% %% � � � , , ,, � �� �� e ee @ @@ l l l Q QQ HHPPP XXX hhhh (((( ��� IFT Instituto de Física Teórica Universidade Estadual Paulista Thesis IFT–006/17 Chiral and Flavor Symmetries in Holographic QCD: Pion Excited States, Strong Couplings of Charmed Mesons and Inverse Magnetic Catalysis Carlisson Miller Cantanhede Pereira Supervisor Dr. Gastão Inácio Krein August 2017 Agradecimentos Durante todos os anos de doutorado, vivi uma experiência incrível, repleta de aprendizados e amizades cultivadas. Olhando para trás, posso recordar dos caminhos e desafios que passei, desde a vida díficil no Maranhão até os problemas mais recentes, e confesso que me supreendo que cheguei até aqui, ao final de um doutorado em Física. Apesar da vida familiar difícil, esta mesma me ensinou a olhar pra frente e ser determinado em meus objetivos e, com certeza, essa conclusão de mais uma etapa da minha vida foi fruto dessa grande lição. Essa tese é dedicada principalmente aos meus pais, Antônio Pedro Pereira e Marta Célia Braga Cantanhede que, sem formação escolar suficiente para ter uma condição de vida melhor, tiveram garra e determinação e foram capazes de darem uma boa educação para meu irmão e para mim, mesmo quando nossas situações financeiras eram precárias. Toda a luta dos meus pais em garantir o básico para que eu estudasse e ainda fosse uma pessoa honrada é um dos exemplos que levo sempre comigo e me faz seguir em frente. Somado a eles, acrescento minha avó que, como lavradora, nos ensinou o valor de se conquistar as coisas. Também sou grato as greves dos professores durante meu ensino médio; através delas, tive a oportunidade de frequentar as bibliotecas municipais e desenvolver meu interesse pela Física. E claro, aos bons professores do meu ensino médio, que viram em mim interesse e me presentearam com meu primeiro livro de Física. Mas, definitivamente, foi na Universidade Federal do Maranhão que decidi meu rumo. E, neste caso, conheci dois professores que traçaram meu destino, os professores Drs. Manoel Messias e Rodolfo Casana. Se hoje sou capaz de aprender alguma coisa em Física, eu devo isso a eles, pois eles transformaram aquele estudante inexperiente em um Físico. Estes professores, com suas visões apuradas em identificar alunos interessados, se preocupavam em me ajudar e me ajudaram em diversas situações. Sou muito grato a eles por todas as oportunidades e, sem dúvida, essa tese é reflexo do bom trabalho deles. Nesta parte, eu aproveito para agradecer aos diversos amigos que contribuíram direta ou indire- tamente para o meu doutorado, entre eles posso citar os amigos do IFT: Renato, Pablo, Ernany, Guilherme, Fernando, Paulo, Natália, Ana, Henrique, Patrice, Stefano, George, Daniel e tantos outros. Também menciono aqueles fora do IFT como o Wagner e o Taciano, que dividiram todo i esse tempo residência comigo e se tornaram bons amigos. Ainda nesse estágio final, conheci uma pessoa muito importante para a mim, Rosane Rocha, que com todo seu carinho tornou esses últimos meses menos penosos e mais divertidos. Também agradeço enormemente ao Professor Dr. Jorge Noronha da USP que me apresentou, pela primeira vez, a correspondência holográfica e esteve presente em muitas discussões durante esses anos. Sua gentileza em compartilhar seus conhecimentos somaram muito para atingir a formação que tenho hoje. Outro professor importante na minha formação foi o Professor Dr. Nick Evans da University of Southampton (UK) que, sem hesitar, aceitou meu convite para um doutorado sanduíche e me deu a oportunidade de expandir meus conhecimentos na área e, como resultado, ele é parte desta tese. Por fim, e mais importante, quero agradecer a dois personagens que, de fato, fizeram essa tese realmente acontecer, o Professor Dr. Gastão Krein e o Dr. Alfonso Bayona. Tenho enorme agradecimento ao Prof. Gastão por ter me aceitado como seu aluno de doutorado e confiado em trabalharmos neste tema. Além de ter compartilhado comigo seus conhecimentos em QCD e física hadrônica, ele confiou e me deixou livre para colaborar com outros grupos de pesquisa. Se hoje sei fazer alguma pesquisa em Física foi, sem dúvida, graças as suas dicas e referências, que aprecio muito. E tudo isso foi complementado pelo Dr. Alfonso Bayona, que veio ao IFT no momento certo e desde então atuou como um excelente co-orientador para mim. Ele teve a paciência de me ensinar detalhamente o que é holografia, deu diversas dicas de como fazer pesquisa, me apresentou a comunidade científica da área e sempre que precisei dele, ele esteve disposto em trabalhar e ensinar coisas. Por isso, serei eternamente grato a eles. Este trabalho contou com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). ii Esta tese é resultado de três trabalhos científicos publicados na Physical Review D: • Strong couplings and form factors of charmed mesons in holographic QCD - Phys.Rev. D96 (2017) no.1, 014017 • Inverse Magnetic Catalysis in Bottom-Up Holographic QCD - Phys.Rev. D94 (2016) no.7, 074034 • Decay constants of the pion and its excitations in holographic QCD - Phys.Rev. D91 (2015) 065024 iii Resumo Existem poucas dúvidas de que a QCD seja a teoria correta das interações fortes. As dificuldades em resolver a teoria em baixas energias no regime fortemente acoplado e não perturbativo tem deixado sem respostas muitas questões importantes, tais como a natureza do confinamento e o mecanismo de hadronização. Diversos métodos têm sido usados para estudar suas propriedades e consequências a baixas energias. Esses métodos incluem a QCD na rede, as equações de Dyson- Schwinger, a teoria de perturbação chiral e os modelos de quarks. Recentemente, a dualidade gauge/gravidade tem fornecido uma nova maneira de acessar o regime fortemente acoplado de uma teoria de calibre via uma teoria de gravidade dual, em especial da QCD através de modelos holográficos. Tais modelos são usualmente denominados modelos holográficos para a QCD, ou apenas modelos AdS/QCD. Nesta tese investigamos importantes problemas de interesse atual em física hadrônica envolvendo as quebras das simetrias chiral e de sabor usando modelos holográficos para a QCD. Estes problemas são: (1) o desaparecimento das constantes de decaimento leptônicas dos estados excitados do pion no limite quiral; (2) os efeitos da quebra de simetria de sabor no acoplamentos do méson ρ aos mésons charmososD andD∗ e seus fatores de forma eletromagnéticos; (3) os efeitos de um campo magnético e da temperatura sobre o condensado quiral, sinalizando uma catálise magnética inversa. Palavras Chaves: Cromodinâmica Quântica, Quarks e Glúons, Simetria Quiral, Simetria de Sabor, Catálise Magnética Inversa, Dualidade Calibre/Gravidade, Holografia Áreas do conhecimento: Teoria de Partículas e Campos, Física Hadrônica, Física Nuclear. iv Abstract There is little doubt that QCD is the correct theory for the strong interactions. The difficulties in solving the theory at low energies in the strongly interacting, non-perturbative regime have left unanswered many important questions, such as the nature of confinement and the mechanism of hadronization. Several approaches have been used to study its properties and consequences at low energies. These include lattice QCD, Dyson-Schwinger equations, chiral perturbation theory and quark models. More recently, the gauge/gravity duality has provided a new way to access the strongly coupled regime of a gauge theory via a dual gravity theory, in special of QCD through holographic models. Such models are usually named as holographic QCD models, or just AdS/QCD models. In this thesis, we investigate three problems of contemporary interest in hadronic physics involving the chiral and flavor symmetries holographic QCD models. These problems are: (1) the vanishing of the leptonic decay constants of the excited states of the pion in the chiral limit; (2) the effects of the flavor symmetry breaking on the strong couplings of the ρ meson to the charmed D and D∗ mesons and the their electromagnetic form factors; (3) the effects of a magnetic field and temperature on the chiral condensate, signalizing inverse magnetic catalysis. Keywords: Quantum Chromodynamics, Quarks and Gluons, Chiral Symmetry, Flavor Symmetry, Inverse Magnetic Catalysis, Gauge/Gravity duality, Holography Research areas: Theory of Particles and Fields, Hadron Physics, Nuclear Physics. v Contents 1 Introduction 1 2 Introduction to QCD 5 2.1 QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Chiral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1.1 Pion leptonic decay constant, PCAC and Gell-Mann-Oakes-Renner relation (GOR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2.1 Wilson loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Gauge/Gravity duality 23 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 The Anti-de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Matching the degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 The holographic dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Massive scalar field in AdS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 One-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.2 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Finite temperature in Gauge/Gravity duality . . . . . . . . . . . . . . . . . . . . . . 37 3.6 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6.1 Adding Flavors in AdS/CFT correspondence . . . . . . . . . . . . . . . . . . 45 4 The Holographic QCD model 49 4.1 Bottom-Up approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Holographic description of quark-antiquark potential . . . . . . . . . . . . . . . . . . 52 i Contents 4.3 DCSB implementation in holographic QCD . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Vacuum Structure of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.2 The Flutuations around the vacuum . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Exploring the Action up to second order . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.1 Field equations and Holographic currents . . . . . . . . . . . . . . . . . . . . 63 4.4.2 The 4d effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.3 Leptonic decay constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Exploring the Action up to third order . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Decay constants of the pion and its excitations in Holographic QCD 73 5.1 Pions and their excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.1 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Extended PCAC and Generalized GOR . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Fixing the free parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 Strong couplings and form factors of charmed mesons in Holographic QCD 89 6.1 Mesonic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Strong Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.3 Electromagnetic Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.1 Low and high Q2 behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.4 Results and comparison with experimental and lattice QCD . . . . . . . . . . . . . . 102 7 Inverse Magnetic Catalysis in Dynamic Holographic QCD 113 7.1 D3/D7 Brane Configuration - Review . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 The Dynamic Holographic QCD model . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3 Dynamic Holographic QCD at finite temperature . . . . . . . . . . . . . . . . . . . . 123 7.4 Magnetic Field in Dynamic Holographic QCD . . . . . . . . . . . . . . . . . . . . . . 127 8 Conclusions 135 9 Appendix 139 9.1 Some algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.2 Kaluza-Klein Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 ii Contents Bibliography 143 iii 1 Introduction Quantum Chromodynamics (QCD) is at present universally accepted as the theory of the strong interaction. The strong interaction is responsible for the bulk of the visible matter in the universe. This matter is contained in atomic nuclei. The fundamental degrees of freedom in the theory are the quarks and gluons that carry color charges. But quarks and gluons are not observed in isolation, they are permanently confined in the interior of the hadrons. Hadrons are strongly interacting subatomic particles; the most well known hadrons are the proton, neutron, and the pion. The existing hadrons are classified into baryons and mesons. Baryons have half-integer spin, and mesons have integer spin. Up today, six types (or flavors) of quarks are known: the up (u) and down (d) quarks, which exist in ordinary matter which is composed of protons and neutrons, and the strange (s), charm (c), bottom (b), and top (t) quarks, which are produced in high energy particle collisions, they are much heavier than the u and d quarks and decay quickly. At high energies, QCD is very well understood, it correctly describes strongly interacting matter under the extreme conditions reached in high energy particle collisions, where quarks and gluons are probed at short relative distances. Our knowledge of the theory under such conditions is due to a property known as asymptotic freedom [1, 2]. This property asserts that the quark-gluon interaction becomes weak at high energies, or at small relative distances. It allows the use of systematic and well-controlled weak-coupling expansions developed in quantum field theory. However, not much is known on how precisely quarks and gluons become permanently confined in the interior of hadrons. This feature is known as the confinement of the color charge, which means that the interaction becomes stronger as quarks and gluons try to separate. Here, new theoretical methods that go beyond weak-coupling expansions are required. At this energy regime, another prominent QCD phenomenon is present: dynamical chiral symmetry breaking (DCSB), which is responsible for more than 98% of the visible mass in the universe. The Higgs particle gives masses to the quarks, but these masses alone explain less than 5% of the masses of the protons and neutrons. The rest of the masses is explained as coming from the strong quark-gluon interactions that lead to dynamical chiral symmetry breaking. One interesting new development in the last decades has been the discovery of the AdS/CFT 1 1 Introduction correspondence, or gauge/gravity duality [3, 4, 5] (see Ref. [6] for a review), which relates quantum gauge field theories in d dimensions to theories of quantum gravity in (d+ 1) or higher dimensions. In particular, in the limit that the QFT is strongly coupled with a large number of gauge degrees of freedom, the corresponding quantum gravity theory is a classical theory of gravity. Thus, problems in strongly coupled QFT’s can be mapped to problems in higher dimensional classical theories of gravity which may pose an easier challenge. Models based on gauge/gravity duality have been applied to investigate nonperturbative low-energy observables of phenomenological interest in QCD, especially in hadron physics. In particular, interesting holographic models incorporating features of DCSB and its explicit breaking by quark masses have been introduced to study pion [7, 8] and kaon [9] physics, including their electromagnetic form factor. The spectrum of scalar glueballs have been studied in this new approach utilizing a different class of models [10]. Light baryons have also been studied in the gauge/gravity duality through their identification with Skyrmions in the AdS space [11]. Finally, it is worth mentioning that applications of the gauge/gravity have been employed to study problems in condensed matter physics [12]. Thus, the exploration and integration of different areas of physics has been a especially attractive feature of the gauge/gravity duality. In this thesis we study three problems of current interest in hadron physics using the framework based on the gauge/gravity duality. The first problem is related to the phenomenon of DCSB. QCD predicts that the leptonic decay constants of the excited states of the pion vanish in the chiral limit when chiral symmetry is dynamically broken [13]. Although chiral symmetry is not an exact symmetry in the real world, as the masses of the u and d quarks are not zero, their masses are much smaller than the typical strong interaction scale (ΛQCD, to be defined in the coming chapter) so that it is natural to expect that the leptonic decay constants of the excited states of the pion are very much suppressed in nature. The essential point behind the suppression of the decay constants, as we shall discuss in this thesis, is the dynamical breaking of chiral symmetry in QCD and the (pseudo) Goldstone boson nature of the ground-state pion. The second problem we tackle in this thesis is related to the consequences of the breaking of flavor symmetry due to the quark masses on strong hadron coupling constants. This issue is of great contemporary interest in connection with the interactions of the heavy-light charmed D mesons with light hadrons and atomic nuclei [14, 15, 16]. Of particular interest in this thesis are the effects of SU(4) flavor symmetry breaking on effective meson couplings due to the different values of the current quark masses in the QCD Lagrangian. This issue has been discussed in the context of different approaches and models, which include the constituent quark model [17, 18], QCD sum rules [19], Dyson-Schwinger equations [20, 2 21], and lattice QCD [22]. Finally, we study of QCD in the presence of an external magnetic field. It has recently attracted much attention because of its phenomenological relevance and of many interesting theoretical features as a possibility of new phases in the QCD phase diagram [23] and more recently its influence on the dynamics of the chiral condensate [24], known as the inverse magnetic catalysis (IMC). The thesis is organized as follows. In chapter 2 we briefly review some important aspects of QCD. We discuss its origins, passing by the phenomenology of hadrons that led to the establishment of the quark model. We discuss its similarities and differences with quantum electrodinamics (QED). Furthermore, we present the QCD Lagrangian, discuss its symmetries and present the relevance of its intrinsic scale, ΛQCD. We also discuss in more detail the chiral physics from the QCD Lagrangian and how it affects the hadronic spectrum. Moreover, we discuss the phenomenon of quark-gluon confinement phenomenon and introduce the Wilson loop, a quantity relevant for discussing color confinement in the context of pure-gauge theories. Chapter 3 contains the general holographic framework used in the thesis. We present arguments to motivate the duality by a geometrization of the renormalization group, in which the AdS space emerges naturally. The holographic dictionary is explained, which is very important to make connection with the QFT of interest. The connection with physical observables in the QFT, done via correlation functions, is described in this chapter, as well as the holographic prescription to evaluate such correlation functions. To illustrate this prescription, we will apply it to a scalar field theory. The inclusion of the thermal effects in the duality is also discussed. We conclude the chapter discussing briefly the real origin of this duality, through of the Dp brane dynamics, which makes clear the limit of applicability of the duality that is used in the present context. We also discuss briefly how to include flavor degrees of freedom in the duality. In chapter 4, we start discussing briefly the two main approaches commonly used to study QCD via the gauge/gravity duality: the top/down and bottom-up approaches. Then, we present the background which satisfies a confinement criterium, the hard wall background. We introduce the bottom-up holographic QCD model used in this thesis and show how the model incorporates DCSB. By exploring the action of the model up to second order, we show how to evaluate vacuum expectation values of operators of interest, to obtain a 4d effective action for mesons and their leptonic decay constants. Finally, we end the chapter exploring the action up to third order to obtain the couplings for 2 pseudoscalar and vector mesons and for the 3 vector mesons. Chapters 5 and 6 contain our results using the bottom-up holographic QCD model to study, 3 1 Introduction respectively, the vanishing of the leptonic decay constants for the excited states of the pion and the effects of SU(4) flavor symmetry breaking on three-meson couplings. In chapter 5, we start identifying the pion in the model and obtain the pionic excitation spectrum. We show explicitly that the model reproduces the partially conserved axial current (PCAC) and Gell-Mann-Oakes- Renner (GOR) relationship. In the numerical part, we show that our results are in complete agreement with the QCD prediction. In chapter 6, we extend the bottom-up holographic QCD model to include four flavors. The mesons in the SU(4) flavor representation are identified and we explore the action up to third order to evaluate the strong coupling constants. Also, the elec- tromagnetic (EM) form factors are defined using a generalized vector meson dominance (GVMD) freamework. We present how the SU(4) flavor symmetry breaking numerically impacts the values of three-meson couplings. We also present results for the EM form factors for the pseudoscalar and vector mesons, comparing them with experimental and lattice results. In chapter 7 we apply the gauge/gravity duality at finite temperature and in presence of an external magnetic field. We introduce a different model, the dynamic holographic QCD model, which is also a bottom-up approach. We start reviewing the D3/D7 brane configuration and show how the model generates a chiral condensate dynamically. We incorporate the effects of temperature in the model to reproduce a second order chiral phase transition. An external magnetic field is included in the model. We show that the model gives inverse magnetic catalysis (IMC), in that the critical temperature for chiral restoration decreases with magnetic field. We also show that the chiral condensate presents a nonmonotonic behavior. Both effects are similar to what is observed the lattice QCD simulations. Finally, our conclusions are presented in chapter 8. 4 2 Introduction to QCD For a long time, it has been assumed that some observed elementary particles are not elementary at all. For example, in Ref [25], it was suggested that the pion is composed by a nucleon and an antinucleon. The discovery of kaons and hyperons in the 1950s, led to models in which such particles were taken as fundamental objects and others as composite objects. A good description for this scenario was achieved by the model proposed by Sakata [26], in which the proton, the neutron and the Λ-hyperon were taken as fundamental particles. In this model, the states p, n,Λ served as basis to build the scheme of unitary symmetry SU(3) and conducted to the classification of the pseudoscalar and vector mesons, although it had some problems in the description of baryons. Some years later, another model appeared, the Eightfold Way, proposed independently by Gell-Mann [27] and Neeman [28]. It is based in the classification of hadrons via an SU(3)-group and the great advantage of that model is that the mesons and the baryons could be described simultaneously by the model. A great success of the model was the experimental verification of its prediction of the Ω− hyperon, confirming the validity of this symmetry. Initially, the theoretical ideas on the internal structure for the hadrons arose in the papers of Gell-Mann [29] and Zweig [30]. They showed that the SU(3) octet symmetry can be understood on the basis of a fundamental triplet of some “hypothetical” particles, named by Gell-Mann as quarks 1 that carry fractional electric charges. Such particles had specific quantum numbers to be consistent with the hadrons known at the time; they are presented in Table 2.1. As said, initially the quarks were considered as mathematical objects in the understanding of hadrons, but years later it became clear that, in fact, hadrons should be considered as bound systems of quarks. In this picture, the mesons would consist of a quark-antiquark pair and the baryons known at that time would consist of three quarks M = q̄q, B = qqq. (2.1) 1In that time, Gell-Mann was uncertain on the term he intended to coin, until he found the word quark in James Joyce’s book Finnegans Wake 5 2 Introduction to QCD flavour charge isospin strangeness baryon charge u 2/3 I = 1/2, I3 = 1/2 0 1/3 d −1/3 I = 1/2, I3 = −1/2 0 1/3 s −1/3 I = 0 −1 1/3 Table 2.1: Quantum numbers of the quarks in the Eightfold Way Despite the great success of the quark model, countless attempts to find these particles (quarks) as free and with such fractional electric charges failed, see [31]. This impossibility in which quarks cannot be seen as experimentally free particles is the hypothesis of quark confinement. Differently to hadrons, quarks have a new quantum number, the “color” [32, 33], and it was established that just colourless particles could be observed as free particles. In 1969, another great step was given in the understanding of the quark structure of hadrons: the introduction of the parton hypothesis [34]. In the deep inelastic scattering (DIS) experiments, an energetic lepton interacts with the target hadron via exchange of a virtual photon and, by analyzing the cross section, the parton structure of hadrons can be observed. The experimental observations also revealed that interacting partons are particles that carry the same quantum numbers of quarks predicted by Gell-Mann’s model. In particular, the proton would consist of three valence quarks. In addition, in the two experiments H1 and ZEUS at the HERA laboratory, detailed investigations in the momentum distributions of the proton revealed that quark-partons carry about half of the momentum of the fast proton and the remaining half of this momentum would be carried by a new neutral particle inside the proton, they are now known to be the gluons. The HERMES experiment at HERA laboratory has conducted extensive and detailed studies of the gluon contribution to the proton spin. It is important to point out that the gluons were theoretically predicted in the mid-1970s. Their properties can be also investigated through other processes as: Drell-Yan pair (µ+µ−) production in hadron collisions, hadron production with large pT , hard hadron jets and, in particular, e+e− annihilation. Nowadays it is well understood that the quarks of the Eightfold Way (u, d and s) are not the only existing quarks. In fact, from experimental verification, three new species of quarks or flavors, were discovered: the c-quark, the b-quark and the t-quark. The masses of the Eightfold Way quarks are considerably smaller than of those new quarks. Therefore, when dealing with processes with small momentum transfers (soft processes) involving light hadrons we can focus our attention on 6 the light quarks which realize the lowest representation of the SU(3)flavor group. The current understanding of hadrons as bound states of quarks is based on quantum chro- modynamics (QCD), the underlying theory of strong interactions, which is a non-Abelian gauge theory [35]. Quantum electrodynamics (QED) is characterized by the presence of electrically char- ged fermions (electrons) and neutral photons and has as basic processes the emission and absorption of photons by electrons, as illustrated in Fig. 2.1a. In QCD, on the other hand, each quark has a color charge and, from experimental observation, quarks appear in three color charges, each quark can transform into another by emitting or absorbing one of the 8 possible colored gluons. As the gluons carry color charge, in addition to quarks emmiting/absorbing gluons as in Fig. (2.1b), gluon-gluon interactions like in Fig. (2.1c) and Fig. (2.1d). The latter processes make the theory more complicated compared to QED. Figure 2.1: QED interaction vertex (a): an electron emits/absorbs a photon; QCD interaction vertices: gluon emission/absorbsion by a quark (b) or by a gluon (c) and gluon-gluon interactions (d) One of the great differences between QED and QCD comes from interactions showed in (c) and (d) which affect the behavior of the effective charge. In fact, at low energies, the QCD effective charge increases significantly and nonperturbative physics takes place. On the other hand, at high energies, the effective charge decreases significantly, signalizing that the theory is asymptotically free [1, 2]. This makes possible the use of perturbation theory. Perturbative QCD (pQCD) is a very powerful theoretical framework that allows to study QCD analytically and generates all the results obtained in the parton model and, in addition, predicts possible deviations from it. For a detailed analysis about the pQCD calculations and their comparisons with results from experimental data, the reader is directed to Refs. [36, 37, 38]. In soft processes, when there is growth of the effective charge, perturbative methods, based on an expansion in the coupling strength, cannot be applied and a study from first principles employing 7 2 Introduction to QCD analytical methods based on quark and gluon degrees of freedom becomes extremely difficult. But, despite of this, some progress has been made using the Dyson-Schwinger equations of QCD [39]. The theoretical difficulties in the study of hadron physics from first principles also motivated the construction of effective models such as chiral effective theories [40, 41], heavy quark effective theories (HQET) [42, 43], non-relativistic QCD (NRQCD) [44], and soft-collinear effective theory (SCET) [45]. A powerful first-principles numerical method to study QCD is the formulation of the theory on lattice (Lattice QCD). Lattice QCD has been used for computations of static or low energy quantities involving a few hadrons, like masses or near threshold form factors. It is also a powerful technique for examining the thermal behaviour of QCD. More recently, a different class of models, based on the gauge/gravity duality [46, 47, 48, 49], has raised great interest in the exploration of nonperturbative aspects of QCD. We will discuss this at length in the next sections. In the nonperturbative regime of QCD, its properties are dominated by two emergent phenomena: confinement, namely, the theory’s elementary degrees of freedom (quarks and gluons) cannot be isolated, and dynamical chiral symmetry breaking (DCSB), which is a remarkably effective mass generating mechanism, responsible for the mass of more than 98% of visible matter in the Universe. We will discuss these phenomena shortly ahead. 2.1 QCD Lagrangian The combination of the successes of the quark and parton models motivated the adoption of Yang Mills theory with the color symmetry elevated to a SU(3)c local symmetry group. The QCD Lagrangian is written as L = ∑ flavors ψ̄fi (iγµDµ −M)ijψfj − 1 4F µν,aF aµν . (2.2) Before describing the components of this Lagrangian, it is important to explain the notation em- ployed • The greek indices (µ, ν, . . .) are Lorentz indices. • f represents the flavor index. • i, j, k in the quark fields ψ are colors indices in the fundamental representation (i, j, k = 1, 2, 3). • a, b, c in the gluon fields Aaµ are colors indices in the adjoint representation (a, b, c = 1, 2, ..., 8). 8 2.1 QCD Lagrangian In the Lagrangian (2.2), M is quark mass matrix and the covariant derivative is defined as (Dµ)ij = ∂µδij + ig(ta)ijAaµ. (2.3) Here, g is the coupling constant, ta are the generators of SU(3) and they define the so-called structure constants through of the commutation relations [ ta, tb ] = ifabctc. (2.4) It is possible to demonstrate that these structure constants satisfy the following Jacobi identity fabefecd + fcbefaed + fdbeface = 0. (2.5) In the fundamental representation, these matrices are Gell-mann matrices, given by t1 = 1 2  0 1 0 1 0 0 0 0 0  , t2 = 1 2  0 −i 0 i 0 0 0 0 0  , t3 = 1 2  1 0 0 0 −1 0 0 0 0  , (2.6) (2.7) t4 = 1 2  0 0 1 0 0 0 1 0 0  , t5 = 1 2  0 0 −i 0 0 0 i 0 0  , t6 = 1 2  0 0 0 0 0 1 0 1 0  , (2.8) (2.9) t7 = 1 2  0 0 0 0 0 −i 0 i 0  , t8 = 1√ 12  1 0 0 0 1 0 0 0 −2  , (2.10) (2.11) and are normalized as Tr ( tatb ) = 1 2δ ab. (2.12) The covariant derivative in Eq. (2.3), implies in the quark-gluon interaction term Lint = −gψ̄iAµa(ta)ijγµψj . (2.13) Therefore, quarks interact with gluons in a way similar to electrons interacting with photons. A new feature here is that the quark can change its color charge from i to j by emitting or absorbing a gluon, coupling through an SU(3)c generator (ta)ij , as shown in Fig. 2.2. 9 2 Introduction to QCD Figure 2.2: The color of a quark can change from i to j by a gluon of color a, coupled through the SU(3) generator (ta)ij . In the term that refers to gluon dynamics (gauge term) in (2.2), the tensor strength F aµν is given by F aµν = ∂µA a ν − ∂νAaµ − gfabcAbµAcν . (2.14) Note that, apart from the standard partial derivative terms in FµνaF aµν , there are non-linear terms (AbµAcν)2 and ∂µA a νA µbAνc. Therefore, the eight gluons do not come in as a simple repetition of the photon in QED theory, they have three and four-gluon self-interactions. We depicted such interaction terms schematically in Fig. 2.1. As a gauge invariant theory, the Lagrangian (2.2) is invariant under local symmetry transforma- tions ψ → U(x)ψ, (2.15) Aaµt a → U(x)AaµtaU †(x) + ig−1(∂µU(x))U †(x) (2.16) where U(x) is an element of SU(3)c . It is well known that a renormalization procedure shows that the strong coupling in QCD, in fact, runs with the energy or momentum involved in a process. Namely, for large values of a typical momentum transfer Q in, e.g. a scattering process, one has that αs(Q2) = 1 b0 ln(Q2/Λ2 QCD) , b0 = 1 6π (11Nc − 2Nf ), (2.17) at the one-loop level. This procedure introduces in the theory a dimensionful parameter ΛQCD, which represents the scale at which the coupling constant becomes large and the physics becomes nonperturbative. Therefore, ΛQCD sets the scale for strong interaction physics. In a particular renormalization scheme (MS-sheme) [50], it is found that ΛQCD ∼ 250 MeV, for three flavors. This 10 2.1 QCD Lagrangian scale is widely responsible to sets the mass scale for the proton and neutron masses, and hence, the mass scale of the baryonic mass in the Universe. In the same way as the coupling constant runs, so do the quark masses, i.e. they depend on the renormalization scale. Because quarks are confined, there is no true quark mass pole, as in QED, for example. A quark mass pole is well defined only in the context of perturbation theory. When one quotes a value of a quark mass, this value refers to a particular renormalization scheme. As we have mentioned before, the nonperturbative QCD physics is dominated by chiral dynamics and confinement. Both topics will be discussed in the next sections. 2.1.1 Chiral Dynamics In the above paragraph, we have briefly discussed the strong interaction scale ΛQCD through the running of the QCD coupling. Now we can clarify the notion of light and heavy quarks. Briefly speaking, light quarks are the ones with masses much smaller than ΛQCD, and heavy quarks with masses much larger than ΛQCD. Clearly, from Table 2.2, the up and down quarks are qualified as light quarks, whereas the charm, bottom, and top as heavy quarks. The strange quark is more subtle, it appears neither light nor heavy. In some cases, it can be regarded as light, in others, as heavy. Quark flavor up down strange charm bottom top Masses 1.5-4 MeV 4-8 MeV 100 MeV 1.25 GeV 4.25 GeV 175 GeV Table 2.2: Quark masses in the MS renormalization scheme at a scale of µ = 2GeV . For our purposes in this section, we focus on the light quarks, which are the most relevant to the real world. To understand the physics related with such quarks, it is convenient to consider a theoretical limit in which their masses are exactly zero, known as chiral limit. This limit is a good approximation as a starting point to discuss the physics of the real world. In the chiral limit, the spin of the quark can either be in the direction of motion, which we call a right-handed quark and denote its field by ψR, or in the opposite direction of the motion, we call it a left-handed quark, with the corresponding field denoted by ψL. In this way, each flavor can be arranged as ψ = ψL ψR  , ψR,L = 1± γ5 2 ψ. (2.18) 11 2 Introduction to QCD The QCD Lagrangian in Eq. (2.2) can, therefore, be written as L = ∑ flavors ψ̄fL,i(iγ µDµ)ijψfL,j + ψ̄fR,i(iγ µDµ)ijψfR,j − 1 4F µν,aF aµν , (2.19) and, as consequence, it presents the following symmetry ψL → ψ ′ L = ULψL, ψR → ψ ′ R = URψR, (2.20) where UL,R are Nf×Nf unitary matrices 2. This symmetry is commonly known as Chiral Symmetry and the group of chiral transformations is denoted by U(Nf )L × U(Nf )R. Since U(Nf ) = U(1)×SU(Nf ), we have two U(1) symmetries, one for the left part UL(1) = eiαL and another for the right part UR(1) = eiαR . From now on, we focus on the two SU(Nf ) symmetries, the U(1) parts, one related to baryon number and the other to axial symmetry (which is anomalous), will not be discussed here. According to Noether’s theorem, the SU(Nf )L × SU(Nf )R chiral symmetry leads to the the following conserved currents JaLµ = ψ̄Lγµt aψL, JaRµ = ψ̄Rγµt aψR. (2.21) For a better physical interpretation, we construct vector and axial vector currents from the linear combinations Ja,µV = Ja,µR + Ja,µL = ψ̄γµtaψ, Ja,µA = Ja,µR − Ja,µL = ψ̄γµγ5t aψ. (2.22) We can find the charges Qa and Qa5 associated with these currents by integrating the temporal part of the above currents. It is straightforward to check that the charges obey the following algebra [Qa, Qb] = ifabcQc; [Qa5, Qb] = ifabcQc5; [Qa5, Qb5] = ifabcQc. (2.23) Note that, the above relations show that the operatorsQa form a subgroup of the chiral symmetry group, it is known as flavor group. This can be better understood separating the vector and axial parts of SU(Nf )L × SU(Nf )R group. Thus, using (2.22), we have the following vector and axial transformations ψ → ψ ′ = UV ψ, ψ → ψ ′ = UAψ, (2.24) where UV,A are unitary matrices of the chiral group denoted by SU(Nf )V ×SU(Nf )A. In this form, the SU(Nf )V refers to the vector part and flavor group and SU(Nf )A refers to the axial part 3 2 In the present case, we are considering the chiral limit of Nf flavors in order to obtain a general description. 3The SU(Nf )A is not a real group, because two axial transformations do not lead to another axial transformation. 12 2.1 QCD Lagrangian For Nf = 2, the flavor group is known as isospin group and states are labeled by the total isospin I and its third component I3. A classical example of the realization of the isospin group is the proximity of the proton and neutron masses, which have I = 1/2 and can be represented as p = ∣∣∣I = 1 2 , I3 = 1 2〉, n = ∣∣∣I = 1 2 , I3 = −1 2〉. (2.25) As an example of a multiplet with I = 1 we have the three pions π+ = ∣∣∣I = 1, I3 = 1〉, π0 = ∣∣∣I = 1, I3 = 0〉, π− = ∣∣∣I = 1, I3 = −1〉, (2.26) all having nearly identical masses. This shows that the isospin is a very good approximate symmetry in the light hadron spectrum. However, the hadron spectrum does not presents the complete SU(2)V × SU(2)A symmetry. This can be seen as follows: let |h,+〉 be a hadron eigenstate of the QCD massless Hamiltonian H0 QCD with eigenvalue Mh, H0 QCD|h,+〉 = Mh|h,+〉. (2.27) In the Hilbert space, the Qa and Qa5 charges act as generators of the transformation; given as UV = eiα a V Q a , UA = eiα a AQ 5a . (2.28) The SU(2)V × SU(2)A symmetry implies that [H0 QCD, Q a] = [H0 QCD, Q 5a] = 0. (2.29) Acting with the UA operator on the left of Eq. (2.27) and using the above commutation relation, we obtain in the same way: H0 QCDUA|h,+〉 = MhUA|h,+〉. (2.30) As the axial transformation changes the parity of the state, we obtain H0 QCD|h,−〉 = Mh|h,−〉 with |h,−〉 = UA|h,+〉. (2.31) This fact would imply in the existence of degenerate parity doublets in the hadron spectrum. For example, the spectra of vector (JP = 1−) and axial-vector (JP = 1+) mesonic excitations should be identical. But, this is not realized in the hadron spectrum, signalizing that chiral symmetry is broken. This can be understood using |h,+〉 = P̂+ h |0〉, where P̂ + h is the operator that creates a hadron h with positive parity. Then, we have (for an infinitesimal) UA|h,+〉 = UAP̂ + h U † AUA|0〉 = P̂−h e iαaAQ 5a |0〉 ' |h,−〉+ iαaAP̂ − h Q 5a|0〉+ . . . (2.32) 13 2 Introduction to QCD Then, if the vacuum is not annihilated by Q5a, the previous arguments no longer apply and the particles with different parities will have different masses, as in the case of hadron spectrum. This shows that chiral symmetry is dynamically broken by the QCD vacuum. The dynamical chiral symmetry breaking means that the QCD vacuum contains a population of quark anti-quark pairs, which are represented by the nonzero values of vacuum expectation values 〈0|ψ̄ψ|0〉 = 〈0|ψ̄uψu|0〉 = 〈0|ψ̄dψd|0〉 = 〈0|ψ̄sψs|0〉. (2.33) This operator has a precise definition in terms of the full quark propagator, as 〈0|ψ̄(x)ψ(x)|0〉 = −iTr lim y→x+ SF (x− y), SF (x− y) = −i〈0|T ψ(x)ψ̄(y)|0〉. (2.34) This is known as the chiral condensate, or quark condensate. It is often used as an order parameter for the dynamical chiral symmetry breaking. In summary, one has that the vacuum expectation value 〈ψ̄ψ〉 is nonzero, i.e. there are quark condensates in the QCD vacuum. Thus, in QCD, one has that Qa|0〉 = 0, Qa5|0〉 6= 0. (2.35) However, according to Goldstone’s theorem, this leads to the existence of three massless spin-0 pseudoscalar bosons. They are pseudoscalars because Qa5 has odd parity. Clearly, in the real world there are no such massless pseudoscalars. But, there are pions. The pion masses are indeed much smaller than a typical hadron mass, about 140 MeV. The pions are called pseudo-Goldstone bosons because the chiral symmetry is not exact. It is slightly broken by the finite up and down quark masses, which appear in the symmetry breaking term in the Lagrangian (2.2) LM = −ψ̄Mψ = −muūu−mdd̄d. (2.36) If we consider three flavors, the u, d, s quarks, we have now eight axial charges, hence, there should be eight massless spin-0 pseudoscalar bosons. But, as before, because of the symmetry breaking due to quark masses, we have the pseudo-Goldstone bosons: pions (π±, π0), kaons (K±,K0, K̄0) and the η meson. 2.1.1.1 Pion leptonic decay constant, PCAC and Gell-Mann-Oakes-Renner relation (GOR) Let us first consider the weak decay of the pion, illustrated in Fig. 2.3. This process is well described using Fermi’s theory, where the weak interaction Hamiltonian involves a current-current 14 2.1 QCD Lagrangian part, where the currents are a sum of axial and vector components Hwk = iGwk√ 2 (JµV+ + JµA+) ∑ l l̄γµ(1 + γ5)νl + h.c, (2.37) where l runs over the fields of the three charged leptons e, µ and τ ; νl runs over the fields of the associated neutrinos and JµV± and JµA± are the charge changing currents JµV± = JµV1 ± JµV2 , JµA± = JµA1 ± JµA2 . (2.38) A good review on this subject is found in Ref. [51]. As discussed in this reference, the constant Figure 2.3: Weak decay charged pion. Gwk can be determined using beta-decay transitions, like π → π0 +e+νe. Due to parity, the matrix element 〈0|Ja,µA |π〉 controls completely the weak decay of the pion. Also, because of the vector nature of the current, this matrix element is proportional to the pion momentum. Using the fact that the pion is spinless, we have then 〈0|Ja,µA (x)|πb(p)〉 = ifπpµδ abe−ip·x, (2.39) and the proportionality constant, fπ, is the leptonic decay constant. Experimentally, its value is fπ = 93 MeV. Now, let us take the divergence of Eq. (2.39): 〈0|∂µJa,µA (x)|πb(p)〉 = fπm 2 πδ abe−ip·x, (2.40) with q2 = qµqµ = m2 π. The above equation gives an interesting piece of insight: the axial current could be carried by a pion field: Ja,µA (x) = fπ∂ µΦa(x), (2.41) that means the divergence of the axial-vector current is related with the pion field Φ(x) (up to a constant). 15 2 Introduction to QCD This can be better understood by looking at the expression of the axial-vector current in Eq. (2.22). In fact, taking the divergence of this current we find ∂µJ a,µ A = iψ̄{M, ta}γ5ψ. (2.42) Even if all quark masses are equal, there remains a non-zero contribution proportional to the quark mass ∂µJ a,µ A = 2imψ̄taγ5ψ = 2mP a, (2.43) where P a is a pseudoscalar density corresponding to the pseudoscalar mesons. This is the famous PCAC (partially conserved axialvector current) relation: the divergence of the axial-vector current is a pseudoscalar current. This equation will become useful later. Next, if we consider the equal time commutation relation [QAa , P b] = −1/2iδabψ̄ψ, we have for the pion case (a = b = 1) [QA1 , P 1] = − i2(ψ̄uψu + ψ̄dψd), (2.44) for which the vacuum expectation value suggests that QA1 |0〉 6= 0, consistent with nonzero values for the quark condensate, 〈ψ̄ψ〉 6= 0. This strongly confirms the Y. Nambu’s suggestion that the vacuum is not invariant under chiral transformation. Moreover, using the PCAC relation in Eq. (2.43), we have [QA1 , ∂µJ 1,µ A ] = − i2(mu +md)〈ψ̄uψu + ψ̄dψd〉. (2.45) Next, inserting a complete set of pion states ˆ d3p 2Ep(2π)3 |πa(p)〉〈πa(p)| = 1, (2.46) and using Eq. (2.40) with the definition 〈0|QAa (t = 0)|πb(p)〉 = iδabfπEp(2π)3δ3(~p), (2.47) one finds the GOR relation 2m2 πf 2 π = −(mu +md)〈ψ̄uψu + ψ̄dψd〉+O(m2 u,d). (2.48) From this, one sees that the GOR relation establishes a clear connection between quark degrees of freedom and mesonic properties, like the pion mass and its decay constant. 16 2.1 QCD Lagrangian 2.1.2 Confinement As we discussed in the introduction, another important feature of QCD at low energies is the color confinement in that isolated quarks cannot be found in nature. As mentioned in the Introduction, color confinement of QCD is a theoretical hypothesis which is consistent with experimental obser- vations. A conclusive proof from first principles QCD still remains a big challenge for theoretical physicists. In order to make clear this, let us suppose that we have a color singlet state, as a quark-antiquark pair, and try to break it by pulling them apart. As a result, the interaction between them becomes stronger as the distance between them becomes larger. This is very similar to what happens in a spring. Indeed, in a spring the potential energy increases when we stretch it. And, if we stretch it beyond of the elastic limit, it breaks into two pieces. In the quark-antiquark pair case, if we pull them beyond of a limiting distance, a new quark anti-quark pair is generated due to enormous quantity of energy produced. As a consequence, one cannot have quarks as free particles. Figure 2.4: Potential energy between a pair of a heavy quark-antiquark pair when the existence of light quarks are ignored [52]. It is important to mention that the above discussion is to some extent a speculation. For a more complete understanding of this, we need to make precise calculations in QCD at small energy scales, when the QCD coupling gets strong. The best way we know to solve QCD in this regime is to perform numerical simulations on a finite space-time lattice. In this way, it is possible to find 17 2 Introduction to QCD the potential energy between a quark anti-quark pair when there are no dynamical quarks; this approximation is named a quenched approximation. The simulations show that this potential does increase linearly beyond the distance of a few fermis, as shown in Fig. 2.4. Of course, this is not relevant for the real world, because one cannot neglect light quarks. Some authors [53], however, believe that understanding confinement in QCD necessarily requires understanding confinement in pure-gauge Yang-Mills theory. The behavior of some gauge invariant observables can signalize if a given gauge system (with no light quarks) is, or is not, confining. These are the Wilson loop, the Polyakov loop, the ’t Hooft loop, and the vortex free energy. In this thesis, the relevant one will be the Wilson loop. 2.1.2.1 Wilson loop Considering an SU(N) gauge theory and two space-time points xµ and xµ + εµ, where εµ is infini- tesimal, we define the following Wilson link W (x+ ε, x) = exp [ iεµAµ(x) ] , (2.49) where Aµ(x) is in the adjoint representation of the SU(N) group, hence it is anN×N matrix-valued traceless hermitian gauge field. Expanding the Wilson link, we have W (x+ ε, x) = I + iεµAµ(x) +O(ε2). (2.50) Now, let us examine the behavior of the Wilson link under a gauge transformation. Using the gauge transformation of Aµ(x) given in Eq. (2.16), we find W (x+ ε, x)→ I + iεµU(x)Aµ(x)U †(x)− εµU(x)∂µU †(x). (2.51) Since UU † = 1, we have −U(x)∂µU †(x) = +∂µU(x)U †(x) and we can rewrite the above equation as W (x+ ε, x)→ [ (I + εµ∂µ)U(x) ] U †(x) + iεµU(x)Aµ(x)U †(x). (2.52) The first term can be also rewritten as (I + εµ∂µ)U(x) = U(x+ ε) +O(ε2). (2.53) In the second term, we can replace U(x) by U(x+ ε) when we neglect O(ε2) terms. Then, we get W (x+ ε, x)→ U(x+ ε) ( I + iεµAµ(x) ) U †(x), (2.54) 18 2.1 QCD Lagrangian which is equivalent to W (x+ ε, x)→ U(x+ ε)W (x+ ε, x)U †(x). (2.55) Now, let us consider a finite Wilson link. This is specified by a starting point x and n sequential infinitesimal displacement εj . The ordered set of ε’s defines a path P through space-time that starts at x and ends at y = x+ ε1 + . . .+ εn. Then, one can define the Wilson line given by WP (y, x) = W (y, x+ εn) . . .W (x+ ε1 + ε2, x+ ε1)W (x+ ε1, x), (2.56) and this expression is equivalent to WP (y, x) = ˆ P exp [ idxµAµ(x) ] . (2.57) Using Eq. (2.55) and the unitarity property of U(x), we see that, under a gauge transformation, the Wilson line transforms as WP (y, x)→ U(y)WP (y, x)U †(x). (2.58) Now if we consider a path that returns to its starting point, forming a closed and oriented curve C in space-time, we can define the Wilson loop as the trace of the Wilson line for this path W (C) = TrWP (x, x), (2.59) whose expression is given as W (C) = ˛ C exp [ idxµAµ(x) ] . (2.60) Using Eq. (2.58), we see that the Wilson loop is gauge invariant, W (C)→W (C). (2.61) Then, the path integral for the Wilson loop expectation value is written as 〈WC〉 = 1 Z ˆ DUe−βSTrP exp ( i ˛ C dxµA µ ) . (2.62) Now, let us consider a quark-antiquark pair at a (spatial) distance R, at t = 0. We take the quark masses to infinity and let them evolve for a large time T , as illustrated in Fig. 2.5 . To describe this system we consider a theory that consists of an SU(N) lattice gauge field, with spacing a, coupled to a quark field in color group representation r. The action is given by S = β ∑ p ( 1− 1 N Re Tr[U(p)] ) + Smatter, (2.63) 19 2 Introduction to QCD Figure 2.5: Retangular Wilson loop. where U(p) is the plaquette variable, given by U(p) = exp ( ia2FµνFµν +O(a3) ) , (2.64) and the matter action is Smatter = −1 2 ±4∑ x,µ=±1 [ ψ̄(x)(α+ γµ)U (r) µ (x)ψ(x+ µ) ] + ∑ x (mq + 4α)ψ̄(x)ψ(x), (2.65) where γ−µ = −γµ, 0 ≤ α ≤ 1, and U (r) µ (x) denotes the link variable in group representation r, with U−µ(x) = U †µ(x− µ). Now, let Q be an operator which creates a color-singlet two-particle state, with separation R: Q(t) = ψ̄(0, t) [ Γ R−1∏ n=0 U (r) i (ni, t) ] ψ(Ri, t), (2.66) where Γ is some 4 × 4 matrix, constructed from Dirac γ matrices. One can evaluate the term 〈Q†(T )Q†(0)〉 by using the path integral formulation. The quark fields can be integrated out and the leading contribution at large mq is obtained by bringing down from the action a set of terms ψ(α+ γ4)Uψ; some details can be found in [54]. After some effort, one obtains 〈Q†(T )Q(0)〉 = 1 Z ˆ DUDψDψ̄Q†(T )Q(0)e−S ∼ C(mq + 4α)−2T 1 ZU ˆ DUχr[U(R, T )]e−SU , (2.67) where χr[g] is the group character (trace) of group element g in representation r, C is a constant arising from a trace over spinor indices and U(R, T ) is the path-ordered product of links along the rectangular contour with opposite sides of lengths R separated by time T . In the continuum limit, this gives W (C) = P exp [ ˛ C dxµAµ(x) ] . (2.68) 20 2.1 QCD Lagrangian Therefore, the integration in Eq. (2.67) is precisely the Wilson loop. One can them write this same equation as 〈Q†(T )Q(0)〉 ∼ C(mq + 4α)−2T 〈WC(R, T )〉. (2.69) On the other hand, using the usual rules of quantum mechanics in imaginary time the correlator in Eq. (2.67) can also be written as 〈Q†(T )Q(0)〉 = ∑ n,m〈0|Q†(T )|n〉〈n|e−HT |m〉〈m|Q(0)|0〉 〈0|e−HT |0〉 = ∑ n |cn|2e−(V (R)+∆En)T , ∆En = En − V (R). (2.70) The lowest energy eigenstate has energy E0 = V (R) by definition, which is referred to as the static quark potential. The higher energy eigenstates are denoted by their excess energies ∆En. Considering T sufficiently large, the higher-energy terms in the sum are then sufficiently suppressed. Now, by comparing the above relation with that in Eq. (2.69), we have ∑ n |cn|2e−(V (R)+∆En)T ∼ C(mq + 4α)−2T 〈WC(R, T )〉. (2.71) Subtracting the ln(mq + 4α), which is independent of R and therefore irrelevant for our purposes, we have V (R) = − lim T→∞ ln [〈WC(R, T + 1)〉 〈WC(R, T )〉 ] , (2.72) or in a more simplified way 〈W (C)〉 ∼ e−TV (R) , T →∞, (2.73) where V (R) will be referred from now on as the static quark potential. Again, this is the potential for massive color charged sources. Using this criterium, the confinement problem is to show that V (R) has the asymptotic behavior V (R) ∼ σR. (2.74) Thus, the string tension σ can serve as an order parameter for the confined phase, in which σ 6= 0 for all color charge sources. We finish this chapter pointing out that we have assumed the absence of light matter fields which could, through pair creation, screen the charge of the massive sources. We also point out that it is not easy to construct a model incorporating both features dynamically, dynamical chiral symmetry 21 2 Introduction to QCD breaking and quark-gluon confinement. But, many efforts have been made in the last years using different approaches. We mention a few examples: Dyson-Schwinger equations [55], Hamiltonian model involving constituents quarks [56], phenomenological models [57, 58], the Polyakov and Nambu-Jona-Lasinio (NJL) models [59, 60], Quark-Meson (QM) models [61, 62, 63] as well as their extensions with nonlocal interactions [64]. An interesting approach to deal with QCD in the strong coupling limit is to explore the theory in the Large-Nc limit. In the beginning of the 1970’s, ’t Hooft suggested [65] that the theory is simplified when the number of colors Nc is very large. In this case, the expansion parameter would be 1/Nc, and the expectation is that one could study QCD in the limit Nc →∞ and then perform an expansion in 1/Nc. In particular, a new approach based on gauge/gravity duality has been used to incorporate these two properties of nonperturbative QCD to calculate quantities in strongly coupled theories like quantum chromodynamics at low energies. 22 3 Gauge/Gravity duality As we mentioned in the introduction of this thesis, the gauge/gravity duality relates the strongly coupled physics of a system to a classical dynamics of gravity in one higher dimension. Originally, the formulation of the duality proposed by Maldacena [3] related a four-dimensional conformal field theory (CFT) to a geometry of an Anti-de-Sitter (AdS) space in five dimensions, hence it was named as AdS/CFT duality. The original papers show that the AdS/CFT duality was inspired in the studies of solitonic solutions, the Dp branes, in string theories, where the gauge field theories have been realized on hypersurfaces embedded in a higher dimensional space which contains gravity. Since then, the study of the duality has been extended and applied to investigate very different problems, as: the analysis of the strong coupling dynamics of QCD, especially in the study of relativistic heavy ion collisions [66], the physics of black holes [67] and also in different problems in condensed matter physics (holographic superconductors, quantum phase transitions, cold atoms, . . . ) [12]. This extended version of the AdS/CFT duality is named as Gauge/Gravity duality or holographic duality. In the present chapter, we will discuss the Gauge/Gravity duality in more details. It is organized as follows: in section 3.1, we present a motivation of the duality using a geometrization of a simple renormalization group; in section 3.2, we introduce the important holographic dictionary to clarify the duality; in section 3.3, we show how to evaluate correlations functions via the duality and use the holographic dictionary; in section 3.4, we apply the corre- spondence for the scalar field case, with the aim of gaining some insight; in section 3.5, we show how to include temperature effects in the duality and how this affects the background; and, finally, in section 3.6, we show briefly the origin of this duality and discuss how to include flavors in the duality, which is relevant for our purposes in this thesis. 3.1 Motivation We start this part motivating the gauge/gravity duality via the Kadanoff-Wilson renormalization group approach within a geometrical perspective. For instance, let us consider a non-gravitational 23 3 Gauge/Gravity duality system on a lattice whose spacing is a and the Hamiltonian is given by H = ∑ i Ji(x, a)Oi(x, a), (3.1) where x denotes the different lattice sites and i labels the different operators Oi and Ji(x, a) are the sources (currents) of the respective operators at the point x on the lattice. Notice that we have also included a second argument in J i and Oi to make clear they correspond to a lattice spacing a. It is well known in renormalization group approach that we “coarse grain” the lattice by increasing the lattice spacing a and then by replacing multiple sites by a single site with the average value of the lattice variables. In this process, the Hamiltonian structure keeps the form of Eq. (3.1) but each current and operator will acquire a different weight. In this way, for each step of coarse graining, the currents Ji(x, a) change. For example, supposing that we double the lattice spacing in each step, then we would get a succession of currents like: Ji(x, a)→ Ji(x, 2a)→ Ji(x, 4a)→ . . . . (3.2) In this way, the currents Ji acquire a dependence on the the lattice spacing. We can compute this evolution in the currents, writing them as Ji(x, u), where u = (a, 2a, 4a, . . .) represents the length scale at which we probe the system. The evolution of the currents with the scale determines the so-called β-function for each operator, given by u ∂ ∂u Ji(x, u) = βi ( Ji(x, u), u ) . (3.3) At weak coupling regime, the βi’s can be determined in perturbation theory. But, at strong coupling nonperturbative methods must be considered. At strong coupling, the AdS/CFT idea is to consider the length scale u as an extra spatial dimension. In this picture, the succession of lattices at different values of u are considered as slices of a new higher-dimensional space. Moreover, the currents Ji(x, u) are considered as fields living in the space labeled by the usual four dimensions plus the extra dimension u. We can then write Ji(x, u) = φi(x, u). (3.4) Then, all dynamics of the fields φi will be governed by some action in five dimensions. Indeed, the AdS/CFT duality states that the dynamics of these fields is determined by some gravity theory (i.e. by some metric) that needs to be found. Therefore, one can naively consider the holographic duality as a kind of geometrization of the renormalization group of a given quantum field theory, and the microscopic currents of the quantum field theory in the UV regime are identified with the 24 3.1 Motivation values of the bulk fields at the boundary of the extra-dimensional space. Thus, one can state that the quantum field theory lives on the boundary of the higher-dimensional space as illustrated in Fig. 3.1. Figure 3.1: On the left we illustrate the Kadanoff-Wilson renormalization of a lattice system. In the AdS/CFT correspondence the lattices at different scales are considered as the layers of the higher dimensional space represented on the right of the figure. This way, the fields φi of the dual gravity theory must have the same tensor structure of the corresponding dual operator Oi of quantum field theory, in such a way that the product φiOi is a scalar. In this way, a scalar field will be dual to a scalar operator, a vector field Aµ will be dual to a vector operator Jµ and a spin two field gµν will be dual to a second order tensor Tµν operator and that is naturally identified with the energy-momentum tensor Tµν of the quantum field theory. Another characteristic raised by the holographic duality is the matching of the degrees of freedom on both sides of the correspondence. In order to explain this point, let us consider a quantum field theory in a d-dimensional space-time. The number of degrees of freedom of the system is measured by the entropy, which is an extensive quantity. Thus, if Rd−1 is a (d−1)-dimensional spatial region, at fixed time, its entropy is proportional to its volume in d− 1 dimensions. It is written as SQFT ∝ Vol(Rd−1). (3.5) On the other hand, on the gravity side, as we have seen, the theory lives in a (d+ 1)-dimensional space-time. An important question that we could ask is if such a higher dimensional theory can contain the same information as its lower dimensional dual. The key element to answer this 25 3 Gauge/Gravity duality question is the fact that entropy in quantum gravity is a subextensive quantity [68]. Indeed, in a gravitational theory, the entropy in a volume is bounded by the entropy of a black hole that fits inside the volume. In this way, according to the so-called holographic principle, the entropy is proportional to the surface of the black hole horizon (and not to the volume enclosed by the horizon). In formal terms, the black hole entropy is given by the Bekenstein-Hawking formula, as follows SBH = AH 4GN , (3.6) where AH is the correspondent area of the event horizon and GN is the Newton constant. In order to apply it for our purposes, let Rd be a spatial region in the (d+1)-dimension space-time where the gravity theory lives and, at fixed time, the region Rd is bounded by a (d− 1)-dimensional manifold Rd−1 = ∂Rd. Then, according to Eq. (3.6), the gravitational entropy associated with Rd scales as SBH ∝ Area(∂Rd) ∝ Vol(Rd−1), (3.7) which agrees with the QFT behavior in Eq. (3.5). 3.1.1 The Anti-de Sitter space Once motivated the duality from a naive point of view, the challenge is to find the correct geometry corresponding to the specific quantum field theory. In general, this task consists in a very hard problem. However, if the quantum field theory is located at a fixed point of the renormalization group flow, it means that its β-function vanishes and it becomes a conformal field theory (CFT), and the corresponding metric can be easily found. Indeed, considering a quantum field theory in d space-time dimensions, the most general metric in (d+ 1)-dimensions with Poincaré invariance in d-dimensions is ds2 = Ω2(z)(−dt2 + d~x2 + dz2), (3.8) where z is the coordinate of the extra dimension: we use ~x = (x1, . . . , xd−1) and Ω(z) is a function to be determined. Note that using the comparison with the RG, the z coordinate is clearly related to the energy scale as z = u−1. In this way, as the quantum field theory is conformal invariant, ds2 must be invariant under the transformation, as follows (t, ~x)→ λ(t, ~x), z → λz. (3.9) 26 3.1 Motivation Then, by imposing the invariance of the metric in Eq. (3.8) under the transformation given in Eq. (3.9), we obtain that the function Ω(z) must transform as Ω(z)→ λ−1Ω(z), (3.10) which fixes Ω(z) to be Ω(z) = L z , (3.11) where L is a constant, which we will conveniently refer to it as the Anti-de Sitter radius. Thus, by replacing the above function into the metric in Eq. (3.8) we arrive at the following form for ds2 ds2 = L2 z2 (−dt2 + d~x2 + dz2), (3.12) which is the line element of the AdSd+1 space in (d+ 1)-dimensions, that we will denote by AdSd+1 space-time. Notice that for each fixed value of z, this metric becomes a Minkowski space-time. Usually, the boundary of the AdS space-time is referred to be located at z = 0. Notice still that this metric is singular at z = 0. This happens because this coordinate system is not appropriated to write this metric and the singularity disappears changing the coordinate system. Despite of this, we will keep this coordinate system due to its simplicity, but this means that it will be necessary to introduce a regularization procedure in order to define physical quantities in the AdSd+1 boundary. It is worth to remember that the AdS space-time itself arises as solution of the equation of motion of a gravity action of the type I = 1 16πGN ˆ dd+1x √ |g|[−2Λ +R+ c2R 2 + c3R 3], (3.13) with GN being the Newton constant, the ci are constants, g = det(gµν), R = gµνRµν the Ricci scalar and Λ is a cosmological constant. In particular, in the case c2 = c3 = . . . = 0, the action (3.13) becomes the Einstein-Hilbert (EH) action of general relativity with a cosmological constant. In this case the equations of motion are just the Einstein equations Rµν − 1 2gµνR = −Λgµν . (3.14) We can still use some algebraic manipulations and find the following expression for the scalar curvature R = gµνRµν = −d(d+ 1) L2 . (3.15) In this thesis, we will be mostly interested in gauge theories in 3+1 dimensions, which corresponds to set d = 4 in our formulas. Thus, the dual geometry found above for this case is AdS5. This was 27 3 Gauge/Gravity duality precisely the system studied in [3] by Maldacena, who conjectured that the dual QFT is the super Yang-Mills theory with four supersymmetries (N = 4 SYM). 3.1.2 Matching the degrees of freedom Once identified the AdS space-time as the dual of a conformal field theory, we can now evaluate and compare explicitly the degrees of freedom of the both sides of the duality, using the previous arguments in terms of entropy. So, let us consider first the QFT side. We located this system in a spatial box of size R, which acts as an IR cutoff, and we introduce a lattice spacing a which regulates the UV region. In d space-time dimensions, the system has Rd−1/ad−1 cells. In CFT, there is an important concept, the central charge, that appears in the commutator of two components of the stress-energy tensor and essentially counts the number of degrees of freedom in the theory. If the CFT considered is a SU(N) gauge field theory, the gauge fields are N×N matrices living in the adjoint representation of this group. Therefore, in the limit large of N , the theory contains N2 independent components. Thus, in the SU(N) CFT’s, the central charge scales as cSU(N) ∼ N2. In this way, the entropy of the CFT scales as SCFT ∼ Rd−1 ad−1 cSU(N) ∼ Rd−1 ad−1 N 2 (3.16) Next, we count the number of degrees of freedom from the AdS side, NAdS dof . As we have seen before, the gravitational entropy is bounded by the black hole entropy, given in Eq. (3.7), and written as Sgrav. = A∂ 4GN , (3.17) where, in the AdS case, A∂ is the area of the region at the boundary z ≡ ε→ 0 of AdSd+1, at fixed time. Then, we evaluate A∂ by integrating the volume element corresponding to the metric, as A∂ = ˆ Rd−1,z≡ε dd−1x √ g = (L ε )d−1 ˆ Rd−1 dd−1x. (3.18) The last integral is the volume of Rd−1 and it is infinite. As done in the CFT side, we regulate it by enclosing the system in a box of size R and we get ˆ Rd−1 dd−1x = Rd−1. (3.19) Thus, the area of the A∂ is given by A∂ = (RL ε )d−1 , (3.20) 28 3.2 The holographic dictionary and, the entropy in Eq. (3.17) can be written as Sgrav. = 1 4GN (RL ε )d−1 . (3.21) In addition, Newton’s constant is directly related to the Planck length lP , which is a measure of the quantum fluctuations of the space-time; the relation is GN = (lP )d−1. (3.22) Then, Eq. (3.21) becomes Sgrav. = 1 4 (R ε )d−1( L lP )d−1 . (3.23) In general terms, as the number of degrees of freedom of the gravity theory is proportional to gravitational entropy, Sgrav. ∼ Ngrav. dof , the duality establishes a matching between the degrees of freedom of both sides by identifying the gravitational entropy (3.23) with the CFT entropy (3.16), leading to ( L lP )d−1 ∼ N2. (3.24) Notice that, in the limit L� lP , we have ignored the quantum fluctuations of the space-time and gravity becomes classical, leading to the AdS space-time. This classical gravity limit implies that the degrees of freedom of the dual gauge field theory must be large, making it a large-N gauge field theory. In this way, we conclude that the classical gravity theory is reliable if classical gravity - AdS→ ( L lP )d−1 ∼ N2 � 1, (3.25) The action of our gravity theory in the AdSd+1 space of radius L contains a factor Ld−1/GN and, when the coefficient multiplying such action is large, the dual gravity theory is (semi)classical. In this case the path integral is dominated by a saddle point. 3.2 The holographic dictionary As discussed before, in the duality, the sources of a gauge field theory living in the boundary theory act as fields evolving in the gravitational bulk. In general terms, there is a field in the bulk for each operator that we can write in the boundary theory. So, a lot of fields. However, in most of the cases, we will be only interested in a few boundary operators of the gauge field theory and, 29 3 Gauge/Gravity duality correspondingly, a few bulk fields. Such a mapping between operators (sources) and fields in the bulk is usually known as holographic dictionary. As seen previously, a simplest example maps the source for a scalar operatorO(x) in the boundary to a scalar field φ(x, z) in the bulk : φ(x, z)←→ O(x). (3.26) We will explore this in more details soon where we will show that the mass of the bulk scalar maps into the dimension of the boundary operator. But first, let us also think about fields of more general spin. As we mentioned in the motivation of the duality, we can find the fields in the bulk corresponding to the bulk operators by just considering their tensor structures. In this way, fermionic fields in the bulk are mapped into fermionic operators in the boundary; vector fields in the bulk are mapped into vector operators in the boundary and so on. One of the most important example is a bulk massless gauge field Am(x, z); it maps into a conserved current Jµ in the boundary as Am(x, z)←→ Jµ. (3.27) It is not hard to show that gauge symmetry in the bulk leads to conservation law of the boundary current, ∂µJµ = 0. 1 This will be discussed in detail in this thesis soon. We conclude this part recalling a very important points. Any theory involving gravity necessarily has a bulk metric gAB, and holographically, it is dual to the energy-momentum tensor in the boundary gAB ←→ Tµν , (3.28) where, diffeomorphism invariance in the bulk ensures the conservation of the energy and momentum currents, ∂µTµν = 0. 3.3 Correlation functions Once explained the holographic dictionary, next we want to understand how to extract information about the strongly coupled gauge theory in d-dimensions using the correspondence described here. In order to do this, let us consider the following correlation functions of local operators in the QFT 〈O(x1)O(x2) . . .O(xn)〉. (3.29) 1 Here we have that the m vector index in the bulk runs over one more value than the boundary µ index. This does not cause a problem with the dictionary because you can always use gauge invariance to set Az = 0. 30 3.3 Correlation functions In a gauge field theory, such correlators can be calculated using a generating function, which is obtained by perturbing the lagrangian by a source term as L → L+ J(x)O(x) = L+ LJ , (3.30) with the generating functional written as ZQFT [J ] = 〈 exp[ ˆ LJ ] 〉 = eWQFT [J ], (3.31) where WQFT [J ] is the generating functional of the connected Green’s functions. The connected correlators are obtained from the functional derivatives of lnZQFT [J ], or just WQFT [J ] as 〈∏ i=1 O(xi) 〉 = ∏ i=1 δ δJ(xi) lnZQFT [J ] ∣∣∣ J=0 = ∏ i=1 δ δJ(xi) WQFT [J ] ∣∣∣ J=0 . (3.32) Our main task here is clarify how to incorporate this prescription on the gravity side. In this context, Gubser, Klebanov, Polyakov and Witten [4, 5] have discovered the right procedure to evaluate correlation functions of a field theory via its dual gravity theory, the GKPW prescription. Those authors have followed the ideas obtained in the context of string theory where the open strings are sources for closed strings. In this way, following the holographic dictionary, the QFT is thought to live on the AdS boundary and on the gravitational side the source is restricted to this boundary. Then, the sources will act as boundary conditions for its corresponding fields propagating in the bulk gravity theory. In general, these fields can present some divergences at the boundary and, because of this, a regularization procedure must be considered, as we will show explicitly. In a precise form, the GKPW prescription is the mathematical “backbone” of AdS/CFT corre- spondence and it identifies the partition function of a QFT in the presence of a source J(x) with the partition function of a bulk gravitational theory where the asymptotically Anti-de Sitter boundary value of the fields φ are matched with the sources J(x). This identification is written as ZQFT [J(x)] = Zstr[φ0(x)] = ˆ Dφe−Sstr[φ(x,z)]|φ0(x) . (3.33) In general, the right-hand side of this equation is very difficult to compute, but it simplifies drama- tically in the classical gravity limit (3.25). In this limit, the string theory path integral is dominated by the saddle point approximation, that contains the classical gravity theory. So, we have ZQFT [J(x)] = Zstr[φ0(x)] ∼ e−Sgrav[φ(x,z)]|φ→φ0(x), N � 1. (3.34) 31 3 Gauge/Gravity duality The duality conjecture asserts that this classical gravity action (3.34) is precisely related to the generating functional for a connected Green’s function in Eq. (3.32) as WQFT [J ] = −Sgrav[φ(x, z)]|φ→φ0(x). (3.35) Therefore, the correlation functions of the QFT can now be evaluated as 〈O(x1)...O(xn)〉 = δ(n)Sgrav[φ0] δφ0(xi)...δφ0(xn) ∣∣∣ φ0=0 . (3.36) In most of the cases, the on-shell action (Sgrav) needs to be renormalized as it typically suffers from infinite volume (i.e. IR) divergences due to the integration region near the boundary of AdS. Such divergences are dual to UV divergences in the gauge field theory, which is consistent with the idea that, the duality is also a UV/IR duality. The procedure to eliminate these divergences on the gravity side is well understood and named as ‘holographic renormalization’. A good review on this topic is found in Ref. [69]. 3.4 Massive scalar field in AdS space In order to study the consistency of the duality, let us consider for simplicity a scalar field φ with mass m2 in the background of an AdSd+1 space-time with classical action given by Iφ = −1 2 ˆ dd+1x √ |g| ( (∂nφ)2 +m2φ2 ) . (3.37) The equation of motion derived from this action is 1√ |g| ∂n( √ |g|∂nφ)−m2φ = 0, (3.38) and using the metric (3.12), this equation explicitly becomes zd+1∂z(z1−d∂zφ) + z2∂µ∂ µφ−m2φ = 0. (3.39) Now, due to translational invariance in 4d, let us perform the Fourier transform of φ in the xµ coordinates φ(x, z) = ˆ ddk (2π)d e ikxϕk(z). (3.40) Then, the equation of motion (3.39) becomes zd+1∂z(z1−d∂zϕk)− k2z2ϕk −m2ϕk = 0. (3.41) 32 3.4 Massive scalar field in AdS space Let us focus on the region near to the boundary z ≈ 0. Therefore, taking the limit z → 0 in the Eq. (3.41) we find the solution ϕk(z)→ φ0(k)zd−∆ + φ1(k)z∆, (3.42) where ∆ is given by ∆ = d 2 + √ d2 4 +m2. (3.43) In particular, the powers d − ∆ and ∆ in Eq. (3.42) are roots of the equation m2 = ∆(∆ − d). Now, by performing the inverse Fourier transform, we can write for (3.42) an expansion near the boundary in position space φ(x, z) = φ0(x)zd−∆ + φ1(x)z∆, z → 0. (3.44) We highlight here that, the first solution zd−∆ is the normalizable solution as it leads to finite action (3.37). On the other hand, the second solution, z∆, is the non-normalizable solution, as it leads to a divergent action. Notice still that, d −∆ and ∆ are real as long as m2 ≥ −d2/4. This bound, known as Breitenlohner-Freedman (BF) bound [70], states that a small negative m2 does not lead to an instability in AdSd+1 as it would in a flat space instead, the would be instability is lifted by the harmonic potential generated by the AdSd+1 curvature. If the BF bound is satisfied, we have as consequence that d−∆ ≤ ∆. Then, the term behaving as z∆ in Eq. (3.44) is the dominant one as z → 0. So, let us take the boundary as z ≈ 0 and neglect the subdominant term, for now. We have φ(x, z = ε) ∼ εd−∆φ0(x). (3.45) Notice that, as d−∆ is negative if m2 > 0, the leading term is typically divergent as we approach the boundary at z = ε→ 0. In order to identify correctly the QFT source J(x) with the boundary value of the field φ(x, z), we need to remove the divergences of the latter. We will simply do it by extracting the divergent multiplicative factor from (3.45), i.e. the QFT source J(x) is identified with φ0(x). Equivalently, we define J(x) = φ0(x) = lim z→0 z−(d−∆)φ(x, z). (3.46) Clearly, with such a definition, J(x) is always finite. Conversely, at leading order, we can write φ(x, z) = zd−∆J(x). 33 3 Gauge/Gravity duality In order to interpret the meaning of ∆, let us focus at the boundary action. Remember that as the current J(x) couples with the operator O(x), the action at the boundary is given by Ibdy ∼ ˆ ddx √ |γε|φ(x, ε)O(x, ε), (3.47) where γε = (L/ε)2d is the determinant of the induced metric at the z = ε boundary. Then, using Eq. (3.46), we have φ(x, ε) = εd−∆J(x) and using it in the expansion for Ibdy, we get Ibdy ∼ Ld ˆ ddxJ(x)ε−∆O(x, ε). (3.48) In order to make Ibdy finite and independent of ε as ε→ 0 we must require O(x, ε) = ε∆O(x), (3.49) which means that the operator also scales with ∆. In order words, passing from z = 0 to z = ε means a scale transformation in the QFT in lower dimension. Thus, ∆ must be interpreted as the mass scaling dimension of the dual operator O. Similarly, from the relation φ(x, ε) = εd−∆J(x), it follows that d−∆ is the mass scaling dimension of the source J(x). 3.4.1 One-point function The first correlation function we can find for the scalar field in AdSd+1 space-time is the 1-point function. In addition, it is also interesting to compute the one-point function of an operator O in the presence of the source φ0, given by 〈O(x)〉φ0 = δSgrav[φ] δφ0(x) . (3.50) In other words, 〈O(x)〉 is understood as being the response of the operator to the source φ0. Taking into account the relation between φ and φ0, given in Eq. (3.46), we get 〈O(x)〉φ0 = lim z→0 zd−∆ δSgrav[φ] δφ(x, z) . (3.51) We can compute in a closed form the functional derivative of the classical on-shell action with respect to φ(x, z). In fact, the action Sgrav be represented as Sgrav = ˆ AdS dzddxL[φ, ∂φ], (3.52) with AdS being a (d + 1)-dimensional manifold whose boundary is located at z = 0. Under a general change φ→ φ+ δφ, the classical action Sgrav changes as δSgrav = ˆ AdS dzddx [∂L ∂φ δφ+ ∂L ∂(∂mφ)δ(∂mφ) ] . (3.53) 34 3.4 Massive scalar field in AdS space Now using, in the above equation, the condition δ(∂mφ) = ∂m(δφ) and next integrating by parts and finally using the Euler-Lagrange equations, we find that δSgrav = ˆ M dzddx∂m(Πm(x, z)δφ), Πm(x, z) = ∂L ∂(∂mφ) , (3.54) where Πm(x, z) is the generalized conjugate momentum to φ(x, z). As the boundary is located at z = ε→ 0, we can write δSgrav = ˆ ∞ ε dz ˆ ddx∂z(Πz(x, z)δφ) = − ˆ ∂M ddxΠz(x, ε)δφ(x, ε). (3.55) Thus, from this, it follows that δSgrav δφ(x, ε) = Πz(x, ε) = − ∂L ∂(∂zφ) . (3.56) In this way, the one-point function in Eq. (3.51) is rewritten as 〈O(x)〉φ0 = lim z→0 zd−∆Πz(x, z). (3.57) Using the Lagrangian written in Eq. (3.37) and the asymptotic behavior in (3.44) we find that the one-point function (3.57) is 〈O(x)〉φ0 = φ1(x). (3.58) And, then, returning to Eq. (3.44), the field near to the boundary of the scalar field takes the form φ(x, z) = J(x) zd−∆ + 〈O(x)〉J z∆, z → 0. (3.59) 3.4.2 Linear response A small space and time perturbation in a equilibrium state of a certain physical system consists the basis of the linear response theory. The one point function with a source can be written in the path integral representation as 〈O(x)〉J = ˆ DφO(x)e−SE+ ´ ddyJ(y)O(y). (3.60) Expanding the exponent in this expression in a power series of the source J and just keeping the terms up to linear order, we have 〈O(x)〉J = 〈O(x)〉J=0 + ˆ ddy〈O(x)O(y)〉J(y) + . . . . (3.61) 35 3 Gauge/Gravity duality Defining the two-point function G(x− y) as G(x− y) = 〈O(x)O(y)〉, (3.62) the Eq. (3.61), can be rewritten as 〈O(x)〉J = 〈O(x)〉J=0 + ˆ ddyG(x− y)J(y) + . . . . (3.63) We consider observables such that 〈O(x)〉φ=0 is zero. Notice that, this vanishing can always be achieved subtracting the vacuum expectation value (VEV) without source in Eq. (3.63). In this way, the fluctuations of the observable can be measured by 〈O(x)〉J . So, the linear response is written as 〈O(x)〉J = ˆ ddyG(x− y)J(y). (3.64) Going to momentum space, this expression is rewritten as 〈O(k)〉J = G(k)J(k), (3.65) and we can rewrite the two-point function in momentum space as G(k) = 〈O(k)〉J J(k) . (3.66) In the framework of the AdS/CFT correspondence, we have obtained 〈O(k)〉J in (3.57), the two- point function in momentum space is then: G(k) = lim z→0 zd−∆ Πz(k, z) φ0(k) . (3.67) To exemplify our description, let us consider a spatially constant electric field Ei(ω), oscillating with frequency ω, and acting on a system whose response is the charge current J i(ω). At first order, this is described by Ohm’s law J i(ω) = σij(ω)Ej(ω). (3.68) where σij(ω) is the conductivity tensor for alternating currents. In the language of the previous subsections, we consider an external vector potential Aµ(x), which plays the role of φ(x), and a conserved current J i(x), which corresponds to the operator O(x). In the temporal gauge (At = 0), the electric field Ek is simply given by Ek = −∂tAk. Using the following Fourier decomposition Ak ∼ e−iωt, the electric field becomes Ek = iωAk. Now, by comparing Ohm’s law (3.68) with (3.66), we obtain a simple expression for the electric conductivity tensor of alternating currents σij(ω) = Gij(k) iω . (3.69) 36 3.5 Finite temperature in Gauge/Gravity duality In our context, Gij(k) are the components of the retarded correlator of currents in Fourier space. Notice that the spatial momentum q is set to zero. Our analysis can be easily extended for the fields with higher order spins. Indeed, the relation (3.43) and the near-boundary behaviour (3.44) also apply to a bulk spin-one and spin-two fields. In general terms, in the case of a bulk p-form, the dimension ∆ of the dual operator is the largest root of the equation m2 = (∆− p)(∆ + p− d), (3.70) and the behavior of the fields near to the boundary becomes φµ1...µp(x, z) ≈ Aµ1...µpz d−p−∆ +Bµ1...µpz ∆−p, z → 0. (3.71) In summary, we have seen that the AdS space-time corresponds to the solution dual to the vacuum of a CFT. But, the gauge/gravity duality we have presented here also can be used to study theories with no scale invariance, like QCD. In fact, QCD is not a conformal theory and a direct consequence is that the dual background is no longer AdS space-time, instead, it is a different space-time that is just AdS in the UV regime (near to the boundary). All the efforts will be to find a correct metric in the IR regime. 3.5 Finite temperature in Gauge/Gravity duality Let us next show how to incorporate finite temperature in the gauge/gravity duality. We start recalling that, the partition function in statistical mechanics in the canonical ensemble is given by Z = Tr e−βH (3.72) where, H is the hamiltonian operator, β related with the temperature by β = 1/T , we have take the Boltzmann constant kB = 1. The thermal average of an operator O at the temperature T is OT = Tr [ O e−βH ] Z . (3.73) Using the path integral approach, the average OT is written as OT = 1 Z ˆ [Dψ]〈ψ(x)|O e−βH |ψ(x)〉, (3.74) where |ψ(x)〉 is the state corresponding to the operator ψ̂(x), given by ψ̂(x)|ψ(x)〉 = ψ(x)|ψ(x)〉. (3.75) 37 3 Gauge/Gravity duality The trace in (3.74) is taken the same initial and final state |ψ(x)〉 in the evaluation of the expectation value. Still, note that the hamiltonian operator represents a time evolution and, using this, we can rewrite (3.74) as OT = 1 Z ˆ [Dψ]〈ψ(x, t)|O|ψ(x, t+ i/T )〉. (3.76) It is well known that, thermal correlation functions are evaluated considering an imaginary time evolution and by imposing periodic boundary conditions for bosons and antiperiodic boundary conditions for fermions. Therefore, the Euclidean time tE is periodic, as tE → tE + i T . Thus, the compactification of Euclidean time is equivalent to having T 6= 0. Next, we will show how the temperature of the field theory is related with the Hawking tempe- rature of a black hole [71]. For simplicity, we will assume an Euclidean metric : ds2 = g(z) [ f(z)dt2E + d~x2 ] + h(z)dz2, (3.77) where we assume that the functions f(z) and h(z) are null at z = zh, which is the location of the horizon, and that g(zh) 6= 0. Then, for z ≈ zh, we have f(z) ≈ f ′(zh)(z − zh) , h(z) ≈ h′(zh)(z − zh). (3.78) For the g(z) function, we just take g(z) = g(zh). Then, the Euclidian metric near to the horizon can be rewritten as ds2 ≈ g(zh) [ f ′(zh)(z − zh)dt2E + d~x2 ] + 1 h′(zh) dz2 z − zh . (3.79) In order to write the metric in an elegant way, let us define two new coordinates: a radial ρ and an angular θ, as 1 h′(zh) dz2 z − zh = dρ2 , g(zh)f ′(zh)(z − zh)dt2E = ρ2dθ2, (3.80) which corresponds to defining θ as θ = 1 2 √ g(zh)f ′(zh)h′(zh)tE . (3.81) In these new coordinates, Eq. (3.79) becomes ds2 ≈ g(zh)d~x2 + dρ2 + ρ2dθ2, (3.82) where the two last parts have the metric of a plane. The event horizon is now located at ρ = 0 and in order to avoid problems with any curvature singularity in this point, the θ variable must be 38 3.5 Finite temperature in Gauge/Gravity duality a periodic coordinate with period 2π. In order to find a relation between the value of T and the event horizon of the black hole, first notice that the periodicity under θ → θ + 2π is equivalent to periodicity under tE → tE + 1/T , therefore: 1 T = 4π√ g(zh)f ′(zh)h′(zh) . (3.83) where T has the same form of the Hawking temperature [71]. As a simple application, we use the AdSd+1 black hole case. Its metric is ds2 = L2 z2 [ f(z)dt2E + d~x2 + dz2 f(z) ] , (3.84) where f(z) has the following function f(z) = 1− zd zdh (3.85) with zh located the event horizon. By comparing with the general form of metric in (3.77), the other functions in Eq. (3.84) are given by g(z) = L2 z2 , h(z) = z2 L2 f(z). (3.86) In this case, the derivatives of f ′(z) and h′(z) at the horizon are f ′(zh) = − d zh , h′(zh) = −dzh L2 . (3.87) And, as a consequence, using (3.83), the temperature becomes T = d 4πzh . (3.88) This relation will be very useful in chapter 7 where we will study the Inverse Magnetic Catalysis (IMC) via dynamic holographic QCD. Surprisingly, not only the temperature, but also all the thermodynamical properties of the black hole, turn out to be the same as those of the dual gauge theory. For example, in Ref. [72] the authors have performed a simple but important calculation, namely that of the entropy density of large N N = 4 SYM at strong coupling. As we have mentioned previously, all the holographic description made so far, starting from QFT and getting a gravity theory, is commonly named as bottom-up approach. We will discuss briefly in the following, the origin of this approach: the AdS/CFT correspondence. 39 3 Gauge/Gravity duality 3.6 AdS/CFT correspondence Let us first review some important aspects in the description of the duality. We start pointing out the concept of strings. Inspired in the point particle in special relativity, we can make a classical description of a relativistic string as follows. The motion of a particle in space-time describes a curve, the so-called worldline, and it can be represented by a function xµ = xµ(σ), where σ is the worldline coordinate and it parameterizes the path of the particle. This is shown in Fig. 3.2 This Figure 3.2: Worldline of a point particles way, its action of the particle is proportional to the integral of the line element along the trajectory in space-time, with the coefficient being given by the mass m of the particle S = −m ˆ ds = −m ˆ √ gabdxadxb. (3.89) Next, we consider a relativistic string, which is a one-dimensional object moving in space-time. Its motion will span a surface, the so-called worldsheet, as shown in Fig. 3.3. By analogy with the particle action (3.89), the action for the relativistic string, known as the Nambu-Goto action, is given by Sstr = −Tstr ˆ d2σ √ detgαβ, (3.90) where Tstr is the tension of the string, given by Tstr = 1 2πα′ , α′ = l2s (fundamental length scale), (3.91) and gab is the induced metric on the worldsheet defined by gαβ = ∂αx a∂βx bgab. Its important to mention that in string theories one has open and closed strings, depending on the boundary conditions. Standard methods of quantum mechanics can be used in the quantization of the strings. For the open strings case, their spectrum present a set of massless particles of spin one. For our purposes, the important fact arises when looking at the spectrum of closed strings: in general terms, the closed string spectrum contains states with a finite number of massless 40 3.6 AdS/CFT correspondence Figure 3.3: Worldsheet of a one-dimensional object (string) modes (graviton) and an infinite tower of massive modes with masses of order ms = l−1 s = α′−1/2. Thus, at low energies E � ms, all the possible higher order corrections come in powers of α′E2 from integrating out the massive string modes. This perturbative regime of strings is tractable, but the theory also has a non-perturbative sector and it plays a crucial role in the formulation of the correspondence. In such a regime, solitonic solutions take place and change the physics of the strings. These solitonic solutions are known as Dp-branes and are extended objects in p+ 1 directions (p spatial + time), where open strings can end [73]. Also, the Dp-branes are also dynamical structures that can move and get excited. We can write an action for its dynamics, known as Dirac-Born-Infeld (DBI) action. In fact, for a single brane, the action takes the form SDBI = −TDp ˆ dp+1xe−Φ √ −det(gab + 2πl2sFab) + fermions, (3.92) with TDp being the tension of the Dp-brane, which in terms of the string coupling constant gs and the string length ls, is given by TDp = 1 (2π)pgslp+1 s . (3.93) Finally, the dilation field Φ, in Eq. (3.92), is directly related to the string coupling constant as eΦ ∼ gs. Notice still that TDp ∼ g−1 s , which confirms the idea of the Dp-branes being non- perturbative objects in string theory. In the case of a single Dp-brane, the massless spectrum which represents excitations along the brane is composed by an Abelian gauge field Aµ(x) (µ = 0, . . . , p) and a massless spectrum which represents excitations transverse to the brane is composed by 9−p scalar fields φi(x) (i = 1, . . . , 9− p). Clearly, there are also all the superpartners. Therefore, in this non-perturbative sector, we have 41 3 Gauge/Gravity duality closed and open strings (which end in the Dp-brane) in a flat background. They, of course, interact with each other, as illustrated in the Fig. 3.4 Figure 3.4: Interactions between open-open, closed-closed and open-closed strings Now, let us move on and consider a stack of Nc Dp-branes located in 10 dimensional space-time 2. Now, in this new context, there are now open strings stretching between different branes and in total we have Nc ×Nc types of open strings, depending on which brane their endpoints lie. Those strings stretching between different branes lead to massive vector fields with masses of the order of m ∼ r/ls. However, when the Nc branes are coincident (r → 0), all the vector fields become massless [see Fig. 3.5] and, as consequence, we get Nc ×Nc massless gauge fields Aµ and Nc ×Nc massless Xi scalar fields which transform in the adjoint representation of U(Nc), which can be decomposed into U(Nc) = U(1)× SU(Nc). As a conclusion of this discussion, we see that a stack of Nc Dp-branes realizes an SU(Nc) gauge theory in p+ 1 dimensions. The dynamics of this full system is composed by the brane action (SDBI), associated with the excitations of open strings, the bulk action (Sbulk), associated with the excitations closed strings, and the interaction action (Sint.), associated with the interaction between open and closed strings: S = SDBI + Sbulk + Sint.. (3.94) As all the forms of matter gravitate, and the D-branes are no exception, they can deform the space- time in which they are embedded. Using the equations of supergravity [74] we can find explicitly the metric produced by the presence of Nc Dp-branes. As we want to make connection with QCD, 2We use Nc to make conection with QCD latter and the branes can be named as “color” branes 42 3.6 AdS/CFT correspondence Figure 3.5: Strings stretching between two D-branes. we set p = 3, and this way, the D3-branes in type IIB theory lead to Horowitz metric ds2 = H−1/2(−dt2 + dx2 1 + dx2 2 + dx2 3) +H1/2(dr2 + r2dΩ2 5), (3.95) where t, x1, x2, x3 are coordinates along the branes, and the radial coordinate r2 = y2 1 + . . . + y2 6, with y1, . . . , y6, are the transverse coordinates to D3-branes. The function H(r) is given by H(r) = 1 + R4 r4 , R4 l4s = 4πgsNc. (3.96) An important regime is the limit of large-Nc, when the open and closed strings are decoupled from each other and are noninteracting: Sint. → 0. But the couplings like open-open strings and closed-closed strings remain. Note that the gravitational effects of the D3-branes are controlled by the factor gsNc. In the limit gsNc � 1, the radius characterizing such effect becomes small in string units and, hence, the closed strings feel a space-time everywhere except very close to the hyperplane where the D3-branes are located. In this regime, a description from the closed string perspective is intractable. On the other hand, at low energies of string theory (ls → 0), by expanding the DBI action (3.92) and keeping just the quadratic terms in Fab and ∂aXi we obtain S (2) DBI = ˆ d4xTr ( − 1 4πgs FabF ab − ∑ i 1 2DaX iDaXi + πgs ∑ ij [Xi, Xj ] ) +O(ls), (3.97) where, for simplicity, we have written just the bosonic part. This is precisely the N = 4 super- Yang-Mills theory when we set g2 YM = (2π)gs. (3.98) 43 3 Gauge/Gravity duality So, its known the Yang-Mills coupling is the t’Hooft constant λ = g2 YMNc and it controls the loop expansion of the theory. The limit considered here (gsNc � 1) can be reinterpreted easily as λ� 1. Thus, for this case, the dynamic of the system at low energies is reduced to S ≈ SSUSY + SIIB string in M10 . (3.99) In the opposite limit (λ� 1), the situation reverts, the open string description becomes intractable, since one would need to deal with strongly coupled open strings. So, we have closed strings moving in this curved space, Fig. 3.6 Figure 3.6: Excitations of the system in the closed string description. As said before, the parameter R is thus understood as the length scale characteristic of the range of the gravitational effects of large Nc D3-branes. For r � R, the metric is asymptotically Minkowski where closed strings live. For r � R, in the other limit, so-called near-horizon limit r � R, the metric element (3.95) becomes ds2 = ds2 AdS5 +R2dΩ2 5, (3.100) where the AdS5 space-time is ds2 AdS5 = r2 R2 (−dt2 + dx2 1dx 2 2 + dx2 3) + R2 r2 dr 2, (3.101) is exactly the metric element in (3.12) of a five-dimensional Anti-de Sitter space-time written in terms of r = R2/z. As final conclusion of this part, one has that in the strong gravity region the ten-dimensional metric factorizes into AdS5×S5, where closed strings live. Then, for this case, the dynamics of the system at low energies is reduced to S ≈ SSUGRA AdS5×S5 + SIIB string in M10 , (3.102) 44 3.6 AdS/CFT correspondence where SUGRA means the supergravity action. In this context, Maldacena conjectured that these two descriptions are equivalents. In the low- energy limits, such a conjecture leads to {N = 4 SU(Nc) SUSY theory} ≈ {SSUGRA AdS5×S5}. (3.103) Figure 3.7: Correspondence. We finish this section clarifying the validity limit of the duality. Firstly, remembering that Newton’s constant G in ten-dimensional can be rescaled as G ∼ l8p, with lp being the Planck length and, secondly, as also the Newton’s constant is defined as 16πG = (2π)7g2 s l 8 s , we have l8p R8 ∝ 1 N2 c , l4s R4 ∝ 1 λ , (3.104) where we have omitted purely numerical factors. Thus, the limit considered to get the