University of Cincinnati

UNIVERSITY OF CINCINNATI

Date:______

I, ______, hereby submit this work as part of the requirements for the degree of: in:

It is entitled:

This work and its defense approved by:

Chair: ______

Topics in supersymmetric gauge theories and the gauge-gravity duality

A dissertation submitted to the oﬃce of research and advanced studies of the University of Cincinnati in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Ph.D.) in the department of physics of the College of Arts and Sciences

B.Sc., Sharif University of Technology, Tehran, Iran, June 2001

Mohammad Edalati Ahmadsaraei

June 1, 2007

Dissertation committee chair: Professor Philip C. Argyres Topics in supersymmetric gauge theories and the gauge-gravity duality

Mohammad Edalati Ahmadsaraei

Department of Physics, University of Cincinnati, P.O. Box 210011, Cincinnati, OH 45221-0011, U.S.A. [email protected]

Abstract:

In this thesis we use supersymmetry and the gauge-gravity duality to shed light on some dynamical features of strongly-coupled non-Abelian gauge theories. In the first half of the thesis, we consider singular superpotentials of four dimensional = 1 supersymmetric QCD N and show that they must exist. In particular, using some non-trivial consistency checks, we show that these superpotentials, albeit singular, are perfectly sensible, and can be used to reproduce in a simple way both the low energy effective dynamics as well as some special higher-derivative terms. In the second half of the thesis, we investigate the behavior of timelike, spacelike and Euclidean stationary string configurations on a five-dimensional AdS black hole background which, in the context of the gauge-gravity duality, correspond to quark-antiquark pairs steadily moving in an = 4 supersymmetric Yang-Mills thermal N bath. We find that there are many branches of solutions depending on the quark velocity and separation as well as on whether Euclidean or Lorentzian configurations are examined. Such solutions can, in principle, be used to determine some transport properties of strongly- coupled = 4 thermal bath. In particular, using a recently proposed non-perturbative N definition of the jet-quenching parameter, we take the lightlike limit of spacelike solutions to evaluate the jet quenching parameter in the bath. We show that this proposed definition gives zero jet quenching parameter, independent of how the lightlike limit is taken. In fact, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear, rather than quadratic, in the quark separation. To my parents, Reza and Maryam, with gratitude Contents

Preface...... vi Citationsandcopyrightnotice...... xii Acknowledgements ...... xiii Notationandabbreviation ...... xiv

1 Singular effective superpotentials in supersymmetric gauge theories 2 1.1 Effective superpotentials for SU(2) SQCD ...... 6 1.1.1 Singular superpotentials and the classical constraints ...... 7 1.1.2 Consistency upon integrating out flavors ...... 10 1.2 Singular superpotentials and the Konishi anomaly equations ...... 10 1.2.1 Directdescription...... 11 1.2.2 Seibergdualdescription ...... 12 1.3 Singular superpotentials and higher-derivative F-terms ...... 13 1.3.1 Taylor expansion around a vacuum ...... 15 1.3.2 Feynmanrules...... 16 1.3.3 Differentflavors...... 18 1.4 Singular superpotentials in three dimensions ...... 24

2 Generalized Konishi anomaly and singular superpotentials of SU(N) SQCD 27 2.1 SU(2) singular superpotentials revisited ...... 31

iii 2.2 Singular superpotential for Nf = N + 2 SQCD: non-integrability of GKA equations ...... 36 2.2.1 ComparisontotheSU(2)solution...... 40

2.3 Singular superpotential for Nf = N + 2 SQCD: Seiberg dual analysis . . . . 42 2.3.1 Derivation of the eﬀective superpotential ...... 43 2.3.2 Integrating out the glueball ﬁeld ...... 46

2.3.3 Comparing to the direct result when Nf = 4 ...... 49

3 Singular superpotentials and higher-derivative F-terms of Sp(N) SQCD 51 3.1 Singular superpotentials of Sp(N) SQCD ...... 52 3.1.1 Sp(N) SQCD for a small number of flavors ...... 52 3.1.2 Superpotentials and classical constraints for a large number of flavors 53 3.1.3 Consistency with Konishi anomaly equations: direct description. . . . 55 3.1.4 Consistency with Konishi anomaly equations: Seiberg dual description. 56 3.1.5 Consistency upon integrating out flavors...... 58 3.2 Higher-derivative F-terms in Sp(N) SQCD ...... 58 3.2.1 ThestructureofSp(N)F-terms ...... 59 3.2.2 Feynmanrules...... 61 3.2.3 Differentflavors...... 64

4 String conﬁgurations for moving quark-antiquark pairs in a thermal bath 70 4.1 Equationsofmotion ...... 74 4.2 Timelike Lorentzian solutions ...... 76 4.2.1 Timelike Lorentzian: perpendicular velocity ...... 77 4.2.2 Timelike Lorentzian: parallel velocity ...... 79 4.3 Euclidean strings and their energetics ...... 81 4.3.1 Euclidean: perpendicular “velocity” ...... 81 4.3.2 Euclidean: parallel “velocity” ...... 85

iv 5 Spacelike string conﬁgurations and jet-quenching from a Wilson loop 86 5.1 String embeddings and equations of motion ...... 89 5.2 Spacelikesolutions ...... 93 5.2.1 Perpendicular velocity ...... 93 5.2.2 Parallelvelocity...... 99 5.3 Application to jet-quenching ...... 100 5.3.1 EuclideanWilsonloop ...... 101 5.3.2 SpacelikeWilsonloop ...... 103 5.4 Appendix ...... 106 5.4.1 Euclideanaction ...... 106 5.4.2 Spacelikeaction...... 110 Bibliography ...... 117

v Preface

Quantum field theory which combines the principles of quantum mechanics and special rel- ativity is the language by which we describe modern particle physics. Using this language one can calculate the typical observable quantities in particle physics such as scattering amplitudes and particle lifetimes. In fact, the Standard Model of particle physics is a special kind of a quantum field theory: a gauge theory with a particular gauge group. A (quantum) gauge theory is a quantum field theory with a symmetry (or more accurately, a redundancy) called gauge symmetry. Understanding the dynamics of gauge theories is of utmost impor- tance for at least two reasons. Firstly, gauge theories give a fairly complete description of the physics of elementary particles at energies that we can reach experimentally these days in particle accelerators. Secondly, over the past thirty years or so, it has become clear that we have to think of such gauge theories as effective theories: a low energy approximation to a deeper theory that may fundamentally be different from a gauge theory (for example, string theory). Therefore, the better we understand the dynamics of gauge theories, the more we can constrain the ultimate theory whose effective descriptions are supposed to be the gauge theories we know of at low energies. The weak-coupling regime of gauge theories, though very important, is relatively easy to understand. One just needs to apply the well-studied tools of quantum field theory such as perturbation theory and Feynman diagrams to the problem at our disposal. The difficult part is to understand the strong-coupling behavior of such theories where perturbation theory is no longer applicable as a result of the coupling being strong. For example, crucial dynamical features such as quark confinement, dynamical generation of a mass gap and chiral symmetry breaking which are all so characteristic of the strongly-coupled regime of asymptotically-free gauge theories can not be analytically addressed using conventional tools of quantum field theory (except for lattice simulations which are numerical techniques).

Thesis in a nutshell

Thus, the question is: how can one obtain a better understanding of the strongly-coupled dynamics of gauge theories such as quantum chromodynamics (QCD)? One approach which comprises the ﬁrst half of the thesis is to enlarge the symmetries of the theory to make it supersymmetric. (See [1] for a general introduction to supersymmetry.) The more symmetry available in a system, the easier it is to be solved exactly. A theory with a certain amount

vi of supersymmetry is more constrained than an ordinary non-supersymmetric one, hence generically more amenable to exact analysis. Another interesting approach which is the subject of the second half of the thesis is to use the celebrated gauge-gravity duality , also known as Maldacena’s conjecture, or AdS/CFT correspondence. (For a detailed review of the subject, see [2].) This duality maps the strong-coupling regime of gauge theories to the weakly-coupled supergravity. Although constructing a supergravity solution dual to QCD is extremely diﬃcult, one can nevertheless come up with solutions dual to QCD-like theories, and learn non-trivial lessons. In what follows, along with explaining these two approaches, we present new results on the strongly-coupled dynamics of supersymmetric gauge theories, as well as on certain features of ﬁnite-temperature QCD-like plasmas at strong coupling.

Part I: singular effective superpotentials of SQCD Supersymmetry is a symmetry that unites particles of integer and half integer spin. It was originally introduced in early 70’s and since then has been an active area of research in theoretical particle physics and a high profile target for high energy experimental physicists. One motivation for studying supersymmetry is that it offers a possible way of solving the hierarchy problem, but in the last fifteen years, more reasons for studying supersymmetry have arisen. In particular, one can consider supersymmetric versions of gauge theories as a window on real-world gauge theories in general. From this viewpoint, supersymmetric gauge theories, being more symmetric than ordinary gauge theories, are more tractable, hence allow us to solve exactly for some of their features which would otherwise be very hard or even impossible to pen- etrate. On this basis, we have a powerful tool to think concretely about quantum field theory especially because some supersymmetric theories share many key dynamical features with their non-supersymmetric cousins. In particular, supersymmetric QCD (SQCD) with = 1 supersymmetry in four dimensions exhibits quark confinement and chiral symmetry N breaking. Four dimensional gauge theories with = 1 supersymmetry were studied in detail in N the mid 90’s mainly due to Seiberg’s seminal works [3, 4]. Using the selection rules of =1 N supersymmetry in four dimensions, exact results for supersymmetric gauge theories have been obtained; see [5, 6, 7] for reviews. These results have been verified using a number of consistency checks, mainly consistency upon integrating out massive quarks. (For technical reasons, from now on, we will call a quark a “flavor”.) Having guessed the results for few- flavor cases and checked their correctness, one adds more light flavors to the theory and tries to guess some exact results for the new theory. The lesson one learns about the dynamics of

vii these theories is that depending on the number of flavors Nf , their effective dynamics varies drastically and quantum effects are more pronounced in the few flavor cases. It is natural to ask whether this procedure can be made more deductive and uniform by turning it on its head, and starting instead with the infra red-free theories with many massless flavors. Recently, light has been shed [8] on how quantum effects at low number of flavors are inherited from the large-flavor theory. It was shown that at large number of flavors there are higher-derivative terms (of a special form) which, upon integrating out, descend to lower-derivative operators, until they finally manifest themselves as quantum corrections in the low energy effective theory. These terms were computed in four dimensional = 1 N SU(2) SQCD with an arbitrary number Nf of fundamental flavors using instanton arguments. But, interestingly, for a small number of flavors it was also shown that the higher-derivative terms can be derived by simply integrating out massive modes at tree-level from an effective superpotential. (An effective superpotential is an extremely important term in the Weff action of supersymmetric theories which controls how various fields, at low energies, interact with each other.) One may ask whether it is possible to generalize this description such that the large- flavor effective superpotentials be able to reproduce the effective dynamics of the theory. Such large-flavor superpotentials have been assumed not to exist because, for large enough number of flavors, they are singular [3, 9] when expressed in terms of the light degrees of freedom in the theory. What we find [10, 11, 12] which is the bread and butter of the first half of the thesis is that such superpotentials must exist. The cusp-like behavior of their associated potentials still unambiguously describes their supersymmetric minima. They can be regularized by turning on arbitrarily small quark masses. We then observe that no matter how the masses are sent to zero, these superpotentials always give the correct constraint equation describing the moduli space of vacua. We also show that these superpotentials reproduce both the low energy effective physics as well as the aforementioned higher-derivative terms. The existence of these singular superpotentials, shown to encode the low energy dynamics, is rather generic in = 1 supersymmetric gauge theories with enough massless flavors. N Given an effective superpotential for the large-flavor case (IR free theory), one can always integrate out flavors to derive consistent effective superpotentials for lower flavor cases. This round-about argument assures us that effective superpotentials exist for all numbers of light flavors in these theories. In chapter one, having described the argument for why we believe singular superpotentials must exist as a function of gauge-invariant local chiral fields in SQCD’s, we consider

viii = 1 SU(2) SQCD in four dimensions as the simplest example of such theories. We N show that although the large-flavor superpotentials are singular, they nevertheless correctly describe the moduli space of vacua, are consistent under renormalization group (RG) flow to fewer flavors upon turning on masses, and also reproduce by a tree-level calculation the higher-derivative F-terms computed by Beasley and Witten [8] using instanton methods. At the end of the chapter, we also note that singular superpotentials not only occur in four dimensional SQCD’s, but also in supersymmetric gauge theories in various dimensions. In chapter two, we extend our arguments to SU(N) SQCD by calculating its singular superpotentials using the generalized Konishi anomaly (GKA) equations. We find, however, that the GKA equations are only integrable in the Seiberg dual description of the theory which enable us to derive the singular superpotentials, but are not integrable in the direct description of the theory. The failure of integrability in the direct, strongly-coupled, description may suggest the existence of non-perturbative corrections to the GKA equations. In chapter three, we first derive the singular superpotentials of Sp(N) SQCD, then generalize the higher-derivative F-terms of Beasley and Witten to Sp(N) gauge theories with fundamental matter. We generate these terms by integrating out massive modes at tree level from an effective superpotential on the chiral ring of the microscopic theory. Though this superpotential is singular, its singularities are mild enough to permit the unambiguous identification of its minima, and gives sensible answers upon integrating out massive modes near any given minimum.

Part II: applications of the gauge-gravity duality to QCD-like plasmas As we alluded to earlier, one can also study supersymmetric gauge theories and hopefully ordinary gauge theories as well, in a non-conventional way through the conjectured gauge-gravity duality. The conjecture states that hidden within any supersymmetric gauge theories (and possibly non-supersymmetric theories), there is a quantum theory of gravity. This duality may seem unusual at ﬁrst sight because when we think of a gauge theory, we usually have non-gravitational degrees of freedom in mind. But a closer analysis shows [13] that gauge theories in an appropriate limit resemble string theory which is a theory of quantum gravity. Therefore, the hope is that a better understanding of the string theory and the duality will shed light on a better understanding of the dynamics of the supersymmetric theories via the gauge-gravity duality. The most-studied example is four dimensional = 4 supersymmetric SU(N) Yang- N 2 Mills theory (SYM) which, in the limit of large N and large ’t Hooft coupling λ = gYMN,

ix is described by type IIB supergravity on AdS S5 background [14]. Here, AdS is the 5 × 5 five-dimensional anti-de Sitter space, S5 is the five-dimensional sphere, and = 4 SU(N) N SYM is the maximally supersymmetric gauge theory in four dimensions which contains a vector, six (real) scalars and four (Weyl) fermions all in the adjoint representation of the SU(N) gauge group. The duality maps different modes of IIB supergravity compactified on AdS S5 background to the (chiral primary) operators of =4 SU(N) SYM. In fact, the 5 × N gauge theory operators on the boundary of AdS5 are sourced by supergravity modes and the correlation functions of such operators are computed using the IIB supergravity action on AdS S5 background [15, 16]. 5 × Zero-temperature =4 SU(N) SYM is very different from realistic theories of nature N such as QCD, because it is a superconformal theory with no S-matrix, and, as mentioned above, all the field excitations are in the adjoint representation of the SU(N) gauge group. This theory is even different from supersymmetic cousins of QCD such as = 1 SU(N) N SQCD. But on the other hand, since at finite temperature the superconformal invariance of the theory is broken, and since fundamental matter can be added by introducing D7- branes [17], it is thought that this theory at finite temperature may shed light on certain aspects of strongly-coupled QCD plasmas. In fact, the successful AdS/CFT calculation of the dimensionless ratio of shear viscosity over entropy density of =4 SU(N) SYM plasma N (or more generally, any finite temperature theories with holographic gravity dual) has taught us that the results one obtains using this correspondence might not be very far from the real world physics [18, 19, 20, 21]. Evidence from fits of relativistic heavy ion collisions (RHIC) data to hydrodynamic models indicate that the quark-gluon plasma produced at RHIC is strongly coupled. So how can we calculate transport properties of such a plasma to compare with this experiment? In particular, an interesting quantity which one would like to calculate in this regard is the jet-quenching parameterq ˆ which is a property of the medium and is a measure of the energy loss of the partons to the hot plasma. Since it is strongly coupled, it is not accessible to perturbative QCD techniques. As a transport property, it is ill-suited to lattice simulation. The AdS/CFT correspondence may be a suitable framework in which to calculate such transport quantities at strong-coupling. Assuming finite-temperature = 4 SU(N) SYM plasma in the large N and large ’t N 2 Hooft coupling λ = gYMN, as a good approximation to the finite-temperature quark-gluon plasma, one can then use its supergravity dual [22], which is the AdS-Schwarzschild blackhole whose Hawking temperature equals the temperature of the plasma, to calculate parameters

x like jet-quenching. According to the AdS/CFT dictionary, string configurations on this background can correspond to quarks and antiquarks in an = 4 SYM thermal bath [23, 24, 25, 26]. More N precisely, a stationary single quark can be described by a string that stretches from the probe D7-brane to the black hole horizon. The semi-infinite string solution with a tail which drags behind a steadily moving endpoint and asymptotically approaches the horizon has been proposed [27, 28] as the configuration dual to a steadily moving quark in the = 4 N plasma, and was used to calculate another transport quantity, the drag force, on the quark. A stationary quark-antiquark pair or “meson”, on the other hand, corresponds to a string with both endpoints ending on the D7-brane [23, 24]. A class of such solutions, namely smooth, static solutions (v = 0), have been used to calculate the inter-quark potential in SYM plasmas. Smooth, stationary solutions (v = 0) for steadily moving quark pairs exist 6 [29, 30, 31, 32, 33] but are not unique and do not “drag” behind the string endpoints as in the single quark configuration. This lack of drag has been interpreted to mean that color- singlet mesons are invisible to the SYM plasma and so experience no drag (to leading order in large N) although the string shape is dependent on the velocity of the meson with respect to the plasma. In fact, a prescription for computing the jet-quenching parameterq ˆ using the lightlike limit of spacelike signature of such string configurations has been proposed in [34]. In chapter four, we investigate the behavior of stationary string configurations on a five- dimensional AdS black hole background which correspond to quark-antiquark pairs steadily moving in an = 4 SYM thermal bath. We restrict ourselves to timelike Lorentzian and N Euclidean configurations, and find [35] that there are many branches of solutions, depending on the quark velocity and separation as well as on whether Euclidean or Lorentzian configurations are examined. We then, in chapter five, investigate stationary string solutions with spacelike worldsheet in a five dimensional AdS black hole background, and find that there are many branches of such solutions as well [36]. Using a non-perturbative definition of the jet-quenching parameter proposed in [34], we took the lightlike limit of these solutions to evaluate the jet-quenching parameter in an = 4 SYM thermal bath. We find that this proposed definition gives zero N jet-quenching parameter, independent of how the lightlike limit is taken. In particular, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear, rather than quadratic, in the quark separation.

xi Citations and copyright notice

The huge volume of papers on supersymmetric gauge theories and the gauge-gravity duality, and limitations due to space and time, prevent me from citing all references relevant to the materials presented in this thesis. However, I did my best to properly cite those papers which are of immediate relevance to this thesis. My apologies go to the authors whose papers I may have neglected to cite. Except where I explicitly cite to the literature, this thesis is my original work done as research projects in collaborations with Philip C. Argyres and Justin V´azquez-Poritz during the years of my graduate studies. More concretely, chapters one, two and three, comprising the ﬁrst half of the thesis, are respectively based on the following three papers of mine coauthored with Philip C. Argyres:

1) P. C. Argyres and M. Edalati, “On singular eﬀective superpotentials in supersymmetric gauge theories”, JHEP 0601 (2006) 012, [hep-th/0510020].

2) P. C. Argyres and M. Edalati, “Generalized Konishi anomaly, Seiberg duality and singular eﬀective superpotentials”, JHEP 0602 (2006) 071, [hep-th/0511272].

3) P. C. Argyres and M. Edalati, “Sp(N) higher-derivative F-terms via singular superpotentials”, JHEP 0607, (2006) 021 , [hep-th/0603025].

The second half of the thesis, which includes chapters four and ﬁve, follows respectively the two papers of mine listed below, coauthored with Philip C. Argyres and Justin V´azquez- Poritz:

4) P. C. Argyres, M. Edalati, J. V´azquez-Poritz, “No-drag string conﬁgurations for steadily- moving quark-antiquark pairs in a thermal bath”, JHEP 0701 (2007) 105, [hep-th/0608118].

5) P. C. Argyres, M. Edalati and J. F. V´azquez-Poritz, “Spacelike strings and jet quenching from a Wilson loop”, JHEP 0704 (2007) 049, [hep-th/0612157].

These papers were all published in the peer-reviewed Journal of High Energy Physics (JHEP). Hence, the materials in this thesis which are based on the above papers are all subject to JHEP copyright protection. The permission to present these papers as part of my thesis was granted to me by the copyright holder as part of the author-publisher agreement.

xii Acknowledgements

It is a pleasure to thank my advisor, Philip C. Argyres. I have been privileged to learn many aspects of quantum field theory and string theory through interactive discussions with him, and have benefited from his easy-to-understand and intuitive arguments. His continuous support and encouragement helped me achieve my current intellectual depth of the field for which I am deeply indebted to him, forever. He is more than an advisor to me: he is an excellent mentor and a good friend who I can easily share my ideas with, and rely on his guidance and wisdom. Many others have also contributed to my understanding of the field of whom I would like to especially thank: the late Freydoon Mansouri, Rohana Wijewardhana, Alex Kagan, Paul Esposito, David Tong, Peter Suryani, Louis Witten, Richard Gass, Justin Vázquez- Poritz, Adel Awad and Simeon Hellerman. Also, I would like to thank the string theory graduate students at the University of Cincinnati, Chetiya Sahabandu, John Wittig and Peter Moomaw for many stimulating discussions I have had with them during the years of my graduate studies. Many thanks are due to Frank Pinski, the former head of the department, and Joe Scanio, the current head, for creating a comfortable atmosphere in the department for doing research, and providing financial support. During these years I was also partially supported by the United States Department of Energy for which I am grateful. I would not have been who I am now if it was not because of my lovely parents, Reza Edalati and Maryam Mansouri. From the very first day of the elementary school to this day that I am writing this thesis, they have always been supportive of me with their unconditional love. I really do not know how to properly express my gratefulness to them. All I know is that part of every achievement of mine in any aspect of life belongs to them. Special thanks are also due to my siblings, Masoud and Mahsa for their love and support. Finally, I would like to single out my best friend Azadeh Namakydoust to whom I cannot express my love enough. Her profound wisdom, maturity and kindness has helped me in countless ways to overcome obstacles during all these years that we have been together. I am utterly privileged to have her next to me and be inspired by her kind and caring soul for years to come.

xiii Notation and abbreviation

We use “natural units” where the Planck constant ~ and the speed of light c are equal to unity. Our convention for the metric signature is mostly plus +++ . For convenience, the − ··· most frequently-used notations and abbreviations in this thesis are listed below:

AdS anti-de Sitter space CFT conformal ﬁeld theory GKA generalized Konishi anomaly IR infra red QCD quantum chromodynamics RG renormalization group SQCD supersymmetric QCD SYM supersymmetric Yang-Mills theory vev(s) vacuum expectation value(s)

AdSd d-dimensional anti-de Sitter space ′ α string length-squared

Gαβ induced worldsheet metric amount of supersymmetry N Λ strong-coupling scale in a gauge theory N number of gauge colors

Nf number of fundamental quark flavors Sn n-dimensional sphere effective superpotential Weff tree-level superpotential Wtree θ superspace fermionic coordinate

xiv Part I:

Singular eﬀective superpotentials of SQCD Chapter 1

Singular eﬀective superpotentials in supersymmetric gauge theories

Using the selection rules of = 1 four dimensional supersymmetry, exact results for super- N potentials for supersymmetric gauge theories have been obtained; see [5, 6, 7] for a review. These results have been inferred in field theory by an elaborate series of consistency checks, having largely to do with consistency upon integrating out massive chiral multiplets. The basic strategy for finding these results has been a loose kind of induction in the number of light flavors in which one works one’s way up to larger numbers of light flavors by making consistent guesses. The heuristic picture obtained in this way for SQCD is that quantum effects are more pronounced in the low energy effective action the fewer the number of light flavors. It is natural to ask whether this procedure can be made more deductive and uniform by turning it on its head, and starting instead with the IR free theories with many massless flavors. Since the leading low energy effective action of IR free theories are free, how do they manage to generate the strong quantum effects as flavors are integrated out? Re- cently, Beasely and Witten [8] have shed light on how quantum effects at low number of flavors are inherited from the large-flavor theory. They found that at large number of flavors there are higher-derivative F-terms (of a special form) in the action which, upon integrating out, descend to lower-derivative operators, until they finally become relevant, and manifest themselves as quantum corrections in the low energy effective action (e.g. as a quantum deformation of the moduli space). Beasely and Witten compute these terms in SU(2) SQCD with an arbitrary number

Nf of fundamental ﬂavors by a one-instanton argument. This is done intrinsically on the

2 moduli space, i.e. using only the massless multiplets in the vicinity of an arbitrary non- singular point on the moduli space. But, interestingly, for Nf = 2 and 3 they also show that the higher-derivative terms can be derived by simply integrating out massive modes at tree-level from an effective superpotential defined on a larger configuration space made up of vevs of the local gauge-invariant chiral meson field. This raises the question of whether a similar efficient description of larger-flavor cases can be made in terms of effective superpotentials. Now, such superpotentials are thought to be problematic because, for large enough number of flavors, they are singular [3, 9] when expressed in terms of local gauge-invariant chiral vevs, even away from the origin. Also, these superpotentials do not vanish as the strong-coupling scale of the theory Λ vanishes. Indeed, such an effective superpotential need not even exist [3]: for only if there is a region in the configuration space of the chosen chiral vevs where all of them are light together and comprise all the light degrees of freedom, are we then assured that there is a Wilsonian effective action in terms of these fields in that region. If this condition is satisfied, then the resulting effective superpotential can be extended over the whole configuration space by analytic continuation using the holomorphicity of the superpotential. For large-flavor SQCD, the only region where all the components of the meson and baryon fields become light at the same time is at the origin. But ’t Hooft anomaly matching implies [4] that there must be additional extra light degrees of freedom beyond the meson and baryon fields at the origin. Thus no superpotential written solely in terms of mesons and baryons need exist: near the origin moduli space where the baryons and mesons are becoming light, there is no guarantee that modes of other operators which account for the additional massless degrees of freedom at the origin are not becoming equally light, and so must also be included in the effective action. However, when there are so many massless flavors that the theory is IR free, we know what the light degrees of freedom are near the origin since we have a weakly coupled lagrangian description there. The physics can be made arbitrarily weakly coupled simply by taking all scalar field vevs µ φ << Λ where Λ is the strong-coupling scale (or UV cutoff) ∼h i of the IR free theory. In this limit the physics is just the classical Higgs mechanism, and all particles get masses of order µ or less. The Wilsonian effective description results from integrating out modes with energies greater than a cutoff, which we take to be some multiple of µ. The effective action will then include all local gauge-invariant operators made from the fundamental fields in the lagrangian and which can create particle states with masses below the cutoff. For the purpose of constructing the effective superpotential, the relevant local

3 gauge-invariant operators are those in the chiral ring. It is then just a matter of constructing in the classical gauge theory set of operators which generate the chiral ring. An effective superpotential which is a function of these operators must then exist. To be concrete, consider the simplest example, which will be the focus of this chapter: = 1 supersymmetric SU(2) QCD in four dimensions. This theory has an adjoint vector N (ab) i “gluon” multiplet Wα and 2Nf fundamental “quark” chiral multiplets Qa; a, b are SU(2) color indices. One can show [37, 38] that a complete basis of local gauge-invariant operators in the chiral ring in this theory is comprised of just the glueball S W W α and the ∼ α · meson operators M ij Qi Qj. At a suitably symmetric vacuum, say Qi = µδi , the gauge ∼ · h ai a i bosons and the quarks Qa with i > 2, as well as their superpartners, all get mass µ by the Higgs mechanism. So, since the glueball and meson operators only involve the product of two fundamental fields, they create modes of particle states with mass at most 2µ. (The masses just add since, by taking µ << Λ, we are at arbitrarily weak coupling.) Thus in a Wilsonian effective action found by integrating out modes above 2µ we may consistently keep all components of S and M ij, and since they generate the chiral ring, there must exist an effective superpotential which is a function of only these chiral fields. So far we have argued that an effective superpotential for local gauge-invariant operators in the chiral ring exists and makes sense for SQCD with enough massless flavors that it is IR free. This does not show the existence of such an effective superpotential in the asymptotically free case. In particular, for theories in the “conformal window” where neither the direct nor Seiberg dual description is IR free [4] (e.g. 3 < Nf < 6 for SU(2) gauge group), we have no useful description of the light degrees of freedom at the origin of moduli space. Nevertheless, given an effective superpotential for an IR free theory, we can always integrate out flavors using holomorphicity to derive consistent effective superpotentials in the conformal window. This round-about argument assures us that effective superpotentials exist for all numbers of light flavors in SQCD.

Outline of the chapter In this chapter we illustrate this line of reasoning for the simplest example: four-dimensional =1 SU(2) SQCD with many light fundamental flavors. N The form of the effective superpotential is fixed by the global symmetries, making this a particularly easy case to study. We start in section 1.1 by assuming we can integrate out the glueball degrees of freedom to express the superpotential solely in terms of the meson vevs. The resulting superpotentials, determined by the symmetries, are singular. We show that they are, nevertheless, perfectly

4 1/n ε 1/n Weff ~Pf(M) Weff ~Pf(M) Weff ~Pf(M) + ε M

2 V~|W'| ε

(a): n=1 (b): n>1

Figure 1.1: Sketches of the effective potential as a function of the meson vevs for = 1 SU(2) SQCD N with (a) n := Nf 2 = 1, where the potential is regular, and (b) for n> 1, where the potential has a cusp but can be smoothed− by a small perturbation ε. Red lines denote the moduli spaces (vacua). sensible. The cusp-like behavior of their associated potentials still unambiguously describes their supersymmetric minima. They can be regularized by turning on arbitrarily small quark masses. We then observe that no matter how the masses are sent to zero, these superpotentials always give the correct constraint equation describing the moduli space. The basic point is illustrated in figure 1.1: even though the potential has cusp-like singularities all along the moduli space, it nevertheless has a well-defined minimum. We also show that upon giving large masses to some flavors and integrating them out, we recover the superpotential for fewer numbers of flavors. In section 1.2 we justify the assumption that the glueball field can be consistently integrated out. Using the Konishi anomaly [39, 40], one can derive a system of partial differential equations satisfied by the superpotential as a function of the meson and glueball vevs. We solve these equations, determining the integration functions by matching to the Veneziano- Yankielowicz potential [45] for pure SU(2) SYM. Since in the IR free case we have included all the local chiral light degrees of freedom, by the arguments of this section we expect these differential equations to be integrable and the superpotential to exist. Indeed they are and it does, and matches (upon integrating out the glueballs) the results of section 1.1.

In the asymptotically free cases in the conformal window, 3 < Nf < 6, since there is no argument that it is consistent to describe the effective theory in terms of the local gauge- invariant chiral ring made from the microscopic fields, it is possible that the differential equations for the effective superpotential derived from the Konishi anomaly may not be integrable. In the case of SU(2) SQCD, however, we find that they are integrable. This is presumably an “accident” due to the large global symmetry group of the theory, and need not remain the case for SU(N) with N > 2 [11]. We also check that we get the same superpotential by using the Konishi anomaly equations in both the direct and Seiberg dual

5 descriptions of the low energy theory. We justify the existence of these singular effective superpotentials in IR free theories. By integrating out flavors we can use them to deduce the correct effective superpotentials for few numbers of flavors where quantum effects dramatically alter the form of the superpotential (first deforming the classical moduli space, then lifting it altogether). It was shown in [8] that in a description in terms of only the massless multiplets in the vicinity of an arbitrary non-singular point on the moduli space, these strong quantum effects descend from higher- derivative F-terms which can be calculated using instanton methods. It is therefore a non- trivial, and quite elaborate, check of our singular superpotentials that by expanding them around a generic vacuum and integrating out at tree level the massive modes of the meson field (those that take us off the moduli space), we reproduce the higher-derivative F-terms computed in [8]. We perform this check in section 1.3. Singular superpotentials are a generic feature of gauge theories with a large number of flavors, and are not special just to four-dimensional theories. In the last section of this chapter we argue in an example with three-dimensional = 2 supersymmetry where the N global symmetries are enough to fix a singular form for the effective superpotential, that they satisfy a similar set of consistency checks as do the four-dimensional theories. However, in this case we no longer have an IR free regime as a starting point from which to derive effective superpotentials by integrating out flavors using holomorphicity. Thus the meaning of singular superpotentials is less certain in d< 4.

1.1 Eﬀective superpotentials for SU(2) SQCD

(ab) = 1 SU(2) SQCD has an adjoint vector multiplet Wα containing the gluons and 2N N f i massless quark chiral multiplets Qa in the fundamental representation; i, j =1,..., 2Nf are flavor indices and a, b = 1, 2 are SU(2) color indices. (There must be an even number of flavors for gauge anomaly cancellation [41].) The classical moduli space of vacua is conveniently parametrized in terms of the vevs of the antisymmetric, gauge-singlet chiral meson ˆ [ij] i ab j ab fields M := Qaǫ Qb, where ǫ is the invariant antisymmetric tensor of SU(2). For Nf = 1, the classical moduli space is the space of arbitrary meson vevs M ij , while for N 2, it is f ≥ all M ij satisfying the constraint

ǫ M i1i2 M i3i4 =0, (1.1) i1···i2Nf

6 or, equivalently, rank(M) 2. ≤ The moduli space is modified by quantum effects when Nf < 3. For Nf = 1, there is a dynamically generated superpotential which lifts all the classical flat directions [42],

Λ5 = , (N =1) (1.2) Weff Pf M f where Λ is the strong-coupling scale of the theory and the Pfaffian is defined as Pf M :=

i1i2 i2N −1i2N ǫi ···i M M f f = √det M. For Nf = 2 the superpotential can be written [3] 1 2Nf ··· = Σ Pf M Λ4 , (N =2) (1.3) Weﬀ − f where Σ is a Lagrange multiplier enforcing a quantum-deformed constraint Pf M =Λ4 which removes the singularity at the origin of the classical moduli space.

For Nf = 3 the superpotential is [3]

1 = Pf M. (N =3) (1.4) Weff −Λ3 f The resulting equations of motion reproduce the classical constraint (1.1), which are therefore not modified quantum mechanically. Note that although the superpotential (1.4) apparently diverges in the weak-coupling Λ 0 limit, it actually vanishes on the moduli space since → (1.1) implies Pf M = 0. The negative power of Λ reflects the fact that fluctuations off the classical constraint surface become infinitely massive in the weak-coupling limit.

1.1.1 Singular superpotentials and the classical constraints

For Nf > 3, the classical constraints are also not modified quantum mechanically. However, the complex singularities of the moduli space defined by (1.1) indicate the presence of new massless degrees of freedom there, in addition to the components of Mˆ ij [4]. We argued at the beginning of this chapter that, nevertheless, an effective superpotential 1 for the IR free case (Nf > 5) should exist as a function of the unconstrained chiral meson and glueball vevs, M ij and S. For the moment, let us assume that the glueball can always be conistently integrated out away from the origin, so we can just deal with an effective superpotential depending only on M ij. Then the possible form of the effective superpotential

1It need not be single valued: it is allowed to shift by integral multiples of 2πiS, reﬂecting the angularity of the theta angle.

7 is completely determined by the symmetries up to an overall numerical factor. The only eﬀective superpotential consistent with holomorphicity, weak-coupling limits, and the global symmetries is [9]

Pf M 1/n eff = n , n := Nf 2 > 1 (1.5) W − Λb0 − where b = 6 N is the coefficient of the one-loop beta function. The coefficient in (1.5) 0 − f will be justified below. We will also check below that this superpotential is consistent under integrating out successive flavors, and so its form in the asymptotically free cases (Nf < 6) follow from any IR free case (N 6) by holomorphicity and RG flow. We leave the f ≥ justification of the assumption that S can be integrated out to section 1.2. The fractional power of Pf M in (1.5) implies that this superpotential has a cusp-like singularity at its extrema. The rest of this chapter is devoted to arguing that this superpotential is nevertheless correct. The first issue is how the classical constraint (1.1) follows from extremizing (1.5). Because these superpotentials are singular at their extrema, we cannot just take derivatives. Instead, we deform by introducing regularizing parameters before extremizing. Independent of Weff how the regularizing parameters are sent to zero, the extrema of the superpotentials will give the classical constraints (1.1). We regularize (1.5) by adding a mass term with an invertible antisymmetric mass matrix

εij for the meson fields: 1 ε := + ε M ij. (1.6) Weff Weff 2 ij Varying ε with respect to M kl yields the equation of motion Weff M kl = Λ−b0/n(Pf M)1/n(ε−1)kl. (1.7) −

Solving for Pf M in terms of ε and substituting back gives M kl = Λb0/2(Pf ε)1/2(ε−1)kl, − which in turn implies

i1i2 i3i4 1 −1 i1i2 −1 i3i4 ǫi1...i M M = ǫi1...i (ε ) (ε ) Pf ε. (1.8) 2Nf Λb0 2Nf

The right hand side of the above expression is a polynomial of order n > 0 in the εij. Therefore, no matter how we send ε 0, the right hand side will vanish, giving back the ij → classical constraint (1.1). Furthermore, it is easy to check that any solution of the classical

8 constraints can be reached in this way. It may be helpful to present another, less formal, way of seeing how the classical constraints emerge from the singular eﬀective action. Use the global symmetry to rotate the meson ﬁelds into the skew diagonal form

M1  M2  M ij = iσ , (1.9) .. ⊗ 2  .     M   Nf    so the eﬀective superpotential (1.5) becomes

1/n = nΛ−b0/n M . (1.10) Weﬀ − i i ! Y

The equations of motion which follow from extremizing with respect to the Mi are

1 1 n −1 n Mi Mj =0. (1.11) Yj6=i

Though these equations are ill-deﬁned if we set any of the Mi = 0, we can probe the solutions by taking limits as some of the Mi approach zero. To test whether there is a limiting solution where K of the M vanish, consider the limit ε 0 with M εα1 ,...,M εαK with i → 1 ∼ K ∼ αj > 0 to be determined. Plugging into (1.11), only the ﬁrst K equations have non-trivial limits, 1 α −α lim ε n ( j j ) i =0, i =1,...,K, (1.12) ε→0 giving the system of inequalities nαi < j αj for i = 1,...,K. These inequalities have solutions if and only if K > n, implying that rank(M) 2 which is precisely the classical P ≤ constraint (1.1).

Note that, as in the Nf = 3 case discussed above, the negative power of Λ appearing in the eﬀective superpotential (1.5) is not inconsistent with the weak-coupling limit because the equations of motion (1.1) following from the superpotential imply Pf M = 0, so that (1.1) vanishes on the moduli space.

9 1.1.2 Consistency upon integrating out ﬂavors

Besides correctly describing the moduli space, the eﬀective superpotentials should also pass some other tests. If we add a mass term for one ﬂavor in the superpotential of a theory with

Nf flavors and then integrate it out, we should recover the superpotential of the theory with N 1 flavors. To show that the effective superpotential (1.5) passes this test, we add a f − gauge-invariant mass term for one flavor, say M 2Nf −1 2Nf :

= nΛ−b0/n(Pf M)1/n + mM 2Nf −1 2Nf . (1.13) Weﬀ −

i 2Nf −1 j 2Nf The equations of motion for M and M (i =2Nf 1 and j =2Nf ) put the meson

6 − 6 ij M 0

matrix into the form M = where M is a 2(N 1) 2(N 1) and X a2 2 matrix. 0 X f − × f − × 2Nf −1 2Nf Integrating out X M σ2 by its equation of motion gives ∼ ⊗ c b b = (n 1)Λb0/(n−1)(Pf M)1/(n−1) (1.14) Weff − − where Λ= mΛ6−Nf is the strong-coupling scaleb of the theoryc with N 1 flavors, consistent f − with matching the RG flow of couplings at the scale m. Dropping the hats, we recognize b (1.14) as the effective superpotentials of SU(2) SQCD with N 1 flavors. f −

1.2 Singular superpotentials and the Konishi anomaly equations

The Konishi anomaly implies a differential equation which the effective superpotential should obey when considered as a function the meson and glueball vevs. We outline here the derivation of this equation and show that its solution enables us to determine the dependence of the effective superpotential on the glueball vev, and to justify the assumption that made in section 1.1 that the glueball can be consistently integrated out. Although this is a simple exercise, it gains interest when compared to the SU(N) case where the corresponding generalized Konishi anomaly equations [37, 38] are much more complicated [43], as mentioned at the beginning of the chapter. In the chiral ring the Konishi anomaly [39, 40] for a tree-level superpotential takes Wtree the form ∂ tree j j W i Qa = Sδi . (1.15) h ∂Qa i

10 ˆ 1 α where S is the vev of the glueball superfield S = 32π2 tr(W Wα). (We distinguish an operator from its vev by putting a hat on the operator.) This is a special case of the generalized Konishi anomaly, which is perturbatively one-loop exact [37], and has also been shown [44] to be non-perturbatively exact for a U(N) gauge theory with matter in the adjoint representation as well as for Sp(N) and SO(N) gauge theories with matter in symmetric or antisymmetric representations. For the theory we are discussing here, we will not prove that the Konishi anomaly is non-perturbatively exact, though presumably this can be done along the lines of [44]. Instead, because the global symmetry of the SU(2) SQCD uniquely determines the superpotential as discussed in the previous section, we only need check that the Konishi anomaly equation implies this form of the superpotential. This check serves as evidence for the non-perturbative exactness of the Konishi anomaly equation for the theory under discussion. Had the Konishi anomaly equation been modified non-perturbatively, we would have found a different result for . Weff

1.2.1 Direct description

In the Konishi anomaly equation (1.15), take as our tree-level superpotential

W = m (Mˆ ij M ij), (1.16) tree ij − so that 1 ∂ m = Weff , (1.17) ij −2 ∂M ij is a Lagrange multiplier imposing that M ij are the vacuum expectation values of the meson operators Mˆ ij. Substituting (1.16) into (1.15) and using the fact that the expectation value of a product of gauge-invariant chiral operators equals the product of the expectation values of kj j the individual ones, gives 2mikM = Sδi . Using (1.17) we then obtain a partial differential equation for the effective superpotential,

∂ Weff M kj = Sδj, (1.18) ∂M ik i whose solution is Pf M eff (M,S)= S ln + f(S), (1.19) W Λ2Nf where f(S) is an undetermined function. Upon giving the quarks a mass m and integrating them out, the superpotential reduces to f(S)+ N S[ln S ln(mΛ2) 1]. In the limit m f − − → , Λ 0 keeping Λ fixed, where 6 ln Λ = b lnΛ + N ln m, this becomes the SU(2) ∞ → 0 0 0 f 11 superYang-Mills theory with strong coupling scale Λ0. The superpotential for this theory is the Veneziano-Yankielowicz superpotential [45] W (S)=2S [ln(S/Λ3) 1], implying that VY 0 − f(S)=(2 N )S ln(S/Λ3) 1 . (1.20) − f − Substituting (1.20) into (1.19) gives the effective superpotential as a function of S and M ij . It is easy to see that at its extrema S is massive (except at the origin), justifying the assumption of the last section that it could be integrated out.2 Finally, integrating S out by solving its equation of motion, we arrive at the effective superpotential (1.5).

1.2.2 Seiberg dual description

Viewing our SU(2) theory as an Sp(1) gauge theory, when Nf > 3 the theory has a Seiberg dual description [9] in terms of an Sp(N 3) gauge group.3 The dual Sp(N 3) theory has f − f − a 2Nf dual quark chiral multiplets qi in the fundamental representation as well as a gauge- singlet chiral multiplet ˆ [ij] which is coupled to the dual meson fields ˆ := qaJ qb through M Nij i ab j the superpotential = ˆ ˆ ij. Here J is the invariant symplectic antisymmetric tensor, W NijM ab i, j = 1,..., 2N are flavor indices, and a, b = 1,..., 2N 6 are the gauge indices. This f f − superpotential gives masses to the dual quarks and sets = 0 when ij = 0. The dual Nij M 6 description is IR free when Nf < 4. To determine the effective superpotentials of the dual theory we can either use the global symmetry, weak-coupling limit and the holomorphicity argument, or the Konishi anomaly equations. Both give the same answer; we discuss the Konishi anomaly equations. The ring of local gauge-invariant chiral operators is generated by ˆ, ˆ ij and ˆ [46]. The Konishi S M Nij anomaly equations are qa(∂W /∂qa) = δi . Take as the tree-level superpotential h j tree i i S j = ˆ ˆ ij + m ( ˆ ij ij), (1.21) Wtree NijM ij M −M so that as before, m = 1 (∂ /∂ ij), is a Lagrange multiplier imposing that ij are ij − 2 Weff M M the vacuum expectation values of the scalar operators ˆ ij. We have not included a Lagrange M multiplier for the dual mesons ˆ because our analysis is valid only for points away from Nij 2Naively it appears that S is infinitely massive on the moduli space. But a careful limiting approach to the moduli space as in section 1.1, together with an analysis of the Kahler terms gives a finite mass-squared of order the non-zero vev of M ij . In the IR free case, this result follows immediately just by the classical Higgs mechanism, as discussed at the beginning of this chapter. 3 The SU(Nf 2) Seiberg dual description [4] is more difficult to analyze since it has a smaller global symmetry group.−

12 the origin of the moduli space where the dual quarks are massive. As in the direct description, the Konishi anomaly with (1.21) gives 2 ik = δi . M Nkj −S j The ˆ ij equation of motion gives = m , giving the partial differential equation M Nij − ij ik(∂ /∂ kj)= δi whose solution is M Weff M S j Pf eff ( ,S)= S ln M + f(S). (1.22) W M Λ˜ Nf f(S) is determined as before to be f(S)=(2 N )S[ln(S/Λ˜ 3) 1]. Integrating out S then − f − gives the effective superpotential in the dual description

1 N −2 =(N 2) Λ˜ 2Nf −6Pf f . (1.23) Weﬀ f − M The dual and direct descriptions are equivalent in the IR: the ij are identiﬁed with the M direct theory mesons by ij = 1 M ij, where µ is a mass scale related to the dual and the M µ direct theory strong-coupling scales by

Λ6−Nf Λ˜ 2Nf −6 =( 1)Nf µNf . (1.24) − Rewriting (1.23) in terms of Λ and M ij gives our superpotential (1.5).

1.3 Singular superpotentials and higher-derivative F- terms

In this section we show that the effective superpotential (1.5) passes a different, more strin- gent, test. In [8] a series of higher-derivative F-terms were calculated by integrating out massive modes at tree-level from the non-singular effective superpotentials (1.3) and (1.4) for SU(2) SQCD with N = 2 and 3, and by an instanton calculation for N 3. In this f f ≥ section we show that our singular superpotential for Nf > 3 reproduces these F-terms by a tree-level calculation. As in our discussion of the classical constraints in the last section, the key point in this calculation is to first regularize the effective superpotential (1.5), and then show that the results are independent of the regularization.

13 The higher-derivative F-terms found in [8] in =1 SU(2) SQCD are, for N 3 ﬂavors, N f ≥

4 2 6−Nf −Nf i1j1···iNf jNf δS = d xd θ Λ (MM) ǫ M i1j1 Z k2ℓ2 kN ℓN (M DM i k DM j ℓ ) (M f f DM i k DM j ℓ ), (1.25) × 2 2 · 2 2 ··· Nf Nf · Nf Nf

ij where (MM) := (1/2) ij M M ij, and the dot denotes contraction of the spinor indices on the covariant derivativesPDα˙ . Although these terms are written in terms of the unconstrained meson field, they are to be understood as being evaluated on the classical moduli space. In other words, we should expand the M ij in (1.25) about a given point on the moduli space, satisfying (1.1), and keep only the massless modes (i.e., those tangent to the moduli space). This should be contrasted with our effective superpotential (1.5) which is makes sense only in terms of the unconstrained meson fields. Note that even though (1.25) is written as an F-term (an integral over a chiral half of superspace), the integrand is not obviously a chiral superfield. But the form of the integrand 2 is special: it is in fact chiral, and cannot be written as D (something), at least globally on the moduli space, and so is a protected term in the low energy effective action. These features of (1.25), discussed in detail in [8], will neither be obvious nor play an important role in our derivation of these terms. We will now show how (1.25) emerges from the effective superpotential (1.5). To derive effective interactions for massless modes locally on the moduli space from the effective superpotential for the unconstrained mesons, and which therefore lives off the moduli space, we simply have to expand the effective superpotential around a given point on the moduli space and integrate out the massive modes at tree level. The only technical complication is that, as discussed in section 1.1, the effective superpotential needs to be regularized first, e.g. by turning on a small mass parameter εij as in (1.6), so that it is smooth at its extrema. At the end, we take ε 0. The absence of divergences as ε 0 is another check of the ij → → consistency of our singular effective superpotential.

14 1.3.1 Taylor expansion around a vacuum

The moduli space is deﬁned by the constraint rank(M) 2 (1.1). Without loss of generality, ≤ we can choose the vacuum satisfying (1.1) around which we expand to be

µ  0  M ij = iσ , (1.26) 0 .. ⊗ 2  .     0     with µ a non-vanishing constant, by making an appropriate SU(2Nf ) global ﬂavor rotation. Note that M ij breaks the SU(2N ) global symmetry to SU(2) SU(2N 2). Accordingly we 0 f × f − henceforth partition the i, j ﬂavor indices into those transforming under the unbroken SU(2) factor from the front of the alphabet—a, b=1, 2—and the remaining SU(2N 2) indices from f − ij ij ij the back: u,v,... =3,..., 2Nf . Linearizing (1.1) about (1.26), M = M0 + δM , implies that the massless modes are δM 12 and δM au, while the δM uv are all massive. The δM 12 mode can be absorbed in a rescaling of µ, so we only need to focus on the δM au modes.

ij Expanding (1.25) around M0 and keeping only the massless modes, we generate an inﬁnite number of terms. The leading term is of order (δM)2Nf −2,

4 2 6−N 1−N −1 u v ···u − v − δS d xd θ Λ f µ f µ ǫ 1 1 Nf 1 Nf 1 (DδM DδM ) ∼ 1u1 · 2v1 × Z (DδM 1u − DδM 2v − ), (1.27) ···× Nf 1 · Nf 1 since DM 0 = 0. It suffices to show that this leading term is generated in perturbation theory since the SU(2Nf ) flavor symmetry together with the chirality of the integrand imply that (1.25) is the unique non-linear completion of (1.27); see section 3.2 of [8].4 In order to demonstrate how (1.27) is generated at tree-level from our effective action, we first regularize ε , which we repeat here, Weff → Weff 1 ε := nλ(Pf M)1/n + ε M ij , (1.28) Weff − 2 ij where we have defined the convenient shorthands

n := N 2, λ := Λ−b0/n. (1.29) f − 4We could, in principle, directly generate the higher-order terms in the expansion of (1.25) by a tree-level argument. In fact, a sixth-order term in the Nf = 3 theory is calculated in this way in [8].

15 Now the extrema of ε no longer satisfy the classical constraint equation (1.1), but are Weﬀ deformed as in (1.8). So we must also deform (1.26) as well. It is convenient to choose ε = λε1/nµ(1−n)/ndiag ε,µ,...,µ iσ so that ij { } ⊗ 2 µ  ε  (M ε)ij = iσ . (1.30) 0 .. ⊗ 2  .     ε     An advantage of this choice is that it preserves an SU(2) Sp(2N 2) subgroup of the × f − ﬂavor symmetry. In the limit ε 0 this is enhanced to SU(2) SU(2N 2). Also, the → × f − massless directions around this choice are still δM ua as before.

1.3.2 Feynman rules

We use standard superspace Feynman rules [47] to compute the effective action for the massless δM ua modes by integrating out the massive δM uv modes. This means we need to evaluate connected tree diagrams at zero momentum with internal massive propagators and external massless legs. The massive modes have standard chiral, anti-chiral, and mixed superspace propagators with masses derived from the quadratic terms in the expansion of ε . The higher-order terms in the expansion give chiral and anti-chiral vertices. Weff A quadratic term in the superpotential, = 1 m(δM)2, gives a mass which enters W 2 the chiral propagator as δMδM = m(p2 + m 2)−1(D2/p2), similarly for the anti-chiral h i | | propagator, and as δMδM = (p2 + m 2)−1 for the mixed propagator. Each propagator h i | | comes with a factor of δ4(θ θ′). Even though the diagrams will be evaluated at zero − momentum, we must keep the p2-dependence in the above propagators for two reasons. First, there are spurious poles at p2 = 0 in the (anti-)chiral propagators which will always 2 cancel against momentum dependence in the numerator coming from D ’s in the propagators 2 and D2’s in the vertices. For instance, D2D = p2 when acting on an anti-chiral field, giving a factor of p2 in the numerator which can cancel that in the denominator of the anti-chiral propagator, to give an IR-finite answer. Second, expanding the IR-finite parts in a power series in p2 around p2 = 0 can give potential higher-derivative terms in the effective action, when p2’s act on the external background fields.

16 Expanding ε around M εij gives the quadratic terms Weff 0 ′ ′ ′ ′ ε ε ε ε ijkℓ ε 1/n ε −1 ε −1 i j k ℓ (M + δM)= (M )+ λt ′ ′ ′ ′ (Pf M ) (M ) (M ) δM δM + , (1.31) Weff 0 Weff 0 i j k ℓ 0 0 ij 0 kℓ ···

ijkℓ We will drop for now the numerical tensor ti′j′k′ℓ′ which controls how the ij . . . indices are contracted with the i′j′ . . . indices, though its form will be needed for a later argument. But for our immediate purposes, it suﬃces to note, as we discuss below, that in the ε 0 limit → the tensor structure of our tree diagrams is ﬁxed by the SU(2) SU(2N 2) subgroup of × f − the global symmetry that is preserved by the vacuum. Specializing to the massive modes, for which i,j,k,ℓ u,v,w,x , and using (1.30) { }→{ } then gives the mass m λε−αµβ, where ∼ n 1 1 α := − , β := . (1.32) n n The propagators are then

−1 εα D2 εα 2 δM uv –––– δM wx 1+ p2 , ∼ λµβ p2 λµβ ! 2 −1 εα D εα 2 δM ———– δM 1+ p2 , uv wx ∼ β p2 λµβ λµ ! −1 εα 2 εα 2 δM —– – – δM wx 1+ p2 . (1.33) uv ∼ λµβ λµβ !

We have suppressed the tensor structure on the u,v,w,x indices. { } ε ε The (anti-)chiral vertices come from higher-order terms in the expansion of eff ( eff ). 2 W W Each (anti-)chiral vertex will have a D (D2) acting on all but one of its internal legs. Also, each vertex is accompanied by an d4θ. The ℓth-order term in the expansion of ε has Weff the general structure R

′ ′ ′ ′ ε 1/n ε −1 ε −1 i j i j λ(Pf M ) (M ) (M ) δM 1 1 δM ℓ ℓ , (1.34) 0 0 i1j1 ··· 0 iℓjℓ ··· where we have suppressed the tensor structure which governs the order in which the i′j′ indices are contracted with the ij indices. Thus vertices with m massless legs and ℓ m −

17 massive legs are accompanied by the factors

m massless m massless

λ λ , , (1.35) z }| { ∼ εγℓ,m µκm z }| { ∼ εγℓ,m µκm

ℓ−m massive ℓ−m massive where | {z } | {z } m n +1 m 1 γ := ℓ , κ := . (1.36) ℓ,m − 2 − n m 2 − n Note that it follows from (1.34) that the number, m, of massless legs, δM au, must be even, and furthermore half must be δM 1u’s and half δM 2u’s. This is because these legs each have one index a 1, 2 and the only non-vanishing components of (M ε)−1 with indices in this ∈{ } 0 ij range are (M ε)−1 = (M ε)−1 = µ−1 which have two of these indices. 0 12 − 0 21 Finally, to each (anti-)chiral external leg at zero momentum is assigned a factor of the au (anti-)chiral background ﬁeld δM au(x, θ)(δM (x, θ)) all at the same x. Overall momentum conservation means that the diagram has a factor of d4x. The δ4(θ θ′) for each internal − 4 4 propagator together with the d θ integrals at eachR vertex leave just one overall d θ for the diagram. R R

1.3.3 Diﬀerent ﬂavors

Nf = 3 We start by ﬁrst looking at the Nf = 3 case. Although this case does not involve a singular superpotential, it has the virtue of being simple and yet still illustrates how the potential IR poles cancel, and may help make the use of the Feynman rules clearer to the 6 bewildered reader. Also, although in [8] a tree diagram is computed for Nf =3, it is a δM term (which was useful for comparing to an instanton computation), and not the leading 4 δM term which we will be computing.

The Nf = 3 case is special since it can only involve anti-chiral vertices. There are two diagrams that contribute, shown in figure 1.2(a). The first diagram, consisting of just an amputated 4-vertex with massless legs, vanishes. This can be seen by a symmetry argument: since the diagram comes with no powers of ε, in the ε 0 limit its index structure must → be, by the unbroken SU(4) part of the flavor symmetry, ǫu1v1u2v2 δM δM δM δM . ∝ 1u1 2v1 1u2 2v2 Because there are no derivatives acting on the δM’s, this vanishes under the antisymmetrization of the ui or vi since the δM’s are bosons. Alternatively, it is easy to calculate the index

18 +

(a) (b)

Figure 1.2: Diagrams for (a) Nf = 3, and (b) Nf = 4. structure of the 4-vertex directly by expanding ε directly as in (1.31). Weﬀ Thus only the second diagram in ﬁgure 1.2(a) contributes. Actually, two diagrams like this contribute: the one shown, and one in a crossed channel. (The third channel does not contribute because, as noted above, the 3-vertex with two external legs of the form

δM 1uδM 1v or δM 2uδM 2v vanishes by antisymmetry.) We will evaluate just one channel; the second gives an identical result. The Feynman rules give for the amplitude

4 4 4 us vt ut vs λ N =3 d xd θ1d θ2 δM 1u(θ1)δM 2v(θ1)(J J J J ) A f ∼ − εγ3,2 µκ2 Z 2 −1 εα D εα 2 δ4(θ θ )(J J J J ) 1+ p2 × 1 − 2 sp tq − sq tp β p2 λµβ λµ " #

λ wp xq wq xp (J J J J )δM 1w(θ2)δM 2 x(θ2) × εγ3,2 µκ2 − 2 λ D = ǫuvwx d4xd4θ δM δM 1 µλ −2p2 + (p4) δM δM µ 1u 2v p2 −| | O 1w 2x Z 2 λ D 2 = ǫuvwx d4xd4θ δM δM (δM δM ) µλ −2D (δM δM )+ (p2) µ 1u 2v p2 1w 2x −| | 1w 2x O Z " # 2 λ D2D = ǫuvwx d4xd2θ δM δM (δM δM ) µ 1u 2v p2 1w 2x uvwxZ ǫ 2 2 d4xd2θ D [δM δM D (δM δM )] + (p2) − λµµ2 1u 2v 1w 2x O Z λ = ǫuvwx d4xd2θ δM δM δM δM µ 1u 2v 1w 2x Z ǫuvwx d4xd2θ (DδM DδM )(DδM DδM )+ (p2). (1.37) − λµµ2 1u · 2v 1w · 2x O Z 19 _ χ χ

(a) (b)

Figure 1.3: (a) The sum of all diagrams with purely chiral internal vertices. (b) The sum of all purely anti-chiral diagrams vanishes for Nf > 3.

The first line includes the tensor structure of the vertices and propagator calculated by Taylor expanding ε around M εij as in (1.31). The antisymmetric symplectic tensor J Weff 0 uv and its inverse J uv arise from the structure of M ε in (1.30); it is simply J := 1l iσ , 0 Nf −1 ⊗ 2 where 1l is the (N 1) (N 1) identity. The second line performs a d4θ integration, Nf −1 f − × f − the tensor algebra, the Taylor expansion of the propagator around p2 = 0, and substitutes the Nf = 3 values α = 0, β = 1, γ3,2 = 0, and κ2 = 0 from (1.32) and (1.36). The fourth − 2 line trades an d2θ for a D2 in the first term, and a d2θ for a D in the second term. 2 2 2 The fifth line usesR the identity D D = p on antichiralR fields to cancel the IR pole in the first term, and uses the equation of motion DδM = 0 to leading order in δM to distribute the D’s in the second term. The first term in the last line cancels by antisymmetry, leaving the second term which is the higher-derivative F-term predicted in [8]. The (p2) terms are O potential higher-derivative terms.

Nf = 4 The next case is Nf = 4. This is the first case where we have a singular superpotential (1.5). Since we need a total of six external massless legs, we can only have one diagram (plus its various corssings) with an internal chiral vertex. This is the single diagram shown in figure 1.2(b). There are also a number of purely anti-chiral diagrams which could contribute. We will show, quite generally, that these diagrams vanish in the ε 0 limit, → leaving only the diagram in figure 1.2(b). We now show that the sum of all purely anti-chiral diagrams, represented in figure 1.3(b),

20 2 vanishes for Nf > 3. All but one of the legs of the χ subdiagram has a D by the Feynamn rules. Rewriting the overall Grassmann integration for χ as d4θ = d2θD2 gives the 2 2 2 remaining leg a D . These D ’s combine with the D ’s fromR each anti-chiralR propagator connecting χ to the external-vertices to give a factor of p2 which cancels the p2 in the denominators of those propagators. Thus, all the potential IR poles cancel, leaving no D’s or D’s to act on the massless external background fields on the external legs. So, setting the momenta to zero gives a finite result. But, in the limit that ε 0 an → SU(2N 2) subgroup of the global flavor symmetry is restored. So, the coefficient of the f − leading power of ε will be SU(2N 2)-invariant. Thus the leading term in the p2 0 and f − → ε 0 limit of the sum of all diagrams of the form shown in figure 1.3(b) will be proportional → to − − u1v1···uNf 1vNf 1 ǫ δM 1u δM 2v δM 1u − δM 2v − , (1.38) 1 1 ··· Nf 1 Nf 1 since the completely antisymmetric tensor is the only SU(2N 2)-invariant way of tying f − together the flavor indices of the massless external δM fields. But the expression in (1.38) vanishes since the product of the δM 1ui ’s and that of the δM 2vi ’s are symmetric on their ui and vi indices, respectively. But this is only the leading term in an expansion around p2 = 0. Higher powers of p2 can be brought to act on the external legs, giving derivatives of the external fields in the 2 2 combinations ∂ (δM 1ui δM 2vi ). The higher powers of p come from the Taylor expansion of the (1 + εα 2p2)−1 denominators of the propagators (1.33). Thus each factor of p2 comes | | with a factor of ε 2α. The flavor symmetry of the leading term in the ε-expansion of the | | amplitude ensures that the external ui and vi indices are completely antisymmetrized. This still enforces the vanishing of the amplitudes as long as there are at least two factors of

(δM 1ui δM 2vi ) without derivatives acting on them. Thus, the ﬁrst non-vanishing term will have a factor of p2 acting on N 2 pairs of external legs. f − Now consider any purely anti-chiral internal sub-diagram, χ. Each anti-chiral vertex has 2 4 a D acting on all but one of its legs as well as an d θi. Likewise each internal anti-chiral 2 4 propagator has a D as well as a δ (θi θj ). The deltaR functions and Grassmann integrations − 2 2 leave just a single overall d4θ. The D ’s and D2’s pair up so there is a D D2 = p2 in the 2 numerator of each internalR chiral propagator, and a D acting on all but one of the external legs. This p2 cancels the p2 in the denominator of the anti-chiral propagator in (1.33), leaving −1 the IR-ﬁnite factor proportional to εαµ−βλ .

If the purely chiral sub-diagram has P internal propagators, E external legs, and Vℓ,0 ℓ-

21 legged vertices, this implies that the whole sub-diagram gives an eﬀective vertex proportional

(−P + ℓ Vℓ,0) (P α− γ V ) (−Pβ− κ V ) −E+(n+1)/n 1/n λ ε ℓ ℓ,0 ℓ,0 µ ℓ 0 ℓ,0 = λε µ , (1.39) plus terms vanishing as p2 0. On the right side we have substituted the values of α, β, → γℓ,0, and κ0 from (1.32), (1.36), and used the identities

P +1= Vℓ,m, 2P + E = ℓVℓ,m, (1.40) Xℓ,m Xℓ,m where Vℓ,m be the number of vertices with a total of ℓ legs of which m are massless external legs. They follow from the topology of connected tree diagrams. (We have set the number m of massless external legs to zero because our sub-diagram is internal, so only connects to massive propagators.) Now we can compute the dependence on ε of the purely anti-chiral amplitude in ﬁgure 1.3(b) with N factors of p2: it will have an overall factor of ε to the power of

n +1 n 1 2αN + E +(N 1)α (N 1)γ = (2N N + 1) − , (1.41) n − f − − f − 3,2 − f n where the first term is from the N factors of p2, the second from the χ internal diagram (1.39) with E =(N 1) legs, the third from the (N 1) anti-chiral propagators attaching f − f − χ to the external 3-vertices, and the fourth from the (N 1) 3-vertices themselves each f − with 2 massless legs. We have used the values of α and γ3,2 from (1.32) and (1.36) on the right-hand side. Thus the power of ε is non-negative when N (N 1)/2. The minimum ≥ f − value of N = N 2 needed for the amplitude not to vanish by antisymmetry is greater than f − (N 1)/2 for N > 3. Thus, for N > 3 the sum of all the diagrams of the form shown in f − f f figure 1.3(b) vanish as ε 0. We evaluated the special N = 3 case above and saw explicitly → f that it does not vanish. It remains to evaluate the single diagram in figure 1.2(b). It is a special case of the class of diagrams shown in figure 1.3(a): purely-chiral internal diagrams with anti-chiral external 3-vertices. It is easy to evaluate the overall structure of these amplitudes. 2 The Feynman rules imply that there is a D acting on all but one of the Nf 1 legs of the χ − 2 internal sub-diagram. Rewriting the overall Grassmann integration for χ as d4θ = d2θD 2 gives the remaining leg a D . Thus each mixed propagator connecting theR χ sub-diagramR 2 to the external anti-chiral 3-vertices will have a D acting on it. Unlike the purely anti- chiral propagator, the mixed propagator (1.33) has neither an IR pole nor any D2’s in the

22 2 numerator. Thus each D will act on a pair of external massless legs. To leading order in 2 2 2 the δM’s, D δM = 0 by equation of motion, so we can replace D (δM )=(DδM)2. Thus, the massless external background ﬁelds must appear as

4 2 d xd θ (DδM 1u DδM 2v ) (DδM 1u − DδM 2v − ). (1.42) 1 · 1 ··· Nf 1 · Nf 1 Z As before, the leading term in the ε 0 limit must be invariant under the SU(2N 2) → f − subgroup of the ﬂavor symmetry that is not broken by the vacuum, and so the ui and vi u v ···u − v − indices must be contracted with the totally antisymmetric tensor ǫ 1 1 Nf 1 Nf 1 . It is easy to compute the dependence of this amplitude on λ, µ, and ε. With E = N 1 f − external legs, we get from the χ internal sub-diagram a factor, (1.39),

λ ε1−Nf +(n+1)/nµ1/n; (1.43) while the N 1 anti-chiral 3-point vertices with 2 massless legs contribute a factor, (1.35) f − and (1.36), (λε−2+(n+1)/nµ−1+1/n)Nf −1; (1.44) and the N 1 mixed propagators at p2 = 0 give the factor, (1.33), f −

λ−1ε2−(n+1)/nµ−1/n 2(Nf −1). (1.45) | | Combining all these factors with (1.42), and recalling that n = N 2, gives f −

− − 4 2 2−Nf 1−Nf −1 u1v1···uNf 1vNf 1 d xd θλ µ µ ǫ (DδM 1u DδM 2v ) (DδM 1u − DδM 2v − ), 1 · 1 ··· Nf 1 · Nf 1 Z (1.46) which is ε-independent. This expression, up to a numerical factor, coincides with (1.27), the SU(2) SQCD higher-derivative F-terms of [8].

Since this was the only diagram contributing in the Nf = 4 case, and since there is only a single diagram in that case, there can be no cancellation of its coefficient. This shows that the Nf = 4 singular superpotential indeed reproduces the corresponding higher-derivative global F-term in perturbation theory. With some more work, this argument could be turned into a calculation of the value of the coefficient of the higher-derivative term. But since the normalization of the higher-derivative F-terms was not determined in [8], we are content to have simply shown that the coefficient is non-zero.

23 + + +

Figure 1.4: Diagrams for Nf = 5 ﬂavors.

N 5 As we go higher in the number of flavors, however, the number of diagrams f ≥ contributing to each amplitude increases. For instance, just among the class of internally purely-chiral diagrams illustrated in figure 1.3(a), there are are four super Feynman diagrams in the case of Nf = 5 flavors. As sketched in figure 1.4, we have one diagram with a single internal vertex, and three different combinations of a diagram with two internal vertices. Although we have shown above that the leading contribution of the sum of these diagrams has the right structure to reproduce the predicted higher-derivative F-term, since now multiple diagrams contribute, we must show in addition that no cancellations occur that could set the coefficient of the higher-derivative term to zero. This seems quite complicated, as it depends on the signs and tensor structures of the vertices. Some sort of symmetry argument is clearly wanted, but still eludes us. In addition, there are now also other classes of diagrams which are neither purely anti- chiral (as in figure 1.3(b)) or internally purely chiral (as in figure 1.3(a)). It is not clear whether these mixed diagrams will also contribute to higher-derivative amplitudes of the form (1.46) or not.

1.4 Singular superpotentials in three dimensions

It is worth mentioning that the method we developed here to get the moduli space of the theory from the singular superpotential (1.13) is not unique to four dimensions. In fact, as we will show below, the method can be used to obtain the moduli space of three dimensional supersymmetric gauge theories (with four supercharges) from singular superpotentials, wherever one is allowed to write such singular superpotenials. See, for example, [48, 49] for discussions of = 2 supersymmetric gauge theories in three dimensions. N Consider an = 2 SU(2) supersymmetric gauge theory in three dimensions with 2N N f light ﬂavors Qi , transforming in the fundamental representation where i = 1, , 2N and a ··· f a = 1, 2. Classically, the moduli space of the theory has a Coulomb branch as well as a

24 Higgs branch for N = 0. The Coulomb branch is parameterized by the vacuum expectation f 6 values of U = eΦ where Φ is a chiral superfield. The scalar component of Φ is φ + iσ, where φ R/Z is the scalar in the vector multiplet of the unbroken U(1) and σ σ +2πr is the ∈ 2 ∼ scalar dual to the gauge field. The Higgs branch is parameterized by the vacuum expectation values of V ij = ǫ Qi Qj. For N = 1, V ij is unconstrained while for N 2, V ij is subject ab a b f f ≥ to rank (M) 2, or equivalently ≤ ǫ V i1i2 V i3i4 =0, (1.47) i1...i2Nf just as in the four-dimensional case. The quantum global symmetry of the theory is SU(2N ) U(1) U(1) under which f × A × R the fields parametrizing the Coulomb and the Higgs branch transform as

SU(2Nf ) U(1)A U(1)R

U 1 2Nf 2(1 Nf ) (1.48) − − V ij 2(2N )2 0 . ∧ f

For N 2, the quantum Higgs branch is the same as the classical Higgs branch, i.e. it is f ≥ described by (1.47). We will be interested in the Higgs branch of the moduli space only for Nf > 2 where the global symmetry of the theory requires one to consider the singular superpotential [48] = (1 N )(U Pf V )1/(Nf −1). (1.49) W − f Although this superpotential is singular, it describes the moduli space perfectly for points away from the origin. (There are additional light degrees of freedom at the origin, which are not captured in (1.49).) To show this, we have to ﬁrst deform (1.49), then send the deformation parameters to zero at the end. In close analogy to what we did in four dimensions in section 2, we deform as follows:

1 ζ,η = W + ζU + η V ij, (1.50) W → W 2 ij

kl where ζ and ηij are some invertible parameters. The equations of motion for U and V yield

1/(N −1) ζ = U 2−Nf Pf V f , 1/(N −1) V kl = (U Pf V ) f (η−1)kl. (1.51) −

25 Solving the ﬁrst for U and substituting the result into the second gives an equation for V which can be solved to obtain

V kl = (ζ Pf η)1/2 (η−1)kl. (1.52) − Multiplying the above equation by itself and contracting the result with ǫ , we arrive i1...i2Nf at ǫ V i1i2 V i3i4 = ǫ ζ Pf η (η−1)i1i2 (η−1)i3i4 . (1.53) i1...i2Nf i1...i2Nf The right hand side of (1.53) is a polynomial of order N 2 > 0 for η and of order one f − ij for ζ. Therefore, independent of how we send ǫij and ζ to zero, the right hand side of (1.53) will vanish and we obtain ǫ V i1i2 V i3i4 =0, (1.54) i1...i2Nf which is exactly (1.47), the constraint equation describing the moduli space. This example gives some evidence that singular superpotentials can also perfectly-well describe the moduli space in supersymmetric gauge theories in three dimensions with four supercharges. A similar argument should also work to describe the moduli space for various (2, 2) supersymmetric gauge theories in two dimensions. However, unlike the situation in four dimensions, there is no range of flavors in these lower-dimensional theories where the theory is IR free. This makes a rigorous justification for the existence of the effective superpotentials of these theories harder to come by. In certain cases, like the example discussed above, the lower-dimensional theory can be obtained by compactification of a four-dimensional theory on a circle.

26 Chapter 2

Generalized Konishi anomaly and singular superpotentials of SU(N) SQCD

As discussed in chapter 1, despite much progress in the effective dynamics of four dimensional = 1 SQCD, the behavior of the effective superpotential for a large number of flavors N N f (large compared to the number of colors N) is not well-understood. This is because, firstly, when the number of flavors increases there are typically additional light degrees of freedom at the origin of the moduli space that one needs to include in the effective description. Secondly, the effective superpotentials become singular when expressed in terms of the local gauge- invariant light degrees of freedom away from the origin; more precisely, the potentials derived from such effective superpotentials have cusp-like singularities at their minima [10]. Thirdly, when the theory is asymptotically free, the dependence of these effective superpotentials on the strong coupling scale of the theory Λ is such that they apparently diverge in the the weak coupling limit Λ 0. Because of these problems the physical meaning of such → superpotentials is thought to be problematic. We have argued in chapter 1 that effective superpotentials for the light gauge-invariant degrees of freedom away from the origin must nevertheless exist [10]. Furthermore, direct computation in the case of SU(2) SQCD shows [10] that these superpotentials, although singular, are nevertheless physically sensible, and reproduce both the low energy physics as well as certain higher-derivative terms [8] in an intrinsic description on the moduli space away from the origin. In this chapter we extend our arguments to SU(N) SQCD by computing its singular

27 effective superpotentials. Unlike the SU(2) case, the SU(N) case has a smaller global symmetry group, making it harder to find the superpotentials. We deal with this by solving a system of differential equations for the effective superpotentials [43] derived from the generalized Konishi anomaly (GKA) equations [37]. The complexity of this system increases with the number of massless fundamental flavors Nf , but we are able to solve them in the first interesting case, Nf = N + 2. However, there are some subtleties involved in applying the GKA equations in this case: the GKA equations are not integrable when applied to the (strongly-coupled) direct description of the theory, but are integrable when applied to the (weakly-coupled) Seiberg dual description of the theory. Below, we will explain these subtleties in more detail and discuss the issues that they raise concerning the possible non-perturbative exactness of the GKA equations. We leave the technical details of the calculations to subsequent sections. The indirect argument for the existence of the effective superpotential referred to above goes as follows: Wilsonian effective superpotentials are assured to exist only if there is a region in the configuration space of the chosen chiral fields where all of them are light together and comprise all the light degrees of freedom. If this condition is satisfied, then the resulting effective superpotential can be extended over the whole configuration space by analytic continuation using the holomorphicity of the superpotential. For a large enough number of flavors the theory becomes IR free and the only region where all the components of the chosen chiral vevs become light at the same time is at the origin. We know what the light degrees of freedom are near that point since we have a weakly-coupled lagrangian description there. The physics can be made arbitrarily weakly coupled simply by taking all scalar field vevs φ << Λ where Λ is the strong coupling scale (or UV cutoff) of the IR free theory. h i In this limit the physics is just the classical Higgs mechanism, and all particles get masses of order φ or less. The Wilsonian effective description results from integrating out modes h i with energies greater than a cutoff, which we take to be some multiple of φ . The effective h i action will then include all local gauge-invariant operators made from the fundamental fields in the lagrangian and which can create particle states with masses below the cutoff. For the purpose of constructing the effective superpotential, the relevant local gauge-invariant operators are those in the chiral ring. It is then just a matter of constructing in the classical gauge theory a set of operators which generate the chiral ring. We will refer to this set as the classical chiral operators of the theory. In a weakly-coupled SU(N) SQCD a basis of local gauge-invariant operators in the chiral ring (the classical chiral operators) is comprised of just the glueball, meson, and baryon

28 operators [37, 38]. An effective superpotential which is a function of these operators must then exist. For Nf > N + 1, the quantum moduli space is also the same as the classical one [3], but effective superpotentials (singular or not) for these cases have not been found before. Also, this theory has an equivalent description in the IR in terms of a “Seiberg dual” SU(N N) SQCD with N (dual) fundamental quarks and anti-quarks and a set of singlet f − f scalars coupled to the dual mesons through a superpotential [4]. Note that the above argument does not directly show the existence of such effective superpotentials in the asymptotically free case. In particular, for theories in the “conformal 3 window” where neither the direct nor Seiberg dual description is IR free ( 2 N < Nf < 3N for SU(N) gauge group), we have no useful description of the light degrees of freedom at the origin of moduli space. Nevertheless, given an effective superpotential for an IR free theory, one can then successively add mass terms to the effective superpotential and integrate out massive flavors to derive consistent effective superpotentials in the conformal window. This then assures us that effective superpotentials exist for all numbers of light flavors in SQCD.

Our method for deriving the effective superpotentials for the Nf > N + 1 theory will be to integrate the generalized Konishi anomaly (GKA) equations [37] following the approach of [50, 43]. The resulting equations become very complicated [43] for large numbers of flavors, so we are not able to solve them directly in the IR free case N 3N, and then integrate f ≥ out flavors as in the above argument.

For Nf = N + 2, however, the GKA equations simplify to a first order matrix differential equation simple enough that we can analyze it. We show that the GKA equations for the effective superpotential are not integrable in this case for N > 2. This is not in direct conflict with the general arguments advanced above: for Nf = N + 2 and N > 2, the theory at the origin is strongly coupled in terms of its microscopic fields, so an effective description in terms of the chiral ring operators made from these fields simply need not exist. However, this failure of integrability presents a sharper puzzle in light of the following: we can nevertheless calculate an effective superpotential by using the fact that for Nf = N +2 with N 4 the Seiberg dual description is an IR free SU(2) gauge theory. By applying the ≥ GKA equations to the Seiberg dual description we derive the effective superpotential of the theory in terms of the dual chiral fields. It is given in equation (2.56) below, where we have used the map [4] between direct and dual chiral operators to interpret this as an effective i superpotential in terms of the classical chiral operators of the direct theory—the mesons Mj , ij baryons Bij and B˜ , and the glueball S.

29 We have thus found the effective superpotential in terms of the classical chiral operators. This raises the question of why were the GKA equations not integrable in the direct theory in the first place? We interpret this failure of integrability as indicating that the GKA equations get non-perturbative corrections. It remains to characterize more precisely the nature of the non-perturbative corrections to the GKA equations. One possibility is that there exists a non-perturbatively modified set of GKA equations in terms of the classical chiral operators (i.e. glueball, mesons, and baryons in our case). Another possibility is that there are additional operators in the chiral ring which are independent of the classical chiral operators; when included, they could render the GKA equations integrable, in much the same way that the extra singlet field in the Seiberg dual description does. It has not been ruled out that such additional fields could be seen in the semi-classical description as higher-derivative chiral operators.1 (Note that the non-trivial higher-derivative chiral operators constructed in [8] are not candidates, since they are Q-exact when extended off the moduli space to the configuration space of chiral vevs.) Finally, both possibilities—a non-perturbative deformation of the GKA equations and the inclusion of additional chiral fields—could occur together. Once the issue of non-perturbative corrections to the GKA equations is raised, it applies equally well to the GKA equations derived in the Seiberg dual description. It is an open question whether the effective superpotential derived below in the dual description is correct or not for N 6. For even though, when N 6, the dual description is weakly coupled at f ≥ f ≥ the origin of moduli space, it becomes strongly coupled an arbitrarily small distance away from the origin since the superpotential term in the dual theory destabilizes the free fixed point at the origin [4].

Outline of the chapter The remainder of the chapter carries out the computations described qualitatively above, and is organized as follows. To illustrate the method in a simple case first, and for later comparison to the Seiberg dual description of the SU(N) case, in section 2.1 we consider the case of SU(2) gauge group with N 4 flavors. We show how f ≥ the GKA equations written in terms of SU(2) mesons and baryons gives an effective superpotential matching that found in chapter 1 where we worked instead with the single antisymmetric meson field appropriate to an Sp(1) description of the theory (i.e. one which makes the enlarged global symmetry group of the SU(2) compared to the general SU(N) theory manifest). In section 2.2 we apply the GKA equations to SU(N) SQCD with Nf = N +2

1We thank S. Hellerman for discussions on this point.

30 and N 4. We show that the resulting equations for the effective superpotential are not ≥ integrable for N 6, but that they are integrable and match the SU(2) result of section f ≥ 2.1 for Nf = 4. In section 2.3 we apply the GKA equations to the Seiberg dual of the theory in section 2.2, and solve for the effective superpotential. The form of this superpotential is complicated: integrating out the heavy glueball gives an effective superpotential of the form √det Mf(X) where X = MBM˜ T B/det M, but a closed-form expression for f(X) is not found. Instead, we show that f obeys a nonlinear first order matrix differential equation (2.72). A power series expansion of the solution to order X4 is computed in (2.67). We then compare this result to the SU(2) effective superpotential of section 2.1 when Nf = 4, and show that they agree at least to order X4.

2.1 SU(2) singular superpotentials revisited

We show how to calculate the effective superpotential of SU(2) SQCD with N 4 using f ≥ the generalized Konishi anomaly equations, following [50, 43]. = 1 SU(2) SQCD has N massless quark and anti-quark chiral multiplets, Qi and N f a ã Qi , transforming in the 2 and 2 of the gauge group, respectively. Here a =1, 2 is the color index and i = 1,...,Nf the flavor index. The apparent global symmetry of the theory is SU(N ) SU(N ) U(1) U(1) . Since 2 and 2 are equivalent representations, though, f × f × B × R SU(2) SQCD actually has the larger symmetry group SU(2N ) U(1) , which is large f × R enough to determine the effective superpotential uniquely [10]. Here, since we are looking to the generalization to SU(N) which only has the smaller symmetry group, we will analyze the SU(2) case in terms of the Q’s and Q˜’s keeping only the SU(N ) SU(N ) U(1) U(1) f × f × B × R symmetry manifest. The classical moduli space is parameterized by the vevs of the meson M and the baryon , ˜ chiral superfields defined by B B M i := Qi Qã, ij := ǫabQi Qj, ˜ := ǫ QãQ˜b, S := tr(W αW )/(32π2), (2.1) j a j B a b Bij ab i j α where we have also defined the glueball chiral superfield S. These fields can be assigned the charges R(S) = 2 and R(M) = R(B) = R(B˜) = 2(N 2)/N under the non-anomalous f − f U(1)R symmetry. The meson and baryon vevs cannot take arbitrary values but are subject

31 to constraints following from (2.1),

[ijM k] = M i ˜ = [ij k]ℓ = ˜ ˜ = M [i M j] ij ˜ =0, (2.2) B ℓ [jBkℓ] B B Bi[jBkℓ] k ℓ − B Bkℓ where the square brackets denote antisymmetrization. These constraints imply that M, , B and ˜ all have rank less than or equal to 2 and, up to ﬂavor rotations, take the form B

m1 b b ˜ M =  m2  , =  b  , =  b  , (2.3) B − B − e 0 0 0            e  with m m = bb and 0 the (N 2) (N 2) matrix of zeros. For N 3, the quantum 1 2 f − × f − f ≥ moduli space is also the same as the classical one [3]. e For SU(2) SQCD with fundamental ﬂavors M, , ˜, and S are thought to generate all B B non-trivial local gauge-invariant operators in the chiral ring of the classical theory [37, 38].

When Nf = 4 or 5 the theory is strongly coupled, and has new massless degrees of freedom at the origin of moduli space, so the chiral ring might be deformed or enlarged from the classical answer. But for N 6, where the theory is IR free, the classical description is as f ≥ accurate as we like (in the vicinity of the origin of the moduli space). So we will make the assumption that we can write our effective superpotentials in terms of just S, M, , and ˜. B B However, for N 4, the global symmetries allow infinitely many terms in the effective f ≥ superpotential, making it hard to guess its correct form. So, instead, we use the generalized i Konishi anomaly (GKA) equations to derive the effective superpotential. If Fr (Φ, Wα) are holomorphic functions transforming in the same representation of the gauge group as a chiral i superfield Φr (i is a flavor index and r an index for the gauge representation), then the GKA equation [37] is i ∂ tree i 1 α s ∂Fs W F = (W Wα) , (2.4) ∂Φj r 32π2 t ∂Φj r t which can be interpreted as the anomalous Ward identity coming from the field transformation δΦi = F i. Here is the classical superpotential. The GKA equation is perturba- r r Wtree tively one-loop exact [37]. It has also been shown [44] that it does not get non-perturbative corrections for a U(N) gauge theory with matter in the adjoint representation as well as for Sp(N) and SO(N) gauge theories with matter in symmetric or antisymmetric representations. For the theories we are discussing here, its non-perturbative status is not known; however, as we show below, there is strong evidence, at least for SU(2), that the GKA

32 equations are actually non-perturbatively exact. Consider now SU(2) SQCD with the classical superpotential

= mi (Mˆ j M j )+ b ( ˆij ij)+ bij(˜ˆ ˜ ). (2.5) Wtree j i − i ij B − B Bij − Bij

Here mi , b and bij are Lagrange multipliers constraining thee operators Mˆ j , îj and ˜ˆ to j ij i B Bij j ij ˜ have M i, and ij as their vacuum expectation values, respectively. (Whenever we need B eB to distinguish an operator from its vev, we put a hat on the operator.) We are looking for the effective superpotential as a function of the vevs S, M, , and ˜. It follows from Weff B B (2.5) and the nature of the Legendre transform [51, 52, 5] that

i ∂ eff 1 ∂ eff ij 1 ∂ eff m = W , bij = W , b = W , (2.6) j − ∂M j −2 ∂ ij −2 ∂ ˜ i B Bij e where the factors of 2 come from the antisymmetry of the baryons. We now use the GKA equations to determine the dependence of the Lagrange multipliers on M, , ˜, and S. First set F i = Φi = Qi in (2.4) yielding B B r r a Mm = S +2 b, (2.7) B where we are using a matrix notation on the flavor indices (so that, e.g. the last equation stands for M i mk = Sδi +2 ikb ). A similar equation, k j j B kj mM = S +2b ˜, (2.8) B i i ã follows from taking Fr = Φr = Qi . Two more independente equations follow from taking i ˜b i i i ab i i ã Fr = ǫabQi and Φr = Qa, and from taking Fr = ǫ Qb and Φr = Qi , giving

˜m = 2M T b, m = 2bM T . (2.9) B − B − e We will carry out subsequent calculations at a generic point on the configuration space of S, M, and ˜ where they are all invertible matrices. Note, however, that when N is B B f odd, and ˜, being odd rank antisymmetric matrices, are never invertible. We get around B B this problem by restricting ourselves to an even number of flavors only. Once we have found the superpotential for even Nf ’s, we can add a mass term for one flavor and integrate it out to get the effective superpotential for the odd N 1 flavors. f − Therefore, multiplying (2.7–2.9) by appropriate inverses and substituting for the La-

33 grange multipliers m, b, and b using (2.6) gives a set of partial diﬀerential equations for

eff W e ∂ Weff = S ( ˜M −1 + M T )−1 ˜M −1 , ∂ ij B B B ij ∂ B h iij Weff = S M −1 ( ˜M −1 + M T )−1 , ∂ ˜ij B B B B h i i ∂ eff −1 ˜ −1 T −1 ˜ −1 −1 W j = S M ( M + M ) M M . (2.10) ∂M B B B B − j i h i Integrate the first equation in (2.10) to find

S = ln det I + M −T ˜M −1 + G(M, ˜,S), (2.11) Weff − 2 B B B where M −T =(M T )−1, I is the N N identity matrix, and G is an undetermined integration f × f function. Comparing the second equation in (2.10) with the derivative of (2.11) with respect to ˜ gives ∂G/∂ ˜ = 0. Also, comparing the derivative of (2.11) with respect to M to the B B third equation in (2.10) gives ∂G/∂M = SM −1, so that − G = S ln det (M/Λ2)+ H(S), (2.12) − for some undetermined function H(S). The Λ-dependence was determined by dimensional analysis, where Λ is the strong-coupling scale of the SU(2) SQCD. Equivalently, the global flavor symmetry implies that = (X, det M,S) where Weff Weff X := M −T ˜M −1 . Plugging this functional form into (2.6–2.9) gives simple matrix differ- B B ential equations leading to (2.11–2.12). H(S) is determined up to a constant by the U(1) symmetry. Since R( ) = 2, H R Weff must be linear in S, plus a logarithmic piece to cancel the U(1)R transformation of the S ln det M term, giving − H(S)=(2 N )S[α ln(S/Λ3)], (2.13) − f − for some undetermined constant α. We can determine α by matching to the Veneziano- Yankielowicz superpotential [45], W (S)=2S[1 ln(S/Λ3 )], for pure SU(2) SYM. It is a VY − YM short exercise to integrate out the mesons and baryons in (2.11) and match strong coupling

34 scales to find α = 1. We therefore find that the effective superpotential is

S = ln (det M)2det (I + M −T ˜M −1 ) +(2 N )S(1 ln S)+(6 N )S ln Λ. (2.14) Weff − 2 B B − f − − f h i Since S is massive we can integrate it out by solving its equation of motion, ∂ /∂S =0 Weff to find

1/(Nf −2) (M, , ˜)=(2 N ) ΛNf −6det M det (I + M −T ˜M −1 ) . (2.15) Weff B B − f B B q This superpotential reproduces all known low energy aspects of SU(2) SQCD. The easiest way to see this is to convert it to a description which makes the full global flavor symmetry i manifest. As we mentioned earlier, SU(2) SQCD with Nf fundamental Qa and Nf anti- fundamental Qã can be equivalently described in terms of 2N doublets I , I =1, , 2N i f Qa ··· f with i = Qi and Nf +i = ǫ Q˜b. Hence M, and ˜ are combined into an antisymmetric Qa a Qa ab i B B 2N 2N matrix V IJ = ǫab I J , f × f QaQb M V = B . (2.16) M T ˜ ! − B After a bit of algebra2 it is seen that our singular effective superpotential (2.15) can be written in terms of this new variable as

1/(Nf −2) = (2 N ) ΛNf −6√det V , (2.17) Weff − f making the SU(2Nf ) global symmetry manifest. Indeed, (2.17) coincides with the singular effective superpotential found in chapter 1, and so it satisfies all the checks discussed there: it gives rise to the correct moduli space, is consistent under integrating out flavors, and reproduces all the higher-derivative F-terms found in [8]. The success of this calculation can be taken as evidence that the GKA equations are non-perturbatively exact for SU(2) SQCD.

2 B 0 1 x −1 −1 T 1 x Write V = ˜ with x := M, y := ˜ M . Use the identity det = det (1 xy), so 0 B y 1 B −B y 1 − det V = det det ˜det (1 xy) = (det M)2det ( y−1x−1 + 1), which gives (2.15). B B − −

35 2.2 Singular superpotential for Nf = N+2 SQCD: non- integrability of GKA equations

= 1 SU(N) SQCD has N massless quark chiral fields Qi and N massless anti-quark N f a f ã chiral fields Qi transforming in the fundamental and anti-fundamental representations, respectively. Here i =1,...,Nf is the flavor index and a =1,...,N is the color index. When

Nf = N + 2 the classical moduli space is parameterized by the gauge-invariant vevs of the glueball, meson, and baryons deﬁned by

1 Sˆ := tr(W αW ), 32π2 α ˆ i i ã Mj := QaQj , 1 Bˆ := ǫ ǫa1···aN Qk1 QkN , ij N! ijk1···kN a1 ··· aN 1 îj ijk1···kN a1 aN B˜ := ǫ ǫa ···a Q˜ Q˜ . (2.18) N! 1 N k1 ··· kN The global symmetry of the theory is SU(N ) SU(N ) U(1) U(1) . The U(1) charges f × f × B × R R are R(S) = 2, R(M)=4/Nf , and R(B) = R(B˜)=2N/Nf . The classical moduli space is described by the constraints that M, B, and B˜ satisfy by virtue of their definitions,

B M k = M i B˜kj = B B = B˜[ijB˜k]ℓ = B˜ijB M −1[iM −1j] det M =0. (2.19) ik j k [ij k]ℓ kℓ − k ℓ Square brackets denote antisymmetrization; antisymmetrization on n indices consists of n! terms (i.e. with out a factor of 1/n!). They imply that by appropriate ﬂavor rotations M, B, and B˜ can be put in the form

m ˜ M =   , B =  b , B =  b , (2.20) b b    −   −       e where m is an N N matrix and b, b are numbers satisfying bb = dete (m). The classical × and the quantum moduli spaces are the same [3], but at the origin there are extra light e e degrees of freedom. At points away from the origin, the only light degrees of freedom are components of M, B, and B˜. At the origin, SU(N) SQCD with Nf = N +2 is an interacting superconformal ﬁeld theory for N = 2 and 3. For N 4 it becomes strongly coupled, but ≥ has an IR free dual description in the IR [4].

36 In this section we will try to construct an effective superpotential in terms of these fields which correctly describes the moduli space of vacua for points away from the origin. As in the last section, we will use the GKA equations to systematically derive . In fact, the Weff GKA equations were used in [43] to construct a set of coupled partial differential equations for the effective superpotentials of SU(N) SQCD. They have been integrated [43] for Nf = N and Nf = N + 1 where the results are in agreement with those in [3]. Unfortunately, the GKA equations are quite complicated for N N + 2 flavors. We will show below how to f ≥ simplify the GKA equations when Nf = N + 2. We briefly recap the derivation of the equations for from the GKA equations [43]. Weff The strategy is the same as in the SU(2) case discussed in the last section: start with the tree level superpotential

îj = mi (Mˆ j M j)+ bij(Bˆ B )+ b ( ˜B B˜ij), (2.21) Wtree j i − i ij − ij ij − where e i ∂ eff ij 1 ∂ eff 1 ∂ eff m = W , b = W , bij = W , (2.22) j j ˜ij − ∂Mi −2 ∂Bij −2 ∂B j ˜ e are Lagrange multipliers enforcing that Mi , Bij and Bij be the vevs of the meson, baryon, and anti-baryon operators, respectively. There is no need to introduce a Lagrange multiplier for S because we are considering points away from the origin and S is massive for these points. We get two relations among the Lagrange multipliers by taking Fr to be the quark or antiquark field in the GKA equation (2.4), and two more by taking it to be proportional

ijkℓ2···ℓN a2 aN to ǫ ǫaa ···a Q˜ Q˜ , and similarly with the Q’s. The resulting GKA equations are 2 N ℓ2 ··· ℓN M i mk = (S + bkℓB )δi 2bikB , k j ℓk j − kj mi M k = (S + b B˜ℓk)δi 2bikB˜ , k j kℓ j − kj m[jB˜kℓ] = 2bhgM −1[jM −1kM −1ℓ]det M, i e g h e i i −1g −1h −1i m[jBkℓ] = 2bhgM [jM kM ℓ]det M. (2.23)

Note that the right sides of the last twoe equations, though they are written using M −1, are actually polynomial in M. Now, the global ﬂavor symmetry implies that

M T BMB˜ = S f(X,S−2det M), where X := , (2.24) Weﬀ det M and we are using matrix notation for the meson and baryon ﬁelds. This symmetry argument

37 is not entirely straightforward. The U(1)B baryon number symmetry implies that for each B there must be an accompanying B˜ in each term. Since is an SU(N ) SU(N ) Weff f × f singlet, all the flavor indices must be contracted in each term. Contractions with the totally antisymmetric epsilon tensors can always be reduced to products of det M and Pf B Pf B˜. · The only other way to contract indices of B and B˜ is with an M as BMB˜ (or its transpose), and since these in turn must be contracted, another factor of M must be included. There are four ways of doing this—M T BMB˜ and its three cyclic permutations—but upon making a flavor singlet expression a trace must be taken, so the cyclic order does not matter. Finally, the product of Pfaffians of baryons is not independent of X and det M, since Pf B Pf B˜ = · det (M T BMB˜)/det M. q Alternatively, one can derive this directly from the GKA equations. Use them to deduce (2.26) and a similar relation for b, multiply these by B˜ and B, respectively, then substitute the second into the first. One finds that bB˜ depends on B and B˜ only through X. Since b ∂ eff /∂B˜, it follows that the dependence of eff on B and B˜ is solely through X. Note ∼ W e W that this symmetry argument is no longer effective when Nf > N + 2. For then B and B˜ e have more than two indices, and the analog of X is no longer a matrix, but has N N 1 f − − upper and N N 1 lower (antisymmetrized) indices. These objects can be contracted f − − in many inequivalent ways to make flavor singlets. This is the source of the difficulty in integrating the GKA equations for Nf > N + 2. The first two equations in (2.23) imply

= S W (X) S ln det M +2S(ln S 1), (2.25) Weff − − where W is to be determined. The GKA equations (2.23) imply a matrix differential equation for W (X) as follows. Contract the ijkℓ indices in the last equation in (2.23) with three M’s, giving 2(2 − N )b det M = tr(mM)M T BM + M T mT M T BM M T BMmM. Substitute for Mm using f − the first equation to get e N 2 4 2 b det M = S + − tr(bB) M T BM M T BbBM. (2.26) − N − N e

38 Derivatives of (2.25) with respect to B and B˜ together with (2.22) imply

˜ T T ˜ T 2 b det M = S (MBWX M + MWX BM ), T T T 2 b det M = S (WX M BM + M BMWX ), (2.27) where WX := ∂W/∂X. Worke at a generic point in parameter space where M, B, B˜ and X are all invertible matrices. This is only possible if Nf , and therefore N, is taken to be even, as in the discussion in the previous section. Substitute (2.27) in (2.26) and multiply on both left and right by B˜ to obtain

BG˜ (X)= GT (X)B,˜ (2.28) − where 1 N 2 2 G(X) := W X + 1+ − tr(W X) X + XW X, (2.29) X 2 N X N X and we have used that BX˜ is antisymmetric. On the other hand, from the deﬁnition of X (2.24) it follows that BX˜ = XT B˜, which implies

BG˜ (X)= GT (X) B˜ (2.30) since G is a function of X alone. (2.30) and (2.28) imply G(X) = 0, which, after being multiplied from the right by X−1, reads

1 N 2 2 W + 1+ − tr(W X) I + XW =0, (2.31) X 2 N X N X where I is the N N identity matrix. This is the matrix diﬀerential equation for W (X). f × f The trace of (2.31), tr(W X)= [2 tr(W )+N +2]/N, allows us to eliminate tr(W X) X − X X from (2.31), giving NW = (N +2X)−1[(N 2) tr(W ) 2]. The trace of this equation X − X − allows us to eliminate tr(WX ) in turn, giving the following diﬀerential equation for W (X):

2(N +2X)−1 W = . (2.32) X (N 2) tr((N +2X)−1) N − − When N = 2, we deﬁne the matrix Y := (N 2)(N +2X)−1, and substitute it into (2.32) 6 − to obtain ∂W 1 Y i = δi , (2.33) ∂Y j k N tr(Y ) j k −

39 where we have explicitly written the indices to avoid any confusion. This diﬀerential equation is not integrable, as it is easy to check that ∂2W/∂Y k∂Y i = ∂2W/∂Y i∂Y k. This shows that l j 6 j l the GKA equations for are not integrable for even values of N > 2. Weﬀ

2.2.1 Comparison to the SU(2) solution.

For N = 2 we are integrating the same GKA equations as we did in section 2.1, though in terms of the baryons and anti-baryons B and B˜ instead of their Hodge duals and ˜. Thus B B the GKA equations must be integrable in this case, and, of course, give the same answer we found in section 2.1, namely, equation (2.15) with Nf = 4. However, there is a subtlety in comparing these two computations, which we will now explain. It will play an important part in our discussion of the results of integrating the Seiberg dual GKA equations in the next section. For N = 2, (2.31) is indeed integrable, and integrates to give W (X)= 1 trln(1 + X). − 2 Integrating out S from (2.25) then gives

Λ = 2√det Mdet (1 + X)1/4, (2.34) Weﬀ,SU(2) − where Λ is the strong coupling scale of the gauge theory. On the other hand, the SU(2) eﬀective superpotential for Nf = 4 found in section 2.1 is

Λ = 2√det Mdet (1 + M −T ˜M −1 )1/4. (2.35) Weff,SU(2) − B B Since the SU(2) baryon fields and ˜ as defined in (2.1) are Hodge-dual to the SU(N) B B baryons B, B˜ defined in (2.18), and since rank(M)= Nf = 4, it follows that

1 M −T ˜M −1 = tr(X) X, (2.36) B B 2 − so that the eﬀective superpotential reads

Λ = 2√det Mdet (1 + 1 tr(X) X)1/4. (2.37) Weff,SU(2) − 2 − Apparently the two answers, (2.34) and (2.37), do not agree for general X. The resolution of this paradox is that X is not a general 4 4 complex matrix, but × satisfies some constraints by virtue of its definition (2.24). In particular, X can be thought of as the product of two antisymmetric matrices (M T BM and B˜). Such a matrix, though

40 not necessarily either symmetric or antisymmetric, has only half the degrees of freedom that a general matrix, Y , of the same rank would have. To see this, recall that an appropriate similarity transformation G−1YG with G ∈ GL(2n, C) will diagonalize Y . Thus the 2n eigenvalues of the general rank 2n matrix Y are all its GL(2n, C)-invariants. More properly, a basis of 2n independent symmetric polynomials of these eigenvalues generates all invariants. This basis can conveniently be taken to be tr(Y p) for p = 1,..., 2n. If, on the other hand, X = AB is the product of two antisymmetric matrices A and B, then a GL(2n, C) transformation acts as −1 −1 −1 −T T G XG = G ABG = G AG G BG. We can always choose a G = G0 so that GT BG = J where J := I iσ is the “unit” skew-diagonal matrix. This condition does 0 0 n ⊗ 2 not fix G, since if H Sp(2n, C) (i.e. H GL(2n, C) satisfies HT JH = J), then G = G H ∈ ∈ 0 will also give GT BG = J. For such G we have G−1XG = H−1A′H−T J where A′ is the an- ′ −1 −T tisymmetric matrix A = G0 AG0 . Now, a Gram-Schmidt othogonalization argument but with respect to the skew product defined by J shows that H can always be chosen to bring H−1A′H−T to skew diagonal form H−1A′H−T = diag a ,...,a iσ . Thus X has only { 1 n} ⊗ 2 n independent (double) eigenvalues, and a basis of generators of the GL(2n, C)-invariants of X can be taken to be tr(Xp) for p = 1,...,n. This is half the number of independent invariants of the general rank 2n matrix Y , and implies, in particular, that tr(Xp) for p > n satisfy additional identities allowing them to be expressed in terms of products of traces with p n. ≤ For example, for Nf =4(n = 2) the new independent cubic and quartic identities are easily found to be

0 = 8tr X3 6tr X2 tr X + (tr X)3, − 0 = 8tr X4 4tr X3 tr X 2(tr X2)2 + tr X2(tr X)2. (2.38) − −

p ′p When Nf = 4, the identities (2.38) are easily checked to imply that tr(X ) = tr(X ) for p = 1,..., 4, where X′ := 1 tr(X) X. Thus all invariants made from X′ are the same as 2 − those for X, and in particular, the two forms of the eﬀective superpotential (2.34) and (2.37) are equivalent.

41 2.3 Singular superpotential for Nf = N+2 SQCD: Seiberg dual analysis

The IR-equivalent Seiberg dual description [4] of =1 SU(N) SQCD with N = N + 2 is N f a i SU(2) SQCD with Nf (dual) quarks qi in the fundamental and Nf (dual) anti-quarksq ã in the anti-fundamental, and a set of gauge singlets ˆ i coupled through the superpotential Mj =q ˜i qa ˆ j, (2.39) W a j Mi where i =1,...,Nf and a =1, 2 are flavor and color indices, respectively. This superpotential breaks the global symmetry of the dual theory down to SU(N ) SU(N ) U(1) U(1) . f × f × B × R The dual meson, baryons, and glueball are defined to be

ˆij ˆ i :=q ˜i qa, ˆ := ǫ qaqb, ˜ := ǫabq˜i q˜j, := tr(wαw )/(32π2). (2.40) Nj a j Bij ab i j B a b S α This dual theory is IR free when N 6(N 4), and there are free quarks, anti-quarks and f ≥ ≥ gluons at the origin of the moduli space. The U(1) charges are R( ) =2, R( )=4/N , R S M f and R( ) = R( ) = R( ˜)=2N/N . The chiral ring of the dual theory ( , , , ˜) is N B B f S M B B related to that of the direct theory (S,M,B, B˜) by [5]

S = , M = µ , B = iµ−1ΛNf −3 , B˜ = iµ−1ΛNf −3 ˜, (2.41) −S M B B where µ is a matching scale deﬁned by

Λ˜ 6−Nf = µNf Λ6−2Nf , (2.42) with Λ and Λ˜ being the dynamical scales of the direct and dual theories, respectively. The gauge-invariant form of the classical F-term equations are

= = ˜ =0, (2.43) N BM MB where we are using a matrix notation for the ﬁelds. The D-term equations give

= ˜ = = ˜ ˜ = ˜ =0, (2.44) B∧N N ∧ B B ∧ B B ∧ B B ⊗ B−N∧N which are the same as the constraints (2.2) of SU(2) SQCD discussed in section 2.1 with the substitution M . The space of solutions to the classical constraints (2.43–2.44) has →N

42 two branches: either = = ˜ = 0 and rank( ) > N, or, up to flavor rotations, N B B M m =0, = , = b , ˜ = b , (2.45) N M   B   B   b b    −   −       e where m is an N N matrix and bb = 0. e × This classical moduli space is clearly not the same as the moduli space (2.19–2.20) of the e direct SU(N) theory. However, the dual theory classical constraints (2.43–2.45) are expected to be modified quantum mechanically by strong coupling effects. Because even though the dual theory is free at the origin of moduli space, arbitrarily small vevs for destabilize M the free fixed point by giving masses to the dual quarks through the superpotential coupling (2.39), so the theory flows to strong coupling for large enough rank of . Strong coupling M effects are argued in [4] to generate the constraints (2.19) of the direct theory. In particular, the branch of (2.43–2.45) with rank( ) > N is lifted, and the bb = 0 M constraint on the other branch is deformed to bb = det (m). As an example—which will be e useful later—of how the classical constraints are modified by strong coupling effects, note e that if the singlet vev is given a generic value (e.g. by constraining it with a Lagrange M multiplier term as we will do below) with = ˜ = 0, the superpotential (2.39) gives mass B B to all the dual quarks. The dual theory thus flows to SU(2) SYM in the IR with glueball vev =(Λ˜ 6−Nf det )1/2 generated by a Veneziano-Yankielowicz superpotential S M

=2 1 ln Λ˜ (Nf −6)/2/√det , (2.46) Weff,VY S − S M h i where Λ˜ is the strong-coupling scale of the dual theory. The scale of the glueball appearing this superpotential is determined by one loop matching Λ˜ 6 = Λ˜ 6−Nf det . Integrat- YM M ing out of (2.46) then generates an effective superpotential for the singlet field given by S 2Λ˜ (6−Nf )/2√det , thus lifting the rank( )= N region of the classical moduli space. M M f

2.3.1 Derivation of the eﬀective superpotential

We now wish to find an exact effective superpotential for the dual theory which reproduces the strong quantum effects described in the last two paragraphs. As in previous sections we

43 start with a tree level superpotential

îj = ˆ i ˆ j + mi ( ˆ j j)+ bij( ˆ )+ b (˜ ˜ij), (2.47) Wtree Nj Mi j Mi −Mi Bij − Bij ij B − B where e i ∂ eff ij 1 ∂ eff 1 ∂ eff m = W , b = W , bij = W , (2.48) j −∂ j −2 ∂ ij −2 ∂ ˜ij M i B B are the by-now familiar Lagrange multipliers. We have onlye specified the vevs of ˆ , ˆ, and M B ˜ˆ, because they completely parametrize the moduli space. B Using the GKA equations (2.4) in precisely the same way as before, we find a set of equations similar to those, (2.7–2.9), found in section 2.1,

= +2b , = +2 ˜b, ˜ T = 2 b, T = 2b , (2.49) MN S B NM S B BM − N M B − N together with the equation coming from the variation of the singlet field : e M = m. (2.50) N − Eliminating by plugging (2.50) into (2.49) gives a set of partial differential equations for N the effective superpotential

m = +2b , m = +2 ˜b, ˜ T =2mb, T =2bm. (2.51) −M S B − M S B BM M B e e To solve for the effective superpotential we multiply the third equation in (2.51) from the right by and from the left by to obtain ˜ T =2 mb = 2( I +2b )b , B M MBM B M B − S B B where in the second equality we used the first equation in (2.51) to eliminate m. Solving M this quadratic equation for b , we find B 1 1 b = + 2 4 ˜ T , m = 2 4 ˜ T , B 4 −S S − MBM B M 2 −S − S − MBM B 1 p 1 p ˜b = + 2 4 ˜ T , m = 2 4 ˜ T , (2.52) B 4 −S S − BM BM M 2 −S − S − BM BM p p wheree the second line comes from solving similar quadratic equations for ˜b and m . B M The matrix square roots in the above expressions need some explanation. In order to e make sense of them, consider a region in the configuration space where the magnitude of each element in the matrix ˜ T is much smaller than 2. We can then expand the MBM B S the square root as a power series, 2 4 ˜ T = √I(I 2 −2 ˜ T + ), and S − MBM B S − S MBM B ··· p 44 by analytic continuation we extend the result to include all points on the parameter space. However, for this to be a definition of the square root, we still need to determine √I. In general √I = 2P I where P can be any projection matrix (P 2 = P ). Which P should − we use? The following argument shows that we have to take √I = I. As we saw in the ± discussion surrounding (2.46), we expect to generate a non-zero at points in the parameter S space where = 0 but = ˜ = 0, so this is a suitable region to evaluate the square roots. M 6 B B At such points (2.51) implies that m = m = and ˜ T = T = 0. But this is M M −S BM M B only consistent with (2.52) if √ 2I = I implying that √I = I. S S ± It is straightforward to integrate (2.52) for the effective superpotential to get

2 ˜ T 4 −6 T −1 eff = S ln det S − MBM B − S S ln det (Λ˜ ˜ )+ ln det (Λ˜ ) W 4 p 2 4 ˜ T + ! − 4 MBM B S M S − MBM B S 1 1 + tr p2 4 ˜ T + (4 N ) α ln( /Λ˜ 3) . (2.53) 2 S − MBM B 2 − f S − S p h i We used the U(1) symmetry to fix the dependence up to an undetermined integration R S constant α. Evaluating at = ˜ = 0 gives, after a somewhat delicate cancellation, Weff B B

Nf ( = ˜ =0)=2 α + (1 α) ln Λ˜ (Nf /2)−3/√det . (2.54) Weff B B S − 4 − S M h i Comparing to the answer (2.46) expected from the strong coupling analysis, fixes α = 1. Note that this limiting form (2.54) is already a check that the effective superpotential (2.53) is consistent with the quantum modified constraints. In particular the ln det term in S M (2.54) serves to lift the whole rank = N region of the classical moduli space (2.43–2.45), M f in accordance with the expected quantum constraints (2.19–2.20). So, our final result for the effective superpotential for the Seiberg dual theory is

2 ˜ T 4 T eff = S ln det S − MBM B − S S ln det ˜ + ln det W 4 p 2 4 ˜ T + ! − 4 MBM B S M S − MBM B S 1 + tr p2 4 ˜ T + (4 N )S (1 ln )+(6 N ) ln Λ˜. (2.55) 2 S − MBM B − f 2 − S − f S p Using the chiral ring mappings (2.41), this can be expressed in terms of the fields of the direct theory. Note that since S = , the definition of the branch of the square root made above, −S √ 2I = I, now becomes √S2I = SI. We make this minus sign explicit by changing the S S −

45 signs of all square roots and keeping the convention √S2I =+SI. We then ﬁnd

s √s2 +4x + s det x 1 s √ 2 weﬀ = ln det 4 tr s +4x +(Nf 4) [1 ln s] , (2.56) 4 " √s2 +4x s! det M # − 2 − 2 − − where we have deﬁned the shorthands

w := ΛNf −3 , s := ΛNf −3S, x := MBM˜ T B, (2.57) eff Weff to avoid having to write factors of Λ. This is the effective superpotential for four dimensional =1 SU(N) SQCD with N = N + 2. N f

2.3.2 Integrating out the glueball ﬁeld

Away from the origin of moduli space, the glueball s is expected to be massive. Solving its equation of motion, ∂weﬀ /∂s = 0, gives s = s∗(M,B, B˜) where s∗ is deﬁned implicitly by

2 s +4x + s∗ det x s2(Nf −4) = det ∗ . (2.58) ∗ 2 4 s∗ +4x s∗ ! det M p − p Substituting this in weff gives the effective superpotential as a function of meson and baryons only, 1 x w ∗ = 2s 1+ tr 1+4 1 . (2.59) eff |s − ∗ 4 s2 − r ∗ The relation (2.58) gives s∗ as a complicated function of M, B, and B˜, which makes it difficult to deduce the equations of motion for these fields from weff .

Before solving (2.58) for s∗, we can extract some of the constraints that M, B, and

B˜ must satisfy on the moduli space. Since weff is singular, as shown in [10] it must be regularized in order to extract its physical predictions. The idea behind the regularization is to deform by introducing some regularizing parameters µ, β, β˜ as follows Weff ˜ w wµ,β,β = w + µi M j + βijB + β˜ B˜ij. (2.60) eff → eff eff j i ij ij Generic small values of µ, β, and β˜, will fix M, B, and B˜ to some values which will be shifted from the moduli space (the extrema of weff ) by positive powers of the regularizing parameters. Thus as µ,β, β˜ 0, M,B, B˜ will approach some point on the moduli space. → { } However, the specific point reached on the moduli space will depend on how the regularizing

46 parameters scale to zero: as different orders of limits of µ,β, β˜ 0 are taken, the whole → moduli space will be scanned. (Note that some vanishing limits of the regularizers can also send M, B, or B˜ to infinity; this is unavoidable since the moduli space itself stretches off to infinity.) This regularization procedure should be compared with the trick we used of introducing

Lagrange multipliers (2.47) to derive equations for weff from the GKA equations. With the replacement of the Lagrange multipliers with the regularizing parameters, m, b, b { } → ˜ µ,β, β , the identification of the Lagrange multipliers as derivatives of weff , (2.48), is now { } µ,β,β˜ e interpreted as the regularized equations of motion. Thus, extremizing weff using (2.56) gives

4βB = s √s2 +4x, 2Mµ = s + √s2 +4x, − 4B˜β˜ = s s2 +4y, 2µM = s + s2 +4y, (2.61) − p p where y := M −1xM. (2.61) is equivalent to (2.52), the original differential equations—with the operator mappings (2.41)—we integrated to get weff in the first place. Then, reversing the manipulations which led from (2.51) to (2.52) gives

Mµ = s 2βB, µM = s 2B˜β,˜ BM˜ T = 2µβ, M T B = 2βµ.˜ (2.62) − − − − These show that independent of how µ,β, β˜ 0, we always end up with BM˜ T = M T B = → 0. This implies in particular that x = 4βµT βµ˜ , and so also always vanishes with the − regularizing parameters. Also, if as µ,β, β˜ 0 M, B, and B˜ remain ﬁnite, then s 0 as → → well. This shows that the extrema of the superpotential (2.56) satisfy the constraints BM˜ T = T M B = s = 0, which are, indeed, part of the constraints (2.19) describing the Nf = N + 2 SQCD moduli space. The remaining constraints should also follow from the eﬀective superpotential. However, they are much harder to derive, as they require solving for s∗ in

(2.58). Now the argument of the last paragraph shows that s∗ = 0, but, because of the singular nature of the superpotential, it is incorrect to simply plug this value into (2.59) to

find weff . Instead, s∗ should be found for generic M, B, and B˜, and then the extrema of the resulting effective superpotential can be analyzed by regularizing it, as above.

The equation (2.58) can be solved systematically for s∗ by assuming that (all the eigenvalues of) ξ := x/s2 1 and expanding in powers of this parameter. The leading order ∗ ≪ 2 solution is s∗ = det M, which can be checked to be consistent with the assumption that

47 ξ 1. Change variables to || || ≪ s2 x σ2 := ∗ , X := , (2.63) det M det M so that (2.58) and (2.59) become

√σ2 +4X + σ 2 σ2(Nf −4) = det X det =2−2Nf det √σ2 +4X + σ , (2.64) √σ2 +4X σ ! − 1 w = 2√det M σ + tr √σ2 +4X σ . (2.65) eﬀ − 4 − Expand the right side of (2.64) in powers of X/σ and solve it consistently order-by-order in a power series expansion σ2 =1+ to get ··· tr X (tr X)2 3tr(X2) (tr X)3 3tr X tr(X2) 5tr(X3) σ2 = 1 + + (2.66) − 2 − 8 4 − 12 4 − 3 9(tr X)4 27(tr X)2 tr(X2) 27 tr(X2)2 5tr X tr(X3) 35 tr(X4) + + + (X5). − 128 32 − 32 − 2 8 O Plugging this into (2.65) gives

tr X (tr X)2 tr(X2) 5(tr X)3 3tr X tr(X2) tr(X3) w = 2√det M 1+ + + + eﬀ − 4 32 − 8 384 − 32 6 h 49(tr X)4 21(tr X)2 tr(X2) 9tr(X2)2 5tr X tr(X3) 5tr(X4) + + + 6144 − 256 128 24 − 16 + (X5) . (2.67) O i 4 This is the answer, to order X , for the eﬀective superpotential for Nf = N + 2 SQCD found by integrating the Seiberg dual GKA equations and integrating out the glueball.

Alternatively, we can derive a differential equation satisfied by weff as a result of integrating out s. Two of the equations of motion that we originally integrated to get the superpotential (2.61) become, when written in terms of σ and X,

4βB = √det M σ √σ2 +4X , − 2Mµ = √det M σ + √σ2 +4X . (2.68) On the other hand, upon integrating out s, we have seen that weﬀ takes the form

weﬀ = √det M f(X) (2.69)

48 for some function f. (This form just follows from the symmetries.) Two of the equations of motion following from this form of weﬀ are

4βB = √det M ( 4Xf ′ ) , 2Mµ = √det M ( 4Xf ′ f + 2tr(Xf ′)) , (2.70) − − where f ′ is the matrix derivative df/dX. Equating and adding (2.68) and (2.70) gives

1 σ = f + tr(Xf ′), (2.71) −2 while equating and multiplying (2.68) and (2.70) gives

1= f ′ (f 2tr(Xf ′)+4Xf ′) . (2.72) −

This is a nonlinear first order (matrix) differential equation for weff with the glueball integrated out. We do not know how to integrate this equation in closed form when rank(X) > 1. But it is straightforward to check that a series expansion of the solution to (2.72) with boundary condition f(0) = 2 reproduces (2.67). − It may be clarifying to note that (2.72) has a one-parameter family of solutions. For if f(X) is a solution, then so is −2 f(a)(X) := af(a X) (2.73) for any a C∗. By (2.71) this implies that σ(X) changes to aσ(a−2X); but it is easy to check ∈ that the σ equation of motion (2.64) does not have this symmetry. The algebraic equation (2.64) determining σ has more information than the differential equation (2.72), and so picks out a single instance of the family of solutions (2.73), namely the one obeying the boundary condition f(0) = 2. (For example, 2tr(√2X) solves (2.72) but does not satisfy (2.64), − − so is not a physical solution.)

2.3.3 Comparing to the direct result when Nf = 4

Is (2.67) the correct superpotential? A basic check is to see whether its extrema (computed by appropriately regularizing, as explained above) reproduce the moduli space given by the constraints (2.19). This seems a very diﬃcult check to perform since we do not have a closed analytic form for weﬀ .

49 However, there is one case where we can carry out a non-trivial check. When Nf = 4, N = 2, so in this case we should reproduce the superpotential (2.34) found in sections 2.2.1 for the SU(2) theory. Expanding (2.34) in powers of X, we ﬁnd

tr X (tr X)2 tr(X2) (tr X)3 tr X tr(X2) tr(X3) Λ = 2√det M 1+ + + + Weff,SU(2) − 4 32 − 8 384 − 32 12 h (tr X)4 (tr X)2 tr(X2) tr(X2)2 tr X tr(X3) tr(X4) + + + 6144 − 256 128 48 − 16 + (X5) . (2.74) O i Though it does not coincide with the expansion (2.67) starting at order X3, we must bear in mind the identities (2.38) that traces of powers of X satisfy starting at cubic order. One finds that the difference between (2.74) and (2.67) is proportional to these identities, and so vanishes. Thus the effective superpotential found by integrating the dual GKA equations matches the correct result at least to quartic order in X.

50 Chapter 3

Singular superpotentials and higher-derivative F-terms of Sp(N) SQCD

A special kind of higher-derivative terms were calculated in [8] for four dimensional =1 N SU(2) SQCD with N 2 fundamental flavors using one-instanton methods. In chapter 1 f ≥ we computed these terms by integrating out massive modes at tree-level from an effective superpotential. These superpotentials are more singular than those normally considered: the potentials derived from them have cusp-like singularities at their minima. However, these singularities are mild enough that they unambiguously define the moduli space of vacua, and can be dealt with analytically by means of a simple regularization procedure.

Outline of the chapter In section 3.1 we first compute the singular effective superpotentials of four dimensional =1 Sp(N) SQCD with matter in the fundamental representation, N then we show that they correctly describe the moduli space of vacua, are consistent under RG flow to fewer flavors upon turning on masses, and are consistent solutions to the Konishi anomaly equations [39, 40]. Then, in section 3.2, we generalize the results of [8] to Sp(N) SQCD by expanding the superpotential around a generic vacuum and integrating out the massive modes of the meson field at tree level to find new higher-derivative F-terms.

51 3.1 Singular superpotentials of Sp(N) SQCD

3.1.1 Sp(N) SQCD for a small number of ﬂavors

Consider a four dimensional = 1 Sp(N) SQCD gauge theory with 2N massless quark N f i chiral fields Qa transforming in the fundamental representation, where i = 1,..., 2Nf and a = 1,..., 2N are flavor and color indices, respectively. (The number of flavors must be even for global anomaly cancellation [41].) The anomaly-free global symmetry of the theory is SU(2N ) U(1) under which the quarks transform as (2N , (N N 1)/N ). f × R f f − − f The classical moduli space of vacua is the space of vevs of holomorphic gauge-invariant ˆ [ij] i ab j chiral fields. For Sp(N) SQCD these are the anti-symmetric meson fields M = QaJ Qb, where J ab is the invariant antisymmetric tensor of Sp(N). (We distinguish vevs from operators by hatting operators.) For Nf < N the classical moduli space is the space of arbitrary meson vevs M ij, while for N N it is the set of all M ij subject to the condition f ≥ rank(M) 2N, or equivalently ≤

i1i2 i2N+1i2N+2 ǫi ···i M M =0. (3.1) 1 2Nf ···

Quantum mechanically there is a dynamically generated superpotential [9]

(N +1 N )(Λb0 /Pf M)1/(N+1−Nf ), for 0 < N N, − f f ≤ b eff = Σ(Pf M Λ 0 ), for N = N +1, (3.2) W  − f  (Pf M/Λb0 ), for N = N +2, − f   i1i2 i2N −1i2N where the Pfaffian is defined as Pf M := ǫi ···i M M f f = √det M, b0 = 3(N + 1 2Nf ··· 1) N is the coefficient of the one-loop beta function, Λ is the strong-coupling scale of − f the theory, and Σ is a Lagrange multiplier. These superpotentials encode the low energy behavior of the gauge theory: for N N all the classical flat directions are lifted, for f ≤ Nf = N + 1 instantons deform the classical moduli space, while for Nf = N + 2 the classical moduli space is not modified.

52 3.1.2 Superpotentials and classical constraints for a large number of ﬂavors

For Nf > N +2 the classical constraints are not modiﬁed, though there are new light degrees of freedom at singular subvarieties of the moduli space when the theory is asymptotically free, Nf < 3N +3. These singular subvarieties are commonly refered to as the “origin” of the moduli space. The only eﬀective superpotential (for points away from the origin) consistent with holomorphicity, weak-coupling limits, and the global symmetries is [9]

Pf M 1/n eff = n , n := Nf N 1 > 1. (3.3) W − Λb0 − − The fractional power of Pf M implies that the potential derived from this superpotential has cusp-like singularities at its extrema. We will devote the rest of this section to arguing that, nevertheless, these singular superpotentials are physically perfectly sensible. The first thing to check is to see whether these singular superpotentials describe the moduli space of vacua. Because these superpotentials are singular at their extrema we cannot just naively extremize them. We get around this problem by first deforming the superpotentials using some regularizing parameters εij, extremizing them, then taking the ε 0 limit at the end. Independent of how the regularizing parameters are sent to ij → zero, the extrema of the superpotentials must reproduce the classical constraint (3.1). The superpotentials (3.3) indeed pass this check, as we now show. We regularize (3.3) by adding a mass term with an invertible antisymmetric mass matrix

εij for the meson fields 1 ε := + ε M ij. (3.4) Weff Weff 2 ij We have chosen to deform by a linear term in M ij because it is simple, it smooths the Weff singularity, and it does not dominate at large M, so does not introduce additional “spurious” extrema. We could have chosen a different deformation. Varying ε with respect to M kl Weff yields the equation of motion

M kl = Λ−b0/n(Pf M)1/n(ε−1)kl. (3.5) −

Solving (3.5) for Pf M in terms of εij and substituting back, we obtain

M kl = Λ−b0/(N+1)(Pf ε)1/(N+1)(ε−1)kl. (3.6) −

53 Multiplying N + 1 copies of (3.6) together, and contracting the result with ǫ , we arrive i1...i2Nf at

N+1 i1i2 i2N+1i2N+2 ( 1) −1 i1i2 −1 i2N+1i2N+2 ǫi1...i M M = − ǫi1...i (ε ) (ε ) Pf ε. (3.7) 2Nf ··· Λb0 2Nf ···

The right hand side of the above expression is a polynomial of order n > 0 in the εij. Therefore, in the ε 0 limit it vanishes independently of how we take the limit and we ij → have

i1i2 i2(N +1)−1i2(N +1) ǫi ...i M M c c =0, (3.8) 1 2Nf ··· which is exactly the classical constraint that we wanted. Furthermore, it is easy to check that all solutions of the classical constraints can be reached by taking ε 0 appropriately. ij → Note that the negative power of Λ appearing in (3.3) is not inconsistent with the weak coupling limit because the constraint equation (3.8) which follows from extremizing the singular superpotential implies that Pf M = 0, thus vanishes on the moduli space for Weff any finite value of Λ as well as in the Λ 0 limit. → We present another way of seeing how the classical constraints emerge from the singular superpotential which might make it clearer why these superpotentials have unambiguous extrema. Use the global symmetry to rotate the meson fields into the skew diagonal form

M1  M2  M ij = iσ , (3.9) .. ⊗ 2  .     M   Nf    so the eﬀective superpotential (3.3) becomes w = nΛ−b0/n( M )1/n. The equations of eﬀ − i i motion which follow from extremizing with respect to the Mi areQ

1 1 n −1 n Mi Mj =0. (3.10) Yj6=i

Though these equations are ill-defined if we set any of the Mi = 0, we can probe the solutions by taking limits as some of the Mi approach zero. To test whether there is a limiting solution where K of the M vanish, consider the limit ε 0 with M εα1 ,...,M εαK with i → 1 ∼ K ∼ αj > 0 to be determined. Note that different non-zero values αj corresponds to different deformations in (3.4). Substituting into (3.10), only the first K equations have non-trivial

54 limits, 1 α −α lim ε n ( j j ) i =0, i =1,...,K, (3.11) ε→0 giving the system of inequalities nαi < j αj for i = 1,...,K. These inequalities have solutions if and only if K > n, implying that rank(M) 2N which is precisely the classical P ≤ constraint (3.1).

3.1.3 Consistency with Konishi anomaly equations: direct description.

In the previous section the eﬀective superpotential of the theory was determined by the global symmetry, weak-coupling limits, and holomorphicity. In this section we use the Konishi anomaly equations to derive the same superpotentials (3.3). We will show, using both the direct description and Seiberg dual description [4, 9] of the theory, that the solution to the Konishi anomaly equations coincides with our singular superpotentials. This is a consistency check on these superpotentials. As mentioned in the previous chapter, it has been shown [46] that for a SYM theory ˆ 1 α with the Sp(N) gauge group, the glueball superﬁeld S = 32π2 tr(W Wα) generates all the local gauge-invariant chiral operators in the chiral ring of the theory. When we add matter multiplets to a SYM theory we also need to include local gauge-invariant matter generators. Following the arguments of [37, 38] it follows that Sˆ and Mˆ ij comprise all the local gauge- invariant chiral generators in the chiral ring. In the chiral ring the Konishi anomaly for a tree-level superpotential Wtree is

∂Wtree j j i Qa = Sδi . (3.12) h ∂Qa i The above set of equations are perturbatively one-loop exact and do not get non-perturbative corrections. See [44, 10, 11] for discussions on the non-perturbative exactness of the Konishi anomaly equations. As our tree-level superpotential, we take

= m (Mˆ ij M ij), (3.13) Wtree ij −

ij ij where mij is a Lagrange multiplier enforcing Mˆ to have M as their vevs. It follows from the form of the above tree-level superpotential and the nature of the Legendre transform

55 [5, 51, 52] that 1 ∂ m = Weff . (3.14) ij −2 ∂M ij Substituting (3.13) into (3.12) and using the fact that the expectation value of a product of gauge-invariant chiral operators equals the product of the expectation values, gives kj j 2mikM = Sδi . Using (3.14) we then obtain a set of partial differential equations for the effective superpotential ∂ Weff M kj = Sδj, (3.15) ∂M ik i whose solution is Pf M eff (M,S)= S ln + f(S), (3.16) W ΛNf where the stong-coupling scale of the theory Λ has been inserted to make the quantity inside logarithm dimensionless. The function f(S) is determined by the U(1)R symmetry to be

f(S)= nS ln(S/Λ3) 1 . (3.17) − − (The constant term in the brackets, which can be absorbed in a re-deﬁnition of Λ, was determined by matching to the traditional normalization of the Veneziano-Yankielowicz superpotential [45] after giving masses and integrating out all the quarks.) The glueball S is massive away from the origin so can be integrated out. Substituting (3.17) into (3.16) and integrating S out by solving its equation of motion, we arrive at the eﬀective superpotentials (3.3).

3.1.4 Consistency with Konishi anomaly equations: Seiberg dual description.

In this subsection we use the Konishi anomaly approach for the dual description as well as the Seiberg duality dictionary to rederive once again our singular effective superpotentials. For N > N + 2 the theory has a Seiberg dual description as an Sp(N N 2) SQCD f f − − a gauge theory [9] with 2Nf (dual) quark chiral fields qi , i = 1 . . . 2Nf , in the fundamental representation and a gauge-singlet elementary field ˆ [ij] coupled to the (dual) meson fields M ˆ a b ˆ ˆ ij ij := qi Jabqj through the superpotential = ij . The dual description is IR free N 3 W N M when Nf < 2 (N + 2). The ring of local gauge-invariant chiral operators for the dual theory is generated by the

56 dual glueball superfield ˆ, ˆ ij and ˆ . As our tree-level superpotential we take S M Nij = ˆ ˆ ij + m ( ˆ ij ij), (3.18) Wtree NijM ij M −M where m = 1 (∂ /∂ ij) is the Lagrange multiplier associated with the dual description ij − 2 Weff M (not to be confused with Lagrange multiplier of the direct description). It imposes the constraint that ˆ ij have ij as their vevs. The superpotential W = ˆ ˆ ij gives masses M M NijM to the dual quarks and sets = 0 when ij = 0, which is why we have not included Nij M 6 Lagrange multipliers for the dual mesons ˆ . Nij The Konishi anomaly equations for a tree-level superpotential in the dual theory is (∂ /∂qa)qa = δi . Substituting (3.18) gives 2 ik = δi . Using the ˆ ij equa- h Wtree i j i S j M Nkj −S j M tion of motion, = m , we can eliminate and arrive at Nij − ij Nkj ∂ ik Weff = δi , (3.19) M ∂ kj S j M whose solution is Pf eff ( ,S)= ln M + g( ), (3.20) W M S ˜ Nf S Λ where Λ˜ is the strong-coupling scale of the dual theory. g( ) is determined as before to be S g( )= n [ln( /Λ˜ 3) 1]. Integrating out then gives the effective superpotential in the S − S S − S dual description 1/n = n Λ˜ 3n−Nf Pf . (3.21) Weff M The dual and the direct theories describe the same physics in the IR regime. Both theories have the same global symmetries and, away from the origin, they have the same moduli space and the same light degrees of freedom. They should also have the same effective superpotentials. Thus, relabeling (3.21) in terms of the direct theory degrees of freedom, we should recover the singular superpotential of the direct theory. In fact, using the Seiberg duality dictionary, the ij are identified with the direct theory mesons through M ij = 1 M ij, where µ is a mass scale related to the dual and the direct theory scales by M µ

Λ3(N+1)−Nf Λ˜ 3n−Nf =( 1)nµNf . (3.22) − Using this, upon rewriting (3.21) in terms of Λ and M ij we indeed ﬁnd the direct theory superpotential (3.3).

57 3.1.5 Consistency upon integrating out ﬂavors.

Besides correctly describing the moduli space, the effective superpotentials should also pass some other tests. If we add a mass term for one flavor in the superpotentials of a theory with Nf flavors and then integrate it out, we should recover the superpotential of the theory with N 1 flavors. To show that the effective superpotential (3.3) passes this test, we add f − a gauge-invariant mass term for one flavor, say M 2Nf −1 2Nf ,

= nΛ−b0/n(Pf M)1/n + mM 2Nf −1 2Nf . (3.23) Weﬀ −

i 2Nf −1 j 2Nf The equations of motion for M and M (i =2Nf 1 and j =2Nf ) put the meson

6 − 6 ij M 0

matrix into the form M = where M is a 2(N 1) 2(N 1) and X a2 2 matrix. 0 X f − × f − × 2Nf −1 2Nf Integrating out X M by its equation of motion gives ∼ c b b = (n 1)Λb0/(n−1)(Pf M)1/(n−1), (3.24) Weff − − where Λ = mΛ3(N+1)−Nf is the strong-couplingb scalec of the theory with N 1 flavors, f − consistent with matching the RG flow of couplings at the scale m. Dropping the hats, we b recognize (3.24) as the effective superpotentials of Sp(N) SQCD with N 1 flavors. f −

3.2 Higher-derivative F-terms in Sp(N) SQCD

So far we have seen that our singular effective superpotentials (3.3) correctly describe the moduli space of vacua. In this section we will use these superpotentials to derive the form of certain higher-derivative F-terms in these theories. This derivation can be taken as a prediction for the result of instanton calculations in the four dimensional = 1 Sp(N) N SQCD with large number of flavors. In [8], Beasley and Witten showed that on the moduli space of the four dimensional =1 SU(2) SQCD with N 2, instantons generate a series of higher-derivative F-terms N f ≥ (also called multi-fermion F-terms). As F-terms they are protected by non-renormalization theorems, and so should be generated at tree-level in perturbation theory from an exact low energy effective superpotential. Indeed, they also calculated these F-terms by integrating out massive modes at tree-level from the non-singular effective superpotentials of SU(2) supersymmetic QCD with Nf = 2 and 3 flavors. In chapter 1, it was shown that singular effective superpotentials of SU(2) supersymmetic QCD can reproduce the corresponding

58 F-terms for Nf > 3, as well. We will show in this section that the singular superpotentials of Sp(N) SQCD (3.3) likewise generate higher-derivative F-terms by a tree-level calculation. As in our discussion of the classical constraints in the last section, the key point in this calculation is to ﬁrst regularize the eﬀective superpotentials (3.3), and then show that the results are independent of the regularization. The SU(2) F-terms of [8] have the form

4 2 6−Nf −Nf i1j1···iNf jNf δS = d x d θ Λ (MM) ǫ M i1j1 Z k2ℓ2 kN ℓN (M DM i k DM j ℓ ) (M f f DM i k DM j ℓ ), (3.25) × 2 2 · 2 2 ··· Nf Nf · Nf Nf

i1i2 i2N+1i2N+2 satisfying ǫi ···i M M = 0, and keep only the massless modes (i.e.those 1 2Nf ··· tangent to the moduli space). We will refer to such terms as being “on vacuum” (in analogy to states being on mass-shell). They should be contrasted with our effective superpotentials (3.3) which are “off vacuum”. Even though (3.25) is written as an integral over a chiral half of superspace, it is not obvious that the integrand is a chiral superfield. But the form of the integrand is special: 2 it is in fact chiral, and cannot be written as D (something), at least globally on the moduli space, and so is a protected term in the low energy effective action [8].

3.2.1 The structure of Sp(N) F-terms

To derive on-vacuum effective interactions from an off-vacuum term, we simply have to expand around a given point on the moduli space and integrate out the massive modes at tree level; see figure 3.1. The only technical complication is that the effective superpotential needs to be regularized first, e.g. by turning on a small mass parameter εij as in (3.4), so that it is smooth at its extrema. At the end, we take ε 0. The absence of divergences as ij → ε 0 is another check of the consistency of our singular effective superpotentials. What ij → we will actually compute is just the leading F-term in an expansion around a generic point on the vacuum in terms of the massless modes of the meson (those tangent to the moduli

59 Figure 3.1: The massless tangent modes Mau (red arrow), and the massive transverse modes Muv (blue arrow) after the meson field Mij has been expanded around a given point on the moduli space. space). As mentioned in section 3.1, the moduli space of vacua for Sp(N) SQCD with N N +2 f ≥ flavors is given by the constraint rank(M ij) 2N. (3.26) ≤ At a generic point on the moduli space, the vev of the meson field M ij = M ij breaks the 0 h i SU(2Nf ) flavor symmetry. We can use flavor rotations to bring the generic vev into the form

ab ij µ M0 = , (3.27) 0!

ab where µ = µIN iσ2 is a skew-diagonalized antisymmetric matrix and µ is a complex ⊗ ij parameter. Note that the above form for M0 breaks the SU(2Nf ) flavor symmetry to Sp(2N) SU(2N 2N). Accordingly, we partition the i, j, . . . flavor indices into two sets: × f − Sp(2N) indices a, b, . . . 1,..., 2N from the front of the alphabet, and SU(2N 2N) ∈ { } f − indices u,v,... 1,..., 2N 2N from the back. Linearizing the meson field around ∈ { f − } ij ij ij (3.27), M = M0 + δM , subject to the constraint (3.26), implies that the massless modes are δM ab and δM au, while the δM uv are all massive, as in figure 3.1. The δM ab modes can be absorbed in a change of µ, so we only need to focus on the δM au modes. We will find that the leading F-term has the form

δS d4x d2θλ−n(µµ)−Nf Pf (µ) ǫu1v1···un+1vn+1 (µc1d1 DδM DδM ) ∼ c1u1 · d1v1 × Z (µcn+1dn+1 DδM DδM ), (3.28) ···× cn+1un+1 · dn+1vn+1 where we have deﬁned

n := N N 1, and λ := Λ−b0/n =Λ(Nf −3N−3)/n. (3.29) f − −

60 Supersymmetry together with the ﬂavor symmetry then uniquely determine the completion of this leading order term to all orders to be

4 2 3N+3−N −N i j ···i j δS = d x d θ Λ f (MM) f ǫ 1 1 Nf Nf M M (3.30) i1j1 ··· iN jN Z kN+1ℓN+1 kN ℓN (M DM i k DM j ℓ ) (M f f DM i k DM j ℓ ). × N+1 N+1 · N+1 N+1 ··· Nf Nf · Nf Nf This follows by an identical argument to one in [8]. Indeed, (3.30) is a straightforward generalization of (3.25) and has many similar properties, including that it is an F-term globally on the moduli space. To generate the leading term (3.28), we first regularize w ε = nλ(Pf M)1/n + eff → Weff − 1 ε M ij , and choose ε = λε1/nµ2/ndiag ε, 1,..., 1 iσ so that 2 ij ij { } ⊗ 2

ab ε ij µ (M0 ) = , (3.31) εuv! where εuv = εI iσ isa 2(N N) 2(N N) skew-diagonalized matrix. An advantage Nf −N ⊗ 2 f − × f − of this choice is that it preserves an Sp(2N) Sp(2N 2N) subgroup of the ﬂavor symmetry. × f − In the ε 0 limi, this symmetry is enhanced to Sp(2N) SU(2N 2N). Also, the massless → × f − ε ij ua directions around this choice of (M0 ) are still δM as before.

3.2.2 Feynman rules

We use standard superspace Feynman rules [47] to compute the eﬀective action for the massless δM ua modes by integrating out the massive δM uv modes. This means we need to evaluate connected tree diagrams at zero momentum with internal massive propagators and external massless legs. In order to evalute these diagrams for the theory under discussion, we closely follow [10] where the superspace Feynman rules for SU(2) SQCD have been explained in detail. Generalizing these rules for Sp(N) SQCD is easy: the massive modes will have standard chiral, anti-chiral, and mixed superspace propagators with masses derived from the quadratic terms in the expansion of ε , while higher-order terms in the expansion give Weﬀ chiral and anti-chiral vertices. A quadratic term in the superpotential, = 1 m(δM)2, gives a mass which enters W 2 the chiral propagator as δMδM = m(p2 + m 2)−1(D2/p2), similarly for the anti-chiral h i | | propagator, and as δMδM = (p2 + m 2)−1 for the mixed propagator. Each propagator h i | | comes with a factor of δ4(θ θ′). Even though the diagrams will be evaluated at zero −

61 momentum, we must keep the p2-dependence in the above propagators for two reasons. First, there are spurious poles at p2 = 0 in the (anti-)chiral propagators which will always 2 cancel against momentum dependence in the numerator coming from D ’s in the propagators 2 and D2’s in the vertices. For instance, D2D = p2 when acting on an anti-chiral field, giving a factor of p2 in the numerator which can cancel that in the denominator of the anti-chiral propagator, to give an IR-finite answer. Second, expanding the IR-finite parts in a power series in p2 around p2 = 0 can give potential higher-derivative terms in the effective action, when p2’s act on the external background fields. Expanding ε around (M ε)ij gives the Weff 0 quadratic terms

′ ′ ′ ′ ε ε ε ε ijkℓ ε 1/n ε −1 ε −1 i j k ℓ (M +δM)= (M )+λ (t ) ′ ′ ′ ′ (Pf M ) (M ) (M ) δM δM + , (3.32) Weﬀ 0 Weﬀ 0 2 i j k ℓ 0 0 ij 0 kℓ ···

ijkℓ where the numerical tensor (t2)i′j′k′ℓ′ controls how the ij . . . indices are contracted with the i′j′ . . . indices. We will drop this tensor for now, though its form will be needed for a later argument. For our immediate purposes it suﬃces to note that in the ε 0 limit the tensor → structure of our tree diagrams is ﬁxed by the Sp(2N) SU(2N 2) subgroup of the global × f − symmetry that is preserved by the vacuum. Specializing to the massive modes, for which i,j,k,ℓ u,v,w,x , and using (3.31) { }→{ } then gives the mass m λε−αµβ, where ∼ n 1 N α := − , β := . (3.33) n n The propagators are then

−1 εα D2 εα 2 δM uv –––– δM wx 1+ p2 , ∼ λµβ p2 λµβ ! 2 −1 εα D εα 2 δM ———– δM 1+ p2 , uv wx ∼ β p2 λµβ λµ ! −1 εα 2 εα 2 δM —– – – δM wx 1+ p2 , (3.34) uv ∼ λµβ λµβ !

where have suppressed the tensor structures on the u,v,w,x indices. { } ε ε The (anti-)chiral vertices come from higher-order terms in the expansion of eff ( eff ). 2 W W Each (anti-)chiral vertex will have a D (D2) acting on all but one of its internal legs. Also, each vertex is accompanied by an d4θ. The ℓth-order term in the expansion of ε has Weff R 62 the general structure

′ ′ ′ ′ i1j1···iℓjℓ ε 1/n ε −1 ε −1 i j i j ′ ′ ′ ′ 1 1 ℓ ℓ λ (tℓ)i j ···i j (Pf M0 ) (M0 )i1j1 (M0 )i j δM δM , (3.35) 1 1 ℓ ℓ ··· ℓ ℓ ···

i1j1···iℓjℓ where the numerical tensor (tℓ)i′ j′ ···i′ j′ controls how the i1j1 iℓjℓ indices are contracted 1 1 ℓ ℓ ··· with the i′ j′ i′ j′ indices. Thus vertices with m massless legs and ℓ m massive legs are 1 1 ··· ℓ ℓ − accompanied by the factors

m massless m massless

λ λ z }| { , z }| { , (3.36) ∼ εγℓ,m µκm ∼ εγℓ,m µκm

ℓ−m massive ℓ−m massive where | {z } | {z } m n +1 m N γ := ℓ , κ := . (3.37) ℓ,m − 2 − n m 2 − n Note that it follows from (3.35) that the number, m, of massless legs δM au must be even. This is because these legs each have one index a 1, 2, 2N and the only non-vanishing ∈{ ··· } components of (M ε)−1 with indices in this range are (M ε)−1 = (M ε)−1 = µ−1. Finally, 0 ij 0 ab − 0 ba to each (anti-)chiral external leg at zero momentum is assigned a factor of the (anti-)chiral au background ﬁeld δM au(x, θ)(δM (x, θ)) all at the same x. Overall momentum conservation means that the diagram has a factor of d4x. The δ4(θ θ′) for each internal propagator − 4 4 together with the d θ integrals at eachR vertex leave just one overall d θ for the diagram. Before going onR to the cases where the eﬀective superpotentials areR singular, we start by

first looking at the Nf = N+1 and Nf = N+2 cases. These cases have regular superpotentials and are simple enough to show the details of the calculations. Although the superpotentials are regular in these cases we nevertheless expand them around the modified vacuum (3.31), and then take the limit ε 0 at the end. The purpose of doing the calculations around M ε → 0 (rather than M0) is to familiarize the reader with how the calculations will be implemented for singular superpotentials where expanding around the modified vacuum is necessary.

63 Figure 3.2: The diagram reproducing the F-term for Nf = N + 1 ﬂavors.

3.2.3 Diﬀerent ﬂavors

Nf = N+1 This case is special since the superpotential is of a diﬀerent form (3.2), involving ε the Lagrange multiplier ﬁeld Σ. Expanding (3.2) around M0 , we have

ε = [Pf M ε Λ2(N+1)] δΣ + [(Pf M ε)(M ε)−1] δM ijδΣ Weﬀ 0 − 0 0 ji ε ε −1 ε −1 ε −1 ε −1 + (Pf M0 ) (M0 )ℓk (M0 )ji +(M0 )jℓ (M0 )ki h +(M ε)−1(M ε)−1 δM ijδM kℓδΣ+ 0 jk 0 iℓ ··· = [Pf M ε Λ2(N+1)] δΣ µN ǫ δM uviδΣ µN−1ǫ J δM auδM bvδΣ+ , 0 − − uv − uv ab ··· where we have just expressed the terms which are relevant in reproducing the multi-fermion

F-term for Nf = N +1. Since the superpotential includes the additional field Σ, we cannot use the coefficients for various superspace Feynman diagrams as expressed in (3.34) and (3.37). Instead, reading the appropriate terms off the ε expansion, the propagator between δΣ Weff and δM uv is accompanied by a factor of ǫuv/[p2 +(µµ)N ], a vertex of δM auδM bvδΣ comes with a factor of ǫuvJ abµN−1, and the δΣ vertex with a factor of (Pf M ε Λ2(N+1)). Evaluating 0 − the diagram in figure 3.2, we have

Pf M ε Λ2(N+1) δS d4x d4θ 0 − ǫuvJ ab δM δM . (3.38) ∼ µµN−1 au · bv Z In the ε 0 limit, Pf M ε vanishes leaving us with → 0 1 δS d4x d2θ ǫuvJ ab (DδM DδM ), (3.39) ∼ µµN−1 au · bv Z 2 2 where we have traded a d2θ for a D and used the equation of motion D δM = 0 to leading order in δM to distributeR the D’s amongst δM’s. The above expression for δS is the higher-derivative F-term for Nf = N + 1.

64 Figure 3.3: Diagrams with four massless external anti-chiral legs for Nf = N + 2. (a) The amputated 4-vertex which does not have the right structure. (b) This diagram reproduces the multi-fermion F-term.

Nf = N + 2 In order to reproduce the F-term for Nf = N + 2 ﬂavors we need four massless anti-chiral legs. There are only two such diagrams, shown in ﬁgure 3.3. Diagram (a) with an amputated 4-vertex (m = l = 4) does not have the right structure to be an F-term because, in the ε 0 limit, it contributes to the action the term →

4 4 λ abcd uvwx a′b′ c′d′ d x d θ A ′ ′ ′ ′ ǫ J J µ2−N a b c d Z δM δM δM δM , (3.40) × au bv cw dx

abcd where Aa′b′c′d′ is a non-vanishing tensor which determines how ab indices are contracted 2 ··· with a′b′ indices. Even if we traded a d2θ for a D and distributed the D’s among ··· 2 δM au’s, we would still need another D .R Also, the coefficient in the integrand of (3.40) does not match that of (3.28) for Nf = N + 2. This term is probably just a correction to the Kähler potential, though we have not ruled out the possibility that it is a new global F-term different from (3.30). Diagram (b) consists of two external anti-chiral vertices and one anti-chiral propagator, and gives

4 4 4 ab us vt ut vs λ δS d x d θ1d θ2 δM au(θ1)δM bv(θ1)J (J J J J ) ∼ − εγ3,2 µκ2 Z 2 −1 εα D εα 2 δ4(θ θ )(J J J J ) 1+ p2 × 1 − 2 sp tq − sq tp β p2 λµβ λµ !

λ wp xq wq xp cd (J J J J )J δM cw(θ2)δ M dx( θ2). (3.41) × εγ3,2 µκ2 −

65 Using the values α = 0, β = N, γ = 0 and κ = 1 N, and substituting them in (3.41), 3,2 2 − we obtain

2 λ D δS d4x d4θ δM δM 1 λµN −2p2 + (p4) δM δM ∼ au bv µ2−N p2 −| | O cw dx Z 2 λ D2D = ǫuvwxJ abJ cd d4x d2θ δM δM (δM δM ) µ2−N au bv p2 cw dx uvwx ab cd Z ǫ J J 2 2 d4x d2θ D δM δM D (δM δM ) + (p2) − λµN µ2 au bv cw dx O Z h i λ = ǫuvwx d4x d2θ δM δM δM δM µ2−N au bv cw dx Z ǫuvwx d4x d2θ (DδM DδM )(DδM DδM )+ (p2), (3.42) − λµN µ2 au · bv cw · dx O Z where in the second line in (3.42) we have traded an d2θ for a D2 and used the identity 2 2 2 2 2 D D = p on antichiral fields to cancel the IR pole.R We then traded an d θ for a D in 2 the third line and used the equation of motion D δM = (δ ) to distribute the D’s in O M R the fifth line. In the last two lines, the first term in the expansion does not have the right structure the multi-fermion F-term, but the second term is, up to some numerical factor, the multi-fermion F-term in (3.28) for Nf = N + 2. All other diagrams in the expansion vanish in the limit p 0. →

Nf = N + 3 This is the ﬁrst case where we have a singular superpotential. The F-term for

Nf = N + 3 has six external anti-chiral massless legs so we have to look for those Feynman diagrams with only six external anti-chiral legs. There are five different possibilities (plus their crossed-channels); see figure 3.4. Among these diagrams, The four diagrams in figure 3.4 (a) either do not have the right structure to be a multi-fermion F-term, or have zero coefficient. For example, the second graph from the left in figure 3.4 (a) comes with zero coefficient because it has vertices with an odd number of massless legs. The rest of diagrams in figure 3.4 (a) are probably corrections to the Kähler term, though we have not ruled out the possibility that some of them might contribute to new classes of global F-terms different from (3.30). The only diagram with the right structure is 5 (b): three external anti-chiral vertices

66 Figure 3.4: Diagrams with six massless external anti-chiral legs for Nf = N + 3. (a) Diagrams which do not have the right structure. (b) The only diagram contributing to the F-term (3.28). with one chiral internal vertex. Evaluating this diagram gives

−1 α 2 α 2 4 4 uvwxyz u′v′ w′x′ y′z′ ab λ ε ε 2 δS d xd θB ′ ′ ′ ′ ′ ′ J J J J δM auδM bv 1+ p ∼ u v w x y z εγ3,2 µκ2 λµβ λµβ Z ! −1 α 2 α 2 λ ε ε 2 λ cd 2 1+ p J D (δM cwδM dx) × εγ3,0 µκ0 λµβ λµβ εγ3,2 µκ2 ! −1 α 2 α 2 ε ε 2 λ ef 2 1+ p J D (δM eyδM fz), (3.43) × λµβ λµβ εγ3,2 µκ2 !

uvwxyz ′ ′ with B ′ ′ ′ ′ ′ ′ being a tensor contracting uv indices to u v indices. Substituting u v w x y z ··· ··· the values α = 1 , β = N , κ = N , κ =1 N , γ = 3 and γ = 1 into (3.43) and taking 2 2 0 − 2 2 − 2 3,0 2 3,2 2 the limit ε 0, we obtain → ǫuvwxyz δS d4x d2θ J abJ cdJ ef (DδM DδM ) ∼ λ2µ3µN au · bv Z (DδM DδM )(DδM DδM ), (3.44) cw · dx ey · fz where we have used the fact that in the ε 0 limit the ﬂavor symmetry group is enhanced → to Sp(2N) SU(2N 2N). This expression coincides with (3.28) for N = N + 3. Since × f − f this was the only diagram contributing in the Nf = N + 3 case, there can be no cancellation of its coeﬃcient. This shows that the Nf = N + 3 singular superpotential indeed reproduces the corresponding higher-derivative global F-term in perturbation theory.

N N + 4 As we go higher in the number of ﬂavors, however, the number of diagrams f ≥ contributing to each amplitude rapidly increases. For instance, just among the class of internally purely-chiral diagrams illustrated in ﬁgure 3.5, there are four superspace Feynman

67 Figure 3.5: Diagrams which have the right structure to give a higher-derivative F-term for Nf = N + 4.

diagrams in the case of Nf = N + 4 flavors each with the right structure to contribute to (3.28). But since now multiple diagrams contribute, we must show in addition that no cancellations occur that could set the coefficient of the higher-derivative term to zero. This seems quite complicated, as it depends on the signs and tensor structures of the vertices. Some sort of symmetry argument is clearly wanted, but still eludes us. In addition, there are now also other classes of diagrams which are neither purely anti- chiral (as in figure 3.4(a)) or internally purely chiral (as in figure 3.5). It is not clear whether these mixed diagrams will also contribute to higher-derivative amplitudes of the form (3.28) or not.

68 Part II:

Application of the gauge-gravity duality to quark-gluon plasma Chapter 4

String conﬁgurations for moving quark-antiquark pairs in a thermal bath

The AdS/CFT correspondence provides a powerful tool with which to study the strong- coupling behavior of certain non-Abelian gauge theories in terms of semi-classical supergravity descriptions [14, 15, 16]; see [2] for a detailed review of the subject. The most-studied example is four dimensional =4 SU(N) supersymmetric Yang-Mills theory (SYM) which, N 2 in the limit of large N and large ’t Hooft coupling λ = gYMN, is described by type IIB supergravity on AdS S5. Since at finite temperature the superconformal invariance of this 5 × theory is broken, and since fundamental matter can be added by introducing D7-branes [17], it is thought that this model may shed light on certain aspects of strongly-coupled QCD plasmas. For example, for any strongly-coupled large N gauge theory with a gravity dual, the dimensionless ratio of the shear viscosity over entropy density has been found to be 1/4π [18, 19, 20, 21] in rough agreement with some hydrodynamic models of RHIC collisions [53, 54]. More generally, the RHIC experiment has raised the issue of how to calculate the transport properties of relativistic partons in a hot, strongly-coupled gauge theory plasma. For example, one would like to calculate the friction coefficient and jet-quenching parameter which are measures of the rate at which partons lose energy to the surrounding plasma [55, 56, 57, 58, 59, 60, 61]. Using conventional quantum field theoretic tools one can calculate these parameters only when the partons are interacting perturbatively with the surrounding plasma. The AdS/CFT correspondence may be a suitable framework in which to study

70 strongly-coupled QCD-like plasmas. In fact, attempts to use the AdS/CFT correspondence to calculate these quantities have been made in [27, 28, 34] and were generalized in various ways in [62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 29, 75, 76, 77, 31, 78, 32, 33, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89]. Finite-temperature =4 SU(N) SYM theory is equivalent to the near-horizon limit of N type IIB supergravity on the background of a large number N of non-extremal D3-branes stacked at a point [22]. From the perspective of five-dimensional gauged supergravity, this is the background of an AdS black hole whose Hawking temperature equals the temperature of the gauge theory. According to the AdS/CFT dictionary, string configurations on this background can correspond to quarks and antiquarks in an = 4 SYM thermal bath N [23, 24, 25, 26], where the quark bare mass is determined by the radial location of the string endpoints on a probe D7-brane. A stationary single quark can be described by a string that stretches from the probe D7-brane to the black hole horizon. The semi-infinite string solution with a tail which drags behind a steadily moving endpoint and asymptotically approaches the horizon has been proposed [27, 28] as the configuration dual to a steadily moving quark in the = 4 plasma, N and was used to calculate the drag force on the quark. A stationary quark-antiquark pair or “meson”, on the other hand, corresponds to a string with both endpoints ending on the D7-brane [23, 24, 17]. A class of such solutions, namely smooth, static solutions (v = 0), have been used to calculate the inter-quark potential in SYM plasmas. Smooth, stationary solutions (v = 0) for steadily moving quark pairs exist 6 [29, 30, 31, 32, 33] but are not unique and do not “drag” behind the string endpoints as in the single quark configuration. This lack of drag has been interpreted to mean that color-singlet mesons are invisible to the SYM plasma and so experience no drag (to leading order in large N) although the string shape is dependent on the velocity of the meson with respect to the plasma. (These timelike Lorentzian string solutions are reviewed in section 4.2, below.) On the other hand, a prescription for computing the jet-quenching parameterq ˆ using the lightlike limit of spacelike Lorentzian configurations has been proposed in [34].(See footnote 14 of [29] and [86]) In this chapter, we restrict ourselves to timelike Lorentzian and Euclidean configurations, and address spacelike Lorentzian configurations in the next chapter.

Concretely, we consider a smooth stationary string in the background of an AdS5 black

71 hole with metric r4 r4 r2 r2R2 ds2 = h dxµdxν = η − 0 dx2 + (dx2 + dx2 + dx2)+ dr2. (4.1) 5 µν r2R2 0 R2 1 2 3 r4 r4 − 0

R is the curvature radius of the AdS space, and the black hole horizon is located at r = r0. Since we are interested in (timelike) Lorentzian as well as Euclidean conﬁgurations, we have introduced the factor η; for Lorentzian signature η = 1 while for Euclidean signature it − equals +1.

1 We put the endpoints of the string on a probe D7-brane at radius r7. The classical dynamics of the string in this background is described by the Nambu-Goto action

η S = dσdτ η G, (4.2) 2πα′ Z p µ ν where G = det (Gαβ), Gαβ = hµν ∂αX ∂βX is the induced metric on the string worldsheet, and ∂ := ∂/∂ξα, with ξα = τ, σ being the worldsheet coordinates. Here Xµ run over the α { } ﬁve coordinates x , x , x , x ,r . { 0 1 2 3 } The steady state of a quark-antiquark pair with constant separation and moving with constant velocity either perpendicular or parallel to the separation of the quarks can be described (up to worldsheet reparameterizations) by the worldsheet embedding

[v⊥] : x0 = τ, x1 = vτ, x2 = σ, x3 =0, r = r(σ),

[v||] : x0 = τ, x1 = vτ + σ, x2 =0, x3 =0, r = r(σ), (4.3) where the ﬁrst is for the velocity perpendicular to the quark separation, and the second parallel to it. In both cases we take boundary conditions

L L 0 τ T, σ , r( L/2) = r , (4.4) ≤ ≤ − 2 ≤ ≤ 2 ± 7 with r(σ) smooth.2

1The background (4.1) can be lifted to ten dimensions on a 5-sphere, where it is the near-horizon geometry of a stack of N non-extremal D3-branes. The probe D7-brane wraps an S3 inside the S5 and fills the entire 5 AdS5 background down to a minimal radius r7 [17, 90, 91, 92, 93, 94]. We assume no motion on the S , and so use the five dimensional perspective. 2The endpoints of a string on a D-brane satisfy Neumann boundary conditions in the directions along the D-brane, whereas the above boundary conditions are Dirichlet, constraining the string endpoints to lie along fixed worldlines a distance L apart on the D7-brane. The correct way to impose these boundary conditions is to turn on a worldvolume background U(1) field strength on the D7-brane [27] to keep the string endpoints a distance L apart.

72 Outline of the chapter: In section 4.1 we derive the equations of motion describing configurations corresponding to quark-antiquark pairs, in both Lorentzian as well as Euclidean signature, which are stationary and smooth. As argued in [30], smooth string configurations cannot drag behind their endpoints as they dip down from the D7-brane. Among these no- drag configurations, we concentrate on two simple geometries, where the common velocity of the quark pair is either perpendicular or parallel to their separation. We examine the timelike Lorentzian solutions of these equations in some detail in section 4.2. In the case that the meson velocity is perpendicular to the quark separation L, we review the known solutions. Up to a maximum L L (v) which decreases with increasing velocity, ≤ c there are two branches of Lorentzian solutions: one “long” whose radial turning point is closer to the horizon than the “short” solution. For L > Lc(v), quark-antiquark pairs only exist as free states described by two disconnected strings. These two branches, as well as the complete screening length, have been discussed in [29, 30, 31, 32, 33]. The long string solution which makes it closer to the horizon has been argued to be unstable [30, 33] and presumably decays into the shorter string configuration which shares the same boundary conditions. This is supported by the fact, shown in section 4.3, that the energy of the Euclideanized version of the long string configuration is greater than that of the short string. When the meson velocity is parallel to the quark separation, the long string configurations are not smooth, but instead develop a cusp at the midpoint for a range of velocities. The tip of the cusp is lightlike. On the other hand, the short string solution is always smooth. In section 4.3 we examine the Euclidean string configurations in detail. Their main interest lies in the fact that their relative thermodynamic stability can be determined by comparing their actions (energies). However, we find that not all Euclidean solutions have Lorentzian counterparts. Euclidean configurations for which the velocity is parallel to their separation share some of the same characteristics as the timelike Lorentzian configurations. Namely, there tend to be two different branches of solutions, except when v > 1, for which there is only one. (There is no restriction to v < 1 for Euclidean strings; v is more properly thought of as a slope parameter, rather than a velocity.) Also, analyzing the equations of motion derived from the Nambu-Goto action shows that there is a maximum distance between the quark and the antiquark past which only free quarks exist, but this may not be the case once the couplings of lightest supergravity modes to the worldsheet (described by the Nambu-Goto action) are considered [95].

73 On the other hand, for Euclidean conﬁgurations with velocity perpendicular to their separation, the discussion of the various branches of string solutions becomes more involved. Firstly, some of the branches no longer have a maximum value of L. Secondly, the number of branches now depends on the velocity. In particular, for low velocities there are actually four branches of solutions, while only two branches exist for higher velocities. The solutions in which the string dips closer to the black hole horizon are less energetically favorable.

4.1 Equations of motion

According to the AdS/CFT correspondence [2] strings ending on the D7-brane are equivalent to quarks in a thermal bath in four dimensional ﬁnite-temperature =4 SU(N) SYM theory. N The standard gauge-gravity dictionary is that

4 ′2 4 ′2 2 ′ N = R /(4πα gs), λ = R /α , β = πR /r0, m = r7/(2πα ), (4.5)

2 where gs is the string coupling, λ := gYMN is the ’t Hooft coupling of the SYM theory, β the inverse temperature of the SYM thermal bath, and m the quark mass at zero temperature. In the semi-classical string limit, i.e. g 0 and N , the supergravity approximation s → → ∞ in the gauge-gravity correspondence holds when the curvatures are much greater than the ′ string length, ℓs := √α . Furthermore, in this limit the mass of the quark is identified with the energy (in some units) of the associated string which, for a static configuration, is just proportional to the value of the Nambu-Goto action (4.2). Now, the string embeddings (4.3) considered here (and elsewhere in this chapter) depend on three additional parameters: T , L, and v. According to the standard AdS/CFT dictionary, these are the time for which the quarks are propagating, their separation at a given time, and their common velocity, respectively– all in the plasma rest frame. Also, it is easy to see from (4.1) and (4.3) that, for Lorentzian (η = 1) solutions, the proper velocity − V of the string endpoints on the r = r7 surface in the AdS black hole background is related to the velocity v in the four dimensional field theory by

r2 V = 7 v. (4.6) r4 r4 7 − 0 Note that real string solutions must have thep same signature everywhere on the worldsheet. Thus, a string worldsheet will be timelike or spacelike depending on whether V , rather than

74 v, is greater or less than 1. In particular, the string worldsheet is timelike for V < 1 which in terms of v translates into v < 1 (r /r )4. (In Euclidean signature, v is more properly − 0 7 thought of as an angular parameter,p and V is likewise a proper version of this parameter as measured in the Euclidean AdS black hole metric. This will be discussed more in section 4.2.) With the embeddings (4.3) and boundary conditions (4.4), the action becomes

ηT L/2 r4 γ2r4 r4 γ2r4 [v ] : S = dσ − 0 + − 0 r′2, ⊥ γ πα′ R4 r4 r4 Z0 s − 0 ηT L/2 r4 r4 r4 γ2r4 [v ] : S = dσ γ2 − 0 + − 0 r′2, (4.7) || γ πα′ R4 r4 r4 Z0 s − 0 where r′ := ∂r/∂σ and 1 γ2 := . (4.8) 1+ ηv2 Thus, γ is the usual relativistic γ-factor for Lorentzian signature (η = 1), while it is always − a positive number less than 1 for Euclidean signature (η = +1). We ﬁnd for the equations of motion

′2 1 4 4 4 2 2 4 [v⊥] : r = 2 2 4 4 (r r0)(r γ [1 + a ]r0), γ a r0R − − γ2 (r4 [1 + a2]r4) [v ] : r′2 = (r4 r4)2 − 0 , (4.9) || a2r4R4 − 0 (r4 γ2r4) 0 − 0 where a2 is a real integration constant. Although we have written a2 as a square, it can be either positive or negative. Using (4.9), the determinant of the induced worldsheet metric becomes

1 4 2 4 2 [v⊥] : G = η 4 2 4 4 (r γ r0) , γ a r0R − 1 4 4 2 [v||] : G = η 2 4 4 (r r0) . (4.10) a r0R −

Thus, the sign of G is the same as that of ηa2 (since the other factors are squares of real quantities). In particular, for Euclidean signature (η = +1) all real worldsheets have G> 0, and so we must take a2 > 0. For Lorentzian signature (η = 1), the worldsheet is timelike − (G< 0) for a2 > 0 and spacelike for a2 < 0. The reality of r′ implies that the right sides of (4.9) must be positive in all these diﬀerent

75 cases. This positivity then implies certain allowed ranges of r. There can only be real string solutions when the ends of the string, at r = r7, are within this range. The edges of this range are (typically) the possible turning points, rt, for the string. We will describe the possible values of rt for the timelike Lorentzian and Euclidean cases in the following sections. Given these turning points, (4.9) can be integrated for a string solution which goes from z7 to the turning point zt and back to give

2aγ z7 dz [v⊥] : L/β = , 4 4 2 2 π z (z 1)(z (1 + a )γ ) Z t − − z 2a 7 p dz z4 γ2 [v||] : L/β = − , (4.11) 4 4 2 πγ z (z 1) z (1 + a ) Z t − p − 2 p where we have used r0 = πR /β. Also, in (4.11) we have rescaled z = r/r0 and likewise zt := rt/r0 and z7 := r7/r0. These integral expressions determine the integration constant a2 in terms of L/β and v. Also, we can evaluate the action for the solutions of (4.9) to be

ηT √λ z7 (z4 γ2) dz [v⊥] : S = − , 4 4 2 2 γβ z (z 1)(z γ [1 + a ]) Z t − − ηT √λ z7 p z4 γ2 [v ] : S = dz − , (4.12) || γβ z4 [1 + a2] Zzt s −

2 ′ where we have used R /α = √λ. For ﬁnite z7, these integrals are convergent. They diverge when z , and need to be regularized by subtracting the self-energy of the quark and 7 → ∞ the antiquark [23, 24].

4.2 Timelike Lorentzian solutions

Turning now to timelike Lorentzian (η = 1) string conﬁgurations, we see from (4.10) that − the integration constant a2 must be positive. An analysis of (4.9), bearing in mind (4.6), easily shows that real solutions can exist only for v < 1 z−4 and as long as the string is − 7 at radii satisfying p

4 4 2 2 [v⊥] : r /r0 > γ (1 + a ),

4 4 2 2 [v||] : r /r0 > max γ , 1+ a . (4.13) 76 LΒ 0.3

0.2

0.1

a 4 8

Figure 4.1: L/β as a function of a for timelike Lorentzian string conﬁgurations with velocity perpendicular to the quark separation, z7 = 2, and γ = 1 (dark blue), 1.5 (light blue), 2 (green), 3 (black), and 3.8 (red).

We will ﬁrst brieﬂy review the case in which the velocity of the quark-antiquark pair is perpendicular to their separation [29, 30, 31] and then consider the parallel case.

4.2.1 Timelike Lorentzian: perpendicular velocity

4 4 Equation (4.13) implies that the radial turning point of the string is at z := (r/r0) = γ2(1 + a2). It also implies that, for a given velocity parameter v, the minimum D7-brane radius z7 := r7/r0 reached by the probe must also be set to be greater than this value. (z7 should also be set greater than the critical value zc 1.02, below which the D7-brane dips 7 ≈ into the horizon, changing the topology of the space [90, 91, 92, 93, 94].)

For a given choice of z7, (4.11) can be numerically integrated to give L/β as a function of a, as shown in ﬁgure 4.1 for a few sample velocities with the choice z7 = 2. (A similar plot has already been presented in [30].) The qualitative features of the plots are not sensitive to the particular value of z7, though one should bear in mind that the range of allowed velocity 2 parameters v depends on z7, since the string endpoints become spacelike for γ > z7 , as can be seen from (4.6). Figure 4.1 illustrates the fact that, for each value of L that is less than a critical value

Lc(v), there are two corresponding values of a, and Lc(v) decreases with increased velocity v. We shall refer to the branch of string conﬁgurations with smaller (larger) a for a given

L as the long (short) conﬁgurations. For L > Lc there is no connected string solution: Lc corresponds to a complete screening length past which quarks and antiquarks only exist as free states [72, 29, 30, 31], given that one ignores the couplings of the lightest supergravity modes to the Nambu-Goto action (which is certainly the case in this chapter); see [95] for

77 L

z=1 z=4

Figure 4.2: Timelike Lorentzian string configurations with velocity perpendicular to the quark separation, L/β = 0.1, and γ = 1 (blue), 1.5 (green), 2 (black), and 2.1 (red). For each velocity there are two string solutions, one long and one short. The black hole horizon (solid black line) is at z = 1 and the minimal radius reached by the probe D7-brane (dotted line) is at z = z7 = 2. furthur discussions on this issue. We have plotted in figure 4.2 both long and short string configurations for fixed L/β and various velocities. The radial direction is horizontal, the x2-direction is vertical and the velocity is orthogonal to both. The black hole horizon is represented by the solid black line at z = 1 and the probe D7-brane corresponds to the dashed line at z = z7. For zero velocity (blue curves), the long string configuration almost touches the black hole horizon. As the velocity is increased so the string worldsheets become more nearly lightlike (V 1, → or γ 4), the long and short string configurations shorten and lengthen, respectively, → and approach a common limiting shape (between the red curves). They coincide when the velocity reaches some γ = γ (γ 2.112 for the specific values of L/β and z in the figure). c c ≈ 7 This is the value of the velocity parameter where Lc = L; for greater velocities there are no string solutions for this given L and z7. A general qualitative property of these solutions is that for any fixed L and z7 there is no lightlike limit of these timelike string configurations: the limiting L = Lc is reached before V = 1. We will see in section 4.3 that the Euclidean counterparts of the long string solutions are not energetically favored. This indicates that this branch of solutions is not stable: a long string state presumably decays into the corresponding short string which has the same boundary conditions. The instability of the long string configurations was hypothesized in [30] and has been argued for dynamically in [33].

78 L0 Β

0.2

0.1

a 2 4

Figure 4.3: L0/β as a function of a for a timelike Lorentzian string with velocity parallel to quark separation, z7 = 2, and γ = 1 (dark blue), 1.5 (light blue), 2 (green), 3 (black), and 3.8 (red). Strings corresponding to points in the shaded region have cusps.

4.2.2 Timelike Lorentzian: parallel velocity

We will now look at the situation for which the velocity is in the same direction as the quark-antiquark separation. Recall that reality of r′ and the equation of motion (4.9) implied (4.13); that is, the allowed region is z4 > max γ2, 1+ a2 . Unlike the perpendicular case, { } this boundary is not always a smooth turning point of the string. In particular, (4.9) implies that r′ = 0 when z = 1+a2, which is a smooth turning point (the string reaches a minimum); but r′ = when z4 = γ2. This latter behavior signals the development of a cusp at the ∞ string midpoint. As we discussed in section 4.1, z4 = γ2 is also the place where the string worldsheet changes from timelike to spacelike signature. Since, by (4.10), real string solutions cannot change their worldsheet signature, we conclude that whenever γ2 > 1+ a2 this cusp is unavoidable. 3 It is not obvious when γ2 > 1+ a2 is satisﬁed, since a2 is determined in terms of v through (4.11). In ﬁgure 4.3 we integrate (4.11) for various velocities to give an indication of how a depends on L and v. Figure 4.3 actually shows the quark separation L0 := γL in

3Without this physical argument, one might imagine that the r′ = vertical tangent is a signal not of a cusp, but just that the string solution should be extended to include∞ a smooth but self-intersecting closed loop. The worldsheet embedding (4.3) we have used does not allow for this extension, and so one might think that the cusp could be avoided by using a diﬀerent embedding. For example, instead of using a parameterization in which x1 = vτ + σ as in (4.3), which forces the string to vary monotonically in the x1 direction, one might use a diﬀerent parameterization with, say, r = σ and x1 = vτ + x(σ) for some undetermined function x(σ). This would, in principle, allow the string to cross itself and form a smooth loop. However, reworking our calculations in this alternative parametrization gives equations of motion completely equivalent to (4.9). Thus, this possibility is not realized, and the cusps cannot be avoided, in agreement with the physical argument.

79 L0

z=1 z=4

Figure 4.4: Timelike Lorentzian configurations with velocity parallel to quark separation, L0/β =0.1, and γ = 1 (blue), 1.5 (green), 2 (black) and 2.5 (red). the quark rest frame rather than the separation L in the plasma restframe. There are two branches of solutions for a when L0 < L0c(v), none when L0 > L0c(v), and L0c(v) decreases for increasing v. This is qualitatively similar to the perpendicular velocity case. The shaded region in figure 4.3 is for γ2 > 1+ a2, in which case the string solutions have a cusp. Note that this region lies to the left of the maximum of the constant-v curves in the figure. This means that the large-a (short string) solutions never have cusps but that, depending on the values of L0 and v, the long string solutions may. Typically, for given L0 the long string solutions for small enough v (close to v = 0) and large enough v (near where

L0c(v) approaches L0) are smooth, while at intermediate v there are cusps.

This is illustrated for z7 = 2 and L0/β = 0.1 in figure 4.4. There the γ = 0 (blue) and γ = 2.5 (red) long strings have no cusp, while the intermediate velocity (green and black) strings do. 4 Just as in the case of perpendicular velocity, as v is increased the long and short strings approach one another until they coincide at a critical value of the velocity parameter (γ 2.6173 for the values of the parameters in the figure), beyond which there ≈ are no more connected solutions. Note that this critical velocity parameter is short of the 2 lightlike worldsheet limit γ = z7 so that, as in the case of perpendicular velocity, no lightlike worldsheet limit of the connected timelike configuration exists at fixed L0 and z7.

4The appearance of a kink—for which there is a finite opening angle—rather than a cusp in the green and black long strings in figure 4.4 is misleading: the cusp behavior is apparent with sufficient resolution.

80 4.3 Euclidean strings and their energetics

Real Euclidean (η = +1) string conﬁgurations must have positive integration constant a2, by (4.10). An analysis of (4.9) easily shows that real solutions can exist for any v (since now 1 γ > 0 for all v) as long as the string is at radii satisfying ≥

4 4 2 2 [v⊥] : r /r0 > max 1, γ (1 + a ) ,

4 4 2 [v||] : r /r0 > 1+ a . (4.14)

Nothing special happens in Euclidean signature as the “velocity” parameter v 1. Indeed, → v is more properly thought of as an angular parameter in Euclidean space, though we will still refer to it as the velocity parameter. Note that there are Euclidean solutions which are not Wick rotations of Lorentzian ones. Lorentzian and Euclidean equations of motion (4.9) are related to each other simply by taking v2 v2. However, this does not mean that the corresponding solutions are → − simply related by Wick rotations since, under v2 v2, the behavior of the turning points → − can change qualitatively. In particular, timelike Lorentzian solutions with perpendicular 4 2 2 4 4 velocity always have rt = γ (1 + a )r0 > r0 and so the string never reaches the horizon. On the other hand, for a < v there is a branch of Euclidean solutions which have the radial turning point on the black hole horizon r = r0. This branch of solutions has no physical Lorentzian counterpart. Other examples of Euclidean string conﬁgurations with no physical Lorentzian counterpart are easy to come by. For instance, the Wick rotation of a steadily moving, purely radial Euclidean string stretched between a probe D7-brane and the black hole horizon fails to exist in Lorentzian signature, since there is an intermediate radial point below which the string travels faster than the speed of light.

4.3.1 Euclidean: perpendicular “velocity”

A numerical plot of L/β as a function of a for various velocities is shown in ﬁgure 4.5. We have set z7 = 2 as an example, though the plot is qualitatively unchanged for other values of this parameter. Since Wick rotation amounts to changing the sign of v2 in the equations, the conﬁguration with v = 0 is exactly the same for Lorentzian and Euclidean signatures.

In particular, there are no solutions for L > Lc, and for a given value of L < Lc there are two string conﬁgurations. For v > 0, however, the story changes dramatically. Firstly, there

81 LΒ Hv=0.0L LΒ Hv=0.15L 0.4 0.4

0.2 0.2 a 2 4a 2 4

LΒ Hv=0.6L LΒ Hv=1.5L 0.4 0.4

0.2 0.2

2 4a 2 4a

Figure 4.5: L/β as a function of a with z7 = 2 for Euclidean configurations with v =0, 0.25, 0.5 and 1, for which the velocity is perpendicular to the quark separation. The red and blue curves represent solutions with a < v and a > v, respectively. is no longer a maximum value of L. Secondly, the number of branches of solutions depends on L as well as the velocity. For intermediate velocities, two new branches of configurations emerge which have no Lorentzian counterparts. This is shown in the upper right region of figure 5 for v = 0.25. One new branch, which is denoted by the red curve, has a < v and exists for all values of L/β. For small and large values of L/β, there is only one branch of blue solutions but for intermediate values of L/β there are actually three branches. For sufficiently large v, only one branch of blue solutions occurs. This is illustrated in the lower left region of figure 4.5 for v = 0.5. For larger values of v there are no qualitative changes, as illustrated in the lower right of figure 4.5 for v = 1. (Nothing special happens at v = 1 in Euclidean signature.) To better illustrate this, the four different string configurations for v =0.25 and L/β = 0.25 are plotted in figure 4.6(a). Only the a < v configuration, represented by the red curve, actually touches the black hole horizon. Only two of the branches of configurations remain for all L/β as the velocity is further increased, as illustrated in figure 4.6(b) for v =0.5. Which of these states is the physical one for a given set of parameters can be determined by comparing their energies. The intuition that the blue curve represents the energetically favorable solution, since it does not stretch as far towards the black hole, is born out by a calculation of the energies. The energy of the Euclidean string configurations is given by

82 HaL HbL HcL

L L

z=1 z=10 z=1 z=1.02 z=1 z=10

Figure 4.6: Euclidean string conﬁgurations with perpendicular velocity, L/β =0.25 and z7 = 2. (a): Four solutions when v =0.25, with a values of approximately 0.237 (red), 0.290 (black), 0.423 (green) and 1.172 (blue). (b): Two solutions when v =0.5 with a values approximately 0.397 (red) and 1.503 (blue).

S/T , where S is the Nambu-Goto action given by (4.12) and T is the time interval. It is more illuminating to discuss the energy difference E between these configurations and some standard string configuration. A simple and natural fiducial configuration to choose is that of two disconnected strings moving at “velocity” v which stretch from the probe D7- brane to the black hole horizon. So in our discussion below we will measure energies in comparison to these straight string configurations, which therefore have energy E = 0 by definition. Eβ/√λ versus L/β is plotted in figure 4.7 for various velocities for the aforementioned configurations. The case of vanishing velocity has already been considered in [25, 26]. As before, the red curve represents the string configuration with a < v, which reaches the black hole horizon. There are multiple configurations with a > v for a given L (blue curves), depending on the velocity. The energy of the fiducial straight string configuration is given by the E = 0 line. As can be seen from figure 4.7, for L less than a critical value, the energetically favorable state is represented by the blue curve. This is the string configuration that remains the furthest from the black hole horizon and is the Wick rotated counterpart of the timelike Lorentzian short string solution with perpendicular velocity that was discussed in section 4.2. As the distance between the quark and antiquark increases to the critical value, the subtracted energy of this configuration becomes positive. At this point, it is energetically favorable for the string to separate into two straight strings (green line). Note that the long string configuration (red curve) is always less energetically favorable than the short strings (blue curve), which agrees with the claim that the corresponding Lorentzian configurations are unstable [30, 33].

83 E Hv=0.0L E Hv=0.15L 0.4 0.4

0.2 0.4 LΒ 0.4 LΒ -0.4 -0.4

E Hv=0.6L 0.4

0.2 0.4 LΒ -0.4

Figure 4.7: Energy in units of √λ/β versus L/β for Euclidean conﬁgurations with perpendicular velocity, z7 = 2, and v = 0, 0.1, and 0.5. The red and blue curves represent solutions with a < v and a > v, respectively, while the green line is the subtracted energy of two straight strings.

It is tempting to identify the transition from the short string solution (blue) to the two straight string solution (green) as the transition in the field theory from a bound quark and antiquark pair to free quark pair due to complete screening by the thermal bath. However, this interpretation is problematical. The reason is that, as mentioned earlier, the straight Euclidean string is not the Euclidean rotation of any straight Lorentzian string solution (since at any nonzero v such a straight Lorentzian string becomes lightlike before it reaches the horizon, and so fails to exist as a solution). A physically acceptable Lorentzian free quark solution is the dragged string solution of [27, 28], so it may be more appropriate to compare the energy of the Euclidean short string solution (blue) to that of the Euclidean rotation of a pair of dragged strings instead. See [30] for a discussion of the issues involved in making this comparison, and [95] for screening in = 4 SYM plasma in general. So the N transition between the blue and green configurations illustrated in figure 4.8 gives at best an upper bound on the critical L at which complete screening by the SYM thermal bath occurs.

84 L0 Β 0.5

0.4

0.3

0.2

0.1

a 2 4

Figure 4.8: L0/β as a function of a for Euclidean string conﬁgurations with parallel velocity, z7 = 2, and v = 0 (blue), 0.5 (light blue), 1 (purple), 2 (light red), and 5000 (red).

4.3.2 Euclidean: parallel “velocity”

For Euclidean string configurations with parallel “velocity”, (4.14) shows that the radial 2 1/4 turning point is at r = (1+ a ) r0. Such solutions with V < 1 are Wick rotations of the timelike Lorentzian solutions with corresponding turning point (e.g. all those outside of the shaded region in figure 4.3). In contrast to the configurations with perpendicular velocity, there is always a maximum L regardless of the magnitude of the velocity. Also, there are no string configurations that reach the black hole horizon. L0/β versus a for various velocities is shown in figure 4.8. In Euclidean signature, L0 := γL measures the shortest distance between the “worldlines” of the endpoints of the strings. Since arctan(V ) measures the angle between these worldlines and the constant-x planes, in the limit V v the 1 ∼ → ∞ worldlines coincide and L 0. The curves in figure 4.8 ascend from v =0 to v = . For 0 → ∞ V < 1 there are two solutions for each value of L < Lc. The short string configurations correspond to the part of the curves to the right of the peak in figure 4.8, while the long configurations correspond to the left side. For V > 1 there is only one solution and, as v , L /β z4 1/(2z2). Thus, L increases as the boundary worldlines are oriented →∞ c → 7 − 7 c more along the xp1 direction.

85 Chapter 5

Spacelike string conﬁgurations and jet-quenching from a Wilson loop

As we mentioned in chapter 4, results coming from RHIC have raised the issue of how to calculate transport properties of ultra-relativistic partons in a strongly coupled gauge theory plasma. For example, one would like to calculate the friction coefficient and jet- quenching parameter, which are measures of the rate at which partons lose energy to the surrounding plasma. With conventional quantum field theoretic tools, one can calculate these parameters only when the partons are interacting perturbatively with the surrounding plasma. The AdS/CFT correspondence [2] may be a suitable framework in which to study strongly coupled QCD-like plasmas. According to the AdS/CFT dictionary, the endpoints of open strings on a black hole background [22] can correspond to quarks and antiquarks in the SYM thermal bath [23, 24, 25, 26]. For example, a quark-antiquark pair or “meson”, corresponds to a string with both endpoints ending on a probe, say, D7-brane. As we described in detail, in the previous chapter, smooth, stationary no-drag solutions for steadily-moving quark-antiquark pairs exist. A particular no-drag string configuration with spacelike worldsheet [29, 86] has been used to evaluate a lightlike Wilson loop in the field theory. It has been proposed that this Wilson loop can be used for a non-perturbative definition of the jet-quenching parameterq ˆ [34]. The purpose of this chapter is to do a detailed analysis of the evaluation of this Wilson loop using no-drag spacelike string configurations in the simplest case of finite-temperature = 4 SU(N) SYM theory. As in chapter 4, we use the Nambu-Goto action to describe N the classical dynamics of a smooth stationary string in the background of a five dimensional AdS black hole. We put the endpoints of the string on a probe D7-brane with boundary

86 conditions which describe a quark-antiquark pair with constant separation moving with constant velocity either perpendicular or parallel to their separation.

Outline of the chapter In section 5.1, we present the string embeddings describing smooth and stationary quark-antiquark configurations, and we derive their equations of motion. In section 5.2, we discuss spacelike solutions of these equations. We find that there can be an infinite number of spacelike solutions for given boundary conditions, although there is always a minimum-length solution. In section 5.3, we apply these solutions to the calculation of the lightlike Wilson loop observable proposed by [34] to calculate the jet-quenching parameterq ˆ, by taking the lightlike limit of spacelike string worldsheets [29, 86]. We discuss the ambiguities in the evaluation of this Wilson loop engendered by how the lightlike limit is taken, and by how self-energy subtractions are performed. Technical aspects of the calculations needed in section 5.3 are collected in the appendix to this chapter. We also do the calculation for Euclidean-signature strings for the purpose of comparison.

Conclusions. We find that the lightlike limit of the spacelike string configuration used in [34, 86] to calculate the jet-quenching parameterq ˆ is not the solution with minimum action for given boundary conditions, and therefore gives an exponentially suppressed contribution to the path integral. Regardless of how the lightlike limit is taken, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear in its width, L. Quadratic behavior in L is associated with radiative energy loss by gluons in perturbative QCD, and the coefficient of the L2 term is taken as the definition of the jet-quenching parameterq ˆ [34]. In the strongly coupled = 4 SYM theory in which we N are computing, we findq ˆ = 0. We now discuss a few technical issues related to the validity of the dominant spacelike string solution which gives rise to the linear behavior in L. Depending on whether the velocity parameter approaches unity from above or below, the minimum-action string lies below (“down string”) or above (“up string”) the probe D7- brane, respectively. The down string worldsheet is spacelike regardless of the region of the bulk space in which it lies. On the other hand, in order for the up string worldsheet to be spacelike, it must lie within a region bounded by a certain maximum radius which is related to the position of the black hole horizon. The lightlike limit of the up string involves taking

87 the maximal radius and the radius of the string endpoints to infinity simultaneously, such that the string always lies within the maximal radius. Therefore, even though the string is getting far from the black hole, its dynamics are still sensitive to the black hole through this maximal radius. In the lightlike limit, the up and down strings with minimal action both approach a straight string connecting the two endpoints. This is the “trivial” solution discarded in [34], though we do not find a compelling physical or mathematical reason for doing so. If the D7-brane radius were regarded as a UV cut-off, then one might presume that the dominant up string solution should be discarded, since it probes the region above the cut-off. However, this is unconvincing for two reasons. First, if one approaches the lightlike limit from v > 1, then the dominant solution is a down string, and so evades this objection. Second, and more fundamentally, in a model which treats the D7-brane radius as a cut-off one does not know how to compute accurately in the AdS/CFT correspondence. For this reason we deal only with the = 4 SYM theory and a probe D7-brane, for which the AdS/CFT correspondence N is precise. A spacelike string lying straight along a constant radius is discussed briefly in [86]. This string also approaches the “trivial” lightlike solution in [34] as the radius is taken to infinity. As pointed out in [86], this straight string at finite radius is not a solution of the (full, second order) equations of motion, and should be rejected. We emphasize that our dominant string solutions are not this straight string, even though they approach the straight string as the D7-brane radius goes to infinity, and are genuine solutions to the full equations of motion. To conclude, the results in this chapter show these solutions to be robust, in the sense that they give the same contribution to the path integral independently of how the lightlike limit is taken. Furthermore, though not a compelling argument, the fact that these solutions do not exhibit any drag is consistent with the fact that they giveq ˆ = 0. Therefore, for the non- perturbative definition ofq ˆ given in [34], direct computation ofq ˆ = 0 within the AdS/CFT 6 correspondence for = 4 SU(N) SYM would require either a compelling argument for N discarding the leading contribution to the path integral, or a different class of string solutions giving the dominant contribution. On the other hand, this computation may simply imply that at large N and strong ’t Hooft coupling, the mechanism for relativistic parton energy loss in the SYM thermal bath gives a linear rather than quadratic dependence on the Wilson loop width L.

88 5.1 String embeddings and equations of motion

We consider a smooth and stationary string in the background of a ﬁve dimensional AdS black hole with the metric r4 r4 r2 r2R2 ds2 = h dxµdxν = − 0 dx2 + (dx2 + dx2 + dx2)+ dr2. (5.1) 5 µν − r2R2 0 R2 1 2 3 r4 r4 − 0

R is the curvature radius of the AdS space, and the black hole horizon is located at r = r0.

We put the endpoints of the string at the minimal radius r7 that is reached by a probe D7-brane. The classical dynamics of the string in this background is described by the Nambu-Goto action 1 S = dσdτ √ G, (5.2) −2πα′ − Z with

µ α ν β G = det [hµν (∂X /∂ξ )(∂X /∂ξ )], (5.3) where ξα = τ, σ and Xµ = x , x , x , x ,r . { } { 0 1 2 3 } The steady state of a quark-antiquark pair with constant separation and moving with constant velocity either perpendicular or parallel to the separation of the quarks can be described (up to worldsheet reparametrizations), respectively, by the worldsheet embeddings

[v⊥] : x0 = τ, x1 = vτ, x2 = σ, x3 =0, r = r(σ),

[v||] : x0 = τ, x1 = vτ + σ, x2 =0, x3 =0, r = r(σ). (5.4)

For both cases, we take boundary conditions

0 τ T, L/2 σ L/2, r( L/2) = r , (5.5) ≤ ≤ − ≤ ≤ ± 7 where r(σ) is a smooth embedding. The endpoints of strings on D-branes satisfy Neumann boundary conditions in the directions along the D-brane, whereas the above boundary conditions are Dirichlet, constraining the string endpoints to lie along ﬁxed worldlines a distance L apart on the D7-brane. The correct way to impose these boundary conditions is to turn on a worldvolume background U(1) ﬁeld strength on the D7-brane [27] to keep the string endpoints a distance L apart.

Thus at ﬁnite r7, it is physically more sensible to describe string solutions for a ﬁxed force on

89 the endpoints instead of a fixed endpoint separation L.1 Our discussion of spacelike string solutions in the next section will describe both the force-dependence and the L-dependence of our solutions. In the application to evaluating a Wilson loop in section 5.3, though, we are interested in string solutions (in the r limit) with endpoints lying along the given 7 →∞ loop, i.e. at fixed L. According to the AdS/CFT correspondence, strings ending on the D7-brane are equivalent to quarks in a thermal bath in four-dimensional finite-temperature =4 SU(N) super N 4 ′2 Yang-Mills (SYM) theory. The standard gauge/gravity dictionary is that N = R /(4πα gs) 4 ′2 2 and λ = R /α where gs is the string coupling, λ := gYMN is the ’t Hooft coupling of the SYM theory. In the semi-classical string limit, i.e. g 0 and N , the supergravity s → → ∞ approximation in the gauge/gravity correspondence holds when the curvatures are much ′ greater than the string length ℓs := √α . Furthermore, in this limit, one identifies

2 ′ β = πR /r0, m0 = r7/(2πα ), (5.6) where β is the (inverse) temperature of the SYM thermal bath, and m0 is the quark mass at zero temperature. It will be important to note that the velocity parameter v entering in the string worldsheet embeddings (5.4) is not the proper velocity of the string endpoints. Indeed, from (5.1) it is easy to compute that the string endpoints at r = r7 move with proper velocity

r2 V = 7 v. (5.7) r4 r4 7 − 0 We will see shortly that real string solutionsp must have the same signature everywhere on the worldsheet. Thus a string wroldsheet will be timelike or spacelike depending on whether V , rather than v, is greater or less than 1. Thus, translating V ≶ 1 into corresponding inequalities for the velocity parameter v, we have

timelike 2 both v < 1 (γ > 1) and z7 > √γ, string worldsheet ⇔ (5.8) either v 1 (γ2 < 0) and any z , spacelike ≥ 7 string worldsheet ⇔  or v < 1 (γ2 > 1) and z < γ.  7 √ 1 We thank A. Karch for discussions on this point.

90 Timelike (v<1) and spacelike (v>1) worldsheets

r = √γ r0 Only spacelike worldsheets

r = r7 probe D7-brane

Event Horizon r = r0

Figure 5.1: Both timelike and spacelike worldsheets can exist above the radius r = √γr0 (blue line) for v< 1 and v> 1, respectively. On the other hand, only spacelike worldsheets exist in the region between the blue line and the event horizon, given by r0

Here we have defined the dimensionless ratio of the D7-brane radial position to the horizon radius, r 1 z := 7 , and γ2 := . (5.9) 7 r 1 v2 0 − Furthermore, since the worldsheet has the same signature everywhere, this implies that timelike strings can only exist for r > √γr0, but spacelike strings may exist at all r, as illustrated in figure 5.1. In this respect, r = √γr0 plays a role analogous to that of the ergosphere of a Kerr black hole, although in this case it is not actually an intrinsic feature of the background geometry but instead a property of certain string configurations (5.4) in the background geometry (5.1). With the embeddings (5.4) and boundary conditions (5.5), the string action becomes2

T L/2 r4 γ2r4 r4 γ2r4 [v ] : S = − dσ − 0 + − 0 r′2, ⊥ γ πα′ R4 r4 r4 Z0 s − 0 T L/2 r4 r4 r4 γ2r4 [v ] : S = − dσ γ2 − 0 + − 0 r′2, (5.10) || γ πα′ R4 r4 r4 Z0 s − 0 2These expressions for the string action are good only when there is a single turning point around which the string is symmetric. We will later see that for [v⊥] there exist solutions with multiple turns. For such solutions the limits of integration in (5.10) are changed, and appropriate terms for each turn of the string are summed.

91 where r′ := ∂r/∂σ. The resulting equations of motion are

′2 1 4 4 4 2 2 4 [v⊥] : r = 2 2 4 4 (r r0)(r γ [1 + a ]r0), γ a r0R − − γ2 (r4 [1 + a2]r4) [v ] : r′2 = (r4 r4)2 − 0 , (5.11) || a2r4R4 − 0 (r4 γ2r4) 0 − 0 where a2 is a real integration constant. Here we have taken the ﬁrst integral of the second order equations of motion which follows from the existence of a conserved momentum in the direction along the separation of the string endpoints. Since a is associated with this conserved momentum, a is proportional to the force applied (via a constant background | | U(1) ﬁeld strength on the D7 brane) to the string endpoints in this direction [27]. Although we have written a2 as a square, it can be either positive or negative. Using (5.11), the determinant of the induced worldsheet metric can be written as

1 4 2 4 2 [v⊥] : G = 4 2 4 4 (r γ r0) , −γ a r0R − 1 4 4 2 [v||] : G = 2 4 4 (r r0) . (5.12) −a r0R −

Thus, the sign of G is the same as that of a2 (since the other factors are squares of real − quantities). In particular, the worldsheet is timelike (G< 0) for a2 > 0 and spacelike (G> 0) for a2 < 0. The reality of r′ implies that the right sides of (5.11) must be positive in all these diﬀerent cases, which implies certain allowed ranges of r. Therefore, there can only be real string solutions when the ends of the string, at r = r7, are within this range. The edges of this range are (typically) the possible turning points rt for the string, whose possible values will be analyzed in the next section. Given these turning points, (5.11) can be integrated to give

L 2 aγ z7 dz [v⊥] : = | | , 4 4 2 2 β π zt (z 1)(z γ [1 + a ]) Z − − z L 2 a 7 p dz z4 γ2 [v||] : = | | − , (5.13) 4 4 2 β π γ zt (z 1) z [1 + a ] | | Z − p −

2 p where we have used r0 = πR /β. Also, in (5.13) we have rescaled z = r/r0 and likewise zt := rt/r0 and z7 := r7/r0. (The absolute value takes care of cases where z7 < zt.) These

92 integral expressions determine the integration constant a2 in terms of L/β and v. Also, we can evaluate the action for the solutions of (5.11):

T √λ z7 (z4 γ2) dz [v⊥] : S = − , 4 4 2 2 ± γβ z (z 1)(z γ [1 + a ]) Z t − − T √λ z7 p z4 γ2 [v ] : S = dz − , (5.14) || ± γβ z4 [1 + a2] Zzt s − where we have used R2/α′ = √λ. The plus or minus signs are to be chosen depending on 2 the relative sizes of z7, zt, and γ , and will be discussed in speciﬁc cases below. For ﬁnite z7, these integrals are convergent. They diverge when z and need to be regularized by 7 → ∞ subtracting the self-energy of the quark and the antiquark [23, 24], which will be discussed in more detail in section 5.3.

Note that, in writing (5.13) and (5.14), we have assumed that the string goes from z7 to the turning point zt and back only once. We will see that more complicated solutions with multiple turning points are possible. For these cases, one must simply add an appropriate term, as in (5.13) and (5.14), for each turn of the string.

5.2 Spacelike solutions

Positivity of the determinant of the induced worldsheet metric (5.12) implies that the integration constant a2 < 0 for spacelike configurations. It is convenient to define a real integration constant α by α2 := a2 > 0. (5.15) − As remarked above, α is proportional to the magnitude of a background U(1) field strength on the D7-brane. We will now classify the allowed ranges of r for which r′2 is positive in the equations of motion (5.11). These ranges, as well as the associated possible turning points of the string depend on the relative values of α, v and 1.

5.2.1 Perpendicular velocity

The conﬁgurations of main interest to us are those for which the string endpoints move in a direction perpendicular to their separation. As we will now see, the resulting solutions have markedly diﬀerent behavior depending on whether the velocity parameter is greater or less

93 z=2

z=1 HcL0 HcL0 HaL1 HbL1 HcL1 HaL2

Figure 5.2: Spacelike string solutions with ﬁxed L/β = 0.25, γ =20 (v 0.99875), z7 = 2, and with low values of n (the number of turns at the horizon). The horizon is the solid≈line at z = 1, and the minimum radius of the D7-brane is the dashed line at z = 2. than 1.

1 z−4 1 z−4. − 7 A case-by-case classification of the possible turning points of the v⊥ equation in (5.11)p gives the following table of possibilities: parameters allowed ranges 0 <α 1 region can reliably describe quarks, this eliminates the left column of allowed ranges. Thus, the only viable configurations are those in the right column with α < v, which all have turning points at 1 or γ2(1 α2). This − restricts the D7-brane minimum radius to lie between these two turning points which, in turn, gives rise to string configurations with multiple turns. In order for the D7-brane to be within the allowed range 1 < z4 < γ2(1 α2), the 7 − parameter v must be at least v2 > 1 z−4. For a given z , v, and α satisfying these − 7 7

94 inequalities, we integrate (5.13) to obtain L/β. There are two choices for the range of integration: [1, z ] and [z ,γ2(1 α2)]. The first one is appropriate for a string which 7 7 − decends down to the horizon and then turns back up to the D7-brane; we will call this a “down string”. The second range describes a string which ascends to larger radius and then turns back down to the D7-brane; we will call this an “up string”. Given these two behaviors, it is clear that we can equally well construct infinitely many other solutions by simply alternating segments of up and down strings. In particular, there are three possible series of string configurations, which we will call the (a)n, (b)n, and (c)n series. An (a)n string starts with a down string then adds n 1 pairs of up and down strings, thus ending − with a down string; a (b)n string concatenates n pairs of up and down strings— for example, starts with an up string and ends with a down string; and a (c)n string starts with an up string and then adds n pairs of down and up strings, thus ending with an up string. n counts the number of turns the string makes at the horizon, z = 1. In particular, for the (a)n and (b) series, n is an integer n 1, while for the (c) series, n 0. Examples of these string n ≥ n ≥ configurations appear in figure 5.2. If the separation of the ends of the up and down strings are Lup and Ldown, respectively, then the possible total separations of the strings fall into three classes of lengths

L = nL +(n 1)L , (a),n down − up L(b),n = nLdown + nLup,

L(c),n = nLdown +(n + 1)Lup. (5.16)

Figure 5.3 illustrates the systematics of the L(a,b,c),n dependence on α. Here we have chosen z7 = 2, so the minimum value of v for the solutions to exist has γ = 4. The leftmost plot illustrates that, for small γ, L = L L for all α. Thus, for each n 1, (c),0 up ≪ down ≥ L L L , and are virtually indistinguishable in the figure. As γ increases, L (a),n ≈ (b),n ≈ (c),n up and L begin to approach each other for most α, except for α near α := 1 z4/γ2, down max − 7 where Lup decreases sharply back to zero. p This behavior implies that, for every fixed L and v, there is a very large number of solutions3 in each series but that the minimum value of n that occurs decreases as v increases. In detail, it is not too hard to show that the pattern of appearance of solutions as v increases for fixed L is as follows: if for a given v there is one solution (i.e., value of α) for each

3Although n does not formally have an upper bound, as n increases the turns of the string become sharper and denser. Therefore, for large enough n the one can no longer ignore the backreaction of the string on the background.

95 LΒ HΓ=4.2L LΒ HΓ=7L LΒ HΓ=20L 0.5 1 1

0.25 0.5 0.5

Α Α Α 0.25 0.5 0.5 1 0.5 1

Figure 5.3: L/β as a function of α for spacelike string conﬁgurations with perpendicular velocity, z7 =2 and γ =4.2, 7, and 10. Green curves correspond to the (a)–series, blue to (b)–series, and red to (c)–series. Only the series up to n = 20 are shown; the rest would ﬁll the empty wedge near the L/β axis. Note that the scale of the γ =4.2 plot is half that of the other two.

(a,b,c)n–string with n > n0, then as v increases first two (c)n0 solutions will appear, then the (c)n0 solution with the greater α will disappear just as a (b)n0 and an (a)n0 solution appear. Also, α((a)n) < α((b)n) < α((c)n). For example, in figure 5.3, when L/β = 0.25 and γ =4.2, there are (a,b,c) solutions for n 2. Increasing v to γ = 7 (for the same L), n ≥ there are now (a,b,c) solutions for n 1. Increasing v further to γ = 20, there are now in n ≥ addition two (c)0 (i.e. up string) solutions. Figure 5.2 plots the string solutions when z7 = 2, L/β =0.25, and γ = 20, for low values of n. Note that if one keeps the D7-brane U(1) field strength, α, constant instead of the endpoint separation, L, then there will still be an infinite sequence of string solutions qualitatively similar to that shown in figure 5.2. In this case the endpoint separation L increases with the number of turns. The action for spacelike configurations is imaginary because the Nambu-Goto Lagrangian is √ G = i√G. Ignoring the i factor (which we will return to in the next section), the − ± ± integral of √G just gives the area of the worldsheet. Dividing by the “time” parameter T in (5.14) then gives the length of the string: ℓ = iS/T . Figure 5.4 plots the lengths of the ± various series of string configurations for increasing values of the velocity parameter. There are negative lengths because the length of a pair of straight strings stretched between the D7-brane and the horizon has been subtracted, for comparison purposes. It is clear from the

ﬁgure that the (c)0 up strings are the shortest for any given L less than a velocity-dependent critical value. Furthermore, for L small enough, they are also shorter than the straight strings. In particular, the shorter (larger α) of the two up strings has the smallest ℓ of all. As

96 { HΓ=6L { HΓ=15L { HΓ=100L 3 6 6

2 4 4 1 2 2 LΒ 1 2 LΒ LΒ 2 4 2 4 -1 -1 -1

Figure 5.4: Spacelike string lengths ℓ in units of √λ/β as a function of endpoint separation L/β and z7 = 2, for γ = 6, 15 and 100. The gray line along the L/β axis is the (subtracted) length of a pair of straight strings stretched between the D7-brane and the horizon. Note that the scale of the γ = 6 plot is half that of the others. v 1, the critical value of L below which the up string is the solution with the minimum → action increases without bound. In this case, any of the other spacelike strings will decay to this minimum-action conﬁguration. Therefore, it is this conﬁguration which must be used for any calculations of physical quantities, such as the jet-quenching parameterq ˆ. v > 1

A case-by-case classiﬁcation of the possible turning points of the v⊥ equation in (5.11) when v > 1 gives the following table of possibilities: parameters allowed ranges 0 <α< 1 < v 1 z4 < ≤ ∞ 0 < 1 <α 1. In the above table we have written “0 z4” when there is no turning ≤ point in an allowed region before the singularity at z = 0. In these cases, a string extending towards smaller z will necessarily meet the singularity. As before, we are only interested in string solutions that extend into the z > 1 region. This eliminates the left column of allowed ranges, with the possible exception of the v = α > 1 crossover case, for which the string

97 iLv8 Β 2

Αv 1 2

Figure 5.5: Lv8/β as a function of α/v for spacelike string configurations with perpendicular velocity, z7 = 2, and v =1.005 (red), 1.05 (green), and 1.2 (blue). might inflect at the horizon and then extend inside. However, if it does extend inside, then it will hit the singularity. Therefore, we can also discard this possibility as being outside the regime of validity of our approximation. Thus, the only viable configurations are those given in the right column, which all have turning points at either 1 or γ2(1 α2), or else go off to − infinity. Since we want to identify the quarks with the ends of the strings on the D7-brane, we are only interested in string configurations that begin and end at z7 > 1, and so discard configurations which go off to z instead of turning. Thus the v > 1 ranges compatible →∞ with these conditions all have only one turning point, describing strings dipping down from the D7-brane and either turning at the horizon or above it, depending on α versus v. Indeed, it is straightforward to check that for any L there are two v > 1 solutions, one with α > v and one with α < v. L/β as a function of α is plotted in figure 5.5. (The rescalings by powers of v are just so the curves will nest nicely in the figure.) The α < v solutions are long strings which turn at the horizon, while the α > v solutions are short strings with turning point z4 = (α2 1)/(v2 1). The norm of the action for these t − − configurations (which is proportional to the length of the strings) is likewise greater for the α < v solutions than for the α > v ones. If, instead, one keeps the D7-brane U(1) field strength α constant, then there is at most a single string solution with a given velocity v.

98 5.2.2 Parallel velocity

For completenes, though we will not be using these confurations in the rest of the chapter, we briefly outline the set of solutions for suspended spacelike string configurations with velocity parallel to the endpoint separation. A case-by-case analysis of the equations of motion (5.11) gives the following table of allowed ranges for string solutions: parameters allowed ranges 0 < (α, v) < 1 1 α2 z4 < 1 1 1 region, since that is where we can reliably put D7-branes. This eliminates the left-hand column of configurations. Recall that the signature of the worldsheet metric for a string with v < 1 changes at z = √γ. A string that reaches this radius will have a cusp there. This is qualitatively similar to the timelike parallel solutions with cusps [35] described in chapter 4, except that in the spacelike case the strings extend away from the horizon (towards greater z). Thus, the string solutions corresponding to the ranges in the right-hand column all either asymptote to z =1, go offto infinity or have a cusp at z = √γ. The first two cases do not give strings with two endpoints on the D7-brane at z = z7. Therefore, the only potentially interesting configurations for our purposes are those with 1 < z7

99 5.3 Application to jet-quenching

We will now apply the results of the last two sections to the computation of the expectation value of a certain Wilson loop W [ ] in the SYM theory. The interest of this Wilson loop is C that it has been proposed [34] as a non-perturbative definition of the jet-quenching parameter qˆ. This medium-dependent quantity measures the rate per unit distance traveled at which the average transverse momentum-squared is lost by a parton moving in plasma [55]. In particular, [34] considered a rectangular loop with parallel lightlike edges a distance C L apart which extend for a time duration T . Motivated by a weak-coupling argument, the leading behavior of W [ ] (after self-energy subtractions) for large T and Lβ 1 is claimed C ≪ to be 1 W A( ) = exp[ qˆ T L2], (5.17) h C i −4 where W A( ) is the thermal expectation value of the Wilson loop in the adjoint represen- h C i tation. We will simply view this as a definition ofq ˆ.4 Note that exponentiating the Nambu- Goto action gives rise to the thermal expectation value of the Wilson loop in the fundamental representation W F ( ) . Therefore, we will make use of the relation W A( ) W F ( ) 2, h C i h C i≈h C i which is valid at large N. Self-energy contributions are expected to contribute on the order of T L0 and, since this is independent of L, their subtraction does not affect the L-dependence of the results. The subtraction is chosen to remove infinite constant contributions, but are ambiguous up to finite terms.5 However, there may be other leading contributions of order T L−1 or T L. For example, as we discussed in the conclusions, a term linear in L would be consistent with energy loss by elastic scattering. Therefore, one requires a subtraction prescription. We will assume the following one: extract 2−3/2qˆ as the coefficient of L2 in a Laurent expansion of the action around L = 0. Thus, concretely,

α qˆ W [ ] exp T + −1 + α + α L + L2 + . (5.18) C ∼ − ··· L 0 1 4 ··· Implicit in this is a choice of ﬁnite parts of leading terms to be subtracted, which could aﬀect

4This diﬀers by a constant factor from the deﬁnition written in [34] since here it is expressed in the reference frame of the plasma rather than that of the parton. 5Note that [96] shows that the correct treatment of the (BPS) Wilson loop boundary conditions should automatically and uniquely subtract divergent contributions; it would be interesting to evaluate our (non- BPS) Wilson loop using a possible generalization of this prescription instead of the more ad hoc one used here and throughout the literature.

100 the value ofq ˆ; we have no justification for this prescription beyond its simplicity. We will see that this issue of L-dependent leading terms indeed arises in the computation ofq ˆ using the AdS/CFT correspondence. There is a second subtlety in the definition ofq ˆ given in (5.18), which involves how the lightlike limit of is approached. In the AdS/CFT correspondence, we evaluate the C expectation value of the Wilson loop as the exponential exp iS of the Nambu-Goto action { } for a string with boundary conditions corresponding to the Wilson loop . If we treat as C C the lightlike limit of a sequence of timelike loops, then the string worldsheet will be timelike and the exponential will be oscillatory, instead of exponentially suppressed in T as in (5.18). The exponential suppression requires either an imaginary action (of the correct sign) or a Wick rotation to Euclidean signature. The authors of [34] advocate the use of the lightlike limit of spacelike strings to evaluate the Wilson loop [29, 86]. Below, we will evaluate the Wilson loop using both the spacelike prescription and the Euclidean one. Our interest in the Euclidean Wilson loop is mainly for comparison purposes and to help elucidate some subtleties in the calculation; we emphasize that it is not the one proposed by the authors of [34] to evaluateq ˆ. (Though the Euclidean prescription is the usual one for evaluating static thermodynamic quantities, we are here evaluating a non-static property of the SYM plasma and so the usual prescription may not apply.) In both cases we will find that, regardless of the manner in which the above ambiguities are resolved, the computed value ofq ˆ is zero.

5.3.1 Euclidean Wilson loop

Euclidean string solutions [35] are reviewed in the appendix. Here we just note their salient properties. In Euclidean signature, nothing special happens in the limit V 1 (v → → 1 z−4). When V = 1 there are always only two Euclidean string solutions: the “long − 7 string”,p with turning point at the horizon z = 1, and the “short string”, with turning point above the horizon. The one which gives the dominant contribution to the path integral is the one with smallest Euclidean action. For endpoint separation L less than a critical value, the dominant solution is the short string. This is the string conﬁguration that remains the furthest from the black hole horizon [35]. We are interested in evaluating the Euclidean string action for the short string in the

101 small L limit (the so-called “dipole approximation”). However, there is a subtlety associated with taking this limit since it does not commute with taking the z limit, which 7 → ∞ corresponds to inﬁnite quark mass. Recall that the quark mass scales as r7 in string units; introduce a rescaled length parameter

1 r ǫ := = 0 (5.19) z7 r7 associated with the Compton wavelength of the quark. Then the behavior of the Wilson loop depends on how we parametrically take the L 0 and ǫ 0 limits. For instance, if → → one keeps the mass (ǫ−1) fixed and takes L 0 first, then the Wilson loop will reflect the → overlap of the quark wave functions. On the other hand, if one takes ǫ 0 before L, then → the Wilson loop should reflect the response of the plasma to classical sources. The second limit is presumably the more physically relevant one for extracting theq ˆ parameter. We perform the calculation in both limits in the appendix to verify this intuition. In the L 0 limit at fixed (small) ǫ, the action of the short string as a function of L → and ǫ is found in the appendix to be

√ 2 3 πT λ L 1 4 8 π L 1 1 4 8 L5 S = 1+ ǫ + (ǫ ) ǫ + (ǫ ) + 4 6 . (5.20) √2β2 ǫ2 4 O − β2ǫ4 3 − 6 O O β ǫ (In fact, this result is valid as long as L 0 as L ǫ or faster.) The main thing to note → ∝ about this expression is that it is divergent as ǫ 0. This is not a self-energy divergence → that we failed to subtract, since any self-energy subtraction (e.g. subtracting the action of two straight strings extending radially from z = z7 to z = 1) will be independent of the quark separation and so cannot cancel the divergences in (5.20). (In fact, it inevitably adds an ǫ−1L0 divergent piece.) This divergence as the quark mass is taken infinite is a signal of the unphysical nature of this order of limits. The other order of limits, in which ǫ 0 at fixed (small) L, is expected to reflect more → physical behavior. Indeed, the appendix gives

βSˆ β L3 = 0.32 +1.08 0.76 + (L7). (5.21) T √λ − L − β3 O

Here Sˆ is the action with self-energy subtractions. This result (which is in the large mass, or ǫ 0, limit) is ﬁnite for ﬁnite quark separation L. The L−1 term recovers the expected → Coulombic interaction. Since there is no L2 term in (5.21), the subtraction prescription

102 (5.18) implies that the Euclidean analog of the jet-quenching parameter vanishes. For the sake of comparison, we also compute the long string action in this limit with the same regularization in the appendix, giving

Sˆ L2 long = +2.39 + (L4). (5.22) T √λ β3 O

This does have the leading L2 dependence, giving rise to an unambiguous nonzeroq ˆ. But it is exponentially suppressed compared to the short string contribution (5.21), and so gives no contribution to the eﬀectiveq ˆ in the T limit. →∞

5.3.2 Spacelike Wilson loop

We now turn to the spacelike prescription for calculating the Wilson loop. We will show that a similar qualitative behavior to that of the Euclidean path integral shown in (5.21) and (5.22) also holds for spacelike strings. In particular, the leading contribution is dominated by a confining-like (L) behavior with no jet-quenching-like (L2) subleading term, and only an exponentially suppressed longer-string contribution has a leading jet-quenching-like behavior. The analogous results are recorded in (5.23) and (5.24), below. Since G < 0 for spacelike worldsheets, the Nambu-Goto action is imaginary and so − exp iS = exp A , where A is the positive real area of the string worldsheet. The sign { } {± } ambiguity comes from the square root in the Nambu-Goto action. For our stationary string solutions, the worldsheet area is the time of propagation T times the length of the string. Thus, with the choice of the plus sign in the exponent, the longest string length exponentially dominates the path integral, while for the minus sign, the shortest string length dominates. Only the minus sign is physically sensible, though, since we have seen in section 5.2 that the length of the spacelike string solutions is unbounded from above (since there are solutions with arbitrarily many turns). Thus we must pick the minus sign, and, as in the Euclidean case, the solution with shortest string length exponentially dominates the path integral. As we illustrated in our discussion of the Euclidean Wilson loop, the physically sensible limit is to take the quarks infinitely massive (z ) at fixed quark separation L. In the 7 →∞ spacelike case, however, there is a new subtlety: a priori it is not obvious that the lightlike limit V 1 will commute with the z limit. Since V = v(1 z−4)−1/2, the lightlike → 7 → ∞ − 7 limit is v 1 when z . We will examine four different approaches to this limit, shown → 7 →∞

103 in ﬁgure 5.6.6

z7 2 HdL + (a) limz7→∞ limV →1

HaL (b) limz →∞ limv→1− 1.5 7

+ (c) limz7→∞ limv→1 HbLHcL (d) lim + limz →∞ v v→1 7 1 2

Figure 5.6: The shaded region is the set of (v,z7) for which the string worldsheet is spacelike and outside the horizon. The curved boundary corresponds to lightlike worldsheets. The various approaches to the lightlike z7 = limit discussed in the text are shown. ∞

Limit (a): limz7→∞ limV→1+ .

This is the limit in which we take the lightlike limit at fixed z7, then take the mass to infinity. Recall from (5.8) that a spacelike worldsheet requires either v 1 (γ2 < 0) for any ≥ 2 2 z7, or v < 1 (γ > 1) and z7 < √γ. Since, at fixed z7, V = 1 corresponds to γ = z7 , we necessarily have v < 1. Thus, only the v < 1 spacelike solutions discussed in section 5.2.1 will contribute. Recall that for these solutions 1 z4 γ2(1 α2), where the integration ≤ ≤ − constant is in the range 0 <α

− Limit (b): limz7→∞ limv→1 .

Another approach to the lightlike limit takes v 1 from below and then takes z . → 7 → ∞ Then the conditions for a spacelike worldsheet are automatically satisfied. Again, only the v < 1 spacelike solutions discussed in section 5.2.1 contribute, but now the γ2(1 α2) z4 − ≥ 7 condition places no restrictions on α. In particular, this limit will exist at fixed L. The behavior of figure 5.3 as v 1 suggests that, at any given (small) L, all the series of string → solutions illustrated in figure 5.2 occur. The lengths of these strings follow the pattern

6See [76] for a related discussion of the lightlike limit.

104 plotted in ﬁgure 5.4. Actually, the analysis given in the appendix shows that the short (c)0

“up” string does not exist in the limit with ﬁxed L. Thus the long (c)0 “up” string dominates the path integral, with the (a)1 “down” string and all longer strings relatively exponentially suppressed.

The result from the appendix for the action of the (c)0 string as a function of L is

βSˆ π L = 1.31 + . (5.23) T √λ − 2 β This result is exact, in the sense that no higher powers of L enter. The constant term is from the straight string subtraction. The linear term is consistent with energy loss by elastic scattering.

For comparison, the next shortest string is the (a)1 down string solution. Computations in the appendix determine its action to be

βSˆ L2 long =0.941 + (L4), (5.24) T √λ β2 O which shows the jet-quenching behavior found in [34]. However, since the contribution from this conﬁguration to the path integral is exponentially suppressed, the actual jet-quenching parameter is zero.

Limit (c): limz7→∞ limv→1+ .

When v > 1, the string worldsheet is spacelike regardless of the value of z7. Thus, we are free to take the order of limits in many ways. Limit (c) takes v 1 from above at ﬁxed z → 7 and then takes z . We saw in section 5.2.1 that there are always two string solutions 7 →∞ for v > 1: a short one with α > v, which turns at z = z := γ2(1 α2), and a long one with t − α < v, which turns at the horizon z = 1. In the appendix we show that in the (c) limit, the short string gives precisely the same contribution as the (c)0 up string did in the (b) limit.

Similarly, the long string contirbution coincides with the (a)1 down string. This agreement is reassuring, showing that the path integral does not jump discontinuously between the (b) and (c) limits even though they are evaluated on qualitatively different string configurations. (The (b) and (c) limits approach the lightlike limit in the same way, see figure 5.6.)

105 Limit (d): limv→1+ limz7→∞.

Limit (d) approaches the lightlike limit in the opposite order to the (c) limit. Somewhat unexpectedly, the results for the string action in the (d) limit are numerically the same as those found in the (b) and (c) limits. This is unexpected since the details of evaluating the integrals in the (c) and (d) limits are substantially diﬀerent. We take this agreement as evidence that the result is independent of how the lightlike limit is taken. (Note that there are, in principle, many diﬀerent lightlike limits intermediate between the (c) and (d) limts.)

5.4 Appendix

5.4.1 Euclidean action

For the Euclidean string solutions Wick rotate x ix in (5.1), and adopt the rotated 0 → 4 boundary conditions (5.5) with x x . Then the Euclidean version of the [v ] embedding 0 → 4 ⊥ (5.4) becomes

x4 = τ, x1 = vτ, x2 = σ, x3 =0, r = r(σ). (5.25)

One then ﬁnds [35] that the integral expressions (5.13) and (5.14) for the quark separation and string action stay the same except for the replacement

1 γ2 γ2 := . (5.26) → E 1+ v2

Real Euclidean string conﬁgurations must have the integration constant a2 be positive, to have positive G. Then an analysis of the Euclidean string equations of motion [35] shows that real solutions can exist for any v as long as the string is at radii satisfying

4 2 2 z > max 1, γE(1 + a ) . (5.27) (These are for the string conﬁgurations with endpoint “velocity” perpendicular to their separation.) For a > v and v suﬃciently large, there is a unique Euclidean solution with turning point 4 4 2 2 at z = zt := (1+ a )/(1 + v ) > 1. We call these the “short string” solutions. For a < v there is a branch of Euclidean solutions which have the radial turning point on the black

106 hole horizon z = 1. These are the “long string” solutions. The solution with the smallest energy dominates the path integral. The energy of the Euclidean string conﬁgurations is given by E = S/T , where S is the Nambu-Goto action and T is the time interval. For L less than a critical value, the energetically favorable state is the short string [35]. The Euclidean rotation of strings whose endpoints have lightlike worldlines are those with Euclidean worldsheet (5.25) with V = 1. By (5.7) this is when v = 1 z−4. But − 7 since nothing special happens to the Euclidean string conﬁgurations at this velocity,p we will do our computations below at arbitrary v, and specialize to the lightlike value at the end.

−1 We are interested in evaluating the action for this string in the small L and small ǫ := z7 (large mass) limit. These two limits do not commute, so we evaluate them separately in the two diﬀerent orders.

L 0 at ﬁxed (small) ǫ From (5.13) with γ γ , the L 0 limit corresponds to → → E → taking z z . So introduce a small parameter δ deﬁned by t → 7 z4 1 z4 := 7 = . (5.28) t (1 + δ)4 ǫ4(1 + δ)4

Thus δ replaces the parameter a. Changing variables to y = ǫ(1 + δ)z, (5.13) and (5.14) can be rewritten in terms of δ as

L 2 1+δ dy 1 γ2 ǫ4(1 + δ)4 = ǫ(1 + δ) − E , 4 4 4 4 β π 1 (y 1)(y ǫ (1 + δ) ) Z p− − βS 1 1+δ dy [y4 γ2 ǫ4(1 + δ)4] = p − E . (5.29) 4 4 4 4 T √λ γEǫ(1 + δ) 1 (y 1)(y ǫ (1 + δ) ) Z − − Systematically expanding in small δ gives seriesp expressions in terms of integrals of the form

1+δ y4mdy Jnm := , (5.30) 4 1 4 4 1 +n 1 (y 1) 2 (y ǫ ) 2 Z − − ∞ 1 n+ 2 which have a series expansion of the form n=0 cnδ , but whose coeﬃcients cn(ǫ) lack closed-form expressions. Nevertheless, the JmnP are uniformly convergent for ǫ< 1, so we can

107 expand the integrands in power series in small ǫ to ﬁnd

L 2ǫδ1/2 = 1+ 1 (1 γ2 )ǫ4 + (ǫ8) β π 2 − E O n + δ 1 + 1 (33 49γ2 )ǫ4 + (ǫ8) + (δ2) , 12 24 − E O O 1/2 o βS δ 1 2 4 8 = 1+ 2 (1 γE)ǫ + (ǫ ) T √λ γEǫ − O n + δ 5 + 17 (1 γ2 )ǫ4 + (ǫ8) + (δ2) . − 4 24 − E O O o Eliminating δ between these two expressions order-by-order in δ then gives the action as a function of L and ǫ: βS π L 1 π3 L3 1 1 = + (ǫ6) (2 γ2 )+ (ǫ4) + (L5). (5.31) √ 2γ β ǫ2 O − 6γ β3 ǫ4 − 2 − E O O T λ E E 4 −1/2 The lightlike limit corresponds to taking γE =(2+ ǫ ) , giving

βS π L 1 1 π3 L3 1 1 = + ǫ2 + (ǫ6) + (ǫ4) + (L5). (5.32) √ √2 β ǫ2 4 O − 3√2 β3 ǫ4 − 2 O O T λ

Note that because of the nice convergence properties of the integrals in (5.30), the order of limits as ǫ 0 and δ 0 does not aﬀect this result. The limiting case where ǫ 0 with → → → δ ﬁxed (and small) corresponds to taking L ǫ. Thus the result (5.32) is valid for all limits ∝ ǫ 0 with L 0 as L ǫ or faster. → → ∝

ǫ 0 limit at fixed L To keep L fixed, examination of (5.29) shows that we need to scale → δ as ǫ 0 keeping ǫ(1+ δ) fixed. So change variables in (5.29) from δ to ℓ := ǫ(1+ δ). →∞ → Since for fixed ℓ < 1 the integral is convergent, we can take the ǫ 0 limit directly, and → then expand in powers of ℓ to get

L 2 ∞ dy 1 γ2 ℓ4 2 Γ[ 3 ] 3 5γ2 = ℓ − E = 4 ℓ 1+ − E ℓ4 + (ℓ8) . (5.33) 4 4 4 1 β π 1 (y 1)(y ℓ ) √π Γ[ ] 10 O Z p− − 4 Similarly, the S integralp is

βS 1 ℓ/ǫ dy [y4 γ2 ℓ4] = lim − E 4 4 4 T √λ ǫ→0 γEℓ 1 (y 1)(y ℓ ) Z − − 1 √π Γ[ 3 ] 1 2γ2 = p4 1 − E ℓ4 + (ℓ8) , (5.34) ǫγ − ℓγ Γ[ 1 ] − 2 O E E 4 108 where we used the fact that only the leading term at small ℓ diverges as 1/ǫ, and so the ǫ 0 limit can be taken directly in all the other terms. → Since S is divergent as ǫ 0, we regulate the action by subtracting the action of a pair → of straight strings with the same boundary conditions. (See, however, [76] for a discussion of an alternative regularization procedure.) The straight string solutions have embeddings

x = τ, x = vτ, x = L/2, x =0, r = σ, (5.35) 4 1 2 ± 3 with boundary conditions 0 τ T and r σ r . The action evaluated on these two ≤ ≤ 0 ≤ ≤ 7 solutions is then

1/ǫ 4 2 3 βS0 1 z γE 1 √π Γ[ 4 ] 1 1 1 2 = lim dz 4− = 1 2F1[ 2 , 4 , 4 ,γE], T √λ ǫ→0 γE 1 s z 1 ǫγE − γE Γ[ ] − − Z − 4 where F is a hypergeometric function. Thus the regularized action Sˆ := S S is 2 1 − 0 βSˆ √π Γ[ 3 ] 1 1 2γ2 = 4 F [ 1 , 1 , 1 ,γ2 ] + − E ℓ3 + (ℓ7) . (5.36) √ γ Γ[ 1 ] 2 1 − 2 − 4 4 E − ℓ 2 O T λ E 4

Eliminating ℓ order-by-order between (5.33) and (5.36) gives

ˆ 3 4 3 2 3 βS Γ[ 4 ] β Γ[ 4 ] 1 1 1 2 1 4 2 L 7 = 2 + 2F1[ 2 , 4 , 4 ,γE]+Γ[ 4 ] (2 5γE) 3 + (L ). (5.37) T √λ −π γE L √2πγE − − − β O

2 The lightlike limit corresponds to γE =1/2, giving

βSˆ β L3 = 0.32 +1.08 0.76 + (L7). (5.38) T √λ − L − β3 O

ǫ 0 limit at fixed L: long string For comparison purposes, we also compute the → contribution to the Wilson loop from the long Euclidean string solution. This is the solution with turning point at the horizon, zt = 1. The integral expression for L is convergent as ǫ and, to keep L fixed and small, we just need to keep a fixed and small. So expanding →∞

109 in small a gives

L 2aγ ∞ dz = E (5.39) 4 4 2 2 β π 1 (z 1)(z γ [1 + a ]) Z − − E 3/2 1/2 1 2 3 3 5 2 3 2 3 7 9 2 = aγE 2 π pΓ[ ] 2F1[ , , ,γ ] γ 2F1[ , , ,γ ] 4 4 2 4 E − 5 E 2 4 4 E h + 3 a2γ2 F [ 3 , 7 , 9 ,γ2 ]+ (a4) . 10 E 2 1 2 4 4 E O i The same expansion of the regularized action gives

βSˆ 1 ∞ dz z4 γ2 z4 γ2 long = − E − E 1 (5.40) 4 4 2 2 T √λ γE 1 √z 1 z γ [1 + a ] − ! Z p − p− E = a2γ 2−1/2π3/2Γ[ 1 ]−2 Fp[ 3 , 3 , 5 ,γ2 ] 3 γ2 F [ 3 , 7 , 9 ,γ2 ] E 4 2 1 4 2 4 E − 5 E 2 1 2 4 4 E h + 9 a2γ2 F [ 3 , 7 , 9 ,γ2 ]+ (a4) . 20 E 2 1 2 4 4 E O i Eliminating a order-by-order between these two expressions gives

ˆ 2 βSlong L 5√π 1 2 3 3 5 2 2 3 7 9 2 −1 4 = 2 Γ[ 4 ] 5 2F1[ 2 , 4 , 4 ,γE] 3 γE 2F1[ 2 , 4 , 4 ,γE] + (L ). (5.41) T √λ β 8√2γE − O 2 The lightlike limit is γE =1/2, giving

βSˆ L2 long = +2.39 + (L4). (5.42) T √λ β2 O

5.4.2 Spacelike action

Here we calculate the regulated action of spacelike strings in the various limits described in section 5.3, keeping L ﬁxed and small. Speciﬁcally, this requires expanding the quark separation (5.13) and string action (5.14) in terms of the appropriate small parameter, and then eliminating that parameter to obtain Sˆ as a function of L. Recall from the discussion in section 5.2 that for spacelike strings the integration constant a is imaginary, so we replace a2 = α2 with α2 > 0. Also, recall that γ2 := (1 v2)−1 and ǫ := z−1. Then our main − − 7 equations (5.13) and (5.14) become

L 2αγ 1/ǫ dz = , (5.43) 4 4 2 2 β π zt (z 1)(z γ (1 α )) Z − − −

110 and

βSˆ 1 1/ǫ (z4 γ2) dz 1/ǫ z4 γ2 = − dz − , (5.44) √ 4 4 2 2 4 T λ γ ( zt (z 1)(z γ (1 α )) − 1 z 1 ) | | Z − − − Z r −

p 4 2 2 where zt is either z t = 1 or z := γ (1 α ), depending on which conﬁguration we are t − considering, and we have subtracted the action of two straight strings to regulate the action. These expressions are valid for string solutions which have a single turning point, which will be the only ones we evaluate.

The (b) limit: the (c)0 “up string” solutions The (c)0 string solutions have a turning point at z4 = γ2(1 α2) > z4. Since in this limit we first take v 1− (γ + ) before t − 7 → → ∞ taking ǫ 0, (5.43) becomes → 1/2 2 1/4 L 2αγ γ (1−α ) dz = . (5.45) 4 2 2 4 β π 1/ǫ (z 1)(γ (1 α ) z ) Z − − − p The upper limit of this integral gives a contribution that scales as α(1 α2)−3/4γ−1/2, while − the lower limit gives α(1 α2)−1/2ǫ. Therefore, to keep L fixed in this limit requires that − either the contribution from the upper limit or from the lower limit remains finite and non- zero. In order for the upper limit to remain finite and non-zero, it is required 1 α2 γ−2/3. − ∼ However, then the lower limit contributes γ1/3ǫ which diverges as we take γ . Thus, ∼ →∞ this scaling does not keep L finite. If, instead, we demand that the lower limit remain finite, we must take 1 α2 ǫ2. This then implies that the upper limit contributes (γǫ3)−1/2 − ∼ ∼ which vanishes in the γ limit, and so L remains finite. →∞ We have thus found that there is a single (b) limit of the (c)0 up strings which keeps the quark separation finite. This limit keeps

ǫ2 δ2 := , (5.46) 1 α2 − fixed and gives L δ for small δ. Since for fixed L and ǫ and γ this limit keeps α ∼ → ∞ fixed away from α = 1, then from the discussion in section 5.2.1 we see that this solution corresponds to the long (c)0 string (see figure 5.3). In particular, the short (c)0 string does not contribute. Plugging (5.46) into (5.45), changing variables to y = (γǫ/δ)−1/2z, and expanding the (z4 1)−1/2 factor for large z gives a series of hypergeometric integrals which are finite in −

111 the γ limit, giving →∞ L 2 2 = lim √δ2 ǫ2 1+ 1 ǫ4 + (ǫ8) = δ. (5.47) β ǫ→0 π − 5 O π Similarly, the regularized action (5.44) becomes

βSˆ 1 √γǫ/δ (γ2 z4) dz 1/ǫ γ2 z4 = − dz − , (5.48) 4 2 2 −2 4 4 T √λ γ ( 1/ǫ (z 1)(γ ǫ δ z ) − 1 z 1 ) Z − − Z r − which, upon making the same changep of variables and taking the large γ limit, becomes

βSˆ √π Γ[ 1 ] = 4 + lim (δ + ǫ) 1+ 1 ǫ4 + (ǫ8) = 1.31 + δ. (5.49) 3 ǫ→0 10 T √λ − 4 Γ[ 4 ] O − Eliminating δ between (5.47) and (5.49) gives

βSˆ π L = 1.31 + . (5.50) T √λ − 2 β

The (b) limit: the (a)1 “down string” solution The (a)1 string descends from the D7-brane and turns at the horizon, so that the quark separation (5.43) is given by

L 2αγ 1/ǫ dz = . (5.51) 4 2 2 4 β π 1 (z 1)(γ (1 α ) z ) Z − − − In the γ limit, L is kept ﬁnite forp ﬁnite α. Taking the limit directly gives →∞ L 1 Γ[ 1 ] α = 4 . (5.52) β 2√π Γ[ 3 ] √1 α2 4 − Similarly, the limit of the regularized action gives

βSˆ √π Γ[ 1 ] 1 long = 4 1 . (5.53) T √λ 4 Γ[ 3 ] √1 α2 − 4 − Eliminating α between these two expressions and expanding in small L yields

ˆ 3/2 3 2 2 βSlong π Γ[ 4 ] L 4 L 4 = 1 2 + (L )=0.941 2 + (L ). (5.54) T √λ 2 Γ[ 4 ] β O β O

We now discuss the (c) and (d) limits. The (c) and (d) limits take v 1 with v > 1. In →

112 this range it is convenient to deﬁne

1 γ˜2 := γ2 = , (5.55) − v2 1 − so the v 1+ limit takesγ ˜2 + . The spacelike string solutions for v > 1 were discussed → → ∞ in section 5.2.1, where we found that there are two solutions: a short string with turning point z4 =γ ˜2(α2 1) and a long string with turning point at the horizon z = 1. t − t

The (c) limit: the short string solution For this solution, the integral expression for the quark separation (5.43) takes the form

L 2 1/ǫ dz = αγ˜ . (5.56) 4 4 2 2 β π γ˜1/2(α2−1)1/4 (z 1)(z γ˜ (α 1)) Z − − − The (c) limit takesγ ˜ first before takingp ǫ 0. Examination of (5.56) shows that, in →∞ → this limit, L remains finite if one takes α 1+ in such a way that → δ2 := ǫ6γ˜2[1 ǫ4γ˜2(α2 1)], (5.57) − − remains fixed. Eliminating α in favor of δ in (5.56) and changing variables to y = ǫz gives

L 2 1 ǫ (1 ǫ4y−4)−1/2 dy = 1 γ˜−2δ2ǫ−6 + ǫ4γ˜2 − . (5.58) − − 2 4 −2 2 −6 β π − (1−γ˜ 2δ2ǫ 6)1/4 y y 1+˜γ δ ǫ p Z − For small ﬁxed ǫ, expanding the numerator of the integrandp in a power series, performing the integrals, and taking theγ ˜ limit yields →∞ L δ δ = lim 1+ (ǫ4) = . (5.59) β ǫ→0 π O π Similarly, in the same limit, the regularized action

βSˆ 1/ǫ γ˜−1(z4 +˜γ2) dz 1/ǫ dz z4 +˜γ2 = , (5.60) 4 4 2 2 4 T √λ γ˜1/2(α2−1)1/4 (z 1)(z γ˜ (α 1)) − 1 γ˜ z 1 Z − − − Z r − becomes p βSˆ δ √π Γ[ 1 ] δ = lim 4 + (ǫ) = 1.31. (5.61) √ ǫ→0 2 − 4 Γ[ 3 ] O 2 − T λ 4

113 Eliminating δ between (5.59) and (5.61) gives

βSˆ π L = 1.31 + . (5.62) T √λ − 2 β

The (c) limit: the long string solution The turning point for the long string is at the horizon, so −1 L 2αγ˜ ǫ dz = , (5.63) 4 4 2 2 β π 1 (z 1)(z +˜γ (1 α )) Z − − from which it follows that L is kept ﬁnitep in theγ ˜ limit for ﬁnite α. Taking the limit →∞ directly then gives L 1 Γ[ 1 ] α = 4 . (5.64) β 2√π Γ[ 3 ] 1 α2 4 | − | Similarly, the limit of the regularized action givesp

βSˆ √π Γ[ 1 ] 1 long = 4 1 . (5.65) 3 2 T √λ 4 Γ[ 4 ] 1 α − ! | − | p Eliminating α between (5.64) and (5.65) and expanding in powers of small L gives

ˆ 3/2 3 2 2 βSlong π Γ[ 4 ] L 4 L 4 = 1 2 + (L )=0.941 2 + (L ). (5.66) T √λ 2 Γ[ 4 ] β O β O

Note that this calculation is essentially identical to that of the (a)1 string in the (b) limit.

The (d) limit: the short string solution The (d) limit takes ǫ 0 first, thenγ ˜ . → →∞ Examination of (5.56) shows that, in this limit, L remains finite if one takes α 1+ in such → a way that δ :=γ ˜−1/2(α2 1)−3/4 (5.67) − remains fixed. Eliminating α in favor of δ in (5.56) and changing variables to y =γ ˜−1/3δ1/3z, the ǫ andγ ˜ limits can be taken directly to give

L 2 ∞ y−2dy 2 Γ[ 3 ] = δ = 4 δ. (5.68) 4 1 β π 1 y 1 √π Γ[ ] Z − 4 p

114 The regularized action can be written as

βSˆ 1/ǫ γ˜−1(z4 +˜γ2) dz 1/ǫ dz z4 +˜γ2 = . (5.69) 4 4 4/3 4 T √λ (˜γ/δ)1/3 (z 1)(z (˜γ/δ) ) − 1 γ˜ z 1 Z − − Z r − To evaluate this expression aspγ ˜ (after ǫ 0), split the ranges of integration into → ∞ → z < γ˜1/2 and z > γ˜1/2. After the pieces of the integrals for z > γ˜1/2 which are divergent at ǫ 0 are canceled, the remainder is easily seen to vanish in theγ ˜ limit. The integrals → →∞ for z < γ˜1/2 are evaluated to give

2 2 βSˆ Γ[ 1 ] Γ[ 3 ] = 4 + 4 δ (5.70) T √λ −4√2π √2π in theγ ˜ limit. Eliminating δ between (5.68) and (5.70) gives →∞ 2 βSˆ Γ[ 1 ] π L π L = 4 + = 1.31 + . (5.71) T √λ −4√2π 2 β − 2 β

The (d) limit: the long string solution For the long strings with v > 1, recall that α< 1 and the turning point is at the horizon, so that

L 2αγ˜ ∞ dz = , (5.72) 4 4 2 2 β π 1 (z 1)(z +˜γ (1 α )) Z − − where we have already taken the ǫ 0p limit since the integral is convergent. L is kept small → and ﬁnite asγ ˜ if α is kept small and ﬁxed. Then the above integral can be evaluated →∞ by splitting the range of integration into z4 > γ˜2(1 α2) and z4 < γ˜2(1 α2). The upper − − range is easily seen to give a vanishing contribution in theγ ˜ limit, while the lower → ∞ range gives

1/2 L 2α γ dz(1 + z4γ˜−2(1 α2)−1)−1/2 1 Γ[ 1 ] α = lim − = 4 . (5.73) 2 4 3 2 β γ˜→∞ π 1 √1 α √z 1 2√π Γ[ ] √1 α Z − − 4 − Similarly evaluating the integral for the action gives

βSˆ √π Γ[ 1 ] 1 long = 4 1 . (5.74) T √λ 4 Γ[ 3 ] √1 α2 − 4 −

115 Eliminating α between (5.73) and (5.74) and expanding in powers of L gives

ˆ 3/2 3 2 2 βSlong π Γ[ 4 ] L 4 L 4 = 1 2 + (L )=0.941 2 + (L ). (5.75) T √λ 2 Γ[ 4 ] β O β O

Note that this calculation gives the same result as that of the (a)1 (down) string in the (b) limit, and the long string in the (c) limit.

116 117 Bibliography

[1] J. Wess and J. Bagger, Supersymmetry and supergravity, second edition, Princeton University Press.

[2] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Large N ﬁeld theories, string theory and gravity, Phys. Rept. 323 (2000) 183, [hep-th/9905111].

[3] N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857, [hep-th/9402044].

[4] N. Seiberg, Electric-magnetic duality in supersymmetric nonabelian gauge theories, Nucl. Phys. B 435 (1995) 129, [hep-th/9411149].

[5] K. A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1, [hep-th/9509066].

[6] P. C. Argyres, Lectures on supersymmetry, http://www.physics.uc.edu/argyres/661.

[7] M.E. Peskin, Duality in supersymmetric Yang-Mills theory, [hep-th9702094].

[8] C. Beasley and E. Witten, New instanton eﬀects in supersymmetric QCD, JHEP 0501 (2005) 056, [hep-th/0409149].

[9] K. A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in

supersymmetric Sp(Nc) gauge theories, Phys. Lett. B 353 (1995) 471, [hep-th/9505006].

[10] P. C. Argyres and M. Edalati, On singular eﬀective superpotentials in supersymmetric gauge theories, JHEP 0601 (2006) 012, [hep-th/0510020].

[11] P. C. Argyres and M. Edalati, Generalized Konishi anomaly, Seiberg duality and singular eﬀective superpotentials, JHEP 0602 (2006) 071, [hep-th/0511272].

118 [12] P. C. Argyres and M. Edalati, Sp(N) higher-derivative F-terms via singular superpotentials, JHEP 0607, (2006) 021 , [hep-th/0603025].

[13] G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461.

[14] J. M. Maldacena, The large N limit of superconformal ﬁeld theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231; Int. J. Theor. Phys. 38 (1999) 1113, [hep- th/9711200].

[15] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from non- critical string theory, Phys. Lett. B 428, 105 (1998), [hep-th/9802109].

[16] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998), [hep-th/9802150].

[17] A. Karch and E. Katz, Adding ﬂavor to AdS/CFT, JHEP 0206 (2002) 043, [hep- th/0205236].

[18] P. Kovtun, D. T. Son and A. O. Starinets, Holography and hydrodynamics: Diﬀusion on stretched horizons, JHEP 10 (2003) 064, [hep-th/0309213].

[19] A. Buchel and J. T. Liu, Universality of the shear viscosity in supergravity, Phys. Rev. Lett. 93 (2004) 090602, [hep-th/0311175].

[20] P. Kovtun, D. T. Son and A. O. Starinets, Viscosity in strongly interacting quantum ﬁeld theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601, [hep-th/0405231].

[21] A. Buchel, On universality of stress-energy tensor correlation functions in supergravity, Phys. Lett. B609 (2005) 392, [hep-th/0408095].

[22] E. Witten, Anti-de Sitter space, thermal phase transition, and conﬁnement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505, [hep-th/9803131].

[23] S. Rey and J. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C22 (2001) 379, [hep-th/9803001].

[24] J. M. Maldacena, Wilson loops in large N ﬁeld theories, Phys. Rev. Lett. 80 (1998) 4859, [hep-th/9803002].

119 [25] S. J. Rey, S. Theisen and J. T. Yee, Wilson-Polyakov loop at ﬁnite temperature in large N gauge theory and anti-de Sitter supergravity, Nucl. Phys. B527 (1998) 171, [hep-th/9803135].

[26] A. Brandhuber, N. Itzhaki, J. Sonnenschein and S. Yankielowicz, Wilson loops in the large N limit at ﬁnite temperature, Phys. Lett. B434 (1998) 36, [hep-th/9803137].

[27] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaﬀe, Energy loss of a heavy quark in N =4 super Yang-Mills plasma, JHEP 0607 (2006) 013, [hep-th/0605158].

[28] S. S. Gubser, Drag force in AdS/CFT, Phys. Rev. D 74 (2006) 126005, [hep-th/0605182].

[29] H. Liu, K. Rajagopal and U. A. Wiedemann, An AdS/CFT calculation of screening in a hot wind, [hep-ph/0607062].

[30] M. Chernicoﬀ, J. A. Garcia and A. Guijosa, The energy of a moving quark-antiquark pair in an =4 SYM plasma, JHEP 0609 (2006) 068, [hep-th/0607089]. N [31] E. Caceres, M. Natsuume and T. Okamura, Screening length in plasma winds, JHEP 0610 (2006) 011, [hep-th/0607233].

[32] S. D. Avramis, K. Sfetsos and D. Zoakos, On the velocity and chemical-potential dependence of the heavy-quark interaction in N = 4 SYM plasmas, Phys. Rev. D 75 (2007) 025009, [hep-th/0609079].

[33] J. J. Friess, S. S. Gubser, G. Michalogiorgakis and S. S. Pufu, Stability of strings binding heavy-quark mesons, JHEP 0704 (2007) 079, [hep-th/0609137].

[34] H. Liu, K. Rajagopal and U. A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97 (2006) 182301, [hep-ph/0605178].

[35] P. C. Argyres, M. Edalati, J. V´azquez-Poritz, No-drag string conﬁgurations for steadily-moving quark-antiquark pairs in a thermal bath, JHEP 0701 (2007) 105, [hep- th/0608118].

[36] P. C. Argyres, M. Edalati and J. F. V´azquez-Poritz, Spacelike strings and jet quenching from a Wilson loop, JHEP 0704 (2007) 049, [hep-th/0612157].

[37] F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 0212 (2002) 071, [hep-th/0211170].

120 [38] N. Seiberg, Adding fundamental matter to ‘Chiral rings and anomalies in supersymmetric gauge theory’, JHEP 0301 (2003) 061, [hep-th/0212225].

[39] K. Konishi, Anomalous supersymmetry transformation of some composite operators in sQCD, Phys. Lett. B 135 (1984) 439.

[40] K. Konishi and K. I. Shizuya, Functional integral approach to chiral anomalies in supersymmetric gauge theories, Nuovo Cim. A 90 (1985) 111.

[41] E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324.

[42] I. Aﬄeck, M. Dine and N. Seiberg, Dynamical supersymmetry breaking in supersymmetric QCD, Nucl. Phys. B 241 (1984) 493; Dynamical supersymmetry breaking in four dimensions and its phenomenological implications, Nucl. Phys. B 256 (1985) 557.

[43] S. Corley, Notes on anomalies, baryons, and Seiberg duality, [hep-th/0305096].

[44] P. Svrˇcek, On non-perturbative exactness of Konishi anomaly and the Dijkgraaf-Vafa conjecture, JHEP 0408 (2004) 036, [he-pth/0308037].

[45] G. Veneziano and S. Yankielowicz, An eﬀective lagrangian for the pure N=1 supersymmetric Yang-Mills theory, Phys. Lett. B 113 (1982) 231.

[46] E. Witten, Chiral ring of Sp(N) and SO(N) supersymmetric gauge theory in four dimensions, [hep-th/0302194].

[47] S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Superspace, or one thousand and one lessons in supersymmetry, Benjamin/Cummings 1983, [hep-th/0108200].

[48] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three dimensions, Nucl. Phys. B 499 (1997) 67, [hep-th/9703110].

[49] J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three dimensions, Nucl. Phys. B 500 (1997) 163, [hep-th/9703100].

[50] A. Brandhuber, H. Ita, H. Nieder, Y. Oz and C. Romelsberger, Chiral rings, superpotentials, and the vacuum structure of = 1 supersymmetric gauge theories, Adv. Theor. N Math. Phys. 7 (2003) 269, [hep-th/0303001].

121 [51] K. A. Intriligator, R. G. Leigh and N. Seiberg, Exact superpotentials in four-dimensions, Phys. Rev. D 50 (1994) 1092, [hep-th/9403198].

[52] K. A. Intriligator, ’Integrating in’ and exact superpotentials in 4-d, Phys. Lett. B 336 (1994) 409, [hep-th/9407106].

[53] E. Shuryak, Why does the quark gluon plasma at RHIC behave as a nearly ideal ﬂuid?, Prog. Part. Nucl. Phys. 53 (2004) 273, [hep-ph/0312227].

[54] E. Shuryak, What RHIC experiments and theory tell us about properties of quark-gluon plasma?, Nucl. Phys. A750 (2005) 64, [hep-ph/0405066].

[55] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiﬀ, Radiative energy loss and p(T )-broadening of high energy partons in nuclei, Nucl. Phys. B 484 (1997) 265, [hep-ph/9608322].

[56] R. Baier, D. Schiﬀ and B. G. Zakharov, Energy loss in perturbative QCD, Ann. Rev. Nucl. Part. Sci. 50 (2000) 37, [hep-ph/0002198].

[57] U. A. Wiedemann, Gluon radiation oﬀ hard quarks in a nuclear environment: Opacity expansion, Nucl. Phys. B 588, 303 (2000), [hep-ph/0005129].

[58] A. Kovner and U. A. Wiedemann, Eikonal evolution and gluon radiation, Phys. Rev. D 64, 114002 (2001), [hep-ph/0106240].

[59] M. Gyulassy, I. Vitev, X. N. Wang and B. W. Zhang, Jet quenching and radiative energy loss in dense nuclear matter, nucl-th/0302077.

[60] A. Kovner and U.A. Wiedemann, Gluon radiation and parton energy loss, [hep- ph/0304151].

[61] P. Jacobs and X. N. Wang, Matter in extremis: ultrarelativistic nuclear collisions at RHIC, Prog. Part. Nucl. Phys. 54 (2005) 443, [hep-ph/0405125].

[62] A. Buchel, On jet quenching parameters in strongly coupled non-conformal gauge theories, Phys. Rev. D 74 (2006) 046006, [hep-th/0605178].

[63] C. P. Herzog, Energy loss of heavy quarks from asymptotically AdS geometries, JHEP 0609 (2006) 032, [hep-th/0605191].

122 [64] J. Casalderrey-Solana and D. Teaney, Heavy quark diﬀusion in strongly coupled N =4 Yang Mills, Phys. Rev. D 74 (2006) 085012, [hep-ph/0605199].

[65] E. Caceres and A. Guijosa, Drag force in charged N = 4 SYM plasma, JHEP 0611 (2006) 077, hep-th/0605235.

[66] J. J. Friess, S. S. Gubser, G. Michalogiorgakis, Dissipation from a heavy quark moving through N=4 super-Yang-Mills plasma, JHEP 0609 (2006) 072, [hep-th/0605292].

[67] J.F. V´azquez-Poritz, Enhancing the jet quenching parameter from marginal deformations, [hep-th/0605296].

[68] S.-J. Sin and I. Zahed, Ampere’s Law and energy loss in AdS/CFT duality, Phys. Lett. B 648 (2007) 318, [hep-ph/0606049].

[69] E. Caceres and A. Guijosa, On drag forces and jet quenching in strongly coupled plasmas, JHEP 0612 (2006) 068, [hep-th/0606134].

[70] F.-L. Lin and T. Matsuo, Jet quenching parameter in medium with chemical potential from AdS/CFT, Phys. Lett. B 641 (2006) 45, [hep-th/0606136].

[71] S. D. Avramis and K. Sfetsos, Supergravity and the jet quenching parameter in the presence of R-charge densities, JHEP 0701 (2007) 065, [hep-th/0606190].

[72] K. Peeters, J. Sonnenschein and M. Zamaklar, Holographic melting and related properties of mesons in a quark gluon plasma, Phys. Rev. D 74 (2006) 106008, [hep-th/0606195].

[73] N. Armesto, J. D. Edelstein and J. Mas, Jet quenching at ﬁnite ‘t Hooft coupling and chemical potential from AdS/CFT, JHEP 0609 (2006) 039, [hep-ph/0606245].

[74] Y. h. Gao, W. S. Xu and D. F. Zeng, Wake of color ﬁleds in charged N = 4 SYM plasmas, [hep-th/0606266].

[75] J. J. Friess, S. S. Gubser, G. Michalogiorgakis and S.S. Pufu, The stress tensor of a quark moving through N = 4 thermal plasma, Phys. Rev. D 75 (2007) 106003, [hep- th/0607022].

[76] M. Chernicoﬀ, J. A. Garcia and A. Guijosa, The energy of a moving quark-antiquark pair in an N=4 SYM plasma, JHEP 0609 (2006) 068, [hep-th/0607089].

123 [77] T. Matsuo, D. Tomino and W.-Y. Wen, Drag force in SYM plasma with B ﬁeld from AdS/CFT, JHEP 0610 (2006) 055, [hep-th/0607178].

[78] E. Nakano, S. Teraguchi and W. Y. Wen, Drag force, jet quenching, and AdS/QCD, [hep-ph/0608274].

[79] P. Talavera, Drag force in a string model dual to large-N QCD, [hep-th/0610179].

[80] J. J. Friess, S. S. Gubser, G. Michalogiorgakis and S. S. Pufu, Expanding plasmas and quasinormal modes of anti-de Sitter black holes, [hep-th/0611005].

[81] M. Chernicoﬀ and A. Guijosa, Energy loss of gluons, baryons and k-quarks in an N=4 SYM plasma, [hep-th/0611155].

[82] Y. Gao, W. Xu, and D. Zeng, Jet quenching parameters of Sakai-Sugimoto model, [hep- th/0611217].

[83] E. Antonyan, Friction coeﬃcient for quarks in supergravity duals, [hep-th/0611235].

[84] S. Gubser, Comparing the drag force on heavy quarks in N=4 super-Yang-Mills theory and QCD, [hep-th/0611272].

[85] S. S. Gubser, Jet-quenching and momentum correlators from the gauge-string duality, [hep-th/0612143].

[86] H. Liu, K. Rajagopal and U. A. Wiedemann, Wilson loops in heavy ion collisions and their calculation in AdS/CFT, JHEP 0703 (2007) 066, [hep-ph/0612168].

[87] J. Casalderrey-Solana and D. Teaney, Transverse momentum broadening of a fast quark in a N = 4 Yang Mills plasma, JHEP 0704 (2007) 039, [hep-th/0701123].

[88] A. Yarom, The high momentum behavior of a quark wake, [hep-th/0702164].

[89] A. Yarom, On the energy deposited by a quark moving in an N=4 SYM plasma, [hep- th/0703095].

[90] J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik and I. Kirsch, Chiral symmetry breaking and pions in non-supersymmetric gauge/gravity duals, Phys. Rev. D 69, 066007 (2004), [hep-th/0306018].

[91] D. Mateos, R. C. Myers and R. M. Thomson, Holographic phase transitions with fundamental matter, Phys. Rev. Lett. 97 (2006) 091601, [hep-th/0605046].

124 [92] A. Karch and A. O’Bannon, Chiral transition of N = 4 super Yang-Mills with ﬂavor on a 3-sphere, Phys. Rev. D 74 (2006) 085033, [hep-th/0605120].

[93] T. Albash, V. Filev, C. V. Johnson and A. Kundu, A topology-changing phase transition and the dynamics of ﬂavour, [hep-th/0605088].

[94] T. Albash, V. Filev, C. V. Johnson and A. Kundu, Global currents, phase transitions, and chiral symmetry breaking in large N(c) gauge theory, [hep-th/0605175].

[95] D. Bak, A. Karch and L. G. Yaﬀe, Debye screening in strongly coupled N=4 supersymmetric Yang-Mills plasma, [hep-th/07050994].

[96] N. Drukker, D. J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006, [hep-th/9904191].

125