Nonperturbative formation
of fermion Zpoint and Cpoint functions
Filippus Stefanus Roux
Thesis submitted in conformity
with the r.quirements for
the Degree of Doctor of Philosophy,
Graduate Department of Physics in
the University of Toront O-
@Copyright by Filippus Stefanus Roux 2000
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Nonperturbative formation of fermion %point and 4point tiinctions
Filippus Stefanus Roux (Ph-D, 20)
Department of Physics, Vniversita, of Taronto
A better understanding of nonperturbative dynamics may help to clarify nature beyond the ele!ctrOweak
scale. To this end we analyzed the nonperturbative formation of 2-point and 4point functions- We considered
both gauge exchange dynarnics and instanton dynamics for this purpose.
The nonperturbative formation of 2-point functions through gauge exchange dynamics is fairly well
understwd. Our contribution in this instance is to show through a next-tdeading order analysis that the
most attractive Channel hypothesis, which relies on leading order analyses, may be misleading.
After the necessary dective action and gap equations were derived, we have shown that 4point fimctions
cannot be generated nonperturbatively through gauge exchange dynamics in the limit of a large number of
colors. This surprising result foilows from the fact that the diagram with one gauge exchange dressing a
4-point function is not present in the gap equation because the vacuum diagram h m which it wouid be
generated is not 4particle irreducibie.
For the purpose of our intended instanton analyses we deriveci a diagrammatic language for instanton
dynamics. The necessary effective actions for 2-point and 4point functions were derived on the basis of this
d iagrmat ic language- When restricted to the one-instanton amplitudes, the 2-point eftective action could
be shown to lead to the Carlitz and Creamer gap equation, Near a continuous phase transition it turns out
however that it is not the oneinstanton amplitudes, but the instanton-anti-instanton amplitudes with two
n-point functions that give the leading contribution. We used these instanton-anti-instanton amplitudes in
stability analyses to determine the critical couplings for which the vacuum at the origin becomes unstable.
A parameter i s introduced to handle the theoretical uncertainty in the contributions of the fermion loops.
We computed the critical couplings for the nonperturbative formation of 2-point fimctions via instanton
dynamics and found that unless the contributions of fermion loops are fairly large instanton dynamics play
an insignificant role compared to gauge exchange dynamics in the nonperturbative formation of 2-point func-
tions. The critical couplings for the nonperturbative formation of 4point functions via instanton dynamics
were computed and found to be smdler than those for 2-point functions, provideci that the contributions of
the fermion loops are not too small. We found a mechanism that may Iower the criticai couplings for Cpoint
functions with respect to those for 2-point functions in a mode1 for the dynamitai origin of quark and lepton
masses.
Acknowledgments
For ail the help and guidance, advice and support, and sometimes putting up with m y obgtinacy, 1
sincerely thank my supenrisor, Bob Holdomc
Thank you to Tibor Tonna, for all 1 could learn fiom you and for your unixdiable skeptiüsm which
helped to raise the q d t y of my work.
1 also give thauks to M i . Luke, Craig Burrell, Christian Bauer, Hael C o k and Sean Flemming for
their help and contributions to m y knowledge and understanding-
To my parents, brothers and theh fimilies, for ail their love and support, 1 give my dearest thanks.
My friends - the Olivier's in Texas, the co1111~iunity of South Afncam in Ottawa, the ceii group members
here in Toronto, aü those who are stiU in South AiSca and those in other parts of the world - in various
ways you aii helped me to persevere; you shared m y joys during this remarkable perïod in my Me and you
supporteci me during the difiicuit times- 1 thank you with all m y heart.
Finaüy, and moBt jubilantly, to the Provider and Sustainer of my abilities and opportuities .--
Contents
1 The dynamïcal origin of scales 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The scale of the problem 1
. . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Understanding strong dynarnics . .. . . . . .. 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A n o d &fermion order paramder 7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Instanton dynamics 8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Chiral symmetry breaking via instantom 9
. . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Chiral symmetry breaking via 4point tunctions 12
2 Nonpert urb'ative analyses 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 What is dynaraical symrnetry breaking? 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Effective action formalism 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 CJT effective action 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stabiity analysis 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Linearization 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Critical coupling 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The gap equation 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Landaugauge 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The dynamitai mass function 26
3 The MAC hypothesis 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Motivation and history of MAC 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The effective action 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and discussion 35
4 &point functions via gauge exchanges 39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definitions and notation 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Ppoint effective action 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The amputated 4pht functiom 41
4.1.3 Typesoffermionpairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
. . . . . . . . . . . . . . . . . . . . . ........... 4.1.4 Simple4pointfunctions .. 43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Direct derivation of the gap equatiom 45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 TheT-terms 45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.1 The master equations 45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.2 Simple 4-point fun~tioxl~ 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.3 T-term expressions 48
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 4particIe irreducibIe diagrams 49
. . . . . . . . . . . . . . . . . . . . . . ............. 4.2.3 Thegapequatios .. 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Derivation of the &ective action 52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Topological equation 52
4.3.2 Paircycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.1 dfenrtion pairs 55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 ssfermion pairs 56
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.3 &fermion pairs 58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.4 d&fermïon pain 59
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The effective action 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Gap equations via the effective action 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Gap equations in the large Nc limit 63
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and conclusions 66
5 Instantonsi 69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A topological defect 70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 TheBPSTinstanton 72
5.3 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The theta-vacua 75
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Zeromodes 76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The U(l)A anomaly .. 76
6 Instanton effective action 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Notation 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fermion zeromodes 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Gauge fields . . . . . .. 85
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 A diagrammatic language 88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The effective actions 89
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 The 2-point &&dive action 89
. . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 The Carlitz and Creamer gap equation 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 The 4point &ective action 90
..................................... 6.6 S ~ a n d c o n c l u s i o n s 91
7 %point functions via instantons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The 2-point effective action
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The nondynamîcal terms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The dynamïcal term
. . . . . . . . . . . . . . . . . . . . . . 7.3-1 The combinatonal factor and color averaging
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 The mass insertions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The remaining fermion loops
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Results and discussion
7.5 Sirmmary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 4-point tiinctions via instantons 105
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The effective action 105
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The breaking pattern 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Nondynamical terms 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fermion loops 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Solving for Cos 108
. . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Expression for the nondynamical terms 108
8.4 Dynamicalt erm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
. . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 The instanton-anti-instanton amplitude 109
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 The combinatorial factor 110
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 The 4-point function insertion 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Results and discussion 112
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 The4pointfunctioncOefficient 112
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions 114
9 A d y n d c a l mode1 117
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The fermion representations 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 P r o p d sequence of symmetry breakings 119
9.2.1 Highsde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Flavot scale 119
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Electroweak scale 120
vii
10 Conclusions 123
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The achievements 123
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The chailenges 126
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The future 128
A Wick rotation
B Gsoup theory factors 133
C Twdoop beta bction 137
List of Figures
1.1 Leading instanton diagrams: a) one4nstanton. b) instanton-anti-instanton with two mas
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inserti0 ns
2.1 ExampIes of cycles of lines: a) a 4-cycle. b) a trivial Zcycle and c) a trivial 1-cycle- ..... 2.2 Vacuum diagram with one gauge boson exchange and two mass insertions . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 2.3 Fermion seIf-energy diagram with one mass insertion-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The dynamid mass function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Leading order diagram
3.2 Next-tdeading order diagrams that are leadhg order in l /Nc+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Next-to-leading order diagram that is leading order in l / N f
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The leading order fermion self-energy diagram
. . . . . . . . . . . . . . . . . . . . . . . . 4.1 The five connectecl. amputatecl 4-point functions
. . . . . . . . . . . . . . . . . 4.2 Four mes of fermion pairs that can connect 4po i . t functions
4.3 The definition of the simple 4point functiom (U's) in terms of the pairings and chiralities
of the extenial lines . The four groups indicate the four different types of fermion pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Master equation
4.5 A pair cycle with eight fermion pairs - i.e . an 8.cycle . The shaded rectangles denote simple
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . subdiagrams-
4.6 Three vacuum diagrams that are not 4PI . The shaded circles denote 4fermion vertices (J's
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0rC's)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Avacuumdiagramthatisnot QPI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Color structures of Co-
. . . . . . . . . . . . . 4.9 The ody two 4PI vacuum diagrams with one-gauge-boson exchanges
4.10 The 4PI gauge interaction a.) vacuum diagrams and b.) &point diagrams at leadhg order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inl/N,
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Vacuum diagrams with two 4point functions
5.1 The shape of the f-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
How the dBerent vertices are connected by the different propagators .............. 85
The shape of ~ ( a ) for N, = 3 .................................... 87
The instanton-anti-instanton diagram with two mass M o n s . . . . . . . O . . . . . , . . . 95
The criticai couplings for the noaperturbative formation of 2-point functions. fn>m instanton
........... dynamicsforNc=3. w i t h 2 s Nf Nj andwhere A=1.4.231 and2.55 100
The critical couplings for the nonperturbative formation of 2-point bctions. h m instanton
dymmicsfor Nc=4. w i t h 2 5 Nf 5 Nj andwhere A = 142.31and2.55. . . . . . . . . . . 101
An example of a pair cycle . . . . . . . . . . . . . . . . . O . . - . . . . . . . . . . . . . . . . 106
. . . . . . . . . . The instanton-anti-instanton diagram with two 4-point function insertions 110
The critical couplings for the nonperturbative formation of 4-point functiom. for Ne = 3.
. . . . . . . . . . . . w i t h 3 5 Nf 5 N; and where A=l.1.347.1.6and 2.5. . . . . . . . .. 114
The critical couplings for the nonperturbative formation of &point fuactions. for Nc = 4.
with 3 5 f i 5 Nj and where A = 1.1.347.1.6 and 2.5. . . . . . . . . . . . . . . . . . . . . . 115
. . . . . . . . . . . . The contour in the upper right-hand quadrant of the complex po.piane 129
A beta function with an infrared h e d point . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
. . . . . . . . The shape of the running coupling as a fundion of Iogarithmic distance scale. 139
Chapter 1
The dynamical origin of scales
The scale of the problem
Nature's playground is demarcated by various d e s . These d e s determine some of the qualitative and
even quantitative aspects of phenomena in nature. The color of the sky is for example determined by the
sizes of particles in the atmosphere. At the subatomic Ievel, the diversity of the quark masses is refiected
in the observed intricacies of hadronic phenomenology. Unless they are fundamental, these scales must be
determined by mechanisms in nature. It is one of the remaining challenges of elementary particle physics
to explain the origin of the various scale parameters that appear in the standard model.
According to the standard model the quark and lepton masses (with the exception of the neutrino
masses) are all explaineci in terms of just one scale - the electroweak scale. They ali receive their values
from the product of the Higgs vacuum expectation value (VEV), v = 246 GeV, and the Yukawa couphgs,
which give the strengths of the coupling between the Higgs field and the quark or lepton. In this way
the smallness of the electron m a s with respect to the top m a s is refiected in the relative sizes of their
respective Yukawa couplings. The origin of these Yukawa couplings is not explaineci within the standard
model. One may therefore be curious as to why the Yukawa coupling of the top quark is so large (of O(1))
or why that of the electron is so smail.
Another intriguingly small parameter in the standard model is the mass of the Higgs field. This is one
of the parameters that determined the value of the Higgs VER the other one is the coupling coetlicient
of the four scalar vertex. The m a s of a scalar field in a non-supersymmetric theory is not protected by
a symmetry. This means that by setting the scalar rnw in the Lagrangian equal to zero, a nonzero rnass
term can stiU be generated by radiative corrections. The leading correction in this case is quadtaticdy
divergent and needs to be regularized with the aid of a natural uitravidet cutoff - the Planck scale. As
a result the natural value for the Higgs scalar mass is near the Planck scale - 16 orders of magnitude
Iarger than the observed value for the E@p VEV! It requires unnatutal cancellations among the radiative
corrections to give the Higgs scalai. a mass that is in accordance with the obsend VEV. This unnaturd
fine tu- is an embarrassment to theoretid particle physics and is r e f d to as the himmhiy prioblem.
The solution of this hierarchy problem has been high on the agenda of elementary particle physics ever
since the advent of the standard model. Let's consider some of the attempted solutions for this problem.
One solution for the hierarchy problem is to assume that there are no elementary scalar fields. The
electroweak symmetry then has to be broken dynamidy, accompanied by the formation of composite order
parameters. Known examples of such proceses are superconductivity [lj and chird symmetry breaking
[2]. The latter has first been demonstrated through the formation of 2-fermion condensates as a r e d t
of 4-fermion interactions [3]. Later it was shown that chiral symmetry breaking also occurs in QED in
the quenched ladder approximation [4]. In Chapter 2 we show that 2-fermion condensates are generated
dynamically in non-Abelian gauge theories such as QCD when the coupling grows strong enough. The idea
that electroweak symmetry breaking may also happen through the formation of such 2-fermion condensates,
led to the so d e d technicolor theories [5]. In these theories electroweak symmetry breaking is accomplished
through the dynamical generation of fermion masses of order a TeV. The basic idea is that there are new
technifermions that carry electroweak quantum numbers and feel a QCD-like strong force - technicolor.
Just Lke in QCD this force grows strong as the coupling nuis to lower scaies. Eventuaiiy it grows strong
enough to break the chiral symmetry, but this happens at a much higher scale than in QCD. In QCD
the scaie of the pion decay constant is: f, = 93 MeV. The quivalent scale in technicolor, as set by the
technipion decay constant is F, = 246 GeV. The technipions carry electroweak quantum numbers and as
a result they break the electroweak symmetry. The pattern of breaking is exactly as it appears in the
standard modei. Three of the technipions are eaten by the three intermediate gauge bosons whereby they
become massive. Naturalness is retained because the chiral symmetry protects f d o n masses until it is
broken due to strong dynarnics.
But what about the quark and lepton masses? In the standard mode1 they are given by the Yukawa
coupling terms involving the scalar Higgs field- Without scalar fields a new mechanism is n e c m to gen-
mate these fermion masses. The chiral symmetry breaking must somehow be fed down to the lighter quarks
and leptons to produce their mass spectrum. This could perhaps be accomplished via gauge interactions
that are broken at some high scale, d e d the flavor scale. The original idea was to embed technicolor into a
bigger gauge symmetry, eztended technicolor (ETC) [6], that breaks down to technicolor at the flavor d e ,
which therefore came to be known as the ETC scale. Below the ETC scale the heavy ETC bosons give
rise to 4fermion interactions. Among these are some that couple technifermions to quarks and leptons.
Once the technifermions condense, below the TC scale (which is near the electroweak sale), the &fermion
interactions act like mass terms for the quarks and leptons. Their masses are therefore given in terms of
1.1. THE SCALE OF THE PROBmM 3
where ATc and AErC are the TC and ETC d e s , respectively- Because Arc < the fermion masses
would be smaller that ATC-
S i c e the ETC force couples all f&om to each 0th- it also allows flavor changing neutmi c u m b
(FCNC's), for which there are strong experimentd bi t s obtained h m kaon phenomenoIogy [;Il. The limits
on FCNC's require the ETC scde to be 100 to 1000 times Iarger than the dectroweak scale- Although the
resulting ratio of s d e s can reproduce the observed fermion masses of the lighter quarks and leptons, it is
too large to generate the masses of the heavier f d o m such as the top quark. Some mechanism is necessary
to give the diversity in fermion masses. This could appear through the anomalous dimensions of the 4
fermion operators in a theory with a slowly nuining or 'walking' coupling constant [a). The enhancement
that 4fermion operators receive fkom anornaIous dimensions under these circumstances may help to give
larger masses for some of the quarks. However, it seems to be insuffiCient to acmunt for the large mass of
the top quark-
Other experimental constrains are presented in the form of oblique parameters: S and T (and to a lesser
extent U) [9]. The T parameter measures the amount of isospin breaking and the S parameter measures
the nuniber of extra particles- The observational limits on these parameters fiom precision electroweak
measuements [7j are quite severe. However, it is difEcult to estimate the theoretical values of these
parameters, because it requires nonperturbative calculations. Mhermore, T is highly model dependent.
The rdatively model independent S bas been estimateci, using an approach [IO] inspired by the Pagels-
Stokar formula for the pion decay constant [II]. Although these constraints do not rule out the technicolor
theories they do make it difficult to produce a sp&c model that solves aU problems. One thing is clear
though: technicolor m o t be just a scaled-up version of QCD-
These difEculties do not imply that the idea of dynamical symmetry breaking is wrong. Alternatives
to the technicolor idea are still being explored- Among them are the top color condensate idea [12] and
gauged flavor symmetries [13]. In this thesis we pursue a better understanding of strong dynamics to
improve our ability to formulate dynamitai models for the origin of the fermion masses and electroweak
synimetry breaking.
An alternative to the strong dynamics approach is supersymmetry [14] - This is a symmetry that can exist
between fermions and bosons. The requirements are that they corne in the same representations and that
they have the same numbers of degrees of freedom, Supersymmetry protects the scalar mass by c o r n b i g
each scalar with a fermion into one supermultiplet. The fermion mass is protected by a chiral symmetry
and via the supersymmetry so is the scalar mas . There exists a model d e d the minimal super~ymmetric
standard model (MSSM) [15] that gives a satishctory explanation for the electroweak breaking. in this
mode1 every fermion or boson field tbat exists in the standard model has an extra field of the opposite type
associateci with it. These extra fields are d e d superportners. The known fields are combiied with their
superpartners into supermultiplets that transform as irreducible representations of the supersymmetry.
Since it is not observed at 10- energy d e s , supersymmdry must be broken spontaneously at or above -
the electroweak scale- This breakhg process gives the superpartners masses which explains why we do not
see them. Some remnants of the superpartners may SUrYive if R-parity (an artifkt of supersymmetry)
is conserved. These remnants are d e d the lrghtest supersynmctnc parfide and are candidates for the
dark matter that is b e l i d to exist throughout the universe. Experimentally one wouid expect to see
superpartners in coliider events of sufficiently high energries. Searches for these particles have been ongoing
for quite a while, So fhr nothing has been found. If a more interesting idea would come dong now, chances
are that the interest in low energy supersymnietry would dwindle-
As a matter of fact an interesthg new idea did appear recently. It was proposeci that the hierarchy
problem caa be solved by assuming relativeiy large curled up extra dimensions [16]. The observable universe
is localized on a domain wall, with a thickness given by the electroweak scaie. Only gravity sees the extra
dimensions. The gravitational coupling can be much stronger leading to a much srnalier Planck scale. The
reason why we observe a small gravitational coupling is because over s m d distances gravity sees more
dimensions and therefore falls of quicker than the l/$ that is observed at larger distances. The problem
with this idea is that it simply replaces the ptevious hiefafchy problem with a new one relating to the ratio
between the electroweak scale and the sizes of the new extra dimensions- For only one extra dimension this
ratio is exactly the same as the ratio between the conventional Planck scale and the electroweak sale-
A variation on this theme of large extra dimensions is to have two domain walls separateci by some
distance [l?]. Due to a ciifference in the tensions of these domain walls, gravity would prefer to be localized
around one of these wak. The other one, where we live, would then bave weak gravi@. This tirne no
hierarchy problem remains because aii the scales could naturaily be of the same order as the electroweak
scale. The remaining problem now is to explain all those small quantities that are usually explained in
terms of scales, hi& above the eiectroweak scale. These include the neutrino masses and the observed
limits on FCNC's and proton decay- One way to explain the extremely small numbers without introducing
extremely hi& scales is to have Werent dynamics localized on Merent domain walls. There is a natural
exponential suppression of the &ects of such dynamics on neighboring domain waüs.
1.2 Understanding strong dynamics
At this moment in tirne it is still completely uncertain how nature unfolds above the electroweak scale. We
do not know whether nature is supersymmetric above this scale; whether there are large extra dimensions
or whether perhaps some completely different mechanism is concealed beyond the electroweak scale. It
therefore makes sense to pursue al1 possibiities and to consider alternatives to supersymmetry and extra
dimensions. Such au alternative is d y n d c a l symmetry breaking. If nature is not supersymmetric and
does not have extra dimensions, then we need to elMinate the elementary scalar fields fiom the standard
model to avoid llnnatural fine tuning- The only alternative seems to be a dynamitai symmetry breaking
process. In light of this it would be helpfd to have a better understanding of nonperturbative dynamics.
The lack of a dynamical theory of el- symmetry bteaking that can compete on equal footing with
the MSSM is perhaps indicative of our d&Qent understanding of nonperturbative dynamics. An important
feature of nonperturbative dynamics is the formation of composite order parameters - fermion n-point
functicns that are generated nonperturbativeiy through strong dynarnics. In this thesis the nonperturbative
formation of 2-point and 4point functions is inyestigated.
Unlike calcuiations in perturbative quantum field theory, exact quantitative calculations are not possible
in nonperturbative quantum fidd theory, The reason is the lack of small parameters in terms of which one
can make asymptotic expansions. As discussed above, nonpei.turbative dynamics is important enough that
one cannot just discard it and only consider perturbative calculations- There are s p d c instances where
nature employs nonperturbative dynamics and to understand these cases we have to h d ways to analyze
them. Fortunately there are ways, which have been developed over t h e , that can help to shed light on the
sub ject . These indude f00ls like the Ward-Takabashi identities [la], Schwinger-Dyson equations 119) and
renormalization group equations (201- But perhaps the most powedul formalism that is available to pedorm
nonperturbative analyses is the &&ive action formalism [2]. The variational properties of the effective
action can be used to derive Schwinger-Dyson and other gap equations for the n-peint Green functions of
the underlying theory. The &ective action can be expresseci in terms of these n-point Green functions with
the procedures of De Dominicis and Martin (211. Often such n-point functions do not respect some of the
symmetries of the underlying theory. They can therefore not be generated perturbatively. However, it may
be possible to generate them nonperturbatively, in which case they act as order parameters that signal the
dynamical breaking of these symmetries.
Much work has been done on the nonperturbative formation of fermion 2-point functions because of the
role they play in chiral symmetry breaking. We review this in Chapter 2. This is not only relevant to QCD
but also to physics beyond the standard model. Cornwall, Jackiw and Tomboulis (CJT) [22] derived an
effective action for the 2-point function (or the fuii fermion propagator) through the use of bilocal sources
for the fermion fields. One can obtain a gap equation for the dynamical mass fuuction in the fermion
propagator Çom the CJT effective action. This follows fkom the stationarity of the effective action with
respect to the full propagator near its physical value. Under certain circurnstances the gap equation can
be simplified by neglecting the momentum region below the dynamical mass scale; linearizing the resulting
gap equation in term of the dynamical mass function; and assuming that the one-gauge-exchange diagram
dominates all other diagrams, which is the ladder or rainbow approximation. Fkom the resulting hearized
gap equation in the ladder approximation one can solve the dynamitai mass function analyticaliy.
The CJT effective action can also be used to study the stability of the symmetric vacuum [23] - Le. to
do a stabilio analysis. For this purpose we interpret the effective action as an effective potential. For smail
couplings the global minimum of the effective potential is located at the origin, where the order parameter
is zero. When the coupling is increaseà the giobal minimum remains at the origin until the coupling cmases
the critical value where the phase transition occurs- At this point the global minimum can either jump
discontinuously to a nonzero value of the order parameter, indicating a first order phase transition, or it
can move continuously (though non-dflcally) to a nonzero value of the order parameter, indicating a
continuous phase transition (second order or higher). For a first order phase transition a locai minimum
must develop as the coupling is increased- This becornes deeper as the coupling is increased M e r until it
is below the minimum at zero and so becornes the global minimum when the coupling crosses the critical
d u e . A continuous phase transition appears when the global minimum 'rok' out of the origin. This
Mplies that the minimum at the origin must flatten out and become unstable.
To investigate fust order phase transitions one must be able to analyze the dective potential at arbi-
trarily large values of the order parameter. Unfortunately, it is not possible to make a reliable fruncation of
the series of diagr- in the CJT &ective potential a t k g e d u e s of the order parameter. For a continuous
phase transition on the other hand, the order parameter wodd be arbitrarily smali close enough to the
transition point. Hence, the order parameter can be used as an expansion parameter, allowing one to make
reliable truncations of the series of diagrams. For this reason we restrict ourselves to the investigation of
continuous phase transitions.
The effective potential consists of terms with positive powers of the order parameter- Near the origin the
nondynamical kinetic part of the C3T &ective potential is second order in the dynamical mass function.
Hence, to destabiie the potential at the origin the dynamitai terms must be of second order or smaller.
Terms with higher powers of the order parameter WU become insignificant compareci to the kinetic term
when the order parameter approaches zero. One can therefore drop ail contributions that are higher than
second order in the order parameter. For an attractive interaction the dynamical tenns will have the
opposite sign to that of the nondynamical terms- When the dynamical tenas exceed the nondynamïcal
terms in size the potentiat becomes unstable at the origin. In this way one can determine the critical
coupling for the nonperturbative formation of 2-point bctions. Note that one c m also use the Schwinger-
Dyson equation to determine the criticai coupling (241.
This critical coupling depends on the representatiotts of the initial fermions as weil as the channel.
The term &tanne1 is used to distinguish the different possible bound States that can condense. It denotes
the irreducible representation under which the bound state transfonns. Not al1 channels are attractive.
The size of the criticai coupling gives an indication of the attractiveness of the channel- Cornwall 1251
and Raby, Dimopolous and Susskind [26] proposed the hypothesis that the most attractive channel in the
ladder approximation indicates which one will be the channe1 of the condensate - the true vacuum. This
hypothesis is called the most attractive Channel (MAC) hypothesis. There is justifiable skepticism about
to validity of the MAC hypothesis- This issue is studied in Chapter 3.
1.3. A NOVEL 4-FERMION ORDER PARAMETER 7
1.3 A novel 4fermion order parameter
It was pointeci out recently [13,27J that 4point functiom may play an important role in flavor scde physics,
high above the eiectroweak d e . The situation at such high d e s is that fermions are (eftectiveiy) massless
so that a chiral symmetry exists. There would not be any 2-point condensates that involve feRniom with
electroweak quantum numbers above the eiectroweak scaie, because t h e would break the electroweak
symmetry. The same applies to any other n-point function that breaks the electroweak symmetry. There
are however some &point functiom involving fermions with el&& quantum numbers that are dowed
at scales above the electroweak scale. One cau c i i s t i i h five such &point hc t ions on the basis of the
chiralities of th& extemal linesr three are chirality preserving and two are chidity changing-
The chirality preserving 4point functions can respect any chiral symmetq-, but the largest chiral sym-
metry that the chirality changing 4point functions can be invariant under is a chiral isospin symmetry,
SU(2) x s U ( 2 ) ~ - These 4point functions would not respect any larger chiral symmetries. Hence, they
act as order parameters for the partial breaking of such chiral symmetries.
The significance of this is that such a breaking proces may be present a t some new scale high above
the electroweak scale- The surviving chiral iaospii symmetry can contain the SU(2)L of the electroweak
çymmetry, which means that this mechauism can exist for standard mode1 fermions. It is therefore not nec-
essary to intmduce new non-standard mode1 fermions to produce fiauor s d e physics. As a result dynamical
models may become simpler.
The 4point functions which are generated at the new scale will result in &ective 4fermion interactions
(operators) in the low energy theory, and thus the same d e r parometers responsâble for b d - n g flavor
symmetn-es can be mponsible for feedàng down the TeV masses to give the quark and lepton masses. The
différence with respect to ETC models is that now a larger class of possible operators exists and their
variety in size and structure, which is a result of strong dynarnics, can produce the variety in the fermion
mass spectrum.
To illustrate this we consider a strong SU(Nc) gauge interaction for an even number Nf > 2 of massless
fermions in the fundamental representation of the gauge group. The global symmetry is SU(lVf)L x
SU(Nf)R x U(1)v . (The subscript V indicates a vectorial symmetry.) The formation of chirality changing
4point functions now signals the breakdown of this symmetry to a chiral isospin (or electroweak) symmetry
and a vector 'family' symmetry:
The flavor structure of this chirality changing 4-point function is
where the upper case superscripts denote family, the lower case superscripts denote isospin and E,, and ébd
are 2 x 2 anti-symrnetric matrices.
For these 4-point fimctions to act as order parameters they must appear at higher scales than 2-point
condensates do. One can imagine that for certain combiitions of number of colors, N,, and number of
flavors, Nt, the criticai couphg for the nonperturbath formation of 2-point functions may be larger than
that of 4point functions. In an asymptotidy free theory the 4point t c t i o m onsuid then be formed a t a
higher scale than the 2-point functions. The separation between these scale may be larger for large numbers
of flouors than it would be in QCD-like theories where the coupling runs relativeiy quickiy. Under these
circumstances one wodd have the breaking pattern of (1.2) a t the higher scale where the &point functions
form and lower down, when 2-point functions form, the chiral isospin symmetry would break to a vectorial
symmetry as is found in the electroweak symmetry breaking process.
One can start the anaiysis by assuming that the Eavorable situation exists for 4point functions to forni
at a much higher scale than 2-point functions. This means that one can assume that f e o m rem&
massless. It aüows one to formulate an effective action for these &point functiom, making the chiraiities of
the fermions explicit- Thus one can distinguish the above mentioned five dBerent Ppoint functions. Such
an analysis is provided in Chapter 4 for gauge exchange dynamics. There it is shown that in the limit of
a large number of colors [28, 291, none of the five &point functions can be generated nonperturbatively.
Since that analysis excludes instanton dynamïcs, the question naturally arises whether 4point functions
can perhaps be generated through instanton dynamics.
1.4 Instanton dynamics
Within a non-Abelian gauge theory there exist two dynamical mechanisms that can contribute to the
nonperturbative formation of n-point functions: gauge exchanges and instantons [30,31]. Ifgauge exchanges
cannot generate certain n-point functions nonperturbatively then perhaps instantons can. A thorough
analysis should strictly speaking include both these types of dynamics. However, such a full treatrnent is
beyond our capability at the moment. Instead we have to contend with separate analyses for each of these
types of dynamics. In this thesis we present both types of analyses for the 2-point and 4-point functions.
However, the emphasis d l be on the instanton analyses. We provide a brief review of instantons in
Chapter 5. More detailed reviews of iastantons are found in Reference [32, 291.
Unlike in the 2-point function case, calculations of the nonperturbative generation of 4-point functions
through instantons have received very üttle if any attention so far. At present there does not seem to exkt
a formalism in which one can treat this problem. Therefore we provide an effective action formalism for the
calculation of the instanton contribution to the nonperturbative formation of 4point functions in Chapter 6.
Because such a formalism requires as part of its derivation a consistent treatrnent of 2-point functions, it
also provides the means to caldate the instanton contribution to their nonperturbative formation. Our
1.5, CEZRAL SYMMETRY BREAKING VIA INSTAM'ONS 9
formalism pravides a consistent framework for deriving the gap equations of the devant n-point functions
through the stationarity of the &&%ive action. The instanton contributions to the ~ ~ v e action are
represented as dhgrams that are generated with Feynman des. These des iaclude a nonlocal instanton
vertex for the fermion zeromdes and a fermion propagator in the instanton background- The nonlocai
instanton vertex difiers h m the usual dective 't Hooft vertex 1311 in the sense that, uniilce the latter, the
former is MZid at distance d e s smaller than the instanton size.
The sizes of the actual n-point hctions help to determine whether the expansion in terms of diagrams
converges. If there is some maximum size for the n-point hct ions for which the expansion still converges
and the actual global minimum of the effective action fies beyond the point where the n-point function
reaches this maximum size, the uSefulIless of this formalism wouid be Iimited. However, in the case of a
continuous phase transition the n-point function wouid have a dliciently small value in the interesting
region near the transition. As explained before, we restrict ourseives to continuous phase transitions so
that we can use the n-point function as an expansion parameter and make reliable truncations of the series
of diagrams.
A semi-ciassical approximation is applied to the action h m which the instanton &ective action is
derived. This implies that the theory behaves like a fkee theory apart from the presence of a background
gauge field. In addition, it is assumeci that the instanton ensemble is fairly dilute - the average size of
the instantons, P, is much d e r than the average separation between them, R. On the basis of this
diluteness we drop terms in our derivation that are suppressed by factors of the average size over the
average separation @IR). The instanton vertex for the fermion zeromodes provides the dynamics through
which the fermion n-point functions are generated. Therefore it represents the important dynamics, whiie
the (p/R)-effects are of secondary importance.
1.5 Chiral symmetry breaking via instantons
We use the instanton dective action formalism to compute the critical couplings for the nonperturbative
formation of Zpoint functions. This caldation, together with its results, appears in Chapter 7. It is
presented in cornparison to the equivalent critical coupling found for gauge exchange dynamics.
A theory with an SU(N,) gauge symmetry and Ns fermions in the fundamental representation is
asymptotically free, provideci that Nf < 5.5N,. Close to this threshold the coupihg is bounded above due
to an infrared 6xed point in the beta function (se Appendix C). As Nf becornes srnalier the h e d point
d u e increases. Beiow a critical value for Nf the coupling can grow strong enough to break the chiral
symmetry down to the vector symmetry:
As a resuit the fermions acquire a dynamical mass, which acts as an order parameter for the spontaneous
breaking of the chiral symm- This much is weU understood, but the respd5ve parts played by gauge
exchanges and instantons in this process SU neeàs chdication-
It has been known for some time that gauge exchanges can geirerate a dynamical mas- In Chapter 2 we
show that to leadhg order (ladder approximation) in Landau gauge the criticai coupling associateci with
gauge exchange dynamics is
Assuming a two-loop beta function, one can show that the critical number of flavors, below which chiral
symmetry breaking occurs, is [33]
It is known that instanton dynamics can also break the chita1 symmetry [34,35]. Attempts to investigate
this p o s s i b i have been hampered by the presence of an infiareci divergence in the integral over the
instanton size- In th& recent investigation of the critical number of flavors for chiral symmetry breaking
via instantons, Appelquist and Selipsky [36] avoided the probiem of h h r e d divergences by introducing
the foliowing two modifications of the Carlitz and Cfeamer gap equation 1351:
a They introduced a m a s dependence for the fermion determinant that is not only valid for the small
mass iimit [31] but also for the large mass Iimit [37]. This function simply rnakes a sharp transition
fiom the small mass behavior to the large mass behavior. The effect is to introduce a fermion mass
scaie below which the fermions are integrated out-
0 In Reference [36] ail coupliags in the instanton expression are allowed to run according to the twdoop
beta function. For those values of Nj where the beta function has a fixed point the coupling remains
fairly constant untii it reaches the fermion m a s d e . Fermions are integrated out below the fermion
mass scale. This causes the beta function to revert back to normal asymptoticaliy fiee nuining, which
gives a suppression of the integraud in the bfkared region because of the way the instanton amplitude
depends on the coupling-
The result of their investigation was that the critical number of fiavors for instantons is about 4.77NC.
Rom this they concluded that the instanton contribution to chiral symmetry breaking is comparable to
that of gauge exchanges
In Chapter 7 we reinvestigate chiral symmetry breaking through instautons, but our investigation differs
from that of Reference [36] in two essential aspects:
0 The instanton effective action formalism, presented in Chapter 6 , is used to determine the conditions
under which the symmetric vacuum becomes unstable. Although, our formalism reproduces the
Carlitz and Creamer gap equation in the s m d mass iimit for SU(2) ( s e Section 6.5.2), we are not
interested in the shape of the mrrpp hc t ion , but instead in the critical couplings and the critical
number of fiavors-
We concentrate on continuous phase transitions, such as found for chiral symmetry breakhg through
gauge exchanges The &ktive action fom- makes it clear that the instanton contribution re-
sponsible for this type of transition is not the one-instanton amplitude of Figure 1-la but instead the
instanton-anti-instanton amplitude with two masi insertions, shown in Figure 1-lb.
Figure 1.1: Leading instanton diagrams: a) one-instanton, b) instanton-mti-instanton with two mass
insertions.
The latter point follows from the discussion in Section 1.2- Looking at the one-instanton diagram,
shown in Figure Lia, one sees that this term always cornes with as many powers of the order parameter
as it takes to close off al1 the zeromode lines on the instanton vertex. For Nj flavors there would be Nf
factors of the order parameter. This means that for Nf > 2 the one-instanton term cannot cornpete with
the kinetic term near the origin. As a resuit the one-instanton amplitude is irrelevant in a stabiiity anaiysis
for continuous phase transitions. The leading contribution to the dynamic part of the potential that is
second order in the order parameter is the instanton-anti-instanton term with two mass insertions shown
in Figure 1-lb. Therefore only this term is considered in our calculation. It tums out that this term is not
injkred divergent
Unfortunately it is very difficult the evaiuate this term exactly for an arbitrary number of flavors- It
is therefore necessary to make an estimate of the size of this contribution. As a result we end up with
an uacertainty in the contribution of a typical f d o n loop in the diagram. This uncertainty can be
parameterized in terms of an O(1) parameter, A, for each loop. We provide the results in terms of this
parameter.
The main result of our stability analysis is presented in terms of an expression for arbitrary Nf and
CIFAPTER 1. THE DYNAMICAL ORIGIN OF SCALES
Ne from which one can compute the criticai couplings for c h i d symmetry breaking via instantons- We
compute the criticai wuplings for N, = 3 and 4, using a feftr din i t values for A. Numericd values for the
criticai couplings are provideci for numbers of flavors h m 2 up to the respective criticai numbers of flavoxs-
We find that unies X is e l y large (on the order of 2.3 and larger) instanton dynamics give insignificant
contributions to the nonperturbative formation of 2-point fimctions.
1.6 Chiral symmetry breaking via &point functions
The second caldation that we perforrn within the instanton eBective action framework is to determine
the critical couplings for the nonperturbative formation of 4point hc t ions via instanton dgnamics- This
is presented in Chapter 8- We use the same generic model that was used for 2-point functions in this
calculation. Eùrthermore, the procedure for the caicuiation of these d u e s follow the same basic procedure
that was followed for the 2-point fmctions, with a few exceptions. We are again lead to introduce the
parameter A for the uncertainty in the fermion Ioop contributions.
Since the 4-point function has a much more complicated momentum dependence than the 2-point
function one cannot simply take the Fourier transform of the 4po i . t kernel as is done in the tx ie of 2-point
functions- We are therefore forced to choose a specific shape for the 4point function. A consequence is
that the values of the criticai couplings may be slightly overestimated. In spite of this we still h d criticai
couplings that are smaüer than those for the 2-point functions, irrespective of the d u e of A.
As an application, we consider a more complicated model in Chapter 9. This gives us the opportunity
to discuss the model that originaiiy sparked the interest in chirality changing 4point hc t ions [13]. Unfor-
tunately, due to the additionai comphcations, a full dculation in this model would require a rederivation
of the 4point &ective action- However, this does not prevent us from using our newly acquired knowledge
about the nonperturbative formation of &point functions to study this model, We are in parti& inter-
ested to see if chirality changing 4point hctions would form before 2-point condensates do. Indeed, we
do find that the interplay of the difterent gauge interactions in this model produces an enhancement of the
4-point amplitude over the 2-point amplitude. The reason for this is that whiie the formation of 2-point
functions are suppressed by repulsive gauge exchange interactions the 4point functions do not feel this
suppression. This is because chiraiity changing 4point functions cannot be a f f i by gauge exchanges in
the large Nc iimit, as found in Section 4.5.
Chapter 2
Nonperturbative analyses
Here we provide the tools and techniques for the analyses of this thesis- For more information about
dynamicai symmetry bteaking and the nonperturbative techniques used to investigate it see Refe~ences [38,
39, 231. We start with a general discussion of dynamicai symmetry breaking. Then we briefly review the
effective action formalism. This is foilowed by a derivation of the CJT effective action [22] which is baseà
on the procedures of De Dominicls and Martin [21].
There are two ways in which cne can use the &ective action to gain information about dynamicai
symmetry breaking in a theory:
a One can use the stationarity of the effective action with respect to an order parameter (n-point Green
function) to derive a gap equation for this order parameter. This gap equation can then be solved
to determine the function of the order parameter (for example the dynamical m a s function) for
any value of the coupling constant. When the coupling constant approaches the critical value at a
continuous phase transition the function goes to zero- In this way one can determine the critical
coupling at which the order parameter vanishes.
a The effective action can also be use as an effective potential to andyze the stability of the symmetnc
vacuum at the origin. This is referred to as a stabdity andysis. In this way one can directly determine
the critical coupling. In cases with more than one breaking channel a stability analysis can be used
to determine which one of the possibilities is most attractive and therefore the preferred breaking
chamel-
Depending on the complexity of the gap equation it may be time consuming to compute the order parameter
fmction. If one is only interested in the critical çoupling then the stabiüty analysis is an easier approach.
Most analyses presented in this thesis are stability analyses.
In Section 2.4 we apply a stability analysis to the CJT dective action. This will give us the opportunity
CEAPTE. 2- NONPErrrrniRArrVF: ANALYSES
to discuss the linearization of &&ve potentials and gap equations;
0 to defme the kenid, which we use in the most attractive channel analysis in Chapter 3;
0 to introduce the ladder approximation;
and to determine the criticai couphg for chiral symmetry breaking through gauge exchange dynamics-
For completeness we aiso discuss the derivation and solution of the gap equation in Section 2.5. This
will give us the opportunity
to discuss the choice of gauge;
to compute the generic shape of the dynamical mass function;
and to m n h the critical couphg found in the stability analpis.
2.1 What is dynamical symmetry breaking?
One of the remarkable features of quantum field theory is the phenornenon of the spontaneous breaking of
a symmetry. This implies that a symmetry of the ciassicai Lagrangian becornes hidden because the vacuum
of the theory is not invariant under this symmetry. According to Noether's theorem every continuous
symrnetry is associated with a conserved quantim However, when the vacuum is not invariant under this
symmetry, it carries this quantity. The conserveci quantity can therefore flow into the vacuum and appear
to be not conserved-
To understand how a vacuum can exist that is not invariant under a symmetry of the theory, we consider
a theory with elementary scaiars. It is possible that the potential of such a theory has the shape of a Mexican
hat - the minimum is not located at the origin but at some nonzero value of the scalar field. This value
is the vacuum expectation value (VEV) of the scalar field. The potential must still be invariant under
the symmetry. This means that the minimum m u t be degenerate - Le. there are an infinite number
of continuously ~ 0 ~ e ~ t e d points with the same m;nimal value. Symmetry tramdonnations relate these
points to one another. Nature would pick oniy one such point. Around this point the potential wodd not
be symmetric. Thus the symmetry is hïdden. For a global symmetry this is caiied the Nambu-Goldstone
phase of the symmetry. When the global symmetry is m a n i f i - not spontaneously broken - it is said to
be in the Wigner phase. Spontaneously broken global symmetries give rise to massless scalar fields d e d
Goldstone bosons [Ml. On the other band spontaneously broken (local) gauge symmetries cause some of
the gauge bosons to become massive through the Higgs mechanism [41].
In a theory that respects the Lorentz symmetry, aii nonzero VEV's must transform as scalars- However,
this does not mean that only fundamental scalars can have nonzero VEV's. In a theory without hindamental
2.2. EFFECTNE: ACTION FORMALLSM 15
scalars it is possible that the vacuum appears at some nonzero VEV of a composite scalar a d . This
composite may for instance be a biiinear of fermion fidds, For this to happen there must be dynamics
in the theory that are stmng enough to bind the fermion fields into a massless composite- This process is
therefore referred to as dyramid symmetry breaking-
How does one know that a symmetry in a given theory is spontaneously broken? Symmetry breaking is
related to a change of phase- The phases of a system are governeci by control parameters. By 'dialing' such
a control parameter the system can go through a phase transition, One can observe a phase transition by
observing what happens to certain order pammeters f39J. 'ïhese are quautities that change non-anaiytically
across the phase transition as functions of the control parameters. Such a non-analytical change in the
value of an order parameters indicates a change in the 'order' of the system. The value of the control
parameter at which the transition occurs is referred to as the critical value.
There are various types of phase transitions. When the phase transition is marked by a discrete jump in
the value of the order parameter it is d e d a first order phase transition. A second order phase transition
occurs when the order parameter is continuous over the transition, but the first derivative with respect to
the control parameter is discontinuous, In generai an n-th order phase transition is when a discontinuity
appears after (n - 1) derivatives of the ordet parameter.
In partide physics the order parameters are the VEV's of certain operators. The operator can be a
fundamental scalar field, in which case its VEV is d e d an dementary order parameter- If the operator
contains a product of fields the VEV is caiied a composite order parameter. The control parameter in this
case is the strength of the interaction - the coupling constant.
Any operator that t ramdom nontriviaüy under a specific symmetry of a theory is a candidate for an
order parameter that wouid indicate the breaking of that symmetry. One can wnsider the energy potential
of the theory as a function of such an operator, This potential function must remain unchangeci when
symmetry transformations are perfonaed on the operator. If it is zero the VEV of the operator would
not be able to transfo= nontrivially, even though the operator itself trandorms nontrivially- If this VEV
is nonzero, the vacuum would be degenerate, implying that the symmetry is spontaneously broken. The
point where the VEV î k t becornes nonzero indicates a phase transition. We say that the order parameter
signals the breaking of the symmetry.
2.2 Effective action formalism
The effective action provides a powerful formalism to study order parameters in quantum fidd theory. It
is stationary with respect to variations in the order parameters around their physical values. One can
therefore use variational techniques to compute the values of these order parameters as functions of the
control parameters. In this way one can observe the appearance of a phase transition - spontaneous
symrnetry breaking - and determine the critical value of the control parameter - the criticai coupling,
In this thesis we use an & ' v e action fonnalism to analyze %fermion and 4fennion order parameters.
Here we present the fundamentah of this formalism. The derivations and calcuiatious of this thesis are
mostly done in Euclidean space- In Appendix A we provide a discussion of the Wick rotation by which
expressions in Minkowski space are analytically continued to Euclidean space.
The Euclidean partition functional Z is detined in terms of a path integrai
where D is the functional meamne over dl the fields in the action, and the fidl action is
Here SE is the Euclidean action of the theory under investigation- S, denotes the source terms that are
added to the action for each field in the theow For example, the source tenns for the fermion fields are
The SJ and SK in (2.2) denote extra nonlocal source terms that contain products of fields- The nodocal
2-fermion source term is
and the nonlocal 4fermion source term is
The nonlocal source terms are helpful in the derivation of efkctive actions for %fermion and 4fermion
order paramet ers.
The generating functional W is definecl by
where the functional dependence on the nodocal sources is shown explicitly. The generating functional is
the sum of ail comected vacuum diagrams that can be constructecl by using the Feynman rules of a gauge
theory, together with the nonlocal sources as 2-fermion and 4fennion vertices.
Formaüy the 2-point function (full propagator) is given by
and the &point fwiction is given by
The d k t i v e action is obtained fiom the generating functiond via two Legendre trdormations- The
first Legendre transformation,
replaces the functional dependences on K by a functiod dependences on the fidl fermion propagator S,
defined in (2.7). By settiag J = 0, one obtains the CJT efllective action [22]:
~ c J T E S J = I'*[S, J = O].
In Section 2.3 we present a derivation for the expression of IIcJT[S] in terms of S.
The second Legendre transformation removes the K source dependence in favor of a fundional depen-
dence on &point functionç. The details of this step is poBtponeà untii Section 4.1.1.
It is easy to show that
In the physical case that we are interesteci, all sources are equal to zero. Setting K = O in (2.11), one fin&
that the efféctive action obeys a stationarity condition:
Given an expression for the efféctive action in terrns of S as the only unknown, the stationarity condition
provides us with a gap equation which one can solve to find S. If the e f f d v e action contains more than
one such unknown - say for instance both S and G - then there would be a stationarity condition for
each of these quantities. These would lead to coupled gap equations which must be solved simultaneously
to find both the quantities.
The challenge is therefore to find expressions for the effective actions in terms of the composite order
parameters.
2.3 CJT effective action
Here we derive the CJT effective action [22], however, we do not follow the derivation of Reference [22].
Instead one of the procedures of De Dominicis and Martin [21] is used. Their procedures employ topological
relationships that exist among certain features or elements of vacuum diagrams to express the sum of al1
vacuum diagrams in terms of n-point functions.
We consider a gauge theory with the foilowing Euclidean action
where C, is the gauge fixing and ghost terms. In this derivation we use the source K as a 2-fermion vertex
and to simplify notation we set J = 0.
Figure 2.1: Examples of cycles of lines: a) a Pcycle, b) a trivial 2-cycle and c) a trivial l-cyde.
Foiiowïng (2 11, we start by dethhg a few usefd concepts. We define a cycle of lines in a vacuum diagram
as a set of iines with the propew that when we cut aU these lines the vacuum diagram separates into a
number of 2-point diagrams qua1 in number to the n u b e r of lines in the set. When one and only one of
these 2-point diagrams consists of only one 2-fermion vertex the cycle of lines is called tritrial with respect
to this 2-fermion vertex. Al1 other cases are called nonhiururd When a cycle of lines contains m lines we
rder to it as an m-cycle.
Next we define the (connected) 2-purticle inducible (2PI) vacuum diagrams as aü those vacuum dia-
grams that contain only nontrivial l-cycles and trivial 2-cycles, with respect to the 2-fermion vertex, K.
(Note that the oniy trivial 1-cycle, Tk{SoK), is thus excluded.) The s u m of di 2PI vacuum diagrams is
denoted by W2PI. One can now readïiy define the (connected) 2PI 2-point diagrams in terms of the 2PI
vacuum diagrams through a functional derivative with respect to the 2-fermion vertex, K. The sum of al1
amputated 2PI 2-point diagram is given by:
where we show the dependence on the propagator to indicate that lines in these diagrami represent the
bare propagator So. One can construct the sum of al1 2-point diagrams from the s u m of ail 2PI 2-point
diagrams, EISo], by simply replacing the bare propagator by the fidi propagator S. Thus C[S] denotes the
sum of aü 2-point diagrams. Note that although the diagrams in C[SJ are 2PI with respect to S, they are
not necessarily 2PI when S is replaceci by the sum of ail 2-point diagrams (which S consists of), containhg
So. One can generate new diagrams by connecting di6erent axnputated 2PI 2-point diagrarns via the bare
propagator.
Now we introduce the topological equation [21] :
2.3. CJT EFFECTIVE ACTION 19
where Nw is the number of cycles of lines in a vacuum &gram, Nli,,,, is the total number of lines in
the diagram and Nakel is the number of 2PI skeletons- A 2PI skeleton is what remains of a 2PI vacuum
diagram after one has cut out aU 2-fermion vertices,
The topological eqgation (2.15) holds for every vacuum diagram separateiy. One can use this topologid
equation to construct a resummation equation for the sum of all vacuum diagrams:
The resltmmation equation will make it possible to mite the surn of al1 vacuum diagraum in terms of the
n-point functions, Each term on the right-hand side of this equation is formed by counting each vacuum
diagram as many times as the number of the corresponding dements (cycles, lines or skeletons) that appear
in it. Adding the three terms, one would then, according to (2.16), get back the sum of aU vacuum diagrams,
W.
One fonns wwde8 by constructing ail cycles:
The minus sign in the sum cornes fiom the fermion loops. To form W1ke, one closes the surn of ail 2-point
diagrams (minus the bare propagator) off by S;':
W,,, = -lk {sr' (S - 4 ) ) . (2.18)
The minus sign in fkont is for the fermion Ioop. Finally WHkel is formed by replacing d the Sa's inside the
sum of ail 2PI vacuum diagrams, WZPI, by the full propagator S and add to this îk{SK), because K is
also a 2PI skeleton and Tk{SoK), is excluded fkom the d a t i o n of 2PI vacuum diagrams- Therefore we
have:
W'ret = W~PI [SI + a {SKI - (2.19)
When we place these terms into (2.16). The result is
One can simplifi. this by using the following expression that relates E and S
which represents the recursive resummation of self-energy diagrams. Then we have
W = Tk {in (S-') + in (so)) + Tr {(Sc' - S-') S) + W ~ P I M + l k {SK) - (2.22)
The Legendre transformation for 2-point functions removes the last term in (2.22), leaving us with an
effective action for 2-point functions:
20 C&APTER 2. NONPERTURBATIVE ANALYSES
Dropping the constant term -Th{h(So)) one obtains the famous expression of the CJT &ective action
P l 9
&JT[S) = (In (S-') ) - B { (S-' - Sc') S) + wapr[S] - (2.24)
One can classify the terms of the effective action as either dynamical or nondynamid Terms that
depend on the coupling constant are the dynamitai terms and those that are independent of the coupling
are the nondyndcal terms. T h d o r e the dynamical terms are aii those that vanish when the coupling
constant is set to zero. The Grst two kinetic terms in (2-24) are nondynamical terms. They consist of one-
loop diagrams that are only composecl of the full fermion propagator, S(p), and the inverse bare propagator:
Sc1@) = -i+ . The dynamical term in (2.24), denoted by Wipr[q, are constructeci with ali the mal
Feynman nùes of a gauge theory, except that the fermion propagator is given by the fidi propagator-
2.4 Stability analysis
In Euclidean space the full fermion propagator can be parameterized as foUows
in terms of A(p), which is the inverse of the wavefunction renormalization, and C(p) , the dynamical mass
fuLlctiona. The propagator is also multiplieci with ideatity matrices in color and flavor space. However, we
shaii not show them explicitly. In general Z(p) aiso carries implicit color indices. One can distinguish two
situations.
O The first is where C ( p ) is a singlet under color. Then one c m write it as the product of an identity
matrix and a mass fundion without color indices.
The other situation is where C(p) is a nonsingiet, which imphes that the color symmetry is sponta-
neously broken.
Although it is often assumed to be a singiet in color space, we also consider in Chapter 3 situations where
C@) transforms under some nonsinglet represeatation. For this reason we keep the representation of E(p)
generd in this chapter.
ONote that C(p) is not the same as CM, defined in (2.14)- Using (2.21) one can show that they are reiated through
Z[S] = C(p)A(p) + i j [ l - A(p)]- (2.26)
For A@) = 1 they are equal.
It is o h convenient to expand the &'ive action (or the gap equation) around a speafic value for
the full propagator, S = SI, For this purpose one can use the following functional Taylor expansion
where Tk{-) contains traces over aü indices (cdor, flavor and Dirac) and integratiom over all the momenta
pi- This trace contracts each functional derivative with one of the propagator ter- in the pmduct.
If X(p) remains constant over a large range offl before starting to fa11 off, then the dominant contribution
to the dynamitai tenu in the & i v e action wii l be fkom the region where 9 > E2 @). As a result the
region where 3 < C2@) can be neglected, Under this assumption, one can malce an expansion in powers
of C ( p ) / p . When we -and the full fermion propagator in this way we have
In view of this we consider an expansion [42] of the CJT &&%ive action around
so that
Computing the first three terms of (2.27) using (2.29), we find
First we consider the nondynamical terms of (2.24) up to second order in E(p), which we denote by rfin:
The dynamical term gives
where
and
The stationarity of rm[S] with respect to variations of Ab) giws us a way to eliminate the WipI-term
1421. For this purpose we generalize to the situation where Ab) carries d o r and flavor indices- Dropping
aü the terms suppressed by factor of C(p)/p, we obtain
where trD(-) denotes the trace over Dùac indices that is left intact because A@) does not have Dirac
indices. Using (2.36) and dropping al1 the terms that are independent of C(p) , we see that oniy the iast
term of (2.32) and the last term of (2.33) survive. The resulting expression for the effective action that is
second order in C(p) is,
where tr,{-) is a trace over the color indices. The kemel function in (2.37) is defineci by
where 8 is the angle between the Euclidean 4-vectors, p and q. Note that A@), C(p) and K(p, q) only
depend on the magnitudes of these 4-vectors.
The expressions in (2.37) and (2.38) provide the starting point for the most attractive channe1 analysis
that is presented in Chapte. 3.
We now proceed with the stability analysis to determine the critical coupiing for chiral symmetry breaking.
The procedure of Merence 1231 is used for this purpose. The caldation is done in Landau gauge (( = 0)
so that A b ) = 1, as shown in Section 2.5.1. We make a redefinition of the integration variables and do a
Fourier transformation. For this purpose we define
where p is the renormalization scale. The kinetic term in (2.37) then becornes
In spite of the fact that a resummation of diagrams has been done to restrict W2pr to contain only 2PI
diagrams, it still contains an infinite nwnber of diagrams. Therefore one cannot calculate thîs term exactly.
Figure 2.2: Vacuum diagram with one gauge boson exchange and two mass insertioxis,
One must in some way truncate the series of diagrazns. One such truncation is based on the assumption that
the coupling is still small enough that subleading order 2PI diagrams are suppressed. When one restricts
this series of diagrams to just the first diagram which is leading order in the coupling constant one obtains
what is d e d the krdder or minbow approximation- This 1-g order diagram is shown in Figure 2.2 and,
in Landau gauge, its Euclidean amplitude is given by
where T,O and Tg denote the generators in the representation of the two fermion fields respectively, and
k = p - q. The trace tr{-) runs over color, flavor and Dirac indices.
In the general case the 2-point function can be formed from two fermion fields of different representations.
We keep the representations of the fermion fields generai. To simpw the color trace oce can use the
following identiw,
which is derived in Appendix B. Here CF denotes the second Casimir constant and r i , rz and R denote
the representations of the two fermions and the condensate, respectively. The constant FG is the channel
factor, which indicates the relative attractiveness of the different channeIs.
Next we linearize (2.41) as explained in Section 2.4.1. The dect is to make the replacement
The term that is second order in the mass function is
where cr = g2/47r and
The angular integral gives
so that we have
CHAPTER 2. NONPERTURBATNE ANALYSES
We apply the redefinitions, (2.39), with the addition of q = p exp(s) , to (2.47)- As a r e d t it becornes
In terms of the kinetic term (2.40) and the dynamid term (2.48) the effective potential giws
One can see that for u(w) = O the potential is zero. When o(w) is small but nonzero the sign of the
potential would indicate whether the vacuum at the origin is stable, If it is negative the origin does not
contain a global minimum anymore and therefore must be unstable. This happens when
Of the four channels considered in Appendix B - the singlet, adjoint, symmetric tensor and antisymmetric
tensor - we see that the singlet channel gives the srnailest critical coupling
This analysis is an elegant way to show that there &ts a critical coupling above which the vacuum at
the origin is unstable. In Section 2.5.2 the same critical coupling is found from the solution of the gap
equat ion.
2.5 The gap equation
2.5.1 Landau gauge
Here we derive the gap equation for the dynamid mass function and show that Landau gauge is a con-
venient gauge to work in. Taking the functional derivative of Cc ~ T [ S ] in (2.24) with respect to the full
propagator S one obtains a gap equation for the fuil propagator:
We again use the ladder approximation and consider ody the first diagram which is shown in Figure 2.3.
The Eudidean Feynman ampütude for this diagram in arbitrary gauge is
2.5. TIiE GAP EQUATION
Figure 2.3: Fermion self-energy diagram with one mass insertion.
where Tf and Ti denote the generators in the npiesentation of the two fermion fields respectively, is the
gauge fixing parameter and k = p - q, We simplifv the color trace again with the use of the identity in
(2.42) and substitute the result into our gap equation (2.52) to obtain
One can use this gap equation to determine the two functions that parameterize the full fermion prop
agator. First we separate the gap equation into two coupled gap equations for these two functions. For the
dynamical mass function we get
and for A(p) we get
-s-l (p) Ab) = ;W.{; }
The contraction of the Dirac trace with the gauge propagator in (2.56) gives
where 8 is the angle between the Euclidean 4-vectors, p and q, and
The angular integral in (2.57) gives
CEAPTER 2. NONPERTURBATIVE ANALYSES
So the expression in (2.56) becornes
Fkom (2.60) one sees that the second term is proportional to the gauge fixing parameter, c. Conse~uently,
to one-loop in Landau gauge, where = O, we have that A(p) = 1- It is thedore convenient to work in
Landau gauge. We henceforth restrict o d v e s to this gauge, except in Chapter 3 where we h d it more
convenient to work in Feynman gauge (e = 1).
Results fiom these types of anaiyses (gap or stability analyses) are known to be sensitive to the choice of
gauge. The reason is that we stitl use the bare gauge vertices together with the full f d o n propagator. A
better treatment muid be to replace the bare locai vertices with nonlocal dressed vertices, [43]- However,
numerid solutions of these gap equations, using such nonlocal vertices do not mer siPnificantly from
those for a bare vertex in Landau gauge (44- In this thesis, therefore, we restrict ourselves to the usual
analysis and use only the bare vertices.
When we substitute = O and A(p) = 1 into the gap equation for Co,), (2.55), we get
Next we proceed to solve this gap equation to find the dynamical mass function and at the same t h e
confirm the critical coupiing found in Section 2-4.2.
2.5.2 The dynamical mass function
Here we linearize the gap equation for C(p) in (2.61) in the same way that we have lin- the effective
action in Section 2.4.1. Since the m a s function is monotonically decreasing this introduces an infrard
cutoff at the scde where
Po = ~(Po) . (2.62)
The linearized gap equation is aven by
One can dehe a dimensionless function in tenns of the dynamical mass function,
where p is some arbitrary UV cutoff scale. When one replaces the dynamitai mass function in (2.63) with
t his dimensionless function the resulting expression is
2.5. THE GAP EQUATION 27
The angular integral in (2.65) is the same one that is evaiuated in (2.46)- Using that resuit in (2.65) we
To simplify the notation we combine aii the constants in fiont of the integral into a single constant:
Therefore the integral equation can now be interpteted as an eigenvaiue equation
Next we d e a redefinition of the variables:
Then we have
a
where g(t) = m(pexp(t) ) . One can now derive a dinerential equation for g(a)
from the integrai equation in (2.70). The solutions of this dinerential equation are of the form
g (a) = Acos(fla) + B &(Ba) (2-72)
where
To find expressions for A, B and f i we impose (2.62) and solve the boundary value problem by substituting
the solution back into the integral equation in (2-70)- With the use of (2.64) and b's definition in (2.69)
the condition in (2.62) leads to
The integral equation in (2.70) gives:
X = - - exp(b - a) [(A - BB) cos(Pb) + (B + AB) sin(@)] 2
Because this e x p d o n must apply for any value of a, the 6rst and last tetms of (2.75) must vanish
separately- For the last term this implies that
which in turn implies that the Grst term gives
The value of B follows fiom (2-76) and (2.74):
For auy given b the equation in (2.77) has a series of solutions &, which gives the series of eigendues,
A, = 2/(1+ 8;). Their associateci eigenfunctions are given by (2.72) with (2.76), (2.78) and fl = 8.- The
first eigenfunction, fio, in this Series gives the dynamitai mas function. It is expressed as
and is shown in Figure 2.4 for b = -2.
Figure 2.4: The dynasnicai mass function.
One can mite (2.77) in the form
where n is an arbitrary integer, which can be set equal to 1. The right-band side of (2.80) is always finite.
So when b -+ oo we must have that /3 + O. This means that in this limit
Using the definition of b in (2.69) and the definition of @
that
(2.81)
in (2.73) together with (2.62) and (2.67) we find
where
The expression in (2.82) te& us how the order parameter C(po) changes zm a function the control parameter
a. As a approaches a, from above, the order parameter becornes drastically smder uatil it YaniShes
identically at a = a,. This confinns that the phase transition is located at the criticai coupiing (2.83),
found in Section 2-4.2.
One c m also see that the phase transition remains continuous for any arbitraxy number of derivatives
in the coupling. It is therefore not a phase transition of any finite order [45& Miransky and Y m w a k i [46]
called this type of phase transition a conformal phase transition and explained it in terms of the breakdown
of the conformal symmetry due to the appearance of a new scale, given by the fermion mas.
Chapter 3
The Most Attractive Channel
hypot hesis
In this chapter we investigate the most attractive channel (MAC) hypothesis which has played an important
role in attempts to understand physics beyond the standard model. The work discussed here has been
reported in Reference [47].
3.1 Motivation and history of MAC
As we have seen, dynamical symmetry breaking is a mechanism that may be usefui in understanding the
breaking of the electroweak symmetry and the origin of the fermion masses. However, dynamical symmetry
breaking requires strong dynamics and is therefore an inherently nonperturbative dect. Unfortunately,
nonperturbative caldations are difficult . In view of this Cornwali [25] and Raby, Dimopolous and Susskind [26] made a dynamical assumption
in the late 70's to aid the formulation of models. Here is the staternent of this dynamitai assumption as
quoted from Reference [26]:
When the gauge forces between fermions in a given channel are attractive, bound States WU
form when the running couphg becornes large enough. If the coupling becornes even larger,
these composites become massless. Rirther increase in the couphg will cause the vacuum
to rearrange and a spontaneous symmetry breaking condensate will form [3]. The nature of
the condensate, its transformation properties under symmetry groups are determineci by the
quantum numbers of the most attractive channel (MAC).
The leading order analysis (ladder approximation) of Section 2.4.2 showed that the most attractive
channe1 is determineci by a group theory -or - the Channel a r (see Appendix B),
Here CF(rn) denotes the second Casimir constant of the irreducible representation, r,, where ri and r2
are respectiveiy associated with the two fermions and R with the condensate. The cbannel factor gives the
largest positive d u e for the singlet Channel (Appendix B). So if a singlet can fonn, it would be the most
attractive channel. This is in agreement with QCD.
The dynamical assumption together with the chamel factor became known as the Most Attractive
Channel or MAC hypothesis. Nummus models have since been proposed on the basis of this hypothesis.
However, more than two decades later we stiii do not have a working dynarnical theory of fermion mass or
electroweak symmetry breaking that is based on the MAC hypothesis.
The reason for this may be that the MAC hypothesis is too restrictive. It dictates the chamel of the
condensate and therefore to a large extent what the breaking pattern looks like. As a result extremely
complicated models were proposed to which the moose diagrams bear witness. If one can lift these restric-
tions the breaking patterns in chiral theories would be more open. Channeis that are repulsive according
to MAC may instead be attractive.
It may appear strange that anyone wouid pay any attention to a leading order analysis when the
phenornenon under investigation is clearly nonperturbative. The justification for the MAC hypothesis is
based on two aspects- One is that a t leading order the critical coupling constant for the formation of a
bilinear condensate, still appears to be smaii enough that one can neglect the higher order contributions.
The criticai value is found to be
for the singlet channel. Since each additionai loop in a loop expansion comes with an additional factor
of a/47r, one can argue [48) that t d y strong interactions appears when a sü 47r. Compared to this, the
coupling in (3-2) is stiU fairIy smaii. One may however ask to what extent the critical value of a! may
change beyond a leadiag order analysis.
The other justification for the MAC hypothesis came from the next-tdeading order (NLO) analysis that
was performed by Appelquist, Lane and Mahanta in the late 80's [49]. They considered a gauge coupiing
that does not nin (Le. a walking theory) by choosing the number of flavors such that the one-loop beta
function vanishes: llC~ - W f T ( R ) = O. To deviate the complicated calculations they further made the
assumption that the NLO kemd is proportional to the leading order kemei up to logarithms. They also
implicitly assumeci that the gauge symmetry does not break by leaving the gauge bosons massless. Their
caldations indicated the large& corrections among the various channeis to be on the order of 20%.
Here we present a NLO analysis that M e r s in the f0110wing aspects h m that of Reference [49]. We
aüow the possibility of gauge symmetry breaking by including a possible nonzero gauge boson mas. The
3.2. TEE EFFECTIVE A m O N 33
MAC hypothesis claims to diffiierentiate befween different chaunels, some of which imply that the gauge
symmetry itself is spontaneously broloen, leading to nonzero gauge boson masses. For consistency a possible
nonzero gauge boson mass should therefore be included in the fermion gap equation.
M e t m o r e , we make no assumptiom about the shape of the NLO kernel. Instead we consider the
kernel at the dominant momentum region. We do not restrict ourselves only to walking theories. However,
ody diagrams that are leading order in l/Nc or l/Nf are considered.
3.2 The effective action
The calculation is performed in the conventional way starting with the CJT &ective action [22] that we
discussed in Section 2.3. It is given by
where the traces include integration over the 4momentum, S denotes the full propagator (2.25), which
contains the dynamid mass function, C(p) .
The goal is to determine the strength of attraction or repuision of the different c h e l s . This is done
using a stability analysis. We are not interested in the actual values of the critical couplings associateci with
any channel but rather in their relative strengths- For this purpose we only need to consider the kemd of
the dynamical term-
The phase transition d t e d with chiral symmetry breaking is believed to be continuous. Therefore,
near the transition the order parameter, C(p) , would be very small. As a result only those terms that are
second order in C ( p ) are important. In the fashion of Section 2.5.2 we assume that only the momenta larger
than po = C(po) would have significant contributes. So we expand in terms of X(p ) /p and keep only the
terms that are quadratic in Ch). The result is the effective action derived in Section 2.4.1:
The first term in (3.4) cornes from the nondynamical terms in CJT &ective d o n (3.3). The second term
in (3.4) is the dynamical term. Here K@, k) is the kernti function of the dynamical tena, defined in (2.38).
It determines whether the massless solution becomcs unstable.
Figure 3.1: Leading order diagram.
The massless adution d make rgiT = 0. If the kemel fhction is large enough it d d o w massive
solutions that wii i make l$lT negative and that are for this -n more stable than the massless solution.
The aim of this anaiysis is t h d o r e not to find the shape of the mass function (Le- to solve a gap equation)
but instead to determine whether there exist solutions that are more stable than the rnassless solution.
Although a gap analysis cari determine the massive solutions it does not reveal whether nature would
prefer such solutions-
Figure 3.2: Next-tdeading order diagrams that are leaciing order in 1/&.
We calculate the kernel to second order in the coupling constant. At leading order there is only the one
diagram shown in Figure 3.1- The crosses denote mas insertions. The NLO diagrams that are of leading
order in 1/N, are shown in Figure 3-2 and the single NLO diagram that is of leading order in 1/% is shown
in Figure 3.3- The NLO diagrams are three loop àiagrams. Two of these loops are regularized by the mass
function, which is çoft in the ultraviolet. We regularize the momentum integral of the remaining loop with
dimensional regularization and renormalize it using modifiecl minimal subtraction, MS-bar- In the case of
a broken gauge symmetry there are additional diagrams involving the Goldstone bosons. These diagrm
are however suppressed by factors of 1/N,.
Figure 3.3: Next-tdeading order diagram that is leading order in 11%.
Considering the expansion of the kemel function in terms of a, we statt by expanding W&,[So](p, q),
where the fermion h e s denote the bare propagator So. AU the diagrams in Figure 3.1, 3.2 and 3.3 are
proportionai to the channe1 factor of (3.1). Apart fiom the influence of the gauge boson mass, the channe1
factor gives the only influence of the condensing channel. One can therefore write
where Kl and K2 respectively denote the leading order and NLO terms. The i is introduced for convenience.
Note that Kl is independent of the propagator because it contains no interna1 fennion lines. In Section 2.4.1
3.3. RESULTS AND DfSCUSSION 35
we defined Sr@) = So(p)/A(p)- According to (2.60) one can expand l/A(p) as
where is the gauge h h g parameter and Al (p) comes h m the leading fermion self-energy diagram, shown
in Figure 3.4. Now we expand the b e l function, given in (2.38), to second order in a:
where R is a normalization constant, which is defined such that Ki = Ki at the dominant momentum
region, In this way R gives the size of the NLO term relative to the Ieading order tenn. In terms of the
quantities in (3.7) 'R is given by
Figure 3.4: The leading order fermion self-energy diagram.
The leading order term in arbitrary gauge is
where M is the gauge boson mass and p and q are the magnitudes of the momenta that flow through the
two mass insertions respectivelyY When the gauge boson is massless, the expression in (3.10) becornes
The kernel function is symmetric under an interchange ofp and q and it has its largest value somewhere
on the diagonal line where p = q. For the case of a massless gauge boson, M = 0, the dominant momentum
region is determined by the shape of the dynamitai mass function, C(p) . When the gauge boson is massive,
M # 0, the dominant momentum region is determineci by the gauge boson mas. This can easily be seen
from the expression for Ki. Away fiom the diagonal line Ki decreases as min(l/#, l/rf). In Merence [49]
it was assumed that K2 behaves like Ki, which made their analysis d e r - it allowed them to evaluate
the kernel where p » q. Howewx, we found that K2 does not decrease as min(l/3, l/q2) away fiom p = q.
3.3 Results and discussion
The value of 7Z is caldateci for the dominant momentum regions in the massless M = O and massive case
M # O. We evaluate the diagrams in Feynman gauge = 1. In the massless case the dominant momentum
Table 3.1: Numerid values for 7Z
region implies that p = q = p where p is the reaormalization scale- For this case we h d that
Here CA denotes the second Casunir constant of the adjoint representation, CF denotes the second Casimir
constant of the fundamental representation and T(R) is 1/2. The first term comes fiom the diagrams in
Figure 3.2 and the second term cornes fiom the diagram in Figure 3.3. The last tenn arïses fiom the second
term in (3.9).
For the massive case the dominant momentum region gives M = p = Q = p- Here we consider
the breakdown of the gauge symmetry SU(Nc) to SU(Ne - 1) where the fermion mass is forrned in the
symmetric tensor representation. The oniy fermion that becomes massive is a singlet under SU(Nc - 1).
The gauge boson associated with the broken diagonal generator will have a m a s which is slightly larger
than the other gauge bosons that became massive. For Nc = 3 the ratio of these manses is @. Ushg
this value we find
7Z = -0.26CA - 0.045Nf T(R) - 0.12C~. (3.13)
In Table 3-1 we provide numerical d u e s for the expressions in (3.12) and (3.13) for Nc = 3 and where
either Nf = 3 or Nf = 16:
The second entry in Table 3.1, where the gauge boson is massless and the number of fermions is large,
is closest to the case considered in Reference [49]. We found a larger contribution than the authors of
Merence [49] did, which may be due to the fact that we did not make the wumption that K2 decreases
as min(l/8, l/q2) away fkom p = q.
Note that the value of 72 is fair1y large in the massless case but it becomes smaller and eventually
negative as the number of fermions is increased. The value of R starts off negative in the massive case and
becomes even more negative when the number of fermions is increased. Fûrthermore the magnitude of 'R
in the massive case indicates that the NLO term dominates over the leading order term. The results of our
NLO anaiysis show that in the presence of a nonzero gauge boson m a s the NLO term becomes as large as
the leading order term with the opposite sign.
3.3- ltESULTS AND D I S C U ' O N 37
if any d e n c e couid be given ta this obsenmtion by itself, it would indicate that what would be the
most attractive chamel at leading order becornes the mast repuisive Channel at the NLO and vice versa.
In actual fact this impiies the breakdm of perturbation theory- Hence identifving the most attractive
Channel on the basis of the leading order contribution to the kernel function (i.e. using the value of the
Channel factor) is misleading.
The analysis that is performed here resembles that of a purely vectQr gauge interaction- It is known
[50] that in such a case the gauge symmetry cannot break itself. However, additional gauge interactions
may turn this into a chiral theory. The analysis also applies to situations where the source of the qauge
boson mass is something other than the strong dynamics in question,
Assiiming that the gauge interaction does break itself, we at least have a situation that is consistent
with the appearance of a massive gauge boson- It is for such a massive gauge boson that the perturbative
expansion seems to be le& reliable.
The problem of gauge dependence also plagues the usuai analysis, but there is no reason to expect that
the additional contributions present in a gauge invariant treatment would cancel those that we have found-
Our results are SuffiCient to caU into question the use of the MAC hypothesis as a test of whether or not
gauge symrnetries break. More powerful techniques are needed to study the symmetry breaking pattenis
of interest for the construction of realistic theories of mass and flavor,
Our exploration of the effects in the NLO kernel indicates that there is iittle reason to beIieve the
leading order most attractive channel hypothesis- Although this result runs counter to conventional wWlom
concerning MAC, it is not terribly surprising - when the coupling is large, higher order d e c t s can be
important. In Chapter 9 we present a mode1 [13] for the dynamitai origin of quark and lepton masses,
which relies on breaking patterns that do not foiiow the MAC hypothesis.
Chapter 4
4-point funct ions via gauge exchanges
More than three decades ago De Dominicis and Martin published a set of papers [21] in which they provide
procedures for the derivation of gap equations and e f f ' v e actions for various n-point Green functions-
Alt hough their analysis was presented in the framework of non-relativistic statistical physics the procedures
can be readily applied to quantum field theory. This was already shown many years ago by Cornwall, Jackiw
acd Tomboulis [22] for the 2-point case. Although they used a mecent procedure to obtain it, the famous
C JT eEective action is precisely the quantum field theory generalization of the De Dorninicis-Martin effective
action for 2-point fimctions. We follow the De Dominicis-Martin p r d u r e for &point functions to derive
the effective action and gap equations in the context of gauge theories with massless fermions.
In this chapter we only use the part of their analysis deaihg with &point functions. There are two
parts to the De Dominich-Martin procedure, which can be regarded as two independent ways to derive
the required gap equations. One part provides a dùect derivation of these equations. The other part is a
derivation of the efkctive action, fiom which one can then reproduce the same gap equations by imposing a
stationarity condition. Although the direct method is perhaps a quicker method to obtain the expressions
for the gap equations, it is the other method which actually makes a statement about the vacuum of the
theory by showing that these gap equations extremize the &&ive action.
The aim of this exercise is to determine whether a 4point function can act as a fundamental order
parameter for (partial) chiral symmetry b r e g . For this to be the case the &point function under
consideration must be a non-singlet of the chiral symmetry. While a 2-point function cannot be a singiet
under a chiral symmetry a 4point function in general can be such a singlet and it may be argueci that
nature would always prefer the singlet over any non-singlet.
It ttunis out that if one distinguishes the difFerent chiral structures that a 4-point function can have, one
h d s that some of these chiral structures carmot be a singlet. Because chirality is conserveci in a massless
theory these chual structures would retain their identity. Furthemore, one fin& that iinlike the 2-point
function some of these non-singIet 4-point functions can stili be invariant under a chiral isoepin symmetry-
This is obvious1y relevant to the electroweak m e t x y We t h d o r e resfnct ourseives to this case where
the chiral symmetry is only partially broken down to a chiral isospin m e t r y -
To analyze this order parameter one needs to make the Chvality expliut in the expressions of the &ective
action and the gap equations. This is responsible for the complications in the derivations presented in this
chapter. This work appears in Reference [27j-
4.1 Definitions and notation
4.1.1 The &point effective action
We start fkom the CJT &e&ve action in the ptesence of 4fermion sources,
as deriveci in Section 2.3. For the purpose of the discussion in this chapter we assume a general masIess
propagator,
As a result the first two nondynamical terrns in the CJT effective action are iInimportant. We therefore
drop them and only consider the dynamical term WZpIM- This term, which consists of all connected 2PI
vacuum diagrarns, will be treated as a generating functional fkom which we derive the dective action for
4point functions, We henceforth drop the subscript and refer to WM. The 2PI vacuum diagrams are
constructed in terms of the usual Feynman rules of a gauge theory, with the exception that fermion lines
represent the (massless) fuii propagator and the nonlocal 4fermion source is an additional 4-fermion vertex.
The absence of a fermion mass aliows us to distinguish five different 4point Green functions on the
basis of their chiral structures. For this purpose we replace the usual &fermion source term in (2-5) with
the foiiowing five noniocal &fermion source terms:
The sources with the subscript C are associated with chirality changing Green functions, while the other
three sources are chirality preserving. Four of the source terms have the same chiralities on both $-fields, as
well as on both &fields. Thus the symmeetry associated with the identicai incoming pair and the identicd
outgoing pair introduces the symmetry factor of f . The color and flavor indices of the fermion fields are
contracted on the sources. We often keep such indices implicit to simplify notation.
The 4-point Green fuoctiom are given by the functional derivatives of the generating functional with
respect to the appropriate sources,
Both the sources and the Green functions carry implicit color and &vor indices. The [Jj is a shorthand
for the functional dependence on ail five sources.
The CJT effective action was obtained from the original generating tunctional in Section 2.2 thtough
a first Legendre transform which replaced the dependence on the 2-point source K with a dependence on
the full propagator S. Now we introduce the second Legendre transform
which gives the &point &ecfive action. This Legendre transform replaces the dependence on the five
4point sources with a dependence on the five &point Green functions.
4.1.2 The amputated &point functions
It is more convenient to write the effective action as a frtnctional of the omputated parts of the 4-point
Green functions. These objects are denoted by the symbol C and we SM henceforth refer to them as
&point functions to distinguish them from the Green functions, G's. So a &point function is given by
while the Green function is
The functionai dependence of the efkctive action on the Green fmctions is replaced by a functional depen-
dace on the 4point functions:
wl= wl- (4-8)
The five 4point functions, C's, are defined as folIows (see Figure 4.1). When ail four of the extemal iines of
a 4-point function have the s a m e chirality, say left-handed (right-handed), w e denote the 4point function
by CL (CR). These two 4point functions are chirality preserving. The only other chirality preserving
Cpoint function, which we denote by Co, has one incoming and one outgoing line of each chirality. The
remaining two 4point functions are chirality changing. Their incorning and outgoing lines have opposite
chiralities. When the Chuality of their incoming lines are left-handed (Bght-handed) we denote them by
cc cc,$,.
Figure 4.1: The five connected, amputated 4point functions
The 4point &&%ive action obeys s t a t i o d t y conditions with respect to the Green functions, which
are of the form
However, because the full propagator is kept Gxed, these stationafity conditions are equivalent to similar
stationarity conditions with respect to the &point functions:
mcl -= 0, - 61'[q - - mc] - -0, -=O wq and -- - O. WC] 6C& acc m o CR &CL
The derivation of the effective action involves finding a diab-atic representation in which the five terms
that are removed through the Legendre t rdormat ion are the oniy ones with explicit sources. AU other
source terms end up inside subdiagraxns which are replaceci by explicit C's, so that we have a formula for
I' that contains no reference to J.
4.1.3 Types of fermion pairs
All the &point functions considered here (that is, di the C's, U's and T's, which are defineci below) are
amputated. Therefore, when two of these 4point fwictions are connected by two fermion lines, two fermion
propagators are to be inserted between the two 4point functions. We s h d leave this step implicit in our
notation, and so really means C~SSC&. Since S is chllality preseMng the product of two 4point
functions is only possible if the directions and chualities of the fermion lines that are to be connected are
compatible. This may depend on the type of 4point function (a Cc can for example never by directly
connected to another Cc) or on the type of fermion pairs.
We define four types of fermion pairs depending on the directions and chiralities of the fermion lines
that co~mect two adjacent 4point functions. Each type of fermion pair is denoted by two letters. The first
is associated with the directions and the second with the chiralities of the fermion lines. We use s and d to
denote 'same' ami 'different' respectively. The subscripts ss, sd, ds, or dd WU appear on objects or brackets
to indicate that the objects or the enclosed expressions have a specific type of fermion pairs. These fermion
Figure 4.2: Four types of fermion pairs that can connect 4point functions-
pairs are iilustrated in F i e 4.2- Often only one of these letters will appear in the subscript. In such a
case the letter denotes the directions and not the chiralities, with the latter king clear from the context.
4.1.4 Simple &point functions
A 4point diagram is said to be simple with respect to a partidar painng of extemal lines when it cannot
be separated into two parts, each having a pair of the original extemal lines, by cutting two interna1 fermion
lines in the Ppoint diagram. We denote the sum of all 4point diagrams that are simple with respect to
one specific pairing of extemal Iines by U and refer to it as a simple 4point function. We use T to denote
the sum of all &point diagrams that are simple with respect to all but a specific pairing of external lines.
Then the Ppoint function, C, is the çwn of a U and a T associated with a specific pairing:
(For a while we shaü not distinguish the different chiralities.) There are three ways of pairing off four
extemal lines. One can therefore define three U's and three T's- Any particular &point diagram is either
non-simple with respect to only one of these pairings or it is simple with respect to aü three pairings. It
thus foiiows that C can be written as the sum of three T's (one for each possible pairing) and K, which is
the sum of all diagrams that are simple with respect to aU pairings:
The T's are the various bubble chains that can be resummed in c l d form.
Once we distinguish between left-handed and ngbt-handed fermions we obtain the five C's defineci in
Section 4.1.2 (see aho Figure 4.1). The chirality preserving 4point functions are Co, CR and CL and the
chirality changing Ppoint hinetions are Cc and CC-
CHAPTER 4. PPOINT FUNCTIONS VIA GAUGE EXCHANGES
Figure 4.3: The definition of the simpIe &point functioas (U's) in temvl of the pairings and chiralities of
the external Lines. The four groups indicate the four dinerent types of fermion pairs.
Each simple 4point function is defineci with respect to one type of fermion pair. The four categories of
U's are shown in Figure 4.3. In m a r y : for the s s - f b o n pairs we have W., a, U: and U'; there is
only one for the sd-fermion pairs, namely q; those for the d s f d o n pairs are UA, ~f and LI:; and the
ddfennion pairs are Ud, U: and a. Note that Ui and appear hice each in Figure 4.3. There is a T
for every LI. They are related by (4.11) for each speafic pairing of the extemal iines of the partidar C.
It will be useful to d&e a set of bubble sumsa that are formed as the sum of sequences of the same U
or C. There are only some of these U's and C's that can form such sequences. The bubble sums invoiving
C'S are
Those involving U's are
Although one can also define such an object for the sdfermion pairs, it does not appear often enough to
merit its definition. The factor of that appears in (4.13) and (4.16) removes the overcounting due to the
aNote that the first term in these 'bubble surns' is the identity element- One can think of th& identity elemeat as an
operator that implies a direct connection of whatever is on either side of it, when placed inside another diagram.
4.2. DIRECT D W A T I O N OF THE GAP EQUATIONS 45
identical fermion lines in the cases with ssférmion pairs. When we encounter this symmetry again it will
be referred to as the ss-~ynmetry,
There are other types of transformations that play an important role in some of the derivations of the
subsequent sections, especidy in Section 4.4. For this purpose we define the following trandormations,
wbich are denoted by superscripts, Assume that G is an object with four extemai lines paired off as (1,2)
and (3,4). Then one can define the following three fransformations:
The last one is r e f ' to as a back--fiont transformation.
4.2 Direct derivation of the gap equations
In this section, foilowing the procedure of [21], we present a derivation of the gap equations for the &point
functions. This derivation comprise of finding expressions for the T' s and K's that form the C's accorciing
to (4.12). We do not need the 4fermion sources for this derivation, they will appear again in Section 4.3.
W e start by considering the T-terms for which one can write doseci form expressions-
4.2.1 The T-terms
4.2.1.1 The master equations
The expressions for the T-terms are derived with the aid of muter equations. Each master equation
expresses a 4point function (C) in terms of some simple 4point function (U) and itself, and leads to an
expression for the U in terms of the C. The T-term then follows as the clifference between the C and this
W. We have four types of fennion pairs, three of which have four master equations each. The remaining
fermion pair type has ody one master equation, and we start our discussion with this equation.
The only C that can have the sd-fermion pair type Co. The associateci simple &point function is
and the master equation is (see Figure 4.4):
Rom (4.22) it then directly follows that
The T-tenu in the gap equation for Co is now given by
CHAPTER 4- 4-POINT FUNCTIONS VL4 GAUGE EXCHANGES
Figure 4.4: Master equation
For the other three Srpes of fermion pairs we fùst present aii the master equations. The master equations
for the 8 s - f d o n pairs are:
The factor, 5, is due to the s+symmetry mentioned in Section 4.1.4. Nad we have the master equations
for the &fermion pairs:
The first two equation, (4.29) and (4.30), and the last terms in the last two equation, (4.31) and (4.32),
may seem ambiguous. Note, however, that corresponding external Lines of different terms in an equation
must have the same directions and chiralities. As a result one can see that (4.29) and (4.30) are related by
a bd-to-front transformation, as defined in (4.21) and that the fermion lines which connect Co and UL
are left-handed in (4.31) and right-handed in (4.32).
The final set of master equations are for the dd-fermion pairs:
The last term in each of (4.35) and (4.36) define the chirality assignments on the extemal lines and thence
the definition of the other terms foiiow unambiguously.
4.2. DIRECT DERNATION OF THE GAP EQVATIONS
4.2-1.2 Simple 4-point fiinctions
The master equations are manipulateci to give the U's expressed in terms of only the C's. The expressions
of the U's are required for the pair cycle terms of the dective action which are derived in Section 4.3.2.
Fkom the tweive expressions in (4.25) - (4.36) one can write down simpler expressions using the definitions
in (4.13) - (4.18). F i we consider the s s f d o n pairs in detail. W1th the use of the definitions in (4-13)
and (4.16) the expressions in (4.25) and (4.26) become
Using the same definitions one can express the ne& two equations, (4.27) and (4.28) as
and
where we have used the identities in (4.37) and (4.38). Rom (4.39) and (4.40) one can now mite down the
expressions for u'. and U::
II' and
One can eiiminate RF between (4.38) and (4.39) to h d an expression for a,
and similady one can elimuiate L: between (4.37) and (4.40) to find an expression for Us,
In the derivation of these equations we have used the identity
The &fermion pairs and d6fennion pairs foiiow the same steps. The equivaient expressions for the
&fermion pairs are:
Rom these we derive the expressions for the foliowing U's:
The last m o equations, (4.52) and (4.53), are tefateci through a back-to-kont transformation- The expres-
sions for the dd-fermion pairs are:
and fkom them foilow the expressions for the remabhg U's:
Here the Grst two equations, (4.58) and (4.59), are related through a back-tdiont transformation.
4.2.1.3 T-term expressions
The final step in the derivation of the T-terms in the gap equation is to use equations of the fom T = C- U
to find their expressions- The expressions for the U's are those provided in (4.23), (4.41) - (4.44, (4.50)
- (4.53) and (4.58) - (4.61). This step is quite straight forward so we merely quote ail the relevant
expressions. ssfermion pairs:
4.2- DIRECT DERIKATION OF THE GAP EQUATIONS
&fermion pairs:
4.2.2 4particle irreducible diagrams
The K-term contains al1 4-point diagrams that are simple with respect to di pairings of externa1 lines.
We want to express th& s u of diagrams in terms of the C7s. This means that we have to recast the
set of diagrams in the K-term such that ail the 4poinf subdiagrams inside these diagrams are containeci
inside C's. This process is neatly performed by constructing aii the 4particle irreducible (4PI) diagrams,
usïng the C7s as 4-fermion vertices, together with the full fermion propagator and the other vertices and
propagators of a gauge theory. F i t we give the forma1 d a t i o n of a 4 . 1 diagram. Then we dari& this
definition-
The 4PI 4point diagratm are defined by first def'ming 4PI vacuum diagrams. The former are then
obtained from these vacuum diagrams by removing one 4fermion vertex. The definition of a 4PI vacuum
diagram uses the concept of a pair cycle, which is a loop of pairs of fermion lines that connect simple
subdiagrams to form a vacuum diagram. An example is shown in Figure 4.5. A pair cycle with p pairs is
referred to aç a p-cycle. Cutting the lines of aii p pairs in a p-cycle one separates the vacuum diagram into
p discomected simple 4-point diagrams. If one and only one of the simple subàiagrams in a p-cycle is just
a 4-fermion vertex, the p-cycle is said to be a trivial p-cycle.
C&APTER 4- PPOlNT EZfNCTIONS VIA GAUGE EXCHANGES
Figure 4.5: A pair cycle with eight fermion - i.e. an 8-cyde. The shaded rectangles denote simple
subdiagrams-
The formal definition of a 4PI vacuum diagram is as follows: dl the pair cycles in a IPI vacuum tiiagrom
are either n ~ n t ~ u i a l 1-cycles or ~uial2-cycles- We deviate slightly fiom the definition in [21] in that their
definition include trivial 1-cycles as 4PI diagrams. The only diagram with a trivial one cycle is shown in
Figure 4.6a This diagram has the undesirable feature that, upon removing the 4fefmion vertex, it gives
the discomecteci propagators which are excluded from C's definition as shown in (4.6).
Let's cl- this definition by considering the reqdrements for 4PI diagrams and giving motivations
for these requirements. For 2PI vacuum diagrams one simply requires that it should not be possible to
separate the diagram into two parts by cutting two diBetent fermion lines in the diagram. Generalizing
this d e to four fermion hes for 4PI diagrams is not enough, because then no 4PI diagram would contain
a 4 fermion vertex, So one must add that if by cutting four different fermion lines in a diagram it separates
into two parts with one and only one part being a 4-fermion vertex, then the diagram is stiU4PI. Thus we
d o w trivial twtxycles.
Figure 4.6: Three vacuum diagrams that are not 4PI- The shaded circles denote &fermion vertices (J's or
C's) .
A vacuum diagram that consists of two C's with their lines connecteci to each other, as shown in
Figure 4.6b, produces the 4fermion vertex upon removal of one of the C's. The latter 4point diagram is
by definition not part of K, and thus The diagram in Figure 4.6b is excluded fkom the definition of triviai
4.2. DIRECT DERNAZION OF THE GAP EQUATIONS
We could have triecl to define 4PI vacuum diagrams as al1 vacuum diagrams that cannot be separated
into two disconnected parts by cutting four fermion lines, uniess one and only one of the parts is just
the 4fenaion vertex. However, this d&tion stiU includes the diagram consisting of three C's, shown in
Figure 4.6~. This diagram must be excluded because it becornes a diagram in T when one removes one of
the C7s. Thus we see why the definition of 4PI vacuum diagrams must exdude aii diagrmm with 3-cycles
and higher. Hence the formal definition as provided above.
One can now obtain al1 the 4PI 4-point diagrams fiom the 4PI vacuum diagrams by cutting out (or
opening up) one 4ferrnion vertex. As an example: if a 4PI vacuum diagram contains five &fermion vertices
then one can form five 4PI &point diagrams h m it, ff two or more of these 4PI &point diagrams are
identical the original vacuum diagram would have a symmetry factor which cancels the overcounting.
The amputateà 4PI Cpoint diagrams that are constructeci from the C's, the full fermion propagators
and the other vertices and propagators of a gauge theory, are exactly those which are contained in the K-
term. To see this, remember that the C's represent the sets of al1 connected amputated 4-point diagrams
with the appropriate chiralities on their externai lines. By replacing the C's inside the 4PI diagr- with
these sets of &point diagrams, one generates all the 4point diagrans which are simple with respect all
pairings of external l ine~ .~ Amputating these simple diagratm, one obtains the set of diagratm that forms
K.
Eere W ~ P I [ ~ denotes the sum of all4.I vacuum diagrarns containhg the 4point functions C as 4-fermion
vertices.
4.2.3 The gap equations
The expansions for the five 4point functions in terms of the T's and K's are:
Note that there are dways two of the T7s with subscript d and only one T with subscript S. The superscript
T, defineci in (4.19) indicates that the two incoming or two outgoing lines are interchangecl. There is only
b ~ o t e that, aithough a 4PI +point diagram implies a diagram that is simple with respect to ail *rinp of extemal lines,
the converse is not necesearily true.
52 CHAPTER 4- 4-POINT FUNCTIONS VLA GAUGE EXCEANGES
one of the equations where the two T's with s u W p t d are diffetent - the equation for Co. The distinction
is that for the chiralities in a pair are opposite and for Ti the chiralities in a pair are the same. (This
is the same as for @ and U& shown in F i e 4.3.)
When distinguishing the chiralities, one has five difterent &point functions on which WIPI depends.
The K-term in each specific expression in (4-75) - (4.79) is generated by taking the functional derivative of
Win with respect to the appropriate 4point function. (For K (Kt) one must take the functionai derivative
with respect to Ct (C).) The expressions for aii the T's are provided in (4.24) and (4.62) - (4.73). These
expressions, together with the defmitions in (4-13) - (4.15), now complete the gap equations in (4-75) - (4.79).
Next we derive the eftective action, which leads to the same gap equatioas.
4.3 Derivation of the effective action
Here an expression is d e r i d for the effective action in terms of the 4-point hinctions (Cs)- This derivation
requins the use of the 4 - f d o n sources, J's, which are shown in (4.3).' We start with the generating
functional, W[JI, defined in Section 4.1.1. It consist of al1 the connected 2PI vacuum diagrams that can
be generated with these sources treated as 4fermion vertices, the massless full fermion propagator, and
other vertices and propagators of a gauge theory. The dective action is obtained by performing a Legendre
t r d o r m , which replaces the source dependence by a dependence on the &point functions. The challenge
is therefore to express W in terms of C's, which only implicitly depend on the J's.
The problem is to avoid overcounting when formiag vacuum diagrams containhg the C's. This problem
is solved in [21] by adding and subtracting various sets of diagrams in such a way that the overcounting is
eiiminated and the original sum of vacuum diagrams is retained. In the sections below we show how this
is achieved with the aid of a topological equation [21]. This topological equation is the 4point function
quivalent of the topologicai equation for 2-point functions (2.15) encountered in Section 2.3.
4.3.1 Topological equation
The topological equation,
which is proven in [21], relates the numbers of certain elements or features (such as lines or pair cycles)
that are present in vacuum diagrams. The quantities on the right-hand side, which we shaU define shortly,
indicate the number for the dinerent elements. =In [21] nonlocal vertices, denoted by v's, are used to repreeent dl interactions in the theory, We replace the v's by sources,
J's, and keep them separate fiom the gauge interactions in our analysis-
4.3. DERNATION OF THE E2?FECTIVE ACTION 53
One can use this topo10gical equation to construct a riesummatim equation for the sum of al1 vacuum
diagrams, similar to the way it was done for 2-point functions in (2.16). The resummation equation is
Each terni on the right-hand side of this equation is formed by counting each vacuum àiagram as many
times as the number of the corresponding elements that appear in it. Adding the terms according to (4.81),
one recovers the sum of ail vacuum diagrams, W. Sïce each of the terms on the right-hand side has an
unambiguous expression in terms of C's, one can write the sum of al1 vacuum diagrams in terms of the C's,
using this resummation equation
This general procedure is now applied to the case where one disfinguishes the chiralities 03 the fermion
lines- Care must be taken to easure that any overcounthg that may result fkom the symmetries associatecl
with the chiralities is removed. Except for the Co, aii &point functions have the same chiralities on both
incoming, as well as on both outgoing f h o n lines. There is then an snsoQated symmetry faetor of ), as
revealed in the expression in (4.3)-
Wé first consider the first three terms in the topological equation, while the last three terms are discussed
in Section 4.3.2. The vacuum diagram shown in Figure 4-7, is used as an example. The number of times that
the associateci element appears in this diagram, is determineci for each term in the topologid equation.
Figure 4.7: A vacuum diagram that is not 4PI.
A 4PI skeleton is what remains of a 4PI vacuum diagram after one has cut out ail the 4fennion vertices-
A vacuum diagram that is not 4PI does in general contain 4PI skeletons. In (4.80) counts the number
of such skeletons. The two gauge exchanges in our example are the only 4PI skeletons, because in each
case, if one replaces the rest of the diagram with one 4fermion vertex, the result is a 4PI diagram. By
doing this replacement on any other 4point subdiagram one h d s that there are no other 4 . 1 skeletons.
So for our example we have Ngkcr = 2-
The term, W,tcr, is generated by the sum of aii 4PI vacuum diagrams, W ~ P ~ M , where aii possible
connecteci Ppoint diagrams (those inside the C's) replace the 4fermion source vertices, J's. This gives
The C's now play the role of 4fermion vertices, in place of the J's.
CaAPTER 4. 4-POïNT FUNCTIONS VIA GAUGE EXCEANGES
The number of articulateci quartets, Nae, counts the nwnber of sets of four fermion Iines in a vacuum
diagram that one can cut to separate the vacuum diagram into two disconnected parts- There are three
such sets in our example- Two sets connect the tum gauge archanges to the rest of the diagram and one
set connects the 4fermion vertex to the rest of the diagram- Thus we have Nad = 3 for o u . example-
One can generate the term, Wad, for the set of aU vacuum diagr- by closhg off the Lines of one
4point diagram by those of another &point diagram for aii possible connectecl 4poht diagmms- The
result is just the trace over the product of two C's,
The symmetry lactors appear as follows: if the term is made of4point fimcti*ons that have identical incoming
or outgoing ünes then the same fytor of $ which appears in (4.3) accompanies this term - this L the case
for the first t h terms in (4.83); if the two &point functions in a term are the same (as in the last three
terms of (4.83)) we need an extra fhctor of a. The third term in (4.80), NVed, is just the number of 4fermion vertices in a vacuum diagram. Our
example contains only one such vertex, so Nved = 1.
The tena, is generated by closing the four lines of ail unamputated 4point diagrams (including
disconnected ones) off with the appropriate source J, which gives a trace over the product of a J with the
Green function, G. This generates the five source terms that are removed by the Legendre transform, (4.5):
Here we again have factors of f which appear due to identical incoming and outgoing lines.
4.3.2 Pair cycles
A pair cycle is a loop of pairs of fermion lines that connect simple subdiagrams to form a vacuum diagram.
(See Figure 4.5.) The sum over pair cycles makes up the last three terms in the topological equation (4.80).
The first term in the sum, C 1, just counts the number of such pair cycles in a vacuum diagram. The next
term, C Np, counts the number of pairs in each pair cycle of the vacuum diagram and the third term in
the sum, Np,,, counts the pairs of pairs in each pair cycle of the diagram. One can see that for 1-cycles
and 2-cycles these three terms add up to zero. Therefore it is only for pair cycles with three or more pairs
that these terms make a nontrivial contribution to the topological equation. In our example there is only
one pair cycle with more than two pairs and it is a 3-cycle. This gives x ( 1 - N, + Nw) = 1 - 3 + 3 = 1
for our example. Adding al) the nwnbers for our example according to (4.80) then gives 1 as it should.
One can define the fourth term in (4.81), WC, as the trace of a power series of simple 4-point functions
(U's) with the appropriate symmetry factors. A product of n simple 4-point functions forms a chah of n
fermion pairs and the trace of this product closes off the chah into a pair cycle. The next term Wp can be
4.3. DERZXATION OF THE EFFECCTW3 ACTION 55
generated fkom the previous one W;, by induding a parameter g with each U and then taking the derivative
with respect to g. This counts the number of pairs in each pair d e , and the second derivative gives twice
the number of pairs of pairs. Half of the latter gives the last tenn, W& Since the U's can be expressecl
in terms of the 4point functions (C's), this procedure leads to the required expressions for the three pair
cycle terms. The pair cycles found in our analysis are more complicated than those in (211 because there
are several ways to fom them due to the distinguished chikatities.
We foilow the a p p d of Section 4.1.4, which is to divide the difilerent U's into four groups depending
on their types of termion pairs. The pair cycle terms are expressed as
for each type of fermion pairs. The aim is to find d the dinerent pair cycles by identifying all the U's
that can form such cycles. Some U's c m form pair cycles on their own. The two pairs of extemal lines on
these U's are compatible, which implies that two of these U's can be interconnectecl. The other U's have
incompatible pairs, Although they cannot form pair cycles on their own, they can be wmbined with other
U's to form a group of U's that can be used to fonn pair cydes. Such groups of U's are denoted by M's-
We shali discuss each of the types of fermion pairs in turn, starting with the simplest case-
The dfermion pair has only one simple 4point function: Ut. There are no simple &point function groups,
M's, for this type of fermion pair. The pair cycle is formed by dehing the foliowing quantity:
Setting g = 1 in (4.86) one k d s the sum of aü pair cycles with this type of fermion pair. The nmber of
pairs per cycle is obtained by taking the derivative of Az(g) with respect to g, whiie a second derivative
gives twice the number of pairs of pairs- The sum of pair cycle tenus, (4.85)' with dfermion pairs are
then given by
Fkom (4.22) one can show that
Using (4.88) and (4.23), (4.87) becornes
56 CaAPTER 4. #POINT FUNCTIONS VIA GAUGE EXCaIANGES
This is similar to the d t in [21]. The subsequent cases are al1 treated according to the same steps, For
each pair cycle one can define
where the subscrïpt f denotes the specific type of simple 4point function or simple 4point function group
for that pair cycle. Some manipulations are necessary to get these expressions into a form that only consist
of C's.
4.3.2.2 sefermion pairs
There are two simple 4point functions that can directly form pair cycles in the sefermion pairs: U ' and
u:. There is aIso a simple 4point function group that can form pair cycles:
Here the pair cycles are formed by
where M(g) denotes M.@), g ~ ' or gU'. The factor of ) is necessary to remove an overcounting which
occurs as a result of the ss-symmetry as discussed in Section 4.1.4. For U: and U: the expressions are
only slightly more complicated than in the s&. We define separate quantities:
and
+' 8 [l- flif];Li: [l- fUf]yU.L)
Ekom (4.39) and (4.40) it can be shown that
and
4.3, D W A T I O N OF THE EFFECTNE ACTION
Using (4.95) and (4.41) one fin& that (4.93) becomes
Similady, fiom (4.96) and (4.42), (4-94) becornes
Due to the intricate g-dependence in (4.91) the pair cycle expression that contains M, is more complicated:
Here we used the identity d 1
-B = - ( B ~ - B ) , (4.100) 47 9
where B = 11- gA]-l for a &point function, A. Using the identities in (4.37) - (4-40) and (4.45)' one can
show that
Now one can add aii the
a s =
- -
a ' s for the s s - f d o n pairs:
The resulting expression is relatively simple thanks to extensive cancellations among the Werent H's.
The remaining two types of fermion pairs foliow exactly the same procedure and the expressions that
are obtained are a h very similar to these- W e quote di the devant expressions without unnecessary
discussion.
CKAPTER 4. 4POINT FUNCTIONS VLA GAUGE EXCX!Uh*GES
4.3.2.3 &fermion pairs
The two simple 4-point functi011~ that can directly form pair cycles in the &fermion pairs are U2 and ~2 and the simple &point function group th& can form paU cycles is
The pair cycles are formeci by
where M@) denotes M*(g), gUf or gU,f. There is a s)mmetry with respect to a kont-tdxdc t r d o r -
mation of these pair cycles. The factor of ) is necessary to remove the overcounting which occurs as a
result of this symmetry. For U: and we haw the fouowing expressions for (4.90), using (4.104):
and
One can show, fiom (4-48) and (4.491, that
and
[l - U:]d1 U , = CL - CoRCo.
Using (4.107) and (4.50) one fin& that (4.105) becomes
Similarly, from (4.108) and (4.5l), (4.106) becomes
4.3. DERNATION OF THE EFFECTNE ACTION
The pair cycle expression that contains Md. is:
where we again used the identity in (4.100). The identities in (4.46) - (4-49) lead to
Now one can add aü the E's for the dfermion pairs:
dere too we have a relatively simple expression as a result of cancellations among the different H's.
4.3.2.4 dd-fermion pairs
There is only one simple Ppoint function, @, which can dïrectly form a pair cyde in the dd-fermion pairs-
The simple 4poi . t function group for this type of fermion pairs is
Because looks different under a fkont-to-back transformation we s h d use it twice. These two pair cycles
are then related through a back-to-fiont transformation and each must be multiplied by ). The pair cycie
for the simple 4point function group of (4.114) has a front-teback symmetry. The pair cycles are therefore
formed by the expression in (4.104) where M(g) denotes &f&(g) or g q .
Substituting U: into (4.90), using (4.104), gives
1 1 4 = n { - 2 ~ ( i - ~ ) , - 2 [ i - ~ ~ ; ' ~
(4.1 15)
Rom this expression one can use two diffeteat sets of expressions to get two different results which are
related by a back-tefront transformation. F i one can use (4.55) and (4.56)' to show that
or one can use (4.54) and (4.57), to show th&
Substituting (4.116) and (4.59) into (4.115) one hds
and substituting (4.117) and (4.58) into (4.115) one h d s
where we placeci a t on the H in the latter expression to show that the chiraüties on the fermion lines are
interchanged. The pair cycle terms that contain Mda are:
where we used the identity in (4-100). The expressions in (4.54) - (4.57) then gives
Adding al1 the H's for the d&fermion pairs, one fin&
4.3.3 The effective action
Now one can constnict the expression for the pair cycle terms of the dective action (4.85) by adding the
expressions in (4.89), (4.lO2), (4.113) and (4.122)
4.4- GAP EQUATIONS VL4 THE EFFECTIVE ACTION
One can then combine the different parts in (4.82), (4.83), (4.84) and (4.124) to construct the generating
functionai, W - Le. the set of di connected vacuum diagrams- Next one can perform the Legendre
kansform, (4.5), which removes (4.84) to produce the expression of the full &&ive action. With ail the
explkit J-dependence removecl, the &&ve action becornes a fuactional of the &point functions:
This result for the effective action is the main result of this chapter. It c=an be applied to arbitrary models,
and it provides the means to determine wbich solution of the gap equations of Section 4.2 represents the
true vacuum of the theory-
4.4 Gap equations via the effective action
The gap equations for the 4-point functions (4.75) - (4.79) are obtained by taking the functional derivatives
of the effective action (4.125), as in (4-IO), and then amputating the resdt. A key step in this process
is to take the functional derivatives of the pair cycle terms which are provided in (4.89), (4.102), (4-113)
and (4.122). It is necessary to take special care of the assignments of external lines that result from these
functional derivatives- For the case of Co this is just
where we use 1,2,3,4 and a, 6, c, d to distinguish extenial fermion Lines. For the other four cases, Cc, CA, CR and CL, we have
The latter functional derivative gives rise to four terms. Often these fout terrns may be related to each
other through some interchanges of external lines. These transformations are defineci in (4.19) - (4.21).
62 CHAPTER 4- 4-POINT FUNCZ7ONS V U GAUGE EXCEANGE$
Next we calculate the functional derivatives of the H's in (4.89), (4.102), (4-113) and (4.122) with
respect to Co, CC and CR. Those of Cc and CI are relateà to these in an obvious aay. In each case we
shail h d that the functional derivative of the pair cycles of a specific type of fermion pairs with respect
to a specific 4point h c t i o n reproduces a specific T (or more t h one if they are related by one of the
transformations in (4.19) - (4.21)).
We start with Co. Ody the pair cycles with the s&, & and ddfermion pairs depend on Co- Their
functional derivatives are
The RF transformations shown in (4-129) and (4-130) are equd to the RF transformations applied to each
individual object in reversed order in each of the tenns- This reproduces the three T's that appear in the
gap equation for Co, (4.24), (4.66) and (4.73).
Next we eonsider CC- Only the pair cydes in the ss- and d&fermion pairs depend on CC:
The (x4) cornes from the ss-symmetry which generates four identicai terms. The (1 + lm + lT + lRFT)
denotes different exchanges of externa1 lines. Because the diagram is symmetric under the combined RF-
transformation, the expression reduces to the final expression with a factor of 2- Thus we reproduce the
three T's that appear in the gap equation for Cc, (4.62) and (4.70). The gap equation for CC ïs quite
similar. One merely interchanges the chiralities to get the necessaty expressions.
4.5. GAP EQUATfONS IN THE LARGE N' LIMlT
For CR only the * and &fermion pairs need to be c ~ n s i d ~
1 -1
= c. + ZR, [l - -&L,C~&] 8 - 2 = TA
So haüy we reproduced the three T's that appear in the CR gap equation - (4.64) and (4.68). Those for
would be identical apart fiom replacing R with L.
Next we find the fundional derivatives of Wad with respect to the various Ppoint fiinctions.
mvt -- mart m a r t -Co, -=CR mart and -- m a r t -cc, -=ch, -- ~ C C ~ C O ~ C R &CL
- CL* SC&
Finally the fundional derivatives of Waka with respect to the various 4point functions gives rise to the
4PI 4-point diagrams which appear as the K-tcnns in the gap equations:
r n s k c l -- -= cl m a k t 1 m a r t e l m a k c l - Kc7
-KR and - - -Ko, -- KA. =- ~ C R ~ C L - KL-
SC; (4.136)
Coilecting the diffefent terms for each specitic 4-point function accordhg to (4.81)' using (4.85) and the
fkst line of (4-123), one reproduces the gap equations exactly as in (4.75) - (4.79). The stmcture of these
equations is Merent fiom what a naive extension of the CJT analysis would suggest, and this is made
evident in the foUowing section where we solve the gap equations in the large Nc huit,
4.5 Gap equations in the large Nc limit
In this section we use the dective action and gap equations preseated in the previous sections to extract
information about the 4point functions. Consider an SU(N,) gauge theory with NI fermions in the
fundamental representation. We use a large N, expansion [28] to simpiify the structure of the gap equations,
Knowledge of the leading N,-dependences of the C's can be obtained from the sequences of &point fundions
of the form [If Cl-' that appear in the gap equations. In these sequences adjacent C's are interconnected
with two fermion lines, which can form color loops, and each color Ioop gives a factor of Nc- In the large
N, limit these sequences would only be convergent and nontrivial if the C's themselves are of 0(1/N,).
This is true for ail types of C's.
VIA GAUGE EXCHANGES
Figure 4.8: Cdor structures of Co.
The next step is to separate each of the different C's into color stmctures. If the color indices on the
extemal lines of C are a, b,E and d, then one can connect these indices intemally in the following two wayss
baë6bz or 60'6k-. Co does not have any symmetry with respect to interchanges of extemai iines and so in
this case the two color structures represent two different objects,
Co = Cov + Cos- (4.137)
These color structures are shown in Figure 4.8, where the dashed lines indicate how the color indicea are
connected intemallyy The connected indices have the same chirality for Cov and the opposite chiraüty
for Cos- The two color structures for the other C's are just transposed versions of the same object- By
'transposed' we mean that the two Iules with the outgoing fermion arrows are interchanged, or equivalently
the two incoming lines are interchanged. We denote a particular color structure of these C's with a hat,
and to reconstruct the original C we mite
The l/Nc expansion is obtained by substituting (4.137) and (4.138) into the various expressions, paying
attention to the color loops formed by the various color structure orientations of the C's.
Figure 4.9: The o d y two 4PI vacuum diagrams with one-gauge-boson exchanges
A significant simpliûcation occurs due to the fact that the diagrams in Wap~ at leading order in I/N,
contain no more than one 4point function each. The two vacuum diagrams that are leading order in l/N,,
as well as being leading order in a, are shown in Figure 4.9.d Because the fermions are massless the single
&point function that appears in the second diagram in Figure 4.9 is chirality preserving. Ali the other
diagrams in W4PI involving a 4point function at leading order in l/Nc also look simiiar to this diagram.
The single gauge boson is just replaced by the set of 4PI planar graphs. We denote the sum of aü these --
d~hese are the only 4PI diagrams at leading order in the gauge coupling, to al1 orders in LIN,.
4.5. GAP EQVATIONS IN TBE LARGE Nc LlMIT 65
vacuum diagrams by the diagram in F i 4AOa and the 4point diagram that is obtained after removing
the 4-point function by the diagram in F i 4.10b.
Figure 4.10: The 4PI gauge interaction a) vacuum diagrams and b.) 4-point diagr- at leading order in
l/Nc-
Note that Ha, in (4,102) and Had in (4.89) are suppressed because their fermion ioopg cannot form cdor
loops. Next one notes that Hd. in (4.113) and Hdd in (4.122) decouple fkom each other because they do not
share the same C7s. Each of Hd, and Ha contain one of the respective color structures of Co- Hence, the
expression for the effective action at leaàing order in l/Nc splits into two parts. These two parts decouple
fkom each other because they do not share the same C7s. One part is
where 2 = [1+ c~&' and WdPI,' contains the diagram in Figure 4.10a with Cos as the Cpoint function.
The other part is
where fi = [l + c&', = [l +CL];' and W4Pr,2 contains the diagram in Figure 4-lOa with CL or CR as
the 4-point hrnction. The gap equations associated with (4.139) are
& = T ~ , %=T: and C ~ s = c + K o (4.141)
and those for (4.140) are
C R = T ~ + K R , CL = T ~ + K ~ and COV=Ti. (4.142)
The various T7s are provided in (4.24) and (4.62) - (4.73)' and the various K's appear in (4.75) - (4.79).
The gap equations for Cc and @ do not have K-terms, because we have seen that in the large Ne limit
the 4PI vacuum diagrarns are independent of cc and &L. The expression for Cov in (4.142) also does not
have a K-term, b e c a w the color structure of the would-be K-term does not match- It shows up in the
Cos-expression in (4.141) instead.
Consider the first gap equations in (4.141)- Rom the expression for rd, given in (4.70), we have:
The only solution for this expression is cc = O, hmpective of what the vaiue of Cos is. This happens
because the equation does not have a K-term.
Usïng the expression for c, (4.72), the gap equation for Cos in (4.141) becomes:
Here Xo, which is a contribution fiom the K-term, denotes the 4.1 p k a r graphs in Figure 4-lob, with
the chiralities of the extenial fermion lines the same as Co. For & = @ = O, (4-144) gives
which Mplies that Cos is generated perturbativeiy by the sum of 'ladder' diagrams, with each 'ning' given
by the set of 4PI planar graphs.
A Ppoint function with the color structure of Cos may have a nontrivial flavor structure which would
prevent it from being generated perturbatively, i.e. its gap equation would not have a K-term:
Io this case it is only possibie to generate a nonperturbative Cos indirectly through Cc and @, if the
latter two were nonzero.
Consider next the last gap equations in (4-142). Using the expression for Ti, (4.66), the gap equation
for Cov becomes
where R = [l + CR]dl and i = [1+ C&'. This is very similar to (4.1431, and again the only solution
is Cov = O. As for ER (or CL), by usiog the expression for ~ f , given in (4.68), we obtain fiom the gap
equation in (4.142)
Here XR denotes the 4 . 1 planar graphs in Figure 4.10b where the chiralities of the extemal fermion Lues
are those of a CR- Since C o v = O, thk gives
Hence, just like Cos, CR h generated perturbatively by a 'ladder' sum of 4PI planar graphs.
4.6 Summary and conclusions
We derived expressions for the 4point effective action and gap equations with the procedures of Mer-
ences [21]. Then we solved the gap equations in the Iimit of a large number of color and found that there
are no nontrivial nonperturbative solutions.
4.6. SUMMARY AND CONCLUSlONS 67
The absence of nontrivial nonperfurbative solutions for the 4point gap equations seems couterintuitive
in the light of the situation wbich is so well known for 2-point functions, as discussed in Chapter 2. The
naive generaiization of the linearized ladder gap equation for the dynamical mass to 4-point functions would
be of the form
Cc = ECc (4.150)
where E is the one-gauge-exchange knd. Equations of this type were considered in [51, 52).= The reason
we do not obtain such an equation can be traced to the absence in w4PI of the diagram consistimg of two
4-point functiom and a single gauge boson exchange, shown in Figure 4.11a Its absence is independent
of the 1/N, expansion and is due simply to the fact that this diagram is not 4PL It is implicitly included
in the T-term shown in Figure 4 .6~ (since the chirality preserving &point hct ions implicitly have single
gauge boson exchange contributions), and we have seen that the T-ternis by themselves cannot generate
nonperturbative solutions. Note that the diagram inmIving a triple gauge vertex, shown in Figure 4-llb,
is included in W4PI, but this contribution is subleading in l/Nc-
Figure 4.11: Vacuum diagrams with two 4point functions.
In light of these results it is of interest to consider the possible effects of instantons. Due to the chirality
changing nature of instanton dects, they may lead to the dynamid generation of the chiraiity changing
4point functions, & and c&. We note that the ünes of a 't Hooft-vertex operator [31] can be cl&
off with chiraüty changing 4point functions, and that the diagram containing chirality changing 4point
frinctions only (no chirality preserving 4point functions) would be leading in il&. Thus it may be the
case that at 1eadi.g order in 1/N,, Cc and CL are generated dynamically by instanton effects. Moreover,
Cos would then be affecteci as in (4-146).
Hence, we h d a qualitative ciifference between the 2-point function case and the 4point function case,
in the large Nc limit. None of the 4point functions can be generated dynamidy by gauge exchanges,
and only the chirality changing 4point hinctions, Cc and CL, can be dynamicaiiy generated via indanton
effects, as we show in Chapter 8. As noted in Section 1.3, these particular &point functions are relevant
to the symxnetry breaking pattern in (1.2), which is of special interest for models of fiavor physics which
preserve electroweak symmetria
=An dedive infiard cutoff, which is necessary for nontrivial solutions and which appears naturally in the 2-point case,
had to be postulateci in Refefences [SI, 521.
C&APTER 4- 4-POINT FUNCTFONS VI;A GAUGE EXCEANGES
Chapter 5
Instantons
In the early 1970's interest in non-Abelian gauge theories wete piqued when 't Hooft [53j showed that these
theories can be renormalized even when they are spontaneously broken through the Higgs mechanism [41].
This meant that gauge theories were viable candidates for the weak interaction. Shortly afterwards, it
was discovered that non-Abelian gauge theories have an added richness that appears due to theit nontriv-
ial topological properties. Certain topological d e f ' d e d magnetic monopoles [54] were argued to be
important for confinement 1551 - A topological gauge structure that later came to be cailed an instanton was discovered by Belavin,
Polyakov, Schwartz and Tyupkin [30]- These were interpreted as tunneling events in Euclidean space [56].
The tunneling amplitude was calculateci by 't Hooft in a semi-classicai approximation (311- He also explained
the U(I) axial m o d y [57] in terms of instanton dynamics and discovered the existence of zeromodes in
the spectnun of the Dirac operator in an instanton background. The idea that the instanton ensemble
can be descnbed as a gas was put forward in 1978 [58]- The iiquid mode1 [59] came almg in 1982 when
Shuryak [60] noted that, aithough dilute enough to ailow a semi-classical approach, the instanton ensemble
in QCD is not so dilute that one can neglect gauge interactions among d i f f m t instantons. The latter are
not included in the dilute gas approximation.
The relevance of instantons for chiral symmetry breaking and, by implication, for the nonperturbative
formation of 2-point functions has been realized long ago [34, 351. Subsequently rnany authors elucidated
various aspects of this mechanism for chiral symmetry breaking. For reviews see Reterences 1321. CarIitz
and Creamer derived a gap equation for the dynamical m a s function using the dilute gas approximation
[35]. A challenge to ail these analyses is the appearance of infrared divergences in the integrai over the
size of the instanton. Appelquist and Selipsky found a way to avoid these infrared divergences and used
the Carlitz and Creamer gap equation to caldate the instanton generated dynamical mass function - see
Section 1.5. In this chapter we provide an introductory discussion of the nature of instantons.
5.1 A topological defect
One can see h m instantmm arise in gauge fields by considering configurations that mhïmize the Euclideau
gauge action. For this purpose we rewrite the action as foliows
- 1 where GP, = S~YoBGzB.
The first term in (5-1) is proportional to a topo10gical i n d a n t :
cailed the Pontryagin indez When the integral in (5.2) is eduated over the entire Qimensional Euclidean
space the result, Y, is always an integer.
The second term in (5.1) is always positive, therefore the mioima in the action occur for cases when the
field strengths are (anti) self-dual
The gauge action for these self-dual
The Pontryagin index can be written as an integral of a total derivative. Using the divergence theorem,
this becomes an integral over the boundaty:
is calleci the Chern-Simons currenk The boundary of 4-dimensional Euclidean space where 1x1 + ou is a
3-sphere S3. Intuitively one might think that this integral should vanish because the field strength tensor
GE, vanishes in the Limit of large z. However, although G;, is zero for lz( + oo the gauge field Al, is not
necessarily zero there. In fact it approaches a pure gauge at infinity
where Ta is a generator of the gauge group and U denotes a gauge transformation. In terms of this pure
gauge the Pontryagui index becomes
Using this expression one can explain why the Pontryagiu hdex is nontrivial even though the field strength
is zero and why it is always an integer when evaluated over the boundary at infinity. For this discussion we
5.1. A TOPOLOGICAL DEFECT
restrict ourseives to the case of SU(2). The existence of instantons in larger gauge groups resuits from the
fm that they contain SU(2) as a subgmup- The topology of the space of all SU(2) trdormations is that
of a 3-sphere S3, just iike the boundary of edimensional Euclidean space. S i A; is uniquely determined
on the boundary by a gauge transformation through (5.71, it defines a mapping from the boundary into the
group space - Le. fiom S3 into S3. A trivial mapping is one where the entire boundary is mapped to the
same point in the group space- For such a case A: is zen, on the boundary- ki general the mapping can
wrap the group space severai times around the boundary. Consequently A: cannot aiways be continuously
deforniecl to the trivial case, Note that the number of times the group space is wrapped around the
boundary wodd always be an integer, In this way the mappings can be groupai into dinerent classes so
that difkent mappings of the same ùass can be continuody deformeci into one another, but mappings
of one class cannot be continuousiy deformed into mappings of another ciass. Each class has a un'nding
number associated with it that tells how -y times the group space is wrapped around the boundary-
This winding number, which is an integer, is precisely the Ponfryagin index calculateci with (5.8). Now
because we have shown that this expression can be deriveci h m the one containhg gauge fields, we see
that (5.2) will be nonzero for nontrivial winding numbers evea though the field strength is zero on the
boundary. Moreover due to the fkt that these winding numbers are always integers the result of (5.5) with
(5.6) is always an integer provided that it is evaluated on a boundary where the field strength is zero.
It is perhaps useful to note that although the integral for the Pontryagin index (5.8) is evaluated only
on the bou~dary, it tells us something about the fields in the interior of that boundary. One can see this
from the following. Consider the case where the field strength is zero everywhere on the boundary and
in the interior. One may ask whether it is possible in such a case to d e h e A; in terms of a pure gauge
iU+&CJ such that the Pontryagin index evaluated on the boundary is nonzero? The answer is no. One can
attempt to do this by continuhg the gauge tramdonnation function U(x) for which the Pontryagin index
is nonzero, to the interior. However one will find that there will be at least one point in the interior where
this U(x) is singular. One can regard this as a point where Ai is forced to deviate h m being a pure gauge.
At this point the field strength would therefore not be zero. This point represents a topologid defect.
Such singular functions are not considerd to be physical, therefore one would rather expect to have a finite
region in the interior where the field strength is nonzero. The topological defect would in effect be srneareci
out over this region to give a more weii behaved fimction- It is this interior region that is teferreci to as an
instanton. A single instanton is a region with a Pontryagin index of v = 1 and an anti-instanton is a region
with a Pontryagin index of u = -1. We s h d o h use the generic term 'instantons' to refer to both kinds.
Formally there is of course nothing that prevents us fiom changing the boundary. One can shrink
this boundary so that it is not a t lzl -+ oo anymore and see what happens. If one were to shrink this
boundary so that it starts to penetrate the region where the field strength is nonzero one may notice that
the Pontryagin index deviates from being an integer. Of course to evaluate the Pontryagia index for this
purpose one cannot use (5.8) anymore but wül have to use (5.5) with (5.6) instead. Eventudly as this
boundary becornes very smdl the Pontryagin index approaches zero (provided of course th& the boundary
does not encloses a pointlike topologid defa) , NOW imagine that we have severai distinct, well separated
regions where the fidd strength is nonzero - in other words, we potentially have several instantons. If we
change the boundary until it only encloses one of these regions in such a way that the field strength is still
zero on the boundary then the resulting Pontryagin index must again be an integer- This time however,
the integer indicates the winding number of the e n d 4 region and not of the entire Euclidean space- So
one can see that the net Pontryagin index of a certain region, even in the case where it is not an integer,
is the algebraic s u m of the Pontryagin indexes of all the instantons in that region. In other words, it gives
the net nurnber of instantons in the e n c l d region.
Using (5.2) one can consider an infinitesimal contribution to the number of Uistantons, This would be
where dV4 is an idinïtesimal 4-volume in Euclidean space. This leads to the deilnition of an instanton
number density:
We s h d encounter this quantity again when we consider the U(l)A anomdy in Section 5.6
5.2 The BPST instanton
Obviously there are infiaitely many functions for A: that have a winding number of, Say, v = 1. Not dl
these fiinctions wodd be self-duai and thetefore mhimize the Euclidean gauge action. It is however possible
to fhd an expression for A: that would have v = 1 and be self-dual. The resulting expression is the famous
BPST instanton [30]. To find this expression one imposes the requirement that on the boundary
A;Ta = iutB,u. (5.11)
To get v = 1 we choose
where ü denotes the Pauli matrices; x is a 3-vector for the three spatial coordinates; zo is Euclidean time
and 1x1 is the magnitude of the Euclidean position 4vector. This lads to
on the boundary. Here Tbrv is the 't Hooft symbol[31] which is given by
eopv for p ,v=l ,2 ,3
Iiapv = sa, for u=O
a for p = O.
5.3. T u m m m v G
One can express Ag away h m the boundary in terms of the anzats
where f + 1 as z? + 00- When one inserts this anzats into the selfkiuality condition (5.3) one fin& that
This has the solution f = 9/($ + 9) where p is an arbitrary scale parameter. The expression for the
BPST instanton is t h d o r e given by
This expression is obviously sensitive to gauge transformations. One specific gauge that is convenient for
calculations is the singdar gauge. In tenns of it the expression for the BPST instanton becornes
where ii,,, M e r s h m in that the signs of the two 6's are interchanged. The singular gauge has
the effect that it localizes the topologid defect at the origin of the instanton, causing a singuiarity at
that location. The instanton can be shifted to any position in Euclidean space. In addition, if the gauge
group is larger than SU(2) the instanton bas a color orientation which denotes how the SU(2) subgroup is
embedded into the larger group. The position, size (p) and color orientation of the instanton are parameters
that distinguish instantons without affecting their winding numbers. These parameters are cailed collective
coordinates and they play an important role in the calculations of instanton amplitudes.
One of the most important aspects of instantons is how they &ect the vacuum. To discuss the nature
of the vacuum it is helpful to treat the Euclidean tirne coordinate separate from the rest and change the
shape of the boundary. We consider two points dong the Eudidean tirne axis, to and t l , and then de6ine
the boundary in terms of the following three 3-surfaces:
0 1: the 3-dimensional space at to;
0 II: the 3-dimensional space at t l , and
the region at spatial infinity that stretches from to to t ~ .
Together these three regions describe a closed boundary, which is topologically equivalent to a 3-sphere.
One can determine whether this boundary encloses an instanton by evaluating the Pontryagin integral over
this boundary. As stated before, one can have various Werent gauge transformation functions, U(x) , that
have the same Pontryagin index- Such fimctiom can be continuously deiormed into one another by applying
appropriate gauge trdormations, Being only interesteci in the Pontryagin index of these fuactions, we
oniy consider those functions that are constant over regions II and IXI and only vary over region L As a
result A: is zero on regions II and IïL The Pontryagin index is cmmpletely determineci by the integral over
region 1. This integral therdore becornes
and is now d e d the Chern-Simons chanacteristic.
What we have achieved is to reduce the boundary to and we now consider mappings from @ to the
group space, which is S3. The important requirement is that aii points a t spatial infinity in @ are mapped
to the same point in the group space- The mapping is d&ed by the gauge transformation function U(x).
This leads to different functions for At, expressed as a pure gauge in the 3-dimensional space.
As before the cliffixent winding numbers (or Chern-Sions characteristics as they are now d e d ) in-
dicate how many times the entire group space is repeated to cover the 3-dimensional space, R3. Smce
the field strength G,, associateci with such a gauge field is zero, this gauge fieid defines a minimum of
the Euclidean gauge action. Gauge fields with different winding numbers denote different minima in this
action. Moreover, since a local minimum defines a (false) vacuum of the theory under investigation, one
can regard such gauge fields with different winding numbers as dinerent Mcua of the theory.
In view of this one can interpret the instanton as a tunnelhg event that takes the system fiom one
vacuum state to another. To understand why this is so, imagine that on region I we have a field configuration
with winding number v = 1 and on region II we have another field configuration with winding number
v = O. On region III the function is stili constant. When we evduate the Pontryagin integral over the entire
boundary we find a combineci winding number of v = 1. Thus we know that there must be an instanton
between the two regions, 1 and ïL This instanton is therefore an event in Euclidean time that takes the
system from a vacuum with a certain winding number to one with another winding number that diffm
from the previous by 1. This defines a tunneling event in Euciidean tirne.
To assess the tunneIing rate 't Hooft calculated the one-instanton vacuum-tevacuum tunnelhg ampli-
tude in a semi-classicai approximation [31]. The initiai calculation of 't Hooft was done only for SU(2), but
it was soon generalized to arbitrary SU(N) by Bernard [61]. Some errors that appeared in 't Hooft's papa
were noted and corrected [62], but it is perhaps the independent caldation done by Lüscher [63] that
helped to dear up all these errors. This instanton amplitude plays a significant role in virtualiy all instan-
ton calculations. For example, it appears in the Carlitz and Creamer analysis of chiral symmetry breaiàng
via instantons [35] and in the instanton liquid mode1 [59] used to analyze the instanton contributions in
QCD phenomenology. It will also appear in the instanton effective action formalism that is presented in
Chapter 6 for the analysis of 2-point and 4point order parameters. We postpone any further discussion of
the instanton amplitude until Chapter 6 where the instanton etfktive action formalism is deriveci,
Here we only note that the tunneling amplitude is proportional to
Rom this one can see th& the tunnelhg amplitude is truly nonperturbative, because it is not possible to
expand (5.20) as a power Sefies in a.
5.4 The theta-vacua
Because of the t u ~ e l i n g the vacua with definite winding numbers cannot be true vacua of the theory.
The true vacua shodd be superpositions of these separate ~ c u a To discuss the true Mcua we again
consider gauge fields with different Chern-Simons characteristics on region 1. Recall that this gauge field
is a pure g a g e everywhere in this 3-dimensional space- Therefore, it is possible to change the Chern-
Simons characteristics of this gauge fieid through a contïnuous gauge transformation, U(x) . This means
that U(x) can take the system fiom one vacuum (local minimum) to another. S i c e U(x) is a pure gauge
transformation it must commute with the H d t o n i a n of the theory. Therefore all the eigenstates of the
ffamiltonian, including the true vacuum, must also be eigenstates of U(x) . Moreover, since U(x) is unitary
its eigenvalues are of the form exp(i8). The only way to form a true vacuum that is an eigenstate of U(x),
with the eigenvalue exp(iO), is
where In) denotes the local minima (false vacua), each with winding number n. We see that there are stü l
various minima dependhg on the vacuum angle 8. The presence of this fiee parameter is related to one of
the wisolved mysterïes in particle physics. We will see in a moment that the presence of instantons means
that fermions can change their chirality. It also means that CP is not conservecl. It is aiready known that
CP is not conserved in weak interactions, but fhis has not been observeci in strong interactions. If CP is
not conserved then there is a new term that is allowed in the QCD Lagrangian:
Oniy if 8 = O would CP be conserved. The upper experimental Mt on 0 obtained from the dipole moment
of the neutron is 0 < IO-^. The reason why it is so s m d is not known. This is referred to as the strong
CP problem [a].
5.5 Zeromodes
It was 't Hooft who first discovered [31] that the Dirac operator has zeromodes in its spectrum as a result
of instantons- This means that the equation
has nontrivial solutions- For an (anfi) instanton these zeromodes are le&-handed (right-handed), The
expression of such a zeromode can be determineci for the BPST instantons. In singular gauge it is
where p is a scale parameter for the size of the instanton, C( is the renormalization d e a and z? denotes an
Euclidean dot-product: z,z,. The Euclidean Dirac matrices, r,, are here defineci so that { T ~ , %) = 2& -1,
tr(rp7,,} = 46,'" and 7; = 7,. The UD denotes a Dirac spinor with isospin index a- (The isospin
inda is implicit in qb"(z).) This spino, is normdized such that Ca Üaua = 1 and it obeys the identity
Ca uav, = f (1 + 7 5 ) .
The Fourier transforms of the zeromodes are given by
and
is expressed in tenns of modifieci Bessel functions [65]. The shape of this f-function is shown in Figure 5.1.
5.6 The U(l)a anomaly
men one considers a gauge theory of fermions in the mass1ess M t then the classical Lagragian has a
U ( N f ) x U(Nf) symmetry. One can also express this as SU(Nf) x SU(N') x U(l)v x U(l)Ay where the
two U(1)'s are respectiveiy associated with the following transformations
At energy scales large compareci to the masses of the u and d quarks one would expect to see this symmetry
in the hadron spectrum. The U( l ) v is associated with the conservation of baryon number which is observed - -
aThe p is included here to get a mas8 dimension of $ for the fermion zeromodes.
5.6- THE U(l)A ANOMALY
Figure 5.1: The shape of the f -function.
to be a good symmetry- The approxhate chiral symmetry SU(2)L x is spontaneously broken in
QCD. The pio.?s are the pseudo-Goldstone bosons associated with this broken approximate symmetry-
That is why they are so light. However, the axial symmetry is not observed. According to this symmetry
there should be a doubling in the spectrum - each hadron should have a partmer of approxïmately the
same mass but with opposite parity. This is not observed,
What is happening here? At the end of the 1960% tbis conundrum was soive when it was discovered
[57, 661 that the axial current associated with the U ( l ) A is not co~l~erved in a quantum theory:
The U(1)* symmetry is said to be anomdous.
The product of the field tensor and its dual which appears on the right-h.md side of (5.30) looks fiuniIiar.
We recognize it as the instanton number density function (times 2Nc) that we had in (5.10). ft is shown
in (5.2) that when we integrate this quantity over 4-dimensional Euclidean space we have the Pontryagin
index, which tells us what the net number of instantons is (instantons and anti-instontons cancel each
other's winding numbers).
The expression in (5.30) therdore irnplies that the violation of the axial symmetry is related to the
number density of instantons. When considering the chiralities of the zeromodes, one can understand
why this is so. Each instanton introduces Mt-handed zeromodes whïie anti-instantons give right-handed
zeromodes. Therefore if there is an excess of instantons or of anti-instantons then there would be an excess
of the associated kind of zeromode. A zeromode with a certain chirality cause5 fermions with that chirality
to be turned into fermions with the oppotsite chirality- This means that chiraüty is not c o d and that
is why the U(l)A is anomaious-
Chapter 6
Instanton effective action
Here we derive the effective actions for investigating the nonperturbative formation of 2-point and 4point
functions through instanton dynarnics. O u . aim is to find expressions for these effective actions in terms
of the full fermion propagator S and the amputated 4-point functions, C. Given a diagrammatic language
one can follow the De Dominicis-Martin procedures [21], as applied in Section 2.3 and Section 4.3 to dezïve
expressions for the 2-point and 4point &&&ive actions, r2[S, Jj and r4[S, q, respectively. The problem
is that we do not have a diagrammatic language for instanton dynamics. The largest part of this chapter
will therefore be devoted to the formulation of such a diagrammatic language. The work in this chapter is
reporteci in Reference [61].
6.1 Notation
We consider here the instanton dynarnics in an SU(N,) gauge theory with Nj massless (Dirac) fermions
in the fundamental representation. The amplitude for the tunnelhg between vacua with different winding
numbers is given by a path integral in which the functionai integral over gauge fields includes those with aU
possible winding numbers, as we saw in Section 5.4. There we also saw that the true vacua of a quantum
gauge theory are formeci fkom linear combinations of aU the classicai gauge configurations. However, this
description stU neglects all the quantum vacuum fluctuations in the gauge field. A true vacuum in the
quantum theory would be a superposition of the classical gauge configurations that m h i m k the gauge
action, together with aii the quantum vacuum fluctuations whose gauge action is close to these minima.
Expecting such vacuum fluctuations to be srnail, one can expand these gauge fields as follows
where AF;) denotes the classicai backgrouud field configurations, each with a specinc winding number
(Pontryagh index); and AK , the quantum fluctuations around the classicai background. The gauge
coupling is incorporateci into the gauge fields and appears explicitly in the gauge kinetic term.
AU terms in the action with more thlui two quantum fields (gauge, ghoat or fermion fields) are dropped.
The resdting action, which we denote by &, behaves like a free semi-clojsid action: the quantum fields
only interact with the background configuration and not with each other- If the background configuration
contains only one instanton the tenns in the Lagrangian with o d y one quantum fiedd would fidl away. This
is because a background configuration with a single BPST instanton obeys the equations of motion- For
background configurations with ensembles of instantons one also has tenns that give interactions between
Merent instantons- These are referred to as classicai instanton interactions and they play an important
role in instanton liquid models [59].
We foilow the usual procedure discussed in Section 2.2, and add to the semi-classical action, &, source
terms for dl the fields in the theory - gauge, ghost and fermion fields. However we shall only explicitly
need the f d o n sources, S,, shown in (2.3), in our derivation. We also add the nonlocal source terms,
SK and SJ, shown in (2.4) and (2.5), respectively. In the diagrammatic fotmalism developed here, these
nonlocal sources are treated as 2-fermion and 4fermion vertices respectively.
The partition functional Z and generating functional W are defined as in Section 2.2, in terms of the
full action, which is given by
Formdy the propagator and 4point Green functions are defined as in (2.7) and (2.8). The 2-point effective
action r2[S1 JI and &point &ective action IT4[S, C] now respectiveiy foliow fiom two consecutive Legendre
transformations, ('2.9) and (429, as described in Sections 2.2 and 4.1.1.
F i t , however, we need to formulate the diagrammatic language, starting from the partition functional,
2. Xn order to simpiifjr our expressions we replace the fermion fieids in the nonlocal source terms by
functional derivatives with respect to the local sources, using the shorthand
d - -6 6=- and 6 = - sri-
Then these terms are pulled out of the functional integral:
The derivation is presented using the partition functionai without the nonlocal sources,
up to the point where the nonlocal sources start to play a role.
6.2. FERMION ZEROMODES
6.2 Fermion zerornodes
We first consider the fermion fields. One can split the action into a fermion part and a pure gauge part
that contains no fetmion fields:
The fermion terms in the Lagrangian in
where the Dirac operator,
Zo are
contains oniy the background gauge fie~d, AL:)- In the presence of instantons (i.e. when the background gauge fieid has a nontrivial winding number)
the Dirac operator possesses fermion zeromdes. It is singular and cannot be inverted. One can avoid this
problem by performing a projection onto the subspace of non-zeromodes and inverting the operator on
this subspace only. For one instanton with Nf mass1ess fetmions there are Nf zeromodes. Considering a
specific configuration (size, position and color orientation) of this instanton, one can separate the subspace
spanned by the zeromodes from the rest which make up the non-zeromodes- The Dirac operator andda tes
the zeromodes, therefore we have
where the operator PN projects onto the non-zeromodes, whiie Po = 1 - PN is the projection operator onto
the zeromodes; furthermore = Po+, QN = PM*, m = Poq, VN = PNq, etc.
For several instantons and anti-instantons the wotdd-be zeromodes of each one are not necessarily
zeromodes of the total configuration. In fact the would-be zetomodes mix and give a spectrum of eigenvalues
around zero. In our approach we keep aii the would-be zeromodes separate from the rest of the fermion
modes. In what foliows we shaü proceed to d them 'zeromodes' even though they are not always actuai
zeromodes of the total configuration.
Now we defme the zeromode and non-zeromode subspaces for any specific background configuration
with multiple instantons. The zeromodes of aü the instantons and anti-instantons span a subspace of the
space of aii fermion fields. We separate this subspace and refer to it as the space of aü zeromodes. The
r e m m g part is caiied the space of ali non-zeromodes. Note that the splitting of the space of fermion
fields into zeromodes and non-zeromodes depends on the background configuration-
For ensembles of instantons and anti-instantons the Dirac operator gives a s m d but nonzero overlap
between the zeromodes of neighboring instantons and anti-instantons. Therefore, for instanton ensembles
we have, in addition to the tenms in (6.9), the terms
+Il -(A) Ibo @*iA)++o P#,
INSTANTON EFFECTNE ACTION
(6.10)
where the superscripts (I) and (A) indicate that the zemmode belon- respeCtidy to an instanton or
an anti-instanton, The cross tenns of (6.10) are respansible for the dynamic~ that transfer zeramodes
to neighboring instantons. They play a cruciai role in the nonperturbative formation of fermion n-point
functions through instantons. Without tbe uoss t m m there would be no fermion interconneetion between
instantons and anti-instantons other than through the interference between zeromodes and non-zeromodes
via the Dirac operator.
These interference terms, - $09 I ~ N + @O (6.11)
give contributions to the fermion propagator that mix zeromodes and non-zeromodes. However, to I h g
order in P/R (p is the average instanton size and R is the average instanton separation) the non-singular
part of the fermion propagator does not have terms that comect zeromodes to non-zeromodes [Ml. Tbis
is related to the fact that, to leading order in p/R, the complete fermion determinant factotorizes into a
zeromode part and a non-zeromode part [68,69]-
Our aim is now to find a way to invert the Dirac operator, and arrive at propagators that correspond to
the various types of fermion modes, while avoiding the singularities. The remainder of this section addresses
this issue-
According to the preceding discussion one can express the partition functional of (6.5) as
Here, V+ is the functional measure over the fermion fields and Dg is the functional measure over the gauge
fields (quantum plus background) and the ghost fields. Using do = b/bm, & = 4/66, we pulled the cross
terms of (6.10) out of the functional integrai over fermion fieids and represented them as one cross term.
It must remain under the gauge functional integral because it still depends on the background gauge fields
through the Dirac operator in (6.8).
Integrating over the non-zeromodes, we h d
where V+, is the measure ovet the fermion zeromodes and detN(@ )
the Dirac operator associateci with non-zeromodes, calcuiated on the
is the determinant of the part of
subspace of non-zeromodes. The
non-zeromode propagator SN is defineci by
9 SN = SNp = PN and SNPo = PoSN = O, (6.14)
The next step is to pedorm the integration over the fermion zeromodes. Consider f h t the case of only
one instanton. At the end we shall generaüze the remit to multiple instantorm. We write the two zeromode
terms in (6-13) as a sum over hvors, Nf
where ¬es the Grassmann variables for the zeromodes and @ denotes the function of the fermion
zeromodes, given in (5.24).
P e r f o h g the functional integral over zeromodes, we get
-0 where d[c , p] denotes the functionai measure over the Grassmann variables of the fermion zeromodes,
The subscripts i on the brackets indicate that the encloseci sources, Grassmann variables and zeromode
functions are ail associated with the same flavor- The result is an object that, iTom a diagrammatic point
of view, behaves like an n-point propagator (n = 2Nf).
It wodd be more convenient to turn this n-point function into an n-fermion v&ex. We achieve this by
adding the following source terms to the full action in (6.2):
Each zeromode source q-, has associated with it a new source, %, and similady an is associated with 7&,.
(Integrations over the relevant spacetime coordinates are understood-)
The addition of these terms is strictly formai. The two types of sources serve merely as variables with
respect to which the functional derivatives are taken. In essence, both types of sources are associateci with
zeromodes and are in that respect not difterent. However, the way they are attachecl to propagators helps
to determine which propagator is to be connected between specific vertices. We discuss the Merent types
of propagators in more detail below.
The introduction of these source terms changes the partition functional. So now we d e h e a new
partition functional Zl which indudes these sources. Upon setting ql = = O we recover Zo.
The idea is now to amputate the n-point n-point function of (6.16) by writing
Here we used the functional derîvatives, 61 = -ci/& and & = 6/613i. As in (6.16). the subscrïpts i on the
right-hand side indicate that both the functionai derivatives and the zeromode functioas inside the brackets
are associated with the same flavor-
84 CZL4PTER 6. IMSTANTON EFFECTIVE ACTION
Now we have
The generalization of (6-19) for multiple instantons and anti-instantons is straightforward- The only ciiffer-
ence is that the number of zeromodes inmeases- When the background configuration contaias n instantons
and m anti-instantons the product of aU zeromodes in (6.19) runs over (n + m) Nj insfead of Nf factors,
Now we aiiow the cross term,
to operate on the propagators
exp (VI Som + VOSO- + %~SAWN) - The vertex, do@ zo, connects zeromodes of instantons to anti-instantons while the bare propagator, So,
is associated with zeromodes of a specific instanton or anti-instanton. One can use the vertex to get a
propagator that connects the zeromodes of Mkrent instantons:
In Figure 6.1 we show which vertices are connecteci by the difirixent propagators. The Sz = -So(i@)So
propagates zeromodes between instantons and anti-instantons (the negative sign is due to the Grassrnaun
nature of the sources). The bare propagator So connects zeromodes between instantons and source vertices
and SN connects source vertices to each other.
To simplifv the notation we combine al1 the propagators into one,
and c d it (Sm) the modd propagator. Its various parts, as shown in (6.23), depend on the background
configuration in a complicated way- However, the expressions for these parts are known [68,69, 701.
The resulting expression of the partition functionai is
where n, denotes the total number of zeromodes.
We have now completed the derivation of the fermion propagator. Ali fermion fields have been suc-
cessfully integrated out and wc have a combineci propagator for the fermion fields. In addition we have a
nonlocal 2Nf-fermion vertex that describes the interactions of fermion zeromodes-
6.3. GAUGE E!IELDS
Figure 6.1: How the difterent vertices are connected by the difilerent propagators.
6.3 Gauge fields
At this point the oniy remaining quantum fieids are the ghost and gauge boson fields that appear in
the gauge action Sg- In the semi-classical appmKimation the functional measure Dg can be split into a
functionai masure over the quantum fieids, Dg, and the masure over the coliective coordinates, in terms
of which the background field configuration is specified. We denote the collective coordinates by C and the
Jacobian that accompanies their measure by J(C).
If the ciassicai background configuration contains several instantons the gauge action would contain
classical interactions among instantons. After integrating over the color orientations of the instantons the
surviving part of these classical instanton interactions is of O([plRI6) [n]. Under the assumption that the
instanton ensemble is fairly dilute we drop the classical instanton interactions fkom our analysis.
The gauge action S, now ody contains kinetic tenns for gauge bosons and ghosts and a classical action
term, (5.4), for each individuai instanton. The functionai integral of Sg leads to the 't Hooft amplitude
without fermions [3l].
The fermion contribution in the 't Hooft amplitude can be separateci into a zeromode and a non-
zeromode part. The non-zeromode part cornes kom the non-zeromode detenninant which, in our derivation,
first appeared in (6.13).
To leading order in (plR) the non-zeromode part of the fermion determinant factorizes into separate
non-zeromode detenninants for each instanton 168, 691- For n instantons and m anti-instantons we have:
where @ denotes a Dirac operator with only one instanton in the background field configuration.
In Reference 1311 't Hooft provides two ways to treat the fermion zeromodes:
1. The first case treats the situation where the fermions in the theory have nonzero masses that are
small compareci to the characteristic scale determinecl by the size of the instanton. The eigenvaiues
of the 'zeromodes' are not zero but are proportional to the f d o n masses- The instanton amplitude
contains a factor of the fermion m a s for each zeromode- For Nf fermions with mass m, the zeromodes
give a factor of (m/lr)Nf, whaa p is the renormalization sede.
If the fermions are exactly massiess the one-instanton amplitude muid vanish uniess there is extra
chirality changing dynamics that can take cate of these zeromodes. The reason is as folIom. In the
presence of fermions each instanton is like a chiraüty changîng scattering event due to the zeromodes.
It can be regardeci as a vertex with incoming and outgoing fermion lines, as we saw in Section 6.2.
A vacuum amplitude does not have external iines, therefore to have a contribution to the effective
action fkom one instanton, there must be a way to close off ail these lines. Moreover, because the
incoming and outgoing hes have opposite chiralities the dynamics that closes off the lines must be
chiraiity changing.
In Reference [31] this extra dynamics is ptovided by gauge invariant fermion bilinear sources. In the
case where one considers the nonperturbative formation of %point functions, the latter can provide
the necessary chirality changing dynamics. One can see that this would reproduce the result found
in the case where fermions have small masses.
In our analysis it is assumed that the fermions are massless and t hat they only receive mass tbrough the
nonperturbative generation of a dynamical mass fimction. The chirality changing dyaaxnics is provided by
the nonlocal 2-fermion or &fermion sources or, in the dective action language, by the 2-point or 4point
functions.
Now we are ready to proceed with the integration of the gauge fields. We s h d first discuss the integration
over quantum gauge fields for one instanton and then at the end generalize the result for muitiple instantons.
Apart fiom the fermion dynamics that must account for the fermion zeromodes and an integral over the
coilective coordinates, the 't Hooft vacuum-to-vacuum tmeling amplitude for one instanton is given by
where Af is the product of the same determinant and gauge functional integrai evaiuated in the absence
of any instantons- Hence N is a collective coordinate independent normalization constant. The functional
integration of (6.26) has been done in Refefence [31], using Pauli-Villars renormdization [72]. The result
can be expresseci as
where p is the scale parameter for the size of the instanton, p is the renormalization scale and bo =
) ( l l ~ , - 2Nf) . The dependence on the gauge coupling in (6.27) is containeci in
6.3. GAUGE FIELDS 87
The fktor (pp)b~ appearing in (6-27) can be incorporateci into the exponent part of y(a). This then leads
to one-loop running as a fimction of p for the coupling in the exponent in y(a)- The instanton expression
in (6.27) can be computed to higùer orders which muid cause the couphg in the monomial part of ~ ( a ) to
run as a function of p as d, but always at one order l m than the coupling in the exponent. To di orders
one would expect that both couplings nui according to the exact beta fundion- So, assuming that the
twc~loop beta function is an accurate enough representation of the exact beta functionP one can allow both
couplings in y(a) to run according to the two.1oop beta function [36j. (See Appendix C for a discussion
of the twdoop beta function.) This is what we shall assume a d therefore we make the replacement
r(a(p))(w)b -+ ~ ( a ( p ) ) . ~ The shape of ~ ( a ) is shown in Figure 6.2- Its peak is located at a = n/N,.
Figure 6.2: The shape of ?(a) for Ne = 3.
The prefactor in (6.27) is
where we use the one-ioop values: B = 0.2917, C = 1.5114 in MS-bar [Ml- Here K includes a color volume
factor that cornes fkom the integral over aii color orientations- This integrai only gives the volume of
the relevant factor group, SU(Nc)/G, where G is the subgroup that leaves the instantons invariant. This
prefactor does not include the eftect of the gauge dependence in the r e m d g dynamics. The complete
integral over d o r orientations can be seen as a product of a color volume factor and an averaging over color
O T h e terms beyond the second term in the beta function are renormalization scheme dependent and there exists a renor-
malkation scheme in which the two-loop beta function ia the exact beta function 1731. bWe reaüy mean a ( l / p ) , but do not want to be stuck with this CurnbeRorne notation. Therefore we write o(p) and
apologize for the inconsistency.
88 C&APTER 6. INSZWVTON EFFECTlVE ACTION
orientations- The latter treats the gauge structure of the remaining dynamics. It gives a tensor structure
that indicates how color indices are contracted. We disCuas the color averaging in more detail in Chapter 7.
We now write (6.24) for the one instanton case as
where the object VI plays the role of a nonlocal vertex:
A simiiar expression exists for anti-instantons, which we denote by VA. The only Merence is that the
zeromode functiom, and 3, have the opposite helicities.
Generalizing (6.30) to configurations with many instantons, we get a factor of A for each instanton. As
a result there is a VI or VA, for each respective instanton and anti-instanton in the background configura-
tion. At the same time each instanton cornes with a complete set of coilective coordinates. The resulting
expression is
where each VI and VA introduces its own integral over its associatecl collective coordinates. The under-
standing is that exp(Tj'S,,,qt) is part of the integrand of the overail coilective coordinate integral.
W e now recover the original partition functional with the nonlocal sources using
6.4 A diagrammat ic language
In References [27,21,22] the diagrammatic language of an underlying gauge theory was exploitecl to express
the effective action as a fundional of the relevant n-point functions. For instanton physics the necessary
diagrammatic language is provided by the Feynman rules that can be derïved fiom the expressions for Z
in (6.32) and (6.33). These include rules for:
a the 2-fermion and 4fermion sources, K and J , that are represented by nonlocal 2-fermion and 4-
fermion vertices in the diagrans;
a the nonlocal 2Nj-point instanton vertices, VI and VA;
the modal propagator Sm, represented by the f d o n lines, and
an integral over the collective coordinates of all the instantons in the diagram.
6.5. THE EFFECTIVE ACTIONS 89
According to these d e s Z[K,Jl is the sum of aii vacuum diagrams in the presenœ of the coiiective
coordinate independent nodocal sources K and J, The modal propagator depends on al1 the coiiective
coordinates of the background configuration, As a result disconnected dbgrams do not factorize. However,
if one assumes that the instanton ensemble is W1y dilute, the overd collective coordhate integral can be
separated into a product of integrais - one for each connecteci part of the diagram- This is because the
modal propagator depends only weakly on the collective coordinates of fat away instantons [TOI. Under the
above assumption each connected diagram only depends on the instautons that are directiy involveà, so
that Z exponentiates. Therefore W = -In(Z) consists of the sum of all (coiiective coordiaate integrated)
connected vacuum diagrams.
6.5 The effective actions
Now we have succeeded in our fi& goal: to formulate a diagrammatic ianguage for Uisfantons. The
rernaining steps in the derivation of the dective actiom are the same as those describeci in Section 2.3 and
Section 4.3. Here we only outline speci6c points of interests for the case at hand.
6.5.1 The 2-point effective action
When we apply the procedure used in Section 2.3 to the case of instanton dynamics, we h d an expression
that is very simiiar to the one in (2.24):
r2 [s, JI = n {h (s-l) - ~n (s;') ) + n { (si1 - s-l) s} + w,,, [s, JI, (6.34)
The clifferences are that the bare propagator that appears in (2.24) is here replaced by the modal propagator
Sm; the traces, Tk{-), in (6.34) include an integration over the coilective coordinates. The Feynman rules
used to construct the 2PI vacuum diagrams in Wzpr[S, JI, Wer fiom those mentioned in Section 6.4 only
in that there is no 2-fennion noniocal source vertex anymore and fermion lines represent the full fermion
propagator S. After the collective coordinate integration the term 3k{ln(Gf )) becomes a constant which
is removed through nonnalization.
Taking the functional derivative of rz[S, Jj with respect to the full propagator, one obtains a gap
equation for the fiill propagator:
The couective coordinate integral of Sm1 in momentum space equais the inverse bare propagator multipiied
by a momentum dependent scalar form factor,
We s h d assume that A'@) = 1.
90 CHAPTER 6- INSTANTON EE'FECTNE ACTION
6.5.2 The Carlitz and Creamer gap equation
As a first application of the %point instanton &&ive action formalism, we derive the Catlitz and Creamer
gap equation [35] in the srnail mass limit. W e restrict W2p1 to be the sum of the oneinstanton and
oneanti-instanton amplitudes. Because these two axnpiitudes are equal we have
where rkin is the kinetic term given in (2.37):
and D(p) denote the expression for a mass insertion:
Here we used the expressions in (5.25) and (5.26) for the Fourier transform of the zeromodes, and --O 9 (p) . The f -function is defmed in (5.27).
Now we perform a functional derivative on (6.37) with respect to the dynamical mas. This gives
where
Rom (6.40) fdows that
For Ne = 2 the gap equation in (6.42) becomes the Carlitz and Creamer gap equation [35] to leading order
in smali mass.
6.5.3 The 4-point effective action
The instanton effective action for 4-point functions is identicai in form to that of (4.125). The oniy
ciifference in this case is that the sum of al1 4PIC vacuum diagrams, Wwr, which appears in the expression
of the effective action, is calculated with Feynman ruies that differ from those mentioned in Section 6.4 in
that there are no nonlocal source vertices anpore. Insteaci the 4-fermion vertices now represent 4point
=Sec definition of 4PI in Section 4.2.2.
6.6. SUMMARY ANP CONCLUSIONS
functions. Fhthermore, fermion iines represent the fidl fermion propagator- W e still have the 2Nj-point
Mantons vertices, VI and VA and each diagram must stiiî be integrated over the coUective coordinates of
all instantons in the diagram- The other terms in the 4point aective action consist of loops of 4point
functions connecteci by pairs of fermion lines.
6.6 Summary and conclusions
The instanton &ective action formaiism presented here makes it possible to investigate the nonperturbative
formation of 2-point and 4point functions- In this way one can determine whether, and under what
circumstances, these n-point functions can act as order parameters that signal the (partial) breaking of
a chiral symmetry- Our effkctive actions are derived using the diagrammatic procedures developed in
Reference [21]. These diagrammatic procedures require as input a diagrammatic language - Feynman rules in terms of which the diagrams can be expresseci. We have presented here such a diagrammatic
language for instanton dynamics- The key features of it are:
the nonlocal SNr-point instanton vertices; and
the modal propagator that propagates the different fermion modes, including the zeromodes, while
avoiding the singularities of the Dirac operator.
One success of this formalism that is already apparent without doing any speciiic caldations is the
fact that the 2-point effective action reproduces the Carlitz and Creamer gap equation [35] in the smaU
mass limit.
It is perhaps useful to compare this formalism to the instanton liquid rnodel [59]- The most obvious
Merence is the fact that the instanton iïquid fomalism is a statist id physics approach whereas the for-
malism here is a quantum fieid theory approach. The instanton liquid model contains statistical quantities
like average instanton size and average instanton separation. The effective action formdism, on the other
hand, is formulated in tenns of the resrtmmation of Feynman diagrans containing the instanton dynarnics
in terms of vertices and propagators-
Another obvious difference is the fact that the instanton liquid approach specifically includes the classical
instanton interactions, which we dropped. Due to this it might appear that o u . formalism neglects an
important part of the physics that govem the behavior of instantons. The point is that the instanton liquid
model addresses a different regime than what we intend to adàress through our formalism. Instanton liquid
models are mostly employed in the low energy phenomenology of QCD. In this regime the coupling constant
is fairly large and the insmton ensemble, although dilute enough to use a semilclassical approximation,
is not dilute enough to neglect instanton interactions. For example, Diakonov and Petrov pointed out in
Reference [Tl] that the coefficient of the instanton interaction term is large in the absence of fermions.
One of the crucial aspects in theh anaiysis is the stabiüzation of the instanton ensemble. The instanton
interactions, which are on average repuïsive, proviàe the mecbanism for this stabiiization.
In contrast to thk, the &&ve action formaüsm attempts to answer questions about the generation of
order parameters, These are expected to appear at hi& energy d e s when the coupling constant is stïU
fairly smaü. Eùrthermore, if the phase transitions associateci with these symmetry breakings are continuous
- second order or higher - the order parameters would be s d close to the transition point and are
therefore ideal expansion parameters. Tenns containhg small powers of the order parameters will dominate
the analysis. The diluteness f m r (plR) is related to the tunnelhg amplitude, which contains positive
powers of the order parameter. One can thus see that terms with factors of diluteness corne with high
powers of the order parameter and are t h d o r e suppressed in the region where these order parameters are
smd.
Yet it must be acknowledged that the diluteness terms do represent important physics. This can be seen
from the fact that without them the &&ive potential d d not be bounded bdowSd The dynamics given
by the instanton vertices is purely attractive and by increasing the size of the order parameter one lowers
the energy of the potential. The repulsiveness of the diluteness terms are required to make the potential
bounded from below, This aspect is however irrelevant for the investigation into the nature of the second
order phase transitions that interests us-
It is in principle possible to include the diiuteness terms in the effective action formalism. One simply
ne& to derive 'Feynman niles' for these types of interactions. However, the resuiting formalism would be
awkward. One of the reasons for neglecting them is because they depend on statistical quantities. Their
inclusion would therefore lead to a mixed statistical physics-quantum field theory formaüsm.
d ~ t is a weiî known fact [74] that the CJT &ective potentid is unbounded beîow, even without dynamics. This is a problem
related to the nonlocality of the Pfermioa sources.
Chapter 7
2-point funct ions via instantons
In Chapter 8 we consider the nonperturbative formation of 4-point hinctions and calculate the associateci
critical coupiings. The aim is to determine whether these nonperturbatively formed &point functions
are preferred over nonperturbative fomed 2-point functions. The purpose of this chapter is therefore to
compute the critical coupiings associated with the nonperturbative formation of 2-point function through
instant on dynamics.
The strength of instanton dynamics is compareci to that of gauge exchange dJFnamics by comparing the
respective critical couplings. We use the instanton effective action formalism presented in Chapter 6. This
analysis appears in Reference [75].
7.1 The 2-point effective action
The 2-point effective action [21, 221 in the instanton formaiism of Chapter 6 is given by
where the fermion modal propagator, Sm, is defined in Section 6.2. The trace over the inverse modal
propagator contains an integral over al1 the collective coordinates iuvolved in the diagram- The general
form of the result of this integration is the h e r s e bare propagator, sr1 = +, mdtipiied by a momentum-
dependent scalar form factor, which we shall assume to be equal to 1 - see (6.36).
The general expression for the full propagator is given in (2.25). However, due to the Chvality changing
nature of the instanton vertices, the instanton diagrams that are second order in the dynamical mas
function, do not contribute to A(p). The result is that to leading order in the dynamical mass function we
have A(p) = 1. Therefore the fidi fefmion propagator S here reduces to
where C(p) is the dynamical ma^ fundion-
The nondynamical terms in (7.1) consist of one-1wp diagrams that are only composed of the full fermion
propagator S and the fermion modal propagator, Sm.
The dynamitai term Wipr[S] in (7.1) is constructed with Feynman rules for:
the 2N'-point hitanton vertices, Vr and VA, given in Section 6.3;
the fidl fermion propagator S, given in (7.2); and
0 an integral over the coiiective coordinates (sizes, positions and color orientations) of ail instantons in
the diagram.
7.2 The nondynamical terms
In terms of (7.2) the nondynamical terms in (Tl), which we denote by I'&=, becornes
Following the steps of Section 2.4.1, we introduce an infrared cutoff at the momentum scale where
po = E(po) and linearize the nondynamical term. Close to a continuous phase transition point C@) is very
mail so that po k: O. The remit, according to (2.37), is
Here we assume that the dynamid mass function is in the singlet representation, so that the color trace
over the two mass functions gives a factor of Nc-
Now we follow Section 2.4.2: we make a redehition of the integration variables and do a Fourier
transformation. For this purpose we defme
where p is the renormalization scale. As a result we have
which is equivalent to (2.40).
7.3. THE DYNAMICAL TERM
Anti-instanton vertex
Mass insertion
vertex
Figure 7.1: The instanton-anfi-instanton diagram with two mass insertions.
7.3 The dynamical term
The instanton-anti-instanton term that is quadratic in the dynamicd mass Iooks as foilows: each mas
insertion closes off two lines of an instanton vertex- The remaining lines are d connected between the two
vertices. (See Figure 7.1.) The resulting diagram is expressecl as
where pl and pz denote the sizes of the two instantons; the combiiatorial factor 3 counts the number of
terms associated with this diagram after color averaging; D(pl) and D(p2) are the parts of the expression
due to the two m a s loops and G(pr, m) is the part of the expression due to the remaining fermion loops.
7.3.1 The combinatorid factor and color averaging
To count the nurnber of terms we start by considering aii the possible ways that the propagators can close
off the lines of the two instanton vertices to give a diagram with two mass Ioops- Then we perforrn the
color averaging and add up ali the resulting terms-
R e c d that each iine of an instanton vertex has a specific flavor associated with it - there is one
incorning and one outgoing line for each flavor. Because the propagator is flavor diagonal it must connect
h e s of the same flavor. Hence, there are Nf distinct ways to connect one of the two mass insertions, each
leaving the other mass insertion and the remaining fermion lines with only one way to be connected.
Next we perform the averaging over the color orientations of the two respective instantons. The color
orientations are represented by gauge transformations that operate on the zeromodes:
In the m a s loops they cancei because there we have a g and a g-l from the same instanton. The gauge
transformatiooas of the remaining iines can be combiied in& relative gauge transformations of one instanton
with respect to the other: gc'h = gr or 9;lg1 = 9;'. Thus the two color averaging integrais bemme one
trivial (= 1) and one nontrivial averaging integrd. The nontrivial integral contains (Nt - 1) kctors of gr
and g;l. The integrai over d color orientations can therefore be written as
where dg is the Haar measure for the gauge group and Np = (* - 1)- The color averaging can be done by
the techniques of Creutz [76]. The r d t of the averaging is a series of tensor structures Ti, each muitiplied
by a coefficient ai that depends on Nf and Ne. General expressions for these coefficients are provided in
Reference [77]- The tensor structures indicate how color indices of the same instanton are interconnectecl.
There are Np! tensot structures and each has Np! terms. The tensot structures t herefore have the foilowing
form Np!
n=l
where ri, denote the n-th term in the i-th tensor structure- Each of these terms has the foliowing form
where aj and dj are color indices of one instanton and bj and are color indices of the other instanton.
Dserent terms of a tensor structure are obtained by permuting all the (aj, q)-pairs- These are therefore
mutual permutations of the color contractions on both instanton vertices- Difterent tensor structures are
obtained by permuting di a,% by themselves. These are relative permutations of the color contractions of
one instanton vertex relative to those of the other vertex- Al1 tensor structures that are associateci with
the same conjugacy class in the permutation group have the same coefficient.
The color contractions determine how the fermion lines between the two instantons are closeci off into
Dirac traces and thereby determine the Dirac structure of the momentum integrals. The evaluation of
al1 these momentum integrals is too challenging to attempt. However, we expect that the sues of the
momentum integrals do not ciif£'' much and that the variations would average out. Therefore we set them
aii equal and estimate their typical size in Section 7.3.3. As a result we only need to add aii the coefficients
of aU the tensor stmctures multipliecl by the number of terms associateci with each tensor structure. It can
be shown that for permutations of Np = (Nf - 1) elements,
where ai denotes the coefficients of the tensor structures. To get the total count we have to multiply (7.12)
by Nf for the number of ways to connect the mass loops and by Np! = (Nf - l)! for the number of terms
7.3. THE DYNAMICAL TERM
in each tensor structure- So the resdting combinatonal factor is
Nj!(Nc - l)! 3= (Nc + Nf - 2)!'
7.3.2 The mass insertions
Using (7.2), (5.25) and (5.26) we obtain the following e x p d o l l s for the mass loops
where we linearized the expression, as in Section 2.4.1, Next we make a redefinition of the integration
variables and do a Fourier transformation as in (7.5)- In addition we define
As a resuIt we have
where
7.3.3 The remaining fermion loops
The rest of the fermion lines between the two instanton vertices combine into
The fermion lines pair up into fermion loops due to the color contractions that appear after the averaging
over color orientations. Considering one such bop, we fkid
where in the h t Line we explicitly show the color indices on the zeromodes. By definhg
one can write (7.18) as
We shall not attempt to evaluate (7.21) exactly7 but instead estimate its size- It turns out that the p
integrals are dominated in the region where pi f i . Therefore, on the side of the diagonal (pl = pz) where
pl > we replace pl in the arguments of the f -functions with pi. One can see that this replacement would
not have a big effect when fi ks pz. Fùrthermore, replacing the smaller p by the larger p in the arguments
of the f -functions, we are closer to the actual situation than with the larger p replaced by the smder p.
Hence, (7.20) becomes
For the region where p > k we set = p inside the f -functions and for p c k ne set 4- = k.
Upon evaluating the angular integrai we arrive at
one can wrïte (7.23) as
The function, R(y), determines the momentum dependence under the remajning momentum integrals
in (7.21). The Dirac delta function replaces the momentum dependence of the integrand of one of the
momentum integrah with a combiiation of the other rnomenta upon evaluation of this particular momentum
integral. In this way all the momentum integrals are interconnecteci. if one replaces this connecting
integrand with a constant that is set by the value of the integrand at the dominant momentum s d e , the
remahhg momentum integrals d decouple. One can now evaiuate each momentum integral separately.
This step does not take into account the angular integrations that would result fiom the contractions
between Merent rnomenta To assess the contributions fiom these angular integral we note that (7.21)
has (Nf - 2) integrals over k+ This will give (Nt - 3) anguiar integrals. (One can arrange it that each ki
at most contracts with its two neighbors). In this way one can determine the factors of 2 ~ . Numerically
one h d s that the size of the integral over R(y) is s i. It is important to note that the uncertainty in this
7.4. RESULTS AND DISCUSSION 99
estimate can have a profound effect on the final results. If the overd unCettainty is only an O(1) factor
then the & ' would not be signifiant for small critical cauplings. However, if the uncertainty in the
contribution of each fermion h o p is an O(1) factor then the overall uncertainty would be this factor raised
to the power of (NI - 1). Thh would have a sis;nificant &ect on the final resdt. To take this uncertainty
into account we paramet& the mceftainty of each fermion loop by a p a r a m e A. Therefore we have an
extra overd factor of ~~f -l. Together with the definitions in (7.24) out estimate for the sizes of G(pl, pz)
for pr > is
W e see that for a large number of flavors the value of G(pi,-) for p2 < fi is severely suppressed with
respect to the d u e on the diagonal (pz = pi).
7.4 Results and discussion
The integrand of (7.7) is invariant under an interchange of pl and pz. Therefore, one can split the pz-
integration into two regions, divideci by the value of pl. First, we consider the part where pz < pi . When
we substitute (7.26) into this part we find
One can see that the &-integral is dominated by the region near the pl-boundary. For this reason
one can set y(a(p2)) = y(a(pi)). The integrand of the pl-integral has its dominant region near some
value p, proportional to p. Because the couphg runs slow for a large number of flavors one can set
~ ( 4 ~ 1 ) ) = r(a(~m)) = 70-
Next we use (7.15) to redefhe the MnabIes in (7.27) and we substitute in (7.16). The result is
Now we add the part where p2 > pl. This is the same as (7.28) but with wl and w2 interchanged. Then
we evaluate one of the w-integrals. The result is
The maximum value of the part of the integrand in the brackets is at w = O with F(0)* =s 0.3. W e notice
that this term does not have an infrared divergence,
Comparing the result in (7.29) with the nondynamid term in (7-6) one can see that the potential
becornes unstable at the orîgin when
This can be expressed as
E270 > 1,
where
is the coefficient for the formation of 2-point functions via instanton dynamics.
Figure 7.2: The critical couplings for the nonperturbative formation of 2-point hctions, from instanton
dynamics for Nc = 3, with 2 5 Nf 5 N j and whae X = 1.4,2.31 and 2.55.
The critid coupling for instanton dynamics a: is the one for which
It necessarily has to be smaller than x/NC.
Using the expression in (7.33), together with (7.32), one can cornpute ai for any value of A, Nc and for
Nf 2 2. Numerical estimates of a: for Ne = 3 and 4 are shown in Figures 7.2 and 7.3, respectively- In both
7.4, REsms AND DCSCUSSION
Figure 7.3: The critical couplings for the nonperturbative formation of 2-point functions, from instanton
dynamics for Nc = 4, with 2 5 Nf 5 Nj and where A = 1.4,2.31 and 2.55.
cases we show the results for X = 1.4,2.31 and 2.55. We chose these values to facilitate cornparisons of these
cases with each other, with the critical couphg for gauge exchange dynamics and with those for the 4-point
functions, which are considered in Chapter 8. The criticai couplings are computed for 2 5 NI 5 Nf , where Nj is the criticai numbers of flavors. For cornparison we incikate the values of the critical aoupluigs nom
gauge exchange dynamics by the dashed lines.
The critical numbers of flavors, Nj, is set by the point beyond which the dynarnics carmot destabiie
the vacuum at the origin. There are tao rnechanisms that can introduce such a limitation. One is where
the critical coupling reaches the value of r/Nc, which is where the instanton amplitude has its strongest
contribution (see Figure 6.2). If the strength of this contribution is not enough to destabilize the vacuum
a larger coupling won't be able to do it either because then the instanton contribution becomes d e r .
This is what sets the critical number of flavors when X = 1.4 for both Nc = 3 and Ne = 4.
The other mechanism appears when the number of flavors becomes very large - close to the point where
asymptotic freedom is lost. In this region the running of the coupling is governeci by an infrared 6xed
point. We computed the values of these fixed points under the assumption that the running of the coupling
is given by the twdoop beta function - an assumption which becomes more accurate as the coupling
becomes smder. (See Appendix C for a discussion of twdoop fixed point running.) When there is a
nontrivial inframi 6xed point in the beta function the maximum coupling that can be reached is the fixed
point coupling. Assuming a two-1oop beta b c t i o n the fixed point mupling is given by
where b and c are tespectively the first and second coefficients in the beta function- The critical numbers
of flavors NF is reached when the fixed point couplïng given by (734) falls bdow the eritieal mupling,
obtained fiom solving (7.33). Therefore for d u e s of NI above Nj the coupihg is unable to reach the
critical value so that dural symmetry breakhg does not occur. This mechanism set the values for NF when
X = 2.31 and 2.55 for both Ne = 3 and N, = 4-
Ekom the results in Figures 7.2 and 7.3 we make the following observations:
r Considering the cases for X = 1.4, we see that if X is srnaIl4 rises very quickly for increasing number
of flavors. In this case we would conclude that instanton dynamics play an insigniscant role in chiral
s y m m e t r y breaking via 2-point condensates.
r For large d u e s of X the value of 4 decreases after an initial increase, with increasing number of
flavors- In this way it can reach values that are significantly s d e r than the critical coupling for
gauge exchange dynamics.
r With increasing number of colors, af decreases, but at a slower rate than the gauge exchange critical
coupling, as can be seen by cornparhg the two lower curves with the gauge exchanges values in
Figures 7.2 and 7.3.
Due to uncertainties in the calculation we cannot make def i te statements about the relative importance
of instanton and gauge exchange d y n d c s - It would be reasonable to concur with Appelquist and Selipsky
[36], in stating that the contributions of instantons and gauge exchanges to chird symmetry breaking are
comparable. However, one can use the results h m lattice gauge calculations [78] and instanton liquid
mode1 calculations for QCD with a large numbers of flavors [79], which indicate that the critical number
of flavors for Nc = 3 is about Nj = 6. In our caldation this value for the critical number of flavors
corresponds to a A that is slightly larger than 1.4. In view of this we see that the critical coupling for
instanton dynamics quickly becomes Iarger than the critical coupling for gauge exchange dynamics. Hence,
based on the results in References [78, 791 we conclude that instanton dynamics do not give a significant
contribution to the nonperturbative forraati011 of 2-point functions.
We have used the instanton effective action formahm of Chapter 6 to compute the critical couplings for
the nonperturbative formation of 2-point functions. For this analysis we restricted ourseives to continuous
phase transitions for chiral symmetry breaking, which Mplies that we only consider the instanton-anti-
instanton amplitude with two mass insertions. We found that this amplitude does not s&er h m an
infi.ared divergence- The d t i n g criticd coupiings indicate that, des s the contributions of the f d o n
loops are fairly large, instanton dynamics do not give a signiscant contribution to the nonperturbative
formation of 2-point functiom-
Chapter 8
4-point funct ions via inst antons
Here we compute the critical coupling for the nonperturbative formation of &point functions through
instanton dynamics. For this purpose we use the same generic model that we used thus hr in our analysis,
including in Chapter 7 where we computed the critical couplings for the nonperturbative formation of
2-point functions. We consider instantons in SU(Nc) acting on N' (Dirac) fermions in the fundamental
representation. h this model it is found that Ppoint functions are indeed generated nonperturbatively
through instanton dynamics-
8.1 The effective action
The full expression of the &point effective action is provided in Chapters 4. In symbolic notation one can
express it as
I'r = rpc + W~PI. (8-1)
It consists of nondynamical pair cycle terms, r,, which are loops of 4point functions connecteci via pairs
of fermion lines, as shown in Figure 8.1 (see also Section 4.3.2), and W4PIi which is the sum of aü 4PI
vacuum diagrans (see Section 4.2.2). The Feynman rules in terms of which these 4PI vacuum diagrams
are constructed, are discussed in Section 6.5.3.
As in Section 4.1.1, it is assumeci here that the fermions are massless. Therefore the Chvality changing
part of the fidi propagator is set to zero. In addition, A(p) is set equal to 1- The full propagator is thus
set equai to the bare propagator. This aüows us to use the 4point function formalism of Chapter 4.
The full effective action is tw complicated to be solved. For this r-n we use the large N, arguments
presented in M i o n 4.5 to simplify the expression. When those terms that are suppressed by factors of
l/Nc are dropped, the effective action separates into two decoupleci expression. The order parameters that
F i e 8.1: An example of a pair cycle-
interest us - the chiraiïty changing &point functions - appear in only one of them. It is given by
rl =n {Ah 2 (1 - &2ec + z a) + 1.n (1 + COS) - COS
whem 2 = [l +cos];' and WeIVi eonsists of those 4PI diaprams that contain only the 4point functions
that are chiraüty changing. The notation in (8.2) is defined in Sections 4.1.3, 4.1.4 and 4.5. Ail the
nondyromicol 4PI diagrams are suppressed by factors of l/Nc- Hence, the W;rpr,i-term is now purely
dynamical, and one can distinguish between the nondynamical pair cyde terms in (8.2) and the dynamical
term, w4~1.1-
The smallness of the order parameters near a continuous phase transition is used in Section 8.3 to
make an expansion to leading order in the order parameters. This further simplifies the expression for the
effective action.
8.2 The breaking pattern
There are at least two potential breaking patterns for the chiraüty changing 4-point functions that one can
consider:
The first is where the chiral ftavor symmetry breaks down to a vector flavor symmetry and a discrete
group (52, 801:
s u ( N f ) s x S U ( N ~ ) R + SU(Nf)v x 2 4 (8-3)
a The second is where the chual flavor symmetry breaks down to a chiral isospin symmetry and a vector
family symmetry:
La a theory that attempts to explain the dynamics high above the electroweak scale and which reduces to
the standard mode1 at lower energy scales, there has to be a chiral isospin symmetry that survives down
to the dectroweak scale- The breaking pattern which is more interesthg in terms of this framework, is the
one that contains a surviving chiral isopsin symmetry- For this mason we shail henceforth ody consider
the second breaking pattern, given in (8.4).
The flavor structure of the Cc that signals this breaking pattern is given by
where A, B, C and D are ïsospii indices associated with SU(2)t x SU(2)R; a, b, c and d are M y
indices associated with SU(Nf/2); and &C and BD are 2 x 2 totally anti-symmetric matrices The order
parameter, Cc, is symmetrized with respect to its impiicit color indices, As explainecl in Section 4.5, the
chirality changing C's contain two orientations of the color structures, &S. However, because their oniy
&ect is to cancel the gauge transformations that represent the color orientations of the zeromdes, the
color indices can be ignoreci. We therefore consider one color structure and multiply it by a factor of 2.
The number of 'flavors' that run in the fennion loops is the number of f d e s , N, = Nf /2, and not
Nj. The isospin indices are connected through the 4point functions. Therefore the isospin states of all
fermion pairs in a pair cycle are relate& There are two fennion lines per pair and there are two isospin
states per fermion line. That gives an overail factor of 4 for ail pair cycles.
8.3 Nondynamical terms
8.3.1 Fermion loops
The nondynamical terms consist of big loops of Ppoint functions that are connected by pairs of fermion
lines. An example is shown in Figure 8.1. These pairs of fermion lines form f d o n loops. In momentum
space the order parameter (4-point function) depends on three momenta: one for each of the two fermion
loops on either side of the &point function and a third that flows around the big loop. The fact that
the order parameter depends on these three momenta makes the stability analysis quite complicated. To
alleviate this complication we assume that the function of the order parameter is constant in al1 three
momenta,
Q@, 4, k) = b? q, k) = a=, (8.6)
up to a scale p, beyond which it is zero. The chiral preserving 4point function is allowed to depend on the
big loop momentum
Cash Q, k) = aob) - 03-71
The dependence on the big loop momentum in ~ ( p ) is required for consistency as we SM see below. Under
these assumptions the fermion loops are not affectecl by the 4point functions, except for the ultraviolet
108 CHAPTER 8. 4POINT FUNCTIONS VIA INSTANTONS
cutoff that is set to be p. The expression for a f d o n loop in Euciidean spaœ is therefore given by
There is an extra faetor of ), which cornes h m )(1 f y5), because the fermions have s p d c chiralities
There are three i>nknowlls in the &i?ctive action in (8.2): &, CC and Cos- One can produœ gap equations
for them by taking functional derivatives of the eftective action with respect to these three unknnams. The
gap equations for cc and are similar in form. These order parameters are thedore equai, as we have
assumed in (8.6).
One ean use the gap equation for Cw to fmd an expression for it in terrns of and CL- This gap
equation is fairly simple because the diagrams in Wipr that contain Cos are aii suppressed by extra factors
of 1/N, and are therefore dropped in . The resulting gap equation is
Note that the factor of 4 that the pair cycles are multiplieci with, is removed by the functional derivative
with respect to Cos, due to the assignment of specific isospin indices on the external lines of the Cos.
Substituting (8.6) and (8.7) into (8.9) and denoting each fermion loop by I,@) we find
The solution of this quadratic equation that is consistent with ~ ( p ) + O when a, + O is:
We see here that the big Ioop momentum dependence of ~ ( p ) is required for consistency-
8.3.3 Expression for the nondynamid terms
One c m now substitute (8.11) into the pair cycle terms in (8.2) to obtain
Here we note that the II{-} involves an integrai over the big loop momentnm. The definition of Cc in (86)
introduces a hard cutoff p for this integral. We change the iategration variable to a dimensionless variable,
y =$IP2, substitute (8.8) into (8.12) and set
The result is
At this point we use the fact that the order parameter z is very s m d close to the transition point to
simpiify rtin. Expanding In z and evaluating the Ieadïng term numericaliy, we find
where A = 0.009936 = 0.01. By defining
we replace A in terms of T, so that the kinetic term becomes
8.4 Dynamical term
8.4.1 The instanton-anti-instanton amplitude
We are only interested in those contributions in that c m compete with the pair cycle terms at
the origin. The leading contribution for this, which we denote by ~i$)~ is the instanton-anti-instanton
diagram that contains two chirality changing 4point functions. This diagram looks as follows: one &point
function, c,!., doses off 4 lines of the instanton vertex while the other, CC, doses off 4 lines of the anti-
instanton vertex. The remahhg (2Nr - 4) lines are all c o ~ e c t e d between the two instanton vertices. (See
Figure 8.2.) The diagram is expresseci as
where pl and denote the sizes of the two instantons; the combiatorial factor 3 4 counts the number of
terms associated with this diagram; D4 (pi) and &(pz) are the parts of the expression due to the 4point
function insertions and G(pi, f i ) is the part of the expression due to the remaining fermion loops.
The combinatorial factor and the 4-point functïon insertions are determined in Sections 8.4.2 and 8.4.3,
respectively. The expression for the remaining fermion loops, G(p1, pz), has been estirnateci in Section 7.3.3
Instanton vertex
Figure 8.2: The instanton-anti-instanton diagram with two 4point function insertions.
for the 2-point function case. The oniy ciifference here is the number of fermion lines- Here we have
(2Nf - 4) lines, which gives us
assuming that fi > m. The uncertainty in the contribution of each fermion loop is parameterized by A.
Note that the value of X would be the same for the 2-point and 4point cases. It may however vary as a
function of the number of flavors.
8.4.2 The combinatorial factor
To determine the combinatorial factor, we foilow the same procedure that was used for the 2-point functions
in Section 7.3.1. Care must be taken to count the number of ways that one can connect the two 4fermion
vertices, but after t hat it is only the number of lines between the two instantons which decreases to (Nt - 2)
that is different fiom the 2-point function case- As notai in Section 8.2, the two color structures give rise
to a factor of 2 for each 4fermion vertex.
One must also keep in mind that there are two terms in the flavor structure, as shown in (8.5). Due
to the isospin subgroup, the flavors are arrangeci into Nt /2 uptype and Nj /2 down-type flavors. The two
incoming lines of the Pfermion vertex must be connecteci to opposite types of flavors and the same for the
outgoing lines.
In total there are Nf(Nf +2)/2 ways to connect the two 4fermion vertices on the two instanton vertices.
Each leaves the remaining lines between the two instanton vertices wïth oniy one way to be contracted. The
color integration and subsequent summation of tensor coefficients give ( N e - l)!/(Nc + Nt - 3)! because
there are ody (Nf - 2) elements in the permutation group comparecl to the (NI - 1) elements for the
2-point case in Section 7.3.1. Moreover, we have (NI - 2)! terms in each tensor structure. Together this
8.4. DYNAMICAL TERM
8.4.3 The &point function insertion
The expression for the &point function insertion is given by
and the same for ci. h momentmm space (8.21) becornes
where we used (5.25), (5.26) and (8.6) and where = q +p, m = -q t p , f i = k - p and p4 = -k - p .
Over the range of angles that dominates the integral (those around B = n/2) the f -funetions do not
vary much. Therdore we drop the anguiar dependences fiom the arguments of the f -fimctions. Then one
can eduate the anguiar integrai. The resuiting expression becornes
where
W e made use of the angdar integrid,
Introducing dimensionless variables
112
we obtain
CHAPTER 8. 4-POINT FUNCI1ONS VIA INSTANTONS
- - -
where M(pp/2) captures the p
One can determine the ultraviolet and infrared behavior of M(-) and use that to appraximate this
function. In the ultraviolet M t , p + O, we have
f (z) = 1 + 0(z2). (8.28)
In this limit
Substituting this into (8.27) with u = pp/2, one obtains
where A is the same constant that we encountered in the kinetic term, (8.15).
For the infiarecl h i t we take the cutoff to infinity: f i + m. Together with (8.6) this implies that
Therefore
Now we use (8.30) and (8.32) to define an approxhate h c t i o n for D4(p). At the same time we use (8.13)
and (8.16) to replace a, with s. The 4po i . t function insertion is therefore approxïmated by
8.5 Results and discussion
8.5.1 The &point fwiction coefficient
for
for
We now substitute (8.19), (8.20) and (8.33) into (8.18). As in the 2-point function case the resulting
integrand is symmetric under exchange of pl and pz. One can therefore split the integration region dong
the line where pl = pz and evaiuate only the one half for which f i > pt and then multiply the result by 2.
Fhrthermore, according to (8.33), the integrais spüt into dinetent regions separateci at p = T / p - A11 these
integrais are finite. Because the coupling runs *1y slowly one can assume that ~ ( a ) is constant a t a value
of 70 = d@/cc)) - Evaiuating the two p integrah, we obtain the expression
We now substitute (8.34), together with the kinetic term (8.17), into the 4point dective action:
As was done for the 2-point function case we express the stability condition of the vacuum at the origin in
ter- of an inequality- The vacuum at the origin becornes unstable when:
where E4 is the coefEcient for the nonperturbative formation of 4-point functions via instanton dynamics.
The expression for this coefEcient is:
The expressions in (8.36) and (8.37) provide the means to compute the criticai couplings for the nonper-
turbative formation of 4point functiom via instanton dynamics for a.ii values of Nc, X and for N' 3 3-
Using (8.36) and (8.3'1) we computed the critical couplings for Nc = 3 and 4, with 3 5 N' 5 NF and
we show the curves for X = 1,1.347,1.6 and 2.5. These couplings are shown in Figures 8.3 and 8.4. For
conparison we also show the critical couplings for 2-point condensates via gauge exchange dynarnics.
Here we make the following observations:
For very smaii values of X - Le. when it is smaiier than approximately 1 - the instantan critical
coupling rises quickly, however, not as quickiy as the 2-point couplings, shown Figures 7.2 and 7.3. If
X is small we conclude that instanton dynamics is unable to form 4point functions for large numbers
of flavors.
If X is on the order of 1.3 to 1.6 we see that instanton dynamics give moderate cr i t id coupiings up to
large numbers of fiavors. These cases can be cornpareci with the 2-point case with A = 1.4. For these
values of A we conclude that &point condensates can be formed by instanton dynamics while 2-point
condensates cannot. Depending on the value of A the 4point condensates may even be favored over
the formation of 2-point condensates via gauge exchange dynamics.
Figure 8.3: The critical couplings for the nonperturbative formation of 4point functions, for N, = 3, with
3 5 NI ( N; and where A = 1,1.347,1.6 and 2.5.
a Comparing the case when the value of X is on the order of 2.5 with the comparable values in the
2-point cases, we see that for smail numbers of flavors the Cpoint critical couplings are significantly
smaller than those for the 2-point condenstates. However, this ciifference becornes srnaüer for larger
numbers of flavors.
With the uncertainty of this calculation we see that it is feaçible that the formation of &point condensates
may be favored over the formation of 2-point condensates.
8.6 Conclusions
Contrary to the situation with gauge exchange dynamics as found in Section 4.5, instanton dynamics can
generate nontrivial chirality changing 4point functions nonperturbatively. We see t h d o r e that instantons
do provide the dynamics that can break chiral symmetries as sigaaUed by the formation of these Ppoint
functions - at least for smali numbers of flavors. Uncertainties in the calculation, however, make it difficult
to make specific statements for larger numbers of flavors.
Here we computed the associated critical couplings. This is done with the formalism presented in
Chapters 4 and 6. We restrict ourselves to continuous phase transitions for chiral symmetry breaking.
Consequently we only consider the instanton-anti-instanton amplitude with two insertions of &point func-
tions. The resulting amplitude does not d e r from any infrared divergences. However, it is difficuit to make
8.6- CONCLUSIONS
Figure 8.4: The critical couphgs for the nonperturbative formation of 4point functions, for N, = 4, with
3 5 Nj 5 NfC and where X = 1,1.347,1.6 and 2.5,
an exact estimate of the contributions of the fermion loops. The couplings are assumed to run accordhg
to the two-loop beta function.
These criticai couplings are compareci with those for 2-point functions that are computed in Chapter 7.
It is found that the &point function couplings are, even within the uncertainties, smaller than those for
the 2-pcint functions.
It deserves to be mentioned that the uncertainties in the values for the critical couplings of the 4point
functions exceed those for the 2-point functions in the following ways. Due to the 2-point fitnction's simpler
momentum dependence, one can perform a stability analysis without assitming some shape for the function
of the order parameters, On the other hand, it is necessary to make an assumption about the 4-point
function's momentum dependence. The effect of this is to overestimate the value of the &point critical
coupiing. There are also uncertainties associated with the approximation of D4(p), which tend to increase
the amplitude and t h d o r e underestimate the critical coupling. These uncertainties are however much
smailer than the uncertainties associated with the fermion loops. Although they may have a signifiant
effect when the critical coupiings become large (close to r /NC) they would be barely noticeable for smali
couplings.
Chapter 9
A dynamical model for fermion
masses
Here we have the culmination of the investigation into the nonperturbative formation of 4point fimctions.
This investigation began in Chapter 4 where the 4point effective action is derived for gauge exchange
dynamics. Solutions of the associateci gap equations in the large Nc limit indicate that in this limit no
4point function is generated nonperturbatively through gauge exchange.. In Chapter 6 we present a
diagrammatic language for instanton dynamics and use it to formulate both 2-point and 4point &&ive
actions for instanton dynamics, We are not only interesteci in whether 4point functions can be generated
but also whether they are generated at a higher scale than 2-point functions are. The critical coupiings for
the nonperturbative formation of 2-point functions, which we want to use for cornparSon, are calcuiated
in Chapter 7. Then in Chapter 8 we compute the critical couplings for the nonpertwbative formation of
Cpoint functions. A comparison of the respective critical couplings indicate that the instantons critical
coupiings for 2-point functions are generdy larger that those for 4-point functions. In this chapter we
apply this knowledge to a model for the dynamitai origin of quark and lepton masses.
It is tc be expected that our results depend on the model under consideration. In Chapter 7 we used
a generic model for the 2-point functions to allow comparison with the generic gauge exchange critical
coupling. The same model is used in Chapter 8 for the 4point functions to aiiow cornparison with the
2-point functions. In this chapter we consider the model [13] that originaiiy sparked the interest in chirality
changing &point functions. We s h d not perform actual calculations in this model because it wouid require
a rederivation of the effective actions. Instead we provide qualitative discussions based on what we have
learned from our analyses of the nonperturbative generation of n-point functions.
Table 9.1: The representations of the fermions under aii symmdries.
Su(N1) su(N2) u ( 1 ) ~ su(2)~ s u ( 2 ) ~ U(l), u(l)v 2,
strong strong strong weak weak weak anornatous -
9.1 The fermion representations
The representations of the fermions in this model are s u m m a r k d in Table 9-1
AU non-anornaIous continuous symmetries are gauged to avoid Goldstone b m n s and the strengths of
these gauge interactions are as indicated in Table 9.1. The fVst three symmetries oppose the formation
of singlet bilinear condensates in each other. The first symmetry, SU(Ni), is regarded as the 'color'
gauge interaction and it gives rise to QCD at some lower d e . The SU(N2), on the other hand, is a
(sideways) family symmetry. Note that there are in total 2N2 families - Nz of the +'s and N2 of the
$'S. The SU(2)L becornes part of the eiectroweak symmetry at lower scaIes and the SU(~)R, fiom which - hypercharge emerges, malces the model left-right symmetric- The Abelian symmetry, U(l)z7 wodd bave
been an undesirable non-anomatous global symmetry. The number of flavors that each gauge interaction
sees is given at the bottom ofeach column-
We indude in Table 9.1 the global symmetry U ( l ) v (fermion number), which is anomalous under the
chiral isospin symmetry. Because chirai isospii is ody weakly gauged the U(l)v is ody weakly anoma-
lous and is assumed to remain unbroken, so that no n-point functions are ailowed that wouid break this
symmetry.
The discrete symmetry Zn is what remains of the anomalous U(l)A and the anomalous U(l)v . The
value of n is the largest cornmon divisor of 8N1, 8N2 and 2N1 N2. To suppress the formation of masses
NI,& > 2. For Nl =N2 = 3 wehaven=6andfor NI =N2 =4wehaven=32.
Note that, unles some gauge symmetry breaks, no masses are aIlowed to form. Any mass that does
form at some scale above the electroweak d e , signalhg the breaking of some gauge interactions, must
respect the electroweak symmetry.
9.2. PROPOSED SEQUENCE OF SYMMETRY BREAKINGS
9.2 Proposed sequence of symmetry breakings
In thïs mode1 the foliowing sequence of symmefxy breakings is conjectwed to happen as we go down in
scaie. There are three d e s of interest: a high scale, An; the flavor scale, Aj; and the eiectnnueak scale,
A,,- Aithough the &.or and e lec trod scaies are assumed to be well separateci, At > A,, the high
s d e could be close to the fiavor scale. The reason for making a distinction between the high and fiavor
scaies is because no continuous symmetnes are broken at the high scale.
9.2.1 High s d e
At the high scale 4point functions are generated that signal the breakhg of 2, down to Z4, Zz or nothing,
depending on what n is- These 4point functions must be suiglets under aU the continuous symmetries,
For Nl = N2 # 4 the following 4point functions are d o d
For Nl # 4 and N2 = 4 we have the following 4point functîons in addition to those in (9.1):
9.2.2 Flavor scaie
Two things happen at the flavor scale. One is that right-handed Majorana neutrino condensates are
generated. They are of the form ( @ ~ < 1 ~ ) , (&%) and (@ReR). The appearance of the fvst two masses
is not what one would expect to find if one were to follow the MAC hypothesis. However, as we found in
Chapter 3 the MAC hypothesis rnay be misleading and it is therefore not unmasonable that the symmetric
tensor channel may turn out to be the most attractive chruinel. These three condensates break SU(Nl) and
presumably also the right-handed isospin symmetry, SU(~)JZ , to form a smaUer symmetry (QCD) together
wit h hypercharge:
SCf(N1) X S u ( 2 ) ~ + SU(N' - 1) X u(l)y. (9-3)
120 CHAPTER 9- A DYNAMICAL MODEL
Since it is pumible for these condensates to form singiets under we have to assume that the
condensates are f o d in the adjoint channei, which is also not in accordance with the MAC hypothesis-
The srnaller symmetry group SU(NL - 1) is to be QCD, This implies that NI = 4. Here we have the typical
Pati-Salam (811 scenario where the leptons are separated h m the ~uarks. We see that, together with
the unbroken left-handed isosp'i symmetry, part of the surviving symnietry is the electmumk symmetry
SU(2) x U(1) y of the standard model- The k t that this symmetry survives means that no D i masses
are dowed to form.
The first two Majorana condensates break the flavor :pymmetry, SU(N2), down to a smaUer symmetry:
As a resuit of this breaking two f d e s decouple fiom the M y gauge interaction. These two fàmiiies
becorne the two lighter famiiies in the standard model- These Majorana condensates also break
The other occurrence is that 4point hctions are generated that break U(l)2, as w d as the hvor
symmetry:
SU(N2) -+ SU(N2 - 2). (9-5)
These Cpoint functions are of the form of the Cc's and CC% in (9.2).
This breaking would cause four h n i i i e s to decouple from the family gauge interaction. The 4point
functions break more of the SU(N2) generators than the Majorana condensates do, but the gauge boson
masses caused by the 4-point function are assumed to be srnalier than those generated by the Majorana
condensates- Therefore the additional two families that decouples as a result of the formation of the 4point
functions, do so at a much lower scale and give rise to two heavier faxnilies than the 6rst two that decoupled
as a result of the Majorana condensates.
The remaining SU(N2 - 2) symmetry is assumed to behave Iike technicolor and the fermions that
transform nontrividy under it are like technifermions. They acquire mass and are integrated out above
the electroweak scale.
9.2.3 Electroweak scale
At the electroweak scale Dirac masses are generated for the fourth family. The masses are of the form
($,v,bL) and (qL&). They give the usual electroweak symmetry breaking
9.3 The dynamics at the high scale
W e are only interestecl in the processes that take place at the hi& scale- Here we use the knowledge that
we have accumulated about the nonpettutbative formation of n-point functions to discuss the proposed
nonperturbative formation of 4-point functions at the high d e - The key question is whether any 4point
function wodd form before the fermion masses do. Note that within the context of this mode1 it is not
necessary that such a separation should edst at the high d e provided that the electroweak symmetry
remains tmbroken, It is thetefore possible to have Majorana 2-point functions and 4point functions at the
same scale- We do however expect to h d some enhancement of the 4point function amplitudes ovex those
for 2-point functions, simply because the 2-point function are protected by g a g e symmetries, while the
4-point functions are not.
The most favorable masaes are:
We shall compare the criticai couplings for the formation of these masses with that of the chirality changing
4-point fundions in (9.1):
Only gauge interactions that are marked as 'strong' in Table 9.1 are considered in this analysis-
Xn the effective action formalism, which is discussed in Section 8.1, the dynamical contributions of these
gauge interactions are balanced against the nondynamical terms to determine the criticai couplings. If we
normalize these equations with respect to the nondynamicai terms we have expressions of the form:
where IVI, Wz and W3, respectively refer to SU(Ni), SU(N2) and U(l)s.
In view of the result of Chapters 7 and 8, we s h d assume that the value of X is on the order of 1.3 to
1.6. For these values the instanton contribution to the nonperturbative formation of 2-point functions is
srnall compared to the gauge exchange contribution, and we therefore neglect instanton contributions to
the nonperturbative formation of 2-point functions in this aaalysis. The gauge exchange contribution is of
the form
Wi = &ai (9.8)
where Bi is a coefficient which is normalized with respect to the nondynarnical term and which only depends
on the nurnber of flavors.
On the other hand, the assumed range of d u e s for X is large enough to give instanton dynamics
the ability to form Cpoint functions. We also saw in Chapter 4 that gauge exchange dynamics does not
contribute to the nonperturbative formation of &point functions for large numbers of colors. Therefore we
neglect gauge exchange dynamics for the nonperturbative formation of4poïnt functions, As a result 4-point
functions only feel the non-Abelian gauge interactions. When normaized with respect to the nondynamical
term, a contribution due to instanton dynamics is of the form
where Es is the instanton d c i e n t that depends on the parameter A, the number of colors and the number
of flavors and ~ ( a ) gives the dependence of the instanton amplitude on the coupling constant ai (see (6.28)).
One can now write down the critical coupling equations for the four cases under consideration:
0):
From these expressions it is clear that the 4point function amplitude could be enhanceci relative to the
others because, contrary to the other three cases, (IV) does not contain any repulsive contributions. With
everything else the same, the three 2-point functions will each have a net number of one attractive contri-
bution while the &point functions have two attractive contributions.
We have presented here the dynamical model for fermion masses of Reference [13]. According to this model
a variety of Ppoint functions are nonperturbativeiy generated at a high scale. We used our knowledge to
discuss the feasibiiity of the nonperturbative formation of chiraiity changing 4point functiom at a scale
that is higher than the scale where 2-point condensates would form. We found that there is an enhancement
that would favor such a separation of scale,
There are several instances in the modei, as pteseated in Section 9.2, where it is assumed that conden-
sates form that are not in accordance with the MAC hypothesis. However, we saw in Chapter 3 that the
MAC hypothesis may be misleading. For that reason we do not restrict ourselves to this d e .
The benefit of this model is that it provides various different 4fermion operators [13] that can feed the
masses of the electroweak scale down to give a d e t y of fermion masses. In this way it may be possible to
explain the observed diversity in the quark and lepton masses.
Chapter 10
Conclusions
10.1 The achievernents
In this thesis the nonperturbative formation of &point functions is shown to be a feasible mechanism for
physics at scales hïgh above the electroweak scale. The interest in this mechanism is founded on the fact
that chirality changing 4point fimctions can be invariant under a chiral isoBpin symmetry, but a t the same
time be order parameters for the partial breaking of larger chiral symmetries. This makes them naturd
candidates for the protagonists in physics at scales high above the electroweak scale. In cornparison, the
alternative of having 2-point functions involving new non-standard modd fermions, now seems to be an
unriecessary complication. Moreover, Ppoint funtions, generated at a high scale, would survive in the form
of 4fermion operators a t lower scales. These may feed down the electroweak scale masses to give masses
to quarks and leptons. The diversity in the quark and lepton masses may in this way be produced by
the diversity of 4point functions, replacing the collection of fiee parameters that is fouad in the standard
model.
Less prominent, but stiU significant, is our analysis of the feasibility of foirming brlinear condensates in
channels not favored by the MAC hypothesis. In Chapter 3 it is indeed found that the MAC hypothesis
may be misleading. The most attractive channel accordhg to a next-teleadhg order analysis appears to
be in contradiction with those of the leading order analysis. Tbis -plies that such analyses are not as
credible as may have been believed to be. In a previous next-to-ladhg order analysis of the validity of the
MAC hypothesis 1491 it was found that the next-to-leading order contributions make ordy a small (20%)
correction to the leading order resuits. This would imply that the MAC hypothesis is not under any threat
from next-to-leading order contributions, which seems to contradict our findings. The ciifference can be
explained by the clifferences between the respective analyses. In R.efierence [49] it is impiicitly assumed that
the gauge interaction does not break, by keeping the gauge bosons massless. Obviously, if any channe1
124 CEAlTER 1 O- CONCLUSIONS
other than the singlet Channel is favored the gauge interaction would be spontaneously broken. We made
room for this p o s s i b i i by dowing the gauge boson to become massive. Another dif5erence appears in
an approximation made in Refetence [49] to alleviate the complexity of the dcuiations. The authors of
Reference [49] considered the kernel in the limit where the momentum through one mass is much iarger
than the momentum through the other. This would have been valid if the leading and next-to-leading
order kernels decrease at the same rate away fiom the dominate part where both momenta are comparable,
Unfortunately, as we found through a numerid investigation, this is not the case. We therefore did our
calculations in the dominant region of the ketnel where the momenta are equal, Our finding opens the door
for breaking patterns that violate the MAC hypothesis. Several of these are found in the dynamitai mode1
presented in Chapter 9-
The f'rrst order of business in our 4point function investigation was to see whether they can be generated
nonperturbatively through gauge exchange dynamics- Previous analyses [SI, 521 used an intuitive gap
equation for 4point fimctions to investigate their nonperturbative formation. Ground breaking t~ork on
the derivation of 4point effective actions [21] enabled us to do a more thorough anaiysis. Our efforts to
this end was rewarded by a surprising finding. Our investigation indicated that the one-gauge-exchange
diagram, which provided the dynamics in the analyses of References [51, 521, is not present in the gap
equations that follow from the dective action, because the required vacuum diagram is not 4particle
irreducible- Mhermore, in the limit of a large number of colors, N,, there are oniy perturbative gauge
exchange contributions at leading order in 1/N,. As a resuit we found that in this lirnit there are no
nonperturbative solutions for any of the 4point functions. These gap equations capture only the gauge
exchange dynamics. Therefore, this result naturally leads one to ask what the contribution of inherently
nonperturbative effects such as iastantons would be.
Due to a lack of the necessary formalism the next phase of our 4-point function investigation concentrated
on the derivation of a diagrammatic language for instanton dynamics in terms of which one can derive an
effective action. The resuiting diagrammatic language contains:
0 an instanton vertex for the zeromodes, which is reminiscent of the 't Hooft vertex [31], apart from
the fact that it is nodocal;
0 a fennion modal propagator, which selectively propagates each type of mode that exists in an instanton
background with its own propagator; and
r an integrai over ail the coiiective coordinates of the instantons in the diagram.
This d iag rma t i c ianguage can only be applied under circumstances when the instanton ensemble is fairly
dilute. This is the case near a continuous phase transition, for which the order parameter approaches zero,
because the terms that are suppressed by the diluteness factor carry extra powers of the order parameter.
Section 6.6 contains a discussion of the difikences between the iristanton dective action formalism, which
we derived hem, and the instanton liquid mode1 [59].
Any thorough derivation of a &point effective action would indude a consistent treatment of 2-point
functions- For our situation this implies that we were able to derive a 2-point dective action, which is
the equivalent of the CJT effective action for instanton dynamics. As an independent check we could
show that the Carlitz and Creamer gap equation 1351 follows fiom this 2-point effective action- Being
equipped with such a 2-point effective action induced us to pause our purely &point function investigation
for a while to ad* another interesthg problem - chiral symmetry breaking by 2-point functions via
instantons. This is not entirely unrelated to our 4-point function investigation, because it will give us the
opporhmity to eventually compare the criticai couplings of these respective n-point fkdions. The role of
instanton dynamics in chiral symmetry breaking has been analyzed before [36] through a numerical solution
of the Carlitz and Creamer gap equation. Apart from using our instanton effective action to do a stability
analysis insteaà of solving a gap equation, our investigation difFers fiom the one in References [36] in that
we considered a different instanton amplitude than those that give rise to the Carlitz and Creamer gap
equation. Equipped with an efilective action for instanton dynamics, we were able to notice that it is not
the one-instanton amplitude that gives the leading contribution near a continuous phase transition, but
rather the instanton-anti-instanton amplitude with two m a s insertions. This point is disamed in detail in
Section 1.5. Unfortunateiy we could not evaluate the amplitude exactly for an arbitrary number of flavors.
Nevertheless we made an estimate of the size and absorbeci the uncertainty into a parameter. Balancing
this against the kinetic term, we obtained expressions from which one can compute critical couplings for
the nonperturbative formation of a dynamical mass via instanton dynamics for any number of colors and
for two or more flavors. We computed some nunerical values and found, on comparing them with the
critical coupliag for gauge exchange dynamics, that unless the fermion loops make large contributions to
the amplitude under investigation, instanton dynamics are dominated by the gauge exchange dynatnics
when the number of flavors is large-
W e did the equivalent computations for &point hc t ions , again considering the instanton-anti-instanton
amplitude, but with two 4point function insertions this tirne. in the same way we obtained expressions
for the critical couplings for the nonpei-turbative formation of 4-point functions via instanton dynamics
for any number of colors and for three or more flavors. We are interested to know whether &point order
parameters can appear a t s d e s that are higher than those where the 2-point order parameters wouid
appear. In an asymptotically free theory, this would require the critical couplings for the 4point functions
to be smaller than those for the 2-point functions. W e found this to be that case for instanton dynamics,
irrespective of the uncertainties in the computations. However, depending on the size of the contribution
of each fermion loop, the 4-point critical couplings may or may not exceed the 2-point critical coupling for
the gauge exchange dynamics. It is nevertheless fwible that the actual amplitude for the &point function
CHAPT. 10. CONCLUSIONS
case is enhanced leading to d e r critical couplings- One may also wonder whether there are some model
dependent mechanisms that can give enhancements of the 4point function amplitudes over the 2-point
function amplitudes- Upon considering a more redistic model for the dynamical origin of the quark and
lepton masses we found that such an enhancement is indeed possible. The complexities in that model
prevented us fiom doing exact caiculations, however we could identify an enhancement mechaniSm that
may give rise to srnalier criticai couplings for the &point hctions.
The challenges
It is fair to Say that our investigation was at various stages hampered by the complexity of the problem
at hand. This is to be expected in any nonperturbative analysis of this nature. Fortunateiy, we could
overwme or sidestep these obstacles to the extent that we could accompiish our goals. Lets consider some
of these obstacles.
The MAC analysis required the calcdation of a collection of three loop diagrams. lii each case two of
the integrals over the magnitudes of momenta were left unevaluated because of the unknown dynamical
mass function that depends on them. The remainïng integrals that give the kemel function were stiii too
challenging to be solved analytically. We could however obtained the necessary results through a numerical
evaluat ion.
The Ppoint function investigation presented several obstacles. For massless fermions there are several
chiral structures for &point functions and it turns out that only some of them - the chirality changing
ones - act as natural order parameters. It was therefore necessary to distinguish the different types
of 4point functions, which compounded our derivation of the 4point d c t i v e action and the associateci
gap equations. We nevertheless went through this process and were rewarded by the interesthg insight
mentioned above. As is often the case with nonperturbative calculations, the solution of the gap equatiom
was hampered by the complexity of the expressions - the effective action and the gap equations consist
of infinitely many terms. We tnincated the series of terms by considering a large number of colors Ne and
dropping terms that are suppressed by factors of l/Ne. In this way we could determine whether 4point
functions could be formed nonperturbatively via gauge exchange dynamics in this limit.
On proceeding to instanton dynamics, we were confronteci by the lack of a diagrammatic language in
terms of which one can derive the instanton effective action. To address this problem we started firom
fmt principles - the path integral, as first introduced by Feynman [82]. A challenge that presented a
formidable obstacle w a ~ the presence of zeromodes. For massless fermions these zeromodes prevent the
inversion of the Dirac operator to give a fermion propagator. We finaUy solved this problem through the
following steps:
First we split the space of all fermion fields into (would-be) zeromodes and non-zeromodes. The Dirac
10.2- TEE CHALLENGES
operator can be inverteci in the non-zeromode space to give a non-zeromode propagator,
Integrating out the zefornodes, we obtained a noniocal n-point function, where n is twice the number
of flavors.
We turned this n-point function into an n-fermion vertex by amputating the hes. The amputation
procedure introduces bare propagators for the propagation of the zeromodes-
0 Fiaily we brought in the &kt of the Dirac operator, which allows zeromodes of one instanton to
'jumpy to a neighboring anti-instanton. This gave rise to another type of propagator.
Combining all the different types of propagators, we obtained a modal propagator which 'knows' how
to propagate aii the different types of modes among the diff't types of vertices in our diagrammatic
language for instanton dynamics.
The calculations done in the instanton effective action formalism, brought forth their own obstacles.
Both the 2-point and 4point calculations requird a careful caldation of the combinatorial factors assm
ciated with the color contractions, which r d t from the color averaging, and the many possible ways of
connecting the n-point functions. The color averaging is part of the coïiectïve coordinate integrals that
must be performed for each diagram. Techniques to hande such color integrals are available [76], but they
are still extremely cornplex, In similar analyses [83] the results of such color integrations are often estimated
using large N, arguments. In our case however the number of flavors, NI, usuaily exceeds the number of
colors so that parts of the result that wouid have been suppresseà in the large Ne lirnit are now in actual
fact enhanceci due to a large NI. We had no choice but to compute the fidi contribution- Thanks to the
work in Reference 1'771, together with some extra analysis we were able to h d a closed form expression
for the total contribution, provided that the mrious terms receive equd contributions fiom the ensuïng
momentum integrais.
The latter point presented another obstacle, The various color contractions give rise to ail possible
Dirac gamma structures. We were unable to evaluate the momentum integrals associateci with al1 these
2Nf-loop diagrarns for an arbitrary Nt. One can however, estimate the typical size of such a diagram.
Moreover, we beiïeve that deviations among the different types of diagrams would average out. Therefore,
to overcome this obstacle, we set these diagrams equal and add up aU the terms to get the closed form
expression mentioned above. The biggest source of uncertainty in this estimate is the contribution of each
fermion loop. Each of them can vary by a factor of 0(1), which, for a large number of flavors can lead to
a large overail uncertainty. To handle this uncertainty we introduced a parameter A for each loop. In this
way we couid study how the results wouid change when we use different d u e s of A.
The calculation of the critical couplings for the 4-point functions gave another problem. For the 2-point
functions it was possible to obtain expressions for the critical coupiings without having to know the shape
of the dynamitai mas as a fundion of momentum. Due to its more compficated momentum dependenœ,
the 4point function codd not be treated in the same manna. We used an anzats for the 4point t c t i o n
which is constant in aii momenta up to a cutoff scale, beyond which it is zero- This is more or less what
one would expect to find for instanton dynamics, for which contributions are restricted to a specific s d e .
Our inability to employ the exact shape of the actual order parameter that would appear at the amtinuous
phae transition implies that we overestimate the value of the critical coupiing.
The fiiture
Theoretical investigations usually provoire more questions than they tend to answer- The investigation
reported in this thesis is no exception to this tendency. In Chapter 3 we pdormed a next-tdeading
order analysis of the MAC hypothesis using only gauge exchange dynamics. On the other hand, when
we performed our instanton analysis for the 2-point functioas we only considered cases that do not break
the gauge interaction. It would be interesthg to extend the instanton analysis to indude cases where the
2-point functions form in non-singiet channeis and to see how the instanton amplitudes compare to the
leading order gauge exchange amplitudes used in the MAC hypothesis.
We assumed massless fermions in our derivation of the Ppoint effective action in Chapters 4 and 6. Yet
we found that in the frarnework of a simple generic mode1 the critical couplings for the nonperturbative
formation of 4-point functions are comparable to those for 2-point functions. In light of this it would be
useful to do a more thorough analysis in which both 2-point and 4-point functions are analyzed in the same
effective action. One can stiU make the chiralities of the fermions explicit by separating chirality changing
and chirality preserving parts of the 2-point functions. The resulting expression for the effective action
would be more complicated than the expression for the 4-point effective action obtained in Chapter 4,
however, one may be rewarded for this d o r t by a better understanding of the interplay between 2-point
and Cpoint functions.
These are but a few of the many aspects that one can proceed to investigate on the theoretical front
with the purpose of understanding nonperturbative dynamics better. However, it may require additional
input from the experimental side before we would obtain a complete understanding of nature beyond the
electroweak scale. If Ppoint functions play as important a role as conjectured in this thesis then they
should make discernible contributions to processes, which we hope to see at the Large Hadron Collider in
the not too distant future.
Appendix A
Wick rotation and pseudo-Euclidean
space
Figure A.1: The contour in the upper right-hand quadrant of the complex po-plane.
When momentum integrais are encountereà in the calculation of Feynman amplitudes it is conventional
to use Wick rotations [84], which analytidy continue these momenta from Minkowski to Euclidean space.
Given that the ié's in the propagators are positive the poles lie below the positive side of the m-axïs. The
contour integral that encloses the upper right-band quadrant of the complex po-plane would therefore not
contain any poles. (See Figure A.1.) Hence, provided that the integrand, f @o + ih), Mnishes dc i en t ly
fast toward infiniw, we have that
The hat, which denotes the Euciidean quantities, is only used in this appendix. In the rest of this thesis the
hats are dropped. The Wick rotation thus changes the Minkowski innerproduct into a negattue Euclidean
130 APPENDM A- WlCK ROTATION
So, for a Minkowski metric of the form g,, = diag(1, -1, -1, -1) we find that the Wick rotation gives:
We see therefore that the resuiting space is in actuai fact a pseud+Euclidean space. The work that is
reportai in tbis thesis is al1 done in this pseud+Euciidean space- This comprise not only momentum
integrals but also manipulations on the Lagragian level. It is therefore necessary to d&e the notation
and conventions that we use in Euclidean space. Elenceforth, we drop the 'pseudo' prefix and only d e r to
Euclidean space as is conventional-
AU 4vectors transform as foliows under the Wïck rotation:
Here a, is an arbitrary 4vector. This implies that
In Euciidean space we define the square of a 4vector as
The Dirac gamma matrices can be d&ed such that they aiso transfonn according to (A.4), but then
they will obey the identities: {%TV) = -26," and 7: = -Tr. M e a d we find it convenient to dehe them
such that their identities are without the minus signs. The Wick rotation for Dirac gamma matrices are
therefore defined by
$' = -?O- -y,, = ro; ,-' = ri = +yi* (A-7)
These Euclidean gamma matrices obey
For the purpose of integrais over position or rnomentum space, it is important to note that position
vectors transform as contravariant vectors, ZN, while momentum vectors ttaflsform as covariant vectors,
p,. Therefore the Wick rotation of the phase space measure over position space produces a -i,
and for momentum space an i is obtained,
Now we apply these d e s to the kinetic terms of the various types of fields. F i we note that due to
the position phase space integrai
etpw) = er~(@- (A-13)
For scalar fields we simply have
Fermion fields m o n n as follows:
and
A
Note that $ is the Hemitiau aonjugate of 6. The fermion kinetic term bemmes
Gauge boson fields transform as usual vectors do:
For the field strength tensor, abc b AC F,O,=apA;-&A;+f Ap "
(A- 16)
(A. 17)
(A.18)
we have the foilowing traflsformations under the Wick rotation
Therefore, the gauge kinetic term has the following transformation under Wick rotation
Note that in the instanton analysis we absorb the gauge coupling g in the gauge field. As a result a factor
of llg2 appears in fiont of the gauge kinetic term- One can see that in aii three cases the kinetic terms are
negative definite in Euclidean space.
The fermion vertex becomes
q&b = (i4) 4 = ig$&.
As a result the covariant derivative remains the same
132 APPENDIX A- W C K ROTATION
One can now determine the Euclidean Feynman rules that follow fiom these terms in the Lagrangian.
The bare (massless) fermion propargator is
To be consistent with this we define the full fermion propagator by
where A@) denotes the inverse of the wavefunction renormalization and CG) is the dyaamical mass func-
tion,
The expression for the gauge propagator is
and the Euclidean Feynman d e for the fermion vertex is given by
where Ta denotes a symmetry group generator.
Appendix B
Group theory factors for bilinear
condensates
In any gap or stability analysis concerneci with the formation of a bilinear condensate one would h d that
diagrams contain the combinat ion
T,OCT,O (B-1)
where C is the dynamical mass and Tf and T,O denote the symmetry generators in the representations of
the two fermions that bind together to generate the b i e a r condensate. Often one would simply assume
that the condensate is a singlet under the gauge interaction so that the dynamïcai mass can be factored
out, leading to:
TaTaC = CFC (B-2)
where CF is the second Casimir constant for the representation of the fermions, assiiming that they are
the same.
In general however the bilinear can be in any representation and when the representations of the two
fermions are dSerent the singiet representation may not even be available. To determine the constant that
the dynamitai mass is multiplied with for this general case we proceed as follows [23]. Fit we perform a
charge conjugation on one of the fermions. The result is that its generator turns into that of its conjugate
representation (Ta + -Ta*). Then we have
where we show the color indices explicitly. The superscript - denotes the complex conjugate. The tensor
product of generators can now be written in terms of the generators of the product representation, which
are given by
TFe2 = 11 8 Tg* + Tf 12 = &(Tg*) jq + (TT)ipdjq. (B-4)
134 APPENDIX B. GROUP THEORY FACTORS
Consider the product of two such generators
The product representation is in general reducible in tenns of a number of irreducible representations. As
a result one can write the Mt-band side of (B.5) as
where ri indicates the d3Ferent irreducible representations that are contained in the product representation
and li is an identity matrix with the dimension of the i-th irreducible representation. The representation
of the dynamid mass ( b i e a r condensate) wodd be one that is contained in this product representation.
When contractai with this product of generators, the dynamical mass extracts from it the information of
its own representation. Thedore, when we contract the dynamical mass with (B.5) we have
where CF(R) is the second Casimir constant for the representation R of the dynamical mas. The irreducible
representations of the two fermions are denoted by ri and r2 respectively. Note that the last term on the
Ieft-hand side of (B.7) is what we had in (B.3). Therefore, using (B.7), one can write (B.3) as
Here we defined the channel factor FG which appears in Chapters 2 and 3.
As an illustration we consider a few examples of this channel factor. Consider fust the case where the
representations of the two fermions are both (Ne-dimensional) fundamental representations of the gauge
group SU(N,). The irreducible representation of the dynamical mass for this case is contained in the prod-
uct of the fundamental and conjugate fundamentai representations (Nc x z). This product representation
contains two irreducible representations: the 1-dimensional singlet and the adjoint representation, which is
(IV: - 1)-dimensional.
The second Casimir constant for the fundamental and conjugate fundamental representations is
For the singlet representation it is zero: CF(l) = 0; and for the adjoint representation it is: CF(G) = N,.
So, the two channe1 factors are:
Singlet channd:
Adjoint -el:
Note that for the singlet
Next we consider the
)Fo = Cp(Nc), whkh with (B.2).
case where the representations of the two fefmions are the fundamental and the
conjugate fundamental representation, mspectively- This time the pmduct representation consists of two
fundamental (or two conjugate fundamental) representations (Nc x Ne). This product representation atso
contains two irreducible representations: the symmetric a d antisymmetric fensor representations.
The dimension of the symmetric tensor representation is )N,(N, + 1) and its second Casimir constant
The antisymmetric tensor representation has a dimension of )N.(N. - 1) and its second Casimir constant
These two channd factors are: - Symmetric tensor channei:
tensor chamel
(B. 14)
We notice that the adjoint channe1 and the symmetnc tensor channel have negative channel fhctors. This
means that these channeb are repulsive (assiiming that the kinematics does not change the sign - see
Chapter 3). We aIso note that the singlet channel is more attractive than the antisymmetric tensor channel.
APPENDLY B. GROUP THEORY FACTORS
Appendix C
Two-loop beta function for gauge
coupiings
In the instanton analysis we use twdoop beta hc t i ons to describe the running of the gauge coupiings. As
a result of the form of the instanton amplitude the running appears as a function of p - the scale parameter
for the instanton size. In this appendur we consider the two-loop beta function and obtain an expression
for the running in tenns of p. We consider a generic gauge theory with an SU(N,) gauge symmetry and
with Nf Dirac fermions in the fundamental representation.
The renormalization group equation for the wupling as a function of p can be expressed as
whese
and
In the region where
we have b > O and the gauge theory is asymptotically Zree. There is a part of this region near the boundary
where c < O. This happens for
This implies that a. > O. The beta fundion will thedore have a zero at a,, as shown in Figure C.1. This
impIies that there is a nontriuid in f ra4 jùed point given by a, [85]. If, at some energy scale, the gauge
coupling is below this fixed point d u e , it wiil move toward this value at lower d e s , but wiii not exceed
it. In the infrard region the coupling wili thdore remai. (almost) constant at the fixed point value.
Provided that there is no other dynamitai & i such as chiral symmetry breaking, this theory is sale
invariant or confonnd in the hkared,
Figure G.1: A beta function with an infrard fixed point.
Here we solve the differentiai equation in (C.1) in the region bounded by (C.5) and (C.6) to find an
Mplicit expression for the running coupling. F i we define new variables
4 ~ ) r =pu and a(r) = - a,
where
The differentiai equation in (C.1) then becornes
We are interested in the case where a(r) < 1 (a(p) < a,). The solution of this equation gives an implicit
equation for a(r)
One can choose the reference point ro such that [33l
Figure C.2: The shape of the running couphg as a function of logarithmic distance scale.
The resuiting value at the refefence scale is a(ro) = 0.7822. Moreover, one can deîme the scale such that
ro = 1. Then we have
This ùnplicit expression can be inverteci numerically- The shape of the resulting u(r) is shown in Figure C.2
as a function of in(r). Note that the infrared Limit is toward large positive values of In(r). The shape
of a(p) as a function of In(p) can be found by a simple scaiing of the two axis: a + a = a,a and
W.) -+ W) = hW-
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