The series in (2.2b) must stop after the second term. In pertur-

bation theory these requirements have a unique solution for g2

For instance, we get that the rest term in (2.2a) is R'

In the dimensional renormalization scheme also the mass parameter

was cut-off-dependent (of course, one cannot define such a thing

as a physical quark mass parameter, which would have heen cut-off

independent) :

Again ve define by

and

where the series in (2.5b) stops efter the a1 term. 21-23)

In OCD the parameters al, 61, 62 are known

(2.5b)

(2 .6 )

552

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMIW?

We emphasize that the new parameters gR and 5 are better than the previous ones, because for any theory for which the perturbation expansion is indeed an asymptotic expansion, they are completely

finite and non-trivial. traction point p.

tions; (2.2b) and (2.5b):

Of coutse, they still depend on the sub- At infinite p they coincide with other defini-

at finite p they are finite because we can solve Eqs.

(2.7b)

Here PO and mo are integration constants. under the renormalization group. Thus, po is a true parameter that fixes the gluon couplings, and has the dimensions of a mass. each quark in the system we have a mass parameter mo. clear from the derivations, ~0 and mo actually tell us how the theory behaves at asymptotically large energies and momenta.

They are invariant

For As must be

The value of po for QCD is presumably of the order of the p

mass, and mo will be a few MeV for the up and down quarks, 100 MeV or so for the strange quark, etc. drop the terms containing 8 2 , and the quark masses will be put equal to zero.

For simplicity, we will often

3. THE !tENORMALIZATION GROUP EQUATION

Now let us consider the Green's functions of the theory. For definiteness, take only the two-point functions (dimensionally renormalized)

553

G. ‘t HOOFT

24). They satisfy a renormalization group equation .

G D L (K,yc ,$)=O (3.2)

R Here f3 is the truncated, finite f3 function for the constant g i

as it occurs in Eq. (2.2b), but y(g2) is still an infinite series. We wish to do something about that also. Let us first make clear how to interpret Eq. (3.2). Consider the p versus g2 plane. Sup- pose we choose a special curve in that plane, where g2 depends on U such that

R

R

(3.3)

then it follows from (3.2) that

or

That implies that, if we stay on one of tbe curves ( 3 . 3 ) , then (3.2) reduces to (3.5) which can easily be integrated. But the integration constant will still depend on k 2 and on the curve

554

CAN WE MAKE SENSE OUT OF "QUANTUM CHAOMOOYNAMICS"?

chosen, that is, on the constant ~ 0 , which we get in solving (3.3),

see Eq. (2.7a). Thus we get

with

and for dimensional reasons, G(k2,po) can only depend on the ratio

k2/d.

For our purposes it is now important to observe the following. The coefficients z o and z1 are clearly very important as g2 + 0, but the terms zpgi + ... in (3.7) can simply be absorbed in a redefinition of the coefficients al, ... in (3.1). That way we get new, improved functions GR that can be written as

In a typical operators

example where we study the time ordered product of two

- ~ ( o ) y ( o ) a d i i ; ( x ) v ( ~ )

corresponding to the Q channel, we have

555

G. 't HOOF1

(3.9)

4. ANALYTIC STRUCTUXE FOR COMPLEX g2

In Eq. (3.8) we can write (neglecting for simplicity the 8 2

terms)

This is a function of one single parameter

Complex x corresponds to either complex k2, real g2 or complex g2, real k2.

at real g2, complex k2 (Fig. 1). The singularities are at kZ real and negative (i.e. Minkowskian). That is when x - real + ( 8 1 / 2 ) *

(2n t l)ni, n integer. Choosing now k2 real and positive we find the same singularities at

Now, on physical grounds, we know what we should expect

They are sketched in Fig. 2.

556

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS"?

R Fig. 1 Expected analytic structure of G for complex k2. The wavy line is a cut (in Baryonic channels this cut starts away from the origin, in mesonic channels, since we have put mf = 0, the cut starts at zero because the pion is massless). The dotted crosses are singularities to be expected in the second Riemann sheet (resonances).

557

G. 't HOOFT

Fig. 2 Resulting analytic structure for complex g i . The single cut of Fig. 1 now reproduces many times on semi circles. These semi-circles are only slightly distorted due to the 8 2 term in Eq. (2.7a). The arrow shows the region where perturbation expansion is done. The cut on the left is due to the Z-factor in ( 3 . 8 ) .

The conclusion of this section is obvious: we find such a bad accumulation of singularities at the origin that the analytic continuation procedure given in the introduction will never work. We

must look for a more powerful technique.

558

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS?

5. BOREL RESUMMATION

We now assume that our Green's functions can be written as a Laplace transform of a special' type

F(z) .can be found perturbatively: If

then

where the

(5.3)

6 function is understood to be included in the integral (5.1). One can check this trivially by inspection. The importance of this is that the series (5.3) converges much faster than (5.2). Contrary to (5.2) it may very well have a finite radius of convergence (this is at present believed to be the case for all renormalizable field theories). If F(z ) can now be analytically continued to all real positive z, and if, for some g2, the integral (5.1) converges, then the series (5.2) is called Borel summable. Be will call F ( z ) the Borel function corresponding to the Green's function G (g2). R

First, we wish to find out where we can expect singularities Let us illustrate an interesting feature in the case of in F(z ) .

an over-simplistic "field theory", namely field theory at one space- time point. Remember that field theoretical amplitudes can be written as functional integrals, with a certain number of integration variables at each space-time point"). If we have one space-time point and one field, then there is just one integration to do

559

G. 't HOOFT

(5.4)

where V(x) - V3x3 + V4x4 + ... and the factor g in front is just for convenience. parameter. Note the minus signs in the integrand. This anticipates that we shall always consider Field theory in Euclidean space-time. Let us rescale the fields, and the action,

g2 comes out this way as the usual perturbation

Our integral becomes

Comparing this with the Bore1 expression (5.1), we immediately find F(z) :

Thus at given z we must find all solutions of S'(A) = z, which we can call Ai(z). The result of the integral is


This outcome reveals the singularities in the z plane: find the solutions of

we must

( 5 . 9 )

to be recognized as the classical field equation of this system.

At a solution z of (5.9) we have

(5.10)

- and so we find a square root branch point at z = z.

Consider now multidimensional integrals of the same type. The

integral (5.7) then corresponds to an integral over the contours in

A space defined by the equation S'(1) = z. only at those values of z that are equal to the total action S' of

a solution of the equation

+ You get singularities

(5.11)

because those are the contours that shrink to a point (in the case

of a local extremum) or have crossing points (saddle points).

Again, Eq. (5.11) is nothing but the classical Lagrange equation for the fields 1. Conclusion: to find singularities in F we have to search for finite solutions of the classical field equations in

Euclidean space-time. Their rescaled action S' corresponds to singularity points

these singularities are branch points.

in the z place for the function F. In general,

In our actual four-dimensional

561

G. 't HOOFT

field theories that are supposed to describe strong (or weak and electromagnetic) interactions; such solutions indeed occur, and are called "instantons", because they are more or less instantaneous and local in the Euclidean sense . Their action (in the

e - 1 0 ) case of QCD) is S ' = 8.rr2n, where n counts the "winding number" and so we may expect singularities in the complex z plane at z =

* 8n2n.

2 5 , 2 6 , 6 , 1 0 )

6 . UNIVERSALITY OF THE BOREL SINGULARITIES

The student might wonder whether the conclusions of the previous sections were not jumped to a little too easily. nected Green's functions in field theories are not just multi(infinite) dimensional integrals but rather the ratio of such integrals with some source insertion and an integral for the vacuum, and then often differentiated with respect to those source insertions. these singularities and/or create new ones? functions perhaps not have their own singular points?

The con-

Do all these additional manipulations not alter or replace Do different Green's

Let us for a moment forget the renormalization infinities, to Then the answer to these equa- which we devote a special section.

tions is reassuring. Multiplications, divisions, exponentiations can be carried out, after which we shall always find the singularities back in the same place as they were before, possibly with a different power behaviour. Bore1 transforms, let us formulate some simple properties.

To understand this general property of

Let

Then, if

562


then Fx .

And let

then

Here the E symbols are there just to tell us to leave the 6 symbols out of the integration. It is easy to show that if (6.3) is solved iteratively, then the series converges for all z, as long as F1 stays finite between E and z .

Now note that we may choose the contours ( 0 , z ) so that they avoid singularities. Only if z is a singularity of either F1 or Fz, or both, then FJ(z) in Eq. (6.2) will be singular. Also F~(z) in Eq. (6 .3 ) is only singular if F (z) is singular. Note, however, that if a singularity lies between 0 and z then the contour can be chosen in two (or more) ways, and, in general, we expect the outcome to depend on that. Thus if we start with pure pole singularities, they will propagate as branch points in the other Bore1 functions.

In quantum field theories, the Green's functions are related through many Schwinger-Dyson equations and Ward-Slavnov-Taylor identities. divisions in these equations without displacement, they must occur in all Borel-Green's functions at the same universal values of z

Since singularities survive the multiplications and

563

G. 't HOOFT

(unless miraculous cancellations occur; clude that possibility). These singularities will, in general, be of the branch-point type.

I think one can safely ex-

In particular, those singularities that we obtained through the solutions of the classical equations, will stay at the same position for all Green's functions.

7. SINGULARITIES IN F(z) DUE TO INSTANTONS

Let us consider these classical equations in Euclidean space- time for the various field theories. First take theory, for simplicity, without mass term. Rescaling the fields and action the usual way

'Q'=JX'4 s r - i S'

A (7.1)

we have

It turns outz5) that a purely imaginary solution exists for the equations 6 SIC4 = 0 (here, %indicates derivative in the Euler sense), namely

(7.3)

Here p is an arbitrary scale parameter (after all, our classical ac-

tion is scale invariant).

In spite of this solution being purely imaginary, it is important to us because it indicates a singularity in F(z) away from the positive real axis. The corresponding value for S ' is

564

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMOOYNAMICS"?

2 S ' t 46 'It ( 7 . 4 )

So the singularity occurs at z =.-16n2, indeed away from the positive real axis. Such singularities are relatively harmless, since F is only needed for positive z. tinuation procedure sketched in the Introduction to improve convergence for the series in z.

We may invoke the analytic con-

Now, let us turn our attention to Quantum Chromodynamics. Here we have a real solution in Euclidean space:

(7.5)

where are certain real ~oefficients~'~') and p is again a free a w

scale parameter. One finds for the action

Thus z = 8a2 is a singularity on the positive real axis. In fact, we can also have n instantons far apart from each other, so we also expect singularities*) at z * 81~*n. Now, Green's functions are obtained from F ( z ) by integrating from zero to infinity, over the positive real axis. Do the singularities on the real axis give un- surmountable problems? I think not, although the correct prescription will be complicated. A clue is the following. The single- instanton contribution to the amplitudes has been computed directly

*) A more precise analysis suggests that only those multi-instantons with zero total winding number (that is, as many instantons as anti-instantons) will give rise to ordinary singularities that limit the radius of convergence of F(z) . The others give dis- continuities rather than singularities.

565

G. 't HOOFT

in the small coupling constant limit. A 1ike27-3 0 )

typical result goes

That is already a Green's function, the one we would like t o obtain after integrating

A function F ( z ) that yields (7.7) exists

Indeed, a "singularity" at z = BIT'.

functions will show the same exponential in their g dependence, the universality theorem of the previous section is obeyed. is important is that by first computing (7.7) one can short-circuit the problem of defining an integration over such singular points. Thus the instanton singularities at the right-hand side on the real axis will not destroy our hopes of obtaining a convergent

theory. The reason is that the physics of the instanton is understood. The situation is less clear for the other type of singularities that we discuss in the next section.

We see that, since all Green's

What

8. OTHER SINGULARITIES IN F,

In principle, the instanton-singularities in F can also be understood within the context of ordinary perturbation expansion, by a statistical treatment of Feynman diagrams31). derivation here, but the following argument has been given. previous section, we have never bothered about the renormalization procedure that is supposed to make all diagrams finite.

We do not show the In the

Suppose we

566


had a strictly finite theory, with bounded propagators, bounded integrals and all that.

bounded by a pure power law as a function of their order n. The

only way that factors n! can arise is because there are n! diagrams

at nth order and they may not cancel each other very well. This is how in the statistical treatment the instanton singularity occurs.

But in realistic four-dimensional renormalizable field theories, the

power law for individual Feynman diagrams no longer holds. A simple example is quantum-electrodynamics. We consider the diagrams of the

type shown in Fig. 3.

Individual diagrams in such a theory are then

Fig. 3 Fourth member of a subclass of diagrams discussed in this section.

It is the class of diagrams with n electron bubbles in a row, which in itself closes again a loop. It is well known that each electron

bubble separately behaves for large k2 as

and each propagator as (k')''. the k variable behaves as

Thus, for large k2 the integrand in

567

G. 't HOOFT

where a is some fixed power. tractions to make the integral converge, and in order to obtain physically relevant quantities, such as a magnetic moment, the leading coefficient a becomes 3 or larger. by a new variable x, then (8.2) becomes proportional to

After having made the necessary sub-

Let us replace log k2

Thus the integral over x will grow as n + 03 like

A more precise analysis shows that C should be proportional to the first non-trivial f3 coefficient

In the expansion for F ( z ) the factor n! is removed, as usual. It is clear that a new singularity develops at

It seems to be a universal phenomenon for all field theories, and not related to any instanton solution. Our definition for fil was positive for asymptotically free theories and negative otherwise. So, the singularity is at negative real z and therefore harmless if our theory is asymptotically free, but for non-asymptotically free theories such as QED and real axis.

we have singularities on the positive Since a detailed understanding of the ultraviolet

568

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS"?

1

-6 -3 instantons

behaviour of non-asymptotically free theories is lacking, there may exist no cure for these singularities then. with the instanton singularities.

This is in contrast

An important observation has been made by G. Parisi3'). The ultraviolet behaviour of A$4 and QED are well understood in the limit N + =, where N is the number of field components. A systematic study of the singular point (8.6) is then possible. Parisi found in A$4 theory a conspiracy between diagrams such that the first singularity at a = 3 cancels. have as (8.2) but with a > 3, after all necessary sudtractions. At present, it is not understood whether this conspiracy is accidental for A$' theory with N components, or whether it is a more general phenomenon. It does seem that only the first singularity may be subject to such cancellations.

In total the integrals do be-

? 4 6 8

renormalms

In Figs. 4 and 5 we show the complex planes for the Borel variables z in theory and in QED, respectively. The singularities discussed in this section are called "renormalons" for short.

The situation for QCD is more complex. Not only do we have the renomalons at points on the negative real axis but also there are such singularities on the positive real axis. They are due to the infra-red divergence of the theory. The mechanism is otherwise the

G. ‘t HOOFT

Fig. 5 Singularities for QED. Here the units are 3n, if a i s the original expansion parameter.

I R divcrpcncies

Fig. 6 Bore1 z plane for QCD. The circles denote IR divergences that might vanish or become unimportant in colour-free channels.

the same as discussed for the ultraviolet singularities (Fig. 6) .

An interesting speculation is that these infra-red singularities are only surmountable in colourless channels, but the integra-

tion over these singularities becomes impossible in single quark-

or gluon- channels. It is likely that these singularities are re-

lated to the quark confinement mechanism.

570


9. SECOND BOREL PROCEDURE

Many features of the singularities in the complex z plane of the Bore1 functions F(z ) are still uncertain and ill-understood. But from the foregoing we derive some hopes that it will be possible to obtain F(z) for 0 5 z < 00 for asymptotically free theories, such as QCD. The only thing to be investigated then is how the integral in

behaves at-. almost certainly: no. Consider massless QCD and its singularities in complex g2 plane as derived in Section 3. there are singularities when

Does the integral converge? The answer t o this is

According to Eq. ( 3 . 3 ) R

where the real number may be arbitrarily large. in Eq. (9.1) we find that

Substituting that

g ( d + &(2a+4)) dz 2

- S-kz) 0 e (9 .3 )

must diverge. not give rise to divergences. than any exponential of z. Note that (9.3) contains an oscillating term, It is likely then, that at z -f m, F(z) does not only grow very fast, but also oscillates with periods 4/81 or fractions thereof.

We had assumed that the singularities at finite z did So F(z) must diverge at large z worse

Can we cure this disease? We have no further clue at hand which could provide us with any limit on the large z behaviour of F. But there is a way to express the unknown Green's functions in terms of

57 1

G. 't HOOFT

Let us treat the divergent a more convergent integral than (9.3).

integral (9.1) on the same footing as the divergent perturbation ex-

pansions which we had before.

integral

We consider a new, better converging

(9.4)

We may hope that this has a finite region of convergence, from which we can analytically continue. Note the analogy between (9.1) and

(9.4) on the one hand, and (5.2) and (5.3) on the other. The inte-

gral relation between W and G, analogous to (5.1), is

Now, remembering that instead of varying g2 we could vary k2, replacing

R So that, ignoring the Z factor that distinguishes G from G (see Eq. ( 3 . 8 ) ) , one gets

(9.7)

Now we can easily prove that, if our theory makes any sense at all, there may be no singularities in W(s) on the positive real axis, and

the integral (9.7) must converge rapidly.

(9.4) makes sense, then our problems are solved.

f 0 1 lows :

Thus, if the integral The proof goes as

572


C(k2) satisfies a dispersion relation: it is determined by its imaginary part. We have, at k2 = -a2, a > 0,

where p is usually a positive spectral function. into (9.7) we get

Substituting (9.8)

Thus, p(a) and W(w2) are each other's Fourier transform. verse of (9.9) is

The in-

(9.10)

A possible singularity at a2 = 0 is an artifact of our simplifica-

tions and can be removed. It is important to observe that (9.10) severely limits the growth of W(s) at large s so that ( 9 . 7 ) is always convergent.

Conclusion: this section results in an improvement on per- The physically relevant quantities can be ex- turbation theory.

pressed in terms of integrals of the type (9.4), which converge better than the original ones of type (9.1). this improvement is sufficient, i.e., whether (9.4) actually converges in some neighbourhood of the origin.

It is not known whether

Even if the important open questions mentioned in these lectures cannot be answered we think that refinement of these techniques will lead to an improved treatment of strong coupling theories.

573

G. 't HOOFT

REFERENCES

3)

4)

5)

F.J. Dyson, Phys. Rev. 85 (1952) 861.

L.N. Lipatov, Leningrad Nucl. Phys. Inst. report (1976) (unpublished).

E. Brbzin, J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. D15 (1977) 1544 and (1977) 1558.

G. Parisi, Phys. Letters 66B (1977) 167.

C. Itzykson, G. Parisi and J.B. Zuber, Asymptotic estimates in

The singularities I discuss in Section 8 of my lectures were Quantum Electrodynamics, CEN Saclay preprint.

assumed to be absent in this paper.

G. 't Hooft, Phys. Rev. Letters 37 (1976) 8 .

A.M. Polyakov, Phys. Letters 59B (1975) 82 and unpublished work.

C. Callan, R. Dashen and D. Gross, Phys. Letters 63B (1976) 334.

R. Jackiw and C. Rebbi, Phys. Rev. Letters 37 (1976) 172.

See S. Coleman's lectures at this School.

K.G. Wilson, Phys. Rev. D10 (1974) 2445.

J. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975). L. Susskind, lectures at the Bonn Summer School (1974).

S.D. Drell, M. Weinstein, S. Yankielowicz, Phys. Rev. D14 (1976) 487 and DL4 (1976) 1627.

G. 't Hooft, Marseille Conf. on Renormalization of Yang-Mills fields and applications to particle physics, June (1972) (unpublished).

H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346.

D.J. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343.

G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189.

C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566.

J.F. Ashmore, Lettere a1 Nuovo Cimento 4 (1972) 289.

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CAN W E MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

20) G. 't Hooft, Nuclear Phys. B61 (1973) 455.

32)

D.R.T. Jones, Nuclear Phys. B75 (1974) 531.

A.A. Belavin and A.A. Migdal, Gorky State University pre- preprint (January 1974).

W.E. Caswell, Phys. Rev. Letters 33, (1974) 244.

S. Coleman, lectures given at the "Ettore Majorana" Int.

Note: School of Subnuclear Physics, Erice, Sicily (1971).

group equation, which can be avoided according to later formulations on the renormalization group (Ref. 20).

we drop the inhomogeneous parts of the renormalization

S. Fubini, Nuovo Cimento 34A (1976) 521.

A.A. Belavin et al., Phys. Letters 59B (1975) 85.

G. 't Hooft, Phys. Rev. D14 (1976) 3432.

F.R. Ore, "How to compute determinants compactly", MIT preprint (July 1977).

A.A. Belavin and A.M. Polyakov, Nordita preprint 7711.

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C.M. Bender and T.T. Wu, Phys. Rev. Letters 27 (1971) 461; Phys. Rev. D7 (1972) 1620.

G. Parisi, private communication.

575

CHAPTER 8

QUANTUM GRAVITY AND BLACK HOLES

Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . 578 ‘Quantum gravity”, in Z h d s in Elementary Particle Theory, eds. H. hllnik and K. Dietz, Springer-Verlag, 1975, pp. 92-113 . .. . . ‘Claesical N-particle cosmology in 2 + 1 dimensions”, C7ass. Quantum Gmv. 10 (1993) S7SS91 ... .. .... .. . . . ... . ... ........... .. . . “On the quantum structure of a black hole”, Nucl. Phys. B266

with T. Dray, “The gravitational shock wave of a massless particle”, Nucl. Phye. B26S (1985) 173-188 .... . . . . . . . . . . . . . . . . . . . . .. . . . . . . , . . . . %matrix theory for black holes”, Lectures given at the NATO Adv. Summer Inst. on New Symmetty Principles in Quantum Field Theory, eds. J. F’rohlich et al., Car-, 1992, Plenum, New York, pp. 275-294 .............................................................. 645

584

606

(1985) 727-736 .......................................................... 619

629

577

CHAPTER 8

QUANTUM GRAVITY AND BLACK HOLES

Introduction to Quantum Gravity [8.1]

The years 1970-1975 gave us a remarkable insight in the general features common to all elementary particle theories. They must contain vector fields in the form of gauge fields, and spinors and scalars that must form representations of the vector field gauge group. The spectrum of all possible particle types that populate our universe will be represented in the most economical way possible, and if more types exist that are as yet unidentified then there are two corners to search for them: either they are very weakly interacting or they have very high masses.

The strength of these results is their completeness: we know exactly how to enumerate all possibilities. There are two weaknesses however. One is that there will be a built-in imprecision as explained earlier, because of the divergence of the loop expansion. We might try to avoid this by postulating asymptotic freedom, but this gives too stringent a restriction that can easily be made undone by suspecting as yet unknown particles in the hidden corners as just mentioned. The other weakness is that the number of different viable theories is still too large for comfort. Why did Nature make the choice she did?

On first sight the gravitational force makes things much worse. Including gravity maka our theory hopelessly nonrenormalizable; infinite series of uncontrollable counter terms undermine our ability to do meaningful calculations. But worse still, we do not at all understand how to formulate precisely the first principles for such theories. We can write down mathematical equations, but upon closer examination these turn out to be utterly meaningless. This problem persists up to this day, in spite of the commercials from the superstring adherents.

It is not hard to speculate on radically new theoretical ideas. I have toyed with discrete theories for space-time myself. But then one opens a Pandora’s box full of schemes and formalisms that are completely void when it comes to making any firm predictions. In my opinion the art is not to obscure our view by clogging the

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scientific journals with speculations. The problem is to deduce from hard evidence features that have to be true for the world at Planckian time and distance scales.

What do we know about the perturbative expansion with respect to Newton’s gravitational constant IC ? In the first paper of this chapter I address this question. It is shown that pure gravity, without any matter coupled to it, is one-loop renormalizable. This means that uncertainties due to nonrenormalizability and arbitrariness of counter terms are down by terms of order IC’. An important lesson learnt here is that the “field variable” g,,(xlt) is not quite observable directly. When we do particle scattering experiments admixtures to this tensor of the form R,, and Rg,, are both necessary for renormalization and undetectable experimentally. Note that these admixtures are of higher order in IC and therefore infinitesimal. There is as yet no clash with causality.

If matter fields are added such admixtures do become visible when compared to similar admixtures in the other fields. Consequently these theories are not even oneloop renormalizable.

Introduction to Classical N-Particle Cosmology in 2 + 1 Dimensions [8.2]

The previous paper treated quantum gravity in a way I would now call “conventional”, which means that we assumed the problem to be to find a consistent prescription for calculating elements of the scattering matrix, under circumstances where we can still speak of a simply connected space-time, locally smooth and asymptotically flat.

But gravity raises a couple of conceptual problems when one tries to do better. One simple demonstration of this is to consider an extremely simplified version of gravity: gravity in 2 space- and onetime dimension. In such a world there are no gravitons, but the gravitational force does exist, and even though the Newtonian attraction between stationary objects vanishes the problems raised by the curvature of space-time are considerable. My aim was originally to obtain a theory that can be formulated exactly and

has two limits: one is a we& coupling limit in which ordinary (scalar) fields are weakly coupled to gravity and the perturbative approach makes sense; the other is a limit where Planck’s constant vanishes such that we have a finite number of massive point particles gravitating in a closed universe.

But the system of classical point particles gravitating in a universe that is closed due to this gravitational force exhibits a number of features that predict evil for attempts to “quantize” it. For one, it may not be possible to define an S matrix. The reason is that under a wide class of initial conditions the universe does not end up in an asymptotic state where particles fly away from each other in classical orbits. Instead of this the universe may end up in a “big crunch”. I discovered this in attempting to disentangle a paradox presented in a paper by J. R. Gott. He had argued that under certain conditions a pair of particles in this 2+1 dimensional

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universe may be surrounded by a region where a test particle may follow a closed timelike trajectory in space-time, and even move backwards in time. I could not believe that a sound physical system should admit such an event, and my intuition turned out to be correct: before the closed timelike curve gets a chance to come into being a “big crunch” ends it all. In a complete universe the region described by Gott is an illegal analytic extension; it is shut off from the physical world by a shower of particles crunching together.

The paper reproduced here explains what happens. I should add that I later found the transitions described in Figure 8 to be strictly speaking not complete. Other transitions are possible where polygons, as described in the paper, disappear altogether, and also sometimes an edge shrinks to zero but the system continues without the topological transition. These new possible transitions do not affect any of the conclusions however.lg

Introduction to On the Quantum Structure of a Black Hole [8.3]

The previous paper demonstrates some of the difficulties that will form considerable obstacles when one wishes to formulate “quantum cosmology”. The big crunch as an asymptotic state is extremely complex. Numerical experiments later showed that this crunch is in some sense “chaotic”, and so we will not have an easy time enumerating the possible asymptotic states in a quantum mechanical “measuring process”.

But one could argue that “quantum cosmology” is not our most urgent project. I think it is legitimate to ask first for a theoretical scheme describing scattering proceases in regions of the universe that are surrounded by asymptotically flat space time. This may not be possible in 2+1 dimensional worlds, but should be an option in 3+1 dimensions. Any finite number of particles with limited total energy will be surrounded by asymptotically flat spacetime sufficiently far away. So should a scattering matrix exist there?

This seem to be plausible, at first sight. But now a new problem emerges, and it is an enormous one. We have to have at least four space-time dimensions. We better avoid taking more than four because then none of our ordinary models is even renormalizable. But in four dimensions the gravitational force, already before quantization, exhibits a fundamental instability: gravitational collapse can occur, and black holes can form.

Black holes will be a new state of matter, and it will be impossible to avoid them. They are regular solutions of the gravitational equations applied exclusively in those regions of spacetime where we may expect that we know exactly how to formulate what happens. An event horizon may open up long before matter fields or particles here had a chance to interact in a way not covered by the Standard Model. Therefore it doesn’t help to postulate that perhaps “black holes do not exist”. They exist as legitimate solutions of the classical theory, and the real problem is to ask what they have to be replaced with in a completely quantized theory.

”G. ’t Hooft, Class. Quantum Gravity 10 (1993) 1023.

At first sight a black hole seems to be just an “extended solution”, of the kind I could have covered in my B d lectures, Chapter 4.1. But if that were 80 it should have been possible to construct all their excited quantum modes just by performing standard field theory in a black hole background. This calculation has been done, or rather, that is what the authors thought. The outcome is a big surprise: not at all a reasonable spectrum of black hole states seems to emerge. What one finds is a continuous spectrum, as if black holes were infinitely degenerate. An immediate clash arises with the determination of the black hole entropy, which can be derived from its thermodynamical properties and it is finite.

So the background field calculations for black holes are incorrect. It turna out not to be hard to guess why they were incorrect: the gravitational interactions between ingoing and outgoing objects were neglected, in the first approximation. Under normal circumstances, as in conventional theories, one may always presume that such interactions can be postponed to later refinements, but gravity is not a conventional theory. We argue that these mutual interactions become infinitely strong and therefore may not be neglected even in the zeroth approximation. The big question is how to do things correctly then. One can do better, but a completely satisfactory approach is still missing.

I do not expect that a “completely satisfactory” formalism will be found very soon, because it can probably not be given without a completely satisfactory theory of quantum gravity itself. But what we can do is ask very detailed questions, insist on seneible answers, and hope that these answers, even if incomplete, also furnish a better insight in the problem of quantizing gravity itself. Another way of phrasing our point is this: if the gravitational force gives us so many problems when it becomes strong, why not concentrate entirely upon the question on how to formulate the theory where it is as strong as possible, which is near black holes?

The quantum black hole paradox began to intrigue me around 1984 and I began to study it in great detail. I am quite convinced that I made substantial progress in understanding its nature and its relevance for the entire theory of quantum gravity. I first learned that the real paradox already occurs at our side of the horizon, and that the “back reaction”, i.e. the mutual (mostly gravitational) interactions between outcoming Hawking particles and objects falling into the hole somewhat later, is of crucial importance to resolve the problem. I found out that in contrast to a wide- spread misconception, the resolution of the problem will not require fundamental nonlocal interactions, but that it is strongly connected to very fundamental issues in the interpretation of quantum mechanics. I also found that one probably has to give up the idea that the quantum state of the black hole as it is seen by the outside observer can be described independently of the adventures of an observer who cro88e8 the horizon to enter into the black hole. The inside region does not make my sense at all to the outside observer. One should never try to describe “super observers”, a hypothetical simultaneous registration of what happens both inside and outside.

I was annoyed by the lack of interest in this problem and my work on it by most of the particle physics community and the general relativists, two groups who for a

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long time were unaware of the mutual incompatability of their views. But now that I am compiling my papers, early 1993, all this has changed. The literature is now being flooded with papers dealing with precisely the problem that kept me busy already for so long. I cannot say that these are already at a stage of affecting my present ideas, but an acceleration is certainly taking place, and the chances that new breakthroughs will occur (probably not by renown physicists but more likely by some young student), are now better than ever.

The following section is part of one of my first papers on black holes. The remainder of the paper is less relevant for the arguments that follow, and was therefore omitted. To understand the WKB approximation in Eq. (3.5) it is necessary to replace the coordinate r temporarily by U according to

r - 2M = e‘,

because in this new coordinate the wave packets are much more regular than before and the WKB procedure much more accurate.

Introduction to the Gravitational Shock Wave of a Massless Particle [8.4]

So what is the gravitational interaction between ingoing and outgoing particles near a black hole horizon? This turns out to be sizeable even if the particles have very low energies as seen by a distant observer. It is large if the time interval between the ingoing and the outgoing object, as seen by the distant observer is of order M logM in Planck units or larger. That is still small compared to the total expected lifetime of the black hole. To find this result we need to compute the gravitational field of one of them. First take coordinates in which this is weak and therefore easy to compute. Then transform to any coordinate frame one wishes. It often becomes strong.

The calculation is done in the next paper, written together with Tevian Dray. The result is that there is a plane associated to the particle’s trajectory in space- time (the “shock wave”) such that geodesics of particles crossing this plane are shifted. The fact that the shift depends on the transverse distance makes it visible. The effect is much like a sonic boom. A fragile detector might break in pieces if it encounters such a shock wave.

Introduction to the S-Matrix Theory for Black Holes [8.5]

Because of the associated shock wave any object falling into a black hole has an effect upon the outgoing “Hawking” particles. The effect of these shifts may seem to be minute, until one realizes that the phases of the wave functions of particles coming out very late are extremely sensitive to precisely shifts of this nature. Thus we can now understand why information going into a black hole is always transmitted onto

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particles coming out, be it extremely scrambled. We can use this to construct a scheme for a unitary black hole S matrix.

As long as we concentrate on effects at moderately large distances across the horizon all interactions that contribute to this S matrix, gravitational and otherwise, are known. So many details of the S matrix we were searching for can be derived. Unfortunately, to see that this matrix will be unitary, and to determine the dimensionality of the Hilbert space in which it acts, we need more information concerning short distance interactions. Our hope was that there are still elements of the theory of general relativity that we have not yet used, and that they would be sufficient to remove most of our present uncertainties. Our feeling is that a lot of further inprovements should be possible along these lines, but they have not (yet?) been realized. The paper gives a good impression of the way I think about the problem at present. The algebra at the end should be seen as speculative and somewhat mysterious in its lack of unitarity. The idea is that it should inspire others to find improved methods along similar lines.

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CHAPTER 8.1

QUANTUM GRAVITY G. 't Hooft

University of Utrecht, T h e Netherlands

1. I n t r o d u c t i o n

The g r a v i t a t i o n a l f o r c e i s by f a r t h e weakest e l e m e n t a r y i n t e r - a c t i o n between p a r t i c l e s . It is s o weak t h a t o n l y c o l l e c t i v e f o r c e s between l a r g e q u a n t i t i e s o f matter are o b s e r v a b l e a t p r e s e n t , and i t i a e lementary because i t a p p e a r s t o obey a new symmetry p r i n c i p l e i n n a t u r e : t h e i n v a r i a n c e under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s .

l a t i v i t y , p h y s i c i s t s have t r ied t o " q u a n t i z e g r a v i t y " ' ) , and t h e f i rs t t h i n g t h e y r e a l i z e d i s t h a t t h e t h e o r y c o n t a i n s n a t u r a l u n i t s o f length (L), t i m e ( T ) and mass (MI. I f

Ever s i n c e t h e i n v e n t i o n of quantum mechanics and g e n e r a l r e -

2 K = 6.67 m3/kg Sec

i s t h e g r a v i t a t i o n a l c o n s t a n t , t h e n

L = = 1.616 m

T = G = 5.39 s e c 2 and M = = 1.221 lo2' eV/c

= 2.177 g

But t h e n t h e t h e o r y c o n t a i n s a number o f o b s t a c l e s . F i r s t there are t h e c o n c e p t u a l d i f f i c u l t i e s : t h e meaning o f s p a c e and t i m e i n Ein- s t e i n ' s g e n e r a l r e l a t i v i t y as a r b i t r a r y c o o r d i n a t e s , i s very d i f f e r e n t from t h a t of s p a c e and time i n quantum mechanics . The m e t r i c t e n s o r g y v , which used t o b e a lways f i x e d and f l a t i n quantum f i e l d t h e o r y , now becomes a l o c a l dynamical v a r i a b l e .

Advances have been made, from d i f f e r e n t d i r e c t i o n s 2 B 3 s 4 ) , t o d e v i s e a language t o f o r m u l a t e quantum g r a v i t y , b u t t h e n t h e n e x t problem arises: t h e t h e o r y c o n t a i n s e s s e n t i a l i n f i n i t i e s such t h a t a f i e l d t h e o r i s t would s a y : it is n o t r e n o r m a l i z a b l e . T h i s problem, d i s c u s s e d i n d e t a i l i n s e c t i o n 12, may b e v e r y s e r i o u s . It may v e r y wel l imply t h a t t he re e x i s t s no well de te rmined , l o g i c a l , way t o combine

Trendr in Elementary Particle Theory, Edited by H. Rollnik and K. Dietz (Springet-Vtrlag. 1975).

gravi ty wi th quantum mechanics from first p r i n c i p l e s . And then one i s led t o t h e ques t ion : should g r a v i t y be quan t i zed a t a l l ? After a l l , such quantum e f f e c t s would be small, t o o small perhaps t o b e e v e r measurable. Perhaps t h e t r u t h i s very d i f f e r e n t , bo th from quantum theory and from general r e l a t i v i t y .

what happens a t a l eng th s c a l e L and a t ime s c a l e T i a incomplete, and we would l i k e t o improve i t . We c la im that i t i s very worthwhile t o t r y and improve ou r p i c t u r e s t e p by s t e p , as a p e r t u r b a t i o n expansion i n K. In the fo l lowing it i s shown how t o apply t h e techniques of gauge f i e l d theory and ga in some remarkable r e s u l t s .

Whatever one should , o r should not do, o u r p re sen t p i c t u r e of

The s e c t i o n s 2-1) d e a l w i t h the convent iona l t heo ry of gene ra l r e l a t i v i t y , seen from the viewpoint o f a gauge f i e l d t h e o r i s t . I n sec t ion 5 it i s i n d i c a t e d how q u a n t i z a t i o n could be c a r r i e d out i n p r inc ip l e , bu t i n p r a c t i c e we need a more s o p h i s t i c a t e d formalism t o ease c a l c u l a t i o n s .

T h i s formalism, t he background f i e l d method215), i s exp la ined i n sec t ion 6-11 . I n t h e s e s e c t i o n s we mainly d i s c u s s gauge t h e o r i e s , and gravi ty i s ha rd ly mentioned; g r a v i t y i s j u s t a s p e c i a l ca se here .

Back t o g r a v i t y i n s e c t i o n 12, where w e d i s c u s s numerical r e s u l t s . It i s shown there why only pure g r a v i t y is f i n i t e up t o t h e one-loop cor rec t ions .

2 . Gauge Transformations

The under ly ing p r i n c i p l e of the theory of gene ra l r e l a t i v i t y is invariance under gene ra l coord ina te t r ans fo rma t ions ,

t’” = f ” ( X ) . ( 2 . 1 )

It i s s u f f i c i e n t t o cons ide r i n f i n i t e s i m a l t r ans fo rma t ions ,

XI’’ = xp + rl”(x) , n i n f i n i t e s i m a l . (2.2)

Or, i n o t h e r words, a func t ion A(x) i s transformed i n t o

A’(x) = A(x + n ( x ) ) = A(x) + n*(x)a,A(x) . ( 2 . 3 )

If A does not undergo any o t h e r change, t h e n i t i s c a l l e d a s c a l a r . We Call t h e t r ans fo rma t ion (2:j) simply a gauge t r ans fo rma t ion , genera ted

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A by the ( i n f i n i t e s i m a l ) gauge f u n c t i o n n ( x ) , t o b e compared w i t h Yang- Mills i s o s p i n t r a n s f o r m a t i o n s ,genera ted by gauge f u n c t i o n s Aa(x).

F o r t h e d e r i v a t i v e of A(x) w e have

a,,A’(X) = aPA(x) + n x DP a ~ ( x ) + t-,’a,apA(x) , (2.4)

where 11’

ming t h e same way, i. e. s t a n d s f o r a ox, t h e u s u a l convent ion . Any o b j e c t A t r a n s f o r -

JP P P

w i l l b e c a l l e d a covec tor . We sha l l a l s o have c o n t r a v e c t o r s B p ( x ) ( n o t e t h a t the d i s t i n c t i o n i s made by p u t t i n g t h e i n d e x u p s t a i r s ) , which t r a n s f o r m l i k e

by c o n s t r u c t i o n such t h a t

t r a n s f o r m s as a scalar. S i m i l a r l y , one may have t e n s o r s w i t h a n ar- b i t r a r y number of upper and lower i n d i c e s .

F i n a l l y , t h e r e w i l l be d e n s i t y f u n c t i o n s w(x) t h a t t r a n s f o r m l i k e

They e n a b l e us t o write i n t e g r a l s of s c a l a r s

Iw(X) A ( X ) d b ( X ) j

which are comple te ly i n v a r i a n t under l o c a l gauge t r a n s f o r m a t i o n s (under c e r t a i n boundary c o n d i t i o n s ) .

F o r the c o n s t r u c t i o n of a complete gauge t h e o r y it is of i m - p o r t a n c e t h a t t h e gauge t r a n s f o r m a t i o n s form a group. O f c o u r s e t h e y do,

and hence we have a J a c o b i i d e n t i t y . L e t u ( i ) be t h e gauge t r a n s f o r - mat ions g e n e r a t e d by n P ( i J x ) . Then i f

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3. The Metr ic Tensor

In much t h e same way as i n a gauge f i e l d theo ry6) , we ask f o r a dynamical f i e l d tha t f i x e s the gauge of t h e vacuum by htwing a nonvanishing vacuum expec ta t ion va lue . (Cont rary t o t h e Yang-Mills case it seeme t o be imposs ib le t o cons t ruc t a reasonable "symmetric" theory . To t h i e end we choose a two-index f i e l d , g,,(x), which i s symmetric i n i t s i n d i c e s ,

%U = gul, (3.1)

and i t s vacuum expec ta t ion va lue i s

(3.2) <gu,,(x) 'o = uu

(our me t r i c corresponds t o a pure ly imaginary time coord ina te ) .

With g,,,, , o r i t s i n v e r s e , g" , we can now de f ine l eng ths and time- in t e rva l s a t each po in t i n space-time:

f a l * = g*va%u J

and g,,,, can be used t o r a i s e o r lower i n d i c e s :

AM g""~" , A,, = %,,A" , e t c . ! 3 . 3 )

Just as i n t h e Yang-Mills case , w e can now d e f i n e cova r i an t d e r i v a t i v e s :

D,,A = a A ( t h e d e r i v a t i v e of a s c a l a r t ransforms as a v e c t o r ) , ' DpAv = aUAu - l'aUvAa ,

D'B" = a,,~" + r" B" . va

( 3 . 4 )

The f i e l d P t u i s c a l l e d t h e C h r i s t o f f e l symbol and i s y e t t o be def ined . First w e write down how i t should t r ans fo rm under a gauge t r ans fo rma t ion , such that t h e above-defined cova r i an t d e r i v a t i v e s be r e a l t e n s o r s :

" = riu + (o rd ina ry terms f o r 3-index t e n s o r ) + a,,a,,n A . (3.5) rlJV

We see t h a t no harm is done by making t h e r e s t r i c t i o n t h a t

r;,, = rt,, , (3.6)

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because t h e symmetric p a r t of l' on ly is enough t o make ( 3 . 4 ) cova r i an t .

W e now d e f i n e the f i e l d l' by r e q u i r i n g

= 0 * (3 .7)

( f rom which fo l lows : DiguV

number of equa t ions . By w r i t i n g Eq. ( 3 . 7 ) i n full w e f i n d t h a t i t i s easy t o s o l v e

= 0 ) . We see t h a t we have e x a c t l y t h e r igh t

Note tha t rA is no t a cova r i an t t e n s o r . UV

The cova r i an t d e r i v a t i v e may be used j u s t l i k e o r d i n a r y de r iva - t i v e s when a c t i n g on a product :

D ( X Y ) = ( D X ) Y + XDY , (3 .9)

bu t two cova r i an t d i f f e r e n t i a t i o n s do no t n e c e s s a r i l y commute:

D,,DVAa # D,D,,A, . (3.10)

Ins t ead , w e have:

D,,D,Aa - DvD,,Aa = RBavU AB (3.11)

w i t h

The comma d e n o t e s ' o r d i n a r y d i f f e r e n t i a t i o n . S ince t h e 1 . h . s . of eq . (3 .11) i s c l e a r l y cova r i an t and A B is an a rb i - t r a r y v e c t o r , RBav,, t ransforms a s an o rd ina ry t e n s o r , i n c o n t r a s t w i t h rBap. It is c a l l e d t h e Riemann o r cu rva tu re t e n s o r (see the s t anda rd t e x t books). I n d i c e s can be r a i s e d o r lowered fo l lowing (3 .3 ) . Without p u t t i n g i n any f u r t h e r dynamical equa t ion , one f i n d s t h e fo l lowing i d e n t i t i e s ,

RaBtjy 3 E -

R a 0 ~ 6 = Ry6a0

R a B ~ 6

RaBy6;p + R a B 6 ~ ; ~ + R a ~ w ; ~

' RaydB + R a ~ ~ y = 0 , (3 .13)

= O .

The semicolon denotes covariant differentiation. Further, we define*)

which satisfy, according to (3.13):

The metric tensor also enables us to define a density function Gee (2.711,

In the quantum theory we shall encounter a fundamental problem: instead of g,, we could go over to a new metric g;,, with, for instance,

So in a curved space there is some arbitrariness in the choice of metric (Section iP).We bypass this problem here.

4. Dynamics

The question now is whether we can make the fields g propagate. UV Indeed we can, because we can construct a gauge invariant action integral

where U is to be identified with the usual gravitational constant:

For simplicity we shall take the units in which

2 ( 4 . 3 ) C

1 6 1 1 K = 1

At a later stage one could put U back in the expressions to find that the expansion in numbers of closed loops will correspond to an expansion

*)Note that there is a sign difference compared to some earlier papers.

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w i t h r e s p e c t t o K .

One can a l s o add o t h e r f i e l d s i n t he Lagrangian , f o r i n s t a n c e

(4 .4 )

where 4 i s a s c a l a r f i e l d . We s h a l l n o t r e p e a t h e r e t h e u s u a l arguments t o show t h a t v a r i a t i o n of t h e Lagrangian ( 4 . 4 ) r e a l l y l e a d s t o t h e familiar g r a v i t a t i o n a l i n t e r a c t i o n s between masses, and t o u n f a m i l i a r i n t e r a c t i o n s between o b j e c t s w i t h a g r e a t v e l o c i t y ( " g r a v i t a t i o n a l magnetism"). The e q u a t i o n f o r t he g r a v i t a t i o n a l f i e l d w i l l b e

(4 .5 )

where T i s t h e u s u a l energy-momentum t e n s o r ( E i n s t e i n ' s e q u a t i o n ) . The a c t i o n ( 4 . 1 ) has much i n common w i t h t h e a c t i o n

CtV

i n Yang-Mills t h e o r i e s . As w e s h a l l i n d i c a t e i n the n e x t s e c t i o n , a massless g r a v i t o n w i t h h e l i c i t y 5 2 w i l l p r o p a g a t e . N o t i c e t h a t w e have been l e d t o E i n s t e i n ' s t h e o r y o f g r a v i t y a lmost a u t o m a t i c a l l y . It seems t o be t h e s i m p l e s t c h o i c e i f w e ask f o r a t h e o r y w i t h i n v a r i a n c e under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s .

5. Q u a n t i z a t i o n

The f i r s t t h i n g w e must do i s make a s h i f t

and c o n s i d e r A as t h e quantum f ie lds . Here t h e problem mentioned i n t h e i n t r o d u c t i o n p r e s e n t s i t se l f : what i f w e s ta r t w i t h

uv

The answer i s t h a t as l o n g as w e take as o u r e l e m e n t a r y f i e l d any l o c a l f u n c t i o n of t h e gUv, t h e o b t a i n e d p h y s i c a l a m p l i t u d e s w i l l b e t h e same. The t r a n s f o r m a t i o n from one f u n c t i o n ' ( f o r example, g,,,,) t o t h e o t h e r ( f o r example, g p v ) w i l l b e accompanied by a J a c o b i a n , o r c l o s e d l o o p s of f i c t i t i o u s p a r t i c l e s ( s e e the Z i n n - J u s t i n l e c t u r e s on gauge t h e o r i e s ) ,

590

But the propagators of t h e s e p a r t i c l e s are cons t an t s or pure polynomials i n k, because t h e t r ans fo rma t ion i s l o c a l . I f we now t u r n on t h e d i -

mensional r e g u l a r i z a t i o n procedure, which has t o b e used i n o rde r t o get gauge i n v a r i a n t r e s u l t s , then t h e i n t e g r a l s over polynomials,

vanish7). Th i s is the reason why i t makes no d i f f e r e n c e whether we s t a r t from Eq. ( 5 . 1 ) o r Eq. ( 5 . 2 ) . Non-local func t ions of g a r e not allowed. These non-local t r ans fo rma t ions would g ive rise t o f i c t i t i o u s pa r t i c l e s that do c o n t r i b u t e i n t h e c u t t i n g rules ') , and they are outlawed once uni ta r i ty has been e s t a b l i s h e d for t h e choice ( 5 . 1 ) o r ( 5 . 2 ) . By choosing a convenient gauge, comparable wi th the Coulomb gauge i n QED, it is indeed not d i f f i c u l t t o e s t a b l i s h t h a t t h e theory i s u n i t a r y , i n a H i l b e r t space wi th massless p a r t i c l e s wi th h e l i c i t y 2 2 .

IJV

We can work out the b i l i n e a r p a r t o f t h e Lagrangian i n Eq. ( 4 . 1 ) i n terms of the f i e l d s A *

,,V*

&= - $ a A ) 2 1 6 $ a I J A a a ) 2 + 1 L2 + h ighe r o r d e r s , ~r aB 2 I J

with 1 2 IJ aa - aaApa *

L,, = - a ~

( 5 . 3 )

( 5 . 4 )

For p r a c t i c a l c a l c u l a t i o n s i t seems t o be convenient t o choose the gauge

The Lagrangian i n t h i s g a u g e , x + & C , has as a k i n e t i c term

where W i s a ma t r ix b u i l t from 6-functions. The propagator i e t hen

( 5 . 5 )

(5.7)

Just i n o r d e r t o show the d ive rgen t c h a r a c t e r of t h e compl ica t ions involved, we show he re t h e Faddeev-Popov ghost') f o r t h i s gauge, ob ta ined by sub jec t ing L,, t o an i n f i n i t e s i m a l gauge t ransformat ion:

59 1

The Lagrangian (4.1), expanded i n powers of A w i t h t h e gauge- PV '

f i x i n g term ( 5 . 5 ) and t h e ghos t term ( 5 . 8 ) , form a p e r f e c t quantum t h e o r y . It i s , however, more compl ica ted t h a n n e c e s s a r y , because gauge i n v a r i a n c e i s g i v e n up r i g h t i n t h e b e g i n n i n g by a d d i n g t h e bad terms (5 .5 ) and ( 5 . 8 ) . The background f i e l d method, d i s c u s s e d i n t h e next s e c t i o n s , i s much more e l e g a n t because gauge i n v a r i a n c e i s e x p l o i t e d i n a l l s t a g e s of t h e c a l c u l a t i o n s , t h u s s i m p l i f y i n g t h i n g s a l o t .

6 . A P r e l u d e f o r t h e Background F i e l d Technique: Qauge I n v a r i a n t Source I n s e r t i o n s

The methods d e s c r i b e d i n t h i s and t h e f o l l o w i n g s e c t i o n s are not o n l y s u i t a b l e f o r quantum g r a v i t y , b u t have a very wide a p p l i c a b i l i t y , i n p a r t i c u l a r i n gauge t h e o r i e s , f o r i n s t a n c e f o r c a l c u l a t i o n s of r e - n o r m a l i z a t i o n group c o e f f i c i e n t s . F i r s t , i t i s convenient t o in t roduce t h e concept of a gauge i n v a r i a n t s o u r c e i n s e r t i o n . T h i s i s a n a r t i f i c i a l term i n the Lagrangian of t h e form

where J(x) i s a c-number s o u r c e f u n c t i o n and R(x) i s some gauge inva- r i a n t combiqa t ion of f i e l d s , c o n t a i n i n g a l i n e a r p a r t and q u a d r a t i c o r h i g h e r o r d e r c o r r e c t i o n s . L e t us g i v e some examples:

i ) I n a gauge t h e o r y w i t h Higgs mechanism:

(6.1)

where i s t h e Higgs f i e l d and D 4 i s i t s c o v a r i a n t d e r i v a t i v e . We take U apJp = 0.

I f < + > = F ;

O I F + @ ,

592

then (6.1) becomes

The first t e r m emits o r absorbs s i n g l e n e u t r a l v e c t o r p a r t i c l e s . The other terms are h i g h e r o r d e r c o r r e c t i o n s , e m i t t i n g two or th ree p a r t i c l e s a t once. I n g e n e r a l one can f i n d f o r a l l p h y s i c a l p a r t i c l e s a similar gauge-invariant s o u r c e t h a t produces them predominant ly and one by one.

ii) I n a p u r e gauge t h e o r y ( i n momentum r e p r e s e n t a t i o n ) :

J:(k)

where kuJE(k h i g h e r terms

= 0 . I n t e g r a t i o n o v e r k, p a n d f o r q i s unders tood . The are de termined by r e q u i r i n g gauge i n v a r i a n c e under

This i m p l i e s

A source i n s e r t i o n t h a t s a t i s f i e s t h e s e c o n d i t i o n s can be w r i t t e n i n a closed form, i n terms of a n a n t i s y m m e t r i c t e n s o r s o u r c e J E v ( x ) :

( 6 . 5 ) J;,,(X) T [exp g A A ( x ' ) dx'A]ab Guv(x) b , p a t h

b C where Ga = a Aa - a A' + g fabc A,, A,, , and A A s t a n d s f o r t h e m a t r i x uv U V P

A:' = 'abcAbX '

The i n t e g r a l i s a l o n g a p a t h from i n f i n i t y t o x. The symbol T s t a n d s for time o r d e r i n g a l o n g the p a t h . Expanding (6 .5) g i v e s

X Jauv(X) @ E v ( X ) + g fabc J Ab,(xI) dx' GEv(x) + ...) (6 .6)

p a t h

iii) In g r a v i t y one c a n do a similar t h i n g : a s o u r c e J""(.x) s a t i s f y i n g

can b e coupled t o A i n eq . (5.1), and one can add h ighe r o r d e r cor-

r e c t i o n s t o r e s t o r e gauge inva r i ance . The d e t a i l s are not ve ry r e l e v a n t

f o r what fo l lows , bu t n o t e t h a t we r e s t r i c t o u r s e l v e s t o t h e perturba- t ive theory .

uv

The ampli tude w i t h which a gauge i n v a r i a n t sou rce emits a s i n g l e

p a r t i c l e , o b t a i n s h ighe r o rde r c o r r e c t i o n s , see Fig . I .

+ + etc. - t ( - )

Fig. 1

A sou rce can emit a s i n g l e p a r t i c l e i n d i f f e r e n t ways

The S-matr ix can now r e a d i l y be ob ta ined i n a gauge-independent way by cons ide r ing vacuum-vacuum t r a n s i t i o n s i n t h e presence o f a gauge- invar ian t sou rce . Then t h e e x t e r n a l l e g s are amputated and put on mass-she l l . From Fig. 1 i t w i l l be c l e a r t h a t i n p r a c t i c e one can j u s t as w e l l c a l c u l a t e t h e amputated Green 's f u n c t i o n s d i r e c t l y and mul t ip ly them wi th some renorma l i za t ion f a c t o r 2 . This f a c t o r 2 however may be gauge-dependent. The c o r r e c t f a c t o r can b e ob ta ined by normal iz ing t h e imaginary p a r t of t h e two-point f u n c t i o n 8 ) .

7. The Background F i e l d Method

The background f i e l d method, u s e f u l f o r gauge t h e o r i e s , i s p r a c t i c a l l y i n d i s p e n s i b l e f o r quantum g r a v i t y . L e t t h e f u l l Lagrangian be

& ( A ) I a i n V ( A ) t J R ( A ) - t C 2 ( A ) +.tF*-' , (7.1)

where &"'(A) i s t h e complete gauge i n v a r i a n t Lagrangian f o r a l l f i e l d s Ai. A s f i e l d combinat ion, and J i s a c-number sou rce func t ion . C ( A ) i s the gauge- f ix ing f u n c t i o n and z'-' d e s c r i b e s t h e a s s o c i a t e d Feynman-deWitt- Faddeev-Popov ghost' >2 The gauge t r ans fo rma t ion law w i l l be w r i t t e n as

desc r ibed i n t h e p rev ious se 'c t ion , R ( A ) i s a gauge inva r i an t

A l l i r r e l e v a n t i n d i c e s have been suppressed .

594

A: = Ai + s:j A a A j + t: Aa (7 .2)

where Aa(x) i s t h e i n f i n i t e s i m a l g e n e r a t o r . t and s are c o e f f i c i e n t s b u i l t from numbers and t h e space- t ime d e r i v a t i v e a

We now i n t r o d u c e t he n o t i o n of a c l a s s i c a l f i e l d A'', which is a f u n c t i o n of t h e c-number s o u r c e s J. U s ~ a l l y ~ * ~ ) one d e f i n e s i t s J dependence by r e q u i r i n g t h a t Acl s a t i s f i e s the c l a s s i c a l e q u a t i o n s of motion. This i s s u f f i c i e n t as l o n g as we are i n t e r e s t e d i n diagrams w i t h a t most one c l o s e d loop . I n these n o t e s we s h a l l make t h a t r e s t r i c t i o n a l s o , b u t i t must be k e p t i n mind t h a t f o r a p p l i c a t i o n s of these t e c h n i q u e s a t s t i l l h ighe r o r d e r s i t w i l l be more convenient t o add quantum c o r r e c t i o n s t o t h e e q u a t i o n of motion. A t t h i s stage we r e q u i r e Acl a l s o t o be i n t h e gauge

lJ*

C(AC1) = 0 . ( 7 . 3 )

Next, we per form a s h i f t :

A = A C 1 + O , ( 7 . 4 )

where now 4 i s t h e new quantum f i e l d , and we r e w r i t e t he Lagrangian i n terms of 0 :

Here zl is l i n e a r i n 4; z2 i s q u a d r a t i c i n + and U(g3) i s of higher order i n +.

Now, s i n c e Acl sat isfies the e q u a t i o n of motion

&a - = o 6AC1

and C(AC1)= 0, a l l terms l i n e a r i n + c a n c e l :

( 7 . 6 )

(7 .7)

595

So t h e s o u r c e J i s n o t coupled t o terms l i n e a r i n @, and t h e r e are no v e r t i c e s w i t h o n l y one @ - l i n e . T h e r e f o r e , t h e @ - l i n e s can only go around i n l o o p s , and i f w e c o n f i n e o u r s e l v e s t o one-loop diagrams t h e n we can n e g l e c t t h e terms Cf(@'). One loop diagrams now o n l y consis t of a +-loop, w i t h b i l i n e a r i n s e r t i o n s o f c l a s s i c a l s o u r c e s depending on Acl and J. But remember t h a t Acl i s a f u n c t i o n of J, which can b e ob-

t a i n e d by s o l v i n g t h e c l a s s i c a l e q u a t i o n s o f motion by i t e r a t i o n . This i t e r a t i o n p r o c e s s cor responds t o a d d i n g a l l p o s s i b l e trees t o t h e s i n g l e l o o p . Thus we reproduce t h e o r i g i n a l Feynman r u l e s , w i t h t h e o n l y change tha t l o o p l i n e s are c a l l e d @ - l i n e s , and tree l i n e s are c a l l e d Acl-lines.

8. The BackRround F i e l d Gauge

The r e l e v a n t p a r t o f t h e Lagrangian ( 7 . 5 ) is

T h i s is j u s t a n o r d i n a r y gauge f i e l d Lagrangian w h e r e Z j n v = 5 t i s i n v a r i a n t under what w e sha l l c a l l gauge t r a n s f o r m a t i o n s of t y p e 8 :

(8.21

c l The gauge i s f i x e d by t h e f u n c t i o n C ( A t $I).

Now there i s a n o t h e r i n v a r i a n c e of xinv , a l s o broken by t h i s C term. We c a l l t h i s a gauge t r a n s f o r m a t i o n o f t y p e C:

The power o f t h e p r e s e n t f o r m u l a t i o n i s tha t we can go o v e r t o a d i f f e r e n t gauge f u n c t i o n C ( A c l , @ ) which b r e a k s t h e Q-gauge invariance (as i s n e c e s s a r y ) b u t p r e s e r v e s gauge i n v a r i a n c e o f t y p e C . F o r example: i f @a i s t h e quantum p a r t o f the v e c t o r f i e l d A a , t h e n t h e c h o i c e

lJ ?J

Ca = Due@," , (8.4)

where D,, i s the cova r i an t d e r i v a t i v e i n the c l a s s i c a l ( C ) sence :

c lear ly p re se rves C-invariance. Such a gauge i s a l s o p o s s i b l e i n t h e Case of g r a v i t y . If

we can take

where

( 8 . 5 )

Again, D i s t h e c o v a r i a n t d e r i v a t i v e i n t h e c l a s s i c a l s ence . Ir The one-loop ( i r r e d u c i b l e ) v e r t e x func t ions ob ta ined i n t h i s

gauge w i l l be C- invar ian t a l s o . It fo l lows t h a t they s a t i s f y not only the Slavnov i d e n t i t i e s t h a t d e s c r i b e t h e Q gauge symmetry, bu t i n addi t ion t h e much s imple r Ward i d e n t i t i e s which are t h e d i r e c t genera- l i z a t i o n s of t h o s e i n quantum e lec t rodynamics .

O f course , t he new Feynman r u l e s i n t h i s background f i e l d gauge are independent of ou r o r i g i n a l choice of t h e gauge C ( A ) i n eq . (7 .1 ) . T h i s imp l i e s t h a t w e can now a l s o drop the gauge cond i t ion (7 .3 ) f o r the c l a s s i c a l f i e l d s s i n c e it can be r ep laced by ano the r , a r b i t r a r y , gauge cond i t ion .

9. A Simple Example o f t h e BackRround F i e l d Gauge: Pure Yang-Mills F i e l d s

Although we are mainly i n t e r e s t e d i n quantum g r a v i t y , i t is much more i n s t r u c t i v e t o i l l u s t r a t e ou r methods i n s imple r f i e l d t h e o r i e s . Let us cons ide r pu re Yang-Mills f i e lds . The i n v a r i a n t Lagrangian i s

.,.inv - Ga Ga - - r i ,,v , , v '

w i th G , " ~ = a,Af - a Y P + g fabcA:A; .

(9 .1 )

597

The gauge i n v a r i a n c e i s

a b c A:' = A lJ t g f a b c A A~ - a,,Aa .

We l e a v e aside t h e gauge i n v a r i a n c e s o u r c e s ( sec t . 6 ) .

We s h i f t

(where D = a t g f A c l )

Using

w e g e t

(9.2)

( 9 . 3 )

(9.4)

(9.51

t t o t a l d e r i v a t i v e .

A convenient background f i e l d gauge i s

( 9 . 6 ) 1 a 2 zc = - 3 C2 = - 7 (DIJQ,) . The ghos t Lagrangian i s t h e n

up t o i r r e l e v a n t i n t e r a c t i o n s w i t h Qa . lJ

O f c o u r s e t h e g h o s t i s also C - i n v a r i a n t . Note t h a t C- invar iance permits us t o write t h e i n t e r a c t i o n s o f @a and qa i n a very condensed way. It is

lJ t h i s f e a t u r e t h a t p r e v e n t s o v e r p o p u l a t i o n o f i n d i c e s i n t h e c a s e of g r a v i t y .

The one-loop i n f i n i t i e s can be sub t r ac t ed by a C-invariant counter

term i n the Lagrangian. The o n l y candida te i s

598

The index a s i m u l t a n e o u s l y governs t h e i n f i n i t i e s o f t h e two, t h r e e and f o u r p o i n t v e r t i c e s , and i s t h e r e f o r e d i r e c t l y p r o p o r t i o n a l t o t h e Callan - Symanzik & - f u n c t i o n 9 ) . To f i n d t h i s & - f u n c t i o n one t h e r e f o r e only needs t o i n v e s t i g a t e t h e two p o i n t f u n c t i o n , c o n t r a r y t o t h e convent ional f o r m u l a t i o n where a l s o th ree p o i n t f u n c t i o n s had t o be c a l c u l a t e d l o ) i n o r d e r t o e l i m i n a t e t h e Callan-Symanzik y- f u n c t i o n .

10. A Master Formula f o r a l l One-Loop I n f i n i t i e s

From the p r e c e e d i n g i t w i l l be c l e a r tha t any one-loop amplitude* can be o b t a i n e d from a Lagrangian b i l i n e a r i n a set o f f i e l d s 4:. One can t h e n r e a r r a n g e t h e c o e f f i c i e n t s i n such a way

.Z=Q r - -$(a,,$i + Ntj+j)gpV(aV$i + N S ~ ~ J +

where N , g and x are a r b i t r a r y f u n c t i o n s of s p a c e

I t h a t

1 pi Xi jb j1 , (10 .11

time x . F u r t h e r ,

In d e a l i n g w i t h g r a v i t y 3 ) we found it very convenient f i r s t t o c a l c u l a t e the i n f i n i t i e s of t h i s g e n e r a l Lagrangian , i n terms o f N , g and X . O f course , t h e background m e t r i c i s a l lowed t o b e curved . Af te rwards one may s u b s t i t u t e t h e de t a i l s such as: t h e way N and X depend on t h e background f i e l d s ; and t h e f a c t t h a t t he i n d i c e s i , j a c t u a l l y s t a n d f o r p a i r s o f L o r e n t z - i n d i c e s . I n gauge t h e o r i e s t he o b j e c t s N are most ly background gauge f i e l d s and i n g r a v i t y t h e N c o n t a i n t h e C h r i s t o f f e l symbols r.

I n ref. 3 ) , we u s e d d i m e n s i o n a l r e g u l a r i z a t i o n and c o n s i d e r e d the Pole8 a t n .* 4 ( n i s t h e number o f space- t ime d i m e n s i o n s ) . From power counting arguments one e a s i l y deduces t h a t t h e r e s i d u e s of these p o l e s can c o n s i s t o n l y of a l i m i t e d number of terms. T h i s number is even more r e s t r i c t e d i f w e u s e t h e o b s e r v a t i o n t h a t , whatever N , g o r X are , there i a a C-gauge symmetry:

dp i s i n v a r i a n t under

f provided a Feynman-like'background f i e l d gauge i s chosen.

599

N;(x) N,, - a U A + A N ,, X'(X) = X + AX - XA

where A i s a n i n f i n i t e s i m a l , a n t i s y m m e t r i c

N U A (10.2)

m a t r i x .

A s t r a i g h t f o r w a r d c a l c u l a t i o n y i e l d s , t h a t a l l one-loop i n f i n i t i e s 3 ) as n + 4 are absorbed by t h e counter -Lagrangian

+ 1 x * - L % T r { $i Y'UY,,v '4 1 2 Rx 1 A Z =

8 n 2 (n-4)

t - i R R v v I + & R 2 1 } 120 ,,U

where

Y v v = aPNw - aVNu + NPNv - NUN,,

and

(10.4)

T r I = number o f f i e l d s . R and R are t h e Riemann c u r v a t u r e t e n s o r s f o r t h e background m e t r i c . R a i s i n g and l o w e r i n g i n d i c e s by use o f t h e background gpU i s unders tood .

U W

The formula (10.3) can a l s o be used t o c a l c u l a t e a s imi la r 9) master formula i n t h e c a s e o f Fermions .

11. S u b s t i t u t i n g E q u a t i o n s o f Motion.

I n some c a s e s t h e r e s u l t i n g formula (10.3) i s n o t t h e end o f t h e s tory. One may s t i l l make use o f t h e i n f o r m a t i o n tha t t h e background f i e l d s are n o t a r b i t r a r y , b u t s a t i s f y e q u a t i o n s of motion. I f , f o r i n s t a n c e , one f i n d s a c o n t r i b u t i o n i n A& p r o p o r t i o n a l t o

and the e q u a t i o n o f motion i s

D2 Acl- = V(AC1) , (11.21

t h e n one may r e p l a c e (10.1) by

- Acl V(AC1) . (11.3)

Addition of infinitesimal terms in the Lagrangian that vanish if the equation of motion is fulfilled, corresponds exactly to making an infinitesimal field renormalization: if the equation of motion is

then the terms in question must be

and we have

A t = A + EB is a field renormalization. Note that all terms in A& are always infinitesimal, because we neglect everything that comes from two-loop diagrams.

12. Some Numerical Results. Conclusions.

The master formula (10.3) has been applied to calculate the 3 ) infinity structure in different cases. For pure gravitation we used

the gauge (8.57 and found from the gravitons

where 1. = 1/8n2(n-4) , E

and from the ghosts

Here g,,,, and R,,,, are the classical metric and curvature .(At this level it is never neceseary to consider quantum fields in the counter terms. We are also not interested in the renormalization of the source terms). From power counting one would expect a third possible term of the form

601

RaB~6 . RaBy6

2 It indeed o c c u r s , b u t we made use of t h e i d e n t i t y

R " B Y ~ - 4 R ~ ~ R ~ " + R 2 35- (RuByG

t o t a l d e r i v a t i v e ,

which i m p l i e s t h a t terms o f t h e form ( 1 2 . 3 ) can be

So a l l t o g e t h e r we have

3a11)

e l i m i n a t e d .

( 1 2 . 4 )

However, we s t i l l have t h e e q u a t i o n s o f motion o f t h e background f i e l d (see p r e v i o u s s e c t i o n ) , which is

Rllv 0 ; R = O (12.6)

T h i s means t h a t t h e i n f i n i t y v a n i s h e s i n pure g r a v i t y . It is i n t e r e s t i n g t o o b s e r v e tha t t he f i e l d r e n o r m a l i z a t i o n t h a t cor responds t o t h e e l i m i n a t i o n o f t h i s i n f i n i t y ( s e e p r e v i o u s s e c t i o n ) i s o f an unusual t y p e :

T h i s was t h e r e a s o n f o r o u r remark i n t h e end o f s e c t . 3 , Note that e q . ( 1 2 . 4 ) was e s s e n t i a l f o r t h i s r e s u l t .

Next w e s t u d i e d p u r e g r a v i t y w i t h a ( m a s s l e s s ) Klein-Gordon field $. The c l a s s i c a l e q u a t i o n s of motion are

(12.8)

There i s one t y p e of c o u n t e r t e r m which i s a l lowed by power c o u n t i n g and cannot be e l i m i n a t e d by s u b s t i t u t i o n of t h e e q u a t i o n s o f motion:

The n e x t t h i n g t h a t h a s been t r i e d i s p u r e g r a v i t y w i th i n a d d i t i o n

602

12 1 Maxwell f i e l d s . The Lagrangian i s

FpV = a,,Av - avA,, .

The e q u a t i o n s o f motion are

From which

R = - " T a = 0 . 2 a

In p r i n c i p l e one may e x p e c t

Expl ic i t c a l c u l a t i o n shows however:

So we see t h a t t h e r e are some c a n c e l l a t i o n s :

a 2 = a 3 = O .

( 12.10)

(12 .11)

(12 .12)

(12 .14)

Indeed, t h e y o c c u r i n a miraculous way duri . ig t h e c a l c u l a t i o n and have n o t ye t been e x p l a i n e d . I n t h e coupled Eins te in-Yang-Mil l s sys tem 13) t h e s e c a n c e l l a t i o n s p e r s i s t and many new c a n c e l l a t i o n s o c c u r . S t a r t i n g from t h e obvious g e n e r a l i z a t i o n of t h e Abel ian Lagrangian ( 1 2 . 1 0 ) , one f i n d s

A & = l ~ { [ # + R,,~R"" + f 2 c2 $ I ( 12.15)

603

where f i s t h e gauge c o u p l i n g c o n s t a n t ,

(12.16)

F i v e o t h e r c o e f f i c i e n t s e a c h happen t o v a n i s h . The second term i n (12 .15) i s o f t h e r e n o r m a l i z a b l e t y p e . It r e n o r m a l i z e s t h e gauge c o u p l i n g c o n s t a n t and f i x e s t h e Callan-Symanzik 6 - f u n c t i o n .

F i n a l l y , a l s o t h e combined E i n s t e i n - D i r a c sys tem h a s been i n - 14) v e s t i g a t e d and nonrenormal izable i n f i n i t i e s have been found a l s o .

The f a c t tha t i n a l l t h e s e sys tems where matter i n some form i s added t o p u r e g r a v i t y i n f i n i t i e s o f t h e n o n r e n o r m a l i z a b l e dimension s u r v i v e r e a l l y means t h a t t h e s e t h e o r i e s cannot be r e n o r m a l i z e d i n the p e r t u r b a t i o n expans ion . I n t h e c a s e o f pure g r a v i t y t h e i n f i n i t i e s have been shown t o be non-phys ica l up t o t h e one-loop l e v e l . No c a l c u l a t i o n s have been performed t o i n v e s t i g a t e r e n o r m a l i z a b i l i t y i n t h e o r d e r o f two l o o p s . The c a l c u l a t i o n s o f Nieuwenhuizen and Deser show t h a t f lmiraculous’ l c a n c e l l a t i o n s o f t e n o c c u r . Perhaps t h i s i s a n i n d i c a t i o n o f a new sor t of symmetry t h a t w e are n o t aware o f . I n v e s t i g a t i o n o f t h i s symmetry could r e v e a l new r e n o r m a l i z a b l e models w i t h g r a v i t y .

Even so, a r e n o r m a l i z e d p e r t u r b a t i o n expans ion would o n l y b e a small s t e p forward . A t very small d i s t a n c e s t h e g r a v i t a t i o n a l e f f e c t s must b e l a r g e , because o f t h e dimension of t h e g r a v i t a t i o n a l c o n s t a n t , 8 0 t h e expans ion would break down a t small d i s t a n c e s anyhow. We have t h e i m p r e s s i o n t h a t n o t o n l y a b e t t e r mathemat ica l a n a l y s i s i s needed, b u t a l s o new p h y s i c s . What w e l e a r n e d ( s e e e q . ( 1 2 . 7 ) and t h e remarks i n t h e end o f s e c t . 3 ) i s t h a t h s u c h a t h e o r y t h e m e t r i c t e n s o r might no t a t a l l b e such a fundamental concept . I n any c a s e , i t s d e f i n i t i o n i a n o t unambiguous.

REFERENCES

1. R. P. Feynman, Acta Phys. Polon. 2, 697 (1963) S. Mandelstam, Phys. Rev. a, 1580, 1604 (1968) E. S. F r a d k i n and I. V. T y u t i n , Phys. Rev. - D2, 2841 (1970) L. D. Faddeev and V. N. Popov,, Phys. L e t t e r s 258, 29 (1967)

2 .

3.

4 .

5 .

6. 1. 0. 9.

10.

11.

12.

13.

14.

B. S. D e W i t t , & R e l a t i v i t y , Groups and Topology, Summerschool of Theor. P h y s i c s , Lea Houches, France, 1963 (Gordon and Breach, New York, London) B. S. D e W i t t , Phys. Rev. 162, 1195, 1239 (1967) G . * t Hooft and M. V e l t m a n , CERN p r e p r i n t TH 1723 (1973), t o be publ . i n Annales de 1 ' I n s t i t u t Henr i Poincar6 D. Chris todoulou , proceedings o f t h e Academy o f A t h e n s . 2 (20 January 1972) D. Chris todoulou , CERN p r e p r i n t TH 1894 (1974) J. Honerkamp, Nucl. Phys. 848, 269 (1972); J. Honerkamp, Proc . Marseille Conf.,l9-23 June 1972, CERN p r e p r i n t TH 1558 G. ' t Hooft , Nucl. Phys. U, 167 (1971) G. ' t Hooft and M. Veltman, Nucl. Phys. E, 189 (1972) G. I t Hooft and M. Veltman, DIAGRAMMAR, CERN r e p o r t 73-9 (1973) 0. 't Hooft , Nucl. Phys. m, 455 (1973) 0. I t Hooft , Nucl. Phys. &, 444 (1973) H. D. P o l i t z e r , Phys. Rev. Letters z, 1346 (1973) D. J. Gross and F. Wilczek, Phys. Rev. Let ters E, 1343 (1973) R. Bach, Math. Z. 2, 110 (1921) C. Lanczos, Ann. Math. 2, 842 (1938) S. Dcaer and P. van Nieuwenhuizen, P h y s . R e v . L e t t e r s , z , 2 4 5 ( 1974); Phys. Rev. G, 401 (1974) S. Deser, Hung-Sheng Tsao and P. van Nieuwenhuizen, Phys. Rev. z, 3337 (1974) S. Deser and P. Van Nieuwenhuizen, Phys. Rev. U , 411 (1974)

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Class. Quantum Grav. 10 (1993) S79491.

CHAPTER 8.2

Classical N-particle cosmology in 2 + 1 dimensions

G ’t Hooft Institute for Theoretical Physics, University of Utrecht, PO Box 80 006, NL 3508 TA Utrecht, The Netherlands

AbshcL In 2 + 1 dimensional cosmology particlea arc topological defects in a universe that is nearly everywhere flat. We use a time dependent triangulation procedure of space to formulate the laws of evolution and to construct phase space of this system. A theorem is derived from which it follaws that many configurations may have a big bang in their past or a big crunch in their future. The dimensionality of phase space for N particles is 4N - 11 + 129, where g is the genus of 2-space. This suggests to consider ZN - 6 + 69 pairs of canonical variables and one time variable. The quantum version of this model is speculated about.

1. Introduction

In 2 + 1 dimensional General Relativity [1,2] Einstein’s equations for the vacuum,

imply also that the entire Riemann tensor R;,,, vanishes. Hence the vacuum is flat. A particle at rest, at the position z = a, produces a curvature proportional to 6*(2 - a). From that the global structure of space-time surrounding the particle can easily be seen to be a cone, and it is described by excising a wedge out of the plane (see figure l), after which one identifies points at each side of the wedge (for instance the two arrows in figure 1 are identified). The angle of the wedge, the conical deficiency angle 0, is proportional to the mass M of the particle:

where G is Newton’s constant, which we will choose to be one from now on.

Purticle P a ) b )

Figure 1. (U) Excised wedge near a particle P (shaded). (b) Diagrammatic notation.

0 1993 IOP Publishing Lld

606

If we wish to use Cartesian coordinates to describe locations in the neighbourhood of this particle we have to attach strings to each of these locations, and then the coordinates depend on how a string attached as indicated in figure 1 can also be described by the coordinates of 2' with string as drawn, provided that

2' = a+ Q(2 -a) (1.3)

where a is the location of P and 52 is the rotation

?b describe a moving particle, one has to Lorentz transform this space-time. The relationship between the two coordinate frames then becomes

R O (;J = L{ ( 0 1) ("7) + (:)p = A ( ; ) + (bbu)

where L is the h r e n Q transformation that gives the particle the velocity w from its rest frame. A is a new Lorentz transformation and b and bo are some shift vectors. Note that now the particle, its mass and its velocity, are determined by just giving an element of the Poinard group. However not all elements of the Poincarf? group specify a particle because the particle's space-time trajectory is given by the points satisfying

( Z ' , t ' ) = ( 2 , t ) (1.6)

an equation that for generic elements of the PoincarC group has no solutions, as we will see.

Consider now WO particles, moving with respect to each other. Following a path around both we find that points I and I' are identified by the product of two transformations of the type (1.5) (see figure 2).

F i p n 3. Tko moving panicles, P and Q.

The lines in figure 2 indicate the separation between the different coordinate frames used. The relation between the different coordinate frames for the point 2,

indicated as z and x', is obtained by the product of the transformations (1.5) for the

607


two particles P and Q. Let us now consider the center of m m pame. This is the coordinate frame that is chosen such that

In general one can indeed find a Lorenk frame such that A,, has only non- trivial elements in the first 2x2 block, as indicated, so that it is a pure rotation. It is natural to define the corresponding rotation angle to be the center of mass energy. Also one can find a vector a such that it acts as the origin of this rotation so that it can be interpreted as the location of the particle. But the new thing is the additional rime shifr 61 as we encircle the two particles. It is not hard to veriQ that 61 can be identified as being the total angular momentum [2], The angles over which one rotates are additive, and so are the time shifts 6t. Thus one recovers the consenation laws of energy and angular momentum.

A complication is now that one can no longer define the trajectory of the Center of mass as being the set of points invariant under (1.7). If the time shift is non-zero there are no invariant points. One has to go to the center of mass frame to locate the center of mass.

It is not difficult to derive the center of mass energy. Since the trace of a Lorentz matrix is Lorentz invariant one finds the cosine of the deficiency angle for the center of mass by taking the trace of the product of the Lorentz matrices corresponding to the two individual particles. The outcome of that (simple) calculation is [2]:

COS A m c m = COS Aml cos mn2 - sin nml sin 7rm2 cosh y (1.8)

where m1,2 are the masses of the particles 1 and 2; mcm is the Center of mass energy; y is the Lorentz boost parameter connecting the rest frame of 1 with that of 2. Note that in this, expression we have the trigonometric functions of half the deficiency angles in stead of the angles themselves. This is because it is more convenient to work with the SL(2, R) representation of the Lorentz group than it is with the S O ( 2 , l ) representation.

We see that there only is a center of mass if the condition

cosnm, cos7rn2 + 1 sin7rm1 sin7rm2 cosh7 <

is satisfied. If this condition is nof met P and Q are said to be a Gott pair [3]. The identification of the Cartesian coordinate frames connected by a loop around the particles is a Lorentz boost rather than a rotation. Such a pair can therefore be surrounded by a closed timelike curve (CTC). If no other particles occur it cannot be avoided that the universe is surrounded by an unphysical boundary condition [4, 51, even though CTC will nof occur if we go sufficiently far to the past or to the future

An open universe has an acceptable boundary if the total energy (corresponding to the sum of all deficiency angles in one coordinate frame divided by 27r) is less than 1. A universe closes if its energy is exactly 2. It then has the topology of a sphere (S2, g = 0). But a torus ( g = 1) or even 2-dimensional spaces with higher genus are also possible. The total energy vanishes for g = 1 and is negative in higher genus spaces. This is not in contradiction with positivity of the rest masses of the

161.

608

particles because when they move there can be negative energy in the surrounding gravitational fields as we shall see.

S M Carroll et 41 [5] proposed to consider a universe where the following happens. At t = 0 both Ml and M2 decay simultaneously each into a pair of light particles with masses:

Ml -+ ml m2 M2 - m3 m,. (1.10)

One easily finds that the two particles ml and m2 created by the decay of M, move away from each other at some angle A( 1 - Ml) (see figure 3). With respect to each other they can never violate condition (1.9) because their Center of mass energy is Ml. But ml and m3 can meet each other head-on, and thus their relative boost parameter y can exceed the value (1.9). Thus they may form a Gott pair. It turns out that this may only happen if

tg2j?rM1 > tg2nm + 1 (1.11)

so that we must have f < M, < 1. This implies that the universe must be closed. So we add an ‘antipode particle’ X with mass M , = 2 - 2Ml. The problem raised by these authors was the question whether closed timelike curves can then also arise in this universe.

X

M

A

Fwurc 3. MI - ml m2. F@rc 4. Ihc CFG u n h i t y at t = 0.

2. ”iangulatlon and Cauchy surfaces

We now propose an approach to 2+1 dimensional gravity/cosmology that stam with construction of Cauchy cross sections of space-time. These are purely spacelike surfaces that are timesrdered, and together span the entire universe. We make optimal use of the fact that in between the particles space-time is flat. The ideal method is ‘triangulation’, which implies that we consider Cauchy surfaces that are built from entirely flat triangular 2-simplexes, glued together [7]. In practice however it is easier to take polygons rather than triangles, because in the generic case the vertex

Classical N-particle cosmology in 2 -+ 1 dimensions

points will not connect more than three simplexes together. The general picture of such a Cauchy surface is sketched in figure 5, although in practice often the polygons will look somewhat more complex Indeed, ultimately the most economic description will be using just one polygon, with rather elaborate prescriptions as to how the edges are to be sewn together.

All particles in our theory must be at vertex points, but vertex points where no particles sit are of course also allowed. In general we will put a particle at a 1-vertex; at such a point the adjacent sides of the corresponding polygon are glued together leaving the appropriate deficiency angle.

If all particles are at rest with respect to each other then the situation is easy to visualize. At all vertices we can take the angles to add up to exactly 2n, unless there is a particle present, and then the angles add up to 2n - 27rm. If the total mass exceeds 1, then the Cauchy surface closes, so that there must be further particles at the ‘antipodes’. The total mass then automatically adds up to 2.

F @ n 5. Cauchy surface built from polygons. The heavy dots are particles.

But if the particles move the situation is more complex. The polygons become deformed as a function of time, and in general the boundary between one polygon and another will involve a Lorenk transformation.. It is now very instructive to formulate the rules that the moving polygons have to obey, if we insist that at the seams between all pairs of polygons space-time remains locally flat.

We consider in each polygon (which is actually a 3-simplex in space-time) a Cartesian coordinate frame. In general the sides all move in this frame, and their lengths change. Now in principle we could allow that on each polygon time runs at different speeds (as long as time runs forward), but we will impose the simplifying extra requirement that time runs equally fast at each polygon. In that case the evolution is uniquely determined, and furthermore the rules at the seams [7] become very simple:

1) The lengths of matching edges of two adjacent polygons, as measured in the coordinate frame of each, are equal. This is not a completely trivial statement because the matching goes with a Lorentz transformation.

2) The velocities with which the matching edges of two adjacent polygons move (always in a direction orthogonal to the orientation of the edge) in the coordinate frame of each, are the same, but the signs may differ. In general the signs are such that if the edge of one polygon recedes, the matching edge in the other one recedes also, because in the other case the matching becomes trivial.

610

3) The identification of points at two matching edges is such that a point moving in an orthogonal direction on one edge, remains identifled with a point moving orthogonally on the matching edge, as seen in the corresponding frames.

4) The vertices between three polygons move in such way that rules 1, 2 and 3 remain valid. When new vertices are created (one vertex may split into several), special attention should be paid to whether they represent flat space or particles. Often this comes out all right automatically because of energy-momentum conservation at such space-time points.

The proof that polygons describing localfy flat space can only move according to the above rules is not difficult. We will not here repeat the arguments already given

It is important to note that when a panicle moves relatively to the coordinate frame chosen for the polygon that it is in, its velocity vector is only allowed to be in the direction of the bisector of the cusp, see figure 6.

in 171.

a b 1 C

F#gurc 6. (0) Particle at r s t . 'Ibe shaded mgion is excised from Bat space and the two edges are glued together. The two dots are points to be identified. (b) Particle moving downwards ( a m ) . We get the Lorenu contraction of picture (q), therefore the angle widens. Since the two dots in ( U ) are at the same height, they still will be at the same height in (4, and after the Lorentz transformation there will be no relative time shift. (c) Diagrammatic notation for a particle.

If < is the particle's rapidity, we see that a moving particle is Lorentz contracted by a factor cosh<. Therefore it gets a widened deficiency angle p:

tg 40 = tg( lrn) cosh f . (2.1)

The boost parameter q describing the velocity thq with which the adjacent edges of the polygon widen or retract, is given by

thq = sin ifl thf. (2.2)

Later it will turn out to be useful to eliminate f or p out of these expressions. One then obtains

coshq cos40 = coslrm

sinh < sin lrrn = sinh q.

( 2 . 3 ~ ) (2.36)

When three polygons meet at a vertex, see figure 7(a), and if there is no particle present precisely at.this vertex (in the generic case there is none), then we can use the fact that space-time is locally flat here (note that although we talk about three

61 1

Classical N-particle cosmology in 2 + 1 dimenswns

different polygons, they could equally well be different regions of just one polygon, meeting at this particular vertex). The three coordinate frames can be seen as three points in a hyperbolic space (figure 7(b)) . Since the sides of two adjacent polygons recede or widen with the same boost factor 7 in each coordinate frame, the distance between these two frames is the boost given by 27. Hence h hyperbolic space the lengths of the sides of the triangle formed by the frames I, I1 and 111 are 27,. The angles A, in this triangle correspond to the angles a, of the three polygons at the vertex, as follows:

A, = A - 0,. (2.4)

These observations make it easy to understand the relations at each vertex as they were reported in ref. 7: writing

and all cyclic permutations These are nothing but the trigonometric properties of the triangle of figure 7 ( b ) in hyperbolic space.

I I

It will be important to observe the ranges of values the angles can take: not more than one of the three angles a, is allowed to exceed A. It is not hard to convince oneself that if there would be two angles ai > A then there would be points in space-time that occur at two different spots on our Cauchy surface, which iS not allowed. Now suppose one only knew the three boost parameters 7, then from (2.9) the three angles can be determined, except for an ambiguity a # 2n - a. But because the relative signs of sina, are fixed by (2.6), and no more than one a is allowed to exceed A, this ambiguity can be resolved completely. So the angles are completely determined by boasts 7, .

612

G 'I Hoofi

3. Evolution

The polygons of the previous section must be seen as fixed time cross sections of 3-simpkxes. At any given time i they together form a Cauchy surface. We now may ask haw such a Cauchy surface evokes. There are several kinds of mutations that can take place at a given moment. First, an edge connecting two polygons (or different regions of a single polygon) may shrink to zero and be replaced by another edge, see figure &a). It is also possible that a particle hits an edge of the polygon it is h. It then crosses over into the next polygon (figure 8(b)). When the edge a particle is on shrinks to zero it hops into the adjacent polygon (figure 8(c). Finally, if one of the angles of a polygon is greater than A then that vertex may hit another edge and cause a 'vacuum cross-over' (see figure 8(d)).

In all these cases the angles and boost parameters of the newly opened edges are all fked by the trigonometric equations, in particular (2.9) and (2.10). These !ix all properties of the edge #1, if the boosts q2, q3 and the relative angle a1 of the other edges are known. It is easily understood why all these properties are fixed: if the boosts and the angles of the transformations between frames I and 11, frames I1 and IV, frames I1 and 111, and frames 111 and IV are well known, then of course the transformation leading from I to IV (see figure 8(a)) is determined by that. And so are all other new edges in figure 8(a)-(d).

a ) 4 I X I V

111

b) /'

d )

4

F h r e 8. (U ) Exchange; (b) cmpt-over, (c) hop; (d) vacuum cmu-avcr.

In practice one needs no more than one single polygon, with identillcation rules

613


at its edges. An example of a five-particle universe is pictured in figure 9(a). The identifications among the various edges are indicated by arrows.

Figure 9. ( 0 ) Polygon representing a universe at given time 1. (b) Diagrammatic notation of the same universe.

It is convenient to indicate the identification rules for the edges by drawing a diagram, obtained by drawing the polygon on an S, sphere. One can then glue the edges together. The seams obtained are shown in the diagram (figure 9(b)).

4. Crunch and bang theorem

An important property of 2+1 dimensional universes was indicated in [7]. It can actually be formulated in terms of a theorem:

Theorem. Let at one moment i = 1 , all edges of all polygons describing a closed or open universe be contracting, which means that all Lorentz boosts 77 at all edges are negative. (We will refer to this condition as the Crunch codifion), then this condition will remain satisfied at all times 1 > i,.

Note that the crunch condition implies that at all vacuum vertices (the 3-vertices containing no particle, so that (2.6)-(2.10) apply), the angles ai are convex: ai < K, because due to (2.6) all sinai have the same sign and because of the remark at the end of section 2. Where the particles are, the polygons have angles a = 2~ - p, where p is given by (2.1) or (2 .3~) . Thus a > K if rn < 4, and a < A if rn > 5. So where we have particles with mass rn < the polygons are not convex. See figure 9(a). Evidently, we will not need figure 8(d).

The proof of the theorem can be given by checking all possible mutations, figure 8(a)-(c), one by one. Consider figure 8(a). The angles all start out being convex (< n). If 77 at the newly opened edge were positive all four adjacent angles would have to be > K, which is clearly not allowed. We see right away that that is geometrically impossible. So this transition keeps the Crunch situation as it was; the newly opened edge has 7) < 0 just like the others.

Next consider the cross-over, figure 8(b). This transition is only possible for a particle with mass < 4 (otherwise it could only be involved with the hop transition,

61 4

G ‘1 Hoop

figure 8(c)). Again the angles start out being all convex. First look at the lower vertex that appears. If the new q that arises there would not be negative the two adjacent angles would come out larger than A, which is not allowed. We a n then repeat the argument for the q of the edge attached to the particle after it entered the other polygon. One could argue alternatively that the particle as it enters gets an additional boost in the direction of the new polygon it is entering. This way also one finds unambiguously that the q of the particle (see equation (2.2)) must again be negative. Thus at this transition also the Crunch condition remains valid.

The case of the hop transition, figure 8(c), is again easy to check if the mass is 3 f. Then again it is the same geometrical argument that tells us that it is impossible for the two new angles at the vertex both to exceed A. If the mass is < 4 this argument does not work. But careful analysis shows the theorem still to hold there. It is again geometry. The particle is entering a polygon at a convex point, where the two edges are contracting. The particle must outrun the edges and hence it must be receding also. It must have a negative q. We herewith checked all possible transitions. The Crunch condition remains valid forever.

Reversing the sign of all q coefficients we see that if one has at t = t1 a situation such that all polygons are expanding at all sides, then this expansion must have existed at all times t < t,. This is why we call this theorem the ‘Crunch and Bang theorem’. If the Crunch condition is satisfied the universe must end in a Big Crunch. If the converse or ‘Bang condition’ is satisfied there must be a Big Bang in the past.

The theorem applies to the CFG universe. At the moment that the two particles Mi decay simultaneously into particles m1 all polygons satisfy the Crunch condition. The theorem tells us that the universe will continue to shrink forever. The particles will all approach each other with smaller and smaller impact parameters until they crunch. Before the crunch occurs there cannot possibly be a closed timelike curve simply because we were dealing with Cauchy surfaces all along. The purported CTC only is obtained when analytic extensions of space-time are considered beyond the Big Crunch. But we showed that the crunch singularity is an essential one; all particles meet each other there. Analytic extensions beyond that point are illegal.

The question could be asked whether perhaps the occurrence of a crunch is frame-dependent. But since the particles approach each other with ever decreasing impact parameters, and since in practice we found that at each mutation the crunch accelerates, we found that this question must be answered in the negative. There is no timelike path for which the moment of the crunch, in terms of its eigen time, can be postponed.

Of course the theorem can be time-inverted.

5. Degrees of freedom

An important question arises in these constructions. We would like to know exactly how to characterize the ‘physical’ degrees of freedom of 2+ 1 dimensional cosmologies. This is particularly of importance if one wishes to ‘quantize’ such models. Several quantization schemes have been proposed. It would be illustrative if a scheme could be devised that starts off with the degrees of freedom of a universe containing a finite number of particles, replacing the Poisson brackets of this finite dimensional phase space by wmmutators, and it would be of importance to consider the large distance, low mass limit of such a model where it should approach an ordinary Fock space

61 5

Classical N-particle cosmology in 2 + 1 dimenswns

consisting of a small number of non-interacting scalar particles, which we certainly know how to quantize. This question calls for the most economic way to represent a 2+1 dimensional universe with N particles at a given time.

Consider a universe of topology S,, and containing N particles. Take coordinates in the rest frame of one of the panicles, say M,. We now construct a Cauchy surface at time 1, corresponding to a moment T in the eigen time of M,. We spread its borders further and further out until points would be included that can be connected by timelike curves. A prescription can be worked out corresponding to moving the borders of our polygon about until a situation is reached that the edges touch each other without any time shifts A typical situation is then shown in figure 9. The diagrammatic description of figure 9(6) is most appropriate. The construction allowed us furthermore to postulate that one particle is at rest = 0. The direction in which the cusp of this particle is rotated is then also immaterial.

The remaining N - 1 particles are connected by a tree graph having L = 2( N - 1) - 3 lines and V = ( N - 1) - 2 vertices. All diagrams of this sort satisfy

2L = 3V+ ( N - 1). (5.1)

There is one boost parameter at each line, and these fix all angles at the vertices via the identities (2.6)-(2.10). This gives us

L = 2 N - 5 ( 5 4

dimensionless degrees of freedom. Now these also fix the cusp angles p where the particles are, via equation ( 2 . 3 ~ ~ ) ~ and all other angles including the cusp of panicle Mu were determined. Now since the polygon must close into itself, all external angles, ?r - ail must add up to 27r. This gives us one more constraint, and so we are left with

P = 2 N - 6 (5.3)

degrees of freedom that determine the angles between the velocifies of the particles. They can be compared with the momentum variables of the particles involved.

Then the lengths of all lines are arbitrary, giving us again 2 N - 5 degrees of freedom with the dimension of a length. Now not only the velocity of particle M, was fixed to be zero, also its position with respect to the other particles is completely determined now by the geometrical exercise of closing the polygon Therefore we find

Q = 2 N - 5 (5.4)

degrees of freedom with the dimension of a length. It may seem to be surprising that Q is one bigger than P and that it is odd. This surprise disappears if one realizes that time 1 was introduced completely arbitrarily. If we look at the total number of degrees of freedom for cosmologies, where time is an undetermined parameter running over a certain domain of real numbers, we find that cosmologies span a

P + Q - 1 = 4 N - 12 (5.5)

dimensional phase space.

616

In an earlier stage of this research the author thought that 2-spaces with nonvanishing genus should not be allowed, because Euler’s theorem would limit the total energy to be zero or negative. Now the total energy is obtained by adding up all deficiency angles of all particles and the angular deficits at all vertices where pofygons meet Indeed, moving particles all contribute here by amounts pi, all having the same sign. But the deficits at the vertices count in the other direction. If for instance the Crunch condition of the previous section is satisfied then it is easily seen that equation (2.8), with y > 1, gives

cosa1 < cos(a* + (lj) * c(li > 27r. (5.6)

Thus, the vacuum vertices in general have negative curvature and hence they tend to cancel and reverse the contributions of the particles. We may interpret this as a negative contribution to the energy by the gravitational field. Due to this phenomenon total curvature can easily be negative, so that we can have a torus (g = 1, total deficit angles zero) or even higher genus surfaces. In these spaces the particles cannot be at rest; they have to move and the situation is far from stationary.

Careful study of the torus gave us that here

P + Q = 4N + 1.

P + Q = 4N + 129 - 11.

(5.7)

(5.8)

In the general case the number of degrees of freedom is

The one extra positional degree of freedom is again the time parameter. In these less trivial space parameters with dimensions of angles and boosts on the one hand and those with dimensions of lengths and times are not so easy to distinguish anymore.

An open question seems to be how exactly to characterize the topological shape of the spaces (5.3)-(5.8). Our difficulty here is that the transitions of figures 8(0)- (d) are mappings of these spaces into themsehres; we would like to have definitions of Poisson brackets that are continuous under these mappings. Only then Poisson brackets may be replaced by commutators if one wishes to quantize the theory.

It is natural to view the parameters counted by equation (5.3) as being the ‘momenta’, they would turn into ordinary momenta of particles in Fbck space in the large N limit, an ordinary non-interacting scalar particle model. The parameters counted by (5.4) are the coordinates plus a time variable. Considering the complicated topology of phase space it is however unclear to the present author whether and how a self consistent Fbck space can be set up describing a cosmology with a fixed number of particles. An open question is also whether or not these particles, interacting only gravitationally can be pair created or destroyed. Classically (in the h 0 limit) as well as in the large IV limit (where we have ordinary Fock space of free scalar particles), this does not happen. Existing quantization schemes [S] suggest that topology change occurs but are inconclusive as to whether particle pair creation and annihilation take place.

Acknowledgments

The author thanks R Jackiw, S Deser and A Guth for extensive correspondence leading to this work. He thanks H B Nielsen and G Gibbons for discussions concerning the last section.

617

Classical N-parricle cosmology in 2 + 1 dimensions

References

(11 Stanrszkiewin A 1963 Acta Phy& pdon U 734 Got1 J R and Alpert M 1984 Gar RcL Grm. 16 243 Giddings S, Abbot J and Kuchar K 1984 Gor RcL Grm 16 751

[2] k r S, Jackiw R and 't Hooft G 1984 Ann Plrys. 152 220 [3] Golt J R 1991 Phys Rn! Lcn 66 1126 [4] Deser S, Jackiw R and '1 Hoof1 G 1992 Phys. Rn! Lcn 68 267 [5] Carroll S M, Farhi E and Guth A 1992 Phys Rev. Lm 68 263 (and A Guth penonal communication) (61 Culler C 1992 Phya Rcv. D 45 487 ( ~ e e also On A 1991 Phys Rn! D 44 R2214) [7] ' 1 Hooft G 1992 Clarr Qumtnun Grm. 9 1335 [8) Witten E 1988 NucL Phys B 311 46

Carlip S 1989 NucL Phys. B 324 lOa; 1990 Physics, Geommy and ToporogY (NATO AS1 Series B, Physics) 238 ed H C Iw (Nm York Plenum) p 541

61 8

Nuclear Physics B256 (1985) 727-736 Q North-Holland Publishing Company

CHAPTER 8.3

ON THE QUANTUM STRUCTURE OF A BLACK HOLE

Gerard ’t HOOFT

Institute for Theoretical Physics, Princetonplein 5, PO Box 80.006, 3508 TA Utrecht, 7he Netherlands

Received 26 October 1984 (Revised 30 November 1984)

The assumption is made that black holes should be subject to the same rules of quantum mechanics as ordinary elementary particles or composite systems. Although a complete theory for reconciling this requirement with that of general coordinate transformation invariance is not yet in sight, a number of observations can be made and a general framework is suggested.

1. Introduction

In view of the fundamental nature of both the theory of general relativity and that of quantum mechanics there seems to be little need to justify any attempt to reconcile the two theories. Yet the essential academic interest of the problem may be not our only motive. It seems to become more and more clear that “ordinary” elementary particle physics, in energy regions that will be accessible to machines in the near future, is plagued by mysteries that may require a more drastic approach than usually considered: the so-called hierarchy problems, and the freedom to choose coupling constants and masses. A variant of these problems also occurs in quantum gravity. Here it is the mystery of the vanishing cosmological constant. It would not be the first time in history if a solution to these mysteries could be found by first contemplating gravity theory; after all, the very notion of Yang-Mills fields was inspired by general relativity. In this paper we will present an approach in which the cosmological constant problem is acute and so, any progress made might help us in particle physics also.

We are not yet that far. The aim of the present paper is to set the scene, to provide a new battery of formulas that may become useful one day.

To many practitioners of quantum gravity the black hole plays the role of a soliton, a non-perturbative field configuration that is added to the spectrum of particle-like objects only after the basic equations of their theory have been put down, much like what is done in gauge theories of elementary particles, where Yang-Mills equations with small coupling constants determine the small-distance structure, and solitons and instantons govern the large-distance behavior.

Such an attitude however is probably not correct in quantum gravity. The coupling constant increases with decreasing distance scale which implies that the smaller the distance scale, the stronger the influences of “solitons”. At the Planck scale it may

619

G.’t Hoofi f Black hole

well be impossible to disentangle black holes from elementary particles. There simply is no fundamental difference. Both carry a finite Schwarzschild radius and both show certain types of interactions. It is natural to assume that at the Planck length these objects merge and that the same set of physical laws should cover all of them.

Now in spite of the fact that the properties of larger black holes appear to be determined by well-known laws of physics there are some tantalizing paradoxes as we will explain further. Understanding these problems may well be crucial before one can proceed to the Planck scale.

At present a black hole is only (more or less) understood as long as it is in a quantum mechanically mixed state. The standard picture is that of the Hawking school [l , 31, but a competing description exists [2] which this author was unable to rule out entirely, and which predicts a different radiation temperature. The question which of these pictures is right will not be considered in this paper; we leave it open by admitting a free parameter A in the first sections.

Whatever the value of A, the picture is incomplete. If black holes show any resemblance with ordinary particles it should be possible to describe them as pure states, even while they are being born from an implosion of ordinary matter, or while the opposite process, evaporation by Hawking radiation, takes place. As we wiH argue, any attempt to “unmix” the black-hole configuration (that is, produce a density matrix with eigenvalues closer to 1 and 0) produces matter in the view of a freely falling observer. I t is this “matter” that we will be considering in this paper.

We start with the postulate that there exists an extension of Hilbert space comprising black holes, and that a hamiltonian can be precisely defined in this Hilbert space, although a certain amount of ambiguity due to scheme (gauge) dependence is to be expected. Although at first sight this postulate may seem to be nearly empty, it is directly opposed to conclusions of the Hawking school [l]. These authors apply the rule of invariance under general coordinate transformations in the conventional way. We believe that, although our present dogma may well prove wrong ultimately, it stands a good chance of being correct if we apply general coordinate transformations more delicately, and in particular if the distinction between “vacuum” and “matter” may be assumed to be observer dependent when coordinate transformations with a horizon are considered.

Another point where the conventional derivations could be greeted with some scepticism is the role of the infalling observer. If the infalling observer sees a pure state an outside observer sees a mixed state. In our view this could be due simply to the fact that the “infalling observer” makes part of the system seen by the outsider: he himself is included in the outside Hilbert space. Clearly the very foundations of quantum mechanics are touched here and we leave continuation of this intriguing subject to philosophers, rather than accepting the simple-minded conclusion that transitions will take place between pure states and mixed states [ 1).

Nowhere a distinction may be made between “primordial” black holes and black holes that have been formed by collapse. Now we will make use of space-time

620

G ’ t Hm/r f Black hole

metrics that contain a future and a past horizon, and at first sight this seems to be a misrepresentation of the history of a black hole with a collapse in its past. However, just because no distinction is made one is free to choose whichever metric is most suitable for a description of the present state of a black hole. It may be expected that in a pure-state description of a black hole a collapsing past will be very difficult to describe if this collapse took place much longer ago than at time t = -M log M in Planck units.

From now on a black hole will be defined to be any particle which is considerably heavier than the Planck mass and which shows only the minimal amount of structure (nothing much besides its Hawking radiation [3]) outside its Schwamchild horizon.

The first part of this paper gives a number of pedestrian arguments indicating what kind of quantum structure one might expect in a black hole. Although they should not be considered as airtight mathematical proofs this author finds it extremely hard to imagine how the conclusions of sect. 2 of this paper could be avoided:

(i) The spectrum of black holes states is discrete. The density of states for heavy black holes can be computed up to an (unknown but finite) overall constant.

(ii) Baryon number conservation, like all other additive quantum numbers to which no local gauge field is coupled, must be violated.

The first of these conclusions is related to the expected thermodynamic properties first proposed by Bekenstein [4]. The second has been discussed at length by him and also by Zeldovich IS].

In sect. 3 a naive model (“brick wall model”) is constructed that roughly reproduces these featyres. Being a model rather than a theory it violates the fundamental requirement of coordinate invariance at the horizon so it cannot represent a satisfactory solution to our problem, but in its simplicity it does show some important facts: it is the horizon itself, rather than the black hole as a whole that determines its quantum properties.

In the second part of the paper*) we formulate a much more precise and satisfying approach. A set of coordinate frames and transformation laws is proposed. One simplification is made that presumably is fairly harmless: we assume that, averaged over a certain amount of time, ingoing and outgoing particles near a black hole are smeared equally over all angles 0, (p. Certainly this is true for a Hawking-radiating Schwarzschild black hole. Furthermore, ingoing things may be chosen to be spherically smeared. As a result we may take the gravitational interactions between ingoing and outgoing matter (our most crucial problem) to be rotationally invariant. This assumption appears to be quite suitable for a first attempt to obtain an improved theory, and in fact it makes our whole approach quite powerful.

As stated before, how to interpret the coordinate transformations physically is yet another matter. The unorthodox interpretation that we proposed in ref. [2] is not ruled out (in fact it fits quite well in our description) but we do not insist on it. Rather, we allow the reader to draw his own conclusions here.

*) not reproduced in this book.

621

G.’t Hooft f Black hole

It is important to note that never space-times are considered that contain a “conical singularity” at the origin or elsewhere. Even in the formalism with A = 2 there is no such singularity. This is because the “identification” of the points x with the points --x is Minkowski space of ref. [ 2 ] only is made in the classical limit, not in the quantum theory.

Quantization of the space-time metric gpv itself is not considered explicitly, since we attempt to deduce the black-hole’s properties from known laws of physics much beyond the Planck length. This may be incorrect: it could be that oniy by considering the full Hilbert space of all metrics a workable picture emerges. Our attitude is to first keep everything as simple as possible and only accept such complications when they clearly become unavoidable.

2. The black hole spectrum

A black hole can absorb particles according to the well-known laws of general relativity: the geodesics of particles with an impact parameter below a certain threshold will disappear into the horizon.

Conversely, black holes may emit particles as derived by Hawking. They radiate as black bodies with a certain temperature:

in units where the gravitational constant, the speed of light, Planck’s constant and Boltzmann’s constant are put equal to one. As is explained in sect. 1 there may be reasons to put into question the usual argument that A = 1. There is an alternative theory with A =2, but the precise value of A is of little relevance to the following argument.

If a black hole is to be compared with any ordinary quantum mechanical system such as a heavy atomic nucleus or any “black box” containing a number of particles and possessing a certain set of energy levels, which furthermore can absorb and emit particles in a similar fashion, then a conclusion on the density of its energy levels, p ( E ) , can readily be drawn.

Imagine an object with energy AE being dropped into a black hole with mass E, so that the final mass is E + AE, where in Planck units

Let R be the bound on the impact parameter; usually

R - 2 E . (2.3)

Then the absorption cross section U is

u = r r R 2 . (2.4)

622

G.’t Hoofr Black hole

From Hawking’s result we conclude that the emission probability W is

w = TR’P,, e - p n A E , (2.5)

where P A E is the density of states for a particle with energy A E per volume element, and PH is the inverse of the temperature TH.

Now if the same processes can be described by a hamiltonian acting in Hilbert space, then we should have in the first case

(2.6) (T = [ ( E + A E J T I E , A E ) ~ ’ ~ ( E + A E ) ,

where T is the scattering matrix, and in the second case

W = I(E, AE~TIE+AE)I~P(E)PAE, (2.7)

by virtue of the “golden rule”. The matrix elements in both cases should be equal to each other if PCT invariance

is to be respected (the expressions hold for particles and antiparticles equally). So dividing the two expressions we find

p ( ~ ) = c e4m’-’E2. (2.9)

Now this result could also have been concluded from the usual thermodynamical arguments [4]. The entropy S is

S = 47rA-I E’+ const, (2.10)

from which indeed (2.9) follows. We note that E’ in (2.9) and (2.10) measures the total area of the horizon.

The constant in (2.9) and (2.10) is not known. Could it be infinite? We claim that this can only be the case if there exists a “lightest” stable black hole. Just compare eqs. (2.6) and (2.7) if E + A E represents the lightest black hole, and E and AE are ordinary elementary particles. If p( E + AE) is infinite but p( E) finite then, since (T

must be finite, W vanishes. Since this object fails to obey the classical laws of physics (it ought to emit Hawking radiation), it cannot be much heavier than the Planck mass. Now it is very difficult to conceive of any quantum theory that can admit the presence of such infinitely degenerate “particles”. They are coupled to the graviton in the usual way, so their contributions to graviton scattering amplitudes and propagators would diverge with the constant C, basically because the probability for pair creation of this object in any channel with total energy exceeding the mass threshold is proportional to C.

Clearly the above arguments assume that several familiar concepts from particle field theory such as unitarity, causality, positivity, etc. also apply to these extreme energy end length scales, and one might object against making such assumptions. But we considered this to be a reasonable starting point and henceforth assume C

623

G.'; H w f t 1 Black hole

to be finite. An interesting but as yet not much explored possibility is that C, though finite, might be extremely large. After all, large numbers such as MPlanck/ q,,oton are unavoidable in this area of physics. One might find interesting links with the 1/N expansion suggested by Weinberg [6], who however finds physically unacceptable poles in the N -* 00 limit.

It is rather unlikely that C is exactly constant. One expects subdominant terms in the exponent. Also, we have not taken into account the other degrees of freedom such as angular momentum and electric (possibly also magnetic) charge. These can be taken into account by rather straightforward extrapolations.

Other additive quantum numbers cannot possibly be conserved. The reason is that the larger black hole may absorb baryons or any other such objects in unlimited quantities. If indeed it allows only a finite number of quantum states then any assignment of baryon number will fail sooner or later. Indeed one of the paradoxes we will have to face is that starting with a theory that is invariant under rotations in baryon number space one must end up with a theory where this invariance is broken [5] .

3. The brick wall model

When one considers the number of energy levels a particle can occupy in the vicinity of a black hole one finds a rather alarming divergence at th-e horizon. Indeed, the usual claim that a black hole is an infinite sink of information (and a source of an ideally random black body radiation of particles [3,7]) can be traced back to this infinity. It is inherent to the arguments in the previous sections that this infinity is physically unacceptable. Later in this section we will see that, as is often the case with such infinities, the classical treatment of the infinite parts of these expressions is physically incorrect. The particle wave functions extremely close to the horizon must be modified in a complicated way by gravitational interactions between ingoing and outgoing particles (sects. 5 and 6). Before attempting to consider these interactions properly we investigate the consequences of a simple-minded cut-off in this section.

It turns out to be a good exercise to see what happens if we assume that the wave functions must all vanish within some fixed distance h from the horizon:

cp(x)=O if x ~ 2 M + h , (3.1)

where M is the black hole mass. For simplicity we take cp(x) to be a scalar wave function for a light ( m 4 1 a M ) spinless particle. Later we will give them a multiplicity 2 as a first attempt to mimic more closely the real world.

In the view of a freely falling observer, condition (3.1) corresponds to a uniformly accelerated mirror which in fact will create its own energy-momentum tensor due to excitation of the vacuum. As in sect. 1 we stress that this presence of matter and energy may be observer dependent, but above all this model should be seen as an

624

G.’t Hmft 1 Black hole

elementary exercise, rather than an attempt to describe physical black holes accurately.

Let the metric of a Schwarzschild black hole be given by

ds2=- ( I-- ”;”) dr2+ ( I-- 2 y ) - 1 d r 2 + r 2 d n ’ (3.2)

Furthermore, we need an “infrared cutoff” in the form of a large box with radius L:

q ( x ) = O if x = L . (3.3)

The quantum numbers are I, l3 and n, standing for total angular momentum, its z-component and the radial excitations. The energy levels E ( / , I,, n) can then be found from the wave equation

As long as M > 1 (in Planck units) we can rely on a WKB approximation. Defining a r3dial wave number k(r, I, E) by

k2 = r2 (( 1 -?)-I E2-r-’I(I+1)- m2 r(r -2M) (3.5)

as long as the r.h.s. is non-negative, and k2=0 otherwise, the number of radial modes n is given by

L m = j drk(r,I,E)

2M+h (3.6)

The total number N of wave solutions with energy not exceeding E is then given by der

a N = I ( 2 f + l ) d f ~ n = g(E)

= /2:+h dr( 1 - 7 ) 2M -’

where the I-integration goes over those values of I for which the argument of the square root is positive.

What we have counted in (3.7) is the number of classical eigenmodes of a scalar field in the vicinity of a black hole. We now wish to find the thermodynamic properties of this system such as specific heat etc. Every wave solution may be occupied by any integer number of quanta. Thus we get for the free energy F at some inverse temperature p,

(3.8)

625

G.'r HooJ / Black hole

or

(3.9)

and, using (3.7),

TDF = I dg( E) log ( 1 - e-PE)

(3.10)

Again the integral is taken only over those values for which the square root exists. In the approximation

m2<2M/p2h, L*2M, (3.1 1 )

we find that the main contributions are

(3.12)

The second part is the usual contribution from the vacuum surrounding the system at large distances and is of little relevance here. The first part is an intrinsic contribution from the horizon and it is seen to diverge linearly as h + 0.

The contribution of the horizon to the total energy U and the entropy S are

(3.13)

(3.14)

We added a factor 2 denoting the total number of particle types. Let us now adjust the parameters of our model such that the total entropy is

S=4h- 'M2, (3.15)

as in eq. (2.10), and the inverse temperature is

p =877h-'M. This is seen to correspond to

(3.16)

(3.17)

626

G.’I Hoofr J Black hole

Note also that the total energy is U = i M , (3.18)

independent of 2, and indeed a sizeable fraction of the total mass M of the black hole! We see that it does not make much sense to let h decrease much below the critical value (3.17) because then more than the black hole mass would be concentrated at our side of the horizon.

Eq. (3.17) suggests that the distance of the “brick wall” from the horizon depends on M, but this is merely a coordinate artifact. The invariant distance is

d r = 2 h M h = p . r = 2 M + h J r = 2 M d n = J -11 - 2 M l r 90a

(3.19)

Thus, the brick wall may be seen as a property of the horizon independent of the size of the black hole.

The conclusion of this section is that not only the infinity of the modes near the horizon should be cut-off, but also the value for the cut-off parameter is determined by nature, and a property of the horizon only. The model described here should be a reasonable description of a black hole as long as the particles near the horizon are kept at a temperature as given by (3.16) and all chemical potentials are kept close to zero. The reader is invited to investigate further properties of the model such as the average time spent by one particle near the horizon, etc.

The model automatically preserves quantum coherence completely, but it is also unsatisfactory: there might be several conserved quantum numbers, such as baryon number*. What is wrong, clearly, is that we abandoned the principle of invariance under coordinate transformations at the horizon. The question that we should really address is how to keep not only the quantum coherence but also general invariance, while dropping all global conservation laws.

References

[ l ] S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D14 (1976) 2460;

[2] G. ’t Hooft, Ambiguity of the equivalence principle and Hawking’s temperature, 1. Georn. Phys. 1

[3] S.W. Hawking, Particle creation by black holes, Comrn. Math. Pbys. 43 (1975) 199;

The unpredictability of quantum gravity, Cambridge Univ. preprinl (May 1982)

(1984) 45

J.B. Hartle and S.W. Hawking, Path-integral derivation of black hole radiance, Phys. Rev. D13 (1976) 2188; W.G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14 (1976) 870

[4] J.D. Bekenstein, Black holes and the second law, Nuovo Cim. Lett. 4 (1972) 737; Black holes and entropy, Phys. Rev. D7 (1973) 2333; Generalized second law of thermodynamics in black-hole physics, D9 (1974) 3292

[ 5 ] J.D. Bekenstein, Non existence of baryon number for static black holes, Phys. Rev. a5 (1972) 1239, 2403 ; Ya.B. Zeldovich, A new type of radioactive decay: gravitational annihilation of baryons, Sov. Phys. JETP 45 (1977) 9; A.D. Dolgor and Ya.B. Zeldovich, Cosmology and elementary particles, Rev. Mod. Phys. 53 (1981) 1

*) One may postpone this difficulty by inserting explicitly baryon number violating interactions near the horizon.

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G.’r Haaft f Black hole

[6] S . Weinberg, in General relativity, an Einstein centenary survey, ed. S.W. Hawking and W. Israel

[7] W.G. Unruh and R.M. Wald, What happens when an accelerating observer detects a Rindler (Cambridge Univ. Press, 1979) 795

particle, Berkeley preprint

628

Nuclear Physics B253 (1985) 173-188 @ North-Holland Publishing Company

CHAPTER 8.4

THE GRAVITATIONAL SHOCK WAVE OF A MASSLESS PARTICLE

Tevian DRAY' and Gerard 't HOOFT

fnstituut voor Theoretischc Fysicq Ph'ncetonplein 5, Postbus SO.OW, 3508 TA Utrecht. n e Netherlands

Received 20 August 1984

The (spherical) gravitational shock wave due to a massless particle moving at the speed of light along the horizon of the Schwanschild black hole is obtained. Special cases of our procedure yield previous results by Aichelburg and Sex1 [ I ] for a photon in Minkowski space and by Penrose [2] for sourceless shock wavui in Minkowski space. A new derivation of the (plane) shock wave of a photon in Minkowski space [ I ] involving explicit calculation of geodesics crossing the shock wave is also given in order to clarify the underlying physics. Applications to quantum gravity, specifically the possible effect on the Hawking temperature, are briefly discussed.

1. Introduction

There are various reasons why one may be interested in exact expressions for the gravitational field surrounding a particle whose mass is dominated by kinetic energy rather. than rest mass. For instance the first non-trivial gravitational effects to be seen in particle-particle interactions at extreme energies may be due to such fields. Our understanding of quantum gravity may be helped by considering these field configurations. A specific case of interest is the gravitational back-reaction and self-interaction of matter entering or leaving a black hole (Hawking radiation). At the black hole horizon the relative velocity of these particles approach that of light.

Aichelburg and Sexl [l] considered the gravitational field of a massless psrticle in Minkowski space, and showed that the resulting space-time is a special case of a gravitational impulsive wave* [ 2 ] which is also an asymmetric plane-fronted gravitational wave [3]. Penrose [2] also gives explicit examples of sourceless gravitational impulsive waves in' Minkowski space. In this paper we first summarize the properties of the shock wave due to a massless

particle in Minkowski space. We do this by presenting a new derivation of the results of Aichelburg and Sexl [ l ] involving explicit calculation of (null) geodesics crossing the shock wave. This enables the physical properties of such shock waves to be easily exhibited. We then determine, for a particular class of vacuum solutions to the Einstein

held equations, the (necessary and sufficient) conditions for being able to introduce

' Supported by the Stichting voor Fundamenteel Ondenbek der Materie. ' Note that Penrose [2] reserves the term gravitational shock wave for a metric which is C' whereas

the metrics we consider are only Co. We will nevertheless use the term "shock wave" for what are. in the terminology of [2], impulsive waves.

629

T. Dray, G. ’ 1 Hoofr / Grauiraiional shock wave

Fig. I . The horizon shift (eq. ( 1 5 ) ) due to the field of a massless particle moving in the v-direction along the horizon of the Schwarzschild black hole. The amount of the shift depends on 8.

a gravitational shock wave via a coordinate shift*. These conditions include both constraints on the metric coefficients and on the form of the shift. In Minkowski space they reduce to the plane-fronted wave of Aichelburg and Sex1 [ l ] and, of course, to Penrose’s results [2] for sourceless waves. However, for Schwarzschild black holes we obtain something new: there is a (spherical) shock wave at the horizon due to a massless particle at the horizon. (See fig. I . )

Throughout this paper we think of the massless particle as the limit of a fast-moving particle with negligible rest mass**; this limit is given explicitly for the Minkowski case. Fig. 1 can thus be interpreted as describing an ordinary particle with small mass falling into the black hole from the left, as seen by an outside observer (on the left) at very late times; the particle is then seen close to the horizon and boosted to high energies.

The paper is organized as follows: in sect. 2 we summarize the situation for the (plane) shock wave due to a massless particle in Minkowski space and discuss the general physical features of such a wave. These results are based on a calculation of the null geodesics in such a space-time, which is given explicitly in appendix A. In sect. 3 we give the conditions, derived in appendix B, for a shock wave to be possible starting from a given “background” space-time. After showing that these conditions reduce to the correct ones [ 1,2] in Minkowski space we then obtain the (spherical) shock wave at the horizon of the Schwarzschild black hole due to a massless particle there. In sect. 4 we discuss our results.

2. Shock waves: an example

Aichelburg and Sex1 [ 13 (cf. eq. (A.37)) have shown that the gravitational field of a massless particle in Minkowski space is described by the metric

d s2= -du^(dv*+4p ln(p2)6(C) dzi) + dx2 + d 3 , ( 1 )

This is just the scissors-and-paste approach of Penrose [2] applied to more general space-times. We assume that the particle has no electric charge and no angular momentum. However, for an elementary particle for example we do not expect the results to differ significantly from those we derive here.

++

630

T. Dray, G. ‘ t Hoof? f Gravitational shock wave

where p” = 2 + y. The particle moves in the 6 direction with momentum p. By calculating geodesics which cross the shock wave, which is located at U = 0, we obtain the following two physical effects of such a shock wave (see appendix A): geodesics have a discontinuityd6 at U = 0 and are refracted in the transverse direction. The shift A 6 is given by (cf. eq. (A.26))

4GP P i c’ ’ -- In -

which, for a photon, is

where we have put the units back in and where we have used E = pc = hu, where U is the frequency of the photon and Ipl is the Planck length. po is the value of p when the geodesic reaches U = 0. This shift is illustrated (for nonzero m and x = 0) in fig. 2; for m = 0 the shift occurs as a discontinuity at U = 0.

Note that the presence of a length scale in the argument of the logarithm is merely a reflection of our choice of units and has no physical meaning. It represents a constant shift in 6 which can be transformed away by a suitable redefinition of 6 (eq. (A.] 1)). Furthermore, by the same procedure, the value of po for which A6 = 0 (here po= f p , ) can be chosen arbitrarily far from the photon (po large). In any case, only the diference in A 6 for nearby geodesics is physically relevant.

There is also a refraction effect described by (cf. eq. (A.36))

4GP cot a +cot p = 3, c Po

which, for a photon, is 41gl U

c o t a + c o t p = - CPO

This is illustrated (for x = 0) in fig. 3, where the angles a and p are defined.

a) Yo < 1 b) yo> 1 Fig. 2. The path of a null geodesic in the 512. 8) plane as described by eq. (2) for m < 1, PO* m, and (a)

p0< 1, (b) po> I . The near region N and the far region F, as well as the shift AI?, are indicated.

631

T. Dray, G. ‘ I Hoofr Gruvitationu/ shock wove

?“ T Y Y=Y, I

(a) r (b) y = -y+y,

Fig. 3. The “spatial refraction” of null geodesics as described by eq. (3) for the two speciat cases (a) a = p, and (b) a = jlr.

Eqs. (2) and (3) are the central results for these shock waves and describe physical effects which should also occur in more general situations. Note that if the shift (2) were constant it could be removed by a coordinate translation and would therefore not be physically observable (cf. the discussion after eq. (2)). Also, a shift linear in the transverse distance p would not be observable since it could be removed by a Lorentz rotation of one of the flat half-spaces with respect to the other. However the shift (2) is logarithmic in p and leads to physically observable effects. The relative shift for nearby observers goes as the first derivative ( l / p ) while the relative refraction goes as the second derivative ( l/pz) of the shift*. See fig. 4.

Fig. 4. Four synchronized clocks were originally situated at rest at the corners of a rectangle. A fast particle approaches from the left. The situation is shown when the shock wave has passed two of the clocks. The one closest to the trajectory of the particles has been shifted to the right with respect to the other; its clock now runs behind the other. They are also moving towards the trajectory of the fast particle at different speeds (arrows). Only their relarioe velocity, which is always away from each other,

is locally observable.

* Note that a local observer can only detect the second derivative ( l /p2) of the shift.

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1 Dray, G. ' t Hooft Gravitational shock waue

3. General result

Consider a solution of the vacuum Einstein field equations of the form

dg2=2A(u, u)du du+g(u, u)h,(x') dx'dx'. (4)

Under what conditions can we introduce a shift in U at U = 0 so that the resulting space-time solves the field equation with a photon at the origin p = 0 of the (xi) 2-surface and U = O? As shown in appendix B the answer is that at U = 0 we must have

A.v = 0 = g.0 9

A - Af-&f= 32?rpA2S(p), g L?

( 5 )

where f=f(x') represents the shift in U, Af is the laplacian off with respect to the 2-metric h,, and the resulting metric is described by (B.2) or (B.4). Eqs. ( 5 ) represent our main result. We now turn to specific examples.

For a plane wave due to a photon in Minkowski space we have

d t2=-du dv+dX2+dy2, ( 6 4

and thus

The conditions on the metric are trivially satisfied, and the condition on the shift f is

Af= -16~pS(p ) , (7)

where p2=x2+y2. The solution of this equation, unique up to solutions of the homogeneous equation, is

(8) f = -4p In p2 ,

which agrees precisely with Aichelburg and Sex1 [ l ] (cf. eqs. (2) and (A.26)). For a sourceless plane wave in Minkowski space we set p = O to obtain

A f = O , (9)

which agrees with Penrose [2]. For a spherical wave in Minkowski space we write the metric in the form

dt2=-du du+~(u-u)2(d02+sin2 Odq?), (104

so that

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7: Dray. G. ' f Hooft 1 Gravitational shock wave

But the derivatives of g are not identically zero at U = 0. Thus, there are no spherical waves (of this form) in Minkowski space.

Physically this might seem mysterious because one expects spherical shock waves to arise in, e.g., the debris of a violent explosion. On closer inspection one concludes that there must be non-zero curvature behind such shock waves. However, note that Penrose [2] does exhibit the existence of sourceless spherical shock waves in Minkowski space but having a different form than our ansatz (eq. (B.2)) .

We now turn to a more interesting example, namely the Schwarzschild metric which in (null) Kruskal-Szekeres coordinates takes the form

so that

g = r 2 .

r is given implicitly as a function of U and U by

so that all u-derivatives of r are proportional to U. Thus, the conditions on the metric coefficients A and g are satisfied at U = 0.

Furthermore, since g,uu = A the condition on f becomes

where K = 29m4p e- ' and where we have arranged the coordinates so that the photon is at 8 = 0 = U.

We now solve eq. (12) by expanding f in terms of spherical harmonics &( 6 , ~ ) . We see immediately that only spherical harmonics with m = 0 contribute; expressing these in terms of'the Legendre polynomials S ( x ) leads to

r + f P,(COS e l .

f = K ? I ( , + 1)+ 1

We can obtain an integral expression for f by using the generating function for the Legendre polynomials, namely

634

T. Dray, G. ’ I Hooft 1 Gravitational shock wave

and the fact that

1 + ; r’ e“” cos ( G s ) ds =

+ 1) + ,

where r = es, t o finally obtain

ds . cosh s -cos O ) ’ / *

We have not attempted to perform the integration explicitly. We note that the homogeneous equation (eq. (12) with p = 0) has no solution. In the limit of small B eq. (15) in appropriate coordinates reduces to eq. (8), with a well-determined value of the integration constant.

4. Discussion

The surprisingly simple geometric shape of a gravitational shock wave of massless particles in flat space can help us obtain a better understanding of gravitational interactions among particles at extreme energies. It is easy to argue that at extremely high energies interactions due to this shock wave will dominate over all quantum field theoretic interactions, simply because the latter will be postponed by an infinite time shift (due to the logarithmic singularitity in eq. (2), see fig. 4). This implies that cross sections at such energies will be entirely predictable.

A problem arises if two such particles are considered, both accompanied by their shock waves, that meet and collide. The result of such a collision will be curved shock waves which obey the vacuum Einstein field equations only if space-time after the collision in the region between both shock waves is curved, so that we then have to deal with the full complexity of general relativity. We have here a limiting case of the general problem of black hole encounters which has been studied in detail by D’Eath [4] and Curtis [5 ] .

On physical grounds the Schwarzschild result, eq. (13), should not be surprising. The flat space result (e.g. eq. (8)) can be obtained [ I ] by infinitely boosting a (massive) source particle. Now take an r = const observer in the usual Schwarzschild coordinates. Put a (nearly) massless particle at the horizon and wait. The observer will see a particle with an increasingly large boost! It is only natural to expect a similar result in both cases. The spherical nature of the wave in the Schwarzschild case (as opposed to the plane wave in Minkowski space) is merely a reflection of the spherically symmetric nature of the “boost” relating an r = const observer to Kruskal-Szekeres coordinates. Physically, one expects any (weak) plane wave approaching the black hole to become gradually more spherical, as seen by an outside observer, as it comes closer to the horizon.

Returning to the picture of a particle of small mass falling into the black hole (see discussion after fig. I ) one expects small increase of the Schwarzschild radius

635

T. Dray, G. ’ I Hoofr 1 Gravitational shock waue

of the black hole, together with a slight expansion of its furture horizon. This expansion then grows exponentially with the Schwarzschild time coordinate. This is what our “shock wave” here actually describes. Eq. (15) is in closed form the extent of the horizon expansion. One might speculate what effect this expansion has on the quantum nature of the vacuum and, in particular, Hawking radiation. We believe that the gravitational interaction between infalling matter and Hawking radiation, crucial for a deeper understanding of the quantum properties of black holes themselves [6], should be described using our expression for the horizon expansion.

Finally, we are aware of the analogy with the electric field of a charged particle moving at the speed of light, which is similar to the gravitational field described here. Our gravitational shock wave can be compared to a limiting case of Cherenkov radiation.

We thank Paul Shellard for bringing the work of D’Eath [4] to our attention, which then led us to the previous work of Aichelburg and Sex1 [ 13 and Penrose [2].

The computer calculations of the Ricci tenser were performed while one of us (T.D.) was a visitor at Queen Mary College, London. He is deeply indebted to Malcolm MacCallum and Gordon Joly for hospitality and assistance.

A PHOTON IN MINKOWSKI SPACE

Consider the linearized field of a point mass in Lorentz gauge:

d s 2 = - I - - dT2+ I + - (dx2+dy2+dZ2) , ( 23 ( 3 with rn 4 R. This is the field of the particle as seen in its rest frame. Boost this rest frame with respect to coordinates ( 1 , x, y, z) via

T = t cosh

Z = - f sinh p + z cosh p , - z sinh p ,

and simultaneously set

rn = 2 p e-@

for some constant p > 0. Introduce null coordinates

u = r - - z ,

v = t + z . (A.4)

636

T Dray, G. ' t Hooft 1 Gravitational shock waw

ne momentum of the particle is

pa = m[(cosh p)S: +(sinh /3)S3, (A.5)

and thus

lim p a = p( S: + 8; ) = 2pSz ; (A.6)

in the limit the particle is massless and moves (at the speed of light) in the u-direction; p is its momentum and is kept finite (possibly large).

8-03

Writing the metric in (U, U, x, y) coordinates we obtain

with

The key idea is to notice that

m-0 fim ds2=-du (A.9) ( U +O.u.x,y) fixed

which is flat although the coordinate U' satisfying

4p du dv' = du -- 14 (A.lO)

suffers a discontinuity at U = 0 due to the absolute value sign. To make this somewhat more precise, introduce coordinates ( 2, i3, x, y ) by*

m2Z In (2R) G = u +

PR 9

4pZ In (2R) C = U +

R * (A.11)

Note that (2, 6, x, y ) is obtained from (U, U, x, y) by adding ( 4 2 In (2R)IR)p" . Then R'=x'+y2+(;G-%) 2 ,

4P (A.12)

lim ds2 = -dG di3+dx2+dy2. (A.13) m+O

(u#O.qx.y)lixcd

The motivation for these coordinates is as follows: in the limit, Z I R acts like a @-function and reproduces the effect of the absolute value sign, while dR/ R = d u / u Furthermore, In R is finite at

-0. The factor 2 is chosen for convenience. We use geometric units in which G = c = h = I ; all quantities are dimensionless.

637

T. Dray, G. ’ t Hoop / Gravitational shock wave

The metric (A.13) is flat. It remains to investigate its behavior near U =0, which, in the limit, is just U = 0. To do this we consider the behavior of (null) geodesics crossing U = 0 for m # 0 and then take the limit as m goes to zero”. We do this both in a “near” region, which collapses to U = 0 in the limit - this is just the rest frame of the particle - and.a “far” region, where U remains non-zero in the limit.

The (linearized) geodesics of the metric (1) are given by

T = E ( 1 + $) , y z - zi, = L( 1 -$) ,

(A.14)

where the dot denotes derivatives with respect to the affine parameter A along the geodesic. We have assumed x = 0 without loss of generality; the constants E, L, M denote the energy, angular momentum, and rest mass of the test particle, respectively. In what follows we consider only “null” geodescis; i.e. we set M = O(rn2).

Expanding y , Z and T in powers of m and considering only the terms linear in m we have

Y = Y o + m Y , ,

2 = Zo+ mZ, ,

T = T o + m T l ,

and eqs. (A.14) now become

To= E ,

i,;+z:= E 2 ,

Yo20 - Z O Y O = L ,

j o j , + zoz, = 0 ,

TI = 2 E / Ro ,

2L RO

Y o ~ l - ~ l ~ o + Y l ~ o - ~ o ~ l = --,

where R : = y i + Z i .

(A.15)

(A.16)

Penrose and MacCallum [7] describe some properties of such geodesics without actually calculating them. Penrose and Curtis have performed similar calculations of null geodesics crossing a shock wave, but as far as we know these have not been published [a].

T. Dray, G. 'I Hoofr 1 Gravitational shock wow

However, since

m 2P

u = - ( T- Z) ,

o =2( T + Z ) , (A.17) m

we must require

Zo=-To=-E (A.18)

if U, and thus 6, is to remain finite in the limit as m goes to zero. The second and third of eqs. (A.16) now yield

YO=(),

yo= - L I E ,

Z , = O .

and the fifth of eqs. (A.16) implies

Thus

m m 2 E U=--+--

E P P %'

v=4p-, RO

Using eq. (A. 18) these can be integrated directly to give

mE m2

P P U = - A -- In (Zo+ R,,) ,

U = -4p In (Zo+ R,) ,

where we have ignored an irrelevant integration constant, and thus

We now separate into a near region N and a far region F as follows:

N = { ~ A ~ c ~ / G } ,

F= {&S mlkl <a}.

(A.19)

(A.20)

(A.21)

(A.22)

(A.23)

(A.24)

639

T. Dray, G. ’ I hoof^ / Gravitationtll shock waw

Note further that

lim C=O, A * - a 0

iim v = -4p In y i , **+a0

mE

P lim ;=-A.

* - t m (A.25)

n u s , there is a total shift in v^ given by

do^= -4p In y : . (A.26)

Note that, in the limit as m goes to zero, A is infinite everywhere in F. Furthermore, in this limit U is identically-zero in N, whereas 6 is a good affine parameter in F along the geodesic.

The shift (A.26) thus occurs, for small m, “essentially” only in N! Thus, in the limit as m goes to zero, the shift (A.26) occurs at U = O and represents a finite discontinuity in 6 along null geodesics! This can also be seen by calculating

(A.27)

thus showing that in the limit as m goes to zero U* is constant in F, i.e. for non- zero U . This is just a reflection of the fact that, in the limit, F is flat. This is illustrated in fig. 2.

We now turn to the behaviour of y. We must solve the last of eqs. (A.16), which, on inserting eqs. (A.19) and (A.20) becomes

~ 1 2 0 - Zoj, = -2 L/ Ro . The homogeneous equation clearly has the solution

(A.28)

Y : = M o (A.29)

for any constant A ; it remains to find a particular solution. Multiplying eq. (A.28) by Zo yields

EZy , -R ,R , j l = 2 L E / R o . (A.30)

But noticing that

(A.31)

suggests an ansatz for y , as a power series in Ro. We thus obtain the particular solution

(A.32)

T. Dray, G. ' t Hooji 1 Gravitational shock wave

The general solution to eq. (A.28) is thus

y, = -*+Azo , Yo

and there fore L E Y = - - + m [ - 2 +

(A.33)

(A.34)

We are interested in the behaviour of y in the far field F for m small. We obtain

This behaviour is illustrated in fig. 3. In general we have

4P cot a +cot p =- Yo

(A.35)

(A.36)

for the angles a and p as defined in fig. 3. At this point several comments are in order. We have nor considered all geodesics

which cross the shock wave, but only a sufficient number to determine how to glue the two flat half-space together. That this is sufficient follows from the existence of coordinates in which the metric is in fact continuous (see appendix B).

Using the results of appendix B we see that

lim ds2=-d6(d6+4p lny iS (6 ) d6)+dx2+dy2 rn-0

( 1 -2e(u))+4p In 4 S ( u ) du

(A.37)

which of course reduce to.(A.I3) and (A.9) respectively for U # 0. The first of (A.37) is just the result of Aichelburg and Sexl [ I ] (their eq. (3.9)), but the second of (A.37) disagrees,with their eq. (3.10). Although this is at first disconcerting, a more careful analysis reveals the source of the discrepancy: we have taken a limit different from theirs. This can be seen by noting that for m + 0, U # 0, our original coordinates (1, z ) are related to their coordinates ( t ; 2 ) by infinire scale factors.

Equivalently, note that the original Minkowski space given in (U, U, x, y ) coordinates is"pushed to infinity" in the resulting space-time given in (6, 6, x, y ) coordinates. Specifically, {U # 0; I u ~ <CO} corresponds to ( 6 # 0; 6 = -(sgn 6 ) 4, although the Source located at {U = 0; Iul c a}, corresponds to {6 = 0; 161 < 00). The corresponding statement for the ( c 2, ji, Z) coordinates of Aichelburg and Sexl [ I ] would be somewhat different.

Finally, note that although we have linearized both the metric (A.l) and the geodesics (A.14) the result is in fact exact. Had we begun (as in [I]) with the exact

641

T Dray, G. ' t Hooft Gravitational shock wave

Schwarzschild metric in isotropic coordinates and expanded in powers of m only the linear terms we consider would have survived.

Appendix B

CALCULATION OF THE RICCl TENSOR

We start with the metric

d?=2A(u, U) du du+g(u, u)h,(x') dx'dx', (B.1)

which is assumed to satisfy the Einstein vacuum equations. We introduce a shock wave by keeping (B. 1) for U -= 0 but replacing U by U + f ( x ' ) for U > 0*:

ds2 = 2A( U, U + Of) du(du + O f i dx') + g( U, U + 6f)hy dx' dx' , (B.2)

where 8 = O( U ) is the usual step function. Changing to coordinates (I?, G, ii) defined by

A u = u ,

G=u+t)f,

9 03.3)

(B.4)

2' =x i

we obtain ds2=2A(I?, 6) du*(d$-d(I?)f dI?)+g(I?, G)h, di 'dk',

where 6 = 6( U) is the Dirac delta "function". We note that the metric ds2 given in (B.2) and (B.4) is in fact conrinuous, i.e.

there exisr coordinates (U, 0, 2') such that the metric coefficients are continuous. A possible choice is given implicitly by

I?=a,

A G = 0 + 6 $ - f U B 2 7 hmnfmfn , g

where

This is not quite the standard ansatz gab = ( I - @ ) g i b + @:b. However, in the examples considend here the corresponding Ricci tensors differ at worst by a term in R,, proportional to O( 1 - e), which is not physically relevant.

642

T. Dray, G. ' I Hooft 1 Graoitarional shock warn

However, the coordinates (B.5) are extremely unwieldy both for mathematical Computations and for preserving physical intuition. We will thus proceed as follows. Noting that the coordinate transformation between (0, 0, 3') and (U, U, x i ) coordinates is continuous, we will calculate the Ricci tensor for the metric (8.2). However since the metric (8.4) is much easier to work with, we will formally transform to ( U , U, x ) coordinates, calculate Rd6, and then transform back to obtain Rob. Direct calculation yields*

c A ri

where Rf' is the Ricci tensor derived from h,, A is the Laplace operator associated with hi,, and 6 = S(u^).

Blithely ignoring the 6' term we transform to (U, U, xi) coordinates and insert the vacuum equations (obtained by setting f= 0) to get

R,, = Rt#=O,

R,, = R,j;=O,

i iit R,=Rg=-hg+ fa, A

we note that Taub [9] has given a systematic presentation of space-times with distribution-valued CUrVIturc tensors.

643

T. Dray, G. 'I Hooft 1 Gravitational shock wave

R,, = R;; i- 2 R;&

The stress-energy tensor for a massless particle located at the origin p = O of the ( x i ) 2-surface and at U = 0 is

Tab =4pS(p)S(u)StG&, (B.9)

where p is the momentum of the particle. Thus, the only non-zero component is

Ti, = 4pA2S(p)S(u) . (B.lO)

Inserting (8.8) and (B. 10) into the Einstein field equations, partially integrating the S' term, noting that e.g. A,G( U = 0) = Oe A,"( U = 0) = 0 yields precisely eqs. ( 5 ) .

The calculation above was first done by hand and then checked using the algebraic manipulation computer system SHEEP, As a further check on the validity of working in the singular coordinates (U^, 6, 2') (eq. (B.4)) SHEEP was also used to calculate the Ricci tensor directly in (U, U, x ' ) coordinates (eq. (B.8)), thus checking the original calculation of 't Hooft for the Schwarzschild case. The same answer, namely eqs. (9, was of course obtained in all cases.

References

[I] P.C. Aichelburg and R.U. Sexl. 1. Gen. Rel. Grav. 2 (1971) 303. [ 2 ] R. Penrose, in General relativity: papers in honour of J.L. Synge, ed. L. O'Raifeartaigh (Clarendon.

Oxford, 1972) 101; in Battelle rencontres, ed. C.M. De Witt and J.A. Wheeler (Benjamin, New York, 1968) 198; in Differential geometry and relativity, ed. M. Cahen and M. Flato (Reidel, Dordrecht, 1976) 271

[3] J. Ehlers and W. Kundt, in Gravitation: an introduction to current research, ed. L. Witten (Wiley, New York, 1962) SSf;, H.W. Brinkman, Proc. Natl. Acad. Sci. (US) 9 (1923) I

[4] P.D. DEath, Phys. Rev. D18 (1978) 990 [S] G.E. Curtis, J. Gen. Rel. Grav. 9 (1978) 987,999 [6] G. 't Hooft, J. Geom. Phys. 1 (1984) 45 [7] R. Penrose and M.A.H. MacCallum. Phys. Reports 6 (1973) 270 [8] P.D. D'Eath, private communication [9] A.H. Taub, J. Math. Phys. 21 (1980) 1423

644

CHAPTER 8.5

S-MATRIX THEORY FOR BLACK HOLES

G. 't Hooft Institute for Theoretical Physics

Princetonplein 5, P.O. Box 80.006 3508 TA Utrecht, The Netherlands

ABSTRACT

We explain the principles of the l a w s of physics tha t we believe to be applicable f o r the quantum theory of black holes. In particular, black hole formation and evolution should be described in terms of a scattering matrix. This way black holes at the Planck scale become indistinguishable from other particles. This S-matrix can be derived from known l a w s of physics. Arguments a re put forward in favor of a dlscrete algebra generating the Hilbert space of a black hole with its surrounding space-time including surrounding particles.

1. INTRODUCTION

There is quite a bit of controversy (and confusion) regarding the nature of physical l a w governing a black hole. Some of the difficult ies have the i r origin in the deceptively clean picture given by the "classical" (here th i s means "non-quantum mechanical") solutions of Einstein's equations of gravity in the case of gravitational collapse. The metric tensor describing the fabric of space-time appears to be smooth and well-behaved in the vicinity of a region we ca l l the "horizon", a surface beyond which there a re space-time points from which no information can reach the outslde world. I t seems tha t one should be able t o apply standard techniques from particle theory here to derive what a dis tan t observer can perceive and, naturally, th l s exercise has been done'.

There is one innocent-looking assumption tha t most practitioners then make. One observes that clouds of particles may venture into the "forbidden region", from which they can no longer escape or even emit any signal towards the outside world, and so one assumes tha t the corresponding s t a t e s in Hilbert space may be treated the way one always does in quantum mechanics: the unseen modes a re averaged over. Operators describing observations in the outside world a re assumed to be diagonal in the sector of Hilbert space that is not seen, and hence in all computations one is obliged to sum over all unseen modes.

The immediate consequence of t h i s practice is that the outside world alone is not anymore described by a single wave function but by a density matrix2. Even if one starts with a "pure" wave function, sooner or l a te r one f inds the system to be in a mixed state. I t 1s as if par t

645

of the wave functions "disappeared into the wormhole" ; information escaped. as if the system were linked to a heat bath.

With large black holes one can perform Gedanken experiments, and consider observers who move semiclassically in the neighborhood of a black hole horizon. If we attach some sense of "reality" to these observers the correctness of the above assumptions seems to be an inescapable conclusion.

But what if the hole is small, so that classical observers a re too bulky to enter? O r let us a s k a questlon that is probably equivalent t o this: suppose one keeps track of a l l possible s ta tes a black hole can be in, is it then still impossible to describe the hole in terms of pure quantum s t a t e s alone? W i l l the very tiny black holes evolve according to conventional evolution equations in quantum physics or is the loss of information a fundamental new feature, even fo r them?

There is a big problem with any theory in which the loss of "quantum information" is accepted as a fundamental item. This is the fact tha t all effective l aws become fuzzy. I t is not d i f f icu l t t o construct a n example of a theory in which pure s ta tes evolve into mixed s ta tes . Consider a system with a Hamiltonian that depends on a f r ee physical parameter a ( for instance the fine s t ruc ture constant). A s t a t e I @ ) o at t=O evolves into the state

at time t . The expectation value fo r an operator 0 evolves into

(1.11

(1.2)

and from the @ dependence one can recover the information tha t the system remained in a pure quantum state. But now assume tha t there Is an uncertainty in a . We only know the f i r s t few decimal places. There is a distribution of values fo r a , each with a probability P(a) . Our theory now predicts f o r the "expectation value" of 0 :

( O ) , = Ida (0)t,a H a ) Tr p ( t ) O ; (1 .3)

(1.4)

This is an impure density matrix, of the kind one obtains in doing calculations with black holes. The outcome of a by now standard calculation is a thermal distribution of outgoing particles. A thermal distribution is always a mixed state.

In our example we clearly see what the remedy is. The ex t ra uncertainty had nothing to do with quantum mechanics; the Hamiltonian w a s not yet known because of our incomplete knowledge of the l a w s of physics (in th i s case the value of a 1. By doing ex t ra experiments or by working harder on the theory we can establish a more precise value f o r a , and thus obtain a more precise prediction fo r

Returning with th i s wisdom to the black hole, what knowledge w a s incomplete? Here I think one has a situation that is common to all macroscopic systems: because of the large number of quantum mechanical s t a t e s it w a s hopelessly difficult to follow the evolution of Just one such s ta te precisely. One was forced to apply thermodynamics. The outcome of our calculations with black holes got the form of thermodynamic expressions because of the impossibility, in practice, to follow in detail the evolution of any particular quantum s ta te .

But th i s does mean that our basic understanding of black holes at present I s incomplete. In a statistical system such as a vessel containing an ideal gas, we have In prfnclple a quantum theory that is precise enough to study pure quantum states. In particular, if we dilute the gas so much that a single atom remains. the thermodynamic

( O ) , .

646

description w i l l no longer be correct, and we must use the rea l quantum theory. Similarly, if we want to understand how a black hole behaves when it reaches the PIanck m a s s , we expect the thermodynamic expressions to break down.

The importance of a good quantum mechanical description is tha t it would enable us to link black holes with ordinary particles. The Planck region may well be populated by a lot of different types of fundamental particles. the i r "high energy l i m i t " w i l l probably consist of particles s m a l l enough and heavy enough to possess a horizon and thus be indistinguishable from black holes. What we want is a consistent theory tha t covers all of th i s region. If we had a "conventional" Schrodinger equation in th i s region, it would be relatively straightforward (at least conceptually) t o extrapolate to large distance scales using renormalization group techniques, and recover the "standard model" (or morel )

There have been many proposals concerning the nature of our physical world near the Planck length. we have seen "supergravity", "string theory", "heterotic strings". e t cetera. My problem with these ideas is tha t they seem to be ad hoc. The models a re "postulated" and then afterwards the authors t ry to argue why things have to be th i s way (basically the argument is that the new model is "more beautiful" than anything else known).

I t would be a lot sa fer if we could derive the only possible correct sett ing of variables and forces, directly from the presently established l aws of physics. In these lectures we w i l l argue tha t it is possible to do th i s , or at least to make a good s t a r t , by doing Cedanken experiments with black holes. The reason why black holes should be used as a s t a r t i ng point in a theory of elementary particles is tha t anything tha t 1s t iny enough and heavy enough to be considered an en t ry in the spectrum of u l t r a heavy elementary particles (beyond the Planck mass), must be essentially a black hole.

Black holes are defined as solutions of the classical , i.e. unquantited. Einstein equations of General Relativity3. This implies tha t we only know how to describe them reliably when they are considerably bigger than the Planck length and heavier than the Planck m a s s . What was discovered by Hawking' in 1975 is tha t these objects radiate and therefore must decrease in size. I t is obvious tha t they w i l l sooner or l a te r enter the domain tha t we presently do not understand.

Curiously, it is not easy to see why Hawking's derivation of the thermal black hole radiation would not be exactly correct. Even in a functional integral expression for t h i s calculation one might still expect wormhole configurations through which quantum information leaks towards a mystical "other universe"'. W e w i l l now decide to be merciless: topologically non-trivial space-times a r e forbidden (unt i l fur ther notice) so that , a t least a t the microscopic level, pure quantum mechanics can be restored. More precisely, what we require is first of all some quantum mechanically pure evolution operator, and secondly tha t t h i s operator be consistent with all we know of large scale physics, in particular general relativity.

A t f i r s t s igh t these requirements a re in conflict with each other. General relativity predicts unequivocally that gravitational collapse is possible, and th i s produces a horizon with all its difficult ies. However, we claim tha t a pure quantum prediction tha t naturally blends into thermodynamic behavior in the large scale l i m i t is not at all lmposslble, but it is t rue tha t the requirement f o r th i s to happen is extremely restrictive. Combining it with all we already know about large scale physics may well yield an unambiguous theory. Anyway, we know f o r sure tha t the amendments needed a t the horizon all re fer t o Planck scale physics. As long as th is physics is not completely understood i t will

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also be impossible to refute our theories on the ground of inconsistencies with known physics.

So t h i s is our program. We assume that , as f o r all spatially confined systems, there exists such a thing as a "scattering matrix". One then t r i e s to reconcile th i s scattering matrix with the l aws of physics already known. We w i l l find that th i s scattering matrix, to some extent, can be derived. More precisely: the exact quantum behavior a t large dis tance sca le s (the distance scales reached in present particle experiments) can b e derived uniquely.

The problem is an apparent acausality. If we apply linearized quantum field theory in the black hole background it seems ununderstandable how information that is thrown into the black hole can reemerge as information in the outgoing s ta tes . This is because the outgoing radiation originates at t = -m and the ingoing matter proceeds unt i l t = +m , so the information had to go backwards in time. We simply claim tha t precisely for th i s reason linearized quantum field theory is inappropriate here. One must take gravitational ( if not other) interactions between in- and outgoing matter into account. One way to interpret what happens then is to assume that there is a fundamental symmetry principle , because matter inside the horizon is unobservable. One can then perform. a transformation that transforms away the singularity at t=+m , and produces one at t = - m .

I t is of crucial importance to note tha t what we a re deriving is not only the (quantum) behavior of the black hole i tself . I t is the entire system, black hole plus a l l surrounding particles, tha t we a re talking about. Using our (assumed) knowledge of physics at large distance scales we derive the properties of the black hole and al l other forms of matter at energies larger than the Planck energy.

In ordinary quantum field systems behavior at s m a l l distance, or equivalently, at high energies, determines the behavior at large distances and low energies. In the present case the interdependence goes both ways, or, in other words, the whole construction w i l l be over- determined. We expect stringent constraints of consistency, which, one might hope, may lead to a single unique theory. The point is tha t the symmetry principle Just mentioned af fec ts matter in an essential way, and thus may perhaps continue to be of relevance at the low energy domain.

This is the motivation of t h i s work. It may lead to "the unique theory". Even though our work is f a r from finished, we w i l l be able to show that there w i l l be a remarkable role fo r the old s t r ing theorys. The mathematical expressions we derive a re so similar to those of s t r ing theory tha t perhaps some of its results w i l l apply without any change. But both the physical interpretation and the derivations will be very different. As a consequence, the mathematics is not identical. One important difference is the s t r ing constant (determining the masses of the excitations), which in our case turns out t o be imaginary'.

In the usual s t r ing theory one uses the obvious requirements of un i ta r i ty and causality to derive tha t the s t r ing is governed by a local Lagrangean on the s t r ing world sheet. To derive similar requirements f o r the s t r ings born from black holes is f a r from easy. This is presently what is holding us back from considerations such as tachyon elimination and anomaly cancellation tha t so successfully seem to have given us the superstring scenario. What we advertise is a careful though slow process establishing the correct demands fo r a fu l l black hole/string theory. If successful, one w i l l know exactly the rules of the game and the ways how to select good from false scenarios and models.

2. QUANTUM HAIR

Classical black holes a re characterized by exactly three parameters3:

the mass M , the angular momentum L , and the electrfc charge Q . If magnetic monopoles exist in nature then there w i l l be a fourth parameter. namely magnetic charge Q, , and if besides electromagnetism there a r e other long range U(1) gauge fields then also the i r charges correspond to parameters fo r the black hole.

However, the existence of long range U(ll gauge fields other than electromagnetism seems to be rather unlikely. Then, since L , Q (and Qm ) a re all quantized, the number of different values they can take is limited, and indeed one can argue convlncingly (more about this in Ref') that the black hole can be in much more different quantum s t a t e s than the ones labeled by L and Q (and Qm 1, or in other words, the mass M must be a function of much more variables than these quantum numbers alone.

An interesting attempt to formulate new quantum numbers fo r black holes was initiated by Preskill. Krauss, Wilczek and others?. They took as a model field theory a U(1) gauge theory in which the local symmetry undergoes a Higgs mechanlsm vfa a Higgs field with charge N e . I n addition one postulates the presence of particles with charge e . I n such a theory there exist vortices, much like the Abrikosov vortex in a super conductor. These vortices can be constructed as classical solutions with cylindrical symmetry, at which the Higgs field makes one fu l l rotation if one follows It around the vortex.

The behavior near the vortex of particles whose charge is only e is more complicated. One finds that because of the magnetic flux in the Abrikosov vortex the fields of these particles undergo a phase rotation when they flow around the vortex, in such a way tha t an Aharonov-Bohm effect is seen. The Aharonov-Bohm phase is 2n/N , or, if we take a particle with charge ne , t h i s phase w i l l be 2nn/N .

The importance of t h i s Aharonov-Bohm phase is tha t it w i l l be detectable f o r any charged particle, at any distance from the vortex, in such a way that we w i l l detect its charge modulo N . This is surprising because there is no long range gauge field present!

An observer who can only detect large scale phenomena may not be able to uncover the chemical composition of the particle, but he can determine I t s charge modulo N . A l l he needs is a vortex, which to him w i l l look Just like a Nambu-Goto string.

Even if a particle were absorbed by a black hole, its electric charge would still reveal i tself . Thus, charge modulo N is a quantum number tha t w i l l survive even fo r black holes. I t must be a s t r i c t ly conserved charge.

One can then formallze the argument using only s t r ings and charges modulo N , without ever referring to the original gauge field. Then there may exist many kinds of strings/vortices, so tha t the black hole may have a rich spectrum of these pseudo-invisible but absolutely conserved charges.

W i l l t h i s argument allow us to specify all quantum numbers f o r a black hole? There are several reasons to doubt this. One is tha t an extremely large number of different kinds of s t r ings must be postulated, which seems to be a substantial departure from the Standard Model at large distance scales.

Secondly, it is not at all obvious that it w i l l be possible to do Aharonov-Bohm experiments with black holes. One then has to assume f f r s t that black holes indeed occur in well-defined quantum s ta tes , j u s t like atoms and molecules. So th i s argument that black holes have quantum ha i r is ra ther circular.

In my lectures there is no need for the mechanism advertised by Preskill e t al. I t is neither necessary nor likely tha t all quantum s t a t e s can be distinguished by means of some conserved quantum number(s). In my other lectures I use jus t the assumption tha t quantum s t a t e s exist , and nothing else. No large-scale s t r ings a re needed.

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3. DECAY INTO SMALL BLACK HOLES

Due to Hawking radiat ion the black hole looses energy, hence also m a s s . The in tens i ty of the radiat ion will be proportional t o T' , where T is the temperature, and the to ta l a r e a of the horizon, which f o r the Schwarzschlld black hole is 4nR2 ; R = 2M . Since one expects'

T = 1/8nM , (3.1)

t h e m a s s loss should obey

The cons tan ts C , C' depend on the number of independent par t ic le types at the corresponding mass scale , and t h i s wil l vary s l i g h t l y with temperature; the coeff ic ients wil l however s t a y of order one (as long as M stays considerably la rger than the Planck mass).

Ignoring t h i s s l igh t m a s s dependence of C' , one f i n d s

(3.3) M ( t ) = C" ( t o - t ) * + ,

where to is a moment where the thing explodes violently. Conversely, t h e lifetime of a n y given Schwarzschild black hole with mass M can be estimated t o be

ti = M3/3C' . (3.4)

Now t h i s is the time needed f o r the complete disappearance of the black hole. One may a l so ask f o r the average lifetime of a black hole i n a given quantum mechanical s t a t e , i .e. the average time between two Hawking emissions.

A rough estimate reveals t h a t the wavelength of the average Hawking par t ic le 1s of the order of the black hole rad ius R , and t h a t t h i s is also t h e expected average spa t ia l dis tance between two Hawking par t ic les . Therefore the lifetime of a given quantum state is of order R , i .e. of order 1 / M in Planck uni ts .

I n the language of par t ic le physics t h i s implies t h a t t h e rad ia t ing black hole is a resonance state tha t in an S matrix would produce a pole at the complex energy value

E = M - C, i/M , (3 .4)

where C, is again a constant of order one. This corresponds t o t h e value

EZ = MZ - 2C3 iMp12 (3.5)

for t h e Mandelstam variable s . We see t h a t all black hole poles are expected t o be below the real axis of s at a universal average d is tance of order one in uni t s of the Planck mass squared.

It is not altogether unreasonable to assume t h a t a black hole is Jus t a pole i n the S matrix like any other t iny physical object.

*As was pointed out by t h i s author', the derivation of t h i s formula requi res a n assumption concerning the interpretat ion of quantum wave func t ions f o r par t ic les disappearing into the black hole. Though plausible , one can imagine t h i s assumption to be wrong, i n which case t h e black hole temperature wil l be d i f fe ren t from (3.1).

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4. THE S-MATRIX ANSATZ AND THE SHIFTING HORIZON

The problem with linearised quantum field theory in the black hole background 1s that the ingoing particles then seem to be independent of the outgoing ones. Hilbert space is then a product space, IS) = I#),, x IS)out . If we were to describe a black hole that obeys an overall Schr6dinger equation then these in- and out-spaces cannot be allowed to be independent of each other. In contrast, one would expect the existence of an S-matrix:

I#)our = s 1S)i. P (4.1)

and with this mapping of in- to out-states the degrees of freedom pictured in Fig. la are replaced by the ones of Fig. lb or Fig. Ic.

matter Out

mat ter i n

a)

r v t t e r In

b )

Fin. 1 - a) Linearised quantum field theory produces a Hilbert space that is the product of two factors: I#) = l # ) ~ n x l # ) o u ~ Ingoing partlcles form the factor space (bl. and outgoing ones form { I # ) o u t ) (c). x and y are the Kruskal Coordinates for the Schwarzschild metric.

A s stated earlier. the reason why superimposing in- and out- particles as in Fig. la is incorrect is the breakdown of linearised quantum field theory at distances closer than a Planck length from the horlzon. Gravitational interactlons there become super strong. W e can obtain the black hole representations of Fig. l b and Fig. lc by adopting the following elementary procedure:

I ) Postulate the existence of an S matrix, and il) take interactions between in- and out-states into account, in particular the gravitational ones.

We can look upon this procedure a s a new and more precise formulation of the general coordinate transformation from Kruskal coordinates3 to Schwarzschlld coordinates, or from f la t space-time to Rindler' space- t ime. There is no direct contradiction with anything we know about general relativity or quantum mechanics, but because of the crucial role attributed to the interactions the picture is only somewhat more complicated.

651

f u t u r e

'.. .. t n f a l 1 i n g ".. .. p a r t I c 1 e t~ o r I z o n,.."'

o r 1 z o n

p a s t e v e n t h o r I z o n

Fig. 2. The horizon displacement.

The most important ingredient of the gravi ta t ional in te rac t ions is the horizon shift". Consider any par t ic le fa l l ing in to the black hole. I t s g rav i ta t iona l f ie ld is assumed to be so weak t h a t a l inear ised descr ipt ion of i t , dur ing the infa l l , is reasonable. The curva tures induced in the Kruskal frame a r e in i t ia l ly much smaller than t h e Planck length. Now perform a time t ranslat ion ( for the ex terna l observer) . A t t h e or ig in of Kruskal space t h i s corresponds t o a Lorentz t ransformation :

x + T - l x ; y + T y . (4.2)

The 3 f a c t o r s here can grow very quickly, exponentially with ex terna l time. The very t iny in i t ia l curvatures soon become subs tan t ia l , but a r e only seen as s h i f t s i n the y coordinate, because y has expanded such a lot .

The r e s u l t is a representation of the space-time metric where t h e ingoing par t ic le en ters along the y ax is , with a velocity t h a t h a s been boosted t o become very close to the speed of l ight . I t s energy i n t e r m s of t h e boosted coordinates h a s become so huge t h a t the curva ture became s izable . I t is described completely by saying t h a t two halves of t h e conventional Kruskal space a r e glued together along the y a x i s with a shift 6y , depending expl ic i t ly on the angular coordinates 6 and (p .

The ca_lculation of the funct ion 6y(6 , (p) is elementary. In Rindler space, 6 y ( x ) is simplyt1

6y(; ; ) = - 4c,p log("x2) + c , (4 .3)

where G, is Newton's constant , p is the ingoing par t ic le momentum, and C is a n a r b i t r a r y constant .

In Kruskal space _the angular dependence of 6y is a b i t more complicated than the x dependence i n Rindler space. I t is found by inser t ing Einstein 's equation, which is RIlv = 0 , everywhere except where the par t ic le comes in. S ta r t ing with a n a r b i t r a r y 6y as a n Ansatz, one f i n d s Einstein's equation to correspond t o

(1 - A,,,v) 6 y ( 6 , 9 ) = 0 , (4 .4)

where Ad, , is the angular Laplacian. A t the angles 6,,(p, where the par t ic le e n t e r s we simply compare wlth the Rindler r e s u l t (4.3) t o obtain

(1 - Ad,.) 6 ~ ( 6 , ( p ) = K Pin 62(6,(p;6,,(po) . (4 .5)

where K is a numerical constant re la ted to Newton's constant . This equation can be solved:

652

where 6, is the angular separat lon between (6,(p) and (bo,(p0) . Other expressions f o r f a r e

f = K~ r m - * ' d z ( C O S ~ ~ - C O S Z ) - ~ e-'fl ; 61

andi2

'-4 + 'id3 [-c0se1) f = 2 c o s h (fnv'3)

(4.6)

(4.7)

(4.8)

where P is a Legendre funct ion with complex index (conical func t ion) . From (4.7) one sees d i rec t ly tha t f o r a l l angles 61 f is positive.

5. SPACE-TIME SURROUNDING THE BLACK HOLE

The hor l ton s h l f t discussed in the previous section is a n essent ia l lngredient i n the S-matrix construction. Wlthout it we would not be able to perform t h i s task. Now we a re . W e will discuss t h i s construct ion in t h e next chapter . F i r s t one has to understand what t h e relevant degrees of freedom are and where in space-time they live. Here, par t ly ant ic ipat ing on o u r resu l t s , we observe that the outgoing configurat ions w i l l depend on what goes in , and with a sens i t iv i ty t h a t depends exponentlally w i t h dt/4M , where d t is the time interval as seen by the d i s t a n t observer. However, the par t ic les g o i n g out l a t e r than a

h o r i z o n

s i n g u l a r i t y

ingofng m a t t e r w t t h momentum 6 p

O n s e t o f horizon

Fig. 3

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lapse of time of the order of At = 4 M log ( M / M p , ) , can no longer be described at all. We w i l l assume, f o r simplicity, t h a t these par t ic les wi l l be completely determined ( in the quantum mechanical sense) by e a r l l e r events , or in other words, one is not allowed t o choose these states a n y way one pleases. Simply counting s t a t e s (as one can der ive from the f i n i t e black hole entropy"l) one notes tha t , given a l l i n g o i n g par t ic les , the outgoing par t ic les can be chosen f ree ly only dur ing a n amount of time At , o r vise versa.

Any i n g o i n g par t ic le wil l not only a f fec t the outgoing par t ic les a f t e r a t i m e lapse of order At , but a l so all t h a t come out l a t e r than tha t . These l imitat ions i n choosing in- and outgoing s t a t e s a r e Jus t what they would be f o r any macroscopic system such as a f i n i t e s i z e box containing a g a s o r liquid, connected to the outside world f o r instance v i a a t iny hole.

6. CONSTRUCTION OF THE S-MATRIX

The argument now goes as followss. 1. Consider one par t icu lar in-s ta te and one par t icu lar out-s ta te .

Assume t h a t someone gave us the amplitude defined by sandwiching the S-matrix between these two s ta tes :

( in lout) . (6.1)

Both t h e in- and the out-s ta te a r e described by giving a l l par t ic les i n some conveniently chosen wave packets. The ingoing wave packets look l ike

i where i r u n s over all par t ic les involved, and f in are smooth funct ions. We assume them to be sharp ly peaked in t h e angular coordinates so t h a t we know exactly where the par t ic les e n t e r into the horizon (so the angufar coordinates and the radial momenta of all par t ic les a r e sharp ly defined). Similarly the outgoing wave packets a r e

Now let u s consider a s m a l l change i n the ingoing s t a t e : {in) + t in ' ) . This brings about a sharply defined s m a l l change i n t h e d is t r ibu t ion of the rad ia l momenta pin(6.(p) on the horizon:

pin + Pin + 6Pin(o,(P) . (6.4)

This 8pi, now produces an (extra) horizon s h i f t ,

8y(R) = J f ( R - R ' ) 6Pin(R') , (6.5)

where f i s the Green funct ion computed in the previous sect ion and R s t a n d s s h o r t f o r (19,cp) ; R-n' stands f o r the angle between n and R' .

The horizon s h i f t (6.5) does not af fec t the thermal na ture of t h e Hawking radiat ion, but i t does change the quantum s t a t e s . A l l out-wave func t ions are sh i f ted . (6 .3) is replaced by

( the e f f e c t of t h e s h i f t on f is of lesser iinportance). The s h i f t 6y

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is assumed to be so small that the outgoing particle is not thrown over the horizon. This requires tha t we consider only those outgoing particles tha t are already sufficiently f a r separated from the horizon, or: they a r e not the ones tha t emerge later than the time interval A t , a s defined in the previous section, see Fig. 3. This is why the time interval At w a s necessary. Now such a restriction w i l l also imply tha t we w i l l have to reconsider the definition of inner products in our Hilbert space, and th i s w i l l imply that the operator exp(-ipAut6y) might not be unitary. We w i l l temporarily ignore th i s important observation.

We observe tha t the S-matrix element (6.1) is replaced:

Here, pout(n) is the total outgoing momentum at the angular coordinates n . What we have achieved is that we have been able to compute another matrix element of S . Now simply repeat t h i s procedure many times. We then f ind all matrix elements of S t o be equal t o

where N is one common unknown factor. Apart from an overall phase, N should follow from unitarity.

The derivation of (6.8) ignores all interactions other than the gravitational ones. W e w i l l be able to do better than tha t , but le t us f i r s t analyze th i s expression.

What is unconventional in the S-matrix (6.8) is the fac t tha t the in- and out-states must have been characterized exclusively by specifying the total radial momentum distribution over the angular coordinates on the horizon. If there a re more parameters necessary to characterize these s ta tes , these extra parameters w i l l not f igure in the S-matrlx. But t h i s would mean that two different s ta tes I A ) and IB) could evolve into the same s ta te , so these ex t ra parameters w i l l not be consistent with unitarity. We cannot allow f o r other parameters than the total momentum distributions (unless more kinds of interactions a re taken into account).

Thus, if fo r the time being we only consider gravitational interactions. the in-states can be given as Ipin(n)) and the out-states as lpout(n)) . The operators pin(S,cp) commute f o r different values of 6 and cp , and their representations span the entire Hilbert space; the same f o r pout(S,cp).

The canonically conjugated operators uin(n) , uout (RI a r e defined by the commutation rules

[p{n(n),u*"(n')l = -iaZ(n-n') (6.9)

(and similarly f o r the out operators), or,

(uln(n)lptn(n)) = C exp i.Jd2n ~ ~ ~ ( n ) u ~ ~ ( n ) ,

where C is a normalization constant. In terms of the U operators the S-matrix is

(6. lo')

(uouttn) lutntn) ) = ppoutnpin exp(-iptnu in +fPoutuout -1PoutfPin) s

(6.11) which is a Gaussian functional integral over the functions pout and p i n . Since the inverse f 1 of f is ~-'(1 - A Q ) , the outcome of

655

t h i s functional integral is

(O,,~(R) lufn(n)) = C exp[-frc-*JdZR uln(n) (1 -4) uout(R)) =

The last term In the brackets 1s something like a m a s s term and becomes subdominant if we concentrate on s m a l l subsections of the horizon. Therefore it w i l l be ignored from now on. Eq. (6.12) seems to be more fundamental than (6.8) because it is local in R .

Fourier transforming back we get

We reobtain (6.8) written as a functional integral. I t is Illuminating to redefine

Pout = P- ; PIn = p* ; u0.t = x- ; U i n = -x+ , (6.14)

and to replace the angular coordinate R by a transverse-coordinate x , so tha t if we define the transverse momentum components p 0 one can write

Suppose now tha t both the in-state and the out-state a r e wrlt ten as se t s containing a f in i te number of particles, havirng not only fixed longitudinal momenta but also transverse momenta pf . We then hav_e,_to convolute the amplitude (6.15) with transverse wave functions exp(iplxl) fo r each particle i . It becomes

This functional integral is very s i m i l a r to the functional integral f o r a s t r ing amplitude, lncluding the integration over Koba-Nielsen variab1esl3, except fo r the unusual imaginary value fo r the s t r ing constant:

T = 8nCHi . (6.17)

7. ELECTROMAGNETISM

What happens if more interactions a r e included? The simplest t o handle turn out to be the electromagnetic forces. Suppose tha t the particles tha t collapsed to form the black hole carried electric charges. The angular charge distribution was

Pin(R) . (7.1)

A s in the previous section, we consider a s m a l l change in t h i s sett ing, so

p i , m + PlnlR) + Gp&) * (7.2)

The B p l n ( f l ) produces an extra contribution to the vector J potential at the horizon which is not difficult to compute .

where ro is the radius of the horizon, and A(R) must satisfy

AQA(n) = 6pin(n) . (7.4)

The field (7.3) is only non-vanishing on the plane x=O , where it cause8 a sudden phase rotation for all wave packets that go through. An outgoing wave undergoes a phase rotation

eiQ*(*) ; A(R) = ro-I A(R) . (7.5)

This rotation must be performed for a l l outgoing particles with charge Q , A l l together the outgoing wave is rotated as follows:

IPouc(n)*Pout(~)) + (7.61

exp -1 ld211sd%' f1tn-Q') pout(n)bpln(n') x l ~ o u ~ ~ Q ) , ~ o u ~ ( ~ ) ) ,

where f1(Q-n') is a Green function that satisfies

Ap f1(n-ll') = - K e B 2 ( n - n ' ) . (7.7)

K, is a numerical constant. And using arguments identical to the ones of the previous section

we repeat the infinitesimal changes to obtain the S-matrix dependence on p&) and pout(n) :

( ~ o u t ( n ) , ~ o u t ( Q ) lPin(Q)*Pin(Q)) = (7.8)

e-fl~pout(RI f(n-n' 1 pin(n' )d2CW2n'

-r~~p,.~cn) r,(n-n' 1 ptn(nD )d2mZn' e

Now let us replace pout(f l )pin(~') by

-4 (~out(n)-~in(~)) [pout(n' )-pin(Q' 1) . (7.9)

This differs from the previous expression by two extra terms in the exponent. one depending on pout(Q) only and the other dependlng on pin(R) only. These would correspond to external "wave function renormalizatlon factors" that do not describe interaction between the in- and the out-state. So we ignore them.

The electromagnetic contribution in (7 .8 ) can then be written as a functional integral of the form

Now it may also be observed that the charge distribution p is actually a combination of Dirac delta distributions,

'The unit e of electric charge of the ingoing particle 1s here included in p i n .

657

p ( ~ ) = ~ ~ i ~ z ( ~ - n i ) ; oi = nie . (7.11)

Therefore, i f we add a n integer multiple of 2n/e t o the f ie ld rP the integrand does not change. In other words: @ is a periodic var iable .

Adding (7.10) to (6.15) we notice t h a t the f ie ld 4 a c t s exact ly as a f i f t h , periodic dimension. Hence, electromagnetism emerges n a t u r a l l y as a Kaluza-Klein theory.

8 . HILBERT SPACE

In t h i s sect ion we br ief ly recapi tulate the nature of the Hilbert space i n which these S matrix elements a r e defined. As explained i n Section 4, the states whose momentum and charge dis t r ibut ion over the horizon were given by p(R1 and p ( Q ) include all par t lc les i n the black hole's vicini ty . But ( i f f o r simplicity we ignore electromagnetism) we can a l s o form a complete basis in terms of s t a t e s f o r which the canonical operators x(n) a r e given. These x ( R ) (eq. 6.12) may be interpreted as the coordinates of the horizon. Apparently. the precise shape of the horizon determines the state of the surrounding particlest

Furthermore, t h e . in-horizon and the out-horizon do not commute. Therefore, the positions of the f u t u r e event horizon and the pas t event horizon do not commute with each other . If we def ine a "black hole" as a n object f o r which the location i n space-time of the f u t u r e event horizon is precisely determined, we can define a "white hole" as a s t a t e f o r which the pas t event horizon is precisely determined. The white hole is a linear superposition of black holes (and vice versa) ; operators f o r white holes do not commute with the ones f o r black holes. In o u r opinion t h i s resolves the i ssue of white holes in general re la t ivi ty .

Obviously, it is important t h a t the horizon of the quant ized black hole is not taken t o be simply spherical ly symmetric. In a black hole with a h is tory t h a t is not spherical ly symmetric, the onset of the horizon, i.e. the point(s1 i n space-time (at the bottom of Fig. 3) where f o r the f i r s t time a region of space-time emerges from which no timelike geodesic can escape to 9* , has a complicated geometrical s t r u c t u r e . I t s mathematical construct ion has the charac te r i s t ics of a caus t ic . One might conjecture t h a t the topological de ta i l s of t h i s caus t ic specify the quantum state a black hole may be In.

The f a c t t h a t the geometry of the ( fu ture or past) horizon should determine the quantum state of the surrounding par t ic les gives rise t o in te res t ing questions and problems. In ordinary quantum f ie ld theory t h e Hilbert space describing par t ic les in a region of space-time is Fock space; an a r b i t r a r y , f in i te , number of par t ic les with specif ied positions o r momenta together define a s ta te . But now, close t o the horizon, a state must be defined by specifying the total momentum enter ing (or leaving) the horizon at a given sol id angle R . Apparently we are not allowed t o specify f u r t h e r how many par t ic les there were, and what their o ther quantum numbers were. Together all these possibf l l t ies form j u s t one state. So, our Hilbert space is set up d i f fe ren t ly from Fock space. The difference comes about of course because we have s t r o n g gravi ta t ional interact ions that we a r e not allowed t o ignore.

The best way t o formulate the specif icat ions of our bas i s elements here is t o assume a la t t ice cut-off in the space of sol id angles (one "lattice point" f o r each uni t of horizon sur face area somewhat bigger than AX (the Planck dis tance squared) , and then to specify t h a t there should be exactly one ingoing and one outgoing particle at each BZ . The momenta are given by the operators pi , , (Q) and p o U t ( n ) (and t h e charges by pi , (Q) and p,,,(Q) I . The in- and out-operators of course do not commute.

One may speculate t h a t s ince AX is extremely small , the to ta l i ty

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of all these particles may be indistinguishable from an ordinary Dirac sea f o r the large-scale observers.

Also one may notice tha t the way conventional s t r ing theory deals with in- and outgoing particles is remarkably similar. Before integrating over the Koba-Nielsen variables the s t r ing amplitudes a l so depend exclusively on the distribution of total in- and outgoing momenta (see concluding remarks in Sect. 6).

If Ji lbert space is constructed entirely from the operators p ( ; ) and x ( x ) then these operators a re hermitean by cons t ry t ion . But we have also- seen tha t in terms of ordinary Fock space p ( x ) , and hence also x ( x ) a re probably not hermitean. There a re different ways to approach th i s hermiticity problem, but we sha l l not elaborate here on th i s point.

9. RELATION BETWEEN TERMS IN THE HORIZON FUNCTIONAL INTEGRAL AND BASIC INTERACTIONS IN 4 DIMENSIONS

In principle one can pursue our doctrine to obtain more precise expressions f o r our black hole S matrix by including more and more interactions tha t we actually know to exist from ordinary particle theory. We should be certain to obtain a result tha t is accurate apar t from a limitation in the angular resolution, because particle interactlons a r e known only up to a certain energy. In th i s section we indicate some qualitative results.

The details of our "presently favored Standard Model" may well change in due time. We w i l l denote anything used as an input regarding the fundamental interactions among in- and outgoing particles near the horizon. at whatever scale. by the words "standard model".

Suppose the standard model contains a scalar f ield. The e f fec ts of t h i s f ield w i l l be f e l t by slowly moving particles at some distance from the horizon. But at the horizon itself these e f fec ts a r e negligible. Consider namely a particle such as a nucleon, surrounded by a sca l a r f ield such as a pion field. Close to the horizon th i s particle w i l l be Lorentz boosted to tremendous energies. The scalar f ield configuration w i l l become more and more flattened. But unlike vector or tensor f ie lds , its intensity w i l l not be enhanced (it is Lorentz invariant) . So the cumulated effect on particles traversing it will tend to zero.

However, one effect due to the scalar f ield w i l l not go away. Suppose our standard model contains a Higgs field, rendering a U(1) gauge boson massive. This means that the electromagnetic field surrounding a f a s t electrically charged particle w i l l be of shor t range only. One can derive that the field equation (7.4) w i l l change into

(AQ-M, ' )A(~) = 6p,,(R) . (9.1)

One may say tha t the ingoing charge density pi,(n) is screened by charges coming from the Higgs particles.

This implies tha t the equations f o r the @ field in Sect. 7 w i l l obtain a m a s s term:

Note tha t t h i s m a s s term breaks explicitly the symmetry 4 + 4 + A This explicit symmetry breaking may be seen as a resu l t of the f in i te and constant value of the Higgs field at the origin of Kruskal space-time .

Next, we may ask what happens if our standard model exhibits confinement. This means tha t a t long distance scales no effect of the gauge field is seen and all allowed particles a re neutral.

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Confinement is usual ly considered t o be the dually opposi te of the Higgs mechanism: Bose condensation of magnetic monopoles. A magnetic monopole is a n object to which the end point of a Dirac s t r i n g is at tached. A Dirac s t r i n g is a s ingular i ty in a gauge t ransformation such t h a t t h e gauge transformation makes one f u l l rotat ion if we follow a loop around the s t r i n g .

W e must know how to describe the operator f ie ld of a monopole at the horizon. Suppose a monopole entered at the sol id angle Rl . This means t h a t a Dirac s t r i n g connects t o the black hole at t h a t point. The outgoing charged par t ic les undergo a gauge rotat ion t h a t ro ta tes a f u l l cycle if we follow a closed curve around nl (an anti-monopole may neut ra l ize t h i s elsewhere on the horizon).

The gauge jump f o r the vector potential f ie ld A can be ident i f ied with t h e periodic f i e l d 0 of Sect 7 . So adding a n en ter ing monopole t o t h e in-s ta te implies t h a t th i s f ie ld @ is sh i f ted by a n amount h(n) where A makes a f u l l cycle when followed over a loop around nl. This is an operation t h a t is called disorder operator i n statistical physics and f ie ld theory. This operator, 0, , is dual to the or ig ina l f i e l d 0 . W e f i n d t h a t the dua l transformation e lec t r ic i ty f--) magnetism corresponds t o the dualiky between @ and .

Thus, if we have confinement, a m a s s term wil l r e s u l t i n t h e equat ions f o r 0, . I t explicitly breaks the symmetry @, -t @, + C . And t h i s b a r s the transformation back to 0 . Therefore, i f confinement occurs , the f i e l d 0 i s no longer wel l -def ined, we have on ly 0, , I t s m a s s wil l be the glueball m a s s .

In Table 1 we list pecul iar i t ies of the mapping from 4 to 2 dimensions. The generators of local symmetry t ransformations i n 4 dimensions correspond to the dynamic var iables i n 2 dimensions. Thus one expects t h a t if t h e s tandard model includes a gravi t ino ( requi r ing a supersymmetry generator of spin $1 then a fermionic f ie ld var iable wil l emerge In 2 dimensions.

But the above are merely qual i ta t ive features . They should be turned into precise quant i ta t ive r u l e s and pr inciples , for which f u r t h e r work is needed.

10. OPERATOR ALGEBRA ON THE HORIZON

A fundamental shortcoming of the procedure described above is t h a t the dimensionality_ of H i l b e r i space is in f in i te from the start. The func t ions p ( x ) and x ( x ) generate a n inf in i te s e t of bas i s elements. Y e t t h e black hole entropy, as calculated from Hawking radiat ion, is f in i te . Indeed, we have not yet been able to reproduce Hawking rad ia t ion from o u r S-matrix. This is because we have ignored t h e t ransverse components of the gravi ta t ional s h i f t s , and the s t r i n g func t iona ls we produced t h u s f a r only allow f o r inf ini tes imal s t r i n g exci ta t ions. We s h a l l now t r y to improve our descr ipt ion of t h e bas i s elements of Hilbert space. This we do by se t t ing up a n operator algebra. F i r s t we consider the algebra generated by the amplitudes we have.

Our s t a r t i n g point here is t h a t s t a t e s i n H i l b e r t space are uniquely determined by specifying any one of th? following f o u r funct ions: th_e dis t r ibut ion of ingoing momenta p , ( x ) ,_ t h e outgoing momenta p - ( x ) , the conjugated operators x ' ( x ) , or x - ( x ) . They obey the algebra

(10.1)

(10.2)

(10.3)

(10.4)

(10.5)

0 S p i n 1: A , ( x . t ) local gauge generator: A( x , t mod 2n/e

0 S p i n 0: 4(x, t )

Table 1

STANDARD m 3 D n IN 3+1 DIMENSIONS INDUCED 2 DIMENSIONAL FIELD THEORY ON BLACK HOLE HORIZON

Scalar var iable (spin 0): @(n) mod 2n/e

No f i e l d at a l l

S p i n 2: B U " ( X , t ) local gauge generator: u " ( x , t )

0 H i g g s mechanism: "spontaneous" mass M A for vector f i e l d

Str ing var iables ( s p f n I ) : X , ( n )

exp l i c i t symmetry breaking; @(n) ge t s same mass MA .

Spin B: gravi t ino local gauge generator s p i n f

Spin f fermion (7)

Conflnernent i n vector f i e l d A, 0 must be replaced by disorder op. OD ; its symmetry broken.

0 Non-Abelian gauge theory ~~ - ~~

only sca l a r s 0 , corresponding t o Cartan subalgebra

0 spin 4: feraions I no f i e l d at a l l

The algebraic relations among &I and #(;'I a re slightly more subtle because of the quantization of electric charge and the ensuing periodic boundary conditions on 0 . We w i l l disregard these from here on,

The relations (10.1-5) are not infinitely accurate. This is because we neglected any gravitational curvature in the sideways directions, This is fine as long as transverse distance scales are kept considerably larger than the Planck scale. One may convince +one_self that this implies neglecting higher orders in the derivatives a x - / a x . Is there any way to obtain a more precise algebra? I t is natural to search for an algebra that is invariant under Lorentz transformations. One might hope that such an algebra could generate the correct dearees of freedom at the Planck scale (In particular quantized degrees of freedom).

I t was proposed in Ref6 that H i l b e r t space on the horizon may be generated by the operator algebra of fundamental surface elements,

661

e b ax' axy

aua a r b I#"(;) = E - - . (10.6)

- where the transverse coordinates x were replaced by more a rb i t ra ry sur face coordinates U', . The relations (10.1-5) may be used in the case U = x , when the derivatives a re s m a l l . This means

The commutation ru les can then be rewritten in the form

c [ZS"(a),ZS"(a')1 = tT E'""ArsA(a)s2(a-a~ , (10.8) A

which is written in such a way that it remains true in all coordinate frames. T is a constant ( 's tr ing constant') equal to 8n in Planck units. In stead of (10.1-51 we can take th i s to be the equation tha t generalizes to a rb i t r a ry surfaces. I t has the advantage of being l inear in W .

Now (10.8) is not a closed algebra, because the l e f t hand side still contains a summation. A complete algebra is obtained a s follows.

Let K be I times the self dual part of W :

I t has three independent components:

K, = i (w23 + ~ ' 4 ) ; K, = i(w3' + ~ 2 4 ) ; K, = i (W'2 + ~ 3 4 ) . (10.10)

Now from (10.8) we derive that these obey a complete commutator algebra,

[K,(a),Kb(a')l = f T ~ & ~ K ~ ( ~ ) 6 2 ( ~ - ~ ' ) . (10.11)

Apart from a complication to be mentioned shortly, t h i s is a local and complete algebra of the kind we were looking for . A t first sight it seems to generate an infinite dimensional Hilbert space because the operators K , like the W , are distributions . But let us introduce tes t functions f(u) , g(u) and define operators

then these sa t i s fy commutation rules:

(10.13)

Let us now res t r ic t to tes t functions f(;) tha t can only take the values 0 or 1 . Then sa t i s fy the commutation rules of ordinary angular momentum operators. Note that fo r such an f the integral (10.12) is nothing but a boundary integral:

L,"' = IT-' $(x2dx3 + x'dx') , e tc . ,

6 f

(10.14)

where 6 f stands fo r the boundary of t_he support of f . We conclude tha t f o r every closed curve 6f on U space we have three 'angular momentum' operators L e t f ) tha t s a t i s fy the usual commutation ru les and

662

addition ru les f o r angular momenta. Given such a bunch of closed curves fi we can characterize the contribution of that part of the horizon to Hilbert space by the usual quantum numbers I i and mi . These a re discrete and so, in some sense, we seem to come close to our a i m of realizing a discrete Hilbert space for black holes. We note an important resemblance with the loop variable approach to quantum gravityi5.

Unfortunately, there is a snag. The operators L, a r e not hermitean. If we take x i to be hermitean and x' anti-hermitean then In the deflnition (10.6) W i j a re hermitean and W i 4 anti-hermitean. Therefore, Lat correspond to the anti-self dual par t s of Wuv. The commutation ru les between L, and Lat a re non-local (they follow from (10.1-5)). The operators Lz are hermitean, but not necessarily positive (they a re only nonnegative fo r time-like surface elements). If we may assume the smallest surface elements to be timelike we can still build our surface using quantum numbers 1 , and mi but the s t a t e s we get are not properly normalized (it is fo r finding the norms of the s ta tes tha t we need hermitean conJugation). If

$ { 1 I , mi)

are the basis elements constructed using the self dual operators Li , and

4(lt .mi)

the basis elements generated by the anti-self dual L i t , then we have

(10.15)

but the $ themselves, or the 4 themselves, a r e not orthonormal. Now rememkr our realization earlier that actually the operators

x+(G) and x - l x ) are not hermitean, when we pass from the "horizon Hilbert space" to ordinary Fock space, because the s h i f t operators may move particles behind the horizon. I t is conceivable tha t t h i s w i l l lead to hermiticity conditions altogether different from (10.15).

But it is f a r from clear whether or not we actually obtained a complete representation of our Hilbert space.

REFERENCES

1.

2.

3.

4.

5.

S.W. Hawking, Commun. Math. Phys. 43 (1975) 199; J .B. Hartle and S . W . Hawking, Phys.Rev. D13 (1976) 2188; W.G. Unruh, Phys. Rev. D14 (1976) 870; R.M. Wald, Commun. Math. Phys. 45 (1975) 9 S . W . Hawking, Phys. Rev. D14 (1976) 2460; Commun. Math. Phys. 87 (1982) 395; S.W. Hawking and R. Laflamme, Phys. Lett. B209 (1988) 39; D.N. Page, Phys. Rev. Lett. 44 (19801 301, Gen. Rel. Grav. 14 (1987) 299; D.J. Gross, Nucl. Phys. B236 (1984) 349 C.W. Misner, K.S. Thorne and J.A. Wheeler. "Gravitation", Freeman, San Francisco, 1973; S . W . Hawking and G.F.R. E l l i s , "The Large Scale Structure of Space-time", Cambridge: Cambridge Univ. Press, 1973; E.T. Newman e t al. J. Math. Phys. 6 (1965) 918; B. Carter, Phys. Rev. 174 (1968) 1559; K.S. Thorne, "Black Holes: the Membrane Paradigm", Yale Univ. press, New Haven, 1986; S. Chandrasekhar, "The Mathematical Theory of Black Holes", Clarendon Press, Oxford University Press S. Coleman, Nucl. Phys. B310 (1988) 643; S.B. Giddings and A. Strominger,Nucl. Phys. B321 (1989) 4&1; ibid. B306 (1988) 890 P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. E56 (1973) 109; M.B. Green, J .H . Schwarz and E. Witten, "Superstring

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6.

7.

8. 9. 10. 11.

12. 13

14.

15.

Theory", Cambridge Univ. Press; D. J . Gross, e t a l , Nucl. Phys. B 256 (1985) 253 G. ' t Hooft, Phys. Scrlpta T1S (1987) 143; fbid. T36 (1991) 247; Nucl. Phys. B33S (1990) 138; G. 't Hooft, "Black Hole Quantization and a Connection to String Theory" 1989 Lectures, Banff NATO A S I , Part 1, "Physics, Geometry and Topology, Series B: Physlcs Vol. 238. Ed. H.C. Lee, Plenum Press, New York (1990) 105-128; C. ' t Hooft, "Quantum gravity and black holes", in: Proceedings of a NATO Advanced Study Insti tute on Nonperturbative Quantum Field Theory, CargBse, July 1987, Eds. G. ' t Hooft e t a l , Plenum Press, New York. 201-226 L. Kraus and F. Wilczek, Phys. Rev. Lett. 62 (1989) 1221; J. Preskill, L.M. Krauss, Nucl. Phys. B341 (1990) 50; L.M. Krauss. Gen. Rel . Grav. 22 (1990); S. Coleman, J.Preskll1 and F. Wllczek, preprint IASSNS-91/17 CALT-68-1717/ HUTP-91-AO16 C. 't Hooft, J. Ceom. and Phys. 1 (1984) 45 W. Rindler, Am.J. Phys. 34 (1966) 1174 T. Dray and G. 't Hooft. Nucl Phys. 8253 (1985) 173 W.B. Bonner, Commun. Math. Phys. 13 (1969) 163; P.C. Alchelburg and R.U. Sexl, Gen.Re1. and Gravitation 2 (19711 303 C . Lousto. private cornmunlcation 2. Koba and H.B. Nielsen, Nucl. Phys. B10 (1969) 633 , Ibid. B12 (1969) 517; B17 (1970) 206; 2. Phys. 229 (1969) 243 J . D . Bekenstein, Phys. Rev. D7 (1973) 2333; R.M. Wald, Phys. Rev. D20 (1979) 1271; G. ' t Hooft, Nucl. Phys. B256 (1985) 727; V.F. Mukhanov. "The Entropy of Black Holes", In "Complexity, Entropy and the Physics of Information, SFI Studies in the Sciences of Complexity, vol IX, Ed. W. Zurek, Addlson-Wesley, 1990. See also M. Schiffer, "Black Hole Spectroscopy", Sao Paolo preprlnt IFT/P - 38/89 (1989) A. Ashtekar, Phys. Rev. D36 (1987) 1587; A. Ashtekar et al , Class. Quantum Grav. 8 (1989) L185; C. Rovelli. C las s . Quant. Grav. 8 (1991) 297, ibid. 8 (1991) 317

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CHAPTER 9

EPILOGUE

[9.1] “Can the ultimate laws of nature be found?”, Celsius-Linnd Lecture, Feb. 1992, Uppala University, Sweden, pp. 1-12 . . .. . .. , ... . 666

665

CHAPTER 9.1

CAN THE ULTIMATE LAWS OF NATURE BE FOUND ?

Celsius/LinnB Lecture, Uppsala University, Sweden

by

Gerard ' t Hooft

Institute for Theoretical Physics Princetonplein 5, P.O. Box 80.006

3508 TA UTRECHT, The Netherlands

Summary: Physicists a r e probing ever smaller structures In the world of fundamental particles. Each time we dig deeper, we f ind different kinds of particles ruled by different types of forces. Will there be an end to this , can there be an "ultimate law?" One thing we know f o r sure: at the distance scale of c m (the "Planck length") particles w i l l start feeling each others gravitational force, and this force is st ranger and probably more fundamental than any other. Tiny "black holes" will make it impossible to consider distances smaller than the Planck length. If there is an ultimate l a w of physics, it should probably be formulated a t this distance scale.

This lecture is commemorated to two great scientists from Uppsala. Both of them have lef t their marks in history. The f i r s t time I encountered the name of C a r l von LinnB w a s when I lived with my parents in the Hague, close to the sea shore. I used to s t ro l l along the beach and collect shel ls . Back home I tried to identify my finds, and found that many of them had beautiful Latin names with a mysterious "L" behind them. The books did not even bother to explain what the "L" meant. I t was supposed to be obvious. More than half of all shell species ever found on our beach were named and classified by the Linnaeus school and still c a r r y the names given by him.

Anders Celsius not only rationalized the temperature scale, but is also known f o r quite a few important researches in astronomy. He managed t o check by observations Newton's prediction that the ear th is flattened, the length of one a rc unit of the meridian being larger near the pole than near the equator. Like Linnaeus, he went to Lapland to do his research.

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Before talking about "The Ultimate Laws of Physics" I have to explain what possibly can be meant with such a phrase. I t is probably not only in my country that physicists who talk about such things a re accused of being arrogant, if not a bit crazy, by colleagues in other f ie lds . Those who introduced the pretentious words "Theory of Everything" probably deserve such criticism. We a re now giving an impression of searching for a Stone of Wisdom, and we seem to pretend that Physics might provide fo r ultimate answers to all questions.

But of course this Is not at all what we a re trying to do. What we are after is most easily explained by giving a simple example of the kind of theories one might stumble upon when doing particle physics. In the ear ly seventies it became fashionable among some physicists and mathematicians to play simple games on computers. Shortly a f t e r that the desk top computers made these games accessible to anyone. One such game w a s "Conway's Came of Life"'. I t went as follows (see Fig. 1):

0 = o = I

Fig. 1. Evolution of a particular pattern in Conway's Game of Life, at three consecutive times. I t w i l l propagate diagonally over the lattice.

On a rectangular infinite lattice we have "cells", each of which carr ies one b i t of information: the cell is said to be "alive" if t h i s bit of information is a one; i t is "dead" if it is a zero. Then there is a clock. A t every tick of the clock the contents of each cell is being updated. For each cell the new status , at time t + l , depends on the contents of i tself and its nearest eight neighbors a t time t (Fig. 2).

Fig. 2. The cell in the center is updated depending on what w a s there before, and on what was in the eight surrounding cel ls drawn here

The rule is as follows:

- I f exactly 2 neighbors are alive, the cell in the center will s tay

- I f exactly 3 neighbors are alive, the cell in the center will l i ve . - In all other cases the cell in the center will d ie .

a s i t was.

In Fig. 1, I show a pattern that, a f t e r a while, re turns into i tself , but at a different place: it moves! One can start with an ini t ia l configuration where several of these moving patterns are sent towards each other so that they w i l l collide, and then study what happens. The

667

r e s u l t s may become ra ther complex, and without a computer it is hopeless to calculate.

The point I want to make is tha t t h i s system is a "model universe". Imagine t h a t , given a large enough latt ice and a patient enough computer, one can have "intell igent" creatures bui l t out of these building blocks. They w i l l investigate the world they a r e in , and perhaps ultimately discover the three fundamental "laws of physics" on which t h e i r universe is based.

The question I wish to address now is: could it possibly be tha t the universe we a re in ourselves has such a simple s t ruc tu re , based on such a simple universal Law, a Law tha t is invariable and absolute? I s it conceivable that we w i l l ever be able to discover tha t Law if it exists?

Theoretical physicists a r e studying the universal l a w s t ha t govern our world, but it seems tha t we a r e still very f a r away from the "smallest possible s t ruc tu res" , i.e. anything like the uni t ce l l s in Conway's model. The l aws of physics tha t we did uncover could possibly be considered as "effective l a w s " , l a w s tha t describe regular i t ies , cor re la t ion phenomena, of pa t te rns at very much la rger distance scales and time scales than the smallest possible. I t should not come as a su rp r i se t h a t these l a w s seem to be much more complicated than the Conway l a w s . If there is a smallest distance and time sca le , what could the laws of physics there be like? One might suspect t ha t these w i l l be something much more clever and beautiful than Conway's game. What can be s a i d about them?

In a certain way physicists have already come close to finding universal l a w s that govern the t in ies t known $articles of matter. These laws a r e known as "The Standard Model" . The model is f a i r l y complicated, and it cannot predict the reaction of par t ic les and f ie lds under all conceivable circumstances, but it seems to be a very good description of what is happening nearly everywhere in our universe. I will now give a rough description of t h i s model and its laws.

The basic en t i t i es a r e "Dirac fermions" , elementary par t ic les tha t show a cer ta in amount of spinning motion. This amount of spinning motion is measured in multiples of a fundamental constant of nature, Planck's constant divided by 2n : These particles a re sa id to have "spin 4". We start with particles tha t can only move with the speed of l ight. For these par t ic les the axis of rotation can be seen to be always parallel t o t he i r velocity vector, and one can derive tha t t h i s statement is unique and independent of the velocity of the observer. This is why one can distinguish unambiguously particles t ha t spin "to the l e f t " from par t ic les t ha t spin "to the r igh t " , with respect to t h i s axis.

The forces these fermions exer t onto each other a r e now described by introducing another se t of particles, the gauge bosons. These a re par t ic les with spin one. they can e i ther be considered as the "energy quanta" of various kinds of e lec t r ic and magnetic f ie lds , or one can view these particles themselves as the transmitters of these forces. When t h a t picture is used one describes the force as being the consequence of an exchange of a gauge boson between two fermions: one fermion emits a boson and the other fermion absorbs it. If the mass of the boson is negligible then the efficiency of t h i s exchange process is inversely proportional to the square of the distance: the Coulomb force law. If t he boson has a certain amount of m a s s when at r e s t , then the force it transmits will range only up to a distance inversely proportional to tha t m a s s , and decrease very rapidly beyond tha t

668

distance. An important feature of the gauge boson force between two fermions

is t ha t before and a f t e r the emission of a gauge boson the fermion keeps the same helicity. This means that a lef t rotating particle remains left rotating, and a right rotating particle remains right rotating. This is of special importance for the neutrinos: neutrinos only exist as left rotating objects. Right handed neutrinos have never been observed (at least not for sure) . In contrast , anti-neutrinos only come rotating towards the right, not left . By emitting a gauge boson a neutrino may tu rn into an electron, but then the electron must rotate towards the l e f t .

I g e n e r a t i o n I 1 generation I I [ g e n e r a t i o n 1 1 1

W e Fig. 3. quarks. rotating

THE STANDARD MODEL Fig. 3.

based on Su(2),,,k x u ( l ) e m X SU(3)stiong Right handed neutrinos, which m y exfst, are indicated in dotted boxes.

can now make a l ist ing of all fermions and gauge particles. See Here we see that the fermions are divided into leptons and

The left rotating objects a r e indicated by an “L“ and the right ones by an “ R ” . For the anti-leptons and anti-quarks, which are

not shown, the L and R a re interchanged. Of all fermions only the quarks are sensit ive to the forces of the eight gauge bosons in the box called “ S U ( 3 ) “ . This force is very strong and so all objects containing quarks w i l l be strongly interacting with other objects.

A l l left rotating fermions are sensitive to the forces from the S U ( 2 ) gauge bosons. Finally, all left rotating, and all electrically charged r ight rotating fermions feel the U(1) force fields.

Now the standard model a lso contains a spin 0 particle, the “Higgs particle”. It also transmits a force, but an important difference with the gauge boson forces is that if a fermion exchanges a Higgs boson it has to f l i p from lef t rotating to right rotating or vice versa. Another difference is that the spin 0 particle can disappear s t ra ight into the vacuum. This implies that some fermions can make spontaneous transit ions from left to right and vice versa. Technically, t h i s is the way one may

669

introduce mass fo r these fermions. A particle with mass moves slower than the speed of light and it turns out that fo r such particles the rotation axis cannot be kept parallel to the velocity vector. So this mass can only be there if the particle has the abil i ty to change i t s spin direction relative to its velocity vector. One f inds that m a s s of a fermion is proportional to the coupling strength of that fermion to the field of the-Higgs particle.

-\ a

"

i

C

. . 3

t . . I 6 I-, ' . - -- '. I a... .

.7\. - Fig. 4. "Neutral current event":

e- U,, * e- ZO U,, + e- up In a beam of n e u t r i n o s moving upwards, an e l e c t r o n ( a r r o w ) is thrown i n an upwards d i r e c t i o n , leav ing a t r a c e made v i s i b l e i n a bubble chamber. ( p h o t o from H.Faissner , Aachen)

The gravitational force could be added to our description of the Standard Model by postulating a spin 2 particle, the graviton. The rules for this force a re very s t r i c t ly prescribed by Einstein's Theory of General Relativity, but only in a s f a r a s it ac t s collectively on many particles. The details of multiple graviton exchange between individual p a r t i c l e s a re not completely understood, mainly because any effects due to such exchanges would be so tremendously weak that no experimental verification will be possible in any foreseeable future .

The above is a qualitative description of the rules according to which the particles in the Standard Model move. To formulate these rules in more precise mathematical terms, a s we were able to do fo r Conway's Game of Life, would lead us way beyond the scope of this lecture. In some sense the rules are nearly as precise.

Much of t h i s information became known in the early 70's. A dramatic new prediction of this scheme at that time were the effects due to exchange of the newly predicted Zo component of the gauge fields. One of the f i r s t pictures made of an electron h i t by a muon-neutrino is shown in Fig. 4.

I would like to point out the similarities between the Standard model and an ideal "Theory of Everything": a l l material objects in the universe have to move according to its rules. But of course the model is f a r from perfect. One reason is that we expect the model not to provide the equations of motion for matter under a l l conceivable circumstances. One can imagine energies, temperatures and matter densities that are s o large that the si tuation cannot be mimicked in any accelerator. Although our model could in principle be used to calculate what will happen,

670

there a re several indications that one should not take such predictions seriously. A second reason is that the mathematics of the model is less than perfect. Even i f we had infinitely powerful computers to our disposal, some phenomena cannot be computed with any reasonable precision. One has to rely on certain perturbative approximation schemes that sometimes f a i l catastrophically.

Finally, in contrast with models such a s Conway’s, our interactions depend on a number of “constants of Nature” that are to be given as real numbers. The precision with which these real numbers are. known w i l l always be limited. There a re essentially 20 independent numbers:

3 gauge coupling constants, corresponding to the strength with which the various kinds of gauge bosons couple to the fermions. The S U ( 3 ) coupling constant is much larger than the one for S U ( 2 ) and U(1).

Then there are several interaction terms between the Higgs field and the fermionic f i e lds (Yukawa terms). Many of them correspond to the masses of the various fermions: 3 lepton masses: me , mu and m, . 6 quark masses: m,, , m,, , m, , m, , m, and mb . 4 quark mixing angles, determining fu r the r details of the decay of

exotic particles. 1 topological angle 0, , a peculiarity relevant only for the strong

interactions; as far as is known it is very close to zero. 2 self-interaction parameters for the Higgs field. One of these

determines the Higgs m a s s MH , the other determines the Higgs-to-vacuum transit ions. In combinations with the other constants it produces the gauge boson masses.

This adds up to 19 “constants of Nature“, which are incalculable; they have to be determined by experiment. We could then add Newton’s gravitational constant G, , but this could be used to f ix the as yet arbi t rary scale f o r m a s s , length and t i m e . Str ic t ly speaking there is also the so-called cosmological coupling constant which is also incalculable, but it may perhaps be set to be identically zero: it would be the 20th parameter.

In principle, the Standard Model can be applied at any length and time scale. Indeed, we have nearly scale-invariance. This means that when we study particles and their scattering properties under a magnification glass, we see very nearly the same as without the magnification glass. Only the masses of particles seem to become much smaller if we magnify them3. but also the (non-gravitational) coupling strengths w i l l not be exactly the same. This is a subtle effect . When we consider a certain exchange process, fo r example the emission of a photon by a fermlon, one might discover in the magnification glass that this process actually took place in several steps: there could have been a very short-lived particle created temporarily, to be reabsorbed immediately after. This implies that the effective coupling strength one experiences without the magnification glass, is actually the resul t of a more complicated effect of various couplings at the microscopic level. One f inds that any coupling constant g is changed by a s m a l l amount, typically of order g3 , when viewed at different scales.

And so it may happen that a coupling constant changes gradually when we go to widely different time- and length scales. Calculations show that the U ( 1 ) force becomes gradually stronger at very small distance scales, the SUE!) force becomes somewhat weaker and the S U ( 3 )

67 1

force goes down more rapidly. Since these changes a re f a i r l y slow, one has to go to tremendously s m a l l length scales before all three couplings become about equal in strength. In the simplest version of the Standard Model t h i s happens at about cm, way beyond anything one can see with whatever microscope or accelerator.

A consequence of the variations of the coupling strengths at very s m a l l length scales is that Coulomb's Law of the electric attraction between opposite charges is no longer exactly valid. In stead of

F = C Q l Q z / r 2 ,

one gets a very t iny modification of the power 2. For the gravitational coupling constant the story is very

different. Gravity acts upon the m a s s of a particle, not its charge. If we go to shorter distance scales, one would have to locate the position of a particle with increasing precision. Due to the quantum mechanical uncertainty relation

Ax Ap h (= Planck's constant/2n) ,

such particles w i l l tend to move around with very large momentum p = mv and since the velocity v is limited to that of l ight, the masses m of our particles w i l l increase. Therefore, the gravitational force w i l l no longer behave as a l/r' force but ra ther become a l/r4 phenomenon.

There will be a scale where the gravitational force between individual particles will become stronger than any of the other forces. A t t h i s scale the gravitational force w i l l be the dominant one. I t is easy to calculate where this happens. Using the three fundamental constants of nature, namely

- the speed of l ight, c = 2 997 924 km sec-', - -

we find a fundamental unit of length,

Planck's constant, Newton's constant,

h = h/2n = 1.054 588 x lo-'' erg sec, G = 6.672 x lo-* cm3 g-' sec-',

1% = 1.6 x cm ;

one uni t fo r t ime ,

' = 5.4 x 10-44 sec ; J 7 - and a uni t fo r m a s s :

= 22 pg = 1.2 x loz2 M e V = 1.2 x 10" TeV . J G

These uni ts are called the Planck length, Planck time and Planck m a s s or energy.

I t is f o r s t ructures at these distance, time and m a s s scales t ha t

672

our Standard Model needs to be thoroughly revised. Fluctuations of the gravitational force here become uncontrollable. In any perturbative approach the higher order corrections become larger than the lower order ones. Technically what this implies is that any conventional candidate theory of "Quantum Gravity" becomes unrenormalizable: ill-defined effects from the higher order corrections cannot be absorbed into redefinit ions of the interaction terms. This unrenormalizability problem is something one can t ry to cure and attempts of a l l so r t s have been made. But the real problem of quantum gravity is not the unrenormalizability; it is the f ac t that the perturbative expansion has a dimensionful expansion parameter (Newton's constant) and that henceforth perturbative corrections a t length scales much smaller than the Planck length w i l l be divergent. The only cure to that would have to be a theory in which no time and length scales exist smaller than the Planck uni ts . Certainly the space-time continuum w i l l have to be either drast ical ly different from what we a re used to, or be abandoned altogether.

Although I do not believe that curing the unrenormalizability problem w i l l help us much in understanding quantum gravity, let us look where some of those attempts led us anyway.

The f i r s t observation was that theories with higher symmetries a re usually better convergent than others. A symmetry that was considered extremely promising was called "super symmetry4" : a relationship between fermions (particles with spin integer + 4) on the one hand and bosons (integer spin) on the other. If one t r i e s to construct an interaction with t h i s symmetry a s a basis one finds as a force carrying particle the g r a v i t h o , a particle with spin 9 . I t s supersymmetric partner must be the graviton. and so one obtains a theory in which the gravitational force has a natural place. Super symmetry greatly reduced the renormalization ambiguities, but did not eliminate the problems completely.

The most curious idea is called "String Theory'". According to th i s theory all point like particles must be replaced by s t r ing like objects, e i ther open s t r ings having two end points each, or l i t t l e closed loops. Such objects can interact for instance by joining end points together followed by a rupture somewhere else, or by exchanging loops of s t r ing. The big advantage of the scheme as it w a s much advertised w a s that in a quantum theory of s t r ings one did not have to specify where and when exactly the joining and rupture take place. The transit ions are gradual, and the result ing expressions fo r the amplitudes, in particular the higher order corrections, seemed to become much less singular and divergent than corresponding expressions in conventional particle theory. The mathematics of these exchanges is very s t r i c t , and it turned out t h a t the exchange of the lightest closed s t r ing loops reproduces all by itself the gravitational force. I t w a s a discovery by Michael Green in England and John Schwarz in Caltech that calculations in s t r ing theory could be made consistent to a high degree provided that at those very s m a l l distance scales space-time exhibits 26 dimensions,

To describe also particles with half-integer spin such as the electron and the quarks it w a s necessary to introduce, again, super symmetry, but only on the s t r ing i tself . A very ingenuous idea came from the Princeton school, namely that all fermionic excitations (all those that cause half-odd-integer contributions to the spin) run along the s t r ing only in one preferred direction. In that direction space-time must be 10-dimensional; in the other directions excitations can take

673

place in 26 dimensions. This s t r ing is called the "heterotic s t r ing" and the nice thing about i t is that it has a cork-screw nature, making a distinction between left and right, just like in the real world.

To explain that the real world only has 4 "visible" dimensions one would have to assume that the 22 (or 6) residual dimensions are "compactified". This means that in those directions space is rolled up very t ightly like a tube. That there could be such compactified dimensions had already be proposed by Theodore Kaluza and Oscar Klein in the ear ly 20's and is hence called the "Kaluza-Klein Theory". In this theory the momentum of a particle in one of the compactified directions corresponds to electric charge, and gravitational f ields in space-time in those directions are to be interpreted a s the electro-magnetic and other similar fields.

The most intriguing aspect of this theory w a s that it seems to admit no free parameters as constants of nature a t a l l . This means that everything should be calculable and it w a s suggested that its laws a re as precisely defined a s in Conway's model. But t h i s was not quite true. The theory, a s i t w a s formulated, requires the application of perturbative expansions ju s t l ike the older particle field theories, and here, like always, one has the problem that the calculations will not converge ultimately. The real problem, as I see it, w a s that the theory is formulated in terms of continuous st r ing coordinates as fundamental variables. They form a highly infinite manifold, and a re vastly different from the nice, discrete zeros and ones in Conw-ay's cellular automaton. Maybe one should be able to calculate everything w i t h s t r ing theory, but presently we can' t .

I t is conceivable that s t r ing theory can be saved. Some suspect t ha t it is a topological theory, which means that not the details of the continuum a re relevant, but only the ways boundaries a re connected together. Maybe so , but we do not understand how to turn such ideas into practical prescriptions. A t the very best the theory is incomplete. Important pages of the manual are missing.

Rather than trying to postulate and guess what our world should be like at the Planck length, one could t r y to deduce this from more rigorous observations and arguments. One such observation is the following: One cannot measure any distance with more precision than the Planck length, 1.6 x cm. Suppose namely that one t r i e s to detect s t ruc tu res of such a s m a l l size. Now in particle theory we have the well-known "uncertainty principle":

Ax-Ap 2 $ h ,

where Ax is the uncertainty in position, A p the uncertainty in the momentum mv of a particle, and h Planck's constant divided by 2n . If Ax must be a s small as the Planck length, then Ap must be a s big as the Planck mass times the velocity of light c . Both the measuring device and the thing measured must possess s t ructures as s m a l l as Ax and therefore contain momenta as large a s Ap . Now if two objects with th i s much momentum collide against each other they ca r ry a total amount of energy which in the center of mass frame exceeds the Planck energy, loi6 TeV. If they approach each other at a distance smaller than the Planck length the mutual gravitational interaction will become so strong that a tiny black hole will be formed. The s ize of a black hole is proportional to its m a s s , and this one willstherefore have to be again larger than the Planck length. A black hole can be viewed as being a

674

conglomeration of pure gravitational energy and its attraction to matter is so s t rong that whatever object enters its immediate neighborhood w i l l be swallowed without leaving a trace. In particular, the "thing we wanted t o measure" w i l l disappear into the hole.

Thus, whenever we wish to describe potential theories concerning s t ruc tu res with sizes smaller than the Planck length we run into the fundamental problem of black hole formation. This problem is a very special and unique one in theoretical physics, and therefore I have advocated fo r some time now that we should make theoretical studies of t iny black holes as best as we can . By doing Gedankenexperimenten with these black holes we should be able to find out at least how to formulate purely logical restrictions of self consistency of whatever theory one might imagine to apply a t the Planck scale. Some beautiful r e su l t s have already been obtained in doing this.

In the by now conventional theory of gravity (Einstein's theory of general relativity) any conglomeration of a certain minimal amount of matter within a certain s m a l l enough volume contracts due to its own gravitational f ield, and this contraction must lead t o an implosion that cannot be averted. Whatever kind of information there w a s inside the imploding cloud of matter, the residual black hole w i l l c a r ry exactly three numbers that enl is t all properties that can ever be measured or transmitted: its total mass (or energy), its total electric charge (positive or negative), and its total angular momentum. Nothing can ever come out of the black hole, not even light, which is why it is black .

But that w a s the conventional or "classical" theory. I t w a s Stephen Hawking's great discovery early 1975 t h a t one can say more than tha t i f one applies what is known about quantum theory. He found that black holes do indeed e m i t matter, and one can compute precisely the probability f o r the black hole to emit j u s t anything. The emission goes exactly as if the surface of the black hole is hot: it is the radiation emitted by any object with a certain definite temperature. Hawking computed the temperature and found it to be inversely proportional to the black hole mass. The t inier the black hole, the l ighter it is and therefore the hotter it is.

Hawking's discovery seems to te l l us that black holes a re much more like ordinary physical obJects than the ideal black holes as descrlbed in Einstein's general relativity. They emit radiation and therefore they decay, Just like any heavy radio-active nucleus. This gave us the idea that the t iniest black holes may perhaps become indistinguishable from the heaviest "elementary particles". After a l l , particles carry gravitational f lelds j u s t like black holes do, they decay radio actively j u s t like black holes do, and If one tries to describe the gravitational f feld surrounding an elementary particle one finds the same expression as for a black hole.

The idea that particles and black holes may ultimately be the same things s e e m s to be so natural that the consequences of such a n important assumption w i l l come as a surprise. The assumption namely does not fit with the technical details of Hawking's calculation. He found a very important difference between black holes and ordinary quantum mechanical obJects. The laws of quantum mechanics and particle theory can be applied perfectly to any small region in the neighborhood of a black hole, but not at all to the black hole itself! We have become used to the f a c t t ha t there a re uncertainty relations in quantum mechanics that render it impossible f o r a physicist to predict exactly how a particle w i l l behave under given circumstances. in stead, the particle physicist

7

8

9

675

can give very precise s a t i s t i c a l predictions: given a large number of particles and an experiment t ha t is repeated many times, he can predict with ever increasing precision what the distribution of the outcome will be. For this he uses parameters called “constants of nature“. The worrisome thing with black holes is that these parameters themselves w i l l be shrouded by uncertainty relations. Even our predictions of the s t a t i s t i c s w i l l be fuzzy.

To me it is clear that this is a consequence of our understanding of the physics of black holes being incomplete. Not only incomplete but also wrong! The prediction of an experiment repeated many times should be sharp! Now here we have a perfect example of a paradox in our understanding of physical law, a paradox of the kind that in the past led to the discovery of fundamental new theories such as quantum mechanics and relativity.

Not every physicist agrees with this view. Hawking, like several others, c l a i m that the fuzziness comes about because there is an inf ini te c lass of “universes“ and we simply do not know which universe we are in . Each of these universes has sl ightly different constants of nature.

But t h i s is an agnostic argument I f ind diff icul t t o accept. In t rying to devise an “improved” theory I hi t upon very strange theories indeed. Requiring the black hole to behave a s a whole in accordance with the laws of quantum mechanics I found s t r iking similari t ies between black holes and s t r ings. Their difference in appearance is merely superficial . Mathematically these objects a r e s tar t l ingly similar.

Conceptually both theoretical constructs a r e prohibitively diff icul t . I t seems to be absolutely necessary tha t the theory to be constructed should be of a “topological” kind, as explained ear l ier . One can argue that the Hawking black hole can only support a limited amount of information. I t s internal structure must, somehow, be discrete. As if it contained an a r r ay of zeros and ones, exactly like Conway’s “game of l i f e “ . But then we must realize that if you look at a black hole from close you just see empty space. Therefore one might suspect that indeed space and t i m e themselves can carry only limited amounts of information, and as such may show much more similarity with Conway’s game of l ife than ever expected.

Unfortunately it is impossible - and it w i l l continue to be impossible fo r any foreseeable amount of time - to perform experiments that directly reveal the s t ructure of space-time at the Planck length. We a re condemned to do only thought experiments and a re asked to sor t out all possible theories by logical reasoning alone. This is a long and painful process without any guarantee of success. A poem by Chr. Morgenstern, a s noted e a r l i e r by M. Veltman, should serve a s a warning here (with English t r a n s l a t i o n ) :

Es gibt viele Theorien Die s i ch jedem Check entziehen, Dfese aber kann mann checken: Elend wird s i e dann verrecken.

Many theories are presented For which no t e s t can be invented. But one t e s t you will appreciate: In boredom they’ 11 disintegrate.

676

References

' J . H . Conway, 1970, unpublished; M. Cardner, Sci. Amer . 224 (19711, Feb, 112; March, 106; April, 114; ibid. 226 (19721, Jan . , 104; S . Wolfram, Rev. Mod. Physics 55 (1983) 601.

There a r e many tex t books on the Standard Model. See f o r instance: I .J .R. Aitchison and A . J . G . Hey, "Gauge Theories in Particle Physics", Adam Hilger, 1989. More pedestrian: N. Calder, "The Key to the Universe", BBC, London 1977.

So t h i s is opposite to what one should expect from dai ly l i f e experiences: the m a s s of a sand grain will seem to be much la rger when inspected through a magnification glass.

4See for instance P.C. West, Introduction to Supersymmetry and Supergravity, World Scientific 1986.

5See for instance M . B . Green, J . H . Schwarz and E. "Superstring Theory", Cambridge University Press 1987.

'See f o r instance "Black Holes: the Membrane Paradigm, ed. by K.S. Thorne, R . H . Price and d.A. Macdonald, Yale University Press, 1986.

J

Witten,

.'G. ' t Hooft, Nucl. Phys. B335 (1990) 138 G. ' t Hooft, Physica Scripta T36 (1991) 247 G. ' t Hooft, in "EPS-8, Trends in Physics 1991, Proceedings of the 8th General Conference of the European Phys. Soc., Sept. 4-6, 1990, par t I , p. 187.

S.W. Hawking and R. Penrose, Proc. Roy. Soc. London A 314 (1970) 529 R. Penrose, Phys. Rev. Lett 14 (1965) 57

Hawking, Phys.Rev. D13 (1976) 2188 W.G. Unruh, Phys. Rev. D14 (1976) 870 S.W. Hawking, Phys. Rev. D14 (1976) 2460 S . W . Hawking and G. Gibbons, Phys. Rev. D1S (1977) 2738 S.W. Hawking, Commun. Math. Phys. 87 (1982) 395

8S.W. Hawking, Proc. Roy. Soc. London A 300 (19671 187

'S.W. Hawking, Commun. Math. Phys. 43 (1975) 199J.B. Hartle and S . W .

677

INDEX

Abbot, J. 618 Abers, E.S. 198, 512 Achiman, Y. 186,200 Adler-Bell-Jsckiw anomaly 142, 176, 192,

302, 324 Adler, Stephen L. Aichelburg, P.C. 629, 644 Aitchiin, I.J.R. 677 Albright, C.H. 184, 185, 200 Alpert, M. 618 Altarelli, Guido 200 anomalous dimension 224 anomaly cancellation condition 366 anti-instantons 332 Appelquiet-Carrazone decoupling 367 Appelquist, Thomas 182, 200, 374 Arnowitt, R.L. 103 Ashmore, J.F. 249, 574 Ashtekar, Abhay 664 asymptotic freedom 175, 227, 231,

142, 287, 351

242-246, 376, 435-438, 548

Bach, R. 605 background field method

592-604 background gauge 194,596-599 Bais, F. Alexander 544, 546 Balachandran, A.P. 202 Balian, R. 199 Barber, M.N. 249 Barbieri, R. 154 Bardeen, William 202, 351 Barnett, R.M. 199 Bars, Itahak 202 Becchi, C. B6g, M.A. 198,200 Bekenstein, J.D. 621, 627, 664

194, 206, 585,

15, 194, 202, 351

Belavin, Alexander A.

Bell, John S. 27, 142, 192, 202, 287, 320,

Bell-Tkeiman transformation 15, 95-100,

Bender, I. 199 Bender, C.M. 575 Berends, h i t s A. 190,201 Bernard, C.W. 203 Big Crunch 614, 615 BigBang 615 Bjorken, James D. black hole

decay 650 entropy 581, 623, 626 quantum 9, 580,583,619-664 spectrum 622

black hole Hilbert space 658 Bloch walls 285, 488 Bludman, Sidney A. 198 Bogoliubov, N.N. Bollini, C.G. 249, 574 Bonner, W.B. 664 Borel, &mile 7 Borel procedure, second 571-573 Borel resummation 378, 428-435,

Borel singularities 562-571 Bose condensation 8, 516, 540 boundary conditions in a box bounds on planar diagrams Brbzin, Edouard 574 brick wall model 621, 624-627 Brinkman, H.W. 644 Brodsky, Stanley J. 7 Brout, Robert 17, 26, 296

5, 203, 286, 320, 351, 575

351

103, 109, 115, 325

185, 200, 442

30, 62, 142, 249

55S562

493495 379, 390-393

678

Cabibbo, Nicola 200 Cahill, Kevin 202 Calder, N. 677 Callan, Jr., Curtis G.

canonical transformations 83-94 Carazzone, J. 374 Carlip, Steve 618 Carroll, Sean M. 609, 618 Carter, B. 663 Casher, A. 374, 452 Caswell, W.E. 203, 575 Cauchy surface 609-614 carnality 26, 66-72 Ceva, H. 513 Chandraeekhar, S . 663 Chang, Ngee Pong 201 charmonlum 17!4-182 Cheng, T.P. 185, 200 chiral charge Q6 335-351 Chisholm rule 144 Chodos, A. 199 Christ, Norman 197 Christodoulou, D. 189, 201, 605 Christoffel symbol 587 Chriatos, G.A. 300, 351 Clark, T. 202 closed timelike curve 608 Coleman, Sidney

286, 319, 351, 374, 476, 512, 574, 575,663

4, 197, 202, 249, 286, 351,453,513, 574

190, 197, 199, 201, 202,

collective coordinates 312 Collins, J.C. 203 combinatorial factors 160-162 combinatorics 120, 121, 151, 160-162 complex coupling constant 556-558 compoeite propagators 393 confinement (of quarks)

456-576 oblique 5404543

constants of nature 671 Conway, J.H. 677 con way'^ Game of Life 667 Cmte, N. 453 Cornwall, J.M. 199 Coulomb phase 510 counterterms 26, 122, 145-154, 218, 225,

Creutz, M. 441, 513 Crewther, Rodney 300, 320, 351 crunch and bang theorem 614 Curtis, G.E. 635, 644 Cutkosky’s rule 68 Cutler, C. 618 cutting equations 61-73, 128

5, 191, 281-285,

235-241

Dashen, Robert F. 574

D’Eath, Peter D. decalan, C. 441 de Groot, Henri 452 A I = 1/2 rule 182, 183 De Rtijula, Alvaro 182, 189, 200 Deser, Stanley viii, 201, 605, 617, 618 de Wit, Bernard 202 DeWitt, Bryce 13, 198, 202, 206, 605 difference equations for diagrams 410-413 Dimopoulos, Saves 358,374 Dirac condition (for electric and magnetic

Dirac spinor algebra 156-159 Dirac, Paul A.M.

disorder operator (or parameter)

disorder loop operator 495499 dispersion relations 26, 72-73 divergences 143

overlapping 146-148 Dolan, Louise 203 Dolgor, A.D. 627 Dray, Tevian viii, 582, 629-644 Dreitlein, J. 201 Drell, Sidney D. 442, 574 drsesed propagators 79-82, 166-173 Drouffe, Jean-Michel 199 dual transformation 285, 504-508

201, 286, 296,351, 513,

635, 636, 644

charges) 5, 293, 295, 537-540

5, 295,353, 374, 512, 546

491 458, 464,

earth-moon-symmetry 188 Ehlera, Jurgen 644 Eichten, E. 200 Ellis, John 200 Ellis, G.F.R. 663 Englert, Francois Epstein, Henri 142 9apu 319 eta problems (see &o U( 1) problem)

Euler’s theorem 385 exceptional momenta 413-418 Eyink, G.L. 249

17, 27, 200, 287, 296

191-194, 324, 326

Faddeev, Ludwig D.

Faissner, H. 670 Farhi, Edward 618 Fayet, Pierre 189, 201 Feinberg, F.L. 200 Ferrara, Sergio 203 Feynman diagrams 14 Feynman, Richard P. 13, 604

13, 198, 202, 604 Fhddeev-Popov ghost 24, 102-108

679

Feynman rulea

F i c k h , S.I. 103 fictitious symmetry 339 floating coupling constant 408 flux, electric 178, 496, 498-502

magnetic 496, 498 Radkin, Efim S. 198, 604 Fkampton, Paul H. 199 F'rbre, J.M. 200 Ftitzsch, Harald Fkohlich, Jorg 201 Fkonedal, C. 197 Fubini, Sergio 575 Fqjikawa, K. 351 functional (or path) integrda 52-55, 83

Gaillard, Mary K. 200 Gardner, M. 677 Gastmenrr, h y m o n d 190, 201 gauge condition 23, 103 gauge invariant source insertions 592 gauge transformation law 18, 22 Gell-Mann, Murray 3, 192, 197-199, 202,

249, 287, 351, 374 Gell-Mann-Uvy sigma model 358, 362 Georgi-Gleshow model

290-294,465, 508 Georgi, Howard

286,512 Genraie, Jean-Loup 295,320 ghosts 22, 24, 93, 116

Giambiagi, J.J. 249, 574 Gibbons, Gary 617, 677 Giddings, S. 618, 663 Glaser,V. 142 Glaahow, Sheldon L.

global symmetries 19, 21 Goddard, P. 663 Goldhaber, A.F. 542, 546 Goldman, T. 200 Goldetone, JeErey 521,546, 663 Goldstone particles 324, 519 Gott, J. Richard I11 579, 618 gravitational shock wave 62- Great Model Rush 175 Green, Michael B. Green function 42

basic 408, 409 Gribov, V. 7 Griseru, Marc T. 201-203 Gross, David d.

Gupta, V. 185, 200

32-40, 105, 106, 155-158, 380-383,392, 444,445

192, 199, 202, 287, 351

177, 182, 185, 269,

177, 182, 19S201, 203,

Fadd~v-Popov 24, 102-108

177, 182, 199,200, 286, 351, 512

663, 673, 677

199, 201, 202, 286, 295, 351, 441, 453, 513, 574,605, 663

Guralnik, G.S. 27, 296 Guth, Alan 617,618

Hagedorn, R. 196, 203 Hagen, C.R. 27, 296 Halpern, M.B. 202, 203 Hen, M.Y. 452 Harari, H. 199 Harrington, Barry J. 200, 203 Hartle, Jim B. 627,663, 677 Hasenfrats, Anna 299 Hasenfratz, Peter 299, 546 Haaslacher, B. 201 Hawking radiation 620, 660 Hawking, Stephen

677 Hepp, Khus 142 Hey, Antony J.G. 677 Higgs mechanism

Higgs, Peter W. Hilbert space

9, 620, 627, 647, 663,

2, 8, 176, 177, 460462, 521

3, 12, 26, 176, 296, 512

huge 341 large 342, 496 physical 31, 343, 496

homotopy classes 264-268, 271, 272, 281,

Honerkamp, J. horizon shift 630,633, 634, 652 horror 281,283 Hudnall, W. 202

284,495 202, 203, 206, 605

Iliopoulos, John impulsive wavw 629 indefinite metric 78 instantons 6, 270, 321 Isler, Karl viii Itzykson, Claude 199, 574

198, 199, 201, 512

J/$J particle 15, 17S182 Jackiw, Roman viii, 10, 27, 142, 192, 202,

203, 286, 287, 319, 320, 351, 374, 513, 546, 574,617, 618

Jacobian 13, 85 Jacobs, L. 441, 513 JafTe, R.L. 199 Jarlskog, Cecilia 184, 185, 200 Jevicki, A. 202 Johnson, K. 199 Jones, D.R.T. 96, 203, 575 Julia, Bernard 199, 286

Kadanoff, L.P. 513 Kalitsin, Stilyan viii KiillBn-Lehmann representation 33,5&58,

81

Kaluea-Klein theory 658, 674 Kibble, Th. W.B. 13, 27, 296, 512 Kluberg-Stern, H. 203 Koba, Z. 664 Kogut, John 199, 201, 202,286,462, 674 Komen, Gerbrand 452 Koplik, J. 442 Komer, J. 199 Krueloel-Smkeres coordmatw 634 Kuchar,K. 618 Kummer, W. 197 Kundt, W. 644

Laflemms, R. 663

large&-time equation 63 Lautrup, Benny 165 Le Guillou, J.C. 674 Lee, Benjamin W. Lee, T.D. Lee, C. 351 Lehmsnn,H. 612 Lepege,G.P. 7 Leutwyler, Harry 192, 202,287 Levin, D.N. 199 LBvy,Maurice 374 Li, L.F. 197, 201 Linde, Audrei 187, 201, 203 Lipatov, L.N. 574 LleweUyn Smith, Chrii H. 185, 198-200 LoUet0,Carlce 664 Low, Francis 3, 197, 249 Lowenstein, J.H. 202

Lanc!4oS,C. 605

26, 198, 200, 374, 612 189, 196,201,203, 255, 286

Ma, E. 201 Ma, S.K. 201 MacCaUum, M.A.H. 638,844 Macdodd, D.A. 677 macromxpic variables 514,536 magnetic mnopolee 5,269,270,283,

Maiaui, Lucian0 199,200,512 Mandebtam, Stanley 198,199, 202, 287,

Mani, H.S. 186, 200 M a r g u h , M. lM, 203,286 Maxwell equStiOM 12, 95 Meieenercdht 523 metonspectrum 460-463 meta-alor (M tdmialor) 368 meta-quarka (or quinks) 358 metric termor g,,,, 579, 587-589 microampic variables 514,530 Migdal, A.A. 203, 575 M ~ , Robert M M , P e t e r 199 Miener, CluuIea W. 663

288496,526

604

2, 12, 26, 198

Mohapatra, Rabindra N. 189, 201 monopoles (magnetic) 5, 269, 270, 283,

288-296, 625 Mukhanov,V.F. 664 multiloop diagrems 1.34-140

n-dimensional integrale 140-142, 163-167,

Nambu, Yoichiro 199,288,296,452 naturalnew 352-374 naturalnew breakdown maw scale (NBMS)

Nauenberg, Michael 452 Neveu, Andre 201,296,442 Newman, E.T. 663 Nicoletopodos, P. 200 Nieleen, Holger B. 178, 199, 286, 295, 493,

non-local tramformatione W 9 4 Nuseinov,S. 442

oblique confinement 540-543 Okun,Lev 262 Oleaen, P o d 178, 199, 286, 295,452,493,

onsloop W t i w 127,236-241, 699804 Ore, A. 200 Ore, F.R. 299, 351, 576 overlapping divergences 146-148

Page, DonN. 663 Paia, Abraham 185, 200, 201 Parmiuk, 0.S. 142 Parhi, Giorgio 202, 296,442,612, 674,

575 Park,S.Y. 200 Pati, Jogwh C. 184,200,201,296 Patraecioiu, A. 195, 203 Pauli, Wolfgang 30,212,249 Penrose, Floger 629,630,633,636,638,

644 Perl, MartinL. 200 Peny,M.J. 203 Petennan, An& 3, 142,208,249 phantomsolitone 528 phyeicalparticles 22 planar field theory 247 P h c k d e 8, 584, 619, 652 Plancklength 10,666 Pohlmeyer, K. 612 Politmr, H. David 182,200,441, 574,606 Polyakov, Alexander M. 5, 252, 351, 513,

674,5’76 Popov, Vidor N. 13, 198,604 Powell, J.L. 200 Prasad, M.K. 253 Prentki, Jacques 186,200

216

354

613,617, 664

513

681

Preskill, J. 649, 664 Price, R.H. 677 Primack, Joel 201 propagator 34, 37, 38

bare 79 composite 393 dressed 79-82, 168-173

pseudo-gauge transformation 497

quantum chromodynamics (QCD, color

quantum hair 648, 649 quark confinement 5, 191, 281-285,

456-576 oblique 54Ck543

Quinn, Helen R. 201, 203

rainbow digrams 377 Rajaraman, R. 201 Rapidis, P. 200 Rebbi, C. 286,319,351,374,441,513,546,

574,663 regularization 25, 122-142

regulators, Pauli-Villars 30, 231

theory) 4, 176-179, 231, 281

dimensional 122-154, 212, 213

space-time dependent 307, 308 unitary 59

21C215 renormalhation 25, 122, 142-154,

dimensional 194, 220-228,315, 316

equation 230, 41-28, 553-556 order by order 150, 151

renormalom, infrared 570 ultraviolet 380, 569, 570

Riemann curvature 588, 606 Rindler, W. 664 Rivasseau, V. 441 Root, R.G. 201 h n e r , J. 200 RQSS, Douglas A. 187,197,201 Fbuet, A. Rovelli, Carlo 664 Rupertsberger, H. 202

renormalization group 4, 205-249

15, 194, 202, 351

Sekita, B. 295, 320 Salam, Abdua 184,186,198,200,201,203,

296, 513 Sarkar, Sarben 202, 203 scattering matrix 25, 43-51, 100, 594

for black holes 645-664 Ansatz 651-653

Schechter, J. 185, 200, 202 Scherk, Joel 190, 201 Schroer, Bert 197, 201 Schwartz, Albert S . Schwarz, John H. 190,201, 663, 673, 677

5, 350, 351

Schwinger, Julian Schwinger’s model 179, 191 Senjanovic, P. 202 Segrb, G. 197 Sexl, R.U. Shaw, R. 198 Shei, S A . 351 Shellard, Paul 636 Shirkov, D.V. 249 Siegel, W. 203 Sinclair, D.K. 201 sineGordon model 256 Singer, M. 200 Sirlin, A. 198 skeleton expansion 387-389 Skyrme, T.H.R. 351 Slavnov-Taylor identities 108, 115, 118,

SlaVnOV, Andrei 27, 351, 442 solitons 191, 260-264,476, 477 Sommerfield, Charles 253 Soni, A. 200 sources 35, 37, 44, 161

physical 47, 592 sphalerons 299 spherical model 377, 439-441

Standard Model 16, 647, 669 Staruszkiewicz, A. 618 Stora, Raymond 15, 202, 351 Strathdee, J. 203, 513 Streater, R.F. 512 string theory 648 strings 26Ck264, 269 Strominger, A. 663 Stueckelberg, E.C.G. supersymmetry 8

relaxed 189 Susskind, Leonard 199, 202, 286,295, 358,

374, 452, 574 Sutherland, D. 202 Suzuki, M. 201 Swieca, J.A. 203 Symanzik, Kurt

W a s h i , Y. 351, 442 n u b , A.H. 644 Taylor, John C. 351, 442 technicolor 358 theory of everything 670 Thorn, C.B. 199, 663 Thorne, Kip S. 663, 677 Tiktopoulos, G. 199 Tjia, M. 0. 184, 185, 200 topological operators 475 topological stability 255, 258 topological singularities 534

198, 199, 295, 452

629, 633, 636, 641, 644

151, 152, 194, 325

stability, against SC8hg 259

3, 142, 208, 249

4, 195, 197, 230, 249, 442

682

Tsao, Hung-Sheng 201, 605 tunneling 275-277 Tuttle, W.T. 442 Tyupkin, Yuri S. Tyutin, I.V. 198, 604

5, 15, 351

U(l) dilemma 321, 337-339 U(l) problem 300, 323-375 Ueda, Y. 200 unitarity 25, 74-77, 100 unitary regulators 59 unitary gauge 526-528, 532 Unruh, W.G. 627, 628, 677

vanDam,Henk 13 van Damme, Ruud 249,374,442 van Kampen, Nicolaaa G. van Nieuwenh-n, Peter 201-203, 605 Veltman, Martinus J.G. 2,13,28-173,187,

197, 198, 201, 249, 351, 574 vertex 37 vertices 34 Villars, F. 30, 212, 249 Vinciarelli, P. 200, 201 vortex 5, 178, 261, 264, 282, 283, 523 vortices, 261

vii, 249, 452

Wdd, Robert M. 628, 664 Ward, J.C. 198, 351, 442 Ward identitiea

wave function renormalization 117-1 19 Weinberg, Steven 27, 197-203, 286, 295,

95-100, 108, 142, 324 anomalous 345, 346

512, 628

Weinstein, M. 574 Weisskopf, Victor F. 199 West, P.C. 677 Wheeler, John A. 663 Wick, G.C. 196, 203, 286 Wightman, Arthur S. 512 Wilczek, Frank

664 Willemsen, J.F. 201 Wilson, Kenneth G.

441, 574 Wilson loop operator 488 Witten, Edward Wolfenstein, Lincoln 184, 200 Wolfram, Steve 677

Wu, Tai Tsun 575

199, 441, 574, 605, 649,

7, 197, 199, 200, 286,

351, 618, 663, 677

wu, C.C. 201

Yang-Mills theory 2, 12 Yang, Chen Ning 2, 12,26, 198, 296 Yankielowicz, S . 574 Yildis, Asim 200, 203

Z ( N ) symmetry 486-492 Zee, Antony Zeldovich, Ya.B. 621, 627 zero eigenstates 311 Zimmermann, W. 202 Zinn Justin, Jean 26, 198, 574 Zuber, Jean-Bernard 203, 574 Zumino, Bruno

185, 199, 200, 286

185, 187, 200, 201, 203, 295, 296, 493, 513

683

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Under the Spell of the Gauge Principle (Advanced Series in Mathematical Physics, Vol 19)

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