GAUGE UNIFICATION OF BASIC FORCES, PARTICULARLY OF GRAVITATION WITH STRONG INTERACTIONS

GAUGE UNIFICATION OF BASIC FORCES, PARTICULARLY OF GRAVITATION

WITH STRONG INTERACTIONS

Abdus Salam

International Centre for Theoretical Physics Trieste, Italy

and Imperial College London, England

INTRODUCTION

The presently accessible range of physical phenomena appears to be governed by the four familiar types of basic forces, mediated either by spin-one or spin-two quanta (TABLE 1).

The spins of the mediating quanta, spin-one for weak, electromagnetic (EM), and strong forces, and spin-two for strong and gravitational forces, appear to correspond to two of the deepest and most elegant theoretical structures based on the gauge principle that we know of. These structures are shown in TABLE^.

Whereas the theme of a Yang-Mills unification of weak, EM, and strong forces (motivated by the shared characteristic of all of these forces being mediated by spin- one gauge particles) has been fairly well emphasized, comparatively less attention has been paid to the spin-two characteristic of the strong force, its resemblances to gravity, and the possible unification of these two forces by use of Einstein-Weyl gauge ideas. It is my principal purpose to motivate such a unification, though in the first part of the paper I shall also briefly review the spin-one unification aspects of weak, EM, and strong interactions. Tentatively, then, I shall be proposing a “Puppi” type of tetrahedral interrelation of fundamental forces, with the strong force playing a pivotal role due to its mediation through both spin-one and spin-two quanta (FIGURE 1).

I will first give a summary of the important points concerning the linkages represented by this tetrahedron.

TABLE 1

Spin of

Quanta Force Mediating

Effective Associated Coupling Characteristic Strength Mass (GeV)

EM 1 - a= 10-2 Weak I - , I + GF = 10-5GeV-Z 102 Strong. 1 - (gluons),2+ G~ = i ~ e v - ~ 1 Gravitational 2 + GN = 10-37GeV-2 1019

Tensor dominance, ’ with its relationship to Pomeron physics, is a signal of the role of spin-two mediating particles in strong interactions. The dual-model theories of strong in- lcractioii\ (opparcntly) need both open-string (zero-slope limit * Yang-Mills spin-one theory) and closed-string (zero-slope limit 4 Einstein spin-two theory) sectors to ensure a consistent, unitary, and renormalizable formulation.

12

Salam: Gauge Unification of Basic Forces 13

TABLE 2

Spin of

Quanta Gauge Theory’ Mediating Gauge Theory Generalization

Maxwell 1 Internal symmetry group U(1) corresponds to electric charge conservation

Einstein 2 group of general coordinate transformations; links up with the notion of space-time curvature

Weyl 2 a rederivation and generalization of Einstein’s theory, gauging the relativistic spin group SL(2.C); links up with space-time torsion of Cartan

(by Yang-Mills-Shaw) to any internal symmetry group, for example, SU(n) or chiral SULW x SUR(n)

(by Isham, Salam, and Strath- dee3) to any (internal symmetry-containing) generalization of SL(2,C), for example, SL(6,C), that emphasizes relativistic marriage of spin- group [SL(2,C)] and internal symmetry SU(3)

Each one of the above gauge theories can be extended by “grading” the appropriate Lie algebra, that is, by adding on anticommuting charges. For example, Maxwell’s spin-one gauge boson may be augmented with a spin-1/2 gauge fermion; or Einstein’s spin-two gauge boson may be augmented with a spin-312 gauge fermion (supergravity theory). In this manner, a gauging of graded Lie structures removes the final distinction between “matter” (conventionally fermions) and (mediating) quanta (conventionally bosons). All fundamental fields in this view are gauge fields. I will later refer to supergravity when I discuss possible renormalizability of gravity theory.

FIGURE 1.

/ \ / \

/ \ / spin -two \

I \

Gauge Unification of Weak and Electromagnetic Forces

Prediction and verification of the existence of neutral currents imply that such a (gauge) unification is likely with the minimal gauge group SU(2) x U(1) or SU,(2)

The characteristic mass (energy) beyond which the distinction between these two forces may be expected to disappear lies beyond 102 GeV.

x SU,(2) x U(1).

14 Annals New York Academy of Sciences

The gauge unification, together with the comparative rarity of AS=l weak transitions, makes the existence of charm almost compulsive.

The most direct test of gauge ideas will, of course, be the production of W* particles and one or more weak partners to the photon (P), hopefully in the 1980s.

Semidirect tests of the linkage between weak and electromagnetic interactions are the symmetry restoration effects in weak interactions, which could be produced by use of strong external electric and magnetic fields.6 For example, it has been suggested that the Cabibbo angle may be expected to be switched off in reactions like

K- +95Mo-93NbA+n

(so that the A-hyperon lifetime is very considerably enhanced) in the nuclear environment provided by 93Nb and 35Ar, assuming that the internal electromagnetic fields inside these nuclei are stronger than the critical transition fields.

Gauge Unifcation of Strong with Weak and Electromagnetic Forces

The next hypothesized linkage is the spin-one mediated gauge unification of strong forces with the weak and the EM. Clearly, the most important signal of such a unification will be the disappearance of the distinction between leptons and quarks. This phenomenon must occur; the question is: Beyond what characteristic energy?

Pati and this author’ have suggested a theory of quark-lepton unification based on the idea that the 12 quarks (which carry four flavors and three colors) combine with the four known leptons in a multiplet of an SU(4) x SU(4) color internal symmetry group. (The fourth color is the lepton color, lilac.) The quarks, ultimately indistinguishable from leptons, must in this model carry integer charges. A spin-one gauging of this theory (flavor and color charges) yields an estimate of the characteristic energy at which the quark-lepton unification should begin to become directly manifest. We estimate this energy to be L lo5 GeV (other gauge theorists, working with fractionally charged quarks, which are permanently confined, estimate unification energies to be much higher, > los GeV). Each one of our quarks can decay into a lepton (plus pions or kaons or a lepton-antilepton pair) with a lifetime of about sec for quarks of mass J 4 GeV. Likewise, the proton, the three-quark composite, must decay into three leptons (plus pions) with a lifetime of the order of 1029-1030 years. (All of these lifetime estimates are correlated with the estimate of the characteristic energy. If the characteristic energy is higher than los GeV, the proton will live longer.)

In addition to the possibility of proton decay as a signal of quark-lepton unification, there are other indirect signals in the model. These signals relate to uL/uT in eN, pN, dileptonic events in vN, and asymmetric production of leptons versus antileptons in NN collisions.

Clues on Unification of Gravity and Strong Interactions

The S-matrix physicists, who have postulated tensor dominance in strong interactions and hypothesized that the Pomeron lies on a spin-two trajectory, have always believed in the important role of spin-two mesons in strong interaction physics. The dual-model physicists have, likewise, discovered that they must utilize

Salam: Gauge Unif icat ion of Basic Forces 15

both the open-string (zero-slope limit = Yang-Mills spin-one gauge theory) and the closed-string (zero-slope limit = Einstein theory) sectors in their search for a consistent theory of strong interactions. (Previously, the higher dimensions needed for dual models and the Symmetries that arise from them were identified with flavor quantum numbers; recently, there has been some shift toward associating these symmetries with color.)

From a gauge theory viewpoint, one can go further. Let us assume that strong interactions are mediated by a strongly interacting spin-two object (generically called f meson: not t o be confused with the spin-two particle a t 1290 MeV) that obeys a n Einstein equation with the Newtonian constant GN= G e V 2 replaced by the strong constant Gs = 1 B e W 2 . We further assume that quarks interact with the f mesons; their normal gravitational interactions are mediated by a (generally covariant) f-g mixing term [the field gr” ( x ) describing normal weak gravity]. This mixing term alsogives mass to the f meson.

This simple version of a two-tensor f-g theory was formulated by Isham er (I/. and, independently, by Wess and Zumino.* In this early.formulation, f quanta were assumed to interact directly with hadrons, and g quanta were postulated to interact directly with leptons. Clearly, with the huark-lepton unification ideas expressed above, this simple version of the theory with f and g tensors so sharply distinguished will need revising. This revision can be accomplished - but I will not discuss this aspect of the theory in this paper or the very difficult problem of reconciling within one structure magnitudes as diverse as GN and G s . Rather, my major and humbler concern is to show how the postulate of a n Einstein equation for the strong gravity field f, with all of the connotations of space-time curvature and torsion being important in strong interactions, manifests itself in physical phenomena, particularly in the limit that the f-g mixing term is neglected.

The claim is that there are two immediate manifestations of this Einstein gauge formulation of strong gravity:

Weak gravity possesses classic solitonic solutions of the Schwarzschild and deSitter type that trap and confine particles. Likewise, strong gravity possesses solitonic solutions (which represent hadrons) that confine (quarks) a t least on the classic level.

Quantum mechanically, Hawking’ has recently shown that the solitonic solutions of (weak) gravity are not black holes from which nothing can escape. He shows that (some of) these solitonic solutions represent black bodies, which radiate all species of particles with a thermal spectrum. The exciting aspect of Hawking’s work is that the temperature is defined in terms of the parameters of the Einstein field equations and their solitonic solutions. Specifically, temperature is proportional to the inverse of (4r times) the Schwarzschild radius.

In strong gravity, for hadrons, we shall see that the strong Schwarzschild radii are of the same order of magnitude as are the Compton radii of hadrons. Ex- tending Hawking’s ideas, one can define a temperature14 in hadronic physics in terms of radii of appropriate hadronic solitons, which controls the thermal emission of particles in (for example) d V o r NNcollisions.

The text that follows will be divided into two parts. The first part (pp. 16-21) is concerned with the Yang-Mills unification of strong, weak, and EM interactions. I will describe the model of Pati and myself and discuss its predictions with respect to proton and quark decays and to manifestations of spin-one strongly interacting color gluons. The second part (pp. 21-36) will describe the use of the spin-two Einstein-Weyl equation for strong gravity. I will search for clues to a partial confinement of quarks in the context of the solitonic solutions of the strong-gravity


equation and also use Hawking’s ideas to give a precise meaning to the concept of hadronic temperature. The first part is a summary of work reported elsewhere ’* lo; the second part describes some new work, particularly related to the possibility of confinement of quarks by use of strong-gravity ideas.

YANG-MILLS GAUGE UNIFICATION OF STRONG, WEAK, AND ELECTROMAGNETIC INTERACTIONS

All of the material in this part has been described in detail in the AachenConfer- ence lecturelo by Pati and this author. I will give a brief summary, emphasizing the gauge unification aspects.

The Scheme and the Fermion Number

We work conservatively with 12 quarks and four leptons. [If further quark flavors and further leptons (colors) are discovered, our fundamental internal symmetry group and the corresponding representations or their number will grow, but nothing basic changes.]

The quark-lepton unification hypothesis is implemented by postulating that all matter belongs to the following fundamental fermionic multiplet, which consists of the 4 x 4 representation of the basic group SU(4) x SU(4) color :

F=

P P P ye

n n n e-

A x x CI-

C C C UP

1 1 1 1

red yellow blue lilac

- colors -

- UP

- down

- strange

- charm

flavors.

We define (an unconventional) baryonic number for quarks (B= 1) and a leptonic number L =Le + L’ = 1 for leptons. The fermion number F for all 16 particles equals F= 1 = B+ Lc + L’. Note that only the total fermion number F has any absolute significance: none of the individual numbers, B, Le, or L’ , is significant in terms of conservation for the whole multiplet.

The electric charge operator is a sum of SU(4) I x SU(4) 1 color generators. We make a choice that assigns the following charges to quarks and leptons (though one can also construct a theory with Q = Qnavor alone):

I 0 1 1 0

I 0 1 1 0


2/3 213 213 0

-1 /3 -113 -113 - 1

-1 /3 -113 -113 - 1

2/3 2/3 213 0

Q = Qnavor + Qoolor = I -213 1/3 113 0

-2/3 1/3 1/3 0

-2 /3 1/3 1/3 0

-2 /3 1/3 1/3 0

+

Note that with this assignment, leptons with fermion number F = 1 (the same as quarks) are absolutely defined as objects that carry zero (we, w,,) and negative charges ( e - , p - ) .

Spin-One Yang-Mills Gauging of SU(4) Il,,guor x SU( 4 ) I l c o l o r ; Basic Model

We gauge flavor for weak and EM interactions and SU(4) lcolor for strong and EM forces. The important point is that the photon has partners in both the flavor and the color sectors thet correspond to the split of charge Q into Qflavor + Qco,?r.

The gauging scheme may thus be represented by the form shown in FIGURE 2, with EM occupying the pivotal position. In detail, the flavor gauges give W:,,,Zo and the flavor piece of the phot_on. The color gauges give an octet of strong color vector gluons ( V & , V&, GQ, U, V")that couple quarks with quarks, a triplet (plus an antitriplet) of exotics pRl,XqLr X g that couple quarks with leptons, and a singlet 9' that couples with the current ( E - 3 L ) . Among the eight color gluons is the rather special object 0, the color partner of the photon.

We give masses to all of these gauge particles (except the photon) through the standard Higgs-Kibble spontaneous symmetry-breaking mechanism. This mass- giving mechanism also mixes the weak W* s with the octet Vs and the triplet Xs, so that the final unification scheme looks like the schematic picture shown in FIGURE 3. To link up with the concept of characteristic energy beyond which the distinction between quarks and leptons should disappear, it is the masses of the exotics that determine this characteristic energy.

To summarize, the Higgs-Kibble mechanism indicates that the photon is composed of flavor and color pieces, mixes V* with W* - , leading to decays of the octet of strong gluons V * , V", . . . , and mixes the exotics X* with W* - , leading to well-defined quark and, in turn, proton decays.

Mass Scales

There is a natural mass scale for masses of the exotics. It is provided by the rate of the decay K-e+ F . From the present rarity of this mode, we infer that mx S lo5 GeV. This relation, in turn, determines (within the model) the lifetime of a quark

EM (elactromagnatiam)

FIGURE 2.

For example, heavy leptons + b quarks, if substantiated, may require SU(5) lfiavor x Su(5) (color.


flavour gauges colour gauges

I octet of vs , wr9zo

\ triplet of x s ,' \ singlet SO ,

\ / /

\ \

/ /

0 . . \ \

\ 7 - -_ - - /

FIGURE 3. Higgs-Kibble particles mixing W* s with Vs, Xs, and So and leading to weak decays of quarks and gluons.

for decays into a lepton plus pions (or a lepton-antilepton pair and ofprotondecay). Alternatively, we could have fixed rnx through any one of the three inputs K-e+ p, quark - lepton transition rate, and proton - three lepton transition rate; the other two processes would provide a test for the assumptions on which the model is based.

Though we have this natural mass scale for the exotic masses, regretfully there is no natural mass scale for the masses of the quarks or the strong (octet of) gluons. These masses could lie in any one of the three ranges: light < 2 GeV (the charmed quark is presumably 1.5 GeV heavier); medium, between 2 and 7 GeV (SLAC range); and heavy, PEP-Petra range of energies.

It is important to stress that these masses are of quarks and gluons outside the nucleonic environment. Inside such an environment, with its high hadronic matter density and hadrostatic pressure, the expectation value of the appropriate Higgs fields can have made a transition to zero. Thus, quarks and gluons could be very light ( 5 300 MeV) inside the nucleonic environment, as the parton model appears to suggest, though they are heavy outside. This difference of effective masses inside and outside, first discovered by Archimedes in the context of hydrostatic pressure, would cause a partial confinement of quarks and gluons in the sense that the tun- neling probability of their crossing through the hadronic surface and penetrating the mass barrier is depressed. In all subsequent remarks, I shall accept this partial confinement as a fact of quark dynamics (exact confinement being the limit when the quark-gluon mass outside is infinite).

Production and Decays of Quarks and Proton Decays

Free quarks may be produced in the following reactions, for example:

e+ + e - - q + q

For quark decays, there are important selection rules in the simplest (basic) version of the model, which I summarize.

Assuming fermion number, F, conservation, A F = O (but A B # O , U # O . F=B+L) , a quark can become a lepton but not an antilepton. The quark- antilepton transition requires AF=2. We have assumed, in the simplest version of our model, that this decay mode is suppressed compared to the A F = O decay mode. [If this restriction is relaxed (as would, for example, be the case in a super- symmetric version of the model), q-I and q-; (and even AF=4 transitions qq-qq) may become competitive with q-I (AF=O) . ]

The simplest (basic) model further strongly restricts the types of quark decays


allowed. The yellow and blue quarks are sharply distinguished from the red quarks in that the former (yellow and blue) go predominantly into neutrinos and not charged leptons. Thus,

4y.B - v, + pions, v p + K+ pions

q,-e+pions, p+K+pions

-and also e+ Y + Y, p + v + V.

The lifetimes (which vary as m i 3 ) range between lo- '* and sec (or shorter) for light to medium quark masses.

Because quarks are presumably point particles so far as electromagnetism is concerned, one is tempted to ascribe the Per1 (p.e) events at SLAC to decays of red quarks of mass = 1.95 GeV

e+ +e--qR +&-e+p+neutrinos.

Note that quarks resemble heavy leptons, in that they are not absorbed in ordinary matter; their only distinction from heavy leptons lies in their scattering (nuclear versus pure electromagnetic) characteristics.

Regarding nucleon dissociation in vN and NN collisions, it is important to remark that whereas partial confinement will make dissociation amplitude (tun- neling through the masc barrier) small, the net mass from final quark decays will mainly reside in the neutrinos that yellow and blue quarks decay into. The red quarks in their decays will, however, contribute to dileptons in VN collisions. Finally, in NN collisions, we expect the nucleon dissociation mechanism to give a sizeable antileptonAepton asymmetry beyond the dissociation threshold. This expectation is based on the assumption that A F = O , W Z 2 selection rules hold (or, more accurately, assuming that the two baryon number-violating amplitudes W = 0 and AF=2, which give q-I and q-I transitions, respectively, are not of the same magnitude).

The most characteristic prediction of the model is proton instability, which (with A F = 0) is a triple violation of baryon-lepton number, AB= - AL = 3. It is this high degree of forbiddenness (effective constant G i = where GB is the effective quark-lepton transition constant, =I0 ', computed wi th in the model, assuming that m, = 105GeV) that is responsible for the inordinately long life of the proton. The predominant decay mode is:

proton-3v+ T + , AF=O.

The most recent reported experiment on proton decay is that of Reines el al.." performed in 1967 (and reanalyzed in 1974). in a South African mine 3000 m deep; a signal of five possible events of proton - p + was recorded, setting a lower limit of 1030 years on the proton lifetime. In the basic model (W=O), this particular decay mode (proton - p + +4v) can only proceed with muons predominantly carrying a rather small fraction of proton rest energy (E,<150 MeV).? To study the

t Even though the discussion above has stressed A F = O transitions, it is important to-remark that one can extend the basic model such that AF=2, 4-1, 1-4 and AF=4, 44-44 transitions are also allowed. In such a case, proton - p + + ro (AF=4) would be a possible decay channel, in addition to r + + 3 v ( A F = O ) .


predominant decay 3v+ K + , Zatesepyn has proposed the use of a 100-ton scintillator to detect the following chain from decays of protons in the scintillator itself:

proton- T + - p + - e + . 1 1 1 -

3v v v + v

A geochemical experiment similar to that used for double /3 decay has been suggested by Rosen.12 This experiment involves examining for rare gas isotopes 22Ne, 38Ar, 84*86Kr, and 132Xe occluded in ancient ores. The sensitivity of the experiment” Rosen has suggested has increased recently to proton life estimates as high as years with the discovery of a new dye laser-based technique by Hurst et al. 21 that detects one atom in an environment of 1019.

Gluon Story

There are eight strong colored gluons 1 - responsible for (part of) the strong force in all models of gauge unification of strong, EM, and weak interactions. In the so-called standard quantum color dynamics model with fractionally charged quarks, the SU(3)IC,,,, symmetry is assumed to be an exact symmetry, and all gluons, electrically neutral, are massless. To keep them invisible, the “dogma of complete confinement” has been formulated, which asserts that both gluons and quarks are permanently confined; in fact, all color will forever confine.

In our model, described above, gluons are integer charged and massive, and they must be produced in all types of collisions (though “partial” confinement due to the Archimedes effect and the need to penetrate the surface barrier of hadrons may depress their production cross-section at present energies). In this context, in addition to any dynamic barrier factors, there is also an exact theorem due to Roy and Rajasekaran and Pati and Salam,” which states that in a gauge theory, lepton production experiments (eN, v N ) are ineffective in producing color. More precisely, in all such experiments, the production rate of color/flavor is governed by a kinematic gauge factor

where mu is the mass of the photon’s partner in the gluon octet, and Iq2 I momentum transfer.

is the

The decay modes of the color gluons are characteristic:

Charged members of the octet V&, V& go into

= (v,e) : ( v ,p ) : hadrons

in the ratio l : l : 3 , with lifetimes of about sec, providing another source of dileptons in v + N - p + +X, in addition to charm. Also, in semileptonic decays

of V’ , one does not expect single production of Ks (V* - Kev, K p v but - KKeu, KKjw, and so on), because the gluons are SU(3) lnavor singlets. This would distinguish such decays from charm decays.

Finally, the neutral gluon (partner to the photon) 0, expected to be produced in e+ e- collisions, would exhibit the following characteristic decays:

L , e )


mu = 1-2 GeV 0-e+ +e- 2-5 keV 2-5 keV

- p + + p - 2-5 keV 2-5 keV ‘**YI 4*Y, q!?l 1-3 MeV 0.1-1 MeV -3m, 5 ~ , p r , K K 0.2-5 MeV 0.1-1 MeV - 3x, 4 ~ , 6~ 0.1-0.5 MeV 0.1-0.5 MeV

mu =4GeV

On the basis of the e+ + e- width of SLAC structures between 4 and 7 GeV (barring the region 3.1-3.2 GeVZ2), it has been suggested that the gluon is either light ( < 2 GeV) or heavy (>7 GeV), so that its mass lies either in the Frascati-Orsay- Novosibirsk or the Pep-Petra region (though this conclusion does not take into account possible mixing of other color states with gluons).

Summary

In summary, the signals for the Yang-Mills gauge unification of strong, weak,

SU(2) x U(l) or left-right symmetric SU,(2) x U,(1) model of weak and EM

Proton decay into three leptons (plus pions). Production and decays of quarks in vN, iN, and NN collisions. In the latter

experiments, we expect the lepton/antilepton ratio to deviate significantly from unity above the nucleon dissqciation threshold, provided that either one of the transitions q - / ( M = O ) or q- / (hF=2) dominates over the other.

In eN and vN experiments, uL /a, # 0 and should scale in x.

SPIN-TWO ASPECTS OF STRONG FORCES, STRONG GRAVITY, AND POSSIBLE

and EM interactions in accordance with our ideas are:

interactions.

ORIGIN OF (PARTIAL) CONFINEMENT AND HADRONIC TEMPERATURE

Introduction

Because I will be discussing (partial) confinement in this part, I will restate two present dilemmas of strong interaction quark physics in this respect.

The parton model gives a picture of essentially free quarks and gluons existing inside hadrons. This (at first surprising) feature of quark dynamics, however, has analogies elsewhere in physics. For example, electrons in metals behave essentially as free particles, notwithstanding the relatively strong electric potentials inside. Likewise, in the theory of nuclear matter, particularly when one attempts to reconcile shell and collective particle pictures of nucleonic interactions, there are dynamic dilemmas of a similar type. In quark dynamics, the “free” behavior of quarks and gluons has been (brilliantly) ascribed to asymptotic freedom of quark- gluon forces, namely, the statement (true of non-Abelian Yang-Mills spin-one theories, and as we shall see, possibly also of strong gravity) that the closer the quarks and gluons come to each other, the weaker the effective strength of the force with which they influence each other. (Parenthetically, it must be remarked that contrary to a general climate of opinion and belief in the subject, the gluon, or Higgs, masses need not affect the issue of asymptotic freedom.)

The second significant fact about quarks and gluons is the Archimedes effect. Quarks and gluons, according to the parton model, are light inside a hadronic environment and heavy outside. There is partial confinement if the mass outside is finite, exact confinement if it is infinite. Because (primeval) fractionally charged quarks appear excluded as physical entities (from experiments with deep seabed


oysters and moondust), such quarks, if they do exist, must be permanently confined. For integer-charge quarks (particularly if they decay fast into leptons), there is no known experimental evidence that would argue for their permanent, as distinguished from their partial, confinement .$

Now, what is the origin of exact confinement, if such indeed is the real physics of the situation? A truly vast amount of intellectual effort has gone into theoretically achieving what I shall call the “Tokamak-like” confinement of color (quarks and gluons) by use of the agency of (non-Abelian) spin-one gluon theories. And one must admit that the basic idea is truly seductive. Assume that the strong color gauge group SU(3) lcolor represents an exact symmetry of nature, so that the color gluons (electrically neutral in a fractionally charged quark theory) are massless, producing long-range forces. Assume that the infrared effects that ac- company such massless gluons are so singular for color-carrying initial or final states that an infinitely rising long-range potential of the type Y = kr or A$ builds up for colored states. In such an event, colored quarks and gluons will be permanently confined inside color singlet hadronic states. Particle physics, on the experimental level, would come to an end, within our generation, for never shall the quark (or the gluon) state be accessible for direct experimentation. In favor of such a rising potential may also be adduced the well-known fact that such potentials would also facilitate, theoretically, the emergence of rising Regge trajectories.

So much for the conjecture. Now, the first bope of carrying this exact confinement conjecture to a proof lay in examining the infrared behavior of non- Abelian Yang-Mills theories in perturbation theory (the infrared slavery hypothesis). Unhappily, it is by now conclusively known that concerning per- t urbation calculations the (infrared) behavior for non-Abelian Yang-Mills quantum color dynamics (QCD) is no more singular than for the familiar Abelian gauge theory of quantum electrodynamics (QED). In any perturbation calculation (or for any summation of perturbation diagrams to a given order), there seems no hope of uncovering infrared slavery or the origin of exact confinement, if any. One could still retain the hope that nonperturbative approaches would succeed where perturbation theory failed in providing an infinitely rising potential of the type kr or M. Numerous attempts have, in fact, been made in this direction but without conspicuous success.

I suggest that rather than look further along the direction of spin-one (Tokamak-like) confinement, one may attempt to exploit the confining properties of an Einstein-like spin-two equation. Classically, Schwarzschild and de Sitter solutions of such equations trap and confine only too well, giving also expressions for the surfaces of confinement in terms of the parameters of the theory. The goal (recently realized by Hawking) is that quantum mechanics may temper this inexorable trapping, this inexorable confinement, to give just the right degree of partial confinement when one works with strong gravity, where the typical (strong) gravitational scale of sizes agrees with hadronic Compton wavelengths and quantum effects are particularly relevant. One will still require the spin-one color aspects of strong interaction physics, but they will be needed more to provide saturation (i.e., why do three, but not two, quarks form a partially confined bound system?), rather than to provide the origin of confinement. It is relevant to remark that there have been great advances made since 1974 in field theory in

If “color” and magnetic monopolarities are.related to each other (as has been surmised), and if magnetic monopoles and the related gluons have masses in the ratio (Y- ’ (’t Hooft’s theorem), quarks (which carry monopolarity) may be heavier than 200 GeV, even for the light gluon case. An awful prospect for experimentation!


curved spaces since Hawking first announced his quantum mechanical results. Some of the techniques developed are extremely powerful, as I shall briefly indicate. 1 feel a personal tinge of regret that few of the advances have come from the community of particle physicists, who have, by and large, unfortunately ignored these ideas (see the review article by Isham15).

f-g Two-Tensor Theory of Strong and Weak Gravitation

To motivate the discussion, consider the simplest version of a unified (gauge) theory of strong and gravitational interactions.* We start with two tensors,Pv ( x ) and g P y ( x ) , and postulate the Lagrangian:

R O and R ( g ) are the Einstein-Lagrangian expressions, Gs - 1 GeV-2, GN - G e V 2 ; C, is a mixing Lagrangian of the form:

m:vv -gpy) Cf"' -gKX ) ( g c p g X y -glAgpv) (2)

and is designed to give a mass (mf) to the strong graviton. Ignoring for the present the subtleties of quark-lepton unification, Smaller gives a quark-f direct interaction of effective strength Gs and a lepton-g direct interaction of strength GN.

g - 1 + mNqjg), that the two fields f and g mix and that the equations of motion describe one massless and one massive spin-two physical quantum associated with each of the two fields g and f. More precisely, the physical fields bear a close resemblance to the photon and its partner, the Zo in the unified EM and weak gauge theory approach. Thus,

Now, one can show, at least in a linear approximation (f- 1 +

1 1 1 1 w3 wo field GN GS e g3 go

The true massless * Gpv* - gpv + - f p U * -A(photon)=s - + -

The true massive =)

field u p , - gpv 1

However, because Gs > > GN, for all purposes, the g field represents the true graviton, and the f field represents the strongly interacting f meson.

Note that the theory as formulated here is fully generally covariant. Regarding the f meson, however, we are interested in the flat space-time limit of the g field (GN = 0) with


(They show that there is no cosmologic counter term, contrary to common belief.) Defining Zo;l~loop = [ 1 - 23/96 GN/s2 L2] , they show that the renormalized Newtonian constant GNR bears to the unrenormalized constant the relation

GN

1 - - 96*

GNR = 23 L2

Here, L is the ultraviolet cutoff. Thus, Zone loop > 1, a statement characteristic of asymptotically free field theories.

This is admittedly just a one-loop argument. One must now derive Callan- Symanzik-like equations (if one can) to show that an appropriate renormalization (Z) can be defined for all orders and that it always exceeds unity. Fradkin and Vilkovisky have claimed to have done this. Irrespective of their detailed con- siderations, however, I believe their result for the following reason.

Consider gravity for what it is, a nonpolynomial Lagrangian theory, and parameterize gr’ in ths nonpolynomial form:

gr’ = (eXpK4)”’ K~ = 8rGn 4 is a 4 x 4 symmetric matrix of 10 fields.

(This parameterization implies that we are not permitting det g to vanish.) Then, the two-point propagator

( gr’gPK ) = [ (expcc4)”” (expcc4)P“ 1

1 A

Now, grr exhibits the invariance K-AK, 4- - 4, or, in terms of the propagator,

1 A2

K-AK, (44)- -(44);thatis,ineffect,$dA2$.

In other words, as A-0 (i.e., as $ -0, or, equivalently, as we approach ultraviolet energies), the effective coupling K~ -0. And this is just the hallmark of asymptotic freedom.

To come back to the issue of high-energy behavior, presumably here we must borrow the techniques of the dual-model physicist, who, with his closed-string sector, incorporates into his formalism essentially what are Reggeized solutions of Einstein’s equation and thereby secures acceptable high-energy behavior for the S-

Salarn: Gauge Unification of Basic Forces 33

matrix elements. [As remarked before, he also needs the open-string (Yang-Mills) sector for this renormalizability to take c l lec t . ] There is also hope, from extended supergravitytt theories, that the mass shell S-matrix elements in these theories may prove renormalizable after all and combine spin-1 and spin-2 structures in a natural manner.

However, even if such a hope fails, I feel (regretfully) that there has not been a proper understanding of the work performed by Isham, Strathdee, and myself regarding the regularizing role of Einstein’s gravity theory. Following Landau, Klein. Pauli, deWitt, Khriplovitch, Deser, and others, we attempted to prove the conjecture made by these authors that gravity realistically regularizes all infinities, including its own. We claim to have demonstrated this conjecture by use of Efimov-Fradkin nonpolynomial techniques. Specifically, we computed the self- mass of an electron in a Dirac-Maxwell-Einstein theory and showed that to the lowest order in a, it equals

The conventional logarithmic infinity of the Dirac-Maxwell theory is recovered if GN is set equal to zero. (Numerically, llog GNmf I = 105, so that 6m,/m, = 1 is approximately equivalent to the relation Q I log GNm,2 I -- I).$$

In general, nonpolynomial field theory techniques are ambiguous, and one must use a principal value prescription in defining certain integrals. This prescription has been the main stumbling block in a general acceptance of nonpolynomial techniques. The paper, at whose neglect 1 do feel sore, is the last paper in our series and is entitled “Is quantum gravity ambiguity-free?”I9 In this paper, we proved what we consider to be a most crucial theorem. By considering the complete expression for the two-point function, we proved that there is one nonpolynomial theory in which the (principal value) ambiguities of other nonpolynomial theories simply do not occur, and this theory is gravity. Gravity escapes this blight because it has the distinction of being a gauge theory. (And, for this “gauge” reason, we also conjectured that though our exact result is for the two-point function, it is likely to hold also for the n-point function.)

At this point I will briefly describe the basis of the proof. Write, as before, g”” = (expK+)”’, gap.= (exp - ~4),,”, where LEinstein has the form *ggg ag ag. It is well known that to define the propagators in the theory. one must add a gauge-fixing term to CEinstein and make computations with CEinstein + Cgauge.fixin We choose a special type of gauge, the conformal gauge, which gives for the free + propagator the expression:

Here, D ( x ) is the free scalar field propagator, and c is the gauge parameter. As in every gauge theory, the final mass shell S-matrix elements are expected to be independent of the gauge parameter (c).

Ashmore and Delbourgo have computed the nonperturbative expression for the two-point function [g‘B(x), gT6(0)] + and given its complete expression; our in- terest lies in its asymptotic behavior, when .$ -0. This looks like the following:

tt In my opinion, if “supergravity” has immediate physical applications, these applications

$$ A similar relation has been derived by Terazawa el dZ4 must relate to strong supergravity.


By use of the Euclidean ansatz, this expression has the form: exp [ ~ ~ ( l - c ) l /RZ] . The origin of the ambiguity that besets nonpolynomial theories in general can now be made manifest. When R2-0 (and if no gauge constant c is present), exp K ~ / R - + OD. To define this propagator, one must go to the complex K~ plane, continue to negative K ~ , that is, consider non-Hermitian Hamiltonians (with K~ <0) (so that exp K ~ / R ~ -0 as r-O), and then continue back to the physical value K~ >O. I t is this continuation in that introduces the principal value ambiguity in expressions like log K* , which occur in the theory.

In gravity theory, this is not the case. Here, the gauge parameter c comes to our rescue. By working with gauges where E > 1, the effective parameter KZff = ~ ~ ( 1 - c) can always be taken to be negative. And because at the end of the calculation, on the mass shell, the theory must be independent of c. this particular choice of c> 1 for calculational purposes is of no consequence. There is never an ambiguity in this theory.

To conclude, we claim that the gauge invariance of gravity theory permits us to use ambiguity-free nonpolynomial techniques and thereby secure a realistic regularization in gravity-modified field theories, 44 with the Newtonian constant GN providing a realistic cutoff. To conclude this defense of the Einstein structure, I believe that there simply has not been enough work performed to explore the deep questions posed by this most elegant of theories. And in this regard, one wishes to understand both the one-tensor g"(x) theory and the two- (or many) tensor theories [which contain gw' ( x ) and p ' ( x ) ] for all of the problems posed in this section. The structure and the invariances of the two-tensor theory are very dif- ferent from the invariance of the one-tensor theory, and we need a deeper understanding of the new problems that arise in this regard.

Summary

I have attempted to make a case for the use of both the Einstein-Weyl spin-two and the Yang-Mills spin-one gauge structures for describing strong interactions. By emphasizing both spin-one and spin-two aspects of this force, I hope we can achieve a unificatien of this force, on the one hand, with gravity theory and, on the other, with EM and weak interactions. The question arises as to whether these two structures (Einstein's and Yang-Mills') themselves be subsumed into one single structure. On the formal level, this may be possible by use of the ideas of extended supergravity theory or, alternatively, by use of a formalism developed by Isham, Strathdee, and the author that works with a gauge theory of groups of the type SL(6,C) or SL(8,c) x SL(8,C), where some of the redundant components of the (16-component) vierbein L, are used to describe spin-one fields in addition to the spin-two fields. In either case (besides the space-time curvature associated with the Einstein structure), it is the concept of space-time torsion, allied with internal symmetries, that appears to play a fundamental role in giving a unified description of physical phenomena.

How much longer can we treat internal symmetries as something decreed from the outside? In my opinion, there is no problem deeper or more urgent of con- sideration than an attempt to comprehend the nature of internal symmetries and their associated charges, the flavors, the colors, and the like, from a deeper fun-

$$ We must still examine whether the mixedf-g theory permits an imposition of two separate conformal gauges of the type we used in the proof above.


damental principle. At the present time, we are treating the flavor (or the color) charges as pre-Copernican epicycles, new ones to be invoked and added on when the old set fails to please and satisfy. We need to know the deeper significance of these charges, just as Einstein understood the deeper significance of the gravitational charge through the concept of space-time curvature.

Because Einstein was the only physicist successful at comprehending the nature of a charge, one’s first thought is to seek the significance of flavors and colors within the ideas of extended curvature, extended torsion, or the topologic concepts associated with space-time and its possible extensions into higher dimensions (both bosonic and fermionic). [The fermionic extension embodied in the notion of superspace probably has the edge so far as extensions of the space-time concept are concerned. As Freund has argued, for fermionic dimensions, one may not have to worry about the problems of physical measurements. Alternatively, one may have to associate a size of the order of Planck length (10-33cm) with these new (bosonic) dimensions, as argued a long time ago by Kaluza and Klein and recently by Scherk, Cremmer, and Sch\\ar/ .]

To go back to Einstein’s comprehension of gravitational charge in terms of space-time curvature, let us recall that Einstein was much impressed by the em- pirically determined equality of gravitational charge with inertial mass. He postulated from this equality the strong equivalence principle, which asserted that all forms of (binding) energy (nuclear, EM, weak, or gravitational) contribute equally to gravitational and to inertial mass. As opposed to this principle, there was advocated, particularly by Brans and Dicke, the so-called weak equivalence principle, which maintained that this equality holds for nuclear, EM, and weak forms of energy but not completely for the gravitational.

It is helpful to remind ourselves of the recent tests undertaken to discriminate between the strong and the weak equivalence principles. The point is that for laboratory-sized objects, the ratio of the gravitational binding energy to the total energy is = l:l@’. Because the best tests of the equivalence principlezs achieve an accuracy no greater than one part in 1OIz, one needed planet-sized objects (e.g., the earth, with its ratio of gravitational binding energy to total energy of 4.6 x 1O-Io) to differentiate between the strong and the weak equivalence principles. The test would consist of measuring departures from Kepler’s Law, of equilibrium distances of the earth and the moon from the sun. As you are aware, the test was conducted recently by two g r o ~ p s . ~ ~ ~ ~ ~ It consisted of echo delays of laser signals sent from the earth and reflected from the moon. The experiment, accurate to lunar-laser ranging measurements of f 30 cm, has unequivocally supported Einstein. The weak equivalence principle appears to be untenable.

1 wish to draw two morals from this. First, a conceptually deeper theory, a theory of more universal applicability, scores even at the quantitative level. Sec- ond, Einstein, in formulating his theory, generalized the single-component field theory of gravity to the theory of a 10-component field g”‘. Instead of a one- component gravitational charge, he (profligately) introduced a 10-component entity (the stress tensor). He was not afraid of inventing myriads of components, myriads of (gravitational) charges, because he knew the deeper principle behind his construct. For me, the moral is clear: Nature is not economical of structures, only of principles of universal applicability. The biologist has long comprehended this moral; we, in physics, must not lose sight of this truth.

[ W I I. M ~ I . I ) IN PROOF: Strathdee and the author have solved exactly the Klein- Gordon equation for a test quark in the deSitter background fieldf,,, , produced by a source quark, given in the text = % (1 +a) - I (1 + P r 2 ) ] Preprint, 1977, International Centre for Theoretical Physics, Trieste. 1977. Phys. Left. To be


published). The exact eigenvalue spectrum consists of a discrete set with no con- tinuum; that is, there is exact confinement. In view o f the deSitter (closed universe) character of the fPy field, this should have been expected: hadrons are closed microuniverses in the strong f-gravity field.]

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GAUGE UNIFICATION OF BASIC FORCES, PARTICULARLY OF GRAVITATION WITH STRONG INTERACTIONS

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