UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto De F´Isica Gleb Wataghin

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¸cão de massa dinâmica dentro do formalismo das equa¸cõesde Schwinger-Dyson

CAMPINAS 2020 Clara Teixeira Figueiredo

Aspects of dynamical mass generation within the formalism of Schwinger-Dyson equations

Aspectos da gera¸cão de massa dinâmica dentro do formalismo das equa¸cões 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¸cão do t´ıtulo de Doutora em Ciências, na área de F´ısica.

Thesis presented to the “Gleb Wataghin” Institute of Physics of the University of Campinas in partial fulﬁllment of the requirements for the degree of Doctor of Science, in the area of Physics.

Orientador: Arlene Cristina Aguilar

Este exemplar corresponde a` versao˜ final 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, 1991- 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 assistance 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 SãoPaulo 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¸cãode 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, Márcia 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 Renéfor 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¸cão de uma massa dinâmica para o gluon na região não-perturbativa de QCD usando o formalismo de equa¸cõesde Schwinger-Dyson. Apre- sentamos uma análise geral que preserva a transversalidade da autoenergia do gluon, exigida pela simetria de gauge não-Abeliana da teoria, e resulta em um propagador de gluon finito na região infravermelha, corroborando os resultados de várias simula¸cões de QCD na rede. Esse estudo érealizado dentro do formalismo conhecido como Pinch Tech- nique e sua correspondência com Background Field Method, e faz uso das identidades de Ward satisfeitas pelos vértices não-perturbativos e de uma identidade especial, chamada identidade de seagull. O resultado dessas considera¸cões éque o gluon pode adquirir uma massa dinâmica somente quando polos longitudinalmente acoplados são incorporados aos vértices da teoria. Tais polos representam excita¸cões de estado ligado não-massivas, que podem ser estudadas dentro do contexto de equa¸cões de Bethe-Salpeter. Trabalhos anteriores sobre a dinâmica desses estados ligados consideram a possibilidade de polo somente no vértice de três-gluons, negligenciando efeitos de poss´ıveis polos nos demais vértices. Aqui, estudamos o impacto do setor de ghost na equa¸cão dinâmica que descreve a cria¸cão desses polos. Essa análise revela que a contribui¸cão do polo associado ao vértice ghost- gluon ésuprimida na gera¸cão dinâmica de massa do gluon. Adicionalmente, estudamos a equa¸cãoda massa do gluon no gauge de Landau, levando em conta a sua estrutura não- linear completa, diferentemente de trabalhos prévios, nos quais essa equa¸cãoélinearizada ao considerar o propagador de gluon como uma fun¸cão externa. Com isso, a indeter- mina¸cãona escala da massa gluônica encontrada nas análises anteriores éeliminada. A renormaliza¸cãomultiplicativa da equa¸cãoda massa érealizada de acordo com um mé- todo aproximado, inspirado no tratamento dado àmassa dinâmica dos quarks. A massa gluônica resultante épositiva-definida e monotonicamente decrescente e o propagador de gluon, constru´ıdo a partir dessa massa, estáde acordo com os dados de simula¸cões de redes de grande volume.

Palavras-chave: QCD não-perturbativa. Equa¸cões de Schwinger-Dyson. Gera¸cão de massa dinâmica. Abstract

In this work, we study the generation of a dynamical mass for the gluon in the nonperturbative 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, contrary 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 resulting 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 generation. 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)

IR Infrared

PT Pinch Technique

QCD Quantum Chromodynamics

QED Quantum Electrodynamics

QFT Quantum ﬁeld theory

rhs Right hand side

SD (SDE) Schwinger-Dyson (Schwinger-Dyson equation)

STI Slavnov-Taylor identity

UV Ultraviolet

WI Ward identity

WTI Ward-Takahashi identity Contents

General Introduction 13

1 General Aspects of QCD 17 1.1 QCDLagrangian ...... 19 1.2 Asymptotic freedom in QCD ...... 21

2 Schwinger-Dyson Equations 24 2.1 Functional formalism ...... 25 2.2 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 BackgroundFieldMethod ...... 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 Seagullidentity ...... 56 4.3 Ward-Takahashi identities for PT-BFM vertices ...... 57 4.4 Gluon self-energy at the origin ...... 59 4.4.1 Renormalization ...... 63 4.5 Circumventing the complete seagull cancellation ...... 65

5 Bethe-Salpeter Equations 70 5.1 DerivationoftheBSE ...... 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)...... 117 Y 7.4.4 Function 1 + G(q2)...... 119 7.5 Numerical analysis of the mass equation ...... 119 7.6 Summary ...... 124

8 Conclusions 126

Bibliography 132 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 particles [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 addition, 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 introduction 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 functional 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 Chapter3, 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 dynamical 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 equation numerically. In Chapter6, 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 Chapter7, 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 e and d, s, and b of 1 e. 3 − 3 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 intriguing 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]

= + + + , (1.1) LQCD LYM LGF LDirac LGhost where

1 = Ga Gµν , (1.2) LYM −4 µν a 1 = (∂µAa )2 , (1.3) LGF −2ξ µ µ = ψ¯(iγ Dµ mq)ψ , (1.4) LDirac − =c ¯a( ∂µDac)cc , (1.5) LGhost − µ with being the Yang-Mills Lagrangian, the Dirac Lagrangian describing the LYM LDirac interaction between fermionic fields, the gauge fixing Lagrangian, and the LGF LGhost ghost Lagrangian. The origin of the latter two terms is explained in Chapter2 from the Faddeev-Popov quantization procedure. a The field strength tensor Gµν appearing in Eq. (1.2) is given by

a a a abc b c G = ∂µA ∂νA + gf A A , (1.6) µν ν − µ µ ν

a where g is the coupling constant and Aµ 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

a a λ Dµ = ∂µ igA , (1.7) − µ 2 with λa being the Gell-Mann matrices that generate the SU(3) symmetry group [48]. These matrices obey the following relations:

λa λb λc , = if abc , tr(λaλb) = 2δab , (1.8) 2 2 2 1.1. QCD Lagrangian 20

µ a b ab (0) ab (0) (0) i(γ pµ + m) SF (p) = δ SF (p) S (p) = F p2 m2 + i − ab a b (0) ab (0) (0) pµpν 1 ∆µν (p) = δ ∆µν (p) ∆ (p) = i gµν (1 ξ) µν − − − p2 p2 + i ab (0) ab (0) i a b D (p) = δ D (p) D(0)(p) = p2 + i

Figure 1.1: Feynman rules for the quark, gluon, and ghost propagators at tree-level.

where f abc is the structure constant of the SU(3) group. In the adjoint representation, ac the covariant derivative Dµ of Eq. (1.5) is given by

ac ac abc b Dµ = δ ∂µ + gf Aµ . (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 a the fields Aµ and ψ are given by

1 Aa Aa + Dacαc , (1.10) µ → µ g µ λa ψ ψ + iαa ψ , (1.11) → 2 with αa being the transformation parameters. From the QCD Lagrangian, we can obtain the Feynman rules of the theory. The

ab (0) ab (0) ab (0) tree-level expressions for the quark, SF , gluon, ∆µν , and ghost, D , propagators are given in Fig. 1.1. In addition, in Fig. 1.2 we present the tree-level expressions for a (0) abc (0) the interaction vertices of the theory: the quark-gluon, Γµ , ghost-gluon, Γµ , three- abc (0) abcd (0) gluon, Γµαβ , and four-gluon, Γµνρσ , vertices. Throughout this thesis, we often extract the coupling g and the color structures from the vertices, deﬁning

λa Γa (q, r, p) = ig Γ (q, r, p) , µ 2 q,µ abc abc iΓµ (q, r, p) = gf Γµ(q, r, p) ,

abc abc iΓµαβ(q, r, p) = gf Γµαβ(q, r, p) . (1.12) 1.2. Asymptotic freedom in QCD 21

µ,a q a λ (0) Γ(0) (q, r, p) = γµ ig Γq,µ(q, r, p) q,µ p r 2

µ,a q amn (0) (0) gf Γµ (q, r, p) Γµ (q, r, p) = rµ p r −

n m

µ,a q (0) (0) amn Γµαβ(q, r, p) = gαβ(r p)µ + gµβ(p q)α gf Γµαβ(q, r, p) − − p r +gµα(q r)β β,n α,m −

µ,m ν,n Γmnrs (0) = f msef ern (g g g g ) 2 mnrs (0) µνρσ µρ νσ µν ρσ ig Γµνρσ (q, r, p, t) mne esr − − +f f (gµσgνρ gµρgνσ) mre esn − σ, s ρ, r +f f (gµσgνρ 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

2 3 dαs(µ) αs αs β(αs) = µ = b + b + , (1.13) dµ π 1 π2 2 ···

2 where αs = g /(4π). Through a one-loop calculation, it was found that for the SU(N) group, we have [1,2] 11 1 b = CA nf , (1.14) 1 − 6 − 3 where CA is 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,

2 2π αs(Q ) = , (1.15) − Q2 b1 ln 2 ΛQCD where ΛQCD is the QCD scale, deﬁned as

2 2 2π ln ΛQCD = ln µ + 2 . (1.16) αs(µ )b1

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.

2 2 When Q = ΛQCD, 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).

The experimental measurements of αs shown in Fig. 1.3 are in agreement with higher 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 observables. 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 Chapter3. 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

4 Z[J] = [φ] exp iS[φ] + i d xJi(x)φi(x) , (2.1) D Z Z

4 where S = d x is the action of the theory and Ji(x) the external sources associated to L the scalar fieldsR 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 [φ] is defined as D

[φ] [φi] . (2.2) D ≡ i D Y 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, 1 δnW [J] 0 T (φi(x ) φj(xn)) 0 = . (2.4) h | 1 ··· | i in−1 δJ (x ) δJ (x ) i 1 j n J=0 ···

For example, for the propagator of the ﬁeld φi, we have

δ2W [J] Di(x z) = 0 T (φi(x)φi(z)) 0 = . (2.5) − h | | i −δJ (x)δJ (z) i i J=0

In order to obtain only the contribution from 1PI diagrams, one must calculate the 2.1. Functional formalism 26

Legendre transform of W [J][51, 52], i.e.,

cl 4 cl Γ[φ ] = W [J] d xJi(x)φ (x) , (2.6) − i Z

cl cl where Γ[φ ] is the generator of proper vertices, often called eﬀective action, and φi

(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

cl cl respect to Ji (keeping φi ﬁxed) and with respect to φi (Ji ﬁxed), we obtain

cl cl δW [J] i δZ[J] δΓ[φ ] φi (x) = = − ; cl = Ji(x) . (2.7) δJi(x) Z[J] δJi(x) δφi (x) −

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

cl cl δφi (x) 4 δφi (x) δJk(y) δijδ(x z) = = d y − δφcl(z) δJ (y) δφcl(z) j Z k j δ2W [J] δ2Γ[φcl] = d4y , (2.8) − δJ (y)δJ (x) δφcl(z)δφcl(y) Z k i j k ! where we have used the relations of Eq. (2.7). Thus, we have

δ2Γ[φcl] δ2W [J] −1 = . (2.9) δφcl(x)δφcl(z) − δJ (x)δJ (z) i i i i

Now, let us consider the following change of variables in Eq. (2.1)

0 φi(x) φ (x) + fi(x) , (2.10) → i where is inﬁnitesimal and fi(x) is an arbitrary function. Then, the generating functional becomes

0 0 0 4 δS[φ ] 4 0 Z[J] = [φ ] exp iS[φ ] + i d x fi(x) + i d x Ji(x)[φ (x) + fi(x)] , (2.11) D δφ0 (x) i Z Z i Z

0 where we have used [φ] = [φ ], because fi does not depend on the ﬁelds. We can D D 2.2. Functional formalism in gauge theories 27 perform a Taylor expansion around = 0 in the equation above,

4 Z[J] = [φ] exp iS[φ] + i d x Ji(x)φi(x) D × Z Z 4 0 δS[φ] 0 0 2 1 + i d x + Ji(x ) fi(x ) + ( ) . (2.12) × δφ (x0) O Z i Thus, the zeroth order term of the expansion already produces the original Z[J], so the

0 4 0 1 order term must evaluate to zero. In particular, choosing fi(x ) = δ (x x), we have −

δS[φ] 4 0 = [φ] + Ji(x) exp i S[φ] + d xJi(x)φi(x) . (2.13) D δφ (x) Z i Z Then, we obtain

δS δ i + Ji(x) Z[J] = 0 , (2.14) δφ (x) − δJ(x) i 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 functional of the free electromagnetic ﬁeld. The Lagrangian in this case is given by

1 µν = FµνF , (2.15) LEM −4 where the ﬁeld strength tensor is given by

Fµν = ∂µAν ∂νAµ . (2.16) − 1 This particular function was chosen for simplicity, but one could keep fi(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 to vanish, which would result in Eq. (2.14) again. 2.2. Functional formalism in gauge theories 28

This Lagrangian is invariant under the local transformation

1 Aµ(x) Aµ(x) + ∂µα(x) , (2.17) → e 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

µ F (A) = ∂ Aµ 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,

δF (Aα) 1 = [α] det δ[F (Aα)] , (2.19) D δα Z

α α 1 where A denotes the gauge transformed ﬁeld Aµ = Aµ(x) + e ∂µα(x). We immediately have δF (Aα) ∂2 = , (2.20) δα e 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

2 iS[A] ∂ iS[A] µ [A]e = det [α] [A]e δ[∂ Aµ ω(x)] . (2.21) D e D D − Z Z Z 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( i d4x ω2(x)/2ξ). Then, we have − R ∂2 (∂µA )2 [A]eiS[A] = N(ξ) det ( [α]) [A]eiS[A] exp i d4x µ , (2.22) D e D D − 2ξ Z Z Z Z where we have performed the integration in ω using the delta function appearing in 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 in [A] will cancel when calculating any correlation function, after performing the D 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., µ 2 4 1 µν (∂ Aµ) S[A] = d x FµνF . (2.23) −4 − 2ξ Z 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 µν µ = FµνF + ψ¯(iγ Dµ m)ψ , (2.24) −4 − where Dµ = ∂µ + ieAµ is 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

4 1 µ 2 −1 ν µ S[A, ψ, ψ] = d x A (∂ gµν (1 ξ )∂µ∂ν)A + ψ(iγ Dµ m)ψ . (2.25) 2 − − − Z Then, for the generating functional, we have

4 Z[J, η, η] = [u] exp i S[A, ψ, ψ] + d x(JµAµ + ηψ + ψη) , (2.26) D Z Z where Jµ is 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 [u] is defined as D

[u] = [A] [ψ] [ψ] . (2.27) D D D D

However, we are interested only in the contribution of 1PI diagrams, so we must look 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],

4 Γ[A, ψ, ψ] = W [J, η, η] d x(JµAµ + ηψ + ψη) , (2.28) − Z we obtain the following relations:

δW δW δW = Aµ , = ψ , = ψ , δJ µ δη δη − δΓ δΓ δΓ = Jµ , = η , = η . (2.29) δAµ − δψ δψ −

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 2 −1 ν = [∂ gµν (1 ξ )∂µ∂ν]A eψγµψ . (2.30) δAµ − − −

Now, setting Z = eiW , we can write the expression analogous to Eq. (2.14) as

−iW [J,η,η] δS δ δ δ iW [J,η,η] e i , i , i + Jµ e = 0 , (2.31) δA − δJ − δη δη µ ν so that we ﬁnd

2 −1 δW δW δW δ δW Jµ + [∂ gµν (1 ξ )∂µ∂ν] e γµ ie γµ = 0 . (2.32) − − δJ − δη δη − δη δη ν

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

2 −1 δΓ 2 −1 ν δ Γ = [∂ gµν (1 ξ )∂µ∂ν]A (x) ie Tr γµ , (2.33) δAµ(x) − − − δψ(x)δψ(x) ψ=ψ=0 " #

where we have used δ2Γ −1 δ2W = , (2.34) δψδψ −δηδη as in Eq. (2.9).

2From here on, we drop the superscript “cl” in order to avoid cluttering the notation. 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

δ2Γ ∆−1(x y) = . (2.35) µν − δAν(y)δAµ(x) A=ψ=ψ=0

Thus, we ﬁnally ﬁnd

−1 2 −1 4 ∆ (x y) = [∂ gµν (1 ξ ) ∂µ∂ν] δ (x y) µν − − − − 2 4 4 µ + ie Tr d u d v [γ SF (x u)Λν(y, u, v)SF (v x)] , (2.36) − − Z where we have used that the derivative of an inverse matrix, M −1, can be obtained from

δM −1 δ = M −1 MM −1 , (2.37) δAν(y) − δAν(y) with δ2Γ M = , δψ(x1)δψ(x2) and the appropriate integrals in the relation implied. In Eq. (2.36), we have deﬁned the fermion propagator as

δ2Γ −1 SF (x y) = , (2.38) − δψ(y)δψ(x) ψ=ψ=0

and the electron-photon vertex as

δ3Γ = eΛν(y, u, v) . (2.39) ν δA (y) δψ(u) δψ(v) A=ψ=ψ=0

Thus, Eq. (2.36) is the SDE for the photon propagator, which can be written as

−1 2 −1 4 ∆ (x y) = [∂ gµν (1 ξ ) ∂µ∂ν] δ (x y) + Πµν(x, y) , µν − − − − where Πµν is the photon self-energy,

2 4 4 µ Πµν(x, y) = ie Tr d u d v [γ SF (x u)Λν(y, u, v)SF (v x)] . (2.40) − − Z 2.4. SDE for the gluon propagator in QCD 32

1 1 − − = +

q q

k + q

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

d4k Π (q) = ie2 Tr [γµS (k)Λ (k, k + q) S (k + q)] . (2.41) µν (2π)4 F ν F Z Then, we ﬁnd the usual form for the photon propagator SDE,

−1 2 −1 qµqν i∆ (q) = q gµν (1 ξ ) + Πµν(q) . (2.42) µν − − − q2

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 propagators 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 and terms presented in Eqs. (1.3) and (1.5), one needs LGF LGhost to apply the Faddeev-Popov procedure to the Yang-Mills Lagrangian, , deﬁned in LYM Eq. (1.2). Then, applying the gauge condition

F a(A) = ∂µAa (x) ωa(x) , (2.43) µ − 2.4. SDE for the gluon propagator in QCD 33 in the gauge transformation of Eq. (1.10), we obtain

δF a(Aα) 1 = ∂µDac , (2.44) δαc g µ

α ac where A is the gauge transformed field and Dµ was defined in Eq. (1.9). One can see that a in the case of QCD the derivative is no longer independent of the gauge field Aµ. 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,

1 det ∂µDac = c c exp i d4x c( ∂µDac)c . (2.45) g µ D D − µ Z Z These fields are Grassmann variables (usually employed for fermion fields), but are scalars (spin 0) under Lorentz transformations. Thus their quantum excitations cannot be physical 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 Lagrangian must be renormalized LQCD according to the prescription:

1/2 g = Zg gR, ξ = Zξ ξR, ψ = Z2 ψR,

a 1/2 a a 1/2 a c = Zc cR ,Aµ = ZA AµR, m = Zm mR, (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

R = R + R + R + R , (2.47) LQCD LYM LGF LDirac LGhost where

R R 1 µ 2 1 ν abc a µ ν + = ZA A ∂ gµν 1 ∂µ∂ν A Z gf (∂µA )A A LYM LGF 2 a − − Z ξ a − 3 µ b c Q 1 Z g2f abef cdeAµAνAc Ad , − 4 4 a b µ ν 2.4. SDE for the gluon propagator in QCD 34

a R µ λ µ = Z ψ(iγ ∂µ Zmm)ψ iZ F gψγµ ψA , LDirac 2 − − 1 2 a R a 2 a abc a µ c = Zc c ∂ c Z gf c ∂µ(A c ) , (2.48) LGhost − − 1 b

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]

1/2 1/2 3/2 2 2 Z1F = ZgZ2ZA ,Z1 = ZgZcZA ,Z3 = ZgZA ,Z4 = Zg ZA . (2.49)

Then, we can proceed to obtain the SDE for the gluon propagator using the renormalized 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

δS[φ] 4 δS [φ] + Ji(x) exp i S[φ] + d xJi(x)φi(x) =: + Ji . (2.50) D δφ (x) δφ Z i Z i Deriving the gluon propagator SDE requires taking two derivatives of the action in relation a to the gauge ﬁeld, Aµ, as in the case of the photon propagator. According to Eq. (2.13), we must have δS QCD + J a = 0 , (2.51) δAa (x) µ µ a a where Jµ is the source of the Aµ ﬁeld and the QCD action is given in terms of its La- grangian, S [A, ψ, ψ, c, c] = d4x R . (2.52) QCD LQCD Z In order to obtain the contribution of 1PI diagrams to the complete gluon propagator, we must use the generating functional ΓQCD,

2 ab −1 δ ΓQCD ∆µν(x y) = ν µ . (2.53) − δA (y)δAa (x) b A=ψ=ψ=0

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]

2 δ ΓQCD 2 1 ab ν µ = ZA ∂ gµν 1 ∂µ∂ν δ δ(x y) δA (y)δAa (x) − − Z ξ − b Q ade 4 ν ρ −1 ρ d e Z gf d z A (y)A (z) A (z)(∂µc (x))c (x) − 1 h b c i c aZ λ 4 ν ρ −1 ρ Z F ig γµ d z A (y)A (z) A (z)ψ(x)ψ(x) − 1 2 h b c i c Z ade 4 ν ρ −1 ρ σ e + Z gf d z A (y)A (z) A (z)A (x)∂µA (x) 3 h b c i {h c d σ i Z ρ σ e ρ σ e A (z)A (x)∂σA (x) A (z)∂σA (x)A (x) − h c d µ i − h c d µ i} Z g2f afgf gde δbf Ad (x)Ae (x) + δbe Af (x)Ad (x) − 4 { h µ ν i h ν µ i bd ρ e + δ gµν A (x)A (x) δ(x y) h f ρ i} − Z g2f afgf gde d4z Aν(y)Aρ(z) −1 Aρ(z)Aσ(x)Ad (x)Ae (x) , (2.54) − 4 h b c i h c f µ σ i Z where the brackets represent correlation functions. Then, the resulting SDE for the gluon propagator in Euclidean space is given by

ab −1 ab (0) −1 ∆ (x y) = ZA ∆ (x y) µν − µν − + Z g2 d4x d4y Γacd (0)(x, x , x )Dde(x y )Dfc(y x )Γbef (y, y , y ) 1 12 12 µ 1 2 2 − 1 2 − 1 ν 1 2 Z 2 4 4 a (0) b + Z F g d x d y Γ (x, x , x )SF (x y )SF (y x )Γ (y, y , y ) 1 12 12 µ 1 2 2 − 1 2 − 1 ν 1 2 Z g2 + Z d4x d4y Γacd (0)(x, x , x )∆βγ(x y )∆αδ(x y )Γbef (y, y , y ) 3 2 12 12 µαβ 1 2 de 2 − 1 cf 1 − 2 νγδ 1 2 Z g2 + Z d4x Γabcd (0)(x, y, x , x )∆αβ(x x ) 4 2 12 µναβ 1 2 cd 1 − 2 Z g4 + Z d4x d4y Γacde (0)(x, x , x , x )∆αλ (x y )∆βσ(x y )∆γρ(x y ) 4 6 123 123 µαβγ 1 2 3 cm 1 − 3 dl 2 − 2 ek 3 − 1 × Z Γbklm (y, y , y , y ) × νρσλ 1 2 3 g4 + Z d4x d4y d4z Γacde (0)(x, x , x , x )∆αρ(x y )∆βλ (x y )∆σκ(y z ) 4 2 123 123 12 µαβγ 1 2 3 ck 1 − 1 dm 2 − 3 lq 2 − 2 × Z Γklm(y , y , y )∆γδ(x z )Γbpq (y, z , z ) , (2.55) × ρσλ 1 2 3 ep 3 − 1 νδκ 1 2

ab a abc abc abcd where D is the complete ghost propagator and Γµ,Γµ ,Γµνα, and Γµναβ represent the full quark-gluon, ghost-gluon, three-gluon, and four-gluon vertices, respectively. In addition, 2.4. SDE for the gluon propagator in QCD 36 we have deﬁned the following notation for the integration measure

j 4 4 d x1...j =: d xi , (2.56) i=1 Y and analogously for the variables yi and zi. Finally, we apply the Fourier transform to obtain the equation in momentum space,

6 ab −1 ab (0) −1 ab ∆µν(q) = ZA ∆µν (q) + (di)µν(q) , (2.57) i=1 X where

Z d4k (d )ab (q) = 3 g2 Γacd (0)(q, k, k q)∆βγ(k + q)∆cf (k)Γbef ( q, k + q, k) , 1 µν 2 (2π)4 µαβ − − de αδ νγδ − − Z Z d4k (d )ab (q) = 4 g2 Γabcd (0)∆ρσ(k) , 2 µν 2 (2π)4 µνρσ cd Z d4k (d )ab (q) = Z g2 Γadc (0)(q, k q, k) Dde(k + q)Dcf (k)Γbfe( q, k, k + q) , 3 µν 1 (2π)4 µ − − ν − − Z 4 4 Z d k d k 0 0 0 (d )ab (q) = 4 g4 1 2 Γacde (0) ∆ρρ (k ) ∆σσ (q + k + k )∆λλ (k ) 4 µν 6 (2π)4 (2π)4 µρσλ cf 2 dh 1 2 eg 1 Z fbgh Γ 0 0 0 ( k , q, k , q + k + k ) , × ρ νλ σ − 2 − − 1 1 2 4 4 Z d k d k 0 0 0 0 (d )ab (q) = 4 g4 1 2 Γacde (0) ∆γγ (q + k + k ) ∆ζζ (k + k )∆ββ (k )∆αα (k ) 5 µν 2 (2π)4 (2π)4 µαβγ ef 1 2 nm 1 2 dh 2 cg 1 Z bfn mhg Γ 0 0 ( q, q + k + k , k k )Γ 0 0 (k + k , k , k ) , × νγ ζ − 1 2 − 1 − 2 ζβ α 1 2 − 2 − 1 4 ab 2 d k a (0) b (d ) (q) = Z F g Γ (q, k, k q) SF (k + q)SF (k)Γ ( q, k + q, k) . (2.58) 6 µν 1 (2π)4 µ − − ν − − Z 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 ab coming from (d6)µν(q) in Eq. (2.57), we obtain the SDE presented in Fig. 2.2, where the gluon self-energy is given by 5 µν µν Π (q) = di (q) . (2.59) i=1 X 2.5. Slavnov-Taylor identities 37

1 1 − + 1 + 1 ∆µν− (q)= 2 2

(d1) (d2)

1 1 + + 6 + 2

(d3) (d4) (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 (q) = iξδab. (2.60) µν −

We can interpret this result by separating the full gluon propagator into transverse and 2.5. Slavnov-Taylor identities 38 longitudinal parts,

ab ab qµqν ∆ (q) = iδ Pµν(q)∆(q) + E(q) , (2.61) µν − q2 where Pµν(q) is the transverse projector,

2 Pµν(q) = gµν qµqν/q , (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 ab ab qµqν ∆ (q) = iδ Pµν(q)∆(q) + ξ . (2.63) µν − q4 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 ghost-gluon vertex SDE, one deﬁnes the quantity called ghost- cab gluon scattering kernel, Hµν ,

bac cab iH (y, x, z) = igf gµνδ(x y)δ(y z) − νµ − − δ2 δ2Γ −1 gf bde , (2.64) − δAc (z)δca(x) δAd(y)δce(y) µ ν which is related to the ghost-gluon vertex by

3 cba δ Γ ν bac Γµ (z, y, x) = i c a b = i∂y Hνµ (y, x, z) , (2.65) − δAµ(z)δc (x)δc (y) − where the subscript y in the derivative speciﬁes the variable of diﬀerentiation. In momentum space, Eq. (2.65) reads

cba ν bac Γµ (r, q, p) = q Hνµ (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

r r

µ µ q q Hνµ(q,p,r) = + ν ν

p p

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

µ 2 −1 µ −1 µ r Γαµν(q, r, p) = ir D(r) ∆ (q)P (q)Hµν(q, r, p) ∆ (p)P (p)Hµα(p, r, q) , (2.67) − α − ν where we have extracted the color structure,

abc abc H (q, p, r) = gf Hµν(q, p, r) . (2.68) µν −

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 truncate 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., µ q Πµν(q) = 0. (3.1)

Thus, it is convenient to write the complete gluon propagator, in general covariant gauges, as in Eq. (2.63),

2 qµqν ∆µν(q) = i Pµν(q)∆(q ) + ξ , (3.2) − q4

2 where Pµν(q) is the transverse projector, given in Eq. (2.62) and ∆(q ) is the gluon

2 propagator scalar function deﬁned in terms of the self-energy, Πµν(q) = Pµν(q)Π(q ), through the relation (Minkowski space)

∆−1(q2) = q2 + iΠ(q2) . (3.3)

−1 It is useful to write the form of the inverse gluon propagator, ∆µν (q), deﬁned as

−1 ρν ρ ∆µν (q)∆ (q) = gµ , (3.4) where

q q ∆−1(q) = i P (q)∆−1(q2) + ξ−1 µ ν , (3.5) µν µν q4 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. 2.2 at one-loop of perturbation theory, the gluon self-energy transversality is broken. The sum of these two diagrams (in Minkowski space) is given by

2 (1) (1) λ 2 q Π (q) d + Π (q) d = 84qµqν 75gµνq ln − , (3.6) µν |( 1) µν |( 2) 36 − µ2 where iC g2 λ = A , (3.7) 16π2 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,

2 2 where the tree-level value for Πµν(q) is recovered at the oﬀ-shell momentum q = µ < 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

2 (1) (1) (1) 13λ 2 q Π (q) d + Π (q) d + Π (q) d = (gµνq qµqν)ln − , (3.8) µν |( 1) µν |( 2) µν |( 3) − 6 − µ2 which is indeed transverse. Thus, we conclude that one cannot simply ignore the contribution 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 µ q Πµν(q) = 0 . (3.9) 6 (d1)+(d2)+(d3)

Therefore, the gluon self-energy transversality only emerges after the inclusion of all diagrams 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

δF a 1 Z[J] = [A] det exp iS(A) + i d4x (F a)2 + J aAaµ , (3.10) D δωb −2ξ µ Z Z

a a where Aµ is the gauge ﬁeld and ω is the gauge transformation parameter,

a 1 a abc b c δA = ∂µω f ω A . (3.11) µ g − µ

For better visualization of the next steps, we have factored out the gauge fixing term, F a, from the definition of the action. The basic idea of the BFM is to split the gauge field into a classical background field a a (Bµ) and a fluctuating quantum field (Qµ), i.e.,

Aa Qa + Ba . (3.12) µ → µ µ

a a We treat the classical part Bµ as a fixed field configuration and the fluctuating part Qµ as the integration variable of the functional integral, where

Z[J] Z[J, B] , → F a(A) F a(Q, B) . (3.13) → e e Then, applying this change of variable in the generating functional, we obtain

a δF 4 1 a 2 a aµ Z[J, B] = [Q] det b exp iS(Q + B) + i d x (F ) + JµQ , (3.14) D " δω # −2ξ Z e Z e e where the background field is not coupled to the source [32]. The term inside the determinant corresponds to the derivative of the gauge fixing term under the infinitesimal gauge transformation

a 1 a abc b c c δQ = ∂µω f ω (Q + B ). (3.15) µ g − µ µ

Note that the gauge fixing term, F a(Q, B), can depend on both the quantum and background fields. e We can also define the corresponding connected, W , and 1PI, Γ, generating functionals

f e 3.2. Background Field Method 44 as in the previous Chapter (see Eqs. (2.3) and (2.6)),

W [J, B] = i ln Z[J, B] , Γ[Q, B] = W [J, B] d4xJ a(x)Qa (x) , (3.16) − − µ µ Z f e e f a where Qµ is the quantum field argument of the background field effective action (remember that, in this case, it is the expectation value of the field),

a δW Qµ = a . (3.17) δJµ f Now we choose a special gauge condition, which preserves gauge invariance in terms of the background ﬁeld B, given by

a µ a abc b cµ F (Q, B) = ∂ Qµ + gf BµQ . (3.18)

e Then, it is possible to demonstrate that Z[J, B] and W [J, B] are invariant under the transformations [33] e f

a abc b c 1 a δB = f ω B + ∂µω , (3.19) µ − µ g δJ a = f abcωbJ c . (3.20) µ − µ

In addition, for this background ﬁeld gauge condition, we have that Γ[Q, B] is invariant under e

a abc b c 1 a δB = f ω B + ∂µω , (3.21) µ − µ g δQa = f abcωbQc . (3.22) µ − µ

a a Notice that Bµ carries the local gauge transformation, whereas Qµ transforms as a matter a field in the adjoint representation. Using that δQµ = 0 when the expectation value of the a field is zero (Qµ = 0), we have that Γ[0,B] is an explicitly gauge invariant functional of Ba. µ e The functional Γ[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 e field, it can be shown that the 1PI Green’s functions obtained from derivatives of Γ[0,B]

e 3.3. New Green’s functions and identities 45 satisfy Abelian WTIs. The relation between the conventional eﬀective action, Γ, and the background action, Γ, can be obtained from the change of variables Qa Aa Ba in Eq. (3.14), which leads µ → µ − µ to e

Z[J, B] = Z[J] exp i d4xJ aBaµ , (3.23) − µ Z e where Z[J] is calculated with the corresponding gauge ﬁxing term

F a = ∂µAa ∂µBa + gf abcBb Acµ. (3.24) µ − µ µ

Following the steps of Ref. [33], one can verify from Eq. (3.23) that

Γ[0,B] = Γ[B] . (3.25)

e Therefore, the BFM effective action, Γ[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 e a the field Qµ is set to zero. The above correspondence between background and conventional effective actions implies 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 quantum gluons (namely QQ), we also have ∆µν(q), formed by one quantum external leg

e 3.3. New Green’s functions and identities 46

∆µν(q) =

∆µν(q) = g

∆µν(q) = d

Figure 3.1: Gluon propagators appearing in the PT-BFM formalism. White circles il- lustrate full propagators and the small external gray circles represent background gluon ﬁelds.

and one background one (QB), and ∆µν(q) comprising two external background gluon legs (BB). In Fig. 3.1, we represent these propagators diagrammatically, where the small b gray circles in the end of the external legs correspond to background gluons. Expressions completely analogous to Eq. (3.3) also hold for the QB and BB propagators, i.e.,

∆−1(q2) = q2 + iΠ(q2) , (3.26)

−1 2 2 2 ∆e (q ) = q + iΠ(e q ) . (3.27)

b b Besides the new gluon propagators, we also have vertices with background gluon legs. The Feynman rules for these new vertices of the PT-BFM formalism are presented in Fig. 3.2. We emphasize that background gluons can only appear as external legs of diagrams, never inside an integration loop. Additionally, along this thesis, we extract the coupling g and the color structures from all vertices whenever possible. For that, we use Eq. (1.12) and the following vertex deﬁnitions (all momenta entering)1:

amn a m n iΓBµQα Qβ (q, r, p) = gf Γµαβ(q, r, p), amn iΓ n a m (p, q, r) = gf Γ (q, r, p), c Bµc¯ eµ 2 amnr ΓQa QmQnQr (q, r, p, t) = ig Γ (q, r, p, t), µ α β γ − eµαβγ 2 amnr ΓBaQmQnQr (q, r, p, t) = ig Γ (q, r, p, t), µ α β γ − µαβγ 2 amnr ΓBaQmBnQr (q, r, p, t) = ig Γ (q, r, p, t), µ α β γ − eµαβγ 1 Green’s functions with only one backgroundb leg are represented with a “tilde”, whereas the ones with two background gluons are represented with the “hat” symbol. 3.3. New Green’s functions and identities 47

µ,a q (0) Γ (q, r, p) = gαβ(r p)µ gf amnΓ(0) (q, r, p) µαβ − µαβ −1 p r + gµβ(p q + ξ r)α e − −1 β,n α,m e + gµα(q r ξ p)β − − µ,a q amn (0) (0) gf Γµ (q, r, p) Γµ (q, r, p) = (p r)µ p r −

n m e e

µ,m ν,n mnrs (0) mse ern 2 mnrs (0) Γµνρσ = f f (gµρgνσ gµνgρσ) ig Γµνρσ mne esr − − +f f (gµσgνρ gµρgνσ) − σ, s ρ, r e mre esn e +f f (gµσgνρ gµνgρσ) − µ,m ν,n mnrs mse ern −1 2 mnrs (0) Γµνρσ = f f gµρgνσ gµνgρσ + ξ gµσgνρ µνρσ − ig Γ mne esr −1 − +f f gµσgνρ gµρgνσ ξ gµνgρσ − − σ, s ρ, r b mre esn b +f f (gµσgνρ gµνgρσ) − µ, a ρ, r

2 mae ern (0) (0) ig f f Γµρ Γµρ = gµρ − m n e e

µ, a ρ, r

2 arnm (0) arnm (0) mae ern mre ean ig Γµρ Γµρ = gµρ (f f + f f ) − m n b b

Figure 3.2: Feynman rules for the new vertices of the PT-BFM scheme [29].

2 mae ern ΓcnBaQr cm (q, r, p, t) = ig f f Γµρ(q, r, p, t), µ ρ¯ − 2 arnm ΓcnBaBrcm (q, r, p, t) = ig Γ (q, r, p, t). (3.28) µ ρ ¯ − µρ e b To exemplify the advantage of dealing with the PT-BFM functions, instead of the conventional Green’s functions, let us take a look at the STI obeyed by the conventional 3.3. New Green’s functions and identities 48 three-gluon vertex. From Eq. (2.67), we have

α 2 −1 σ −1 σ q Γαµν(q, r, p) = iF (q ) ∆ (r)H (r, q, p) ∆ (p)H (p, q, r) , (3.29) σµ ν − σν µ 2 which involves the gluon propagator, ∆(q ), the ghost-gluon scattering kernel, Hνµ, de- picted in Fig. 2.3, and the ghost dressing function, F (q2), deﬁned as

F (q2) D(q2) = i , (3.30) q2 where D(q2) is the ghost propagator. Note that at tree-level F (0)(q2) = 1.

On the other hand, the BQQ vertex, Γαµν, which has one background and two quantum gluons, satisﬁes an Abelian-like WTI when contracted with the background gluon e momentum,

α −1 −1 q Γαµν(q, r, p) = i[∆ (r) ∆ (p)] , (3.31) µν − µν e which only involves conventional gluon propagators. We will see how this type of identity enables the truncation of the gluon SDE in the PT-BFM framework in the next Section. These new Green’s functions are related to the conventional functions by a set of Background-Quantum identities (BQI) [29, 58, 59]. In particular, the new gluon propagators obey the following BQIs:

∆−1(q2) = 1 + G(q2) ∆−1(q2) , (3.32) −1 2 2 2 −1 2 ∆e (q ) = 1 + G(q ) ∆ (q ) . (3.33) b 2 The auxiliary function G(q ) is the scalar term that multiplies gµν in the Lorentz decomposition of a special Green’s function Λµν(q), which is diagrammatically represented in Fig. 3.3,

q q Λ (q) = g G(q2) + µ ν L(q2) . (3.34) µν µν q2

The two point function Λµν(q) is related to the ghost-gluon scattering kernel by

2 σ Λµν(q) = ig CA D(q k)∆ (k)Hνσ( q, q k, k), (3.35) − µ − − Zk 3.3. New Green’s functions and identities 49

Figure 3.3: Diagrammatic representation of the Λµν(q) two-point function.

where the integral is deﬁned in dimensional regularization,

ddk , (3.36) ≡ (2π)d Zk Z with d = 4 the dimension of space-time. To isolate the functions G(q2) and L(q2) of Eq. (3.34), one performs the projections [60]

2 1 2 µ µ ν G(q ) = q Λ q q Λµν , (d 1)q2 µ − − 2 1 µ ν 2 L(q ) = dq q Λµν q Λµν . (3.37) (d 1)q2 − − Then, from Eq. (3.35), we see that the evaluation of 1 + G(q2) depends on the knowledge of the ghost-gluon scattering kernel Hνµ, whose Lorentz decomposition can be written as [61, 62]

Hνµ(q, p, r) = A1gµν + A2qµqν + A3rµrν + A4qµrν + A5rµqν , (3.38) where the momentum dependence, Ai Ai(q, p, r), has been suppressed for compactness. ≡ In particular, in the Landau gauge, G(q2) is related to the ghost dressing function,

2 2 F (q ), and the longitudinal part of Λµν(q), L(q ), by [63, 64]

F −1(q2) = 1 + G(q2) + L(q2) . (3.39)

In addition, L(q2) gives a small contribution compared to 1 + G(q2) in the entire range of momenta. In particular, L(q2) 0 both for IR and UV limits [60, 63–65]. Therefore, we → have 1 + G(q2) F −1(q2) , (3.40) ≈ with this relation being exact for q2 0. → 3.4. SDE for the gluon propagator in PT-BFM 50

Figure 3.4: SDE for the QB gluon propagator, ∆µν(q), within the PT-BFM scheme neglecting the contribution from quark ﬁelds. The red circles represent the full PT-BFM e vertices, whereas the blue one represents the conventional three-gluon vertex. White circles correspond to fully dressed propagators.

3.4 SDE for the gluon propagator in PT-BFM

In Fig. 3.4 we show the SDE for the QB gluon propagador in the quenched approximation [19]. This propagator may be written in terms of the self-energy, Πµν(q), as (Minkowski space) −1 e −1 (0) ∆ (q) = ∆ (q) Πµν(q) , (3.41) µν µν − h i where e e e 6

Πµν(q) = (ai)µν . (3.42) i=1 X Then, using the BQI of Eq. (3.32), onee can see that, in the Landau gauge, the conventional gluon propagator, ∆µν(q), can be expressed as

6 2 q Pµν(q) + i (ai)µν ∆−1(q2)P (q) = i=1 . (3.43) µν 1 + G(qP2)

Thus, we can obtain the conventional QCD gluon propagator, ∆µν(q), from the SDE for

2 ∆µν(q) using the auxiliary function G(q ). Interestingly enough, using the WTIs satisﬁed by the full vertices with one leg in the e background, it has been shown that [34, 66]

ν ν ν q Πµν(q) = q Πµν(q) = q Πµν(q) = 0 . (3.44) (a1)+(a2) (a3)+(a4) (a5)+(a6)

e e e

3.4. SDE for the gluon propagator in PT-BFM 51

Thus, transversality is forced independently for each of the blocks of Fig. 3.4. As an example, we can show the transversality of the ﬁrst group of diagrams, composed by the sum of diagrams (a1) and (a2) in Fig. 3.4.

Considering ﬁrst (a1) and contracting it with the external background gluon momentum, we have

ν ab 1 (0)aex αρ ν be0x0 βσ q Πµν(q) = Γµαβ ∆ee0 (k) q Γνρσ ∆xx0 (k + q) (a1) 2 k Z h i 1 (0)aex αρ be0x0 −1 −1 βσ e = Γµαβ ∆ee0 (k)gf e ∆ρσ (k + q) ∆ρσ (k) ∆xx0 (k + q), (3.45) 2 − Zk where in the second line we have used Eq. (3.31). Using the conventional Feynman rule for the three-gluon vertex at tree-level (see Fig. 1.2), we obtain

ν 1 2 q Πµν(q) = CAg [gαβ(2k + q)µ + gβµ( k 2q)α + gµα(q k)β] (a1) 2 k − − − Z e ∆αβ(k) ∆αβ(k + q) . (3.46) × − Performing the contractions, we ﬁnd

ν 1 2 α α q Πµν(q) = CAg (2k + q)µ [∆α(k) ∆α(k + q)] (a1) 2 k { − Z e (2k + q) ∆α(k) ∆α(k + q) . (3.47) − α µ − µ After some algebraic manipulation in the integral above, one can arrive at

ν 2 α α q Πµν(q) = CAg qµ∆α(k) qα∆µ(k) . (3.48) (a1) − Zk e ν Now, contracting the mathematical expression for diagram (a2) with q , we easily ﬁnd

ν 2 α α q Πµν(q) = CAg qµ∆α(k) qα∆µ(k) , (3.49) (a2) − k − Z e where we used the Feynman rule for the BQQQ vertex given in Fig. 3.2. Therefore, the transversality of the ﬁrst block of diagrams in Fig. 3.4 is proved, i.e.,

ν ab q Πµν(q) = 0. (3.50) (a1)+(a2)

e 3.4. SDE for the gluon propagator in PT-BFM 52

In order to verify the transversality of the other two groups, one can follow the same procedure, using the appropriate WTIs, i.e.,

µ −1 −1 q Γµαβ(q, r, p) = i∆ (r) i∆ (p), αβ − αβ µ −1 2 −1 2 q Γµ(q, r, p) = iD (r ) iD (p ), e − µ mnrs mse ern mne esr q Γµαβγe(q, r, p, t) = f f Γαβγ(r, p, q + t) + f f Γβγα(p, t, q + r) mre ens e + f f Γγαβ(t, r, q + p) . (3.51)

The second identity in the above equation is used to prove the transversality of the second block of diagrams of Fig. 3.4 (orange dashed rectangle). To verify the transversality of the last group (blue dotted), one needs to use the ﬁrst and last relations of Eq. (3.51). As a result, we can truncate the SDE for the QB propagator without violating the transversality of the gluon self-energy, as long as we consider all diagrams within the chosen blocks. Note, however, that this fact does not imply that the contributions from the neglected groups are necessarily small. Nonetheless, being able to truncate the SDE while preserving, by construction, the symmetry of the theory is an advantageous accom- plishment. 53

Chapter 4

Dynamical Mass Generation and Seagull Cancellation

One of the interesting QCD nonperturbative phenomena is the dynamical mass generation, both for quarks and gluons. The idea that gluons acquire a momentum dependent dynamical mass due to their self-interactions was originally proposed in the early 80s [21, 67, 68]. Its validity was tested in a series of phenomenological models [69–71], but only gained general acceptance recently after a series of SDEs works on dynamical mass generation, performed in the Landau gauge [13, 14, 17, 34], which later were corroborated by large-volume lattice simulations, both for SU(3) [8–11, 35] and SU(2) [4–7]. Since then, this behavior has been studied by several distinct methods, including the functional renormalization group formalism [72] and the refined Gribov-Zwanziger approach [73]. The idea that a generic gauge boson may acquire a dynamical mass while remaining massless at the level of the fundamental Lagrangian was proposed by Schwinger [74, 75], and is known as the Schwinger mechanism. In this Chapter, we present a generalized theoretical framework, published in [76], in which we verify how the Schwinger mechanism can be triggered to generate a dynamical gluon mass in the context of QCD. For that, we first show how the IR saturation of the scalar form factor of the gluon propagator, ∆(q2), can be interpreted by means of a dynamical mass. Then, we introduce the so-called seagull identity, valid in dimensional regularization, which is important to cancel all the quadratic divergent integrals appearing in the gluon propagator SDE. Next, we present the Abelian-like Ward identities (WI) satisfied by the PT-BFM vertices, which are used together with the seagull identity to demonstrate that, in the absence of massless poles in 4.1. General considerations 54

6 ] -2

4 ) [GeV 2 (q

0 10-3 10-2 10-1 100 101 q2 [GeV 2]

Figure 4.1: The lattice data for the gluon propagator, ∆(q2), in the Landau gauge [8], renormalized at µ = 4.3 GeV, together with the ﬁt with a running gluon mass given in Eq. (6.22) (blue continuous line) and the simple massive propagator, ∆−1(q2) = q2 + m2 (green dashed line). In the deep IR, we have ∆(0) = m2, where m2 is a nonvanishing constant.

the structure of the nonperturbative PT-BFM vertices, the gluon does not acquire a mass. To circumvent this problem, we allow for the possibility that the vertices are endowed with longitudinally coupled poles, and, by doing so, we are able to trigger a dynamical gluon mass generation.

4.1 General considerations

The starting point to study dynamical gluon mass generation is to analyze the behavior of the gluon propagator, ∆(q2), in the nonperturbative region of QCD. The central idea here is to notice that an IR ﬁnite ∆(q2), like the one typically found in the lattice simulations, reveals a strong signal of dynamical gluon mass generation [14]. To see that, in Fig. 4.1 we show the lattice data of Ref. [8] and, from there, it is clear that the scalar function, ∆(q2), freezes in a constant value in the IR, i.e.,

∆−1(0) = m2 , (4.1) 4.1. General considerations 55 where m2 = 0. Although a simple massive gluon propagator, ∆−1(q2) = q2 + m2, repro- 6 duces the IR saturation as shown in Fig. 4.1 (see green dashed curve), it is well known that

2 2 a hard mass term for the gauge boson of the type m Aµ in the Yang-Mills Lagrangian is forbidden by gauge invariance. Therefore, we need a dynamical mass generation procedure which does not modify the Lagrangian. The most natural way to accommodate a ﬁnite gluon propagator in the IR, without breaking gauge invariance [21], is to suppose that ∆(q2) can be written as (Euclidean space) ∆−1(q2) = q2J(q2) + m2(q2) , (4.2) where J(q2) would correspond to the kinetic term and m2(q2) to the dynamical gluon mass [38, 77]. This mass term should display the following features: (i) saturate in a nonvanishing value in the IR, (ii) go to zero in the UV, in order to recover the perturbative behavior, and (iii) guarantee that the physical observables, which depend on ∆(q2), display a smooth transition between the IR and UV regimes. A self-consistent way to generate a dynamical gluon mass without violating gauge invariance can be found in the generalization of the Schwinger mechanism [74, 75]. Ac- cording to Schwinger, gauge invariance does not necessarily imply a massless gauge boson particle, as long as a strongly coupled theory can produce such a mass dynamically. The essential observation is that, although gauge symmetry prohibits a mass term, at the level of the fundamental Lagrangian, the gauge boson can acquire a mass if a pole appears in its vacuum polarization at zero momentum transfer. In order to understand such propositon, let us consider the gluon propagator in general covariant gauges of Eq. (3.2). Its scalar factor, ∆(q2), can be expressed in terms of the gluon self-energy, Π(q2), according to Eq. (3.3). It is also convenient to deﬁne the dimensionless vacuum polarization denoted by Π(q2) as Π(q2) = q2Π(q2). Then, we can rewrite Eq. (3.3) in Euclidean space as

∆−1(q2) = q2[1 + Π(q2)] . (4.3)

Thus, if Π(q2) acquires a pole when q2 = 0 with positive residue m2, i.e., Π(q2) = m2/q2, 4.2. Seagull identity 56 we have ∆−1(0) = m2 . (4.4)

Therefore the propagator represents a massive particle while remaining massless in the absence of interactions (g = 0). As will be demonstrated later, the origin of such a pole can be purely dynamical, without introducing new terms in the Lagrangian. From the theoretical point of view, the generation of a dynamical gluon mass presents some diﬃculties due to the appearance of quadratic divergences of the type

−1 2 2 ∆ (0) = c1 ∆(k) + c2 k ∆ (k) , (4.5) Zk Zk in the gluon SDE. This type of quadratic divergence is often called seagull divergence because such terms are typical from diagrams (a1) and (a2) of Fig. 3.4, with diagram (a2) being known as seagull diagram. Such divergent integrals must be regulated appropriately. However, although several regularization procedures have been employed in the literature, they do not present completely satisfactory results [14, 21, 34]. Nonetheless, it is possible to eliminate this problem by means of the so-called seagull identity.

4.2 Seagull identity

In order to derive the seagull identity, let us consider the class of functions

2 µ(k) = f(k )kµ, (4.6) F

2 where f(k ) is an arbitrary function. Because µ( k) = µ(k), in dimensional regular- F − −F ization, we have

µ(k) = 0 . (4.7) F Zk Now we require f(k2) to vanish rapidly as k2 , such that the integral (in spherical → ∞ coordinates, with y = k2)

∞ 2 1 d −1 f(k ) = dyy 2 f(y) (4.8) d d k (4π) 2 Γ 0 Z 2 Z 4.3. Ward-Takahashi identities for PT-BFM vertices 57 converges for positive values of d below a certain value to be denoted d∗. Thus, the integral is well deﬁned for d within the interval (0, d∗) and can be analytically continued outside it. Note that, due to the translational invariance of the dimensional regularization scheme, we have

µ(k + q) = µ(k) = 0 . (4.9) F F Zk Zk We can Taylor expand µ(k + q) around q = 0, F

ν ∂ µ(k) 2 µ(q + k) = µ(k) + q F + (q ) , (4.10) F F ∂kν O so, from Eq. (4.9), we obtain

∂ µ(k) qν F = 0. (4.11) ∂kν Zk Since in the above equation q is arbitrary and the integral has to be proportional to the tensor gµν, we ﬁnd ∂ µ(k) F = 0 , (4.12) ∂kµ Zk which is the seagull identity [76]. Such identity was originally written as [78]

∂f(k2) d k2 + f(k2) = 0 , (4.13) ∂k2 2 Zk Zk which can be obtained from Eq. (4.12) by using the relation

∂f(k2) ∂f(k2) = 2k . (4.14) ∂kµ µ ∂k2

4.3 Ward-Takahashi identities for PT-BFM vertices

An additional comment which is important in this Chapter is related to the distinction between Takahashi and Ward identities. Up until now we have been identifying the Abelian (and Abelian-like) identities simply as WTIs, however, in the context of QED, 4.3. Ward-Takahashi identities for PT-BFM vertices 58

the Takahashi identity for the photon-electron vertex, Γµ, is given by

µ −1 −1 q Γµ(q, r, q + r) = S (q + r) S (r) . (4.15) F − F

2 Now, under the assumption that Γµ does not contain poles of the type 1/q , the limit q 0 of the above equation gives → ∂S−1(r) Γ (0, r, r) = F , (4.16) µ ∂rµ which is known as WI. Therefore the WI is the correct IR limit of the WTIs if and only if the vertex does not contain poles of the type 1/q2. In this Section, we assume that none of the vertices entering in the SDE of Fig. 3.4 contain such poles, and therefore the corresponding Takahashi identities given by Eq. (3.51) can be converted in WIs (similar to the one of Eq. (4.16)) when we take the limit q 0. → In particular, for the study of mass generation, we are interested in ∆(0), or, given the BQI in Eq. (3.32), ∆(0). Therefore, it is relevant to obtain the limits of Eqs. (3.51) as the momentum of the external background gluon, to be denoted by q, approaches zero. e Then, Taylor expanding both sides of the second equation in Eq. (3.51) around q = 0, we obtain

µ 2 µ ∂ −1 2 q Γµ(0, r, r) + (q ) = iq D (q + r) + (q ) . (4.17) − O − ∂qµ O q=0 e Thus, we have that the ghost-gluon vertex Γµ satisﬁes the WIs

∂D−1(r2) e ∂D−1(p2) Γµ(0, r, r) = i ; Γµ(0, p, p) = i . (4.18) − − ∂rµ − ∂pµ e e We can do the same with the ﬁrst identity of Eq. (3.51), in order to obtain the WIs for the three-gluon vertex Γµαβ,

e −1 −1 ∂∆αβ (p) ∂∆αβ (r) Γµαβ(0, p, p) = i ; Γµαβ(0, r, r) = i . (4.19) − ∂pµ − − ∂rµ e e For the case of the BQQQ vertex, the derivation of its WI is a bit more intricate, but 4.4. Gluon self-energy at the origin 59

Figure 4.2: One-loop dressed gluonic contribution to the QB gluon self-energy, Πµν(q).

e after some manipulations, one obtains [76]

mnrs mne esr ∂ mre ens ∂ Γ (0, r, p, r p) = f f + f f Γαβγ(r, p, r p); µαβγ − − ∂rµ ∂pµ − − mnrs mne esr ∂ mre ens ∂ Γ e (0, r, p, r + p) = f f + f f Γαβγ( r, p, r + p) . µαβγ − − − ∂rµ ∂pµ − − e (4.20)

The derivations above depend on the vertices involved admitting a Taylor expansion, i.e., they must not contain poles in q = 0. Later, in Sec. 4.5, this assumption will be removed in order to obtain a dynamical gluon mass, but for now, we use the WIs derived under such supposition.

4.4 Gluon self-energy at the origin

The WIs derived in the previous Section can be used to write the gluon self-energy,

Πµν(q), deﬁned in Eq. (3.42), in the limit q 0. As we will see, Πµν(0) triggers the → seagull identity, given by Eq. (4.12). In order to compute the self-energy at the origin, we e e set q = 0 directly in the diagrams of Fig. 3.4. Thus, the resulting tensorial structures can only be saturated by gµν, so we have (color indices are suppressed whenever possible)

Πµν(0) = Π(0)gµν . (4.21)

e e Then, let us start by obtaining the contribution to the self-energy from diagrams (a1) (1) and (a2) of Fig. 3.4 (green block); we will denote such contribution as Π (0). For con- venience, we show these two diagrams again in Fig. 4.2, where we specify the momentum e convention. 4.4. Gluon self-energy at the origin 60

Figure 4.3: One-loop dressed ghost diagrams contributing to Πµν(q).

e Therefore, we have (1) d Π (0) = a1(0) + a2(0), (4.22) where e

1 2 (0) αρ βσ µ a (0) = CAg Γ (0, k, k)∆ (k)∆ (k)Γ (0, k, k) , (4.23) 1 2 µαβ − ρσ − Zk 2 α a (0) = iCAg (d 1) ∆ (k) . e (4.24) 2 − − α Zk Using Eq. (4.19), we can rewrite Eq. (4.23) as

αβ i 2 (0) ∂∆ (k) a (0) = CAg Γ (0, k, k) 1 −2 µαβ − ∂k Zk µ i 2 ∂ (0) αβ α = CAg Γµαβ(0, k, k)∆ (k) 2(d 1) ∆α(k) , (4.25) −2 k ∂kµ − − − k Z h i Z where in the last line we have used the tree-level expression for the conventional three- gluon vertex to obtain ∂ (0) Γ (0, k, k) = 2(d 1)gαβ. (4.26) ∂kµ µαβ − −

Thus, the second term of Eq. (4.25) cancels exactly against the expression for a2(0) in Eq. (4.24), which leads us to

(1) (1) 2 ∂ µ (k) (1) 2 d Π (0) = CAg (d 1) F ; (k) = ∆(k )kµ. (4.27) − − ∂kµ Fµ Zk e Then, according to the seagull identity presented in Eq. (4.12), Π(1)(0) = 0. Next, we look at the ghost-loop diagrams of Fig. 4.3, which have been extracted from e 4.4. Gluon self-energy at the origin 61

Figure 4.4: Two-loop dressed gluonic contribution to Πµν(q).

e the orange block of Fig. 3.4. At q = 0, we have

2 2 2 µ a (0) = CAg kµD (k )Γ (0, k, k), (4.28) 3 − Zk 2 2 a (0) = idCAg D(k )e. (4.29) 4 − Zk

From Eq. (4.18), we can rewrite a3(0) as

2 ∂ 2 2 a (0) = iCAg kµD(k ) d D(k ) , (4.30) 3 − ∂kµ − Zk Zk where the second term cancels with a4(0). Therefore, we obtain

(2) (2) 2 ∂ µ (k) (2) 2 d Π (0) = iCAg F ; (k) = D(k )kµ , (4.31) − ∂kµ Fµ Zk e which implies Π(2)(0) = 0. Finally, the last block is shown in Fig. 4.4 and contains the two-loop dressed gluonic e (3) contribution, Πµν (q). At q = 0 such diagrams are mathematically expressed as

e 1 aab(0) = g4Γ(0)amnr ∆γτ (k)∆βσ(`)∆αρ(k + `)Γbrnm(0, k, `, k `), 5 −6 µαβγ µτσρ − − Zk Z` γτ λδ αβ µ a (0) = i µαβγ ∆ (k)∆ (k)Y (k)Γ (0, k, k)e, (4.32) 6 N δ τλ − Zk e where we have deﬁned 3 2 4 µαβγ = C g (gµαgγβ gµβgγα) , (4.33) N 4 A − and αβ αρ βσ Y (k) = ∆ (` + k)∆ (`)Γσρδ(`, ` k, k) . (4.34) δ − − Z` 4.4. Gluon self-energy at the origin 62

αβ The quantity Yδ (k) corresponds to the subdiagram nested inside diagram (a6) in Fig. 4.4

(the one containing the blue conventional vertex Γσρδ). Such subdiagram was studied in [38] and, due to the Bose symmetry of the vertex Γσρδ, it can be writen as

αβ α β β α 2 2 1 1 δ αβ Y (k) = (k g k g )Y (k ); Y (k ) = kαg Y (k). (4.35) δ δ − δ d 1 k2 β δ − Using Eq. (4.19) again, we obtain

γδ αβ ∂∆ (k) a (0) = µαβγ Y (k) . (4.36) 6 −N δ ∂kµ Zk

For diagram a5, we employ the WI of Eq. (4.20) and arrive at

1 aab(0) = g4Γ(0)amnr ∆γτ (k)∆βσ(`)∆αρ(k + `) 5 −6 µαβγ Zk Z` bre emn ∂ bne erm ∂ f f + f f Γτσρ(k, `, k `). (4.37) × ∂kµ ∂`µ − −

(0)amnr Then, using the explicit expression for Γµαβγ , after some manipulation we obtain

2 ∂ ∂∆γδ(k) a (0) = µ ∆γδ(k)Y αβ(k) Y αβ(k) . (4.38) 5 −Nαβγ 3 ∂kµ δ − δ ∂kµ Zk Zk The second term cancels against Eq. (4.36) and, making use of Eq. (4.35), we have

(3) (3) 4 2 ∂ µ (k) (3) 2 2 d Π (0) = i(d 1)g C F ; (k) = ∆(k )Y (k )kµ. (4.39) − A ∂kµ Fµ Zk e Therefore, we have demonstrated from Eqs. (4.27), (4.31), and (4.39) that, in the absence of poles of the type 1/q2 in the PT-BFM fundamental vertices, the self-energy at q2 = 0 vanishes,

3 Π(i)(0) = 0, i = 1, 2, 3 Π(0) = Π(i)(0) = 0 . (4.40) ⇒ i=1 X e e e Consequently, from Eq. (3.26), we also have

∆−1(q2) = q2 + i Π(1)(q2) + Π(2)(q2) + Π(3)(q2) ∆−1(0) = 0 , (4.41) ⇒ h i e e e e e 4.4. Gluon self-energy at the origin 63 which implies that, as long as 1 + G(0) is ﬁnite1, the inverse of the conventional gluon propagator is also zero (see BQI given by Eq. (3.32)),

∆−1(0) ∆−1(0) = = 0 . (4.42) 1 + G(0) e This fact ensures a massless gluon in the absence of poles at q = 0.

4.4.1 Renormalization

It is important to verify if our conclusion about the vanishing of ∆−1(0) in the absence of poles remains true after renormalization. In the quantum sector, we have

−1 2 −1 2 −1 ∆R = ZA ∆(q ); DR = Zc D(q ); gR = Zg g, (4.43)

µ µ µαβ µαβ mnrs mnrs ΓR = Z1Γ ;ΓR = Z3Γ ;ΓR µαβν = Z4Γµαβν , (4.44) where the renormalization constants were deﬁned in Eqs. (2.46) and (2.49). On the other hand, for the functions particular to the PT-BFM scheme, the relevant two point functions are renormalized as

−1 −1 ∆R = Z ∆; ∆R = ∆; GR = ZGG, (4.45) B Z b b e e with the renormalization constants satisfying [33]

−1/2 1/2 1/2 −1/2 1/2 −1 −1 Zg = ZB ; = Z ZB ; ZG = Z ZB = ZcZ = Z . (4.46) Z A A 1 Z A

In addition, for the vertices involving one background gluon, we deﬁne the renormalization constants as

µ µ µαβ µαβ mnrs mnrs ΓR = Z1Γ ; ΓR = Z3Γ ; ΓR µαβν = Z4Γµαβν . (4.47)

1 −1 In the Landaue e gauge,e from Eq.e (3.40), wee havee 1 + G(0) = Fe (0). Sincee F(0)e is known to saturate at a non-vanishing value for ξ = 0, the ﬁniteness of 1 + G(0) is assured in this case. 4.4. Gluon self-energy at the origin 64

The Abelian STIs obeyed by these vertices, presented in Eq. (3.51), impose the following conditions to these renormalization constants

Z1 = Zc; Z3 = ZA; Z4 = Z3. (4.48)

e e e Looking at the terms that compose the ﬁrst block of diagrams in the gluon SDE, we have

(1) 2 2 2 dΠ (q ) = a1(q ) + a2(q )

2 1 (0) αρ βσ µ α = g CA Γ ∆ (k)∆ (k + q)Γ + (d 1) ∆ (k) e 2 µαβ σρ − α Zk Zk 2 2 1 (0) αρ βσ e µ α = Zg ZAgR CA Γµαβ∆R (k)∆R (k + q)ΓR σρ + (d 1) ∆R α(k) 2 k − k 2 R 2 Z R 2 Z = Zg ZA a1(q ) + a2(q ) e 2 (1) 2 = Zg ZA d ΠR (q ). (4.49)

e Similarly, one can show that [76]

(2) 2 2 2 2 R 2 R 2 2 (2) 2 dΠ (q ) = a3(q ) + a4(q ) = Zg Zc a3(q ) + a4(q ) = Zg Zc dΠR (q ) , (3) 2 2 2 4 −1 3 R 2 R 2 4 −1 3 (3) 2 dΠe (q ) = a5(q ) + a6(q ) = Zg Z3 ZA a5(q ) + a6(q ) = ZgeZ3 ZA dΠR (q ) . (4.50) e e Combining these equations and using the relations between the renormalization functions from Eqs. (2.49) and (4.46), we obtain

∆−1(q2) = q2 + i Z Π(1)(q2) + Z Π(2)(q2) + Z Π(3)(q2) . (4.51) R Z 3 R 1 R 4 R h i e e e e −1 Thus, it becomes clear that the renormalized propagator ∆R (0) vanishes in the absence of poles in q2 in the fundamental vertices. e Speciﬁcally, the constant can be ﬁxed in the momentum subtraction (MOM) scheme Z [79, 80], which imposes that the gluon propagator recovers its tree-level value at the

−1 2 2 renormalizarion point µ, thus, ∆R (µ ) = µ . Consequently, we have

i e = 1 Z Π(1)(µ2) + Z Π(2)(µ2) + Z Π(3)(µ2) . (4.52) Z − µ2 3 R 1 R 4 R h i e e e 4.5. Circumventing the complete seagull cancellation 65

Then, we obtain

−1 2 2 (1) 2 (2) 2 (3) 2 ∆R (q ) = q + i Z3ΠR (q ) + Z1ΠR (q ) + Z4ΠR (q ) 2 h i q (1) 2 (2) 2 (3) 2 e i Z ΠRe(µ ) + Z ΠRe (µ ) + Z ΠRe (µ ) , (4.53) − µ2 3 1 4 h i e e e −1 such that indeed ∆R (0) = 0.

4.5 Circumventing the complete seagull cancellation

In the previous Section, we have veriﬁed that the inverse of the gluon propagator is zero at q2 = 0, in the absence of poles in the vertices, due to the seagull identity. The generation of an IR ﬁnite gluon propagator relies on the non-Abelian implementation of the Schwinger mechanism, which requires the introduction of longitudinally coupled massless poles in the fundamental vertices. Then, we now assume that the full nonperturbative vertices contain 1/q2 poles2, so that they can be divided into two parts, one in the absence of poles (superscript “np”) and another that contains the contribution from the poles (superscript “p”), thus,

np p Γµαβ(q, r, p) = Γµαβ(q, r, p) + Γµαβ(q, r, p),

np p e Γµ(q, r, p) = Γeµ (q, r, p) + Γeµ(q, r, p), mnrs np,mnrs p,mnrs Γµαβγe(q, r, p, t) = Γeµαβγ (q, r, p,e t) + Γµαβγ (q, r, p, t) . (4.54)

e e e In Fig. 4.5 we present a schematic representation of the decomposition for the three-gluon vertex. The same diagramatic image can be extended to the ghost-gluon and four-gluon vertices. The requirement that the poles must be longitudinally coupled is dictated by the fact that they should act like dynamical Nambu-Goldstone bosons and decouple from on-shell amplitudes. Note, however, that unlike Nambu-Goldstone bosons, the origin of such massless poles is not related to the spontaneous breaking of any global symmetry, but happens for purely dynamical reasons due to the strong coupling of QCD.

2We only consider poles in q2 because it is the relevant channel for the gluon SDE, where the vertices will be inserted. Poles associated with r and p can also emerge since the vertices also satisfy STIs with respect to rµ and pµ; however, such STIs do not appear in our analysis of the gluon propagator of Fig. 3.4.

4.5. Circumventing the complete seagull cancellation 66

⑥

☛❀ ❛

❡ ❡

♥✂ ✂

✐

③

✁ ✁

④

❂

✰

✷

✄

♣

✗❀ ❝

✖❀ ❜ ⑤

Figure 4.5: Decomposition for the full BQQ three-gluon vertex (red circle) into no-pole, np p Γµαβ (yellow), and pole parts, Γµαβ (green). Analogous decompositions hold for Γµ and Γmnrs. eµαβγ e e e For the case of the three-gluon vertex, demanding the pole part to be longitudinally coupled implies in α0α µ0µ ν0ν p P (q)P (r)P (p)Γαµν(q, r, p) = 0 , (4.55) where analogous relations hold for the other twoe vertices. Therefore, we can write the pole parts of the vertices of Eq. (4.54) as

q Γp (q, r, p) = µ C (q, r, p), µαβ q2 αβ q e Γp(q, r, p) = µ Ce (q, r, p), µ q2 gh q Γp,mnrse(q, r, p, t) = µ Cemnrs(q, r, p, t). (4.56) µαβγ q2 αβγ e e In order to preserve the symmetry of the theory, the Takahashi identities must also be satisﬁed by the vertices in the presence of poles. Thus, combining Eq. (3.51) with Eqs. (4.54) and (4.56), we have

µ np −1 −1 q Γ (q, r, p) + Cαβ(q, r, p) = i∆ (r) i∆ (p), µαβ αβ − αβ qµΓnp(q, r, p) + C (q, r, p) = D−1(r2) D−1(p2), e µ e gh − µ np,mnrs mnrs mse ern q Γµαβγ (q,e r, p, t) + Cαβγe(q, r, p, t) = f f Γαβγ(r, p, q + t) mne esr e e + f f Γβγα(p, t, q + r) mre ens + f f Γγαβ(t, r, q + p). (4.57)

Note that, although the conventional three-gluon vertex Γαβγ(q, r, p) may contain poles 4.5. Circumventing the complete seagull cancellation 67 in q2, the right hand side (rhs) of the third STI in Eq. (4.57) does not, because q never appears alone in the arguments of the vertices. From these STIs, we can derive expressions analogous to the WIs of Eqs. (4.18), (4.19) and (4.20), by Taylor expanding both sides of Eq. (4.57) around q = 03. One can easily see that the zeroth order terms vanish,

mnrs Cαβ(0, r, r) = 0; C (0, r, r) = 0; C (0, r, p, p r) = 0 . (4.58) − gh − αβγ − − e e e On the other hand, the terms of order qµ furnish

−1 np ∂∆αβ (r) ∂ Γ (0, r, r) = i Cαβ(q, r, r q) , µαβ − − ∂rµ − ∂qµ − − q=0 −1 e np ∂∆αβ (p) ∂ e Γ (0, p, p) = i Cαβ(q, p q, p) , (4.59) µαβ − ∂pµ − ∂qµ − − q=0 e e for the three-gluon vertex,

∂D−1(r2) ∂ Γnp(0, r, r) = i C (q, r, r q) , µ − − ∂rµ − ∂qµ gh − − q=0 ∂D−1(p2) ∂ Γenp(0, p, p) = i C e(q, p q, p) , (4.60) µ − ∂pµ − ∂qµ gh − − q=0 e e for the ghost-gluon vertex, and

np,mnrs mne esr ∂ mre ens ∂ Γ (0, r, p, r p) = f f + f f Γαβγ(r, p, r p) µαβγ − − ∂rµ ∂pµ − − ∂ e Cmnrs(q, r, p, q r p) , − ∂qµ αβγ − − − q=0 np,mnrs mnee esr ∂ mre ens ∂ Γ (0, r, p, r + p) = f f + f f Γαβγ( r, p, r + p) µαβγ − − − ∂rµ ∂pµ − − ∂ e + Cmnrs( q, r, p, q + r + p) , (4.61) ∂qµ αβγ − − − q=0 e for the four-gluon vertex. Now we can repeat the computations of the previous Section by replacing the vertices at q2 = 0 appearing in the self-energy diagrams by their new WIs of Eqs. (4.59), (4.60) and (4.61). Then, the derivatives which already appeared in the absence of poles will

3 2 2 Note that, because we have factored out qµ/q in Eq. (4.56), the poles 1/q no longer appear in Eq. (4.57), allowing the Taylor expansion of the equations. 4.5. Circumventing the complete seagull cancellation 68 trigger the seagull identity and vanish as before, whereas the contributions from the derivatives of the C functions will survive. Therefore, for the ﬁrst contribution to the self-energy, we obtain e

(1) 1 2 (0) αρ βσ ∂ dΠ (0) = CAg Γ (0, k, k)∆ (k)∆ (k) Cσρ(q, q k, k) . (4.62) −2 µαβ − ∂qµ − − Zk q=0 e e From the pole contribution for the (red) ghost-gluon vertex appearing in Fig. 4.3, we have

(2) 2 (0) 2 2 ∂ dΠ (0) = CAg Γ (0, k, k)D (k ) C (q, q k, k) . (4.63) − µ − ∂qµ gh − − Zk q=0 e e Finally, for the diagrams appearing in Fig. 4.4, we ﬁnd

(3) ab ab γτ λδ αβ ∂ dΠ (0)δ = i µαβγδ ∆ (k)∆ (k)Y (k) Cτλ(q, k q, k) N δ ∂qµ − − Zk q=0 ig4 ∂ +e Γ(0)amnr ∆γτ (k)∆βσ(`)∆αρ(k + `) Cbrnme(q, k q, `, k + `) . (4.64) 6 µαβγ ∂qµ τσρ − − − Zk Zl q=0 e brnm Thus, in principle, if at least one of the functions Cσρ, Cgh, and Cτσρ appearing in Eqs. (4.62), (4.63), and (4.64) acquires a non-null value4, automatically, one has e e e

3 Π(0) = Π(i)(0) = 0, 6 i=1 X e e leading to a ﬁnite gluon propagator in the IR, i.e.,

∆−1(0) = m2 = 0 . (4.65) 6

Let us now review the main conclusions of the current Chapter. The quadratic divergences that plague the study of dynamical mass generation can be shown to vanish by virtue of the seagull identity. Such identity, together with the WIs satisﬁed by the vertices, is also responsible for the vanishing of ∆−1(0) in the absence of poles. Therefore, in order to be able to obtain massive solutions for the gluon propagator, we must re- e quire the vertices to contain massless poles in the background gluon momentum channel. These poles are responsible for generating a pole in the gluon vacuum polarization, which

4In Chapter6, we will see that the generation of a dynamical gluon mass hinges on the existence of a pole at least in the three-gluon vertex, i.e., Cσρ = 0. 6 e 4.5. Circumventing the complete seagull cancellation 69 triggers the Schwinger mechanism allowing for a dynamical gluon mass. We know that poles in the Green’s functions of a QFT identify the presence of bounds states [47]. In the next Chapter, we will present a brief introduction to the formalism of BSEs, which are the equations that describe two-body bound state amplitudes. Using this framework, in Chapter6, we check the viability of the dynamical generation of these poles as massless bound state excitations. 70

Chapter 5

Bethe-Salpeter Equations

In Chapter2, we briefly reviewed the SDEs, which can be understood as the equations of motion of the Green’s functions of a QFT. Bound-states in QFTs, however, are studied under the context of BSEs, which are integral equations describing the dynamics of two- body bound states. The BSE is a general nonperturbative quantum field theoretical tool that can be written down in any field theory. The form of this equation was first derived in 1951 by Bethe and Salpeter [40], although it had originally been proposed by Nambu in 1950, but without derivation [81]. The BSE was also derived from field theory independently by Gell-Mann and Low [82] and Schwinger [83], using different approaches. Bound states and resonances are identified by the occurrence of poles in the Green’s functions of the QFT. It is well known that bound states are not accessible in perturbation theory because a finite sum of perturbative diagrams can never create a pole in the scattering matrix [84]. On the other hand, the required pole can arise from a divergence of the sum of an infinite subset of diagrams. This property makes the generation of bound states a nonperturbative problem in QFT. From the study of BSEs, we can obtain valuable information about bound states by accessing the dynamical properties of the system, such as the bound state amplitude and its mass spectrum. In this Chapter, we will focus on the derivation of the BSE for the simple case of two scalar particles interacting by the exchange of a third scalar. We start by deriving the inhomogeneous BSE for our model. Then, we obtain the homogeneous BSE for the Bethe-Salpeter amplitude (BSA), which is more commonly used in the studies of bound states. Finally, we present a numerical treatment for solving these equations, which will 5.1. Derivation of the BSE 71 be used in Chapter6. The reader who is already familiarized with the formalism of BSEs can skip this Chapter and go directly to Chapter6, where we apply the BSEs within the context of dynamical gluon mass generation.

5.1 Derivation of the BSE

Because of the presence of poles in the integration contours, BSEs are usually studied after implementing an analytic continuation into Euclidean space (known as Wick rotation), which allows one to avoid these troublesome poles [85, 86]. Therefore, in our analysis, we must transform the generating functional deﬁned in Eq. (2.1) to Euclidean space. Let us start with the Lagrangian for the free scalar ﬁeld in Minkowski space,

1 µ 2 2 = ∂µφ∂ φ m φ . (5.1) L 2 − Then, to transform to Euclidean space, we make the substitutions

4 4 µ E µ ix x , d x id x ; ∂µφ∂ φ ∂ φ∂ φ . (5.2) 0 → 4 → − E → − µ E

Thus, we write 1 = ∂Eφ∂µφ + m2φ2 , (5.3) LE 2 µ E so the generating functional is given by

Z[J] = [φ] exp S [φ] + d4x J(x)φ(x) , (5.4) D − E E Z Z

4 where S = d x E and J is the source of the scalar ﬁeld φ. From now on everything E E L in this SectionR is in Euclidean space, so we drop the subscript E in order to simplify the notation.

We consider a model of two scalar particles φ1 and φ2 exchanging a third scalar particle

φ3. The Lagrangian of such theory is given by

3 1 µ 2 2 g 2 2 = ∂µφi∂ φi + m φ + φ φ + φ φ . (5.5) L 2 i i 2 1 3 2 3 i=1 X 5.1. Derivation of the BSE 72

For the generating functional, we have

3 4 Z[J ,J ,J ] = [φ] exp S[φ] + d x Ji(x)φi(x) . (5.6) 1 2 3 D − " i=1 # Z Z X where Ji are the external sources corresponding to the ﬁelds φi and φ represents the collective of ﬁelds as in Chapter2. The equivalent of the SD relation of Eq. (2.14) in Euclidean space is given by

δS δ Ji(x) Z[J] = 0 . (5.7) δφ (x) δJ(x) − i

So, taking the functional derivative with respect to φ1, we ﬁnd

δ δZ[J ,J ,J ] ∂2 + m2 + g 1 2 3 J (x)Z[J ,J ,J ] = 0 . (5.8) − 1 δJ (x) δJ (x) − 1 1 2 3 3 1

We follow the steps of Chapter2, and take another derivative with respect to J1(y), turning oﬀ the sources for φ1 and φ2. Then, using Eq. (2.9), we obtain [87]

∂2 + m2 D (x, y) + d4z Σ (x, z )D (z , y) = δ4(x y) , (5.9) − 1 1 2 1 2 1 2 − Z where D1 is the propagator for the ﬁeld φ1, deﬁned as (i = 1, 2)

δ2Γ −1 D (x, y) = , (5.10) i δφ (x)δφ (y) i i and Σ1 is the self-energy,

2 4 4 Σ (x, z ) = g d zd z ∆s(x, z)D (x, z )Λ(z, z , z ) , (5.11) 1 2 − 1 1 1 1 2 Z with ∆s being the propagator for the exchanged ﬁeld φ3 and Λ the 1PI vertex function of the theory. Additionally, Eq. (5.9) can be written as

∂2 + m2 + Σ D (x, y) = δ4(x y) , (5.12) − 1 1 1 − 5.1. Derivation of the BSE 73

Figure 5.1: Diagrammatic representation of the four-point Green’s function,

G(x1, y1; x2, y2), which describes the propagation of the two scalar particles φ1 and φ2, represented by the continuous lines.

where we use the notation

4 Σ1D1(x, y) = d z2Σ1(x, z2)D1(z2, y) . (5.13) Z In order to obtain the BSE for our case, we must study the propagation of the two particles φ1 and φ2, so we are are interested in the four-point Green’s function, G(x1, y1; x2, y2), which is diagrammatically represented in Fig. 5.1 and is deﬁned as

G(x , y ; x , y ) = 0 T φ (x )φ (x )φ†(y )φ†(y ) 0 (5.14) 1 1 2 2 h | { 1 1 2 2 1 1 2 2 } | i 4 −1 δ Z[J1,J2,J3] = Z [0, 0,J3] . (5.15) δJ2(y2)δJ1(y1)δJ2(x2)δJ1(x1) J1=J2=0

Thus, we need to take more functional derivatives with respect to the sources.

Then, we go back to Eq. (5.8) and take one functional derivative with respect to J1 again and two extra with respect to J2. After setting the sources equal to zero, we ﬁnd [87]

δ ∂2 + m2 + g G(x , y ; x , y ) = δ4(x y )D (x , y ) . (5.16) − 1 δJ (x ) 1 1 2 2 1 − 1 2 2 2 3 1

It is useful to rewrite Eq. (5.16) as

∂2 + m2 + Σ G(x , y ; x , y ) = δ4(x y )D (x , y ) − 1 1 1 1 2 2 1 − 1 2 2 2 δ + Σ g G(x , y ; x , y ) . (5.17) 1 − δJ (x ) 1 1 2 2 3 1

Then, we multiply both sides by D1(z1, x1), so we can use Eq. (5.12), and integrate over 5.1. Derivation of the BSE 74

Σ1 δ D1(x1, z1) G(z1, y1; x2, y2) = G + K G δJ3(z1)

Figure 5.2: Functional derivative of the four-point Green’s function, G(x1, y1; x2, y2). The continuous line represents the scalar particles φ1 and φ2 and the white circle represents the self-energy, Σ1, for the ﬁeld φ1.

x1, obtaining

G(x1, y1; x2, y2) = D1(x1, y1)D2(x2, y2) δ + d4z D (x , z ) Σ g G(z , y ; x , y ) . (5.18) 1 1 1 1 1 − δJ (z ) 1 1 2 2 Z 3 1 Now, note that in Eq. (5.15), we have used the deﬁnition of the four-point function in terms of the generating functional Z, which includes the contribution from both connected and disconnected diagrams. We expect that, in analogy to the case of the two-point function, this equation can be written in terms of two-particle irreducible (2PI) contributions.

It is possible to show that the functional derivative of G(x1, y1; x2, y2) appearing as the last term in the integral of Eq. (5.18) can be written as the sum of a term that cancels with the Σ1 term inside the integral and another that involves a four-point 2PI kernel, which we call K [87]. The diagrammatic decomposition of this term is given in Fig. 5.2[88]. The kernel K appearing in Fig. 5.2 contains an inﬁnite sum of Feynman diagrams, but its explicit form does not interest us at the moment. This decomposition allows us to write the four-point Green’s function, G(x1, y1; x2, y2), as [87]

G(x1, y1; x2, y2) = D1(x1, y1)D2(x2, y2)

4 + d z1234 D1(x1, z1)D2(x2, z2)K(z1, z3; z2, z4)G(z3, y1; z4, y2) , (5.19) Z where we employ the measure notation introduced in Eq. (2.56). The equation above is the so called inhomogeneous BSE for two scalars interacting. It can also be written in the abbreviated form,

G(1,2) = D1D2 + D1D2KG(1,2) , (5.20) whose diagrammatic representation in given by Fig. 5.3. 5.1. Derivation of the BSE 75

G = + K G

Figure 5.3: The inhomogeneous BSE for two scalars interacting.

Here, we are interested in the BSE in momentum space. Due to the translational invariance of the theory, it is useful to deﬁne the relative space-time coordinates as

x := x x , y := y y . (5.21) 1 − 2 1 − 2

Then, we can use a partition parameter η inside the interval 0 η 1 to deﬁne 1 ≤ 1 ≤

X := η1x1 + η2x2 , (5.22) with η = 1 η . Thus, we obtain 2 − 1

x = X + η x , x = X η x . (5.23) 1 2 2 − 1

Analogously, we also have

Y := η y + η y , y = Y + η y , y = Y η y . (5.24) 1 1 2 2 1 2 2 − 1

Then, considering momentum conservation and translational invariance, we can write the functions of interest in terms of the relative (p) and total (P ) momentum of the pair

φ1 and φ2, so that

p = η P + p , p = η P p , (5.25) 1 1 2 2 − where pi is the momentum of φi, and

P = p + p , p = η p η p . (5.26) 1 2 2 1 − 1 2 5.2. The Homogeneous BSE 76

With these deﬁnitions, we Fourier transform Eq. (5.19) and obtain

G(p, q; P ) = D (η P + p)D (η P p)δ4(q p) 1 1 2 2 − − + D (η P + p)D (η P p) K(p, k, P )G(k, q, P ) , (5.27) 1 1 2 2 − Zk which is the inhomogeneous BSE in momentum space.

5.2 The Homogeneous BSE

Now, we focus on studying the bound states amplitude, known as BSA, and the homogeneous BSE that it satiﬁes; a detailed derivation of this equation was given by Nakanishi in [89]. First, note that, in order to represent a bound state, the four-point Green’s function G, deﬁned in Eq. (5.14), must have a pole when the total momentum equals the bound state mass. Then, we can insert a complete set of two-particle states P, α inside Eq. (5.14), where P is the total momentum of the state and α all other | i quantum numbers,

G(x , y ; x , y ) = 0 T φ (x )φ (x ) P, α P, α T φ†(y )φ†(y ) 0 . (5.28) 1 1 2 2 h | { 1 1 2 2 } | i h | { 1 1 2 2 } | i P,α X The contribution from a particular bound state B is given by

† † G(x , y ; x , y ) = 0 T φ (x )φ (x ) PB PB T φ (y )φ (y ) 0 , (5.29) 1 1 2 2 h | { 1 1 2 2 } | i h | { 1 1 2 2 } | i P X

2 2 where we have deﬁned PB = P, αB . Because the on-shell condition P = M holds | i | i for the bound state with mass M, this sum over states can be represented as the integral over the mass shell in momentum space as

= d4P θ(P )δ(P 2 M 2) . (5.30) ··· 0 − ··· P X Z We, then, deﬁne the Bethe-Salpeter (BS) wave function and its conjugate as

χB(x , x ; P ) = 0 T φ (x )φ (x ) PB , 1 2 h | { 1 1 2 2 } | i † † χ¯B(x , x ; P ) = PB T φ (x )φ (x ) 0 . (5.31) 1 2 h | { 1 1 2 2 } | i 5.2. The Homogeneous BSE 77

χB χ¯B G = + R P 2 + M 2

Figure 5.4: Pole contribution to the four-point Green’s function, G(p, q, P ).

In addition, the translational invariance of the theory implies that

−iP ·X χB(x1, x2; P ) = e χB(x, P ) ,

iP ·X χ¯B(x1, x2; P ) = e χ¯B(x, P ) . (5.32)

Considering its Fourier transform, we obtain the BS wave function in momentum space,

4 −ik·x χB(x; P ) = d k e χB(k; P ) , Z 4 ik·y χ¯B(y; P ) = d k e χ¯B(k; P ) . (5.33) Z Then, Nakanishi demonstrates that the contribution to G from the intermediate state

PB , after adding the contribution from the anti-particle states, can be written as (Eu- | i clidean space) [89],

χB(p, P )¯χB(q, P ) 2 2 G(p, q; P ) = |P =−M + R (5.34) P 2 + M 2 where R represents the terms regular in the P 2 = M 2. We represent Eq. (5.34) dia- − grammatically in Fig. 5.4. Now, we can insert Eq. (5.34) into the abbreviated expression for G of Eq. (5.20), so that we have χ χ¯ χ χ¯ B B + R = D D + D D K B B + R . (5.35) P 2 + M 2 1 2 1 2 P 2 + M 2 Multiplying both sides by (P 2 + M 2) and taking the limit P 2 M 2, all regular terms → − vanish and we obtain

χBχ¯B = D1D2KχBχ¯B . (5.36) 5.2. The Homogeneous BSE 78

Γ = K Γ

Figure 5.5: Homogeneous BSE for the BSA, Γ(p, P ).

Comparing both sides of Eq. (5.36), we arrive at

χB = D1D2KχB , (5.37) the explicit form of which reads

χB(p, P ) = D (η P + p)D (η P p) K(p, k, P )χB(k, P ) . (5.38) 1 1 2 2 − Zk It is more convenient to consider the amputated BSA,

−1 −1 Γ(p, P ) = D (η P + p)χB(p, P )D (η P p) . (5.39) 1 1 2 2 −

Then, we ﬁnd the homogeneous BSE

Γ(p, P ) = K(p, k, P )D (η P + k)Γ(k, P )D (η P k) , (5.40) 1 1 2 2 − Zk whose diagrammatic representation is given in Fig. 5.5. In order to apply Eq. (5.40) to a particular bound state, one needs to identify the proper 2PI scattering kernel K, which depends on the dynamics of the theory and is given by an inﬁnite sum of diagrams, so that some truncation is required. The most com- mon approximation in the literature is the so-called ladder approximation, which consists in keeping only diagrams with a single exchanged particle [90–94]. In this case, the ladder approximation for Eq. (5.40) is given in Fig. 5.6, where ∆s represents the propagator of the exchanged particle φ3. In QCD, the ladder approximation is widely used phenomeno- logically to calculate hadron masses, because it respects chiral symmetry breaking, thus being an important truncation to understand the dynamical mass generation [24, 95–102]. 5.3. Numerical treatment 79

D1 Γ = ∆s Γ

Figure 5.6: Ladder approximation for the homogeneous BSE. Continuous lines represent

φ1 and φ2 particles, whereas the dashed line represents the exchanged particle φ3.

5.3 Numerical treatment

Note that Eq. (5.40) can be converted to an eigenvalue problem,

Γ(p, P ) = (p, k, P )Γ(k, P ) , (5.41) K Zk where (p, k, P ) = K(p, k, P )D (η P + k)D (η P k) , K 1 1 2 2 − so that Γ is an eigenfunction to . Then, one can introduce a parameter λ in the equation, K

Γ(p, P ) = λ (p, k, P )Γ(k, P ) , (5.42) K Zk where, in order to recover the original integral equation, one has to look for nontrivial solutions corresponding to the eigenvalue λ = 1. Equivalently, λ can be given in terms of some parameter extracted from the deﬁnition of (p, k, P ). For example, one can extract K the coupling g2, so that

Γ(p, P ) = g2 0(p, k, P )Γ(k, P ) , (5.43) K Zk with 0 = /g2. In this form, this eigenvalue problem can be solved numerically, finding K K a solution for the BSA for specific values of g2. The simplest case is the one where the bound state is massless, i.e., P 2 = M 2 = 0. − This is in fact the problem we face in Chapter6, so we briefly discuss the numerical method to obtain a solution for this case here. First, we can convert Eq. (5.42) to spherical coordinates. Because P 2 vanishes, we only need the modulus of the two independent Euclidean momenta p and k and the angle between them, θ, to describe the system. So, 5.3. Numerical treatment 80 we define the variables1

x = p2, y = k2, p k = √xy cos θ, (5.44) · with the radial and angular integration variables deﬁned within the intervals, y [0, ) ∈ ∞ and θ [0, π]. ∈ Then, we are left with a homogeneous Fredholm equation of the second kind, so Eq. (5.42) can be rewritten in the form

π b Γ(x) = λ dθ dy (x, y, θ)Γ(y) , (5.45) K Z0 Za where a = 0 and b = , but in practice one chooses a ﬁnite b 1 and a 1. In order ∞ to deal with this equation, we can expand the unknown function, Γ(x), in terms of an appropriate function basis and later determine the expansion coeﬃcients.

Here we opt to use the Chebyshev polynomials of the ﬁrst kind, Tm, as our basis, such that [103] n c Γ(x) = 0 + c T (x), (5.46) 2 m m m=1 X where the argument x is deﬁned in the range [ 1, 1] and −

Tm(cos φ) = cos(mφ) . (5.47)

Note that there are n + 1 coeﬃcients in this expansion, therefore, in order to determine them, we need a grid for x with n + 1 points. We can choose this grid to be related to the n + 1 zeros of the polynomial Tn+1(x), which are located at

π(i 1/2) zi = cos − , i = 1, , n + 1 . (5.48) n + 1 ···

The natural range for the arguments of Chebyshev polynomials is [ 1, 1], however, we − are interested in the behavior of the BSA amplitude within the range [a, b]. Therefore, we want to a grid for x within this interval, which can be done by deﬁning

1 1 xi = (b a)zi + (b + a) , (5.49) 2 − 2 1Here x and y are new variables not related to the position of the particles. 5.3. Numerical treatment 81

where zi are the zeros deﬁned in Eq. (5.48). Then, Eq. (5.46) becomes

n c Γ(x) = 0 + c T (x0) , (5.50) 2 m m m=1 X 0 where x represents the grid formed by xi and x is calculated using this grid according to

0 2xi (b + a) x = − = zi . (5.51) i b a − Then, using the expansion of Eq. (5.50) in Eq. (5.45), we obtain

n π b n c0 0 c0 2y (b + a) + cmTm(x ) = λ dθ dy (x, y, θ) + cmTm − . (5.52) 2 K 2 b a m=1 0 a " m=1 # X Z Z X − We can write this equation in matrix form as the generalized eigenvalue problem,

Ac = λBc , (5.53)

where c is the column vector formed by the coeﬃcients ci, while the matrix elements of A and B are given by

π b 2y (b + a) Aij = δTj(zi ) ,Bij = δ dθ dy (xi , y, θ)Tj − , (5.54) +1 K +1 b a Z0 Z0 − with i, j = 0, 1, , n and δ = 1, unless j = 0, in which case it is 1/2. ··· We can rewrite Eq. (5.53) as Mc = 0 , (5.55) where M = A λB. Eq. (5.55) represents a homogeneous system of equations for the − coeﬃcients ci. In order to obtain nontrivial solutions for the vector c, we need to ﬁnd the eigenvalues λ for which det(M) = 0. After that, we choose one of the possible values of

λ and assign a predetermined value to c0 (usually c0 = 1), so that we can determine all the other expansion coefficients cm of Eq. (5.50) by solving the resulting reduced system given by Eq. (5.55). Finally, with these coefficients, we can obtain the function describing the BSA according to Eq. (5.50). Note that if λ is related to the coupling, as in Eq. (5.43), we can only find nontrivial solutions for the BSA for specific values of g2. Then, one should choose the value that 5.3. Numerical treatment 82 can actually represent the physics of the system described. As a final observation, note that, because Eq. (5.45) is linear, the solution obtained can be rescaled by an arbitrary constant. In the next Chapter, we use the concepts presented here as part of our study on dynamical gluon mass generation. There, we will start from the BSEs for the three- gluon and ghost-gluon vertices to check if QCD allows for the dynamical generation of the massless bound states that will ultimately drive to a nonvanishing gluon mass. To obtain a solution for such bound states, we will use a generalization of the numerical method just described. 83

Chapter 6

Massless Bound State Excitations

In Chapter4, we verified that, in order to generate a dynamical gluon mass from the Schwinger mechanism, the vertices entering the SDE for the gluon propagator must acquire massless poles. The appearance of these poles can occur for purely dynamical reasons. In particular, in a pure Yang-Mills theory, one supposes that, for sufficiently strong binding, longitudinally coupled massless bound state excitations may be produced dynamically [104–108]. As mentioned before, these colored excitations act like dynamical Nambu-Goldstone bosons, with the difference that they do not originate from spontaneous symmetry breaking. Such excitations serve to provide the required pole in the gluon self- energy to trigger the Schwinger mechanism, thus generating a mass for the gauge boson without affecting gauge invariance. In this Chapter, we analyze the conditions necessary to generate these massless poles. In Chapter5, we mentioned that bound states are identified as poles in the Green’s functions of the theory; therefore, one can describe the poles required for triggering the Schwinger mechanism as massless bound state excitations. These bound states are longitudinally coupled and, thus, do not appear in the physical spectrum of QCD (we recall that they carry color charge). However, they can still be studied by means of the BSEs, so that we can verify whether the QCD dynamics is sufficiently strong to generate them. The analysis presented here is an extent of the studies of [42, 43], where simplified versions of the BSE were derived. In these previous works, it is assumed that the dom- inant effect for gluon mass generation comes from the pole in the three-gluon vertex. Therefore, the system of BSEs responsible for describing the formation of the massless bound state excitations is simplified by taking into account a single dynamical equation, 6.1. Relation between the gluon mass and the poles 84 which involves only the contribution from the pole in the three-gluon vertex. We improve such approximation by deriving a coupled system that includes the poles generated in the three-gluon and ghost-gluon vertices. We start our analysis by obtaining the relation between the running gluon mass and the function that describes the pole in the three-gluon vertex. Then, we use the concepts of Chapter5, and derive a system of two coupled BSEs that describes the formation of the massless bound state excitations associated with the three-gluon and ghost-gluon vertices. Finally, we solve this system numerically and discuss the results obtained. The discussion and results presented in this Chapter have been published in [44].

6.1 Relation between the gluon mass and the poles

The saturation of the gluon propagator in the deep IR suggests the physical parametriza- tion of Eq. (4.2). According to the discussion of Chapter4, the mass term of the propagator will come from the pole part of the vertices. Therefore, after replacing the inverse gluon propagators appearing in the rhs of the modiﬁed STI for the three-gluon vertex, given in Eq. (4.57), as sums of kinetic and mass terms as given in Eq. (4.2), we can asso- ciate the mass functions with the pole part of the vertex, so that in the Landau gauge we have 2 2 2 2 Cαβ(q, r, p) = m (p )Pαβ(p) m (r )Pαβ(r) , (6.1) − where we recall that, frome momentum conservation, p = q r. − − Going back to Fig. 4.5, the green vertex can be understood as an interaction between the colored massless excitation (the propagator of which is represented by the double lines) and two gluons. This eﬀective vertex, shown in isolation in Fig. 6.1, is the diagrammatic representation of the BSA Cαβ, which can be decomposed as

e Cαβ(q, r, p) = Cgl gαβ + C2 qαqβ + C3 pαpβ + C4 rαqβ + C5 rαpβ , (6.2)

e e e e e e where we have omitted the dependence of the form factors with the momenta. In order to obtain a relation between the dynamical gluon mass and the form factor 6.1. Relation between the gluon mass and the poles 85

m, α r q

a p n, β

Figure 6.1: BSA Cαβ, which represents the pole in the three-gluon vertex.

Cgl, let us contract both sides of Eq. (6.1) with two transverse projectors, e α0α β0β 2 2 2 2 α0 σβ0 P (r)P (p)Cαβ(q, r, p) = [m (p ) m (r )]P (r)P (p) . (6.3) − σ e Then, from the decomposition of Eq. (6.2), we ﬁnd

α0α β0β 2 2 2 2 α0 σβ0 P (r)P (p)[C gαβ + C qαqβ] = [m (p ) m (r )]P (r)P (p) . (6.4) gl 2 − σ e e Thus, we obtain the important relations

C (q, r, p) = m2(p2) m2(r2) , C (q, r, p) = 0 . (6.5) gl − 2 e e Taylor expanding both sides of the ﬁrst equation in (6.5) around q = 0, we arrive at

dm2(r) C0 (r2) = , (6.6) gl dr2 e where 0 2 ∂Cgl(q, r, r q) Cgl(r ) = lim − 2− . (6.7) q→0 ( ∂(r + q) ) e e 0 Therefore, we have established a link between Cgl and the momentum dependent gluon mass, which upon integration gives1 e

q2 2 2 2 0 m (q ) = m (0) + dy Cgl(y) . (6.8) Z0 e Notice that Eq. (6.8) enables us to obtain the running gluon mass directly from the form factor Cgl related to the pole in the three-gluon vertex.

1Notice that the prime refers to the derivative with respect to z = (r + q)2 and not with respect to the integratione variable y (see Eq. (6.7)). 6.2. BSEs for the massless bound-state excitations 86

= + 1 + 2 K K

(a) (b)

+ +

Figure 6.2: The complete BSE for the BQQ three-gluon vertex. The red circles represent the vertices that may contain massless poles and white ones fully dressed propagators. The gray blobs represent the corresponding fully dressed interaction kernels.

6.2 BSEs for the massless bound-state excitations

From Eq. (6.8) and Eqs. (4.62), (4.63) and (4.64), it becomes clear that the existence of brnm a gluon mass depends on the functions Cαβ, Cgh, and Cτσρ and their derivatives. If they were to vanish, we would again obtain m2(0) = 0. Therefore, we must verify whether the e e e dynamical formation of the massless bound states required for triggering the Schwinger mechanism is viable in QCD. In order to authenticate the existence of such bound states, brnm we must derive the BSE for the BSAs Cαβ, Cgh, and Cτσρ and check for the existence of nontrivial solutions for the resulting coupled system of integral equations. e e e To derive this system, we follow the methodology explained in [105, 106]. First, we look to the inhomogeneous BSE for the BQQ three-gluon vertex presented in Fig. 6.2. Note that the vertices in which the background gluon enters (carrying momentum q) are fully dressed and may contain the massless poles. Then, we divide these red vertices appearing on both sides of the equation presented in Fig. 6.2 into a “regular” part (which does not contain poles in q2) and another one containing the pole in q2 (as in Fig. 4.5).

Next, the BSE for Cαβ is obtained from equating the pole parts of both sides. Because all of the red vertices appearing in Fig. 6.2 may contain poles in q2, this e procedure leads to an equation for Cαβ which depends on all the other C functions. These functions, on the other hand, also have their own BSEs, resulting in a complicated e e system of several coupled integral equations, containing combinations of the numerous form factors composing these vertices. In order to simplify this system into something 6.2. BSEs for the massless bound-state excitations 87

m, α m, α m, α r r r q =0 q =0 k q =0 k (A) = 1 + 2 + a a K a K · · · k + q k + q p p p n, β n, β n, β

m m m r r r q =0 q =0 k q =0 k (B) = 3 + 4 + a a K a K · · · k + q k + q p p p n n n

Figure 6.3: Coupled system for Cαβ and Cgh: (A) BSE for the three-gluon vertex pole function C ; (B) BSE for the ghost-gluon pole function C . αβ e e gh e e more manageable we need to consider some simpliﬁcations. In previous works [42, 43], it is assumed that the main contribution for the dynamical gluon mass originates from the pole in the three-gluon vertex, so that the terms coming from the pole parts of the remaining vertices are numerically subleading and, therefore, are neglected. This assumption greatly simpliﬁes the problem given that only diagram

(a) of Fig. 6.2 would contribute to the equation for Cαβ. So, in this case, one has a single integral equation instead of a coupled system of several equations. e In our analysis presented here, we also study the eﬀect of the pole in the ghost-gluon vertex, Cgh, in the generation of a dynamical gluon mass [44]. Thus, diagram (b) of Fig. 6.2 also contributes to the equation for C . Then, the BSE for the BSA C is given in line e αβ αβ (A) of Fig. 6.3, where the ellipsis refer to the neglected terms, corresponding to diagrams e e (c) and (d) of Fig. 6.2.

Note that the second term in the BSE for Cαβ involves the BSA Cgh. The dynamics of this vertex is governed by its own BSE, which can be derived from the BSE for the e e ghost-gluon vertex, forming a coupled system of two equations, as presented in Fig. 6.3, where line (B) represents the BSE for Cgh. Taking the limit in which the momentum of the background gluon vanishes, we obtain e 6.2. BSEs for the massless bound-state excitations 88

k r k ρ, b α, m ρ, b α, m ρ, b α, m ρ, b α, m r r r k k

+ r k + k + r + q K1 ≈ − r + q r + q k + q r + q k + q r + q k + q k + q

σ, c β,n σ, c β,n σ, c β,n σ, c β,n

k r k b α, m b α, m b α, m

r r k

r k + k + r + q K2 ≈ − r + q k + q r + q k + q r + q k + q

β,n β,n c β,n c c

Figure 6.4: One-loop dressed approximation for the kernels and contributing to K1 K2 the BSE for Cαβ. Blue circles represent full conventional vertices (no background gluons), whereas white circles represent fully dressed propagators. e the coupled system

amn abc γρ δσ bmnc f lim Cαβ(q, r, p) = f lim Cγδ(q, k, t)∆ (k)∆ (t) 1ραβσ( k, r, p, t) q→0 q→0 − K − ( Zk e e + C (q, k, t)D(k)D(t) bmnc( k, r, p, t) , (6.9) gh − K2αβ − Zk ) e amn abc γρ δσ bmnc f lim Cgh(q, r, p) = f lim Cγδ(q, k, t)∆ (k)∆ (t) 3ρσ ( k, r, p, t) q→0 q→0 − K − ( Zk e e + C (q, k, t)D(k)D(t) bmnc( k, r, p, t) , (6.10) gh − K4 − Zk ) e where we deﬁned t = k + q. From Eq. (4.58), we know that the zeroth order terms in the Taylor expansion of Eqs. (6.9) and (6.10) vanish, so the derivative terms become the leading contributions. In addition, because we are interested in the dynamically generated 0 gluon mass from Eq. (6.8), we want to obtain an equation for Cgl, so we contract both sides of Eq. (6.9) with P αβ(r) in order to extract the g contribution from C (see αβ e αβ Eq. (6.2)). e Before proceeding further, we approximate the four-point i kernels appearing in K Fig. 6.3 by their lowest-order set of diagrams. The one-loop dressed approximation for these four kernels are diagrammatically represented in Figs. 6.4 and 6.5, in which we keep 6.2. BSEs for the massless bound-state excitations 89

k r r ρ, b m ρ, b m ρ, b m r

k k r k + k + r + q 3 ≈ − K k + q r + q k + q r + q r + q k + q

σ, c n σ, c n σ, c n

k r b m b m r

r k ≈ K4 − r + q k + q r + q k + q

c n c n

Figure 6.5: One-loop dressed approximation for the kernels and contributing to K3 K4 the BSE for Cgh. Blue circles represent fully dressed conventional ghost-gluon vertices, whereas white circles again represent full propagators. e vertices and propagators fully dressed. After performing the contractions of Eq. (6.9), one can verify that the contribution from the first diagram of in Fig. 6.4 (tree-level four- K1 gluon vertex) vanishes. Moreover, the other two diagrams entering the one-loop dressed expansion of furnish equal contributions. The same is also true for the two diagrams K1 that contribute to in the bottom panel of Fig. 6.4. K2 We also perform an additional simplification by considering simple Ansätze for the dressed vertices of Figs. 6.4 and 6.5,

(0) Γµαβ(q, r, q r) = f (q + r)Γ (q, r, p), − − gl µαβ (0) Γµ(q, r, q r) = f (q + r)Γ (q, r, p) , (6.11) − − gh µ where Γ(0) is the standard tree-level expression for the corresponding vertex, given in the table of Fig. 1.2, and the form factors fgl and fgh are functions of a single kinematic variable. Then, from all the considerations above, and using the usual conversion rules to Eu- clidean space, i.e.,

2 2 2 2 r r ; k k ;(r k) (rE kE); → − E → − E · → − · 6.2. BSEs for the massless bound-state excitations 90

i ; ∆(k2) ∆( k2 ); D(k2) D( k2 ) , (6.12) → → − − E → − − E Zk ZkE we obtain

0 2 8π 0 2 2 C (r ) = CAαs C (k )∆ (k)∆(k + r) (k, r) gl 3 gl N1 Zk 1 e + e C0 (k2)D2(k)D(k + r) (k, r) , (6.13) 4 gh N2 Zk 0 2 0 2 2 C (r ) = 2πCAαs C (ke )∆ (k)D(k + r) (k, r) gh gl N3 Zk 1 e + e C0 (k2)D2(k)∆(k + r) (k, r) , (6.14) 2 gh N4 Zk e where

(r k)[r2k2 (r k)2] (k, r) = · − · f 2 (k + r) 8r2k2 + 6(r k)(r2 + k2) + 3(r4 + k4) + (r k)2 , N1 r4k2(k + r)2 gl · · (r k)[r2k2 (r k)2] (k, r) = · − · f 2 (k + r), N2 r4 gh (r k)[r2k2 (r k)2] (k, r) = · − · f 2 (k + r), N3 r2k2 gh (r k)[r2k2 (r k)2] (k, r) = · − · f 2 (k + r). (6.15) N4 r2(k + r)2 gh

Note that in the limit r 0, = = 0, so that C0 (0) = 0. In order to obtain → N 3 N 4 gh C0 in the same limit, let us ﬁrst write Eq. (6.13) in spherical coordinates, gl e e α C ∞ y π C0 (x) = s A dy y C0 (y)∆2(y) dθ sin4 θ cos θ (x, y, z)∆(z)f 2 (z) gl 12π2 gl x N gl gl Z0 r Z0 α C ∞ y π e + s A dy y2Ce 0 (y)D2(y) dθ sin4 θ cos θD(z)f 2 (z) , (6.16) 12π2 gh x gh Z0 r Z0 e where we have introduced the variables x r2, y k2 e z (k + r)2, and used ≡ ≡ ≡ 1 ∞ π = dy y dθ sin2 θ , (6.17) (2π)3 ZkE Z0 Z0 with θ being the angle between r and k. Additionally, the function is deﬁned as N gl 1 (x, y, z) = z + 10(x + y) + (x2 + y2 + 10xy) . (6.18) N gl z 6.3. Numerical analysis 91

The Taylor expansion of 1/z around x = 0 is given by

1 1 √xy = 1 2 cos θ , (6.19) z x + y − x + y where we have used that z = x + y + 2√xy cos θ. Then, using the Taylor expansion for a function of z,

f(z) = f(y) + [2√xy cos θ + x]f 0(y) + 2xy cos2 θf 00(y) + (x3/2) , (6.20) O we ﬁnd

C α ∞ C0 (0) = A s dy y3C0 (y)∆2(y)f (y) f (y)∆0(y) + 2∆(y)f 0 (y) gl 8π gl gl gl gl Z0 C α ∞ e + A s dy y3Ce0 (y)D2(y)) f (y)D0(y) + 2D(y)f 0 (y) , (6.21) 96π gh gh gh Z0 e which does not vanish and depends on the structure of the form factors fgl(y) and fgh(y), along with the gluon and ghost propagators.

6.3 Numerical analysis

Before proceeding to the numerical analysis, let us ﬁrst note that, despite appearances,

0 2 Eq. (6.13) is not linear in Cgl, because the ∆(q ) appearing inside the integrals depends on the dynamical gluon mass through Eq. (4.2), which in turn is related to C0 by virtue e gl of Eq. (6.8). However, here we treat ∆(q2) as an external input. In doing so, the equation e is linearized and can be solved using a generalization of the numerical method described in Sec. 5.3. Taking these considerations into account, we have that the system given by Eqs. (6.13) and (6.14) is composed of two coupled homogeneous linear integral equations. Such type of system has some important properties:

Trivial solutions for C0 and C0 are always valid solutions for the system; • gl gh Because the equationse are homogeneous,e if S0 and S0 are nontrivial solutions, • gl gh C0 = a S0 and C0 = a S0 , with a being an arbitrary constant, are also valid gl gl gh gh e e solutions; e e e e 6.3. Numerical analysis 92

8 3

2.5 6

2 4 1.5

2 1

0 0.5 10-3 10-2 10-1 1 10 102 10-3 10-2 10-1 1 10 102

Figure 6.6: Left panel: The ﬁt for the gluon propagator, ∆(q2), given by Eq. (6.22), together with the lattice data of Ref. [8]. Right panel: The ﬁt for the ghost dressing function, F (q2), given by Eq. (6.24), with the corresponding lattice data. Both quantities are renormalized at µ = 4.3 GeV.

In particular, this system is an eigenvalue problem, where αs assumes the role of • 0 0 the eigenvalue, so nontrivial solutions for the “eigenvectors” Cgl and Cgh exist only when α assumes speciﬁc values. s e e

In order to proceed with the numerical analysis of this system, we ought to specify some of the functions that appear in Eqs. (6.13) and (6.14). For the gluon propagator and the ghost dressing function (recall Eq. (3.30)), we use the ﬁts for the lattice SU(3) data, renormalized at the momentum scale µ = 4.3 GeV, presented in Fig. 6.6. The ﬁt for ∆(q2) is given by [65]

13C g2 q2 + ρ M 2(q2) ∆−1(q2) = M 2(q2) + q2 1 + A 1 ln 1 , (6.22) 96π2 µ2 where 4 2 2 m0 M (q ) = 2 2 , (6.23) m0ρ2 + q 2 with the ﬁtting parameters being g1 = 5.68, ρ1 = 8.55, ρ2 = 1.91, and m0 = 520 MeV. For the ghost dressing function, F (q2), we have [77]

2 2 2 2 9CAg q + ρ (q ) F −1(q2) = 1 + 2 ln 3M , (6.24) 192π2 µ2 6.3. Numerical analysis 93

2 1.4

1 1.3

0 1.2 -1 1.1 -2

0.01 0.10 1 10 100 0.01 0.10 1 10 100 1000

Figure 6.7: Left panel: SU(3) lattice data (evaluated with various β and volumes) for

2 the form factor fgl(q ) in the symmetric conﬁguration [109, 110]; the continuous line corresponds to the ﬁt obtained in [43]. Right panel: The ghost-gluon vertex form factor

2 fgh(q ) in the symmetric conﬁguration obtained from solving its SDE [111]. where 4 2 2 m0 (q ) = 2 2 , (6.25) M m0ρ4 + q 2 and the ﬁtting parameters are g2 = 8.65, ρ3 = 0.25, ρ4 = 0.64.

For the form factors fgl and fgh, we use the curves shown in Fig. 6.7, which are motived by a considerable number of lattice simulations and studies in the continuum. In particular, the three-gluon vertex is known to be suppressed with respect to its tree- level value, reverting its sign in the deep IR (an eﬀect known as zero crossing), and ﬁnally diverging at the origin [112–119]. The reason for this negative divergence can be traced back to the “unprotected” logarithms coming from the massless ghost loops, in contradistinction to the contributions originating from gluon loops, which are “protected” by the gluon mass [114]. This behavior can be observed in the left panel of Fig. 6.7,

2 which shows the lattice data for fgl(q ) in the symmetric conﬁguration [109, 110], which is deﬁned as

q2 = p2 = r2 , q p = q r = p r = q2/2 . (6.26) · · · −

Note that the lattice data of Refs. [109, 110] have been properly normalized by dividing

2 out its coupling value, αs(µ ) = 0.32, employed in this data set. The ﬁt for these data, 6.3. Numerical analysis 94 shown as the blue continuous curve in the left panel of Fig. 6.7, is given by [43, 109]

q2 + m2 q2 2(q2 µ2) f (q2) = σ 1 + b ln 1 + c ln + e M − , (6.27) gl µ2 µ2 (q2 + 2)(µ2 + 2) M M where m = 0.124 GeV, σ = 1.16, b = e = 5.30, and c = 5.40. 1 − On the other hand, for the ghost-gluon vertex we use the solution for the one-loop dressed approximation of its SDE in the symmetric conﬁguration [111], shown in the right panel of Fig. 6.7,

a q2 q2 egh f (q2) = 1 + gh ln d + , (6.28) gh c + (q2 + b )2 gh k2 gh gh 0 where a = 8.62 10−4 GeV2, b = 4.06 10−1 GeV2, c = 9.39 10−2 GeV4, d = 34.3, gh · gh · gh · gh 2 2 egh = 5.36, and k0 = 1.0 GeV . This form factor recovers its tree-level value in both IR and UV limits, with a characteristic peak appearing at the intermediate region of momenta. Such peak also appears in other kinematic conﬁgurations, such as the soft gluon (when the gluon momentum approaches zero) and soft ghost (when the ghost momentum approaches zero) limits [120]. Using the functions mentioned above, we can obtain the numerical solution for the

0 2 0 2 coupled system given by Eqs. (6.13) and (6.14), involving Cgl(q ) and Cgh(q ), employing a generalization of the method explained in Sec. 5.3 for the case of two coupled BSAs. e e Although the numerical calculation is more intricate in this case, the idea of expanding the unknown function as Chebyshev polynomials of the first kind and finding its coefficients after determining the system’s eigenvalue remains the same.

0 2 Then, from solving the coupled Eqs. (6.13) and (6.14), we obtain the solutions Sgl(q ) and S0 (q2) presented on the left panel of Fig. 6.8, where S0 (q2) is the solution for the gh gl e BSA related to the pole in the three-gluon vertex and S0 (q2) to the one corresponding e gh e to the ghost-gluon vertex. These solutions were obtained for the eigenvalue correponding e 2 to αs(µ ) = 0.43. This result indicates that the QCD dynamics is strong enough to generate massless poles for both the three-gluon and ghost-gluon vertices, however, one

0 2 can see their “strengths” are very different, with Sgh(q ) being considerably suppressed with respect to S0 (q2) (at peak value the latter is about five times the former). gl e As a consequence of treating the coupled system of Eqs. (6.13) and (6.14) as a set of e linear equations, the physical scale of our problem will be fixed by demanding that the 6.3. Numerical analysis 95

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0.01 0.10 1 10 100 1000

0 2 0 2 Figure 6.8: Unnormalized solution for the BSAs, Sgl(q ) and Sgh(q ), obtained from solving the coupled system given by Eqs. (6.13) and (6.14) (left panel), and the corresponding e e normalized curves (right panel). value of m2(0) is consistent with the value of ∆−1(0) obtained by the lattice. To do that,

0 2 0 2 we must normalize the numerical solutions Sgl(q ) and Sgh(q ), by fixing a free constant a, such that C0 (q2) = aS0 (q2) and C0 (q2) = aS0 (q2) are also solutions of the homogeneous gl gl gh e gh e system. The normalization procedure is implemented with the help of Eq. (6.8). However, e e e e it is important to notice that for m2(q2) to be interpreted as a running gluon mass, it must have a non-vanishing value in the deep IR and drop sufficiently fast in the UV. Then, from Eq. (6.8), we see that fulfilling these requirements forces the normalization constant a to be negative, i.e., C0 = a S0. Furthermore, the requirement m2( ) = 0 in Eq. (6.8) i −| | i ∞ implies e e ∞ m2(0) = a dy S0 (y) , (6.29) | | gl Z0 where m2(0) = ∆−1(0) can be obtained from thee gluon lattice data presented in left panel of Fig. 6.6, where we have m2(0) 0.14. This value can then be used to find ≈ a 0.015 and obtain the normalized curves for C0 (q2) and C0 (q2) given in the right | | ≈ gl gh panel of Fig. 6.8. e e Using Eq. (6.29) together with Eq. (6.8), we have that the gluon mass function can be

0 2 obtained from Cgl(q ) as

e ∞ 2 2 0 m (q ) = dy Cgl(y). (6.30) − 2 Zq e The resulting running mass is presented as the blue continuous curve of Fig. 6.9. For 6.3. Numerical analysis 96

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00 0 5 10 15

0 Figure 6.9: The dynamical gluon mass obtained by integrating the BSA Cgl obtained as solution of the coupled system of Eqs. (6.13) and (6.14) (blue continuous line), compared e 0 to the one obtained considering Cgh = 0 in Eq. (6.13) [43] (dashed gray line).

e comparison, we also plot the result obtained in [43] when neglecting the eﬀect from the

0 2 pole in the ghost-gluon vertex, i.e., Cgh = 0, as the dashed gray curve (αs(µ ) = 0.45 in this case). We observe that the presence of a pole in the ghost sector implies a slightly e faster running of the gluon mass, which can be accurately ﬁtted through the formula [39]

2 2 2 m (0) m (q ) = 2 2 1+p , (6.31) 1 + (q /m2) with m2 = 2.25 GeV and p = 0.24, as opposed to m2 = 2.52 GeV and p = 0.1 in the absence of the ghost contribution in the BSE. Our analysis presented here reveals that the contribution of the massless pole associated with the ghost-gluon vertex, Γµ, is suppressed with respect to that originating from the corresponding pole of the three-gluon vertex, Γ . Therefore, the latter contribution, e µαβ C0 (q2), is ultimately decisive for the generation of a dynamical gluon mass and its run- gl e ning with the momentum. This is a nontrivial result, since there is no obvious a-priori e argument that would imply suppression of the ghost sector, and we can only verify its impact by carrying out the full analysis of the BSE system as it has been done here. Moreover, we have veriﬁed that the dynamics of QCD indeed enables the generation of poles in both vertices. Finally, it is important to mention that the precise value found for α(µ2) in this analysis depends on the truncation scheme implemented in the kernels entering in the BSE shown 6.3. Numerical analysis 97 in Fig. 6.3. Moreover, if we had treated the system of Eqs. (6.13) and (6.14) as a system of nonlinear equations, we would no longer have an eigenvalue problem and, most likely, the gluon mass would be generated provided that the strength of the entire kernel exceeds a critical value, as typically occurs in the dynamical quark mass generation [121–123]. 98

Chapter 7

Gluon Mass Equation

In the previous Chapters, we have seen that the generation of a dynamical gluon mass explains the IR saturation of the gluon propagator observed in lattice simulations. We have demonstrated that such mass generation relies on the appearance of poles in the fundamental vertices. In addition, we have veriﬁed that indeed the dynamics of QCD may generate these poles, provided that the running coupling furnishes the right amount of strength to trigger the Schwinger mechanism that allows for a momentum dependent gluon mass, m2(q2). Although the theoretical derivation of the gluon mass equation is well established [38], its actual complete treatment is still pending because of the technical complexities involved in such analysis. In particular, a linearized approach has been used in previous studies [38, 39], introducing an indeterminacy in the scale of the corresponding mass, which is later ﬁxed using the available lattice data for the gluon propagator, in a similar way as it was performed in the BS treatment discussed in the previous Chapter. In this Chapter, we present our novel analysis of the gluon mass equation in the Landau gauge, which takes into account its full nonlinear structure [45]. We implement the multiplicative renormalization of this equation using an approximate method, which is extensively employed in the studies of the quark propagator SDE [121–123]. Our presentation starts with some general considerations about the system that de- termines the gluon dynamical mass function. Then, we review the steps for the actual derivation of the mass equation, and its renormalization treatment. In the sequence, we discuss the main ingredients needed in order to proceed to the numerical analysis. Some of these ingredients depend on the structure of the three-gluon vertex, which was studied 7.1. General Considerations 99 in [124] using the results for the form factors of the ghost-gluon scattering kernel published in [111]. Finally, we present a summary of the main conclusions of this Chapter.

7.1 General Considerations

From Eq. (3.43), we know that the conventional gluon propagator, ∆(q2), can be obtained from the diagrams involved in the SDE for the QB propagator, ∆(q2), shown in Fig. 3.4. In particular, in the presence of a dynamical gluon mass, ∆(q2) can be e written in Minkowski space as ∆−1(q2) = q2J(q2) m2(q2). Replacing the left hand side − of Eq. (3.43) with this expression for the gluon propagator, we obtain

6 2 q Pµν(q) + i (ai)µν 2 2 2 2 i=1 q J(q ) m (q ) Pµν(q) = . (7.1) − 1 + G(qP2)

Note that the diagrams (ai), which were presented in Fig. 3.4, contain poles in the structure of the vertices, with such poles being responsible for triggering the Schwinger mechanism as discussed in Chapter4. Because the pole parts of the vertices are the ones responsible for mass generation, one can split Eq. (7.1) into a system of two equations, where the equation for the mass will involve the pole parts of the vertices, whereas the one for the kinetic term will involve their no-pole parts. Then, we will have a coupled system of the generic type

2 2 2 J(q ) = 1 + J (q , m , ∆), K Zk 2 2 2 2 m (q ) = m(qe , m , ∆), (7.2) K Zk e 2 2 2 2 2 2 where q J (q , m , ∆) 0, as q 0, and m(q , m , ∆) = 0 in the same limit. Although K → → K 6 the derivation of this system is theoretically well-deﬁned, its complete treatment has yet e e to be performed, mainly because of technical complexities associated with the equation governing J(q2). In previous works, this problem was circumvented by solving the homogeneous integral equation for m2(q2) in isolation [38, 39]. In particular, the gluon propagator appearing inside the kernel m of Eq. (7.2) was treated as an external quantity, with its form K being ﬁtted from the data of large-volume lattice simulations (similarly to what was e 7.2. Derivation of the mass equation 100 done in Chapter6). In doing so, the gluon propagator is not decomposed according to Eq. (4.2), i.e., ∆−1(q2) = q2J(q2) m2(q2) in Minkowski space, and the nature of the − coupled system is distorted, with the equation for the mass being converted to a single linear integral equation. As a consequence, this becomes an eigenvalue problem, as in

2 Chapter6, which means that the equation has a solution for specific values of αs(µ ), instead of a continuous interval of values, and that we have a family of possible solutions (since a function given by a constant multiplied by a solution is also a solution). This adds an ambiguity because the final physical scale for the mass is fixed “by hand” using the lattice value for ∆−1(0), without a known connection to the fundamental theory. Therefore, in a more complete treatment, we must consider the nonlinearity of the mass equation, by applying the decomposition of Eq. (4.2). The multiplicative renormalization of the equation must also come into play. For that, we use an approximate method, inspired in what has been done for the quark gap equation, where the renormalization constants are replaced by appropriately chosen momentum-dependent functions [121– 123, 125].

7.2 Derivation of the mass equation

Before proceeding any further in the details of our analysis, let us ﬁrst derive the gluon mass equation from the SDE presented in Fig. 3.4. Note that the transversality of both sides of Eq. (7.1) implies that we can obtain the mass equation by isolating the cofactors

2 of qµqν/q on the two sides (as it has been done in [38]). The reason for choosing the

2 contributions proportional to qµqν/q instead of gµν is related to the fact that the pole parts of the vertices are longitudinally coupled (proportional to qν). Thus, denoting by p (ai )µν the contributions coming from the pole parts to the diagrams, we have

q q (ap) = µ ν ap(q2) . (7.3) i µν q2 i

Because diagrams (a2) and (a4) of Fig. 3.4 do not contain nonpertubative vertices with poles, they will not contribute to the mass equation.

Then, we start by considering the pole contribution to diagram (a1) of Fig. 4.2, given 7.2. Derivation of the mass equation 101

Figure 7.1: Contribution from the pole part of the three-gluon vertex to the (a1) diagram of the QB gluon SDE shown in Fig. 3.4.

p 1 2 (0) αρ βσ p (a )µν = g CA Γ (q, k, k q)∆ (k)∆ (k + q)Γ ( q, k, k + q) , (7.4) 1 2 µαβ − − νρσ − − Zk e whose diagrammatic representation is shown in Fig 7.1. Following the same arguments p that led to Eq. (6.1), we know that the pole part of the three-gluon vertex, Γνρσ, is associated to the gluon mass through the WTI e

ν p 2 2 q Γ (q, k, k q) = m (k)Pρσ(k) m (k + q)Pρσ(k + q) . (7.5) νρσ − − − e In addition, after contracting with the transverse projectors as in Eq. (6.3), we have

q P αρ(k)P βσ(k + q)Γp (q, k, k q) = ν [m2(k) m2(k + q)]P αρ(k)P β(k + q) . (7.6) νρσ − − q2 − ρ e So, after appropriate shifts in the integration variable, we ﬁnd

2 p qµqν CAg 2 2 2 2 αρ (a )µν = m (k ) (k + q) k P (k)Pαρ(k + q)∆(k)∆(k + q) , (7.7) 1 q2 q2 − Zk consequently,

2 p 2 CAg 2 2 2 2 αρ a (q ) = m (k ) (k + q) k P (k)Pαρ(k + q)∆(k)∆(k + q) . (7.8) 1 q2 − Zk From the results of Chapter6, we know that the contribution from the ghost sector is suppressed in the gluon mass scale dynamics. Basically, the eﬀect of the gluon mass in the ghost propagator, D(q2), or equivalently in its dressing function, F (q2), is indirect. For example, if in the absence of a gluon mass F (q2) ln(q2), the appearance of a mass ∼ 7.2. Derivation of the mass equation 102

Figure 7.2: Contribution from the pole part of the three-gluon vertex to the (a5) diagram of the QB gluon SDE shown in Fig. 3.4.

will induce the qualitative change of the type F (q2) ln(q2 + m2). However, this occurs ∼ without the need to add a pole in the structure of the Γµ vertex, which clearly makes F (q2) ﬁnite. In fact, Γ also changes in the presence of a mass indirectly because of µ e its dependence on the gluon propagators and other vertices, which appear in its SDE. e Therefore, we can neglect the eﬀect from the (a3)µν ghost diagram to the gluon mass equation.

Now, let us study the contribution from diagram (a5)µν given in Fig. 7.2,

1 (ap)ab = g4Γ(0)amnr ∆γτ (k+q)∆βσ(`)∆αρ(k+`)Γp,brnm( q, k+q, `, k `) . (7.9) 5 µν −6 µαβγ ντσρ − − − Zk Z` e In order to the pole part of the four-gluon vertex be longitudinally coupled, it must satisfy the condition

P λ(q)P γτ (k + q)P βσ(`)P αρ(k + `)Γp,brnm( q, k + q, `, k `) = 0 . (7.10) ν λτσρ − − − e λ λ After writing Pν (q) explicitly in Eq. (7.10), we note that the gν term must be equal to λ 2 the qνq /q term. This fact can then be used to rewrite Eq. (7.9) as

g4 q (ap)ab = Γ(0)amnr ν ∆γτ (t)∆βσ(`)∆αρ(k + `)qλΓp,brnm( q, t, `, k `) , (7.11) 5 µν − 6 µαβγ q2 λτσρ − − − Zk Z` e where t = k + q. brnm The four-gluon vertex Γλτσρ satisﬁes the WTI of Eq. (3.51), where both sides can be separated into pole and no-pole parts, which generates two separate equations for Γnp,brnm e ντσρ e 7.2. Derivation of the mass equation 103

p,brnm and Γντσρ . We are interested in the identity obeyed by the pole part, namely

e µ p,mnrs mse ern p mne esr p q Γµαβγ (q, r, p, t) = f f Γαβγ(r, p, q + t) + f f Γβγα(p, t, q + r)

mre ens p e + f f Γγαβ(t, r, q + p) . (7.12)

Using Eq. (7.12), it is possible to write Eq. (7.11) in the form

i q q 3 (ap)ab = C2 g4δab µ ν I (q2) , (7.13) 5 µν 4 A q2 j j=1 X with

(qγgαβ qβgαγ) I (q2) = − P γτ (t)P βσ(`)P αρ(s)∆(t)∆(`)∆(s)Γp (t, `, t `) , 1 q2 τσρ − − Zk Z` (qαgβγ qγgαβ) I (q2) = − P γτ (t)P βσ(`)P αρ(s)∆(t)∆(`)∆(s)Γp ( s, t, ` q) , 2 q2 ρτσ − − Zk Z` (qβgαγ qαgβγ) I (q2) = − P γτ (t)P βσ(`)P αρ(s)∆(t)∆(`)∆(s)Γp (`, s, k) , (7.14) 3 q2 σρτ − Zk Z` where s = k + `. One can show, after appropriate shifts of momenta and relabeling of dummy Lorentz indices, that the three Ii terms of Eq. (7.14) are equal. Then, we can use the longitudinally p coupled condition for the vertex Γλρσ, i.e.,

P λ(k)P αρ(` + k)P βσ(`)Γp (k, ` k, `) = 0 , (7.15) ν λρσ − − to obtain

3 2 qτ γτ Ij(q ) = 3(qαgβγ qβgαγ) P (k + q)∆(k + q) − q2 × j=1 k X Z kλ P βσ(`)P αρ(k + `)∆(`)∆(k + `) Γp (`, k `, k) . (7.16) × k2 σρλ − − Z` Thus, the integral in ` is a function of k only, with two free Lorentz indices, so that

2 2 I(q ) = Ij(q ) can be written as P 2 qτ γτ 2 αβ 2 α β I(q ) = 3(qαgβγ qβgαγ) P (k + q)∆(k + q) A(k )g + B(k )k k . (7.17) − q2 Zk 7.2. Derivation of the mass equation 104

Figure 7.3: The (a6) diagram contributing to the QB gluon SDE shown in Fig. 3.4.

Note that the resulting integral is symmetric under the exchange α β, whereas its ↔ prefactor is antisymmetric, therefore, I(q2) = 0 regardless of the expressions for A(k2)

2 p ab and B(k ). Consequently, diagram (a5 )µν does not contribute to the gluon mass function, since p 2 a5 (q ) = 0. (7.18)

ab Finally, we can evaluate the contribution from graph (a6)µν presented in Fig. 7.3,

ab ab γτ λδ αβ (a ) = iδ µαβγ ∆ (k + q)∆ (k)Y (k)Γντλ( q, k + q, k) , (7.19) 6 µν N δ − − Zk e αβ where µαβγ and Y (k) were deﬁned in Eqs. (4.33) and (4.34). Both Γντλ and Γσρδ N δ (which appears inside the deﬁnition of Y αβ) can be split into pole and no-pole parts. δ e Then, the product of these vertices is given by

np np p np np p p p ΓντλΓσρδ = ΓντλΓσρδ + ΓντλΓσρδ + ΓντλΓσρδ + ΓντλΓσρδ , (7.20)

e e e e e where only the last three terms contribute to the gluon mass equation. However, the terms proportional to Γp (`, ` k, k) vanish in the Landau gauge due to its longitudinally σρδ − − 1 p np coupled condition , so only the term ΓντλΓσρδ gives a non-zero contribution to the mass function. e p Therefore, we can write (a6 )µν as

p γτ λδ αβ p (a )µν = i µαβγ ∆ (k + q)∆ (k)Y (k)Γ ( q, k + q, k) , (7.21) 6 N δ ντλ − − Zk 1 np p np The limit ξ = 0 must be taken carefully for the term proportionale to ΓντλΓσρδ, since Γντλ contains a term proportional to ξ−1. However, after a complete analysis it is possible to verify that the total contribution of this term also vanishes [38]. e e 7.2. Derivation of the mass equation 105

αβ where Yδ (k) now only contains the contribution of the no-pole part of Γσρδ, i.e.,

Y αβ(k) = ∆αρ(` + k)∆βσ(`)Γnp (`, ` k, k) . (7.22) δ σρδ − − Z` Then, using again Eq. (7.6), we obtain

p qν 2 2 γ λδ αβ (a )µν = i µαβγ m (k) m (k + q) ∆ (k + q)∆ (k)Y (k) 6 N q2 − λ δ Zk qµqν = ap(q2) , (7.23) q2 6 where in the last line we have used that the result of the integral in k depends only on the momentum q and has three free Lorentz indices, which upon contraction with µαβγ N will result in terms proportional to qµ. Thus, we have

µ p 2 q 2 2 γ λδ αβ a (q ) = i µαβγ m (k) m (k + q) ∆ (k + q)∆ (k)Y (k) . (7.24) 6 q2 N − λ δ Zk Then, using the deﬁnition for Y (k2) of Eq. (4.35), after shifting the integration variable,

p 2 we can write a6 (q ) as

2 4 p 2 3i CAg 2 2 2 λδ a6 (q ) = 2 m (k) (k + q) k [Y (k + q) + Y (k)] ∆λδ(k + q)∆ (k) 4 q k − 2 4 Z 3i CAg 2 2 γ λδ + q gγδ 2qγqδ m (k)[Y (k + q) Y (k)] ∆ (k + q)∆ (k) . (7.25) 4 q2 − − λ Zk Therefore, the full mass equation can be obtained by adding up the non-null contri-

p 2 p 2 butions given by a1 (q ) of Eq. (7.8) and a6 (q ) of Eq. (7.25), which leads to

i m2(q2) = ap(q2) + ap(q2) 1 + G(q2) 1 6 2 i CAg 2 2 2 2 αρ = m (k ) (k + q) k P (k)Pαρ(k + q)∆(k)∆(k + q) 1 + G(q2) q2 − × Zk 3 2 1 iCAg [Y (k + q) + Y (k)] × − 4 2 4 2 3 C g (q gγδ 2qγqδ) + A − m2(k2)P γ(k + q)P λδ(k)∆(k)∆(k + q) 4q2 1 + G(q2) λ × Zk [Y (k + q) Y (k)] , (7.26) × − whose diagrammatic representation is given by Fig. 7.4. 7.2. Derivation of the mass equation 106

        2 qµ    +  m (q) = 2   qν q [1+G(q)]  µ µ  ×   ν ×                   Figure 7.4: Diagrammatic representation of the operations leading to the all-order gluon mass equation of Eq. (7.26). White circles represent fully dressed propagators in the np p Landau gauge, the blue circle corresponds to Γσρδ, and the green one to Γντλ.

e We can write Eq. (7.26) in Euclidean space using the standard rules of Eq. (6.12) along with

2 2 2 2 2 2 2 2 m (q ) = m ( q ); GE(q ) = G( q ); YE(q ) = iY ( q ) . (7.27) E E − E E − E E − − E

Thus, we obtain (suppressing the E subscript as usual)

2 2 2 CAg 1 2 2 2 2 αρ m (q ) = m (k ) (k + q) k P (k)Pαρ(k + q)∆(k)∆(k + q) −1 + G(q2) q2 − × Zk 3 2 1 + CAg [Y (k + q) + Y (k)] × 4 2 4 2 3 C g (q gγδ 2qγqδ) A − m2(k2)P γ(k + q)P λδ(k)∆(k)∆(k + q) − 4q2 1 + G(q2) λ × Zk [Y (k + q) Y (k)] . (7.28) × −

We can write Eq. (7.28) in a compact form as

2 2 2 CAg 1 2 2 m (q ) = m (k )∆(k)∆(k + q) m(q, k) , (7.29) 1 + G(q2) q2 K Zk so that the kernel m is given by K

αρ β + 2 2 m(q, k) = P (k)P (k + q) (q, k) (k + q) k gαβ K ρ K − − 2 + (q, k) q gαβ 2qαqβ , (7.30) K − where we have deﬁned

+(q, k) = (k + q) + (k) 1 , −(q, k) = (k + q) (k) , (7.31) K Y Y − K Y − Y 7.3. Renormalization of the gluon mass equation 107 with (k) being related to the quantity Y (k2) deﬁned in Eq. (4.35) according to Y

3 2 2 (k) = CAg Y (k ) Y −4 ρ 1 2 k µρ αν np = CAg ∆(`)∆(` + k)P (`)P (` + k)Γ (k, `, ` k) . (7.32) −4 k2 αµν − − Z`

7.3 Renormalization of the gluon mass equation

The renormalization of Eq. (7.29) is carried out multiplicatively by introducing the appropriate renormalization constants of Eqs. (4.43), (4.44) and (4.45). We again use the MOM scheme, in which the propagators assume their tree-level values at the renormalization point µ, and an analogous condition is imposed on the vertices at special momentum configurations, such as the symmetric one defined for the three-gluon vertex in Eq. (6.26). Then, using the relations of Eqs. (2.49) and (4.46), we find that the net effect of renormalization amounts to replacing bare quantities by renormalized ones on both sides of Eq. (7.29), along with the modification

+(q, k) = Z [ (k + q) + (k)] Z , KR 4 Y Y − 3 −(q, k) = Z [ (k + q) (k)] , (7.33) KR 4 Y − Y whose effective result is illustrated on the top of Fig. 7.5. Nonetheless, the actual implementation of multiplicative renormalization at the level of SDEs is a particularly complicated issue. Thus, we make use of an effective approach to this problem inspired in the treatment carried out in several studies of the SDE for the quark propagator [121–123]. In these analyses, the multiplicative renormalization is performed by replacing the quark-gluon vertex renormalization constant, Z1F , by an appropriate function (q), which varies depending on the particular details and approx- C1 imations. Then, let us proceed to implement this effective renormalization approach to the gluon mass equation. In order to simplify the structure of the vertices, we consider a single form factor for each vertex, proportional to their tree-level expressions. Moreover, the form 7.3. Renormalization of the gluon mass equation 108

(k) k Y         2 qµ  Z4   +  (A) m (q) = 2  Z3  qν q [1+G(q)]  µ µ  ×   ν ×            k + q   k + q      eﬀective renormalization

(k) k Y       q   2 µ  4(k)  (B)  +  m (q) = q2[1+G(q)]  3(k) C  qν  µ C µ  ×   ν ×            k + q   k + q      Figure 7.5: Diagrammatic representation of the renormalized gluon mass equation. White (colored) circles represent fully dressed propagators (vertices).

factors are evaluated in the symmetric conﬁguration, so we have

(0) Γαβµ(q , q , q ) (s)Γ (q , q , q ) , 1 2 3 → C3 αβµ 1 2 3 Γabmn(p , p , p , p ) (s)Γabmn (0)(p , p , p , p ) , (7.34) αβµν 1 2 3 4 → C4 αβµν 1 2 3 4

(0) abmn (0) where Γαβµ and Γαβµν are given in Fig. 1.2 and s is the symmetric point associated with the totally symmetric conﬁguration, whose kinematics for the case of the three-gluon and four-gluon vertices are given respectively by

2 2 2 s q = q = q := s , qi qj = for (i = j) , (7.35) 1 2 3 · −2 6 and [126]

2 2 2 2 s p = p = p = p := s pi pj = for (i = j) . (7.36) 1 2 3 4 · −3 6

In addition, it is important to note that, unlike the dynamical quark mass, m2(q2) is not a renormalization group invariant (RGI), i.e., it is not a µ-independent quantity. Then, to proceed with our discussion, we introduce the dimensionless RGI quantity m2(q2) = m2(q2)/m2(0). Using Eqs. (2.49) and (4.46), it is useful to deﬁne the following 7.3. Renormalization of the gluon mass equation 109

RGI combinations [39, 60],

1/2 −1 G(s) = g∆ (s) [1 + G(s)] , R (s) = g∆3/2(s) (s) , R3 C3 (s) = g2∆2(s) (s) , (7.37) R4 C4 which depend only on the squared momentum s, because of the chosen kinematics con- ﬁguration. Using Eq. (7.37), we can write the terms appearing in the renormalized version of Eq. (7.29) schematically as

2 2 −1 −1 Z g ∆ [1 + GR(q)] = Z G , 3 R R 3 C3R R R3 4 4 −1 −1 Z g ∆ R [1 + GR(q)] = Z G , (7.38) 4 R R C3 4 C4R R R3R4 where we have suppressed all momentum dependence and have used Y g2∆2Γ , with ∼ 3 2 2 Γ3 representing the three-gluon vertex. Then, the equation for the RGI quantity m (q ) can be written in a schematic form as

2 2 −1 −1 m (q) m (k) Z G + Z G , (7.39) ∼ 3 C3R R R3 4 C4R R R3R4 Zk where we have neglected the kinematic factors irrelevant for our renormalization analysis. The point of Eq. (7.39) is to use the fact that, because m2(q2) is RGI, i.e., dm2/dµ = 0, the rhs must also display the same property. Since the i are RGI by themselves, we R must have d(Z −1) d(Z −1) 3 C3R = 4 C4R = 0. dµ dµ

−1 The simplest solution to enforce Zi to be an RGI quantity is to implement the substi- CiR tutions Z R and Z R. This step is illustrated at the bottom panel of Fig. 7.5. 3 → C3 4 → C4 At the level of Eq. (7.33) this implies (setting s = k)

+ + (q, k) (q, k) = R(k)[ (k + q) + (k)] R(k) , KR → Keff C4 Y Y − C3 − − (q, k) (q, k) = R(k)[ (k + q) (k)] . (7.40) KR → Keff C4 Y − Y 7.3. Renormalization of the gluon mass equation 110

µ µ µ µ

q1 q1 q1 q1

Z = 3 + + + 4 5 · · · q2 q3 q2 K K q3 q 2 q3 q2 q3

α β α β α β α β (c0) (c1) (c2) µ µ µ µ

p1 p1 p1 p1

Z = 4 + + + 5 6 · · · p2 p4 p2 K K p3 p p3 4 p2 p4 p2 p4 p3 p3

α β σ α β σ α β σ α β σ (d0) (d1) (d2)

Figure 7.6: The SDEs for the three-gluon and four-gluon vertices.

7.3.1 Z3 and Z4 from the SDEs for the vertices

The above substitutions can be understood in the context of the SDEs satisﬁed by the three-gluon and four-gluon vertices, presented in Fig. 7.6. We denote the kernel with n incoming gluons by n (suppressing all color and Lorentz indices) and its renormalized K n n R R 2 form by , with = Z n. Therefore, we can deﬁne the RGI quantity n = ∆2 n. Kn Kn A K K K Then, the SDEs of Fig. 7.6 can be written schematically as b

Γ = Z Γ(0) + Γ + g∆1/2Γ + ... , 3 3 3 3K4 K5{ 4} Zk Zk Z` Γ = Z Γ(0) + bg−1∆−1/2Γb + Γ + ... , (7.41) 4 4 4 K5{ 3} 4K6 Zk Zk Z` b b where we have dropped the R subscript for the renormalized functions and Γ3 and Γ4 represent the renormalized three-gluon and four-gluon vertices, respectively. From Eq. (7.37), we have g∆1/2 = −1, thus, we obtain C4 C3R4R3

Γ(0) = Z Γ(0) + Γ(0) + Γ(0) −1 + ... , C3 3 3 3 C3 3 K4 C3 4 R4R3 K5 Zk Zk Z` Γ(0) = Z Γ(0) + Γ(0) b −1 + Γ(0)b + ... . (7.42) C4 4 4 4 C4 3 R3R4 K5 C4 4 K6 Zk Zk Z` b b 7.4. Main ingredients of the numerical analysis 111

The above equations can be rearranged as

Z Γ(0) = Γ(0) Γ(0) + Γ(0) −1 ... , 3 3 C3 3 − C3 3 K4 4 R4R3 K5 − Zk Z` Z Γ(0) = Γ(0) Γ(0) b −1 + Γ(0)b ... , (7.43) 4 4 C4 4 − C4 3 R3R4 K5 4 K6 − Zk Z` b b with appearing in both terms of the rhs of the ﬁrst equation, and in both terms on C3 C4 the rhs of the second. Then, neglecting the higher-order contributions on the rhs of both equations leads to the substitution Z Γ(0) Γ(0) and Z Γ(0) Γ(0), in accordance 3 3 → C3 3 4 4 → C4 4 with Eq. (7.40).

7.4 Main ingredients of the numerical analysis

In order to solve the gluon mass equation, we switch to spherical coordinates and introduce the variables x = q2, y = k2, and z = (k + q)2 = x + y + 2√xy cos θ in Eq. (7.29). Then, deﬁning

1 ∞ π = , := dy y dθ sin2 θ , (7.44) (2π)3 Zk Zy,θ Zy,θ Z0 Z0 we have

α C 1 m2(x) = s A z−1∆(y)∆(z)[ (x, y, z) + (x, y, z)]m2(y) , (7.45) 2π2 x [1 + G(x)] K1 K2 Zy,θ where

(x, y, z) = (y)[ (z) + (y)] (y) (z y) 3z xs2 , K1 {C4 Y Y − C3 } − − θ (x, y, z) = (y)[ (z) (y)] 3zx x(z + y)s2 . (7.46) K2 C4 Y − Y − θ 7.4.1 Kinetic term of the gluon propagator

As mentioned before, in previous studies involving Eq. (7.45), this equation was linearized by treating the gluon propagators appearing inside the integral as external inputs. Here, instead, we maintain the nonlinear aspect of the equation by using the Euclidean relation of Eq. (4.2), i.e., −1 ∆(y) = yJ(y) + m2(y) . (7.47) 7.4. Main ingredients of the numerical analysis 112

1.5 1.7

1.25 1.6

1 1.5

0.75 1.4

0.5 1.3

0.25 1.2

0 1.1

-0.25 1

-0.5 0.9 0.03 0.1 1 10 0 0.5 1 1.5 2 2.5 3

Figure 7.7: Left panel: The (k) = J (k) employed in the ﬁrst iteration. Right panel: The C3 0 (k) given by Eq. (7.62), with the parameter d varying in the range (4.0 10.0) GeV2. C4 1 −

However, in order to replace ∆(y) and ∆(z) in Eq. (7.45) by Eq. (7.47), we need an initial Ansatz for J, which can be improved during the iterative procedure. For such Ansatz we use the functional form employed in our recent study of the longitudinal part of the three-gluon vertex [124],

C λ τ q2 + η2(q) 1 q2 J(q2) = 1 + A s 1 + 1 2 ln + ln , (7.48) 4π q2 + τ µ2 6 µ2 2 with 2 η1 η (q) = 2 . (7.49) q + η2 The renormalization point is set at µ = 4.3 GeV throughout, but the ﬁtting parameters will change during the iterative procedure, as we discuss later. The initial seed J is shown in the left panel of Fig. 7.7 for J(q2) J (q2), with the 0 → 0 parameters given in Table 7.1. This function retains the preeminent qualitative features known about the kinetic term, due to its relation with the three-gluon vertex [45, 114]. In particular, as the momentum q2 decreases, J(q2) departs gradually from its perturbative value, reverses its sign, at a point called zero crossing, and ﬁnally diverges logarithmically at the origin. As explained in [45, 114], this divergence is a consequence of the massless ghost loops appearing in the SDEs for the gluon propagator and for the three-gluon vertex, which result in a logarithmic divergence. 7.4. Main ingredients of the numerical analysis 113

2 2 2 4 2 J(q ) λs τ1 [GeV ] τ2 [GeV ] η1 [GeV ] η2 [GeV ] 2 J0(q ) 0.220 9.870 0.910 17.480 1.180 2 J1(q ) 0.243 2.638 0.265 6.451 0.388 2 J2(q ) 0.220 3.503 0.263 8.261 0.454 2 J3(q ) 0.220 2.8 0.201 6.489 0.363

2 Table 7.1: The ﬁtting parameters for Ji(q ), whose functional form is given by Eq. (7.48).

2 2 2 J0(q ) is the initial Ansatz presented in the left panel of Fig. 7.7, while J1(q ), J2(q ), and 2 J3(q ) are the solutions shown in the top right panel of Fig. 7.12.

7.4.2 Three- and four-gluon vertices

We also need to establish the expressions used for the renormalization functions (s) C3 and (s). From Eq. (7.34), we have deﬁned (s) and (s) as the form factors of the C4 C3 C4 tree-level structures of the three-gluon and four-gluon vertices, respectively, evaluated at the symmetric conﬁguration. In particular, we know that the three-gluon vertex can be divided into two parts, one p that contains the longitudinally coupled poles, Γαµν, and another that captures all the np remaining contribution, Γαµν. Then, the STI of Eq. (3.29) obeyed by the full vertex is realized as the sum of two pieces, namely

α np 2 2 2 α 2 2 α q Γ (q, r, p) = F (q )[p J(p )P (p)Hαµ(p, q, r) r J(r )P (r)Hαν(r, q, p)] , (7.50) αµν ν − µ α p 2 2 2 α 2 2 α q Γ (q, r, p) = F (q )[m (r )P (r)Hαν(r, q, p) m (p )P (p)Hαµ(p, q, r)] . (7.51) αµν µ − ν

Then, the no-pole part of three-gluon vertex can be cast in the form

αµν αµν αµν Γnp (q, r, p) = ΓL (q, r, p) + ΓT (q, r, p) , (7.52)

αµν where the “longitudinal” part, ΓL (q, r, p), saturates the STI of Eq. (7.50) and its cyclic αµν permutations, whereas the totally “transverse” part, ΓT (q, r, p), satisﬁes

αµν αµν αµν qαΓT (q, r, p) = rµΓT (q, r, p) = pνΓT (q, r, p) = 0 . (7.53)

These longitudinal and transverse parts can be decomposed in the Ball-Chiu basis 7.4. Main ingredients of the numerical analysis 114 introduced in [61] as 10 αµν αµν ΓL (q, r, p) = Xi(q, r, p)`i , (7.54) i=1 X αµν where the tensors `i are given by

`αµν = (q r)νgαµ , `αµν = pνgαµ , `αµν = (q r)ν[qµrα (q r)gαµ] , 1 − 2 − 3 − − · `αµν = (r p)αgµν , `αµν = qαgµν , `αµν = (r p)α[rνpµ (r p)gµν] , 4 − 5 − 6 − − · `αµν = (p q)µgαν , `αµν = rµgαν , `αµν = (p q)µ[pαqν (p q)gαν] , 7 − 8 − 9 − − · αµν ν α µ µ ν α `10 = q r p + q r p , (7.55) and 4 αµν αµν ΓT (q, r, p) = Yi(q, r, p)ti , (7.56) i=1 X with

tαµν =[(q r)gαµ qµrα][(r p)qν (q p)rν] , 1 · − · − · tαµν =[(r p)gµν rνpµ][(p q)rα (r q)pα] , 2 · − · − · tαµν =[(p q)gνα pαqν][(q r)pµ (p r)qµ] , 3 · − · − · tαµν =gµν[(r q)pα (p q)rα] + gνα[(p r)qµ (q r)pµ] + gαµ[(q p)rν (r p)qν] 4 · − · · − · · − · + rαpµqν pαqµrν . (7.57) −

αµν (0) αµν (0) At tree-level, ΓL (q, r, p) = Γ (q, r, p) (see deﬁnition in Fig. 1.2), therefore, the only nonvanishing form factors are

(0) (0) (0) X1 (q, r, p) = X4 (q, r, p) = X7 (q, r, p) = 1 .

In addition, due to the Bose symmetry of the three-gluon vertex, we ﬁnd the following relations between diﬀerent form factors [61]

X4(q, r, p) = X1(r, p, q) ,X5(q, r, p) = X2(r, p, q) ,X6(q, r, p) = X3(r, p, q) ,

X7(q, r, p) = X1(p, q, r) ,X8(q, r, p) = X2(p, q, r) ,X9(q, r, p) = X3(p, q, r) , (7.58) which reduce the number of independent form factors from the original ten to only four, 7.4. Main ingredients of the numerical analysis 115

2 2 2 namely X1, X2, X3, and X10. Then, at the symmetric point (q = r = p = s), we have

X (s) = X (s) = X (s) := (s) . (7.59) 1 4 7 C3

The Xi may be obtained from Eq. (7.50) using the so-called gauge technique, which

2 2 “solves”the STI, expressing the Xi in terms of J(q ), the ghost dressing function F (q ), and

three of the ﬁve form factors of the ghost-gluon scattering kernel Hνµ of Eq. (3.38). This

was recently done in [124] using the results for the form factors of Hνµ obtained in [111]. There, it was veriﬁed that the “Abelian approximation”, obtained by turning oﬀ the ghost sector2, is numerically close to the full nonperturbative evaluation, as can be observed

from the results obtained in [124] for the form factor X1 presented in Fig. 7.8. Notice

that the Abelianized X1(q, r, p) (cyan surface) displays the same qualitative behavior as the full nonperturbative curves X (q, r, p) (colorful surfaces), which depend on the angle b 1 θ between the momenta q and r. Therefore, in order to simplify our analysis here, we use the Abelianized form factor, X (q, r, p) X (q, r, p), which is given by 1 → 1 b 1 X (q, r, p) = [J(r) + J(q)] . (7.60) 1 2 b In fact, Eq. (7.60) is exactly the result for the vertex with three background gluons, which satisﬁes Abelian STIs when contracted with any of the gluons momenta [26, 127–129]; but, in that case, the additional replacement J(q) J(q)[1 + G(q)]−2 must be carried → out. Then, from Eqs. (7.59) and (7.60) in the symmetric conﬁguration, we have

(k) = J(k) , (7.61) C3

as represented in the left panel of Fig. 7.7 for the initial seed J0(k). Because J(k) will change during the iterations, so will (k). C3 On the other hand, there is very limited information on the nonperturbative properties of the four-gluon vertex in the available literature. In particular, lattice simulations for this vertex have yet to be carried out. Hence, we will base our Ansatz for (k) in some C4 general characteristics observed in [119, 125, 126, 130, 131].

2 This is done by setting the quantities related to the ghost sector to tree-level, i.e., Hνµ(q, p, r) = gνµ and F (q2) = 1. 7.4. Main ingredients of the numerical analysis 116

2 2 Figure 7.8: X1(q , r , θ) for θ = 0 (top left), π/3 (top right), and 2π/3 (bottom left), together with the “abelianized” X1 (bottom right).

b The main feature observed in diﬀerent kinematic conﬁgurations described by a single (0) momentum scale is the presence of a peak in the form factor accompanying either Γ4 or its transversely projected counterpart. Such typical peak is located in the region of a few hundred MeV. We can model this qualitative behavior of (k) by [45] C4 λ d k2 k2 + 4m2 (k) = 1 + 1 1 ln 0 , (7.62) C4 4π − (k2 + d )2 µ2 2

2 2 2 where λ = 0.28, d2 = 0.26 GeV , and m0 = 0.14 GeV . We present the curves corresponding to diﬀerent values of d1 on the right panel of Fig. 7.7. The red shaded area is cre- ated varying the value of d1, which controls the height of the peak, in the range of (4.0 10.0) GeV2, while all other parameters in Eq. (7.62) are kept ﬁxed. − 7.4. Main ingredients of the numerical analysis 117

2 2 Figure 7.9: X3(q , r , 0) for θ = 0 (top left), π/3 (top right), and 2π/3 (bottom left), together with the “abelianized” X3 (bottom right).

b 7.4.3 Subdiagram (y) Y The last quantity appearing in the kernels of Eq. (7.46) is (y), which has been deﬁned Y in Eq. (7.32). In order to evaluate its expression numerically, we decompose the three- gluon vertex appearing in Eq. (7.32) as the sum of a longitudinal and a transverse part, αµν as in Eq. (7.52). Keeping only ΓL , we obtain in spherical coordinates

αsCA 2 (y) = s ∆(t)∆(u) Y (t, y, ω) , (7.63) Y 8π2 ω K Zt,ω 7.4. Main ingredients of the numerical analysis 118

0.7 3

2.8 0.6 2.6

2.4 0.5 2.2

2 0.4 1.8

0.3 1.6

1.4 0.2 1.2

1 0.1 0 2 4 6 8 10 0 1 2 3 4 5

Figure 7.10: Left panel: The numerical solution for (k) obtained from Eq. (7.63) (circles), Y and the corresponding ﬁt given by Eq. (7.65) (continuous). Right panel: The inverse of the auxiliary function, 1 + G(q), whose ﬁt is given by Eq. (7.66).

where we have used the same deﬁnition of Eq. (7.44) for the integral and deﬁned the

2 2 2 variables y = k , t = ` , u = (k + `) = y + t + 2√yt cω. Additionally, we have

t (u + t y) Y (t, y, ω) = tX + 6X (u + y t) 3X + X + − [X 2X ] , (7.64) K − 6 7 − − 9 u 3 2u 4 − 1 with Xi = Xi(y, t, ω). To proceed further, instead of using the Abelianized approximation as in Eq. (7.60), we employ the nonperturbative results obtained in [124]. The solutions for X1 and X3, for certain values of the angle between the two independent momenta, are given in Figs 7.8 and 7.9, respectively (remember that, by virtue of Eq. (7.58), X4 and X7 can be obtained from X1, while X6 and X9 can be found from X3). Then, using this set of numerical data, we obtain the solution shown in the left panel of Fig. 7.10. It can be accurately ﬁtted by

k2 + η2(k) k2 Ck (k) = 3παsC A ln + B ln 1 + + D , (7.65) Y A µ2 µ2 1 + (k2/ν2)γ

2 4 where η (k) is given by Eq. (7.49), and the ﬁtting parameters are η1 = 0.0103 GeV , η = 0.184 GeV2, A = 0.015, B = 0.0095, C = 2.158 GeV, D = 0.039, ν2 = 2.422 GeV2, 2 − and γ = 1.074. The curve presented in the left panel of Fig. 7.10 is obtained for αs = 0.27. 7.5. Numerical analysis of the mass equation 119

7.4.4 Function 1 + G(q2)

Finally, the last ingredient entering the mass equation of Eq. (7.45) is the PT-BFM auxiliary function 1 + G(q2). As one can see from Eqs. (3.35) and (3.37), this function depends on the form factors of the ghost-gluon scattering kernel, Hνµ, of Eq. (3.38). Under some simpliﬁcations, one can obtain a solution for 1 + G(q2) that can be ﬁtted by [124]

2 2 2 9CA 2 2 q + ρ2η (q) 1 + G(q ) = 1 + [αg + A exp ρ q /µ ] ln , (7.66) 48π 1 − 1 µ2 2 4 2 where η (q) is also given by Eq. (7.49), but now with η1 = 0.30 GeV , η2 = 0.33 GeV . The 2 remaining parameters are αg = 0.21, A1 = 0.77 GeV , ρ1 = 0.78, and ρ2 = 0.50. Then, the inverse of 1 + G(q2) is shown in the right panel of Fig. 7.10.

7.5 Numerical analysis of the mass equation

Going back to Eq. (7.45), we can take the limit x 0 and establish the following → constraint 3C α ∞ m2(0) = A s [1 + G(0)]−1 dy m2(y) (y) , (7.67) − 8π K0 Z0 where

0 0 (y) = (y) y2∆2(y) 2 (y) y2∆2(y) (y) . (7.68) K0 C3 − C4 Y Setting (y) = (y) = 1 reduces the kernel to C3 C4 K0

0 (y) = (1 2 (y)) y2∆2(y) , (7.69) K0 − Y which is in accordance with the expression used in [38]. Then, it is useful to check the effect of the functions (y) and (y), as defined in C3 C4 Eqs. (7.61) and (7.62), on the structure of the kernel of Eq. (7.68) in comparison to the case of Eq. (7.69), where both form factors were set to tree-level. Such contrast can be observed in the left panel of Fig. 7.11, where one notices an enhancement in the negative support of the kernel of Eq. (7.68) in comparison with the one of Eq. (7.69). Therefore, Eq. (7.67) indicates that a positive-defined m2(y) can be accommodated more comfortably with the i(y) of Eqs. (7.61) and (7.62) than for the case where (y) = (y) = 1. C C3 C4 7.5. Numerical analysis of the mass equation 120

1 0.3

0 0.25

-1 0.2

-2 0.15

-3 0.1

0.05 -4

0 -5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 7.11: Left panel: The kernel αs (k) deﬁned by Eq. (7.68) for (i) (y) K0 C3 and (y) given by Eqs. (7.61) and (7.62), respectively (red continuous), and (ii) for C4 (y) = (y) = 1 (blue dotted). For both cases we have used αs = 0.27. Right panel: C3 C4 2 2 The values of m (0), obtained from solving Eq. (7.45) for a ﬁxed J(q ), as function of αs.

2 The blue star denotes the αs that reproduces the lattice value m (0) = 0.14.

In addition, we note that, by using the nonlinearized mass equation, we can obtain solutions for a continuous interval of values for αs, in contrast to the linearized case [38, 39], in which solutions exist only for a specific value. In order to establish this, we vary αs while keeping the form of J(q2) fixed, so that we obtain a continuous family of m2(q2), as can be observed in the right panel of Fig. 7.11. However, these solutions for m2(q2), by means of Eq. (7.47), result in gluon propagators that are not in accordance with the lattice results for ∆(q2). In fact, we will see later that, in order to approach the lattice data, the values of αs must be chosen from a rather narrow interval. Taking into account all the considerations described above, we can finally proceed to the numerical analysis. The solution for m2(q2) is obtained by an iterative procedure consisting of the following steps: (i) First, we employ the numerical fit of Eq. (6.22) to the gluon lattice data of [8], which we now denote by ∆L(q). (ii) Then, we start the iteration with initial seeds for m2(q2) and J(q2). For the mass we

2 use a random function, while for the kinetic term we use J0(q ), as deﬁned in Eq. (7.48) and Table 7.1. (iii) With these starting ingredients, we solve Eq. (7.45) iteratively, adjusting the value 7.5. Numerical analysis of the mass equation 121

2 −1 of αs such that m (0) = ∆L (0). The solution is accepted when the relative diﬀerence 2 2 −5 2 2 between two successive results for m (q ) is below 10 ; we denote this solution by ms0 (q ). 2 2 2 (iv) We use the solution ms0 (q ) and J0(q ) in Eq. (7.47) to obtain our approximation for 2 ∆(q ), which is then compared with ∆L(q). (v) In order to improve the result of the previous step, we determine a new J(q2) from J(q2) = [∆−1(q) m2 (q2)]/q2, which is then fed into Eq. (7.45), so we can obtain a new L − s0 solution for m2(q2) following the prescription of step (iii). (vi) Then, steps (iii)-(v) can be repeated, saving those combinations of m2(q2) and J(q2) which best reproduce ∆L(q). The outcome of the procedure described above is shown in Fig. 7.12 for three slightly diﬀerent functions for J(q2). The curves plotted represent the best results obtained for each case, which occur when αs = 0.272 (blue dashed dotted curves), αs = 0.278 (red dashed), and αs = 0.289 (yellow dotted). As mentioned in step (iii), these values of αs

−2 are essentially determined from the requirement that ∆L(0) = m (0), which forces αs to be within a small interval, compatible with the values of αs used in the literature [109, 110, 132]. As an important conclusion, we observe, from the top panels of Fig. 7.12, that small

2 variations in the J(q ) can be compensated by minor adjustments in the value of αs, producing basically the same solution for m2(q2). Also, note that the dynamical gluon mass on the top left panel of Fig. 7.12 is positive- deﬁned and monotonically decreasing in the entire range of momenta. The ﬁt, shown as the continuous purple line, in the same graph is given by

4 2 2 m0 m (q ) = 2 2 2 2 2 , (7.70) µ1 + q ln [(q + µ2)/λ ]

4 4 2 2 2 2 where the fitting parameters are m0 = 0.107 GeV , µ1 = 0.756 GeV , µ2 = 0.266 GeV , and λ2 = 0.123 GeV2. The fit of Eq. (7.70), in addition to describing well the solution for the mass, also captures the behavior of its derivative with respect to q2, denoted bym ˙ 2(q2). In fact, in Fig. 7.13, one can see that the result of the differentiation of the fit is practically identical to the direct numerical differentiation of the data for m2(q2). This fact suggests that this fit is superior to the simpler functional form used previously in Eq. (6.31). In particular, the latter yields a derivative that vanishes at the origin, a feature which is certainly not 7.5. Numerical analysis of the mass equation 122

0.16 1.25

0.14 1

0.12 0.75

0.1 0.5 0.08 0.25 0.06

0 0.04

0.02 -0.25

0 -0.5 0 0.5 1 1.5 2 2.5 3 0.03 0.1 1 10

0 0 0.5 1 1.5 2 2.5 3

Figure 7.12: Top left panel: The numerical results for the dynamical gluon mass, m2(q), for αs = 0.272 (blue dashed dotted), αs = 0.278 (red dashed), and αs = 0.289 (yellow dotted). Top right panel: The corresponding kinetic term, J(q). Bottom panel: The resulting gluon propagator, ∆(q), obtained from Eq. (7.47), along with the lattice data from [8]. In all plots, we employ the same color code.

shared by the actual numerical solution. The importance of capturing the correct behavior of the derivative of the mass is related to Eq. (6.6), in which we see thatm ˙ 2(q2) gives the

0 2 BSA Cgl(q ), studied in Chapter6. We can also comment on the main characteristics of the top right panel of Fig. 7.12, e 2 2 which presents the kinetic term J(q ) for three values of αs. The three curves for Ji(q )

2 have similar behavior to that of the initial Ansatz J0(q ) shown in Fig. 7.7, so that these curves can be fitted by the same functional form of Eq. (7.48), with the corresponding fitting parameters for the three cases given in Table 7.1. In fact, the main differences 7.5. Numerical analysis of the mass equation 123

0.2

0.16

0.12

0.08

0.04

0 0 0.5 1 1.5 2 2.5 3

Figure 7.13: Comparison of the quantity m˙ 2(q) obtained from diﬀerentiating (i) the − numerical data (blue circles), and (ii) the ﬁt given in Eq. (7.70) (red continuous curve).

2 2 between the three Ji(q ) and the initial J0(q ) are the location of the point in which the curve crosses zero and the “bending” displayed in the intermediate region. In particular, the zero crossings are located at q = 78 MeV (blue dashed dotted), q = 96 MeV (red dashed), and q = 90 MeV (yellow dotted).

2 Finally, we compare our results for the gluon propagator for each Ji(q ) with the lattice data of [8] in the bottom panel of Fig. 7.12. We observe a general agreement between our calculated ∆(q2) and the lattice data, with the largest discrepancy occurring in the region of momenta between (0.8 2.5) GeV, where the relative error ranges from − [0.1 0.15] for J (q2) (blue dashed dotted curve), [0.1 0.16] for J (q2) (red dashed), − 1 − 2 and [0.1 0.2] for J (q2) (yellow dotted curve). The relative errors drop considerably − 3 outside this intermediate region, which suggests that this region is more sensitive to the truncations and approximations employed. We can also test the eﬀect of variations in (k) in our solution for the mass. For C4 that we use seven curves for (k), varying the value of the parameter d in Eq. (7.62) C4 1 within the range (4.0 10.0) GeV2, which changes the height and area of the peak as seen − in the right panel of Fig. 7.7. Then, we solve Eq. (7.45) as before. It is interesting to observe the relation between the size of the peak in (k) and the value of αs necessary for C4 obtaining the proper mass solution. Such relation is shown in Fig. 7.14 and we see that as the maximum value of (k) decreases, αs increases, so that changes in the height of C4 the peak of (k) can be counterbalanced with adjustments in the value of αs to generate C4 7.6. Summary 124

1.7

d =10 1.6 1

d =9 1 1.5 d =8 1

d =7 1.4 1

d =6 1 1.3 d =5 1

d =4 1.2 1

1.1 0.25 0.26 0.27 0.28 0.29 0.3

Figure 7.14: The values of αs as a function of the peak height of the function (k) shown C4 2 in Fig. 7.7. The values of d1 are given in units of GeV .

essentially the same solution for m2(q2).

7.6 Summary

Let us now summarize this Chapter and its main conclusions. Using Eq. (7.47), the nonlinearity of the mass equation, given in Eq. (7.45), is enforced due to the introduction of the unknown function m2(q2) in the integrand denominator, thus eliminating the freedom of rescaling the solutions. Then, we use an initial Ansatz for the kinetic term, J(q2), entering the mass equation. Such physically motivated Ansatz is later refined in the iterative process, which allows us to obtain a gluon propagator in accordance with lattice data. We have also used an effective method for the multiplicative renormalization of the equation, inspired by analogous studies of the quark sector of the theory. In this approach, we introduce two renormalization functions in the mass equation, named and , which C3 C4 are related to the form factors of the tree level structures of the three-gluon and four-gluon vertices, respectively, in the totally symmetric kinematic configuration. In particular, due to the IR suppression of the three-gluon vertex, and consequently of , the contribution of the one-loop term in the mass equation is reduced compared to the C3 two-loop, which is enhanced because of the IR peak of . Then, the competition between C4 these terms leads to solutions for the running gluon mass that are positive-defined and 7.6. Summary 125 monotonically decreasing, as can be observed in the top left panel of Fig. 7.12. We have also obtained the derivative of the mass function with respect to q2, as shown in Fig. 7.13. This quantity can be compared with the BSA that controls the

0 2 formation of the massless excitations in the three-gluon vertex, Cgl(r ), through Eq. (6.6). Comparing Figs. 7.13 and 6.8, we verify that the qualitative behavior is similar, although e the corresponding maxima are located at different momenta, approximately 340 MeV and 1 GeV, respectively. Nonetheless, note that the two results were not only obtained by means of different approaches but also use different approximations for the kernels involved

2 and are given for distinct values of the coupling αs(µ ). In addition, the analysis of the BSE of Chapter6 is done by linearizing the equation by treating the gluon propagators appearing inside the integrals as external quantities. Therefore, this qualitative agreement represents a highly nontrivial check of the entire mechanism and the approximations employed here. 126

Chapter 8

Conclusions

We conclude this thesis by emphasizing that the results presented here establish the generation of a dynamical gluon mass as a self-consistent mechanism which can be per- fectly accommodated in QCD. More specifically, we have scrutinized the theoretical aspects of dynamical gluon mass generation using two formalisms. In one, the integral equation governing the momentum evolution of the gluon mass was derived from the SDE for the gluon propagator. In contrast, in the other, the derivation was based on the massless bound-state formalism. It is known that, despite all differences between the ingredients and terminology employed in both frameworks, the mass equation obtained with these two distinct methods are formally equivalent [41]. However, this equivalence only holds when no approximations are employed in either approach. In this thesis, we have seen that, in practice, the use of approximations is unavoidable in both methods. In general, they are carried out in different intermediate steps for each formalism. As a result, different momentum dependences for the gluon mass are generated. Our focus here was precisely to explore these differences in both formalisms and check the self-consistency of the entire gluon mass generation mechanism. To set up the theoretical background required for the development of our entire analysis, in Chapter1, we have presented a brief compilation of the main QCD features. On the other hand, to keep the thesis self-contained, Chapters2 and5 were devoted to the formal derivation of the SD and BS equations, respectively. In Chapter3, we have briefly reviewed the so-called PT-BFM framework. There, we have seen that one of the major advantages of the PT-BFM formalism is how the 127 transversality of the gluon self-energy may be systematically enforced in the SDE [14, 34, 66]. The blockwise transversality of the PT-BFM gluon self-energy arises from the fact that the fully dressed vertices entering the gluon SDE satisfy a simple set of Abelian- like WTIs instead of the complex STIs. Then, in Chapters4,6, and7, we use this set of WTIs, and its corresponding zero momentum limit WIs, to derive the corresponding mass equations in both the SDE and massless bound-state formalisms. More specifically, in Chapter4, using the PT-BFM SDE for the gluon propagator, we have shown that the WIs trigger a set of crucial cancellations between the diagrams belonging to each of the subsets shown in the Figs. 4.2, 4.3, and 4.4. Then, the leftover of this cancellation process assumes precisely the form of the seagull identity. As a consequence, the net effect of the joined action between the WIs and the seagull identity is the total annihilation of all quadratic divergences and finite contributions, leading to a null value for ∆−1(0) = 0 [76]. It is precisely the same synergy between WIs and seagull identity that takes place in QED and cancels all quadratic divergences so that the photon stays massless at all orders. In previous works, the proof of the total annihilation of the seagull divergences had been addressed by only taking into account the one-loop dressed gluonic diagrams of Fig. 4.2[78]. In this thesis, we have extended this analysis to incor- porate also the one-loop dressed ghost contributions and the two-loop dressed diagrams. Therefore, the entire set of dressed diagrams comprising the gluon SDE was treated in a unified way [76]. We have also presented, in Chapter4, a self-consistent way to evade the complete cancellation of the finite contributions for ∆−1(0) by triggering the Schwinger mechanism in QCD. The ultimate goal of this mechanism is to allow the gluon dimensionless vacuum polarization to acquire a massless pole dynamically, leading to a finite value for ∆−1(0) (see Eq. (4.4)). The requirement that enables the gluon self-energy to develop such behavior is the existence of a particular type of nonperturbative vertices. These vertices must contain massless poles and be completely longitudinally coupled. Then, assuming such poles in the structure of the fundamental vertices of the theory, one can see that their net effects are: (i) the circumvention of the seagull cancellation, yielding ∆−1(0) = m2;(ii) the Abelian- like WTIs and non-Abelian STIs of the theory remain intact; and (iii) the massless poles decouple from on-shell amplitudes because they are longitudinally coupled. 128

Thus, in Sec. 4.5, we show how the WIs are rearranged in the presence of poles, producing the cancellations of the quadratic divergences, but leaving ﬁnite residual terms that give rise to ∆−1(0) = m2. It is important to stress that the analysis of Chapter4 was performed without ﬁxing the gauge parameter ξ, which indicates that the generation of a dynamical mass for the gluon is independent of the chosen linear covariant gauge [76]. The main question we have addressed in the Chapter6 is: how can such poles be dynamically produced in QCD? We have seen that the origin of these massless poles is due to purely nonperturbative QCD dynamics. Since these poles appear in the structure of the Green’s functions, one can treat them as massless bound state excitations, whose dynamical formation can be studied within the BSE formalism. In previous works, the presence of massless poles was assumed to be exclusively restricted to the three-gluon vertex [42, 43]. Here, we have extended the previous analyses considering that the poles may also appear in the ghost-gluon vertex. Then, under this hypothesis, we have derived a system of two coupled BSEs, which is responsible for describing the dynamics of the

0 2 0 2 derivatives of the BS wave functions, Cgl(q ) and Cgh(q ), associated with the existence of poles in the three-gluon and ghost-gluon vertices, respectively (see Fig. 6.3). To solve e e this system of equations, we have considered only the one-loop dressed contributions in the expansion of the four-point kernels appearing in the BSEs (see Figs. 6.4 and 6.5). Our full numerical analysis of the BSE system demonstrates that QCD dynamics is indeed strong enough to generate massless poles for both vertices studied. More speciﬁcally,

0 2 0 2 we have obtained nontrivial solutions for Cgl(q ) and Cgh(q ) for αs = 0.43. Moreover, our analysis has revealed that C0 (q2) is rather smaller than C0 (q2), with the relative size gh e e gl between these two quantities being approximately C0 (q2)/C0 (q2) 1/5. Therefore, we e gh egl ≈ verify that the impact of the ghost sector is extremely suppressed in the gluon mass gen- e e eration, where its main effect is a slightly faster running of the gluon mass for momenta larger than 2 GeV [44]. Consequently, the contribution of C0 (q2) is of paramount im- ≈ gl portance to the generation and the momentum evolution of the gluon mass. We would e like to stress that the above result is nontrivial, in the sense that the small contribution of the ghost sector could not be inferred based on any previous observation. Then, after establishing that the generation of the massless poles is a viable mechanism within of the QCD framework, in Chapter7 we put together all concepts explored in the previous Chapters, in order to derive, from the PT-BFM gluon SDE, the integral equation 129 governing the dynamical gluon mass. The novelty in the analysis presented here is related to the nonlinearity of this equation, which has not been taken into account in previous studies, where the gluon propagator is treated as an external input [38, 39]. By introducing the explicit dependence of the gluon propagators on the unknown running mass, we have eliminated the indeterminacy in the gluon mass scale, caused by the linearization of the equation [45]. In addition, the multiplicative renormalization of the mass equation has been implemented using an approximate method extensively employed in the studies of dynamical mass generation for the quark [121–123]. In this approach, the renormalization functions were replaced by appropriate functions related to the structures of the three- and four-gluon vertices. We observe that the interplay between these functions favors the generation of a positive-defined gluon mass as desired. Additionally, one of the ingredients entering the gluon mass equation is the subdiagram (k2), which had been studied previously considering only the tree-level tensorial Y structure of the three-gluon vertex [38]. Here, instead, we have computed (k2) taking Y into account all ten longitudinal form factors entering in the full three-gluon vertex [124]. With all of these ingredients, we are able to obtain positive-defined and monotonically decreasing running gluon masses using values for the coupling constant, αs, within the range (0.27 0.29), which is compatible with the values used in the literature [109, 110, − 132]. In addition, the kinetic terms, Ji, used to obtain these masses present the desired characteristics, which are known due to the profound connection between the kinetic term and the three-gluon vertex [114, 124]. Then, combining the masses and kinetic terms obtained with our numerical method, we can recover the lattice gluon propagator rather well. Let us now, for the sake of completeness, suggest some new directions to the analyses presented here. From the point of view of the BSE approach, one may envisage various improvements. First, the replacement of the full tensorial structure of three gluon vertex by only its tree-level tensors inside the coupled system of BSEs ought to be ameliorated. It can certainly be improved by using the results for longitudinal form factors of this vertex recently obtained in [124]. Secondly, as we have done in our gluon mass equation analysis, it would also be

0 2 preferable to take into account the nonlinearity of the BSE for Cgl(q ), instead of treating

e 130 the gluon propagators entering in it as external quantities. Such a nonlinear approach results in a rather complicated integro-diﬀerential equation, which must be numerically solved. This type of treatment may open up the possibility of obtaining solutions for a continuous interval of values for the coupling constant. Furthermore, in our BS studies, we have explicitly neglected any possible eﬀects stem- ming from poles associated with the four-gluon vertex. Thus, it would be interesting to eventually relax the assumption that these pole terms are numerically suppressed in order to gain some direct information on the actual size of such contributions. Nonetheless, this task is particularly complex, given our limited knowledge about the complete nonperturbative structure of this vertex, due to its rich tensorial structure [130, 131, 133, 134]. In particular, the corresponding BSA depends on four rather than three kinematic variables, and, equivalently, the derivatives as q 0 will depend on two variables instead of one, → which will greatly complicate the structure and treatment of the resulting BSE system. Concerning the SDE based analysis, presented in Chapter7, we recall that the gluon mass equation was solved in isolation, instead of considering the coupled system with the kinetic term J(q2) (see Eq. (7.2)). In fact, due to a series of technical complexities related to the derivation of the equation for J(q2), this equation has not been obtained yet. Evidently, it would be desirable to eventually solve the coupled system of equations for m2(q2) and J(q2) and verify how closely the lattice results for the gluon propagator can be reproduced. We would also like to emphasize a clear indication of the self-consistency of the entire gluon mass generation mechanism explored in this thesis. From Eq. (6.6), we know that

0 2 the derivative of the dynamical mass must be equal to the BSA Cgl(q ). When we compare the derivative of the dynamical mass obtained in Chapter7 with the numerical solution e 0 2 for Cgl(q ) obtained in Chapter6, we observe that, although they are not quantitatively the same, they do present similar qualitative behavior. Notice that one expects these e results do not coincide due to the diﬀerent approximations used when obtaining them (in particular for the three-gluon vertex). However, the similarity in the overall shape of these two curves provides a highly nontrivial check of the methods and calculations employed in the studies presented here. To conclude this thesis, it is important to stress that, although we have restricted our studies to pure Yang-Mills theories, lattice simulations reveal that the phenomenon of the 131 dynamical mass generation also persists in the presence of light dynamical quarks [12, 36, 37]. In addition, although in Chapters6 and7 we have focused on the Landau gauge, analytic studies [135–138] and lattice simulations [35, 139, 140] have indicated that gluon propagator in linear covariant gauges continues to saturate in the IR region. Moreover, a large number of profound phenomenological implications are related to the generation of a gluon mass, such as the notion of a freezing eﬀective charge in the deep IR [21, 60, 141], which was shown to be in agreement with a widespread phenomenological model employed in a successful description of meson properties [80, 142]. Bibliography 132

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