UNIVERSIDADE ESTADUAL DE CAMPINAS
Instituto de F´ısica Gleb Wataghin
CLARA TEIXEIRA FIGUEIREDO
ASPECTS OF DYNAMICAL MASS GENERATION WITHIN THE
FORMALISM OF SCHWINGER-DYSON EQUATIONS
Aspectos da gera¸c˜ao de massa dinˆamica dentro do formalismo das
equa¸c˜oes de Schwinger-Dyson
CAMPINAS 2020
Clara Teixeira Figueiredo
Aspects of dynamical mass generation within the formalism of Schwinger-Dyson equations
Aspectos da gera¸c˜ao de massa dinˆamica dentro do formalismo das equa¸c˜oes de Schwinger-Dyson
Tese apresentada ao Instituto de F´ısica “Gleb Wataghin” da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten¸c˜ao do t´ıtulo de Doutora em Ciˆencias, na ´area de F´ısica.
Thesis presented to the “Gleb Wataghin” Institute of Physics of the University of Campinas in par-tial fulfillment of the requirements for the degree of Doctor of Science, in the area of Physics.
Orientador: Arlene Cristina Aguilar
Este exemplar corresponde `a vers˜ao fi-nal da tese defendida pela aluna Clara Teixeira Figueiredo e orientada pela Profa. Dra. Arlene Cristina Aguilar.
Campinas 2020
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Figueiredo, Clara Teixeira,
F469a FigAspects of dynamical mass generation within the formalism of Schwinger-Dyson equations / Clara Teixeira Figueiredo. – Campinas, SP : [s.n.], 2020.
FigOrientador: Arlene Cristina Aguilar.
FigTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.
Fig1. QCD não perturbativa. 2. Schwinger-Dyson, Equações de. 3. Geração de massa dinâmica. I. Aguilar, Arlene Cristina, 1977-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Aspectos da geração de massa dinâmica dentro do formalismo
das equações de Schwinger-Dyson
Palavras-chave em inglês:
Non-perturbative QCD Schwinger-Dyson Equations Dynamical mass generation
Área de concentração: Física Titulação: Doutora em Ciências Banca examinadora:
Arlene Cristina Aguilar [Orientador] Adriano Antonio Natale
Attilio Cucchieri Márcio José Menon Jun Takahashi
Data de defesa: 20-08-2020
Programa de Pós-Graduação: Física
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-3995-5934 - Currículo Lattes do autor: http://lattes.cnpq.br/6035273307680376
MEMBROS DA COMISSÃO JULGADORA DA TESE DE DOUTORADO DE CLARA TEIXEIRA FIGUEIREDO – RA 123161 APRESENTADA E APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 20 / 08 / 2020.
COMISSÃO JULGADORA:
- Profa. Dra. Arlene Cristina Aguilar– Orientador – IFGW/UNICAMP - Prof. Dr. Adriano Antonio Natale – IFT/USP
- Prof. Dr. Attilio Cucchieri – IF/USP
- Prof. Dr. Marcio José Menon – IFGW/UNICAMP - Prof. Dr. Jun Takahashi – IFGW/UNICAMP
OBS.: Ata da defesa com as respectivas assinaturas dos membros encontra-se no SIGA/Sistema de
Fluxo de Dissertação/Tese e na Secretaria do Programa da Unidade.
CAMPINAS 2020
Acknowledgements
Firstly, I would like to express my deepest gratitude to my supervisor Professor Cristina Aguilar for the several years of dedicated guidance and for the advice provided throughout my time as her student. I have been extremely blessed to have her as my supervisor. I would also like to extend my sincere gratitude to Professor Joannis Papavassiliou for welcoming me in Valencia, supervising my work there, and teaching me so much.
I also wish to thank my colleague Mauricio N. Ferreira for the discussions and assis-tance provided during the research process. Special thanks to the IFGW Professors Jun Takahashi, Marcelo M. Guzzo, Marcio J. Menon, Orlando L. Goulart, and Pedro C. de Holanda for participating in the evaluation committees and contributing with suggestions and questionings. I would also like to thank Professors Adriano A. Natale and Attilio Cucchieri for their disposition to evaluate and contribute to this work as members of the thesis examination committee.
I would like to acknowledge the financial support from S˜ao Paulo Research Foundation (FAPESP) through the Grants No. 2016/11894-0 and No. 2018/09684-3. I also wish to acknowledge the support provided by the Brazilian National Council for Scientific and Technological Development (CNPq) under Grant No. 142228/2016-8. I would also like to acknowledge that this study was financed in part by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.
I would also like to thank the staff of the “Gleb Wataghin” Institute of Physics (IFGW), the University of Campinas (UNICAMP), and the University of Valencia (UV) for offering the structure and resources needed for the realization of this research project.
Thanks should also go to my family and friends who have supported me during these years. In particular, I am sincerely grateful to my husband, Paulo Henrique, for the love
and support in any circumstance. I am also deeply indebted to my parents, M´arcia and Ricardo, for their constant care and encouragement. In addition, I would also like to thank my brother, Pedro, for his fellowship and assistance whenever needed.
Many thanks to my friend Ren´e for our conversations and his constant disposition to aid in both personal and academic matters. I would also like to extend my thanks to my friends at the University of Valencia for making my time there so pleasant and at the University of Campinas for all the conversations and encouragement during these years.
Finally, I praise God for his grace, sustaining me every day, and for placing all of these people in my life.
Resumo
Neste trabalho estudamos a gera¸c˜ao de uma massa dinˆamica para o gluon na regi˜ao n˜ao-perturbativa de QCD usando o formalismo de equa¸c˜oes de Schwinger-Dyson. Apre-sentamos uma an´alise geral que preserva a transversalidade da autoenergia do gluon, exigida pela simetria de gauge n˜ao-Abeliana da teoria, e resulta em um propagador de gluon finito na regi˜ao infravermelha, corroborando os resultados de v´arias simula¸c˜oes de QCD na rede. Esse estudo ´e realizado dentro do formalismo conhecido como Pinch Tech-nique e sua correspondˆencia com Background Field Method, e faz uso das identidades de Ward satisfeitas pelos v´ertices n˜ao-perturbativos e de uma identidade especial, chamada identidade de seagull. O resultado dessas considera¸c˜oes ´e que o gluon pode adquirir uma massa dinˆamica somente quando polos longitudinalmente acoplados s˜ao incorporados aos v´ertices da teoria. Tais polos representam excita¸c˜oes de estado ligado n˜ao-massivas, que podem ser estudadas dentro do contexto de equa¸c˜oes de Bethe-Salpeter. Trabalhos ante-riores sobre a dinˆamica desses estados ligados consideram a possibilidade de polo somente no v´ertice de trˆes-gluons, negligenciando efeitos de poss´ıveis polos nos demais v´ertices. Aqui, estudamos o impacto do setor de ghost na equa¸c˜ao dinˆamica que descreve a cria¸c˜ao desses polos. Essa an´alise revela que a contribui¸c˜ao do polo associado ao v´ertice ghost-gluon ´e suprimida na gera¸c˜ao dinˆamica de massa do ghost-gluon. Adicionalmente, estudamos a equa¸c˜ao da massa do gluon no gauge de Landau, levando em conta a sua estrutura n˜ao-linear completa, diferentemente de trabalhos pr´evios, nos quais essa equa¸c˜ao ´e n˜ao-linearizada ao considerar o propagador de gluon como uma fun¸c˜ao externa. Com isso, a indeter-mina¸c˜ao na escala da massa gluˆonica encontrada nas an´alises anteriores ´e eliminada. A renormaliza¸c˜ao multiplicativa da equa¸c˜ao da massa ´e realizada de acordo com um m´e-todo aproximado, inspirado no tratamento dado `a massa dinˆamica dos quarks. A massa gluˆonica resultante ´e positiva-definida e monotonicamente decrescente e o propagador de gluon, constru´ıdo a partir dessa massa, est´a de acordo com os dados de simula¸c˜oes de redes de grande volume.
Palavras-chave: QCD n˜ao-perturbativa. Equa¸c˜oes de Schwinger-Dyson. Gera¸c˜ao de massa dinˆamica.
Abstract
In this work, we study the generation of a dynamical mass for the gluon in the nonper-turbative region of QCD using the formalism of Schwinger-Dyson equations. We present a general analysis that preserves the transversality of the gluon self-energy, required from the non-Abelian gauge symmetry of the theory, and results in the infrared finiteness of the gluon propagator observed in several lattice simulations. This study is done within the Pinch Technique formalism, and its correspondence with the Background Field Method, and relies on the Ward identities satisfied by the nonperturbative vertices and a special identity named seagull identity. The result of these considerations is that the gluon can only acquire a dynamical mass when longitudinally coupled massless poles are incorpo-rated into the vertices of the theory. These poles act as colored massless bound state excitations and can be studied under the context of Bethe-Salpeter equations. Previous works on the dynamics of such bound states considered only the possibility of a pole in the three-gluon vertex, neglecting effects from possible poles in the remaining vertices. Here, we study the impact that the ghost sector may have on the dynamical equation that describes the creation of such poles. This analysis reveals that the contribution of the pole associated with the ghost-gluon vertex is suppressed. We also study the gluon mass equation in the Landau gauge, taking into account its full nonlinear structure, con-trary to what has been done in previous works, in which this equation was linearized by considering the gluon propagator as an external input. This eliminates the indeterminacy in the scale of the mass found in these previous analyses. In addition, our treatment of the multiplicative renormalization of the mass equation is carried out according to an approximate method inspired in several works about the quark gap equation. The re-sulting dynamical gluon mass is positive-defined and monotonically decreasing with the emerging gluon propagator matching rather accurately the data from large-volume lattice simulations.
Keywords: Nonperturbative QCD. Schwinger-Dyson Equations. Dynamical mass gener-ation.
List of publications related to this Ph.D.
Articles:
1. A. C. Aguilar, D. Binosi, C. T. Figueiredo, and J. Papavassiliou. Unified description of seagull cancellations and infrared finiteness of gluon propagators. Phys. Rev., D94 (4), 045002, 2016.
2. A. C. Aguilar, D. Binosi, C. T. Figueiredo, and J. Papavassiliou. Evidence of ghost suppression in gluon mass scale dynamics. Eur. Phys. J., C78 (3), 181, 2018. 3. A. C. Aguilar, M. N. Ferreira, C. T. Figueiredo, and J. Papavassiliou.
Nonpertur-bative structure of the ghost-gluon kernel. Phys. Rev., D99 (3), 034026, 2019. 4. A. C. Aguilar, M. N. Ferreira, C. T. Figueiredo, and J. Papavassiliou.
Nonperturba-tive Ball-Chiu construction of the three-gluon vertex. Phys. Rev., D99 (9), 094010, 2019.
5. A. C. Aguilar, M. N. Ferreira, C. T. Figueiredo, and J. Papavassiliou. Gluon mass scale through nonlinearities and vertex interplay. Phys. Rev., D100 (9), 094039, 2019.
Proceedings:
1. C. T. Figueiredo and A. C. Aguilar. Mass generation and the problem of seagull divergences. Journal of Physics: Conference Series, 706, 052007, 2016.
2. J. Papavassiliou, A. C. Aguilar, D. Binosi, and C. T. Figueiredo. Mass generation in Yang-Mills theories. EPJ Web Conf., 164, 03005, 2017.
3. C. T. Figueiredo and A. C. Aguilar. Effects of the ghost sector in gluon mass dynamics. In 14th International Workshop on Hadron Physics, 2018.
List of Abbreviations
1PI One-particle irreducible 2PI Two-particle irreducible BFM Background Field Method
BQ (BB)Gluon propagator with one (two) background gluon(s) BQI Background-Quantum identity
BQQ (BQQQ)Three-gluon (four-gluon) vertex with one background gluon
BS (BSA/BSE)Bethe-Salpeter (Bethe-Salpeter amplitude/Bethe-Salpeter equation) IRInfrared
PT Pinch Technique
QCD Quantum Chromodynamics QED Quantum Electrodynamics QFT Quantum field theory rhs Right hand side
SD (SDE) Schwinger-Dyson (Schwinger-Dyson equation) STISlavnov-Taylor identity
UV Ultraviolet WI Ward identity
Contents
General Introduction 13 1 General Aspects of QCD 17 1.1 QCD Lagrangian . . . 19 1.2 Asymptotic freedom in QCD . . . 21 2 Schwinger-Dyson Equations 24 2.1 Functional formalism . . . 252.2 Functional formalism in gauge theories . . . 27
2.3 SDE for the photon propagator in QED . . . 29
2.4 SDE for the gluon propagator in QCD . . . 32
2.5 Slavnov-Taylor identities . . . 37
3 PT-BFM Framework 40 3.1 Transversality of the gluon self-energy . . . 40
3.2 Background Field Method . . . 42
3.3 New Green’s functions and identities . . . 45
3.4 SDE for the gluon propagator in PT-BFM . . . 50
4 Dynamical Mass Generation and Seagull Cancellation 53 4.1 General considerations . . . 54
4.2 Seagull identity . . . 56
4.3 Ward-Takahashi identities for PT-BFM vertices . . . 57
4.4 Gluon self-energy at the origin . . . 59
4.5 Circumventing the complete seagull cancellation . . . 65
5 Bethe-Salpeter Equations 70 5.1 Derivation of the BSE . . . 71
5.2 The Homogeneous BSE . . . 76
5.3 Numerical treatment . . . 79
6 Massless Bound State Excitations 83 6.1 Relation between the gluon mass and the poles . . . 84
6.2 BSEs for the massless bound-state excitations . . . 86
6.3 Numerical analysis . . . 91
7 Gluon Mass Equation 98 7.1 General Considerations . . . 99
7.2 Derivation of the mass equation . . . 100
7.3 Renormalization of the gluon mass equation . . . 107
7.3.1 Z3 and Z4 from the SDEs for the vertices . . . 110
7.4 Main ingredients of the numerical analysis . . . 111
7.4.1 Kinetic term of the gluon propagator . . . 111
7.4.2 Three- and four-gluon vertices . . . 113
7.4.3 Subdiagram Y(y) . . . 117
7.4.4 Function 1 + G(q2) . . . . 119
7.5 Numerical analysis of the mass equation . . . 119
7.6 Summary . . . 124
8 Conclusions 126
13
General Introduction
The quantum field theory (QFT) responsible for describing the strong interaction in terms of the fundamental fields of quarks and gluons is called Quantum Chromodynamics (QCD). It is a renormalizable, non-Abelian gauge theory that is asymptotically free in the ultraviolet (UV) region [1,2]. Asymptotic freedom assures that the coupling between the quarks and gluons is very small at high energies, which allows for the perturbative treatment in this region. The predictions of the theory in this high energy limit have been successful when tested in different scattering experiments involving elementary par-ticles [3].
On the other hand, the infrared (IR) region of QCD, characterized by small momenta or large distances, still presents several theoretical challenges. In this limit, the coupling is not small enough to apply the known perturbation theory methods. Therefore, the perturbative treatment is not appropriate for the IR region, which accommodates some intriguing phenomena, such as confinement and dynamical mass generation.
Recently our understanding of the IR sector of QCD has advanced considerably due to thorough studies of the fundamental Green’s functions of the theory, i.e., its propagators and vertices. Although these functions depend on the gauge and renormalization point choices, they capture the essential properties of the perturbative and nonperturbative dynamics of the theory. In addition, when properly combined, they can generate physical observables, such as cross sections, decay rates, and hadron masses.
Two first principle approaches stand out in the studies of nonperturbative QCD: (i) lattice QCD [4–12] and (ii) Schwinger-Dyson equations (SDE) [13–21].
Lattice QCD is a method that consists of the discretization of the Euclidean space-time so that the continuous space-time is transformed into a 4-dimensional lattice. The matter
14 fields (quarks) are defined in the lattice points, whereas the gauge fields (gluons) are defined in the links between one point of the lattice and the other. With the imaginary time of Euclidean space, QFTs become analogous to statistical mechanics so that one can apply Monte Carlo simulations [22]. In these simulations, the symmetry may be compromised but eventually recovered in the continuous limit, i.e., when the size of the lattice is taken to infinity and the spacing between points taken to zero. However, the precision of the obtained results depends on the lattice spacing and volume parameters, which are limited to the computational power available. In addition, dealing with a large disparity of physical scales in the theory (take, for example, the different mass scales of the quarks) present some complications for the simulations.
On the other hand, the so-called SDEs furnish the equations of motion for the Green’s functions of the theory, being the analog of the Euler-Lagrange equations for a QFT. Each n-point Green’s function has its own SDE, which, in turn, involves other Green’s functions. Therefore, if we were able to write all the possible equations, we would end up with an infinite set of coupled integral equations, forming an infinite tower of SDEs [23, 24].
Thus, both nonpertubative tools have their pros and cons. On one side, the numerical results obtained from lattice simulations have to be extrapolated to the continuous limit, and the inclusion of quark interactions with real mass values for these quarks is problem-atic from the computational point of view. On the other side, the fundamental difficulty in working with SDEs is related to the need for a self-consistent truncation scheme for the equations, which must not compromise the fundamental properties of the functions studied [21, 25].
Significant advances were obtained in the last decades due to the synergy between both methods. In this thesis, we focus on the SDEs and, therefore, we must deal with the difficulties associated with the need for an appropriate truncation scheme. For that, we make use of the Pinch Technique (PT) formalism [21, 26–30] and its correspondence with the Background Field Method (BFM) [31–33]. The synthesis of these two methods is known in the literature as PT-BFM scheme [19,20,34].
As mentioned previously, one of the interesting phenomena that occur in the IR region of QCD is the dynamical mass generation. In this work, we focus on the mechanisms that enable the generation of a dynamical gluon mass from the SDE for the gluon propagator. This mass generation results in the IR finiteness of the gluon propagator, expressed in the
15 fact that its cofactor, ∆(q2), saturates at a finite non-vanishing value in the low energy
region.
Such behavior was first proposed by Cornwall [21] and later confirmed in large-volume lattice simulations in the Landau gauge, both for SU(2) [4–7] and SU(3) [8–11]. In ad-dition, lattice simulations in linear covariant gauges (Rξ) reveal that this property is not
particular to the Landau gauge (ξ = 0), but persists for other values of ξ evaluated within the interval [0, 0.5] [35]. Additionally, the inclusion of a small number of dynamical quarks (a process called unquenching) produces a relative suppression in the gluon propagator but preserves the IR saturation [12,36,37]. These results offer a valuable opportunity to explore the nonperturbative dynamics of Yang-Mills theories.
The gluon’s running mass function can be obtained directly from the gluon propagator SDE, which generates a nonlinear integral equation for the dynamical mass. In general, this equation is linearized by considering the gluon propagators appearing in the integrand as external functions, given by a fit of the lattice data [38,39]. However, this linearization results in an eigenvalue problem where nontrivial solutions are possible only for specific values of the coupling constant evaluated at the renormalization point. In addition, this process also introduces an indeterminacy in the scale of the mass. Therefore, in this work, we consider the dynamical gluon mass equation taking its nonlinear structure into account.
Moreover, the study of dynamical mass generation for the gluon involves a number of different ideas, including the appearance of poles in the vertices of the theory. These poles represent longitudinally coupled colored massless bound state excitations, which do not appear in the spectrum of the theory but are responsible for the gluon acquiring a mass. The formation of bound states is also a nonperturbative effect and is studied within the context of Bethe-Salpeter equations (BSE) [40]. Consequently, it is also possible to obtain the running gluon mass from studying the coupled system of BSEs, which describes the dynamics of theses massless bound states [41].
In this work, we obtain the dynamical gluon mass employing two formally equivalent methods [41]. The first one consists of using the coupled system of BSEs for the functions representing the poles in the vertices to find the running mass function. In previous analyses, only the possibility of poles in the three-gluon vertex was taken into account, neglecting the effects of poles in other vertices [42, 43], while here we investigate the
16 impact of considering poles in both the three-gluon and ghost-gluon vertices. The second method is to consider the gluon mass equation obtained from the gluon SDE, including its full nonlinear structure and renormalization effects, in contrast to previous studies where this equation was linearized [38, 39]. The results obtained from both methods have been published in [44, 45].
This thesis is then divided into eight chapters. In Chapter1, we present a brief intro-duction to QCD, revealing its Feynman rules and reviewing its property of asymptotic freedom. Then, in Chapter2, we explain how to derive the SDEs of a QFT using the func-tional formalism. We start with the derivation of the SDE for the photon propagator in Quantum Electrodynamics (QED) and, then, generalize to obtain the SDE for the gluon propagator in QCD. Using the functional formalism, we can also derive the Slavnov-Taylor identities (STI) of QCD, which are the non-Abelian generalization of the Ward-Takahashi identities (WTI). In Chapter 3, we present the difficulty in finding a suitable truncation scheme for the gluon SDE and introduce the BFM, with its new Green’s functions and Feynman rules. Then, we write the SDE for a new gluon propagator that appears within the PT-BFM scheme and present the identity that relates this new propagator with the conventional one. We then pass to the study of dynamical gluon mass generation in QCD. In Chapter4, we show how one can obtain a dynamical gluon mass from the SDE of its propagator by introducing the so-called Schwinger mechanism, which allows for a dynam-ical mass generation for the gauge boson of a QFT as long as the coupling of the theory is sufficiently strong. In the case of QCD, such mechanism is triggered by the existence of longitudinally coupled poles in the vertices, which represent colored massless bound state excitations. In Chapter5, we introduce the concept of BSEs by deriving this equation for a simple scalar model, in addition to presenting a generic treatment for solving this equa-tion numerically. In Chapter 6, the dynamical generation of the bound state excitations is studied in the context of BSEs to verify the possibility of poles in the three-gluon and ghost-gluon vertices. In Chapter 7, we present an analysis of the gluon mass equation where its full nonlinear structure and an effective approach to renormalization are taken into account. Finally, in Chapter8, we conclude with a brief discussion about the results of this work.
17
Chapter
1
General Aspects of QCD
QCD is the theory that describes the strong interaction among quarks and gluons, which are the fundamental constituents of hadrons. In the standard model, there are six flavors of quarks: up (u), down (d), strange (s), charm (c), bottom (b), and top (t). They have spin 1/2 and non-integer electric charge (in relation to the fundamental charge of the electron, e), with u, c, and t quarks having charge of +2
3e and d, s, and b of − 1 3e.
Quarks carry a quantum number not present in QED, the so-called color charge, which exists in three types: red, green, and blue. In nature, the colors are combined to create color neutral (or “white”) particles. This occurs most commonly from the combination of the three different colors in the case of baryons (composed by three quarks) or from the combination of color and anticolor in the case of mesons (composed by a pair quark-antiquark).
Gluons have spin 1 and are the particles that mediate the interactions. They also carry the color charge, more precisely, there are eight kinds of gluons carrying a color-anticolor charge. This fact allows them to self-interact, highlighting the non-Abelian character of QCD.
As a non-Abelian gauge theory, QCD presents unique features that do not appear in Abelian theories, such as QED. First, we have asymptotic freedom, which guarantees that in the UV limit the coupling constant goes to zero, so the quarks behave as if they were free. Thus, at high energies, it is possible to apply the perturbative treatment to deal with the Green’s functions of the theory.
However, the same is not true at lower energies, which accommodate several intrigu-ing phenomena, and one has to study QCD nonperturbatively. Among the most famous
18 features, we have color confinement. In basic terms, the property of confinement is man-ifested in the non-observation of free quarks, which are always confined within hadrons. Contrary to what happens with the electron in QED, when one tries to separate the quarks within a hadron, the force between them does not decrease with distance. So an infinite amount of energy would be needed to break the quarks apart, which would at some point produce new hadrons instead of free quarks. More generally, color confinement requires all asymptotic particle states to be color neutral. The full description of the confinement mechanism is still an open problem.
Another important phenomenon occurring in the nonperturbative region of QCD is the dynamical mass generation, both for quarks and gluons. In the case of quarks, dynamical mass generation explains the emergence of the proton mass around 1GeV, whereas the current masses of its constituent quarks are of the order of a few MeV [46].
In fact, because these quark masses are small, there exists an import approximate symmetry called chiral symmetry, which would be exact if the masses of the current quarks were exactly zero. The dynamical breaking of this exact symmetry implies in the quarks acquiring a dynamically generated mass and the appearance of a Goldstone boson. The chiral boson can be identified with the pion, which is not massless because the symmetry is only approximate, but it indeed has a much smaller mass than the other mesons (around 140 MeV). Such dynamical chiral symmetry breaking can only be studied nonperturbatively.
The nonperturbative mechanism that generates a dynamical mass for the gluon differs from the one for the quark because it is not connected to the breaking of any symmetry. The dynamical gluon mass generation is the main object of study of this thesis. Therefore, we detail such mechanism and its consequences in future chapters.
In this Chapter, we start to set up our notation and conventions and, for that, we present the QCD Lagrangian and Feynman rules. Then, we show how the asymptotic freedom is encoded in the perturbative behavior of the QCD coupling constant.
1.1. QCD Lagrangian 19
1.1
QCD Lagrangian
We know that QFTs can be described from their Lagrangians. The QCD Lagrangian can be written as [47] LQCD=LYM+LGF+LDirac+LGhost, (1.1) where LYM =− 1 4G a µνG µν a , (1.2) LGF =− 1 2ξ(∂ µAa µ)2, (1.3) LDirac = ¯ψ(iγµDµ− mq)ψ , (1.4) LGhost = ¯ca(−∂µDµac)c c, (1.5)
with LYM being the Yang-Mills Lagrangian, LDirac the Dirac Lagrangian describing the
interaction between fermionic fields, LGF the gauge fixing Lagrangian, and LGhost the
ghost Lagrangian. The origin of the latter two terms is explained in Chapter 2 from the Faddeev-Popov quantization procedure.
The field strength tensor Ga
µν appearing in Eq. (1.2) is given by
Ga
µν = ∂µAaν − ∂νAµa + gfabcAbµAcν, (1.6)
where g is the coupling constant and Aa
µ the gauge fields. In addition, ξ is the gauge
fixing parameter, c (¯c) the ghost (antighost) field, ψ ( ¯ψ) the quark (antiquark) field, mq
the mass of the quark in question, and γµ the gamma matrices. The covariant derivative
Dµ appearing in Eq. (1.4) is defined as
Dµ = ∂µ− igAaµ
λa
2 , (1.7)
with λa being the Gell-Mann matrices that generate the SU(3) symmetry group [48].
These matrices obey the following relations: λa 2 , λb 2 = ifabcλc 2 , tr(λ aλb) = 2δab, (1.8)
1.1. QCD Lagrangian 20 a b Sab(0) F (p) = δabS (0) F (p) SF(0)(p) = i(γ µp µ+ m) p2− m2+ i a b ∆ab(0) µν (p) = δab∆(0)µν(p) ∆(0) µν(p) =−i gµν − (1 − ξ) pµpν p2 1 p2+ i a b Dab(0)(p) = δabD(0)(p) D(0)(p) = i p2+ i
Figure 1.1: Feynman rules for the quark, gluon, and ghost propagators at tree-level.
where fabc is the structure constant of the SU(3) group. In the adjoint representation,
the covariant derivative Dac
µ of Eq. (1.5) is given by
Dac µ = δ
ac∂
µ+ gfabcAbµ. (1.9)
The gluon self-interaction term of Eq. (1.6) reveals the non-Abelian character of QCD. The gauge fields are allocated to the adjoint representation of SU(3), whereas the fermion fields belong to the fundamental representation. The infinitesimal transformation laws for the fields Aa
µ and ψ are given by
Aaµ→ A a µ+ 1 gD ac µ α c , (1.10) ψ → ψ + iαaλa 2 ψ , (1.11)
with αa being the transformation parameters.
From the QCD Lagrangian, we can obtain the Feynman rules of the theory. The tree-level expressions for the quark, SFab(0), gluon, ∆abµν(0), and ghost, Dab(0), propagators
are given in Fig. 1.1. In addition, in Fig. 1.2 we present the tree-level expressions for the interaction vertices of the theory: the quark-gluon, Γa(0)µ , ghost-gluon, Γabcµ (0),
three-gluon, Γabcµαβ(0), and four-gluon, Γabcdµνρσ(0), vertices. Throughout this thesis, we often extract
the coupling g and the color structures from the vertices, defining
Γa µ(q, r, p) = ig λa 2 Γq,µ(q, r, p) , iΓabc µ (q, r, p) = gfabcΓµ(q, r, p) , iΓabc µαβ(q, r, p) = gfabcΓµαβ(q, r, p) . (1.12)
1.2. Asymptotic freedom in QCD 21 µ, a q r p ig λa 2 Γ (0) q,µ(q, r, p) Γ (0) q,µ(q, r, p) = γ µ µ, a m n q r p gf amnΓ(0) µ (q, r, p) Γ(0)µ (q, r, p) =−rµ q µ, a α, m β, n r p gf amnΓ(0) µαβ(q, r, p) Γ (0) µαβ(q, r, p) = gαβ(r− p)µ+ gµβ(p− q)α +gµα(q− r)β ρ, r σ, s µ, m ν, n −ig2Γmnrs(0) µνρσ (q, r, p, t) Γmnrs(0) µνρσ = f msefern(g µρgνσ − gµνgρσ) +fmnefesr(g µσgνρ− gµρgνσ) +fmrefesn(g µσgνρ− gµνgρσ)
Figure 1.2: Feynman rules for the quark-gluon, ghost-gluon, three-gluon and four-gluon interaction vertices at tree-level; we assume all momenta entering.
However, the Feynman rules can only be applied in the weak coupling limit, i.e., where the perturbative treatment is valid. In this case, one can calculate the quantum corrections to propagators and vertices at higher orders. However, unlike QED, these diagrammatic expansions must take into account the self-interacting gluonic vertices and the ghost-gluon interactions.
1.2
Asymptotic freedom in QCD
The existence of asymptotic freedom in QCD can be verified by studying the β function of the theory. This function expresses the rate at which the renormalized coupling constant changes as the renormalization scale, µ, is increased. The calculation of the β function of QCD was first done simultaneously by Gross and Wilczek [1] and Politzer [2]. From these computations, it was demonstrated that the coupling parameter of QCD decreases as the distance between quarks decreases, which implies that quarks and gluons interact weakly
1.2. Asymptotic freedom in QCD 22 at short distances.
The expression for the β function of a non-Abelian gauge theory is given by
β(αs) = µ dαs(µ) dµ = α2 s π b1+ α3 s π2b2+· · · , (1.13)
where αs = g2/(4π). Through a one-loop calculation, it was found that for the SU(N)
group, we have [1, 2] b1 =− 11 6 CA− 1 3nf , (1.14)
where CAis the eigenvalue of the quadratic Casimir operator in the adjoint representation
and nf is the number of fermions. Using this one-loop approximation for the β function,
one can obtain the perturbative behavior of the coupling constant,
αs(Q2) =− 2π b1ln Q2 Λ2 QCD , (1.15)
where ΛQCD is the QCD scale, defined as
ln Λ2QCD = ln µ2+
2π αs(µ2)b1
. (1.16)
For the color group SU(3), CA= 3, so from Eq. (1.14), one can conclude that, for the
known number of quark flavors (or more generally up until nf ≤ 16), we have asymptotic
freedom, i.e., the coupling decreases when the momentum increases. Such feature results in the quarks being basically free at short distances (high energy limit). This explains, for example, why the protons behave approximately as three free point particles in high energy scattering experiments. Thus, asymptotic freedom allows us to use the coupling constant as a perturbation parameter in the UV limit.
When Q2 = Λ2
QCD, the denominator of Eq. (1.15) becomes zero, therefore we have
a pole at this point, the so-called Landau pole. This indicates that, for Q . ΛQCD the
perturbative treatment cannot be used to describe the dynamics of hadrons, because the coupling is no longer small enough to be used as a perturbation parameter. Typically, the value of ΛQCD is found to be within the 200− 400 MeV range, or ∼ 1 fm in distance
terms (confining region).
1.2. Asymptotic freedom in QCD 23
Figure 1.3: Summary of measurements of αs as a function of the energy scale Q obtained
from [46]. The respective degree of QCD perturbation theory used in the extraction of αs is indicated in brackets (NLO: next-to-leading order; NNLO: next-to-next-to leading
order; res. NNLO: NNLO matched with resummed next-to-leading logs; N3LO:
next-to-NNLO)
order calculations of Eq. (1.15). Note, however, that the coupling constant in itself is not a physical observable, but it enters in predictions for experimentally measurable observ-ables. Thus, the experimental value of the strong coupling constant is inferred from such measurements and is subject to experimental and theoretical uncertainties [46].
We conclude this Section by emphasizing that the perturbative treatment in QCD is only valid for the UV limit. For the low momenta region, where the coupling constant becomes large, we need nonperturbative tools. In this thesis, we use one of the main first principles tools for treating the IR region of QCD, the SDEs, which are introduced in the next Chapter.
24
Chapter
2
Schwinger-Dyson Equations
In this Chapter, we introduce the SDEs, which we will use to study the nonperturbative region of QCD. They were first derived in QED by Dyson [49] and Schwinger [50], and can be understood as the equations of motion describing the dynamics of the Green’s functions of the theory. They form an infinite system of nonlinear integral equations that couple all the existing propagators and vertices. Thus, it is not possible to exactly solve the entire system of SDEs, and it is often needed to truncate this system employing an Ansatz to some of the Green’s functions. We will look into a self-consistent way of performing such truncation on Chapter 3. Here we focus on the formal derivation of the SDEs of a quantum field theory from the generating functional.
In order to do that, first, we give a brief outline of the functional formalism and how it can be used to obtain the correlation functions of a quantum field theory. Then, we review the gauge fixing procedure proposed by Faddeev and Popov, which is necessary for the quantization of gauge theories. With the tools presented, we derive the SDE for the photon propagator of QED. In the sequence, we move to QCD and introduce the SDE for the gluon propagator. Finally, we establish some important identities derived from the non-Abelian gauge symmetry of QCD.
2.1. Functional formalism 25
2.1
Functional formalism
In the functional approach we introduce the generating functional, Z[J], defined by the path integral
Z[J] = Z D[φ] exp iS[φ] + i Z d4xJi(x)φi(x) , (2.1)
where S =R d4xL is the action of the theory and J
i(x) the external sources associated to
the scalar fields φi(x). The sources are set to zero at the end of calculations, serving the
sole purpose of functional differentiation. In addition, J and φ represent the collection of sources and fields, respectively. Therefore, the integral measure D[φ] is defined as
D[φ] ≡ Y
i
D[φi] . (2.2)
From Z[J], one can obtain both connected and disconnected diagrams. Connected diagrams can be further divided into improper and proper. Improper diagrams are those that can be split into two by removing a single line, whereas proper diagrams, also called one-particle irreducible(1PI), cannot be split into two by removing a single line.
When using the path integral formalism, the complete information can be obtained by considering only connected diagrams. To eliminate the contribution from disconnected graphs, we consider the connected generating functional, W [J] [51],
W [J] =−i ln Z[J] . (2.3)
Then, we can obtain the connected n-point Green’s function by deriving W [J] successively n times, h0|T (φi(x1)· · · φj(xn))|0i = 1 in−1 δnW [J] δJi(x1)· · · δJj(xn) J=0 . (2.4)
For example, for the propagator of the field φi, we have
Di(x− z) = h0|T (φi(x)φi(z))|0i = − δ2W [J] δJi(x)δJi(z) J=0 . (2.5)
2.1. Functional formalism 26 Legendre transform of W [J] [51,52], i.e.,
Γ[φcl] = W [J]− Z
d4xJi(x)φcli (x) , (2.6)
where Γ[φcl] is the generator of proper vertices, often called effective action, and φcl i
(called classical field ) represents the vacuum expectation value of the field operator φi in
the presence of the sources. Calculating the partial derivative of the above equation with respect to Ji (keeping φcli fixed) and with respect to φcli (Ji fixed), we obtain
φcli (x) = δW [J] δJi(x) = −i Z[J] δZ[J] δJi(x) ; δΓ[φ cl] δφcl i (x) =−Ji(x) . (2.7)
We often have to convert the Green’s functions of Eq. (2.4) into functions that only take into account 1PI diagrams. For the case of two-point Green’s functions, we can show that δijδ(x− z) = δφcl i (x) δφcl j(z) = Z d4yδφ cl i (x) δJk(y) δJk(y) δφcl j(z) =− Z d4y δ2W [J] δJk(y)δJi(x) δ2Γ[φcl] δφcl j(z)δφclk(y) ! , (2.8)
where we have used the relations of Eq. (2.7). Thus, we have δ2Γ[φcl] δφcl i (x)δφcli(z) =− δ2W [J] δJi(x)δJi(z) −1 . (2.9)
Now, let us consider the following change of variables in Eq. (2.1)
φi(x)→ φ0i(x) + fi(x) , (2.10)
where is infinitesimal and fi(x) is an arbitrary function. Then, the generating functional
becomes Z[J] = Z D[φ0] exp iS[φ0] + i Z d4x δS[φ 0] δφ0 i(x) fi(x) + i Z d4x Ji(x) [φ0i(x) + fi(x)] , (2.11)
where we have used D[φ] = D[φ0], because f
2.2. Functional formalism in gauge theories 27 perform a Taylor expansion around = 0 in the equation above,
Z[J] = Z D[φ] exp iS[φ] + i Z d4x Ji(x)φi(x) × × 1 + i Z d4x0 δS[φ] δφi(x0) + Ji(x0) fi(x0) +O(2) . (2.12)
Thus, the zeroth order term of the expansion already produces the original Z[J], so the order term must evaluate to zero. In particular, choosing fi(x0) = δ4(x0− x), we have1
0 = Z D[φ] δS[φ] δφi(x) + Ji(x) exp i S[φ] + Z d4xJi(x)φi(x) . (2.13) Then, we obtain δS δφi(x) −iδJ(x)δ + Ji(x) Z[J] = 0 , (2.14)
which is the Schwinger-Dyson (SD) relation. By analogy, one can understand this relation as the generalization os the Euler-Lagrange equation for a classical field (δS/δφ = 0). We can take derivatives of Eq. (2.14) with respect to the fields to obtain the corresponding SDEs. In addition, we can expand these equations in powers of the coupling constant to reproduce the known perturbation theory.
2.2
Functional formalism in gauge theories
Before proceeding to derive SDEs of gauge theories, let us obtain the generating func-tional of the free electromagnetic field. The Lagrangian in this case is given by
LEM =−
1 4FµνF
µν, (2.15)
where the field strength tensor is given by
Fµν = ∂µAν − ∂νAµ. (2.16)
1This particular function was chosen for simplicity, but one could keep f
i(x) arbitrary and would have
an integral involving fi(x) in Eq. (2.13). However, since fi(x) is arbitrary, we would need the integrand
2.2. Functional formalism in gauge theories 28 This Lagrangian is invariant under the local transformation
Aµ(x)→ Aµ(x) +
1
e∂µα(x) , (2.17)
which means that these different field configurations are physically equivalent. Therefore, before proceeding to obtain the generating functional of a gauge theory, we need to make sure we count each physical configuration only once. This can be accomplished by means of the Faddeev-Popov procedure [53].
The procedure starts by imposing a gauge fixing condition F (A) = 0. In particular, we choose a general class of functions
F (A) = ∂µA
µ− ω(x) , (2.18)
where ω(x) is an arbitrary scalar function. Then, we insert the following identity in the definition of the generating functional,
1 = Z D[α] det δF (Aα) δα δ[F (Aα)] , (2.19)
where Aα denotes the gauge transformed field Aα
µ = Aµ(x) + 1e∂µα(x). We immediately have δF (Aα) δα = ∂2 e , (2.20)
so the functional determinant appearing in Eq. (2.19) is independent of Aµ and can be
treated as a constant in the generating functional integral.
In addition, since the Lagrangian is invariant under gauge transformations, we have S[A] = S[Aα]. Thus, after a change of variables, we can write
Z
D[A]eiS[A]= det ∂2 e Z D[α] Z D[A]eiS[A]δ[∂µA µ− ω(x)] . (2.21)
Since this equation is true for any arbitrary ω(x), it must also hold when integrated over all ω(x) with a Gaussian weight function, exp(−iR d4x ω2(x)/2ξ). Then, we have
Z
D[A]eiS[A]= N (ξ) det ∂2 e Z (D[α]) Z
D[A]eiS[A]exp −i Z d4x(∂ µA µ)2 2ξ , (2.22)
2.3. SDE for the photon propagator in QED 29 Eq. (2.21). The normalization constant N (ξ) along with the other factors in front of the integral inD[A] will cancel when calculating any correlation function, after performing the same manipulation on both the numerator and denominator of these functions. Therefore, the net effect of Eq. (2.22) in the generating functional is to add the extra ξ term in the action, i.e., S[A] = Z d4x −14FµνFµν − (∂µA µ)2 2ξ . (2.23)
This extra contribution is known as gauge fixing term.
2.3
SDE for the photon propagator in QED
With the machinery presented in the previous sections, we can proceed to derive SDEs. We start with the example of the photon propagator in QED. For that, we use the QED Lagrangian, given by LQED =LEM+LDirac =−1 4FµνF µν + ¯ψ(iγµD µ− m)ψ , (2.24)
where Dµ= ∂µ+ ieAµis the gauge-covariant derivative and m is que mass of the fermion.
Using the Faddeev-Popov trick, the action we use in the generating functional of QED can be written as S[A, ψ, ψ] = Z d4x 1 2A µ(∂2g µν − (1 − ξ−1)∂µ∂ν)Aν + ψ(iγµDµ− m)ψ . (2.25)
Then, for the generating functional, we have
Z[J, η, η] = Z D[u] exp i S[A, ψ, ψ] + Z d4x(JµAµ+ ηψ + ψη) , (2.26)
where Jµis the source related to the gauge field Aµ, while η and η are the sources associated
to the fermionic fields ψ and ψ, respectively. In addition, the measure D[u] is defined as
D[u] = D[A] D[ψ] D[ψ] . (2.27)
2.3. SDE for the photon propagator in QED 30 to Γ[A, ψ, ψ], instead of Z[J, η, η]. From the Legendre transform of W [J, η, η],
Γ[A, ψ, ψ] = W [J, η, η]− Z
d4x(JµAµ+ ηψ + ψη) , (2.28)
we obtain the following relations: δW δJµ = Aµ, δW δη = ψ , δW δη =−ψ , δΓ δAµ =−Jµ, δΓ δψ = η , δΓ δψ =−η . (2.29)
In Eqs. (2.28) and (2.29) the symbols Aµ, ψ, and ψ represent expectation values of the
fields2.
Then, in order to obtain the photon propagator SDE, we need the functional derivative of the action presented in Eq. (2.25) with respect to the gauge field Aµ, i.e.
δS δAµ
= [∂2gµν− (1 − ξ−1)∂µ∂ν]Aν − eψγµψ . (2.30)
Now, setting Z = eiW, we can write the expression analogous to Eq. (2.14) as
e−iW [J,η,η] δS δAµ −i δ δJν ,−i δ δη, i δ δη + Jµ eiW[J,η,η] = 0 , (2.31) so that we find Jµ+ [∂2gµν− (1 − ξ−1)∂µ∂ν] δW δJν − e δW δη γµ δW δη − ie δ δη γµ δW δη = 0 . (2.32)
Next, we must convert Eq. (2.32) given in terms of W into an expression for Γ. Using the relations of Eq. (2.29), the equation above can be rewritten as
δΓ δAµ(x) ψ=ψ=0 = [∂2gµν− (1 − ξ−1)∂µ∂ν]Aν(x)− ie Tr " γµ δ2Γ δψ(x)δψ(x) −1# , (2.33)
where we have used
δ2Γ δψδψ −1 =−δ 2W δηδη , (2.34) as in Eq. (2.9).
2.3. SDE for the photon propagator in QED 31 Now, we must take the derivative of Eq. (2.33) with respect to Aν, in order to obtain
the SDE for the photon propagator, defined as
∆−1 µν(x− y) = δ2Γ δAν(y)δAµ(x) A=ψ=ψ=0 . (2.35)
Thus, we finally find
∆−1µν(x− y) = [∂2gµν− (1 − ξ−1) ∂µ∂ν] δ4(x− y)
+ ie2Tr Z
d4u d4v [γµS
F(x− u)Λν(y, u, v)SF(v− x)] , (2.36)
where we have used that the derivative of an inverse matrix, M−1, can be obtained from
δM−1 δAν(y) =−M−1 δ δAν(y) M M−1, (2.37) with M = δ 2Γ δψ(x1)δψ(x2) ,
and the appropriate integrals in the relation implied. In Eq. (2.36), we have defined the fermion propagator as SF(x− y) = δ2Γ δψ(y)δψ(x) −1 ψ=ψ=0 , (2.38)
and the electron-photon vertex as δ3Γ
δAν(y) δψ(u) δψ(v)
A=ψ=ψ=0 = eΛν(y, u, v) . (2.39)
Thus, Eq. (2.36) is the SDE for the photon propagator, which can be written as
∆−1µν(x− y) = [∂2gµν − (1 − ξ−1) ∂µ∂ν] δ4(x− y) + Πµν(x, y) ,
where Πµν is the photon self-energy,
Πµν(x, y) = ie2Tr
Z
2.4. SDE for the gluon propagator in QCD 32 −1 = −1 + q q k + q k
Figure 2.1: Diagrammatic representation of the SDE for the photon propagator, ∆µν(q).
White circles represent fully dressed propagators and the black circle corresponds to the fully dressed electron-photon vertex.
Finally, Eq. (2.40) can be Fourier transformed to momentum space, arriving at
Πµν(q) = ie2
Z d4k
(2π)4Tr [γ µS
F(k) Λν(k, k + q) SF(k + q)] . (2.41)
Then, we find the usual form for the photon propagator SDE,
i∆−1µν(q) =−q2 gµν − (1 − ξ−1) qµqν q2 + Πµν(q) . (2.42)
In Fig.2.1, we represent this equation diagrammatically.
2.4
SDE for the gluon propagator in QCD
The derivation of the SDE for the gluon propagator is longer and more cumbersome than that of QED, due to the non-Abelian structure of QCD, which results in additional Green’s functions, such as the triple and four gluon self-interactions vertices and propa-gators and vertices involving ghosts. Naturally, there are SDEs for all the n-point Green’s functions composed by quarks, gluons, and ghosts fields, which involve knowledge of all the propagators and vertices of the theory.
In order derive the LGF and LGhost terms presented in Eqs. (1.3) and (1.5), one needs
to apply the Faddeev-Popov procedure to the Yang-Mills Lagrangian, LYM, defined in
Eq. (1.2). Then, applying the gauge condition
Fa(A) = ∂µAaµ(x)− ω a
2.4. SDE for the gluon propagator in QCD 33 in the gauge transformation of Eq. (1.10), we obtain
δFa(Aα) δαc = 1 g∂ µDac µ , (2.44)
where Aαis the gauge transformed field and Dac
µ was defined in Eq. (1.9). One can see that
in the case of QCD the derivative is no longer independent of the gauge field Aa
µ. Then,
the functional determinant of this derivative adds new terms in the QCD Lagrangian. Faddeev and Popov [53] represented such determinant as a functional integral over a set of anticommuting fields in the adjoint representation, which are called ghosts,
det 1 g∂ µ Dacµ = Z DcDc exp i Z d4x c(−∂µ Dµac)c . (2.45)
These fields are Grassmann variables (usually employed for fermion fields), but are scalars (spin 0) under Lorentz transformations. Thus their quantum excitations cannot be phys-ical particles, because they violate spin-statistics.
After including the contribution from ghosts and fermionic fields, we find the La-grangian of QCD presented in Eq. (1.1). The LQCD Lagrangian must be renormalized
according to the prescription:
g = ZggR, ξ = ZξξR, ψ = Z 1/2 2 ψR, ca= Z1/2 c c a R, A a µ = Z 1/2 A A a µR, m = ZmmR, (2.46)
where the subscript R indicates the renormalized quantity, Zg is the renormalization
constant for the coupling constant g, Zξ the one for the gauge fixing parameter ξ, Zm for
the quark mass, and Z2, Zc, and ZA for the quark, ghost, and gluon fields respectively.
So, we obtain LR QCD=L R YM+L R GF+L R Dirac+L R Ghost, (2.47) where LR YM+L R GF = ZA 1 2A µ a ∂2gµν− 1− 1 ZQξ ∂µ∂ν Aν a− Z3gf abc(∂ µAaµ)A µ bA ν c − Z4 1 4g 2fabefcdeAµ aA ν bA c µA d ν,
2.4. SDE for the gluon propagator in QCD 34 LR Dirac = Z2ψ(iγµ∂µ− Zmm)ψ− iZ1Fgψγµ λa 2 ψA µ a, LR Ghost =−Zcca∂2ca− Z1gfabcca∂µ(Aµbc c) , (2.48)
with Z1F, Z1, Z3, and Z4 being the renormalization constants for the quark-gluon,
ghost-gluon, three-gluon and four-gluon vertices, respectively. These renormalization constants are given by [54] Z1F = ZgZ2ZA1/2, Z1 = ZgZcZ 1/2 A , Z3 = ZgZ 3/2 A , Z4 = Z 2 gZ 2 A. (2.49)
Then, we can proceed to obtain the SDE for the gluon propagator using the renormal-ized Lagrangian in a similar way to what was done in the case of the photon.
In order to simplify our expressions, from now on we denote Z D[φ] δS[φ] δφi(x) + Ji(x) exp i S[φ] + Z d4xJi(x)φi(x) =: δS δφi + Ji . (2.50)
Deriving the gluon propagator SDE requires taking two derivatives of the action in relation to the gauge field, Aa
µ, as in the case of the photon propagator. According to Eq. (2.13),
we must have δSQCD δAa µ(x) + Jµa = 0 , (2.51) where Ja
µ is the source of the Aaµ field and the QCD action is given in terms of its
La-grangian,
SQCD[A, ψ, ψ, c, c] =
Z
d4xLR
QCD . (2.52)
In order to obtain the contribution of 1PI diagrams to the complete gluon propagator, we must use the generating functional ΓQCD,
∆ab µν(x− y) −1 = δ 2Γ QCD δAν b(y)δA µ a(x) A=ψ=ψ=0 . (2.53)
Then, from the SD relation presented in Eq. (2.51) and using the Lagrangian of Eq. (2.47), after a lengthy calculation following the same steps from the derivation of the photon
2.4. SDE for the gluon propagator in QCD 35 propagator, we can obtain [16]
δ2Γ QCD δAν b(y)δA µ a(x) = ZA ∂2gµν − 1− 1 ZQξ ∂µ∂ν δabδ(x− y) − Z1gfade Z d4zhAν b(y)A ρ c(z)i−1 Aρc(z)(∂µcd(x))ce(x) − Z1F ig λa 2 γµ Z d4zhAν b(y)A ρ c(z)i−1 Aρc(z)ψ(x)ψ(x) + Z3gfade Z d4zhAν b(y)A ρ c(z)i−1{hA ρ c(z)A σ d(x)∂µAeσ(x)i − hAρ c(z)A σ d(x)∂σAeµ(x)i − hA ρ c(z)∂σAσd(x)A e µ(x)i} − Z4g2faf gfgde{δbfhAdµ(x)A e ν(x)i + δ be hAf ν(x)A d µ(x)i + δbdgµνhAρf(x)Aeρ(x)i}δ(x − y) − Z4g2faf gfgde Z d4zhAν b(y)A ρ c(z)i−1hA ρ c(z)A σ f(x)A d µ(x)A e σ(x)i , (2.54)
where the brackets represent correlation functions.
Then, the resulting SDE for the gluon propagator in Euclidean space is given by ∆ab µν(x− y) −1 = ZA ∆ab(0) µν (x− y) −1 + Z1g2 Z d4x12d4y12Γacdµ (0)(x, x1, x2)D de(x 2− y1)Df c(y2− x1)Γbefν (y, y1, y2) + Z1F g2 Z d4x12d4y12Γµa(0)(x, x1, x2)SF(x2− y1)SF(y2− x1)Γbν(y, y1, y2) + Z3 g2 2 Z d4x12d4y12Γacdµαβ(0)(x, x1, x2)∆ βγ de(x2− y1)∆ αδ cf(x1− y2)Γbefνγδ(y, y1, y2) + Z4 g2 2 Z d4x12Γabcdµναβ(0)(x, y, x1, x2)∆ αβ cd(x1− x2) + Z4 g4 6 Z d4x123d4y123Γacde(0)µαβγ (x, x1, x2, x3)∆ αλ cm(x1− y3)∆ βσ dl (x2− y2)∆ γρ ek(x3− y1)× × Γbklm νρσλ(y, y1, y2, y3) + Z4 g4 2 Z d4x123d4y123d4z12Γacdeµαβγ(0)(x, x1, x2, x3)∆ αρ ck(x1 − y1)∆ βλ dm(x2− y3)∆ σκ lq (y2− z2)× × Γklm ρσλ(y1, y2, y3)∆γδep(x3− z1)Γbpqνδκ(y, z1, z2) , (2.55)
where Dabis the complete ghost propagator and Γa
µ, Γabcµ , Γabcµνα, and Γabcdµναβ represent the full
2.4. SDE for the gluon propagator in QCD 36 we have defined the following notation for the integration measure
d4x1...j =: j
Y
i=1
d4xi, (2.56)
and analogously for the variables yi and zi.
Finally, we apply the Fourier transform to obtain the equation in momentum space, ∆ab µν(q) −1 = ZA ∆ab(0) µν (q) −1 + 6 X i=1 (di)abµν(q) , (2.57) where (d1)abµν(q) = Z3 2 g 2 Z d4k (2π)4 Γ acd(0) µαβ (q, k,−k − q)∆ βγ de(k + q)∆ cf αδ(k) Γ bef νγδ(−q, k + q, −k) , (d2)abµν(q) = Z4 2 g 2 Z d4k (2π)4Γ abcd(0) µνρσ ∆ ρσ cd(k) , (d3)abµν(q) = Z1g2 Z d4k (2π)4 Γ adc(0) µ (q,−k − q, k) D de (k + q)Dcf(k)Γbf eν (−q, −k, k + q) , (d4)abµν(q) = Z4 6 g 4 Z d4k 1 (2π)4 d4k 2 (2π)4Γ acde(0) µρσλ ∆ ρρ0 cf(k2) ∆ σσ0 dh (q + k1+ k2)∆λλ 0 eg (k1) × Γf bghρ0νλ0σ0(−k2,−q, −k1, q + k1+ k2) , (d5)abµν(q) = Z4 2 g 4Z d4k1 (2π)4 d4k 2 (2π)4Γ acde(0) µαβγ ∆ γγ0 ef (q + k1+ k2) ∆ ζζ0 nm(k1+ k2)∆ββ 0 dh (k2)∆ αα0 cg (k1) × Γbf n νγ0ζ0(−q, q + k1+ k2,−k1− k2)Γmhgζβ0α0(k1+ k2,−k2,−k1) , (d6)abµν(q) = Z1F g 2 Z d4k (2π)4 Γ a(0) µ (q, k,−k − q) SF(k + q)SF(k)Γbν(−q, k + q, −k) . (2.58)
Note that in Eqs. (2.55) and (2.58) we have factored out the coupling constant g from the definition of the vertices.
In this work, we use the so-called quenched approximation, in which we neglect the effects of quark fields, considering that the interactions in the IR region of QCD are mainly dominated by the dynamics of the gluon fields. Then, disregarding the contribution coming from (d6)abµν(q) in Eq. (2.57), we obtain the SDE presented in Fig. 2.2, where the
gluon self-energy is given by
Πµν(q) = 5
X
i=1
2.5. Slavnov-Taylor identities 37 ∆−1µν(q) = + + 16 (d3) −1 + 12 + 12 (d1) (d2) (d4) + 12 (d5)
Figure 2.2: Diagrammatic representation of the quenched gluon propagator SDE. White circles represent full propagators and black ones full vertices. Diagrams (d1)-(d5) define
the gluon self-energy, Πµν(q).
2.5
Slavnov-Taylor identities
In the previous sections, we have seen that the fact that a shift in the integration variable does not change the result of the path integral allows us to derive the SDEs for the Green’s functions of the theory. In the same way, the functional formalism provides a class of identities which result from taking these field transformations to be symmetries of the action. In this case, only the source terms and the terms introduced by the Faddeev-Popov method will change inside the integrand, since the action is invariant. Thus, we can derive constraints from the symmetry of the theory, such as the WTIs of QED.
The generalization of the WTIs, which result from Abelian gauge symmetry, to the case of non-Abelian gauge symmetries is the STIs [55, 56]. The derivation of these identities can be quite intricate, so we will present them here without derivation.
In the case of QCD, the simplest STI is the one satisfied by the gluon propagator, which in momentum space is given by
qµqν∆ab
µν(q) =−iξδab. (2.60)
2.5. Slavnov-Taylor identities 38 longitudinal parts, ∆ab µν(q) =−iδ ab Pµν(q)∆(q) + E(q) qµqν q2 , (2.61)
where Pµν(q) is the transverse projector,
Pµν(q) = gµν− qµqν/q2, (2.62)
and ∆(q) and E(q) are scalars associated with the transverse and longitudinal parts of the gluon propagator, respectively. Using Eq. (2.61) inside Eq. (2.60), we obtain E(q) = ξ/q2,
so that ∆abµν(q) =−iδ ab Pµν(q)∆(q) + ξ qµqν q4 . (2.63)
Therefore, from Eq. (2.63) we see that the longitudinal part of the propagator does not gain radiative corrections. In particular, in the Landau gauge, i.e., ξ = 0, the complete gluon propagator is purely transverse, a fact that we use throughout this work.
In the derivation of the gluon vertex SDE, one defines the quantity called ghost-gluon scattering kernel, Hcab
µν , −iHbac νµ(y, x, z) = igf cabg µνδ(x− y)δ(y − z) − gfbde δ2 δAc µ(z)δca(x) δ2Γ δAd ν(y)δce(y) −1 , (2.64)
which is related to the ghost-gluon vertex by
Γcba µ (z, y, x) =−i δ3Γ δAc µ(z)δca(x)δcb(y) =−i∂ν yH bac νµ(y, x, z) , (2.65)
where the subscript y in the derivative specifies the variable of differentiation. In momen-tum space, Eq. (2.65) reads
Γcba
µ (r, q, p) = q νHbac
νµ(q, p, r) , (2.66)
where r, q, and p are the gluon, anti-ghost, and ghost momenta, respectively. In Fig.2.3, we present the diagrammatic representation of the ghost-gluon scattering kernel, Hνµ.
2.5. Slavnov-Taylor identities 39 + Hνµ(q, p, r) = ν q p µ r p µ ν q r
Figure 2.3: The diagrammatic representation of the ghost-gluon scattering kernel. The tree-level contribution is given by gµν.
Finally, for the three-gluon vertex STI, we have
rµΓ αµν(q, r, p) =−ir2D(r) ∆−1(q)Pµ α(q)Hµν(q, r, p)− ∆−1(p)Pνµ(p)Hµα(p, r, q) , (2.67)
where we have extracted the color structure,
Habc
µν (q, p, r) =−gf abcH
µν(q, p, r) . (2.68)
In addition, one can obtain two other STIs for the three-gluon vertex, by performing cyclic permutations in Eq. (2.67).
With these identities and the gluon SDE, given in Sec.2.4, we can end this Chapter. In the next, we study a framework that allows us to truncate the infinite system of SDEs in QCD by generating new STIs for the vertices, which have a structure similar to the WTIs of QED.
40
Chapter
3
PT-BFM Framework
In the previous Chapter, we have shown that the (quenched) gluon propagator SDE involves ghost and gluon propagators, three- and four- gluon vertices, and the ghost-gluon vertex (see Fig. 2.2). Each of these functions satisfies their own SDE, involving other Green’s functions. Therefore, there is an obvious need to truncate this infinite tower of equations. However, one of the main difficulties in working with SDEs in QCD is to obtain a self-consistent truncation which does not violate the essential symmetries of the theory.
In this Chapter, we briefly introduce the PT-BFM scheme, which we can use to trun-cate the SDEs while preserving gauge symmetry. To do that, first, we discuss the difficulty that arises when performing a naive truncation in the SDE for the gluon propagator. Then, we introduce the basic ideas of the BFM, and we see that new Green’s functions, endowed with special properties, emerge in this formalism. Those functions are not only related to the propagators and vertices of the conventional QCD, but, more importantly, they obey QED-like WTIs instead of the STIs. We conclude by presenting the SDE for the gluon propagator in the PT-BFM formalism, which allows for symmetry preserving truncations.
3.1
Transversality of the gluon self-energy
The gluon self-energy, Πµν(q), is defined as the sum of the (di) diagrams of Fig. 2.2
(see Eq. (2.59)). From the gluon STI of Eq. (2.60), we know that non-Abelian gauge symmetry ensures that the longitudinal part of the gluon propagator does not acquire perturbative and nonperturbative corrections. Therefore, the gluon self-energy must be
3.1. Transversality of the gluon self-energy 41 transverse, i.e.,
qµΠ
µν(q) = 0. (3.1)
Thus, it is convenient to write the complete gluon propagator, in general covariant gauges, as in Eq. (2.63), ∆µν(q) =−i Pµν(q)∆(q2) + ξ qµqν q4 , (3.2)
where Pµν(q) is the transverse projector, given in Eq. (2.62) and ∆(q2) is the gluon
propagator scalar function defined in terms of the self-energy, Πµν(q) = Pµν(q)Π(q2),
through the relation (Minkowski space)
∆−1(q2) = q2+ iΠ(q2) . (3.3)
It is useful to write the form of the inverse gluon propagator, ∆−1
µν(q), defined as ∆−1µν(q)∆ ρν (q) = gµρ, (3.4) where ∆−1 µν(q) = i Pµν(q)∆−1(q2) + ξ−1 qµqν q4 , (3.5)
which is singular in the Landau gauge (ξ = 0). However, because the explicit ξ dependence is present only in the longitudinal term, which does not acquire dressing, this term is canceled by the tree-level gluon propagators appearing in the calculations.
To understand the problem that emerges when truncating the gluon SDE, note that if one considers only diagrams (d1) and (d2) of Fig.2.2at one-loop of perturbation theory, the
gluon self-energy transversality is broken. The sum of these two diagrams (in Minkowski space) is given by Π(1)µν(q)|(d1)+ Π (1) µν(q)|(d2) = λ 36 84qµqν − 75gµνq 2ln −q2 µ2 , (3.6) where λ = iCAg 2 16π2 , (3.7)
3.2. Background Field Method 42 so that Eq. (3.6) is not proportional to the transverse projector given in Eq. (2.62). Note that Eq. (3.6) was obtained after renormalization in the momentum subtraction scheme, where the tree-level value for Πµν(q) is recovered at the off-shell momentum q2 =−µ2 < 0.
At one-loop we know that we must also add the contribution from the ghost diagram (d3). Then, summing up the three one-loop diagrams, we have
Π(1)µν(q)|(d1)+ Π (1) µν(q)|(d2)+ Π (1) µν(q)|(d3)=− 13λ 6 (gµνq 2 − q µqν)ln −q2 µ2 , (3.8)
which is indeed transverse. Thus, we conclude that one cannot simply ignore the contri-bution from the ghost diagram without breaking the fundamental symmetry of the theory already at one loop.
However, at the two-loops level, considering only the contribution from the sum of these three diagrams does not suffice to produce a transverse self-energy. Thus, at all orders, one has that
qµΠ µν(q) (d1)+(d2)+(d3) 6= 0 . (3.9)
Therefore, the gluon self-energy transversality only emerges after the inclusion of all dia-grams di shown in Fig. 2.2.
The SDE formulation based on the so-called PT-BFM formalism [14,34,57] allows us to circumvent this problem. Essentially, this formalism enables the construction of a new SDE for the gluon propagator, where the dressed diagrams are organized in independently transverse groups, leading to a transversality preserving truncation scheme [14, 29, 34]. The reason behind how this happens is related to the emergence of new Green’s functions in the PT-BFM formalism, which obey Abelian WTIs, in contradistinction to the non-Abelian STIs satisfied by the conventional QCD functions [29]. To better understand the ideas mentioned above, let us briefly introduce the BFM [31–33].
3.2
Background Field Method
As discussed in the previous Chapter, the functional formalism can be applied to gauge theories according to the Faddeev-Popov procedure [53]. Then, using the
Yang-3.2. Background Field Method 43 Mills Lagrangian of Eq. (1.2), the generating functional becomes
Z[J] = Z D[A] det δFa δωb exp iS(A) + i Z d4x − 1 2ξ(F a)2+ Ja µA aµ , (3.10) where Aa
µ is the gauge field and ωa is the gauge transformation parameter,
δAaµ = 1 g∂µω a − fabc ωbAcµ. (3.11)
For better visualization of the next steps, we have factored out the gauge fixing term, Fa,
from the definition of the action.
The basic idea of the BFM is to split the gauge field into a classical background field (Ba
µ) and a fluctuating quantum field (Qaµ), i.e.,
Aa µ → Q a µ+ B a µ. (3.12)
We treat the classical part Ba
µ as a fixed field configuration and the fluctuating part Qaµ
as the integration variable of the functional integral, where
Z[J] → eZ[J, B] , Fa(A)
→ eFa(Q, B) . (3.13)
Then, applying this change of variable in the generating functional, we obtain e Z[J, B] = Z D[Q] det " δ eFa δωb # exp iS(Q + B) + i Z d4x −2ξ1 ( eFa)2+ Ja µQ aµ , (3.14)
where the background field is not coupled to the source [32]. The term inside the determi-nant corresponds to the derivative of the gauge fixing term under the infinitesimal gauge transformation δQa µ= 1 g∂µω a − fabcωb(Qc µ+ B c µ). (3.15)
Note that the gauge fixing term, eFa(Q, B), can depend on both the quantum and
back-ground fields.
3.2. Background Field Method 44 as in the previous Chapter (see Eqs. (2.3) and (2.6)),
f W [J, B] =−i ln eZ[J, B] , eΓ[Q, B] = fW [J, B]− Z d4xJa µ(x)Q a µ(x) , (3.16) where Qa
µis the quantum field argument of the background field effective action (remember
that, in this case, it is the expectation value of the field),
Qa µ= δfW δJa µ . (3.17)
Now we choose a special gauge condition, which preserves gauge invariance in terms of the background field B, given by
e Fa(Q, B) = ∂µQa µ+ gf abcBb µQ cµ. (3.18)
Then, it is possible to demonstrate that eZ[J, B] and fW [J, B] are invariant under the transformations [33] δBa µ =−f abcωbBc µ+ 1 g∂µω a, (3.19) δJa µ =−f abcωbJc µ. (3.20)
In addition, for this background field gauge condition, we have that eΓ[Q, B] is invariant under δBa µ=−f abcωbBc µ+ 1 g∂µω a, (3.21) δQa µ=−fabcωbQcµ. (3.22) Notice that Ba
µ carries the local gauge transformation, whereas Qaµtransforms as a matter
field in the adjoint representation. Using that δQa
µ= 0 when the expectation value of the
field is zero (Qa
µ = 0), we have that eΓ[0, B] is an explicitly gauge invariant functional of
Ba µ.
The functional eΓ[0, B] is the effective action used in calculations of the BFM. Then, because this effective action is invariant under a gauge transformation of the background field, it can be shown that the 1PI Green’s functions obtained from derivatives of eΓ[0, B]
3.3. New Green’s functions and identities 45 satisfy Abelian WTIs.
The relation between the conventional effective action, Γ, and the background action, eΓ, can be obtained from the change of variables Qa
µ→ Aaµ− Bµa in Eq. (3.14), which leads
to e Z[J, B] = Z[J] exp −i Z d4xJa µB aµ , (3.23)
where Z[J] is calculated with the corresponding gauge fixing term
Fa = ∂µAa µ− ∂ µBa µ+ gf abcBb µA cµ. (3.24)
Following the steps of Ref. [33], one can verify from Eq. (3.23) that
eΓ[0, B] = Γ[B] . (3.25)
Therefore, the BFM effective action, eΓ[0, B], corresponds to the usual effective action, Γ[B], for the specific gauge condition of Eq. (3.24) when the vacuum expectation value of the field Qa
µ is set to zero.
The above correspondence between background and conventional effective actions im-plies that physical observables calculated using the BFM are equal to those obtained from the conventional formalism, despite having different Green’s functions. Thus, one can use the BFM Green’s functions, which satisfy WTIs, instead of the conventional functions that obey the more complicated STIs. Additionally, the BFM is closely related to the PT formalism [21, 26–30], an algorithm for constructing gauge fixing parameter independent Green’s functions. The synthesis of these two methods generates the formalism known as PT-BFM [19, 20, 34].
3.3
New Green’s functions and identities
We have just seen that in the PT-BFM formalism we have two types of gluon fields, which lead to three different gluon propagators, depending on the nature of their external legs [29]. In addition to the conventional propagator, ∆µν(q), composed by two external
