Conformal Dynamics for Electroweak Symmetry Breaking

Transcription

1 Conformal Dynamics for Electroweak Symmetry Breaking Niels Bohr Institute Faculty of Science University of Copenhagen Ph.D. Thesis Advisor: Francesco Sannino May 009 Thomas Aaby Ryttov

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3 Table of Contents Abstract Acknowledgements v vii 1 Introduction Outline I The Phase Diagram of Nonabelian Gauge Theories 7 Phases of Gauge Theories 9.1 The Ladder Approximation The Supersymmetric Phase Diagram The Conformal Window of Supersymmetric Gauge Theories Sizing The Unparticle World The Supersymmetric Case The Nonsupersymmetric Case The All-Orders Beta Function Conjecture All-Orders Nonsupersymmetric Beta Function IR Fixed Point Comparison with the Ladder Approximation Comparison with Lattice Data Generalization to Multiple Representations Matching to Exact Results and Lattice Data Super Yang-Mills i

4 ii TABLE OF CONTENTS 4.3. Pure Yang-Mills and Comparison with Lattice Data Re-sizing the Unparticle World: A New Universal Ratio II Near-Conformal Technicolor 39 5 Minimal Walking Technicolor The Underlying Lagrangian for Minimal Walking Technicolor Low Energy Theory for MWT Unification Notation and Conventions Studying S U(3) S U() U(1) Unification in Technicolor Minimal Walking Technicolor Traditional Walking and Non-Walking One Family Model Partially Electroweak-Gauged Technicolor The Technicolor Coupling Constant Proton Decay Constructing a Simple Unifying Group Providing Mass to the fermions A New Extension of the Standard Model Unifying Technicolor as Well Comparing with MSSM and a Hint of Dark Matter Ultra Minimal Walking Technicolor From the Conformal Window to Ultra Minimal Technicolor The Model Low Energy Spectrum Linear Lagrangian Non-Linear Lagrangian The TIMP The Electroweak Phase Transition(s) Effective Potential Effective potential in UMT: without EW Effective potential in UMT: with EW

5 TABLE OF CONTENTS iii 8. Analysis of the phase transition: setup Parameter space Decoupled transitions Results Without EW With EW Simultaneous and extra phase transitions Conclusions 113 A Generators for MWT 115 B Generators for UMT 117 C Dark Matter Computations 11 C.1 nd Order Phase Transition C. 1st Order Phase Transition D Zero-Temperature Background-Dependent Scalar Masses 17 E Temperature and Background Dependent Scalar Masses 19 F Gauge Boson Mass Matrices 133 Bibliography 135

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7 Abstract This thesis discusses recent progress in the uncovering of the phase diagram of (non) supersymmetric gauge theories and possible applications for beyond Standard Model physics. We first introduce the phase diagram as a function of the number of colors, flavors and type of fermion representation before providing an estimate of the nonsupersymmetric conformal window using the ladder approximation. We then investigate the phase diagram of N = 1 supersymmetric gauge theories. Based on the supersymmetric analysis we propose an all-orders beta function for nonsupersymmetric gauge theories which allows us to bound the conformal window. We find that our prediction agrees with all recent lattice data. The nature of the conformal window for higher dimensional representations suggests a possible way to construct realistic technicolor models. Two such explicit theories are then provided. The first is the Minimal Walking Technicolor model for which we derive the effective theory and investigate the prospects of gauge coupling unification. With a small modification of the Standard Model fermionic content we find that the coupling constants provide excellent unification. The second model is the Ultra Minimal Walking Technicolor model which consists of fermions belonging to two distinct representations of the gauge group. This model has a natural cold dark matter candidate which can be light enough to be detected at the LHC. We finally study the electroweak phase transition at finite temperature and find a very rich phase diagram. v

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9 Acknowledgements I am deeply thankful to my advisor Prof. Francesco Sannino for his incredible support during my graduate studies. His never ending belief in our projects and his engagement in my work have been without compare. I would also like to thank the rest of the group for many stimulating discussions and for making life and science much more enjoyable for the past three years. This includes P. Channuie, D. Dietrich, R. Foadi, M. T. Frandsen, H. S. Fukano, M. Järvinen, C. Kouvaris and K. Petrov. I am also grateful to Scientific Staff Member C. Grojean for advising and supporting me during my one year stay at the theory division at CERN. Also I would like to thank SLAC for the kind hospitality during my three months visit. Last but not least I would like to thank my family for their love and support all along. vii

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11 Chapter 1 Introduction The quest for a deeper understanding of the Universe we inhabit has driven mankind forward for centuries. Ever since the dawn of time scientists have pursued a set of fundamental laws of Nature that are believed to exist. On this long journey we have overcome many obstacles and for each day developed new and more advanced technologies, just to learn that around the next corner great unforseen and unexpected discoveries would await. Throughout the centuries experiments have solidly demonstrated that Nature and the laws by which it is governed are much more sophisticated and mysterious than our naive intuition would have expected. Perhaps the greatest leap of all has been the realization that the world cannot be described within the framework of classical physics. Instead, today it is in the realm of the quantum world that we seek to describe the elementary building blocks as well as the forces via which they interact and communicate. Our current understanding of the smallest structures of the Universe is encoded in the Standard Model. The electromagnetic, the weak and the strong forces together with the elementary particles are neatly incorporated into this single theoretical framework. The matter sector of the Standard Model is constituted by three families of both quarks and leptons. They all carry spin equal to one half. Besides being of different mass what differentiates quarks from leptons is the fact that the former interact through all the fundamental forces whereas the latter are blind to the strong force. 1

12 Chapter 1 - Introduction In a quantum field theoretical description of the Universe forces are mediated via the exchange of gauge bosons which all carry a spin of one. The force carrier of the electromagnetic force is the photon, the weak force is mediated via the exchange of the three massive gauge bosons W +,W and Z and the eight gluons are the force carriers of the strong force. Despite the fact that all three forces are described within the same mathematical structure of gauge theories they each possess unique and distinct features. For instance QCD, the theory of the strong interaction, exhibits two phenomena known respectively as confinement and chiral symmetry breaking. The former is responsible for binding the quarks into hadronic states such as the proton and the neutron while the latter is the reason for the near masslessness of the pions. Even though both phenomena yet have to be proven in QCD they are both strongly believed to occur. On the other hand it seems a paradox to describe the weak interactions in the language of gauge theories since the associated gauge fields are required to be massless whereas the weak gauge bosons are known experimentally to be massive. The realization that gauge theories indeed provide the correct mathematical structure arose in the fifties and the sixties and culminated with the Weinberg-Salam theory [1, ]. Besides uniquely tying the electromagnetic and the weak interactions into one another the Weinberg-Salam theory also postulates the existence of a single elementary scalar boson coined the Higgs particle. It is via the mechanism of spontaneous symmetry breaking that the Higgs field becomes responsible for the mass generation of the weak gauge bosons as well as the other elementary particles. In empty space the Higgs field has a non-zero value which rearranges the vacuum and thereby causes the electroweak symmetry to break. Simply stated, as the particles move through the new vacuum the Higgs field sticks to them thereby converting them to massive particles. The Higgs particle is the only piece of the Standard Model whose existence yet has to be confirmed by experiments. However, on purely theoretical grounds there are many reasons to believe that the Higgs sector of the Standard Model might be much more rich and complicated than expected in the standard electroweak theory. In essence the elementary scalar Higgs boson might just parameterize our lack of knowledge of a true fundamental theory of the origin of mass. Except for gravity which is mediated by the graviton with spin equal to two.

13 3 The uncovering of the Higgs sector in the Standard Model is without a doubt one of the greatest challenges of modern science both experimentally and theoretically. On the experimental side enormous efforts are invested in the Large Hadron Collider (LHC) experiment at CERN, Geneva. The main objective of LHC is precisely to probe the structure of the Higgs sector and to reveal the true nature of the Higgs particle. On the theoretical side many new models and ideas have been proposed in the past in order to shed light on the elusive Higgs. It is fair to say that to this day no physicist can claim to have the fully correct and adequate answer to this profound puzzle. However, many advances have recently been made for which a great part of this thesis is devoted to. One of the best motivated extensions of the Standard Model is that of technicolor [3,4]. To have a better understanding and clearer view of the technicolor proposal we must first take a closer look at the intrinsic dynamics of the Standard Model per se. In the strong sector we know that a quark-antiquark condensate is formed which triggers the breaking of chiral symmetry. Since the left handed quarks are also charged under the electroweak symmetry the chiral condensate simultaneously induces a breaking of the electroweak gauge group. In other words, the electroweak gauge bosons are massive even without the introduction of an elementary scalar Higgs particle. However, this simple argument cannot constitute a complete and satisfactory explanation since the scale of the chiral condensate is incorrect in order to fully accommodate for the correct masses of the weak gauge bosons. The value of the chiral condensate is simply too small. With the above analysis in mind introducing technicolor is straightforward. The idea is to mimic the physics of QCD by replacing the elementary Higgs boson with a new strongly interacting gauge theory coined technicolor. On top of the new gauge dynamics one introduces a set of fermions charged under the technicolor gauge symmetry as well as the electroweak gauge symmetry. Once the technicolor bilinear condensate forms the electroweak symmetry breaks. By fixing the value of the condensate to 46 GeV one recovers the correct experimental values of the masses of the weak gauge bosons. The idea is simple and relies on principles and phenomena already known to exist in similar physical systems in Nature. Despite the elegance of the technicolor proposal it is only very recent that viable specific models not at odds with experiments have been constructed [5 8]. Typically one is faced with the following two requirements when aiming to construct a realistic model

14 4 Chapter 1 - Introduction To accommodate for the Standard Model fermion masses one requires the existence of extended technicolor interactions which should be consistent with the observed amount of Flavor Changing Neutral Currents. The theory should not be at odds with the Electroweak Precision Tests. As we shall see there is a certain tension between the above two requirements making them the sources of most of the problems technicolor model builders encounter. First, a possible solution to the problem of Flavor Changing Neutral Current suppression is to construct a technicolor theory which exhibits walking dynamics [9 14]. One assumes that just before chiral symmetry breaking is triggered the theory enters a (near) conformal phase raising the scale which controls the amount of Flavor Changing Neutral Currents. In the (near) conformal phase the coupling constant is said to walk, as opposed to run, since it evolves very slowly with the energy scale. In the traditional technicolor scheme in which one arranges the technifermions to be in the fundamental representation of the gauge group in general one needs a large number of flavors to achieve the desired dynamics. On the other hand the theory should not be at odds with the Electroweak Precision Tests. The effects of beyond the Standard Model physics on the electroweak observables are encoded in the oblique parameters [15,16]. Specifically it is the contribution stemming from the new strongly coupled sector to the S parameter which is of crucial importance. A naive estimate reveals that the larger the number of technifermions is the larger the contribution to the S parameter is finally ruling out the model. Comparing with the desire of achieving walking dynamics the tension is therefore clear. Despite the fact that studies have shown a reduction in the S parameter for walking theories [17] it is the better and more complete understanding of the phase diagram of nonsupersymmetric theories that have paved the way for new realizations of viable technicolor theories. Specifically it is the important discovery [5] that only a few flavors are needed in order to be near the conformal window when the fermions belong to the twoindexed symmetric or adjoint representation of the gauge group that has lead the way. In other words the tension between the above two requirements is significantly reduced if one arranges the fermions to belong to certain higher dimensional representations. Shortly after the appearance of [5] a complete investigation of the phase diagram for Recent studies [18, 19] observe a similar behavior.

15 1.1 - Outline 5 any representation obtained in the ladder approximation was conceived [0]. From this it is clear that the nature of the conformal window strongly depends on the type of the fermion representation. Since the estimate is based on an approximation also new and different nonperturbative tools are needed in order to more carefully examine the phase diagram. Based on this need an all-orders beta function was conjectured in [1] which allows for a bound of the conformal window. The analysis is similar to the supersymmetric case for which it is the nontrivial relation between the beta function and the anomalous dimension of the mass together with the unitarity bound of conformal operators that provide the estimate. Whereas the ladder approximation leads the way to the construction of the Minimal Walking Technicolor model [5 7] it is via the all-orders beta function that the Ultra Minimal Walking Technicolor model [8] is conceived. The Ultra Minimal Walking Technicolor model is constituted by fermions transforming according to two distinct representations of the gauge group and while the ladder approximation is inadequate for estimating the conformal window for multiple representations the all-orders beta function is sufficient. After the initial investigations of [5] much work has been done. The complete phase diagram for supersymmetric theories have appeared in [], the nonsupersymmetric phase diagram has appeared in [3] for S p(n) and S O(N) gauge groups and a great effort has been invested by the lattice community to uncover the nonsupersymmetric phase diagram [4 34]. From the point of view of beyond the Standard Model Physics and technicolor theories studies range from LHC phenomenology [35 46], cosmology and dark matter [47 56], unification [57, 58] and the finite temperature electroweak phase transition [59 61]. A recent review can be found in [6]. 1.1 Outline The thesis is divided into two parts. The first part is concerned with the phase diagram of both nonsupersymmetric and supersymmetric gauge theories with matter belonging to arbitrary representations. We begin in chapter by giving an introduction to the phase diagram of strongly coupled theories and briefly describe the ladder approximation. In chapter 3 the conformal window is derived for any representation in the supersymmetric

16 6 Chapter 1 - Introduction case. In chapter 4 we propose an all-orders beta function which allows for a bound of the nonsupersymmetric conformal window. The second part of the thesis is devoted to the construction of various technicolor extensions of the Standard Model. Chapter 5 introduces the Minimal Walking Technicolor model. In chapter 6 we study the possibility of gauge coupling unification using Minimal Walking Technicolor as a template. Chapter 7 introduces the Ultra Minimal Walking Technicolor model while chapter 8 is devoted to the associated finite temperature phase transition. Various appendices provide the reader with notation, explicit computations, etc.

17 Part I The Phase Diagram of Nonabelian Gauge Theories 7

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19 Chapter Phases of Gauge Theories We shall begin our journey by considering an S U(N) gauge theory with N f Dirac fermions in an arbitrary representation r of the gauge group. Depending on the number of colors, flavors and type of representation we are specifically interested in classifying the possible phases which the theory might exhibit in the infrared (IR). To set the notation for now and throughout the thesis we denote the generators of the gauge group in the representation r by Tr a, a = 1,..., N 1. They are normalized according to Tr [ ] Tr a Tr b = T(r)δ ab while the quadratic Casimir C (r) is given by Tr a Tr a = C (r)1. The trace normalization factor T(r) and the quadratic Casimir are connected via C (r)d(r) = T(r)d(G) where d(r) is the dimension of the representation r. The adjoint representation is denoted by G. In Table.1 we list the explicit group factors for the representations used in this thesis. Picking a specific theory we want to investigate what nontrivial long distance physics it leads to. To elucidate the various possibilities we begin by reviewing the two-loop beta function [63, 64] β(g) = β 0 (4π) g3 β 1 (4π) 4 g5, (.0.1) 9

20 10 Chapter - Phases of Gauge Theories r T(r) C (r) d(r) 1 N 1 N N G N N N 1 N+ N (N 1)(N+) N (N+1)(N ) N N(N+1) N(N 1) Table.1: Relevant group factors for the representations used throughout the thesis. However, a complete list of all the group factors for any representation and the way to compute them is available in Table II and the appendix of [0]. where g is the gauge coupling and the beta function coefficients are given by β 0 = 11 3 C (G) 4 3 T(r)N f, (.0.) β 1 = 34 3 C (G) 0 3 C (G)T(r)N f 4C (r)t(r)n f. (.0.3) To this order t Hooft has shown that the two coefficients are universal, i.e. they do not depend on which renormalization group scheme one has used to determine them [65]. For a generic theory the long distance physics depends on the beta function coefficients β 0 and β 1 and hence on the number of colors, flavors and type of representation. First imagine that we are given a specific number of colors N and type of representation r and let us study this theory as a function of the number of flavors N f. For a sufficiently large number of flavors the first coefficient of the beta function is negative β 0 < 0 and asymptotic freedom is lost. The lack of asymptotic freedom determines a distinct class of models, i.e. a particular phase in the phase diagram spanned by the number of flavors, number of colors and type of representation. The critical number of flavors N I f above which asymptotic freedom is lost reads N I f = 11 4 C (G) T(r), (.0.4) which corresponds to a change of sign in the first coefficient of the beta function. One should note that since the first coefficient of the beta function is negative also the second

21 11 Β g Α Μ Β g Α Μ Μ Μ Figure.1: The beta function and the associated running coupling constant µ dg g dµ = β(g) and α = 4π is plotted for two different nonasymptotically free theories. Top panel: The theory develops a Landau pole. Bottom panel: The theory develops a UV fixed point. We shall not distinguish between two such different dynamics. coefficient is negative. Therefore the two-loop beta function increases monotonically and the theory develops a Landau pole. In the top panel of Fig..1 the two-loop beta function together with the associated running coupling constant is plotted. As a side remark we note that it might be possible for certain nonasymptotically free theories to be driven to an ultraviolet (UV) fixed point instead of developing a Landau pole. At least, as the coupling constant grows large perturbation theory is likely to fail and nonperturbative effects will start to become relevant. However, since we are only interested in classifying the possible phases according to the long distance physics we do not distinguish between the two scenarios. In the bottom panel of Fig..1 we plot the beta function as well as the running coupling constant for a nonasymptotically free theory developing a UV fixed point. As we lower the number of flavors we find a regime where the first coefficient is positive while the second coefficient is negative. Therefore according to the two-loop beta function the theory develops an IR fixed point. The critical number of flavors N III f at

22 1 Chapter - Phases of Gauge Theories Β g Α Μ Figure.: The beta function and the associated running coupling constant µ dg dµ theory developing an IR fixed point. Μ = β(g) and α = g 4π for a Β g Α Μ Figure.3: The beta function and the associated running coupling constant µ dg dµ QCD-like theory. Μ = β(g) and α = g 4π for a which the second coefficient changes sign is N III f = 17C (G) C (G). (.0.5) 10C (G) + 6C (r) T(r) In Fig.. we plot the two-loop beta function as well as the associated running coupling constant for a theory developing an IR fixed point. Similar to above the theories developing an IR fixed point determine a distinct phase in the phase diagram. One should note that having reached the IR fixed point the theory is conformal. Hence this phase has been coined the conformal window in the literature. Lowering the number of flavors even further we encounter the last possibility where both coefficients are positive. Here the two-loop beta function decreases monotonically and the coupling constant blows up in the IR. The theories belonging to this phase are characterized by chiral symmetry breaking: As the coupling constant grows large the

23 13 N f Figure.4: Naive phase diagram for nonsupersymmetric theories stemming from the two-loop beta function with fermions in the: i) fundamental representation (black), ii) two-indexed antisymmetric representation (blue), iii) two-indexed symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. Above the upper curve the theories are no longer asymptotically free. In between the upper and the lower curves the first coefficient of the beta function is positive while the second coefficient is negative. Below the lower curve both coefficients are positive. chiral condensate forms and breaking of chiral symmetry is triggered [66 68]. In Fig..3 we plot the two-loop beta function as well as the associated running coupling constant for a sufficiently low number of flavors. It is important to note that we shall not distinguish between whether there exists a confinement/deconfinement phase transition or not, i.e. we shall not pay attention to the possibility that the theory might possess a center group symmetry [69]. It is clear that in general the true critical number of flavors above which a nontrivial IR fixed point is generated does not coincide with the above perturbative estimate since we have no control of the nonperturbative effects. For an asymptotically free theory with a number of flavors above N III f the fixed point value of the coupling constant reads g c = (4π) β 0 β 1. (.0.6) As we lower the number of flavors and approaching N III f the second coefficient of the beta function tends to zero and the fixed point value of the coupling constant grows large

24 14 Chapter - Phases of Gauge Theories signaling the breakdown of perturbation theory. Therefore the estimate of the conformal window stemming from the two-loop beta function is not reliable. Determining the true critical number of flavors above which an IR fixed point is reached will be of our main concern in the first part of this thesis. In Fig..4 we plot the naive phase diagram derived using the two-loop beta function for the fundamental and two-indexed representations. We stress that it is only a naive estimate which we provide in order to give the reader a first sense of the features of the phase diagram..1 The Ladder Approximation Having introduced and motivated the phase diagram in the previous section we are now ready to present various methods for determining the conformal window, the first of which is the ladder approximation [5, 0]. As mentioned above, the formation of the fermion bilinear condensate is associated to the coupling constant reaching a certain critical value g [66 68]. What is of our interest is then to compare g with g c which is the coupling constant associated to an IR fixed point. If g < g c a dynamical mass for the fermions is generated and chiral symmetry breaks and if g c < g the theory flows to an IR fixed point. Hence the condition g = g c yields the boundary of the conformal window. As an estimate of g c one employs the result of the two-loop analysis (.0.6) while g is estimated in a truncated Schwinger-Dyson analysis and is given by [66 68] g = 4π 3C (r). (.1.1) Therefore the critical number of flavors above which a nontrivial IR fixed point is generated is N II f = 17C (G) + 66C (r) 10C (G) + 30C (r) C (G) T(r), Ladder Approximation. (.1.) In Fig.5 we plot the phase diagram for the fundamental and two-indexed representations in the ladder approximation. First we observe that the conformal window is smaller compared to the naive two-loop estimate. The reason for this is clear since we effec-

25 .1 - The Ladder Approximation 15 N f Figure.5: Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-indexed antisymmetric representation (blue), iii) two-indexed symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories are expected to develop an IR fixed point according to the ladder approximation. The dashed curve represents the change of sign in the second coefficient of the beta function. tively put an upper bound on the value of the fixed point coupling constant: only IR fixed points for which the coupling constant is smaller than g are physical. Second we note the remarkable feature that only a low number of flavors for the adjoint and two-indexed symmetric representation is needed in order to be near the conformal window. This has two important implications Such (near) conformal theories are easily accessible on the lattice. They are perfect candidates for walking technicolor theories able to dynamically break the electroweak symmetry. The fact that the simplest theories with fermions in higher dimensional representations are easily accessible on the lattice provides an excellent test of the validity of our analysis. In chapter 4 we will elaborate more on this issue. The relevance of higher dimensional representations to beyond Standard Model physics arises due to the fact that one only needs a low number of flavors to achieve (near) conformal dynamics. Such models give a small contribution to the oblique parameters [15, 16]

26 16 Chapter - Phases of Gauge Theories and will therefore not be at odds with the Electroweak Precision Tests. The construction of these models together with their low energy dynamics will be the subject of the second part of the thesis.

27 Chapter 3 The Supersymmetric Phase Diagram In the previous chapter we analyzed the phase diagram of nonsupersymmetric gauge theories using the ladder approximation with Dirac fermions in a single arbitrary representation of the gauge group as function of the number of flavors and colors. The phase diagram is sketched in Figure.5 with the exceptions of a few isolated higher dimensional representations below nine colors. In the past also other similar studies of the nonsupersymmetric conformal window and its properties have been carried out [70 76]. Below we shall study the conformal window of S U(N) supersymmetric gauge theories with vector-like matter transforming according to a single but generic irreducible representation of the gauge group. The analysis first appeared in [77] for the fundamental representation and in [] for higher dimensional representations. The results are subsequently confronted with the nonsupersymmetric ones. 3.1 The Conformal Window of Supersymmetric Gauge Theories The gauge sector of a supersymmetric S U(N) gauge theory consists of a supersymmetric field strength belonging to the adjoint representation of the gauge group. The supersym- 17

28 18 Chapter 3 - The Supersymmetric Phase Diagram [S U(N)] S U(N f ) S U(N f ) U(1) B U(1) R Φ r N f 1 1 Φ r 1 N f 1 T(r)N f C (G) T(r)N f T(r)N f C (G) T(r)N f Table 3.1: Summary of the local and global symmetries and charge assignments of a generic N = 1 gauge theory with matter in a given representation r of the gauge group. metric field strength describes the gluon and the gluino. The matter sector is taken to be vectorial and to consist of N f chiral superfields Φ in the representation r of the gauge group and N f chiral superfields Φ in the conjugate representation r of the gauge group. The chiral superfield Φ (or Φ) contains a Weyl fermion and a complex scalar boson. With this notation we summarize the symmetries of the theory in Table 3.1. The first S U(N) is the gauge group. The two abelian symmetries are anomaly free with the first one being the baryon number and the second one an R-symmetry. Note that the global symmetry is enhanced from S U(N f ) S U(N f ) U(1) B to S U(N f ) when the representation of the matter field is real or pseudoreal. The exact beta function of supersymmetric QCD was first found in [78 80] and further investigated in [81, 8]. For a given representation it takes the form β(g) = g3 β 0 + T(r)N f γ(g ), (3.1.1) 16π 1 g C 8π (G) γ(g ) = g 4π C (r) + O(g 4 ), (3.1.) where g is the gauge coupling, γ(g ) = d ln Z(µ)/d ln µ is the anomalous dimension of the matter superfield and β 0 = 3C (G) T(r)N f is the first beta function coefficient. For a given representation the loss of asymptotic freedom manifests itself as a change of sign in the first coefficient of the beta function. The critical number of flavors for which this occurs is N I f = 3 C (G). (3.1.3) T(r)

29 3.1 - The Conformal Window of Supersymmetric Gauge Theories 19 Note that compared to the nonsupersymmetric case (.0.4) this value is lowered due to the additional screening of the scalars and the gluinos. In fact the coefficient 3 should be replaced by 11 in the nonsupersymmetric case. 4 Similar to the nonsupersymmetric case it might be possible that an IR fixed point exists since for a certain number of flavors and colors the first coefficient of the beta function is positive while the second coefficient is negative [83]. This situation appears as soon as the second coefficient changes sign. For a given representation this occurs when N III f = 3C (G) C (G). (3.1.4) C (G) + 4C (r) T(r) In general the true critical number of flavors above which a nonperturbative IR fixed point is generated does not coincide with the above perturbative estimate. The former was first found by Seiberg [77] and will be determined below. To show the existence of a nontrivial IR fixed point we will consider the large N limit holding N f = 1 ɛ, ɛ 1 and Ng fixed. In case of the fundamental representation it is also important to take the large N f limit in order to have N f N I f fixed because the N I f trace normalization is a constant. This is in contrast to the two-indexed representations for which the trace normalization factors grow as N. The fixed point is now given by C (r)g = 4π ɛ + O(ɛ ) with C (r) growing as N both for the fundamental and twoindexed representations. The argument above cannot be applied to the case of matter in representations with more than two indices since all these theories are not asymptotically free at large number of colors. In the following we will only consider either the fundamental or the two-indexed representations. Since a fixed point exists, at least at large N, we follow Seiberg [77] and derive some exact results about the theory. The strategy is to first obtain an exact expression for the dimension D of some spinless operator in terms of the number of colors and flavors. We will then use a property of conformal field theory stating that spinless operators (except for the identity) have D 1 in order not to have negative norm states in the theory [84 86]. When this bound is saturated it gives us a relation between the number of colors and flavors at which our conformal description breaks down. There are two ways to obtain the dimension of chiral operators in the theory. First we note that the superconformal algebra includes an R-symmetry and find the following

30 0 Chapter 3 - The Supersymmetric Phase Diagram relation between the corresponding R-charge and dimension D of the operators D R. The bound is saturated for chiral operators D = 3 R and for antichiral operators D = 3 R. Since this R-symmetry must be anomaly free and commute with the flavor symmetries it must be the one assigned in Table 3.1. For the spinless chiral operator Φ Φ we therefore arrive at D(Φ Φ) = 3 R(Φ Φ) = 3 T(r)N f C (G) T(r)N f. (3.1.5) Perhaps an easier way to obtain D(Φ Φ) is to note that at the zero of the beta function we have γ = T(r)N f 3C (G) T(r)N f. Hence from D(Φ Φ) = γ + we end up with Eq. (3.1.5). As discussed above our conformal description of the theory requires D(Φ Φ) 1 with the bound being saturated by free fields. Hence the critical number of flavors above which the theory exists in a conformal phase is therefore N II f = 3 C (G). (3.1.6) 4 T(r) In Fig. 3.1 we plot the phase diagram for the supersymmetric gauge theories with matter in one of the three two-indexed representations - adjoint, two-indexed symmetric and two-indexed antisymmetric - as well as the fundamental representation. Note how the various representations merge into each other when, for a specific value of N, they are actually the same representation. The supersymmetric conformal window displays many qualitative features in common with the nonsupersymmetric one (Fig..5). The nonsupersymmetric window however, is only an estimate which makes use of the ladder approximation. We observe that in the case of the fundamental representation the supersymmetric conformal window extends below the curve defined as where the second coefficient of the beta function changes sign. This does not happen for the adjoint and two-indexed symmetric and antisymmetric representations for any N larger than four. In the nonsupersymmetric case the curve N II f the ladder approximation. stays well above N III f for any N and any representation in

31 3. - Sizing The Unparticle World 1 N f Figure 3.1: Phase diagram for supersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-indexed antisymmetric representation (blue), iii) two-indexed symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories develop an IR fixed point. The dashed curve represents the change of sign in the second coefficient of the beta function. 3. Sizing The Unparticle World Georgi has recently proposed to couple a conformal sector to the Standard Model [87 89]. We find it interesting to provide a measure of how large, in theory space, the fraction of the unparticle world is. We assume, following Georgi, the unparticle sector to be described, at the underlying level, by asymptotically free gauge theories developing an IR fixed point. A reasonable measure is then, for a given representation, the ratio of the conformal window to the total window of asymptotically free gauge theories R FP = N I N min f dn N II N min f dn N I N min f dn, (3..1) where N min is the lowest value of number of colors permitted in the given representation for which the above ratio is computed. Similarly we define for the nonconformal region,

32 Chapter 3 - The Supersymmetric Phase Diagram but still asymptotically free, the following area ratio R NFP = N II N min N I N min f f dn dn. (3..) We now estimate the above fractions within the N = 1 phase diagram as well as for the nonsupersymmetric one. Note that we have already taken the upper limit of integration to be infinity, which effectively reduces the set of representations we are going to consider to those with at most two indices The Supersymmetric Case A straightforward evaluation for the supersymmetric case yields R FP = 3 C (G) N min T(r) dn N min 3 3 N min 4 C (G) T(r) dn C (G) T(r) dn = 1. (3..3) Surprisingly the result is independent on the chosen representation and, of course, R NFP = 1 1/ = 1/. The universality of this ratio is impressive. 3.. The Nonsupersymmetric Case We now determine R FP in the case of nonsupersymmetric gauge theories with only fermionic for the nonsupersymmetric theo- matter. This task requires the knowledge of N I f and N II f ries which are found in (.0.4) and (.1.). We now list the ratios for the fundamental (F) and the two-indexed representations, i.e. Adj (G), two-indexed symmetric (S) and two-indexed antisymmetric (A) R FP [F] = , R FP[G] = R FP [A] = R FP [S ] = (3..4) Remarkably in the nonsupersymmetric case as well the fraction of the conformal window for the representations which are asymptotically free for any number of colors is very

33 3. - Sizing The Unparticle World 3 close to each other. Circa 5% of the nonsupersymmetric asymptotically free gauge theories with fermions in a given representation is expected to develop an IR fixed point. This can be compared with the exact 50% in case of N = 1 supersymmetric vector-like theories. We note that in the nonsupersymmetric case, except for the adjoint representation, the values of the ratios are determined by the large N part of the integration.

34

35 Chapter 4 The All-Orders Beta Function Conjecture Having studied the supersymmetric beta function in the previous chapter we propose an all-orders beta function for nonsupersymmetric gauge theories with or without massless fermions transforming according to arbitrary representations of the underlying S U(N) gauge group as first conjectured in [1]. The beta function at small coupling reduces to the two-loop beta function. Similar to the supersymmetric case the form of the new beta function allows us to bound the phase diagram for a generic nonsupersymmetric gauge theory with fermionic matter in a given representation of the underlying gauge group. The result is simple and we compare it with the phase diagram presented in section.1 obtained using the ladder approximation. We find that the ladder results provide a conformal window systematically smaller than the one presented here. The conformal windows we propose make use of the new beta function and the condition of the absence of negative norm states in a conformal field theory. The actual size of the conformal window may be smaller than the one presented here which can be considered as a bound on the size of the conformal window. The beta function is then generalized to the case of a gauge theory with matter in different representations of the gauge group. We consider the specific case of a single massless Weyl fermion in the adjoint repre- 5

36 6 Chapter 4 - The All-Orders Beta Function Conjecture sentation which corresponds to super Yang-Mills. By directly comparing our expression with the super Yang-Mills result we determine the anomalous dimension of an adjoint fermion. At infinite number of colors a prediction of the beta function was made in [90] for theories with matter in the two-indexed symmetric and antisymmetric representation of the gauge group and for Yang-Mills theory in [91]. Our proposed all-orders beta function coincides with these results at infinite number of colors. The zero flavor limit, i.e. pure Yang-Mills, is a quite interesting case since we can compare the running of the coupling due to the new beta function with lattice data for S U(), S U(3) and S U(4) [9 94]. We find the new beta function to compare well with data, capturing the fact that the results do not depend on the number of colors when plotting the running of the t Hooft coupling. This result and the comparison with data is rather encouraging. We finally determine the ratio between the area of a given conformal window to the associated asymptotically free one and find that it is universal, i.e. does not depend on the specific matter representation. A universal ratio was found earlier in section 3. in the supersymmetric case. The ratio assumes the same value in the supersymmetric and in the nonsupersymmetric case. We then generalize the phase diagram to the case of multiple matter representations simultaneously affecting the gauge dynamics. We determine the size of the new conformal regions and find a remarkably simple formula measuring the ratio of conformal regions with respect to the associated asymptotically free regions. Universality manifests again since the ratios depend only on how many representations are considered but not which ones. Our analysis of gauge theories with fermions transforming according to multiple representations have an immediate implication for beyond Standard Model physics. In fact it will allow us to construct an explicit walking Technicolor model with fermions in the fundamental and adjoint representation of the gauge group not at odds with the Electroweak Precision Tests. This will be investigated in detail in chapter 7.

37 4.1 - All-Orders Nonsupersymmetric Beta Function All-Orders Nonsupersymmetric Beta Function Consider now a generic nonsupersymmetric gauge theory with N f Dirac fermions in a given representation r of the gauge group. For the readers convenience we repeat the two-loop beta function here β(g) = β 0 (4π) g3 β 1 (4π) 4 g5, (4.1.1) where g is the gauge coupling and the beta function coefficients are given by β 0 = 11 3 C (G) 4 3 T(r)N f (4.1.) β 1 = 34 3 C (G) 0 3 C (G)T(r)N f 4C (r)t(r)n f. (4.1.3) To this order the two coefficients are universal, i.e. do not depend on which renormalization group scheme one has used to determine them [65]. The perturbative expression for the anomalous dimension reads γ(g ) = 3 C (r) g 4π + O(g4 ). (4.1.4) With γ = d ln m/d ln µ and m the renormalized fermion mass. It would be great to have the complete expression for the beta function for a nonsupersymmetric theory. This seems to be a formidable task. Inspired by supersymmetry we suggest an all-orders nonsupersymmetric beta function which has a number of interesting properties and predictions which we will compare and test against nonperturbative results found using various methods and models. The first observation is that the perturbative anomalous dimension depends on C (r) which appears explicitly in the last term of the second coefficient of the beta function. We hence write the beta function in the following form β(g) = g3 β 0 T(r) N 3 f γ(g ) ( ), (4.1.5) (4π) 1 g C 8π (G) 1 + β 0 β 0

38 8 Chapter 4 - The All-Orders Beta Function Conjecture with β 0 = C (G) T(r)N f. (4.1.6) It is a simple matter to show that the above beta function reduces to Eq. (4.1.1) when expanding to O(g 5 ). Given that only the two-loop beta function has universal coefficients, i.e. is independent of the renormalization scheme, we will assume the existence of a scheme for which our beta function is complete. 4. IR Fixed Point As we decrease the number of flavors from just below the point where asymptotic freedom is lost, corresponding to N I f = 11 C (G), (4..1) 4 T(r) one expects a perturbative zero in the beta function to occur [83]. From the expression proposed above one finds that at the zero of the beta function, barring zeros in the denominator, one must have γ = 11C (G) 4T(r)N f T(r)N f. (4..) The anomalous dimension at the IR fixed point is small for a value of N f such that N f = N I f (1 ɛ), with ɛ > 0, (4..3) and ɛ 1. Indeed, in this approximation we find γ = ɛ 1 ɛ 1. (4..4) It is also clear that the value of γ increases as we keep decreasing the number of flavors. Before proceeding let us also analyze in more detail the denominator of our beta function.

39 4. - IR Fixed Point 9 At the IR fixed point we have 1 g 8π C (G) 1 ( 5 1 ) 11ɛ. (4..5) For very small ɛ the denominator is positive while staying finite as ɛ approaches zero. The finiteness of the denominator is due to the fact that from the perturbative expression of the anomalous dimension (valid for small epsilon) the fixed point value of g is g 8π = ɛ 3C (r) + O(ɛ ). (4..6) Since a perturbative fixed point does exist we extend the analysis to a lower number of flavors. The dimension of the chiral condensate is D( ψψ) = 3 γ which at the IR fixed point value reads D( ψψ) = 10T(r)N f 11C (G) T(r)N f. (4..7) To avoid negative norm states in a conformal field theory one must have D 1 for nontrivial spinless operators [84 86]. Hence the critical number of flavors below which the unitarity bound is violated according to the conjectured all-orders beta function is N II f = 11 8 C (G) T(r), (4..8) which corresponds to having set γ =. One should note that the analysis above is similar to the one done in chapter 3 for supersymmetric gauge theories. However, the actual size of the conformal window may be smaller than the one presented here which therefore can be considered as a bound of the size of the window. In Figure 4.1 we plot the new phase diagram Comparison with the Ladder Approximation We now confront our bound of the conformal window with the one obtained using the ladder approximation (.1.). Comparing with the result of the all-orders beta function we see that it is the coefficient of C (G)/T(r) which is different. To better appreciate the differences between these two results we plot the two con-

40 30 Chapter 4 - The All-Orders Beta Function Conjecture N f Figure 4.1: Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-indexed antisymmetric representation (blue), iii) two-indexed symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. Between the upper and the lower solid curves the theories are expected to develop an IR fixed point according to the all-orders beta function. The dashed curve represents the change of sign in the second coefficient of the beta function. formal windows predicted within these two methods in Figure 4.. The ladder result provides a size of the window, for every fermion representation, smaller than the bound found with our approach. This is a consequence of the value of the anomalous dimension at the lower bound of the window. The unitarity constraint corresponds to γ = while the ladder result is closer to γ 1. Indeed if we pick γ = 1 our conformal window approaches the ladder result. Incidentally, a value of γ larger than one, still allowed by unitarity, is a welcomed feature when using this window to construct walking technicolor theories. It allows for the physical value of the mass of the top while avoiding a large violation of flavor changing neutral currents [95]. 4.. Comparison with Lattice Data From Fig. 4. it is clear that the bound of the conformal window is larger than the one obtained in the ladder approximation. It is therefore interesting to compare both analytical methods with lattice results. Recently the following theories have been investigated quite extensively via first principle lattice simulations