Comportamento infravermelho do vértice ghost-gluon e suas implicações : Infrared behavior of the ghost-gluon vertex and its implications

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Instituto de F´ısica “Gleb Wataghin”

Antonio Mauricio Soares Narciso Ferreira

Comportamento infravermelho do v´ertice ghost-gluon e suas

implica¸c˜oes

Infrared behavior of the ghost-gluon vertex and its implications

CAMPINAS, SP 2016

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Antonio Mauricio Soares Narciso Ferreira

Comportamento infravermelho do v´ertice ghost-gluon e suas

implica¸c˜oes

Infrared behavior of the ghost-gluon vertex and its implications

Disserta¸c˜ao apresentada ao Insti-tuto de F´ısica “Gleb Wataghin” da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten¸c˜ao do t´ıtulo de Mestre em F´ısica.

Dissertation presented to the “Gleb Wataghin” Institute of Physics of the University of Campinas in par-tial fulfillment of the requirements for the degree of Master of Physics.

Supervisor/Orientador: Arlene Cristina Aguilar ESTE EXEMPLAR CORRESPONDE À VERS ÃO FINAL DA DISSERTA ¸C ÃO DEFENDIDA PELO ALUNO ANTONIO MAURICIO SOARES NAR-CISO FERREIRA, E ORIENTADA PELA PROFA. DRA. ARLENE CRISTINA AGUILAR.

CAMPINAS, SP 2016

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Ficha catalográfica

Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174

Ferreira, Antonio Mauricio Soares Narciso,

1988-F413i FerInfrared behavior of the ghost-gluon vertex and its implications / Antonio Mauricio Soares Narciso Ferreira. – Campinas, SP : [s.n.], 2016.

FerOrientador: Arlene Cristina Aguilar.

FerDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.

Fer1. Schwinger-Dyson, Equações de. 2. Teoria quântica de campos. 3. QCD não perturbativa. 4. Cromodinâmica quântica. 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: Comportamento infravermelho do vértice ghost-gluon e suas implicações

Palavras-chave em inglês: Schwinger-Dyson Equations Quantum field theory

Non-perturbative QCD Quantum chromodynamics Área de concentração: Física Titulação: Mestre em Física Banca examinadora:

Arlene Cristina Aguilar [Orientador] Tereza Cristina da Rocha Mendes Orlando Luis Goulart Peres Data de defesa: 30-09-2016

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MEMBROS DA COMISSÃO JULGADORA DA DISSERTAÇÃO DE MESTRADO DE

ANTONIO MAURICIO SOARES NARCISO FERREIRA - RA 073905

APRESENTADA E APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 30 / 09 / 2016.

COMISSÃO JULGADORA:

- Profa. Dra. Arlene Cristina Aguilar – Orientadora – DRCC/IFGW/UNICAMP

- Profa. Dra. Tereza Cristina da Rocha Mendes – IFSC/USP

- Prof. Dr. Orlando Luis Goulart Peres – DRCC/IFGW/UNICAMP

OBS

_{.: Informo que as assinaturas dos respectivos professores membros da}

banca constam na ata de defesa já juntada no processo vida acadêmica do aluno.

CAMPINAS

2016

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Dedico este trabalho `a minha orientadora, Professora Doutora Arlene Cristina Aguilar, quem me confiou a oportunidade, pela qual serei sempre grato.

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Agradecimentos

Durante o desenvolvimento deste trabalho, tive a felicidade de contar com o apoio de muitas pessoas às quais sou profundamente grato. Gostaria de agradecer em primeiro lugar à minha orientadora, Professora Doutora Arlene Cristina Aguilar, pela oportuni-dade, pelos conselhos, por partilhar de seu extenso conhecimento e por sua dedica¸cão e disposi¸cão a seus alunos. Também agrade¸co ao Professor Doutor Orlando Luis Goulart Peres e ao Professor Doutor Donato Giorgio Torrieri, ambos professores do Instituto de F´ısica “Gleb Wataghin”, por participarem das comissões dos exames de qualifica¸cão e pré-requisito para defesa, desta forma contribuindo com sugestões, cr´ıticas e questionamentos que tornaram esta disserta¸cão mais rica e, espero, mais clara. Ao Professor Doutor Joan-nis Papavassiliou, da universidade de Valência, Espanha, que à distância fez comentários e sugestões valiosas à pesquisa. Aos meus colegas de trabalho Clara Teixeira Figueiredo e Jeyner Castro Cardona, pelas discussões e colabora¸cões. A todo o corpo do Instituto de F´ısica e da Universidade Estadual de Campinas, por oferecer o ensino, estrutura e recursos necessários à minha forma¸cão e pesquisa. Em particular, à equipe do cluster Feynman do Centro de Computa¸cão John David Rogers, pelo suporte a mim dedicado na fase de cálculo numérico deste trabalho e à Secretaria de Pós-Gradua¸cão, pelos servi¸cos diversos e pelo apoio para participa¸cão em encontros cient´ıficos onde muito aprendi. Sou grato ao Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico - CNPq, pelo finan-ciamento concedido a este projeto de mestrado. Finalmente, agrade¸co à minha fam´ılia e amigos, pelo apoio, compreensão e amor, em todas as horas. Especialmente, agrade¸co à minha mãe, Maria de Fátima Quitéria Soares Narciso, meu pai, Antonio Celso Ferreira, meus irmãos, Lu´ıs Marcelo Soares Narciso Ferreira e Luana Carollo Ferreira, e minha amada Betina Grosser Martins.

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As equa¸cões de Schwinger-Dyson alcan¸caram uma posi¸cão de destaque como uma das abordagens principais de estudo não perturbativo da QCD, servindo como campo de prova de primeiros princ´ıpios para hipóteses sobre o comportamento das fun¸cões de Green da teoria. Neste contexto, a avalia¸cão efetiva de vértices vestidos em configura¸cão cinemática geral é uma conquista recente. Apresentamos um extenso estudo do vértice ghost-gluon da QCD, partindo de truncamentos das equa¸cões de Schwinger-Dyson satisfeitos por esta fun¸cão em cinemática geral e preservando quase todos os vestimentos. Particular ênfase é dada à preserva¸cão de propriedades especiais satisfeitas pelo vértice ghost-gluon, no gauge de Landau, tais como o teorema de Taylor e a simetria ghost-anti-ghost. Primeiro, revi-samos a deriva¸cão das equa¸cões de Schwinger-Dyson e das identidades de Slavnov-Taylor por métodos funcionais. Em seguida, consideramos as dificuldades envolvidas em truncar o sistema infinito de equa¸cões preservando as simetrias da teoria e revisamos brevemente o esquema de truncamento PT-BFM, introduzindo a fun¸cão auxiliar 1 + G. Posteriormente, demonstramos que o truncamento da equa¸cão do vértice ghost-gluon, ao n´ıvel de um loop vestido, viola a simetria ghost-anti-ghost. Uma modifica¸cão do mesmo truncamento é desenvolvida visando restaurar a referida simetria. A análise numérica das equa¸cões do vértice ghost-gluon e do propagador do ghost é apresentada em seguida, comparando os resultados obtidos aos existentes na literatura. Nossos métodos são então estendidos ao cômputo do kernel de espalhamento ghost-gluon, que posteriormente é utilizado para refinar os cálculos da fun¸cão 1 + G. Em todos os casos estudados, observamos acordo qua-litativo entre nossos resultados e os descritos na literatura, com diferen¸cas quantitativas discutidas em detalhes. A disserta¸cão é encerrada com um exame cr´ıtico das aproxima¸cões empregadas e sugerimos aplica¸cões futuras para os resultados aqui descritos.

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Abstract

The Schwinger-Dyson equations raised to a foreground position as one of the main frameworks for the non perturbative study of QCD, serving as a first principles test ground for hypothesis on the behavior of Green’s functions. In this context, the effective evaluation of dressed vertex functions in general kinematics is a recent achievement. We present an extensive study of the QCD ghost-gluon vertex, considering different trunca-tions for the Schwinger-Dyson equatrunca-tions, satisfied by this function, in general kinematics. Particular emphasis is laid on the preservation of the Landau gauge special properties of the ghost-gluon vertex, such as the Taylor theorem and the ghost-anti-ghost symmetry. First we review the derivation of Schwinger-Dyson equations, as well as Slavnov-Taylor identities, through functional methods. Next, the difficult task of truncating the infinite system of equations while preserving the symmetries of the theory is considered and the PT-BFM scheme is briefly reviewed, introducing the auxiliary 1 + G function. It is shown then, that the one-loop dressed truncation of the ghost-gluon vertex equation violates the ghost-anti-ghost symmetry of the function. A modification to the truncation is devised in order to restore it. Subsequently, we present numerical solutions of the ghost-gluon vertex and ghost propagator equations, comparing our results to those found in the literature. Our methods are then extended to the evaluation of the ghost-gluon scattering kernel and the latter is used to refine computations of the PT-BFM 1 + G function. For all functions studied, qualitative agreement to previous finds was observed, with quantitative differ-ences that we discuss in detail. We close the dissertation with a critical examination of the approximations employed and suggest further applications of the results here described.

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The following table collects the various abbreviations and acronyms used through-out the dissertation, in alphabetical order, and their meanings. The page of the first occurrence of an abbreviation is included in the rightmost column.

Abbreviation Meaning Page

1PI one particle irreducible function 19

BFM Background Field Method 15

BQI Background Quantum Identity 15

DCSB dynamical chiral symmetry breaking 123

IR infrared 13

MPI message passing interface, standard for parallel com-putation

88

OPE operator product expansion 99

PT Pinch Technique 15

PT-BFM synthesis of PT and BFM 15

QCD Quantum Chromodynamics 12

QED Quantum Electrodynamics 12

QFT Quantum Field Theory 14

RBOD right bare one loop dressed truncation of the ghost-gluon vertex SDE

82

RGI renormalization group invariant 59

RLAOD right left averaged one loop dressed truncation of the ghost-gluon vertex SDE

82

SDE Schwinger-Dyson equation 14

STI Slavnov-Taylor identity 17

UV ultraviolet 13

VEV vacuum expectation value 19

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1

Introduction

The strong nuclear interaction has been successfully described by Quantum Chromo-dynamics (QCD) which is a non-Abelian gauge theory of fermions, called quarks, and gauge fields, referred to as gluons. The fundamental Lagrangian is invariant under the color gauge group SU(3) and it is given by

L = iqQ αD/αβqβQ− 1 4F a µνFaµν, (1.1)

where Q is the quark flavor, α, β = 1, 2, 3 the quark colors and a = 1, 2, · · · , 8 the gluon color. The fermion fields are assigned to the fundamental representation of the color group, whereas the gauge fields belong to the adjoint representation.

The covariant derivative for fields in the fundamental representation,

/

Dαβ = (δαβ∂µ− igλaαβAaµ/2)γµ, (1.2)

where λa

αβ are the Gell-Mann matrices, enforces the gauge invariance of the quark kinetic

energy term through an interaction with the gluon. The field strength tensor,

F_µνa = ∂µAaν− ∂νAaµ+ gfabcAbµAcν, (1.3)

is such that (Fa

µν)2 is also gauge invariant. The non-Abelian character of the theory

man-ifests itself in the gluon self-interaction terms, arising from Eq. (1.3), which largely enrich the dynamics of QCD, as compared to Abelian theories such as Quantum Electrodynamics (QED).

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according to the eightfold way proposed by Gell-Mann. This is accomplished assuming that the group structure is attributed to a new degree of freedom called color. As a consequence the correct spin statistics emerges naturally in this framework [1].

However, understanding the consequences of a non-Abelian gauge symmetry in the context of a complete quantum field theory was found to be one of the most challenging goals in the seventies, and to some extent still is. Among many difficulties, the strong nuclear interaction posed the problem of asymptotic freedom. It was known that deep hadronic scattering phenomena could be understood under the assumptions that hadrons were bound states of partons which interacted very weakly at high momentum transfer. This meant the strong nuclear interaction had to weaken at short distances, a behavior difficult to find in quantum field theories known at the time [1].

QCD started to be taken more seriously when Gross, Wilczek and Politzer showed that non-Abelian Gauge theories can be asymptotically free [2, 3]. Through a one-loop calculation, they found the beta function, β = µ∂g/∂µ, of Yang-Mills theories to be

β(g) = g

3

48π2 [−11CA+ 2NF] , (1.4)

where NF is the number of fermions and CAδab = fabcfdbc = N is the Casimir eigenvalue

of the adjoint representation of SU(N). It was shown that the correct gauge group for QCD is SU(3), since we have three colors. Then, for SU(3), β(g) < 0 for NF ≦ 16, more

than enough room to accommodate asymptotic freedom in QCD with the known number of quark flavors.

From then on, QCD accumulated several successes in the ultraviolet (UV) regime. The infrared (IR), on the other hand, presents remarkable challenges. To begin with, free quarks and gluons are not observed, only their color singlet bound states are, and pertur-bative analyses are unable to derive this behavior, known as confinement. Also, the hadron mass spectrum cannot be simply understood as combinations of quark Lagrangian masses, implying that mass is generated dynamically in QCD. Finally, quantities calculated, using the perturbative framework, suffer from severe IR divergences, like the Landau pole in the coupling constant, which make perturbation theory in the IR inapplicable. In order to understand the strong interaction at the hugely important IR regime, non perturbative methods are then required.

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14

field theory and Schwinger-Dyson equations (SDEs). The former, introduced by Wilson [4], consists of formulating quantum field theory in a discretized Euclidean space-time, with the lattice spacing serving as the UV regulator. With the imaginary time, quantum field theories become analogous to statistical mechanics and Monte Carlo simulations are thus well suited [5].

SDEs, on the other hand, are the exact equations of motion of the Green’s functions of the theory [6, 7], forming an infinite tower of coupled non-linear equations. They were first derived in QED by Dyson, as a resummation of the perturbative series [8], and by Schwinger, through the functional formulation of quantum field theory (QFT) [9], as part of an effort to define QFT beyond the calculation of a few perturbative loops. In their intact form, SDEs may be used to prove some exact results, such as the Taylor theorem for the ghost-gluon vertex [10], which is demonstrated in Chapter 4 of this work. Otherwise, they can be truncated to a finite system of equations with the help of ansatz employed for some Green’s functions and approximate solutions can then be found, a framework with a long and rich history [11, 12]. In the latter approach, difficulties arise regarding the symmetries of the theory in hand, which must be preserved under the truncation. Devising such consistent truncation schemes is a highly non-trivial task, on which an accurate description of the theory from SDE studies hinges decisively.

In order for gauge theories to be effectively quantized, a gauge fixing procedure must be introduced and the physically equivalent field configurations be integrated out of the generating functional [13]. For non-Abelian theories, fixing the gauge in a covariant way leads to the introduction of ghost fields [14], which do not appear in the spectrum, but are vital in order to preserve unitarity of the theory. Accordingly, the addition of a ghost sector to the set of Green’s functions manifests the non-Abelian nature of the theory. It is the non perturbative behavior of the ghost sector of QCD which is the main subject of this dissertation.

The importance of a detailed understanding of the ghost sector of QCD in the non per-turbative regime can be seen in the intricate ways the ghost Green’s functions are related to other quantities of interest. For instance, the gluon self interaction is related to the ghost sector through a generalized Ward identity (WI) relating the full gluon propagator and three gluon vertex to the ghost-gluon scattering kernel and ghost propagator, as will be demonstrated in Chapter 2.

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Also the ghost sector is connected to the gauge symmetry constraints on the gluon propagator in a remarkable way. A symmetry preserving scheme for the gluon SDE was devised, based on the fusion of the Pinch-Technique (PT) and the Background Field Method (BFM), known in the literature as PT-BFM scheme [15]. In this formalism, a special function 1 + G(p) connects the quantum and background Green’s functions through a set of non-trivial Background-Quantum identities (BQIs) [16, 17]. In its turn, the 1 + G(p) function is composed of gluon and ghost propagators and the ghost-gluon scattering kernel.

It is important to reinforce that the auxiliary function 1 + G(p) is also an ingredient in the definition of a renormalization group invariant effective charge for QCD, useful in certain phenomenological studies [18, 19]. Also, in the Landau gauge, it is equivalent to the Kugo-Ojima function, which was connected, in the past, to a possible criterion for confinement [20].

The research presented in this thesis focused on the study of the ghost-gluon vertex and scattering kernel. For that purpose, truncations of the ghost-gluon vertex SDE are considered and the extent to which they preserve the symmetry properties of that function is discussed. An improvement to the common one loop dressed truncation of the referred SDE is devised, in order to explicitly preserve the symmetry of exchange of ghost and anti-ghost momenta, which holds in the Landau gauge. We shall direct our attention for now to the quenched, that is pure glue, version of the theory, since many of the foremost QCD phenomena are manifest even without consideration of quarks.

We then proceed to solve the SDEs numerically, comparing the results to several previous studies, using multiple tools, including SDE and lattice, and apply the resulting ghost-gluon vertex to the solution of the ghost propagator SDE. Then we consider the ghost-gluon scattering kernel, computing some of its form factors, and apply the results to the computation of the 1 + G function and related quantities.

The thesis is presented with the following division. In Chapter 2 we review the deriva-tion of the SDEs through funcderiva-tional methods and extend the discussion in order to derive the generalized WIs, which enforce gauge symmetry by suitable constraints on the Green’s functions. In Chapter 3 an overview of the PT-BFM truncation scheme is provided, focus-ing on the special properties of BFM Green’s functions and the Abelian-like WIs satisfied by them. In that chapter we introduce the 1 + G(p) function and some related functions,

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16

to be computed later on. Then we prepare ourselves to the practical evaluation of the ghost-gluon vertex by first reviewing some of its fundamental properties, in particular the Taylor theorem and the ghost-anti-ghost symmetry, in Chapter 4. There we also discuss the truncation and renormalization of the relevant SDEs. The numerical results for the ghost-gluon vertex and the ghost propagator are presented in Chapter 5, where we also discuss in detail how our results compare to those found in the literature, for QCD as well as for the related SU(2) theory. The numerical study of the ghost-gluon scattering kernel and the application to the evaluation of the 1 + G(p) function is presented in Chapter 6. Finally, a discussion of the approximations employed and the results obtained is followed by the presentation of our conclusions in Chapter 7.

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2

Schwinger-Dyson Equations

The present chapter is dedicated to review the derivation of SDEs through the func-tional formalism. We assume the reader has some familiarity with the path integral approach to quantum field theory, presenting only a short overview of the quantities of interest.

We shall also concern the derivation of Slavnov-Taylor identities (STIs), which, like the SDEs, are all order identities relating the Green’s function of the theory, but stemming from and enforcing gauge symmetry. The STIs form the generalization of the WIs to non-Abelian gauge theories and due to their origin in gauge invariance, it is vital that the identities be satisfied by the ansatz employed in the truncation of SDEs. In addition, one of the reasons for the exposition of STIs in this work, is that the ghost-gluon scattering kernel, to be evaluated numerically in Chapter 6, is related to the three gluon vertex, one of the least known functions of QCD in the IR, through an STI.

First we outline the functional approach, in Section 2.1. Next we extend the formal-ism to gauge theories, indicating the additional features introduced by the gauge fixing procedure. Then the SDEs for the ghost propagator and ghost-gluon vertex are derived in Section 2.3. In that Section, the definition of the ghost-gluon scattering kernel emerges naturally in the derivation of the ghost-gluon vertex SDE. This function will be shown along the dissertation to connect the ghost-gluon interaction to many other functions. The derivation of STIs is illustrated in Section 2.4, where we derive the identity for the gluon propagator and the one for the three gluon vertex. Finally, we consider the implications of the STIs on the renormalization constants in Section 2.5.

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2.1 The functional formalism 18

2.1 The functional formalism

Let us briefly review the functional approach to quantum field theories. For details, see Ref. [6, 7] and references therein.

The fundamental quantity in the functional formalism is the generating functional, Z[J], defined by a path integral,

Z[J] = Z

Dφei(S[φ]+J ). (2.1)

Above, S[φ] = R d4_{xL[φ(x), ∂}

µφ(x)] is the action of the theory, φi denote generic fields

and i is an index denoting field type as well as their possible internal degrees of freedom. The symbol φ, without an index, denotes the collective of fields and the integral measure is Dφ = ΠiDφi. The term

J = Z

d4xJi(x)φi(x) , (2.2)

introduces external sources, Ji, coupled to the fields. They are set to zero at the end of

calculations, serving the only purpose of functional differentiation. J is the collective of the sources.

The Green’s functions of a quantum field theory may be obtained by functional dif-ferentiation of Z[J], h0|T φi1(x1) · · · φin(xn)|0i = Z Dφ φi1(x1) · · · φin(xn)e iS[φ] = 1 Z[0] δ iδJi1(x1) · · ·_iδJ δ in(xn) Z[J] J=0 , (2.3)

where T is the time ordering operator.

The above equation generates connected as well as disconnected contributions. The complete information about these functions can be obtained from the connected Green’s functions alone, derived from the connected generating functional, W [J] = i ln Z[J],

h0|T φi1(x1) · · · φin(xn)|0iconn = 1 in+1 δ δJi1(x1) · · · δ δJin(xn) W [J] Ji=0 . (2.4)

In particular the full propagators are defined as

Di(x − y) = i δ2_{W [J]} δJi(x)δJi(y) J=0 . (2.5)

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The connected Green’s functions can be further decomposed into one particle irre-ducible (1PI) terms. The latter are obtained from the Legendre transform (see Ref. [21] for an introduction) of W [J],

Γ[φ] = −W [J] − Z

d4xJi(x)φi(x) , (2.6)

by functional differentiation,

h0|T φi1(x1) · · · φin(xn)|0i1PI = −i

δ δφ_i1(x1) · · · δ δφ_i_n(xn) Γ[φ] φi=φcl , (2.7)

where the functional Γ[φ] is called the effective action and φi is the vacuum expectation

value (VEV) of the field φi in the presence of the sources J, defined as

φi(x) = − δW [J] δJi(x) = 1 Z[J] δZ[J] iδJi(x) = h0|φi(x)|0iJ . (2.8)

When sources are turned off, this value is usually zero, except for spontaneously broken symmetry theories. The sources are related to the effective action by

Ji(x) = −

δΓ[φ]

δφ_i(x). (2.9)

It is often necessary to convert back and forth between 1PI and connected functions. To relate the two-point 1PI function to the propagator, notice that

δijδ(x − y) = δφ_i(x) δφ_j(y) = Z d4zδJk(z) δφ_j(y) δφ_i(x) δJk(z) = Z d4z δ 2_Γ[φ] δφ_j(y)δφ_k(z) ! δ2_{W [J]} δJk(z)Ji(x) . (2.10)

In operator language, the second functional derivative of Γ[φ] is the inverse of the propa-gator, given in Eq. (2.5), i.e.

δ2_Γ[φ] δφi(x)δφi(y) = δ2_{W [J]} δJi(z)Ji(x) −1 = iD−1_i _{(x − y) .} (2.11)

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2.2 Generating functional for gauge theories 20

To relate higher order functions we can use the chain rule, δ δiJi(x) = Z d4zδφj(z) iδJi(x) δ δφj(z) = i Z d4z δ 2_{W [J]} δJi(x)δJj(z) δ δφj(z) = i Z d4z δ 2_Γ[φ] δφ_i(x)δφ_j(z) !−1 δ δφ_j(z) . (2.12)

In computing high order 1PI functions, we need to deal with derivatives of the inverse propagators. Using the operator identity

δ(O−1_[φ]O[φ])

δφ = 0 , (2.13)

it is simple to show that

δ δφ_i(x) δ2_Γ[φ] δφ_j(y)φ_k(z) !−1 = − Z d4w1d4w2 δ2_Γ[φ] δφ_j(y)φ_m(w1) !−1 δ3_Γ[φ] δφ_m(w1)φi(x)φn(w2) × δ2_Γ[φ] δφn(w2)φk(z) −1 . (2.14)

The above machinery allows us to compute Green’s functions and the relations between them.

2.2 Generating functional for gauge theories

The generating functionals corresponding to gauge theories have some important new features, which we now review. We shall focus on pure gauge theories in this work, ignoring fermion fields.

The Lagrangian for a Yang-Mills theory is,

LYM = − 1 4F a µνFaµν, (2.15) where Fa µν = ∂µAaν − ∂νAµa+ gfabcAbµAcν, (2.16) and Aa

µ is the gauge field. This Lagrangian is invariant under the gauge transformation

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where αa_{(x) is an arbitrary function and the covariant derivative is defined as}

D_µab = δab∂µ+ gfacbAcµ, (2.18)

for fields in the adjoint representation. The fabc _{are the structure constants of the gauge}

group.

In order to meaningfully quantize a gauge theory, it is necessary to fix the gauge. This is done through the Faddeev-Popov procedure [14], which results in the following path integral for a Yang-Mills theory.

Z[jµ] = Z DA det(M) exp i Z d4x_LYM+ LGF+ Aaµ(x)jaµ(x) , (2.19) where LGF = −(F a_[A(x)])2 2ξ (2.20)

is responsible for fixing the gauge. The functional Fa_{[A] is the gauge fixing condition,}

which we choose to be the Lorentz condition,

Fa[A(x)] = ∂µAa_µ(x) . (2.21)

Finally, ξ is called the gauge fixing parameter.

The gauge fixing procedure also introduces the Faddeev-Popov operator, M, defined as Mab_xy = δF a_[A(x)] δαb_(y) = ∂ µ xDabµ,xδ(x − y) . (2.22)

In the above equation, the subscript x in the derivatives specify the variable of differen-tiation and its repetition does not imply summation, for now.

The term det(M) is most conveniently written as a path integral over ghost, c(x), and anti-ghost, c∗

(x), fields, which are independent. The ghost and anti-ghost fields are Grassmann variables, which anti-commute, but are scalars. As such, they violate spin-statics, which is not a problem since they do not appear in the spectrum. Also, the anti-commutation of Grassmann variables implies, c2_{(x) = 0 and c}∗2_{(x) = 0.}

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2.2 Generating functional for gauge theories 22

With the inclusion of ghosts the generating functional translates into

Z[jµ, J, J∗] = Z DADc∗ Dc det(M) exp i Z d4_{x [L}YM+ LGF+ Lghost] + iJ , (2.23)

where the ghost Lagrangian is

Lghost= −

Z

d4y c∗a(x)M_xyabcb_{(y) = −c}∗a(x)∂_xµDab_µ,xcb(x) , (2.24)

and the sources are introduced as

J = Z

d4x[j_µa(x)Aµ_a(x) + J∗a(x)ca(x) + c∗a(x)Ja(x)] , (2.25)

where J∗ _{and J are Grassmann sources.}

At this point, the effective action is defined by extension of Eq. (2.6),

Γ[Aµ, c∗, c] = −W [jµ, J, J∗] −

Z

d4x [Aµ_aj_µa+ c∗aJa+ J∗aca] . (2.26)

Then it follows that δΓ δAµ_a(x) = −j a µ(x) , δΓ δca_(x) = J ∗a_{(x) and} δΓ δc∗a_(x) = −J a_{(x) .} _(2.27)

A couple more comments on the ghost fields are in order. Differentiation with respect to a Grassmann variable will mean left differentiation, that is, the variable to be differen-tiated must be anti-commuted all the way to the left before differentiation. For instance, let C1 and C2 be Grassmann variables. The left derivative with respect to C2 is defined

by ∂ ∂C2 (C1C2) = − ∂ ∂C2 C2 C1 = −C1. (2.28)

For multiple derivatives, the order of differentiation matters, ∂2

∂C1∂C2 = −

∂2

∂C2∂C1

. (2.29)

This creates an ambiguity in the definitions of propagators and vertices through functional differentiation. We adopt the convention that the anti-ghost field must be in the leftmost

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derivatives. For instance, the ghost propagator is defined by

Dab_{(x − y) = i}δ

2_{W [j}

µ, J, J∗]

δJa_(x)δJ∗b_(y), (2.30)

recalling that J is the source for the anti-ghost.

Finally, since ghosts do not show up in the spectrum, c and c∗ _{must always be paired}

in the Green’s functions, i.e. there must be the same number of c and c∗ _fields.

2.3 Derivation of the SDEs

Now we have the tools to derive the SDEs. We follow the expositions in [22, 23]. First we derive the master equations, from which the SDEs for specific functions are generated by functional differentiation. Then we derive the SDEs for the ghost propagator and the ghost-gluon vertex.

2.3.1 Master SDEs

Our starting point, is to consider a change of variables in Eq. (2.1) of the form

φ′_i(x) = φi(x) + εfi(x) , (2.31)

where ε is infinitesimal and fi(x) is an arbitrary function of position only. The Jacobian

of this transformation is unit since fi does not depend on the fields, so that Dφ′ = Dφ.

Then the generating functional reads

Z[J] = Z Dφ′ exp{i(S[φ′ ] + δS[φ] + J + δJ )} , (2.32) where δS[φ] = Z d4x δS[φ] δφi(x) δφi(x) = ε Z d4x δS[φ] δφi(x) fi(x) (2.33)

and similarly for J .

Now, a change of variables does not change the result of the integral, but expanding to first order in ε we find

Z[J] = Z Dφ′ exp{i(S[φ′ ] + J )} 1 + iε Z d4x fi(x) δS[φ] δφi(x) + Ji(x) . (2.34)

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2.3 Derivation of the SDEs 24

vanish. Using differentiation under the integral sign, we obtain

0 = Z Dφ′ exp{i(S[φ′ ] + J )} Z d4x δS[φ] δφi(x) + Ji(x) fi(x) = Z d4x fi(x) δS[φ] δφi(x) δ iδJj + Ji(x) Z[J] , (2.35)

where each field inside the path integral is substituted by the functional derivative with respect to its corresponding source, outside.

Finally, since fi(x) is arbitrary, the integrand of

R

d4_{x must vanish. It follows that}

δS[φ] δφi(x) δ iδJj + Ji(x) Z[J] = 0 . (2.36)

Eq. (2.36) is the master equation for disconnected Green’s functions. Functional differ-entiation generates the corresponding SDEs. Notice that δS[φ]/δφi(x) = 0 produces the

Euler-Lagrange equation for the classical φi(x) field. By analogy, we may think of the

SDEs, as the Euler-Lagrange equations for the Green’s functions of a quantum theory. To obtain the SDEs for connected functions we set Z[J] = e−iW [J] _{in Eq. (2.36),}

eiW [J] δS[φ] δφi(x) δ iδJj e−iW [J] _{= −J}i(x) . (2.37)

Using the Baker-Campbell-Hausdorff lemma,

eABe−A= B + 1

1![A, B] + 1

2![A, [A, B]] + 1

3![A, [A, [A, B]]] + · · · , (2.38) we can rewrite Eq. (2.37) in the form

δS[φ] δφi(x) δ iδJj − δW [J] δJj · 1 = −Ji(x) . (2.39)

This is the master equation for connected functions. The ‘·1’ notation means that the operators to the left act on 1. This notation is to distinguish between an equality between operators and an equation for numbers. Eq. (2.39) corresponds to the latter case.

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and the chain rule, as in Eq. (2.12). Then we obtain δS[φ] δφi(x)  φ_j(x) + i Z dz δ 2_Γ δφj(x)δφk(z) !−1 δ δφk(z)   · 1 = δΓ[φ] δφi(x) , (2.40)

which is the master SDE for 1PI functions.

2.3.2 Ghost propagator SDE

We can now derive definite SDEs for QCD. The complete manipulations are long and we shall show only the most essential steps. We start with the ghost propagator.

To derive the ghost SDE, we first write explicitly the master equation corresponding to the anti-ghost field, φi = c∗b(y). Because ghosts only appear in the action through the

ghost term, Eq. (2.24), we find the simple relation δS

δc∗b_(y) = −∂ µ

yDbcµ,ycc(y) = −∂2ycb(y) − gfbdc∂yµ[Adµ(y)cc(y)] . (2.41)

Then the master equation, Eq. (2.40), reads

− ∂2ycb(y) − gfbdc∂yµ     Ad_µ(y) + i Z d4z δ 2_Γ δAd_µ(y)δφ_i(z) !−1 δ δφ_i(z)   cc_(y)   = δΓ δc∗b_(y), (2.42) where we have already canceled terms containing derivatives of constants.

Differentiating Eq. (2.42) with respect to a ghost VEV, ca_{(x), and then setting the}

VEVs to zero entails δ2_Γ δca(x)δc∗b(y) = −δ ab_∂2 yδ(x − y) − igfbdc∂yµ δ δca_(x) δ2_Γ

δAd_µ(y)δcc_(y)

!−1

. (2.43)

The first term on the right is readily identified as the inverse tree level propagator. The last term, which we write as ∂µ

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2.3 Derivation of the SDEs 26 rewrite it as ∂_yµΣab_µ_{(x − y) ≡ ∂}_yµ   (−igf bdc₎ δ δca_(x) δ2_Γ

δAd_µ(y)δcc(y) !−1_  (2.44) = +igfbdc∂_yµ Z d4w1d4w2 δ2_Γ δAd_µ(y)δφm(w1) !−1 δ3_Γ δφ_m(w1)δca(x)δφn(w2) × δ2Γ δφ_n(w2)δcc(y) −1 . (2.45)

Notice that we have defined the function Σab

µ in the above equation for later convenience.

The next step is to pair up ghost and anti-ghost fields and order them accordingly to the convention of Section 2.2. Repeating the derivation that led to Eq. (2.11), this time for Grassmann fields, we find that the inverse ghost propagator is given by

iD−1

ab (x − y) =

δ2_Γ

δca_(y)δc∗b_(x). (2.46)

Then, we can write

Σab_µ_{(x − y) = gf}bdc Z

d4w1d4w2∆deµν(y − w1)Dcf(w2− y)Γνeaf(w1, x, w2) , (2.47)

where the gluon propagator is defined by

∆ab_µν_{(x − y) = i} δ

2_Γ

δAa_µ(x)δAb_ν(y) !−1

, (2.48)

and the ghost-gluon vertex is

Γµ_abc_{(x, y, z) = −i} δ

3_Γ

δAa_µ(x)δcb_(y)δc∗c_(z) . (2.49)

The ghost propagator equation then reads

D_ab−1_{(y − x) = iδ}ab∂_y2_{δ(x − y) − i∂}_yµΣab_µ_{(x − y)} = iδab∂y2δ(x − y) − igfbdc∂yµ

Z

d4w1d4w2∆deµν(y − w1)Dcf(w2− y)

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where term igfbdc_∂µ

y appearing in the second term of the second line can be identified as

the tree level ghost-gluon vertex.

Finally, we Fourier transform, to obtain

D−1_ab(p) = δab(i/p2)−1₋

Z _d4_k

(2π)4∆ de

µν(k)Df c(p − k)Γ[0] µ_dcb(k, p)Γνeaf(−k, p − k) , (2.51)

where Γ[0] µ_abc_{(k, p) = −gf}abc_pµ _{is the tree level ghost-gluon vertex in momentum space.}

We adopt the convention that the momenta are all entering the vertices, except for the anti-ghost momentum, which we take as outgoing for explicit ghost number conservation. In addition, unless it is necessary to eliminate ambiguity, we denote the functional de-pendence in terms of the gluon and anti-ghost momenta only, as the dede-pendence on the ghost momentum follows from momentum conservation. To be clear about our convention, Fig. 2.1 shows the ghost-gluon vertex in diagrammatic form and its textual notation.

k, µ ↓ p q = p − k = Γ abc µ (k, p) = gfabcΓµ(k, p) a b c

Figure 2.1: Diagrammatic representation for the ghost-gluon vertex and the notation used for it. The grey blob means the vertex is dressed.

Then we have all the ingredients to work out the diagrammatic representation of the ghost equation. It is depicted in panel (A) of Fig. 2.2.

It is interesting to notice that there are multiple SDEs for the same Green’s function. This happens because we can obtain the same function starting from different master equations. For instance, we could have seeded the derivation of a ghost SDE with the master equation for the ghost field, instead.

First, we rearrange the ghost term in the Lagrangian to the form

Lghost = ca∂2c∗a− gfadeAdµce∂µc

∗a_, _(2.52)

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(A)

(B)

−1 −1

=

−

p

_{p − k}

p

k

ν µ

→

−1 −1

=

−

p

_{p − k}

p

k

ν µ

→

x y x y x y x y x y x y

Figure 2.2: Ghost SDE in two different forms. In (A) the anti-ghost leg, position y, has the bare vertex and in (B) the bare vertex is on the ghost leg, position x. (A) and (B) correspond to Eqs. (2.50) and (2.55), respectively. The blobs represent full propagators and 1PI vertices.

reads ∂_x2c∗a(x) + gfabc  Ac µ(x) + i Z d4z δ 2_Γ δAcµ(x)δφj(z) !−1 δ δφ_j(z)   ∂µ xc∗b(x) = δΓ δca(x). (2.53) Differentiating (2.53) with respect to an anti-ghost, c∗b_{(y), and proceeding as before,}

one eventually obtains

D_ab−1_{(y −x) = iδ}ab∂_x2_{δ(x−y)−igf}adc∂_xµ Z

d4w1d4w2∆deµν(x−w1)Df c(w2−x)Γνebf(w1, y, w2) .

(2.54) This is the second form of the ghost SDE. It is almost identical to Eq. (2.50). The only difference is that the bare ghost-gluon vertex is in position x, while the previous equation had the tree level vertex in y. This form of the ghost SDE is represented in panel (B) of Fig. 2.2. In momentum space, the equation reads

D−1_ab(p) = δab(i/p2)−1₋ Z d4_k (2π)4∆ de µν(k)Df c(p − k)Γ[0] µ_adc(−k, p − k)Γνef b(k, p) . (2.55)

The fact that we can choose to undress the right or the left ghost-gluon vertex in the interaction diagram, as represented in Fig. 2.2, will be useful in Chapter 4 to prove an important property of the ghost-gluon vertex, the ghost anti-ghost symmetry.

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2.3.3 Ghost-gluon vertex SDE

Now, we consider the derivation of the ghost-gluon vertex SDE. Differentiating the anti-ghost master equation, Eq. (2.42), with respect to a ghost and a gluon VEVs and then setting VEVs to zero yields

iΓcab_µ (z, x, y) = δ

3_Γ

δAµ_c(z)δca_(x)δc∗b_(y) (2.56)

= −gfbca∂y_µ_{δ(x − y)δ(y − z) − igf}bde∂_yν δ

2

δAc_µ(z)δca_(x)

δ2_Γ

δAd_ν(y)δce_(y)

!−1

.

Notice that this equation can be rewritten as

Γcabµ (z, x, y) = ∂yνHµνcab(z, x, y) , (2.57)

by defining

H_µνcab(z, x, y) = igfcabgµνδ(x−y)δ(y −z)−gfbde

δ2

δAc_µ(z)δca_(x)

δ2_Γ

δAd_ν(y)δce(y) !−1

. (2.58)

The function Habc

µν is called the ghost-gluon scattering kernel and will be in the

fore-ground together with the ghost-gluon vertex itself throughout this thesis. It is related not only to the ghost-gluon vertex, but also to the three gluon vertex and the PT-BFM auxiliary function, 1 + G, as we will see along. In momentum space, its relation to the ghost-gluon vertex reads

Γcab_µ (k, p) = ipνH_µνcab(k, p) , (2.59) where k and p are the gluon and anti-ghost momenta, respectively. The anti-ghost mo-mentum will always be considered outgoing.

Turning back to Eq. (2.56), we recognize that the first term gives the tree level ghost-gluon vertex. We have to develop the second. Using Eq. (2.14) twice, pairing up ghosts

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and anti-ghosts and identifying the inverse propagators, one obtains

Γcab_µ (z, x, y) = igfcab∂_µy_{δ(x − y)δ(y − z) + igf}aed∂_yν Z d4u1d4u2d4w1d4w2∆efνα(y − w1) × Γαf bg(w1, x, w2)Dhg(u1− w2)Γcgiµ (z, u1, u2)Ddi(y − w1) + Z d4u1d4u2d4w1d4w2∆νfeα(y − u1)Γf bg_αµβ(u1, z, u2)∆gh_βγ(u2− w1) × Γγhbi(w1, x, w2)Ddi(y − w2) + Z d4w1d4w2∆efνα(y − w1)Γα µ_{f cbg}(w1, z, w2, x)Did(y − w2) . (2.60)

Two new functions appear in the above equation, the three gluon vertex

Γabc_µνρ_{(x, y, z) = −i} δ

3_Γ

δAµ_a(x)δAν_b(y)δAρ_c(z), (2.61) and the ghost-ghost-gluon-gluon function, defined by

Γabcd_µν _{(x, y, z, w) = −i} δ

4_Γ

δAµ_a(x)δAν_b(y)δcd_(w)δc∗c_(z). (2.62)

Then, we are ready to develop the diagrammatic representation of the vertex SDE, which is shown in Fig. 2.3.

(A) (B) + ↓ k µ, a, z b, x c, y p − k p = c, y = + ↓ k b, x c, y p µ, a, z ↓ k b, x c, y p p − k µ, a, z + l ← l ← b, x µ, a, z ↓ k p l ← ↓ k b, x c, y p µ, a, z µ, a, z b, x c, y p տ l ↓ k µ, a, z ↓ k b, x c, y p l ←

Figure 2.3: Panel (A) shows the ghost-gluon vertex SDE in a compact form. The blob with a cross in the center, in the second diagram, is not 1PI. Panel (B) shows the decomposition of the second diagram in (A) in terms of 1PI vertices. The oval in the third diagram on the right hand side of (B) is the 1PI ghost-ghost-gluon-gluon function. These diagrams correspond to Eq. (2.60).

Once again, there are multiple equations for the same function. Starting the derivation of the ghost-gluon vertex SDE from the ghost master equation instead, Eq. (2.53), we

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obtain

Γcab_µ (z, x, y) = igfbca_{δ(x − z)∂}_µx_{δ(x − y) − igf}cea∂_xν δ

2 δAc_µ(z)δc∗b_(y) δ2_Γ δAe_ν(x)δc∗c_(x) −1 . (2.63) Now, Eq. (2.63) can be converted to a form similar to (2.60),

Γcab_µ (z, y, x) = igfcab_{δ(x − z)∂}_µx_{δ(x − y) + igf}aed∂_xν Z d4u1d4u2d4w1d4w2∆efνα(x, w1) × Γαf bg(w1, y, w2)Dhg(u1, w2)Γcgiµ (z, u1, u2)Ddi(x, w1) + Z d4u1d4u2d4w1d4w2∆νfeα(x, u1)Γf bgαµβ(u1, z, u2)∆ghβγ(u2, w1) × Γγhbi(w1, y, w2)Ddi(x, w2) + Z d4w1d4w2∆efνα(x, w1)Γα µ_{f cbg}(w1, z, w2, y)Did(x, w2) . (2.64)

In this form of the SDE, the bare vertex sits on the ghost leg, in position x, while Eq. (2.60) had the bare vertex on the anti-ghost leg, in y. Eq. (2.64) is depicted in Fig. 2.4.

We have proved then, that we can choose to keep the right or the left ghost-gluon vertex in the interaction diagrams bare. This property will be important in Chapter 4, to truncate the ghost-gluon vertex in a way as to preserve its symmetries.

(A) (B) + ↓ k µ, a, z b, x c, y p − k p = c, y = + + b, x µ, a, z ↓ k p b, x c, y ↓ k µ, a, z l ← p − k p b, x c, y ↓ k µ, a, z l ← p b, x c, y ↓ k µ, a, z l ← p µ, a, z b, x c, y p տ l ↓ k µ, a, z ↓ k b, x c, y p l ←

Figure 2.4: The second form of the ghost-gluon vertex SDE. Panel (A) shows a compact form, with a reducible function. Panel (B) shows the decomposition of the second diagram on the right hand side of (A). The bare vertices are on the ghost legs, in position x, rather than y as in Fig. 2.3.

For the ghost-gluon vertex case, there is yet another possible starting point, the master equation for gluon fields. The resulting SDE is much more complicated, due to the fact

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2.4 Slavnov-Taylor identities 32

that the gluon has more couplings in the Lagrangian, which enriches its master equation. We shall not show the result here, but, by now, the reader can anticipate one of its features from the pattern that arises, the vertices containing the leg corresponding to the master equation used in the derivation are kept undressed. Hence, the third form of the ghost-gluon vertex SDE has the vertices on the ghost-gluon leg bare. See Ref. [22] for the explicit form.

2.4 Slavnov-Taylor identities

As we have seen in the previous section, the fact that a change of variables does not change the result of a path integral allows us to derive identities for the Green’s function. An important class of identities results from taking the field transformations to be symmetries of the action. Since the action is invariant, the changes in the path integrandare only in the terms introduced by the gauge fixing procedure, the source terms and eventually the integral measure. Relations between the Green’s functions can then be derived, as before, by imposing the first order variation with respect to an infinitesimal such change to vanish. In this way, the functional formalism provides a powerful method to derive constraints on Green’s functions resulting from symmetry (see Ref. [24] for a general treatment).

The best known examples of this class of identities are the WI of QED, resulting from Abelian gauge symmetry. The generalization to non-Abelian gauge symmetries consists of the STIs [25, 10]. Their derivation is complicated by the fact that the determinant of the Faddeev-Popov operator for non-Abelian theories,

det(M) , (2.65)

depends on the gauge fields and is not usually invariant under gauge transformations. There are many routes to circumvent this difficulty and we follow the one proposed by Slavnov [25].

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2.4.1 Master STI

Let M−1 _{be the inverse of the Faddeev-Popov operator, defined in Eq. (2.22), such}

that

M−1 ab_xyMbc_yz = δac_{δ(x − z) .} (2.66)

To simplify the notation, we now adopt the convention that repeated continuum subscripts are summed, i.e. integrated. For instance, Eq. (2.66) means

Z

d4y M−1 ab_xyMbc_yz= δac_{δ(x − z) .} (2.67)

Exceptionally, continuum indices of derivatives or covariant derivatives are not summed and we will write integrations over them explicitly, when they happen.

We consider a non-local gauge transformation of the form,

δAa µ= Dµ,xab M −1 bc xyδωyb = Dabµ,x Z d4_{y M}−1 bc xyδωyb, (2.68) where δωb

y = δωb(y) is an arbitrary infinitesimal function. The gauge part of the action is

automatically invariant under (2.68). Additionally, the measureR DA det(M) is invariant under this unusual gauge transformation. We refer to [26] for the proof. Hence, the only changes in the integrand are in LGF and J .

From Eq. (2.22) follows immediately that

δFa[A(x)] = Mab_xyM−1 bc_yzδω_zc = δxzδωza, (2.69) where δxz ≡ δ(x − z). Therefore, δLGF= − 1 ξF a_[A(x)]δ xzδωza, (2.70) which implies δSGF = Z d4_xδLGF = − 1 ξF a_[A(z)]δωa z . (2.71)

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As for the source term, it transforms as

δJ = Z

d4x j_aµ(x)δAa_µ(x) = Z

d4x j_aµ(x)D_µ,xab M−1 bc_xzδω_zc. (2.72)

Expanding Z[J] to first order in δωc

z and setting its variation to zero yields

Z DA det(M)ei(S+J ) Z d4x −1_ξFc[A(x)]δxz + jaµ(x)Dµ,xab M−1 bcxz δω_zc = 0 . (2.73) Since δωc

z is arbitrary, the integrand of the implied z integration must vanish. Therefore

−1_ξFc δ iδj(z) + Z d4x j_aµ(x)D_µ,xab δ iδj M−1 bc_xz δ iδj Z[J] = 0 . (2.74)

This is the master STI for disconnected functions. Notice that the integration over x in the second term remains in the expression, reflecting the non-locality of the Slavnov transformation, Eq. (2.68).

To obtain the STIs for connected functions we write Z[J] = e−iW [J] _{and use the}

Baker-Campbell-Hausdorff lemma to rewrite the master equation as 1 ξF c δW δj(z) + Z d4x j_aµ(x)D_µ,xab δ iδj − δW δj M−1 bc_xz δ iδj − δW δj · 1 = 0 , (2.75)

where we assumed Fc _{to be a linear functional of A, such as the Lorentz condition, to}

simplify the first term. Eq. (2.75) is the master STI for connected functions.

We still do not have a closed expression for M−1_{, but one for M}−1_{· 1 will suffice. To}

find it, notice that the master connected SDE for the anti-ghost field, given by Eq. (2.39), can be written as ∂xµDabµ,x δ iδj − δW δj δ iδJ∗b_(y)− δW δJ∗b_(y) ·1 = −Mabxy δ iδj − δW δj δW δJ∗b y = Ja(x) . (2.76)

Differentiating with respect to an anti-ghost source, Ja_{(z), and multiplying on the left by}

M−1 _{we obtain} M−1 ca zy· 1 = − δ2_W δJa_(z)δJ∗c_(y) = iD ca_{(y − z) ,} _(2.77)

relating M−1_{· 1 to the ghost propagator.}

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case will be enough.

Finally, from now on we will restrict the discussion to the general covariant gauges, where Fa_{[A(x)] = ∂}µ

xAaµ(x), is the Lorentz condition.

2.4.2 Gluon STI

We are in position to derive STIs. The simplest one is that satisfied by the gluon propagator.

Differentiating Eq. (2.75) with respect to a gluon source and then setting sources to zero, we obtain 1 ξ∂ µ x δ2_W δjν b(y)δja(x) = −D bc ν,y δ iδj − δW δj M−1 ca_xy δ iδj − δW δj · 1 . (2.78)

Differentiating the above equation with respect to y leads to 1 ξ∂ µ x∂yν(−i)∆abµν(x − y) = −∂yνDν,ybc δ iδj − δW δj M−1 ca_xy δ iδj − δW δj · 1 = −MbcyzM −1 ca zx· 1 = −δbaδ(x − y) . (2.79)

Consequently, in momentum space, we have

pµpν∆abµν(p) = −iξδab. (2.80)

To interpret this result, separate the Lorentz structure of the gluon propagator in its transverse and longitudinal parts,

∆ab_µν_{(p) = −iδ}ab Pµν(p)∆(p) + E(p) pµpν p2 , (2.81) where Pµν(p) = gµν− pµpν p2 , (2.82)

is the transverse projector and ∆(p) and E(p) are the radiative corrections that the transverse and longitudinal part of the gluon propagator may acquire, respectively. Sub-stituting decomposition (2.81) into Eq. (2.80) we find that

∆ab_µν_{(p) = −iδ}ab Pµν(p)∆(p) + ξ pµpν p2 . (2.83)

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Thus, Eq. (2.83) states that the longitudinal part of the gluon propagator does not get radiative corrections. In particular, in the Landau gauge, ξ = 0, the all-order gluon propagator is purely transverse, an observation that cannot be overemphasized.

Sometimes it is important to know the form of the inverse gluon propagator,

∆−1 µν_ab(p)∆bc_νρ(p) = δacg_ρµ, (2.84)

From Eq. (2.83), it is clear that

∆−1 µν_ab(p) = iδab Pµν(p) 1 ∆(p) + 1 ξ pµ_pν p2 . (2.85)

In the Landau gauge, this quantity is singular, but that is not usually a problem. Since the singularity in ξ = 0 is longitudinal to the gluon momentum and this part is not radia-tively corrected, it is canceled by tree level gluon propagators, present in some particular calculations.

2.4.3 Three gluon vertex STI

The derivation of STIs is increasingly longer as we proceed to higher order functions. For the three gluon vertex, already, the detailed calculations are voluminous and we shall show only the most important steps.

First we differentiate Eq. (2.75) twice with respect to gluon sources. The result is

−1_ξ∂_xµ δ 3_W δjcρ(z)δj_bν(y)δjaµ(x) = δ δjcρ(z) D_ν,ybd δ iδj − δW δj M−1 da_yx δ iδj − δW δj · 1 + δ δjν b(y) D_ρ,zcd δ iδj − δW δj M−1 da_zx δ iδj − δW δj · 1 + O(J) , (2.86) where O(J) denotes terms proportional to the sources, which we do not set to zero yet, but will vanish in the end. Eq. (2.86) contains a lot of information about the three gluon vertex, which we now extract.

First, differentiation of Eq. (2.86) with respect to y and to z results in

−1_ξ∂_zρ∂_yν∂_xµ δ

3_W

δjcρ(z)δj_bν(y)δjaµ(x) = O(J) .

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To convert this result into a statement about the 1PI three gluon vertex, we use Eqs. (2.11) and (2.14) to find that

O(J) = −1_ξ∂_ρz∂y_ν∂_µx Z

d4w1d4w2d4w3∆cfργ(z − w3)∆νβbe(y − w2)∆µαad(x − w1)

× Γdefαβγ(w1, w2, w3) . (2.88)

Having performed all functional differentiations, we may set the sources to zero and Fourier transform. Then, using Eq. (2.83), leads to

pµ₁pν₂pρ₃Γabc_µνρ(p1, p2, p3) = 0 . (2.89)

Notice that Eq. (2.89) simply states that the three gluon vertex remains totally anti-symmetric under the exchange of two momenta in its Lorentz structure, to all orders.

To actually relate Γabc

µνρ to other Green’s function we have to come back to Eq. (2.86)

and differentiate it only with respect to y. Then we obtain

−1_ξ∂_yν∂_xµ δ 3_W δjcρ(z)δj_bν(y)δjaµ(x) = ∂_ρz∂_yν δ δjν b(y) M−1 ca_zx δ iδj − δW δj · 1 + ∂ν_y δ δjν b(y) gfced δ iδjeρ(z) − δW δjeρ(z) × M−1 dazx δ iδj − δW δj · 1 + O(J) , (2.90)

where the covariant derivative was expanded.

We anticipate that we are going to separate, in the end, a component transverse to the momentum pρ₃, conjugate to z. The first term on the right hand side of Eq. (2.90) is clearly going to be longitudinal, so we do not write it explicitly anymore.

The following step is to use Eq. (2.77) for M−1 _{and discard a term which is just}

proportional to a VEV in the term within brackets in Eq. (2.90). One then obtains

−1_ξ∂_yν∂_xµ δ 3_W δjcρ(z)δj_bν(y)δjaµ(x) = gf ced_∂ν y −∆beνρ(y − z)Dad(z − x) +i δ 2 δjν b(y)δJa(x) δ2_Γ δAρ_e(z)δcd_(z) −1) + O(J) + L(p3ρ) , (2.91)

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where L(p3ρ) denotes terms longitudinal to p3ρ.

Using the chain rule, Eq. (2.12), twice, we can decompose the second term in Eq. (2.91) into 1PI functions. The right hand side is then found to be

−1 ξ∂ ν y∂xµ δ3_W δjcρ(z)δj_bν(y)δjaµ(x) = −gf ced_∂ν yDad(z − x)∆beνρ(y − z) − igfced Z d4w1d4w2∆bgνα(y − w1)Dah(w2− x) δ2 δAg_α(w1)δch(w2) δ2_Γ δAρ_e(z)δcd_(z) −1 − igfced Z d4w1d4w2d4w3d4w4∆bgνα(y − w1)Dah(x − w3)Dij(w4− w2)Γαghi(w1, w3, w4) × _δcj_(wδ 2) δ2_Γ δAρ_e(z)δcd_(z) −1 + O(J) + L(p3ρ) . (2.92)

Comparing the last line in Eq. (2.92) to the definition of the function Σµ, in Eq. (2.45),

we see that the last term equals Z

d4w1d4w2d4w3d4w4∆bgνα(y − w1)Dah(x − w3)Dij(w4− w2)Γαghi(w1, w3, w4)Σjcρ(w2− z) .

(2.93) At this point we notice that the only function of the coordinate z in the above integral is Σjc

ρ(w2 − z). Therefore, the Fourier transform of this integral will be proportional to

Σjc

ρ(p3), which, by Lorentz covariance, is necessarily longitudinal to p3ρ, so that we absorb

it into L(p3ρ).

As for the first two terms in Eq. (2.92), their sum is found to be related to iHαρ, by

comparing it to Eq. (2.58). Specifically, we obtain

−1 ξ∂ ν y∂xµ δ3_W δjcρ(z)δj_bν(y)δjaµ(x) = i∂_νy Z d4w1d4w2Dah(w2− x)∆ναbg(y − w1)Hαρghc(w1, w2, z) + O(J) + L(p3ρ) . (2.94)

Then we can decompose the left hand side of Eq. (2.91) into 1PI, trivially, set the sources to zero and Fourier transform. This task entails

1 ξp

µ

1pν2∆adµα(p1)∆νβbe(p2)∆cfργ(p3)Γαβγdef(p1, p2, p3) = −p2ν∆ναbe(p2)Dad(p1)Hαρecd(p2, −p3)

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Using the STI for the gluon propagator, Eq. (2.83), one has that pµ_∆

µν(p) = −iξpν/p2.

Hence, Eq. (2.95) may be written as

−ξp µ 1pν2 p2 1 ∆cγ_ρf(p3)Γabfµνγ(p1, p2, p3) = ip2αDad(p1)Hαρbdc(p2, −p3) + L(p3ρ) . (2.96)

Finally, we separate the part of Eq. (2.96) that is transverse to p3ρ by multiplying it

by Pγ1ρ (p3), leading to pµ₁pν₂P_ργ(p3)Γabcµνγ(p1, p2, p3) = p21pν2Pργ(p3)Dad(p1) 1 ∆(p3) H_νγbdc(p2, −p3) . (2.97)

This is the STI for the three gluon vertex, showing its relation to the ghost-gluon vertex through the ghost-gluon scattering kernel, Hµν. Notice that, in this form, no explicit

dependence on the gauge fixing parameter remains, so that, in particular, it is valid in the Landau gauge, even though our starting point, Eq. (2.86), was seemingly singular for ξ = 0.

Clearly, Eq. (2.97) remains true if we interchange two gluon legs, since derivatives with respect to gluon sources commute.

In fact, it is possible to derive a stronger STI for the three gluon vertex. One way to achieve it is to start from (2.86), without taking any space-time derivatives, in which case there are many other terms to take into account. Otherwise, one can consider the most general Lorentz structure for the vertex, taking into account its anti-symmetry under momenta exchanges, and combine the results with Eq. (2.97), with the momenta interchanged. Then one finds the improved STI,

pµ₁Γµνγ(p1, p2, p3) = p21D(p1) Pα ν(p2) ∆(p2) Hγα(p3, −p2) − Pα γ(p2) ∆(p2) Hνα(p2, −p3) , (2.98)

where we separate in the vertex functions a tree level color structure,

Γabc_µνρ(p1, p2, p3) = gfabcΓµνρ(p1, p2, p3), Hµνabc(k, p) = gfabcHµν(k, p) . (2.99)

Clearly, Eq. (2.98) satisfies (2.97).

Notice that, since we extracted an anti-symmetric color structure, Eq. (2.98) holds under cyclic permutations of the gluon legs only.

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2.5 Implication for the renormalization constants 40

2.5 Implication for the renormalization constants

We close this chapter by deriving an identity between the renormalization constants that we will use later.

The quantities we have dealt with so far are all unrenormalized. We define the multi-plicative renormalization constants as

∆(p) = ZA∆R(p) , Γµνρ(p1, p2, p3) = (Z1A) −1_Γµνρ

R (p1, p2, p3) ,

D(p) = ZcDR(p) , Γµ(k, p, q) = (Z1c)−1ΓµR(k, p, q) , (2.100)

where ZA, Zc , are the renormalization constants of the gluon and ghost propagators

and ZA

1 and Z1c are the ones of the three gluon and ghost-gluon vertices, respectively.

In these equations, the quantities without subscript denote unrenormalized functions, whereas ‘R’ denotes a renormalized quantity. Also we extracted a color factor from the ghost propagator, Dab_{(p) = δ}ab_D(p).

Now, notice that the divergence of function Hµν is the same as that of the ghost-gluon

vertex, Γµ, due to Eq. (2.59). If a regularization scheme is used that retains the gauge

symmetry, such as dimensional regularization [28], we can substitute the definitions in (2.100) into the three gluon vertex STI, Eq. (2.97), to find that

ZA ZA 1 = Zc Zc 1 , (2.101)

a result first obtained by Taylor [10]. A consequence of this identity is that the coupling of the ghosts to the gauge field remains the same as that of the gauge field to itself, under renormalization [10]. Also, we will need Eq. (2.101) to renormalize the ghost-gluon vertex equation later.

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3

The Background Field Method and its

auxiliary function

We slightly detour from the main line of development in this work, that is the study of the ghost-gluon vertex, to introduce the BFM in this chapter. The main motivation is to introduce the auxiliary function 1 + G which is one of the quantities computed in this thesis, as an application of the ghost-gluon vertex results.

The BFM is an unconventional kind of gauge in which gauge symmetry is explic-itly preserved with regards to transformations of an external gauge field. As a result, the Green’s functions satisfy QED-like WIs, which lead to significant simplifications in some calculations. In particular, the BFM WIs enable the truncation of SDEs with their symmetries unharmed, which makes the formalism highly valuable for SDE studies.

In this truncation scheme, a new auxiliary function needs to be evaluated in conjunc-tion with the desired Green’s funcconjunc-tion, which is the 1+G. The 1+G funcconjunc-tion is responsible for relating background and quantum Green’s functions through a set of identities, called the BQIs. As such, knowledge of the 1 + G plays a central role in the symmetry preserv-ing truncation of SDEs that we will present. This function is related to the ghost-gluon vertex through the ghost-gluon scattering kernel and its evaluation will be possible once the latter has been computed.

In Section 3.1 we illustrate the difficulty of truncating the gluon SDE, showing how the transversality of the gluon self energy is rapidly compromised by truncation. In Section 3.2 the BFM is introduced and its special gauge symmetry proved. Next, we show how the WIs of the BFM rearrange diagrammatic contributions to the gluon self energy into independently transverse groups, in Section 3.3. The truncation of the gluon SDE based

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3.1 Gluon SDE 42

on those rearrangements is presented in Section 3.4. Next we introduce the BQIs and the 1 + G function in Section 3.5 as a new feature of the PT-BFM truncation scheme. Finally, we provide a perturbative example of the BQIs in Section 3.6.

3.1 Gluon SDE

Truncation of the infinite tower of SDEs while preserving the fundamental symmetries of a theory is a highly demanding task in the case of non-Abelian gauge theories, due to the complex and delicate interconnections of their Green’s functions. For instance, the latter are constrained to obey the STIs, which are much more complicated than the WIs of Abelian theories and easily violated by naive truncation of the SDEs. We exemplify this issue with the case of the SDE for the gluon propagator.

The gluon SDE [11, 12] is illustrated diagrammatically in Fig. 3.1, with the color structure factored out. Besides the gluon propagators itself, the gluon SDE involves the full ghost propagator and all three fundamental vertices of the theory, each of which function satisfies its own SDE, and contains two-loop dressed diagrams, (a4) and (a5),

that are costly to evaluate. In practice, some simplification of this equation is immensely desirable.

∆

−1_µν

(p) =

−1

+

1₂

+

1₂

(a

1

)

(a

2

)

+

1₆

(a

3

)

(a

4

)

+

1₂

(a

5

)

Figure 3.1: SDE for the gluon propagator. Diagrams (a1)-(a5) define the gluon self energy,

Πµν(p).

The sum of the 1PI diagrams (a1)-(a5), in Fig. 3.1, defines the gluon self energy,

Πµν(p). As we have shown in the previous chapter, the gluon STI, Eq. (2.80), implies that

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the gluon self energy must be transverse,

qµΠµν(p) = 0 . (3.1)

For convenience, we rewrite the propagator in general covariant gauges,

∆µν(p) = −i Pµν(p)∆(p) + ξ pµpν p2 . (3.2)

In agreement with Eq. (3.1), the radiative corrections are inserted only in ∆(p) by

∆−1(p) = p2+ iΠ(p) , (3.3)

where the vacuum polarization Π(p) is defined as Πµν(p) ≡ Pµν(p)Π(p) and is given by

the sum of all diagrams (a1)-(a5), appearing in the Fig. 3.1.

Then, because the gluon SDE is so complicated, we consider truncating the equation at some point. Besides the question of whether the neglected terms are truly subleading, truncation will usually violate the gluon self energy transversality.

To see the level of difficulty, let us consider retaining only the diagrams (a1) and (a2)

in Fig. 3.1, for example. Through a perturbative one-loop level calculation in the Landau gauge, ξ = 0, we obtain for the UV logarithm

Π[1]_µν_(p)|(a1)+ Π [1] µν(p)|(a2)= ig2_C A 16π2 84pµpν − 75gµνp 2 1 36ln −p2 µ2 , (3.4)

which is already not transverse. The expression above was obtained after renormalizing through a momentum subtraction procedure, such that the tree level value for Πµν(p) is

recovered at the off-shell momentum p2 _{= −µ}2 _{< 0.}

However, we know that at one-loop, it is crucial to add the contribution of the ghost diagram (a3), given by,

Π[1]_µν_(p)|(a3) = − ig2_C A 16π2 6pµpν + 3gµνp 2 1 36ln −p2 µ2 , (3.5)

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3.2 Background Field Method 44 Π[1]_µν_(p)|(a1)+ Π [1] µν(p)|(a2)+ Π [1] µν(p)|(a3) = − ig2_C A 16π2 (gµνp 2 − pµpν) 78 36ln −p2 µ2 . (3.6)

Nonetheless, already at two loops retaining only the first three diagrams would not suffice either. The transversality in this case will emerge only when all diagrams are included.

One way to circumvent this difficulty and devise a symmetry preserving truncation of the gluon SDE is provided by the BFM.

3.2 Background Field Method

We proceed to introduce the BFM, following Refs. [29, 30]. Firstly, we define a new generating functional, with the gauge field shifted by A → A + B in the action,

e Z[j, B] = Z DA det eM exp (Z d4x " LYM[A + B] − ( e_Fa_{[A(x), B(x)])}2 2ξQ + j_aµ(x)Aa_µ(x) #) . (3.7) Notice that only A is an integration variable, so that B is a background, i.e. classical field. The gauge action is the usual Yang-Mills one for the combination of the fields, i.e. S[A + B] =R d4_L

YM[A + B], which is invariant under a gauge transformation of the form

δ[Aa_µ+ B_µa] = Dµ[A + B]abθb(x) ≡ [δab∂µ+ gfacb(A + B)cµ]θb(x) . (3.8)

In general we will denote by Dµ[φ]ab the covariant derivative with respect to a gauge

transformation of the field φa µ.

The Faddeev-Popov operator, eM, arises from integrating out the gauge freedom in the path integral. Therefore, it is the derivative with respect to a transformation of the quantum field alone, that leaves the action invariant. Namely,

e

Mab_xy = δ eF[A(x), B(x)] δθb

A(y)

Comportamento infravermelho do vértice ghost-gluon e suas implicações : Infrared behavior of the ghost-gluon vertex and its implications

Comportamento infravermelho do v´ertice ghost-gluon e suas

implica¸c˜oes

Infrared behavior of the ghost-gluon vertex and its implications

Comportamento infravermelho do v´ertice ghost-gluon e suas

implica¸c˜oes

Infrared behavior of the ghost-gluon vertex and its implications

- Profa. Dra. Arlene Cristina Aguilar – Orientadora – DRCC/IFGW/UNICAMP

- Profa. Dra. Tereza Cristina da Rocha Mendes – IFSC/USP

- Prof. Dr. Orlando Luis Goulart Peres – DRCC/IFGW/UNICAMP

OBS

.: Informo que as assinaturas dos respectivos professores membros da

banca constam na ata de defesa já juntada no processo vida acadêmica do aluno.

CAMPINAS

2016

Agradecimentos

Abstract

Contents

1

Introduction

2

Schwinger-Dyson Equations

2.1

The functional formalism

2.2

Generating functional for gauge theories

2.3

Derivation of the SDEs

2.3.1

Master SDEs

2.3.2

Ghost propagator SDE

(A)

(B)

=

−

p

p

p

p − k

p

k

→

=

−

p

p

p

p − k

p

k

→

2.3.3

Ghost-gluon vertex SDE

2.4

Slavnov-Taylor identities

2.4.1

Master STI

2.4.2

Gluon STI

2.4.3

Three gluon vertex STI

2.5

Implication for the renormalization constants

3

The Background Field Method and its

auxiliary function

3.1

Gluon SDE

∆

(p) =

+

+

(a

)

(a

)

+

+

(a

_{.: Informo que as assinaturas dos respectivos professores membros da}

_{p − k}

_{p − k}