STUDIES ON GAUGE THEORIES FROM STRING THEORY

AND SCATTERING AMPLITUDE METHODS

(Spine Title: Holographic Gauge Theories and Tree Level S-Matrix)

(Thesis Format: Integrated-Article)

by

Paolo Benincasa

Graduate Program in Applied Mathematics (Theoretical Physics)

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Faculty of Graduate Studies The University of Western Ontario London, Ontario, Canada

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While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada THE UNIVERSITY OF WESTERN ONTARIO FACULTY OF GRADUATE STUDIES

CERTIFICATE OF EXAMINATION

Supervisor Examiners

Dr. Alex Buchel Dr. Mikko Karttunen

Advisory committee Dr. Gerry McKeon

Dr. Gerry McKeon Dr. Martin Houde

Dr. Volodya Miransky Dr. Jaume Gomis

The thesis by Paolo Benincasa

entitled:

STUDIES ON GAUGE THEORIES FROM STRING THEORY AND SCATTERING AMPLITUDES METHODS

is accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Date- Chair of the Examination Board

n ABSTRACT

In this thesis we investigate different aspects of gauge theories, both at strong and weak coupling, which seem hard to address using standard field theory meth­ ods. Specifically, we focus on both strongly-coupled gauge theory plasmas and the perturbative regime of General Relativity (GR) as well as arbitrary spin particles. In the first case the conjectured gauge/string correspondence is used. We extracted hydro dynamical properties of some non-conformal theories at large N and large 't Hooft coupling. For the first time the bulk viscosity for a non-conformal theory has been computed. In the models analyzed the values for the speed of sound are less than its value in conformal theories. Another interesting feature is that, turning on chemical potentials, gauge theories admitting a dual string theory still belong to the universality class defined by the ratio shear viscosity by entropy density. We also studied the implication of introducing the leading a'-correction in string the­ ory. The consistent hydrodynamical picture obtained is a test for the a'-structure of the theory. All these results hold for static plasmas. We also attempted to study plasmas as dynamical phases. Because of logarithmic singularities, the supergrav- ity approximation turns out to be inconsistent and one would need the full string theory. At weak coupling, we investigated properties of GR and interactive theories of arbitrary spin particles from an S-matrix point of view. At tree level, looking at the singularity structure of the amplitudes, n-graviton amplitudes turn out to be uniquely determined by three-graviton amplitudes. Moreover, we have formulated consistency conditions for the existence of a particular class of high spin theories admitting an S-matrix.

Keywords : String Theory, Gauge/Gravity Correspondence, Thermal Field Theory, Supersymmetric Gauge Theory, Scattering Amplitudes, Classical Theories of Gravity, Gauge Symmetry. iii AUTHORSHIP

This integrated-article thesis contains seven papers which have been published in five different international journals and/or have been posted on the online archive http://arxiv.org. Prof. Alex Buchel, as my current supervisor, is co-author in five of them. Chapter 6 contains a paper in collaboration with Prof. Alex Buchel and Michal Heller and Romuald Janik from Jagellonian University. Chapter 8 is the result of a collaboration with Camille Boucher-Veronneau from Waterloo University (now in Stanford) and Dr. Freddy Cachazo from Perimeter Institute for Theoretical Physics. Chapter 9 contains a paper which is a result of a collaboration with Dr. Freddy Cachazo. Approval from all parts, including co-authors as well as publishers, has been granted to include the results from the aforementioned articles within this thesis.

IV // knowledge can create problems, it is not through ignorance that we can solve them.

Isaac Asimov

In some sort of crude sense, which no vulgarity, no humor, no overstatement can quite extinguish, the physicists have known sin; and this is a knowledge which they cannot lose.

J. Robert Oppenheimer

It takes so long to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them.

Eugene Wigner

v To those who are always with me

VI ACKNOWLEDGEMENTS

This thesis represents the end of another stage of my life. It is indeed simply the achievement of an intermediate step, but nevertheless it has a key role in my growing process, both as (hopefully) a researcher and as a person. It represents the end of my first experience outside my home-country. It represents the end of my path as a student. From now on it is completely up to me. For all these reasons, I would like to thank several people who played an impor­ tant role in this stage of my life. Firstly, I would like to thank my parents, to whom it was not easy to live the idea that one of their sons were in another continent - in the "America"-, and my brother Francesco for their love and their unconditional support. Well, to be completely honest, I should thank my brother also for his constant technological support in remote!! I would like to thank my advisor, Alex Buchel, who decided to accept me as his first graduate student on the basis of one single email I sent him in the fall of 2004. He guided me through the world of research, enthusiastically proposing me projects, prompting me to participate to international conferences since the very beginning of my PhD and trying to make me become an independent researcher, I enjoyed our conversations in our weekly trips to Perimeter Institute, in which I had the chance to keep learning as well as make fun about food habits and my Italy. I really hope I have made treasure of his advice. I also have to thank him for having introduced me to Perimeter Institute, where I have been part of a beautiful string theory group. Without participating to its activities, my PhD experience would not have been the same. I have benefitted very much from the interactions with its members. I am especially grateful to Freddy Cachazo, whom I have had - and I am having - the luck of collaborating with. I am grateful to him for everything he taught me and being so available for discussions. vii I also thank Fiorenzo Bastianelli, who saw me moving my very first steps in the research world, for his constant support after I left the Universita di Bologna. I am in debt with Alex, Freddy and Fiorenzo for having helped me to put some more bricks in building my future: if I can finish my PhD knowing that I will join a really good research group as postdoctoral fellow, I own it to them. I did not really expect that the outcome of my postdoctoral application would have been as great as has been. For interesting discussions about physics and nice time spent in schools and con­ ferences, it is a pleasure to thank Fernando Alday, Michele Arzano, Sujay Ashok, Roberto Balbinot, Ling Bao, Francesco Benini, Francesco Blanco, Camille Boucher- Veronneau, Stan Brodsky, Diego Chialva, Andres Collinucci, Paul Cook, Clau- dio Coriano, Olindo Corradini, Jan de Boer, Eleonora Dell'Aquila, Erik Devetak, Bianca Dittrich, Rossella Ferrandes, Valentina Giangreco, Simone Giombi, Luca Grisa, Jaume Gomis, Amihay Hanany, Michal Heller, Barbara Jacak, Niklas Jo­ hansson, Thomas Klose, Mira Kramer, Josh Lapan, Magdalena Larfors, Paola La Rocca, Xiao Liu, Pierpaolo Mastrolia, Shunji Matsuura, Paul McFadden, Gerry McKeon, Chris Miller, Volodya Miransky, Rob Myers, Suresh Nampuri, Hiroshi Ooguri, Sara Pasquetti, Filippo Passerini, Antonio Polosa, Shlomo Razamat, Slava Rychkov, Massimiliano Rinaldi, David Skinner, Julian Sonner, Simone Speziale, Andrei Starinets, Nemani Suryanarayana, Stefan Theisen, Maciej Trzetrzelewski, Mirian Tsulaia, Linda Uruchurtu-Gomez, Samuel Vazquez, David Vegh, Alessandro Vichi, Amos Yarom. A special thank is to Vic Elias who prematurely left us in 2006.

I am also grateful to Claudio Coriano and the Theoretical physics group at Universita di Lecce for their hospitality during my visit and Graham Shore and Toby Wisemann for their help during my postdoctoral application process. A really special mention is for all my friends in Italy, with whom never changed despite my three years in another continent and who always warmly welcomed me each time I was back. For their never-ending friendship, I really want to thank An­ drea, Carlo, Cecilia, Cristian, Daniela, Daniele, Danilo, Diego, Emanuele, Federica,

viii Prancesca, Fulvio, Gabriele, Gabriella, Gianfranco, Giulio, Jacopo, Karina, Lisa, Lorenzo, Lucia, Luca, Maria Grazia, Marcello, Marco, Mario, Nicoletta, Pasquale, Sebastiano, Simona, Simone, Tommaso. A special thought is for Vincenzo, whose adventure ended in February 2007. I will always remember him. I must also include all those friends who directly shared this experience with me here in Canada: Aurelie, Carlos, Chris P., Chris S., Daniela, David, Dmitriy, Eri, Filip, Igor, John, Manuela, Marco, Muna, Natasha, Pablo, Ramin, Raz, Roman, Samuel, Simona, Soha, Wade. I really thank them for having made my staying in Canada overjoyed, sharing their own experiences with me and helping me in growing up as a person. Last but not least, I really thank my girlfriend Julie for the wonderful person she is and for being always so close to me. She was able to make my last five months unforgettable.

Don't go far off, not even for a day, because- because-I don't know how to say it: a day is long and I will be waiting for you, as in an empty station when the trains are parked off somewhere else, asleep. Pablo Neruda - Sonnet XLV

IX TABLE OF CONTENTS

CERTIFICATE OF EXAMINATION ii

ABSTRACT iii

AUTHORSHIP iv

EPIGRAPH v

DEDICATION vi

ACKNOWLEDGEMENTS vii

TABLE OF CONTENTS x

LIST OF FIGURES xv

LIST OF ABBREVIATIONS AND SYMBOLS xvi

PREFACE xvii

1 Introduction 1 1.1 General introduction 1 1.2 Overview of the thesis 13

2 Literature review 23 2.1 Gauge Theories at strong coupling: hydrodynamics and gauge/string duality 24 2.1.1 Large-TV field theory 24 2.1.2 Type IIB string theory 26 2.1.3 D-branes 30 2.1.4 Gauge/gravity correspondence 32 2.1.5 Relativistic hydrodynamics 37

x 2.1.6 Hydrodynamics and gauge/string correspondence 41 2.1.7 A prescription for two-point Minkowski thermal correlators . . 44 2.1.8 Holographic renormalization 50 2.1.9 The KSS bound and universality 52 2.1.10 Israel-Stewart transport theory, Bjorken flow and gauge/string correspondence 54 2.2 Gauge theories at weak coupling: scattering amplitudes methods ... 57 2.2.1 Spinor-helicity formalism 58 2.2.2 BCFW construction 62

PART 1: GAUGE THEORIES AT STRONG COUPLING 73

3 Sound waves in strongly coupled non-conformal gauge theory plasma 74 3.1 Non-extremal Af — 2* geometry 80 3.1.1 The high temperature Pilch-Warner flow 82 3.2 M = 2* SYM equation of state and the speed of sound 84 3.3 Sound attenuation in J\f = 2* plasma 85 3.3.1 Correlation functions from supergravity 85 3.3.2 Fluctuations of the non-extremal Pilch-Warner geometry ... 86 3.3.3 Gauge-invariant variables 88 Sound wave quasinormal mode for AT = 4 SYM 92 Bosonic mass deformation of the Af = 4 sound wave mode . . 93 Fermionic mass deformation of the J\f = 4 sound wave mode . 94 3.4 Solving the fluctuation equations 95 3.4.1 Speed of sound and attenuation constant to O (5|) in N = 2* plasma , 95 Step 1 96 Step 2 97 Step 3 98 Step 4 100 3.4.2 Speed of sound and attenuation constant to O (8\) in M = 2* plasma 101 Step 1 101 Step 2 102 Step 3 103 Step 4 103

xi 3.5 Conclusion 104 3.6 Appendix: Energy density and pressure in M — 2* gauge theory . . . 105 3.7 Appendix: Coefficients of Eq. (3.52a) 106 3.8 Appendix: Coefficients of Eq. (3.52b) 107 3.9 Appendix: Coefficients of Eq. (3.52c) 108 3.10 Appendix: Structure functions of the solution 3.99 109 3.11 Appendix: Structure functions of the solution 3.121 110

4 Hydrodynamics of Sakai-Sugimoto model in the quenched approx­ imation 116 4.1 Consistent Kaluza-Klein Reduction to 5-dimensions 119 4.2 Fluctuations 122 4.3 Hydrodynamic limit 126

5 The shear viscosity of gauge theory plasma with chemical poten­ tials 132 5.1 The proof 136 5.2 Appendix: Effective bulk action for ip 141

6 On the supergravity description of boost-invariant conformal plasma at strong coupling 147 6.1 Review of boost-invariant kinematics 151 6.2 M = 4 QGP 154 6.2.1 Consistent Kaluza-Klein Reduction 155 6.2.2 Equations of motion 156 6.2.3 Late-time expansion 158 6.2.4 Solution of the late-time series and curvature singularities . . 159 6.2.5 Curvature singularities of the string frame metric 161 6.3 KW QGP 162 6.3.1 Consistent Kaluza-Klein reduction 163 6.3.2 Late-time expansion and solution 164 6.3.3 Quadratic curvature invariants of (6.61) 165 At leading order 166 At first order 166 At second order 166 At third order 167

xn 6.3.4 Higher order curvature invariants of (6.61) 167 6.4 Conclusion 168

7 Transport properties of J\f = 4 supersymmetric Yang-Mills theory at finite coupling 177 7.1 General computational approach and the results 180 7.2 Diffusion constant of the black 3-branes hydrodynamics: the effective action approach 185 7.3 Transport properties of black 3-branes at 0((a')3) order 187 7.3.1 Shear quasinormal mode 188 7.3.2 Sound wave quasinormal mode 189

7.4 Appendix: Coefficients of Jshearfi 190

7.5 Appendix: Coefficients of JSOUndfl 191

PART 2: GAUGE THEORIES AT WEAK COUPLING 198

8 Taming tree amplitudes in general relativity 199 8.1 Preliminaries and conventions 202 8.2 BCFW construction for gravity amplitudes 203

8.3 Vanishing of Mn(z) at infinity 205 8.3.1 Outline of the proof 207 8.3.2 Auxiliary recursion relation 208 8.3.3 Induction and Feynman diagram argument 212 8.3.4 Analysis of the contribution from propagators 217 Propagators in leading Feynman diagrams of Mj 218 Propagators in leading Feynman diagrams of Mj 221 8.3.5 Analysis Of The Special Case J+ = {i} 223 8.4 Ward identities 225 8.5 Conclusions and further directions 227 8.6 Appendix: Proof of auxiliary recursion relations 228

8.6.1 Vanishing of Mn(w) at infinity 229 8.6.2 Location of poles and final form of the auxiliary recursion relations 231

9 Consistency Conditions On The S-Matrix Of Massless Particles 237 9.1 Preliminaries 240

xm 9.1.1 S-Matrix 240 9.1.2 Massless Particles Of Spin s 241 9.2 Three Particle Amplitudes: A Uniqueness Result 242 9.2.1 Helicity Constraint and Uniqueness 243 9.2.2 Examples 245 9.3 The Four-Particle Test And Constructible Theories 246 9.3.1 Review Of The BCFW Construction And Constructible The­ ories 247 9.3.2 Simple Examples 250 General Formulas For Integer Spins 250 Theories Of A Single Spin s Particle 252 9.4 Conditions For Constructibility 254 9.4.1 Behavior at Infinity 255 9.4.2 Physical vs. Spurious Poles 258 9.5 More Examples 258 9.5.1 Several Particles Of Same Integer Spin 259 Spin 1 260 Spin 2 261 9.5.2 Coupling Of A Spin s Particle To A Spin 2 Particle 262 9.6 Conclusions And Future Directions 264 9.7 Appendix: Relaxing Constructibility: Auxiliary Fields 268

10 Conclusion 276

A Copyright agreements 286 A.l Elsevier Limited 287 A.2 Journal of High Energy Physics - Institute of Physics Publishing copy­ right agreement 288 A.3 American Physical Society copyright agreement 288

VITA 311

xiv LIST OF FIGURES

9.1 Factorization of a four-particle amplitude into two on-shell three- particle amplitudes. In constructible theories, four-particle ampli­ tudes are given by a sum over simple poles of the 1-parameter family

of amplitudes M4(z) times the corresponding residues. At the loca­ tion of the poles the internal propagators go on-shell and the residues are the product of two on-shell three-particle amplitudes 249 9.2 The three different kinds of Feynman diagrams which exhibit different behavior as z —> oo. They correspond to the s-channel, t (u)-channel and the four-particle coupling respectively 257

xv LIST OF ABBREVIATIONS AND SYMBOLS

QFT : Quantum Field Theory

QCD : Quantum Chromo-Dynamics

RHIC : Relativistic Heavy Ion Collider

SYM : Supersymmetric Yang-Mills

QGP : Quark Gluon Plasma

MHV : Maximal Helicity Violating

BCFW : Britto-Cachazo-Feng-Witten

BGK : Berends-Giele-Kuijf

KLT : Kawai-Lewellen-Tye

AdSd : d-dimensional Anti-de-Sitter space-time

Sd : d-dimensional Sphere

[A,B]± == AB ± BA Graded algebra

XVI PREFACE

The main subject of this thesis is the study of gauge theories both in a strongly- coupled and weakly-coupled regime. Specifically, in the first part the application of the physical conjecture relating gauge theories and string theory to the investigation of transport properties in strongly-coupled gauge theory plasmas is analyzed. The second part is devoted to the study of recursive relations for tree level scattering amplitudes in general relativity as well as the analysis of these methods to explore the spectrum of consistent theories of interacting arbitrary spin particles where the S-matrix can be defined.

However, not all my work has been included in this thesis, Another subject of investigation has been the worldline formalism and its application to the propaga­ tion of degrees of freedom of antisymmetric tensor fields in a curved background. I decided to not include the results I got in collaboration with Prof. Fiorenzo Bas- tianelli from Universita di Bologna and Dr. Simone Giombi from SUNY-Stony Brook (JHEP 0504 (2005) 010 [arXiv:hep-th/0503155] - JHEP 0510 (2005) 114 [arXiv:hep- th/0510010]) so that the present thesis could appear as homogeneous as possible. I only briefly contextualize and report them in what follows.

It can be shown that field theories can be recovered from the infinite-tension limit of string theories. In particular, in string theory the transition amplitude for a parti­ cle is computed using Polyakov path integrals, which are quantum mechanical func­ tional integrals. Therefore, it was possible to reduce the calculation of the S-matrix of particular processes to the analysis of a quantum mechanical path integral. This first quantization treatment has been completely disentangled from any string the-

xvn ory formulation by defining appropriate quantum mechanical cr-models to describe the dynamics of relativistic particles. This gave rise to what is known as worldline formalism. Such an approach also allows to discuss gravitational interactions by considering, for example, the path integral quantization of spin-0 and spin-1/2 par­ ticles in a curved space-time. The propagation of antisymmetric tensor fields in a gravitational background has also been studied. In this case, the mechanical model to consider is the M = 2 spinning particle. It is a one-dimensional field theory in which the space-time coordinates are viewed as fields on a one-dimensional space (the worldline), and two real (or one complex) anti-commuting variables implement the spin degrees of freedom. It also shows two worldline supersymmetries. We de­ scribed the one-loop effective action of antisymmetric tensor fields, which turns out to be expressed in terms of two parameters, the proper-time and a parameter re­ lated to the gauge fixing of an internal symmetry of the worldline action. The latter parameter restricts the propagation to a single tensor field. The worldline represen­ tation provides a drastic simplification with respect to the heat-kernel methods. It allows computation of the first Seeley-DeWitt coefficients for antisymmetric tensor fields in arbitrary dimensions. They are coefficients of the proper-time expansion of the one-loop effective action. In the case of a spin-1 particle in four-dimension, we were able to compute the trace anomaly which is given by one of those coefficients. In the massive case the Seeley-DeWitt coefficients can be derived from the mass- less ones by shifting the space-time dimensions by one. Furthermore, the worldline representation has been used to calculate the one-loop contribution to the graviton self-energy due to both massless and massive antisymmetric tensor fields.

I hope that the structure of the thesis (i.e. the way the papers have been put together) and the decision to exclude part of the work done will turn out to be the right one and helpful to whom needs to consult this manuscript. I also hope that who is interested in the excluded work can find the above description useful, at least to get a flavor of the results my collaborators and I obtained. Who would like to

xvm get more insights about them can always refers to the two papers mentioned in the previous paragraph and references therein.

Terranova da Sibari, Italy 29 December 2007, 11:45pm

Paolo Benincasa 1

CHAPTER 1

Introduction

1.1 General introduction

The most successful theoretical framework in describing nature at the microscopic level is quantum field theory (QFT). In this type of approach, the states of a system are described by a wave-function in the occupation number space, whose squared absolute value determines the probabilities of the different values of such numbers, and the point particles are seen as excitations of some fields (which are distribution- valued functions). The guiding principle in formulating a theory for the fundamental interactions is the requirement of invariance of the theory under a local gauge trans­ formation. It implies the introduction of gauge fields, which are nothing but fiber bundle connections and a gauge transformation maps a section of a principal bundle to another one. The gauge fields mediates the fundamental interactions, i.e.: upon quantization, matter particles interact with each other by exchanging bosons which are excitations of the gauge fields.

Among the four fundamental interactions (electromagnetism, weak and strong nu­ clear forces, gravity) three of them are well-described by the so-called Standard- Model (SM), which includes the electro-weak theory by Glashow-Weinberg-Salam

[1, 2, 3] and the Quantum-Chromo-Dynamics (QCD). The gauge group is SU(3)C x

SU(2)L x U(l)y, SU(3)C and SU(2)L being the gauge groups underlying the strong 2 and weak interactions respectively while the electromagnetic interaction arises from

U(l)em C SU{2)L x U(1)Y- The matter fields are in the fundamental representation of the gauge group, while the gauge bosons are in its adjoint representation.

One of the biggest success of the Standard Model has been the detection of the gauge bosons W^1 and Z° [4, 5, 6, 7], which mediates the weak interaction, with the same masses that the model predicted. In the model, the masses of W± and

Z° are generated via the Higgs mechanism [8, 9]: the symmetry SU(2)L X U(1)Y is spontaneously broken by the presence of a condensate, the Higgs scalar, in the ground state. At present, the Higgs mechanism still represents an open problem because there is no experimental evidence of the existence of the Higgs boson yet.

Despite its incredible success, the Standard Model shows several problems. Firstly the theory presents a large number of parameters which need to be put by hand, like the list of elementary particles and their masses and the strength of the fun­ damental forces. Another parameter which the theory does not account of is the electro-weak mixing angle which measures the way the electromagnetic and the weak forces combine. Furthermore, the Standard Model does not provide an explanation to the possibility that neutrinos are massive rather than massless, as strongly in­ dicated in recent experiments. As far as the strong force is concerned, there is no real explanation for quark confinement and, at low energies, QCD is analytically untreatable given that in this regime the theory becomes strongly coupled and there is no method which provides a complete and satisfactory picture of the theory in this regime. Finally, the SM cannot be considered as a complete theory of fundamental interactions because it does not deal with gravity. Therefore, given the large set of unanswered questions, it is plausible to look for an alternative approach.

In attempting to investigate the interaction of hadrons, one dimensional objects, like Wilson loops and flux tubes [10], have been observed. This has led to the idea of replacing the concept of the zero-dimensional particle with one-dimensional string 3 as the fundamental object, giving rise to what today is known as String Theory [11, 12, 13, 14, 15]. In String Theory, the spectrum of particles comes from the different states of oscillations of the strings. Among these particles, a massless spin- two particle naturally arises, which turns out to be the mediator of the gravitational interaction. The existence of such a particle which is not present in the hadron spectrum, together with the high number of space-time dimensions of the theory (26 for the bosonic string, 10 for the superstring) and the excellent results obtained by QCD, led string theory to be discarded as hadronic theory and, later, be considered as potential theory for the unification of all the fundamental forces, including gravity. Since the string spectrum shows the presence of a spin-2 particle which mediates gravity, string theory can be viewed as a consistent theory of quantum gravity being UV-divergence free.

The bosonic string is described, in analogy with the point particle, by the Nambu- Goto action on the worldsheet

2 5NG = -^ Jd cnJ-det{(dax») (dpx")^}, (1.1) or, equivalently (at classical level), by the Polyakov action

2 a SP = —^ J d a S=h h ? (dax^) (dux") Vlll/ (1.2) where aa (a = 0,1) are the worldsheet coordinates, the coefficient T — l/2ira' is

2 the string tension (a' — l s, ls being the string length) and hap is the metric on the worldsheet with h being its determinant. Typically, the quantization procedure is applied on the action (1.2) because the presence of the square-root in (1.1) is problematic. It turns out that, leaving the space-time dimension d generic, the quantized theory develops a Weyl anomaly, whose value depends on d. There is no Weyl anomaly for the critical value d = 26 only. The number of space-time dimensions is thus fixed by the requirement of absence of Weyl anomaly. One of the features of the theory is that its spectrum contains bosonic excitations only. 4

Furthermore, the bosonic string theory suffers the presence of a tachyonic state as ground state and, therefore, the vacuum is unstable.

Fermionic degrees of freedom can be incorporated in two different ways, i. e. via the RNS [16, 17] and the GS [18] formalism. While the latter formulation is necessarily in the light-cone gauge, the RNS formulation implements fermionic degrees of freedom introducing worldsheet spinors ^ and worldsheet supersymmetry:

2 a } a SRNs = --^ J d a [V^h f (dax»)(d0x»)+rp daV}v^, (1-3) where l a P°= ft A P = ft j), b ,/]+ = 2^. (1.4)

The quantization of (1.3) also returns a tachyonic state in the spectrum. Moreover, the realization of the fermionic degrees of freedom in (1.3), as well as the space-time supersymmetry, is not manifest. These problems are solved using the so-called GSO projection [19].

In the case of the superstring, the requirement of absence of Weyl anomaly implies that the theory has to be formulated in a ten-dimensional space-time. In principle, it is possible to reduce the theory to an effective four-dimensional theory by com- pactifying the six extra-dimensions, i.e. by formulating the theory on a background of the form M4 x MQ where M4 is the four-dimensional Minkowski space-time and M.Q is a compact six-dimensional manifold, and integrating over the six-compact dimensions.

The consistency of the theory can be tested by checking the absence of gravitational and gauge anomalies. This requirement strongly constrains the theory. It turns out that it leads to five possible types of string theory: type IIA and IIB which have M = 2 supersymmetries and gauge group U(l) (they differ with each other because the two supercharges have opposite chirality in the type IIA and same chirality in type IIB), type I which involves both oriented and unoriented open and closed 5 strings, have J\f — 1 supersymmetry and gauge group 50(32), heterotic with gauge group SO(32) and heterotic with gauge group E8 x E$ [20]. However, these five string theories, as well as the eleven-dimensional supergravity [21], are nothing but different aspects of the same theory [22]. More precisely, it is possible to define a theory with several vacua and the previous different theories are perturbative expansions around different vacua. The five string theories and the eleven-dimensional supergravity (or also M-theory whose low-energy limit is the eleven-dimensional supergravity itself) are related to each other by duality transformations (S,T,U dualities). Specifically, type IIA and type IIB are related by T-duality: when the ten-dimensional Minkowski space Mio is compactified on a Sl, it is possible to define a transformation which inverts the radius of S1 so that type IIA string compactified on S1 with radius

l 2 R is mapped into type IIB compactified on an S whose radius is l P/R, and vice versa. More generally, a T-duality transformation is a mapping between a theory compactified on a space of small volume and another theory compactified on a space of large volume. This same type of T-duality described for type II strings holds between the two heterotic strings. T-duality transformations relate M-theory to type-IIA and E% x E& heterotic strings, but not vice versa. In order to get M-theory from these two string theories, it necessary to take the string coupling constant gs to infinity. Finally, type I and 50(32) heterotic strings are related to each other by an S-duality transformation, i. e. a transformation which inverts the string coupling constant gs —* l/gs determining a weak-strong coupling duality.

The discovery of these dualities allowed to take into consideration also non-perturbative effects in string theory. Consider an open string with Neumann boundary conditions for its endpoints. Applying a T-duality transformation on n compactified dimensions maps this open string into another open string whose zero mode of the compactified coordinate does not carry momentum and has Dirichlet boundary conditions along all the n directions. Therefore, the endpoints of the open string are free to move on a hypersurface of dimension 10—n. Such an hypersurface is called Dp-brane (p — 9—n) 6

[23, 24, 25]. As a 1-form potential naturally couples to the worldline of a particle, (p + l)-form potentials naturally couple to the Dp-brane world-volume. If type-II (A and B) string theories are considered, the massless spectrum in the Ramond- Ramond sector allows only for potentials of odd rank (type IIA - p = 0,2,4,6,8) or even rank (type IIB - p = —1,1,3,5, 7,9). As a consequence, type IIA and type IIB admit Dp-branes having even and odd number of spatial dimension respectively. The open string endpoints can have static degrees of freedom called Chan-Paton degrees of freedom: each endpoint is in a state i or j with i, j'< — 1,. .., N, It turns out that the open string amplitudes are invariant under a U(N) symmetry [26]. When one space-time dimension is compactified on an S1, it is possible to introduce a Wilson line diag (#i ..., 9^) /2ixR which breaks the Chan-Paton gauge symmetry U(N) to [[/(l)] . Being a pure gauge, the Wilson line can be gauged away. As a consequence, the fields pick up a phase diag {e~ldl,..., e~l9N}. This implies that in the T-dual theory the open string endpoint in the i-th Chan-Paton state is located at 6iR' (i = I ..., N): N parallel D-branes are identified and the endpoints of the open strings can belong to different D-branes. In the case of N parallel D-branes sitting at the same point, the presence of the Chan-Paton degrees of freedom, carried by the endpoints of the open string, implies that the actual effective loop parameter expansion is gsN. The dynamics of the Dp-branes is governed by the DBI-action

dP+1(j det m SDBI = - t2ir\p (a>)(p+i)/2 / y- {Vap + {dax ) (d/)Xm) + 2iva'Fafs}, (1.5) with m = p+ 1,... ,9.

Consider now the low-energy limit of (1.5), which is given by the degrees of freedom with energy E <^C l/y/a'. It can be written as

ilp+l 1 (p+1) 2 1 (p 3)/2 (27r)^s(a') / 4(27r)P- 3s(a') " r r 2 i (L6) p+1 a0 m a 4 x / d a FapF + ——j (dax ) (d xm) + 0(F ). J I (27TQ;') J Notice that the action (1.6) can be also obtained from d = 10 C/(l) Yang-Mills 7 theory, by taking the fields obtained from the dimensional reduction to depend on the coordinates on the brane only. This lead to the following relation between the

Yang-Mills coupling constant and the string parameters gs and a':

2 3 glM = (27rr gs(a'r . (1.7)

All this can be generalized to a stack of N parallel Dp-branes whose low energy limit turns out to be given by the dimensional reduction to p + 1 dimension of the ten-dimensional J\f = 1 U(N) Supersymmetric Yang-Mills theory.

Actually, the link in between string theory (and Dp-branes) and gauge theories goes well beyond what we have just described. If one consider p = 3, the low- energy limit for the stack of N D3-branes (1.6) is the four-dimensional U(N) Af — 4

Supersymmetric Yang-Mills theory, with coupling constant g\M = 2-Kgs. Moreover, the near horizon geometry for the N D3-branes is AdS^ x S5. Perturbatively, the first type of description holds when gsN ~ g\uN *C 1, while the second one when gsN ~ g\MN ^> 1 since one needs the radius of curvature R of yWS-space to be large. Given all these facts, in [27] Maldacena conjectured a physical correspondence between M = 4 SU(N) Supersymmetric Yang-Mills theory in four dimensions and type IIB string theory on AdS^ x S5, with the SYM theory sitting on the boundary

4 of AdS5. Notice that the radius of curvature R is proportional to N, (R/lp) ~ JV. Therefore, in order to have large curvature on the string theory side and be able to apply perturbation theory, one needs to take N large. This correspondence can be generalized to d-dimensional conformal field theories and string theories on AdSd+i x M.g-d (f°r a review see [28]) and even to non-conformal gauge theories and string theories on "some" higher dimensional curved background [29]. This conjecture is a realization of the holographic principle [30, 31, 32], which states that a quantum gravity theory can be described in terms of a theory living at its boundary with less then one degree of freedom per Planck area. In order to count the degrees of freedom, one can introduce an IR cut-off in the AdS*, theory, so that the maximum 8

1 2 3 entropy, which is proportional to the area of the region [33], is SAds ~ N /e , e

2 3 being the IR cut-off. On the CFT side, the entropy is SSYM ~ N /e' , with e' as UV cut-off. Notice that a cut-off in the IR region in the gravity theory corresponds to a cut-off in the UV in gauge theory [34].

As we just discussed, the gauge/gravity correspondence related gravity theories and gauge theories in different regimes. The importance of this conjecture is two-fold: it allows one to obtain information about gauge theories by computation in the related string theory, and vice versa. The aim is to understand QCD in the strong coupling regime, where no satisfactory analytic approach is known. Most of the known results are obtained by lattice simulations, which anyway do not provide a complete picture of the strongly-coupled theory. An analytic method is desiderable in order to get a better understanding of QCD. The gauge/gravity correspondence provides such a tool. Anyway, this statement needs to be made a little bit more precise. Indeed, no string theory has been built to be dual to the real-world QCD and it is more likely that it is not possible to construct the string-dual to it. However, several insights can be still obtained. The idea is to investigate gauge theories which share some important features with QCD, like confined phase, mass gap, chiral symmetry breaking, so that a qualitative understanding can be achieved. Indeed, all the results obtained cannot be quantitatively trusted. However, the gauge/gravity analysis may provide a good qualitative description of low-energy QCD phenomena.

In this view, hydrodynamic studies acquire a particular relevance, in relation to the quark gluon plasma, which is a phase in QCD where quark and gluons are "almost" free. In heavy-ion collision experiments at RHIC (and in the near future at LHC as well), where nuclei of gold collide with a center of mass energy of 200 GeV per nucleon, a thermal (deconfined) strongly-coupled state (the Quark Gluon Plasma)

lrThe Bekenstein bound [33] states that the entropy in a given region of space is S < A/AG, A being the area of the region, which implies that the degrees of freedom in the region scales as the area of its boundary. 9 should be produced. The quark gluon plasma should then expand and hadronize, i.e. produce jets of hadrons.

For hydrodynamics we mean a large distance and large time scale regime (distance and time scale being of the same order), in which external perturbations varies slowly in both time and space. In such a regime, a system can be described using macroscopic hydrodynamics, viewed as an effective theory where only slowly-varying (in space and time) modes contributes while all the others have been integrated out. A lot of excitement about this type of analysis arose when it was understood that in Af = 4 SU(N) SYM at finite temperature the ratio shear viscosity by entropy density is a universal constant [35] while for thermal field theories in general such a ratio is bounded from below by the same universal constant [36, 37]

V- > 1- = I reSt°ring the 1 = T4- « 6.08 x 10-" Ks, (1.8)

s 4-7T I measure units J ATTKB and that gauge theories at zero chemical potential which admit holographic string theory dual saturate this bound [38, 39]. A further impulse to this research direction has been given when a prescription to compute equilibrium correlation functions in Minkowski space-time has been given [40]. This prescription provides a concrete tool to investigate both conformal and non-conformal plasmas and, therefore, to strengthen our hope to get insights in the dynamics of the Quark Gluon Plasma. Moreover, recent developments [43, 44] seem to allow a good control of the second order hydrodynamics, at least in the conformal case. These are really important steps towards understanding of the gauge theory plasmas as dynamical phases.

So far we mentioned the importance of having an analytic tool for investigating the strong coupling regime of QCD-like theories. It is also important and interesting to analyze the perturbative regime of some theories. It is indeed true that the weakly- coupled regime of gauge theories is better-understood because perturbative methods are well-developed. However, looking at this regime of gauge theories can provide informations about the structure of the theories themselves. One idea is to look at 10 scattering amplitudes. Consider Yang-Mills theory. It was already noticed that at tree level a helicity amplitude can be written in terms of tree level diagrams which are constructed from tree level MHV amplitudes with some legs continued off-shell [45], This type of representation showed to be a powerful algorithm for calculating gauge theories scattering amplitude and to be connected with some twistor string theory, where the particles involved in an MHV amplitude lie on a straight line in twistor space [46]. Interestingly, it was also proved that tree level scattering amplitudes of gluons satisfies a particular recursive relation: it can be written in terms of sum of products of two amplitudes with fewer external legs and a propagator [47]. Originally, this recursive relation was obtained by looking at the IR behavior of J\f = 4 one-loop amplitudes. However, later a simple and elegant proof has been formulated using the properties of the tree level amplitudes only [48]. The idea is to look at the singularity structure of the amplitudes. This goes back to the old approach of the S-matrix [49], in which the S-matrix is assumed to be an analytic function of the external momenta. There the particles emerge from poles in the S-matrix. The residues of the poles are required to factorize in two contributions, one depending on the momenta and helicities of the initial channel and the other on the momenta and helicities of the final channel, so that two successive reactions, which appear in the S-matrix, are independent.

At tree level, the singularity structure of scattering amplitudes shows only poles in the Lorentz invariants. Considering the scattering amplitudes as functions of the Lorentz invariants is a multi-variable problem which is not easy to attack. The idea [48] is to firstly reduce the problem to a single-variable problem. This can be done by introducing a one-parameter deformation in the momentum space, leaving the amplitudes physical. This means that the deformation is chosen in such a way that it does not break the on-shell condition and momentum conservation. In order to introduce this one-parameter deformation, one necessarily has to go to the complexified momentum space, since the on-shell and momentum conservation 11 constraints in the real momentum space would lead the deformation to be trivial. These two physical requirement do not fix uniquely the deformation. The simplest one can think of, known as BCFW-deformation, is

k p(>) _> pW-zq pCi) _> pV)+zq p(V _> p( \ Vfc ± i,j, (1.9) where only the momenta of two particles, namely i and j, have been deformed. Now, it is possible to look at the one-parameter family of amplitudes and its singularity structure in z. Notice that the real physical amplitude can be obtained by setting to zero the deformation parameter z. The poles appear to be simple poles only, because of the particular form of the deformation (1.9). This is already an amazing result because it implies that the amplitude can be written in terms of the residues of the poles, provided that the deformed amplitude vanishes as the parameter is sent to infinity. The residue of the poles are nothing but the product of two amplitudes with fewer external particle. In Yang-Mills theory, a Feynman diagrams as well as MHV diagrams analysis easily show that the deformed amplitude vanishes at infinity. Therefore, the general structure of the recursive relation is

M p Mn ~ J2 L(l- k)^2MR(Pk,j), (1.10) kdV k where V is the set of poles and the "hatted" quantities are calculated at the location of the poles.

The result is much deeper than what it can appear at first sight. The existence of this type of recursive relations for Yang-Mills indeed provides a really efficient tool for computing tree level scattering amplitudes as well as the discontinuities in the branch-cuts at loop level, those discontinuities being expressed as integrals of products of tree level amplitudes over the Lorentz invariant phase space. However, this is not the end of the story. The recursive relation implies that Yang-Mills amplitudes are uniquely determined by the three particle amplitudes, which are not zero in the complexified momentum space. This is really amusing, giving the fact that Yang-Mills Lagrangian shows both a three- and a four-gluons vertices. 12

Generally, the analysis of the infinite-,? limit of the deformed amplitudes can be tricky, as gravity showed [50, 51, 52]. The reason is that typically it relies on Feynman diagrams in which one cannot account of the cancellations among all the diagrams contributing to the amplitude as well as cancellations in a single diagram itself. Therefore, the outcome of Feynman diagram analysis can be interpreted as an upper bound on the large-z behavior of the deformed amplitude. Recently, a new method to investigate the large-z behavior of the scattering amplitude has been proposed [53]. It no longer relies on Feynman diagram expansion and, therefore, it allows a more precise analysis.

Beside Yang-Mills and General Relativity, it is indeed interesting to seek an S- matrix for arbitrary spin particles, paying attention to what happens in the case of spin higher then two. The eventual existence of higher spin particles (for reviews see [54, 55, 56]) is interesting in itself, because it would imply that it is possible to define consistent interactive theories other than the ones we already know and investigate why at the moment there is no signature of their existence. Another reason comes from string theory: taking the zero tension limit of string theory one obtains a spectrum of particle containing an infinite tower of massless particles in which an infinite number of high spin particles is present. Moreover, there is the idea that symmetries which at string level are broken can be restored in this limit. Therefore, the high energy limit of string theory would be governed by a gauge theory of interactive higher spin particles and a neat formulation of this gauge theory would allow to understand better string theory at high energy. Another motivation is given by the AdS/CFT correspondence, for which each conserved high spin current in CFT is related to massless high spin field propagating in AdS [57]. The BCFW- deformation method may be a powerful tool to investigate such theories, provided they admit an S-matrix. 13

1.2 Overview of the thesis

This thesis is structured in nine more chapters which are organized into two main parts. Part 1 contains studies on the transport properties of strongly coupled gauge theories using gauge/string correspondence. The papers are not organized in a chronological order, but rather in a conceptual one. I chose to first put papers where the supergravity approximation has been used and at the end the only paper where the leading a'-correction has been considered. Part 1 contains Chapters 3-7.

In Chapter 2 the relevant literature is reviewed. In particular, we briefly discuss the conjectured gauge/gravity duality. We also review both first and second or­ der hydrodynamics2 and discuss the prescription for computing retarded thermal correlators proposed in [40] in some detail. We also explain the holographic renor- malization procedure which allows us to obtain a well-defined stress-energy tensor. Finally, we review the KSS bound [36] and discuss the universality class of theories at zero-chemical potential defined by the saturation of the KSS bound. As far as the weakly-coupled regime is considered, we will review and discuss the helicity- spinor formalism and the BCFW construction for scattering amplitudes which will be widely used in Chapters 8 and 9.

In Chapter 3 we study sound wave propagation in strongly coupled non-conformal gauge theory plasma. We compute the speed of sound and the bulk viscosity of TV = 2* supersymmetric SU(N) Yang-Mills plasma at a temperature much larger than the mass scale of the theory in the limit of large N and large 't Hooft coupling. The speed of sound is computed both from the equation of state and the hydrodynamic pole in the stress-energy tensor two-point correlation function. Both computations lead to the same result. Bulk viscosity is determined by computing the attenuation

2More precisely, here we review Israel-Stuart theory [41, 42] for second-order hydrodynamics, which is a phenomenological theory and seems to not account of all the possible terms at this order [43] 14 constant of the sound wave mode.

In Chapter 4 we analyze transport properties of the finite temperature Sakai-Sugimoto model. The model represents a holographic dual to 4 + 1 dimensional supersym- metric SU(NC) gauge theory compactified on a circle with anti-periodic boundary conditions for fermions, coupled to Nf left-handed quarks and Nf right-handed quarks localized at different points on the compact circle. We analytically compute the speed of sound and the sound wave attenuation in the quenched approxima­ tion. Since confinement/deconnnement (and the chiral symmetry restoration) phase transitions are first order in this model, we do not see any signature of these phase transitions in the transport properties.

Chapter 5 deals with strongly coupled gauge theory plasmas with conserved global charges that allow for a dual gravitational description. We study the shear viscosity of the gauge theory plasma in the presence of chemical potentials for these charges. Using gauge theory/string theory correspondence we prove that at large 't Hooft coupling the ratio of the shear viscosity to the entropy density is universal.

In Chapter 6 we study string theory duals of the expanding boost invariant confor- mal gauge theory plasmas at strong coupling. The dual supergravity background is constructed as an asymptotic late-time expansion, corresponding to equilibration of the gauge theory plasma. The absence of curvature singularities in the first few orders of the late-time expansion of the dual gravitational background unambigu­ ously determines the equilibrium equation of the state, and the shear viscosity of the gauge theory plasma. While the absence of the leading pole singularities in the grav­ itational curvature invariants at the third order in late-time expansion determines the relaxation time of the plasma, the subleading logarithmic singularity cannot be canceled within a supergravity approximation. Thus, a supergravity approximation to a dual description of the strongly coupled boost invariant expanding plasma is inconsistent. 15

In Chapter 7 we compute the leading correction in inverse 't Hooft coupling to the shear diffusion constant, bulk viscosity and the speed of sound in the large-iV N = 4 Supersymmetric Yang-Mills theory plasma. The transport coefficients are extracted from the dispersion relation for the shear and the sound wave lowest quasinormal modes in the leading order a'-corrected black D3 brane geometry. We find the shear viscosity extracted from the shear diffusion constant to agree with result of [60]; also, the leading correction to bulk viscosity and the speed of sound vanishes. Our computation provides a highly nontrivial consistency check on the hydrodynamic description of the a'-corrected non-extremal black branes in string theory.

Part 2 contains the analysis of the tree-level structure of General Relativity and, more in general, arbitrary spin particles, paying attention to the possibility of defin­ ing consistent interactive theories involving particles with spin higher than 2 which admit an S-matrix. Part 2 includes Chapters 8 and 9.

In Chapter 8 we give a proof of BCFW recursion relations for all tree-level amplitudes of gravitons in General Relativity. The proof follows the same basic steps as in the BCFW construction and it is an extension of the one given for next-to-MHV amplitudes in [51]. The main obstacle to overcome is to prove that deformed graviton amplitudes vanish as the complex variable parameterizing the deformation is taken to infinity. This step is done by first proving an auxiliary recursion relation where the vanishing at infinity follows directly from a Feynman diagram analysis. The auxiliary recursion relation gives rise to a representation of gravity amplitudes where the vanishing under the BCFW deformation can be directly proven. Since all our steps are based only on Feynman diagrams, our proof completely establishes the validity of BCFW recursion relations. This means that many results in the literature that were derived assuming their validity become true statements.

In Chapter 9 we introduce a set of consistency conditions on the S-matrix of theories of massless particles of arbitrary spin in four-dimensional Minkowski space-time. 16

We find that in most cases the constraints, derived from the conditions, can only be satisfied if the S-matrix is trivial. Our conditions apply to theories where four- particle scattering amplitudes can be obtained from three-particle ones via a recent technique called BCFW construction. We call theories in this class constructible. We propose a program for performing a systematic search of constructible theories that can have non-trivial S-matrices. As illustrations, we provide simple proofs of already known facts like the impossibility of spin s > 2 non-trivial S-matrices, the impossibility of several spin 2 interacting particles and the uniqueness of a theory with spin 2 and spin 3/2 particles. 17

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CHAPTER 2

Literature review

In this chapter, all the necessary background is reviewed. Like the body of this thesis, it is organized in two parts. The first part concerns the hydrodynamical description of gauge theories using the gauge/gravity correspondence. The second one deals with the study of interactive field theories using a novel S-matrix approach.

In the first part, the gauge/gravity correspondence is discussed. Specifically, the relevant feature of large-N field theories, type-IIB string theories and D-branes are discussed in some detail. The gauge/gravity correspondence is reviewed by dis­ cussing the relation between type IIB string theory on AdS$ x S5 and M — 4 Super- symmetric Yang-Mills theory in four dimensions, and mentioning the generalization to the relation of theories on AdS^+x and d-dimensional conformal field theories as well as the application of the conjecture to non-conformal theories. Finally, we review first and second order relativistic hydrodynamics and the application of the gauge/gravity correspondence to the hydrodynamics regime of field theories and its consequences.

In the second part, a self-contained discussion about the spinor-helicity formalism for scattering amplitudes is provided as well as the BCFW-construction is reviewed. 24

2.1 Gauge Theories at strong coupling: hydrodynamics and gauge/string duality

In this section the conjectured correspondence between conformal field theories on d- dimensional space-time and string theories on a AdS^+i x M-s-d background [1, 2, 3], where M.g~d is a (9 — d)-dimensional compact manifold with positive curvature, is discussed. In particular, the starting point is the duality between N = 4 SU(N) supersymmetric Yang-Mills theory on a

2.1.1 Large-N field theory

Most of analytic results concerning QCD at strong coupling were obtained by study­ ing the large-iV limit of QCD (for a review about large-N QCD see [4]). The gauge group in QCD is 5(7(3). It is possible to consider SU(N) as a gauge group and then take the limit N —> oo. Fields in the adjoint representation of the gauge group can be diagrammatically represented in the double-line notation [5], where gluon prop­ agators are drawn as a pair of oriented lines carrying color indices, one line carrying an index in the fundamental representation N and the other line carrying the index in the anti-fundamental representation N. The quark propagators are represented as a single line carrying a color and a flavor indices. These diagrams in the double- line notation (ribbon graphs) can be viewed as open string worldsheet. This implies that two connected three point vertices can be mapped into a four point vertex by contraction and, vice versa, one four point vertex can mapped into two connected three point vertices by an expansion. Contraction/expansion relate higher point ver­ tices to low point vertices (and vice versa). This means that the coupling constant for the n-point vertex can be written as K"-2, K being the coupling constant for the 25 three-point vertex. Let us consider a given ribbon graph having b closed lines and vn n-point vertices. Since the indices associated with each line of the ribbon graphs run from 1 to N, the b closed lines generate a factor Nb. Therefore, the graph under analysis acquires a factor NbK^n^Vn. These closed lines define the boundaries of the ribbon graph and being the lines oriented the boundaries are oriented as well. If an oriented disk is attached to each boundary, the ribbon graph plus the disks form a Riemann surface. Notice that the original diagram and the disks naturally define a cell-decomposition of the Riemann surface. If e is the number of edges in the ribbon graph and g the genus of the Riemann surface, then the number of edges e, the number of vertices v = ^2vn, the number of boundaries b and the genus g of the Riemann surface are related to each other by the Euler characteristic formula:

v-e + b = 2-2g = \- (2.1)

Furthermore the number of edges and the number of n-point vertices are related by Y2 nvn — 2e. Therefore, the overall factor of the graph can be written as

b 2 ]\r KE(«- K _ ]\jb f]y-(e-v)jye-v\ K^{e-v) (2.2) = N^{^N)h-(2-29).

Introducing the 't Hooft coupling A = n2N, a general correlation function of gauge- invariant operators can be written as

oo oo 2 29 C xb {2 2g) 2_29 G = Y,N - J2 9fi ~ ~ = Z> G9(A), (2.3) 9=0 b g=0

Gg(X) is some polynomial in A. Notice that the expression in (2.3) has the form of a close string expansion with 1/N as coupling constant. In the large-JV limit, keeping A fixed, the leading contribution is of order iV2 and comes from the Riemann surface with genus g = 0, i.e. the sphere. This means that the leading contribution comes from all those diagrams which can be drawn on a sphere, and are called planar diagrams. An n-point correlator of gauge invariant operators Ok can be computed by adding a coupling term Sint = ./V J^fc KkOk to the action, extracting the generating 26 functional W containing the connected vacuum diagrams only and differentiating elW n times with respect to «'s

n Gn(0) = (l[Ok) = (iN)- T^- (2.4)

Each insertion of a vertex for the operator O is equivalent to add a vertex in the diagrams. From (2.3), it is easy to see that in the large-iV limit and at fixed 't Hooft coupling A the vacuum diagrams contribute with a factor of iV2 and, therefore, the n-point correlator of gauge invariant operators behaves as ~ N2~n. This means that two-point correlators are of the order one, three-point correlators are of the order 1/N and so on, which again shows that 1/JV can be taken as coupling constant.

The similarity between the 1/JV expansion of gauge theories in the large-iV limit and the loop expansion in string theory suggests a connection between these two types of theories.

2.1.2 Type IIB string theory

A string which contains both bosonic and fermionic excitations is consistently de­ scribed by a worldsheet action endowed with worldsheet spinors and worldsheet supersymmetry (see (1.3) and (1-4)). In the light-cone coordinates a± =

2 S = A ^ , J d a {(d+x) • (cLz) - ^ • d+i>h - V;R • cL*} , (2.5) 47rcr where the spinor ip of (1.3) has been expressed as a two-component Majorana spinor

L i> = K j, 4>U = VVR- (2.6)

The equations of motion for x^ and I/JL/R are

d.d+x" = 0 <9_V>L = 0 - d+V'R, (2.7) 27 which lead to independent expansions for x^ and ip in left- and right-movers

(2.8) -0L = A(&+), V'R = ^RO-)-

Taking a1 G [0, 27r], the spinors ^L/R can consistently satisfy two type of boundary conditions, which define the so-called Ramond sector (R) and the Neveu-Schwarz sector (NS)

Ramond sector: iph(a°, 2TT) = ipR(a°, 2ir)

Neveu-Schwarz sector: I/>L(CT , 27r) = —^R(cr, 27r).

The mode expansions of x£/R and V'L/R for an open string can be written as follows

with { Z Ramond sector . ' , Z + g Neveu-Schwarz sector, and ij)*_k = ipk (Majorana condition). In the case of closed string the modes ^_fc for the left- and right- movers are independent of each other. Therefore, the boundary conditions are applied separately. The different possible pairings of left- and right- movers define four sectors: NS-NS, NS-R, R-R, R-NS. Two of them are bosonic (NS-NS, R-R) and the other two are fermionic (NS-R, R-NS). The vacuum of the theory is defined by acting with the zero modes of the expansions (2.10) on the trivial vacuum |0), while excited states are constructed by applying negative modes in (2.10) on the vacuum of the theory in the different sectors sepa­ rately. The ground state in the NS sector turns out to be tachyonic. This problem is cured via GSO projection [6], which realizes space-time supersymmetry and defines a chiral spinor in the R-sector as well. In particular, after the GSO projection, the ground state in the NS sector is a massless vector, while in the R-sector the left- and right-movers of the GSO-projected spinor can have same or opposite chirality. In the first case, the superstring theory obtained is called type IIB. In the other case, 28 one gets type IIA superstring theory. In the case of closed strings, the states are given by tensor products of left- and right-movers. At low-energy (supergravity), i.e. taking the infinite tension limit, one can reduce the spectrum to massless states only, which are classified on the basis of the transformation properties under the group 50(8). The NS-NS, the NS-R and R-NS, and the R-R sectors are constructed as tensor product of two massless vectors, one massless vector and one massless spinor and two massless spinors respectively. These states can be decomposed according to the irreducible representation of S0(8), so that

NS-NS sector 8W 8„ = 35 © 28 © 1

NS-R sector 8,, 8S = 8S © 56s

R-NS sector 8S ® 8„ = 8S © 56s

R-R sector 8S 8S = p-form fields.

Specifically, the state in the NS-NS sector is decomposed into the symmetric traceless irrep (35), which represents the graviton field g^, the antisymmetric irrep (28), which gives the Kalb-Ramond 2-form B^, and the trace part (1), which returns the dilaton

\PM and a spinor A correspond to the irreps 56s and 8S respectively. Finally, the R-R sector of type IIB string theory is made out of the tensor product of left- and right- movers of GSO-projected spinors with same chirality. Expanding it on a basis of antisymmetrized Dirac-matrices, it is easy to see that it can be decomposed in p-form fields. More precisely, in type IIB only p-forms with p = 2k, k = 0,..., 4 are allowed. These p-forms are not independent of each other but they are related by Hodge duality. Specifically, the 8-form and the 6-form are dual to the 0-form and 2-form respectively, while the 4-form is self-dual.

The low energy limit of type IIB string theory is type IIB supergravity, whose action 29 can be written as

dl x R(10) M SHB-SVCHA = —^- J ° ^9 { - ^2 (dMr) (3 r) + Fl- 1 (2.12) ,,, i i fermionic terms & \TH + H' \2} + 12T2 3 3 J I higher derivative terms J where the capital Latin letters M, N, ... = 0,..., 9 are ten-dimensional space-time indices, H3 = dB2, H$ = dC2 are the field strength of the Kalb-Ramond 2-form in the NS-NS sector and the field strength of the R-R 2-form C2 respectively, F5 — dC$ is the field strength of the self-dual 4-form Cj, and r is a complex scalar defined

- by r = T\ + ir2 = x + ^e * (x is the axion and (j) is the dilaton). Notice that, while the string is charged under the Kalb-Ramond 2-form B^ (B^ directly couples to the string), R-R p-form potentials do not directly couple to the string and, as a consequence, in perturbation theory the string cannot be charged under such potentials.

It is consistent to set some fields to zero, keeping the metric, one of the p-forms and the dilaton as non-vanishing. In this case the type IIB supergravity assumes the following form

(2.13) The action (2.13) will be useful in the AdS/CFT computations. The equations of motions coming from the supergravity action (2.12) ((2.13)) have extended objects as solitonic solutions. Let us split the ten-dimensional coordinates xM into longitudinal and transverse coordinates xM = (x^, xm), with [i = 0,... ,p and m = p + 1,..., 9 as indices running over the longitudinal and transverse directions respectively. These solitonic solutions can be written as:

2 (p 7)/8 l (p+1)/s m n ds = [H(r)] - r]flI/dx^dx ' + [H(r)} 8mndx dx ,

p)/4 e* = gs[H(r)f- , (2.14)

1 Fp+2 = dCp+1, CMp+1 = ([tf(r)]- - 1) £„,. /*P+I > 30

m where r = x xm and H(r) is an harmonic function:

H{r) = l + %, (2-15)

Rp~p being a constant1 related to the charge of the p-brane. This solution saturates the BPS bound (which relates its mass M and the charge N carried by the R-R flux) N M > ^ZT, (2.16) and preserves one half of the supersymmetry.

2.1.3 D-branes

As just observed, perturbative strings do not directly couple to R-R potentials, but one can expect that these potentials are associated to non-perturbative states. Let us consider a (p+ l)-form potential Cp+\. As a 1-form potential couples to a particle worldline, a (p + l)-form naturally couples to a (p + l)-dimensional world-volume:

SfCp+i] = / Cp+i, ^p+i = (p + l)-dimensional world-volume, (2.17) and the correspondent electric and magnetic charges are given by

d Qel = / *dCp+i, Qms = / dCp+x, S = d-dimensional sphere. 8 Js -p JSP+2 (2.18)

This suggests that Cp+\ is carried by an object having p spatial dimensions. These p-branes can be found as solitonic solutions of supergravity equations [7]. However, a particular class of p-brane turns out to be physical objects already embedded in string theory [8, 9]. In order to see this, first consider a closed string and compactify

1 one of the spatial dimension, so that the theory is defined on M9 x S . The mass spectrum of the theory is then given by

2 2R2 o 2 ™ = T^ + ^^ + -(^ + ArR-2), Nh-NR + nw = 0, (2.19) lrThe constant in (2.15) has been written as El p for convenience since it has the dimensions of [length]7~P. 31

1 where R, iVL/R, n and w are respectively the radius of S , the number of left/right movers, the momentum-quantization number and the winding number. It turns out to be invariant under the following transformation (T-duality)

a' R <—> —, n <—> w. (2.20) R This symmetry can be realized at level of left- and right-movers

x {a+) _^ xl{a+)f 4(a_) _> _4(a_), (2.21) where x9 is the coordinate which has been compactified. The transformations (2.21) define a dual coordinate x'9:

9 9 x = xl(a+) + a£(a_) —> x' = x*(a+) - *»(*_). (2.22)

Consider now an open string with Neumann boundary conditions at the endpoints (i.e. dixti(a°, a1) = 0) on Wg x S1. Applying the T-duality transformation (2.21), the dual string does not carry momentum and the dual coordinate (2.22) satisfies Dirichlet boundary conditions (i.e. doXti(a°, a1) = 0). This implies cr°=0,27T that the endpoints of the dual string do not move along x'9, but they are free to move along the other directions, spanning an hypersurface (D-brane). The D-branes are therefore defined as space-time hypersurfaces where open strings can end. In this discussion, we T-dualized only one direction. This means that the endpoints of the dual string span an hypersurface with 8 spatial dimensions (D8-brane). It is possible to T-dualize n spatial dimensions. In this case, the dual string endpoints are free to move along p + 1 = 10 — n dimensions: they span a Dp-brane. In type IIB string theory only Dp-branes with p odd are admitted: p = 2k — l, k = 0,..., 5.

The endpoints of a string can be endowed with static degrees of freedom called Chan-Paton degrees of freedom [10] which can be indicated with an index per end- point (i,j = 1,...,JV). The wave-function for the open string can therefore be expanded in a basis of N x N matrices (Chan-Paton factors). Scattering ampli­ tudes of open strings contain traces of product of such matrices and therefore they 32 enjoy an invariance under global U(N) transformations, with one of the indices at the string endpoint transforming under the fundamental representation N and the other one under the anti-fundamental representation N. Since the vertex operator V? = X^dox^e1^ transforms under the adjoint representation N x N, U(N) be­ comes a local symmetry. If one of the coordinates, namely x9, is compactified and a background constant Wilson line is introduced A^ = <5M)9diag (81,..., 6^) /2TTR, the gauge group U(N) is broken to [U(1)]J\ This Wilson line can be gauged away leaving a phase. The dual coordinate becomes x'9 = 9iR', which implies that the same endpoint, with Chan-Paton index i, can lie on different hyper surf aces, i.e. on different D-branes. If a stack of parallel D-brane is considered, the presence of Chan-Paton degrees of freedom on the endpoints of open strings generates the ef­ fective loop expansion parameter gsN. As already pointed out in the introductory section 1.1, the Dp-brane dynamics is described by the DBI-action (1.5), whose low energy limit coincides with the dimensional reduction of the ten-dimensional U(l) M — 1 Supersymmetric Yang-Mills theory to p + 1 dimensions.

Dp-branes are thought to be linked to the extremal p-branes discussed in sec­ tion 2.1.2. One of the argument is that, as the extremal p-branes, Dp-branes saturate the BPS bound (2.16) and preserve half of the supersymmetries.

2.1.4 Gauge/gravity correspondence

Let us start with reviewing the arguments motivating the conjectured equivalence between Af = 4 SU(N) Supersymmetric Yang-Mills theory in four dimension and the ten-dimensional type IIB string theory [1, 3]. As mentioned in section 1.1, the low-energy limit of a stack of A^ parallel Dp-branes is the dimensional reduction to p + 1 dimensions of the ten-dimensional J\f = 1 U(N) Supersymmetric Yang-Mills theory. In the case of D3-branes, the low energy limit is the four dimensional M = 4 U(N) Supersymmetric Yang-Mills theory: the massive string states are too heavy 33 to be excited and, therefore, only the open string massless excitations are present, which turn out to be in the multiplet of the gauge theory. The string and the Yang-Mills coupling turn out to be related by

TSJ£+« i+x (2.23) 0™ 2TT 9. 2?r However, the presence of the D3-branes in the ten-dimensional Minkowski space Mio deforms the bulk geometry since they source closed strings. The low energy effective action is then obtained by integrating out the massive modes and is given by a contribution from the brane modes, one from the bulk modes and a term of interaction between the two set of modes. As previously explained, the action for the brane modes, in principle, contains the J\f = 4 SYM multiplet and high derivative terms. It turns out that the high derivative contribution is suppressed as well as the eventual term of interaction between bulk and D-brane modes. Moreover, the gravity theory in the bulk becomes free. This means that brane and bulk modes decouple returning two independent systems: the four-dimensional M — 4 SYM and free gravity in the bulk.

Now, we can also consider that, as explained in section 2.1.3, the D3-branes sources an R-R 4-form. This supergravity solution is given by (2.14) with p = 3. The

p 2 parameter R^ takes the value R\ = 47rgsa' N and the line element (2.14) can be written as

2 1/2 ti v 1/2 2 2 2 ds = [H(r)]~ rtlu/dx dx + [H(r)} (dr + r dn ) . (2.24)

Furthermore, the dilaton is constant. As before, let us now consider the low-energy regime. Again, there are two type of low energy modes: the "brane" and the bulk modes. The brane modes are the ones of the near horizon region. The horizon for the metric (2.24) is at r = 0. In the limit r —>• 0, the D3-brane metric reduces to the metric of AdS$ x S5. This can be easily seen by considering that:

„(,.)_! + *f ~> 3 (2.25) 34 and, as a consequence, the near-horizon geometry is:

ds2 = ^rj^dx^dx" + ^f- (dr2 + r2dnl) , (2.26) which is nothing but the AdS^ x S5 metric, with AdS$ and S5 having the same radius of curvature R3. Bulk modes are massless particles, which decouples at low energies from the "brane" modes. Again, we obtain two independent systems: type IIB supergravity on AdS$ x S5 in the near-horizon region, and free gravity.

All this naturally leads to conjecture the equivalence between the four-dimensional

5 U(N) Super Yang-Mills theory and type IIB string theory on AdS5 x S . More pre­ cisely, the original conjecture [1] considers the four-dimensional U(N) Super Yang- Mills theory in the large-A limit. The supergravity approximation is valid for large radius R% of AdS^: R\/a' S> 1. In the weak-coupling limit gs

It is also interesting to look at the symmetries of J\f = 4 SYM and type IIB string 35

5 theory on AdS5 x S . The isometry group of AdS5 is SO (4, 2), which is also the conformal group in four dimensions. Similarly, the isometry group of S5 is 50(6), which is the R-symmetry group of the gauge theory. Moreover, the relation between couplings (2.23) shows the presence of an SL(2, Z) self-duality symmetry [11, 12, 13]

ar + b a,b,c,d eZ\ad-bc=l. (2.28) CT + d'

The relation between these two theories has been made more precise by identifying the generating function of the correlators in gauge theory with the string theory partition function [2]

\e J ' /FT — -^string 4>{x,r)\ = 0(x) (2.29) r=ra where

Matching the quantum numbers related to the global symmetries of the theories on the different sides of the duality identifies operators in J\f = 4 gauge theory with fields on type IIB string theory.

It is interesting to point out that the boundary theory changes depending on the choice of the foliation of the AdS$ space. Specifically, if the global coordinates (r, p, Qi) are considered

ds2 = R\ (- cosh2 pdr2 + dp2 + sinh2 pdfl2) , 3 (2 31) r € [0, 2TT[, p>0, J]^ = 1,

i=i the boundary is at p — 0 and in a arbitrarily small neighborhood of p = 0 the metric (2.31) becomes

2 2 2 2 2 2 ds p^0 = R 3 {-dr + dp + p dn ) (2.32) 36 which is the line element for E1 x S3 and no horizon is present. The boundary theory has a mass gap. In the Poincare patch (t, x\ r)

r2 R2 ds2 = -g {-dPSijdJdd) + -^dr2 (2.33) the boundary is at infinity and its geometry is R3+1, where a horizon is present. The boundary theory has no mass gap.

More generically, the conjecture can be discussed for string theory AdSd+x x -M-9-dt where Aig-d is a compact manifold. The dual gauge theory is conformal (SO(d,2) is both the isometry group of AdSj+i and the conformal group in d-dimensions) and the relation (2.30) becomes

d I d2 A = - + ^JRW + -. (2.34)

Compactifying on a manifold Mg-d other than S5, one obtains a superconformal field theory with less supersymmetries.

All these systems have zero temperature. Finite temperature field theories are stud­ ied by considering a near-extremal D3-brane supergravity background. The Hawk­ ing temperature of the background geometry corresponds to the temperature of the boundary field theory and the entropy is computed from the Bekenstein-Hawking formula A/AG. In [15], it has been argued that the leading term of the free energy density F in the 1/N has the form

F = -f(X)^N2T. (2.35)

The correspondence has also been extended to include non-conformal theories (for a review see [14]) in order to analyze QCD-like theories. One way to define more realistic theories is by considering a conformal theories and introducing a mass- deformation by a relevant or marginal operator. The Af = 4 supersymmetric Yang- Mills theory contains a vector multiplet V and three chiral multiplets $, Q and 37

Q? The mass-deformation may be introduced by adding mass terms to the M = 4 superpotential

{*»} . (2.36) On the supergravity side, one needs to find a solution of the supergravity equations of motion by imposing suitable boundary conditions.

In principle, it is also possible to directly find the string theory dual to a non- conformal theory. One example is given by the so-called cascading gauge theories which are dual to the warped deformed conifold of type IIB string theory (see [16] and references therein).

2.1.5 Relativistic hydrodynamics

Let us consider a flat space-time theory in thermal equilibrium. Depending on its symmetries, the theory is characterized by the related conserved quantities. The simplest possible example is a theory which is invariant only under space-time translation. In this case the system is described by the conservation law for the stress-energy tensor3 T^v(x) d^(x) = 0. (2.37)

Another slightly more complicated example is given by a theory which is also in­ variant under a (7(1) global charge with conserved current ^{x). The complete description of the system is then provided by (2.37) and

d^fix) = 0. (2.38)

Considering fluctuations from the thermal equilibrium, at large length and time scales compared with the microscopic scales of the theory, macroscopic hydrody-

2Here a M = 1 language has been used. 3In what follows, space-time translation invariance is always assumed. 38 namics provides an effective description of the system [17, 18]. The basic assump­ tions are the local thermal equilibrium and the validity of linear response theory. In thermal equilibrium, the stress-energy tensor assumes the form valid for ideal fluids:

T"" = (e + pKu" + prT (2.39) where u*1 is the local four-velocity of the fluid, which is such that u^u^ = — 1, e is the energy density and p is the pressure. Dissipative corrections can be introduced and appear as derivative terms. At the first order in such a derivative expansion, the stress-energy tensor can be written as

T^ = (e + pK«" + wT + T»V (2.40) where T^V is the dissipative part of the stress-energy tensor and, at the moment, it is approximated to be function of a gradient for the velocity u^ only. In the rest frame u% — 0 and it is always possible to set r00 = 0, r°l = 0, so that

The form of the remaining components ru is constrained by rotational symmetry so that the constitutive relation assumes the following form

k k Tij = -(5ijdku - rj ( diUj + djUi - -5ijdku J , (2.42) where C is the bulk viscosity, which couples to the trace of the velocity gradients, and r) is the shear viscosity, which couples to the traceless part of the velocity gradients. In a general reference frame, the dissipative stress-energy tensor can now be written as

W a c -A A*" CVpcrdau + 77 ( dpua + daup - -T}p

m l l 3 d0T + d{T° = 0, d0T° + djV = 0, (2.44) 39 where

f00 = T00 -e, e = (T00),

jnj <^ rpij __ pfiij — ij ok 0i ij nOk (5 dkT + 7] (tfT^ + ^'r - -5 dkT° e +p (2.45)

Considering the near-equilibrium fluctuations, the linearized hydrodynamic equa­ tions (2.44) show two eigenmodes:

1. shear mode. It is the transverse fluctuation of the momentum density T°\ Its eigenvalue turns out to be purely imaginary:

2. sound mode. It is the simultaneous fluctuation of the energy density T00 and longitudinal component of T0t and its dispersion relation is given by

.-2 V (, , K\ 2 wsound = usq - i-^- ( 1 + — 1 q\ (2.47)

with us as speed of sound, which is related to pressure and energy density by the following thermodynamical relation:

< - |. (248)

As far as the conserved global current j^ is concerned, assuming zero chemical po­ tential for the related conserved charge, the spatial density jo is such that (jo) = 0- Because of the assumed validity of the linear response theory and local thermal equi­ librium, in a off(near)-equilibrium state, jo evolution is governed by Fick's second law [19]

2 30j° = DV j°, V = diffusion constant. (2.49)

u,; r+ x As jo oc e~ ' "?' , Fick's second law implies the following dispersion relation:

UJ = -iVq2. (2.50) 40

All these informations can be extracted from the two-point correlation functions of the theory. Specifically, in order to compute a two-point correlator of some operator O, one needs to turn on a (small) source term. Turning on a source coupled to the current perturbates the system, whose (linear) response to such perturbation is given by the retarded correlator:

coordinate-space: GR{x) = -iti(x°)([d(x),6(0)]) (2 51) d4xei{u'x0-^)'d(x0){[d(x),d{0)}}. / The average (...) is taken on the equilibrium ensemble. The source of the stress- energy tensor is the metric g^p. Therefore, in order to calculate the correlation functions of the stress-energy tensor, small perturbations of the flat metric need to be analyzed. The two-point retarded correlators are related to the transport properties through dispersion relations as they appear as poles in the correlators, or via Kubo's formulae. In particular, consider the retarded two-point correlators for the stress-energy tensor

O 0 G*,,p>,0) = -i J d"xe^ ^x )([T^(x),Tpa(0)}). (2.52)

The correlator Gxyxy does not couple to any energy or momentum fluctuations and, therefore, it does not possess any pole. It is related to the shear viscosity through the following Kubo's formula:

7? = — lim — Im G>„ r,.(iu, 0) (2.53)

— lim -— [Gty^yiu, 0) - GlyjXy(u), 0)

The other correlators instead show poles:

/-.R / \ /oo,oo(w,T, q) ,T 2 . .»,/ 3\ G oo,oo(",?) = -2 _ uW + iTg2 - - = uag-t-q +0{q ) (2.54)

which are the dispersion relations for the sound and shear mode respectively (the

as second relation of (2.54) holds for GfX,tx and Gtx,xz well). The coefficients T and 41

V can be explicitly written as r=Wl + ?y V = -H-=« (2.55)

Thus, extracting the poles of the two-point retarded correlation functions of the stress-energy tensor, it is possible to compute bulk viscosity £, speed of sound us (pole in the sound mode) and the shear viscosity r\ (diffusive pole). Notice that the speed of sound can be also extracted from the equilibrium one-point correlator (Tin/) of the stress-energy tensor: it is possible to compute the density energy and the pressure from it and, then, use the thermodynamical relation (2.48). Obviously, the value of the shear viscosity computed via the Kubo's formula (2.53) and the one extracted from the pole in the shear mode of the two-point correlators of the stress- energy tensor need to match, as well as the value of the speed of sound computed via the relation (2.48) and the one extracted from the pole in the sound mode of the two-point correlators of the stress-energy tensor: such matchings can be looked at as consistency checks.

2.1.6 Hydrodynamics and gauge/string correspondence

A fairly recent application of gauge/string correspondence has been the investigation of the properties of strongly coupled gauge theory plasmas [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] (see [38] for a review and references therein). The main motivation for this type of studies lies on the fact that heavy-ion collisions at RHIC and LHC are expected to produce the quark-gluon plasma (QGP). The QGP is a strongly coupled phase of QCD where quarks and gluons are deconfined.

As most of the other properties of QCD in the strong coupling regime, it is not still well-understood because of the lack of efficient methods of investigation: in this regime the perturbative approach breaks down and most of the results come from lattice simulations, while an analytic approach is missed. Actually, gauge/gravity correspondence does provide such an analytic tool. Moreover, it opens a new window 42 on this phases of QCD. As already mentioned, it is true that there is no string theory dual to QCD and that quantitatively such an analysis cannot be trusted at all. However, the study of plasmas in strongly coupled gauge theories which share some features with QCD may shed light on the qualitative behavior of the QGP.

The interest in this area arose when for the first time AdS/CFT correspondence was used to compute the shear viscosity 77 for strongly-coupled thermal Af = 4 SU(N) Supersymmetric Yang-Mills theory (SYM) in the large-N and large 't Hooft coupling limit [20]. It was pointed out that the two-point thermal correlator G*yxy of the stress-energy tensor in J\f = 4 SYM theory is related to the absorption cross-section

4 a(w) = ^L fd xe^°([Txy(x),Txy(0)}) = u J (2.56) = -^Im{^M)}.

Comparing this expression for the cross-section (2.56) to the Kubo's formula (2.53) which relates the shear viscosity r\ to the retarded correlator GJL , it is easy to see that the shear viscosity can be expressed in function of the zero-frequency cross- section:

The cross-section at zero frequency

3 3 2 a(0) = ir r 0R , (2.58) r0 and R being the extremality parameter and the AdS-radius respectively. Since the

2 5 2 Hawking temperature of the background is T = r0/(iTR ) and R = y/2irGic,N/Tr / , from (2.57) and (2.58) the expression for the shear viscosity r\ becomes

77 = ^N2T\ (2.59) 43

Being both the shear viscosity r\ and the entropy density s proportional to the area of the horizon of black three-branes, 7] and s differ only for a constant factor [41, 15].

This amusing result brought a lot of interest in trying to compute transport prop­ erties of strongly coupled gauge theory plasmas using AdS/CFT. For conformal theories, conformal invariance fixes two coefficients, i.e. the bulk viscosity £, which turns out to be zero, and the speed of sound us. In order to see how the conformal symmetry sets us — l/\/3, one needs to keep in mind that the speed of sound us is related to energy density e and pressure p through the thermodynamical relation (2.48). From (2.40) the stress-energy tensor becomes diagonal in the rest frame, where u^ = (1,0,0,0), with T00 = e and Tu = 0 for each i (with i — 1,... ,3 ). Conformal invariance implies that the stress-energy tensor is traceless

T^ = 0. (2.60)

As a consequence, pressure and energy-density are related by

e = 3p. (2.61)

Therefore, the relation (2.48) immediately returns

us = ^=. (2.62)

Most of the transport properties of interest in first order hydrodynamics are related to the poles of the retarded two-point thermal correlation functions. Therefore, one would need to compute correlators in Minkowski space. As explained more exten­ sively in section 2.1.7, analytic continuing the Matsubara correlators to Minkowski ones is practically not feasible because one does not typically have control of the Matsubara correlator on the whole spectrum of Matsubara frequencies. Further­ more, the naive Minkowski version of (2.29) leads to a real function which cannot be identified with a retarded correlator because the latter is complex. In order to overcome all the issues arising, it has been formulated a working recipe [21] for 44 computing two-point Minkowskian correlators. Such a prescription opened a new window on thermal gauge theories in the strongly coupled regime.

2.1.7 A prescription for two-point Minkowski thermal correlators

As explained in section 2.1.4, the prescription relating gauge theory operators O to the correspondent bulk field is given by the equality between the generating functional for correlators of O and the string theory partition function with the condition that the bulk field related to O takes, at the boundary, the value of the field which sources O (2.29). In the limit of large TV and large 't Hooft coupling, the prescription reads

5 (e/aM^^W^W) = e- sugra[

The correlators for the operator O are obtained as functional derivatives of the generating functional with respect to the source

GR(i2irTn,q) = -GM(2nTn,q). (2.64)

In order to be able to analytic continue the Euclidean correlators to the Minkowskian ones, one should know them for all the Matsubara frequencies, and typically this is not possible because of approximations that are often used. Furthermore, the hydrodynamic limit is the limit of low frequencies and low momenta, while their ratio is kept constant, but the smallest Matsubara frequency is u> = 2TTT and such a value cannot indeed be considered low. Therefore, one would need an independent way to compute the Minkowskian correlators, i.e. a Minkowskian prescription equivalent to (2.29) and (2.63). At the moment, such a general prescription is not available. 45

However, a recipe has been formulated [21] for calculating the two-point Minkowski correlators. The definition of this recipe is indeed not theoretically satisfying; on the other hand, in all the known examples it turned out to be successful. One might ask why the Minkowskian equivalent of (2.63),

0 iS, (e^/sM*^* ^)^)) = e sugra[0(a:,r)|r=re=(x,r) be one of the decoupled fluctuations which satisfies a certain equation of motion. The quadratic term of the bulk action Sbnlk for the classical solution (f)(x, r) can be written (up to a factor

-1/16TTG) as

d 2 5bulk ~ I [ dr d x A(r)(drcj>) + .... (2.66) 1 JM Typically, the solution for (x, r) takes the form

d (x,r) ~ Jd qe^fg(r)MQ), (2-67) with the condition at the boundary r = r$

d i(x,ra) ~ J d qe ^{q). (2.68)

The mode equation one obtains from the equation of motion for by inserting (2.67) has two solutions (incoming and outgoing waves) which are finite at the horizon. As the horizon is approached both of them rapidly oscillate with constant oscillation amplitude, so that the simple requirement of the finiteness of fq(r) admits all the possible linear combinations of the two solutions near the boundary and the Minkowskian correlator is not uniquely defined. This reflects the existence of different types of correlators in Minkowski space. One can circumvent this problem considering that there are no emissions from the horizon and, therefore, on physical ground it is reasonable to impose the incoming-wave boundary condition at the horizon r — r#. 46

The boundary action Sa has the form

Sa[0o] = lim (5bulk + 5GH+ c.t.), (2.69) r^rg where Sbnlk is the string theory effective action, SGH is the standard Gibbons-Hawking term, which is needed in order to have a well-defined variational principle [42], and "c.t." indicates contact terms, which are divergent as the boundary is approached. A more extensive treatment of (2.69) will be given in section 2.1.8, where the holo­ graphic renormalization is discussed. At the moment, it is important to stress only that boundary terms can only affect contact terms in the correlators and, therefore, the theory does not need to be renormalized in order to be able to extract the poles of the retarded correlators.4 Introducing (2.68), the boundary action (2.69) reduces to r=rg / d?q

According to the prescription (2.65), the two-point correlator of interest would be obtained by differentiating twice SQ with respect to

r—r r=r G(q) = -F(q,r)\ g -.F(-?,r) 9 . (2.72) r=rH r=rn The function G(q) in (2.72) cannot be a retarded correlator. Notice that the imag­ inary part of F(q, r) does not depend on r being proportional to a conserved flux

lm{f(q,r)} ~ A(r) (/•$./, - /„&/•) = (2.73) A = (r) (f-qdrfq ~ fqdrf-q) , and therefore drlm {T(q, r)} = 0. As a consequence, the imaginary part of ^{q^) in (2.72) completely disappears, leaving a real function. However, a retarded corre­ lator is a complex function and then G(q) cannot represent it. 4It is necessary to renormalize the theory when the one point function of the stress energy tensor (T^) needs to be computed 47

The retarded correlator is identified with

G*(g) = -2F{q,r) . (2.74) r=rg As already anticipated, this recipe lacks a strong theoretical explanation. However, it turns out to be successful in all the cases it has been applied [21, 22, 23, 28, 29, 30, 31]. Summarizing, the recipe to compute two-point retarded correlation functions is

1. find a solution fq(r) of the mode equation such that its boundary value is

Mr) = 1; r=rs 2. at the horizon r = r# impose incoming-wave boundary condition for time-like momenta and regularity condition for space-like momenta;

3. compute the two-point correlator using eq. (2.74).

In order to compute the correlation functions of the stress-energy tensor T^(x), one needs to study the fluctuations hy,v of the (d + l)-dimensional metric. The (d + l)-dimensional background of interest is a black-brane metric which, after the compactification of the ten-dimensional theory on the (9 — d)-dimensional compact manifold Adg-a, has the following form:

p 2 2 0 2 1 2 2 2 ds = -c {r)(dx ) + 4(r) J2 {dx ) + c 3{r)dr , p + 1 = d. (2.75) j=i A hydrodynamics analysis is possible if the boundary theory is invariant under space-time translations. As a consequence of the space-time translation requirement, the fluctuations h^v can be taken to be proportional to the plane-waves: h^, —

% x H

• sound channel. The fluctuations in this channel do not transform under the group 0{p — 1). They are "spin-0" fluctuations:

hzr; (2.76)

• shear channel. These fluctuations transform as a vector under 0(p — 1). They are "spin-1" fluctuations:

hu, hzi, hri, i = l,...,p-l; (2.77)

• scalar channel. This channel contains only the traceless part of hij (i,j — 1,... ,p — 1). It is a "spin-2" fluctuation:

hii--^y2hkk- (2.78) p L ~ fc=i

It is therefore possible to study the different channels separately. A similar classi­

_ u :c +iqz fication also holds for the fluctuations A^ = AA((r)e * ' of the bulk field whose boundary value is, on the field theory side, the field sourcing a (7(1) conserved current:

• diffusive channel. The fluctuations in this channel do not transform under the group 0(p — 1). They are "spin-0" fluctuations:

At, Az, Ar (2.79)

• transverse channel. These fluctuations transform as a vector under 0(p — 1). They are "spin-1" fluctuations:

Au i = l,...,p-l. (2.80)

However, it turns out that not all the fluctuations are physical due to gauge invari- ance. In the same fashion as in cosmology where Bardeen potentials were defined 49

[44], gauge-invariant variables Zk can be introduced (they are defined as gauge- invariant combination of the fluctuations) [43]. The system of coupled ODEs ob­ tained can typically be diagonalized. If Z(r) is one of such gauge-invariant variables, the related ODE can have both incoming and outgoing waves as solutions near the horizon (the most general solution is obviously given by a linear combination of these two solutions). As explained above, one imposes the incoming-wave boundary condition at the horizon because the horizon does not radiate. Near the boundary instead, the solution is a linear combination of the non-normalizable and normaliz- able modes:

Z(r) = Ar-A~ (1 + ...) + Br~A+ (1 + ...), (2.81) where A± are exponents of the ODE as r —> +oo (for the metric (2.75), the boundary value for the radial coordinate r is rg = +oo) and "..." represent terms with higher powers of r. The connection coefficients A and B are not constant: A = A(u>, q, {p}) and B = B(u>, q, {p}), with {p} as the set of other parameters A and B can depend on.

In terms of the gauge-invariant variable Z(r), the quadratic term of the boundary action (2.69) becomes

d Sd ~ lim I du>d qA(LU,q)Z'(r)Z(r) + at., (2.82) where "at." are again contact terms. Applying the recipe [21] discussed above, the two-point retarded correlation function of the related operator O in the boundary theory is given by + c t (2 83) (6d)R ~ 5 '" '

Eq. (2.83) implies that the poles of the retarded correlator (00)n in the dual field theory are the zeros of the function A(u, q, {p}). From eq. (2.81), it is possible to see that setting A — 0 is equivalent to have only the normalizable mode for the solution Z(r) near the boundary. Notice that the condition A = 0 is equivalent to the statement that the poles of the two-point correlators in the dual field theory coincide 50 with the spectrum of quasi-normal frequencies of the black brane background5 [46, 47, 48, 49].

2.1.8 Holographic renormalization

As mentioned in section 2.1.5, there are two methods to compute the speed of sound: from the dispersion relation in the sound mode which appears as pole in a two- point correlation function of the stress-energy tensor, and from the thermodynamical relation (2.48) which relates pressure p and energy density e. In order to use (2.48), one needs to extract pressure p and energy density e from the one-point correlators of the stress-energy tensor (T^) defined at the boundary of the space-time M.d+i- However, it turns out to diverge and, therefore, it needs to be regularized and renormalized [50, 51, 52, 53]. Let 7M„ be the metric of the boundary dM-d+i- The boundary stress-energy tensor is defined as

where 7 is the determinant of the boundary metric 7^. The action S in the def­ inition of the stress-energy tensor (2.84) is the sum of the bulk action S^uik, the

Gibbons-Hawking term SGH, which is needed in order to have a well-defined varia­ tional principle [42], and a boundary counterterm action Sc, which is introduced to obtain a finite result:

= Jbuik + uGH + oc. (2.85)

The Gibbons-Hawking term SGH is a boundary term and, therefore, it does not affect the equations of motion:

ddx 0 2 86 5GH = -rrr- I v^ * ( ' ) 6TTUd+1 JdMd+1 5Z(r) is a quasi-normal mode if it satisfies the incoming-wave condition at the horizon and Dirichlet conditions at the boundary. 51 where 0 = 7^0^ is the trace of the extrinsic curvature 0^ of the boundary:

1

0M„ = — - (V^n,, + Vj,nM), n^ = outward normal vector to the boundary. (2.87) Consider now a foliation of the space-time Ada+i along the radial direction r in time­ like hypersurfaces dM.r which are isomorphic to the boundary dAid- Such foliation along the radial direction defines a natural cut-off so that the renormalized action can be seen as the limit as r goes to its boundary value TQ of the sum of the bulk action ,S£ulk defined on the cut-off space Mr, the Gibbons-Hawking term (2.86) and the boundary counterterm action Sc, which makes the limit r —* rg well-defined:

r r S = lim S = lim (S hulk + SGH + Sc). (2.88) r—>rg r—>rg

The counterterms in Sc are obtained as function of the local metric 7p„ and the other relevant fields in the specific theory on the boundary, as well as one needs terms associated with the conformal anomaly which show a dependence on the position of the boundary. If the theory under analysis is supersymmetric, it is suitable to introduce some finite term as well, so that supersymmetry is preserved.

The renormalized stress-energy tensor is then obtained by evaluating at the bound­

r ary dMd+i the variation of S with respect to the metric of dM.r: 2 5Sr rp rjireg = lim ——— = dMd+1 r^r9 ^f Sj

2 SSrJC ^ ' (0^ - ©7^)

The two terms in (2.89) come from the variation of the Gibbons-Hawking term SGH and the boundary counterterm action Sc, which are boundary terms. This is because the variation of the action is considered on-shell.

Let now £ be a space-like hypersurface of the boundary dA4d+i and let aab be its metric (as usual, a indicates the determinant of the metric <7ab)- Furthermore, let TVs be the time-like Killing vector. The boundary metric can be written in ADM 52 form [54] (see also [55] and references therein):

2 0 2 a 0 b ds d = ^daffa" = -AT| (dx ) + aab (N£dx° + dx ) (N^dx + dx ) . (2.90)

The flow of time in the boundary hypersurface dMd+i is identified by the time-like unit vector u>* which is normal to the space-like hypersurface S. The energy density e and the pressure p are then defined as

(2.91) xx P = V^Nx(Txx)7 .

Once the density energy and the pressure are known using (2.91), the speed of sound can be easily obtained via eq. (2.48).

2.1.9 The KSS bound and universality

As already noticed in [20], the shear viscosity and the entropy density in M = 4 SU(N) SYM theory (at large iV and large 't Hooft coupling) are equal up to the proportionality constant 1/ATT: -, = i- (292) This relation was also checked [26, 28] for the J\f = 2* theory, which is a mass- deformation of Af = 4 SYM [56, 57]. It was conjectured that the value 1/4-7T is actually a lower bound for the ratio shear viscosity by entropy density in quantum field theories [26, 27]: i * h- (293) Using a simple argument based on Heisenberg's uncertainty principle [27], it was found that - > 1. (2.94)

Moreover, it can be shown that in weakly-coupled systems

- > 1. (2.95) s 53

One more argument which supports the conjecture (2.93) is the explicit computation of the leading correction in the inverse 't Hooft coupling for the ratio rj/s in J\f = 4 SYM [58]:

2 l(i+ 135^-1^. (2.96) s Ait \ 8 (2A)3/2y/ Notice that the correction in (2.96) is positive and, therefore, the bound (2.93) is satisfied.

As formulated in [26, 27], the bound (2.93) holds for theories with zero-chemical potential and it is saturated if the theories admit a holographic dual supergravity description. The fact that thermal field theories admitting higher dimensional grav­ itational duals saturate the bound (2.93) was shown in general in [59, 60]. The key observation is that the fluctuation hxy (it decouples from the other fluctuations, it can thus be analyzed independently of them and allows to compute the stress-energy correlator entering the Kubo's formula (2.53)) has to satisfy the equation of motion for a minimally coupled scalar. This can happen only if the background geometry satisfies the following condition [59, 60]

R\-R\ = Q i = l,...,p-l (2.97)

(here no summation on the repeated indices is understood). The condition (2.97) is always satisfied by supergravity backgrounds which are dual to some strongly- coupled thermal field theory [59].

In the cases of thermal field theories with chemical potential, the condition (2.97) in­ deed no longer holds. The presence of chemical potentials related to some conserved C/(l) charges in thermal field theories translates on the supergravity side in gauging the U{\) isometries which corresponds to each field theory conserved charge. This implies Rtt-R\ = oEF(2*v (2-98) 1 fc=l m being the number of chemical potentials in the theory. 54

However, it was observed that in specific cases of thermal gauge theories with non­ zero chemical potential admitting dual supergravity theories, the relation (2.92) was still satisfied [32, 33, 34, 35, 61]. A general proof, which enlarges the universality class defined by (2.92) to theories with non-zero chemical potential, was given in [62] and it will be the subject of Chapter 5.

2.1.10 Israel-Stewart transport theory, Bjorken flow and gauge/string correspondence

In section 2.1.5 only dissipative first-derivative corrections to the ideal fluid have been considered: at first order the dissipative term of the stress-energy tensor is a function of bulk viscosity £ (proportional to the gradient of the velocities) and shear viscosity rj (proportional to the traceless part of the gradient of the velocities).

Dissipative terms can be treated in more generality. Let j^ and TMI/ be a conserved current and the stress-energy tensor respectively. Let S^ be the entropy four-current. These quantities can be decomposed as follows f = jV + e

v) u) T"" = {e + p + IIKM" + (p + U)rT + 2q^u + 2^1^u + if" = n (2.99) = eu^uv + {p + II) A"" + TT"" + 2 q^ + Ltl^ n S» = su^ + IP, where the second and third term in the decomposition of the stress-energy tensor define the pressure tensor P'"', with effective pressure peK = p + LT pi» d^ (p + n)A^ + Tr"", (2.100)

A^B"} indicates symmetrization with respect to the two indices

AU*B») ^ * {AnB* + AvBn} ? (2.101) and the four-vectors ^, q^ and the tensor 7r/"/ are such that

V = 0- u^ = ° V" = °. ^ = 0- (2.102) 55

According to the second law of thermodynamics

dpSv > 0 (2.103)

In [64, 65] the following form for the off-equilibrium entropy four-current S^ has been proposed S" = p(ji, T)— - iif + T^f + Q», (2.104) where /i and T are respectively the chemical potential and the temperature of a close equilibrium state, p(fj,, T) is the related pressure and Q^ is a function of the deviation of jp and Tp from equilibrium. Q^ is taken to be at most quadratic in the dissipative fluxes:

q 2 S" = su» + - - - (f30U + faqrf + /327r'%a) — + a0U^ + a^|, (2.105)

where at = ai(e,j), A = Pi{e,j)-

Taking the four-divergence of (2.105) and considering the conservation laws for j^ and T^, as well as Gibbs' equation

dn r-w fd^-T^d^ (2.106) the second low of thermodynamics assumes the following form

Td^S* = - n d^u" + foil + ify (^v? ) n + aoDtf"

u D^ InT + ^ + faq^ + ^Tdu (^u j Qft + aQDun\ - a^II

puV w

2TdP f YUP^^ + aiD^q^ > 0, (2.107) where D^ = A^d^ is the gradient operator and A^B^ is defined as

A<"B") = A»pAua A B„, - \A^ApaA B„. (2.108) (n(pDa) *p-^cr-n 56

In (2.107), it is possible to recognize the shear tensor (it is the term in the round brackets in the third line) and the bulk term (the trace of the gradient of the veloc­ ities in the fist line).

Israel-Stewart picture assumes that extended thermodynamical forces and thermo- dynamical flows are related by linear relations, the second law of thermodynamics (2.103) turns out to be satisfied, leading to the following transport equations

mri + n = -cax - l(Td» Q|) n - T0D^ AT Vy^ + g,, = -A^T + T^) > ^ (M ^ -ToD^^ + nD^U

P rw A/A/ + TT^ = -v (A/A/S(P?M - |A^u") - rfTdp (^ ) ^

(2.109) where the relaxations times TQ, rq and rn (i.e. the time that the system, subject to a particular dissipative process, takes to relax and equilibrate) have been introduced, together with the coefficients {TJ}^=0:

rn = CA), rq = ATA, TV = 2r,/32 (2.110) r0 = C«o, TI = XTai, TI = 2r?ai.

However, as recently pointed out in [66], the choice to restrict the dissipative cor­ rections only to linear terms is completely arbitrary: when dissipative terms of the second order in the gradients of the velocities are introduced, non-linear contribu­ tions of the same order are allowed. Furthermore, in [66] it is stressed that if the system enjoys conformal invariance such terms are required.

The second order viscous hydrodynamics is of interest in the study of dynamical processes in the quark-gluon plasma. Specifically, in sufficiently high energy central collisions of large nuclei the plasma expands longitudinally and homogeneously along the collision axis. Moreover, the evolution of the system appears to be the same in all the reference frames where the pancakes of produced nuclei move at the speed of 57 light in opposite direction from the collision point. This implies that there is a region (central rapidity region) where the plasma is boost-invariant. Interestingly, near the collision axis the plasma moves with a longitudinal local velocity equal to z/t, where z is the coordinate of the axis. This boost-invariant (one-dimensional) evolution of the plasma is known as Bjorken flow [67]. There have been some attempts to study the Bjorken flow in a conformal plasma using gauge/gravity correspondence [68, 69, 70, 71, 72]. The application of gauge/gravity correspondence to the study of a conformal plasma in the boost-invariant framework is the subject of Chapter 6. Recently, gauge/gravity correspondence allowed a derivation from first principles of the non-linear hydrodynamics (up to the second order in the derivative expansion) [73].

2.2 Gauge theories at weak coupling: scattering amplitudes methods

In recent years a great deal of progress in studying interactions of fundamental par­ ticles has been done by looking at the scattering amplitudes. The main approach for computing them has made use of Feynman diagrams (see for example [74]). Despite the amazing breakthru due to the introduction of the Feynman diagram expansion which led to understand several processes, this method is computational inconve­ nient as the number of particles involved in the process grows since the number of diagrams contributing rapidly increase as well as the number of kinematic invariants. However, it has often been found that cumbersome Feynman diagram calculations where leading to really simple answers, suggesting that simpler representations for the amplitudes can be found, as shown by Berends-Giele recursive relation for gluons [75] and Parke-Taylor formula for MHV amplitudes of gluons [76]. 58

2.2.1 Spinor-helicity formalism

A convenient way to deal with scattering amplitudes is the so-called spinor-helicity formalism (see [77, 78] and references therein). In a four-dimensional Minkowski space-time the isometry group is the Poincare group V = {T, 50(3,1)*}, T and 50(3,1)T being the translation group and the orthochronous Lorentz group re­ spectively. The orthochronous Lorentz group 50(3,1)T is locally isomorphic to

6 5L(2,C). Let Aa and AQ, with a, a = 1,2 respectively transform according to (1/2,0) and (0,1/2) representation of 5L(2, C). The isomorphism between 50(3,1)^

l and 5L(2, C) is implemented by the matrices a^ = (IOOJO^O), a ad being the usual Pauli matrices. Therefore, a rank-(m, n) Lorentz-tensor is mapped into an object with 2(ra + n) spinorial indices:

/1/*1---Mn . fjdxai -UniXn v\ vm AAi\,..{hn _ \a\a\...anan U U U "l-»m °Vl ••• Vn blbt • • - bmbm "1-"m ^ blbl...bmbm' (2.111) The spinor indices are raised and lowered by the two-dimensional Levi-Civita sym­ bols eab, e&h: \a ab \ \ \ b A — e Ab, Aa — €abA (2-112) a a A = e Xb, Xa — ehbX , with

12 i2 ac a e12 = l = ei2, e = -l = e , e ecb = S b. (2.113)

It is possible to define two inner products for spinors, one for each representation of SL(2, C) under which they can transform. They are defined as antisymmetric linear maps from C2 x C2 to C:

2 • let Aa, X'aEC , then: (•,•): C2 x C2 —> C (2.114)

6More precisely, the group 51/(2, C) is isomorphic to the universal covering Spin(3,l) of 50(3,1)T. 59

with (A', A) = -(A, A'} (2.115)

due to the antisymmetry of eab;

2 • let Ad, X'd G C , then: C2 x C2 —> C (2.116) a (Aa, Aa) —> [A, A] = edj,A A , with [A', A] = -[A, A'] (2.117)

due to the antisymmetry of eab.

Let i, j, k and I label four different spinors transforming under the same represen­ tation of SL(2, C). The inner products satisfy the Schouten identity (i,j)(k,l) = (i,k){j,l) + (i,l)(k,j), (2.118) [i,j][k,t\ = [i,k]\j,l] + [i,l][k,j], where the labels have been written down in the brackets in order to simplify the notation. Let now v^ be a Lorentz vector. Applying the mapping (2.111), v^ is

mapped into a bispinor vail which transforms under the (1/2,1/2) representation of SX(2,C):

V/i * aaaV» = Vaa- (2.119) The relation (2.119), implies that

v2 = w% = detKa}. (2.120)

For future purposes, it is interesting to look at light-like vectors. If v^ is a light-like 2 vector, then v = 0 and, as a consequence of (2.120), det{vaa} — 0 which necessarily

implies that vaix = Xa\a- However, the two spinors appearing in the previous relation are defined up to a scale transformation

l 7 (Aa, A4) — (tXa, r Xh) teC*. (2.121) 7The notation C* indicates the complex field without zero: C* = C \ {0}. 60

If Vfj, is a real vector, it is easy to show that the spinors Aa and AQ are not independent of each other, but they are the complex conjugate of each other up to a sign.

The momentum p^ of a massless particle is light-like. From the previous discussion, its bispinorial representation is paa = AaA„, and Aa and A

i*Ma = 0. (2.122)

Writing ipa as a plane wave ipa = aaelXahpaa, the equation (2.122) becomes

a a 0 = Paaa = Xaka = (A, a)~\a. (2.123)

It is non-trivially satisfied if and only if (A, a) = 0, which implies that aa is proportional to A0. Therefore, Aa has negative chirality. On the same lines, it is possible to show that \a has positive chirality.

For future purposes, we write here the scalar product of two massless momenta p = A^A^ and q = A^A^ in terms of the inner product earlier defined

2p-q = {p,q)\p,q]. (2.124)

As spinors provide a representation of the wave-function for chiral spin-1/2 particles, it is possible to represent the wave-function of massless spin-1 particles. Specifically, polarization vectors are mapped into bispinors which are written as

£a" ~ $J]' £ah ~ (^A)' (2'125) /j, and p, being reference spinors which implement the gauge freedom. This can be showed as follows. Let us map the reference spinor /ia into £a. Being the spinors elements of C2, each of them can be expressed as linear combination of a spinor basis. Let jia and Aa be such a basis. Therefore

U - afia + pK, (2-126) 61 and

** ~ & A) (a/x, A) + (a/z, A) e<*+ a(/i, A>Pad" ^'Ut) Similarly for e~. From (2.128), the variation of the reference spinor implies a varia­ tion of the polarization tensor which is proportional to the momentum of the particle

Ka = -zfTV^a, (2.128) a{fi, A) which is the usual gauge transformation for polarization vectors. Furthermore, it is straightforward to check that the transversality condition ep = 0 is satisfied.

It is interesting to notice that under the scaling (2.121) the wave function of massless particles of helicity h scales like t~2h. In the case of arbitrary spin-s particles we assume a representation where the wave-function is represented as a completely symmetric rank-(0,s) tensor in the case of bosons, and as a symmetric rank-(0,s') "tensor" with an additional spinorial index (here s — s' + 1/2). The polarization tensors are represented as products of spin-1 polarization vectors, as it will be shown in Chapter 9.

Scattering amplitudes can be expressed in terms of spinors and helicities

4 4 w (i) W W M = (2TT) (5 ( J^A A ] M(A , A , ht). (2.129)

We use the convention that all the particles in the amplitude are incoming. The amplitudes (2.129) are eigenfunctions of the helicity operator which is defined as a differential operator in the spinorial space. If X is the set of the particles in the amplitude, then:

9 9 VKl: ( A«^ -_ Af-xW ^ M(AW, A<*>, h ) = -2/nM(A«, A«, h,). SA« h d~\i t (2.130) 62

2.2.2 BCFW construction

One of the basic assumptions concerning the scattering amplitudes is its analyticity as function of the Lorentz invariants since the old S-matrix approach [79]. This implies that its singularity structure can show at most poles and branch points. In a perturbative expansion, the tree level approximation shows only poles and branch points appears at loop level. Here we focus on tree level scattering amplitudes. From a complex analysis viewpoint, dealing with scattering amplitudes is not an easy task since it is a problem of several variables (the Lorentz invariants). A drastic simplification would occur if these multi-variable problem can be reduced to a single variable problem.

It turns out that this is indeed possible [80]. The idea is to introduce a one-parameter deformation of the momentum space which preserves both momentum conservation and on-shell condition p2 = 0, and analyze the singularity structure of the ampli­ tude as function of the deformation parameter. However, a real momentum space does not allow to introduce any deformation consistent with the above require­ ments. This problem can be overcome by complexifying the momentum space. The isometry group of the Poincare group is now 5*0(3,1,C) which is isomorphic to £1/(2, C) x SX(2,C) and it is possible to choose the two spinors A and A related to the same momentum as independent of each other by considering one of them as transforming under one copy of SX(2,C) and the other one under the second copy. With this choice, it is possible to introduce a complex one-parameter defor­ mation of the momentum space which does not break momentum conservation and the on-shell condition.

There is no unique way to introduce such a deformation. The simplest deformation [80] is defined by deforming the momenta of two particles, namely i and j, leaving 63 the other ones unchanged:

ij) {j {j) p(i) _> p«)(z) = p®-zq, p -> p \z) = p + zq, (2.131) q being a light-like momentum. The poles in z appear only in correspondence of internal off-shell propagators involving only one of the deformed momenta: 1 1 _ 1 _ 1 _

P P z 2 rs ?si ) (Pr + •••+ Pi{z) + •••+ p3) (Prs ~ zqf ^ ^^

= PI - 2zPrs • q

Equation (2.132) shows that only simple poles appear and they have the form

*» = •*§*-- (2.133) £irs' q The presence of simple poles only allows us to use the power of the residue theorem: if the deformed amplitude M(z) vanishes at infinity, then the amplitude can be expressed only in terms of the residues of its poles

M(*) ~ ]£-!-' (2.134)

V being the set of the poles Z{ of the amplitude. The physical amplitude is then obtained by setting the parameter z to zero in (2.134).

The residues of the poles, that we indicated in (2.134) with Q, have a simple physi­ cal interpretation. As one of the poles, namely zrs, is approached, the contributions containing the correspondent propagator 1/P^s(z) dominates. Furthermore, at the pole location z = zrs the momentum Prs goes on shell so that the amplitude fac- torizes in two physical amplitudes with smaller external legs8, one of which is the internal particle sent on-shell. Thus, the residue of one of these poles is the product of two on-shell physical amplitudes with fewer legs:

MM-E^H (2,35)

3times the propagator containing the pole 64

In this construction the behavior at infinity of the one-parameter family of ampli­ tudes M{z) plays a crucial role. If the condition

lim M(z) = 0 (2.136) 2—>00 is not satisfied, the singularity at infinity would contribute as well:

ML( M(* P^''+^ (2i37)

Unfortunately, there is no physical interpretation of the singularity at infinity and its residue yet. Furthermore, the analysis of the behavior at infinity is still an open issue. At present, it has been carried out by analyzing the large-z behavior of all the Feynman diagrams contributing to the process of interest. However, the result is only an upper bound because it is not possible to take track of all the eventual cancellations among different diagrams and inside a diagram itself. This means that scattering amplitudes which appear to have a divergent large-z limit can actually vanish at infinity.

Important progress has been recently achieved in [81], where the analysis relies only on the Lagrangian structure of the theory and on a special light-cone gauge. Another amusing result of [81] is that, relying only on the Lagrangian structure, it is possible to perform such an analysis for theories in arbitrary space-time dimensions, while before it was constrained in four-dimensions because the spinor-helicity formalism was crucial. 65

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Gauge theories at strong coupling 74

CHAPTER 3

Sound waves in strongly coupled non-conformal gauge theory plasma

The conjectured duality between gauge theory and string theory [1, 2, 3, 4] is use­ ful for insights into the dynamics of gauge theories. For strongly coupled finite- temperature gauge theories in particular, the duality provides an effective descrip­ tion in terms of a supergravity background involving black holes or black branes. A prescription for computing the Minkowski space correlation functions in the gauge/gravity correspondence [5, 6] allows one to study the real-time near-equilibrium processes (e.g. diffusion and sound propagation) in thermal gauge theories at strong coupling [7, 8, 9, 10, 11, 12, 13, 14, 15]. Computation of transport coefficients in gauge theories whose gravity or string theory duals are currently known may shed light on the true values of those coefficients in QCD, and be of interest for hydrody- namic models used in theoretical interpretation of elliptic flows measured in heavy ion collision experiments at RHIC [16, 17, 18]. This expectation is supported by an intriguing observation [13, 15, 19, 20] that the ratio of the shear viscosity to the entropy density in a strongly coupled gauge theory plasma in the regime described by a gravity dual is universal for all such theories and equal to 1/47T. (Finite 't

Hooft coupling corrections to the shear viscosity of J\f = 4 supersymmetric SU(NC) gauge theory plasma in the limit of infinite Nc were computed in [14].)

Previously, it was shown [10] that the dual supergravity computations reproduce 75 the expected dispersion relation for sound waves in the strongly coupled M = A supersymmetric Yang-Mills (SYM) theory plasma

2 u(q) = vsq-i^q + 0{q'), (3.1) where gpxl/2 '• " I BE 1 (3'2) is the speed of sound, P is pressure and £ is the volume energy density. The attenuation constant V depends on shear and bulk viscosities r] and £,

C + tv)- (3-3) £ + P V 3 .

For homogeneous systems with zero chemical potential £ + P = s T, where s is the volume entropy density. Conformal symmetry of the JV = 4 gauge theory ensures that

v* = ^ C = 0. (3.4)

Indeed, in conformal theories the trace of the stress-energy tensor vanishes, implying the relation between £ and P of the form £ = 3P, from which the speed of sound follows.

Non-conformal gauge theories1, and QCD in particular, are expected to have non- vanishing bulk viscosity and the speed of sound different from the one given in Eq. (3.4). Lattice QCD results for the equation of state £ — P(£) suggest that vs « l/\/3 « 0.577 for T ~ 2TC, where Tc w 173 MeV is the temperature of the deconnning phase transition in QCD. When T —> Tc from above, the speed of sound decreases rather sharply (see Fig. 11 in [21], [22], [23], and references therein). To the best of our knowledge, no results were previously available for the bulk viscosity

xOne should add at once that throughout the paper terms "conformal" and "non-conformal" refer to the corresponding property of a theory at zero temperature. Conformal invariance of N = 4 SYM is obviously broken at finite temperature, but we still refer to it as "conformal theory" meaning that the only scale in the theory is the temperature itself. 76 of non-conformal four-dimensional gauge theories2.

In this paper we take a next step toward understanding transport phenomena in four-dimensional gauge theories at strong coupling. Specifically, we study sound wave propagation in mass-deformed Af = 4 SU(NC) gauge theory plasma. In the language of four-dimensional Af = 1 supersymmetry, Af = 4 gauge theory contains a vector multiplet V and three chiral multiplets <3>, Q,Q, all in the adjoint representa­ tion of the gauge group. Consider the mass deformation of the Af = 4 theory, where all bosonic components of the chiral multiples Q,Q receive the same mass mj, and all their fermion components receive the same mass mf. Generically, mf ^ mb, and the supersymmetry is completely broken. When mb = mf = m (and at zero temper­ ature), the mass-deformed theory enjoys the enhanced Af = 2 supersymmetry with V, $ forming an Af = 2 vector multiplet and Q, Q forming a hypermultiplet. In this case, besides the usual gauge-invariant kinetic terms for these fields3, the theory has additional interaction and hypermultiplet mass terms given by the superpotential

W = ^p- Tr([Q,Q]$W^(TrQ2 + TrQ2) . (3.5) 9YM ^ ' 9YM ^ ' The mass deformation of Af = 4 Yang-Mills theory described above is known as the AT = 2* gauge theory. This theory has an exact non-perturbative solution found in [27]. Moreover, in the regime of a large 't Hooft coupling, gYMNc ^> 1, the theory has an explicit supergravity dual description known as the Pilch- Warner (PW) flow [28]. Thus this model provides an explicit example of the gauge theory/gravity duality where some aspects of the non-conformal dynamics at strong coupling can be quantitatively understood on both gauge theory and gravity sides, and compared. (The results were found to agree [29, 30].) Given that the H — 2* gauge theory is non-conformal, and at strong coupling has a well-understood dual

2Perturbative results for the bulk viscosity and other kinetic coefficients in thermal quantum field theory of a scalar field were reported in [24]. Bulk viscosity of Little String Theory at Hagedorn temperature was recently computed in [25] using string (gravity) dual description. The speed of sound of a four-dimensional (non-conformal) cascading gauge theory was computed in [26], also from the dual gravitational description. 3 The classical Kahler potential is normalized according to 2/gYM Tr[#$ + QQ + QQ}. 77 supergravity representation, it appears that its finite-temperature version should be a good laboratory for studying transport coefficients in non-conformal gauge theories.

The supergravity dual to finite-temperature strongly coupled A/" = 2* gauge theory was considered in [31]. It was shown that the singularity-free non-extremal deforma­ tion of the full ten-dimensional supergravity background of Pilch and Warner [28] al­ lows for a consistent Kaluza-Klein reduction to five dimensions. These non-extremal geometries are characterized by three independent parameters, i.e., the temperature and the coefficients of the non-normalizable modes of two scalar fields a and x (see section 2 for more details). Using the standard gauge/gravity correspondence, from the asymptotic behavior of the scalars the coefficient of the non-normalizable mode of the a-scalar was identified with the coefficient of the mass dimension-two operator in the A/" = 4 supersymmetric Yang-Mills theory, i.e., the bosonic mass term m^; the corresponding coefficient of the %-scalar is identified with the fermionic mass term rrif. The supersymmetric Pilch-Warner flow [28] constrains those coefficients so that rrib = rrif. However, as emphasized in [31], on the supergravity side m& and rrif are independent, and for m\, ^ rrif correspond to the mass deformation of M — 4 which completely breaks supersymmetry. In the high temperature limits m^/T

Our goal in this chapter is to compute the parameters of the dispersion relation 78

(3.1) for the Af = 2* plasma. Since the ratio r]/s is known, this will allow us to determine the speed of sound and the ratio of bulk to shear viscosity in the M = 2* theory. We will be able to do it only in the high-temperature regime where the metric of the gravity dual is known analytically, leaving the investigation of the full parameter space to future work. The dispersion relation (3.1) appears as a pole of the thermal two-point function of certain components of the stress-energy tensor in the hydrodynamic approximation, i.e. in the regime where energy and momentum are small in comparison with the inverse thermal wavelength (co/T

Though the general framework for studying sound wave propagation in strongly coupled gauge theory plasma from the supergravity perspective is known [10, 32], the application of this procedure to a non-conformal gauge theory is technically quite challenging. The main difficulty stems from the fact that unlike the (dual) shear mode graviton fluctuations, the (dual) sound wave graviton fluctuations do not decouple from supergravity matter fluctuations. The interactions between various fluctuations and their background coupling appear to be, gauge-theory specific. As a result, we do not expect speed of sound or bulk viscosity to exhibit any universality property similar to the one displayed by the shear viscosity in the supergravity approximation. In fact, we find that the speed of sound and its attenuation do depend on the mass parameters of the J\f = 2* gauge theory.

Let us summarize our results. For the speed of sound and the ratio of the shear to bulk viscosity we find, respectively, -*(•-* (^""IS^M' (3'6) H* (?)*•£ (?)4+- 79 where /?£ « 0.9672, /5^ « 8.001, and the ellipses denote higher order terms in rrif/T and rnij/T, From the dependence (3.6), (3.7) it follows that at least in the high temperature regime the ratio of bulk viscosity to shear viscosity is proportional to the deviation of the speed of sound squared from its value in conformal theory, \ * "K (?•" I) • <3-8> 4 2 where K = 3TT/?^/2 « 4.558 for mb = 0, and K = ?r /?f/16 « 4.935 for m/ = 0.

2 (Note that the result (3.8) appears to disagree with the estimates £ ~ rj (v s — 1/3) [33, 34], later criticized in [24]).

The chapter is organized as follows. In section 3.1 we review the non-extremal (finite-temperature) generalization of the Pilch-Warner flow. We discuss holographic renormalization and thermodynamics of M — 2* gauge theory, and determine its equation of state to leading order in mb/T

4In general K = K(X) is a function of the ratio A = mj,/m/ which we were able to compute in two limits A —> 0 and A —* oo. Assuming K(A) to be a smooth monotonic function, we find that it varies by ~ 8% over the whole range A e [0,+oo). Additionally, for both finite (and small) m,b/T, rrif/T we verified scaling (3.8) by explicit fits. 80

3.1 Non-extremal J\f = 2* geometry

The supergravity background dual to a finite temperature M — 2* gauge theory [31] is a deformation of the original AdS$ x S5 geometry induced by a pair of scalars a and x °f the five-dimensional gauge supergravity (at zero temperature, such a deformation was constructed by Pilch and Warner [28]). According to the general scenario of a holographic RG flow [35], [36], the asymptotic boundary behavior of the supergravity scalars is related to the bosonic and fermionic mass parameters of the relevant operators inducing the RG flow in the boundary gauge theory. The action of the five-dimensional gauged supergravity is s (3-9) d 2 V / ^^9 [\R - 3(da) - (dXf - V] , ArrGrLr5 J Mr, where the potential5 2 dW 8W\ ' V = :W2 (3.10) 16 da + dx ) is a function of a and x determined by the superpotential

W = -e~2a - \e4a cosh(2x). (3.11)

The five-dimensional Newton's constant is

Cio 47T GB = 5 2 (3.12) 2 VO1SB iV

The action (3.9) yields the Einstein equations

Ryu, = 12dftadva + Ad^xduX + 7i9i*»'P (3.13) as well as the equations for the scalars

1&P_ 19P Da •X (3.14) 6 da 2dX 5We set the five-dimensional gauged supergravity coupling to one. This corresponds to setting the radius L of the five-dimensional sphere in the undeformed metric to 2. 81

To construct a finite-temperature version of the Pilch-Warner flow, one chooses an ansatz for the metric respecting rotational but not the Lorentzian invariance6

2 2 2 ds\ = ~c\{r) dt + c 2(r) {dx\ + dx\ + dx\) + dr . (3.15)

The equations of motion for the background become

a.

X" + X'(lnClc!)'-i|^0, (3.16)

4 + 4 (ln^)' + |ClP = 0,

, 4 + 4 (]nc1($) + -c2P = 0, where the prime denotes the derivative with respect to the radial coordinate r. In addition, there is a first-order constraint

{*'? + \u? -\v -\{\nc2y{\nClc2y = v. (3.17)

A convenient choice of the radial coordinate is

x(r) x G [0,1] . (3.18) C2

With the new coordinate, the black brane horizon is at x = 0 while the boundary of the asymptotically AdS*, space-time is at x = 1. The background equations of motion (3.16) become

l 4 + 4c2 {a'f --d2- A(4f + \C2 {X'f = 0, X C2 O &P_ 6(a'f4x + 2(x')24x - 3c2c2 - 6(4)2x = 0. x 12Vc2x da (3.19)

1 dV %{a'fc\x + 2(x')24x - 3c' c - 6(c' fx = 0. x" + -x'x 4 PcZx 2 2 2 dx where the prime now denotes the derivative with respect to x. A physical RG flow should correspond to the background geometry with a regular horizon. To ensure

sThe full ten-dimensional metric is given by Eq. (4.12) in [31]. 82 regularity, it is necessary to impose the following boundary conditions at the horizon,

x^>-0+: < a(x), x(x), c2(x) \ —M <5i,^2,^3 ? , (3.20) where Si are constants. In addition, the condition £3 > 0 guarantees the absence of a naked singularity in the bulk.

The boundary conditions at x = 1 are determined from the requirement that the solution should approach the AdS$ geometry as x —> 1_:

2 1 4 x —• 1_ : {«(*), x(x), C2(x)j—• jo.O.oca-a; )- / !. (3.21)

The three supergravity parameters Si uniquely determine a non-singular RG flow in the dual gauge theory. As we review shortly, they are unambiguously related to the three physical parameters in the gauge theory: the temperature T, and the bosonic and fermionic masses m&, m/ of the TV" = 2* hypermultiplet components.

General analytical solution of the system (3.19) with the boundary conditions (3.20), (3.21) is unknown7. However, it is possible to find an analytical solution in the regime of high temperatures.

3.1.1 The high temperature Pilch-Warner flow

Differential equations (3.19) describing finite temperature PW renormalization group flow admit a perturbative analytical solution at high temperature [31]. The appro­ priate expansion parameters are

2 <5l0c(^) «l, 52oc^«l. (3.22)

Introducing a function A(x) by

A c2 = e , (3.23)

7One can study the system numerically, see Fig. 2 of [31]. 83 to leading nontrivial order in Si, S2 we have [31]

1 2 2 A(x) =ln<&3 - - ln(l - x ) + S\ Ax{x) + S A2(x) ,

3 24 a(x)=81 ai(x), ( ' )

X(x) =S2 X2(x), where a^il-x^^d^Ux2) , (3.25)

2 3 4 2 X2 = (l-x ) / 3F1(f,f;l;x )> (3.26)

zdz ( fz , /&*i\2 (l-y2)2'

(3.27) 2 2 2 5%2\ (l-y )

The constants 7$ were fine-tuned to satisfy the boundary conditions [31]:

2 8-7T 8-3TT . „.

The parameters Si are related to the parameters m&, m/, T of the dual gauge theory via

/rnbV <$!=- 24TT VT/ ' [r(l)]2 ^/ (3 29) 52 27r3/2 r '

2.T=fe(l + ^?+±.?)

Given the solution (3.24), c2 is found from Eq. (3.23), and cx = xc2. The transition to the original radial variable r can be made by using the constraint equation (3.17).

At ultra high temperatures Si —» 0, 52 —> 0, the conformal symmetry in the gauge theory is restored, and one recovers the usual near-extremal black three-brane metric

rlr2 ds\ = (2vrT)2(l - re2)"1/2 (-x2dt2 + dx\ + dx\ + dx\) + _ , (3.30) describing a gravity dual to a finite-temperature M = 4 SYM in flat space. 84

3.2 J\f = 2* SYM equation of state and the speed of sound

To determine the equation of state of TV = 2* gauge theory, one needs to com­ pute energy and pressure, given by the corresponding one-point functions of the stress-energy tensor. In computing the one-point functions, one has to deal with divergences at the boundary of the asymptotically AdS space which are related to UV divergences in the gauge theory. The method addressing those issues is known as the holographic renormalization [37, 38, 39, 40]. Some details of the holographic renormalization for the J\f = 2* gauge theory are given in Appendix 3.6. The method works for arbitrary values of rrib/T, mj/T, once the solution to Eqs. (3.16) is known. In the high-temperature limit, energy density and pressure can be computed explic­ itly

£ =-K2N2T4 1 + ^ (In(vrT) - 1) 5\ - A 82 (3.31) p =-TT2N2T4 1 - ^ ln(TrT) 5\ - * 52 8 7T 7T where 8i, S2 are given by Eqs. (3.29). One can independently compute the entropy density of the non-extremal Pilch-Warner geometry [31], a = WT3 (1 - 4 52 - - 52) , (3.32) and verify that the thermodynamic relation,

S-Ts = -P, (3.33) is satisfied. Alternatively, the free energy can be computed as a renormalized Eu­ clidean action. Then it can be shown [12] that the free energy density T — —P obeys T — £ — Ts for arbitrary mass deformation parameters rrib/T, rrif/T. Finally, using Eq. (3.29) it can be verified that

d£ = Tds . (3.34)

These checks demonstrate that the J\f — 2* thermodynamics is unambiguously and correctly determined from gravity. 85

We can now evaluate leading correction to the speed of sound in J\f = 2* gauge theory plasma at temperatures much larger than the conformal symmetry breaking scales rrif, m,b. Using Eqs. (3.31) and (3.29) we find 1 2 OP opfoey if 64 2 s 2\ where ellipses denote higher order terms in rrif/T and rrib/T. Substituting <5i, ^ from Eqs. (3.29), we arrive at Eq. (3.6). In the next section we confirm the result (3.35) by evaluating the two-point correlation function of the stress-energy tensor in the sound mode channel and identifying the pole corresponding to the sound wave propagation. In addition to confirming Eq. (3.35), this will allow us to compute the sound wave attenuation constant and thus the bulk viscosity.

3.3 Sound attenuation in M = T plasma

3.3.1 Correlation functions from supergravity

We calculate the poles of the two-point function of the stress-energy tensor of the M = 2* theory from gravity following the general scheme outlined in [32]. Up to a certain index structure, the generic thermal two-point function of stress-energy tensor is determined by five scalar functions. In the hydrodynamic approximation, one of these functions contains a pole at u> = u(q) given by the dispersion relation (3.1) and corresponding to the sound wave propagation in J\f = 2* plasma. On the gravity side, the five functions characterizing the correlator correspond to five gauge-invariant combinations of the fluctuations of the gravitational background.

The functions are determined by the ratios of the connection coefficients of ODEs satisfied by the gauge-invariant variables. Moreover, if one is interested in poles rather than the full correlators, it is sufficient to compute the quasinormal spectrum of the corresponding gauge-invariant fluctuation. This approach is illustrated in [32] by taking M = 4 SYM as an example. For a conformal theory such as M = 4 86

SYM, the number of independent functions determining the correlator (and thus the number of independent gauge-invariant variables on the gravity side of the duality) is three. In the J\f = 2* case, the situation is technically more complicated, since we need to take into account fluctuations of the two background scalars. These matter fluctuations do not affect the "scalar" and the "shear" channels8, entering only the sound channel. In the sound channel, this will lead to a system of three coupled ODEs for three gauge-invariant variables mixing gravitational and scalar fluctuations. The lowest or "hydrodynamic" quasinormal frequency9 in the spectrum of the mode corresponding to the sound wave gives the dispersion relation (3.1) from which the attenuation constant and thus the bulk viscosity can be read off.

In this section, we derive the equations for the gauge-invariant variables for a generic finite-temperature Pilch-Warner flow, i.e. without making any simplifying assump­ tions about the parameters of the flow. Then we solve those equations in the regime rrib/T

3.3.2 Fluctuations of the non-extremal Pilch-Warner geometry

Consider fluctuations of the background geometry

a—>aSG + , (3.36)

X -» XBG + $ ,

G where g^ (more precisely, cf , cf°), OI.BG> XBG are the solutions of the equations of motion (3.16) and (3.17). To simplify notations, in the following we use Ci, C2 to denote the background values of these fields, omitting the label "BG".

8See [32] for classification of fluctuations. 9Gapless frequency with the property lim to(q) = 0 required by hydrodynamics. 87

For convenience, we partially fix the gauge by requiring

hfr = hXir = hrr = (J . (3.37)

This gauge-fixing is not essential, since we switch to gauge-invariant variables shortly, but it makes the equations at the intermediate stage less cumbersome. We orient the coordinate system in such a way that the x% axis is directed along the spa­ tial momentum, and assume that all the fluctuations depend only on t,x^,r. The dependence of all variables on time and on the spatial coordinate is of the form

t t i x oc e~ <^ + Q 3) so the only non-trivial dependence is on the radial coordinate r.

The fluctuations can be classified according to their transformation properties with respect to the 0(2) rotational symmetry in the x\ — X2 plane [8, 32]. The set of fluctuations corresponding to the sound wave mode consists of

Due to the 0(2) symmetry all other components of h^ can be consistently set to zero to linear order. It will be convenient to introduce new variables Htt,Htz,Haa,Hzz by rescaling

hu = c.\ Htt , Ihz = c2 Htz , haa = c2 Haa , hzz = c2 Hzz . (3.39)

We also use Ha = Haa 4- Hzz. Expanding Eqs. (3.16) and (3.17) to linear order in fluctuations, we obtain the coupled system of second-order ODEs

2 2 H^ + H'tt {lncjciy-H'^ (lnCl)'-\ L % Htt + u; Hii + 2uqHu c c l \ 2 (3.40) 8 fdV , &P , , 3U^+^>=0 n'

H", + H'u (ln^)' + \uqHaa = 0, (3.41) V cl / C2

2 2 Ka + Ka (Inc^y + (H'„ - H'u) (Inciy + i L - q ^) Haa c c l V 2/ (3.42) 16 (dV . &P , . n 88

1 H»„ + H'„ {\nClct)' + {H'aa-H'tt) (lnC2)' + i J Hzz + 2ujq Htz (3.43)

+ q^-\(Htt-Ha C2

2 2 b" + ' (lnclCjj)' + \a'BG (Hu - Htty + \U - q ^) 1 cl \ c2/ 2 2 (3.44) 1 (d V , d V , . n 6 \<9a2 dadx 2 2 r + V (InClcl)' + IX'BG (HU - #«)' + \ L - q °\ i> (3.45) 1 / d2V , d2V i>\ =0, 2 \dadx dx2 where the prime denotes the derivative with respect to r, and all the derivatives of the potential V are evaluated in the background geometry. In addition, there are three first-order equations

£2 U! H', + In Hi, + q H'tz + 2 ( In ~ 1 H,tz Cl (3.46)

8a; (3a'BG (f) + X'BG VO = ° »

^ + -fa; i^ -9ifM- 8g (3a^G 0 + XBG

2 2 (lnclC2)'^-(lnc^^ + 3 a; Ha + 2coq Htz + q -\ (Htt - Haa) (3.48) (Prp QV \

+ 4 \-~ cf> + ^ VJ -8 (3a^G ^ + X'BG ^) =0.

3.3.3 Gauge-invariant variables

A convenient way to deal with the fluctuation equations is to introduce gauge- invariant variables [32]. (Such an approach has long been used in cosmology [41], [42] and in studying black hole fluctuations [43].) Under the infinitesimal diffeomor- phisms

ar a^-K (3.49) 89 the metric and the scalar field fluctuations transform as

A - V aSG 6 , (3.50)

4> -> ^ - VAXBG 6 , where the covariant derivatives are computed in the background metric. One finds the following linear combinations of fluctuations which are invariant under the dif- feomorphisms (3.50)

Q I Q d Ci \ 0 C ZH =4— Htz + 2 Hzz — Haa I 1 --j— 1 + 2—-| Htt , U) \ UT C2C2 J U! C2

^=0~(m^^aa' (3'51)

^=^"(ln%Faa-

Using Eqs. (3.40)-(3.48), one finds the new variables ZH, Z4, Z^ satisfy the following system of coupled equations

AHZ'H + BHZ'H + CHZH + DHZt + EHZlf, = 0, (3.52a)

A^Z'l + B^ + C^Z* + D^ + E^Z'H + F^ZH = 0, (3.52b)

A^Z^ -f B^Z^ + C^Z^p + D^ZQ + E^pZjj + F^ZH = 0 , (3.52c)

where the coefficients depend on the background values c1; c2, O.BG and \BG (the coefficients are given explicitly in Appendices 3.7, 3.8 and 3.9). Eqs. (3.52) describe fluctuations of the background for arbitrary values of the deformation parameters mb/T, rrif/T.

The analysis of Eqs. (3.52) is simplified by switching to the new radial coordinate

(3.18). Asymptotic behavior of the solutions to Eqs. (3.52) near the horizon, x —> 0+, corresponds to waves incoming to the horizon and outgoing from it, i.e. for each of the gauge-invariant variables we have ZH, Z^.Z^ OC x±luJ. We are interested in the lowest quasinormal frequency of the "sound wave" variable ZH • To ensure that this frequency is indeed the hydrodynamic dispersion relation (3.1) appearing as the 90 pole in the retarded two-point function of the stress-energy tensor of the N = 2* SYM, one has to impose the following boundary conditions [32] • the incoming wave boundary condition on all fields at the horizon: ZH, Z^^Z^ OC

tU} x~ as x —> 0+; • Dirichlet condition on ZH at the boundary x — 0: 2#(0) = 0.

The incoming boundary condition on physical modes implies that

iu iw ZH{x)=x- ZH(x), Z0(ar) = *-*%(*), Zi,(x)=x- Zll,(x), (3.53) where ZH, Z$, Z^ are are regular functions at the horizon. Without the loss of generality the integration constant can be fixed as

ZH = 1. (3.54)

Then the dispersion relation (3.1) is determined by the Dirichlet condition at the boundary [32]

ZH(X) = 0. (3.55) a;->l_ Following [44, 8], a solution to Eqs. (3.52) can in principle be found in the hydrody- namic approximation as a series in small u>, q (more precisely, UJ/T

af provided the background values c\, (%, OIBG, XBG e known explicitly. Here we consider the high-temperature limit (3.22) discussed in section 3.1.1, and expand all fields in series for 5\

l ZH=\2% + % Zl + 51 Z°2j+zq [Zl + 8\ Z\ + 5l Z 2j , Z^sJzl + iqZl), (3.56)

Z^ = 82 I Zl + iq Z\ J , where the upper index refers to either the leading, oc q°, or to the next-to-leading, oc q1 order in the hydrodynamic approximation, and the lower index keeps track of the bosonic, 5i, or fermionic, <52, mass deformation parameter. Eqs. (3.53), (3.56) 91 represent a perturbative solution of Eqs. (3.52) to first order in u>, q, and to leading nontrivial order in Si, S2- From (3.54), the boundary conditions at the horizon are

7° = 1, Z°i = 0, zl = 0. x-> 0+ x^0+ x-»0+ (3.57) zl = 0, z\ = 0, z\ = 0. x^0+ x^0+ x^0+ The Dirichlet condition at the boundary (3.55) becomes

7° = o, 3? = 0, z2° = 0. X->1- X-^l- X->1- (3.58) zl = o, ^ o, zl = 0. a:->l- X->1- X-*\- We also find it convenient to parameterize the frequency as

2 iq' r x2 to = ^(l + /^ + /^2 l + # <^ + $ ^ (3.59)

In the absence of mass deformation {Si — 52 = 0), Eq. (3.59) reduces to the sound wave dispersion relation for the Af — 4 SYM plasma [10]. The parameterization (3.59) reflects our expectations that the conformal J\f — 4 SYM dispersion relation will be modified by corrections proportional to the mass deformation parameters Si,

S2. Our goal is to determine the coefficients (3^, /3%, /3f, 0% by requiring that the perturbative solution (3.56) should satisfy the boundary conditions (3.57), (3.58).

Using the high-temperature non-extremal Pilch-Warner background (3.24), param- eterizations (3.56) and (3.59), and rewriting Eqs. (3.52) in the radial coordinate (3.18), we obtain three sets of ODEs describing, correspondingly: • the pure J\f = 4 physical sound wave mode {Si — 0, Si — 0); • corrections to pure N = 4 physical sound wave mode due to the bosonic mass deformation {Si ^ 0, 5% — 0) ; • corrections to pure M — 4 physical sound wave mode due to fermionic mass deformation {Si — 0, S2 ^ 0).

In the remaining part of this subsection we derive equations corresponding to each of these three sets. 92

Sound wave quasinormal mode for J\f = 4 SYM

Setting 5i = 52 — 0 in Eqs. (3.52), (3.56), (3.59) leads to the following equations

J2 7O J70 2 2 x(x + 1)£J^ + (l - 3a; ) ^ + 4:rZ0° = 0 , (3.60) ax1 ax

x(x2 + l)2 ±f§- - (*2 + l)(3x2 - 1) ^ + 4x(x2 + 1) Z* \JbJu KAJtLi /Q /% -1 \ - 4(^2 -1)2 =r- + -p*(*2 -i)z° = o. V3 ' dx ^3 The general solution of Eq. (3.60) is

2 2 Z0° = Ci(l - x ) + C2 [(a; - 1) Ins - 2] , (3.62)

where C\, C2 are integration constants. The condition of regularity at the horizon and the boundary condition (3.57) lead to

Zl = 1 - x2. (3.63)

Notice that the boundary condition (3.58) is automatically satisfied, as it should be, since (3.59) with 8\ = 0, S2 = 0 is the correct quasinormal frequency of the gravitational fluctuation in the sound wave channel for the background dual to pure J\f — 4 SYM plasma. Given the solution (3.63), the general solution to Eq. (3.61) reads

2 2 Z\ = C3(l - x ) + C4 [{x - 1) Ins - 2] , (3.64) where C3, C4 are integration constants. Imposing regularity at the horizon and the boundary condition (3.57) gives Z\ = 0 . (3.65)

Again, the boundary condition (3.58) at x = 0 is automatically satisfied as a result of the parameterization (3.59). 93

Bosonic mass deformation of the J\f = 4 sound wave mode

Turning on the bosonic mass deformation parameter 8\ (while keeping 52 = 0) and using the zeroth-order solutions (3.63), (3.65), we find from Eqs. (3.52), (3.56), (3.59)

3x\x'"*2 - 1^^^-^^t^^^-^t) (3.66) — x(x2 — 1) a.\ = 0 ,

a dZ\ 2 2 c2(x4 _l) L^_ x(x2 _ 1)(3x2 _ 1) ^ + 4X (X - 1) Z\ dx2 dx 2 2 2 70 , \c„(Jl 2 i\3 °^1 + 192a; Ax(l-x ) ^i + (l + a; )a1 Z% + 16a;(a; - 1)< (3.67) dx da; da i 32(a;4 - l)(a;2 - l)2 ( + 8xl(x2-l) ^ = 0, dx

3a;3 (a;2 - 1) 2^ < 3x2(x2-l)2^ + 3x*Zl dx2 dx (3.68) dZ° - 2v/3xV - l)2 -j± - V^a;2 - 1) 2(x — 1)— a; Q;I = 0, dx dx

x2(x4 - 1)(1 + x2) ^ - *(*4 - l)(3x2 - 1) =1 + 4a;2(a;4 - 1) Z\ dx1 dx -192x2(l + x2) Ax(x2 -l)^-(x2 + 1) a dx x

_2_ x(x2 - l)3 ^ + ~x2{x2 - l)2 Z\ dx v^" "73 (3.69) 2 2 2 (iaj 2 2 - YZ&yflx 2x(a; - l)(3a; + 1) —- - (1 + x ) ax Zl 32 da 1 dAi l + x2)2(a;2-l)3 x{x2 + 3)(x2 - If 71' dx dx - ^=(x2 + 3)(x2 - l)x2 # - -|*V - 1) /?[ = 0, where functions Ai(x), a\{x) are given by Eqs. (3.25), (3.27). 94

Fermionic mass deformation of the M — 4 sound wave mode

Similarly, turning on the fermionic mass deformation parameter 82 and leaving 81 = 0 we get

d2Z?„ „. „ .„ dZ% \2x\x2 - l)2 ^ + 12x2(x2 - l)2 ^ + 9x3 Z° + 8(x2 - l)2 % dx2 -i + wzi + w-ir-^ {3_7Q) - 3x(x2 - 1) X2 = 0 ,

rl27° r17° 3x2(x4 - 1) ?L£r - 3x(a:2 - l)(3x2 - 1) ^ + 12x2(x2 - 1) Z ° ax2 dx 2 -48x2 16x(x2-l) ^-3(l + x2) 7° as X2

+ 32(x4 - l)(x2 - l)2 f^) + 48x(x2 - l)3 ^ + 24x2(x2 - 1) $ = 0,

(3.71)

3 2 2 4 2 3 12x (x - l) -f±2 + 24x (x - 1) ^ + 9x 4 dx ax v (3.72) 2 2 dZl 2 2 dX2 8V3x (x - 1) V> v^x - 1) (x - 1) 3x X2 = 0, dx dx

d'2Z -7. ! dZ\ 3x2(x4 - 1)(1 + x2) ^ - 3x(x4 - l)(3x2 - 1) 12x2(x4 - 1) Z\ dx2 dx 1 -48x2(l + x2 2 2 7 16x(x -l)^-3(l + x )X2

/ 2 3 dZ9 / 2 2 2 - 2v 3x(xK - l) -^ + 4v 3a; (x - l) Z\ ' dx \ J i (3.73) / 2 2 2 2 2 - 32v 3x 8x(x -l)(3x + l)^-3X2(l + x ) 7° ax

2 -»{1+^,V-l)'(^) -18VSx(x" + 3,(x"-l)'^

2 2 2 2 4 - 8\/3(x + 3)(x - l)x $ - 8\/3x (x - 1) ff2 = 0, are where functions A2(x), ^(aO given by Eqs. (3.26), (3.27). 95

3.4 Solving the fluctuation equations

In this section, we provide some details on solving the boundary value problems for the bosonic and fermionic mass deformations of the M = 4 SYM sound wave mode, discussing in particular the numerical techniques involved. We start with solving Eqs. (3.70)-(3.73) subject to the boundary conditions (3.57)-(3.58).

3.4.1 Speed of sound and attenuation constant to O (5|) in J\f = 2* plasma

Here we solve Eqs. (3.70)-(3.73) subject to the boundary conditions (3.57)-(3.58). Notice that the coefficient /3% can be determined by imposing the boundary condition (3.58) on the perturbation mode Z%. The coefficient (3% can then be extracted by solving for the mode Z\ subject to the boundary condition (3.58).

For the purposes of numerical analysis it will be convenient to redefine the radial coordinate by introducing

Near the horizon [x = 0) we have y = x2 + 0(x4), while the boundary (x = 1) is pushed to y —> +oo. The computation proceeds in four steps: • First, we solve Eq. (3.70). Applying the arguments of [32] to Eq. (3.70), we find that the appropriate boundary condition on Z§ at y —> oo is

Z$~y-3/4 as y^oo. (3.75)

Eq. (3.75), along with the requirement of regularity at the horizon, uniquely deter­ mines the solution Z^(x). • Second, we solve Eq. (3.71). The solution is an analytic expression involving in­ tegrals of the solution Z§(x) constructed in Step 1. Again, the regularity at the horizon plus the horizon boundary condition (3.57) uniquely determine Z\{x). The coefficient /?£ is evaluated numerically after imposing the boundary condition (3.58). 96

• Third, we solve Eq. (3.72). The boundary condition

Z\ ~ y~3/4 as y -* oo (3.76) uniquely determines the solution.

• Finally, we solve Eq. (3.73). The regularity at the horizon and the horizon bound­ ary condition (3.57) uniquely determine Z\{x). Then the coefficient 0% is determined numerically by imposing the boundary condition (3.58).

Having outlined the four-step approach, we now provide more details on each of the steps involved.

Step 1

Asymptotic behavior near the boundary of the general solution to Eq. (3.70) regular at the horizon is given by

% = AW* + ••• + KV^ + • • • . (3-77) where A%, B& are the connection coefficients of the ODE. Rescaling the dependent variable as

2$ = (l + i/)-s/4«fo(y), (3.78) we find that the new function g$(y) satisfies the following differential equation

2 d g^ y + 2 dgj, _ 9^ 3 /7 7 y \ = dy* 2y(l + y) dy 16y(l + yf 512y(l + yf 2 ' \4' 4' ' 1 + y) (3.79) Imposing the regularity condition at the horizon, one constructs a power series solution near y = 0 »* = 9°+ (iU+Bis) *+ (-m*+ 8^>2+ofe3) • (380) The integration constant g% is fixed by requiring that, as Eq. (3.75) suggests,

^-0(1) as y^oo. (3.81) 97

For the numerical analysis, we had constructed the power series solution (3.80) to order 0(y19). Then, for a given g^, we numerically integrated g^ from some small initial value y = yin, using the constructed power series solution to set the initial values of g^(yin), g'^iVin)- (This is necessary as the differential equation has a singularity at y = 0.) We verified that the final numerical result is insensitive to

15 the choice of j/;n as long as y,n is sufficiently small (we used yin = 10~ ). Then we applied a "shooting" method to determine g^. The "shooting" method is convenient in view of the asymptotic behavior (3.77) near the boundary: unless g§ is fine-tuned appropriately, for large values of y, g^(y) would diverge, g^ oc y1/2 as y —> oo. We thus find gl fa -0.02083333(4), (3.82) where the brackets indicate an error in the corresponding digit. We conjecture that the exact result is

9% = -^ • (3-83)

(Note that a formal analytical solution to Eq. (3.79) is available. It allows to ex­ press #9, in terms of a certain definite integral. Numerical evaluation of the integral confirms the result (3.83).)

Step 2

The solution to Eq. (3.71) regular at the horizon and satisfying the boundary con­ dition (3.57) is

2 2 Z°2(x) = l(x - 1) 2^(x) - I [(* - 1) Ins - 2] 2*o(x), (3.84) where 3x2(s2 + 3 + 21ns) [' (z»-l)lns-2

2 2 2 6z -16z(z - 1) ^ + 3(* + 1) X2 Zi(z) (3.85)

+ 4(z4-l)(z2-l)2 (^Y + 6^2-l)3^ \ dz J az 3x f 1 x + dz x ^)=-2(iT^^ y0 ^rwiy 2 dX2 2 6z -Wz(z*-l)^- + 3(z + l)X2 Z*(z) (3.86)

2 .2 -.w.,2 i\2 I dX2\ . „ , 2 ,x3 dA2

We explicitly verified that

a \im_l zo(x) = 0(1). (3.87)

Thus the boundary condition (3.58) becomes

lim I*o(a;) = 0. (3.88)

Numerically solving Eq. (3.88) for Z^, we find

02 ~ -^" x 0.9999(5), (3.89) where we factored out the value —4/37T for the coefficient (3% obtained earlier from the equation of state (see Eq. (3.35)).

Step 3

The general solution to Eq. (3.72) has the form

\/3 \ r l fx cos ( 2arc*ann z) Z^ — sin I -— arctanh x J x < Ci — - / z3

dz dz (3.90)

(V3 u \ \n If* ^n (f arctanh ^ + cos —arctanh x x < C2 / cfe—— 5 — x 3 2 / I 6 i0 z 8*V-i)^ + 8(z»-!)£-** 99 where C\, Ci are integration constants. For generic values of these constants we have asymptotically

Z\ -sin f ^arctanh x ] {A% (1 + • • •) + 6^(1 - xf'4 + • • •) f r \ (3'91) + cos I ^arctanh x\ (A%{1 + • • •) + B%{\ - xf'4 + • • •) where A^, B^, A^,, B^ are the ODE connection coefficients. The integration con­ stants Ci, C2 should be chosen in such a way that the boundary conditions for the matter fields At, = 0, A^ = 0 are satisfied.

: To make numerical analysis more convenient, we introduce functions J s(x), Tc{x)

2 3 4 * Z]p{x) = (1 — x ) ' •< sin I *-.,_(-^-arctanh x ] Ta(x) + cos , ( -^-arctanh x, ] J-C(x) ,

(3.92)

Redefining the radial coordinate as in Eq. (3.74) and using Eq. (3.72) we find that

: J s{x)) Tc{x) satisfy the equations

dTa 3 = dy 4(1 + y) s (3.93) cos (^arctanh y 2Fi ; 3; + (y+1) yl/2(1 + y)3/2 i28 U'i rr^J ^"3 ^

dTc 3 Fc = dy 4(1 + y) (3.94) sin (^arctanh y 3 P /^7 7 q 2/^ . 4^ . 1\ ^_ yl/2(1 + y)3/2 .i282FlU'i; 3;r^J+^_3(y+1)^j r Asymptotics of the solutions J s(x), Tc{x) for y —» 00 are ^ - 0(1) + C? (y3/4) , (3.95) ^ -> 0(1) + O (y3/4) .

Accordingly, the initial conditions for the first-order ODEs (3.93) and (3.94) should be chosen in such a way that the coefficients of the leading asymptotics in (3.95) 100 vanish. (This choice guarantees that the matter fluctuations of the mode Z]p do not change the fermionic mass parameter of the dual gauge theory.) Near the horizon, power series solutions to Eqs. (3.93) and (3.94) are r. = ft + (-i»J - ^) v1'2 + \ft v + 0(y"2), (3.96) * = J?+ (i+ ¥+T*) »+ ("I'"+M - Sf*°) *+ofe3) • (3.97) where /s°, /° are integration constants, and g\ is chosen as in Eq. (3.83). We use a "shooting" method to determine /°, /°: these initial values should be tuned to ensure that Tc, Ts remain finite in the limit y —>• oo. We find /°« 0.01964015(5), (3.98) /° m - 0.01743333(5), where the brackets symbolize that there is an error in the corresponding digit.

Step 4

The solution to Eq. (3.73) regular at the horizon and obeying the boundary condition (3.57) reads

b where the functions Izl(x), l zl(x), T%i(x) are given explicitly in Appendix 3.10. We verified that Jim 2|x = 0(1), liml^ = O(l), (3.100) and thus the boundary condition (3.58) translates into the equation

b lim I 7l = 0. (3.101)

To determine the coefficient (3%, we solve Eq. (3.101) numerically using the value of /3£ computed in (3.89). We find

/?[ = /5j « 0.9672(1). (3.102) 101

3.4.2 Speed of sound and attenuation constant to O (S2) in J\f — 2* plasma

We now turn to solving Eqs. (3.66)-(3.69) subject to boundary conditions (3.57)- (3.58). The computation is essentially identical to the one in the previous section, thus we only highlight the main steps.

Step 1

Using the radial coordinate defined by Eq. (3.74), we find that the asymptotic be­ havior near the boundary of the general solution to Eq. (3.66) is given by

Zl = Ay^ + ••• + By1'2logy + ... , (3.103) where A^,, B\ are the connection coefficients of the differential equation. Introducing a new function g^,(y) by

Zl = (l + y)-^g,(y), (3-104) we obtain an inhomogeneous differential equation for g^\

2 d g4> 1 dg^ 1 3105 dy2 + ~y ~Xdy7 ~ 777T-,4y(l +— yf\o 9w 96y(]kw*$\*T$=°- <' >

Near the horizon one can construct a series solution parameterized by the integration constant g^ of the solution to the homogeneous equation

9* = 9l+(± + \gl) y+ (~-~9l) v2 + <%3). (3.106)

The integration constant g^ is fixed by requiring that

30 ~ 0(1) as y-»oo. (3.107)

Using the "shooting" method we obtain

flj « -0.08333333(3). (3.108) 102

We conjecture that the exact result is

(3.109) 12

(Note that a formal analytical solution to Eq. (3.105) is available. It allows to express g^ in terms of a certain definite integral. Numerical evaluation of the integral confirms the result (3.109).)

Step 2

The solution to Eq. (3.67) regular at the horizon and satisfying the boundary con­ dition (3.57) has the form

Z\{x) = 8(x2 - 1) l«o{x) - 8 [(x2 - 1) lnar - 2] 2|o(z), (3.110) where

x2(3 + x2 + 2\nx) [z2 -l)lnz-2 x 3-z°( ) — ~~ 2 2 P1+ f dz 4 2 2 4(1 + x ) Jo0 ~~ z(z - 1)(1 + z )

a 2 da.\ 2 24z -Az{z - 1) ^ + (z + 1) ai (3.111)

+ 4(z4 - l)(z2 - I)2 (*p) + 6z(z2 - l)3 dM dz dz '

X x z°S )~ 2(1 + ^)2 (5\ + f dz 2 3 X Jo z(z + l)

2 2 2 24z -4z(z - 1) ^ + (z + 1) ax 7° (3.112)

+ V-i)(^-i).(f)+^-i)'^

We have verified that lim2^(x) = 0(l), (3.113) X-+\- 1 and thus the boundary condition (3.58) translates into the equation

lim lL(x) = 0, (3.114) X—>1_ 1 103

Solving Eq. (3.88) numerically for (3\ we find 32 ftta TV2 x 1.00000(1), (3.115) where we factored out the value —32/IT2 obtained from thermodynamics (see Eq. (3.35)).

Step 3

Asymptotics of the general solution to Eq. (3.68) near the boundary y —>• oo is given by Eq. (3.103) with different coefficients A\, B\. Rescaling the dependent variable

Z} = (l + y)-1/2G^(y), (3.116) we find that the function G$ satisfies the following differential equation

2 d G^ , 1 dG0 1 y/3 p/33 y dy2 y dy 4y(l + yf 9 96y(l + y)2 A l \2'2' ' I + y (3.117) 1 = 0. ~2v/3y(l + y)2 2(V+I)f-* One can construct a series solution regular near y — 0

G, = G° + f-^| - ^; + 1 Gj) y + 0(y2). (3.118)

The integration constant G^ is fixed by requiring that

Using the "shooting" method and the value of the constant g^ given by Eq. (3.109) we find Gj « 0.059(0). (3.120)

Step 4

The solution to Eq. (3.69) regular at the horizon and obeying the boundary condition (3.57) reads

Z J }1 2(ga )hia; (3 121) IW ^N rzlV) + ^ 4i(*) + ^ W • ' 104

b where the functions lz%(x), l zl(x), Izi(x) are given explicitly in Appendix 3.11. We checked that

lim T7i ~ 0(1), lim T-i ~ 0(1). (3.122) X—>1_ 1 X—>1_ 1 In view of Eq. (3.122), the condition (3.58) becomes

limJ|i=0. (3.123)

To obtain the coefficient /3f, we solve Eq. (3.123) numerically using the value of (5\ given by Eq. (3.115). We find

0[ = fil « 8.001(8). (3.124)

This completes out computation of the coefficients P±, (5^, fl\, 0%: they are given, respectively, by Eqs. (3.115), (3.89), (3.124), (3.102).

3.5 Conclusion

In this Chapter, we considered the problem of computing the speed of sound and the bulk viscosity of H = 2* supersymmetric SU(NC) gauge theory in the limit of large 't Hooft coupling and large Nc, using the approach of gauge theory/gravity duality. The computation can be done explicitly in the high temperature regime, i.e. at a temperature much larger than the mass scale m& and m/ of the bosonic and fermionic components of the chiral multiplets, where the metric of the dual gravitational background is known. Our results for the speed of sound and the bulk viscosity computed in that regime are summarized in Eqs. (3.6), (3.7), (3.8). It would be interesting to extend the computation to the full parameter space of the theory as well as to other theories with non-vanishing bulk viscosity. It would also be interesting to compare our results with a perturbative calculation of bulk viscosity in a finite-temperature gauge theory at weak coupling. 105

3.6 Appendix: Energy density and pressure in J\f = 2* gauge theory

Energy density and pressure of J\f = 2* SYM theory on the boundary dM.5 of a manifold Ai5 with the metric 3.15 can be related to the renormalized stress-energy tensor one-point function as follows

S = v^JVs u^iT^), (3.125)

P = ^Ns {TXXXl)f^ , (3.126) where u^ is the unit normal vector to a space-like hypersurface E in dM.5, a is the determinant of the induced metric on E, and iVs is the norm of the time-like Killing vector in the metric (3.15). The renormalized stress-energy tensor correlation func­ tions are determined from the boundary gravitational action (with the appropriate counterterms added) in the procedure known as the holographic renormalization. Holographic renormalization of J\f = 2* gauge theory on a constant curvature man­ ifold was studied in [12]. Using the results for the renormalized stress-energy tensor one-point functions [12], one finds

1 Q £ = Q^Q-/* i ^ ~ Pu -12*foio + MpuPio ~ IQxU + 36&0 , (3.127)

P = ^c~e4€ (W + 9Pn + 12XoXio - mnPio + 16*& - 36fe) , (3.128) where the parameters P,^,pio,Pn,XotXio are related to physical masses and the temperature for generic values of rrib/T and rrif/T (see section 6.4 of [31]). In the limit mb/T

0 = 2, e^2^r(i-^'-^), 41n2 . „ 8 . Pw = oil Pn = — *i, (3.129) TV TX x ' y - ^ 5 v - 2[r«W4 2 ~ [rmi2 ~ ~ TV 106

Using Eq. (3.129), from Eqs. (3.127) and (3.128) we obtain (to quadratic order in 6i, 62) the energy density and the pressure given in Eqs. (3.31).

3.7 Appendix: Coefficients of Eq. (3.52a)

2 2 2 2 2 AH(x) = Zu d2c 2c\ ( -c2c1q c'1 - 2d2c\q + 3co d2cl J ,

2 2 2 3 2 3 2 2 2 2 BH(x) = LU c2d ( 27LO C 2C1C'2 - 42clq c'2 + §d\cxq 'd2d - %c 2c\q d2V

2 4 2 2 2 CH(x) = to (s^c'^lqW, + 9u ($4 - %q c\d2 c\V + 36q c^f c[c2

2 4 4 2 2 2 4 2 4 4 - 24q c d2 + 16q clc'2c'14V - %q c\d c\V + 3d2c\q dxc2 + 6d c q

2 2 2 2 - Ylq Cldlc\d2 - Vou d2 c\q c\ I ,

,cTP D (x) = 16q2c2 i-c d + c[c J (24c cu2{aBG)'V4 + 364co2d (aBG)'d + 3c2 c u>2c: H x 2 2 x 2 1 2 x da

2 BG 2 BG 2 - m'cf^c', - 36c2cxc^ (a )' - 24c2q c\V{a )> - ^ c\d2^ ,

2 2 2 C C 1 EH(x) = \§q c\ ( -cxd2 + c[c2 j I %CXUJ ^BGVCI + 12U) C'2X'BG 'I I + 3(?2cxu d2 —

2 2 2 2 - c2q c\—dl - SC^CIVX'BG - l2cxd uj ^BGC2 - 2q c\d2—\ .

The prime denotes the derivative with respect to r. 3.8 Appendix: Coefficients of Eq. (3.52b)

2 2 2 2 A^ = 12caCi4 I -c2ciq c[ - 2d26\q + 3u d2c 2 1 ,

2 2 2 B^ = \2c\d2c21 —c2c1q d1 — 2d2c\q + 3uj d2d\ I I 3cxd2 + dxc2 J ,

2 2 2 2 2 Ct, = -§c\

2 2 2 2 2 2 2 2 2 2 - 30c LU c2 q c + ^d2c\dxc2 + 2q c\c d ^ - 48c^ c2a'BGc ^

2 3J + 2 l 2 2 + q'c\c%d^ 2^-^d^x + ^\§q^WBG4~c\d2otBG 4^- ++ 8q8q'cia'c\a'BGBG4—dd^dx x ++ 12q%12q^c4

2 4 2 2 + 18c'2 4u + 192q ct4(a'BG) V, ( d2V dV 82V 2 2 3 2 J2J2 " ' V D^ = -2c\c 2\-d2c2cx-^-dxq + 24c2c 2aBG—to + 3d2 c 'dadx

2 + 8C 2 ~ ^H^ '^ X'BG^ + M4a>BG^BGV dV dV 2 2 2 2 2 - 16d2c2cxa'BG—q - ^c 2c xa'BGq ^BGV - 8d2c2c\q ^BG —

o 2 / 9V , 2\ - 8c2cxaBG-—cxq \ ,

8a'BGVc2 + d2 — J,

A 2 F+ = cxc 2u f 8a'BGVc2 + d2 — J ( -cxd2 + c[c2 j .

The prime denotes the derivative with respect to r. 3.9 Appendix: Coefficients of Eq. (3.52c)

„2„2„' / „ „ „2J o,l„2„2 , o, ,2 / 2 A^ = 12c2qc2l —c2Ciq cx — 2c2cxq + 3u> C2C2 I ,

2 2 2 £,/, = 12cic2c2 ( -^c^ci - 2d2c\q + 3u d2c 2 j f 3ci4 + c[c2 ) ,

4 2 2 C* = 2U?$44?^4 + Qq c'2clc'lC2 - 30C2Vc2yC? - 64C2VC (XBG) 7>

5 2 2 2 - A8C 2UJ%X'BGCI^ ~ fc&A^cx + Qq c\c 2{d2f^ + Uq 44(X'BG)

2 2 2 2 + ^A^BG^k + ^ 4CWBG4^ ~ ^ d2 cf^ + 18c> 4u*

4 2 + 12g (c2) c^ ,

2 2 2 2 2 2 2 ^ = -2c*c (-6c2 c ||v + 94 ci^|-a; - 192c c aBG(?YBGP

4 / 2 2 2 2 + 192c a BGa; x'BGP - 8c ClxSG—c'l(? - 16c2c2c gYSG~

+ 244^WVBG^" ( dV \ 2 E^ = -clc\u (3—c2 + 8C2XBGVj ,

FTP = -c\cxu? f 3—c2 + 8c2xBGV j ( ci4 - 4c2 j

The prime denotes the derivative with respect to r. 3.10 Appendix: Structure functions of the solution 3.99

v y 2 2 A + lnX + 1} ^ ((1l ++ xx ) 1 (lW " ^ 4x2(7 + 7x2 + 2a;4)l r (^2-l)lnz-2 2 3 22+ + 22 x (1 + z ) J ^ /J00 ^-1)z(z* - (1)(11 ++ ,z )f 24^2(1 + ^2) -lQz(z2 - 1) ^ + 3(z2 + 1) X2 4W dz

_3^2_1)3^+6Z2(22_1)2V2Z 07 0

2 2 2 2 + 48z -8z(3z + l)(^-l)^ + 3(l + z ) X2 4W

- 24^2 + 3)(z2 - l)3 ^ - 16(1 + z2)2(z2 - l)3 (^

6x2 2 2 2*fx)- gr, 2x (9 + 6x + ^) /" 1 2z»l(XJ"(l + xa)»^+ (1 + z2)3 ^2 +h 'z(l + z•: 2\4X '

a 2 2 2 24Vlz (l + z ) -16z(z - 1) ^ + 3(^ + 1) X2 ^

_3^2_l)3^2+6/22(z2_1)2z0

2 2 2 2\2 + 48z -8z(3z + l)(z - 1)^ + 3(1 + ^ ) X2 7°*j, A/, 2 3 -24z(z2 + 3)(z2-l ^-16(l + z )V-l) (p- dz 2 2 2 4 6x ,r 2x (9 + 63; + a: ) f , 1 Xk^~ (1 + ^2)2^/*2 . 2 2 az 2 X ((1i ++ xx )3 ^ J0 z(i + z y

2 2 2 dX2 2 24^z (l + z ) -16z{z - 1) ^ + 3(z + 1) X2 *J W7° 2 3 2 -3*(* -l)v ^^p +' "6*V-l)* y *> Z-2°

+ 48z2 -8z(3z2 + l)(z2-l)^ + 3(l + z2)2 J 2; 2 - 24z(z2 + 3)(z2 - l)3 ^dA 2- _16( 1 + zmz 2_lf (**f\ y 3.11 Appendix: Structure functions of the solution 3.121

2 2 2 2 2 .a . , x {'S + 2\nx + x ) aT f2x (x + 3) , 1 . 2 .,, J! ) = 2 2 + ^ (l + x ) fi { 3(14-^)3 ^-3^ + D 4z2(7 + 7x2 + 2x4)\ /"* (^2-l)ln^-2 + 3(1+ x2)3 ;A+7o z(z*-l)(l + z2y X _ r w™. i 2 2 2 2 96\/3z (l + £ ) -4^ -l)^l + (z + 1) ai Zi

-z(z2-l)3d^- + 2z2(z2-l)2 Z° \AiAi + 192z2 -2z(3z2 + l)(z2 - l) ^L + (1 + z2)2 J Zl

- 8z(z2 + 3)(z2 _ 1)3 ^1 _ 16(1 + *2)2(z2 - l)3 (^Y aU4z \ UbAdz J 2 2 4 Ib (x] ^ ,r , 2x (9 + 6x + a: ) /* 1_ J 1(X) 2 2 A + 2 + dZ x ^ "(I + x ) 3(1 + x f & J0 4T^)< 2 2 2 [96\/3/(l + ^ ) -Hz -1) ^ + (z +1) ai] z}

2 2 2 2 - z{z -if ^ + 2z {z -I) Z°x

+ 192z2 -2z(3z2 + l)(z2-l)^i + (l + z2)2a 7° dz 1 da\ - 8z(z2 + 3)(z2 - l)3 ^ - 16(1 + z2f(z2 - l)3 (^

2 2 (x)_ ^ 3r 2x (9 + 6x + x') [* 1 [x) 2 2 dz x * - (i+x ) ft 3(1+^)3 ft y0 ^rr )' (96V3z2(l + z2) \-4z(z2 -l)^ + (z2 + 1) aj ^ d7° — z(z' - f^l 2(z2-l)2Z\ 1 dz + 2z 2 2 2 2 + 192z -2^(3/+ 1)(^ - 1) ^ + (1 + z ) ai Zl

- 8z(z2 + 3)(z2 - l)3 ^i - 16(1 + z2)2(z2 - l)3 (P-X) dz \ dz J J Ill

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CHAPTER 4

Hydrodynamics of Sakai-Sugimoto model in the quenched approximation

Recently Sakai and Sugimoto (SS) [1, 2] introduced a supergravity model which is the holographic dual [3, 4] to four-dimensional, large Nc QCD with massless flavors.

Specifically, they considered five-dimensional SU(NC) maximally supersymmetric Yang-Mills (SYM) theory compactified on a circle with anti-periodic boundary con­ ditions for the fermions1, and coupled to Nf left-handed quarks and Nf right-handed quarks localized at different points on the compact circle. At weak coupling the model can be represented by intersecting DA/D8/D8 brane system in type IIA string theory compactified on a circle. Coincident iVc D4 branes wrap the compact- ification circle, while two stacks (with Nf branes in each) of D8 and D8 branes are localized at different points on this compactification circle. At strong coupling, the wrapped D4 branes are replaced with an appropriate near horizon geometry, while the D8 and D8 branes are treated in the probe approximation2. The probe brane approximation is valid in the low-energy limit, and as long as Nf

One of the most interesting aspects of the SS model is that it provides a simple

1Such boundary conditions completely break the supersymmetry and give masses to adjoint fermions of the 5d SYM theory. 2In other words, the eight-brane backreaction on the DA brane bulk geometry is neglected. 117 holographic realization of the non-abelian chiral symmetry breaking. In [5, 6] the authors studied finite temperature confinement/deconnnement and chiral symmetry restoration in SS model. Rather interestingly, these two phase transitions are not necessarily simultaneous: for small enough separation of quarks on the circle, it was found [5] that there is a phase of the hot SS gauge theory which is deconfined, but with a broken chiral symmetry. As the separation between quarks exceeds certain critical value, both the deconfinement and the chiral symmetry restoration occur at the same temperature. For all range of parameters, all of these phase transitions are of first order. Moreover, each phase, even being thermodynamically unfavorable, i.e., having a larger free energy, appears to exist at arbitrary temperature3.

In this paper we study transport properties of the hot SS gauge theory plasma in the quenched approximation. Since the backreaction of the D8 and D8 branes is neglected, effectively, we study the hydrodynamics of near-extremal D4 branes wrapped on a circle with anti-periodic boundary conditions for the fermions. The latter model was discussed in [12] as the first example of the confining theory con­ structed within gauge theory-string theory correspondence. Since the background geometry satisfies condition of [13, 14], the shear viscosity r\ of the SS plasma satu­ rates the universal viscosity bound proposed4 in [15] hi <"> where s is the entropy density. On the other hand, the speed of sound and the sound wave attenuation is gauge theory specific. Given the dispersion relation for the sound waves

where vsX are the plasma sound speed, and bulk viscosity correspondingly, for the

3This should be contrasted with a model with abelian chiral symmetry breaking [7], where the chirally symmetric phase is believed to exist only for sufficiently high temperature [8, 9, 10, 11] even though the chiral symmetry restoration phase transition is expected to be first order. 4The universality of the shear viscosity in the supergravity approximation was proven in [13, 16, 14]. 118

SS model (in the deconfined phase) we find

vs = -7= , - = 77 • (4.3) V5 7? 15 Notice that transport coefficients (4.3) are not dissimilar from the transport prop­ erties of other examples of strongly coupled near-conformal gauge theory plasma [17, 18] (^2-£)«l, jj - -* (V- "4) ' K"1' ^ In fact, the precise value of K = 2 from (4.3) is exactly the same as for the cascading gauge theory [18]. Also, there is no signature of the confinement/deconfinement phase transition in the sound wave dispersion relation. The latter is not unexpected, given that this phase transition is a first order.

The rest of this paper is the derivation of (4.3). In the next section we discuss effective five-dimensional action of the near-extremal wrapped DA brane system. Interestingly, the resulting five-dimensional supergravity action is very similar to the one obtained in [11, 18]. In section 3 we study fluctuations of the corresponding black brane geometry dual to a sound wave mode of the quenched SS gauge theory plasma. We introduce gauge invariant fluctuations and obtain their equations of motion. These equations of motion are valid beyond the hydrodynamic approximation, and for arbitrary temperature. In section 4 we derive and solve fluctuation equations analytically in the hydrodynamic limit. Imposing Dirichlet condition on the gauge invariant fluctuations at the boundary of the background black brane geometry determines [19] the dispersion relation for the lowest quasinormal frequency (4.2). Finally, using the universality result (4.1) we can extract (4.3).

It would be very interesting to extend our computation beyond the probe approx­ imation. We expect that the speed of sound waves and their attenuation would develop in this case dependence on the number of fundamental flavors Nf, and the dependence on the radius of the compactification circle. Relevant discussion of the related supergravity background was presented in [20]. 119

4.1 Consistent Kaluza-Klein Reduction to 5-dimensions

In this section we derive the 5-dimensional effective action from the type IIA 10- dimensional supergravity action by reduction on S1 x S4.

Consider the type IIA supergravity action in the Einstein frame5:

1/2 (w) 2 2 SUA — R -\v^V»® - e*l \F4\ (4.5) K - "110 JM\QJM\a where $ is the dilaton and F\ is the 4-form field strength, and the following metric ansatz:

10 — ^A^TM^^'MNL (X^U 10 . (4.6) - e-Tfg^dxi'dx" + e2/ Ie8w (dS1)2 + e~2w (dS4)2 , where the capital Latin indexes (M, N, ...) run from 0 to 9 and the Greek indexes

(/a, v,...) run from 0 to 4. Moreover, the fields / and w as well as and F4 do not depend on the coordinates of S1 x S4. The 4-form field strength F<± is given by

1 FA = :AWc (4.7)

where A is a constant and ojs* is the 4-sphere volume form. With such an ansatz, we obtain: {-Gn112 = (-9)l/2 e-"f (a) 1/2

(_Gd0))l/2 |F4|2 = A2e-*U-v>) (_5)l/2 e-ff {gi)l/2 ^ (4.8)

(10 1/2 1/2 1/2 (-G >) (^*)(^$) = (-g) (g4) (d^)(d^), where g4 is the determinant of the metric of the 4-sphere, and the curvature scalar .R(10) is given by:

40 #10) = gf / 2 R(» _ 20g>»(dtlw)(dvw) - jg^(dj)(dj) 12 e- ^-^, (4.9) where R(s) is the curvature scalar with respect to the 5-dimensional metric g^. From (4.8) and (4.9), and integrating over S1 x S4 in the action (4.5), the effective

5We use conventions of [21] and keep only relevant fields. 120

5-dimensional action follows:

2IXVA 5 2 40 1 S* = d x (-gf (5) 2 2 2 (4.10) 2K? R - -f(df) - 20(dw) - ^(«9$) - V "10 JM5 where V4 is the volume of the 4-sphere and

V = A2e^e'"f+Sw-l2e-"f+2u (4.11)

Notice that (4.10) is very similar to the five dimensional effective action of the cascading gauge theory derived in [11, 18]. From (4.10) we obtain the following equations of motion:

, 3 dV n (4.12)

1 dv n (4.13) D"-40^=°' (4.14) 40 1 R% = j djdj + 20 d^wduw + - dpQdvQ + -g^ V. (4.15)

Consider now the following ansatz for the 5-dimensional metric:

ds\ - -c\ dt2 + c2, dx2 + c2 dr2 , (4.16) the correspondent 10-dimensional line element therefore takes the form:

2 2 2 2 2 2 +Sw l 4 2 ds w = e""/ [-el dt + c\ dx + c 3 dr } + e f {dS f + e^~^ (dS ) . (4.17)

A comparison with the finite temperature bulk geometry of the Sakai-Sugimoto model in the deconfined phase which is given by [12, 5]:

9 d

419 ci = ^44^[A(r)]2, ^ )

c2 - Cl [A(r)p ,

<* = gP &DA r~* [&(?)]-*.

2 In what follows we set gs — 1 and #04 = 1 (in this case A = |), so that the relations (4.19) are:

w = Tolnr'

13 f = 1 80lnr' 5 1 (4.20) Cl = re [A(r)]2 .> 5 c2 = r6 , 2 1 — _ "2. c3 r 3 [A(r)]~

We conclude this section with short comments on the thermodynamics of the black brane configuration (4.18). The Hawking temperature T is related to the non- extremality parameter r\ as T-W (4'21)

From (4.18) the entropy density s of the black branes is

s oc r5/2 oc T5 (4.22)

The first law of thermodynamics -dP = dF = -s dT , where P is the pressure and F is the free energy density, then implies that

-P = F = -\ sT =» e = - sT => F = - e, (4.23) 6 6 5 122 where e is the energy density. From the equation of state (4.23) we find that

In the next section we reproduce (4.24) from the dispersion relation for the pole in stress-energy tensor two point correlation function in the sound wave channel, or equivalently [19], from the dispersion relation for the lowest sound channel quasi- normal mode in the non-extremal geometry (4.18).

4.2 Fluctuations

Now we study fluctuations in the background geometry

9/J,U * 9/J.V > "•/tf >

/ - f + Sf, (4.25) w —> w + Sw ,

$ -> $ + <5$, where {g^, f, w, $} are the black brane background configuration (satisfying (4.20)), and {hpvjSf^WjS®} are the fluctuations. We choose the gauge

htr = hXir = hrr = 0. (4.26)

Additionally, we assume that all the fluctuations depend only on (t,x^,r), i.e., we have an 0(2) rotational symmetry in the X\X2 plane.

At a linearized level we find that the following sets of fluctuations decouple from each other

\I^XlXl 1^X2X2) )

{htxi, hXlX3} , (4.27)

\1Hx2i ')'X2X3 J 1

\htt, haa = hXlXl + hX2X2, htX3, hX3X3, Sf, Sw, 5$>}. 123

The last set of fluctuations is a holographic dual to the sound waves in quenched SS gauge theory plasma which is of interest here. Introduce t+i htt=c\htt = e-^ ^ c\Htt,

t+i X3 2 hz =4 kz = e-^ " c-2 H11tzt.. )

U _r2 ? _ -iwt+iqx3 2 TT

ilzz —c2 "'22 — e ^2 "zz > (4.28)

-"ri — -f^aa "T" J^zz > 7 where {#«, if^, i/aa, HZZ,J , Q,p} are functions of a radial coordinate only. Expand­ ing at a linearized level Eqs. (4.12)-(4.15) with Eq. (4.25) and Eq. (4.28) we find the following coupled system of ODEs

• r2 3-1 ' 2 / 2 2 2 0=H" + HL ln ^2 ^ [In Cl]' -^f g ^i/tt+W #« + 2u,g iftz C C c3 l V 2 (4.29) 2 2/ap ap ap -^<^3^3U/ +a^+a¥p

(4.30) o =#Hz" + HLtz In + 4 toq Haa, cic3

C1C2 0 =#1 + H'aa In + (#;,-#;) Ml'+ 2 2°1 #„ c3 (4.31) 4 2 zap ^ ap _ dv

0 =H,y + i?„ In + (^a-if;)[lnc2]' C3 2 + 4 P #z, + 2u;g Htz + q ^(Htt - Haa)) (4.32) Cl V C2 / 2 2 fdV _ ap _ 9P 3 d V df dw d<& ' £ic| 0 =.F" + j?> In , , 2 c + ^/ [^i-^]' + |(^-9 |) ^ 3 (4.33) 3 d2V d2v d2v - — CS F + Q + so 6\dp dfdw a/a* P , 124

£i£| C 0 =0" + J In + l:w'[H,-Hu}'+ i(^-q4 n c3 (4.34) 2 2 1 o 2 d v d v (. o v F + 40 6\dwdf dw2 n + dwd$ P

1*/ 2 2 0 =p" + p' In + ^'[Hii-Htt]' + ^[u -q ^) p c3 (4.35) d2V „, d2V „ d2V -cl( where all derivatives dV are evaluated on the background geometry. Additionally, there are three first order constraints associated with the (partially) fixed diffeomor- phism invariance

In* 0=to H' + Hii)+q[Hi + 2 In* H tz C L (4.36) + u ( ^f'F + 40w'n + &p

C2 0 =q H' In Ht \u H'tz - g H,aa q \^-ff + 4:0w'n + &p) , Cl (4.37) 2 / 2 , 3 2 2 0 =[lnc1t|] ^f - [lnc 2]' H'tt + % I u Hu + 2wg ifta + q °\ (Htt - Ha C C l \ 2 (4.38) ap_.0p„.ap ^ ^0/^ + 40^/ + $y + C"[df T+ dw Q ' d$rJ p - V3 We explicitly verified that Eqs. (4.29)-(4.35) are consistent with constraints (4.36)- (4.38).

Introducing the gauge invariant fluctuations

O ( G C Ci 0 C + 2—--x f/tt, ZH =4— Htz + 2 Hzz — Haa 1 — 2 U) Ul c'2C2 UT Co

Z =T- r H* f [lnc|f (4.39) V rr J-J-U) 0 ' [lnc|]' a°' $' Z§ =p- H \\nc\]> naa ' and a new radial coordinate = £l (4.40) ~ c2 125 we find from Eqs. (4.29)-(4.35), (4.36)-(4.38), decoupled set of equations of motion for Z's : 2 2 2 2 _d ZH (3q (2x -l) + 5u ) dZH ~ dx2 + x(5u>2-q2(3 + 2x2)) ~dx~ 2 2 2 2 2 2 2 2 2 5/3 4 ((-co + q x ) (q (3 + 2 x ) - 5 to ) - 18 q rA x (1 - z ) ) , x 4 41 + - "^ ; '- ZH ( ' ) 2 2 2 2 5/3 2 9 (5u - q (3 + 2 x )) (1 - x ) x rA ( 3 92 + ^ 2 ,f 2~ 2T'"P 2„ (48 Zw + 9 Z» + 52 Z,) , 15 ur (5w2 — g2 (3 + 2 a;2)) 2 2 2 2 2 2/3 _d Z; ldZ;_ 4 (25(l-x )(-W + gV) + 243rAx (l-o: ) ) 2 2 2 8/3 f ~ dx x dx 225 rAx (1 - x ) 9 + n (Zs> + 12 Zm) , 25(l-x2)2 K ! (4.42) 2 2 2 2 2 3 =fZ^ 1 dZw 4 (25 (1 - x ) (-c + qV) + 162 rA z (1 - z ) / ) 2 + 2 2 3 Zw ° dx x dx 225 rAx (l-x f + —; :? (12 2/ - Z$) , 25(l-z2)2 V ; ; (4.43) 2 2 + 2 2 2 2 8/3 2 2 2/3 ddxZ* x1 dZ,dx 445 ( 5 (1 - * ) (""r A+x <(l-xM + )9rA* (1 - * ) )

(4.44) Further decoupling occurs if we introduce

K = 48 Zw + 9 Z$ + 52 Z/ . (4.45)

In this case we find6: d2Z (3q2(2x2-l) + 5uJ2) dZ 0 H H dx2 x{5u2-q2(3 + 2x2)) dx

2 2 2 2 2 2 2 2 2 4 ((-w + q x ) (q (3 + 2 ,T ) -5u )- 18 g rA a; (1 - x f'*\ ZH (4.46) 2 2 2 2 5/3 2 + 9- (5 u> - q (3 + 2 x )) (1 - x ) x rA 4 g2(-3g2 + 5a;2) + 15w2(5w2-g2(3 + 2a;2)) K' 6This set of gauge invariant fluctuations will be sufficient to determine the sound wave dispersion relation. 126

d2K 1 dn 4 (or - q2x2) 0=-^2 + - — + - 2 2^ 5/s K. (4.47) dx x dx 9rAx (l-x )

4.3 Hydrodynamic limit

We study now physical fluctuation equations (4.46), (4.47) in the hydrodynamics approximation, u —> 0, q —*• 0 with - kept constant. Similar to the computations in [17, 18], we would need only leading and next-to-leading (in q) solution of (4.46)

±luJ 2nT and (4.47). We find that at the horizon, x —•> 0+, Zfj oc x ^ \ and similarly for K. Incoming boundary conditions on all physical modes implies that

im iV0 ZH(x) = x- zH(x), K{X) = x~ 1C(x), (4.48) where {ZH,1C} are regular at the horizon; we further introduced

UJ q ft) = (4.49) 2vrT ' ' 2TTT There is a single integration constant for these physical modes, namely, the overall scale. Without the loss of generality the latter can be fixed as

zH(x) = 1. (4.50) 2)^0+ In this case, the pole dispersion relation is simply determined as [19]

zH(x) = 0. (4.51) X->1- The other boundary condition (besides regularity at the horizon and (4.51)) is [19]

K{x) = 0. (4.52) £->U Let's introduce

ZH = ZH,O + i q zH,i, K = K0 + i q Ki, (4.53) where the index refers to either the leading, oc q°, or to the next-to-leading, oc q1, order in the hydrodynamic approximation. Additionally, as we are interested in the 127 hydrodynamic pole dispersion relation in the stress-energy correlation functions, we find it convenient to parameterize

2 ro = va q — i q T (4.54)

where the speed of sound vs and the sound wave attenuation F are to be determined from the pole dispersion relation (4.51)

ZH,O = 0. 2ff,i 0. (4.55) X—*l- !-»!_ Using parameterizations (4.53), (4.54), we obtain from (4.46) and (4.47) the follow­ ing ODEs

0 =x /C0' + K'Q, (4.56)

2 2 (6x -3 + 5^ 4{-3 + 5vs )/C0 2 2 ZH 2 2 ZH 2 2 "'° X(2X -5VS + 3) '° 2x -5vs + 3 '° 15 (2x - 5vs* + 3) ws (4.57) describing leading (a q°), and

0 =x K!{ + /C; - 2vs K.Q , (4.58)

2 2 (6x -3 + 5t;s ) 2 2 l 2 2 ZH,I x(2x -5ws + 3) "' 2x -5vs + 3 2 4 2 4 2 2vs (40 x T + 20 arV - 25 vs + 30^ - 4 x - 12 x - 9) , x(2x2-5^2 + 3)" H,0 2 2 2 4 2 8vs{-2x + 5vs -3 + WT) 8 r (6a; + 9 + 2bvs - 30ws ) /C0 z ' 2 ' 2 ~T2 Hfi H 2 2 z 3 (2x -5ws + 3)' 15 (2 x - 5 w, + 3) vs 2 4 (-3 + 5^ )/Cx 2 2 2 15 (2s -5ws + 3)^ ' (4.59) describing next-to-leading (ex q1) order in the hydrodynamic approximation. Solving (4.56) and (4.58) subject to regularity at the horizon and the boundary condition (4.52) we find

/C0 - 0, Kx = 0. (4.60) 128

Given (4.60), solution to (4.56) subject to regularity at the horizon and the boundary- condition (4.50) is 5^2 + 2x2-3 Za>° = 5^-3 ' (461) which from (4.55) determines (in agreement with (4.24))

„, = -L. (4.62)

With (4.60)-(4.62), solution to (4.59) subject to regularity at the horizon is

2 zH,x =C(l-x ) + ^= (5r - 2) , (4.63) where C is an arbitrary integration constant. From (4.55) we conclude7

r = |. (4.64)

Finally, comparing (4.2) and (4.54), and using (4.1), we obtain from (4.64) result quoted in (4.3).

7Boundary condition (4.50) fixes C = 0. 129

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CHAPTER 5

The shear viscosity of gauge theory plasma with chemical potentials

The gauge theory/string theory correspondence of Maldacena [1, 2] provides a valu­ able insight into the non-perturbative dynamics of strongly coupled gauge theory plasma. Originally, the string theory correspondence was formulated for static prop­ erties of strongly coupled gauge theories. It was pointed out in [3] that computing equilibrium two-point correlation functions of stress-energy tensor one can extract the physics of the near-equilibrium (hydrodynamic) description of strongly coupled hot gauge theory plasma. The computations of the finite temperature transport properties of the J\f = 4 SU(N) supersymmetric Yang-Mills (SYM) theory plasma at large t' Hooft coupling1 [3, 6] were extended to various non-conformal gauge the­ ory plasmas in [7, 8, 9, 10, 11, 12, 13, 14, 15]. One of the most surprising results of these computations was the discovery of the universality of the shear viscosity of strongly coupled gauge theory plasma. Specifically, for all gauge theories which allow for a dual supergravity description, and without chemical potentials for con­ served global charges (if such charges are present), it was shown that the ratio of the shear viscosity r\ to the entropy density s is a universal constant [16, 17, 18]

--7- (51) in units h = k^ = 1.

1Finite 't Hooft coupling corrections to TV = 4 hydrodynamics were discussed in [4, 5]. 133

3 J\f = 4 SYM has a maximum U(l) C S0(6)R abelian subgroup of the R-symmetry for which one can introduce (at most three) different chemical potentials. At strong coupling and finite temperature, the dual supergravity description of this gauge theory plasma is given by a system of near extremal D3 branes with generically three different angular momenta along the five-sphere S5 [19]2. This supergravity solution allows for a consistent Kaluza-Klein reduction to five dimensions, where it is described within M = 2 supergravity coupled to two abelian vectors commonly referred to as the STU model [21]. Within this effective five dimensional description, finite temperature M — 4 SYM plasma is dual to non-extremal black holes carrying generically three different £7(1) charges corresponding to three different S5 angular momenta of the near-extremal D3 branes in the type IIB supergravity description. Even though Af = 4 SYM plasma with chemical potentials violates the assumptions of the universality theorem [16, 17, 18], explicit computation of shear viscosity leads to (5.1) [22, 23, 24].

Expectation that (5.1) might in fact be more universal than anticipated in [16, 17, 18] was strengthened3 by explicit construction of new models of finite tempera­ ture gauge/string theory correspondence with an i?-charge chemical potential [26]. Specifically, it was found that shear viscosity of all strongly coupled Yp'q quiver gauge theory [27, 28, 29, 30] plasmas with a U(l) R-charge chemical potential sat­ isfies the universal relation (5.1). It was further conjectured in [26] that (5.1) holds true even in the presence of chemical potentials.

In this paper we prove that (5.1) is indeed satisfied for generic strongly coupled gauge theory plasma with global charge chemical potentials. We begin with spelling out explicit assumptions (and their implications) under which we obtain (5.1). • First, we consider strongly coupled gauge theory plasma that allow for a dual string theory description. This allows us to use the gauge theory/string theory

2The misprints in the five-form expression for this supergravity solution were fixed in [20]. 3Shear viscosity of the M2-brane plasma also appears to satisfy the universal relation (5.1) [25]. 134

Minkowski-space prescription [31] for computing two-point correlators of the stress- energy tensor. We will work in the regime of large 't Hooft coupling, where the supergravity approximation to string theory is valid. • The gauge theory/string theory correspondence relates a (D + l)-dimensional strongly coupled gauge theory on K75'1 space-time to a particular background of ten- dimensional type IIB supergravity. In all known examples of gauge theory/string theory correspondence it is possible to do a Kaluza-Klein reduction along a com­ pact (8 — jD)-dimensional manifold and relate (D + l)-dimensional strongly coupled gauge theory to an effective (.D + 2)-dimensional gauged supergravity, obtained from a consistent truncation of the full ten-dimensional supergravity. However, we do not believe that this is always possible to do. In fact, the strongest form of the universal­ ity (5.1) proven in [18] does not rely on this assumption4. In this paper we assume that our gauge theory plasma is described by a type IIB gravitational background that allow for a consistent truncation to a (D -f 2)-dimensional gauged supergravity. However, we will not assume that the geometry is asymptotically flat, or that the asymptotically flat region can be "re-attached". Thus, our universality class of (5.1) is necessarily weaker than that of the zero chemical potential case considered in [18]. Since we would like to study gauge theory plasma with finite chemical potentials, these gauge theories must have a set of abelian conserved charges — otherwise we simply would not be able to introduce a chemical potential. In the dual supergravity picture, for each conserved C/(l) charge, there must be a corresponding U(l) isome- try. Introducing a chemical potential for a particular 17(1) charge on a gauge theory side results in gauging corresponding L^(l) isometry on the supergravity side. In the effective (D + 2)-dimensional gravitational description, besides Einstein-Hilbert term and scalar fields, one gets a set of Maxwell fields, one for each gauged isometry

4Both proofs [16, 17] implicitly assume the existence of a consistent truncation. The proof [17] further implicitly assumes that the background is asymptotically flat, as it relies on the universality of the cross section for the graviton scattering from the black hole horizon [32, 33]. The latter universality was derived only for asymptotically flat space-times, and in general it is not known how to "re-attach" the asymptotically flat region in gravitational dual to non-conformal gauge theories (other than those described by fiat Dp branes). 135

(or a chemical potential from the gauge theory perspective). To summarize, we consider strongly coupled (D + l)-dimensional gauge theory plasmas which have a following dual effective (D + 2)-dimensional gravitational description

0 SD+2 = T7T^ / d^V^lR - JCap^d^d^ - T^)F$F»"® lt>7rGD+2 JMD+2 I

(5.2)

a where V, JCa/3 and rab are arbitrary functions of scalar fields (p , and the index

{ ] { (a) in Maxwell fields F$ = d^A „ - dvA fi runs over the set of nonzero chemical potentials. Corresponding to finite temperature (D + l)-dimensional gauge theory with chemical potentials, the effective action (5.2) must admit a black £)-dimensional brane solution, electrically charged under vector fields Aa. The SO(D) invariance of the solution implies that it is of the form

2 2 dsl+2 = - c\{r) {dtf + c|(r) f>^) + 4(r) (dr) (5.3) AP=6l&a\r), r = 4>a{r) for some scalar potentials &a\ • Finally, we assume that the black brane horizon of (5.3) is nonsingular. This implies that there is a choice of the radial coordinate such that as r —• rhorizon

1/2 l/2 c\ -» ax(r - rhorizon) , c2->a2, c3 -> a3(r - rhorizon)~ , (5.4) where constants cti satisfy «i «2 «3 7^ 0 . (5.5)

Notice that given (5.4) the black brane temperature T and the entropy density s are correspondingly

47ra3 4:GD+2 136

5.1 The proof

In this section, we examine the hydrodynamics of the gauge theory plasma at finite chemical potential dual to the generic black hole solution (5.3). In particular, using prescription [31], we compute the retarded Green's function of the boundary stress-

l t energy tensor TM(/(i, x ) (/J, = {t,x }) at zero spatial momentum, and in the low- energy limit u —»• 0:

D Gf2,i2(w,0) = -i Jdtd x e^(t)([T12(i,^,^(0,0)]). (5.7)

Computation of this Green's function allows for a determination of the shear vis­ cosity r\ through the Kubo relation

?? = lim ^ [, 0)) . We find

Gf2>;,0) = -^(l + o(£)), (5.9) where s is the entropy density. Inserting this expression into (5.8) yields the universal ratio (5.1).

We begin the computation of (5.7) by recalling that the coupling between the bound­ ary value of the graviton and the stress-energy tensor of a gauge theory is given by 5g\T%/2. According to the gauge/gravity prescription, in order to compute the re­ tarded thermal two-point function (5.7), we should add a small bulk perturbation 5gi2(t,r) to the metric (5.3)

2 2 l 2 ds D+2 —> ds D+2 + Sgi2(t,r) dx dx , (5.10)

and compute the on-shell action as a functional of its boundary value 5g\2(t). Sym­ metry arguments [34] guarantee that for a perturbation of this type in the back­ ground (5.3) all the other components of a generic perturbation Sg^, along with 137 the gauge potentials perturbations 5A)? and scalar perturbations S

Instead of working directly with Sgu, we find it convenient to introduce the field ip = ifj(t,r) according to

il> = ^ S9i2 = 2C22 59i2 • (5.11)

The retarded correlation function Gf2 12(w, 0) can be extracted from the (quadratic)

r b boundary effective action

D+l d d k \ -MD+2 Sboundaryl^] = / J^x^ 1p\-U>) T(uJ,r) i>\uj) (5.12) horizon where dD+lk ^)= l—y^e-^^r) (5.13) 8M D+2 In particular, the Green's function is given simply by

Gf2ll2(w,0)= lim 2^(W,r), (5.14) 9MD+2~*9MD+2 where T is the kernel in (5.12). The boundary metric functional is defined as

counter ShmmdaryW\ = 8Mr lim (SIM + SGHM + S m) , (5.15)

where Slulk is the bulk Minkowski-space effective supergravity action (5.2) on a cut­

off space -M£)+2 (where MD+2 in (5.3) is regularized by the compact manifold A^£,+2

r with a boundary dM D+2). Also, SQH is the standard Gibbons-Hawking term over

r the regularized boundary dM D+2- The regularized bulk action S[ulk is evaluated on-shell for the bulk metric fluctuations ip(t, r) subject to the following boundary conditions:

(a): „ur linL, ip(t,r) = i>b(t), r 9M D+2^dMD+2 (5.16) (b) : ip(t, r) is an incoming wave at the horizon.

counter r The purpose of the boundary counterterm S is to remove divergent (as dM. D+2 - dM.D+2) and ^-independent contributions from the kernel T of (5.12). 138

The effective bulk action for ip(t, r) is derived in Appendix 6.23. It takes the form

D D+ SkOkM =^r^- f d +*i £D+2 = —I— f d H

107TGD+2 JMD+2 lb7TG£,+2 JMD+2 2 2 cicfc3|^ W) - ^| W) } (5.17)

The second line in (5.17) is the effective action for a minimally coupled scalar in the geometry (5.3), while the third line is a total derivative. Thus the bulk equation of motion for tp is that of a minimally coupled scalar in (5.3). Decomposing ip as

V>(t,r) = e—^(r), (5.18) we find that the equation of motion reduces to wv«, € . r, c^iv 2 2 ln^ 2=0, (5.19) r + ^r + C c3 3 where primes denote derivatives with respect to r.

Let's understand the characteristic indices of ^ as r -» Thm-izon- Consider the following ansatz

x Mr) ~ (Mr ~ rhorizon)) , (5.20) where A is a characteristic, finite, w-independent energy scale associated with the background geometry (5.3). For background geometries dual to conformal gauge theory plasmas the role of A is being played by the temperature. More generally, A could be a scale at which conformal invariance is broken (either explicitly or spontaneously). It is crucial for A to be finite — otherwise the hydrodynamic approximation to a strongly coupled gauge theory plasma would not applicable5. Substituting (5.20) into (5.19) and using the near horizon asymptotics (5.4), we find to leading order in (A(r - rhorizon)) < 1,

(r - rhorizon)^ K + ^il + A) = 0 , (5.21) I «i <*! a| J 5This applicability of the hydrodynamic regime is an implicit assumption of all the studies of strongly coupled gauge theory plasma within gauge theory/string theory correspondence. 139 or a uj . LO A = ±^ 3 = ±i —— , (5.22) ax 4vrT ' where we used (5.6). Given (5.18), for the incoming wave at the horizon we must have

incoming VUO (A(r - rhoriz

Mryncommg „ 1 _ ._u>_ ln((A(r _ rhoHzon)) ? (5.24) valid when both inequalities are satisfied

ln((A(r - rhorizon)) >1 f ln((A(r - rhorizon)) <1 (5.25)

Clearly, for finite A and sufficiently small u> there is an overlap region in (5.25).

In what follows we will need a solution to (5.19), subject to (5.16), to linear order in ^, i.e., in the low-frequency approximation. In this approximation one can neglect the first term in (5.19) and write down the most general solution as

c3(p)dp fair) = Ai(u>) + A2(u)) (5.26) °1{P)C2)D ' where Ai(u) are constants, depending at most linearly on ^. The first boundary condition in (5.16) implies that Ai(u) = l. (5.27)

Substituting (5.4) into (5.26) we obtain the leading near horizon behavior of xjju

i>u>{r) « 1 p ln(A(r - rhorizon)) kl(A(r - Tharizon)) » 1. (5.28) aiOirj

Comparing (5.28) with (5.24) in the hydrodynamic approximation, i.e., for suffi­ ciently small ^, we conclude that

-4.2(w) = i a® u> • (5.29) 140

To summarize, we have to leading order in ^

i(r) = l + i^ / -T^TV (5-3°) Jr Notice that as r —> oo

^w(r) -> 1, <9r^w -> -* ot^ui _ ,J_ f.DJ . (5.31) Ci(r)c2(r)

Once the bulk fluctuations are on-shell (i.e., satisfy equations of motion) the bulk gravitational Lagrangian becomes a total derivative. From (5.17) we find (without dropping any terms)

1 r LD+2 = dtJ + 8rJ , (5.32) where

rt _ 3c?c3 2c a (5.33) c c 3C2 Ci C2 l 2 /2

2c3 c3 Additionally, the Gibbons-Hawking term provides an extra contribution so that

jr_>jr_ M£l ^g^ _ (£ic£y 2> c3 c3

We are now ready to extract the kernel T of (5.12). The regularized boundary effective action for ip is

= SboundaryW TF~7^ / dt d° X [ ^ 1pdrl/j ) + C.t. , (5.35) r lC lb7rCr£)+2 JdM D+2 \ Z J where, as prescribed in [31], we need only to keep the boundary contribution. In (5.35) c.t. stands for (finite) contact terms that will not be important for computa­ tions. Substituting (5.18), (5.30) into (5.35), and using (5.31), we obtain J^(u>, r) in the limit r —» oo lim^)=--^(l + 0(£))

r-KX> 6Z7TUD+2 V \1JJ (5 36) itos

8TT ' 141 where we have recalled the definition of s in (5.6). Using (5.14) to extract the Green's function from ^Fr we find

Gf2>12(^,0)«-|^, (5.37) at least in the low frequency limit to —> 0. This is the result claimed in (5.9), giving rise to the universal ratio of shear viscosity to entropy density (5.1).

5.2 Appendix: Effective bulk action for ip

We begin with collecting some useful expressions. The trace of the Einstein equations derived from (5.2) takes the form

D — 2 D2 R = K^d^dcp + -p- rabF$F^ + — V. (5.38)

Additionally we have

, L>Vl' 1 / \(0) C V^) R^ = - p^* = ^ V ^(V - rabF^F^j , (5.39) where we used superscript (0) to emphasize that the Ricci tensor component and the metric determinant are evaluated on the background (5.3). The second equality in (5.39) makes use of the Einstein equation for R J1X1.

Expanding the effective action (5.2) to quadratic order in perturbation (5.10) and using (5.38), we find

(5.40) where in the second line we used identity (5.39). In (5.40), R^ is a perturbation of 142 the Ricci scalar, quadratic in ip and its derivatives V^R{2) =^^{^ W)2 - ^1W)2} 2 2 2 + \-dt f ^ W) + dr f ^- tw + ^^ V + y/^ ^ Rxlf],

(5.41) where we again used (5.39). From (5.40) and (5.41) we obtain (5.17). 143

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CHAPTER 6

On the supergravity description of boost-invariant conformal plasma at strong coupling

Gauge/string correspondence [1, 2] developed into a valuable tool to study strong coupling dynamics of gauge theory plasma, and might be relevant in understanding the physics of quark-gluon plasma (QGP) produced at RHIC in heavy ion collisions [3, 4, 5, 6]. Besides applications devoted to determine the equation of state of the gauge theory plasma at strong coupling [7, 8, 9, 10, 11], the dual string theory models has been successful in computing plasma transport properties [12, 13, 14, 15, 16, 17, 18], the quenching of partonic jets [19, 20, 21], and photon and dilepton production [22]. Much less studied are the truly dynamical processes in plasma1 .

In a series of papers [25, 26, 27] a framework of constructing the string theory duals to expanding boost invariant conformal plasma2 was proposed. The basic idea of the approach is to set up the asymptotic geometry of the gravitational dual to the boost invariant frame, suggested by Bjorken [31] as a description of the central rapidity region in highly relativistic nucleus-nucleus collisions. One further has to specify the normalizable modes (a string theory duals to expectation values of appropriate gauge invariant operators in plasma) for the metric and other supergravity/string theory fields so that the singularity-free description of string theory is guaranteed.

1One notable example is the analysis of the shock waves generated by a heavy quark moving in gauge theory plasma [23, 24]. 2Some applications of the framework were further discussed in [28, 29, 30]. 148

As explained in [25, 26, 27], requiring the absence of certain curvature singularities in dual gravitational backgrounds constrains the equilibrium equation of state of supersymmetric J\f = 4 Yang-Mills (SYM) plasma (in agreement with [7]), its shear viscosity (in agreement with [32]), and the relaxation time of N — 4 SYM plasma (in agreement with [33]). Despite an apparent success of the framework, it suffers a serious drawback: not all the singularities in the dual string theory description of the strongly coupled expanding H = 4 SYM plasma have been canceled [27]. The subject of this paper is to comment on the gravitational singularities observed in [27].

To begin with, we would like to emphasize the importance of singularities in string theory. As we already mentioned, constructing a string theory dual to expanding boost-invariant plasma implied identification of a set of string theory fields with non-vanishing normalizable modes, dual to vacuum expectation values (VEVs) of the gauge invariant operators in plasma. A priori, it is difficult to determine which operators in plasma will develop a vacuum expectation value 3. In practice, on the dual gravitational side of the correspondence one usually truncates the full string theory to low-energy type IIB supergravity, and often further to a consistent Kaluza- Klein truncation of type IIB supergravity, keeping supergravity modes invariant un­ der the symmetries of the problem. From the gauge theory perspective, such an approximation implies that only certain operators are assumed to develop a VEV. Typically these operators are gauge-invariant operators of low dimension. Of course, the latter assumption might, or might not be correct: luckily, on the gravitational side the consistency of the approximation is severely constrained by requiring an absence of singularities in the bulk of the gravitational background. A typical ex­ ample of this phenomena is thermodynamics of cascading gauge theory [35]: if one assumes that the only operators that develop a VEV at high temperatures in this gauge theory are the same ones as those at zero temperature, all dual gravitational

3For a weak coupling analysis in QGP pointing to a development of a nonzero expectation value for TrF2 see [34]. 149 backgrounds will have a null singularity [36, 37]; on the other hand, turning on a VEV of an irrelevant dimension-6 operator [38] leads to a smooth geometry [39]. Thus, the presence of bulk curvature singularities in gravitational backgrounds of string theory points to an inconsistent truncation.

In [25, 26, 27] the authors assumed a truncation of the full string theory dual to Af = A expanding boost-invariant SYM plasma to a Kaluza-Klein reduction of type IIB supergravity on S5, maintaining only the normalizable modes of the five- dimensional metric (dual to a stress-energy tensor of the plasma), and the dilaton normalizable mode (dual to a dimension-4 operator Tr F2). The absence of singular­

,il Xp ities in five-dimensional Riemann tensor invariant Rr„x R^ ' of the asymptotic late-time expansion of the background geometry at leading and first two subleading orders correctly reproduced the equilibrium equation of state of J\f = 4 SYM plasma and its shear viscosity. Furthermore, removing the pole sing ularities in R^R^^ at third order in late-time expansion determined for the first time the relaxation time of the strongly coupled J\f = 4 SYM plasma [27]. Given the truncation of the string theory, it was not possible to cancel a logarithmic singularity at third

{ {5) xp order in R ^XpR ^ . Turning on an appropriate dilaton mode removes a logarith­

r 9 mic singularity in the five-dimensional string frame metric in ivjtj ™ >fti?Mrmg)nv\P^ However, as we show in the present paper, the full 10 dimensional metric remains singular even in the string frame.

The paper is organized as follows. In the next section we review the boost invari­ ant kinematics of gauge theory plasma. We identify parameters in the late-time expansion of the stress-energy tensor one-point correlation function with the hydro- dynamic parameters of Miiller-Israel-Stewart transient theory [40, 41]. Following the idea that a presence of singularities indicates an inconsistent truncation, we consider in the section 3 the full 50(6) invariant sector of type IIB supergravity, assuming the parity invariance of the M = 4 SYM expanding plasma. Compare to the analysis 150 in [27], this includes one additional SO(6) invariant scalar, dual to a dimension-8 operator. We show that using this additional scalar one can cancel the logarithmic singularity at the third order in late-time expansion either in ten-dimensional Ein­ stein frame Ricci tensor squared TZ^TZ^, or in ten-dimensional Einstein frame Rie-

ll Xp mann tensor squared 1ZiXV\pW ' ^ but not both the singularities. Given the assumed symmetries of the expanding plasma system, there are no additional supergravity fields to be turned on to cancel the singularity in string theory dual to expanding M = 4 SYM plasma. In section 4 we further show that canceling a singularity typically requires additional supergravity (or string theory) fields to be turned on. Using the consistent truncation presented in [38, 43], we consider supergravity dual to superconformal Klebanov-Witten gauge theory [44] plasma. Here, in addition to a dimension-8 operator (analogous to the one discussed in section 3) there is an additional scalar, dual to a dimension-6 operator [38]. We show that turning on the latter scalar can remove logarithmic singularities in the third order in the late-time

lv expansion of the ten-dimensional Einstein frame Ricci scalar 1Z, as well as TZilvW

LvX9 and Tljlv\p'R} . However, we find new logarithmic singularities at the third or­

1 lA2P2 v der in higher curvature invariants such as 7?./il„1A1p17?/' '' 7?.Al2i,2 *ww* i Aap2) as well as logarithmic singularities with different coefficients in (TZ....)8, (7\L..)16 and so on. Canceling the logarithmic singularities in all the curvature invariants appears to require an infinite number of fields. Since such fields are absent in supergrav­ ity approximation to string theory dual of expanding boost invariant plasma, we conclude that such a supergravity approximation is inconsistent. In section 5 we present further directions in analyzing the time-dependent framework proposed in [25, 26, 27]. 151

6.1 Review of boost-invariant kinematics

Quark-gluon plasma is expected to be produced in high-energy collisions of heavy ions. In nucleus-nucleus experiments a central rapidity plateau structure for particle formation has been observed. Specifically, the expansion of the plasma appears to be longitudinal and homogeneous near the collision axis in the case of central collisions. Thus the system is expected to be boost-invariant in the the longitudinal plane. It is convenient to introduce the proper-time r and the rapidity y, defined as

x°=TCoshy, x3 = rsinhy. (6-1)

Then rapidity invariance amounts to independence on y. We will also assume y —> —y invariance.

We will also make another approximation, following [31], namely independence on the transverse coordinates x±. This corresponds to the limit of very large nuclei.

With these assumptions, the whole energy-momentum tensor can be expressed purely in terms of a single function4 e(r), the energy density in the local rest frame. This remaining function should be fixed by gauge theory dynamics.

The proposal of [25] to use the AdS/CFT correspondence to determine e(r) amounts to constructing a dual geometry, by solving supergravity equations with the bound­ ary conditions

where the (5-dimensional part of the) metric is written in the Fefferman-Graham form:

ds5dim. = — -2— (6.3)

4After taking into account energy-momentum conservation and tracelessness. For explicit for­ mulas see [25]. 152

Given any e(r) it is possible to construct such a metric. In [25] it was advocated that the requirement of the non-singularity of this geometry picks out a physically acceptable e(r). In [25, 26, 27] this problem was analyzed in an expansion for large r. At each order the square of the Riemann curvature tensor had poles which could be canceled only by a specific choice of the coefficient of e(r). In the third coefficient there remained a leftover logarithmic singularity in the Einstein frame5 . In this paper we would like to revisit this issue.

The outcome of this calculation is a specific r dependence for e(r):

e(r) = \(l-^T + ^T + ..) . (6.4) T3 \ T3 T3 /

Now we may obtain the physical interpretation of the above result by checking whether it is a solution of some specific phenomenological equations. In the above case, the leading behavior corresponds to perfect fluid hydrodynamics, the sublead- ing one is related to the effect of shear viscosity, while the third one is related to second order viscous hydrodynamics relaxation time. Let us note that the relaxation time depends on the form of phenomenological equations used to interpret (6.4).

Miiller-Israel-Stewart theory [40, 41] of dissipative processes provides a nice frame­ work to study relativistic fluid dynamics. In the Bjorken regime of boost invariant expansion of plasma, the transport equations take the form [42]

. de e + p 1 0=—+ £--$ u/T T T o^+l+Vl+iT-i^Yi-^i (6'5) dr T» 2 \T ft dT\T)) 3ftr' where TV is the relaxation time, e is the energy density, p is the pressure, <& is related to the dissipative part of the energy-momentum, and A = |, (6.6)

5The is also a singularity in the string frame Ricci tensor squared R^ 's rm9> R(l0>strm9)t"' _ 153 with 77 being the shear viscosity. In (6.5) anticipating application to conformal gauge theory plasma we set the bulk viscosity ( to zero.

In the case of the M = 4 SYM plasma we have

2 2 4 2 2 e(r) = ^ iV T(T) , p(r) = \t{r), V(r) = A s(r) = A ^ N T(rf ,

T^T) = r TBoltzmann(r) = r 3T?(T) Mr) ' (6.7) Given (6.7), we can solve (6.5) perturbatively as r —• 00:

(6.8) where A is an arbitrary scale. The constant coefficients {tk,fk} are related to the M = 4 SYM dimensionless transport parameters A ( the ratio of shear-to-entropy density ) and r ( the plasma relaxation time in units of Boltzmann relaxation time ). For the first few coefficients we find:

2 3 *i = -|^. h = ~A r, t3 = --^A r(l + 24r),

2 2 h = 2A(2r - 1), /2 = ^4 (1 - 8r + 24r ), (6.9)

3 2 3 /3 = |^ (-! + 24r - 234r + 1296r ).

From the supergravity computations first done in [27] (and reproduced in the fol­ lowing section of this paper) we find

*>-(£) ^{i-$+G?*+iU+-}. <-> Matching the gauge theory expansion for the energy density (6.7) with that of the dual gravitational description (6.10) we find A ^ 33/4 11 1 C A= A ri0 r ( } 3V%' = ¥JH > = -T8-W8vl- Using the supergravity results (6.42) and (6.46), we find

A=^, r = i(l-ln2). (6.12) 154

A different formulation of second order hydrodynamics in [45] derived from Boltz- mann equations lead to the equation de e + p 1 _ 0=—n + £ _-$ d$ $ 2 11 (6'13) which provides a different definition of the relaxation time r4 . Using (6.13) leads to r™ = 1(1 - In2) quoted in [27]. Of course (6.5) and (6.13) provide different interpretations of the same e(r). Without lifting the symmetry assumptions of the uniform boost invariant plasma expansion we may not rule out one of the (6.5) and (6.13) descriptions.

6.2 Af = 4QGP

Supergravity dual to boost invariant expanding J\f = 4 QGP was discussed in [25, 26, 27]. We extend the previous analysis including all supergravity modes, invariant under the symmetries of the problem. Specifically, we assume that the 50(6) R- symmetry of TV = 4 SYM is unbroken. We further assume that the boost invariant plasma is invariant under separate reflections of all three spatial directions. The latter assumption in particular ensures that the supergravity dual excludes the axion and various 3-form fluxes along extended spacial directions of the boundary — say

RR fluxes with components FTXlX2 within the metric ansatz (6.24). In section 5 we comment why we believe that even allowing for a parity violating supergravity modes would not change our main conclusion, i.e., the inconsistency of the supergravity approximation in dual description of boost invariants expanding plasmas.

Given the symmetry assumptions of the problem as well as the approximation of the full string theory by its low energy type IIB supergravity, the non-vanishing fields we need to keep is the five-dimensional metric, the warp factor of the five-sphere, the self-dual five-form flux and the dilaton. Parity invariance and the symmetries of 155 the boost invariant frame further restricts the five-dimensional metric as discussed below.

6.2.1 Consistent Kaluza-Klein Reduction

Consider Einstein frame type IIB low-energy effective action in 10-dimensions = d 0i n 2 ( 4) SM 4 L ' ^{ ~\ H - JV4 • " with metric ansatz:

2 M N ds w = gMNd£ dt = (6-15) 2 i l/ 6/5 5 = a- (x)gtlu(x)dx' dx + a (x) (dS ) , where M, N, ... = 0,..., 9 and /x, u, ... — 0,..., 4, and (dS5)2 is the line element for a 5-dimensional sphere with unit radius. For the 5-form F5 we assume that

F5 = Th + *.F5, ^5 = -4Q us*, (6.16)

where u>S5 is the 5-sphere volume form and Q is a constant. We further assume that the dilaton is —

With the ansatz (6.15), we find

2 2 2 n = a U- ^(dlna) \+20a-^ (6.17) 1 F| = -8QV"6 4-5! i(8#)'= -!(*)'.», where i? is the Ricci scalar for the 5-dimensional metric g^. The 5-dimensional effective action therefore takes the following form: Sf = 2 hJM ^v^^-^^-f^lna) -^)} , (6. 18) 156 with 2 MK i P(a) = -20a-16/5 + 8QV"8, «§ = -^. (6.19)

It is convenient to introduce a scalar field a(x) defined as

a a(x) = e & , (6.20) so that (6.18) can be rewritten as: Sf = 2 2 hJM ^V^{i2-|(^) -f (5a) -P(«)|, (6.21) where V{a) = -20e-16a/5 + 8Q2e-8a. (6.22)

From (6.21), the Einstein equations and the equation of motion for a are respectively given by: 24 1 1 R^ = -g- {dp,®) {d„a) + - (d^) (d„) + -g^V^a) 5 dV (6.23) 0a= 48^ U = 0.

6.2.2 Equations of motion

For the five-dimensional metric we use the same ansatz as in [25, 26, 27]:

2 p v ds = gpuVdx 'dx = (6.24) = J_ \-e2a^dr2 + e2b^r2dy2 + e2c^dx2} + % , where dx\ = dx\ + dxg. It is easy to see that this is the most general boost invariant geometry written in Fefferman-Graham coordinates subject to a parity invariance along the boost direction6.

Further assuming a = CX(T,Z) and 4> = and the scalar a become:

6 The parity invariance excludes metric components gyz(T,z) and gTy(T,z), in principle allowed by a Fefferman-Graham coordinate frame. 157

• Einstein equations

_2a 2 2 d za + (dza) + (dza) (dzb) + 2 (dza) (dzc) dza dzb dzc +

2 2 8 b + {dTbf - (dTa) {dTb) + 2 (8 c - (dTcf - {dTa) (dTc)) - -dTa + -8Tb

,2a = y(9 «r + -(a 0r--—P(a), T T 3^2 (6.25)

.2a 2 2 4" «9 6 + (d26) + (d2a) (dzb) + 2 (3,6) (d2c) - -dza - -dzb - -dzc + Z £>

2 d b + {dTbf - {dTa) (dTb) + 2 (0T6) (<9Tc) - -dTa + -dTb + -8Tc T T T

(6.26)

,2a 2 2 4 8 zc + 2 (d2c) + (dza) (dzc) + (9,6) (dec) - -dza - -dzb - -dzc +

2 2 d c + 2 (aTc) - (dTa) (dTc) + (dTb) {dTc) + - (dTc) T le2" ,

(6.27)

2 2 2 2 2 2 <9 a + (cU) + <9 6 + (<926) + 2 [<9 c + (d2c) ] - -dza - -9,6 - -dzc +

24 2 2 lP(a) (d2a) - 1 (d2) T 3 £2 (6.28)

«9T<92& - (5T6) (<92a) + (dTb) (dzb) + 2 [dTdzc+ {dTc) (dzc) - (dTc) [dza)\ 1 1 24 1 (6.29) dza + -dzb = —- (dTa) (dza) - - (dT) (<92) ; T T O I dilaton equation:

2 2 0 = z { d z

+ (-dTa + dTb + 2dTc + M (dT) | ; 158

• equation of motion for a

2 2 — -+- = z { d za + (dza + dzb + 2dzc) (dza) - - (dza) . . .*.. (6-31) 2 —e -2a d a + (-8Ta + dTb + 28Tc + -J {dTa)

6.2.3 Late-time expansion

In [25, 26, 27] equations similar to (6.25)-(6.30) were solved asymptotically as a late- time expansion in powers of (r~2/3) (or powers of [r"1^) for the dilaton) introducing the scaling variable

in the limit r —>• oo, with v kept fixed. Such a scaling is well motivated on physical grounds as in this case we find that the boundary energy density e(r) extracted from the one-point correction function of the boundary stress-energy tensor

N2 2a(z,r) N2 2a(u,r) e um \T) = —TT-Z hm r— = — r—^ —, ./o (6.33) 27T2 2^0 Z4 27T2 *-0 V4T4/3 V V would have a late-r expansion appropriate for a conformal plasma in Bjorken regime [27]. Thus we expect

a(T,v) = ao(u) + ^aatu) + ^^(u) + ^(u) + C(r-8/3)

8 3 6(r, <;) = fioCv) + ^h(v) + ^h(v) + ±b3(v) + 0(r- / ) (6.34)

8 3 c(r,<;) = c0(v) + ^Cl(v) + ^c2(v) + ^c3(v) + C(r- / ).

In the equilibrium, both the dilaton and the a-scalar vanish, implying that in (6.25)- (6.29) they enter quadratically for small (ft, a. Thus, given (6.34) we have two possible late-time asymptotic expansions for 0 and a: ttT>v)= ^rM1 v.) , . + -^~M1 . V, ) . + fM1 v) + °(T~S/3) or (6.35) w + + v + (r_T/3) 0(r>V) = 4s ^( ) "^^ 4s ^( ) ° 159 and 111 8/3

or (6.36)

a(r'v) = ^I7I«i(^) + -«2(«) + ^&s(v) + 0(r-7/3) .

For each of the four possible asymptotic expansions, we solve (6.25)-(6.31) subject to the boundary conditions

ai(v),bi(v),Ci(v) = 0, \ i(v) or 4>i(v) = 0, v^° (6.37) cti(v) or &i(v = 0. v—>0 Additionally, we will require the absence of singularities in the asymptotic late-time expansion of the quadratic curvature invariant J'2':

= 421w + ~A^) + ^P(«) + ^ + ^-8/3) • (6'38) Of course, ultimately, all the curvature invariants of the bulk geometry must be nonsingular for the supergravity approximation dual to boost invariant J\f — 4 SYM plasma to be consistent, unless singularities are hidden behind the horizon. However, it turns out that asymptotic expansions (6.34)-(6.36) are fixed unambiguously, given (6.37) and (6.38). Thus non-singularity of the other curvature invariants provides a strong consistency check on the validity of the supergravity approximation.

6.2.4 Solution of the late-time series and curvature singularities

The absence of curvature singularities in XQ determines the leading solution to be [25]

( )8> fe ^ = ^ \VJv3 o = Co-^ln(l + .V3). (6-39)

Notice that v = 31/4 is the horizon to leading order in r. It is straightforward to find that the absence of naked singularities in the bulk (specifically at v — 31/4) in 160

X| 2 w°uld require that a\ = «2 = 0 6c\ = &2 = 0, 2 and 4>\ = 4>2 = 0. We further have [26, 27]

(9-u4)w4 w4 770 , 3- w4 9 — ?r 3 + v4 2 3 + f4 (6.40) w4 and

4 2 4 4 4 8 12 A _(9 + 5v )v (9 + v )v 2 (-1053 - 171^ + 9v + 7v )v a2_12(9-w8) 72(9-w8)+7?0 6(9 - w8)2 2 4 1 , V3-v 3 9l 3-tf ln ln + ^78^S v/^^ +, wo2 - 74'%° 3 + v4 2 2 4 4 4 8 4 IT v (9 + v ) v 2(-9 + 5Av + 7v )v °2~ 288^/3 + 12(9 -v8) + 72(3 + v4) ~ ^ 6(3 + v4)(9 - v8) 1 , VS-v2 1 ,„ „„ 3 .4 + —F^-^ + —(C + 66^)ln 2N1 - (6 41) 8\/3 V3 + v2 72v /o; 3 + w4 l°^iJ

24A/3 V V^ + w2 4(3+ t;4)2 z\^ (\/3 + u2)2 t>2 _ w4 (39 + 7w>4 1 v"3- 62 = - 2c + + C——-^ + ^i-__-i— + __ In 2 4(3 + v4) 24(3 + w4) ,u 2(3 + v4f 8^3 y/Z + v* 3 o, 3-u4 + 4%ln3T^' with [26]

While the equations at the third order for {03,63,03} are a bit complicated, it is possible to decouple the equation for a%. The resulting equation is too long to be presented here. It is straightforward to solve the equation perturbatively as

V = 3 - v4 -• 0+ , (6.43)

1 ^03 VE 4 /2V23V4 2 /23l/4ln2\

1 4 which is sufficient to determine the singularities in I3 at v = 3 / . 161

1//4 The absence of pole singularities in Zg at v = 3 implies that only a3 7^ 0:

"3 = as-° ((i + 8^4) ln I^J - ^4) • (6-45) where 03,0 is an arbitrary constant; it further constraints [27]

C = 2\/3 In2-^L. (6.46) V3 Using (6.44)-(6.46) we find

1 2 3 4 4 /4 2? = finite + U 2 ' 3 / + ~ a3,o) ln(3 - v ), v - 3L . (6.47)

Notice that with 0:3,0 = 0 the logarithmic singularity agrees with the one found in [27]. From (6.47) it appears that the curvature singularities in the bulk of the super- gravity dual to expanding N = 4 SYM plasma can be canceled for an appropriate choice of a3:0. While the Ricci scalar TZ is indeed nonsingular, unfortunately, this is not the case for the square of the Ricci tensor. We find

4 /4 njRT = finite + \ ~ a3 0 ln(3 - v ), v -> 3l . (6.48)

a/ xl pX Thus, it is impossible to cancel logarithmic singularity both in Hllv'RJ and TZ^up\R} ' at order O (T-2) with a supergravity field a.

6.2.5 Curvature singularities of the string frame metric

In [27] it was suggested that the logarithmic singularity in the curvature invariants of the Einstein frame metric (6.15) might be canceled in string frame metric

firing _ ^/2 ^ _ {QAQ)

Unfortunately, this is not possible. Indeed, as in [27], in order to avoid pole sin­ gularities in the curvature invariants of the string frame metric at leading and first three subleading orders, the dilaton can contribute only at order 0(T~2)

T 8/3 6 5 0(r, v) = \k3 In |^4 + ° ( " ) > ( ' °) 162

where k3 is an arbitrary constant.

We find then that

R(W,string)R{10Mring)»v = finite + J_ f ^ ^ _ 16Q ^\ ^3 _ ^ ^ (g^ and7

i2(10^infl)/2(io,rtrtnflWA = finite + J_ ^g gl/2 38/4 + ^ ^ _ ^ \ ^ _ ^ ^

(6.52) as v —> 3_ . While it is possible to remove logarithmic singularities in (6.51) and

(6.52) by properly adjusting a3;0 and fc3, the higher curvature invariants of the string frame metric would remain necessarily singular. Specifically, for string frame fourth order curvature invariants X\ 's r we find

-y-(10,string)[A] _ p(10,string) -n{X0,string)mv^ r>(W,string) jiv p(10,strmp)/i2^2pA fl ix -^3 — ^mviiiv • pA /i2^2 ^

1 2 3 4 4 =finite + ^ (^ 2 / 3 / + ^ a3>o - 3904 fc3) m(3 - t> ), (6.53)

ol/4 as v —> 3_ .

6.3 KW QGP

In the previous section we studied supergravity dual to J\f = 4 SYM plasma assum­ ing unbroken 50(6) global symmetry and the parity invariance along the extended boundary spacial directions. At a technical level, we found that the absence of bulk singularities in the gravitational dual of expanding boost invariant plasma is linked to nontrivial profiles of massive supergravity modes (corresponding to VEV's of irrelevant gauge invariant operators in plasma). This suggests that turning on addi­ tional massive supergravity modes might remove curvature singularities. Within the assumed unbroken symmetries of the M — 4 plasma there are no additional massive

7The result for the Riemann tensor squared here corrects the expression presented in [27]. 163 modes in the supergravity approximation. However, we can test the link between massive supergravity modes and singularities in the supergravity dual of expanding boost invariant conformal plasmas for a slightly more complicated example.

In this section we study supergravity dual to boost invariant expanding Klebanov- Witten superconformal plasma [44]. Assuming that parity along extended boundary spacial directions as well as the global SU(2) x 517(2) x U(l) symmetry of KW plasma at equilibrium is unbroken, the effective supergravity description will contain two massive modes [38, 43] — a supergravity mode dual to a dimension-8 operator ( an analog of a field in the context of J\f = 4 plasma ) and a supergravity mode dual to a dimension-6 operator. We expect that appropriately exciting both modes we can remove curvature singularities in all quadratic invariants of the metric curvature at the third order in late-time expansion. While we show that the latter expectation is correct, we also find that higher curvature invariants will remain singular in this model. In fact, it appears we need an infinite set of massive fields ( which is not possible in the supergravity approximation ) to have a nonsingular metric.

Since the computations for the most part mimic the analysis of the previous section, we highlight only the main results.

6.3.1 Consistent Kaluza-Klein reduction

Consistent KK reduction of the KW gauge theory plasma has been constructed in [38, 43].

The five dimensional effective action is [43]

Ss = ~-s J voW |i?5 - y {dff - 20(dwf - \{dY - PJ , (6.54) where we defined

16, „ 16, 10 1 o 40. f 2w f 12w 2 V = -24e-z - + 4e-^ ~ + =:K e-^f . (6.55) 164

We set the asymptotic AdS radius to one, which corresponds to setting

K = 4. (6.56)

From Eq. (6.54) we obtain the following equations of motion

0 = a°-k%- (6B8) dV 0 = acP- — , (6.59)

l R^v =y djdj + 20 d„wdvw + - d^dA + ^ V . (6.60)

The uplifted ten dimensional metric takes form

2 2 e 6 61 ds w = g^ivWdy" + nUvK + ni(i/) E (4. + l) - ( - ) a=l where y denotes the coordinates of .M5 (Greek indices fi, v will run from 0 to 4) and the one-forms e^, e#a, e<^a (a = 1,2) are given by

if2 \ 1 1 cos e^ = g I # + E 0a #a 1 , eea = ~7=d9a , ea = — sin 9a d

/3 8/3 1 4 w gMdtfdy" = Qf ^ ds , fix = e^ " , 02 - e^ , (6.63) and ds2 is the five-dimensional metric (6.24).

6.3.2 Late-time expansion and solution

We consider the same five-dimensional metric ansatz as in (6.24); we use the late- r expansion of the metric warp factors {a,b,c} as in (6.34). In order to avoid 165 pole singularities in curvature invariants, the dilaton must be set to zero. For the supergravity scalar f{r,v) dual to a dimension-8 operator in KW plasma and for the supergravity scalar W(T,V) dual to a dimension-6 operator in KW plasma we use the asymptotics

f(r,v) = \fs(v) + 0{r-^) \ (6-64) 8 3 W(T,V) = ^w3(v) + O (r- / ) .

Since the massive supergravity modes {f,w} are turned on only at order 0(r~2), the metric warp factors {a^v), b^v), c^v)} are exactly the same as for the M = 4 SYM plasma, see (6.39)-(6.41), (6.42), (6.44) and (6.46). Moreover, analogously to (6.45)

For Ws we find the following equation

„ 5w8 + 27 . 12 , 0 = w'l + , 8 Q, w'3--w3. 6.66 v (f8 — 9) vz Solution to u>3 must have a vanishing non-normalizable mode as v —• 0. Near the horizon (6.43), the most general solution to (6.66) takes form

3/4 w3 = w0,o + w0>1 In (4 3 y) + O (y) . (6.67)

We find that vanishing of the non-normalizable mode of w3 as v —• 0 requires

(6.68) w0,o = wo,i x 27-"»[§)+«(i)+*(| in which case,

7T 6 u w3 = 'L- x w0,i v + 0 (v ) . (6.69)

6.3.3 Quadratic curvature invariants of (6.61)

lv We collect here results for 1Z, HtwW and Tl^pxTZ^^ curvature invariants of the metric (6.61) to the third order in the late-r expansion. At leading order

(o) n -20 + 20, (6.70)

(0) n^-Rr = 80 + 80 , (6.71)

(0) 8(5w16 + 60w12 + 1566w8 + 540w4 + 405) + 136, (6.72) n^xRr^ (3 + V^ where in (6.70)-(6.72) we separated the AdSs and the T1'1 contributions.

At first order

Ui n = o, (6.73)

(1) n^-RT = o, (6.74)

x (i) 1 41472(w4 - 3)v8 n^pXn^ 4 5 VQ. (6.75) 7-2/ 3 (3 + ^ )

At second order

(2) n = o, (6.76)

(2) n^-RT = 0. (6.77)

(2) 4 8 px 1 576(w - 3)v n^\xvp\lwnX n^ C r2/3 [ (3 + v4)5 6912(5v24 - 60x;20 + 2313w16 - 6912v12 + 26487w8 - 18468v4 + 13851) Ao + (3-^4)4(3 + ,4^6 4608(5w16 + 6vu + 162v8 + 54w4 + 405)v10 (3-v4)4(3 + w4)5 (6. 167

At third order

Provided that C is chosen as in (6.46), we find that

I (3) n = o, (6.79) once we use Einstein equations. Also, it is easy to determine that

(3) 100 9 V 3 + v4 n^-RT = -/ ,o x _ * + _ In (6.80) 3 2 7 VVv ' v4 3-v* The non-singularity condition therefore requires that

/3o = 0. (6.81)

Finally, using (6.44), (6.67) and (6.81), we find

1(3) pX 1 2 3 4 4 n^pvpXlwn,n^ (8 2 ' 3 / - 384«;0,i) ln(3 - v )

1 2 3 4 a 5 4 _ (™ 2 ' 3 / In (6 (3e) / ) + 384™0,o) + O ((3 - v )) (6.82)

Thus, choosing 2l/2 33/4 W ,l = (6.83) 0 "48 the logarithmic singularity in (6.82) is removed.

6.3.4 Higher order curvature invariants of (6.61)

Let us define shorthand notation for the contractions of the Riemann tensor. For each integer n we have

n 1 -1 ^[2-] =7?[2 - ] .ftp" ] wi . (6.84)

where

pi/pXu,vn\ — '^pupX (6.85) 168

We further define higher curvature invariants J'2"!, generalizing (6.38):

=4%) + ^ir'(v) + ^if'M + ^if'(«) + o(r-8'3). (6'86)

With a straightforward albeit tedious computation we can extract logarithmic sin- [2nl gularities in I3 . For the first couple invariants we find:

3/4 4 if = - 384 (w0,i - -^ 3 \/2 J In (3 - w ) + finite

[4] 3 4 4 T = - 3072 (wQtl - |- 3 / V2^ In (3 - v ) + finite

] 3/4 4 Xf = - 98304 ( w0li - ^ 3 V2 J In (3 - v ) + finite

16 3/4 4 j| l = - 50331648 U0,i + ~ 3 \/2 j In (3 - v ) + finite

1 1375 1712 3 4 4 if = - 6597069766656 (w0,i + ^ 3 / y/$\ In (3 - v ) + finite

1 if = - 56668397794435742564352 (^ + 5044031582654955520 ^

x In (3 - v4) + finite , (6.87) as v —• 3_ . Interestingly, we find that all lower order invariants, i.e.,

lf\ n = {1,2,3,4,5,6}, i = {0,1,2}, (6.88) are finite as v —> 3_ .

Clearly, given (6.87), logarithmic singularities of the curvature invariant (6.86) can not be canceled within the supergravity approximation.

6.4 Conclusion

In this paper, following [25, 26, 27] we attempted to construct a string theory dual to strongly coupled conformal expanding plasmas in Bjorken regime [31]. In order 169 to have computational control we truncated the full string theory to supergravity approximation, and focused on the well-established examples of the gauge/string dualities: we considered J\f = 4 SYM [1] and superconformal Klebanov-Witten [44] gauge theories. We used non-singularity of the dual gravitational backgrounds as a guiding principle to identify gauge theory operators that would develop a vacuum expectation value during boost-invariant expansion of the plasma. Truncation to a supergravity sector of the string theory ( along with parity invariance in the Bjorken frame ) severely restricts a set of such operators. In the case of the J\f = 4 SYM, there are only two such gauge invariant operators, while for the Klebanov-Witten plasma one has an additional operator. We constructed supergravity dual as a late-time asymptotic expansion and demonstrated that the gravitational boundary stress energy tensor expectation value has exactly the same asymptotic late-time expansion as predicted by Muller-Israel-Stewart theory of transient relativistic ki­ netic theory [40, 41] for the boost invariant expansion. As an impressive success of this approach, one recovers by requiring non-singularity of the background geome­ try at leading and first three subleading orders8 [25, 26, 27] the equation of state for the plasma, its shear viscosity and its relaxation time, in agreement with values extracted from the equilibrium higher point correlation functions. Unfortunately, we showed that logarithmic singularity in the background geometry can not be can­ celed within the supergravity approximation. Moreover, given that the singularities appear to persist in arbitrary high metric curvature invariants, we suspect that re­ laxing the constraint of parity invariance in the Bjorken regime would not help, Indeed, relaxing parity invariance would allow for only finite number of additional (massive) supergravity modes, which, as an example of Klebanov-Witten plasma demonstrates, would only allow to cancel logarithmic singularities in finite number of additional metric curvature invariants.

We would like to conclude with several speculations. 8At the third subleading order, logarithmic singularities of the metric curvature invariants remain. 170

• It is possible that though the full asymptotic late-time expansion of strongly cou­ pled expanding boost invariant plasma is ill-defined within supergravity approxima­ tion, the first couple orders can nonetheless we used to extract transport coefficients, and thus can be of value to RHIC (and future LHC) experiments. To this end we note a curiosity that the relaxation time of the J\f = 4 and the KW plasma came up to be the same. Moreover turning on the scalar fields did not modify the extracted relaxation time. This might point to the universality of the relaxation time, not too dissimilar to the universality of the shear viscosity in gauge theory plasma at (infinitely) strong 't Hooft coupling (see [46] for a conjecture on a bound on relax­ ation times). Second, the relaxation time computed is the relaxation time for the equilibration of the shear modes. If there is substantial bulk viscosity at RHIC, one would also need an estimate for the corresponding relaxation time — this is where non-conformal gauge/string dualities might be useful. • It is possible that the singularity observed in the supergravity approximation is genuine, and can not be cured by string theory corrections. A prototypical example of this is the Klebanov-Tseytlin solution [47]. There, one does not expect a resolution of the singularity by string corrections (while preserving the chiral symmetry and supersymmetry) simply because the corresponding (dual) gauge theory phase does not exist. By the same token, the observed singularity in the string theory dual to strongly coupled boost invariant expanding plasma might indicate that such a flow for a conformal plasma is physically impossible to realize. For example, a Bjorken flow of a conformal plasma might always become turbulent at late times. In fact, at weak coupling, rapidity breaking instabilities were found [48, 49]. Although this was far from a hydro dynamical regime, the leftover logarithmic singularity appearing at the third order might be a manifestation of such an effect.

We hope to address these issues in future work. 171

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CHAPTER 7

Transport properties of J\f = 4 supersymmetric Yang- Mills theory at finite coupling

The correspondence between gauge theories and string theory of Maldacena [1, 2] has become a valuable tool in analyzing near-equilibrium dynamics of strongly coupled gauge theory plasma [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The best studied example of strongly coupled thermal gauge theory plasma is that of the

Af = 4 SU(nc) supersymmetric Yang-Mills theory (SYM). In the large-nc limit, and at large 't Hooft coupling g\Mnc 3> 1, the holographic dual description of the H — 4 plasma is in terms of near-extremal black 3-brane geometry in type IIB supergravity

[20]. In this case one finds [5, 6, 8] the speed of sound cs, the shear viscosity ij, and the bulk viscosity £ correspondingly

c 2 ' = ^' V = ln cT\ C = 0. (7.1)

In the hydrodynamic approximation to near-equilibrium dynamics of hot gauge the­ ory plasma there are several distinct ways to extract the transport coefficients (7.1). First [5], the shear viscosity can be computed from the two-point correlation function of the stress-energy1 tensor at zero spatial momentum via the Kubo formula

iut r, = lim -?- fdtdx e ([Txy(x), Txy(0)}). (7.2)

1 Computation of the thermal correlation functions in the dual supergravity description was explained in [3, 4], 178

Second [6], the diffusive channel two-point retarded correlation function of the stress energy tensor, for example,

i<}z GW*(u>, ?) = -*/ dtdxe^- e{t){[Ttx{x), Ttx(0)]> ex ^ _* (7.3) has a pole at w = -Wq2 , (7.4) where the shear diffusion constant V is

J>=£, (7.5) with s being the entropy density of the gauge theory plasma. From the thermal field theory perspective it is clear that computation of the shear viscosity via Kubo relation (7.2), or from the pole of the stress-energy correlation function (7.3) must give the same result. Such an agreement should persist automatically also on the supergravity side. Thus, we regard above consistency of the hydrodynamic descrip­ tion of the black 3-branes in type IIB supergravity as a highly nontrivial check of the Maldacena correspondence [1] applied to near-equilibrium thermal gauge theories.

The situation with the sound wave propagation in the hydrodynamic limit is similar [8] (though perhaps less dramatic compare with shear viscosity given the conformal invariance of the J\f = 4 SYM). The speed of sound can be computed from the equation of state as

where P and e are correspondingly the pressure and the energy density of the strongly coupled gauge theory plasma which can be extracted from the thermo­ dynamic properties of the black 3-branes [20]. Alternatively, all the transport coef­ ficients (7.1) can be read off from the dispersion relation of the pole in the sound wave channel two-point retarded correlation function of the stress energy tensor, for example,

GUMM = ~i Jdtdxe^-^e^UTuix), T«(0)]> , ' (7.7) 179 as

Again, all these computations point to a consistent picture of a hydrodynamic de­ scription of the supergravity black 3-branes2.

In this paper we prove that consistent hydrodynamic description of black 3-branes persists even once one include leading a! correction to type IIB supergravity from string theory [22, 23, 24, 25], which translates into finite 't Hooft coupling correction on the M = 4 SYM side of the Maldacena duality. To appreciate the nontrivial fact of the agreement we point to some features of a'-corrected description of the black 3-branes:

• including leading order a' correction, the entropy density of the black 3-branes differs from the Bekenstein-Hawking formula which relates the latter to the area of the horizon [26]; • the Hawking temperature of the black 3-branes as well as their equilibrium ther­ modynamic quantities, i.e., the entropy, energy and the free energy, receives a' corrections [26, 27]; • unlike the supergravity approximation [20], the radius of the S5 of the a' corrected black 3-brane geometry is not constant [27],

We find that only properly accounting for all of the above facts one finds a consis­ tent picture of the a' corrected black 3-brane hydrodynamics. Lastly, we strongly suspect that consistency of the hydrodynamics is sensitive to the structure of the a' corrections in type IIB string theory. Thus our computations can be helpful in determining exact structure of such corrections3.

The paper is organized as follows. In the next section we discuss our computational approach and present the results. In section 3 we apply the general computational 2 Consistency of hydrodynamic description of more complicated examples of gauge theory- supergravity correspondence follows from [13, 17, 18, 21, 19]. 3We hope to report on this elsewhere. 180 scheme to the evaluation of the dispersion relation of the shear quasinormal mode in black 3-branes geometry without a' corrections. This was previously discussed in [16], though our approach highlights the use of the effective action rather than equations of motion4. In section 4 we discuss the main computational steps leading to the shear and the sound wave lowest quasinormal modes dispertion relations.

7.1 General computational approach and the results

In the context of gauge theory-string theory correspondence [1] poles of the finite temperature two-point retarded correlation functions of the stress-energy tensor can be identified with the quasinormal frequencies of the gravitational perturbations in the background string theory geometry [16]. Strictly speaking, such an identification has been made in the supergravity approximation to gauge theory-string theory cor­ respondence, but as it is derived from the standard prescription for the computation of the correlation functions [28, 29, 3], we expect it to be valid beyond the super- gravity approximation. In this paper we extract a'-corrected transport coefficients (7.1) from the a'-corrected dispertion relation for the lowest shear quasinormal mode (7.4) and the lowest sound wave quasinormal mode (7.8) in the a'-corrected black 3-brane geometry [26, 27].

We start with the tree level type IIB low-energy effective action in ten dimensions taking into account the leading order string corrections [22, 23, 24, 25]

2 1 = 16^ ' *°" R - \idct)) - ^{F,f + -. + 7 e~i*W + (7.9)

7=^C(3)(«')3, where TI/ rihmnk/~i r* rsPr~

As in [26, 14] we assume that in a chosen scheme self-dual F5 form does not receive order (a')3 corrections. In (7.9) ellipses stand for other fields not essential for the present analysis.

We represent ten dimensional background geometry describing 7-corrected black

3-branes by the following ansatz

ds2 =gW dx»dxu + = (p(r) and for the five-form

^5 = ^5 + *^5 , ^5 = -4WSB , (7.12) where w^s is the 5-sphere volume form. In (7.12) the 5-form flux is chosen in such a way that 7 = 0 solution corresponds to c4 = 1. To leading order in 7, solution can be written explicitly [26, 27]

„4\l/2 Cl=r(l-^\ e-i"(l + a + 4&)

_5 c2 =re 3", (7.13) c3 =—, 7Zm e 5" (1 + 6) , r\\-ra

c4 where to order 0(7 2>

15r( r a = — 7—M - (25 A _ 79^ + 25^ , ' 2r4 fe=7i?(5?-193 + 5>' (7'14)

732r8 V r4

The dilaton

T = T0 (1 + 157) = - (1 + 157) . (7.15) 7T

Next, consider perturbation of the five dimensional metric glJ (7.11)

9$-+9$ + ^*, (7.16) where it will be sufficient to assume that

V = M*> *> r) = e-M(lz V(r). (7.17)

With the metric perturbation ansatz (7.17) we have 0(2) rotational symmetry in the xy plane. The latter symmetry guarantees that at the linearized level the following sets of fluctuations decouple from each other [6, 16]

{hXy}, {hxx — hyy}, (7-18)

\IHXJ ">xzi i^xrS i \^ty> ""yz i flyr f i V.' /

X^tti zzi i^zri ItrrJ • \' '"^)

Scalar channel fluctuations (7.18) were studied in [14] leading (using the Kubo rela­ tion (7.2)) to the following prediction for the shear viscosity to the entropy density ratio 5 = _L(i + i357). (7.21) S 4-7T In this paper we study shear channel (7.19), and the sound channel (7.20) fluctu­ ations. Effective action for the fluctuations (7.19) and (7.20) can be obtained by expanding the supergravity action (7.9) around the background5 (7.11) to quadratic order in h^. Though we can always choose the gauge

5There is a subtlety in evaluating the action with a self-dual 5-form background. The correct way to do this is to assume that F5 has components only along S5 and double that contribution in the lOd effective action 1301. 183 doing so at the level of the effective action for the fluctuations would lead to missing important constraints, i.e., equations of motion coming from the variation of the action with respect to {htr, hxr, hyr, hZT) hrr}. As we explicitly demonstrate on a simple example in the next section, these constraint equations are crucial in decoupling gauge invariant fluctuations. Rather, the correct way is to impose the gauge fixing (7.22) on the level of equations of motion for the fluctuations.

Without loss of generality, for the shear channel we consider metric perturbations

{htx-i hxz, hxr}. Imposing the gauge condition hxr=0 on the equations of motion and further introducing

2 2 htx(r) = r Htx(r), hxz(r) = r Hxz{r), (7.23) we find that the shear channel gauge invariant combination [16]

Zshear = QHtx + U>Hxz (7.24) decouples to order 0(j). The spectrum of quasinormal modes is determined [16] by imposing the incoming wave boundary condition at the horizon r —> rjj", and the Dirichlet condition at the boundary r —> +00 in the 7-deformed black 3-brane ge­ ometry (7.11) on Zshear- The main steps of the computation are discussed in section 4.1. For the lowest shear quasinormal mode (in the hydrodynamic approximation) we find to order 0("f)

2 3 ro = -i Tv q + 0(q ), (7.25) where

rv = ~ + 601 + Otf), (7.26) and we additionally introduced6 ™yk- < = 55S- (7-27) 6As will be clear from the discussion in section 4, tt) and q are the natural dimensionless parameters describing quasinormal modes. 184

From (7.5), (7.15), (7.26), (7.27) we find

=r2? = TX 2 ? 2^ x7„ = i-(l + 1357 + 0(7 )) , (7.28) in precise agreement with (7.21) reported in [14].

There is additional subtlety in computing the lowest quasinormal frequency in the sound channel. As for the shear channel7 we first derive from the effective action for the fluctuations equations of motion, and after that impose the gauge condition

htr — hzr = hrr = 0. (7.29)

As explained in [17], for a general five-dimensional Einstein frame background ge­ ometry with the metric

ds\ = -c\ dt2 + c\ (dx2 + dy2 + dz2) + c\ dr2 , (7.30) with q = Cj(r), in the gauge (7.29), and reparameterizing metric perturbations (7.17) as

ha = Ci Htt, htz = c2Htz, haa = c2Haa, hzz = c2Hzz, (7.31) the gauge invariant gravitational perturbation is given by

ZSOund = 4-Htz + 2Hzz- Haa (l - —2^\ +2^%Htt} (7.32) and thus (in the absence of matter sector) must have decoupled equation of motion. In the absence of 7-corrections, the ten-dimensional Einstein frame reduces to the five-dimensional Einstein frame, so in defining ZS0UTUi we can simply take q = Q. This is no longer the case with 7 7^ 0, as in this case the S5 warp factor oc c| is no longer constant. Indeed, we find that defining Zsound as in (7.31) produces decoupled equation of motion only after q are rescaled as appropriate for the five-dimensional Einstein frame, namely di = c5J3 ^ . (7.33)

7The main computational steps are presented in section 4.2. 185

Again, the spectrum of quasinormal modes is determined by imposing the incoming wave boundary condition at the horizon, and the Dirichlet condition at the boundary in the 7-deformed background geometry (7.11) on Zsound- For the lowest shear quasinormal mode (in the hydrodynamic approximation) we find to order 0(j)

2 3 tv = csq-i Tsound q + C(q ), (7.34) where

|3 (7.35) 2 r,=3+4o7 + 0(7 ).

Given (7.8), (7.15), (7.27), (7.28) we conclude from (7.35) that the bulk viscosity of the strongly coupled J\f = 4 plasma is

C =

Of course, as finite 7-corrections translate into 't Hooft coupling corrections on the gauge theory side of the Maldacena correspondence, from the field theory perspective (given that the latter is conformal) we immediately conclude that c2 = | and £ = 0 independent of the 't Hooft coupling. We showed here that the dual string theory description reproduces this fact as well, albeit in a highly nontrivial fashion which is moreover consistent with shear viscosity computation (7.28) and alternative analysis in [14].

7.2 Diffusion constant of the black 3-branes hydrodynamics: the effective action approach

Consider the shear channel gravitational perturbations {htx, hxz, hxr} in the absence of a' corrections, i.e., setting 7 = 0. This was previously discussed in [6], where equations of motion for the fluctuations {/iix, hxz] in the gauge hxr — 0 were derived as perturbation of the full type IIB supergravity equations of motion. Such an 186 approach becomes technically very difficult in the presence of 7 corrections: one needs to derive equations of motion for the deformed effective type IIB supergravity action (7.9). In the latter case we find it much easier to derive first the effective action describing the fluctuations, and then derive the equations of motion from this action. The effective action for the fluctuations can be obtained by simply evaluating (7.9) to quadratic order in metric perturbations (7.16). The important point we want to stress here is that the gauge fixing condition hxr = 0 can not be imposed on the level of action. If we do this, we obtain two second order ODE's

(coming from variation of the action with respect to {htx, hxz} )

0 =Hl - I--H'tx/ - n "1 „ t»Hxz + qHtx tx 2 3 2 x (1-rc ) / (7.37) 0 =H"Z + xH'xz + x2(i-x2)3/2 y°Hxz + qHtx where H... = H...(x), and all the derivatives are with respect to

1/2 x = ( 1 - -j,AA ) • (7-38)

It is easy to see that given (7.38) equation of motion for

Zahear{x) = qHtx{x) + tvHxz(x) (7.39) does not decouple. On the other hand, if we impose the gauge fixing condition hxr — 0 on the level of equations of motion, we obtain an extra constraint equation coming from the variation of the effective action for the fluctuations with respect to nxr

2 0 = toH[x + qx H'xz. (7.40)

Notice that (7.40) is consistent with (7.37). Given (7.40) we can now obtain the decoupled equation of motion for ZShear x2q2 + m2 tt)2 - x2q2 u 1 — ^shear 1" ^,^2 _ x2q2) ^shear ^ x2n _ #2)3/2 ^^hear • V ' -* )

The incoming boundary condition at the horizon (x —• 0+) implies that

£>shear\%) ~ X Zshear\%) j \'•^^) 187

where zshear(x) is regular at the horizon. Without loss of generality we can assume

Zshear = 1 , (7.43) x-*0+ the spectrum of quasinormal frequencies is then determined by imposing a Dirichlet condition at the boundary [16]

Zshear =0. (7.44)

In the hydrodynamic approximation (to <§C 1 and qd) the solution can be written in the ansatz

2 Zshear = zfh\ar + i <\Z^ear + 0(q ) , (7.45) where z\h\arl z],h\ar are invariant under the scaling tv —» Att), q —> Aq with constant A. Substituting (7.45) into (7.41), and we find [6, 16]

z{0) -1 z(1) ---^r2 (7 46) ^'shear ~ x » ^shear "~ 0 tt) ' which from (7.44) determines the lowest shear quasinormal frequency as [16]

* = -^q2. (7.47)

7.3 Transport properties of black 3-branes at 0((a')3) order

The general computational scheme for deriving equations of motion for the metric perturbation and decoupling the gauge invariant combinations for these perturba­ tions is explained in section 2. The analysis is straightforward, though quite tedious. As in the previous section, computations are simplified using the radial coordinate x, defined by (7.38). Both for the shear mode (7.24) and the sound mode (7.32) gauge invariant combination of metric perturbations we find that the corresponding equations of motion decouple. These equations can be expanded perturbatively in 7, provided we introduce

^shear ^shearfi + 7 Za hear,! (7.48) ^sound —^sound ,0 H~ 7 ^sound,! < ^\t ) • 188

The incoming wave boundary conditions are set up at the level of the leading order in 7, thus it is not a surprise that a natural dimensionless frequency to and a momentum q are introduced (see eq. (7.27)) with respect to To, rather than the a'-corrected Hawking temperature T of the black branes (7.15).

7.3.1 Shear quasinormal mode

For the shear channel fluctuations we find

a-7" i x2q2 + m2 7' | K>2 - x2q2 -^shearfl + ^ _ ^ ^shearfi + ^ _ ^3/2 ^hear,0 • (7.49) n -7" + x'*'+ ">* 7' + *°2 ~x2*2 7 4. 7 u 2 2 2 sftear 1 2 2 3 2 ~^s/iear,! 1" x(tt>/^ -_ xx2,j2q ^) ^stear,' ! "+" ^xQ(l-x _ x2\3/) /2 ^^e<"M ^ ^ear.O where the source Jshear,o is a functional of the zero's order shear mode Zshear,o

4 j _/7( ) ^ Zshearfl ^,(3) « Zshearfi p{2) " Zshearfi p{\) dZshearfi J'shearfl shear.0 —L-ehon^shear r ;7 4A "rr L.i»„^shear , ;1 3Z "•r"

'shear ^shearfl • (7.50)

The coefficients C^ear are given explicitly in appendix A. In the hydrodynamic

approximation we look for the solution for Zshear,i in the following ansatz

im 2 Zshearfi=X- (zfLrfi + ^Lrfi + Oiq )) , (7 51) w 2 ZshearA =x-* (^Lnl + iqzHarA + 0(q )) ,

where z^eari are regular at the horizon, and satisfy the following boundary condi­ tions

z{0) - 1 z(1) = z{0) = 0. (7.52) — shear,1 shearfi *- > ^shearfl shear,I +0+ +0+ Explicit solution of (7.49) subject to boundary conditions (7.52) takes form

r(°) - 1 *M - IiL2 (7 5S1 z l 1 shear,0 ~ ' Zshear0 — X , l '^ 189

(o) 25 2 , 4 Zs//ear,l — -,QX~ \X ^X + 5) ,

*SL,i = - ^t^2 (V (-240 - 1565a;2 - 860rr4 + 695a:6) (7.54)

+ 16m2 (594 - 264a;2 + 43a;4)

%K Imposing the Dirichlet condition on x Zshearfl at the boundary determines the lowest shear quasinormal frequency (7.25).

7.3.2 Sound wave quasinormal mode

For the sound channel fluctuations we find n =7" - (3£2-2)q2 + 3tt)2 2 2 /Jsound sounds x(_3toi + (x + 2)q ) '° qV(x2 + 2) - 2q2ro2(2a;2 + 2) - 4a;2(l - x2)3/2q2 + 3ft)4 Z X2(! _ a;2)3/2((x2 + 2)q2 _ 3n)2) -^° ' 2 2 2 n =7„ _ (3a; -2)q + 3ft)

sound,l a,(_3n,2 + (3.2 + 2)q2) ^™™U qV(a;2 + 2) - 2q2ft?2(2:r2 + 2) - 4x2(l - x2)3/2q2 + 3ro4 Z J X2(1 _ X2)3/2((X2 + 2)q2 _ 3tt)2) —

j _/i(4) ^ Zsoundfl p{i) d Zgoundfi (2) a Zsoundfl ^,(1) dZsoundfi J soundfl ^sound J 4 ' soMnd r/r^ sound /-/T-2 sound j

"" ^ sound ^soundfl • (7.56)

The coefficients C^und are given explicitly in appendix B. In the hydrodynamic

m approximation we look for the solution for Z3(mnd,i the following ansatz

zm ZSound,o -x [zsound0 + iqz d0 + O(q )) , (7-57) w 2 ZSound,i =*-* (zf^ + iqz^! + C?(q )) , where z^„n(ii are regular at the horizon, and satisfy the following boundary condi­ tions

{0) {l) = JO) z z sound, 1 sound.l = 0. (7.58) soundfl *Q+ soundfl x-*0+ x^0+ x->0+ 190

Explicit solution of (7.55) subject to boundary conditions (7.58) takes form

2 2 2 2 (o) = 3tu + (x - 2)q (1) = 2ft.qx , . ^soundfl ^tt)2 — 2fl2 ' soundfl ^ft)2 — 2d2 ' '

5ar2 (°) q4 (2404 + 446a;2 - 4164a:4 + 2006a;6) sound,l 16^m2 _ 2q2)2

- 3ro2q2 (1588 + 183a;2 - 2072a:4 + 1003a;6) + 45m4 (5 - 4a:2 + x4) V

2 / 2 2 (_13344 + 5846x2 A52 X4 + 1?34x6) *£L,i =8q(3mr~2q ) (^ " ° - 3ft)2q2 (-9744 + 5035a:2 - 2604a;4 + 867a:6)

- 36ft>4 (594 - 264a;2 + 43a:4) J . (7.60)

m a Imposing the Dirichlet condition on x Zsouncifl t the boundary determines the lowest sound quasinormal frequency (7.34).

7.4 Appendix: Coefficients of J,shear.O

CfLr =45(1 - x2)4 (7.61)

C^ar = ~8x2(-q2a:2 + TO2) (5(1 " a;2)2(c'^2(280^4 + l0l8x2 + 216) - ^2(72

+ 3811x4 + 154x2)) + 16(1 - x2)5/2(qV - ra2)(45ft>2 + SSqV)] (7.63)

C'^ = -^i-^ + ^Y (-5qV(l - x2)(1105a;6 - 2411a:4 - 802a:2 - 216) - 4tt>2q2a;2(-8792a;6 + 1467a;8 + 8045a;4 + 60a;2 - 1080) - ft>4(-13479a;4 + 3353a;8

+ 2930a:2 + 5876a:6 + 2520) + 16(1 - a:2)3/2(83a:6(4a;2 + l)q6 - m2q4a;4(497a:2 - 372)

- q2tt)4a:2(162x2 + 353) + 45fti6(3a:2 + 2)) J (7.64) 191 c-- - -s^ra] fa2"**"'-1+**>"" 8(>«W(-i+*2> + 4a;2q2(-242to4 + 242tt)4a;2 - 75a;8 + 100a;6) + 10m2(-36tt)4 + 36to4x2 + 45a;8 - 80a;6 + 25a;4) + (-5q4x6(-831a;2 + 274 + 677a;4) + 2ro2qV(-1858a;2 + 4453a;6 - 5295s4 + 3600) - (4320 + 1250x2 + 6097a;6 - 10467a;4)m4)(l - x2)~l/2 J (7.65)

7.5 Appendix: Coefficients of JSound,o

2 2 2 (4) 2q (183a; + 29) + 333tP 1 X ^sound- 3(2q2(x2_1) + 3tt,2) I ) V™)

C^ = 3,(^ + 2)-^-!) + ^ (4qV - 1)i872X* + MSQX4 - 567a;2 - 58) - 6ro2q4(557x6 - 7955a;4 + 2386a;2 - 106) - 9ttJ4q2(4237a;4 - 3875a;2 + 386)-2997tt>6(9a;2-l)J (7.67)

C^nd = (24a;2(q*(a;2 + 2) - 3m2))(2q^(x2 - 1) + 3m2)3 {^17357xW + 64126a;8 - 125343a;6 + 33528a;4 + 4468a;2 + 464)(x2 - l)3q8 - 12w2(12419a;10 - 416676a;8 + 639279a;6 - 205814a;4 - 18792a;2 + 384)(a;2 - l)2q6 - 18tu4(a;2 - 1)(183535a;10 - 1069144a;8 + 1301579a;6 - 428618a;4 + 14112a;2 + 3936)q4 - 27m6(a;2 - l)(359001a;8 - 732179a;6 + 440634a;4 - 71592a;2 - 4864)q2 - 243m8(a;2 - 1)(22327a;6 - 28997a;4 + 6974a;2 + 296) - 16(2q2(a;2 - 1) + 3tv2)(q2(x2 + 2) - 3w2)(l - a;2)5/2(4a;2(a;2 - 1) x (125a;2 - 87)q6 + 4rt)2(273x4 - 86a;2 + 29)q4 - 3m4(159a;2 - 164)q2 - 999m6) J (7.68) 192

^ = 2tf(2q^-l) + 3n,')W + 2)-3»')- (<«««°*" + H24342," - 1565613a;10 - 408216s8 + 1082544s6 - 675120s4 + 31312s2 - 1856)(s2 - l)3q12 - 12w2 (234845s14 - 6242508a;12 + 5011337s10 + 4244306s8 - 6922564s6 + 3818200s4 - 275392s2 + 2176)(s2 - l)2q10 - 18m4(s2 - l)(3115141s14 - 10599760s12 + 3749965s10 + 13178414s8 - 17763088s6 + 9000248s4 - 777360s2 - 16960)q8 - 27m6(-52480 + 19140582s8 + 10819632s4 - 7115740s12 + 3171763s14 - 23084512s6 - 1929885s10 - 981760s2)q6 - 81tt)8(313560s2 + 591463s12 - 1966197s8 - 476182s10 - 2942744s4 + 4503220s6 + 30880)q4 - 243m10(-8416 + 96733s10 - 43280s2 + 303614s4 - 158080s8 - 215771s6)q2 - 2187m12(7105s8 - 7980s6 + 296 + 426s2 + 1353s4) - 16(q2(s2 + 2) - 3tt)2)(2q2(s2 - 1) + 3tt>2) x (1 - s2)3/2(4s2(s2 - l)(2863s8 + 5423s6 - 8425s4 - 1988s2 - 348)q10 - 2m2(4979s10 - 83715s8 + 87128s6 + 16642s4 + 3624s2 + 232)q8 - 3w4(192 + 38586s8 - 110297s6 - 4976s2 - 22460s4) q6 - 9tD6 (20610s6 + 4957s4 - 148s2 - 984)q4 - 27w8(206s4 + 1561s2 + 608)q2 + 8991w10(l + 4s2)) J (7.69) 193

C™und = 24a;4(2q2(a;2 - 1) + 3ro2)3(q2(a;2 + 2) - 3m2)3 V ^'^^ ~ 779)(x2 ~ 1)3 x (x2 + 2)3q16 + 32a:2 W2 (5747a;6 - 31591a;4 + 25688a;2 + 696)(a;2 - l)2(a;2 + 2)2q14

- 16(a;2 - 1) (343159a;18 + 359756a;16 - 2430770a;14 - 19905m4a;12 + 1208858a;12

- 216211m4a;10 + 2130051a:10 - 1974730a;8 + 212468m4a:8 + 789188m4a:6 + 317076a:6 - 704624tt>4a;4 + 46600a;4 - 43376mV + 928tt)4)q12 + 24ro2(x2 - l)(385565x16 - 4651541a;14 + 3705393x12 + 6897979a;10 - 73557m4a:10 - 7874224a;8 + 121796m V + 1319004a;6 + 803308w4a;6 - 855456m4a;4 + 217824a;4 - 96080m V + 1088tt>4)q10 + 36m4(a;2 - 1)(2216157a;14 - 3463628a;12 - 8055387a;10 - 26682m4a;8 + 12127416a;8

- 2103232a;6 - 408776tt)4x6 - 547176a;4 + 579864m4a;4 + 117600m4a;2 + 8480m4)q8 + 54m6(26240m4 + 10762468a;8 - 803584a;4 + 2980096a;12 + 956891a;14 - 462808x6

- 13364213a;10 + 122304ra4a;4 - 290744m4a;6 + 58560m4x2 + 83640mV)q6 - 162m8(15440m4 + 355529a;12 + 1905537a;8 - 2260254a;10 - 303468a;4 + 349906a;6 - 2056m4a;4 - 15120mV + 1736mV)q4 - 486m10(-4208m4 + 150699a;10 + 45576a;4

- 93108a;8 - 111717a;6 + 2684m4x2 + 1524m4a;4)q2 + 4374ro12(-148m4 - 400a;6 + 148tt)V + 125a;4 + 225a;8) - (q2(x2 + 2) - 3m2)(l - a;2)"1/2(8a;4(117035x10

+ 280248a;8 - 600005a;6 - 185858a;4 + 519204a;2 - 114424)(a;2 - l)2q12 - 4m2(a;2 - 1) x (403063a;14 - 5091874a;12 + 4346503a;10 + 5410988a;8 - 6467736a;6 + 1273520a;4 + 35760a;2 - 7424)q10 - 6m4(10631928a;8 + 2466759a;14 + 1592283a;10 - 7034846a;12

- 8874092a;6 + 974760a;4 + 195888a;2 - 1280)q8 - 27m6 (-797920a:8 + 485056x4 + 475731a;12 + 927244a;6 - 918927a;10 - 23040 - 115744a;2)q6 - 27m8(-269131a;8 + 473495a;10 - 1348292a;4 + 908224a;6 + 70400 + 149104x2)q4 - 81m10(314448x4 + 56787a;8 - 321287a;6 - 26560 - 30588a;2)q2 - 729m12(1184 + 2665a;6 - 4027a;4 + 778a;2))) (7.70) 194

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Gauge theories at weak coupling 199

CHAPTER 8

Taming tree amplitudes in general relativity

The analytic computation of scattering amplitudes in gauge theory and gravity has always been a very challenging problem. In principle, this problem is solved by using Feynman diagrams. However, in practice, the fast growth in the number of diagrams makes the calculation impossible.

In some cases, closed formulas have been found for large classes of amplitudes. One of the main tools has been the use of recursion techniques. Many analytic formulas were found or proven by using the Berends-Giele recursion relations introduced in the 80's [1, 2, 3, 4, 5]. One important example are the wonderfully simple formulas conjectured by Parke-Taylor [6] for MHV (Maximally Helicity Violating) tree level amplitudes of gluons.

More recently, a new set of recursion relations for tree level amplitudes of gluons was introduced by Britto, Feng and Cachazo [7]. These recursion relations were inspired by [8, 9] and reproduced very compact results obtained in [10] by studying the IR behavior of Af — 4 one-loop amplitudes. A simple and elegant proof of the relations was later given by the same authors in collaboration with Witten in [11]. The proof is constructive and gives rise to a method using the power of complex analysis for deriving similar relations in any theory where physical singularities are well understood. The BCFW method has been successfully applied in many contexts involving massless particles at tree and loop level [12, 13, 14, 15, 16, 17, 18, 19] as 200 well as for massive particles at tree level [20].

The possibility of the existence of BCFW recursion relations in General Relativity was first investigated in [21, 22]. There it was pointed out that the main obstacle to establish the validity of the recursion relations is to prove that deformed amplitudes vanish at infinity while individual Feynman diagrams diverge. In [21], the desired behavior was checked for MHV amplitudes up to n < 11 under the (—,—) defor­ mation using the BGK formula [23]. In [22], it was shown that the BGK formula vanishes at infinity for any n1 under the (+, —) deformation. Also in [22], a proof based on Feynman diagrams was given for all next-to-MHV amplitudes2 and for all amplitudes up to eight gravitons using the KLT relations [24].

The fact that individual Feynman diagrams diverge very badly in the limit when the deformation parameter is taken to infinity and yet the amplitude vanishes im­ plies that a large number of cancelations must happen. What was shown in [22] is that such cancelations can be made explicit if representations of amplitudes where Feynman diagrams have been re-summed are used.

This is just one more example where Feynman diagrams not only give rise to ex­ tremely long answers which then collapse to very compact expressions but actually imply a completely wrong behavior of the amplitude for large momenta.

A surprising example of this, now at the loop level, is the work of [25, 26, 27] where A/" = 8 supergravity has been shown to possess a remarkably good ultraviolet behav­ ior even though a direct power counting argument indicates that bad divergencies must be present. Also recently, a careful study of the structure of certain one-loop amplitudes in J\f = 8 supergravity shows that even though power counting implies that after a Passarino-Veltman reduction [28] the amplitude should contain boxes,

1 Since the BGK formula has been tested against Feynman diagrams only for n < 11 [23], one cannot make a general statement for actual amplitudes based on BGK. 2Although not mentioned in [22], this technique clearly also works for MHV amplitudes. 201 triangles, bubbles and rational pieces only the boxes can have non-zero coefficients [29, 30]. That this might hold for generic one-loop amplitudes is now known as the no-triangle hypothesis [29, 30]. A striking possibility, which could explain all these properties, is that a twistor string-like construction for this theory could exists [31, 32, 33].

In this chapter we give a complete proof that the miraculous behavior exhibited in next-to-MHV tree level amplitudes of gravitons in [22] actually extends to all amplitudes.

The strategy we follow is exactly the same as the one used in [22] to prove the next- to-MHV case. We use an auxiliary recursion relation to derive a more convenient representation for the amplitudes and then show that they vanish at infinity under the BCFW deformation.

The most important aspect of our proof is that both the auxiliary recursion relations and the vanishing under the BCFW deformation are proven using only Feynman diagram arguments. Since Feynman diagrams are the basic way to define gravity amplitudes, our result completely establishes the validity of the BCFW recursion relations for General Relativity.

The auxiliary recursion relations are obtained by using a deformation that affects the maximum possible number of polarization tensors while keeping propagators linear functions in the deformation parameter. Such a deformation was also introduced in [22]. Quite interestingly, this "maximal" deformation on a given amplitude induces non-maximal deformations on amplitudes with smaller number of gravitons. One of the non-maximal deformations that naturally shows up only affects gravitons of a given helicity. Very interesting results have been obtained in the literature by assuming that under such deformations amplitudes vanish at infinity. More precisely, Bjerrum-Bohr et.al were able to derive MHV expansions for gravity in [34] along 202 the same lines as done for gauge theory by Risager in [35]. Since all non-maximal deformations can be thought of a compositions of the basic BCFW one, our proof validates the assumptions made in [34].

It is also important to mention that at one-loop in gauge theory one can find that compositions of BCFW deformations can vanish at infinity while individual defor­ mations do not. This was actually the motivation for the first use of compositions in the literature in [36].

This chapter is organized as follows. In section 8.2, we follow the same steps as in the original BCFW construction to show the form of recursion relations for gravity amplitudes that can be obtained if one assumes that the amplitudes vanish at infinity under the deformation. In section 8.3 we prove that statement by using auxiliary recursion relations. In section 8.4, we use Ward identities for MHV amplitudes to show how our proof implies the validity of other recursion relations obtained by different deformations. In section 8.5 we give our conclusions and future directions. Part of the proof of the validity of the auxiliary recursion relations is given in the Appendix 8.6.

8.1 Preliminaries and conventions

Tree level amplitudes of gravitons are rational functions of the momenta of the gravitons and multilinear functions of the polarization tensors. It is convenient to encode all the information in terms of spinor variables using the spinor-helicity formalism [37, 38, 39]. Each momentum vector can be written as a bispinor paa =

a6 AaAd- We define the inner product of spinors as follows (A, A') = e AaA'b and [A, A'] = cab\a\';- Polarization tensors of gravitons can be expressed in terms of polarization vectors of gauge bosons as follow

e € 6 e e aa,bb ~~ ^bb ' ad,bb ~~ a,a bb l°--U 203 where polarization vectors of gauge bosons are given by

+ _ fla^a _ _ A /2a a (8.2)

with /jLa and //„ arbitrary reference spinors.

Using the spinor-helicity formalism all the information about a particular graviton is encoded in A, A and the helicity, h, which can be positive or negative. Therefore a given amplitude can be written as

h h n 2 y h h Mn(l \...,n -) = K - 5^\J2^ ^)Mn(l \...,n -), (8.3) where K2 — 87TGN, the label {%) on the spinors is the particle label and the notation (ihi) stands for (X^,X^\hi). In the rest of this paper we will only be concerned

hl hn with Mn(l ,...,n ).

Sometimes it will be convenient to write (ihi) as p^* where pi is the momentum of the ith graviton.

a a Also useful is the following notation: (A|P|A'] = — A Pa,jA . The minus sign in the definition is there so that if Paa is a null vector Hafia one has (A|P|A'] = (A, fj)\p,, A]. This formula has several generalizations. In this paper we only use the one that involves two generic vectors P and Q that are written as sums of null vectors as P = YlsPs and Q = Y^rPr- Then we have

(X\PQ\X') = ^(A,AM)[A(r\A(s)](A^,A'). (8.4)

8.2 BCFW construction for gravity amplitudes

hl hn Consider a scattering amplitude of n gravitons Mn(l ,... ,n ). Construct a one complex parameter deformation of the amplitude that preserves the physical prop­ erties of being on-shell and momentum conservation. The simplest way to achieve 204 this is by choosing two gravitons of opposite helicities3, say i+ and j~, and perform the following deformation

A«(*) - A<*> + z\(j\ \W(z) = W - z\®. (8.5)

All other spinors remain the same. The deformation parameter z is a complex variable. It is easy to check that this deformation preserves the on-shell conditions of all gravitons, i.e., Pk(z)2 = 0 for any k and momentum conservation since p%{z) +

Pj(Z) =Pi+Pj.

The main observation is that the scattering amplitude is a rational function of z

hl hn which we denote by Mn(z). This fact follows from Mn(l ,..., n ) being a rational function of momenta and polarization tensors. Being a rational function of z, Mn(z) can be determined if complete knowledge of its poles, residues and behavior at infinity is found.

We claim that Mn(z) only has simple poles and it vanishes as z is taken to infinity. This means that

Mn(z) = Y^-z^-z z (8-6) a a where the sum is over all poles of Mn(z).

The fact that Mn(z) only has simple poles follows by considering its form as a sum over Feynman diagrams. Choosing a gauge where polarization tensors do not have poles in z, i.e, one in which the reference spinors of the ith and jth gravitons are /j,a = Xj and jla = X^' respectively, the only possible singularities come from propagators. Propagators are functions of momenta of the form

= 2 (8 T) Pi (EfcGx^) ' where I C {1, 2,..., n} is some subset of gravitons with more than one and less than n — 1 elements. 3This is always possible since tree-level amplitudes with all equal helicities vanish and are not of interest for our discussion. 205

Clearly, the only propagators that can depend on z are those for which either i El or j 6 I but not both. Without loss of generality let us assume that i G I. Then the propagator has the form

JW) = /?(0)-*0W0)|t1' (8'8) This shows that all singularities are simple poles. Their location is given by

The proof that Mn(z) vanishes as z is taken to infinity is basically the main result of this paper and it is presented in the next section. Here we simply assume it and continue in order to present the final form the BCFW recursion relations.

The final step is the computation of the residues cj. This is easily done since close to the region where a given propagator goes on-shell the amplitude factorizes as the product of lower amplitudes. Collecting all these results one finds that

M Mn(z) = £ Yl * {{Ki},Pi(zi), -Pi(zj)) -w^Mj ({Kj},Pj(Zj), P^(zj)) x{ } I,J h=± (8.10) where {I, J~} is a partition of the set of all gravitons such that i El and j € J, K% (Kj) is the collection of all gravitons in 1 (J) except for i (j) and h is the helicity of the internal graviton.

The BCFW recursion relation is obtained by setting z = 0 in (8.10). It is important to mention that the value of z% was determined by requiring Px(zj) be a null vector. Therefore the BCFW recursion relations only involve physical on-shell amplitudes.

8.3 Vanishing of Mn(z) at infinity

In the previous section we showed that the validity of the BCFW recursion relations for gravity amplitudes simply follows from the vanishing of Mn(z) at infinity. In 206 this section we provide a proof of this statement.

It is instructive to start by computing what the behavior of Mn(z) for large z is from a naive Feynman diagram analysis4. A generic Feynman diagram is schematically given by the product of polarization tensors, propagators and vertices. We are looking for Feynman diagrams that give the leading behavior for large z. We choose generic reference spinors in polarization tensors such that

++(, IMOAJW^ lA?rf^ ,Rin €i e (*W~^ (yM(;))2 . , UW~^ [X(i)jA]a > (8-11) while all others are independent of z. Note that only vertices that depend on mo­ menta can give z contributions in the numerator. Therefore we should look for Feynman diagrams with the maximum number of z dependent vertices. Such dia­ grams are those for which one has only cubic vertices. For n gravitons there can be a maximum of n — 2 vertices. Each vertex can give at most a z2 dependence5. There­ fore, the leading diagrams will have a £2(n~2) dependence from vertices. Finally we are left with propagators. The z dependence flows in the diagram along a unique path connecting the ith graviton with the jth graviton. Therefore there are n — 3 of them. Each propagator gives a 1/z contribution. Collecting all contributions gives

Mn{z) ~ (?) ^(n_2)) (^) = ^- (8'12)

This implies that Mn(z) ~ 0 for large z only if n < 5. As n increases individual Feynman diagrams diverge more at infinity.

This means that we have to find a better representation of Mn(z) where Feynman di­ agrams have been re-summed into better behaved objects. This is the main strategy of our proof.

4The reason we use the word "naive" is that the argument only takes into account the behavior of individual diagrams and does not consider possible cancelations among them. 5 This and all statements about the general structure of Feynman diagrams can be easily derived from the lagrangian density C = yf^gR with g^ = rj^v + /iM„. 207

The proof is straightforward but it might be somewhat confusing if an overall picture is not kept in mind. This is why we first provide an outline and then give the details.

8.3.1 Outline of the proof

We start by finding a convenient representation of Mn(z). The new representation comes from some auxiliary recursion relations. The auxiliary recursion relations are obtained using a BCFW-like construction but with a deformation under which individual Feynman diagrams vanish at infinity. The way we achieve this is by making as many polarization tensors go to zero at infinity as possible.

Let us denote the new deformation parameter w. Then one has that Mn(w) —> 0 as w —•> oo. The recursion relations are schematically of the form

M h n = £ £ M^Wj)^Mj (wx) (8.13) J, J h=± J where the sum is over some sets J, J of gravitons. These auxiliary recursion rela­ tions actually provide the first example of recursion relations valid for all physical amplitudes of gravitons. However, the price one pays for being able to prove that

Mn(w) —> 0 as w —• oo directly from Feynman diagrams is that the number of terms in (8.13) is very large and many of the gravitons depend on wj. These features make (8.13) not very useful for actual computations.

The next step in our proof is to apply the BCFW deformation to Mn now given by (8.13). Then we have

W>C7A=± * (gl4)

h + E Y,M2(wI(z),z)15f(-Mj (wI(z),z) where the z dependence on the right hand side can appear implicitly through wj(z) as well as explicitly. The first set of terms on the right hand side of (8.14) has 208 both deformed gravitons in J. Therefore, all the z dependence is confined to Mj. We then show that Mj is a physical amplitude with less than n gravitons under a BCFW deformation. Therefore, we can use an induction argument to prove that it vanishes as z —> oo.

For the second set of terms the z dependence appears not only explicitly but also implicitly via wj in many gravitons. Quite nicely, it turns out that one can show that each one of those terms vanishes as z goes to infinity by using a Feynman diagram analysis similar to the one done at the beginning of this section. The reason for this is again the large number of polarization tensors that pick upaz dependence.

There is a special case that has to be considered separately. This is when there is only one positive helicity graviton in Z, i.e., the ith graviton. We prove the desired behavior at infinity in this case at the end of this section.

8.3.2 Auxiliary recursion relation

The auxiliary recursion relations we need are obtained by using a composition of BCFW deformations introduced in [22] and which was used to prove the vanishing of Mn(z) for next-to-MHV amplitudes. The basic idea comes form the analysis of Feynman diagrams we performed above. It is clear that the reason individual Feynman diagrams diverge as z —• oo for n > 5 is that the number of propagators and vertices grow in the same way but vertices give an extra power of z which can be compensated for by two polarization tensors that depend on z only if n is not too large. The key is then to perform a deformation that will make more polarization tensors contribute.

Recall from the outline of the proof that the deformation parameter is denoted by w. The simplest choice is to deform the A's of all positive helicity gravitons and the A's of all negative helicity gravitons. This choice will give l/w2n from the polarization 209

4 tensors. This makes Mn(w) go at most as 1/w even without taking into account the propagators. Propagators are now quadratic functions of w and therefore they contribute 1/w2 each. This last feature is what makes this choice very inconvenient since every multi-particle singularity of the amplitude will result in two simple poles rather than one.

We are then looking for a deformation that gives a w dependence to the largest number of gravitons and at the same time keeps all propagators at most linear functions of w. The most general such deformation depends on the number of plus and minus helicity gravitons in the amplitude. Let {r~} and {k+} denote the sets of negative and positive helicity gravitons in the amplitude respectively. Also let m and p be the number of elements in each. Then if p > m the deformation is

\W(w) = \ij)-w ^2 «(S)A(S), \{k)(w) = \ik) + waik)\(j\ \/ke{k+} se{k+} (8.15) where j is a negative helicity graviton and a^'s can be arbitrary rational functions of kinematical invariants.

If m > p the deformation is

\M(w) = \{i) + w Y^ a(s)A(s), A(fc)H = A(fc)-Wfc)A(i), Vk e {r~}

(8.16) where iisa positive helicity graviton.

The deformation introduced in [22] to prove the case of next-to-MHV amplitudes corresponds to taking all a^ — 1 in (8.15). It turns out that not all choices of a^s' lead to the desired behavior of individual Feynman diagrams at infinity. For example, any choice that removes the w dependence on any single spinor or even on any linear combination of subsets of them will fail. This is usually due to some subtle Feynman diagrams. It is interesting that one has to use precisely the maximal choice. In other words, we have to choose all a^ = 1. Given that this is the choice 210 we use in the rest of the paper, we rewrite (8.15) and (8.16) with a^ — 1 for later reference.

For p > m:

~\W(w) = \U)-w ^ ^(S)- A(fc)H = A(fe) + w\U\ Vfc€{fc+} (8.17) se{k+} and j a negative helicity graviton.

If m > p the deformation is

\W(w) = \® + w J2 A(S)> A(fc)H = A(fc)-w;A«, Vfce{r~} (8.18) se{r~} and i a positive helicity graviton.

The proof that this choice gives Mn(w) —> 0 as w —> oo and more details are given in the appendix. The proof involves a careful analysis of when the w can possibly drop out of propagators. This is basically the point where all other deformations fail.

Here we simply give the final form of the auxiliary recursion relations. Again we have to distinguish cases. If p > m we write Mn as sums of products of amplitudes with less than n gravitons as follows:

+ M + x Mn({r-},{k }) = EE i(K}'fe W}'-^))^ xt^± F* (8.19) xMj {{rj(wj)} , {*+(«*)} , Pj\wi)) where:

• X and J are subsets of the set {1,..., n} such that X U J — {1,..., n}. The sum is over all partitions {I, J} of {1,..., n} such that at least one positive helicity graviton is in X and j G J.

• Pj is the sum of all the momenta of gravitons in X; 211

• {r^ } = X~ is the set of negative helicity gravitons in X;

• {rj(wx)} is the set of negative helicity gravitons in J. The wj dependence is only through X^(wj);

• {kj(wj)} = 1+ is the set of positive helicity gravitons in 1. All of them have been deformed and their dependence on wj is only through

\W(Wl) = X^ + wjX^; (8.20)

• [kj(wx)} is the set of positive helicity gravitons in J. All of them have also been deformed via (8.20).

• The deformation parameter is given by

WI = ^—Pl\Pm • (8>21) L,ker+v\pi\k\ This definition ensures that the momentum

Pj(wj)aa = Plaa + WzX® ^ A^ (8.22) feez+

is a null vector, i.e., PJ(WJ)2 \2 = 0.

Now, if m > p then we write Mn as a sum over terms involving the product of amplitudes with less than n gravitons as follows:

+ + x Mn({r-}Ak }) = EE^(KW},fe M},-^N)4 i /i=± i (8.23) where most definitions are as in the p > m case except that the sets 1 and J are such that i € I and all the negative helicity gravitons and the ith positive helicity graviton are deformed via (8.18) instead of (8.17).

The two rules, (8.19) and (8.23), provide a full set of recursion relations for grav­ ity amplitudes. To see this note that using them one can express any n-graviton 212 amplitude as the sum of products of two amplitudes with less than n gravitons. The smaller amplitudes which depend on deformed spinors and the intermediate null vector P(wj) are completely "physical" in the sense that by construction their momenta are on-shell and satisfy momentum conservation. Therefore they admit a definition in terms of Feynman diagrams again and can serve as a starting point to apply either (8.19) or (8.23), depending on the new number of plus and minus helicity gravitons.

8.3.3 Induction and Feynman diagram argument

Consider any n-graviton amplitude under the BCFW deformation (8.5) on gravitons i+ and j~: \M(z) = \V + z\W, \U\z) = \^ -z\®. (8.24)

Without loss of generality we can assume that Mn has p > m and use (8.19) as our starting point. If m > p we use (8.23) and everything that follows applies equally well.

Note that the choice of deformed gravitons in (8.24) is correlated to that in (8.19) or (8.23).

Our goal now is to prove that by using (8.24) on (8.19) the function Mn(z) vanishes as z is taken to infinity.

Let us consider each term in the sum of (8.19) individually. There are two classes of terms. The first kind is when {i,j} C J. The second kind is when i £ I and jej. 213

Consider a term of the first kind,

h X>z (K|, {#(«*)} , -F^K)) ~Mj {{rj(wx,z)} , {k+(wj,z)} ,Pj (Wl)) .

(8.25) Since both i+ and j~ belong to J, the momentum Pj does not depend on z. Like­ wise from the definition of wj in (8.21) one can see that it does not depend on z. Therefore, the z dependence is confined to the second amplitude in (8.25) which we can write more explicitly as

h Mj (^{rjf} ^k+^wj)} ,{^\wI,z)^},{^\^\wI,z)},P^ (wJ)^ (8.26) where the set J' — J\ {i,j}- It is straightforward to show that

(i) {i) {j {j) {i) A (wx, z) = \ (wj) + z\^\ \ \wT, z) = ~\ (wj) - z\ . (8.27)

The fact that \^{wj) and \^\wx) get deformed exactly in the same way as AW and A^ do is what allows us to use induction for these terms. Note that the amplitude (8.26) is therefore a physical amplitude with a BCFW deformation. The number of gravitons is less than n and by our induction hypothesis it vanishes as z goes to infinity.

To complete the induction argument it suffices to note that the auxiliary recursion relations we are using can reduce any amplitude to products of three graviton am­ plitudes. Finally, recall that the Feynman diagram argument at the beginning of this section showed that amplitudes with less than five gravitons vanish at infinity under the BCFW deformation.

Consider now a term of the second kind,

]TMz(K} ,{#(«*(*),*)} ,-P£(wj(z),z)) -±-x ti F^z> (8.28) h xMj ({rj(Wj(z),z)} , {k}(wj(z))},Pz (wx(z)jZ)) . 214

Recall that for these terms i+ € 1 while j G J• The z dependence we have displayed in (8.28) looks complicated at first since

(8 29) "*<" " E^OTP ' appears to be a rational function of z since Pi{z)ah = Piaa+z\a X& . Note, however, that A^'s with k € T+ do not depend on z and that the z dependence z\a X£ in Pj(z) drops out of the denominator thanks to the contraction with (j\.

Then we find that wj(z) is simply a linear function of z:

^-^-'(E^W) (MO) where u>j is just the undeformed one, i.e., wj(0).

The final step before we proceed to study the behavior for z —> oo using Feynman diagrams is to determine the properties of the internal graviton that enters with op­ posite helicities in the amplitudes of (8.28). The momentum of the internal graviton is given by

Pi{wj(z),z)= ^2pk + pi(wj(z),z)+ ^ Ps{wj{z)). (8.31) kex~ sei+, s+%

The important observation is that the z-dependence can be fully separated as follows

ft(^),,) = ft W + ,A« (- (g^lisi) E A<-» + A«) (8.32) where Px(wj) is the z-undeformed one, i.e., Pj(wj(0), 0).

Note that we have written PJ(WJ(Z), z), which is a null vector, as the sum of two null vectors. For real momenta, this would imply that all three vectors are proportional. However, in this case all three vectors are complex and all that is required is that either all A's or all A's be proportional. We claim that in this particular case all A's 215 are proportional. To see this note that if we write Pz(u>x)aa — Ai A^ , then A^ is

a a proportional to Q — r] Pj(wj)aa for some arbitrary spinor r/ .

We claim that the A spinor of the vector multiplying z in (8.32) is also proportional

{ ] a a to C if rja = \ J . In this case, (d = A^ Pj{wx)ail = A^ PXaA. To prove our claim consider the inner product of the two spinors c = (8.33) (j\Px\i] \ |liW lftW E^feiPsel+ -«

The right hand side of (8.33) vanishes trivially showing that the two spinors are proportional.

Therefore, it follows that we can write Pj{wj(z),z)aa = \a(z)\? where Xa(z) = P) { ] A„ + zpX J for some f3 which is z independent. Note that if z = 0 we recover

Let us turn to the analysis of the amplitudes in (8.28) to show that their product vanishes as z is taken to infinity. In other words, we will see that Mj and Mj may not vanish simultaneously but their product together with the propagator always does.

Consider the first amplitude Mj ({fj} , {kj(wx(z),z)} , — Pj(wj(z),z)). Let the number of particles in the sets {rj} and {kj} be m% and pj respectively6.

The Feynman diagram analysis is very similar to that performed at the beginning of section 8.3. The leading Feynman diagram is again one with only cubic vertices that posses a quadratic dependence on momenta. The number of cubic vertices is the 6Note that if h = + this is a physical amplitude where only the A's of positive helicity gravitons have been deformed. It is interesting to note that this deformation is basically the one introduced by Risager in [35] and later in [34] to construct an MHV diagram expansion for gravity amplitudes. 216 total number of particles7 minus two, i.e, mx + pj — 1. Therefore the contribution from vertices gives at most a factor of z2ljnx+Px~v>. There are pj + 1 polarization vectors that depend on z, giving a total contribution of l/z2^Px+h\ Here we have used that since z enters in —Px{wj(z), z) only through X(z), its polarization tensor gives a contribution of l/z2h. Finally, we need to count the number of propagators that depend on z. It turns out that there are exactly mj + p% — 2 of them giving a contribution of I / zmx+Px~2. This last statement is not obvious since there could be accidental cancelations of the z dependence. Let us continue with the argument here and we will prove that there is no accidental cancelations within the propagators in the next subsection8. Collecting all factors we get

8 34 Mx({r;},{J4Mz),*)},-P£M*),z)) ~ zPxlI+2h- ( - )

The propagator 1/Pj(z) in (8.28) goes as 1/z.

The reader might have noticed that in this argument special care is required when X+ = {i}. We postpone the study of this case to the end of the section. Until then we simply assume that i 6l+ but 1+ ^ {i}.

Consider now the second amplitude in (8.28),

h Mj {{rj(Wl(z),z)} , {k}(wx(z))},Pj (wj(z),z)) . (8.35)

Let the number of gravitons in {rj} and {fcj} be mj and pj respectively.

The cubic vertices give again a factor of z2^Pj+mj~1\ The polarization tensors give a factor of l/z2(pj~h+l\ Here we have taken into account the contribution from the z dependent negative helicity graviton, i.e, the jth graviton, and from the internal graviton, Pjh(wi(z),z). Finally, the propagators contribute again a factor

7 The total number of gravitons in Mj is mj + pj + 1 since — PJ(WJ(Z), z) should also be included. 8More precisely, what we prove in the next subsection is that trivial cancelations in which neither propagators nor vertices depend on z are the only ones that can occur. 217

m 2 of i/zPJ+ J- . Collecting all factors we get

8 36 Mj ({rj(Wl(z),z)} , {#(«*(*))} .Pf WO,*)) ~ zPj-ml-2{h-iy ( - )

Combining all contributions from (8.34), the propagator and (8.36), the leading z behavior of (8.28) is l/zp-m+3.

This shows that all the amplitudes with p > m vanish at infinity.

As stated at the beginning of this subsection, a similar discussion holds for the case of amplitudes with m > p: by repeating the same counting starting from relation (8.23), the behavior at infinity of terms of the second kind turns out to be l/zm~p+3. Terms of the first kind can again be treated by induction.

It is important to mention that the way amplitudes vanish at infinity is generically only as 1/z2. This is because terms of the first kind which are treated by induction vanish as three-graviton amplitudes do, i.e, as 1/z2.

This completes our proof of the vanishing of Mn(z) as z goes to infinity up to the claim made about the number of propagators that contribute a 1/z factor and the exceptional case when J+ = {i}. We now turn to these crucial steps of our proof.

8.3.4 Analysis of the contribution from propagators

One thing left to prove is that in the leading Feynman diagrams contributing to the first amplitude, Mj, there are exactly mj + px — 2 propagators giving a 1/z contribution at infinity while in the second amplitude, Mj, there are exactly mj + Pj — 2 of them. 218

Propagators in leading Feynman diagrams of Mz

Let us start with Mj. The argument here uses similar elements to the ones given in the appendix where we provided a proof of the auxiliary recursion relations.

Consider a given Feynman diagram. A propagator naturally divides the diagram into two subdiagrams. Let use denote them by C and TZ. Without loss of generality, we can always take the graviton with momentum —PJ(WJ(Z),Z) to be in H. In the set of positive helicity gravitons, {kj (wj(z), z)}, there is one that is special; the ith graviton. We consider two cases, the first is when i E C+ and the second when ieK+.

Case A: i e C+

Let i G £+, then the propagator under consideration has the form

Pc(Mz),z) = Pc(wj(0)) + z\® ( - {jlP*\l E A« + A« ] . (8.37)

We are interested in asking when

2 U k] Pc(wT(z),Zy = P,K(0)) + z ( ™^{^ - 01^1*]) (8-38) can be z independent. Therefore we have to analyze under which conditions the factor multiplying z can be zero for a generic choice of momenta and polarization tensors of the physical gravitons subject only to the overall momentum conservation constrain.

Let us write the factor of interest as follows

01W1 E U\Pc\k] - UWi[ E <3\Px\k] = XM'XWPcatPjtT* (8.39) kec+ kei+ with rpab _ jji) a V^ ^(fe) b _ ^(i) b V^ ^(fc) a (8.40) fce£+ fcex+ 219

Here we have to consider two different cases9:

• 1+ \ C+ ^ 0.

• J+ = £+ and £+ ^ {i}.

Let us start by assuming that X+ \ C+ is non-empty and that, say, s e X+ \ C+. The space of kinematical invariants we consider is determined by the momentum and polarization tensors of each of the original gravitons. Consider both objects for the sth graviton

^ = ^unf> P£ = A«AM. (8.41) aa,bb In \(s))2 a a \ J It is clear that if we take {Xa , X& } to {t~xXa , t\£ } with t a fourth root of unity, i.e, t4 — 1 then (8.41) is invariant. Therefore, any quantity that vanishes for t = 1 must also vanish for all four values of t. In particular, it must be the case that (8.39) must vanish for all four values of t. Since momentum is not affected only the tensor Tah changes. Taking the difference between two values of t, say t = 1 and t = i, we

hh S find that !"%=! - T \t=i - A^A( K Therefore, the vanishing of (8.39) implies that of (j\Pc\t](j\Pj\s]=0. (8.42)

This condition is then equivalent to

tT(fr?cfr#c) = 0 or tTy>jfcfafc) = 0 (8.43) but these are constraints on the kinematical space which are not satisfied at generic points.

The second case we have to consider is when T+ = C+ and C+ ^ {i}. Let us introduce the notation jl^ = J2k€i+ ^i • Therefore the condition we want to exclude

9There are actually three cases. The third is when J+ = £+ = {i} but this is part of the special case that is considered at then end of the section. 220 is (j\Pj\i}(j\Pc\p,} - (j\Pc\i}(j\Pi\jj] = 0. (8.44)

Using Schouten's identity we can write this as

U\PxPc\J)[i,fi = 0. (8-45)

The vanishing of either factor10 implies a constraint for the space of kinematical invariants. In the case of the second factor this can easily be seen by choosing s (E T+

4 2 and s ^ i, then using the scaling by t with t = 1 to conclude that (ps + pi) = 0.

This completes the proof that the z dependence cannot drop out of any propagator and therefore all mj + px — 2 of them give a \jz factor in MT if i € C

Case B: i e 1l+

The analysis when i 6 72. is completely analogous except for the fact that there is one case that was not possible before. As we will show, this will correspond to diagrams which give a non-leading contribution.

Consider the analog of (8.38)

2 8 46 Pc{wx{z), zf = P£(u*(0)) + *01 Wl (I^O-'l^l'fcj) • < - )

The new case is when C+ = 0, then the z dependence drops out. Of course, this is not a problem because if the set C+ is empty it means that nothing on the subdiagram C depends on z, including the cubic vertices. Therefore, neither propagators nor cubic vertices contribute. One can then concentrate on the subdiagram 1Z, but this subdiagram has less particles than the total diagram and the same number of z-dependent polarization tensors. Therefore these diagrams go to zero even faster than diagrams where C+ is not empty.

10See section 8.1 for the explanation of the notation in the first factor. 221

Propagators in leading Feynman diagrams of Mj

Let us now study the leading Feynman diagrams contributing to Mj. Again, the propagator divides the diagram in two subdiagrams that we denote L and K. With­ out loss of generality, we can always take the graviton with momentum Pjh(wx(z), z) to be in 1Z. As in the previous discussion we have a special graviton, i.e, the jth graviton. Therefore we have to consider two cases, j £ C and j G TZ.

Case A: j 6 C~

Let us first consider the case j € C. The z dependence of X^'(wx(z),z) is the most complicated of all. This is why we write it explicitly

{JlP ] A"W*), z)& = A^fa + z ( -A£° + ^ p £ A? ) . (8.47) \ 2^er+0l^z|KJae{fc+} J Using this and the fact that the set of labels of all positive helicity gravitons {k+} must be equal to X+ U J+, we find that the propagator of interest has a momentum dependence of the form

2 (J\Pc\k}\ Pciwziz)^) = Pc(wj(0)Y + z (j\Pc\ij - {j\Px\i) Zkei+(J\Pi\k] J' (8.48) We are then interested in asking when this expression can be z independent.

The analysis is similar to the one given for M% so we will be brief. The factor of interest is now

(3\PM £ {j\Pz\k\ - (JlPzli] Y, Ol^l*]- (8-49) fe£Z+ feei+u(j+\£+)

We have to consider two cases:

• J+ \ £+ + 0. 222

• J+ = C+ and T+^{i}.

In the first case we can assume that, say, the 5th graviton is in J+ \ C+. Then by using the argument that any statement about {A(S),A^} must also be true for {r^W.tAW} with t4 = 1 one can show that the vanishing of (8.49) implies a nontrivial constraint on kinematical invariants that is not generically satisfied.

The second case is also similar to one we considered in the analysis of Mj. Here we have that 1+ U (J+ \ £+) = 1+ U 0 = J+. Therefore (8.49) becomes

+ where jla = J2sex+\{i} ^^- Since by assumption T \ {i} ^ 0 we can use Schouten's identity to derive non-trivial constraints on the kinematical invariants which are not satisfied for generic momenta.

Recall that the case when X+ = {i} is special and will be treated separately.

Case B: j € K~

In this case, the propagator of interest can be written as

8 51 Pc(wI(z),zf = Pc(wI(0)f + z(j\PJ\i} (l^^^ifcj) • ( - )

This is again similar to the corresponding case in Mj. The only new case compared to when j € C" is when £+ is empty. Then nothing in £ depends on z and we can consider a Feynman diagram that has less minus helicity gravitons than the original one and therefore it goes faster to zero at infinity than the leading diagrams obtained when C+ ^ 0.

This conclude our discussion about the contribution of the propagators. 223

8.3.5 Analysis Of The Special Case J+ = {i}

Let us now consider the final case. This is when I+ = C+ = {i}. This case is quite interesting since several unexpected cancelations take place. Consider wj(z) given in (8.30). In this case, it is easy to check that wj(z) — wj(0) — z. A consequence of this is that X^(wj(z), z) — X^(z) + wj(z)X^ becomes ^-independent. To see

w this recall that \®(z) = \® + zX^. Therefore A \wT{z),z) = X^(wj). This also implies that Pj(wj(z),z) is z independent. Therefore, the full amplitude M% is z independent.

Recall that we are interested in the behavior of

(8 52) EMXT4TM^(Z)- - h=± ^I[Z) The propagator 1/Pj(z) contributes a factor of l/z.

Now we have to look at

h Mj(z) = Mj ({rj(wT(z),z)} , {k+(wx(z))},P; (wj(z),z)) .

Let us study the z dependence of each graviton carefully. We have that the jth graviton (which has negative helicity) and all positive helicity gravitons in J+ = {kj(wj(z))} behave as

ij X(S) {s) s + X \wx(z),z) = \W(wi) + z J2 ~ > \ (Mz)) = X^ \wj)-zX^ \/s£j .

(8.53) Close inspection of (8.53) shows a striking fact. This deformation is exactly the same as the one that led to the auxiliary recursion relations in the first place, i.e, the deformation given in (8.17) but using z instead of w as deformation parameter and X^\wx) and X^(wx) as undeformed spinors. Finally, recall that Pj(wj(z),z), which also appears in Mj, was shown to be z independent. 224

Now, if h = + we have PJ(WJ) and therefore, Mj(z) is nothing but a physical amplitude under the maximal deformation (8.17). In the appendix, we showed that amplitudes vanish as the deformation parameter, which in this case is z, is taken to infinity if the number of pluses is greater than or equal to the number of minuses minus two. To see that this condition is satisfied in Mj note that since T+ = {i} we have that the total number of positive helicity gravitons in Mj is p — 1 while that of negative helicity gravitons is m — mj + 1. Since the number of external negative helicity gravitons in M% must be at least one, i.e, mx > 1 and recalling that we are studying the case when p > m, we get the desired result.

The next case to consider is when h = —. Since PJ(WJ) is z independent, the deformation (8.53) of Mj is no longer maximal. However, it is possible to show that these terms are identically zero. This is obvious when the on-shell physical amplitude Mj, which has only one positive helicity graviton, has more than two negative helicity gravitons.

Consider now the case when Mj has precisely two negative helicity gravitons. A three-graviton on-shell amplitude need not vanish if momenta are complex therefore this is a potentially dangerous case. Three-graviton amplitudes are given as the square of the gauge theory ones. Therefore we have (8 54) *,(,-*<«*),.-,-/?(«*)) = {{xmJM)(^M,^)) - where as in subsection 8.3.3 we have defined Pj(wx)ad = Ai A^ \

Since this is a physical amplitude, momentum is conserved which means

AW(«*)aAW + AWA« =Ai*>Ar. (8.55)

For real momenta, this equation implies that all X's and all X's are proportional. Therefore three-graviton amplitudes must vanish. For complex momenta, this need not be the case and one can have all X's be proportional with the A's unconstrained. In such a case (8.54) would not vanish. 225

We claim that, luckily in our case of interest, all X's are proportional and (8.54)

{ j) vanishes. To see this note that wj = — {i,s)/(j,s) and X^(wj)a = A« + wjX a ,

a p } therefore (\®(wx), A^) = 0. Contracting (8.55) with A^ we find (A< \ A^)Af = 0. Therefore we must have (X^p\ X^) = 0 which completes the proof of our claim.

From (8.54), this condition implies that Mj is identically zero. Thus, we can con­ clude that the cases of Mj with a non-maximal deformation are not there.

This is the end of our proof. We now turn to some extensions and applications of the BCFW recursion relations that can be obtained by using Ward identities.

8.4 Ward identities

Our proof of the BCFW recursion relations was based on deforming two gravitons of opposite helicities, i+ and j~, in the following way:

X^(z) = X{i) + zX{j), X{j)(z) = X^ -zX{il (8.56)

However, it is known that in gauge theory, deformed amplitudes also vanish at infinity if the helicities {hi,hj) of the deformed gluons are ( —,—) or (+,+) [11]. It would be interesting to prove a similar statement for General Relativity. Here we show that this is indeed very straightforward in the case of MHV scattering amplitudes if one uses Ward identities.

The Ward identity of relevance for our discussion can be found for example in [40] and it is given by

(AW, AM)8 (A(S),A(9)}8' where the notation M^v indicates that the gravitons a and b in this amplitude are the ones with negative helicity.

Consider first the (+,+) case. We use the Ward identity (8.57) to relate it to the 226 usual (+, —) case. For clarity purposes, we explicitly exhibit the dependence of the amplitudes on only four gravitons: {l,m,i,j}. The dependence on the rest of the gravitons (all of which have positive helicity) will be implicit. Then we have

Mrv(^(z),j+(z),l-,m-) = (^jy^y) M^v (^ (z), j~ (z), l~, m+). (8.58)

The MHV amplitude on the right hand-side is deformed as in (8.56), thus it vanishes at infinity by our proof. Since both inner products expressed explicitly in (8.58) do not depend on z, the amplitude on the left hand side of (8.58), where (h^hj) = (+,+), will vanish as z goes to infinity.

Consider now the (—, —) case. Using again the Ward identity (8.57) we have

v + + w + + Mr (i-(z),r(z),l ,m )= ( ( ^p) j ) AC (< (*),r(*),r,m ) (8.59)

{i) Note that (\ (z), \W) does not depend on z since A«(z) = A« + Z\V). Therefore, the amplitude still vanishes in this case.

In [21], a very nice compact formula was conjectured for MHV amplitudes of gravi­ tons by assuming the validity of BCFW recursion relations obtained via a defor­ mation of the two negative helicity gravitons. Our proof and the discussion in this section validates the recursion relations used to construct the all multiplicity ansatz. It would be highly desirable to show that the formula proposed by Bedford et al. [21] does indeed satisfy the recursion relations. The formula is explicitly given by

6 3 w n- r ,+ .+ N (i,2) [Mn-2u, . yT (2|t1 + ...+t..1i».] M„(l ,2 tf)...>l+_2) = —— G{ti,t2,i3) [[—7^—.—rjz-.—v-

+V{ii,...,in-2) (8.60)

an where V(ii,..., in-2) indicates a sum over all permutations of (ij,..., in-2) d

%Uti G(ii,i2,i3) = \ (/0 .... .J L -w. •,) • (8-61) 2 V,(2,%i)(2,i2)(11,12)(12,13)(»i,*3>/ 227

It is also interesting to show why the case (hi, hj) = (—, +) does not lead to recursion relations. Using the Ward identity (8.57) once again we have

MTv(r(z),J+(z),l\m-) = (^f^) Mrv^-(z),r(z),l+,m+) (8.62) The amplitude on the right hand-side vanishes as z goes to infinity. However, (A(i)(z), A(m))8 contributes with a factor of z8 while (\(l\z), A^) is z independent. Either using BGK (together with (8.59)) or directly (8.60), one can show that M™HV(i~(z),j~(z),l+,m+) goes like 1/z2, therefore the amplitude with (hi,hj) = (—, +) behaves as z6 at infinity.

8.5 Conclusions and further directions

In this paper we have proven that tree level gravity amplitudes in General Relativ­ ity are very special. Contrary to what can be called a naive power counting of the behavior of individual Feynman diagrams, full amplitudes actually vanish when mo­ menta are taken to infinity along some complex direction. The naive power counting gives that the amplitudes diverge. This miraculous property implies that tree am­ plitudes of gravitons satisfy a special kind of recursion relations. One in which an amplitude is given as a sum of terms containing the product of two physical on- shell amplitudes where the momenta of only two gravitons have been complexified. These recursion relations, originally discovered in [7] in gauge theory, were proven using the power of complex analysis in [11]. The BCFW construction opened up the possibility for using complex analysis in many other situations. There are only two major difficulties when applying the BCFW construction to a general field theory at any order in perturbation theory. One of them is that complete control of the singularity structure of the amplitude is required. At tree-level this means poles but at the loop level one can also have branch cuts. The other one is to have a good control on the behavior at infinity. In the case of gravity amplitudes this had been 228 the stumbling block. The way we overcome this obstacle was by constructing aux­ iliary recursion relations. These were obtained by exploiting as many polarization tensors as possible in other to tame the divergent behavior of vertices in individual Feynman diagrams while still keeping the linear behavior of propagators. In a sense, the deformation we introduced is the "maximal" choice.

This procedure seems quite general and it would be very interesting to classify field theories according to whether their amplitudes vanish or not at infinity under this maximal deformation.

8.6 Appendix: Proof of auxiliary recursion relations

In the main part of Chapter 8 we used certain auxiliary recursion relations to prove that Mn(z) vanishes as z is taken to infinity under the BCFW deformation. It is therefore very important to establish the validity of the auxiliary recursion relations.

Consider the case when then number of positive helicity gravitons is larger or equal than the number of negative helicity ones, i.e, p > m. The case when m > p is completely analogous. Let us start by constructing a rational function Mn(w) of a complex variable w via the deformation (8.17), i.e,

\(J\w) = \{j) - w J2 ^(S)> A(fc)H = A(fc) + w\{j\ Vfc <= {k+} (8.63) se{k+} where j is a negative helicity graviton and {k+} is the set of all positive helicity gravitons in Mn.

The claim is that Mn{w) vanishes as w is taken to infinity and its only singularities are simple poles at finite values of w. 229

8.6.1 Vanishing of Mn(w) at infinity

Let us prove that Mn(w) vanishes as w —> oo. Consider the leading Feynman diagram that contributes to Mn(w). Such a diagram has n — 2 cubic vertices each contributing a factor of w2. It also has p+1 polarization tensors that depend on w and give 1/w2 each. Finally, we claim that all n — 3 propagators that can possibly depend on w actually do giving each a contribution of 1/w. Putting all contributions together we find that the leading Feynman diagrams go like l/wp~m+3. Therefore, if p > m then Mn{w) —> 0 as w —»• oo.

We are only left to prove that n — 3 propagators depend on w. A similar statement has to be proven in section 8.3.4. The proof there is more involved since it requires the study of many cases. The discussion that follows can be thought of as a warm up for that in section 8.3.4.

Consider a given Feynman diagram. A propagator naturally divides the diagram into two sub-diagrams. Let us denote them by X and J. Without loss of generality, we can always take the jth graviton to be in J. Let us denote the set of positive helicity gravitons in X by T+.

The propagator under consideration has the form 1/Pj(w) with

P^H = P^-w^2U\Px\k] (8.64) ket+ where Px = Pz(0).

The only way the w dependence can drop out of the propagator is that X)fc€i+ 01^1^1 = 0.

Since the jth graviton belongs to J, the condition X}fce:r+0'1-^x1^1 ~ 0 can orny be satisfied if the vector J2kzi+ ^x aa^^ a vanishes. To see this note that there must be at least two gravitons in J, one of them j. Therefore we can use momentum 230 conservation to determine the other one in terms of the other n — 1 gravitons. This allows us to consider all the remaining n — 1 gravitons as independent. In particular, the jth graviton is independent from the ones in X.

Our goal is then to prove that the combination P%aa(J2kei+ ^i ) cannot vanish for generic choice of momenta and polarization tensors.

Consider first the case when the set X+ has only one element, say the sth graviton.

a s 2 Then the vanishing of Pxaa^ implies that of ]C/tej fc,s> where Sk,s — (Pk + Ps) -

Since X must have at least two gravitons, the vanishing of J2keT SktS is a constraint on the kinematical invariants which is not satisfied for generic momenta.

Consider the case when X+ has at least two elements. Let one of them be the sth graviton. Since our starting point is a physical on-shell amplitude, the dependence of the amplitude on the sth graviton can only be through its polarization tensor and its momentum vector,

If we transform {A(S),A(S)} into {t~xX^s\tX^} with t4 = 1, i.e., t is any 4th root of unity, then both e f; and p£l are invariant. This means that any statement we make for t = 1 must be true for the other three possible values of t. In particular, it must be the case that Pjad(Ylkei+ k^s^a + ^d ) vanishes for all four values of t. Since Pjaa does not depend on t the only way to satisfy this condition is if Pi • P^ — 0. This is clearly a condition that is not satisfied for generic momenta and therefore this possibility is also excluded.

Finally, there is one more possibility to consider. If the set T+ is empty then the w dependence drops out. Of course, this is not a problem because if the set X+ is empty it means that nothing on the subdiagram X depends on w, including the cubic vertices. Therefore, neither propagators nor cubic vertices contribute. One can then 231 concentrate on the subdiagram J", but this subdiagram has less particles than the total diagram and the same number of ^-dependent polarization tensor. Therefore these diagrams go to zero even faster than diagrams where X+ is not empty.

8.6.2 Location of poles and final form of the auxiliary recursion relations

Having proven that Mn(w) vanishes at infinity, we turn to the question of the sin­ gularity structure. We claim that it has only simple poles coming from propagators in Feynman diagrams. Again as in section 8.2 where we discussed the BCFW defor­ mation, one has that the poles generated by the w dependence in the polarization tensors can be eliminated by a gauge choice. We pick the reference spinor of each of the polarization tensors of the positive helicity gravitons to be fia = Aa and that of the jth helicity graviton to be /}„ = Y^ke{k+} ^

We have already given the structure of propagators in (8.64) from where we can immediately read off the location of the poles to be

P2(Q)

Finally, we need the fact that a rational function that vanishes at infinity and only has simple poles can be written as Mn(w) = J2a ca/(w — wa) where the sum is over the poles and ca are the residues. The residues in this case can be determined from factorization limits since all poles come from physical propagators.

Collecting all results we arrive at the final form of the auxiliary recursion relation used in the text (8.19):

Mn({r-},{k+}) = ^^M^KUfc+K)},^^))-^ ifcL F* (8.67) + x Mj (feW} , {k j{wj)} , Pj\wx)) . 232

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CHAPTER 9

Consistency Conditions On The S-Matrix Of Mass- less Particles

The power of the constraints that Lorentz invariance imposes on the S-matrix of four dimensional theories has been well known at least since the work of Weinberg [1, 2]. Impressive results like the impossibility of long-range forces mediated by massless particles with spin > 2, charge conservation in interactions mediated by a massless spin 1 particle, or the universality of the coupling to a massless spin 2 particle are examples beautifully obtained by simply using the pole structure of the S-matrix governing soft limits in combination with Lorentz invariance [2, 3].

Weinberg's argument does not rule out the possibility of non-trivial Lagrangians describing self-interacting massless particles of higher spins. It rules out the possi­ bility of those fields producing macroscopic effects. Actually, the theory of massless particles of higher spins has been an active research area for many years (see reviews [4, 5] and reference therein, also see [6, 7] for alternative approaches). Lagrangians for free theories have been well understood while interactions have been a stumbling block. Recent progress shows that in spaces with negative cosmological constant it is possible to construct consistent Lagrangian theories but no similar result exists for flat space-time [8, 9]. Despite the difficulties of constructing an interactive La­ grangian, several attempts have been made in studying the consistency of specific couplings among higher spin particles. For example, cubic interactions have been 238 studied in [10, 11, 12, 13]. Also, very powerful techniques for constructing inter­ action vertices systematically have been developed using BRST-BV cohomological methods [14, 15, 16] and references therein.

In this paper we introduce a technique for finding theories of massless particles that can have non-trivial S-matrices within a special set of theories we call con­ structive. The starting point is always assuming a Poincare covariant theory where the S-matrix transformation is derived from that of one-particle states which are irreducible representations of the Poincare group. There will also be implicit as­ sumptions of locality and parity invariance.

The next step is to show that for complex momenta, on-shell three-particle S- matrices of massless particles of any spin can be uniquely determined. As is well known, on-shell three-particle amplitudes vanish in Minkowski space. That this need not be the case for amplitudes in signatures different from Minkowski or for complex momenta was explained by Witten in [17].

We consider theories for which four-particle tree-level S-matrix elements can be com­ pletely determined by three-particle ones. These theories are called constructible. This is done by introducing a one parameter family of complex deformations of the amplitudes and using its pole structure to reconstruct it. The physical singularities are on-shell intermediate particles connecting physical on-shell three-particle ampli­ tudes. This procedure is known as the BCFW construction [18, 19]. One can also introduce the terminology fully constructible if this procedure can be extended to all n-particle amplitudes. Examples of fully constructible theories are Yang-Mills [19] and General Relativity [20, 21, 22] (the fact that cubic couplings could play a key role in Yang-Mills theory and General Relativity was already understood in [24, 23]).

The main observation is that by using the BCFW deformation, the four-particle 239 amplitude is obtained by summing over only a certain set of channels, say the s- and the u- channels. However, if the theory under consideration exists, then the answer should also contain the information about the t- channel. In particular, one could construct the four-particle amplitude using a different BCFW deformation that sums only over the t- and the u- channel.

Choosing different deformations for constructing the same four-particle amplitude and requiring the two answers to agree is what we call the four-particle test. This simple consistency condition turns out to be a powerful constraint that is very difficult to satisfy.

It is important to mention that the constraints are only valid for constructible the­ ories. Luckily, the set of constructible theories is large and we find many interesting results. We also discuss some strategies for circumventing this limitation.

As illustrations of the simplicity and power of the four-particle test we present several examples. The first is a general analysis of theories of a single spin s particle. We find that if s > 0 all theories must have a trivial S-matrix except for.s = 2 which passes the test. As a second example we allow for several particles of the same spin. We find that, again in the range s > 0, the only theories that can have a nontrivial S-matrix are those of spin 1 with completely antisymmetric three- particle coupling constants which satisfy the Jacobi identity and spin 2 particles with completely symmetric three-particle coupling constants which define a commutative and associative algebra. We also study the possible theories of particles of spin s, without self-couplings and with s > 1, that can couple non-trivially to a spin 2 particle. In this case, we find that only s = 3/2 passes the test. Moreover, all couplings in the theory must be related to that of the three-spin-2 particle amplitude. Such a theory is linearized Af = 1 supergravity.

The paper is organized as follows. In section 9.1, we review the construction of the S- 240 matrix and of scattering amplitudes for massless particles. In section 9.2, we discuss how three-particle amplitudes are non-zero and uniquely determined up the choice of the values of the coupling constant. In section 9.3, we apply the BCFW construction to show how, for certain theories, four-particle amplitudes can be computed from three-particle ones. A theory for which this is possible is called constructible. We then introduce the four-particle test. In section 9.4, we discuss sufficient conditions for a theory to be constructible. In section 9.5, we give examples of the use of the four-particle test. In section 9.6, we conclude with a discussion of possible future directions including how to relax the constructibility constraint. Finally, in the Appendix 9.7 we illustrate one of the methods to relax the constructibility condition.

9.1 Preliminaries

9.1.1 S-Matrix

In this section we define the S-matrix and scattering amplitudes. We do this in order to set up the notation. Properties of the S-matrix, which we exploit in this paper like factorization, have been well understood since at least the time of the S-matrix program [25, 26, 27].

Recall that physically, one is interested in the probability for, say, two asymptotic states to scatter and to produce n — 2 asymptotic states. Any such probability can be computed from the matrix elements of momentum eigenstates

out(Pl • • -Pn-^PaPbjin = (Pi • • • Pn-2 \ S\paPb) (9.1) where S is a unitary operator. As usual, it is convenient to write S = I + iT with

{4) (Pi • • -Pn-2\iT\papb) = 5 lpa+Pb- ^Pi J M(pa,pb -> {pi,P2, • • • ,Pn-2»-

(9.2) 241

M(pa,pb —> {pi,P2,---,Pn-2}) is called the scattering amplitude (see for example chapter 4 in [28]).

Assuming crossing symmetry one can write pa = —pn-i and Pb = — pn and introduce a scattering amplitude where all particles are outgoing. Different processes are then obtained by analytic continuation of

Mn = Mn(pi,p2,...,pn-l,Pn)- (9.3)

Mn is our main object of study. Our goal is to determine when Mn can be non-zero. Up to now we have exhibited only the dependence on momenta of external particles. However, if they have spin s > 0 one also has to specify their free wave functions or polarization tensors. We postpone the discussion of the explicit form of polarization tensors until section V.

9.1.2 Massless Particles Of Spin s

It turns out that all the information needed to describe the physical information of an on-shell massless spin s particle is contained in a pair of spinors {Xa, Aa}, left- and right-handed respectively, and the helicity of the particle [29, 30, 31, 17]. Recall that in a Poincare invariant theory, irreducible massless representations are classified by their helicity which can be h = ±s with s any integer or half-integer known as the spin of the particle.

The spinors {Aa, A„} transform in the representations (1/2,0) and (0,1/2) of the universal cover of the Lorentz group, SL(2,C), respectively. Invariant tensors are

ab ab e , e and (cr^)aa where a^ = (!,

AaA^(A,A'), AdA^ = [A,A']. (9.4) 242

Finally, using the third invariant tensor we can define the momentum of the particle

a 1 a by p^ = \ (a' )aa\ , where indices are raised using the first two tensors. A simple consequence of this is that the scalar product of two vectors, ]f and q^ is given by 2p-g=

9.2 Three Particle Amplitudes: A Uniqueness Result

In this section we prove that three-particle amplitudes of massless particles of any spin can be uniquely determined.

The statement that on-shell scattering amplitudes of three massless particles can be non-zero might be somewhat surprising. However, as shown by Witten [17], three- particle amplitudes are naturally non-zero if we choose to work with the complexified Lorentz group SL(2, C) x SL(2, C), where (1/2,0) and (0,1/2) are completely inde­ pendent representations and hence momenta are not longer real. In other words, if

Xa 7^ ±Aa then p^ is complex.

l Let us then consider a three-particle amplitude M3({A^, M \hi}) where the spinors of each particle, A^ and A^, are independent vectors in C2.

Momentum conservation (pi + p2 + p?)aa = 0 and the on-shell conditions, pf = 0, imply that pi -pj — 0 for any i and j. Therefore we have the following set of equations

(1,2)[1,2]=0, (2,3)[2,3]=0, (3,1)[3,1] = 0. (9.5)

Clearly, if [1, 2] =0 and [2, 3] = 0 then [3,1] must be zero. The reason is that the spinors live in a two dimensional vector space and if A^1) and A(3) are proportional to A*-2-* then they must also be proportional.

This means that the non-trivial solutions to (9.5) are either (1,2) = (2, 3) = (3,1) = 0or[l,2] = [2,3] = [3,l] = 0. 243

] ) Take for example [1,2] = [2,3] = [3,1] = 0 and set \f = a2\£ and ~\f = a3X^.

Then momentum conservation implies that Ai + a2Xa + a3Xa =0 which is easily seen to be satisfied if a2 = —(1, 3)/(2, 3) and a3 = —(1, 2)/(3,2).

The conclusion of this discussion is that three-particle amplitudes, M3({A^, A^, hi}), which by Lorentz invariance are only restricted to be a generic function of (i,j) and [i,j] turn out to split into a "holomorphic" and an "anti-holomorphic" part1. More explicitly

H A M3 = M3 «1, 2), (2, 3), (3,1)) + M3 ([1, 2], [2,3], [3,1]). (9.6)

It is important to mention that we are considering the full three-particle amplitude and not just the tree-level one. Therefore M3 and M£ are not restricted to be rational functions2. In other words, we have purposefully avoided to talk about perturbation theory. We will be forced to do so later in section 9.4 but we believe that this discussion can be part of a more general analysis.

9.2.1 Helicity Constraint and Uniqueness

One of our basic assumptions about the S-matrix is that the Poincare group acts on the scattering amplitudes as it acts on individual one-particle states. This in particular means that the helicity operator must act as

X X M lhl 2k2 h3 /ll /l2 ft3 ( *W~~ *W) ^ ' ^ ) = -2/ilM3(l ,2 ,3 ). (9.7)

Equivalently, KWa + 2K) M*(<1'2); (2'3)'(3,1)} = ° (9-8) on the holomorphic one and as

(\°4-a - 2fc) Mi([l, 2], [2,3], [3,1]) = 0 (9.9)

1 Using "holomorphic" and "anti-holomorphic" is an abuse of terminology since A

It is not difficult to show that if dx = hj — h2 — h3, d2 = h2 — h3 — hi and d3 = ha — hi — h2, then

F=(l,2)d3{2,3)dl(3,l)d2, G= [1,2]-* [2,3]"dl [3,1]"* (9.10) are particular solutions of the equations (9.8) and (9.9) respectively.

Therefore, M% /F and M^/G must be "scalar" functions, i.e., they have zero helic- ity.

Let xi be either (2, 3) or [2, 3] depending on whether we are working with the holomorphic or the antiholomorphic pieces. Also let x2 be either (3,1) or [3,1] and

A x3 be either (1, 2) or [1,2]. Finally, let M be either M£*/F or M3 /G. Then we find that

xdM(Xi,X2,X3)=Q (9n) OXi for i = 1,2,3. Therefore, up to solutions with delta function support which we discard based on analyticity, the only solution for Ai is a constant. Let such a constant be denoted by KH or KA respectively.

We then find that the exact three-particle amplitude must be

(i (i2 li3 M3({A«)A UI}) = %(l,2)*(2,3)*(3)l} + «A[l)2]- [2,3]-*[3)l]-*. (9.12)

Finally, we have to impose that M3 has the correct physical behavior in the limit of real momenta. In other words, we must require that M3 goes to zero when both 3 (i, j) and [i, j] are taken to zero . Simple inspection shows that if di + d2 + d3, which is equal to —hi — h2 — h^, is positive then we must set K& = 0 in order to avoid an infinity while if —hi — h2 — /i3 is negative then KH must be zero. The case when

3Taking to zero (i, j) means that A^ and A^ are proportional vectors. Therefore, all factors (i,j) can be taken to be proportional to the same small number e which is then taken to zero. 245 hi + hi + h% = 0 is more subtle since both pieces are allowed. In this paper we restrict our study to h\ + hi + hs ^ 0 and leave the case hi + hi + /i3 = 0 for future work.

9.2.2 Examples

Let us consider few examples, which will appear in the next sections, as illustrations of the uniqueness of three-particle amplitudes.

Consider a theory of several particles of a given integer spin s. Since all particles have the same spin we can replace h = ±s by the corresponding sign. Let us use the middle letters of the alphabet to denote the particle type.

There are only four helicity configurations:

+ [1,2] M3(l-,2-,3+) = /w,f Jowl „) , Ma3(l+,2r ,3;) = K; ( (2,3)(3,1); ' y-m^r^sj '•"™\J2)3][3,1] (9.13) and

S M3(l-,2-,3;) = ^rs((l,2)(2,3)(3,l)r, M3(l+, 2+,3+) = K'mrs ([1, 2][2,3][3,1}) (9.14) The subscripts on the coupling constants n and «' mean that they can depend on the particle type4. We will use the amplitudes in (9.13) in section 9.5.

A simple but important observation is that if the spin is odd then the coupling constant must be completely antisymmetric in its indices. This is because due to crossing symmetry the amplitude must be invariant under the exchange of labels.

This leads to our first result, a theory of less than three massless particles of odd spin must have a trivial three-particle S-matrix. Under the conditions of constructibility, 4Note that here we have implicitly assumed parity invariance by equating the couplings of conjugate amplitudes. 246 this can be extended to higher-particle sectors of the S-matrix and even to the full S-matrix.

9.3 The Four-Particle Test And Constructible Theories

In this section we introduce what we call the four-particle test. Consider a four- particle amplitude M\. Under the assumption that one-particle states are stable in the theory, M4 must have poles and multiple branch cuts emanating from them at

2 2 2 5 locations where either s = (pi + p2) , t = (p2 + P3) or u = (p3 + pi) vanish .

We choose to consider only the pole structure. Branch cuts will certainly lead to very interesting constraints but we leave this for future work. Restricting to the pole structure corresponds to working at tree-level in field theory.

As we will see, under certain conditions, one can construct physical on-shell tree-level four-particle amplitudes as the product of two on-shell three-particle amplitudes (evaluated at complex momenta constructed out of the real momenta of the four external particles) times a Feynman propagator. In general this can be done in at least two ways. Roughly speaking, these correspond to summing over the s-channel and u-channel or summing over the t-channel and u-channel. A necessary condition for the theory to exists is that the two four-particle amplitudes constructed this way give the same answer. This is what we call the four-particle test. It might be surprising at first that a sum over the s- and u-channels contains information about the t-channel but as we will see this is a natural consequence of the BCFW construction which we now review. 5We have introduced the notation s for the center of mass energy in order to avoid confusion with the spin s of the particles. 247

9.3.1 Review Of The BCFW Construction And Constructible Theories

The key ingredient for the four-particle test is the BCFW construction [19]. The construction can be applied to n-particle amplitudes, but for the purpose of this paper we only need four-particle amplitudes.

We want to study M4({Aa , AJ> , hi}). Recall that momenta constructed from the spinors of each particle are required to satisfy momentum conservation, i.e., (p\ +

P2+P3+P4Y=0-

Choose two particles, one of positive and one of negative helicity6, say i+Si and j~Si, where s, and Sj are the corresponding spins, and perform the following deformation

A(i) (z) = A(i) + zXU); \U)(Z) = \U) - z\(i). (9.15)

All other spinors remain the same.

The deformation parameter z is a complex variable. It is easy to check that this de­ formation preserves the on-shell conditions, i.e., Pk(z)2 = 0 for any k and momentum conservation since Pi(z) + Pj{z) = Pi + Pj-

The main observation is that the scattering amplitude is a rational function of z which we denote by M±{z). This fact follows from M±(\hl,... ,4/l4) being, at tree- level, a rational function of spinor products. Being a rational function of z, M^{z) can be determined if complete knowledge of its poles, residues and behavior at infinity is found.

Definition: We call a theory constructible if M4(z) vanishes at z = 00. As we will see this means that M^(z) can only be computed from M3 and hence the name.

In the next section we study sufficient conditions for a theory to be constructible. 6Here we do not consider amplitudes with all equal helicities. 248

The proof of constructibility relies very strongly on the fact that on-shell amplitudes should only produce the two physical helicity states of a massless particle7. In this section we assume that the theory under consideration is constructible.

Any rational function that vanishes at infinity can be written as a sum over its poles with the appropriate residues. In the case at hand, M^z) can only have poles of the form

(9.16) (KM+M)2 [i,*r] Ui,k) + zQ,k))[i,k] where k has to be different from i and j.

As mentioned at the beginning of this section, M^{z) can be constructed as a sum over only two of the three channels. The reason is the following. For definiteness let us set i = 1 and j — 2, then the only propagators that can be ^-dependent are l/(pi(z) +P4)2 and l/(pi(z) +P3)2. By construction l/(pi + P2)2 is ^-independent.

The rational function M$(z) can thus be written as

Mi1'2)(2) = -^ + -^- (9.17) Z Z\ Z Zu

2 where zt is such that t — (pi{z) + Pi) vanishes, i.e., zt = —(l,4)/(2,4) while zu

2 is where u = (pi(z) + P3) vanishes, i.e., zu = —(l,3)/(2,3). Note that we have added the superscript (1,2) to M±{z) to indicate that it was obtained by deforming particles 1 and 2.

Finally, we need to compute the residues. Close to the location of one of the poles, Mi{z) factorizes as the product of two on-shell three-particle amplitudes. Note that each of the three-particle amplitudes is on-shell since the intermediate particle is

7This in turn is simply a consequence of imposing Lorentz invariance [2]. 249

4hi 3/13 4^4

(1-2) M: 2 = £ A^24 + E -/l P

I'll 2/12 J/n 2/»2

Figure 9.1: Factorization of a four-particle amplitude into two on-shell three-particle amplitudes. In constructible theories, four-particle amplitudes are given by a sum over simple poles of the 1-parameter family of amplitudes M^(z) times the corre­ sponding residues. At the location of the poles the internal propagators go on-shell and the residues are the product of two on-shell three-particle amplitudes. also on-shell. See figure 9.1 for a schematic representation. Therefore, we find that

h ' (9.18) 1 h h M 2 h h J^MM (zu),p 3^-P1 ^))j^7ZS M (zll),p 4\Prt3 (zu)). h ^^z) where the sum over h runs over all possible helicities in the theory under consider­ ation and also over particle types if there is more than one.

The scattering amplitude we are after is simply obtained by setting z = 0, i.e,

1 2) M4({AW,AW,/li}) = Mi ' (0).

Recall that we assumed h\ = s\ and hi = —^2- Let us further assume that /14 = — S4. Therefore we could repeat the whole procedure but this time deforming particles 1

A and 4. In this way we should find that M4({A«, \®, hi}) = M^ \ti).

We have finally arrived at the consistency condition wc call the four-particle test. One has to require that Mi1,2)(0) = MjM)(0). (9.19)

As we will see in examples, this is a very strong condition that very few constructible 250 theories satisfy non-trivially. In other words, most constructible theories satisfy (9.19) only if all three-particle couplings are set to zero and hence four-particle amplitudes vanish. If the theory is fully constructible, this implies that the whole S-matrix is trivial.

9.3.2 Simple Examples

We illustrate the use of the four-particle test by first working out the general form of M.\ ' (0) and M\ ' (0) for a theory containing only integer spin particles8. We then specialize to the case of a theory containing a single particle of integer spin s. It turns out that the theory is constructible only when s > 0. For s > 0, we explicitly find the condition on s for the theory to pass the four-particle test.

General Formulas For Integer Spins

Consider first M\ ' ' (0). In order to keep the notation simple we will denote (A^ (z), •) by (1, •) and so on. The precise value of z depends on the deformation and channel being considered.

(1 ,2) i 4 /l_/ll 4 4 ll /l4 i /l4 ,( ftl M4 (°) =E (^1+^)( . ) ^ ( >A,4)' - -' (A,4)i) - - + h

A r-i A]hi+ti4-h\A p ihi+h-hxtp -i]h+hi-hA v _£_. K(l-hi-h4~h)ll>4\ Y^^lM KM'1! ) X ^2~

(^H /o o\-ft-/i3-ft2/o p \h3-h2+h/ p G\ti2-h3+h , (9.20)

i3+/l2 /l i2+ft3 h <1_,2^3+ft)[3,2]^^[2,A,4]-' - [A)4,3]-' - ) + E(4~3)-

h Here the subscripts on the three-particle couplings denote the dimension of the coupling. The range of values of the helicity of the internal particle depends on the details of the specific theory under consideration. Even though (9.20) is completely including half-integer spins is straightforward and we give an example in section 9.5. 251 general we choose to exclude theories where h can take values such that h+hi+hi = 0 or — h + hi + /13 = 0. The main reason is that formulas will simplify under this assumption.

Note also that we have kept the two pieces of all three-particle amplitudes entering in (9.20). However, recall that we should set either the holomorphic or the anti- holomorphic coupling to zero. As we will now see this condition is very important for the consistency of (9.20).

2 Let us solve the condition Pi^(z) = 0. As mentioned above this leads to zt = — (l,4)/(2,4). Since Pi^Zt), which we denoted by P14, is a null vector, it must

p) (/>) (/,)< be possible to find spinors X^ and \( such that P£4 = A V)aaA \ Clearly, given Pit4 it is not possible to uniquely determine the spinors since any pair of spinors {iA^,i-1A^} gives rise to the same Pi,4- This ambiguity drops out of (9.20) as we will see.

After some algebra we find that

P1A(zt) = A,4 = |^||A^A^. (9.21)

Therefore we can choose

(p) {p) A = a\4, ~\ = p\3, with ap = M. (9.22)

Moreover, it is also easy to get

A -(2'1}A l -[1'2]A fQ2^ Al-(2^)A4' A2_MA3- (9-23)

Using the explicit form of all the spinors one can check that the three-particle amplitude with coupling constant K%+hl+hi+h) in (9.20) possesses a factor of the form (4,4) = 0 to the power —hi — /14 — h. From our discussion in section 9.2, if —h\ — hi — h is less than zero then the coupling K^l+hi+hi+K) = 0. In this way 252 a possible infinity is avoided. Therefore we get a contribution from the term with coupling nf1_hl__ili_h\ whenever h > — (hi + /14).

Now, if — hi — hi — h is positive then ^fx+h +h +h\ need not vanish but the factor multiplying it vanishes. In this case nf1_h _/J4_M must be zero and we find no contributions.This means that the only non-zero contributions to the sum over h can only come from the region where h > — (h\ + h^).

Turning to the other three-particle amplitude, we find that the piece with coupling

nas a ^n_/l2_/l3_i_/l) factor [3, 3] = 0 to the power —h + h (h2 + ^3).

Putting the two conditions together we find that the first term gives a non-zero contribution only when h > max(—(/ii + fo4), (h2 + h3)).

Simplifying we find

1V U K K H \ ) ~ Z^ \ l-hl-h4-h l+h2+h3~h pi I H 3| I h>max(-(hi+h4),(h,2+h3)) '[l,3][l,4]\fcl/ (3,4) \hW (2,4) ^ [3,4] I V(2,3)(2,4)7 V(2,3)(3,4), , (9.24)

Finally, it is easy to obtain M\ ' (0) from (9.24) by simply exchanging the labels 2 and 4.

Next we will write down all formulas explicitly for the case when \hi\ = s for all i.

Theories Of A Single Spin s Particle

Consider now the case h\ — s, h2 = —s, /13 = s and /14 = — s. We also assume that the theory under consideration has a single particle of spin s. This restriction 253 is again for simplicity. If one decided to allow for more internal particles then the different terms would have to satisfy the four-particle test independently since the dimensions of the coupling constants would be different9.

Using (9.24) we find that the first sum contributing to M\ ' (0) allows only for h — s while the second one allows for h — — s and h — s. Using momentum conservation10 to simplify the expressions we find

3 s H /[l,3] (4,2)\ 1 Mi (0) Kl - -^U,2)(3,4)J (1,44)[1,4) ] ^-s x-s V [4,3][1,2] 7 (1,3)[1,3]

2s ^-33«f-3S([l,3](4,2)) —5-

(9.25)

We would like to set all couplings with the same dimension to the same value.

In other words, we define n = nf_s = K^-S- We also choose to study the case

K' = ftf_3s = Ki_Zs = 0. It turns out that if we had chosen K — 0 and K' non-zero the resulting theories would not have been constructible. In section 9.6 we explore strategies for relaxing this condition.

As mentioned above we can write M^'4^(0) by simply exchanging the labels 2 and 4. We then find r(UW_..2/(2,4)3[l,3]Y 1 , .^M]3<4,2) M4 (0) + : ' **\{lt2)(3,4)) <1,4>[1,4] ^\ [4,3][1,2] ^ <1,3>[1,3] [1 3]3(2 4r 1 Mr(o)^fMMT^<1,4>(3,2>; (iMh'A +^ 'V [2,3][1,4'] y <1,3>[1,3]'

Both amplitudes can be further simplified to

MilA(0) = -(-l)V([1'3](2'4))2Sxs2- Mi1A\0) = -(-l)^2([1'3](2'4))25xt2-. stu stu (9.27) 9There might be cases where the dimensions might agree by accident. Such cases might actually lead to new interesting theories. We briefly elaborate in section 9.6 but we leave the general analysis for future work. 10One can easily show that momentum conservation for four particles implies that (a, b)/{a, c) = — [d, c]/[d, b] for any choice of {a, b, c, d}. 254

(1,2) (1,4) (1,2) (1,4) Finally, the four-particle test requires M4 (0) = M4 (0) or equivalently M4 (0)/M4 (0) 1. The latter gives the condition (s/t)2_s = 1 which can only be satisfied for generic choices of kinematical invariants if s = 2. If s ^ 2 the four-particle test 1,2) M) Mi (0) - Mi (0) then requires K = 0 and hence a trivial S-matrix.

9.4 Conditions For Constructibility

The example in the previous section showed that the only theory of a single massless spin s particle that passes the four-particle test is that with s = 2. This theory turns out to be linearized General Relativity. For s — 1, the result is also familiar: a single photon should be free. However, if s = 0 one knows that a single scalar can have a non-trivial S-matrix. The reason we did not find s = 0 as a possible solution in the previous example is that precisely for s = 0 the four-particle amplitude is not constructible. Therefore our calculation was valid only for s > 0.

In this section we study the criteria for constructibility in more detail. Unfortu­ nately, we do not know a way of carrying out this discussion without first assuming the existence of a Lagrangian. The conditions for constructibility will therefore be given in terms of conditions on the, interaction vertices of a Lagrangian. We will also assume that it is possible to perform a perturbative expansion using Feynman diagrams. The starting point of all theories we consider is a canonical kinetic term (free Lagrangian) which for s — 0,1, 2 is very well known and for s > 2 can be found for example in [32, 34, 4].

The first ingredient is the polarization tensors of massless particles of spin s. Polar­ ization tensors of particles of integer spin s can be expressed in terms of polarization vectors of spin 1 particles as follows:

s s

e — e e — e aidi,...,asd,s J.J. aidj ' oidi,...,asds J.J. aidi" ^y.ZOJ i=l t=l 255

For half-integer spin s + 1/2 they are

s s

e €a 1 e = aiai,...,a3asi>~ b\.Y i< i' aid1,...,asds,6 "*& J_ J_ e^ > (9.29) i=l i=l and where polarization vectors of spin 1 particles are given by

with /ia and p,& arbitrary reference spinors.

This explains how all the physical data of a massless particle can be recovered from A, A and h. A comment is in order here. The presence of arbitrary reference spinors means that polarization tensors cannot be uniquely fixed once {A, A, h} is given. If a different reference spinor is chosen, say, // for e^h then

4(^')=e+(/i) + u;AaA4 (9.31) where

UI —

If the particle has helicity h = 1 then it is easy to recognize (9.31) as a gauge transformation and the amplitude must be invariant.

However, one does not have to invoke gauge invariance or assume any new principle. As shown by Weinberg in [2] for any spin s, the only way to guarantee the correct Poincare transformations of the S-matrix of massless particles is by imposing invari­ ance under (9.31). In that sense, there is no assumption in this section that has not already been made in section 9.1. In other words, Poincare symmetry requires that

Mn gives the same answer independently of the choice of reference spinor \x.

9.4.1 Behavior at Infinity

If a theory comes from a Lagrangian then the three-particle amplitudes derived in section 9.2 can be computed as the product of three polarization tensors times a 256 three-particle vertex that contains some power of momenta which we denote by L3. Simple dimensional arguments indicate that if all particles have integer spin then L3 = \h\ + h,2 + /13I. Let us denote the power of momenta in the four-particle vertex by L4.

We are interested in the behavior of M4, constructed using Feynman diagrams, under the deformation of A^) and X^ defined in (9.15) as z is taken to infinity.

Feynman diagrams fall into three different categories corresponding to different be­ haviors at infinity. Representatives of each type are shown in figure 9.2. The first kind corresponds to the (l,2)-channel (s-channel). The second corresponds to either the (l,3)-channel (u-channel) or the (l,4)-channel (t-channel). Finally, the third kind is the four-particle coupling.

Under the deformation X^(z) — X^ + zX^ and X^{z) — W^ — zX^\ polarization tensors give contributions that go as z~Sl and z~S2 respectively in the case of integer

Sl+l 2 S2+l 2 spin and like z~ l and z~ l in the case of half-integer spin. Recall that we chose particle 1 to have positive helicity while particle 2 to have negative helicity. Had we chosen the opposite helicities, polarization tensors would have given positive powers of z at infinity. For simplicity, let us restrict the rest of the discussion in this section to integer spin particles.

For the first kind of diagrams, only a single three-particle vertex is z dependent and gives zLa. Combining the contributions we find 2;L3-Sl_S2. Therefore, we need s1 + s2> L3.

For the second kind of diagrams, two three-particle vertices contribute giving zL3+La. This time a propagator also contributes with z~x. Combining the contributions we get zL3+L'3-si-s2-i^ Therefore we need Sj + s2 > L3 + L'3 - 1.

Finally, for the third kind of diagrams, only the four-particle vertex contributes 257

Figure 9.2: The three different kinds of Feynman diagrams which exhibit different behavior as z —•» oo. They correspond to the s-channel, t (u)-channel and the four- particle coupling respectively. giving zLi. Combining the contributions we find zL4~sl~S2. Therefore we need si + s2 > L4.

Summarizing, a four-particle amplitude is constructible, i.e., M\ ' \z) vanishes as z —* oo if s\ + s2 > L3, si + s2 > L3 + L'z — 1 and s\ + s2 > L\. It is important to mention that these are sufficient conditions but not necessary. Recall that we are interested in the behavior of the whole amplitude and not on that of individual diagrams. Sometimes it is possible that the sum of Feynman diagrams vanishes at infinity even though individual diagrams do not. Also possible is that since our analysis does not take into account the precise structure of interaction vertices, there might be cancellations within the same diagram. In other words, our Feynman diagram analysis only provides an upper bound on the behavior at infinity.

Let us go back to the example in the previous section. There S\ = s2 — 5, L3 =

L'3 = s. Note that sj + s2 > L3 implies s > 0, as mentioned at the beginning of this section. The second condition is empty and the third implies that L4 < 2s. Thus, our conclusions in the example are valid only if s > 0 and four-particle interactions have at most 2s — 1 derivatives. Note that for s = 1 this excludes (F2)2 terms and for s = 2 this excludes R2 terms. We will comment on possible ways to make these theories constructible in section 9.6. 258

9.4.2 Physical vs. Spurious Poles

There is an apparent contradiction when in section 9.3 we used that the only poles of M±{z) come from propagators and when earlier in this section we used that polarization tensors behave as z~s.

The resolution to this puzzle is very simple yet amusing. Recall that polarization tensors are defined only up to the choice of a reference spinor /i or p, of positive or negative chirality depending on the helicity of the particle. The ^-dependence in polarization tensors comes from the factors in the denominator of the form (X(z), n)s or [X(z),p]s. The deformed spinors are given by X(z) = A + z\' (or A(^) = A + zX') where A' (or A') are the spinors of a different particle. Now we see that if JJL is not proportional to A' then individual Feynman diagrams go to zero as z becomes large due to the z dependence in the polarization tensors. In the same way, individual Feynman diagrams possess more poles than just those coming from propagators. Now let us choose JX proportional to A'. Then the z dependence in polarization tensors disappears. We then find that individual Feynman diagrams do not vanish as z becomes large but they show only poles at the propagators. Recall that we are not interested in individual Feynman diagrams, but rather in the full amplitude, which is independent of the choice of reference spinor. Therefore, since M${z) vanishes for large z for some choice of reference spinors it must also do so for any other choice. This means that the pole at infinity is spurious. Similarly, poles coming from polarization tensors are spurious as well.

9.5 More Examples

In this section we give more examples of how the four-particle test can be used to constrain many theories. In previous sections we studied theories of a single particle of integer spin s and found that only s — 2 admits self-interactions. Here we allow 259 for several particles of the same spin. In this section we consider the coupling of a particle of spin s and one of spin 2. The spin s can be integer or half-integer.

9.5.1 Several Particles Of Same Integer Spin

Consider theories of several particles of the same integer spin s. The idea is to see whether allowing for several particles relaxes the constraint found in section 9.3.2 that sets s = 2.

We are interested in four-particle amplitudes where each particle carries an extra quantum number. We can call it a color label. The data for each particle is thus {\(%\)Sl\hi,ai}. As discussed in section 9.2.2, the most general three-particle am­ plitudes possess coupling constants that can depend on the color of the particles. Here we drop the superscripts H and A in order to avoid cluttering the equations and

= K define Kaia2a3 i-sfaia2a3 where faia2a3 are dimensionless factors. The subscript (1 — s) is the dimension of the coupling constant.

Repeating the calculation that led to (9.26) but this time keeping in mind that we have to sum not only over the helicity of the internal particle but also over all possible colors, we find

= J"4 (ly ftq-s / JJaia4aiJaia3a2'A--r ^i_s / ^ Jaia3aj Jaia4a2'5, [\i.oZj aj aj while

K K •^4 ' V") ~ l-s / j Ja1a2a,IJaIa3a4^ + l-s / , }a\a3ai JaIa2a4^ (9.dd) aj aj with

4 3 •S-1l In A\Z3 /In A\%\->3 nl \ S — l A= (2,4) /(2,4) [1,3]V- B= (2,4> f<2,4) [l,3] (1,2)(2,3)(3,4)(4,1) V<1,2)<3,4>; ' (1,2)(4,3)(3,1) V<1,2>(3,4) 4 3 1 3 3 S 1 c_ (2,4) /(2,4) [1,3]V- v= (2,4) f(2A) [l^ ~ (1,2)(2,3)(3,4)(4,1) V(l,4)(2,3)7 ' <1,3)(3,2)<4,1) V(M>(2,3) (9.34) 260

In order to understand why we have chosen to factor out the pieces that survive when s = 1 let us study this case in detail.

Spin 1

Before setting s = 1 it is important to recall that three-particle amplitudes for any odd integer spin did not have the correct symmetry structure under the exchange of particle labels. At the end of section 9.2, we concluded that if no other labels were introduced then the three-particle couplings had to vanish. Now we have theories with a color label. In this case, it is easy to check that in order to ensure the correct symmetry properties we must require faia2a3 to be completely antisymmetric in its indices.

Let us now set s = 1. The four-particle test requires M\ (0) — M\ (0) = 0. First note that the factor in front of B and V are equal up to a sign (due to the antisymmetric property of /). Therefore they can be combined and simplified to give

= — / j Jaiasai Jara4a2 {& + L>) / v Jaia3ai JaIaia2 I Tj 0\/0 Q\/Q A\/A 1\ / V."-^/ where the right hand side was obtained by a simple application of the identity (1,2)(3,4) + (1,4) (2,3) = (1,3)(2,4) which follows from the fact that spinors are elements of a two-dimensional vector space11.

Note that the right hand side of (9.35) can nicely be combined with the other terms to give rise to the following condition

== / _, Jaicnai Ja[aza,2 ' / y Jaiazai J0,10,4(12 <~ / j Ja\Q2ai J010304 "• (y.obj aj aj aj This condition is nothing but the Jacobi identity! Therefore, we have found that the four-particle test implies that a theory of several spin 1 particles can be non-trivial nReaders familiar with color-ordered amplitudes possibly have recognized (9.35) as the t/(l) decoupling identity, i.e., A(l, 2,3,4) + A(2,1,3,4) + A{2,3,1,4) = 0. 261

only if the dimensionless coupling constants faia2a3 are the structure constants of a Lie algebra.

Spin 2

After the success with spin 1 particles, the natural question is to ask whether a similar structure is possible for spin 2. Once again, before setting s — 2 let us mention that like in the case of odd integer spin particles, the requirement of having the correct symmetry properties under the exchanges of labels implies that the dimensionless structure constants, /aia2a3, must be completely symmetric for even integer spin particles.

Imposing the four-particle test using (9.32) and (9.33) we find that the most general solution requires

/ j Jaia^ai J0,10,30,2 — / ^ ia\Cfsa,iJ 0,10,4,0,1 \p.o() ai ai which due to the symmetry properties of fabc implies that all the other products of structure constants are equal and they factor out of (9.32) and (9.33) leaving behind the amplitudes for a single spin 2 particle which we know satisfy the four-particle test.

Note that (9.37) implies that the algebra defined by

£a*£b = fabc £c (9.38) must be commutative and associative. It turns out that those algebras are reducible and the theory reduces to that of several non-interacting massless spin 2 particles. This proves that it is not possible to define a non-abelian generalization of a theory of spin 2 particles that is constructive12. The same conclusion was proven by using BRST methods in [35].

We thank L. Preidel for useful discussions about this point. 262

Finally, let us mention that for s > 2 there is no non-trivial way of satisfying the four-particle test.

9.5.2 Coupling Of A Spin s Particle To A Spin 2 Particle

Our final example of the use of the four-particle test is to theories of a single spin s particle (\&) and a spin 2 particle (G). Here we assume that the spin 2 particle only has cubic couplings of the form (+ H—) and ( h). This means that we are dealing with a graviton. Let the coupling constant of three gravitons be « while that of a graviton to two ^'s be «'. Assume that the graviton coupling preserves the helicity of the ^ particle. This implies that K and K' have the same dimensions. Also assume that there no any cubic coupling of \I/'s13.

We need to analyze two different 4 particle amplitudes: M4(Gi, G2, W3, ^4) and

M4($i,$2,*3,*4)-

Consider first Mj^f, \I>^, \E^, *|) under a BCFW deformation. A Feynman dia­ gram analysis shows that the theory is constructible, i.e., the deformed amplitude vanishes at infinity, for s > 1. This implies that the following discussion applies only to particles \I/'s with spin higher than 1.

Let us consider the four-particle test. We choose to deform (1~,2+) and (T~,4+): < «j«i=WM_._.M ' [l,4][l,2p-»[2,3P[3,4] 2s-2 (9-39) M(M)=(^a.2> [2,4] [l,2][l,4p-2[3,4]2[2,3p-2-

Notice that is obtained from MJ1'2^ by exchanging 2 and 4. Taking the ratio of the quantities in (9.39) leads to:

13This last condition is not essential since such a coupling would have dimension different from that of K and K' and hence it would have to satisfy the four-particle test independently. 263

2 2 where s = P 2 and t = P 4. This ratio is equal to one only if s = 3/2. Thus, the only particle with spin higher than 1 which can couple to a graviton, giving a constructible theory, has the same spin as a gravitino in TV = 1 supergravity.

At this point the couplings K and K! are independent and it is not possible to conclude that the theory is linearized supergravity. Quite nicely, the next amplitude constrains the couplings.

Consider the four-particle test on the amplitude M±(G\, G2, ^3, ^4). Again we choose to deform (1,2) and (1,4):

2 2s 2 r(1>2) _ . ,.2 (l,3) [2,4] + Mf = - (K'Y [l,2]2[3,4]2[2,3]2*-4tu 2 2 (9.41) (1;4) ,(l,3) [2,4p+ (K £ 4 K [l,4]2[2,3p-2 U u where u = Pf3.

Taking their ratio and setting s = 3/2, we get

r(l>4) 1 M(U) T^- )- ™

Requiring the right hand side to be equal to one implies that K' = K. This means that this theory is unique and turns out to agree with linearized J\f = 1 supergravity.

An interesting observation is that the local supersymmetry of this theory arises as an accidental symmetry. The only symmetry we used in our derivation was under the Poincare group; not even global supersymmetry was assumed. It has been known for a long time [36] that if one imposes global supersymmetry, then M = 1 supergravity is the unique theory of spin 2 and spin 3/2 massless particles. The uniqueness of M — 1 supergravity was successively [37] derived from the non-interactive form by using gauge invariances. More recently and by using cohomological BRST methods, the assumption of global supersymmetry was dropped [38]. 264

Finally, let us stress that this analysis does not apply to the coupling of particles with spin s < 1 since the deformed amplitude under the BFCW deformation does not vanish at infinity. This simply means that we need to implement our procedure in a different way. We discuss this briefly in the next section as well as in the appendix.

9.6 Conclusions And Future Directions

Starting from the very basic assumptions of Poincare invariance and factorization of the S-matrix, we have derived powerful consistency requirements that constructible theories must satisfy. We also found that many constructible theories satisfy the conditions only if the S-matrix is trivial. Non-trivial S-matrices seem to be rare.

The consistency conditions we found came from studying theories where four-particle scattering amplitudes can be constructed out of three-particle ones via the BCFW construction. While failing to satisfy the four-particle constraint non-trivially means that the theory should have a trivial S-matrix, passing the test does not necessarily imply that the interacting theory exists. Once the four-particle test is satisfied one should check the five- and higher-particle amplitudes. A theory where all n- particle amplitudes can be determined from the three-particle ones is called fully constructible.

It is interesting to note that Yang-Mills [19] and General Relativity [22] are fully constructible. This means that the theories are unique in that once the three-particle amplitudes are chosen (where the only ambiguity is in the value of the coupling constants) then the whole tree-level S-matrix is determined. In the case of General Relativity it turns out that general covariance emerges from Poincare symmetry. In the case of Yang-Mills, the structure of Lie algebras, i.e., antisymmetric structure constants that satisfy the Jacobi identity, also emerges from Poincare symmetry. In 265 both cases, the only non-zero coupling constants of three-particle amplitudes were chosen to be those of Mz{+ + —) and M${ h). It is important to mention that our analysis does not discard the possibility of theories with three-particle amplitudes of the form M3( ) and M3(+ + +). Dimensional analysis shows that these theories are non-constructible due to the high power of momenta in the cubic vertex. For example, if s = 2 one finds six derivatives. Indeed, for spin 2, Wald [40] found consistent classical field theories that propagate only massless spin 2 fields and which are not linearized General Relativity. Those theories do not possess general covariance and the simplest of them possesses cubic couplings with six derivative interactions. In this class of theories might be the spin 3 self-interaction, which seems to be possible from [41], as well as the recent proposal for spin 2 and spin 3 interaction of [42].

There are some natural questions for the future. One of them is to ask what the cor­ responding statements are if one replaces Poincare symmetry by some other group. In particular, it is known that interactions of higher spins are possible in anti-de Sitter space (see [39] and references therein). It would be interesting to reproduce such results from an S-matrix viewpoint.

The constraints we obtained in this paper only concern the pole structure of the S-matrix. It is natural to expect that branch cuts might lead to more constraints. In field theory one is very familiar with this phenomenon; some theories that are classically well defined become anomalous at loop level. It would be very interesting to find out whether the approach presented in this paper can lead to constraints analogous to anomalies. Speculating even more, one could imagine that since three- particle amplitudes are determined exactly, even non-perturbatively, then it might be possible to find constraints that are only visible outside perturbation theory.

A well known way to handle quantum corrections is supersymmetry. A natural generalization of the results of this paper is to replace Poincare symmetry by super 266

Poincare and then explore consistency conditions for theories involving different supermultiplets.

All of these generalizations, if possible, will only be valid for the set of constructible theories. In order to increase the power of these constraints one has to find ways of relaxing the condition of constructibility. Two possibilities are worth mentioning.

The first approach is to compose several BCFW deformations [43] so that more polarization tensors vanish at infinity and make the amplitude constructible. This procedure works in many cases but it is not very useful for four particles since deforming three particles means that one has to sum over all channels at once and the four-particle constraint is guaranteed to be satisfied. One can however go to five and more particles and then there will be non-trivial constraints.

Some peculiar cases can arise because, as it was stressed in section 9.4, the behavior at infinity obtained by a Feynman diagram analysis is only an upper bound. It turns out in many examples that a Feynman diagram analysis shows a non-zero behavior at infinity under a single BCFW deformation and a vanishing behavior under a composition of BCFW deformations. Using the composition, one computes the amplitude which naturally comes out in a very compact form. When one takes this new compact, but equivalent, form of the amplitude and looks again at the behavior under a single BCFW deformation, one finds that it does go to zero at infinity! This shows that there are cancellations that are not manifest from Feynman diagrams. It would be very interesting if there was a simple and systematic way of improving the Feynman diagram analysis so that it will produce tighter upper bounds. It would be even more interesting to find a way of carrying out the analysis only in terms of the S-matrix.

The second possibility is to introduce auxiliary massive fields such that quartic vertices with too many derivatives arise as effective couplings once the auxiliary field 267 is integrated out. Propagators of the auxiliary field create poles in z whose location is proportional to the mass of the auxiliary field. The theory is then constructible, in the sense that no poles are located at infinity. Once the amplitudes are obtained one can take the mass of the auxiliary field to infinity and then recover the original theory. This gives a nice interpretation to the physics at infinity of some non- constructible theories: the "presence of poles at infinity implies that the theory is an effective theory where some massive particles have been integrated out. The simplest example is a theory of a massless scalar s = 0. Recall that one condition for a theory to be constructible is that the quartic interaction has to have I < 2s derivatives. In the case at hand, with s = 0, this means that the quartic interaction must be absent. Therefore, a scalar theory with a X(f)A interaction is not constructible. In the appendix, we show that this theory can be made constructible by introducing an auxiliary field (and deforming three particles).

A necessary ingredient to carry out the program of auxiliary fields is to find three- particle amplitudes where one or more particles are massive. More generally, it will be interesting to extend our methods for general massive representations of the Poincare group. A good reason to believe that this might be possible is the analysis of [44] where amplitudes of massive scalars and gluons were constructed using a suitable modification of BCFW deformations. In the case of massive particles of higher spins one might try to generate a mass term using the Higgs mechanism.

Finally, there are two more directions that, in our view, deserve further study. The first is the extension to theories in higher or less number of dimensions, including theories in ten dimensions. The second is to carry out a systematic search for theories where several three-particle amplitudes might have coupling constants with different dimensions but that when multiplied to produce four-particle amplitudes produce accidental degeneracies. Such degeneracies might lead to new consistent non-trivial theories which we might call exceptional theories. 268

9.7 Appendix: Relaxing Constructibility: Auxiliary Fields

Our proposal for studying arbitrary spin theories is very general, but it suffers from the fact that some interesting theories are not constructible. In section 9.6, we mentioned several ways of trying to extend the range of applicability of our technique. One of them was the introduction of auxiliary fields. In this appendix we illustrate the idea by showing how the A4 theory, which is not constructible (even under compositions of BCFW deformations), can be thought of as the effective theory of a constructible theory which contains a massive field. The constructibility here is under a composition of two BCFW deformations.

The failure to be constructible of the four-particle amplitude in the A04 theory is understood as a consequence of sending the mass of the heavy auxiliary field to infinity.

Let us start with a massless scalar with a A4 interaction:

£() = \ (drf) (0"0) - ^04. (9.43)

We can remove the quartic coupling by introducing a massive auxiliary field \'-

2 2 W, X) = \ (9,0) (#V) + \ (d,X) (0"X) - \m\x - 9X

It is straightforward to check that (9.43) can be obtained from (9.44) by integrating

2 2 out the field x taking the limit of large g and large mx, and by keeping g /2m = A/4! finite.

The theory (9.44) now has only cubic interactions. Since massless scalar fields do not possess polarization tensors that can be made to vanish at infinity, the theory with only cubic interactions is still not constructible under a BCFW deformation of two particles. This problem is resolved by applying a composition and deforming three particles. 269

Another problem one has to deal with is that the new vertex in (9.44) involves a massive scalar. This implies that the analysis of section 9.2 is not readily applicable. However, in this specific case, the three particle amplitude is simply given by the coupling constant g.

Since we are interested in the scattering of the massless scalars represented by the field 0, we consider only amplitudes where \ appears as an internal particle. This means that an internal propagator takes the form

1 (9.45) P2 - m\'

Let M4(^1, 2,3,4) be the four particle amplitude of interest. From Feynman diagrams, it is easy to see that it is given by

9 46 M4(&,02,03,&) = y!-^ 2> ( ' ) 3=1 ~ Xj where PXj — p^> +p^\ Already from (9.46), one can see that the correct limit leads to the four point vertex of the original theory:

q2 a2 y • JL „ . (9.47) PV ~ ml ml A

Let us apply a three-particle deformation:

1 3 >y-\z) = \M-z( i ' ] \(2) , Iil^l\(4)

A<2|(z) = A(2> + j||# (9.48) [2,3J

A Feynman diagram analysis shows that the deformed amplitude vanishes at infinity as z~l. Taking the t-channel as an example, the deformed propagator in this channel is: 1 Fl4(^) = p14_zMA(DA(2)! (9.49) PUz)-m2' *^~' *" "[2,3]' 270 and its pole is given by z- ~ MJTMIPT (950) The momentum Pu on-shell becomes: Pu{z") = ^-{jtw§xil)~xi2)- (9-51)

As stated at the beginning of the appendix, the three-particle amplitude is just the coupling constant g, so it is easy to reconstruct the result (9.46) and, as a consequence, (9.47). 271

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CHAPTER 10

Conclusion

In this thesis, we investigated both strongly-coupled and weakly-coupled regime of gauge theories. The analysis of the gauge theories at strong coupling has been carried out using the conjecture that relates gauge-theories to string theory [1, 2, 3, 4]. In particular, we focused on the hydrodynamic limit of some theories and studied their transport properties using the prescription to compute thermal retarded correlators in gauge/gravity correspondence [5]. We started with considering two non-conformal theories, given that QCD is non-conformal.

The first model that has been analyzed is the A/" = 2* supersymmetric gauge the­ ory in the limit of temperature larger than then the mass scales of the bosonic and fermionic components of the chiral multiplet, [6] where the dual supergravity back­ ground is known. We computed the speed of sound using two different methods. We used the thermodynamic relation (2.48), which relates the speed of sound to the energy density and the pressure, and we extracted the dispersion relation in the sound-mode from the hydrodynamic pole in the two-point correlation function of the stress-energy tensor. The two results are coincident (see Chapter 3). Notice that the leading term is the speed of sound for conformal theories and the high tem­ perature corrections are negative in sign: the speed of sound in the J\f = 2* gauge theory is less than the speed of sound in conformal theories. We also computed the bulk viscosity for the first time. As expected, the bulk viscosity is non-zero, due to 277 the high temperature corrections. Both the results for the speed of sound and bulk viscosity are gauge theory specific, since they depend on the deformation parame­ ters. It is interesting to point out that the ratio bulk viscosity by shear viscosity is proportional to the deviation of the speed of sound from its value dictated by the conformal invariance:

l 4 558 C ..(' .2 \ \ - if m6 = 0, = -« <- o > K = S ' (1(U) V V 3/ \ 4.935 if m/ = 0.

We noticed a similar behavior for the transport coefficients of another non-conformal theory, the (4 + l)-dimensional SU(N) supersymmetric gauge theory compactified on a circle with antiperiodic boundary conditions for fermions, coupled to Nf right- handed quarks and Nf left-handed quarks [7] (see Chapter 4). This theory is dual to the Sakai-Sugimoto model [8, 9]. The speed of sound and the ratio bulk viscosity by shear viscosity are respectively 1 C 4 V5 V 15 with us < 1/y/Z and (/rj satisfying (10.1) with K ~ 1. Furthermore, the super- gravity background turned out to satisfy the condition in [10, 11] and therefore the KSS bound [12] is saturated.

c Dl The common feature of having us < u ° in the two theories we analyzed may lead to conjecture that speed of sound in all non-conformal theory plasmas is less than its conformal value. This trend has been also confirmed by computations in cascading gauge theories [13, 14].

We also proved in full generality that gauge theories with chemical potential for which there exists an holographic supergravity description admitting consistent trun­ cation belong to the universality class defined by the ratio shear viscosity by entropy density rj/s [15] (see Chapter 5).

All these results apply to static plasmas. We also considered a recent proposal 278 for the analysis of the plasma dynamics in a boost-invariant framework [16, 17, 18] and looked at the Bjorken flow [19]. In the supergravity approximation, two well-known examples, i.e. J\f = 4 SYM and Klebanov-Witten background [20], have been analyzed. The dual supergravity background has been constructed as a late-time asymptotic expansion and computed the expectation value of the stress- energy tensor which turned out to have the same late-time expansion as in [21]. We explicitly computed the leading and subleading terms until the third order, obtaining the equation of state for the plasma, the shear viscosity and the relaxation time which agree with computations from the equilibrium correlators. However, at third order logarithmic singularity in the background geometry appear and they cannot be cancelled even if all the fields admitted by the supergravity approximation acquire non-zero vacuum-expectation value. One possible interpretation for the presence of these singularities in the geometry is that, if they cannot be cancelled by introducing string corrections, the correspondent gauge theory phase does not exist. These would imply that there would be no Bjorken flow for conformal gauge theories.

Finally, we also studied leading a'-correction to the supergravity approximation [22]. Specifically, we considered the black 3-brane geometry in type IIB supergrav­ ity, which is dual to J\f — 4 SYM. Taking into account that the entropy-density for the black 3-brane background in presence of a'-corrections is no longer simply proportional to the area of the horizon as in the Bekenstein-Hawking formula, the equilibrium thermodynamic quantities receive a'-corrections and the radius of the S5 is no-longer constant, the resulting hydrodynamic picture for the dual M = 4 SYM with leading corrections in the inverse 't Hooft coupling is consistent. In particular, the ratio shear viscosity by entropy density rj/s obtained from the two-point corre­ lator of the stress-energy tensor turned out to coincide with a previous computation based on the Kubo formula [23]. As expected from the conformal invariance of the theory, both the speed of sound and bulk vviscosity do not receive any correction. Given the consistency of the hydrodynamic picture at finite 't Hooft coupling, these 279 results provide a highly non-trivial consistency check on the structure of the leading a'-correction to the black 3-brane background in type IIB string theory.

As far as the weakly-coupled regime is concerned, we used the S-matrix approach based on the BCFW recursive relations [24, 25] as method of investigation, which are based on complex analysis theorems. In order to use the BCFW deformation to explore the structure of general field theories, one needs to have complete control of the singularity structure of the S-matrix and on the behavior at infinity. We dealt with the tree level approximation whose singularity structure has only poles. Including loop-corrections implies the introduction of branch points. Because of the presence of poles only, a tree level amplitude can be expressed in terms of the residues of the poles. In principle, there is also a contribution from infinity to take into account. If taking some of the external momenta to infinity along some complex direction makes the amplitude vanish, this contribution at infinity is zero. For this reason, one needs to have a good control of the behavior at infinity. In this work, the analysis at infinity has been performed by using Feynman diagrams, which provide an upper bound only.

As starting point, we considered General Relativity at tree level [26] (see Chapter 8). The usual power counting shows that tree level scattering amplitudes diverge if some external momenta are sent to infinity along some complex direction. In order to show that also in gravity BCFW-like recursive relations exist, we deformed as many external momenta as possible to have a clear vanishing behavior at infinity in a single Feynman diagram. This allowed us to obtain a set of (involved) recursive relations. We then applied the usual BCFW-deformation on the expression provided by these auxiliary recursive relations, which appeared to be a more convenient representation of the amplitude than the Feynman diagram one. The existence of recursive relations has an important consequence: amplitudes with an arbitrary number of external particles are uniquely determined by the (complex) three-particle amplitudes. This 280 was already an astonishing fact in Yang-Mills theory, but it is even more amazing in General Relativity because the Lagrangian shows an infinite set of interaction vertices.

These results suggest that an S-matrix theory for arbitrary particles can be de­ fined by assuming only the analyticity of S-matrix, its invariance under Poincar'e transformations, locality and the existence of one-particle states [27]. The BCFW construction is then used to study the factorization of the S-matrix, define a particu­ lar set of theories named constructible and propose a consistency condition that these theories must satisfy. A constructible theory has been defined as a theory whose four-particle amplitudes can be constructed out of the three-particle ones. The consistency conditions are found by applying two different BCFW-deformations to four-particle amplitudes of constructible theories. When they are not satisfied, the theory has a trivial S-matrix, i. e. there is no interaction. It is interesting to notice that also theories like Yang-Mills and General Relativity can be fully constructed from our previous four assumption. Poincare symmetry gives rise to the Lie algebra in Yang-Mills theory and general covariance in General Relativity.

One natural question is that if this S-matrix approach can reproduce a gauge theory of higher spin particles containing an infinite tower of these particles. This type of gauge theory is interesting because it is supposed to govern the high energy limit of string theory. A high spin theory of this type has been proposed in [28].

It would be also interesting to study the theories that can be defined if the Poincare group is replaced by another group. This would lead to a redefinition of the asymp­ totic states. High spin particles can interact if they propagate in a AdS background [29]. Therefore, a first step could be to replace the Poincare group with the isometry group of AdS. The issue in this case would be the definition of the asymptotic states because a massless particle in AdS can reach the boundary of AdS in a finite time and go back in the bulk. 281

Another issue is related to the constructibility of the theory. Our analysis was based on Feynman diagrams and is not able to take into account the cancellations occurring when when diagrams are summed as well as inside a single diagram itself. Therefore, only an upper bound of the behavior at infinity is provided. An important point would be to relax the constructibility conditions. A big improvement in this direction has been given in [30] where the analysis of the behavior at infinity does not rely on Feynman diagrams and the whole approach is not constrained by the helicity-spinor formalism so that several results can actually be extended to arbitrary space-time dimensions. However, a knowledge of the Lagrangian of the theory is assumed. A further generalization would be to formulate a pure S-matrix approach, without relying on any Lagrangian.

Finally, it would be interesting to go beyond tree level and analyze the further constraints that the branch cuts at loop level might dictate. 282

BIBLIOGRAPHY

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[16] R. A. Janik and R. Peschanski, "Gauge/gravity duality and thermalization of a boost-invariant perfect fluid," Phys. Rev. D 74, 046007 (2006) [arXiv:hep- th/0606149]

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[18] M. P. Heller and R. A. Janik, "Viscous hydrodynamics relaxation time from AdS/CFT," Phys. Rev. D 76 (2007) 025027 [arXiv:hep-th/0703243].

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[23] A. Buchel, J. T. Kiu and A. O. Starinets, "Coupling constant dependence of the shear viscosity in N—4 supersymmetric Yang-Mills theory" Nucl. Phys. B 707, 56 (2005) [arXiv:hep-th/0406264]

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Message 1 of 296 | Next Quota : 23% of 250MB | Move message to folder: H [email protected] _2 Inbox Subject Re: Permission to reproduce &i _3 Conferences From APS ASSOCPUB • J. Drafts 1*1 O mail Date Wednesday, January 23, 2008 6:16 pm QMathrev To Paolo Benincasa 4IVttVttM^- CSent f^ Trash Attachments Benlncasa._5845.pdr 32K f$j Manage Folders Dear Paolo Benincasa Attached is a letter granting your request for permission to use one article from the Physical Review Journals. Please note that the appropriate bibliographic citation, notice of the APS copyright, and a link to the online abstract in the APS journal must be included with the reprinted material. Thank you

Eileen LaManca Publications Marketing Coordinator [email protected] http://librarians.aps.org/ >>> Paolo Benincasa » Dear Publisher, I am writing you to ask you for permission to include the following paper, I am one of the author of, in my PhD thesis (the thesis will be in the integrated article format). The paper has just been accepted for publication in Physical Review D fthe code of the paper is: DN10233) Title: On the supergravity description of boost invariant conformal plasma at strong coupling Authors: Paolo Benincasa, Alex Buchel, Michal P. Heller, Romuald A. Janik Accepted for publication in: Physical Review D (paper code: DN10233) My thesis defence is scheduled on March 28th and it needs to be submitted at latest six weeks in advance. I would like to thank you in advance for your help. Best regards, Paolo Benincasa

Paolo Benincasa PhD Candidate Department of Applied Mathematics University of Western Ontario http://pablowellinhouse.altervista.orLT

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Paolo Benincasa PhD Candidate Department of Applied Mathematics University of Western Ontario

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N§ [email protected] Move message to folder: S Inbox l+l (jj| Conferences Subject Re: Permission request S- Dra"s ' From heller -sMHHMHMMat El Q mail n Mathrev Date Thursday, February 7, 2008 6:00 am ESent To Paolo Benincasa 4|l_HIMHii^Hk ©•Trash gj Manage Folders Dear Paolo, I see no problem in including our paper, that is Title: On the supergravity description of boost invariant conformal plasma at strong coupling. Authors: Paolo Benincasa, Alex Buchel, Michal P. Heller, Romuald A. Janik Journal: accepted for publication by Phys. Rev. D in your PhD thesis. Thus I give you my permission. Best regards, Michal Heller Jagiellonian University Cracow, Poland

On Thu, 7 Feb 2008, Paolo Benincasa wrote: > Dear Michal, > > I am writing you to ask you for permission to include the following paper of ours in my PhD thesis (the thesis will be in the integrated article format). > > I already obtained permission from the Phys. Rev. D, which accepted our paper for publication. However, they require me to obtain permission from at least on of the other authors. > > Title: On the supergravity description of boost invariant conformal plasma at strong coupling. > Authors: Paolo Benincasa, Alex Buchel, Michal P. Heller, Romuald A. Janik > Journal: accepted for publication by Phys. Rev. D > > I would like to thank you for your help. > > Cheers, > > Paolo

> Paolo Benincasa > PhD Candidate > Department of Applied Mathematics > University of Western Ontario > http://pablowellinhoxise.altervista.org

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Previous | Message 2 of 324 | Next Quota : 23% of 250MB ^H | Move message to folder: ff [email protected] Permission to reproduce Q Inbox Subject & t_Conferenr.es From Sarah Ryder <<4_fl______k-: on behalf of; Permissions _/ Drafts WCjmail Date Monday, January 14, 2008 3:18 pm _JMathrev To £*Sent |% Trash Attachments Benincasa 14 0t.pdf _j Manage Polders Dear Dr Benincasa Please find attached the permission to reproduce material from an IOPP publication. Please acknowledge receipt of this email. Best wishes Sarah Ryder Publishing Administrator Email: [email protected]

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—Original Message— From: Paolo Benincasa Sent: 1/9/2008 3:11:59 PM To: [email protected] Subject: Copyright permission for JHEP 0601:103,2006

Dear Publisher,

I am writing you to ask you for permission to include the following paper, I am one of the author of, in my PhD thesis (the thesis will be In the integrated article format).

Title: "Transport properties of N=4 supersymmetric Yang-Mills theory at finite coupling" Authors: Paolo Benincasa, Alex Buchel Published in: JHEP 0601:103, 2006

My thesis defence is scheduled on March 28th and It needs to be submitted at most six weeks in advance

I would like to thank you in advance for your help.

Best regards,

Paolo Benincasa

Paolo Benincasa PhD Candidate Department of Applied Mathematics University of Western Ontario http://pablowellinhouse.altervista.org

PERMISSION TO REPRODUCE AS REQUESTED IS GIVEN PROVIDED THAT: i author(o) la obtoinod

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Message 1 of 331 | Next Quota : 23% of 250MB mg | Move message ?q folder: H [email protected] ^ Inbox permission to reproduce l±S £3 Conferences Sarah Ryder ^HH^HHHHHfe on behalf of; Permissions J. Drafts WCjmall Wednesday, January 16, 2008 3:24 pm QMathrev CSent ^ Trash 2£| Manage Folders Dear Dr Benincasa Please find attached the permission to reproduce material from an IOPP publication. Please acknowledge receipt of this email. Best wishes Sarah Ryder Publishing Administrator Email: [email protected]

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cc # U/01/200816:09 ^ Subject Re: Permission to reproduce

Dear Ms Ryder, thank you very much for your email. With the email which has been forwarded you, I sent a second email which probably you did not receive, where I was asking for permission related to another of my papers published in JHEP. i am pasting it here.

Thank you very much for your quick reply.

Paolo Benincasa

Dear Publisher,

I am writing you to ask you for permission to include the following paper, I am one of the author of, in my PhD thesis (the thesis will be in the integrated article format).

Title: "Taming Tree Amplitudes In General Relativity" Authors: Paolo Benincasa, Camille Boucher-Veronneau, Freddy Cachazo Published in: JHEP 0711:057, 2007

My thesis defence is scheduled on March 28th and it needs to be submitted at most six weeks in advance

I would like to thank you in advance for your help.

Best regards,

Paolo Benincasa

Original Message — From: Permissions Date: Monday, January 14,2008 3:18 pm Subject: Permission to reproduce To:MWbai > Dear Dr Benincasa > > Please find attached the permission to reproduce material from > an IOPP > publication. > > Please acknowledge receipt of this email. > > Best wishes > > Sarah Ryder > > Publishing Administrator > Email: [email protected] > > 309

Paolo Benincasa PhD Candidate Department of Applied Mathematics University of Western Ontario http://pablowellinhouse.altervista.org

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;J [email protected] I Move message to folder: Slnbox t+i Q Conferences Subject RE: Permission request J! Drafts From Freddy Cachazo« Etl Qmail QMathrev Date Tuesday, February 5, 2008 5:41 pm OSent To Paolo Benincasa (§»Trash {9j Manage Folders Dear Paolo, This email is to give you permission for the use of the paper Title: Consistency Conditions on the S-Matrix of massless particles Authors: Paolo Benincasa, Freddy Cachazo arXiv: 0705.4305 [hep-th] in your PhD thesis. Best regards, Freddy Cachazo

—Original Message— ^.^^^^.^^ ' From: Paolo Benincasa [rnailtoi^HMHIHMH Sent: Tuesday, February 05, 2008 11:11AM To: Freddy Cachazo Subject: Permission request Dear Freddy, I am writing you to ask you for permission to include the following paper of ours in my PhD thesis (the thesis will be in the integrated article format). Title: Consistency Conditions on the S-Matrix of massless particles Authors: Paolo Benincasa, Freddy Cachazo arXiv: 0705.4305 [hep-th] I would like to thank you for your help. Cheers, Paolo

Paolo Benincasa PhD Candidate Department of Applied Mathematics University of Western Ontario http://pablowellinhouse.altervista.org

ldil 37:00 311 VITA

Name: Paolo Benincasa

Date of Birth:

Place of Birth:

Post-secondary The University of Western Ontario, London, Ontario, Education and Canada, Jan. 2005 - present Ph.D. Degrees: Alma Mater Studiorum Universita di Bologna, Bologna, Italy 1999 - 2004 M.Sc.

Research and Teaching Assistant: Department of Applied Mathe­ Teaching matics, UWO, London, Ontario, Canada, Jan. 2005 - Experience: present.

Research Assistant: Department of Applied Mathemat­ ics, UWO, London, Ontario, Canada, May 2005 - Aug. 2005, May 2006 - Aug. 2006, May 2007 - Aug. 2007.

Scholarships and Awards:

• Western Graduate Research Scholarship: The University of Western Ontario, London, Ontario, Canada, 2005 - 2008.

• Western Graduate Research Thesis Award: The University of Western Ontario, London, Ontario, Canada, April 2006.

Publications:

• P. Benincasa, A. Buchel, M. P. Heller, R. A. Janik, On the Supergravity 312

Description of Boost-invariant Conformal Plasma at Strong Coupling, Phys. Rev. D 77: 046006, 2008 [arXiv:0712.2025[hep-th]].

• P. Benincasa, Hydrodynamics of Strongly Coupled Gauge Theories from Gravity, Prepared for Cargese Summer School on Strings and Branes: "The Present Paradigm for Gauge Interactions and Cosmology", Cargese, France, 22 May - 3 Jun 2006. Nucl. Phys. B (Proc. Suppl.) 171 (2007) 261.

• P. Benincasa and F. Cachazo, Consistency Conditions on the S-Matrix of Massless Particles, arXiv:0705.4305 [hep-th].

• P. Benincasa, C. Boucher-Veronneau and F. Cachazo, Taming Tree Amplitudes in General Relativity, JHEP 0711 (2007) 057 [arXiv:hep-th/0702032].

• P. Benincasa, A. Buchel and R. Naryshkin, The Shear Viscosity of Gauge Theory Plasma with Chemical Potentials, Phys. Lett. B 645 (2007) 309 [arXiv:hep-th/0610145].

• P. Benincasa and A. Buchel, Hydrodynamics of Sakai-Sugimoto Model in the Quenched Approximation, Phys. Lett. B 640 (2006) 108 [arXiv:hep-th/0605076].

• P. Benincasa and A. Buchel, Transport Properties of J\f = 4 Supersymmetric Yang-Mills Theory at Finite Coupling, JHEP 0601 (2006) 103 [arXiv:hep-th/0510041].

• F. Bastianelli, P. Benincasa and S. Giombi, Worldline Approach to Vector and Antisymmetric Tensor Fields. II, JHEP 0510 (2005) 114 [arXiv:hep-th/0510010].

• P. Benincasa, A. Buchel and A. O. Starinets, Sound Waves in Strongly Coupled Non-Conformal Gauge Theory Plasma, Nucl. Phys. B 733 (2006) 160 [arXiv:hep-th/0507026]. 313

• P. Benincasa, Sound Waves in Strongly Coupled Non-Conformal Gauge Theory Plasma, Presented at the 27th Annual Montreal-Rochester-Syracuse-Toronto Conference on High Energy Physics (MRST 2005), Utica, New York, 16-18 May 2005. Int. J. Mod. Phys. A 20 (2005) 6298 [arXiv:hep-th/0506149].

• F. Bastianelli, P. Benincasa and S. Giombi, Worldline Approach to Vector and Antisymmetric Tensor Fields, JHEP 0504 (2005) 010 [arXiv:hep-th/0503155].

Talks:

31 August 2007 : International School of Subnuclear Physics, EMFCSC, Erice, Italy: Consistency Conditions on the S-Matrix of Massless Particles

25 July 2007 : Universita di Lecce, Lecce, Italy (Invited talk): Transport Properties from String Theory

19 June 2007 : Universita di Bologna, Bologna, Italy (Invited talk): Consistency Conditions on the S-Matrix of Massless Particles

3 April 2007 : Carleton University, Ottawa, ON, Canada (Invited talk): Recursive Methods for Scattering Amplitudes

10 January 2007 : Universita di Bologna, Bologna, Italy (Invited talk): A Holographic Perspective on the Transport Properties of Strongly Coupled Gauge Theories

26 May 2006 : Institut d'Etudes Scientifiques de Cargese, Cargese, France (Cargese Summer School): Hydrodynamics of Strongly Coupled Gauge Theories from Gravity 314

1 April 2006 : MCTP, Ann Arbor, MI, USA (Great Lakes String Conference): Transport Properties of N = 4 Supersymmetric Yang-Mills Theory at Finite Coupling

17 May 2005 : SUNY at Utica/Rome, Utica, NY, USA (27th MRST Conference): Sound Waves in Non-Conformal Strongly Coupled Gauge Theory Plasma

2 May 2005 : Fields Institute, Toronto, ON, Canada (Workshop of Gravitational Aspects of String Theory): Sound Waves in Non-Conformal Strongly Coupled Gauge Theory Plasma