Thermodynamics of Generalizations of Quantum Chromodynamics

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Fantastic Gauge Theories and Where to Find Them: Thermodynamics of Generalizations of Quantum Chromodynamics

Daniel C. Hackett

B.A., University of Virginia, 2012

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulﬁllment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2019 This thesis entitled: Fantastic Gauge Theories and Where to Find Them: Thermodynamics of Generalizations of Quantum Chromodynamics written by Daniel C. Hackett has been approved for the Department of Physics

Prof. Thomas DeGrand

Prof. Ethan Neil

Prof. Anna Hasenfratz

Prof. Paul Romatschke

Prof. Markus Pﬂaum

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁnd that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii

Hackett, Daniel C. (Ph.D., Physics)

Fantastic Gauge Theories and Where to Find Them:

Thermodynamics of Generalizations of Quantum Chromodynamics

Thesis directed by Prof. Thomas DeGrand

Over the past few decades, lattice gauge theory has been successfully employed to study the ﬁnite-temperature phase structure of quantum chromodynamics (QCD), the theory of the strong force. While this endeavor is well-established, QCD is only one strongly-coupled quantum

ﬁeld theory in a larger family of similar theories. Using the lattice toolkit originally invented to investigate QCD, we have begun exploring the phase structures of these cousins of QCD.

This thesis focuses on generalizations of QCD with multiple diﬀerent species of fermions charged under distinct representations of the gauge group. I present the results of the ﬁrst-ever lattice study of the thermodynamics of one such theory, as well as an analytic calculation which predicts the order of the phase transition for all such theories. Dedication

In loving memory of my grandmother Jeanne.

To my parents, Karen and Chuck, for creating me.

For my siblings, Cait, Will, and Kelsey.

To all of the friends, human and animal, who let me know when it was time to go outside. v

Acknowledgements

Foremost among those deserving acknowledgement here are my advisors, Tom DeGrand and

Ethan Neil. In spite of a proliferation of ideas and an admittedly combative learning style on my part, they have managed to not only consistently guide me towards interesting and approachable problems, but also to teach me how to ﬁnd them myself. I could not have asked for better advisors, and count myself incredibly lucky to have learned from them.

Among friends and fellow students, William Jay deserves to be singled out. Notably, our conversations about his work with Tom and Ethan sparked my initial interest in lattice gauge theory. I am further obliged to note that I learned much of the quantum ﬁeld theory and virtually all of the math I know from our work together. I look forward to whenever the Hackett-Jay collaboration enters its next high-bandwidth phase.

I owe the remaining members of the local particle/nuclear theory group (comprised at diﬀer- ent times of various combinations of Paul Romatschke, Shanta DeAlwis, Anna Hasenfratz, Oliver

DeWolfe, Andrea Carosso, Oliver Witzel, Venkitesh Ayyar, and Takaaki Ishii) thanks for years of stimulating and formative (for better and worse) discussion.

Among my collaborators, Ben Svetitsky and Yigal Shamir deserve particular thanks. I have learned a great deal of interesting physics from our many conversations(/arguments) over the past few years, and look forward to continuing to do so in the future.

I sometimes feel that the symmetry (softly broken Z2?) between art and science goes un- derappreciated. It is worth noting that I can trace my career as a physicist directly and un- ambiguously back to reading “The Diamond Age: a Young Lady’s Illustrated Primer” by Neal

Stephenson. vi

Contents

Chapter

1 Introduction & Overview 1

1.1 Notation...... 5

2 QCD & its Extended Family 7

2.1 Quantum chromodynamics...... 7

2.1.1 Physics of quarks and gluons...... 7

2.1.2 QCD as a gauge theory...... 8

2.1.3 QCD Lagrangian...... 9

2.1.4 Asymptotic freedom & conformality...... 12

2.2 Generalizations of QCD...... 13

2.2.1 Number of ﬂavors and quark masses...... 13

2.2.2 Other gauge groups...... 14

2.2.3 Higher-representation fermions...... 16

2.3 Zoology...... 18

2.3.1 Participating irreps...... 18

2.3.2 Theory demographics...... 24

2.4 Ferretti’s Model...... 26

3 Lattice Gauge Theory 28

3.1 Path integrals & Euclidean time...... 28 vii

3.2 How to ﬁt QFTs on computers...... 30

3.3 Actions for lattice gauge theories...... 32

3.3.1 Gauge ﬁelds...... 32

3.3.2 Fermions on the lattice...... 34

3.3.3 Pseudofermions...... 36

3.4 Computing on the lattice...... 38

3.4.1 Monte Carlo integration...... 38

3.4.2 Markov Chain Monte Carlo...... 39

3.5 Hadron Spectroscopy...... 46

3.5.1 Correlators...... 46

3.5.2 Extracting masses...... 47

3.5.3 AWI quark mass...... 49

3.5.4 Systematic errors...... 50

3.6 Gradient ﬂow...... 52

3.6.1 Scale setting...... 53

4 Lattice Thermodynamics of QCD-like Systems 55

4.1 Exotic phase structures...... 55

4.2 Chiral symmetry breaking...... 56

4.2.1 Tumbling & the Most Attractive Channel hypothesis...... 60

4.2.2 Parity doubling...... 62

4.3 Conﬁnement...... 63

4.3.1 Polyakov loops & multiple representations...... 64

4.3.2 Center symmetry...... 67

4.3.3 Partial breaking of center symmetry...... 70

5 Multirep Thermodynamics on the Lattice 72

5.1 Lattice-deformed Ferretti model...... 73 viii

5.2 Theoretical expectations...... 74

5.2.1 Tumbling & separation of chiral transitions...... 74

5.2.2 Polyakov loops & center symmetry...... 74

5.2.3 Order parameters & transition orders...... 76

5.3 Lattice details...... 79

5.3.1 Simulation details...... 79

5.3.2 Data sets...... 80

5.4 Results: Phase structure of limiting-case theories...... 82

5.4.1 A2-only limit...... 82

5.4.2 F -only limit...... 86

5.5 Result: Phase structure of full theory...... 89

5.5.1 Simultaneous transitions...... 89

5.5.2 First-order phase transition...... 92

5.5.3 Critical temperature...... 93

5.6 Discussion...... 95

5.6.1 Phase structure...... 95

5.6.2 Transition order...... 97

5.A Scale setting...... 99

5.A.1 Scales for SU(4)...... 99

5.A.2 Fitting the lattice spacing...... 99

6 Multirep Pisarski-Wilczek 104

6.0.1 Notation...... 105

6.1 Overview and applicability...... 106

6.2 Chiral symmetries & Lagrangian construction...... 108

6.2.1 Field content and single-irrep Lagrangians...... 108

6.2.2 Irrep coupling & constraints from anomaly...... 110 ix

6.3 Lagrangian zoology...... 113

6.4 Single-irrep subsectors...... 118

6.5 Results: Theories without anomaly terms...... 119

6.5.1 β functions...... 119

6.5.2 Solving for ﬁxed points...... 120

6.5.3 Fixed point zoology...... 121

6.5.4 Stability of ﬁxed points...... 122

6.6 Example: Ferretti’s model & its lattice deformation...... 124

6.6.1 Large Nc ...... 126

6.7 Results: Theories permitting anomaly-implementing terms...... 135

6.7.1 1-1 anomaly...... 135

6.7.2 1-2 anomaly...... 137

6.7.3 1-3 anomaly...... 140

6.7.4 2-2 anomaly...... 142

6.8 Discussion...... 147

6.A Constraints due to vacuum stability and χSB pattern...... 148

Bibliography 153

Appendix

A Autocorrelations in MCMC Simulations 165

B Group Theory 168

B.1 Diagnosing complexity classes of irreps of SU(N)...... 168

B.2 Algebraic methods for linear sigma models...... 168 x

C Gradient Flow Phase Diagnostics 172

C.1 Flowed anisotropy...... 172

C.2 Polyakov loops at long ﬂow times...... 174 xi

Tables

Table

2.1 Exhaustive list of irreps of SU(Nc) that may be present in asymptotically free SU(Nc)

gauge theories with vectorlike matter content. The column “relevance” states the

ranges of Nc over which each irrep is well-deﬁned, not the conjugate irrep of a more

fundamental one, and doesn’t break asymptotic freedom. dr is the dimension and Tr

is the trace of representation r; the quadratic Casimir Cr may be obtained using the

identity Cr = TrdG/dr, where dG is the dimension of the adjoint irrep at the same

Nc. The trace of the fundamental representation TF = a is left arbitrary...... 19

3.1 The full set of diﬀerent meson correlators used in these results. All sources are

smeared with radius r0 = 6a, denoted by S, and all sinks are points, denoted by P .

Sign denotes whether the correlator is of cosh form (+) or sinh form (−)...... 47

5.1 Order parameters in various mass regimes. Note that e.g. m4 in the column m4

indicates that the F s have ﬁnite mass...... 76

5.2 Summary of ﬁnite-temperature ensembles...... 81

5.3 Zero-temperature data sets used to compute the scale near the thermal transition.. 81

2 5.4 Best-ﬁt parameters for the t1/a model deﬁned by Eqns. 5.5, 5.7, and 5.8...... 101

5.5 Model parameters at β = 7.75 from direct ﬁt to β = 7.75 data versus predictions for

those parameters from Eqs. (5.7) and (5.8), and the best-ﬁt values in Table 5.4... 102 xii

6.1 Structure of anomaly-implementing irrep-coupling terms in the Lagrangians for all

29 asymptotically-free theories which permit them. Vertex type indicates how many

ﬁelds of irreps 1 and 2 are present in anomaly-implementing terms, e.g., 1-2 indicates

2 a term like φ1φ2 (ignoring any group structure). Lagrangian structure indicates irrep complexity class, number of ﬂavors, and exponent of the determinant/Pfaﬃan for

irrep 1 and irrep 2 in order; complexity class is indicated as 1 when nr = 1 ﬂavor

is present, as these cases all have the same Lagrangian structure. For example, the

2 structure 1 -C2 with S2 as irrep 1 and A2 as irrep 2 indicates a theory with nS2 = 1

ﬂavors of S2-irrep fermions with exponent 2 on the (trivial, for nr = 1) determinant

and n = N D = 2 ﬂavors of complex A -irrep fermions with exponent 1 on the A2 A2 2

2 determinant, yielding an anomaly-implementing term δL ∼ (φS2 ) Det φA2 ...... 117

6.2 The ﬁve (of six total) ﬁxed points which are amenable to concise analytic expression

∗ P using the ansatz solution g = 1/[4 + r dr]. All ﬁxed points found have vF =

vA2 = 0, as necessary for real-valued ur. The ﬁrst four ﬁxed points are “decoupled

product” ﬁxed points. Couplings are in the convention of the β functions, and thus

diﬀerent from those in the Lagrangian by an overall factor of 1/(16π2)...... 126

6.3 Values of couplings at the second multirep ﬁxed point, which is not amenable to

concise analytic expression, for our theories of interest. Again, vF = vA2 = 0. The

numerical values are computed from closed-form expressions and truncated at ﬁve

signiﬁcant digits. Couplings are in the convention of the β functions...... 127

6.4 Distinct Lagrangian structures of interest for the large nF (large Nc) limit; note that

there is always one additional complex irrep for the F irrep. n1 is the number of

one-ﬂavor irreps present, nG is the number of (real) adjoint-irrep ﬂavors, and nC1

and nC2 are how many ﬂavors of up to two distinct complex irreps (other than F )

max are present. nF indicates the largest number of fundamental ﬂavors examined in

this class by the numerical survey of all Lagrangians relevant to 2 ≤ Nc ≤ 20.... 128 xiii

6.5 Nontrivial ﬁxed points for the β functions Eqs. 6.62, 6.63, and 6.64. For all ﬁxed

points, vr = 0 and c = 0. The value of each coupling is provided to six digits of

precision. Nλ<0 indicates the number (out of ﬁve) of negative eigenvalues of the

stability matrix at this ﬁxed point...... 143

6.6 Nontrivial ﬁxed points for the β functions Eq. 6.66. The value of each coupling

is provided to six digits of precision. Nλ<0 indicates the number (out of four) of

negative eigenvalues of the stability matrix at this ﬁxed point; 0∗ indicates that two

of the eigenvalues are exactly zero...... 143

6.7 Nontrivial ﬁxed points for the β functions Eqs. 6.69 and 6.70. For all ﬁxed points,

vr = 0 and c = 0. The value of each coupling is provided to six digits of precision.

Nλ<0 indicates the number (out of six) of negative eigenvalues of the stability matrix

at this ﬁxed point...... 145

6.8 Nontrivial ﬁxed points for the β functions Eqs. 6.73, 6.74, and 6.75. For all ﬁxed

points, vr = 0 and c = 0. The value of each coupling is provided to six digits of

precision. Nλ<0 indicates the number (out of ﬁve) of negative eigenvalues of the

stability matrix at this ﬁxed point...... 146 xiv

Figures

Figure

max 2.1 Top panel shows the maximum number of ﬂavors nr of each irrep permitted in a

single-irrep theory, as a function of Nc. The large spike for A2 is because it is real

for Nc = 4. Note that for Nc = 5, there is exactly one ﬂavor allowed of each of the

irreps G3, G4, and 755. Second panel shows the total number of irreps present, as

well as the number of those irreps which are real; the dashed line indicates the con-

tribution of the adjoint irrep, which is always real. The bottom two panels show the

number of asymptotically free multirep theories vs. Nc. The third panel shows the

total number, while the bottom panel shows the number of distinct theories without

diﬀerentiating between theories with diﬀerent numbers of fundamental fermions nF .

In the bottom panel, the dashed line is at the asymptotic value of 55...... 25

5.1 Columbia plot illustrating expectations for the order of the ﬁnite-temperature phase

transition. The axes are the masses of the two fermion species in the theory, with

m4 on the x-axis and m6 on the y-axis. The upper right corner is the pure-gauge

limit; the lower left corner is the double chiral limit; the upper-left corner is the

F -only chiral limit; the lower-right corner is the A2-only chiral limit. Green ﬁelds

indicate regions of parameter space where the theory is predicted to exhibit a ﬁrst

order transition. Blue lines indicate regions of parameter space where the theory is

predicted to exhibit a second-order transition...... 78 xv

3 5.2 Dependence on κ6 of various quantities in the A2-only theory for β = 8.5 on 12 × 6.

The top panel shows diagnostics of conﬁnement: unﬂowed Polyakov loops and the

ﬂowed anisotropy at t/a2 = 1. (Quantities are normalized by their maximum values

along the slice for ease of comparison of their qualitative behavior.) The middle panel

shows chiral diagnostics, the mass splittings of parity-partner mesons. The bottom

panel shows the plaquette and the AWI fermion mass. Points with closed (open)

circles are deemed conﬁned (deconﬁned) according to the behavior of Polyakov loops

at long ﬂow time. There is a single transition (gray band) from the conﬁned and

chirally broken phase to the deconﬁned and chirally restored phase...... 83

3 5.3 Phase diagram for the A2-only limit for 12 × 6 lattices. Blue dots are conﬁned

and chirally broken ensembles while yellow stars and red Xs indicate deconﬁned and

chirally symmetric ensembles. The blue ﬁeld thus indicates the conﬁned and chirally

broken region of parameter space, while the orange ﬁeld indicates the deconﬁned and

chirally restored region of parameter space. The red Xs mark deconﬁned ensembles

where m6 < 0. The black box indicates the slice through bare parameter space

shown in Fig. 5.2. The circled ensembles have matching zero-temperature ensembles. 84

5.4 Transition temperature Tc for the A2-only theory with Nt = 6 as a function of quark

mass. The three heaviest ensembles (down arrows) are deconﬁned-side ensembles

and thus give upper bounds on Tc. The lightest ensemble (up arrow) is a conﬁned-

side ensemble and thus gives a lower bound on Tc. The scale has been set via

2 t1 ≡ 1/(780 MeV) , corresponding to the scale of QCD, for easy comparison..... 85

5.5 Dependence on κ4 of various quantities in the F -only theory along a slice of constant

β = 9.2 on 123 × 6. The top panel shows diagnostics of conﬁnement: the unﬂowed

fundamental Polyakov loop and the ﬂowed anisotropy at t/a2 = 1. The middle

panel shows diagnostics of chiral condensation, the mass splittings of parity-partner

mesons. The bottom panel shows the plaquette and AWI fermion mass. The peaks

of the Polyakov loop and chiral susceptibilities lie somewhere in the gray band... 87 xvi

5.6 Phase diagram for the F -only limit on 123 × 6 lattices. Symbols and colors are

as in Fig. 5.3, with the addition of hollow diamonds indicating ensembles in the

crossover region, where the diagnostics are ambiguous. The black box indicates the

slice through bare parameter space shown in Fig. 5.5. The circled ensembles have

matching zero-temperature ensembles available...... 88

5.7 Temperature for the crossover of the fundamental-only theory with Nt = 6 as a func-

2 tion of the AWI mass. The scale has been set via t1 ≡ 1/(780 MeV) , corresponding

to the scale of QCD, for easy comparison...... 88

5.8 Behavior of various quantities in the full theory, varying κ6 across the transition while

3 holding β = 7.4 and κ4 = 0.1285 constant on 12 × 6. The gray band brackets the

transition. Points with closed (open) circles are conﬁned (deconﬁned) according to

the behavior of Polyakov loops at long ﬂow time. Top: Unﬂowed Polyakov loops for

both representations and the ﬂowed anisotropy. All diagnostics of conﬁnement show

simultaneous discontinuities. Middle: Mass splittings of parity partner mesons:

scalar vs. pseudoscalar, and vector vs. axial vector. Chiral symmetry restoration

occurs simultaneously for the two representations. Bottom: AWI fermion masses

for both representations, and the plaquette. All quantities jump discontinuously at

the transition...... 90 xvii

5.9 To the left is the phase diagram for Nt = 6 lattices, while to the right is the same

region of bare parameter space for Nt = 8 lattices. Blue dots indicate conﬁned and

chirally broken ensembles. Yellow stars indicate deconﬁned and chirally restored

ensembles with mr > 0 for both species; red Xs are in regions where m4 < 0 or

m6 < 0 or both. The violet dot is a conﬁned and chirally broken ensemble with

m4 < 0. The black box indicates the slice through bare parameter space shown in

Fig. 5.8. In the right ﬁgure, the transition region from Nt = 6 is overlaid in gray,

demonstrating that the transition moves as Nt is varied. The circled ensembles have

matching zero-temperature ensembles available. Ensembles enclosed by diamonds

are where the phase changed when volume was changed (see text)...... 91

3 5.10 Sighting of a tunneling event in the equilibration of a 12 ×6 ensemble at (β, κ4, κ6) =

(7.75, 0.127, 0.128). The gray band highlights the region of the tunneling event. The

top panel shows the plaquette expectation value and the bottom panel shows the real

and imaginary parts of the fundamental Polyakov loop expectation value. As seen

from the behavior of the Polyakov loop in the bottom panel, the ensemble tunnels

from the conﬁned phase to the deconﬁned phase...... 92

5.11 Transition temperature of the multirep theory as a function of κ4 on Nt = 6 and

Nt = 8. Lattice spacings used to determine the temperature are computed using

2 2 the ﬁt to t1/a with t1 = 1/(780 MeV) (for easy comparison to QCD) as discussed

in Appendix 5.A. Axes are matched between Nt = 6 and Nt = 8. Black lines with

error bands indicate the temperature on ensembles on either end of the transition

bands in Fig. 5.9(b). The transition temperature lies in the span indicated by the

double-headed arrows...... 94 xviii

5.12 Columbia plots, by analogy with QCD. In Fig. 5.12(a), the quark masses are in

2 lattice units. In Fig. 5.12(b), quark masses are in MeV [t1 ≡ 1/(780 MeV) ], for

easy comparison to QCD, as deﬁned in Appendix 5.A. Each color and symbol is

associated with a diﬀerent β and Nt. Closed symbols are ﬁnite-temperature quark

masses, while hollow symbols are zero-temperature quark masses from ensembles

on the transition boundary The lattice spacings for the zero temperature quark

2 masses are computed directly from t1/a on that ensemble. The lattice spacings

for the ﬁnite-temperature quark masses are computed from the model described in

Appendix 5.A...... 98 √ √ 5.13 Ratio of the Wilson ﬂow scales t1 and t0 as a function of lattice spacing, plotted √ against lattice spacing measured in units of t1. Above a ∼ 0.8, large lattice spacing

eﬀects begin to contaminate t0...... 100

2 5.14 Lines are predictions for t1/a as a function of κ6 for various κ4 at β = 7.75 by the

model deﬁned by Eqs. (5.5)–(5.8) and the best-ﬁt parameters of Table 5.4. Dots are

2 t1/a data at β = 7.75. Colors are matched between dots and lines at the same κ4. 103

6.1 Dots are the values of the couplings at ﬁxed points from theories in the C2-R2 large

Nc Lagrangian class at diﬀerent values of nF . Fixed points are found separately at

each nF using numerics, and associated using the assumption that couplings will not

change sign as nF varies. Lines indicate this association. Diﬀerent colors correspond

to different fixed points, but no graphical distinction is made between the different

couplings for a given ﬁxed point...... 132

6.2 Eigenvalues of the stability matrix as a function of nF for non-decoupled-product,

non-ansatz fixed points for the large Nc Lagrangian class C2-R2 (top) and 1-1-C2 (bottom). Different colors are associated with different fixed points. Dots are eigen-

values, while lines are associations comprising eigenvalue trajectories, with associa-

tions made as described in the text...... 134 xix

2 2 C.1 Evolution under Wilson ﬂow of the split ﬂow observables t Ess and t Est , shown

3 for three diﬀerent 12 × 6 ensembles on a slice at constant (β, κ6) = (7.4, 0.1285) in

the lattice-deformed Ferretti model (see Chapter5). Also shown is the behavior

of RE − 1, where RE ≡ hEss(t)i / hEst(t)i. The left panel shows typical conﬁned

behavior: the split ﬂow observables are degenerate and RE = 1, indicating isotropy.

The central panel shows some hints of impending deconﬁnement: the split ﬂow

observables separate slightly, and RE − 1 becomes nonzero. The right panel shows

typical deconﬁned behavior: the split ﬂow observables break apart, and RE −1 grows

rapidly in ﬂow time, indicating anisotropy...... 173

C.2 Fundamental Polyakov loops under Wilson ﬂow, depicted for three diﬀerent 123 × 6

ensembles on a slice of constant (β, κ6) = (7.4, 0.1285) in the lattice-deformed Ferretti

model (see Chapter5). Each panel depicts the complex plane. Each line is the

evolution of PF under Wilson ﬂow on a conﬁguration in the ensemble (i.e., each line

is the complex function PF (t) where t is the ﬂow time). In the two left panels, we see

typical conﬁned behavior. In the right panel, we see typical deconﬁned behavior. The

ensemble shown in the central panel sits almost directly on top of the conﬁnement

transition...... 175

C.3 Ensemble-averaged evolution of PF under Wilson ﬂow, depicted for three diﬀerent

3 12 ×6 ensembles on a slice of constant (β, κ6) = (7.4, 0.1285) in the lattice-deformed

Ferretti model (see Chapter5). In the left panel and center panels, we see typical con-

ﬁned behavior: the Polyakov loop either does not increase in magnitude or increases

very slowly. In the right panel, we see typical deconﬁned behavior: the magnitude

of Polyakov loop rapidly approaches its maximum value (max PF = d(F ) = 4).... 175 xx

C.4 The evolution of the Polyakov loop under ﬂow along a slice through bare parameter

D 3 space varying β at constant κ = 0.128 for NF = 2 SU(3) on 12 × 6. Each line is the same slice, but at diﬀerent ﬂow times t/a2. As t/a2 increases, Polyakov loops for

deconﬁned ensembles saturate to their maximum value of Nc, while Polyakov loops

for conﬁned ensembles remain small. The dashed lines indicate the boundaries used

in this study to define confined, ambiguous, and deconfined, for t/a2 = 2.0...... 177 Chapter 1

Introduction & Overview

“You experimentalists are conﬁned to the real line, but we

theorists stride godlike over the complex plane.”

–Tom DeGrand

Many of the particles that we observe in nature, like the everyday proton and neutron or more exotic objects only seen in colliders and cosmic rays, are hadrons. Hadrons are not irreducible objects, but rather composite particles made of quarks and gluons, bound tightly together by the strong nuclear force. The strong nuclear force, one of the four known fundamental forces, is best described by quantum chromodynamics (QCD), a gauge theory wherein the interactions of quarks and gluons are highly constrained by symmetry. As part of our most fundamental understanding of nature, it is imperative to understand how QCD yields the rich phenomenology we observe in particle and nuclear physics. However, QCD is strongly coupled and exhibits complex nonperturbative behavior that cannot be investigated using analytic pen-and-paper calculations.

To investigate QCD (or theories like it), we must instead use lattice gauge theory, a formalism which regulates the infinites inherent to quantum field theories by discretizing spacetime. Soon after its development, it was recognized that the lattice formalism allows us to put QCD on a computer and study it numerically. The past few decades have seen the development of a powerful toolkit of Markov Chain Monte Carlo techniques for simulating quantum field theories, an approach that has enabled us to make enormous progress in understanding the dynamics of the strong force. 2

However, quantum chromodynamics is only one member of an infinite family of similar theories with different gauge groups and fermion content. All of the techniques originally developed to study QCD can be reapplied to learn about to these generalizations. These systems are of interest for a variety of reasons both practical and theoretical. Individually, some of QCD’s cousins appear in models for physics beyond the Standard Model (BSM), particularly ones wherein the Higgs boson and/or dark matter appear as bound states of new fundamental particles. Others may exhibit novel behavior not present in QCD, or provide probes of phenomena currently inaccessible for QCD using lattice techniques, like physics at finite chemical potential. Taken together, branches of this family can be thought of as deformations of QCD and used to make statements about QCD itself.

The strong force behaves in qualitatively different ways at different temperatures. In the low-temperature world around us, the strong force confines quarks and gluons inside hadrons like the proton and neutron. However, the Universe has not always been as cold as it is now. At high temperatures, quarks and gluons break free and form a quark-gluon plasma. Cooling from high temperatures, QCD exhibits a smooth crossover between these two phases. Although this long- speculated transition was only recently experimentally confirmed to exist at the Relativistic Heavy

Ion Collider, it has been a topic of numerical investigation for decades in lattice gauge theory [72].

This thesis explores the dynamics of this extended family of QCD, and in particular their dynamics at finite temperatures. QCD’s thermodynamic behavior is increasingly well-understood, but quantum chromodynamics is not the only theory which exhibits its phase structure. Indeed, analytics and numerics suggest that these phases, with a color-confined phase in one extreme temperature regime and a deconfined phase in another, are typical1 of gauge theories defined by compact gauge groups. However, there exists the tantalizing possibility that some QCD-like theories might exhibit phase structures unlike anything we have yet observed in nature. It could be that some systems possess exotic intermediate phases other than the confined and deconfined ones seen in QCD. Further, while QCD does not possess a true finite-temperature phase transition, this does not need to be the case for its cousins. Beyond being of fundamental interest, via cosmological

1 Excepting theories which exhibit a conformal ﬁxed point in the infrared. 3 constraints the thermodynamics of QCD’s cousins can have implications for what types of gauge theories may be present in the as-yet-unknown UV completion of the Standard Model.

This thesis will focus on one particularly interesting branch of QCD’s family, “multirep” theories wherein quarks are generalized to multiple species of fermion charged under diﬀerent representations of the gauge group. A longstanding idea, the “tumbling” scenario or Most Attractive

Channel hypothesis [132], suggests that species of fermions charged under higher representations will develop chiral condensates at higher scales. This mechanism could allow a multirep theory to dynamically generate some of the many hierarchies of scales we see in nature, without ﬁne-tuning.

Translated to thermodynamics where the temperature provides the scale, this same idea implies novel phases at intermediate temperatures where only some fermion species are conﬁned.

The remainder of this thesis is structured as follows.

Chapter2 begins by motivating QCD as the theory of the strong force, then moves on to review QCD, its Lagrangian, and asymptotic freedom. With QCD established, we will discuss how QCD may be generalized by changing the number of quark flavors and their masses, using a gauge group other than SU(3), and by charging quarks under other representations of the gauge group than the fundamental. We will introduce the particular branch of QCD’s family that will be the main focus of this thesis, “multirep” SU(Nc) gauge theories with quarks generalized to multiple different species of fermion charged under different representations of the gauge group.

Restricting to fermions charged under irreducible representations (irreps) of the gauge group and theories amenable to lattice simulation, we enumerate which irreps may participate and describe the space of possible multirep theories. Because gauge theories lose asymptotic freedom when too much matter is coupled in, we will ﬁnd it is possible to do so exhaustively. Finally, we present

Ferretti’s model, a particular multirep theory that will serve as our go-to example and the primary subject of numerical study in this thesis, and brieﬂy discuss the BSM phenomenology motivating it.

Chapter3 provides a review of lattice gauge theory. Starting with continuum thermal QCD, the chapter walks through the sequence of deformations necessary to ﬁt QCD (or one of its cousins) 4 on a computer. We follow with overview of Markov Chain Monte Carlo, a class of numerical methods used to simulate statistical-mechanical systems, as applied to lattice gauge theory. The chapter

ﬁnishes with an overview of our methods for doing spectroscopy in lattice gauge theory, as well as a brief discussion of gradient ﬂow and its application in scale setting.

Chapter4 discusses the thermodynamics of QCD-like theories on the lattice. After a brief review of chiral symmetry and its spontaneous breaking at low temperatures, we discuss the possibility of exotic phase structures in QCD’s cousins. In particular, we present the tumbling scenario and Most Attractive Channel hypothesis, a longstanding idea [132] that suggests multirep theories should exhibit novel phases where chiral symmetry is broken incompletely. We sketch an indirect diagnostic of chiral symmetry breaking, parity doubling, that provides a means of diagnosing the presence of a chiral condensate in spite of the limitations of the Wilson discretization of fermions.

We discuss the set of phase diagnostics that we use to search for these exotic phases in lattice simulations. In particular, we describe the physics of Polyakov loops, conﬁnement, and center symmetry breaking in the context of multirep theories.

Chapter5 presents a numerical study of the thermodynamics of Ferretti’s model (introduced in Chapter2). This study comprises the first-ever lattice investigation of the thermodynamics of a multirep theory on the lattice. We first present and motivate the lattice deformation of the theory that we actually simulate, and discuss our expectations for its phase structure from the arguments presented in Chapter4. After going over details about our data set and how we generated and analyzed it, we present our findings for the phase structure of Ferretti’s model and its limiting case theories. We find that the theory exhibits the same phase structure as QCD, in direct contradiction to the MAC expectation of separated phase transitions. We further find that the theory exhibits a strongly first-order phase transition. We discuss potential phenomenological consequences of this

ﬁnding.

Chapter6 presents a generalization of an analytic calculation originally performed by Pisarski and Wilczek to predict the order of the chiral transition in QCD [130]. Pisarski and Wilczek’s method looks for stable fixed points in an effective model of chiral symmetry breaking; if none 5 are found, then the calculation predicts the chiral transition is first order. We generalize the calculation to apply to multirep theories, taking the results of Chapter5 as input by assuming that simultaneous transitions for all representations is a generic feature. We apply the method to the infinite family of theories enumerated in Chapter2, working to one loop and leading order in the

expansion. The results suggest that the ﬁrst-order phase transition observed in Chapter5 is an almost completely generic feature of multirep theories with simultaneous transitions, with only a handful of limiting-case theories possibly exhibiting continuous phase transitions.

AppendixA is a discussion of autocorrelations and how they aﬀect uncertainty estimation in datasets generated using Markov Chain Monte Carlo methods.

AppendixB compiles some useful group theory and group representation theory results used throughout the rest of this work.

AppendixC presents a set of phase diagnostics based on the gradient ﬂow used in Chapter5, including the novel long-ﬂow-time Polyakov loop diagnostic.

1.1 Notation

This thesis discusses theories with fermions charged under multiple representations (reps) of the gauge group, which we call multirep theories. Further, we will largely specialize to considering irreducible representations (irreps) of the gauge group. Throughout, we denote by R the number of distinct representations of fermion present in a theory.

Irreps of SU(N) come in three types, which we will call complexity classes: complex, real, and pseudoreal. We indicate complex irreps with C, real with R, and pseudoreal with P. Exactly what is meant by these symbols will be context dependent. Something like r ∈ C should be read as “the irrep r is a complex irrep”. Something like uC should be read as “the quantity u for the complex irrep of fermion under discussion”.

Throughout, we will denote the number of Dirac ﬂavors as N D, and the number of Weyl 6

ﬂavors as N W = 2N D. We will frequently make use of the compact notation   D Nr , r ∈ C, P nr = (1.1)  W Nr , r ∈ R which is useful because the quantity nr is an integer for any theory with vectorlike fermion content. Chapter 2

QCD & its Extended Family

2.1 Quantum chromodynamics

This section is intended to be a quick review of QCD to set up discussion of its generalizations later in the chapter. For more details, the author recommends Refs. [145] and [154].

2.1.1 Physics of quarks and gluons

Quarks and antiquarks resemble the more familiar electron and positron, except instead of carrying negative and positive charge, they carry a color charge or anticharge, respectively. Color charges (unrelated to the optical phenomenon) are either red, green, or blue and anticharges are either antired, antigreen, and antiblue; in this sense, there are three distinct “negative” color charges and three distinct “positive” color charges. An object (e.g. positronium) endowed with one unit of positive charge and one unit of negative charge is electrically neutral. Similarly, one may construct neutral “color singlet” objects with equal and opposite color charges and anticharges. Mesons, like the pions and their heavier relatives, are examples of such objects. Unlike with electric charge, one may also construct a color-singlet object using one charge of each color (or similarly with anticharge). Hadrons like the familiar proton and neutron are such objects, the three constituent

(valence) quarks of which all carry a diﬀerent color charge.

Color-charged objects (i.e. quarks) interact with one another by exchanging gluons. Gluons resemble photons, in that they are massless vector bosons. However, photons do not carry any electric charge, while gluons carry a color charge and a color anticharge. This means that gluons 8 interact with themselves; this fundamental diﬀerence leads to the vastly more complex phenomena exhibited by quarks and gluons than by electrons and photons. A universe containing only photons is boring: without electrically-charged objects to interact with, photons simply propagate freely.

However, a universe containing only gluons exhibits complex behavior without any matter particles to mediate their interactions.

There are three colors and three anticolors, and gluons carry one color charge and one anticolor charge, so naively one might expect that there are nine types of gluon. In fact, there are only eight. The absent gluon is the color singlet gluon, whose color charge is typically written as a superposition like rr + gg + bb. This hypothetical gluon is eﬀectively uncharged under color and would not interact with the other gluons. Instead, it would appear like a second photon which interacts with the quarks. We do not observe such a particle in nature.

In the low-temperature, low-density world around us, we do not ever observe isolated quarks or gluons, or any object with net color charge. Instead, we only observe color-singlet bound states: typically baryons (e.g. nucleons) or mesons (e.g. pions), but occasionally as more exotic states like glueballs. This phenomenon is known as color conﬁnement, or simply conﬁnement.

2.1.2 QCD as a gauge theory

A gauge theory is a quantum ﬁeld theory wherein the structure of interactions between particles is constrained by a local symmetry. Quantum electrodynamics, the theory of the photon and electron, is an example of a simple gauge theory. In QED, the dynamics are invariant under redeﬁnition of the electric charge by a phase. Allowing this phase to vary spatially tightly constrains how electrons and photons may interact.

QCD is the gauge theory which describes how quarks and gluons interact. As discussed above, quarks and gluons look like electrons and photons but with positive and electric charge replaced by colors and anticolors. This can be made mathematically precise. A phase is an element of the symmetry group U(1), so QED is the gauge theory deﬁned by invariance under a local U(1) symmetry. U(1) is an Abelian group, so we say that QED is an Abelian gauge theory. We may 9 instead construct gauge theories analogous to QED with larger, non-Abelian symmetry groups as the gauge group. By replacing U(1) with SU(3) [the set of all 3 × 3 unitary matrices with determinant 1] we obtain QCD. An SU(3) gauge transformation amounts to redeﬁning the colors as a complex mixture of themselves in a spatially-dependent way.

The replacement of U(1) with SU(3) is exactly the replacement we discuss above of negative/positive electric charge with three colors/anticolors. Given some simple scalar object φ with no indices, one may sensible apply a U(1) transformation eiα to it as

φ → φ0 = eiαφ. (2.1)

i In contrast, elements of SU(3) are 3 × 3 matrices Ωj which cannot sensibly be used to transform a

i scalar object. Whatever Ωj is applied to must have an index to contract with like

0 i φj → φj = Ωjφi. (2.2)

In matrix notation this is φ → Ωφ, which for 3 × 3 matrices Ω implies φ is a three-element column vector. In generalizing from objects that can be transformed by U(1) elements to objects that can be transformed by SU(3) elements, we are forced to add an additional “color index”.

2.1.3 QCD Lagrangian

To begin our construction of the QCD Lagrangian, define the Dirac spinor field ψi,α,f to describe the quarks. i ∈ {r, g, b} is a color index. The Dirac index α differentiates the four independent degrees of freedom of the spinor. The flavor index f ∈ {u, d, s, c, b, t} differentiates the

6 different quark flavors observed in nature. This field is typically written with the Dirac index and color indices suppressed as ψf and thought of as a column vector in Dirac space, and separately a

α ∗ β α column vector in color space. The conjugate of ψf is ψi,f ≡ (ψ )i,f (γ0)β . In matrix notation, this † is written as ψf ≡ ψf γ0 and thought of as a row vector in Dirac space and in color space. Under gauge transformations, the quark ﬁeld transforms as

ψf → Ω ψf (2.3) † ψf → ψf Ω 10 where Ω = Ω(x) is a spatially-dependent SU(3) matrix.

i To describe the gluons, we define the algebra-valued gluon or gauge vector field (Aj)µ. The spatial modes are differentiated by the Lorentz index µ ∈ {0, 1, 2, 3}. Like photons, gluons are massless vectors, and so there are only two independent degrees of freedom in these four components.

Gluons each carry a color charge and an anticolor charge, different combinations of which are differentiated by the color indices i and j. A is su(3) algebra-valued, and su(3) is 8-dimensional, so there are only 8 independent color-anticolor combinations. Altogether, the field A has 16 (real) degrees of freedom. Typically this field is written with color indices suppressed as Aµ and thought of as a matrix in color space.

To make the counting of color degrees of freedom more obvious, Aµ is frequently expanded like

i a i (Aj)µ = Aa,µ(τ )j (2.4) where the τ are a basis for su(3), or equivalently, a set of generators of SU(3). The index a runs from 1 ... 8, so there are eight independent vector ﬁelds Aa,µ.

We would like to construct a Lagrangian that is invariant under gauge transformations.

Masses for Dirac fermions are invariant under gauge transformations

mψψ → mψΩ†Ωψ = mψψ (2.5) because Ω†Ω = 1 for unitary matrices. The kinetic term, however, is not invariant:

ψi∂ψ/ → ψΩ†i∂/(Ωψ) = ψi∂ψ/ + iψΩ†(∂/Ω)ψ (2.6)

µ where ∂/ = γ ∂µ. Abstracting, we see that ∂µ is not gauge covariant, but rather transforms as

† ∂µ → ∂µ + Ω (∂µΩ). (2.7)

To write a gauge invariant kinetic term, we need to construct a gauge covariant derivative Dµ which transforms as

† Dµ → ΩDµΩ (2.8) 11

† where Dµ does not act on Ω . With those transformation properties, ψiDψ/ is obviously gauge invariant. The canonical construction of the gauge covariant derivative is

Dµ = ∂µ − igAµ (2.9)

where we have introduced the gauge coupling g. In order for Dµ to have the desired transformation properties, the gluon ﬁeld must transform as

i A → ΩA Ω† + (∂ Ω)Ω†. (2.10) µ µ g µ

Using Eqs. 2.7 and 2.10, we see that the covariant derivative transforms as

† † † † † Dµ = ∂µ − igAµ → ∂µ + Ω (∂µΩ) − igΩAµΩ + (∂µΩ)Ω = Ω∂µΩ − igΩAµΩ (2.11)

† as desired, where in the ﬁnal expression ∂µ does not act on Ω and in the last equality we use

† † Ω(∂µΩ ) = −(∂µΩ)Ω to cancel the terms. Because the transformation properties of ψ and A are so closely tied together, it is diﬃcult to construct gauge-invariant objects involving the two ﬁelds, thereby constraining how quarks and gluons may interact.

The final remaining piece of the Lagrangian is a kinetic term for the gluons. The covariant derivative is a convenient building block for this task: it transforms simply under gauge transformations and is constructed of only derivatives and gluon fields. We may define the gluon field strength tensor Gµν in terms of the covariant derivative as

i G = − [D ,D ] = ∂ A − ∂ A − ig[A , A ]. (2.12) µν g µ ν µ ν ν µ µ ν

We construct the kinetic term from the ﬁeld strength tensor in the usual way, obtaining

1 − Tr GµνG (2.13) 2 µν where the trace is over color.

Compiling, we obtain the QCD Lagrangian

1 L = ψ (iD/ − m )ψ − Tr G Gµnu (2.14) i ij j 2 µν 12 where i and j are ﬂavor indices and mij is a mass matrix with the six quark masses on the diagonal.

We note that the masses are the only free parameters in this theory; via dimensional transmutation, the coupling constant g is determined by the RG scale at which the theory is probed.

2.1.4 Asymptotic freedom & conformality

It is a standard calculation to compute the β function of QCD which governs the running of the gauge coupling g. To two loops, the result is

dg2 g4 g6 = −b − b d ln µ 1 16π2 2 (16π2)2 4 b = 11 − N D (2.15) 1 6 F 38 b = 102 − N D 2 3 F

D D where NF is the number of light ﬂavors. We note immediately that, so long as NF < 16.5 the one-

D loop coeﬃcient b1 is positive, and in nature NF ≤ 6 depending on which quarks should be considered light. Thus, ignoring for a moment the higher-order contributions, we see that the β function is negative overall. This leads to asymptotic freedom: at high momentum scales, g2 grows small and the theory becomes weakly coupled. In this regime, relevant to e.g. collider experiments, QCD can be treated with perturbation theory sensibly. However, the QCD β function gives rise to a Landau pole at ΛQCD ∼ 200 MeV, which indicates the breakdown of perturbation theory and the onset of strong coupling. This gives rise to the phenomenon of color conﬁnement at the low momentum scales of everyday life.

In a thermodynamics context, the temperature sets the characteristic momentum scale. We thus expect nonperturbative conﬁning behavior at low temperatures, and perturbative behavior and the breakdown of conﬁnement at high temperatures.

We defer discussion of the two-loop contribution until Section 2.2.1 directly below. 13

2.2 Generalizations of QCD

QCD is SU(3) gauge theory coupled to six quarks with the masses of the six quarks observed in nature. However, there are many ways in which this speciﬁc theory may be generalized. We will focus on three in particular: changing the number of quarks and the masses of those quarks; changing the gauge group; and changing the representations of the gauge group under which the fermions are charged. In practice, the ﬁeld of lattice gauge theory studies these generalizations exclusively.

2.2.1 Number of ﬂavors and quark masses

The most ubiquitous generalization of QCD is changing the number of quark flavors. At present, it is not possible to simulate full QCD on a computer. As we will discuss in greater detail in Chapter3, lattice gauge theory introduces both a UV cutoff (the lattice spacing a) and an IR cutoff (the lattice extent L, where the volume V ∼ L4). Heuristically, lattice gauge theory cannot treat systems with scales more widely separated than the cutoffs used in the simulation. In top- of-the-line modern simulations, L/a ∼ 102 − 103; meanwhile, the top quark in nature is ∼ 6 × 104 times heavier than the light up and down quarks. Thus, common practice is to simply exclude the heaviest quarks from the simulation; their contributions to the dynamics are suppressed by their large masses, so this is not a poor approximation. To get qualitatively-correct dynamics, a simulation must include the nearly-degenerate lightest up and down quarks. Increasingly commonly, simulations will include the strange quark, whose mass is ∼ 30 times greater than the up and down quarks. Top-of-the-line simulations are beginning to include dynamical charm quarks, whose masses are ∼ 400 times greater than the up and down quarks. [159]

Increasing the number of flavors sufficiently high that we near or enter the so-called “conformal window” can yield theories with qualitatively different dynamics. In the QCD β function

D Eq. 2.15, the coeﬃcient of the two-loop contribution b2 switches from positive to negative at NF ≈ 8,

D long before b1 switches signs at NF = 16.5. This allows for the first two terms in the β function 14 to cancel at some finite value of g2, zeroing the β function. In reality, the β function does not stop at two loops and the story is drastically more complicated; however, if such a Banks-Zaks or infrared conformal fixed point exists, it can lead to a nontrivial theory which is both interacting and scale-invariant in the infrared [22]. Such theories “inside the conformal window” are a topic of considerable theoretical interest. If the Banks-Zaks fixed point does not exist but is “nearby” such that the β function is nearly zero, then the coupling in the theory can run very slowly. The resulting “walking” (slower than running) theories, which live “near the conformal window”, have been the subject of considerable investigation on the lattice in the past few decades in the context of technicolor theories and composite Higgs models [59, 155]. However, such theories will not be a focus of this thesis.

The other ubiquitous generalization is to adjust the masses of the quarks included in the theory. As quarks get lighter, simulating them becomes more computationally costly. With the increasing availability of computational resources, it has recently become feasible to simulate QCD with the up and down quarks at their physical masses1 [159, Quark Masses]. However, historically and commonly still, simulations are run with the light quarks unphysically heavy. To learn about physics at the physical point, one simulates at several diﬀerent masses and extrapolates. For a more fundamental limitation, we are frequently interested in physics in the exact chiral limit where the light quarks are massless. With our present technology, this is impossible: the computational cost diverges as the quarks become massless. Thus, physics with massless quarks can only be accessed using extrapolation from ﬁnite masses.

2.2.2 Other gauge groups

One may also consider gauge groups other than SU(3). In our discussion, we motivated QCD as the result of generalizing QED from U(1) gauge theory to SU(3). However, we could have chosen another gauge group. This can be motivated by practical reasons: the ﬁrst lattice gauge theory

1 In nature, the down quark is roughly twice as heavy as the up quark, so isospin symmetry is broken. As discussed in Chapter3, in lattice gauge theory it is more convenient to simulate with exact isospin symmetry, mu = md. This typically introduces error only on the O(1%) level [2] unless the physical phenomenon of interest is particularly sensitive to isospin-breaking eﬀects, e.g. big bang nucleosynthesis [88]. 15 simulations were of U(1) gauge theory [49], and in the early days it was common to simulate SU(2) gauge theory [50]. Because the gauge ﬁelds in these theories have fewer color degrees of freedom

(1 and 3 for U(1) and SU(2), respectively), simulating them is less computationally demanding.

However, with modern computing resources, this is only infrequently a concern.

Gauge theories with other gauge groups can individually be of interest as components of models of physics beyond the Standard Model (BSM models). As discussed in Section 2.4 below, a study of one such model involving the SU(4) gauge group is a primary focus of this thesis. Other studies have looked at, for example, SU(2) and SU(4) gauge theories as components of models for composite dark matter [73,5, 90] and composite Higgs bosons [89].

There have also been some recent lattice investigations of the Sp(2Nc) gauge group in the context of composite Higgs models [28].

There are also more theoretical reasons to study other gauge groups. For example, one may leave the number of colors Nc arbitrary such that the gauge group is SU(Nc). In strongly coupled systems, the usual perturbative expansion in the coupling constant g2 breaks down at low temperatures where the system conﬁnes. One may instead consider expanding in a diﬀerent

2 parameter, 1/Nc, about the limit Nc → ∞ with λ ≡ g Nc held constant. An extensive literature of lattice studies (see e.g. Ref. [111] for a review) supports the ansatz that any observable O can be expanded in a power series in 1/Nc like

ˆ α 1 hOi = Nc O0(λ) 1 + O1(λ) + ··· (2.16) Nc

α where Nc is some characteristic leading power of Nc and Oi are coeﬃcients encoding nonperturbative physics (and all λ dependence) [158, 162, 164]. Taking the ’t Hooft or large Nc limit Nc → ∞, the leading-order behavior dominates and the expansion truncates to

ˆ α hOi ≈ Nc O0(λ). (2.17)

Large Nc QCD is still nonperturbative (as indicated by the survival of the coefficient O0) and at present remains unsolved. However, the theory simplifies in this limit and one begins to be 16 able to make analytic statements: the leading scaling with Nc can be determined using simple diagrammatic counting rules, with physics at finite Nc given by subleading corrections to this limit.

For example, in the formal Nc → ∞ limit, mesonic resonances become inﬁnitely narrow, and the

OZI rule and quark model become exact [170]. This resembles nature, where mesons are long-lived and the OZI rule and quark model are useful but approximate. Studying the large Nc world can thus tell us about the Nc = 3 world that we live in, providing insight and intuition for why QCD behaves as it does: 1/Nc expansions give a quantitative answer to the question “how typical is

QCD?”

2.2.3 Higher-representation fermions

In this thesis, we will concern ourselves only with irreducible representations (irreps).

The quarks in QCD are charged under the fundamental irrep F of SU(3), meaning that under gauge transformations the quark ﬁeld transforms as ψ → Ωψ. The antiquarks live in the antifundamental irrep F and transform as ψ → ψΩ†. With color and anticolor indices explicit, this looks like

j ψi → Ωi ψj (2.18) i ∗ i j ψ → (Ω )jψ .

The quarks in nature appear to be fermions charged under F and F , but SU(N) has an inﬁ- nite tower of higher representations and we are free to imagine matter ﬁelds charged under these representations instead.

For example, consider a fermion Ψ charged under the Nc(Nc − 1)/2 dimensional two-index

antisymmetric representation A2; abstractly, we can write Ψ → ΩA2 Ψ, where ΩA2 is a dA2 × dA2

SU(Nc) matrix in the A2 representation. More concretely, higher irreps can be constructed from symmetrized and antisymmetrized products of the fundamental and antifundamental representations. In practice, this means that matter ﬁelds charged under higher representations have multiple color and anticolor indices, typically symmetrized and/or antisymmetrized in some way. For ex- 17 ample, we may write an A2-charged fermion ﬁeld with two fundamental color indices as

Ψij = −Ψji (2.19) which transforms like 1 Ψ → (Ω Ω − Ω Ω )Ψ (2.20) ij 2 ik jl il jk kl where Ω are the usual Nc × Nc F -irrep SU(Nc) matrices. Ferretti’s model, to be introduced in

Section 2.4, involves fermions charged under the A2 representation.

As we will ﬁnd in Section 2.3, there are 12 distinct irreps that matter may be charged under in an asymptotically-free SU(Nc) gauge theory with vectorlike matter content. Although we have observed no higher-representation matter in nature, these systems have received signiﬁcant attention on the lattice. To the best of the author’s knowledge, only four irreps have been investigated:

F , A2, S2, and G. Theories with S2 fermions have been investigated in the context of composite

Higgs models [146, 68]. Theories with A2 fermions are discussed in this work in the context of a composite Higgs model, and have previously been considered as models for composite dark matter

[5]. A2 fermions are also interesting for pure theoretical reasons: in SU(3), A2 = F , so A2 fermions provide an alternative large Nc limit [43]. Fermions in the adjoint irrep G have received particular attention. SU(Nc) gauge theory with a single Weyl ﬂavor of adjoint fermion is N = 1 super Yang-

Mills, which has received some study on the lattice [29]. SU(2) gauge theory with a single Dirac

flavor of adjoint fermion is of interest to the condensed matter community as a dual description of the physics at a quantum critical point in certain systems [34]; this theory has also been studied on the lattice [6]. Theories with two Dirac flavors of adjoint fermion have received some lattice attention both for their finite-temperature phase structure [95] and for their infrared-conformal dynamics [67, 69].

The main focus of this work is a ﬁnal generalization to “multirep” theories with fermions in multiple representations. We are not limited to theories with fermions all charged under the same

irrep; in fact, we may have fermions in R distinct irreps of the color group, with nr1 ﬂavors charged

under irrep r1, nr2 ﬂavors charged under irrep r2, etc. One such theory is Ferretti’s model which 18

contains NF fermions charged under the fundamental irrep and NA2 fermions charged under the

A2 irrep of SU(4). We describe this model at length below.

2.3 Zoology

We want to catalog the class of gauge theories obtained by generalizing QCD’s gauge group and irrep content. We will only consider SU(Nc) gauge groups. Irreps of SU(Nc) each fall into one of three complexity classes: complex, real, or pseudoreal. Real and pseudoreal representations are self-conjugate, i.e. r = r.

We are interested primarily in theories that can be put on a lattice and simulated on a computer. Modern lattice gauge theory technology cannot treat chiral gauge theories [82, 61], so we will restrict our attention to gauge theories with vectorlike matter content. In a vectorlike theory, every Weyl fermion charged under some representation r of the gauge group has a partner fermion charged under the conjugate representation r. We get this structure automatically with

Dirac fermions, which encode an r-irrep Weyl fermion and an r-irrep Weyl antifermion. Thus, for most irreps, we will concern ourselves with only whole numbers of Dirac flavors2 . The exception is for fermions in real representations of the gauge group, for which color and anticolor are related by a symmetry (as discussed in Section 4.2). These theories can be put on the lattice with integer numbers of Weyl flavors (or, half-integer numbers of Dirac flavors), so we include them.

2.3.1 Participating irreps

For any Nc, there is an infinite tower of irreps. However, coupling even one flavor of most of these irreps to a gauge field will break asymptotic freedom. We are only interested in asymptotically free theories, so there are only a finite (and, in fact, small) number of relevant irreps for each Nc.

The QCD β function Eq. 2.15 generalizes straightforwardly to arbitrary-irrep matter content.

2 We will still discuss the number of Weyl ﬂavors in these cases, which should be read as N W = 2N D. 19

Name Deﬁnition Relevance Complexity dr Tr

,N = 2 F All P N a C, N > 2

R,N = 4 N(N−1) A2 N > 3 a(N − 2) C, N > 4 2

P,N = 6 N(N−1)(N−2) (N−2)(N−3) A3 6 ≤ N ≤ 15 a C,N 6= 6 6 2

R,N = 8 70,N = 8 20a, N = 8 A4 Nc = 8, 9 C,N = 9 126,N = 9 35a, N = 9

(2),N = 2 G All N 2 − 1 2aN (101), N > 2 R

N(N+1) S2 N > 2 C 2 a(N + 2)

20,N = 4 13a, N = 4 G N = 4, 5 3 C 40,N = 5 22a, N = 5

G4 N = 5 C 45 24a

P,N = 2 (N+2)(N+1)N (N+3)(N+2) S3 2 ≤ N ≤ 4 a C, N > 2 6 2

S4 N = 2 R 5 20a

204 N = 4 R 20 16a

755 N = 5 R 75 50a

Table 2.1: Exhaustive list of irreps of SU(Nc) that may be present in asymptotically free SU(Nc) gauge theories with vectorlike matter content. The column “relevance” states the ranges of Nc over which each irrep is well-deﬁned, not the conjugate irrep of a more fundamental one, and doesn’t break asymptotic freedom. dr is the dimension and Tr is the trace of representation r; the quadratic Casimir Cr may be obtained using the identity Cr = TrdG/dr, where dG is the dimension of the adjoint irrep at the same Nc. The trace of the fundamental representation TF = a is left arbitrary. 20

To two loops, it is dg2 b g4 b g6 = − 1 − 2 d ln µ 2a 16π2 4a2 (16π2)2 11 4 X b = C − N W T 1 3 G 6 r r (2.21) r 34 X 10 b = C2 − N W T C + 2C 2 3 G r r 3 G r r

W D where Nr = 2Nr is the number of Weyl ﬂavors of a irrep r, Tr is the trace of irrep r, a = TF is the trace of the fundamental irrep F , Cr is the quadratic Casimir of irrep r, and CG = 2aNc is the quadratic Casimir of the adjoint irrep G. We are interested in asymptotically-free theories, so we will consider all theories for which the one-loop coeﬃcient b1 is positive. We make the assumption that subleading terms will not spoil asymptotic freedom in all but pathological edge cases.

Gauge theories with conformal ﬁxed points do not conﬁne or break chiral symmetry, and so are not QCD-like in the sense we are interested in. We would exclude them from our consideration, but it remains an unsettled question exactly when a theory becomes conformal. Instead of guessing the location of the conformal window, we will simply consider all asymptotically free theories, and note whenever a theory may plausibly be conformal if it’s relevant to the discussion.

In order for an irrep to be present in an asymptotically free theory, we must be able to couple at least one flavor of the irrep to the gauge field without flipping the sign of b1. For brevity’s sake, we will refer to irreps which satisfy this condition as “asymptotically free irreps”. For complex and pseudoreal irreps, we may only add and remove flavors in units of Dirac fermions; however, real irreps may be present in integer numbers of Weyl fermions. From the one-loop multirep β function, we may define the “cost to asymptotic freedom” for an irrep as

W Tr Pr = . (2.22) 11aNc

W Pr is the fractional contribution of one Weyl ﬂavor of an irrep towards ﬂipping the sign of b1. For

D W D convenience, we may similarly deﬁne Pr = 2Pr , and Pr such that Pr = Pr when r is complex or

W max pseudoreal and Pr = Pr when r is real. Thus, starting from a matterless theory, nr = 1/Pr is the number of ﬂavors (Dirac for complex and pseudoreal, Weyl for real) that can be coupled to the 21 gauge ﬁeld before asymptotic freedom is lost. The condition that an irrep be asymptotically free is Pr < 1.

In order to systematically enumerate asymptotically free irreps, we need some means of parametrizing irreps of SU(N). Dynkin labels provide such a means [74, 150, 167]: for a given Nc, any irrep is specified by a tuple of Nc − 1 integers like (a1, a2,...). A set of Dynkin labels defining an irrep are one-to-one with a Young diagram and a value of Nc: the first Dynkin index a1 is equal to the difference between the number of boxes in the first row of a tableau and the second row of the tableau, etc (including rows with 0 boxes in them, if there are fewer than Nc nontrivial rows in the tableau). As an example, the fundamental irrep F is defined by the set of Dynkin labels starting with a 1 and followed by Nc − 2 zeros. For brevity’s sake and because no irrep we will consider has any an ≥ 10, we will write Dynkin index tuples without commas like (10) [the F of

SU(3)] or (010) [the A2 of SU(4)]. Dynkin labels encode Nc; to specify irreps for arbitrary Nc, we will denote e.g. the F representation as (10), to be read as 1 followed by Nc − 2 zeros.

With this technology, it is straightforward to enumerate all of the asymptotically free irreps.

Given the Dynkin labels (a1 . . . aNc−1) that parameterize an irrep r, we can compute the quadratic

Casimir using the expression

N−1 " m−1 # X 2 X Cr = N(N − m)mam + m(N − m)am + 2n(N − m)anam , (2.23) m=1 n=0 and the dimension using the expression3

N−1  N−1  p  Y  1 Y X  d = (1 + a ) . (2.24) r p!  r  p=1  q=p r=q−p+1 

The identity

Crdr Tr = (2.25) dG finally provides the trace. Evaluating these expressions is a task best left to computers, but we observe that incrementing any Dynkin label ai will increase both Cr and dr, and thus Tr by equation 2.25. Considering that Pr ∼ Tr, if we have some irrep defined by a set of Dynkin labels 3 If q − p + 1 ≥ p, the sum over r is not well-defined. When this occurs, this term should simply be left out of the product over q, rather than evaluated as zero (which would zero the entire expression). 22 ai that is not asymptotically free, then any irrep made by incrementing any of the ai is also not asymptotically free. Thus, to enumerate all asymptotically free irreps for some particular Nc: start with the trivial irrep ai = 0; try incrementing each ai by one, then recursively apply this procedure on the resulting asymptotically free irreps.

All possible irreps that may participate in an asymptotically free SU(Nc) gauge theory with vectorlike matter content are listed in Table 2.1. We are able to make such an exhaustive statement because, for Nc ≥ 16, the only asymptotically free irreps are the fundamental F , and the three two-index higher irreps: the two-index antisymmetric A2, the two-index symmetric S2, and the adjoint G [74]. This follows because the traces of these irreps go like either T ∼ 1 for the F or

T ∼ Nc for the two-index representations, and so Pr ∼ Tr/Nc does not diverge as Nc → ∞. The

2 trace of all other irreps go like Tr ∼ Nc at least, and so Pr diverges at large Nc.

We need to know the complexity class of each irrep: complex, real, and pseudoreal irreps each have diﬀerent chiral symmetry breaking patterns, and real irreps can be included in units of

Weyl ﬂavors rather than Dirac. We discuss methods for diagnosing the complexity class of an irrep of SU(N) in Appendix B.1; here, we simply list the classes for each irrep in Table 2.1. While we ﬁnd many asymptotically free self-conjugate irreps, only three are pseudoreal: the F and S3 of SU(2), and the A3 of SU(6). We also note that, with the exception of the adjoint irrep (which is real for all Nc), irreps are usually only self-conjugate for exactly one Nc and complex otherwise.

In some cases, an irrep exists (in that it is well-deﬁned and non-trivial) but is degenerate with the conjugate of a more fundamental representation. Because we have restricted ourselves to considering theories with vectorlike fermion content, we do not consider an irrep R and its conjugate

R as distinct, so we will say that an irrep does not properly exist in these cases. For example, the

A2 is well-deﬁned in SU(3), but in SU(3) we have A2 = F , so we do not count it as its own irrep.

As a more pathological example of the consequences of this deﬁnition, the A4 is well-deﬁned and non-trivial starting at Nc = 5, but is not the conjugate irrep of a more fundamental irrep (i.e., F ,

A2, A3) until Nc = 8.

The definitions of F , An, and Sn do not depend on Nc, and can be conveniently defined 23 using Young diagrams as in Table 2.1. However, the same is not true for the adjoint irrep G, which is specified with a different Young diagram at every Nc. Instead, G is most conveniently defined using Dynkin labels: in SU(2) it is (2), while for all Nc > 2 it is (101). In several cases, however, the irrep specified by the Young diagram for G at some Nc continues to exist at higher Nc as a different irrep. One of these is S2, which is degenerate with G at Nc = 2 but continues to exist for all Nc. The others (which exist only transiently) we denote as G3 and G4, indicating that they are defined by the Young diagrams for G in SU(3) and SU(4), respectively.

For Nc = 6, the irrep G3 has Pr = 1 exactly. Thus, for an SU(6) gauge ﬁeld coupled to one Dirac ﬂavor of this irrep, the one-loop contribution to the β function is exactly zero; the two- loop contribution is positive, so this theory is not asymptotically free. This theory motivates our condition that Pr be strictly less than one, rather than less or equal.

We encounter only two irreps which are not encompassed by the naming scheme F , An,

Sn, G, and Gn, each of which are relevant to only a single Nc. We name them 204 [read: the

20-dimensional irrep of SU(4)] and 755 (similar).

The top panel of Figure 2.1 shows the maximum number of ﬂavors of each irrep permitted at each Nc, computed as max 1 nr = ﬂoor . (2.26) Pr

Note that these numbers are for single-irrep theories where the relevant irrep is the only one present.

max Note also that the definition nr can refer to Weyl or Dirac flavors at different Nc, depending on whether the irrep is real or complex/pseudoreal, respectively; we choose to examine this quantity because it is the one relevant for determining the multiplicity of possible theories. We exclude the

max fundamental irrep from the plot because it would not fit (nF = 11Nc/2). The irreps A2 and A4 are real for the Nc at which they are “born” (Nc = 4 and 8, respectively) but complex otherwise, accounting for the initial enhancement and subsequent immediate fall by a factor of more than 2 of number of flavors permitted for each. A3 is pseudoreal in Nc = 6, so it does not see a similar initial enhancement. As will be discussed below for A2 and S2, these numbers differ slightly if we require 24 at least two irreps to be present. The second panel in the figure counts the number of available irreps at each Nc, and the number of those which are real. There are at most 7 irreps at any given

Nc (Nc = 4, 5), although no more than 5 may ever may be present simultaneously (as discussed below).

2.3.2 Theory demographics

For a multirep theory to be asymptotically free, the condition must hold that

X X W W nrPr = Nr Pr < 1 (2.27) r r where the sum is over all distinct irreps present in the theory. We know all of the asymptotically free irreps at any given Nc, so enumeration gives us every possible theory. There are too many to list here, but it is worth making some observations about the results.

The bottom two panels of ﬁgure 2.1 shows the number of multirep theories versus Nc for

2 ≤ Nc ≤ 30. At lower Nc, the number varies signiﬁcantly depending on which irreps are available and how many are real. Comparing with the top plot, we can see that the spike at Nc = 4 can be attributed to the large number of A2 fermions possible, the spike at Nc = 6 can be attributed to the declining cost of S2 and the birth of A3, and the spike at Nc = 8 is due to the birth of A4. For

Nc > 9, the number of theories settles quickly into its large Nc asymptotic behavior. The linear ramp in the third panel is due to the linearly-increasing number of F ﬂavors permitted.

Because TF ∼ 1, the number of flavors of F -irrep fermion allowed in an asymptotically free theory diverges linearly in Nc as Nc → ∞. However, the number of flavors of two-index fermions permitted becomes fixed at sufficiently large Nc. Because TG = 2aN, the maximum number of

W Weyl ﬂavors of G-irrep fermion allowed in an asymptotically free theory is Nmax = 5 for all N. In

D the large Nc limit, there may be at most N = 5 ﬂavors of A2-irrep or S2-irrep fermions. The trace

of the A2 irrep TA2 = a(N − 2) and the trace of the S2 irrep TS2 = a(N + 2) asymptote to aN in the large Nc limit; we thus ﬁnd the asymptotic maximum number of ﬂavors for each to be

D 1 11Nc lim Nmax(A2,S2) = lim D = lim = 5.5 (2.28) Nc→∞ Nc→∞ Pr (A2,S2) Nc→∞ 2Nc 25

A2 20 A3 A4 G G3 15 G4 S2 x a S3 m r n 10 S4 204

755

0 7.5 All

s 5.0 p e r r i G

N 2.5

0.0 s e i 2000 r o e h t N

F 200 % s e i r o e 100 h t N 0 5 10 15 20 25 30

max Figure 2.1: Top panel shows the maximum number of flavors nr of each irrep permitted in a single-irrep theory, as a function of Nc. The large spike for A2 is because it is real for Nc = 4. Note that for Nc = 5, there is exactly one flavor allowed of each of the irreps G3, G4, and 755. Second panel shows the total number of irreps present, as well as the number of those irreps which are real; the dashed line indicates the contribution of the adjoint irrep, which is always real. The bottom two panels show the number of asymptotically free multirep theories vs. Nc. The third panel shows the total number, while the bottom panel shows the number of distinct theories without differentiating between theories with different numbers of fundamental fermions nF . In the bottom panel, the dashed line is at the asymptotic value of 55. 26

D where because A2 and S2 are complex we use Pr , and we round the ﬁnal result down to 5. As can

D be observed in the top panel of Figure 2.1, in single-irrep theories, Nmax(A2) achieves its asymptotic

D value at Nc = 24 approaching from above, and Nmax(S2) achieves its asymptotic value at Nc = 20 approaching from below. However, in a theory with at least two distinct irreps of fermion, this occurs at Nc = 22 for both A2 and S2, with the allowed number of A2 reduced prematurely and S2 delayed in increasing.

Approaching the large Nc limit, the combination of possible irreps allowed in a theory stops changing for Nc ≥ 16. Thus, for Nc ≥ 16, the set of permitted chiral symmetry breaking patterns for asymptotically-free theories depends only on the number of ﬂavors of each of the four irreps

F , G, A2, and S2. The permitted combinations of diﬀerent numbers of ﬂavors of the higher-irrep

A2, S2, and G fermions stop changing for Nc ≥ 22. We may thus group all theories with Nc ≥ 22 into 55 different classes, defined by the number of A2, S2, and G fermions, with all theories in each class differing only by the number of F -irrep fermions. This can be observed in the bottom panel of Figure 2.1, which shows the number of theories identifying theories that differ only by number of fundamental flavors.

We observe that there are at most ﬁve distinct irreps present in any asymptotically free theory for Nc < 16. Because there are only 4 irreps for Nc ≥ 16, this is a strict upper bound on the number of distinct irreps allowed in a multirep theory.

One may concern themselves with the scenario wherein an irrep is asymptotically free for some Nc, but its cost is so high that no other irrep may ﬁt alongside it. We do not ﬁnd that this is the case: all irreps described in Table 2.1 may exist in multirep theories.

2.4 Ferretti’s Model

Composite Higgs models circumvent the hierarchy problem by supposing that the Higgs boson is a bound state of some new QCD-like “hypercolor” sector coupled to the Standard Model.

From a survey of plausible hypercolor sectors by Ferretti and Karateev [79], Ferretti identified one as particularly phenomenologically promising: SU(4) gauge theory with 3 Dirac flavors of 27 fundamental hyperquarks and 5 Weyl flavors of two-index antisymmetric hyperquarks [78]. In this model, the Higgs is an exact Nambu-Goldstone boson of the antisymmetric hyperquarks, making it light relative to most of the other bound states of the hypercolor sector. The Higgs potential originates entirely from electroweak and top Yukawa interactions. Separately, Ferretti’s model accounts for the top quark’s large mass by supposing that it is partially composite, i.e., the top quark’s mass is enhanced by linear mixing with a “chimera” baryon made of hyperquarks of both species.

For technical reasons (discussed in Chapter3), the exact fermion content of Ferretti’s model is difficult to simulate on the lattice. So, rather than simulating Ferretti’s model directly, the numerical study of Ferretti’s model presented in Chapter5 examines the “lattice-deformed Ferretti model,” SU(4) gauge theory with N D = 2 Dirac flavors fundamental fermions and N D = 2 Dirac F A2 flavors of two-index antisymmetric fermions. Ferretti’s model and our lattice deformation will be used for specific examples throughout this thesis whenever one is called for.

Although it will not be discussed further in this thesis, it is worth summarizing our ﬁndings for the zero-temperature properties of Ferretti’s model and their phenomenological implications.

To assess its viability as a BSM theory, we carried out a program of many-ensemble ﬁts of lattice observables to models based on chiral perturbation theory and quark models. The resulting models fully constrain the spectrum of pseudoscalar and vector masses and decay constants, as well as baryon masses, up to a ﬁducial scale [7, 11, 14]. Combined with experimental constraints, this allowed us to put lower bounds on the masses of particles predicted by Ferretti’s model. Most recently, we calculated the top partner chimera’s contribution to the Higgs potential, putting severe constraints on the theory’s viability as a BSM model [13]. Chapter 3

Lattice Gauge Theory

This chapter is an overview of lattice gauge theory. To develop this formalism, we will perform a sequence of manipulations of the QCD partition function that eventually allow calculation with the theory on a computer using Monte Carlo methods. Starting with thermal field theory, we move to the path integral formulation to obtain a relationship between statistical mechanics and quantum field theories. Then, we render the theory completely finite by restricting to finite volume, discretizing, and digitizing, allowing the degrees of freedom of the field theory to be represented on a finite computer. As part of this step, we will discuss the particular discretizations of the

QCD Lagrangian that we use in the numerics to follow in subsequent chapters. Armed with a

finite theory and a discrete action, we will be prepared to sketch the Monte Carlo methods used to compute observables for these theories. We will finally discuss our methods for doing spectroscopy using lattice simulations, as well as using the gradient flow to set the scale.

Everything described here is standard; for more detailed discussions, see Refs. [82, 61].

3.1 Path integrals & Euclidean time

Consider the thermal partition function

−H/Tˆ ZT = Tr[e ] (3.1) where Hˆ is the Hamiltonian for the (d + 1) system of interest and T is the temperature. For a quantum ﬁeld theory, Hˆ = H(Φ(ˆ x), Π(ˆ x)), where we mean by Φ(ˆ x) the set of ﬁeld operators and 29

Π(ˆ x) their conjugate momenta. If we can compute it, this object tells us everything we want to know about the system described by Hˆ at ﬁnite temperature T .

As written, ZT is in the canonically-quantized operator formalism. In order to apply Monte

Carlo methods, we must move to the path integral formulation, where everything is phrased in terms of functional integrals. Discretizing spacetime turns functional integrals into products of standard integrals, which can be computed using Monte Carlo methods.

A sequence of standard manipulations moves us to the path integral formulation, which we will sketch brieﬂy here. This argument closely follows one presented in [82]. For a generic

Hamiltonian

Hˆ = Hˆ0 + Uˆ (3.2) where Hˆ0 is the Hamiltonian of the free theory for the ﬁelds Φ and Uˆ is the interaction Hamiltonian, deﬁne

ˆ ˆ ˆ Wˆ = e−U/2e−H0 e−U/2. (3.3)

Applying the Trotter formula

ˆ e−H/T = lim Wˆ 1/T (3.4) →0 we can break apart the exp[−H/Tˆ ] inside the trace into Nt = 1/T pieces. To evaluate the trace, we integrate over the set of eigenstates of the ﬁeld operators Φ(ˆ x)|Φi = Φ(x)|Φi,

Z −H/Tˆ −H/Tˆ ZT = Tr[e ] = d[Φ(x)]hΦ(x)|e |Φ(x)i (3.5) where the eigenvalues Φ(x) are functions and R d[Φ(x)] is a functional integral over them. To ﬁnish, we commute the limit → 0 outside of the integral R d[Φ], insert complete sets of states between all Wˆ , evaluate the resulting matrix elements, and then ﬁnally take the limit → 0. The result is

Z −SE [Φ(x,τ)] ZT = d[Φ(x, τ)]e (3.6)

where the ﬁelds Φ(x, τ) are now functions over dE = (d + 1)-dimensional Euclidean space. Note that the Euclidean temporal direction τ is compact. In order to reproduce the original thermal 30 trace correctly, we must apply periodic boundary conditions in the temporal direction for bosonic

fields and antiperiodic boundary conditions for fermionic fields. The functional SE is the Euclidean action Z I d SE[Φ] = d x dτ (S0[Φ(x, τ)] + U[Φ(x, τ)]) (3.7) 1/T H where L0 is the Euclidean action density for the free theory for the fields Φ and 1/T indicates an integral over the temporal direction, whose extent is 1/T . Through similar manipulations, we may obtain expectation values in the thermal field theory as

1 h i 1 Z −H/Tˆ −SE [φ] hOˆiT = Tr Oeˆ = d[Φ]e O[φ] (3.8) ZT ZT where Oˆ is some operator and O[φ] is the corresponding functional over the ﬁelds. Outside of this section, rather than x, τ we will simply write x for position in dE-dimensional Euclidean space, and d for dE.

3.2 How to ﬁt QFTs on computers

To apply Monte Carlo methods to calculate in some field theory, we must represent that theory on a computer. However, field theories are physical systems with infinitely many degrees of freedom. To fit one on a computer, we must render it completely finite. This requires three different types of deformation, all of which introduce errors which must in principle be controlled for.

We are usually interested in the properties of field theories in infinite volumes. An infinite volume will obviously not fit in a finite amount of computer memory, so we restrict the spatial volume to a box of size L3. There are many choices for the boundary conditions, but the most common choice (and the one used throughout this thesis) is periodic. In the Euclidean action

Eq. 3.7, the integration over the (Euclidean) temporal direction is already restricted to run over

finite length 1/T . 1/T → ∞ as T → 0, so we would need an infinite temporal lattice to study zero-temperature properties of the system. In practice, one usually takes 1/T ≥ L so that the errors due to working at finite temperature are no larger than the finite volume errors. 31

The next necessary step is discretization. The fields integrated over in the path integral are functions Φ(x), with independent values at each point in space x. Regardless of the volume the fields are limited to, they comprise an infinite number of degrees of freedom. To render the theory finite, we restrict x to lie on a grid with lattice spacing a. In a finite volume there are only a finite number of lattice sites, and Φ is only valued on the lattice sites, so there are only a

ﬁnite number of degrees of freedom. The functional integral R d[Φ] becomes a product of standard integrals Q [R dΦ ], but is recovered in the continuum limit a → 0. Discretization introduces x∈V4 x errors in all quantities of interest. Thus, to learn about the continuum ﬁeld theory, we must take the continuum limit a → 0.

Derivatives are no longer well-deﬁned on a discrete grid, and so the continuum action SE must be replaced by a discretized version. How this is done is theory dependent and not unique.

The next section will discuss how to discretize QCD.

There is a final infinity that must be tamed: the degrees of freedom in the discretized theory are real numbers. Real numbers may take on an infinite number of values, and so require an infinite number of bits to represent. Thus, in practice, we must also digitize a theory by restricting the values the field degrees of freedom may take on. In modern simulations, these degrees of freedom are typically represented by double-precision floating point numbers, which represent real numbers sufficiently well that the error due to digitization is completely negligible. However, for reasons both theoretical and practical, it is interesting to consider exactly how few bits are necessary to represent a QFT. Although outside the scope of this thesis, Ref. [85] describes a study wherein we systematically examine the error induced in simulations of SU(2) gauge theory by digitizing the continuously-valued gauge links to a finite set. Our results indicate that each SU(2) gauge link can be represented (without compromising the physics qualitatively) by O(10) bits, a factor of ∼ 100 less than used in modern simulations. 32

3.3 Actions for lattice gauge theories

3.3.1 Gauge ﬁelds

First consider the gluon ﬁeld A. In principle, we could consider a discretization with values of Ax,µ on each site. However, per Wilson [168], a more convenient description of the gluon degrees of freedom is to introduce gauge links which connect sites to their neighbors. Each gauge link is deﬁned as

iaAx,µ Ux,µ = e (3.9) where a is the lattice spacing. Because A is su(Nc) algebra-valued, the gauge links U are SU(Nc) group-valued. In this formalism, a gauge transformation transforms each link as

† Ux,µ → ΩxUx,µΩx+ˆµ (3.10)

where Ωx is a SU(Nc) matrix ﬁeld parametrizing the gauge transformation, andµ ˆ is a vector of length a pointing in the µ direction (to move to the next lattice site).

Almost all operators of interest in lattice gauge theory are gauge invariant. The gauge link description of the gauge ﬁeld allows for a particularly natural way of constructing gauge-invariant degrees of freedom: Wilson loops. Wilson loops are constructed from Wilson lines. To compute a

Wilson line, one moves from site to site through the lattice. Starting with the identity, for each link traversed, one appends a factor of U for links traversed forwards and U † for links traversed backwards. More quantitatively, this is

Y W = U P (3.11) xy (x+ j

† vector to move from the site at xi to the site at xi+1. For negativeµ ˆ, we deﬁne Ux,−µ = Ux−µ,µˆ . Wilson lines transform like their endpoints, for example:

† † Wx,x+ˆµ+ˆν = Ux,µUx+ˆµ,ν → ΩxUx,µΩx+ˆµ Ωx+ˆµUx+ˆµ,νΩx+ˆµ+ˆν = ΩxUx,µUx+ˆµ,νΩx+ˆµ+ˆν (3.12) 33 which generalizes obviously to

† Wxy → ΩxWxyΩy. (3.13)

A Wilson loop is the trace of a Wilson line Tr Wxx along any closed path of gauge links through the lattice Px→x, including paths that loop around through the periodic boundary conditions.

Schematically, any Wilson line ending on the same site it started on will transform as

† Wxx → ΩxWxxΩx. (3.14)

Tracing to obtain a Wilson loop, we see Tr Wxx → Tr Wxx by trace cyclicity.

1 µν The pure gauge action − 2 Tr FµνF is gauge invariant, so whatever discretization of it we construct must be also. The simplest possible Wilson loop is the plaquette

h † † i x,µν = Tr Ux,µÛx+µ,νÛx+ν,µÛx,νˆ (3.15) from which we may define the prototypical Wilson gauge action

β X X SG = Re (1 − x,µν) (3.16) 2Nc x µ6=ν

2 where β = 2Nc/g0 (where g0 is the bare gauge coupling). It is a standard exercise to demonstrate that the Wilson gauge action reproduces the correct

F 2 action in the a → 0 limit,

1 X S = βa4 Tr F F µν + O(a2). (3.17) G 4 x,µν x x

4 P R 4 2 As a → 0, a x → dx and the lattice artifacts (starting at order a ) vanish. Generically, there are many discrete actions that reproduce a continuum action as a → 0.

One may take advantage of this fact to construct improved actions, which reproduce the same continuum physics but improve numerical behavior in some way. In Eqn. 3.17, we see that the leading-order lattice artifacts in the Wilson gauge action go as a2. In some cases it is desirable to improve convergence to the continuum limit. To this end, some improved pure-gauge actions include terms that are functions of higher-order Wilson loops to cancel oﬀ discretization errors up 34 to a higher order in a. We will not employ any such improved gauge actions, but we will discuss an example of this for the fermionic part of the action (clover improvement) in the section below.

Other improved actions are designed to allow access to regions of parameter space that would otherwise be too costly to simulate. We use one such action, the NDS action, in our study of the lattice-deformed Ferretti model in Chapter5.

3.3.2 Fermions on the lattice

The general form of the discretized fermion action for the fermion ﬁelds ψ and ψ is

4 X SF = a ψxDxyψy (3.18) x,y where Dxy is the lattice Dirac operator. Under gauge transformations, the lattice fermion fields transform just as their continuum counterparts do, ψx → Ωxψx. Fermion fields are vectors in color space, so we see that a Wilson line connecting two spatially-separated fermion fields is gauge invariant:

† † ψxWxyψy → ψxΩxΩxWxyΩyΩyψy = ψxWxyψy. (3.19)

Thus, for any gauge-invariant fermion action ψxDxyψy, the Dirac operator Dxy is generically constructed out of Wilson lines Wxy.

The most obvious discretization of the Dirac operator is

1 D = γ [U δ − U δ ] + mδ (3.20) xy 2a µ x,µ x+ˆµ,y x,−µ x−µ,yˆ xy where Dirac and color structure is again left implicit. This discretization yields what are called

“naive fermions”. Notoriously, naive fermions exhibit “doublers”: rather than giving one flavor of fermion, this discretization gives 16 degenerate flavors. QCD with 16 flavors is uninteresting, so we need a discretization without doublers to usefully simulate fermions on the lattice.

There are many diﬀerent formulations of lattice fermions in common use today, each of which takes a diﬀerent approach to solving the problem of doublers. In this work, we will consider only one of these formulations: Wilson fermions. Wilson fermions add an additional term proportional 35 to a, the Wilson term, to the Dirac operator which adds corrections ∼ 1/a to the masses of the

15 doublers. As a → 0, the doublers become inﬁnitely massive and decouple from the dynamics.

Adding the Wilson term and redeﬁning the fermion ﬁelds in the conventional way, we obtain the

Dirac operator for Wilson fermions:

±4 X Dxy = 1 − κ (r − γµ)Ux,µδx+ˆµ,y (3.21) µ=±1 where κ = 1/[2(am0 + 4)] is the “hopping parameter” and almost always r = 1.

The Wilson term breaks chiral symmetry explicitly but vanishes as a → 0, so chiral symmetry is restored in the continuum limit. However, this symmetry breaking has some unfortunate numerical consequences; in particular, both the quark mass mq and the chiral condensate Σ are additively renormalized. This means that we must treat the quark mass as an observable that must be computed, rather than a parameter to set; we describe how to measure it in Section 3.5.3 below. Further, in thermodynamics we are frequently interested in diagnosing whether or not a chiral condensate has formed and chiral symmetry is broken. Because it is additively renormalized, however, we cannot be sure when Σ = 0 and we are thus forced to use indirect diagnostics of chiral symmetry breaking. We will discuss one such diagnostic, parity doubling, in Section 4.2.2.

For all of the simulations in this work, we use clover-improved Wilson fermions, also known as clover fermions. This improved action adds one additional term to the lattice Dirac operator

1 X ∆D = − κc ψ σ Cx,µνψ (3.22) xy 2 SW x µν x x,µ,ν

i where σµν = − 2 [γµ, γν]. The operator C is the “clover”

i † Cx,µν = − 2 (Qx,µν − Qx,µν) 8a (3.23) Qx,µν = x,µ,ν + x,ν,−µ + x,−µ,−ν + x,−ν,µ, recalling that x,µ,ν is the plaquette operator (cf. Eq. 3.15). Given proper tuning of the parameter cSW , this additional term will cancel off lattice artifacts at O(a); in practice, this tuning is difficult, but in practice setting cSW = 1 is sufficient to make O(a) artifacts small (especially in combination with smearing). 36

So far, we have written the fermion action in terms of the “thin” gauge links U. However, more commonly one uses so-called “fat” or smeared gauge links to improve numerical behavior. To construct a smeared link, one takes the gauge link and replaces it by some function of the gauge link and its neighbors. The numerics presented in Chapter5 use nHYP smeared links engineered to maintain locality by constructing each fat link only from gauge links on hypercubes including the original thin link [86].

We are interested in investigating theories with fermions in higher representations of the gauge group. Under gauge transformations, a fermion ﬁeld in irrep r transforms as ψ(r) → Ω(r)ψ(r), where Ω(r) is the gauge transformation in irrep r. The usual gauge transformation is in the F irrep,

Ω(F ) = Ω, while (as discussed in Sec. 2.2.3) higher-irrep gauge transformations are constructed

(r) (r) from products of Ω. To construct gauge-invariant objects of the form ψx Wxyψy , we need to use r-irrep gauge links in Wxy. Higher-irrep gauge links are constructed from the F -irrep gauge links

U similarly to higher-irrep gauge transformations; for example,

1 (U (A2))ij = (U U − U U ) (3.24) kl 2 ik jl il jk for the A2-irrep gauge link (cf. Eq. 2.20). Thus, the generalization of the lattice fermion action to treat higher irreps is straightforward: we take the same action, but replace the fundamental- irrep gauge links with higher-irrep ones. To combine with smearing, one simply constructs the higher-irrep links from fundamental-irrep fat links.

3.3.3 Pseudofermions

In the previous section we discussed the Wilson discretization of the lattice Dirac operator.

However, merely discretizing the Dirac operator is insuﬃcient to simulate fermions on the lattice.

The remaining issue is that the ﬁelds ψ and ψ are not real-valued, but rather abstract Grassmann variables. Grassmann variables are an algebraic construct, versus an object with a well-deﬁned value, and so cannot be put on a computer.

The fermionic part of the path integral can be evaluated analytically. For Nf degenerate 37

ﬂavors, Z P Nf x,y ψxDxy[U]ψy Nf d[ψx]d[ψy]e = (Det D[U]) (3.25) where the determinant is evaluated over the spatial, Dirac, and color indices. We can consider learning about a theory with fermions simply by including Det D[U] as part of each observable,

Z R −SG Nf Nf 1 1 d[U]e Det D[U] O[U] hDet D[U] O[U]iG hOi = d[U]e−SG[U]−SF [U]O[U] = = G,F R −S N N ZG,F ZG d[U]e G Det D[U] f hDet D[U] f iG (3.26) where h· · · iG,F is evaluated with S = SF + SG and h· · · iG is evaluated with S = SG. However, there are two serious practical problems with this approach. First, in practice, the Dirac matrix is a 4NcV × 4NcV matrix, where V is the number of lattice sites; computing the determinant of such a large matrix is computationally expensive. Second, as we will discuss below, Monte Carlo methods work by preferentially sampling the part of field space where e−S is not exponentially small. Suppose we sample field space per S = SG, but we are considering parameters where the fermions contribute significantly to the dynamics (as in almost all interesting cases). Monte Carlo will then sample the wrong part of field space for S = SG + SF , so the fermion determinant

Det DNf ∼ e−SF in the expectation value will exponentially suppress most of our data. The result is serious issues with convergence and signal-to-noise.

A better approach is to treat Det DNf as part of the probability weight defining what part of field space to sample. For this to work, Det DNf must be positive definite. Although without a chemical potential Det DNf is typically real, it can be negative. A standard trick to circumvent

† this issue is to introduce fermions in degenerate pairs: taking advantage of Det D = Det D by γ5- hermiticity, we obtain Det D Det D† = Det DD† > 0 for each pair of degenerate fermions. Because

DD† is a Hermitian matrix with positive eigenvalues, we may use a standard Gaussian integral identity to rewrite

1 Z † † −1 Det DD† = = d[φ, φ†]e−φ (DD ) φ (3.27) Det[(DD†)−1] where we have introduced the complex scalar pseudofermion ﬁeld φ. These are purely a calculational construct: φ carries spinor indices, despite being a boson, and (DD†)−1 is highly nonlocal in space. 38

However, unlike the Grassmann ψ and ψ, the complex scalar φ ﬁelds are something we can put on a computer. Further, although inverting DD† is computationally costly (frequently, the dominant cost), this action can be used to sample the correct part of ﬁeld space.

3.4 Computing on the lattice

Following the discussion in the previous sections of this chapter, we have a fully ﬁnite theory wherein expectation values can be phrased in terms of integrals over real degrees of freedom. We can now use Markov Chain Monte Carlo methods to do importance sampling and compute observables for this theory.

3.4.1 Monte Carlo integration

Monte Carlo refers to the set of methods that uses random sampling to compute otherwise- intractable quantities. The term is better-deﬁned in speciﬁc applications. Suppose we have some probability distribution p(x) where x is some set of variables limited to lie in Ω. We want to R compute the expectation value of some function f(x) with respect to p(x), hfi = Ω dx p(x)f(x). A naive approach is to simply randomly draw values of x evenly distributed over Ω. After drawing

N points and evaluating f(x) at each one, our best estimate of the expectation value is

N Z V X hf(x)i = dx p(x)f(x) ≈ p(x )f(x ), (3.28) U N i i Ω i=1 where the subscript U indicates this is expression is speciﬁc to uniform sampling. In the N → ∞ limit, this is guaranteed to converge to the correct answer. However, in practice we are limited to

ﬁnitely many draws, so our estimate will necessarily have some error. We can estimate it as

V 2 N δf 2 = h(f − hfi)2i. (3.29) N N − 1

Notably, the relative error does not depend on the dimension; Monte Carlo is particularly well- suited to compute high-dimensional integrals like those encountered in lattice gauge theory.

This approach of uniform sampling will converge eventually. However, suppose p(x) is exponentially small for most x and the integral is primarily dominated by a very small region ω ∈ Ω. 39

Our estimate will then only improve when we draw an x that lies in ω; the rest of the time, when x ∈ Ω/ω, we will effectively add zero to the integral. One may consider pathological cases, like approximations of the δ function, where ω is so small that we effectively never sample it. Our √ estimate will sit near zero, and because δ ∼ 1/ N, our error estimate will shrink and the integral will appear to converge. Thus, the entire procedure can fail when run for only finite time, with the underestimated error confidently giving us the wrong answer.

Lattice gauge theory is one of these pathological cases. In lattice gauge theory, the probability weight p(x) is the Boltzmann factor exp[−S]. Consider the Wilson gauge action, where

S ∼ V hi. The plaquette expectation value hi is O(1), but the number of lattice sites V is typically O(104)−O(107). Due to this large enhancement by V , even small upwards ﬂuctuations in the plaquette expectation around the minimum of the action will result in an exponentially-suppressed contribution to the integral. This is typical of lattice actions.

The solution to this issue is importance sampling. Rather than drawing x uniformly across

Ω, importance sampling draws x with probability p(x). The formula for the expectation value then reduces to the particularly simple form

N 1 X hfi ≈ f(x ) (3.30) N t t=1 where the index is labeled t to suggest “Monte Carlo time”, to be discussed below. This is the only viable approach to computing integrals dominated by an exponentially small region of ﬁeld space, which is the case relevant to lattice gauge theory.

3.4.2 Markov Chain Monte Carlo

3.4.2.1 Theory

We will implement importance sampling using Markov Chain Monte Carlo (MCMC). Special- izing to lattice gauge theory, we draw field configurations U. MCMC methods construct a process which stochastically wanders through the space of field configurations U. By enforcing certain properties, this process can be made to pass through configurations U with our desired probability 40 distribution P [U] ∝ exp(−S[U]).

A Markov chain is a sequence of random draws where the probability of the draw depends on the result of the previous draw in the sequence, taking no information from earlier draws in the chain. We will consider discrete Markov chains, where we move forward step-by-step (through what we will refer to as “Monte Carlo time”, a ﬁctitious time direction unrelated to the physical one). Markov chains are generated by Markov processes. More quantitatively, the condition that

Markov processes are memoryless can be phrased as

p[Un+1 = U|Un,Un−1,...,U1] = p[Un+1 = U|Un] (3.31) where p[Un+1 = U| ...] is the probability that the next draw yields U, conditional on the pre- ceding elements in the sequence. With this structure we can deﬁne the more intuitive notation

T [A → B] ≡ p[Un+1 = B|Un = A], the “transition probability” of drawing conﬁguration B given that the current conﬁguration is A.

To obtain a sequence of U distributed per P [U] as desired, we construct a Markov process for which P [U] is a ﬁxed point. This amounts to the condition

X P [B] = P [A] T [A → B], (3.32) A i.e. if we start with some ensemble A of conﬁgurations distributed per P [U] and apply one step of our Markov process to each conﬁguration, the resulting new ensemble B will also be distributed per

P [U]. An obvious property of T is that, if we start with some conﬁguration B and move forwards one step in the Markov chain, we will always arrive at some other conﬁguration A:

X T [B → A] = 1 (3.33) A which we can use to rewrite Eq. 3.32 as

X X T [A → B] P [B] = P [A] T [A → B]. (3.34) A A This necessary condition is known as “global balance”. Any T we can construct that satisﬁes global balance for our P of interest will generate U with the desired probability distribution (up to issues with whether or not T implements an ergodic process). 41

In practice, it is easier to implement the stricter, suﬃcient condition of “detailed balance”

T [A → B] P [B] = . (3.35) T [B → A] P [A]

Note that we only need the ratio P [B]/P [A], so we are free to use P [U] = exp(−S[U]) and do not P need to compute Z = U exp(−S[U]) to obtain a normalized probability distribution. To understand detailed balance, imagine an ensemble of Markov processes (“walkers”) wandering through

ﬁeld space. If these processes wander according to a simpler T , such that T [A → B] = T [B → A], they have no reason to prefer any part of ﬁeld space and so will sample it uniformly. Now, if

T instead satisﬁes detailed balance and P [A] > P [B], then the relative probability of a walker moving from A to B is suppressed relative to that of it moving from B to A. Thus, the walkers will preferentially populate areas of ﬁeld space where P is larger and the sample generated by the

Markov process (consisting of the points along the discrete path followed through ﬁeld space by the walker) preferentially contains high-probability conﬁgurations, as desired for an importance sampling algorithm.

The probability distribution P [U] is a fixed point of any T which satisfies Eq. 3.35. However, upon starting out we will not have a configuration drawn from that distribution. Thankfully, a properly-implement MCMC algorithm has the property that it will (barring pathological cases) eventually converge to the correct distribution, regardless of where the Markov Chain starts in

field configuration space [75]. Thus, to simulate a theory, we start with some convenient initial configuration (for example, the infinite-temperature start where all gauge links are the identity) and generate configurations until we observe them to settle into the correct distribution. Determining when exactly this process, known as equilibration, has completed is a necessary but subtle part of any numerical study performed with MCMC.

3.4.2.2 MCMC algorithm design

The only challenge remaining is to devise an appropriate T . Because T depends on P ∼ exp(−S), how we do this depends on the theory of interest. One example is the heat bath algorithm, which 42 is the most common algorithm used to generate conﬁgurations in pure gauge theory. Heat bath updates the conﬁguration by taking some particular gauge link Ux,µ and replacing it with a new

0 one Ux,µ randomly drawn from the distribution

0 p(Ux,µ = U|{U}/Ux,µ), (3.36)

where by {U}/Ux,µ we mean every gauge link in the ﬁeld other than Ux,µ (i.e., the environment of

Ux,µ). Heat bath can only be implemented for the simplest actions, as it requires analytic control

0 over the distribution p(Ux,µ = U| ...), which is only practically possible when it depends on few links; further, it is only computationally practical for the most local actions, as otherwise it cannot be parallelized eﬃciently. This was done long ago [48] for the pure-gauge Wilson action Eq. 3.16, for which the dependence of the action on a single gauge link U0 is a function of only the 6(d − 1) = 18 other gauge links in the 2(d − 1) = 6 plaquettes that U0 participates in.

For most systems of interest, the analytics required to implement something as sophisticated as heat bath are practically impossible. Thankfully, we have a very robust meta-algorithm to construct T that does not, in principle, require any analytic control over the theory at all. This is known as the “Metropolis algorithm” [120]. The Metropolis algorithm breaks the Markov step into two substeps such that T = TP TA: conﬁguration proposal, corresponding to TP , and the accept/reject step, corresponding to TA. For the proposal step, we devise some procedure that

0 0 proposes a new candidate conﬁguration A with probability TP [A → A ]. Formally, it does not

0 matter what TP [A → A ] is. It does not need to satisfy detailed balance or, in principle, incorporate any information about the action at all. The accept/reject substep then ensures detailed balance is satisfied. We take the original configuration A and proposed configuration A0, then compute the probability of accepting

 0 0 exp(−S[A ]) TP [A → A)] TA = min 1, 0 . (3.37) exp(−S[A]) TP [A → A )]

We finally draw a random number to determine whether to accept or reject. If A0 is accepted, the next configuration in the Markov chain after A is A0; if not, A0 is thrown out and the next 43 configuration is instead simply a repeated copy of A. It is straightforward to demonstrate that with this prescription, T [A → B] = TP TA satisfies detailed balance. Commonly, one proposes

0 0 conﬁgurations symmetrically such that TP [A → A ] = TP [A → A] and TA reduces to the more

0 familiar form TA = min(1, exp(−∆S)), where ∆S = S[A ] − S[A].

The Metropolis algorithm gives us a way to use importance sampling for any action S. In principle, the algorithm will work even if we propose completely random configurations. However, for our results to be correct, we must sample field space sufficiently well in finite time. Data generated using MCMC generically suffer from autocorrelations: if our Markov process moves too slowly through field space, subsequent configurations in the Markov chain will not be statistically independent. As discussed in AppendixA, autocorrelations increase the uncertainty in our estimates and thus slow the convergence of our results. It is thus practically necessary to incorporate information about the action into our proposal algorithm to avoid having a too-high reject rate or proposing configurations too similar to the original configuration.

3.4.2.3 Hybrid Monte Carlo

As we saw in Section 3.3.3, unlike with pure gauge theory, the pseudofermion action used to simulate fermions on the lattice involves the inverse of a very large matrix, DD†, which is highly nonlocal in space. This has two important and interacting numerical consequences. First, due to

† −1 the nonlocality of (DD ) , the value of SF as a function of any particular gauge link formally depends on the values of all other gauge links on the lattice. Second, computing the inverse of

DD† is computationally expensive, which means it is very difficult to evaluate the action. These combine to make theories with dynamical fermions particularly hard to simulate: because the action is nonlocal, we cannot get away from evaluating the entire action even if we only attempt to change a gauge link. Thus, to reduce computational costs, we should try to change the configuration as much as possible between every evaluation of the action. However, if we change the configuration too significantly, it is likely to be rejected, leading to long autocorrelation times. 44

The solution to this problem is Hybrid Monte Carlo (HMC),1 used by nearly all modern simulations incorporating dynamical fermions. HMC uses “molecular dynamics” (so called because of algorithmic similarity to simulations of real molecules) to evolve the entire conﬁguration simultaneously, then a Metropolis-like accept/reject step to guarantee the correct statistics.

The molecular dynamics (MD) algorithm has us introduce a fictitious “MD time” direction tMD (unrelated to the physical time). We reimagine the field variables Ux,µ as position variables, and introduce a set of conjugate momentum fields Π with respect to the fictitious time direction.

Schematically, we can multiply by 1 to rewrite any expectation value as

1 2 R d[U] e−S[U] O[U] R d[U]d[Π] e− 2 Π −S[U] O[U] hOi = R = 1 (3.38) d[U] e−S[U] − Π2−S[U] R d[U]d[Π] e 2 where we exclude fermion and pseudofermion ﬁelds intentionally, as in principle we can write

SF [U] = − log Det D[U] as a function of only the gauge links. We recognize the argument of the exponentials as a Hamiltonian for positions U and momenta Π, so we deﬁne

1 H[U, Π] = Π2 + S[U]. (3.39) 2

1 2 2 To start, Π are drawn from the Gaussian distribution exp(− 2 Π ). We then simply evolve the U and Π ﬁelds using Hamilton’s equations, thereby tracing out a wandering “MD trajectory” through

(Π,U) space. If we imagine we are considering a statistical-mechanical system described by the

Hamiltonian H, the ergodic hypothesis states that the average over a microcanonical ensemble of systems (i.e., the RHS of Eq. 3.38) is equal to the long-time average over a single system. Thus, we can evaluate the RHS of Eq. 3.38 by sampling along the MD trajectory, which in turn gives us the

LHS of Eq. 3.38, which is the ﬁeld-theoretical expectation value we were originally interested in.

A simple and universally-used improvement to molecular dynamics is “refreshed MD”. At some regular interval in MD time, the momenta Π are thrown out and a new set of momenta are drawn from the Gaussian distribution. Heuristically, one can think of this as giving the time- evolving system a “kick” in some random direction at regular intervals. Empirically, this scheme

1 Also known as Hamiltonian Monte Carlo (HMC) in the statistics literature. 2 The analytics involved in taking the necessary derivatives of the Hamiltonian are nontrivial. 45 has been found to reduce autocorrelation times and paranoia about ergodicity.

The molecular dynamics (MD) algorithm is, in principle, suﬃcient on its own to simulate a theory with dynamical fermions. However, in practice we must integrate these equations on a computer. This requires us to discretize MD time into short timesteps of length , breaking some of the assumptions used to validate the use of MD and distorting the statistical distribution sampled by MD away from the correct one. This induces integration errors ∼ in all observables that must be extrapolated away. While the need for an additional extrapolation is inconvenient, more insidiously, these errors can pile up at late times and cause numerical instability.

Hybrid Monte Carlo introduces an accept/reject step that fixes the statistical distortions caused by the finite step size . Each iteration of HMC starts with some gauge field configuration

1 2 U0. We generate a set of conjugate momentum ﬁelds Π distributed per exp(− 2 Π ), evolve the ﬁeld per molecular dynamics for some regular amount of MD time (typically δt = 1), then accept or

0 reject the resulting conﬁguration U with probability PA = min(1, exp(−∆H[Π,U])). Note that this is not the same as using MD to propose updates for the Metropolis algorithm, which would accept or reject based on exp(−∆S[U]) instead. However, it is straightforward to show that this accept/reject step combined with the MD evolution satisﬁes detailed balance for U and S [75].

Unlike molecular dynamics alone, HMC is guaranteed to produce the correct statistical distribution. In the “MD limit” of perfect integration → 0, H is a constant of the motion, so

∆H = 0 and every trajectory is accepted. Away from this limit, ∆H 6= 0, and so trajectories are occasionally either accepted or rejected. However, HMC recovers this condition in a statistical sense, in that h∆Hi = 0. To see this, imagine that the ﬁelds (Π,U) and Hamiltonian H describe our system of interest, rather than U and S. For this system, we can view HMC as using MD to

1 2 propose a Metropolis update for the ﬁelds (Π,U). Drawing Π from the distribution ∼ exp( 2 Π )

1 2 satisﬁes detailed balance trivially: the equilibrium distribution of Π is just P [Π] ∼ exp( 2 Π ) so T [Π → Π0] ∝ P [Π0]. Both steps applied to the ﬁelds Π and U in HMC satisfy detailed balance, so

HMC generates a Markov chain for U, Π, and H. In equilibrium hHi is constant, so h∆Hi = 0. 46

3.5 Hadron Spectroscopy

3.5.1 Correlators

Spectroscopy in lattice gauge theory amounts to measuring Euclidean-time correlation functions (“correlators”) and ﬁtting them to extract masses. To measure the quantities we are interested in, we can restrict our attention to two-point correlators of the form3

Cij;rs(x, t) = hJir(x, t)J js(0, 0)i (3.40) where the J are currents, and bolded positions like x are spatial coordinates only. In this work we will be exclusively concerned with meson correlators, for which the currents are generically

X Jir(x, t) = ψx1,t Γi ψx2,t φr(x1, x2; x, t) (3.41) x1,x2 where Γi is a Dirac matrix with the appropriate quantum numbers to couple into our channel of interest. The function φr(x1, x2; x, t) is used to implement source/sink smearing. The simplest choice is the point source/sink, φP (x1, x2; x, t) = δ(x − x1)δ(x − x2). The point sink frequently does a poor job coupling into our state of interest, resulting in a noisy correlator that is diﬃcult to extract information from. To get a cleaner source, one commonly chooses φS as a product of

Gaussians in x−x1 and x−x2 each with characteristic radius r0. The resulting smeared correlators are not gauge invariant quantity, so to measure it we must ﬁx the gauge. All correlators used in this work are measured in Coulomb gauge. Throughout this work we smear our sources with r0 = 6a and use point sinks.

Once we have computed the object Cij;rs(x, t), we ﬁnally project to zero momentum by summing over all spatial sites to obtain

X Cij;rs(t) ≡ Cij;rs(x, t) (3.42) x

3 Note that we insert one current at fixed position (0, 0) on the lattice. In principle, this is not a necessary restriction; the generic many-to-many two point correlator C(x1, t1; x0, t0) is an interesting object and necessary to compute e.g. correlators for flavor singlet mesons. However, C(x1, t1; x0, t0) is notoriously difficult to compute, 3 naively costing a factor of Ns × Nt more than C(x, t). We do not need it for our purposes, so we immediately restrict our attention to the one-to-many correlator. 47 Channel i = Γsource r = source smear j = Γsink s = sink smear Sign

Pseudoscalar γ5 S γ5 P + Pseudoscalar γ5 S γ0γ5 P − Scalar 1 S 1 P + Vector γ3 S γ3 P + Axial vector γ3γ5 S γ3γ5 P +

Table 3.1: The full set of diﬀerent meson correlators used in these results. All sources are smeared with radius r0 = 6a, denoted by S, and all sinks are points, denoted by P . Sign denotes whether the correlator is of cosh form (+) or sinh form (−).

which we will ﬁt to extract masses. Table 3.1 lists the full set of diﬀerent types of correlators we

ﬁt in this study.

The methods used to actually compute correlation functions on lattices are standard but baroque, and thus not worth reproducing here. We note, however, that the generalization to computing correlation functions in higher irreps of the gauge group is straightforward: one simply constructs the Dirac operator out of higher-representation gauge links.

3.5.2 Extracting masses

In order to extract masses, we need a functional form to ﬁt our correlators to. Inserting a complete set of energy eigenstates and using O(t) = eHtˆ Oe−Htˆ , we ﬁnd

e−mnt C0(t) = h0|Jir|nihn|Jjs|0i (3.43) 2mn where we have suppressed indices on C for clarity, and recalling that we are projecting to zero momentum where the energy of the nth eigenstate En = mn. The matrix elements will be zero for states with the wrong quantum numbers, so we resolve a tower of increasingly massive states with the quantum numbers of our channel of interest. This reveals the ﬁrst numerical problem we will have to cope with: we are typically interested in measuring the mass of the lowest-lying state in this tower, but C(t) includes large contributions from excited states at short times t.

Our argument above is valid for zero-temperature field theory, but unless our lattice has infinite extent in the temporal direction, we are working at finite temperature. To account for the 48

ﬁnite size of our system correctly, we re-express our correlation function Eq. 3.40 as a thermal trace like X 1 h ˆ i C(t) = Tr Jˆ (x, t)Jˆ (0, 0)e−HLt (3.44) Z ir js x T where Lt = aNt = 1/T is the temporal extent of the lattice. With some manipulation, one ﬁnds schematically that [61]

∗ −Lt C(t) = C0(t) + C0(Lt − t) + (Thermal contamination ∼ e ). (3.45)

There are two important features to note, here. First, there are thermal artifacts proportional to matrix elements hm|J|ni where neither m nor n is zero, unlike in our zero-temperature expansion

Eq. 3.43. These vanish as we take the temporal extent of the lattice Lt → ∞, but compromise our ability to extract accurate spectroscopic results from ﬁnite-temperature lattices. Second, the state propagating forwards in Euclidean time from t = 0 is now accompanied by a complex-conjugate state propagating backwards in time from the far end of the lattice t = Lt. For all of the mesonic

∗ correlators we will examine C0(t) = ±C0(t) (see Table 3.1 for the signs for correlators studied in this work). This allows us to rewrite Eq. 3.45 more usefully as

X e−mnt ± e−mn(Lt−t) C(t) = h0|J |nihn|J |0i + ··· . (3.46) ir js 2m n n

Armed with Eq. 3.46, we are almost equipped to fit our correlator data. We make several decisions that will apply to all of the spectroscopy we compute in this work. First, we will assume that thermal artifacts can be neglected. We will further not attempt to fit any excited states, and instead simply fit correlators in the regions where the excited state contributions have exponentially P decayed. This allows us to truncate n after the lowest-lying state. We thus arrive at our model to fit for meson correlators, e−mnt ± e−M(Lt−t) C(t) = Z B (3.47) ir js 2m where Zir and Zjs are the matrix elements. Table 3.1 lists all of the correlators we will fit to this model, including what physical channel we expect them to couple to and the sign to resolve the ± 49 in the expression. We note that if Jir = Jjs, then Zir = Zjs and the fit can be run with one fewer parameter. However, this requires that the source and sink smearing be the same.

We are now equipped to measure meson masses. In summary, the procedure is: compute the correlator C(t) coupling to the channel of interest over an ensemble of gauge conﬁgurations.

Average the correlator, including computing the covariance matrix between C(t) at diﬀerent t.

Using a nonlinear least squares ﬁtter, ﬁt the averaged correlator data to the model Eq. 3.47. The

ﬁt, if successful, yields estimates of the matrix elements Zir and Zjs and the meson mass m.

We will consider only “folded correlators”: all of the meson correlators we would like to ﬁt are cosh-like or sinh-like, which permits us to average the ﬁrst and second half of the correlator together.

The resulting folded correlator has length Lt/2 + 1. This procedure incorporates information from the entire correlator and, practically, ﬁts to such correlators tend to be more numerically stable.

3.5.3 AWI quark mass

As discussed previously, the quark mass for Wilson fermions receives a (large) additive renormalization. Thus, we must treat it as an observable to measure. To do so, we use the axial Ward identity (AWI) quark mass [also known as the “partially conserved Axial current” (PCAC) quark mass]. The axial Ward identity reads

(r) (r) ∂µh0|Aµa (x)Or(0)|0i = 2mrh0|Pa (x)Or(0)|0i, (3.48)

where a is an isospin index. Applying the identity with Or as the pseudoscalar current, we can relate the two correlators h i X −MPS t −MPS (Lt−t) C55(t) = hJ5S(0, 0)J5P (x, t)i = Z5SZ5P e + e x (3.49) h i X −MPS t −MPS (Lt−t) C505(t) = hJ5S(0, 0)J05P (x, t)i = Z5SZ05P e − e x as

∂tC505(t) = −2mqC55(t) (3.50) 50 from which we can derive the relation

mq Z05P = 2 Z5P (3.51) mπ which allows us to write a model for the two correlators with the masses mq and MPS and the matrix elements Z5S and Z5P as shared parameters. We can then extract an estimate of the quark mass by performing a simultaneous ﬁt to both correlators (making sure to include the correlations between C55 and C505).

3.5.4 Systematic errors

Fitting a meson correlator requires the choice of several metaparameters. The most important are tmin and tmax, which define the range of points tmin ≤ t ≤ tmax of the correlator to include in the fit. As discussed above, at small t correlators suffer from excited state contamination. Moving tmin away from t = 0 allows these excited states to decay away, better exposing the ground state at the cost of throwing away the information in the early-t points. At late times (towards the middle of the lattice, where t ∼ Lt/2), the ground state has decayed to an exponentially small value, and noise overtakes the signal. Attempting to fit this noise frequently causes numerical issues, so it is important to trim tmax back out of the noisy region.

In practice, one does not find that there is a single correct choice of tmin and tmax. Instead, there are usually many tmin and tmax that produce reasonable-looking fits, yielding similar but not identical measurements of e.g. masses. In the absence of better statistical technology, which fit to take as the correct one is a matter of art and frequently done by hand. However, this would be intractable with the large datasets involved in this study. Instead, we have adopted a procedure of selecting a best fit algorithmically, and then computing a systematic error associated with our choice of best fit.

The first step in our procedure is to compute a grid of fits over all tmin < tmax, up to a few constraints (e.g. we never include the point t = 0, which is maximally contaminated by excited states, and tmax − tmin must always be greater than the number of fit parameters). The result is 51 a pool of fits. We then trim failed fits out of the pool, where we diagnose failure by looking for

fits where the fitter threw an error or returned NaNs, or where the fit χ2 is infinite, δM/M > O(1), and/or δM/M ≈ 0. The result is a pool of viable fits, from which we must pick a best one.

To deﬁne a best ﬁt, we use the “MILC criterion” following [35]

Q · (dof) CMILC = 1 . (3.52) P 2 2 ( m δm /m) We note immediately that this quantity and its properties are based on heuristic, and have no rigorous statistical meaning. In the expression, m and δm are the parameters (and corresponding standard errors) that we are interested in extracting from the fit (typically masses). The quantity Q is the goodness-of-fit statistic4 derived from the χ2 distribution. The exact definition of Q depends

−χ2 on the number of degrees of freedom in the ﬁt, but it goes roughly as e , meaning CMILC will be exponentially small for bad ﬁts. The remaining factors in CMILC mostly help to disambiguate

ﬁts with similar Q. The factor of (dof) encourages selection of ﬁts which include more points

(i.e., greater tmax − tmin), while the sum over the standard errors encourages selection of ﬁts which determine the parameters more precisely.

After choosing the ﬁt with the best CMILC, we further trim the pool of viable ﬁts to include

fits where CMILC is at minimum some factor of that of the best fit, typically 0.1. We say the remaining fits are all equally viable choices. The values of the parameters estimated by the fit will vary over the pool of viable fits. For each parameter, we estimate a systematic error σsys as the half difference between the value at the 85th-percentile and the value at the 15th-percentile. For zero-temperature fits the range is usually safe, but the percentile definition is more robust against the more frequent outliers that occur when fitting data from small, noisy finite-temperature lattices.

Throughout the rest of this work, when an error is quoted on e.g. a mass, it is σstat + σsys.

This scheme can be extended straightforwardly to higher-dimensional metaparameter spaces.

For example, extracting the AWI quark mass involves ﬁtting two correlators simultaneously. Each of the two correlators has its own tmin and tmax that can be gridded over separately. However, in

4 This is sometimes also rendered as p and called the p value; however, there is another definition which has P = 1 − Q. To be unambiguous, Q ∈ [0, 1], and Q → 1 indicates a good fit. 52 practice, results in this paper were computed with using the same tmin and tmax for both correlators in our AWI fits, thus keeping metaparameter space conveniently low-dimensional.

3.6 Gradient ﬂow

Gradient flow is a smearing operation that evolves gauge fields in a fictitious “flow time” direction [123, 114]. The differential equation defining the procedure is

˙ 2 Vx,µ(t) = −g0{∂x,µS[V (t)]}Vx,µ(t) (3.53)

where Vx,µ(0) = Ux,µ, the unflowed gauge links. The defining equation takes an action S as input, so one must be chosen before the su(3)-valued differential operator ∂x,µ can be evaluated and the expression resolved into something concrete that may be integrated on a computer. Commonly, and exclusively in this work, one uses the Wilson gauge action (Eq. 3.16). The resulting concrete realization of gradient flow is called the Wilson flow, defined by the flow equation

˙ 2 X Vx,µ(t) = −g0τa Tr[τa (x,µ,ν + x,µ,−ν)]Vx,µ(t) (3.54) ν6=µ where is the plaquette operator Eq. 3.15, evaluated with ﬂowed gauge links V (t), and τa are the

a anti-hermitian generators of su(N). The structure τa Tr[τ ··· ] arises from ∂x,µ and acts to project out the traceless antihermitian part of the sum. This diﬀerential equation is straightforward and numerically inexpensive to integrate.

Gradient flow has a number of interesting numerical properties, and new uses are still being discovered for it. As discussed immediately below, gradient flow is most commonly used to set the scale for lattice simulations [116]. It has also been used to compute the renormalized coupling in lattice gauge theories [115], as well as renormalized observables like the Polyakov loop [128]. Gra- dient flow “looks like” RG flow; this resemblance is more than aesthetic, as recent work has related the two [41, 153]. As we will discuss later in SectionC, the behavior of certain observables under

Wilson ﬂow has recently been found to provide useful diagnostics of phase in ﬁnite-temperature lattice simulations [55, 54, 166]. 53

3.6.1 Scale setting

In a confining system all ratios of dimensionful numbers are fixed quantities, which allows us to quote results in physical units by taking a single dimensionful number from experiment to set an overall scale. In the numerical studies presented in this work, we will use scales computed using the Wilson flow as our fiducial quantities.

We can deﬁne a general ﬂow scale t∗ via

2 ht E(t)i|t=t∗ = C∗ (3.55)

µν where E(t) is some discretization of the gluonic action density Tr FµνF at ﬂow time t, and C∗ is a dimensionless constant deﬁning the scale. In QCD, for t0 one sets C0 = 0.3 and for t1 one sets

2 C1 = 2/3 [152]. Evaluating ht Ei on a lattice and ﬁnding where condition 3.55 is satisﬁed yields

2 √ t0/a ; the physical scale enters by ﬁxing t0 ' 0.142 fm [25]. There is another family of scales w∗ based on the deﬁning relation

2 t∂ ht E(t)i| 2 = C , (3.56) t t=w∗ ∗ but we do not use them in this work.

We are interested in using Wilson ﬂow to set the scale in generalizations of QCD where

Nc 6= 3. To translate quantities to physical values for comparison with QCD, we need some way of matching our definition of t0 at arbitrary Nc with the definition at Nc = 3. Large-Nc scaling arguments provide a translation procedure [60, 91]. The observable ht2E(t)i can be used to define the renormalized coupling at the scale t [42, 81],

2 2 128π 2 gwf(t) ≡ 2 ht E(t)i. (3.57) 3(Nc − 1)

2 The usual large-Nc scaling argument holds the ’t Hooft coupling λ0 = g0Nc constant in Nc. Thus,

2 g ∼ 1/Nc at leading order, regardless of renormalization scheme. We can immediately read oﬀ

2 2 from Eq. 3.57 that, for g ∼ 1/Nc to hold, ht E(t)i ∼ Nc at leading order. Comparing Eq. 3.55, we conclude that a reasonable choice is C∗ ∝ Nc. 54

We are particularly interested in SU(4), the gauge group of Ferretti’s model. Applying our scaling for C∗ to translate from SU(3) to SU(4), we deﬁne the scales t0 and t1 via

2 ht0E(t0)i = 0.4 (3.58) 8 ht2E(t )i = . (3.59) 1 1 9

Assuming that our Nc scaling has produced an equivalent quantity in SU(4), we may take the value

2 t0 = (0.142 fm) from SU(3) [25].

µν There are many possible discretizations for the observable E = FµνF . The simplest uses the plaquette for Fµν. In this work we will exclusively use the clover deﬁnition where Fµν = Cµν, with C as deﬁned in Eq. 3.23 above. Chapter 4

Lattice Thermodynamics of QCD-like Systems

4.1 Exotic phase structures

We know from lattice studies of quantum chromodynamics at zero density and finite temperature that QCD has two distinct phases: a high-temperature deconfined and chirally symmetric phase, and a low temperature confined and chirally broken one. In QCD, these phases are separated by an analytic crossover occurring near T ∼ 150 MeV [151]. We have two different stories for discussing this phase transition: confinement and the spontaneous breaking of center symmetry, and separately, condensation and the spontaneous breaking of chiral symmetry.

Confinement refers to whether color charges may exist in isolation. In the low-temperature confined phase, quarks and gluons exist only in hadronic bound states. Attempting to pull a hadron apart will simply induce pair creation, breaking the hadron apart into more hadrons rather than exposing isolated color charges. In the high-temperature deconfined phase, quarks and gluons exist in a deconfined plasma. Exposed color charges roam freely, and the free energy cost of separating two quarks to infinity is dramatically less due to screening of color charges by the plasma.

Meanwhile, condensation refers to whether a chiral condensate has formed and spontaneously broken chiral symmetry, thereby dynamically generating constituent quark masses and rendering the pions (nearly) massless via the Goldstone mechanism.

In QCD, all of the quarks confine simultaneously. Further, the chiral and confinement transitions are found to coincide. However, there are no firm theoretical arguments that either of these must be the case in generalizations of QCD. This suggests several obvious possibilities for phase 56 structures unlike QCD’s:

• Chiral condensation could occur at diﬀerent temperatures for diﬀerent irreps of fermion.

The tumbling scenario, discussed at length below in Section 4.2.1, suggests that chiral

condensation in multirep theories will occur in a cascade across a hierarchy of exponentially-

separated temperatures [132, 118].

• Chiral condensation and conﬁnement could occur at diﬀerent temperatures for the same

species of fermion. Previous work with dynamical fermions appeared to indicate that this

occurs for SU(3) with two Dirac ﬂavors of adjoint fermions [95], but it is likely that the

theory explored in that work is infrared conformal rather than conﬁning (See Ref. [134]

and Refs. [59, 155] for reviews).

• The conﬁnement transitions of diﬀerent representations can be separated: if center symme-

try breaks in several stages, there may exist phases where some representations of charge

are deconﬁned while others remained conﬁned. We motivate this possibility in Section 4.3

below.

We will check for all of these (interacting) possibilities in our numerical study of Ferretti’s model in Chapter5. To do so, we must generalize the phase diagnostics that we use in QCD and other single-irrep theories to look for exotic phases in multirep theories.

4.2 Chiral symmetry breaking

Chiral symmetry is a global symmetry relating diﬀerent quark ﬂavors. We can catalog the symmetries making up the chiral symmetry group by examining the QCD Lagrangian. Consider the kinetic term for nF quarks

δL = ψjiDψ/ j (4.1)

iθ where j is a ﬂavor index. This Lagrangian is obviously invariant under the symmetry ψj → e ψj; this symmetry, known as the vector U(1)V , remains unbroken and gives rise to conservation of 57

iαγ baryon number. It is similarly invariant under ψj → e 5 ψj; this symmetry, known as the axial

U(1)A, is broken at the quantum level by the axial anomaly. These are the full symmetries of the

Lagrangian when nF = 1, but there are additional symmetries associated with rotating the ﬂavors amongst themselves when nF > 1. To see them, we can break the fermion ﬁelds apart into their

1 left- and right-handed parts as ψj = ψLj + ψRj where (ψj)L,R = PL,R ψj and PR,L = 2 (1 ± γ5) are the usual projectors, as

δL = ψLjiDψ/ Lj + ψRjiDψ/ Rj. (4.2)

i As written, it is obvious that the Lagrangian is independently invariant under ψLj → (UL)jψLi

i and ψRj → (UR)jψRi where UL,R ∈ SU(nF )L,R. Altogether, the chiral symmetry group of QCD Lagrangian in the chiral limit where all quarks are massless is

SU(nF )L × SU(nF )R × U(1)V × U(1)A (4.3)

where nF is the number of ﬂavors one wants to consider light, usually taken to be either 2 or 3.

The quarks in the universe around us are light but not massless, and it is presently impossible to simulate with massless quarks. We should thus consider the eﬀects of moving away from the chiral limit. The mass term for Dirac fermions is

δL = −ψimijψj = −ψLimijψRj − ψRimijψLj (4.4)

where mij is the (diagonal) mass matrix. Again, this is obviously invariant under the U(1)V , but breaks U(1)A explicitly without the additional Dirac structure in D/. Specializing to the case of nF † degenerate ﬂavors of mass mq such that mij = mqδij, ψLψR → ψLULURψR is only invariant if UL =

UR. Thus, giving the fermions a degenerate mass explicitly breaks SU(nF )L × SU(nF )R → SU(nF )V .

1 nF If the nF fermions each have diﬀerent masses, this further breaks SU(nF )V → [U(1)] , i.e., the ﬂavors may not be rotated amongst each other.

We can understand the QCD phase transition near the chiral limit mq → 0 in terms of spontaneous chiral symmetry breaking (χSB). Below some critical temperature, a Lorentz scalar

1 This is the case in nature, but mq ΛQCD for the light ﬂavors so the breaking is weak. 58 bilinear develops a nonzero vacuum expectation value

hψiψji = hψLiψRji + hψRiψLji = Σδij 6= 0. (4.5)

We call this process chiral condensation, and the resulting Σ the quark condensate. Comparing

Eqs. 4.5 and 4.4, we see that the chiral condensate breaks chiral symmetry in the same way as a

Dirac mass term. Said another way, the Dirac mass term acts like an external magnetization for the spontaneous breaking of chiral symmetry. Altogether, the χSB pattern is

SU(nF )L × SU(nF )R × U(1)V × U(1)A → SU(nF )V × U(1)V (4.6)

where we include U(1)A even though it is strongly explicitly broken to point out that, were it not for the axial anomaly, it would have been spontaneously broken and yielded another Goldstone boson.2

We performed the analysis above for the F -irrep quarks of QCD speciﬁcally, but the same arguments work for any generalization of QCD where the fermions are in some complex representation of the gauge group (regardless of Nc). We will also be interested in real and pseudoreal representations; in particular, relevant to Ferretti’s model, the A2 of SU(4) is a real representation.

Fermions charged under self-conjugate (i.e., real and pseudoreal) representations have additional symmetries relating color and anti-color [126]. These symmetries allow us to rewrite the fermion

D W D kinetic term for N Dirac fermions in terms of N = 2N left-handed Weyl ﬁelds ξi as

δL = ξiDξ/ i (4.7) where i ∈ 1,..., 2N D labels the Weyl ﬂavors. This Lagrangian is invariant under ξ → V ξ where

D D D V ∈ SU(2N ); the chiral symmetry is thus enlarged from SU(N )L × SU(N )R × U(1)V to SU(2N D). Translated to this basis, the Dirac mass term for N D degenerate ﬂavors is [62]

1 αβ δL = 2 m (ξi,αJijξj,β + ξi,αJijξl,β) (4.8)

2 The corresponding particle in nature is the ﬂavor singlet η0 meson. 59 where we have made the Weyl spinor indices α and β explicit. For real representations, we may

3 R D choose Jij = δij. In this case, the mass term transforms under SU(2N ) as

T † ∗ δL ∼ ξiξi + ξiξi → ξiV V ξi + ξiV V ξi (4.9) which is only invariant if V ∈ SO(2N D). We thus ﬁnd that the χSB pattern for N D Dirac ﬂavors of real-rep fermions is

SU(2N D) → SO(2N D) (4.10) where we note that we are free to generalize to N W = 2N D odd. For pseudoreal representations, we have that   0 −1 P   J =   (4.11) 1 0 where each block is an N D × N D matrix. The condition that the mass term is invariant is thus

T V J PV = J P, which we recognize as the requirement that V ∈ Sp(2Nf ). Thus, we ﬁnd that the

χSB pattern for pseudoreal-rep fermions is

SU(2Nf ) → Sp(2Nf ). (4.12)

This analysis generalizes straightforwardly to multirep theories. For each distinct irrep of fermion in a multirep theory, there is a completely independent chiral symmetry. Each distinct irrep in a theory has its own independent axial U(1)A would-be symmetry, but each of the associated axial currents is anomalous. However, for a theory with fermions charged under R diﬀerent irreps, it is possible to construct R − 1 linear combinations of the axial currents that are not anomalous

(see for example Refs. [46, 62]). These R − 1 non-anomalous U(1)A symmetries are spontaneously broken, yielding an additional R−1 Goldstone bosons. Schematically, the χSB pattern of a general

SU(Nc) multirep theory is just the product of the chiral symmetries of each single-rep sector, times

3 D If one is concerned with enforcing that a U(N )V vector symmetry holds, one should instead take J R = |J P|, where J P is Eq. 4.11. See Sec. 6.A for further discussion of this point. 60 the R − 1 unbroken U(1)A symmetries:

Y D D Y W Y D R−1 SU(Nr ) × SU(Nr ) × U(1)V × SU(Nr ) × SU(2Nr ) × [U(1)A] r∈C r∈R r∈P Y D Y W Y D → SU(Nr ) × U(1)V × SO(Nr ) × Sp(2Nr ) (4.13) r∈C r∈R r∈P where r ∈ C denotes that r is a complex representation, etc. We make explicit in Eq. 4.13 that the

U(1)V symmetries for complex irreps are unbroken for consistency: the equivalent symmetries for real and pseudoreal irreps are embedded within the unbroken SO(N W ) and Sp(2N D)[65].

4.2.1 Tumbling & the Most Attractive Channel hypothesis

The tumbling scenario, as originally proposed by Raby, Susskind, and Dimopolous in Ref. [132], posits a plausible mechanism to dynamically generate exponentially-separated scales. The most general conception of the tumbling scenario considers an asymptotically-free chiral gauge theory with R different species of Weyl fermions charged under representations {r} = {r1, r2, . . . , rR} of the gauge group (note that here, unlike in the analysis of Sec. 2.3, r and its conjugate r are thought of as distinct). One imagines starting far in the UV where all species of fermion are deconfined and chirally symmetric. Running the scale down from the UV, the fermions become increasingly attracted to one another via interaction with the gauge field. When some critical degree of attraction is achieved for some species, a Lorentz-scalar bilinear condensate will form breaking chiral symmetry for that species. We know that in theories with fermions charged only under F and F , condensation happens only once and is simply the usual chiral transition forming a color-singlet condensate where F × F → 1. However, in general, the condensate does not have P to be a color singlet and can be charged under some other rep rc in i,j ri ⊗ rj. A color-charged condensate will Higgs the gauge field, rendering some of the gauge bosons massive and “breaking” the gauge symmetry down to a subgroup. The fermion species that participate in the condensate gain dynamically-generated mass (cf. QCD, where the constituent quark mass ∼ the scale of chiral symmetry breaking). We continue to run the scale down, and this process repeats for the new 61 theory defined by the unbroken gauge symmetry and remaining massless fermions. This continues until no more condensation is possible.

The Most Attractive Channel (MAC) hypothesis says that the condensate that forms ﬁrst is the one for which the force of attraction between fermions into that condensation channel is greatest. We can make this statement semi-quantitative. Any bound state (and thus, any bilinear condensate) formed out of fermions from reps ri and rj will be charged under some composite rep rc of the gauge group in r1 ⊗ r2. For example,

A2 ⊗ F = ⊗ = ⊕ = G3 ⊕ A3 (4.14) so any bound state of F and A2 will be charged under either the irrep G3 (see Table 2.1) or the irrep A3 [note A2 = F and A3 = 1 in SU(3)]. We use the tree-level approximation of single gluon exchange to extract the leading g2 and representation dependence of the potential for two fermions charged under reps ri and rj to condense into a bound state charged under rep rc as

2 Vrirj →rc ∼ g [Cc − Ci − Cj] (4.15) where Cr is the quadratic Casimir of rep rr. Thus, MAC says the condensation that occurs ﬁrst is the one deﬁned by the channel rirj → rc for which the attraction ∼ −V is maximized.

If tumbling occurs as hypothesized by Ref. [132], it will dynamically generate a hierarchy of exponentially separated scales, one for each condensate that is formed on the way down from the

UV. To demonstrate this, we will specialize to the simpler case of a theory with vectorlike fermion content. Such a theory has real fermion content (as deﬁned by Ref. [132]), in that every rep is accompanied by its conjugate, i.e., for each rep r in the set of distinct irreps in the theory {r}, r also appears in {r}. For such a theory, the Most Attractive Channel is always condensation of r∗ and r∗ into the singlet, where r∗ is the rep with the greatest quadratic Casimir. This follows obviously from Eq. 4.15: if Cr∗ is the greatest Casimir, then even if the singlet is available in every

∗ other channel, min V = −C1 − C2 > −2Cr∗ unless r1 = r2 = r . This drastically simpliﬁes the analysis, as the gauge group is never Higgsed; instead, we simply form a hierarchy of color-singlet condensates ordered by quadratic Casimir. 62

Consider a theory with vectorlike fermion content in two irreps, r1 and r2, such that C1 > C2.

Running the scale µ down from the UV, we imagine r1r1 → 1 ﬁrst occurs at scale µ1, and then r2r2 → 1 at scale µ2. We are only interested in demonstrating that µ1 and µ2 are exponentially separated, so we make the rough approximation of neglecting the fermionic contribution to the running of g2. Following Ref. [100], we will estimate that condensation occurs for rep r at the scale

µr deﬁned by g(µ )2 r C ∼ 1. (4.16) 4π r

Plugging in for the running of g2 at one loop, we obtain

2πCr/b1 µr = Λe (4.17)

4 where Λ is the UV scale from which we start running, and b1 is the one-loop coeﬃcient of

W the multirep β function Eq. 2.21 evaluated with all Nr = 0, per our approximation. We see immediately that the condensation scale of irrep r is exponentially sensitive to Cr. The ratio of condensation scales is

2π µ1 (C1−C2) = e b1 (4.18) µ2 and so we see that the separation is exponential in C1 − C2.

This argument can be made more precise using solutions of the Dyson-Schwinger equation in the “ladder approximation” [138], but the result is qualitatively the same.

4.2.2 Parity doubling

To determine whether chiral symmetry is broken for an irrep, we should check for the presence of the corresponding chiral condensate. However, we are using Wilson fermions, which break chiral symmetry explicitly. This gives the chiral condensate an additive mass renormalization and thus makes the condensate diﬃcult to both measure and interpret. Instead, we will indirectly diagnose the presence of a chiral condensate using parity doubling.

4 In the TF = 1/2 convention. 63

When chiral symmetry is broken, the pseudoscalars5 (pions) are pseudo Goldstone bosons

2 and the GMOR relation MPS ∝ mq holds. Their parity partner scalar states have a mass ∼ ΛQCD. Meanwhile, in the chirally symmetric phase at ﬁnite temperature T we expect all mesons degenerate with masses originating from the lowest-lying Matsubara frequency M ∼ πT . Thus, we expect to see a large mass splitting between the pseudoscalar and scalar mesons when a chiral condensate has formed, and degeneracy when one has not. The vector and axial vector parity partner mesons show similar behavior.

As discussed above, in a multirep theory, we do not expect one irrep to be sensitive to whether a chiral condensate has formed in another irrep. Thus, we will assume that we can diagnose chiral condensation in each distinct irrep separately by looking for parity doubling in the spectrum of mesons for that irrep.

4.3 Conﬁnement

While the use of Polyakov loops as a diagnostic of conﬁnement in gauge theories with fermions in only a single representation is well-understood, the situation is more subtle in multirep theories.

In a single-rep (or zero-rep/pure gauge) theory, there are typically only two phases: the confined phase where center symmetry holds (at least approximately), and the deconfined phase where center symmetry is spontaneously broken to the trivial group Z1 = {1}. Given this phase structure, a single Polyakov loop in any representation can serve as a diagnostic of confinement. However, in a theory with fermions in multiple representations, it may be possible for charges of some irreps to be confined while charges of other irreps are deconfined. To understand what types of phases may exist and how they may be identified, it is necessary to cautiously consider the physics of Polyakov loops in higher representations in the context of multirep theories.

5 Except for the ﬂavor singlet meson whose mass is enhanced by explicit breaking of an axial U(1)A. 64

4.3.1 Polyakov loops & multiple representations

Usually, Polyakov loops and center symmetry are discussed hand-in-hand, with the use of the

Polyakov loop as a phase diagnostic motivated by its role as an order parameter of center symmetry breaking. However, it is possible to motivate Polyakov loops with an entirely physical story and without mention of center symmetry.

Given some field configuration, we can diagnose confinement of a charge of irrep r by inserting a separated static pair of color “test charges” (one charge r, one anti-charge r) and measuring the strength of attraction between them. A static r-irrep test charge at spatial position x loops through the temporal extent of the lattice. Along the way, this charge has its colors rotated by the r-irrep thermal Wilson line Nt (r) Y (r) L ≡ U (4.19) x (x,t),tˆ t=1 where U (r) is the time-direction r-irrep gauge link at site (x, t). The periodic boundary conditions (x,t),tˆ in time allow us to close the thermal Wilson line into a Wilson loop. We thus take the trace to construct the gauge-invariant r-irrep Polyakov loop

(r) Pr(x) = Tr Lx . (4.20)

The Polyakov loop for an r test anti-charge is constructed analogously, except with U (r) → (U (r))†

∗ [and thus Pr = (Pr) ]. Gauge links of arbitrary irrep r can be constructed from appropriately symmetrized and antisymmetrized combinations of F and F irrep gauge links U ≡ U (F ) and

U † ≡ (U (F ))† = U (F ), so the thermal Wilson line for irrep R can always be written in terms of the fundamental one L ≡ L(F ) and its Hermitian conjugate L† ≡ (L(F ))† = L(F ). For example (as discussed previously in Section 3.3.2), the A2-irrep gauge link is

1 (U (A2))ij = U i U j − U iU j , (4.21) kl 2 k l l k from which it follows that the A2-irrep Polyakov loop is

Nt (A ) Y (A ) 1 P (x) ≡ Tr L 2 = Tr U 2 = (Tr L )2 − Tr(L2 ) . (4.22) A2 x (x,t),tˆ 2 x x t=1 65

Similary, one ﬁnds that the S2 irrep Polyakov loop is

1 P (x) = (Tr L )2 + Tr(L2 ) (4.23) S2 2 x x and the G irrep Polyakov loop is

2 PG(x) = |Lx| − 1. (4.24)

We can probe the free-energy cost Frr(d) to separate a test r charge and r anti-charge by a distance d by measuring

∗ −Frr(d)/T hPr(0)Pr(d)i = hPr(0) [Pr(d)] i ∼ e . (4.25)

Taking the limit d → ∞ we expect factorization,

∗ ∗ 2 −Frr(∞)/T hPr(0) [Pr(d)] i → hPr(0)ihPr(d)i = |hPri| ∼ e (4.26) and the last equality follows from taking a volume average. We may equivalently measure |hPri| ∼ exp[−Frr(∞)/2T ].

A gauge field coupled to fermions can pair produce charges from the vacuum in the adjoint irrep G (gluon) and whatever representations r1, r2,... that the dynamical fermions are charged under. In the zero-temperature phase where all irreps of charge are confined, we may only separate to infinity an r-r pair of static test charges for finite energy cost if it is possible to pair-produce charges that can then form color-singlet “static-light hadronic” states with the r and r test charges

[72]. For example, in pure gauge theory only gluons are dynamical, so only G-G pairs can be pair produced from the vacuum. An F charge may not form a color-singlet state with any number of G charges, so an F -F pair may not be separated to inﬁnity for ﬁnite energy cost: as r → ∞,

FF F (∞) → ∞ and thus |hPF i| → 0. Conversely, a G charge may form a color-singlet state with another G charge, so a G-G pair can be separated to inﬁnity for ﬁnite energy cost: as d → ∞, the

G-G pair of test charges splits into two GG static-light states, and so FGG(∞) and |hPGi| remain ﬁnite.

In the infinite temperature phase where all irreps are deconfined, the gluon plasma can screen exposed color charge, so it takes finite energy to separate an r-r pair to infinity. Pair production is 66 energetically costly, so Frr(∞) is less when r is deconfined than when r is confined and a static-light state exists.

If there are dynamical r and r quarks, a static-light rr state exists that allows pairs of r-r test charges to be separated to infinity for finite energy cost. In a theory with dynamical charges of multiple irreps (which includes single-rep theories where the fermions are not charged under the adjoint representation), we can also consider “chimera” static-light states constructed from charges of multiple irreps. For example, given dynamical F and F quarks, a static-light state exists for any higher irrep r: the color wavefunction of the static r charge has some number of appropriately symmetrized and antisymmetrized F and F indices, to each of which an F or F charge may attach to form a color-singlet state. Thus, in the presence of dynamical F quarks, Frr(∞) for any r is expected to be finite in the confined phase.

The existence of a static-light state is not guaranteed for an r static charge in a field with dynamical fermions charged under r1, r2, ... (none of which are r). For example, consider using a F -F pair to probe a theory with dynamical A2 or S2 fermions: for even Nc, it follows from simple index counting that it is impossible to construct a color singlet state with exactly one F and any number of two-index quarks. Without any static-light hadron state to pair-produce into, the infinite-separation energy FF F (∞) remains infinite in the presence of dynamical quarks. We thus obtain a “color selection rule” which blocks F -F pairs from being separated for Nc even with dynamical two-index quarks.

Static-light states need not be simple nor small to render Frr(∞) ﬁnite in the conﬁned phase.

Consider again using an F -F pair to probe a theory with dynamical A2 or S2 quarks. For Nc odd, a static-light state can always be constructed for either S2 or A2. For dynamical A2s, the state involves one tensor and (Nc − 1)/2 many A2 quarks to contract with the Nc − 1 indices of the

a bc not taken up by the F charge; in SU(3), this state is abcq Q , which happens to be the meson. For dynamical S2s, an S2 color state cannot have both indices contracted with the same tensor. Thus, the simplest static-light hadron state involves three tensors and (3Nc − 1)/2 many S2 quarks. In

a b1c1 b2c2 d1e1 d2e2 SU(3), this is q ab1b2 c1d1d2 c2e1e2 Q Q Q Q . Thus, the smallest static-light state of F 67 with dynamical S2s is larger than the state with dynamical A2s for any Nc ≥ 3.

4.3.2 Center symmetry

Because Polyakov loops are a diagnostic of conﬁnement, they allow us to relate conﬁnement physics to the spontaneous breaking of the global center symmetry [157]. Given pure-gauge SU(Nc) on the lattice, any topologically trivial Wilson loop is invariant under a (timelike) center transformation: multiply every time-direction link on some time slice by an element z = exp[2πin/Nc] of

the ZNc center of the gauge group SU(Nc), i.e.,

U(x,t),tˆ → zU(x,t),tˆ (4.27) for all x and some ﬁxed t. Any Wilson loop that passes through the center-transformed time slice acquires a factor of z. However, if the loop is topologically trivial, it must pass back through the time slice in the opposite direction, acquiring a factor of z∗. Because z and z∗ are pure phases, they commute through link matrices and cancel. Topologically nontrivial Polyakov loops wind around the time direction, and so transform nontrivially under center transformations as

PF → zPF

2 PA2 → z PA2 (4.28) 2 PS2 → z PS2

PG → PG. We can classify charge irreps by the behavior of their Polyakov loops under center transformations.

For a general representation r,

nr Pr → z Pr (4.29) where we have deﬁned nr, the n-ality or center charge of r. Comparing with 4.28, we read oﬀ

nF = 1, nA2 = 2, and nG = 0. Because z are Ncth roots of unity, the n-ality is deﬁned mod Nc

∗ such that nr = Nc = 0. Further, because Pr = (Pr) , we also have nr = −nr ≡ Nc − nr.

We can now relate conﬁnement physics to center symmetry breaking. Consider pure gauge

SU(Nc). At low temperatures, the F -irrep thermal Wilson line field Lx is disordered [157]. At 68 high temperatures, Lx orders towards zI, the identity multiplied by one of the center elements. In the disordered-twist phase the center transformations are exact symmetries of the ground state, and so for any r with nr 6= 0 we find hPri = 0 by symmetry. We thus obtain infinite Frr(∞), allowing us to identify the disordered-twist phase with the confined phase. Said differently, for any r with nr 6= 0, Pr is an exact order parameter for the spontaneous breaking of center symmetry and thus an exact order parameter for confinement of that irrep in pure gauge theory. Meanwhile, if nr = 0 then Pr is not protected by symmetry and generically nonzero, and so we find that Frr(∞) is finite in the confined phase. However, for any r, Pr → 0 as β → ∞ (i.e., T → 0), so at low temperature we expect that any finite Frr(∞) is large. As discussed above, we expect FGG(∞) to be finite in the confined phase because of the existence of static-light GG states; we can thus identify the non-protection of Pr with the availability of some static-light state involving a static r charge. Finally, in the ordered phase, center symmetry is spontaneously broken and all Polyakov loops acquire a (relatively large) finite expectation value. For any r, we thus find that Frr(∞) is

ﬁnite, so we may identify the ordered-twist phase with the deconﬁned phase.

Pure-gauge actions are constructed out of topologically trivial Wilson loops and so are invariant under center transformations. However, the addition of dynamical fermions to the theory can explicitly break center symmetry. On a lattice, each fermion determinant det Dr for quarks in the r representation can be expanded in the hopping parameter κr as

 ∞  X 1 det[D ] = exp − κj Tr Hj (4.30) r  j r r  j=1 where Hr is the hopping operator constructed out of r-irrep gauge links. This expansion can be interpreted as adding an inﬁnite number of terms to the action corresponding to all possible r- representation Wilson loops. This includes topologically non-trivial loops like the r Polyakov loop, which enters at lowest order in a term like

Nt X Nt X δL ∼ −(κr) (Pr + Pr) = −(κr) Re Pr. (4.31) x x

If nr 6= 0, then such terms are not invariant under center transformations and act as external magnetizations. Thus, the addition of dynamical r fermions results in Pr acquiring a (relatively 69 small) ﬁnite expectation value in the disordered-twist phase. Because Pr is not protected by symmetry, a static-light state involving a static r charge must exist.

In a theory with dynamical rs with nr 6= 1, the center symmetry is not necessarily broken entirely. For example, in SU(4) with A2 fermions, the center symmetry is broken from Z4 → Z2

because PA2 is invariant under the center transformations z = {1, −1}. The remnant Z2 symmetry is suﬃcient to protect PF . Because PF is protected by symmetry, there is no static-light state involving F charges, and we rediscover the color selection rule discussed above.

We can generally relate color selection rules with remnant center symmetries. The path integral in multirep theories involves only a simple product of the fermion determinants, so the hopping expansion will not induce any terms involving products of Polyakov loops of more than one irrep. Because each irrep of dynamical fermion breaks the center symmetry independently, diagnosing the existence of color selection rules reduces to a number-theoretical problem [47]: given

SU(Nc) gauge theory with dynamical fermions charged under irreps r1, r2, ..., the remnant center symmetry is Zp where

p = gcd(Nc, nr1 , nr2 , ...) (4.32) and gcd is the greatest common divisor. The remnant center symmetry will zero Pr for any r with nr not an integer multiple of p. Thus, for any r where nr is not an integer multiple of p, no static-light state involving an r charge exists.

There are only a limited number of independent Polyakov loops in any Nc. In SU(3) only,

∗ A2 = F , so PA2 = (PF ) . Thus, F and A2 are always conﬁned or deconﬁned together in SU(3).

In fact, in SU(3), PF contains all the physical information: there are two real degrees of freedom

(DOF) in the eigenvalue spectrum of L and two real DOF in the complex PF . This is no longer true for Nc > 3, where A2 6= F and L has Nc − 1 > 2 real DOF in its eigenvalue spectrum. For

n any Nc, all Polyakov loops may be expressed in terms of Tr[L ] for integer n ≤ Nc/2. Because

n Tr[L ] with different n are independent, floor[Nc/2] Polyakov loops are necessary to reproduce the eigenvalue spectrum of L. The lowest-lying floor[Nc/2] antisymmetric irreps An (where A1 = F ) are 70 independent and generically include a term like Tr[Ln], so they span a sufficient set. For example,

in SU(4), PF and PA2 are independent; PF has two real DOF and the real PA2 has one real DOF; thus, these two Polyakov loops contain enough information to reconstruct the eigenvalue spectrum of L and all of the physics of the thermal Wilson line ﬁeld.

4.3.3 Partial breaking of center symmetry

The confinement transition we observe in pure gauge theory and QCD corresponds to the spontaneous breaking of center symmetry. If the physics of the Lx field is governed by some three- dimensional magnetic effective field theory with fermion-induced potentials that encourage Polyakov loops to be nonzero, then the most plausible mechanism for separated confinement transitions is partial breaking of center symmetry. We can make this statement stronger by considering the relationship between center symmetry and confinement as discussed above. In the partial breaking

scenario, we imagine an additional intermediate phase(s) between the ZNc disordered phase and the Z1 completely-ordered phase, with symmetry Zp where p is some factor of Nc. In such a phase, charges of rep r will remain confined for any n-ality nR not an integer multiple of p. Charges of all other n-alities are deconfined. Physically, this is simply the statement that if some charge r deconfines that can form a static-light hadronic state with another charge r0, then r0 can be screened by the r charges in the plasma and must deconfine as well. If this final piece of physical reasoning holds, the equivalence of these statements implies that the only combinations of charge

irreps that are allowed to be conﬁned simultaneously are dictated by what subgroups of ZNc exist, and thus cascades of partial breakings of center symmetries are the only permissible conﬁnement phase structures.

Very speculatively, a partially-deconﬁned phase may have been sighted in SU(4) gauge theory with heavy adjoint fermions [122]. In appropriate regimes of parameter space in this theory, the center symmetry is spontaneously but incompletely broken from Z4 → Z2. This amounts to the

PF on a given conﬁguration clustering along the real or imaginary axis. In such a phase, hPF i is

still zeroed by the remnant Z2 symmetry, but hPGi acquires an expectation value (as would hPA2 i 71

and hPS2 i); in this case, we have Zp with p = 2, of which nA2 = nS2 = 2 is an integer multiple). Chapter 5

Multirep Thermodynamics on the Lattice

In this chapter, we investigate the phase structure of SU(4) gauge theory with the gauge ﬁeld simultaneously coupled to NF = 2 ﬂavors of fermion in the fundamental representation (F , quartet,

4) and NA2 = 2 ﬂavors of fermion in the two-index antisymmetric representation (A2, sextet, 6).

This first-of-its-kind lattice investigation of a multirep theory provides an arena to test the tumbling hypothesis (as discussed in Section 4.2.1)[132]. To review, the physical picture is that when a gauge coupling becomes sufficiently strong in the infrared, a scalar fermion bilinear condensate will form, breaking chiral symmetry. In a system with multiple representations of fermions, a weaker gauge coupling is needed to drive condensation for higher representation fermions, since their color charges are greater. Thus, different representations of fermion may condense at different scales.

However, as we will see in this chapter, our simulations find that the theory has only two phases: a low-temperature phase with both species of fermion confined and chirally broken, and a high-temperature phase with both species of fermion deconfined and chirally restored. Separating them is a single phase transition with characteristics of chiral restoration and deconfinement for both fermion species. This phase transition appears to be first order, in agreement with the analytic arguments laid out in Chapter6.

Throughout this chapter, we specialize to the case of SU(4) where dF = 4 and dA2 = 6. In what follows, a quantity labeled m4, P4, etc., corresponds to that quantity as measured for the fundamental fermions, while m6, P6, etc., correspond to the sextet fermions. We set the scale by

2 √ computing the gradient ﬂow scale t1/a , then identifying 1/ t1 ≡ 780 MeV to allow for intuitive 73 comparison with the familiar case of QCD (see Section 5.A). Other choices for scale setting would give diﬀerent physical units; for example, in the context of a composite Higgs model [79, 78], the value of the pseudoscalar decay constant in TeV would be most appropriate to determine the value of the critical temperature. Since the scale setting in the latter case is model dependent, we do not attempt it here.

5.1 Lattice-deformed Ferretti model

As discussed in Section 3.3.2, when treating fermions on the lattice, one wants to interpret the fermion determinant Det D as part of the probability weight for importance sampling. However,

Det D is not necessarily positive, and in order to use pseudofermions we must have some Det M with positive-definite eigenvalues. Although algorithms like Rational Hybrid Monte Carlo [44] can work around this issue to simulate odd numbers of flavors, in practice it is easiest to simply work with even numbers of degenerate flavors.

D To this end, rather than simulating Ferretti’s model with NF = 3 flavors of F -irrep fermion and N W = 5 flavors of A -irrep fermion, we round to the nearest easily-simulable theory. The result A2 2 is the “lattice-deformed Ferretti model”, with N D = 2 flavors of F and N D = 2 (N W = 4) flavors F A2 A2 of A2. This theory is an equally-good probe of the MAC hypothesis, which is insensitive to flavor number, as Ferretti’s model. However, finite-temperature properties like the order of the phase transition in QCD-like theories are known to depend strongly on flavor number. This calls into question how applicable our lattice results are if we are interested in the phase dynamics of Ferretti’s model as a candidate UV completion of the Standard Model. The analytic calculation presented in Chapter6 attempts to alleviate this issue by predicting the order of the phase transition of both

Ferretti’s model and our lattice deformation, ﬁnding that both should be ﬁrst-order. 74

5.2 Theoretical expectations

5.2.1 Tumbling & separation of chiral transitions

Plugging the fermion content of Ferretti’s model into the multirep β function Eq. 2.21, we

find that the first- and second-order coefficients of the beta function are negative. This indicates that this system is likely to be an ordinary confining and chirally broken QCD-like system at zero temperature. Our zero-temperature simulations confirm that this is the case [7, 11].

Because CA2 > CF , we expect the antisymmetric fermions to chirally condense ﬁrst if the tumbling hypothesis holds. In this case, we would like to know how far separated Tc for the chiral transitions should be. In Section 4.2.1, we derived a rough estimate Eq. 4.18 of the separation between the scales of chiral symmetry breaking for two irreps. We can appropriate this to estimate

(A2) (F ) (r) Tc /Tc ; with Tc ≡ µr, we obtain

2π TA2 (CA −CF ) = e b1 2 . (5.1) TF

Plugging in CA2 = 5/2, CF = 15/8, and b1 = 44/3 for SU(4) in the TF = 1/2 convention, we

(A2) (F ) obtain Tc /Tc ∼ 1.3. This number is small for an exponential separation, but very convenient

1 for lattice study. Examining Fig. 5.14 to get an idea for the variation of a ∼ 1/Tc with the bare parameters, we see that this would amount to a separation between the phase transition lines of

∆κ ∼ .001 − .01 in our simulations, which should be easily visible in the resolution we typically work at in bare parameter space.

5.2.2 Polyakov loops & center symmetry

In Section 4.3, we discussed the physics of Polyakov loops in theories with fermions in multiple representations. The relevant Polyakov loops for the lattice-deformed Ferretti model are

P4(x) = Tr Lx (5.2)

1 2 2 P6(x) = 2 (Tr Lx) − Tr(Lx) (5.3)

1 (A2) (F ) If the separation had been much larger such that Tc /Tc & L/a, we would have had to concern ourselves with cutoﬀ eﬀects. 75 where N Yt Lx = Utˆ(x, t). (5.4) t=1

Physically, P6 tells us whether A2 charges are conﬁned in the exact same way that P4 tells us whether F charges are conﬁned.

In SU(3) it is possible to write any higher-representation Polyakov loop in terms of the fundamental Polyakov loop and its complex conjugate. The behavior of higher-representation

Polyakov loops is thus completely determined if the fundamental Polyakov loop is known. In

SU(4), P4 and P6 are a suﬃcient set, so we do not risk missing any interesting dynamics by only measuring these two quantities.

A Polyakov loop is an order parameter if the fermion action preserves enough center symmetry to protect it in the unbroken phase. In an SU(4) gauge theory, A2 fermions break Z4 to Z2. The residual Z2 symmetry is not enough to protect P6, but P4 remains an order parameter when only A2 fermions are present [65]. Physically, this is due to the absence of a static-light hadronic state with only a single F charge but any number of A2s, as demonstrated in Section 4.3. Adding fundamental fermions breaks the Z4 center symmetry completely, so in the full theory neither P4 nor P6 is an order parameter.

We argue in Section 4.3.3 that separated conﬁnement transitions, if they exist, must occur in an order dictated by what is allowed by partial breaking of center symmetry. The center of SU(4) is Z4, whose only subgroup is Z2. Thus, other than the QCD-like case of a single transition where center symmetry is totally broken like Z4 → 1, the only possible phase structure is Z4 → Z2 → 1.

In a Z2-symmetric phase, P4 is still protected by symmetry so the F s are still conﬁned; meanwhile,

P6 is unaware of the residual Z2 and acquires a large expectation value, so the A2s are deconﬁned.

Physically, deconfined A2s do not induce deconfinement of single F quarks due to the color selection rule discussed in Section 4.3: no static-light color singlet may be constructed from a static F and any number of A2s. However, non-color-singlet F diquarks (i.e., states with two color indices) are deconfined [122]. Notably, this order (F confines first cooling from T = ∞) is the opposite order 76 Limit m4 m6 Order parameters

Quenched ∞ ∞ P4, P6 Chiral 0 0 hψ4ψ4i, hψ6ψ6i Chiral A2-only ∞ 0 hψ6ψ6i, P4 Chiral F -only 0 ∞ hψ4ψ4i A2-only ∞ m6 P4 F -only m4 ∞ - Multirep m4 m6 -

Table 5.1: Order parameters in various mass regimes. Note that e.g. m4 in the column m4 indicates that the F s have ﬁnite mass.

of what is predicted by the MAC hypothesis.

5.2.3 Order parameters & transition orders

It is useful to enumerate the various order parameters, and where they are exact. There are four limits to consider: each mass can be either infinity (quenched limit) or zero (chiral limit). In a limit where a representation is chiral, its chiral condensate ΨrΨr is an exact order parameter. In the fully quenched pure-gauge limit, both Polyakov loops are exact order parameters. Whenever m4 is finite, the Z4 symmetry is completely broken by dynamical F s and so neither P4 nor P6 are exact order parameters. Finally, as discussed in Section 4.3, the fundamental Polyakov loop P4 is an exact order parameter in the A2-only limit, even when the mass of the A2 fermions is finite.

Table 5.2.3 enumerates what order parameters are present in what regimes.

Fig. 5.1 is a rough sketch of a “Columbia plot” summarizing the theoretical predictions for the nature of the ﬁnite-temperature transition in the various fermion-mass regimes (with some inputs from our results discussed below). This sketch is made in analogy with the QCD Columbia plot, where the order of the phase transition encountered is plotted as a function of mu = md and ms.

In the pure-gauge limit, the transition is ﬁrst order [53]. The Pisarski-Wilczek stability analysis presented in Chapter6 predicts that the transition in the double chiral limit m4 = m6 = 0 will also be ﬁrst order. First-order transitions are generically robust against small perturbations, so these transitions presumably extend into the regions around m4 = m6 = ∞ and m4 = m6 = 0. 77

High-order Pisarski-Wilczek calculations [23, 24] indicate that the transitions in the massless limits of the fundamental-only and sextet-only theories can be second-order [(m4, m6) = (0, ∞) or (∞, 0)]. We thus assume that both of them will be, as reflected in Fig. 5.1. This is well-justified for the F -only limit, SU(4) with two flavors of fundamental fermion, which we expect to behave similarly to QCD with mu = md = 0 and ms = ∞. In this limit, QCD is believed to exhibit a second order phase transition with O(4) critical exponents (compare the discussion in Ref. [151]).

This guess is less justiﬁed in the A2-only limit, but previous studies of this system (and this one) indicate that its transition is at least not strongly ﬁrst-order [65].

In the single-species limits, nonzero fermion mass for the light species will convert a second- order chiral transition into a crossover. Adding heavy fermions of the other species can leave the second-order transition undisturbed or convert it to first order, as shown, but the phase transition cannot disappear as long as one species is exactly massless, as the condensate for the chiral species will remain an exact order parameter. If either single-species transition were first order, there would be a first order region in the corresponding corner. In the pure A2 theory there will be a true confinement transition for all values of m6 because P4 is an exact order parameter. Because we have already assumed that the transition in the chiral limit of the pure A2 theory is second-order, we extend this assumption to all finite m6; however, it could also be first order.

There are no analytical predictions for the intermediate region, where m4 and m6 are neither light nor heavy; we thus have no predictions for whether the ﬁrst order regions connect, or whether there is an intermediate continuous crossover region. 78

Fundamental only (m6=∞) Fund. only Pure chiral limit gauge Sextet only =0) 4 ( m

? ( m 4 = ∞ )

Double

Chiral fundamental chiral Sextet only (m4=m6=0) chiral limit

Chiral sextet (m6=0)

Figure 5.1: Columbia plot illustrating expectations for the order of the finite-temperature phase transition. The axes are the masses of the two fermion species in the theory, with m4 on the x-axis and m6 on the y-axis. The upper right corner is the pure-gauge limit; the lower left corner is the double chiral limit; the upper-left corner is the F -only chiral limit; the lower-right corner is the A2-only chiral limit. Green fields indicate regions of parameter space where the theory is predicted to exhibit a first order transition. Blue lines indicate regions of parameter space where the theory is predicted to exhibit a second-order transition. 79

5.3 Lattice details

5.3.1 Simulation details

For the fermions, we use a clover-improved Wilson action built from fat gauge links constructed by normalized-hypercubic (nHYP) smearing [87, 86]. We set the clover coeﬃcients for both representations equal to unity, cSW = 1, a choice motivated by results from Ref. [147]. We construct the action for the sextet fermions by constructing A2-irrep smeared links from F -irrep smeared links as described in Section 3.3.2[65]. The action for the gauge sector is the usual plaquette action with gauge coupling β augmented by an nHYP-dislocation suppression (NDS) term [70], constructed from the nHYP-smeared links. Simulating with nHYP-smeared links is known to cause occasional large “dislocations” which can make it diﬃcult to simulate in certain parameter regimes.

The NDS action is designed to suppress these events, opening up larger regions of parameter space to simulation. We ﬁx the NDS parameter γ such that β/γ = 125, leaving β the only free parameter in the gauge sector. Altogether, the simulation parameter space is three dimensional: β, and two hopping parameters κ4 and κ6.

We use HMC to generate gauge conﬁgurations and discard at least the ﬁrst 100 trajectories of each ensemble for equilibration. As we will discuss below, we have observed a large degree of metastability in equilibrating ensembles near the transition line, where an ensemble will sit in one phase before eventually tunneling to the correct one. In some cases, this requires us to discard

> 1000 trajectories for equilibration.

We perform spectroscopy by ﬁtting two-point correlation functions as described in Section 3.5.

However, our lattices are infeasibly short (Nt = 6, 8) to extract useful results from temporal- direction correlators. We instead use two-point correlation functions extending in a spatial lattice direction. The meson screening masses in the scalar, pseudoscalar, vector, and pseudovector channels extracted from the resulting correlators are suﬃcient for our purposes. We compute spectroscopic observables using valence compound boundary conditions (also known as the “periodic plus antiperiodic” or “P+A” trick) to eﬀectively double the spatial extent of our lattices [36,3, 66]. 80

Fermion masses m4 and m6 for the two representations are defined through the axial Ward identity (AWI), as discussed in Section 3.5.3. The axial Ward identity is a statement of current conservation and is thus local and insensitive to finite volume effects as long as we stay in a confined phase, so we use the AWI quark mass computed from spatial-direction correlators on

ﬁnite-temperature lattices to estimate its zero-temperature value.

To determine whether chiral symmetry is broken for an irrep, we use parity doubling as described in Section 4.2.2. To this end, we measure the masses of the parity-partner scalar and pseudoscalar meson states, and the vector and pseudovector meson states.

To diagnose the confinement transition, we examine the Polyakov loops P4 and P6. We also look at several flow-based diagnostics described in AppendixC: the flowed anisotropy as described in Sec. C.1, and the Polyakov loops P4 and P6 at long flow times as described in Sec. C.2. All of our phase diagnostics (unflowed Polyakov loops, flowed anisotropy, Polyakov loops at long flow time) agree everywhere in our data set, within our resolution in coupling space. The flowed anisotropy and flowed Polyakov loops may be used to determine the phase of an ensemble without comparing it with nearby ensembles or picking some arbitrary threshold value, as is required when using unflowed Polyakov loops. Such ensemble-local observables are better suited for automation.

5.3.2 Data sets

In order to search for the various possible phase transitions, we explored a wide region of the three-dimensional bare parameter space. This required an unusually large and heterogeneous data set, summarized in Table 5.2. To render this exploration tractable, we found it necessary to automate much of our data generation and analysis. Section 3.5 describes some of our automated analysis methods; Ref. [16] provides further discussion of some of the automated data generation techniques we used in this study, and Ref. [63] describes a more developed version of the same used in another study.

For the full theory with both ﬂavors dynamical we focused predominantly on β = 7.4 and

β = 7.75, mostly on 123 ×6 and 163 ×8 volumes but with some additional data on 183 ×6 and 243 ×8 81 Theory Volume Subset Ensembles fundamental-only 123 × 6 121 sextet-only 123 × 6 239 full theory 123 × 6 β = 7.4 128 β = 7.75 135 All 409 163 × 8 β = 7.4 22 β = 7.75 35 All 57 183 × 6 49 243 × 8 26

Table 5.2: Summary of ﬁnite-temperature ensembles.

to check for ﬁnite-volume eﬀects. For the single representation theories, we ran only on 123 × 6.

We also made use of zero-temperature data to determine the lattice scale, the fermion masses, and the pseudoscalar-to-vector mass ratio for some bare couplings near the phase transition. Table 5.3 summarizes these zero-temperature data sets.

2 √ √ Ns Nt β κ4 κ6 t1/a MP 4/MV 4 MP 6/MV 6 t1m4 t1m6 16 18 9.4 0.123 - 3.27(2) 0.85(1) - 0.321(2) - 16 18 9.2 0.126 - 2.460(8) 0.742(8) - 0.171(3) - 16 18 9.0 0.13 - 2.10(1) 0.39(2) - 0.027(6) - 16 18 9.2 - 0.115 3.40(3) - 0.940(9) - 0.999(5) 16 18 9.0 - 0.1205 3.13(2) - 0.896(4) - 0.587(3) 16 18 8.8 - 0.124 2.63(2) - 0.842(8) - 0.381(2) 16 18 8.6 - 0.127 2.20(2) - 0.79(1) - 0.250(3) 16 32 7.4 0.131 0.13 1.171(3) 0.765(5) 0.824(3) 0.140(1) 0.240(1) 16 18 7.4 0.132 0.131 2.13(2) 0.60(3) 0.78(1) 0.073(2) 0.216(3) 12 24 7.4 0.1285 0.13175 1.239(8) 0.83(1) 0.79(1) 0.229(3) 0.219(3) 16 18 7.75 0.1295 0.126 1.53(1) 0.74(2) 0.87(1) 0.161(3) 0.398(4) 12 24 7.75 0.128 0.128 1.87(2) 0.80(1) 0.82(1) 0.203(3) 0.304(3) 16 18 7.75 0.124 0.129 1.455(8) 0.888(8) 0.82(1) 0.374(3) 0.271(3) 12 24 7.75 0.129 0.129 2.91(6) 0.74(2) 0.79(2) 0.145(4) 0.249(5) 16 32 7.75 0.13 0.1295 4.19(4) 0.61(2) 0.787(7) 0.0749(8) 0.212(1) 12 24 7.75 0.127 0.1305 3.28(4) 0.82(2) 0.73(2) 0.248(3) 0.169(3)

Table 5.3: Zero-temperature data sets used to compute the scale near the thermal transition. 82

5.4 Results: Phase structure of limiting-case theories

5.4.1 A2-only limit

5.4.1.1 Phase structure

We first consider the gauge theory coupled only to A2 fermions. Figure 5.2 shows the behavior of our various phase diagnostics along a typical slice through bare parameter space. The top panel shows the behavior of the fundamental and sextet Polyakov loops P4 and P6. Physically, P6 is the quantity of interest as it measures whether A2 fermions are confined. However, as discussed above, the F -irrep Polyakov loop P4 is an exact order parameter for the spontaneous breaking of the residual Z2 center symmetry [65] present in the A2-only theory. We also include the flowed anisotropy observable RE, which we expect to be sensitive to any change in phase. We see that all three quantities jump simultaneously and vary only smoothly elsewhere: there is only a single confinement transition in this theory. The middle panel shows the mass splittings of the sextet mesons. The parity partners become degenerate simultaneously, indicating the restoration of chiral symmetry. Within our resolution in κ6, the confinement transition coincides with the chiral transition.

The behavior shown on the slice in Fig. 5.2 is typical for this theory: everywhere we have looked in parameter space, we see a single, uniﬁed conﬁnement and chiral transition, as in QCD.

Figure 5.3 summarizes our ﬁndings for the phase diagram for the A2-only theory in the β-κ6 plane.

For the four points marked by circles in Fig. 5.3, we ran zero-temperature simulations at the same bare couplings in order to determine the lattice scale (see Table 5.3). As we describe in

Section 5.A, we set the lattice scale in each zero-temperature ensemble through calculation of the

2 √ ﬂow scale t1/a . Choosing the ﬁducial value 1/ t1 ≡ 780 MeV gives a physical value to the lattice

−1 spacing a in each ensemble, and hence to the temperature T = (Nta) . As can be seen in Fig. 5.3, one of the ensembles is a blue point on the conﬁned side of the transition, while the other three are orange points on the deconﬁned side. These provide lower and upper bounds, respectively, on Tc at the corresponding m6 values. We plot these temperatures in Fig. 5.4. The transition temperature 83

1.00

0.75

0.50 |P4| 0.25 Normalized |P6| 0.00 RE(1) 1

1.00

a(MS6 MP6) 0.75 a(MA6 MV6) 0.50

0.25

0.00

0.6 1.75

0.4

6 am 6 1.70 m plaq a 0.2 plaq

0.0 1.65

0.124 0.126 0.128 0.130 0.132 0.134

3 Figure 5.2: Dependence on κ6 of various quantities in the A2-only theory for β = 8.5 on 12 ×6. The top panel shows diagnostics of confinement: unflowed Polyakov loops and the flowed anisotropy at t/a2 = 1. (Quantities are normalized by their maximum values along the slice for ease of comparison of their qualitative behavior.) The middle panel shows chiral diagnostics, the mass splittings of parity-partner mesons. The bottom panel shows the plaquette and the AWI fermion mass. Points with closed (open) circles are deemed confined (deconfined) according to the behavior of Polyakov loops at long flow time. There is a single transition (gray band) from the confined and chirally broken phase to the deconfined and chirally restored phase. 84

0.1350 Confined & Broken Deconfined & Restored

0.1325 m6 < 0

0.1300

0.1275

6 0.1250

0.1225

0.1200

0.1175

0.1150

8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00

3 Figure 5.3: Phase diagram for the A2-only limit for 12 × 6 lattices. Blue dots are confined and chirally broken ensembles while yellow stars and red Xs indicate deconfined and chirally symmetric ensembles. The blue field thus indicates the confined and chirally broken region of parameter space, while the orange field indicates the deconfined and chirally restored region of parameter space. The red Xs mark deconfined ensembles where m6 < 0. The black box indicates the slice through bare parameter space shown in Fig. 5.2. The circled ensembles have matching zero-temperature ensembles.

curve must pass below the upper bounds (downward arrows) and above the lower bound (upward arrow). We can compare the transition temperatures seen here to those in more familiar theories: the transition in the pure SU(3) gauge theory occurs near 280 MeV [112], while the crossover in

QCD at physical quark masses occurs near 150 MeV [151].

The order of the transition seen in Fig. 5.2 is not obvious. Without additional volumes to perform a volume scaling analysis, we can make no claim concerning the order of the transition for any value of m6. 85 275

250

225 ) V e

M 200 (

c T 175

150

125 0 100 200 300 400 500 600 700 800

m6 (MeV)

Figure 5.4: Transition temperature Tc for the A2-only theory with Nt = 6 as a function of quark mass. The three heaviest ensembles (down arrows) are deconﬁned-side ensembles and thus give upper bounds on Tc. The lightest ensemble (up arrow) is a conﬁned-side ensemble and thus gives 2 a lower bound on Tc. The scale has been set via t1 ≡ 1/(780 MeV) , corresponding to the scale of QCD, for easy comparison.

5.4.1.2 Eﬀects of the NDS action

A previous study examined the A2-only theory [65], but without the NDS term in the gauge action. This study found that the system exhibited a bulk transition. Bulk transitions are a feature of the lattice theory and induced by dynamics on the lattice scale (versus the thermal transition, which because T ∼ 1/Lt is a finite-volume effect). These phases, an ubiquitous inconvenience in lattice studies of thermodynamics, stay approximately fixed in bare parameter space as Lt varies.

They thus do not survive the continuum limit and are irrelevant to our physics of interest.

The previous study found that the plaquette showed a large discontinuity at a value of κ6 value distinct from (and below that of) the value of κ6 where the thermal transition occurred.

This large discontinuity is absent in our data. In Fig. 5.2, the plaquette shows structure occurring simultaneously with the response of the Polyakov loops, but varies smoothly otherwise.

We were interested in locating the bulk transition in nearby parameter space, both to avoid it and to potentially determine how far the NDS term had shifted it by. We also wanted to ensure 86 that the transition we were observing was, in fact, a thermal transition. To this end, we ran a grid of 44 ensembles covering the region of bare parameter space covered by our 123 × 6 data. We found that the ﬁnite-temperature transition shifts substantially when changing Nt, thus conﬁrming that it is not a softened bulk transition. The NDS term appears to have completely banished the bulk transition, at least from the region of bare parameter space that we have explored.

5.4.2 F -only limit

The other limiting case of our model contains only fundamental fermions and no A2s. Fig- ure 5.5 shows the behavior of our observables along a typical slice through bare parameter space, varying κ4 over the transition at fixed β = 9.2. The top panel shows our confinement diagnostics, the fundamental Polyakov loop and the flowed anisotropy. The A2-irrep Polyakov loop P6 is not physically interesting for this theory. The two quantities change simultaneously and only once, varying smoothly otherwise: there is only a single crossover in each observable. The middle panel shows our chiral diagnostics, the mass splittings of parity-partner mesons. The splittings smoothly go to zero beyond κ4 = 0.127, indicating chiral restoration. Comparing the top and middle panels, we see that the chiral and confinement crossovers overlap. In the bottom panel, we see that the quark masses and plaquette vary smoothly, as expected for a crossover.

The behavior observed on the slice in Fig. 5.5 is typical for this theory: everywhere we have investigated, we observe only a single unified chiral and confinement crossover. Figure 5.6 summarizes our findings for the β-κ4 phase diagram for the fundamental-only theory. As we did for the A2-only theory, we have determined the physical temperature at three points along the transition, this time choosing three points inside the crossover region (points enclosed by circles in Fig. 5.6). See Table 5.3 for the zero-temperature data involved. We plot Tc versus the fermion

2 mass m4 in Fig. 5.7.

2 Note that our rough estimate of Tc(m4) does not correspond to the peak of any susceptibility in the crossover region. 87

1.00

0.75

0.50

Normalized 0.25 |P4|

RE(1) 1 0.00

a(M M ) 0.6 S4 P4 a(MA4 MV4) 0.4

0.2

0.0

0.3 1.94 1.93

0.2 am4

m 1.92 a plaq plaq 0.1 1.91

1.90 0.0

0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129

Figure 5.5: Dependence on κ4 of various quantities in the F -only theory along a slice of constant β = 9.2 on 123 × 6. The top panel shows diagnostics of confinement: the unflowed fundamental Polyakov loop and the flowed anisotropy at t/a2 = 1. The middle panel shows diagnostics of chiral condensation, the mass splittings of parity-partner mesons. The bottom panel shows the plaquette and AWI fermion mass. The peaks of the Polyakov loop and chiral susceptibilities lie somewhere in the gray band. 88

0.134 Confined & Broken Deconfined & Restored 0.132 Ambiguous

m4 < 0

0.130

0.128 4

0.126

0.124

0.122

0.120 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00

Figure 5.6: Phase diagram for the F -only limit on 123 × 6 lattices. Symbols and colors are as in Fig. 5.3, with the addition of hollow diamonds indicating ensembles in the crossover region, where the diagnostics are ambiguous. The black box indicates the slice through bare parameter space shown in Fig. 5.5. The circled ensembles have matching zero-temperature ensembles available.

240

220 ) V e

M 200 (

c T 180

160

0 50 100 150 200 250

m4 (MeV)

Figure 5.7: Temperature for the crossover of the fundamental-only theory with Nt = 6 as a 2 function of the AWI mass. The scale has been set via t1 ≡ 1/(780 MeV) , corresponding to the scale of QCD, for easy comparison. 89

5.5 Result: Phase structure of full theory

5.5.1 Simultaneous transitions

Figure 5.8 shows a slice through bare parameter space in the full theory with both fermion species dynamical, varying κ6 while holding β = 7.4 and κ4 = 0.1285 ﬁxed. The top panel shows the behavior of our conﬁnement diagnostics. The Polyakov loops for both representations and the

ﬂowed anisotropy all jump simultaneously, implying that the two species conﬁne simultaneously.

The middle panel shows the behavior of the chiral diagnostics for both species, the mass splittings of parity partner states. The parity partners of both representations are split significantly at small κ6, but simultaneously become nearly degenerate as κ6 is increased. This indicates that chiral symmetry restoration occurs simultaneously for the two representations. Comparing the top and middle panels, we see that the combined confinement transition and the combined chiral transition coincide. Away from the single jump, all quantities vary smoothly. This behavior is typical throughout the region of bare parameter space that we have investigated: all phase diagnostics jump simultaneously and only once. Thus, we find only two phases: a low-temperature phase where all fermions are confined and chirally broken and a high-temperature phase where all fermions are deconfined and chirally symmetric.

Figures 5.9(a) and 5.9(b) show phase diagrams for the theory at β = 7.75 and β = 7.4 for Nt = 6 and Nt = 8. In these plots, conﬁned and chirally broken regions of parameter space are highlighted in blue while deconﬁned and chirally restored regions of the parameter space are highlighted in orange. The transition thus lies somewhere in the white band in each phase diagram.

Points enclosed by diamonds are where the phase changes when varying Ns from 12 to 18 when

Nt = 6 and from 16 to 24 when Nt = 8. The absence of many such points indicates that the location of the transition is insensitive to ﬁnite-volume eﬀects. Points enclosed by circles have matching zero-temperature data available (see Table 5.3). Comparing the left and right panels in Figs. 5.9(a) and 5.9(b), we see that the transition moves substantially in bare parameter space as we vary Nt.

This behavior is consistent with a thermal transition, and inconsistent with a bulk transition. 90

1.0

0.5 |P4|

Normalized |P6|

RE(1) 1 0.0

a(M M ) 0.6 S4 P4 a(MS6 MP6)

0.4 a(MA4 MV4)

a(MA6 MV6) 0.2

0.0

0.25 1.47 0.20 1.46 am 0.15 4

m am6 1.45 a 0.10 plaq plaq 1.44 0.05 0.00 1.43

0.1314 0.1315 0.1316 0.1317 0.1318 0.1319 0.1320 0.1321 0.1322

Figure 5.8: Behavior of various quantities in the full theory, varying κ6 across the transition while 3 holding β = 7.4 and κ4 = 0.1285 constant on 12 × 6. The gray band brackets the transition. Points with closed (open) circles are confined (deconfined) according to the behavior of Polyakov loops at long flow time. Top: Unflowed Polyakov loops for both representations and the flowed anisotropy. All diagnostics of confinement show simultaneous discontinuities. Middle: Mass splittings of parity partner mesons: scalar vs. pseudoscalar, and vector vs. axial vector. Chiral symmetry restoration occurs simultaneously for the two representations. Bottom: AWI fermion masses for both representations, and the plaquette. All quantities jump discontinuously at the transition. 91

Nt = 6 Nt = 8 0.134 0.134

0.132 0.132

0.130 0.130

6 0.128 0.128 6

0.126 0.126

0.124 Confined & Broken 0.124 Deconfined & Restored

Either mr < 0 0.122 0.122 0.122 0.124 0.126 0.128 0.130 0.132 0.122 0.124 0.126 0.128 0.130 0.132

4 4

(a) Phase diagram for β = 7.75.

Nt = 6 Nt = 8

0.135 0.135

0.134 0.134

0.133 0.133

0.132 0.132 6 6 0.131 0.131

0.130 0.130

0.129 0.129 Confined & Broken 0.128 Deconfined & Restored 0.128

Either mr < 0 0.127 0.127 0.128 0.130 0.132 0.134 0.128 0.130 0.132 0.134

4 4

(b) Phase diagram for β = 7.4.

Figure 5.9: To the left is the phase diagram for Nt = 6 lattices, while to the right is the same region of bare parameter space for Nt = 8 lattices. Blue dots indicate confined and chirally broken ensembles. Yellow stars indicate deconfined and chirally restored ensembles with mr > 0 for both species; red Xs are in regions where m4 < 0 or m6 < 0 or both. The violet dot is a confined and chirally broken ensemble with m4 < 0. The black box indicates the slice through bare parameter space shown in Fig. 5.8. In the right figure, the transition region from Nt = 6 is overlaid in gray, demonstrating that the transition moves as Nt is varied. The circled ensembles have matching zero- temperature ensembles available. Ensembles enclosed by diamonds are where the phase changed when volume was changed (see text). 92

5.5.2 First-order phase transition

The observed transition appears to be strongly ﬁrst order. As shown in the bottom panel of

Fig. 5.8, the plaquette and quark masses show a discontinuous jump at the transition, providing strong evidence that the observed transition is first order. Further, we observe strong metastability in HMC time, characteristic of a first-order transition (an effect related to hysteresis). Specifically, we have sighted several tunneling events in HMC time. Figure 5.10 depicts one such event: we equilibrate a lattice at deconfined bare parameters close to transition, seeding the run with a configuration from a confined ensemble nearby on the other side of the transition. Observables begin with typical “confined” values, and stick at them until trajectory ∼ 1200, when they rapidly tunnel to typical deconfined values (highlighted with a gray band). This behavior extends to spectroscopic observables: when measured before thmc ∼ 1200, both the vector and scalar mesons are split before the tunneling event, and nearly degenerate afterwards. That this tunneling event

3 occurs at tMC ∼ 1200 is very signiﬁcant metastability: typical equilibration times for 12 ×6 lattices in this study are O(100) trajectories or shorter.

Figure 5.10: Sighting of a tunneling event in the 3 equilibration of a 12 ×6 ensemble at (β, κ4, κ6) = (7.75, 0.127, 0.128). The gray band highlights the region of the tunneling event. The top panel shows the plaquette expectation value and the bottom panel shows the real and imaginary parts of the fundamental Polyakov loop expectation value. As seen from the behavior of the Polyakov loop in the bottom panel, the ensemble tunnels from the conﬁned phase to the deconﬁned phase. 93

5.5.3 Critical temperature

We are interested in determining whether the transition temperature Tc is comparable to its value in QCD, how strongly Tc depends on fermionic effects, and how significant are lattice spacing artifacts in Tc. In the full theory, Tc is a function of the fermion masses and 1/Nt. We do not have sufficient data to constrain the location of the transition with any of m4, m6, or a held

ﬁxed. Instead, we examine the behavior of Tc as we interpolate along the transition at ﬁxed β and

1/Nt. For simplicity, we use κ4 to parameterize each transition curve, along which all of m4, m6,

2 and Tc vary. We have estimated the lattice spacing and thus Tc using the ﬁt to t1/a described in

Appendix 5.A.

Tracing along the transition bands in Figs. 5.9(a) and 5.9(b), wherever there are ensembles

2 matched in κ4 on the edges of the transition band, we use our t1/a ﬁt to estimate upper and lower bounds for Tc. The resulting bounds on Tc as a function of κ4 are the horizontal black dashes with error bands in Figs. 5.11(a) and 5.11(b). As indicated by the spanning arrows, the transition temperature must lie between these bounds. Note that there is an uncontrolled systematic error for these bounds: the phases of some ensembles near the transition edge may be misdiagnosed if they have not been equilibrated long enough to tunnel to the correct phase.

As in the A2-only and F -only theories, Tc is comparable to its value in QCD (150 MeV) and SU(3) pure gauge theory (280 MeV). For both Nt = 6 curves, Tc may not remain constant as we vary κ4 to interpolate along the transition. We cannot exclude that the dependence may be a lattice artifact, but as one might expect from fermionic inﬂuence on the transition, Tc appears to depend more strongly on κ4 at β = 7.75 than at β = 7.4. Comparing with the κc curves of Fig. 19 of Ref. [7], we see that at β = 7.4 the transition curve is roughly parallel to κc and thus traces lines of approximately constant quark mass for whichever species is lighter; this slow variation in the masses is consistent with the observed slow variation in Tc. Meanwhile, at β = 7.75 the transition curve moves further away from κc when κ4 ≈ κ6, leading to heavier fermions; when the fermions are heavier, the system becomes more pure-gauge-like and Tc increases. 94

Nt = 6 Nt = 8

190

180

170 )

V 160 e M (

c 150 T

140

130

120

0.129 0.130 0.131 0.132 0.133 0.134 0.129 0.130 0.131 0.132 0.133 0.134 4 4

(a) Tc at β = 7.4.

Nt = 6 Nt = 8

190

180

170 )

V 160 e M (

c 150 T

140

130

120

0.120 0.122 0.124 0.126 0.128 0.130 0.132 0.120 0.122 0.124 0.126 0.128 0.130 0.132 4 4

(b) Tc at β = 7.75.

Figure 5.11: Transition temperature of the multirep theory as a function of κ4 on Nt = 6 and 2 Nt = 8. Lattice spacings used to determine the temperature are computed using the ﬁt to t1/a 2 with t1 = 1/(780 MeV) (for easy comparison to QCD) as discussed in Appendix 5.A. Axes are matched between Nt = 6 and Nt = 8. Black lines with error bands indicate the temperature on ensembles on either end of the transition bands in Fig. 5.9(b). The transition temperature lies in the span indicated by the double-headed arrows. 95

5.6 Discussion

We summarize our results for the phase structure of our theory in two Columbia plots in

Fig. 5.12. The axes in Fig. 5.12(a) are plotted in lattice units, while the axes in Fig. 5.12(b) are plotted in physical units, as deﬁned in Appendix 5.A. Each plot has two kinds of symbols. Open

2 symbols show quark masses from matching zero-temperature simulations: am4, am6, and t1/a are all taken from zero-temperature ensembles run at bare parameters matched to finite-temperature ensembles near the transition. Closed symbols show the quark masses from finite-temperature simulations: am4 and am6 are measured on finite-temperature ensembles along the deconfined side

2 of the transition curve, and t1/a is obtained from the fit described in Appendix 5.A. If there were no lattice artifacts, the open and closed symbols for each (Nt, β) set would coincide. The solid lines in Fig. 5.12(b), which come from the interpolation, lie close to the open symbols, which were measured directly and did not require a fitting function. The curves in Fig. 5.12 where m4 6= ∞ and m6 6= ∞ indicate the masses at which we have examined the phase structure of the full theory in detail and found only a single first-order thermal transition. The points where m4 = ∞ indicate where we have examined the sextet-only theory and found only a single transition. The points where m6 = ∞ indicate where we have examined the fundamental-only theory and found only a single crossover or transition.

5.6.1 Phase structure

Our numerical investigation of the full multirep theory ﬁnds only a single, ﬁrst-order thermal transition. This non-observation of separated chiral phase transitions is in direct contradiction to predictions of the Most Attractive Channel hypothesis as laid out in Section 4.2.1, according to which the sextet fermions should condense before the fundamentals as the temperature is lowered.

This is also potentially in conﬂict with the results of some quenched simulations from the early 80’s

(performed on small lattices with large gauge couplings) [103, 102, 101, 99] which observed Casimir scaling in the potential. However, quenching neglects the back-reaction of the fermions on the 96 gauge dynamics. While our exploration of the three-dimensional parameter space is by no means exhaustive, and separated phase transitions might exist for some values of fermion mass, we have examined the theory at masses ranging from 50 to 400 MeV and ruled them out in this domain.

We similarly ﬁnd that the fundamental-only and sextet-only theories appear to be QCD-like, with a combined chiral and conﬁnement transition.

It is worth asking why we did not see a phase transition. There are two obvious possible explanations. One is that, comparing Sections 5.2.1 and 5.2.2, we find two conflicting predictions: the MAC hypothesis predicts that A2 fermions will confine before F s. Meanwhile, partial breaking of center symmetry only permits F -irrep fermions to confine first if the phase transitions are separated. It may be that this incompatibility does not permit an intermediate phase in this theory.

The second possibility is that we have been naive in identifying the scale in the tumbling scenario with the critical temperature. In a deconfined phase, the universe is full of a hot plasma of color-charged fermions and gluons. In such a plasma, deconfined fermionic charges screen the attraction between any two charges. In the tumbling scenario, we imagine that a chiral condensate forms once the attraction between two species of fermion hits a critical level. Screening by fermions reduces this attraction and thus lowers the critical temperature at which confinement occurs, as we

(PG) (QCD) see comparing pure gauge theory with Tc ∼ 280 MeV and QCD with Tc ∼ 150 MeV. If one species of fermion confines and drops out of the plasma, they will no longer contribute to screening, and so the attraction between the remaining deconfined species will suddenly jump. If any of these species were already close to confining themselves, this sudden loss of screening will cause them to confine as well. To get an idea for the typical size of screening effects, note that removing NF = 2

(PG) (QCD) ﬂavors of F charge from QCD will increase its critical temperature by Tc /Tc ∼ 1.9. We estimated in Section 5.2.1 that the critical temperatures of the F s and A2s are only separated by a factor of ∼ 1.3, comparable to what we expect from the loss of screening.

We observe no bulk transitions in the full theory or in either of its single-species limits.

Comparison to a previous study of the sextet-only theory [65] found that the addition of the NDS 97 term to the action moved the bulk transition in that theory out of the region of interest.

5.6.2 Transition order

In the multirep theory we ﬁnd a strongly ﬁrst-order phase transition. This is consistent with the stability analysis described in Chapter6 appropriate to the limit where m4 = m6 = 0 [84].

The A2-only theory has an order parameter which characterizes its phase, and so it presumably possesses a real phase transition line. We could not determine its order. The F -only theory has no order parameter for conﬁnement and appears to exhibit crossover behavior, as expected from its close resemblance to NF = 2 QCD.

Increasing the quark masses in the single-species theories, we expect eventually to run into the

ﬁrst-order region of the pure-gauge transition. Our exploration of these theories stops short of the quark masses at which this occurs. Similarly, because we have only explored relatively light quark masses, we are unable to determine whether the ﬁrst-order regions surrounding the double-chiral limit and the pure-gauge limit are connected.

The ﬁnite-temperature properties of this model may have important implications for cosmol- ogy if it or something like it happens to be realized in nature as part of a composite Higgs model

[78], in which case it would correspond to a real cosmological phase transition around the TeV scale. First order transitions in the early universe give rise to gravitational waves with distinctive properties [144, 40]. These signals may be accessible to near-future gravitational wave detectors such as LISA [40]. Furthermore, such a transition could produce macroscopic nuggets of quark matter (or the dark-sector analog thereof), a candidate dark matter particle [20, 19]. 98

Nt = 6 = 7.4

Nt = 6 = 7.75

Nt = 8 = 7.4

Nt = 8 = 7.75 0.5 Nt = 6 Fundamentalonly

Nt = 6 Sextetonly

0.4 6 m a

0.3

0.2

0.1

0.0 0.0 0.1 0.2 0.3 0.4 0.5 am4

(a) Columbia plot with quark masses in lattice units

Nt = 6 = 7.4

Nt = 6 = 7.75

Nt = 8 = 7.4

Nt = 8 = 7.75

Nt = 6 Fundamentalonly 400 Nt = 6 Sextetonly ) V e M (

6 300 m

200

100

0 0 100 200 300 400

m4 (MeV) (b) Columbia plot with quark masses in MeV

Figure 5.12: Columbia plots, by analogy with QCD. In Fig. 5.12(a), the quark masses are in lattice 2 units. In Fig. 5.12(b), quark masses are in MeV [t1 ≡ 1/(780 MeV) ], for easy comparison to QCD, as defined in Appendix 5.A. Each color and symbol is associated with a different β and Nt. Closed symbols are finite-temperature quark masses, while hollow symbols are zero-temperature quark masses from ensembles on the transition boundary The lattice spacings for the zero temperature 2 quark masses are computed directly from t1/a on that ensemble. The lattice spacings for the finite-temperature quark masses are computed from the model described in Appendix 5.A. 99

5.A Scale setting

5.A.1 Scales for SU(4)

In order to estimate the transition temperature, we need the lattice spacing. To this end, we generated zero-temperature ensembles at points in bare parameter space along the transition surface and measured the Wilson ﬂow scale t1 [152].

Assuming that our Nc scaling has produced an equivalent quantity in SU(4), we may take the

2 value t0 = (0.142 fm) from SU(3) [25]. Our data set, however, samples the transition at relatively √ large lattice spacings a & t0. In this regime, t0 develops nonlinear lattice-spacing artifacts; in practice, this is the failure of ht2E(t)i to reach the linear regime before it exceeds 0.4, and so we

2 are sampling the “knee” in the typical ht Ei trajectory. Because t1 is larger, it can be measured on larger lattice spacings than t0. In order to use t1 in lieu of t0, we need a conversion factor. p In Fig. 5.13 we examine the behavior of the ratio of t1/t0 as a function of the lattice spacing √ √ over our entire zero-temperature data set [7]. The quantities t1 and t0 are fixed lengths, so their ratio should be some constant independent of lattice spacing. We see, however, that at large lattice √ spacing the ratio is not constant. Making the cut a < 1/ 2t1 (the dashed line in Fig. 5.13), we p √ √ find t1/t0 ' 1.77, and so t1 ' 0.252 fm or equivalently 1/ t1 ' 783 MeV. For simplicity √ we take 1/ t1 ≡ 780 MeV for our fiducial value. Determining the physical temperature of any

2 given ensemble merely amounts to measuring the dimensionless quantity t1/a in order to calculate √ −1 −1 p 2 T = (Nta) = (Nt t1) × t1/a .

5.A.2 Fitting the lattice spacing

We have only a few zero-temperature ensembles in the region of bare parameters relevant to

2 the transition. In order to perform a more detailed temperature analysis, we modeled t1/a as a

2 function of the bare parameters. We do not have any theoretical expectations for the form of t1/a , so this modeling is entirely empirical.

2 To include in our ﬁt, we have 39 zero-temperature ensembles where t1/a > 1 to avoid 100

1.9

1.8 Figure 5.13: Ratio of the Wilson ﬂow scales 1.7 √ √ t1 and t0 as a function of lattice spac- 0 t / 1

t ing, plotted against lattice spacing measured 1.6 √ in units of t1. Above a ∼ 0.8, large lattice

1.5 spacing eﬀects begin to contaminate t0.

1.4

0.4 0.6 0.8 1.0 1.2

a/ t1

discretization eﬀects; of these, 12 ensembles are at β = 7.75. Lattice spacings computed using the

Wilson ﬂow are easily determined to very high statistical precision. Without knowing the true form

2 of t1/a as a function of β, κ4, and κ6, and with our limited dataset, we are unable to produce a

2 2 2 model that can ﬁt t1/a with convincingly small χ . We are thus led to inﬂate the errors on t1/a

2 to 2.5% of the value. Such a model predicts the value of t1/a to within 2.5% for any (β, κ4, κ6) in the region of interest.

2 The form of our model is motivated by several observations about the behavior of t1/a as a function of the bare parameters (β, κ4, κ6). At ﬁxed β, we observe that (1) as κ4 or κ6 is increased,

2 2 t1/a increases monotonically; (2) t1/a varies smoothly as a function of κ4/κ6, in such a way that

2 2 there are smooth curves of constant t1/a ; (3) these curves of constant t1/a are roughly elliptical

2 in shape; and (4) as both κ’s go to zero and the fermions decouple, t1/a settles to a constant. This motivates the functional form

t r2 − r2(β) 1 = exp 0 + C(β) , (5.5) a2 γ(β) where

2 2 2 r ≡ (8κ4) + α(β)(8κ6) (5.6)

2 and α(β), γ(β), r0(β), and C(β) are functions of β only and thus constant at ﬁxed β. An equally reasonable functional form would be a power law rather than an exponential; however, ﬁts to such 101 Parameter Parameter

α0 6.1(8) α1 -4.6(8) γ0 -0.48(6) γ1 0.56(6) R0 10.7(9) R1 -8.2(9) C0 0.94(7) C1 2.4(1.6)

2 Table 5.4: Best-ﬁt parameters for the t1/a model deﬁned by Eqns. 5.5, 5.7, and 5.8.

functional forms produce unreasonably large powers and do not model the data as well. Note that we write the model in terms of 8κ4, 8κ6, and β/8 (below) which are all O(1), so that the size of the

2 2 ﬁt parameters may be compared easily. Physically, r0 quantiﬁes the value of r where fermionic

2 eﬀects are frozen out; α projects the elliptical curves of constant t1/a in the κ4–κ6 plane to circles;

2 γ quantiﬁes how quickly t1/a increases as fermionic eﬀects become strong; and C is the value of

2 2 t1/a in the pure-gauge limit where κ4 = κ6 = 0 [up to a small exp(−r0/γ) correction]. To obtain a concrete realization of the abstract model (5.5), we approximate α(β), γ(β), and

2 r0(β) as linear functions, β α(β) = α + α 0 1 8 β γ(β) = γ + γ (5.7) 0 1 8 β r2 = R + R 0 0 1 8 where α0, α1, γ0, γ1, R0, and R1 are ﬁt parameters. In the pure-gauge limit where κ4 = κ6 = 0, as

2 β → 0, we expect a → ∞ and thus t1/a → 0. We thus demand that C(0) = 0, and model C(β) as a power, β C1 C(β) = C , (5.8) 0 8

2 where C0 and C1 are fit parameters. Fitting the 39 ensembles of our zero-temperature t1/a dataset to the model defined by Eqs. (5.5)–(5.8), we obtain the fit parameters in Table 5.4 with

2 2 χ /31 = 0.87 and Q = 1 − P = 0.67. The resulting model predicts the value of t1/a within 5% for all 39 ensembles included in the ﬁt, and within 2.5% for 28 of these ensembles. Figure 5.14 shows the predictions of the model versus data at β = 7.75.

To cross-check our model, we compare with ﬁts of subsets of the dataset to simpler models. 102 Fit to β = 7.75 only Full ﬁt at β = 7.75 α(7.75) 1.61(5) 1.58(4) γ(7.75) 0.062(2) 0.061(2) 2 r0(7.75) 2.73(5) 2.70(4) C(7.75) 0.87(3) 0.87(3)

Table 5.5: Model parameters at β = 7.75 from direct ﬁt to β = 7.75 data versus predictions for those parameters from Eqs. (5.7) and (5.8), and the best-ﬁt values in Table 5.4.

2 At fixed β, all of α(β), γ(β), r0(β), and C(β) are constant, providing a simplified four-parameter realization of Eq. (5.5). Fitting to 12 ensembles at fixed β = 7.75 yields the model parameters in the second column of Table 5.5 with χ2/12 = 0.78 and Q = 1 − P = 0.67. By comparison, the model parameters in the third column of Table 5.5 are predictions obtained by plugging the best-fit parameters in Table 5.4 into Eqs. (5.7) and (5.8) for β = 7.75. Even though the full-dataset fit includes more than three times as many ensembles, the parameters agree closely; if this did not hold, it would suggest that the model is overfitting the data. For the 12 ensembles at β = 7.75, the predictions of these two fits agree within 1.5%. 103

9 4 = 0.122 8 4 = 0.124 = 0.126 7 4 4 = 0.127

6 4 = 0.128

4 = 0.129

2 5 4 = 0.1295 a /

1 = 0.13 t 4 4 4 = 0.131 3

0 0.122 0.124 0.126 0.128 0.130 0.132 6

2 Figure 5.14: Lines are predictions for t1/a as a function of κ6 for various κ4 at β = 7.75 by the 2 model deﬁned by Eqs. (5.5)–(5.8) and the best-ﬁt parameters of Table 5.4. Dots are t1/a data at β = 7.75. Colors are matched between dots and lines at the same κ4. Chapter 6

Multirep Pisarski-Wilczek

D In Chapter5, we investigated the phase structure of SU(4) multirep theory with NF = 2 fundamental and N D = 2 two-index antisymmetric fermions. We find that, similarly to QCD, this A2 theory has only two phases: a high-temperature chirally symmetric phase, and a low-temperature chirally broken one. However, unlike in QCD, we find that the transition between these two phases is strongly first-order. Further, while the lattice-deformed Ferretti model of Chapter5 is an interesting theory in its own right, it has different flavor content than the system relevant to

Ferretti’s BSM model. Transition order is sensitive to the number of ﬂavors, so we would like some way of “analytically continuing” from the ﬂavor content of our lattice-deformed theory to that of

Ferretti’s model. Furthermore, it is useful to provide independent analytic conﬁrmation of the lattice ﬁnding.

Ferretti’s model is not the only multirep theory which is interesting in a BSM context.

In fact, Ferretti’s model is simply one system chosen from a survey of viable composite Higgs models with partially composite top quarks [79]. Considering the cosmological consequences of (and potential cosmological probes of UV physics allowed by) ﬁrst-order phase transitions as discussed in Section 5.6.2, it is interesting to know what we should expect from these systems.

In this chapter, we generalize the chiral stability analysis method pioneered by Pisarski and

Wilczek for QCD to multirep theories with vectorlike fermion content where all fermions undergo the chiral symmetry breaking transition simultaneously. This analysis generalizes straightforwardly to theories where there are multiple representations of fermion, as long as the chiral symmetry 105 breaking pattern can be written as a direct product of the patterns of the single-representation sectors (up to additional U(1) factors due to non-anomalous axial symmetries). The eﬀective theory for the multirep theory is then simply the single-representation theories coupled together in a way consistent with the symmetries.

The method is mostly insensitive to Nc and takes little input from what representation fermions are charged under other than what their chiral symmetry breaking pattern is, so it is possible to do this exhaustively for the full inﬁnite class of multirep theories enumerated in Chapter2.

Working to one loop and leading order in the expansion, the calculation indicates that first-order phase transitions are a nearly completely generic property of multirep theories with simultaneous phase transitions. This prediction of a first-order phase transition includes both Ferretti’s model and our lattice deformation. There are six edge-case theories for which the calculation does not predict a first order transition, all of which have a single flavor each of two different irreps. The three theories marked “1-1” in Table 6.1 may exhibit continuous phase transitions instead of first-order.

The results are ambiguous for the three theories marked “1-13”.

6.0.1 Notation

a b ab In this chapter, we will denote by Tr the trace of representation r, deﬁned as Tr[τr τr ] = Trδ .

We work with generators normalized such that TF = 1 (note the diﬀerence from the TF = 1/2 convention employed elsewhere in this work).

Recall that we have made the deﬁnition   D Nr , r ∈ C, P nr = . (6.1)  W Nr , r ∈ R

Throughout this chapter, we will commonly use the symbol dr for the coset dimension   2 nr, r ∈ C   d = , (6.2) r nr(2nr − 1), r ∈ P    1  2 nr(nr + 1), r ∈ R 106 where r ∈ C here means that the irrep r is in a complex representation, so we consider the coset

SU(N) × SU(N)/SU(N), etc. Similarly, we denote by Cr the quadratic Casimir of each coset   nr/2, r ∈ C   Cr = 1 , (6.3) nr − 2 , r ∈ R    1  2 (nr + 1), r ∈ P

Throughout this chapter, couplings written in the Lagrangian and couplings written in the

β functions are in diﬀerent conventions. The couplings in the β functions are always redeﬁned by a factor of 1/(16π2) versus the couplings in the Lagrangians. We do not make any notational distinction to avoid clutter.

6.1 Overview and applicability

We are interested in theories with R diﬀerent species of fermion, each charged under distinct representations of the gauge group. We limit our attention to the irreps listed in Table 2.1, but will consider all theories discussed in the zoology of Chapter2. We note that many of these theories are infrared conformal and thus never break chiral symmetry, so our application of Pisarski-Wilczek to these theories will be invalid. However, the exact boundary of the conformal window is still controversial [59, 155], and we wish to include all interesting theories in our analysis. So, we will not attempt to make any cuts on which theories we consider on the basis of conformality. This ensures we will include all interesting theories: in fact, the conformal window provides a convenient buﬀer, as long as we keep in mind that the calculation does not apply to theories where chiral symmetry never breaks.

We immediately specialize to the case where all species of fermion undergo the chiral symmetry breaking transition simultaneously: the only multirep theory whose phase dynamics have been explored on the lattice to date (see Chapter5) has exhibited this behavior, and it may be that this is a generic phenomenon in multirep theories (see Section 5.6.1 for discussion of this point). In practice, the only way to test whether this assumption holds is by lattice simulation of particular 107 theories. If a multirep theory is found to have separated phase transitions, then this calculation will not apply. We make a further assumption that, when there is a simultaneous phase transition in all irreps, we can describe it by simultaneously tuning all condensate mass parameters rr (deﬁned below) to zero.

We only consider the limit where all species of fermion are massless, i.e. the quark mass mr = 0 for all r. This simplifies the analysis dramatically. We also know that first-order transitions are robust against small perturbations, so if we find a first-order chiral transition in the massless limit, it is likely to persist for light masses. Thus, the calculation can predict the order of the transition for a small region of finite-mass parameter space without computing there directly.

A Pisarski-Wilczek stability analysis amounts to analyzing the critical behavior of a three- dimensional Landau-Ginzburg-Wilson (LGW) model of the chiral condensates of the theory of interest—a linear sigma model. The procedure introduced by Pisarski and Wilczek [130] is:

• Following the usual EFT prescription, identify the symmetries of the theory and its spon-

taneous symmetry breaking pattern.

• Construct the most general LGW Lagrangian consistent with those symmetries. Because

we are interested in critical behavior, this Lagrangian includes only relevant and marginal

operators.

• Consider this theory in three dimensions: the ﬁnite-temperature system is compact in the

(Euclidean) temporal dimension, which will trivialize as the correlation length diverges at

criticality.

• Compute the β functions of the theory and identify their ﬁxed points.

• Finally, determine whether any of these ﬁxed points are infrared-stable by examining the

eigenvalues of the stability matrix ∂βgi /∂gj (where gi are the couplings of the theory). If

any of the eigenvalues of ∂βgi /∂gj at a ﬁxed point are negative, that ﬁxed point is infrared-

unstable. If the eﬀective theory has any stable ﬁxed points, the calculation is not predictive: 108

the chiral transition may be second-order, provided that the transition occurs in the basin

of attraction of one of the stable ﬁxed points (but may be ﬁrst-order otherwise); otherwise,

it will be ﬁrst order [57]. However, if no infrared-stable ﬁxed points exist, the calculation

predicts that the chiral transition must be ﬁrst order.

Pisarski-Wilczek analyses take very little information about the speciﬁc irreps of the fermions.

As discussed in Section 6.2.1 below, the chiral symmetry breaking pattern, and thus field content and form of the (anomaly-naive) Lagrangian, is determined purely by whether the irrep is complex, real, or pseudoreal. The only point at which any further information about the specific fermion representation enters is in determining the form of any anomaly-implementing terms per the procedure described in Sec. 6.2.1. Because there are a finite number of irreps that can be present in asymptotically free gauge theories, and because the trace of the representation only enters in determining whether any determinant terms are present, there are a finite number of multirep LGW

Lagrangians. This permits us to perform a stability analysis applicable to the full inﬁnite class of multirep theories, with some numerical assistance.

We will carry out the calculation of the β functions to one-loop order. We are interested in the behavior of the dimensionally-reduced 3D EFT, so we employ the expansion: we expand as usual in small = 4 − d, then set to 1. While more sophisticated methods exist [23, 39] to treat three-dimensionality, results are scheme-independent at one loop, so the expansion provides the same results as more careful treatments. However, this lowest-nontrivial-order approach is known to miss stable ﬁxed points in cases relevant to our analysis [23, 24]. We discuss implications for this calculation in Section 6.8 below.

6.2 Chiral symmetries & Lagrangian construction

6.2.1 Field content and single-irrep Lagrangians

In order to derive the LGW Lagrangian, we must ﬁrst identify the symmetries and symmetry breaking patterns of our theory. Although diﬀerent chiral symmetry breaking patterns may be 109 possible, we will make the assumption that the χSB pattern of an entire multirep theory is the direct product of the χSB patterns of the single-irrep subsectors times the breaking of non-anomalous

U(1)A symmetries, as written in Eqn. 4.13.

Each distinct irrep corresponds to an independent matrix-valued ﬁeld in the EFT; the values that ﬁeld may take on are determined by the chiral symmetry appropriate to the irrep’s complexity

j j class. For complex irreps, we deﬁne the order parameter ﬁeld for irrep r as (φr)i ∼ ψRiψL, where

D D ψ are Dirac fields and i and j are flavor indices. The field φr is an Nr × Nr arbitrary complex matrix field which transforms under chiral rotations like

† 2iαr φr → ULr φr URr e (6.4)

D D where ULr ∈ SU(Nr )L, URr ∈ SU(Nr )R, and αr is the angle of the axial rotation for irrep r. For real and pseudoreal irreps, we deﬁne the order parameter ﬁeld (φr)IJ ∼ ξI ξJ , where ξ is a left-

W W handed Weyl field and I and J are Weyl flavor indices. For real irreps, the field φr is an Nr × Nr

D D symmetric complex matrix ﬁeld, while for pseudoreal irreps, φr is a 2Nr × 2Nr antisymmetric complex matrix ﬁeld. Both cases transform under chiral rotations like

T 2iαr φr → Vr φr Vr e (6.5)

W D where Vr ∈ SU(Nr ) or SU(2Nr ) for real and pseudoreal irreps, respectively.

The most general Lagrangian for each single-irrep subsector, invariant under Eq. 6.4 or Eq. 6.5 and including only relevant and marginal terms, is

1R h † µ i h † i 1 † 2 1 † 2 Lr = Tr ∂µφr∂ φr + rr Tr φrφr + 4 ur(Tr[φrφr]) + 4 vr Tr[(φrφr) ] (6.6) where the traces are over ﬂavor. Note that without accounting for symmetry breaking by the axial anomaly, the Lagrangian governing each single-irrep subsector is of the same form, independent of whether the irrep is complex, real, or pseudoreal [130, 23, 169].

D Note that in the one-ﬂavor case nr = 1 (i.e. N = 1 for complex and pseudoreal representations and N W = 1 for real representations), the traces in the Lagrangian 6.6 become trivial. In 110 this case, the operators associated with the couplings ur and vr are the same, so the Lagrangian simpliﬁes to

1R † µ † 1 † 2 L1 = ∂µφr∂ φr + rrφrφr + 4 ur(φrφr) . (6.7)

For clarity, we will treat the one-ﬂavor case as a fourth type of irrep alongside complex, real, and pseudoreal. We denote it as 1.

6.2.2 Irrep coupling & constraints from anomaly

There is one form of non-irrelevant irrep-coupling term that is consistent with chiral symmetry and which does not break any U(1)A symmetry,

IC 1 † † δLrs = 2 wrs Tr[φrφr] Tr[φsφs] (6.8) where r 6= s are irrep labels. In the most general Lagrangian, there will be one such term for all pairs of distinct irreps r and s.

As written, each sub-Lagrangian of the form Eq. 6.6 is invariant under the full set of R

U(1)A axial symmetries. We can implement symmetry breaking by the axial anomaly using terms constructed from products of determinants and Pfaﬃans. Under transformations Eq. 6.4 or Eq. 6.5, for complex and real irreps, the determinant of φr varies as

2inrαr Det φr → e Det φr, (6.9)

D D W i.e., is invariant under SU(N )L × SU(N )R or SU(N ), but not with respect to U(1)A. For the antisymmetric φr corresponding to pseudoreal irreps, we can construct lower-order terms with the

D D correct properties using the Pfaffian of φr. Written out, the Pfaffian of a 2N × 2N scalar field is a sum over product of N D fields, while the determinant would involve terms with 2N D fields; thus, the Pfaffian is more fundamental. For antisymmetric φr, the Pfaffian varies under transformation

Eq. 6.5 as

D 2iN αr Pf φr → e r Pf φr. (6.10) 111

Note that the Pfaﬃan cannot be used for complex or real irreps: for symmetric φr, the Pfaﬃan is

∗ zero; for arbitrary φr, the Pfaﬃan is only invariant if UrL = UrR in Eq. 6.4, and so is not invariant under the full chiral symmetry.

Whatever terms we add to the Lagrangian must respect the R − 1 good U(1)A symmetries of the theory. Consider an arbitrary linear combination of the axial currents Arµ associated with each irrep,

qrArµ (6.11) where qr are real coeﬃcients [62]. Each axial current is individually anomalous,

2 W g ∂µArµ = N Tr F Fe (6.12) r 64π2 where Tr is the trace of representation r, with TF = 1 in our convention. From Eq. 6.12 we may read oﬀ a singled-out direction in qr space,

W W qe = (N1 T1,N2 T2,...). (6.13)

Note that if we pick any qr perpendicular to qe, then qr∂µArµ = 0, i.e., this linear combination of axial currents is non-anomalous. Thus, our Lagrangian must be invariant under axial rotations where αr ∝ qr for any qr perpendicular to qe. A family of such operators parametrized by a constant g is

Y 2Tr/g Y 2Tr/g Y Tr/g A = (Det φr) (Pf φr) (Det φr) (6.14) r∈C r∈P r∈R which transforms as " w # X N Tr A → exp 2i α r A = exp [2i~α · q] A (6.15) r g e r thanks to the factors of 2 in the deﬁnition of A. The argument of the phase contains an inner product between qe and the axial rotation angles αr, and so is invariant when qe ⊥ αr as desired. To obtain the most fundamental operator which is analytic in the ﬁelds, we choose

N W T N W T N W T g = gcd 1 1 , 2 2 ,..., R R (6.16) n1 TF n2 TF nR TF 112

W where gcd indicates the greatest common denominator, and the factor Nr /nr amounts to a factor of 2 for r ∈ C, P and 1 for r ∈ R. The anomaly-implementing term in the Lagrangian is then

δL = c(A + A∗). (6.17)

P W For a non-irrelevant term, we must have [A] = r TrNr /g ≤ 4. As discussed in the next section, no theories with more than two distinct irreps admit an anomaly-implementing term. For the case of two representations 1 and 2, invariant axial rotations must satisfy W α1 N2 T2 = − W (6.18) α2 N1 T1 For theories with two distinct irreps, we may construct terms that respect condition 6.18 with the general form1

d1 d2 (Det φ1) (Det φ2) + (c.c.) (6.19) where d1 and d2 are positive integers, and determinants should be replaced by Pfaﬃans when the corresponding irrep is pseudoreal. Demanding invariance under arbitrary axial rotations yields the condition W d1 T1 N1 n2 = W . (6.20) d2 T2 n1 N2

The dimension of operator 6.19 is d1n1 + d2n2, so for non-irrelevant operators we must have d1n1 + d2n2 ≤ 4. This restricts the possible values of (d1, d2) to (1, 1), (1, 2), (2, 1), (1, 3), (3, 1), and (2, 2). If condition 6.20 cannot be satisﬁed for any of these values of (d1, d2), or (suﬃciently) if n1 + n2 > 4, then there are no anomaly-implementing terms to add to the LGW Lagrangian.

Note that the traces in Eq. 6.14 and 6.20 are the only point in the calculation where any information enters about the speciﬁc irreps under consideration beyond whether the irrep is complex, real, or pseudoreal.

1 † d d Without the minus sign in Eq. 6.18, these terms would have the general form (det φ ) F (det θ) A2 + (c.c.). 113

6.3 Lagrangian zoology

For theories that do not admit anomaly-implementing terms, the full Lagrangian is

X 1R X IC L = Lr + δLrs r r

~~We may determine which theories require additional anomaly-implementing terms in their~~

Lagrangian by simply enumerating over all theories described in Section 2.3.2 and checking whether the lowest-order operator of the form 6.14 is non-irrelevant. We find only R = 2 theories that admit non-irrelevant terms. In total, we find 29 theories that admit 14 different structurally distinct anomaly-implementing terms; they are enumerated in Table 6.1. For convenience of discussion below, we introduce the notation e.g. 1-1 to describe a two-irrep Lagrangian where the anomaly-

2 2 implementing term ∼ φ1φ2, 1-2 to describe a two-irrep Lagrangian with δL ∼ φ1φ2 (where φ2 may be from the determinant of a 2 × 2 Φ matrix or the square of a determinant of a 1 × 1 Φ), etc.

~~Similarly, we deﬁne e.g. 1-C2 to describe a two-irrep Lagrangian with (n1, n2) = (1, 2) with irrep~~

~~2 2 complex and such that the anomaly-implementing term is ∼ φ2 Det φ2, 1 -C2 similar except the~~

~~2 anomaly-implementing term is ∼ φ2 Det φ2, etc. For 1-1 theories, the Lagrangian is corrected by~~

AI 2 δL1-1 = cφ1φ2 + d(φ1φ2) + (c.c.) (6.22) where we use that determinants and/or Pfaﬃans trivialize to single ﬁelds in the nr = 1 case; note that this case uniquely permits multiple orders of anomaly-implementing terms. For 1-12 and 1-13 theories, the corrections are

AI c 2 δL1-12 = φ1φ2 + (c.c.) 2! (6.23) c δLAI = φ φ3 + (c.c.) 1-13 3! 1 2 114 where c has been redeﬁned from Eq. 6.17 to absorb combinatoric factors. For 1-C2, 1-R2, 1-C3, and 1-R3 theories, the correction is

~~AI AI AI AI δL1-C2 = δL1-R2 = δL1-C3 = δL1-R3 = cφ1 Det φ2 + (c.c.) (6.24) and for 1-P2 and 1-P3 theories, it is~~

~~AI AI δL1-P2 = δL1-P3 = cφ1 Pf φ2 + (c.c.). (6.25)~~

~~2 2 2 2 The corrections for the 1 -C2, 1 -P2, and 1 -R2 are similar but with φ1 → φ1 and c → c/2! (to absorb combinatoric factors). For C2-R2 theories, the correction is~~

~~AI δLC2-R2 = c Det φ1 Det φ2 + (c.c.) (6.26) and for the P2-R2 theory, it is~~

~~AI δLP2-R2 = c Pf φ1 Det φ2 + (c.c.). (6.27)~~

~~While more are possible, these are the only Lagrangian structures present in the family of asymptotically free SU(Nc) multirep theories with vectorlike fermion content as categorized in Chapter2.~~

We ﬁnd no two-irrep theories with anomaly implementing terms for Nc > 9. This occurs because the trace of each representation is a function of Nc, so it becomes impossible to construct a fraction that will satisfy condition 6.20 with small integer d1 and d2. Taking the limit Nc → ∞, the ratios of the traces of the three two-index irreps A2, S2, and G limit to small, ﬁnite fractions:

1 1 T (A2)/T (S2) → 1, T (A2)/T (G) → 2 , and T (S2)/T (G) → 2 . With these limiting values it is possible to satisfy condition 6.20 with small integer d1 and d2, and so it could be argued that it is justiﬁable to include anomaly-implementing terms in the EFT Lagrangian; however, this is not true for any ﬁnite Nc, and beyond the scope of this calculation.

For most R = 2 theories, and for all theories with R > 2, there are no non-irrelevant anomaly-implementing terms that we can add to the Lagrangian. In these cases, all R-many U(1)A are separately good symmetries of the eﬀective ﬁeld theory. Physically, this says that the axial anomaly is not relevant to the physics of the chiral transition. 115

~~For any given Nc, there are ﬁnitely many theories without anomaly-implementing terms in the~~

Lagrangian. As discussed in Sec. 2.3.1, only F , A2, S2, and G may participate in asymptotically- free theories for Nc ≥ 16. For Nc < 16, where irreps other than F , A2, S2, and G may participate in asymptotically-free theories, there are only ﬁnitely-many Lagrangians, even with numbers of

~~flavors all specified. We find that there are 44 structurally distinct Lagrangians with numbers of~~

~~ﬂavors left arbitrary for Nc < 16; they are~~

~~R = 2 : 11, 1C, 1P, 1R, CC, CP, CR, RP, RR~~

~~R = 3 : 111, 11C, 11P, 11R, 1CC, 1CP, 1CR, 1RP, 1RR, CCC, CCP, CCR, CRP, CRR~~

~~R = 4 : 1111, 111C, 111P, 111R, 11CC, 11CP, 11CR, 11RP, 11RR, 1CCC, (6.28)~~

~~1CCP, 1CCR, 1CRP, 1CRR~~

R = 5 : 11111, 1111C, 1111P, 1111R, 111CC, 111CP, 111CR where e.g. 1CCP denotes an R = 4 theory with one single-ﬂavor irrep, two distinct complex irreps, and a single pseudoreal irrep. As discussed below, underlines indicate theories relevant to large-Nc irrep content.

There are infinitely many theories for Nc ≥ 16, but because there are only four permitted irreps, there are only finitely many Lagrangians. Without anomaly-implementing terms, the La- grangian is a function only of the complexity class of the irreps present and numbers of flavors of each irrep; F , A2, and S2 are all complex for Nc > 4, and G is always real. Through simple enumeration, we find that there are only 19 distinct Lagrangians with all numbers of flavors arbitrary for Nc ≥ 16 (up to one-flavor cases yielding distinct Lagrangians). These are a subset of the 44

~~Lagrangians found in Nc < 16 and are indicated with underlines in the list 6.28.~~

~~In Section 6.6.1 below, we will extrapolate our ﬁnite-Nc results to make predictions for arbitrarily large Nc. This is possible because (as discussed in Sec. 2.3.2) all theories for Nc ≥ 20~~

~~D may be grouped into 55 different classes where the number of fundamental flavors NF is left arbitrary and numbers of flavors of higher-irrep (A2, S2, G) fermions are specified. The Lagrangian~~

~~D D takes a diﬀerent form for NF = 1 and NF > 1, so there are theoretically 110 possible Lagrangians 116~~

~~F with ND otherwise arbitrary; enumerating theories for Nc ≥ 20 we ﬁnd that only 95 of these Lagrangians actually appear.~~

~~For convenient methods to calculate with these Lagrangians and ﬁeld content, see Ap- pendixB. 117~~

~~Vertex Type L Structure Nc Irrep 1 Irrep 2~~

1-1/2-2 1-1 4 FA2 6 A3 G 8 S2 A4 1-2 1-12 2 FG 4 G 204 6 A2 S2 7 A2 A3 1-C2 4 A2 F 8 A4 S2 1-P2 6 GA3 1-R2 4 FA2 6 A3 G 8 S2 A4 1-3 1-13 3 FG 5 FA2 9 A2 A3 1-C3 4 A2 F 8 A4 S2 1-P3 6 GA3 1-R3 4 FA2 6 A3 G 8 S2 A4 2 2-2 1 -C2 6 S2 A2 7 A3 A2 2 1 -P2 2 GF 2 1 -R2 4 204 G C2-R2 4 FA2 8 S2 A4 P2-R2 6 A3 G

Table 6.1: Structure of anomaly-implementing irrep-coupling terms in the Lagrangians for all 29 asymptotically-free theories which permit them. Vertex type indicates how many fields of irreps 2 1 and 2 are present in anomaly-implementing terms, e.g., 1-2 indicates a term like φ1φ2 (ignoring any group structure). Lagrangian structure indicates irrep complexity class, number of flavors, and exponent of the determinant/Pfaffian for irrep 1 and irrep 2 in order; complexity class is indicated as 1 when nr = 1 flavor is present, as these cases all have the same Lagrangian structure. 2 For example, the structure 1 -C2 with S2 as irrep 1 and A2 as irrep 2 indicates a theory with nS2 = 1 flavors of S2-irrep fermions with exponent 2 on the (trivial, for nr = 1) determinant and n = N D = 2 flavors of complex A -irrep fermions with exponent 1 on the determinant, yielding A2 A2 2 2 an anomaly-implementing term δL ∼ (φS2 ) Det φA2 . 118

~~6.4 Single-irrep subsectors~~

The Lagrangian Eq. 6.21 can be decomposed as a sum of the sub-Lagrangians of the single- irrep subsectors plus an irrep coupling term. It follows that the β functions of the theory reduce to the collection of single-irrep β functions plus corrections due to irrep coupling. Thus, to set up the full calculation, we will ﬁrst review the results of its previous applications. The relevant results are for the single-irrep theories without determinant terms in the Lagrangian, computed to one loop and leading order in the expansion with = 1.

~~D The β functions for a theory with nr = Nr Dirac ﬂavors of complex-irrep fermions are [130]:~~

~~2 2 2 βur = − ur + (nr + 4)ur + 4nrurvr + 3vr (6.29)~~

~~2 βvr = − vr + 6urvr + 2nrvr . (6.30)~~

~~D For a theory with nr = Nr Dirac ﬂavors of pseudoreal-irrep fermions, the β functions are [169]:~~

~~2 2 3 2 βur = − ur + (2nr − nr + 4)ur + 2(2nr − 1)urvr + 2 vr (6.31)~~

~~5 2 βvr = − vr + 6urvr + (2nr − 2 )vr . (6.32)~~

~~W For a theory with nr = Nr Weyl ﬂavors of real-irrep fermions, the β functions are [23]:~~

~~1 2 2 3 2 βur = − ur + 2 (nr + nr + 8)ur + 2(nr + 1)urvr + 2 vr (6.33)~~

~~5 2 βvr = − vr + 6urvr + (nr + 2 )vr (6.34) where we have redeﬁned all couplings by a factor of 1/16π2 from their deﬁnitions in the Lagrangian.~~

Although obscured by the differing definitions of nr for real and pseudoreal irreps, the β functions for each case are similar up to minus signs if both are written in terms of N W . All of these β functions have a nontrivial fixed point at

~~ 1 (ur, vr) = , 0 (6.35) 4 + dr where dr is the coset dimension Eqn. 6.2. 119~~

~~For each set of β functions there is also a trivial ﬁxed point, which is always unstable. The nontrivial ﬁxed points are unstable for nr ≥ 2, which is all values for which they are applicable.~~

~~In the nr = 1 one ﬂavor case, the β function for the single quartic coupling is~~

~~2 βur = −ur + 5ur, (6.36)~~

~~which has a nontrivial ﬁxed point at ur = 1/5. The stability matrix for this theory is trivially~~

~~calculated, and its value at ur = 1/[4 + d1] = 1/5 is ∂βur /∂ur = 1; the single eigenvalue of the stability matrix is positive, so the ﬁxed point is stable.~~

~~In most cases examined in this chapter, we must necessarily (but not suﬃciently) have vr = 0~~

~~to have a real-valued ur at any ﬁxed point of all three sets of single-irrep β functions βur and βvr .~~

~~Because βvr is linear in vr, when βvr = 0 either vr = 0 or vr takes on some ﬁnite value in terms of ur.~~

~~2 Plugging the ﬁnite expression for vr in, we obtain a function of the general form βur = Aur+Bur+C,~~

2 which has real roots if B > 4AC. For each irrep type, it is straightforward to plug in vr and derive that nr < 2 for ur to be real; however, the β functions are only applicable when nr ≥ 2, and so we

ﬁnd a contradiction. Further, in most cases below vr = 0 even after accounting for corrections due to irrep coupling: except for the 1-3 (Section 6.7.3) and 2-2 (Section 6.7.4) Lagrangians with anomaly- implementing terms, we ﬁnd that irrep-coupling corrections do not correct βv and corrections to

~~βu amount to positive-deﬁnition corrections to C. It is straightforward to check that A > 0 for all three irrep types, so vr = 0 in these cases.~~

~~The coeﬃcient of the linear term in each β function is the classical dimension = 4 − d = 1 of the corresponding coupling constant.~~

~~6.5 Results: Theories without anomaly terms~~

~~6.5.1 β functions~~

For theories that do not satisfy the conditions to have a non-irrelevant anomaly-implementing term, the Lagrangian is simply Eq. 6.21. The couplings of interest are then ur for each representation with nr = 1, ur and vr for each representation with nr > 1, and the R(R − 1)/2 many wrs 120 couplings.

~~For the ur and vr couplings, the β functions are simply those for the single-irrep subsectors~~

~~(Eqs. 6.29–6.34, 6.36), plus irrep-coupling corrections. The correction to βur is simply~~

~~X 2 ∆ [βur ] = dswrs, (6.37) s6=r while βvr does not receive a correction. Note that except the dr dependence this correction is insensitive to whether r is complex, pseudoreal, or real.~~

~~The β function for each wrs coupling is~~

2 X βwrs = −wrs + 2wrs + wrs [ur(dr + 1) + 2Crvr + us(ds + 1) + 2Csvs] + dqwrqwqs (6.38) q6=r,s where we take advantage of the notational convenience wrs = wsr. Note that the ﬁnal term does not appear in theories with only two irreps.

~~6.5.2 Solving for ﬁxed points~~

~~With the recipe given above it is straightforward to assemble the β functions for each La-~~

grangian. Because βvr = 0 is not affected by irrep coupling, we can solve it analytically for each irrep; by the arguments given in Section 6.4, we choose the vr = 0 root to obtain (possibly but not necessarily) real-valued solutions of ur. We thus have R(R + 1)/2 β functions in R(R + 1)/2 couplings to solve for an R-irrep theory. We found in Section 2.3 that R ≤ 5 in any asymptotically free theory, so we have at most 15 β functions. While analytically solving for the roots of this many polynomials is intractable by hand and difficult for computer algebra systems, it is is entirely feasible using numerical root finding.

The Mathematica/Wolfram Language method NSolve can find the full set of 2b complex roots of a set of b many polynomials with numeric coefficients, each of which is at most quadratic in b variables. To use this method to solve for the zeros of sets of β functions, we must specify the numbers of flavors of each irrep present to make the coefficients of the couplings completely numeric. For 2 ≤ Nc ≤ 20 we find this yields 660 Lagrangians with two distinct irreps, 665 three- irrep Lagrangians, 244 four-irrep Lagrangians, and 21 five-irrep Lagrangians. To do this in practice, 121 we use a Python script that constructs the set of β functions for each Lagrangian, then generates a WolframScript script that calls NSolve.

~~6.5.3 Fixed point zoology~~

The search finds 985779 fixed points, too many to list here. Instead, we can discuss some features of the fixed points we observe. We reiterate that at all fixed points, vr = 0 to have real ur, so the only single-irrep-specific coupling that we discuss for each irrep is ur. Many of the fixed points found by NSolve are at complex values of the couplings, which we discard as unphysical. For all two-irrep Lagrangians, six out of the eight fixed points are real. The number of complex fixed points varies from one Lagrangian to another for R > 2, but typically for R = 3, about 40% of fixed points are real, for 20–30 per Lagrangian; for R = 4, ∼ 10% are real, for O(102) per Lagrangian; for R = 5, ∼ 2% are real, for O(103) per Lagrangian.

We observe that the irrep-coupling couplings wrs may only be zero in combinations that decouple entire sectors of the theory from one another: for example, in a theory with three distinct irreps 1, 2, and 3, we may have w12 = w13 = w23 = 0, decoupling the three sectors from one another; or, we may have irrep 3 decoupled from irreps 1 and 2 like w23 = w13 = 0 with w12 6= 0; however, we may not have w23 6= 0, w12 6= 0, but w13 = 0. This is due to the ﬁnal term in Eq. 6.38

for βwrs , which will render the β function nonzero unless the w couplings are zero in an appropriate pattern. Physically, we expect that if two sectors are coupled only indirectly (e.g., w13 = 0 but w12 6= 0 and w13 6= 0), the direct coupling between them will be “activated” by renormalization group ﬂow.

~~Every possible zeroing of the wrs couplings deﬁnes a class of “decoupled product” ﬁxed points.~~

The values of couplings in these ﬁxed points are simply the direct products of the values of the couplings at the ﬁxed points of the decoupled subtheories. For example, with two representations 1 and 2, we may have either w12 = 0 or w12 6= 0; when w12 = 0, the possible values of the remaining

~~∗ ∗ ∗ ∗ ∗ ∗ ∗ couplings are (u1, u2) are (0, 0), (u1, 0), (0, u2), and (u1, u2), where u1 and u2 are the values of u1~~

∗ and u2 at the nontrivial ﬁxed point of the single-irrep theories for irreps 1 and 2, respectively. With 122 three irreps 1, 2, and 3, there will be a set of ﬁxed points wherein w23 = w13 = 0, decoupling irrep

3; u3 takes on values from the single-irrep theory for irrep 3, while u1, u2, and w12 take on values from the two-irrep theory for irreps 1 and 2. For a system of b many β functions and couplings, decoupled-product fixed points make up half of all the 2b complex-valued fixed points; the fraction of these which are real depends on the specific Lagrangian.

~~A subset of the ﬁxed points may be found analytically using the ansatz that all nonzero~~

2 ∗ couplings are equal, gi = g . Inserting this ansatz, we ﬁnd that the β functions for all nonzero couplings become " !# ∗ ∗ ∗ X βg = g −1 + g 4 + dr (6.39) r which, when set to zero, yield the solutions g∗ = 0 (the trivial ﬁxed point) and

~~∗ 1 g = P . (6.40) 4 + r dr~~

The couplings are always real-valued at this fixed point. All fixed points with all wrs positive are of this form, including the fixed points of the single irrep theories (which trivially meet the condition wrs ≥ 0 for all wrs). If some subset of the wrs are zero, then this solution can describe the couplings P of a decoupled subtheory, with r restricted to irreps involved in that subtheory. Many more fixed points, however, have at least one wrs negative; the author has been unable to find an analytic solution for the values of the couplings at these fixed points. Hereafter, we refer to the class of

~~ﬁxed points whose couplings are given by “the ansatz solution” as “the ansatz ﬁxed point”.~~

~~6.5.4 Stability of ﬁxed points~~

Given the β functions, it is trivial to calculate the stability matrix ∂βgi /∂gj. We do so using the sympy Python computer algebra package to construct the β functions and take analytic partial derivatives. To determine whether any theory has a stable ﬁxed point, we plug in the number of

flavors and the values of the couplings at each fixed point into the stability matrix. The resulting stability matrix is fully numeric, so we may numerically solve for its eigenvalues. We find negative

2 This is dependent on the exact definitions of our couplings; a similar (but less simple) ansatz would work if some of the couplings were defined differently by a factor of, e.g., 1/2. 123 stability matrix eigenvalues for all of the 63544 real fixed points across the 1590 theories we examine.

~~Thus, for 2 ≤ Nc ≤ 20, there are no stable ﬁxed points for any asymptotically-free theories whose irrep content doesn’t permit anomaly-implementing terms in its Pisarski-Wilczek Lagrangian.~~

~~While exhaustive and conclusive, this numerical study is our only evidence that most of the~~

fixed points are unstable. It is thus worth backing up the conclusion with additional reasoning for why this should be, where possible. As discussed above, the wrs couplings may only be zero in patterns that break the system apart into decoupled subtheories. The stability matrix is block- diagonal for these decoupled-product fixed points, and a single negative eigenvalue in the stability matrix is sufficient to deem the fixed point unstable. Thus, if any subtheory in a decoupled-product

fixed point is unstable, then the entire fixed point is unstable. It follows that if the couplings for some subtheory are all zero at a fixed point, then the fixed point is unstable, because the trivial

fixed point of the subtheory is unstable. As discussed in Section 6.4, the nontrivial fixed points for the single-irrep theories are unstable for all nr. So, any decoupled-product fixed point with an isolated irrep will be unstable; this includes all decoupled-product fixed points for theories with two or three distinct irreps.

We note that, although vr = 0 for all ﬁxed points of interest, all vr couplings must nevertheless be included in the set gi when computing the stability of ﬁxed points. The β functions for all types of couplings include contributions linear in vr, which will give nonzero contributions to the ∂gi/∂vr rows of the stability matrix when vr = 0. Although we have performed no systematic study of this

~~eﬀect, we have observed stable ﬁxed points in all (of a small number of) analyses with vr and βvr excluded from the stability matrix. This is equivalent to examining a theory with operators like~~

~~Tr |φ|4 excluded from the Lagrangian. This suggests that the Tr |φ|4 operators associated with the vr couplings may be responsible for the generic instability we observe.~~

~~If we had found any stable ﬁxed points, whether or not these were physically relevant would have depended on whether they satisﬁed several additional cuts. It is interesting to note that we~~

~~ﬁnd no ﬁxed points that would have been excluded by any of the cuts we might reasonably make.~~

For example, perturbativity requires g2 < 1 for all couplings. As discussed in Section 6.A, further 124 constraints can be derived by requiring that the potential be stable and the correct χSB pattern be

~~2 realized. This yields the condition that ur > 0 for all r, and that if any wrs < 0 then wrs < urus.~~

~~None of these conditions are violated for any of the couplings in the set of real-valued ﬁxed points.~~

~~6.6 Example: Ferretti’s model & its lattice deformation~~

Although Ferretti’s model and its lattice deformation were covered by the exhaustive survey of the previous section, it is enlightening to work out the calculation for them explicitly. We specialize herein to the case of N D = n ﬂavors of the complex irrep F and N W = n ﬂavors of F F A2 A2 the real irrep A2.

~~To determine which Lagrangian is appropriate, we should check if a non-irrelevant anomaly-~~

implementing irrep coupling term is possible. In SU(4) where TA2 = 2TF , we ﬁnd that the good axial charge ratio is α /α = −N w /N D and thus d = d . Thus, the lowest-order term we can add to F A2 A2 F F A2

dF dA2 our Lagrangian has dF = dA2 = 1. Examining dimension of this term, [(Det φF ) (Det φA2 ) ] = d N D +d N W = N D +N W , we see that it is only non-irrelevant if N D +N W ≤ 4. This condition F F A2 A2 F A2 F A2 is violated for both Ferretti’s model (N D = 3, N W = 5) and our lattice deformation (N D = 2, F A2 F N W = 4) so we do not need to include this term in our analysis. A2 The Lagrangian of interest is thus

~~† µ † 1 † 2 1 † 2 L = Tr[∂µφ ∂ φ] + rF Tr[φ φ] + 4 uF (Tr[φ φ]) + 4 vF Tr[(φ φ) ]~~

~~† µ † 1 † 2 1 † 2 (6.41) + Tr[∂µθ ∂ θ] + rA2 Tr[θ θ] + 4 uA2 (Tr[θ θ]) + 4 vA2 Tr[(θ θ) ]~~

~~1 † † + 2 w Tr[φ φ] Tr[θ θ]~~

~~where there is only one w-type coupling so for brevity we name w ≡ wF,A2 . 125~~

~~For the full Lagrangian Eq. 6.41, we ﬁnd that the β functions are~~

~~2 2 2 1 2 βuF = − uF + (nF + 4)uF + 4nF uF vF + 3vF + 2 nA2 (nA2 + 1)w~~

~~2 βvF = − vF + 6uF vF + 2nF vF~~

~~β = − u + 1 (n 2 + n + 8)u2 + 2(n + 1)u v + 3 v2 + n2 w2 uA2 A2 2 A2 A2 A2 A2 A2 A2 2 A2 F~~

~~β = − v + 6u v + (n + 5 )v2 vA2 A2 A2 A2 A2 2 A2~~

~~2 1 2 βw = − w + w (nF + 1)uF + 2nF vF + 2 (nA2 + nA2 + 4)uA2 + (nA2 + 1)vA2 + w where we have redeﬁned all couplings from their deﬁnitions in the Lagrangian by an overall factor~~

2 of 1/16π . The irrep-coupling term associated with the coupling w has induced corrections to βuF and β , but not β and β . uA2 vF vA2 We find six real-valued fixed points in total, which are enumerated in Tables 6.2 and 6.3. As expected, all have vr = 0. Four are decoupled product fixed points where w = 0; the couplings

~~2 in them are simply given by vF = vA2 = 0 and one choice among uF ∈ {0, 1/[4 + nF ]} and~~

1 uA2 ∈ {0, 1/[4 + 2 nA2 (nA2 + 1)]}. This includes the trivial fixed point. We find two fixed points where w 6= 0 and the F and A2 sectors are coupled. The w > 0 fixed point, listed in the last row of

~~∗ Table 6.2, is the ansatz solution where g = 1/[4 + dF + dA2 ]. The closed form of the w < 0 ﬁxed point, while computable by computer algebra systems, is too long to be worth recording here. In~~

~~Table 6.3, we provide numerical values for the couplings at this ﬁxed point for our two theories of interest.~~

~~All six ﬁxed points satisfy the stability constraints from Section 6.A, and also satisfy the~~

~~2 constraint for perturbativity gi < 1 for all i.~~

The stability matrix ∂βgi /∂gj (where gi ∈ {uF , vF , uA2 , vA2 , w}) is straightforwardly computed from the β functions and not worth reproducing here. At each of the six fixed points, we compute the eigenvalues of the stability matrix. We find that none of the fixed points are stable for

~~any asymptotically free nF ≥ 2 and nA2 ≥ 2, which is all of the theories for which the Lagrangian~~

~~Eq. 6.41 applies. Thus, our calculation indicates that the transition should be ﬁrst order for any~~

~~asymptotically free theory with nF ≥ 2 and nA2 ≥ 2 and with no anomaly-implementing terms. 126~~

~~uF uA2 w~~

~~0 0 0 2 1/(4 + nF ) 0 0 2 0 2/(8 + nA2 + nA2 ) 0 2 2 1/(4 + nF ) 2/(8 + nA2 + nA2 ) 0 2 2 2 2 2 2 2/(8 + nA2 + nA2 + 2nF ) 2/(8 + nA2 + nA2 + 2nF ) 2/(8 + nA2 + nA2 + 2nF )~~

Table 6.2: The five (of six total) fixed points which are amenable to concise analytic expression ∗ P using the ansatz solution g = 1/[4+ r dr]. All fixed points found have vF = vA2 = 0, as necessary for real-valued ur. The first four fixed points are “decoupled product” fixed points. Couplings are in the convention of the β functions, and thus different from those in the Lagrangian by an overall factor of 1/(16π2).

~~The case nF = nA2 = 2 is the only one for which we would need to add an anomaly-~~

implementing term to this Lagrangian (nF = 1 or nA2 = 1 would also permit such a term, but the single-irrep subLagrangian for the one-flavor sector would be different), yielding the C2 − R2 Lagrangian. As we will see in Section 6.7.4 below, the additional anomaly-implementing term does not yield a theory with stable fixed points either. The only case where this theory could exhibit a

~~stable ﬁxed point, discussed in Section 6.7.1 below, is the single-ﬂavors case nF = nA2 = 1.~~

~~6.6.1 Large Nc~~

~~Unlike with Nc < 16, we cannot look for stable ﬁxed points in the large Nc limit by exhaustion.~~

D As Nc → ∞, the allowed number of fundamental flavors NF diverges; thus, there are infinitely many possible theories in the large Nc limit. However, as discussed in Sec. 6.3 above, we may group all theories with Nc ≥ 20 into 55 classes, with the different classes defined by different numbers of

~~A2, S2, and G ﬂavors, and with theories in each class diﬀering by only the number of fundamental~~

D flavors NF = nF . Our survey above found that none of these classes exhibits a stable fixed point for any small number of nF . Thus, to check for stable fixed points in the large Nc limit, we only need to examine whether any of these classes develops a stable fixed point in the large nF limit.

~~In actuality, there are fewer than 55 classes to examine because F , A2, and S2 are all 127~~

~~Theory NF NA2 uF uA2 w~~

~~Ferretti 3 5 0.042519 0.035907 -0.035606 Lattice 2 4 0.075832 0.056288 -0.054615~~

Table 6.3: Values of couplings at the second multirep fixed point, which is not amenable to concise analytic expression, for our theories of interest. Again, vF = vA2 = 0. The numerical values are computed from closed-form expressions and truncated at five significant digits. Couplings are in the convention of the β functions.

~~complex irreps for Nc > 4. The structure of our Lagrangian, up to labels, depends only on the complexity classes and numbers of ﬂavors of the irreps present; so, the Lagrangian for e.g.~~

(nF , nA2 , nS2 , nG) = (nF , 2, 4, nG) is the same as for (nF , nA2 , nS2 , nG) = (nF , 4, 2, nG). Cases where nr = 1 further reduce the number of distinct Lagrangians that must be examined. The contribution of an nr = 1 irrep to the total Lagrangian is independent of complexity class;

~~this means that the Lagrangian for e.g. (nF , nA2 , nS2 , nG) = (nF , 1, 2, 1) is the same as for~~

(nF , nA2 , nS2 , nG) = (nF , 2, 1, 1). With all numbers of ﬂavors left arbitrary, asserting nF > 1 because we are interested in the large nF limit, there are 13 structurally distinct Lagrangians of interest. Their structures are

~~R = 2 : C-1, C-C, C-R;~~

~~R = 3 : C-11, C-1C, C-1R, C-CC, C-CR;~~

R = 4 : C-111, C-11C, C-11R, C-1CC, C-1CR where e.g. CC denotes that two distinct complex irreps are present, and we always write one complex irrep set aside to associate it with the F irrep. With the numbers of ﬂavors of the higher irreps

(nA2 , nS2 , nG) specified but nF > 1 left arbitrary, we find 29 Lagrangians (tabulated in Table 6.4) defined by the number of one-flavor irreps n1, the number of (real) adjoint-irrep flavors nG, and how many flavors of up to two complex irreps are present nC1 and nC2. We choose nC1 ≥ nC2 to avoid double counting from the interchangeability of the two complex irreps. Note that other than the irrep content explicitly specified by n1, nG, nC1, and nC2 in Table 6.4, there is an additional complex irrep present with number of flavors arbitrary for the F irrep. 128

max nC2 n1 nG nC1 nF 0 0 0 2 73 3 55 4 37 5 19 6 9 0 0 2 0 70 2 34 3 15 0 0 3 0 50 2 14 0 0 4 0 30 0 0 5 0 17 0 1 0 0 92 2 54 3 35 4 17 0 1 2 0 52 2 12 0 1 3 0 32 0 1 4 0 14 0 2 0 0 72 2 32 3 14 0 2 2 0 30 0 2 3 0 10 0 3 0 0 50 2 0 0 2 30 3 12 2 1 0 2 10

Table 6.4: Distinct Lagrangian structures of interest for the large nF (large Nc) limit; note that there is always one additional complex irrep for the F irrep. n1 is the number of one-flavor irreps present, nG is the number of (real) adjoint-irrep flavors, and nC1 and nC2 are how many flavors of max up to two distinct complex irreps (other than F ) are present. nF indicates the largest number of fundamental flavors examined in this class by the numerical survey of all Lagrangians relevant to 2 ≤ Nc ≤ 20. 129

~~The study in Sec. 6.5 of all Lagrangians relevant to 2 ≤ Nc ≤ 20 above included all 29 of the~~

max Lagrangians with the structures in Table 6.4, covering 2 ≤ nF ≤ nF as tabulated in Table 6.4 for each Lagrangian. In Sec. 6.5, we found no stable fixed points in any theories from these classes: for all fixed points in these theories, the stability matrix had at least one negative eigenvalue. To determine whether any of these Lagrangians develops a stable fixed point at large nF , we need to look at the stability matrix at each fixed point and determine whether all of its negative eigenvalues will become positive as nF increases. To this end, we take the large nF limit analytically for the subset of fixed points for which we have an analytic solution. We approach the remaining fixed points numerically.

At any decoupled-product ﬁxed point, at least one subtheory will be decoupled from the subtheory containing the F irrep, and so the values of the couplings for this subtheory will be constants in nF . Thus, if this subtheory is unstable at those values of the couplings, it will destabilize the

fixed point for all nF . The 29 Lagrangians in Table 6.4 comprise nearly the set of all subtheories of themselves, missing only the Lagrangian for a single complex irrep. In the study in the previous section, we found that none of the 29 Lagrangians have any stable fixed points, and the single complex irrep Lagrangian is unstable for nF ≥ 2, i.e. all nF to which it applies [130]. It follows immediately that decoupled-product fixed points are unstable for all nF , and a fixed point must have all wrs 6= 0 to feasibly be stable.

~~We have an analytic solution (the ansatz solution) for the class of ﬁxed points (the ansatz~~

~~ﬁxed point) where all wrs > 0: vr = 0 and all nonzero couplings are equal with the value~~

∗ P g = 1/[4 + r d(r)]. We may compute the stability matrix at these ﬁxed points in the large nF limit analytically. We will assume that if there are negative eigenvalues both at the ﬁnite nF where we have looked and in the nF → ∞ limit, then there is no intermediate regime where there are no negative eigenvalues. To test this assumption, note that in the nF → ∞ limit, the ansatz

ﬁxed point for the single-complex-irrep theory is unstable. For couplings gi = {u, v}, the limit goes 130 like  2    nF +4 4nF −1 + 2 2 2 1 0 ∂βg n +4 n +4 nF →∞ i =  F F  −−−−→   (6.42) ∂gj  6    0 −1 + 2 0 −1 nF +4 ij ij where after computing the stability matrix from Eqs. 6.29 and 6.30 we plugged in the ansatz

2 solution v = 0, u = 1/[4 + nF ]. There is a negative eigenvalue, so the ﬁxed point is unstable. As discussed in Section 6.4 above, [130] proved the stronger condition that this ﬁxed point is unstable

~~∗ 2 for all nF ≥ 2, so our assumption holds in this case. Given this example, and that g ∼ 1/nF is monotonic in nF rather than oscillatory, the author believes this assumption is well-motivated.~~

Using computer algebra to examine the limiting behavior of the ansatz fixed point for the remaining 29 Lagrangians, we find in all cases that the fixed point is unstable in the nF → ∞ limit. Using the Python package sympy, it is straightforward to construct the stability matrix for each Lagrangian of interest, plug in the ansatz solution for arbitrary nF , take the nF → ∞ limit, then compute the eigenvalues of the resulting fully-numerical matrix. We find at least 2 negative eigenvalues in all cases. The vr couplings guarantee this instability, as the part of the stability matrix involving vr decouples from the rest of the matrix and limits to −1. This is because all

~~∗ ∗ nonzero oﬀ-diagonal terms in ∂gj βgi involving either βvr or ∂vr go as either ∼ CF g or ∼ Csg~~

~~∗ 2 for some rep s 6= F ; in the large nF limit, g ∼ 1/nF , while CF ∼ nF and Cs ∼ 1. Meanwhile,~~

~~∗ ∂βvr /∂vr = −1 + 6g → −1 for r = F and r 6= F , so the eigenvalues of the block involving the vr couplings are negative.~~

~~We resort to numerics to examine the ﬁxed points that remain after eliminating all ansatz~~

~~ﬁxed points and decoupled-product ﬁxed points. To examine the nF dependence of particular~~

fixed points, we need some way to identify numerically-computed fixed points between two theories in the same class. The values of zeroed couplings are independent of nF , so any fixed point identified between two theories must have the same couplings zeroed. Taking inspiration from this observation, we will make a stronger assumption that the sign of a coupling will not change as a function of nF . Although made without proof, this assumption seems reasonable, is by itself sufficient to fully organize the numerical data, and produces sensible-looking results: as an example 131 of typical behavior, Figure 6.1 shows the values of the couplings at fixed points which have been associated using this assumption.

We are interested in the asymptotic behavior of the eigenvalues of each fixed point that we have not already ruled unstable for each of our 29 Lagrangians. For a fixed point to remain unstable at large nF , we need at least one of its eigenvalues to asymptote to a negative constant as nF increases (an eigenvalue diverging negatively would also suffice, but we do not observe this behavior). Examining plots of the eigenvalues as a function of nF , this appears to be common behavior; Figures 6.2(a) and 6.2(b) show typical examples. However, it is at best intractably tedious to examine every fixed point with this graphical approach. We instead need a numerical diagnostic.

In Figure 6.1, as well as Figures 6.2(a) and 6.2(b), we see that there are ﬁxed points that disappear at small values of nF . The disappearance is due to the couplings at these points becoming complex-valued as nF increases. Examining Fig. 6.2(b) closely, there are several instances where two

fixed points collide and disappear; these are positive and negative roots, which become degenerate and then complex as their disciminant passes through zero. We already know these fixed points are unstable and are instead interested in the large nF limit, so we restrict our numerical analysis to fixed points that survive past nF = 10.

For a theory with b many couplings and β functions, the stability matrix has b eigenvalues, which we find numerically. To learn about the nF dependence of each eigenvalue, we must associate eigenvalues between different nF , just as with the fixed points. To do this, moving from large nF downwards, to group the eigenvalues at some particular nF , for each series, we use linear extrapolation to determine where the next eigenvalue for the series is expected to be at nF then associate the closest eigenvalue to that projection; eigenvalues left unassociated after all series have been extended seed a new series starting at nF , as do eigenvalues too far away from any expected value. The resulting associations are the lines connecting points in Figures 6.2(a) and 6.2(b).

~~With the eigenvalues associated into series, we can apply a simple diagnostic of asymptoticity to the eigenvalues for each ﬁxed point. At smaller nF , the eigenvalues move more rapidly and 132~~

~~0.125~~

~~0.100~~

~~0.075~~

~~0.050~~

~~0.025 Couplings~~

~~0.000~~

~~0.025~~

~~0.050~~

~~5 10 15 20 25 30 35 nF~~

Figure 6.1: Dots are the values of the couplings at fixed points from theories in the C2-R2 large Nc Lagrangian class at different values of nF . Fixed points are found separately at each nF using numerics, and associated using the assumption that couplings will not change sign as nF varies. Lines indicate this association. Different colors correspond to different fixed points, but no graphical distinction is made between the different couplings for a given fixed point. 133 sometimes display nontrivial structure, leading to confusion in associating them into series; to avoid this issue, we consider only the last half of each series where (from inspection of plots like

Figs. 6.2(a) and 6.2(b)) we expect the eigenvalues to vary slowly and smoothly. After this cut, we simply check that each series is going asymptotically constant at large nF by demanding |∂λ/∂n| be monotonically decreasing, which is necessary for asymptoticity and sufficient to exclude oscillatory behavior. We find that all eigenvalue trajectories for all fixed points satisfy this check, indicating that there is a well-defined large nF limit for these fixed points.

Applying this diagonstic, we find that every fixed point has at least one eigenvalue that asymptotes to a negative constant, indicating that all fixed points in all Lagrangian classes that we examine are unstable in the large nF limit. We already found that these fixed points are unstable at all of the finite nF we have examined; given the smooth asymptotic behavior we observe for all eigenvalues, we may reasonably conclude that these fixed points are unstable for all nF .

Having excluded the existence of stable fixed points as nF → ∞, we conclude that no theory in the Nc ≥ 20 regime has a stable fixed point. We discovered in Section 6.3 that there are no theories that require anomaly-implementing terms in their Lagrangians for Nc > 9. This calculation thus predicts that all multirep theories with Nc > 9 exhibit first-order phase transitions. 134

~~1.00~~

~~0.75~~

~~0.50~~

~~0.25~~

~~0.00~~

~~0.25~~

~~Stability matrix eigenvalues 0.50~~

~~0.75~~

~~1.00 5 10 15 20 25 30 35 nF~~

~~(a) Eigenvalues for large Nc Lagrangian class C2-R2.~~

~~1.0~~

~~0.5~~

~~0.0~~

~~Stability matrix eigenvalues 0.5~~

~~1.0~~

~~5 10 15 20 25 30 nF~~

~~(b) Eigenvalues for large Nc Lagrangian class 1-1-C2.~~

Figure 6.2: Eigenvalues of the stability matrix as a function of nF for non-decoupled-product, non- ansatz fixed points for the large Nc Lagrangian class C2-R2 (top) and 1-1-C2 (bottom). Different colors are associated with different fixed points. Dots are eigenvalues, while lines are associations comprising eigenvalue trajectories, with associations made as described in the text. 135

~~6.7 Results: Theories permitting anomaly-implementing terms~~

~~6.7.1 1-1 anomaly~~

In the 1-1 case, there is one ﬂavor (nr = 1) of each of two distinct irreps of fermion. Thus, each single-irrep subsector has no chiral symmetry. The only symmetry in the theory is the single non-anomalous U(1)A, as discussed in Section 6.2.2, and so the χSB pattern for the theory is simply

~~U(1)A → 1. (6.43)~~

~~This is the symmetry breaking pattern associated with the O(2) spin model, also know as the XY model. We can write down an LGW Lagrangian for this theory straightforwardly,~~

1-1 2 2 1 4 L = |∂µφ| + r|φ| + 4 u|φ| , (6.44) where φ is a complex scalar ﬁeld. In three dimensions this theory is known to exhibit a continuous phase transition, so we can conclude immediately that a stable ﬁxed point exists. Note that, because the non-anomalous U(1)A involves a superposition of axial currents from each of the two irreps, breaking it will amount to a simultaneous transition for both irreps.

~~For these theories, it is useful to see how the machinery we apply to all other theories in this study can produce the same result. Following our Lagrangian construction prescription, the 1-1~~

~~Lagrangian can be written as~~

~~1−1 2 2 2 2 1 4 1 4 1 2 2 L = |∂µφ1| + |∂µφ2| + r1|φ1| + r2|φ2| + 4 u1|φ1| + 4 u2|φ2| + 2 w|φ1| |φ2|~~

† † 2 2 †2 †2 + c(φ1φ2 + φ1φ2) + d(φ1φ2 + φ1 φ2 ) (6.45) where the couplings c and d are associated with the two non-irrelevant types of anomaly-implementing operators. Note that [c] = 2, so the term c[φ1φ2 +(c.c.)] is an oﬀ-diagonal mass term. Diagonalizing this theory, we obtain

~~1-1 2 2 2 2 1 4 1 4 1 2 2 LD = |∂µφ+| + |∂µφ−| + r+|φ+| + r−|φ−| + 4 u+|φ+| + 4 u−|φ−| + 2 w±|φ+| |φ−|~~

~~1 2 † † 1 2 † † (6.46) + 2 x+|φ+| (φ+φ− + φ+φ−) + 2 x−|φ−| (φ+φ− + φ+φ−)~~

2 2 †2 †2 +d±(φ+φ− + φ+ φ− ) 136 where the fields φ+ and φ− have masses s r + r r − r 2 r = 1 2 ± 1 2 + c2 (6.47) ± 2 2 and the four-point couplings of the diagonal theory (u+, u−, w±, d±, x+, and x−) are related to the couplings of the original off-diagonal theory (r1, r2, c, u1, u2, w, and d) in a straightforwardly- calculable way. The off-diagonal theory no longer depends on c, at the cost of generating new four-point operators that were not present in the original theory.

~~Note that c is generated by the axial anomaly so we cannot tune it to zero, nor do we have~~

2 any reason to expect it to be small. When c > 0, r+ > r−, so we expect the + mode to always be more massive; we cannot tune φ+ and φ− massless simultaneously. Tuning r− → 0, the running of r+ is given by

~~βr+ = −2(1 − u+)r+. (6.48)~~

Tuning r− → 0 from above, we have r+ > 0. Thus, for any perturbative values of u+ (i.e. u+ < 1), this β function drives r+ more and more massive as we tune r− towards the massless limit and the system attempts to go critical. This justiﬁes integrating out the φ+ mode, exactly analogously to what is done to the heavier modes in the two-ﬂavor QCD case [133]. The resulting Lagrangian with φ+ decoupled is simply

~~L1−1 = |∂ φ |2 + r |φ |2 + 1 u |φ |4 (6.49) D/ µ − − − 4 − − recovering the LGW Lagrangian Eq. 6.44 that we wrote down from inspection of the SSB pattern.~~

There are three 1-1 theories (listed in Table 6.1) which the analysis of this section applies to. However, there are a larger number of two-irrep theories with nr = 1 for both irreps which do not permit non-irrelevant anomaly-implementing terms in their Lagrangians. The Lagrangian for these theories was analyzed as part of the exhaustive study of 2 ≤ Nc ≤ 20 above, wherein no stable fixed points are found. The difference between that analysis and the one of this section is that, when there are no relevant anomaly-implementing terms, it is possible to simultaneously tune 137 both r1 and r2 to zero. When this is possible, we find that the phase transition must be first order; if not, we find that it may be second order.

~~6.7.2 1-2 anomaly~~

~~For Lagrangians with structure 1-C2 and 1-R2, the anomaly-implementing term in the La- grangian is~~

~~AI δL = c[φ1 Det φr + (c.c.)] (6.50) where φr is a 2 × 2 matrix ﬁeld. For Lagrangians with structure 1-P2, the term is instead~~

~~AI δL = c[φ1 Pf φr + (c.c)] (6.51)~~

~~2 where φr is a 4 × 4 matrix ﬁeld. For Lagrangians with structure 1-1 , the term is~~

~~c δLAI = [φ φ2 + (c.c)] (6.52) 2! 1 2 where c was redeﬁned to absorb combinatoric factors. In all cases, the coupling c has dimension 1.~~

~~When the only dimensionful couplings in the theory are the masses r1 and rr (r2), then by~~

~~dimensional analysis all terms in the mass beta functions βr1 and βrr must be proportional to either~~

~~r1 or rr. In that case, tuning r1 and rr to zero is suﬃcient to zero βr1 and βrr . However, in the~~

2 present case of dimensionful c, βr1 and βrr may contain terms proportional to c , i.e., the c coupling contributes an additive renormalization to the condensate masses. For the 1-C2 Lagrangian, we ﬁnd

~~2 βr1 = −2r1 + 2u1r1 + 4wrr + 4c (6.53) 2 βrr = −2rr + 5urrr + 4vrrr + wr1 + 2c ; for 1-R2, we ﬁnd~~

~~2 βr1 = −2r1 + 2u1r1 + 3wrr + 3c (6.54) 2 βrr = −2rr + 4urrr + 3vrrr + wr1 + 2c ; 138 and for 1-P2, we ﬁnd~~

~~3 2 βr1 = −2r1 + 2u1r1 + 6wrr + 2 c (6.55) 1 2 βrr = −2rr + 7urrr + 3vrrr + wr1 + 2 c where couplings subscripted with 1 are for the nr = 1 irrep and with r are for the nr = 2 irrep.~~

~~For the 1-12 Lagrangian, we ﬁnd~~

~~2 βr1 = −2r1 + 2u1r1 + wr2 + c (6.56) 2 βr2 = −2r2 + 2u2r2 + wr1 + 2c where couplings subscripted with 2 are for the ﬁeld which appears squared in δLAI.~~

~~Because [c] = 1, there are no corrections proportional to c in the β functions for the four-point~~

~~4 couplings. Consider some φ theory with a three-point coupling λ3 and a four-point coupling λ4.~~

~~By dimensional arguments, the superﬁcial degree of divergence of a diagram correcting λ4 is~~

~~D4 = [λ4] − V3[λ3] − V4[λ4], (6.57)~~

~~where V3 and V4 are the numbers of three- and four-point vertices in the diagram, respectively~~

[154]. Plugging in [λ4] = and [λ3] = 1 + /2, we find that when ≥ 0 any diagram with at least one three-point vertex is (superficially) convergent. Only the divergent parts of diagrams contribute to β functions, and diagrams with three-point vertices do not diverge, so the c coupling does not appear in diagrams contributing to the renormalization of the four-point couplings at any order. It follows that the fixed points of the four-point couplings will be the same as the equivalent

~~Lagrangians without anomaly-implementing terms, as discussed in Section 6.5 above. None of the~~

fixed points in those theories were stable, so the only way stable fixed points can exist for these theories is if the c coupling stabilized an existing fixed point via off-diagonal terms in stability matrix. As we argue below, this does not occur.

~~We expect terms in βc to be exactly linear in c and independent of r1 and rr. The superﬁcial degree of divergence of a diagram correcting λ3 is~~

D3 = [λ3] − V3[λ3] − V4[λ4], (6.58) 139 from which we ﬁnd that V3 ≤ 1 in any superﬁcially divergent diagram for any . Thus, βc is at most linear in c. The lowest-order divergent diagrams with three external legs in our Lagrangian are logarithmically divergent, and we see from Eq. 6.58 that adding any additional vertices can only make a diagram less divergent. This includes two-point mass vertices, so we do not expect any r1 or rr dependence in these integrals. It follows from dimensional analysis that βc must be linear in c.

~~For the 1-C2 and 1-P2 Lagrangians, we ﬁnd~~

~~3 βc = − 2 c + urc − vrc + 2wc, (6.59) for the 1-R2 Lagrangian we ﬁnd~~

~~3 1 βc = − 2 c + urc − 2 vrc + 2wc, (6.60) and for the 1-12 Lagrangian we ﬁnd~~

~~3 βc = − 2 c + u2c + 2wc. (6.61)~~

By the arguments above, we do not expect any r1 or rr (r2) dependence and only linear c dependence at all orders. If c 6= 0, βc = 0 provides an additional constraint on ur, vr, and w incompatible with those from the remaining set of β functions (i.e., the system is overdetermined); thus, all ﬁxed points have c = 0 exactly. Because the four-point β functions are not corrected by c, the stability

~~matrix ∂βgi /∂gj is zero if gi 6= c and gj = c. Because c = 0 at any ﬁxed point and βc is linear in~~

c, ∂βgi /∂gj is zero if gj 6= c and gi = c. Thus, the stability matrix is block-diagonal at any ﬁxed point, with ∂βc/∂c isolated. The block corresponding to the four-point couplings is the same as in the theory without an anomaly-implementing term; as we discovered in Section 6.5, this block has at least one negative eigenvalue for every theory under consideration, so there are no stable ﬁxed points. 140

~~6.7.3 1-3 anomaly~~

For the 1-3 Lagrangians (except the 1-13 Lagrangian), the structure of the anomaly implementing terms are the same as those for the 1-2 Lagrangians Eqns. 6.50 and 6.51, except φ is now a 3 × 3 ﬁeld for complex and real irreps and a 6 × 6 ﬁeld for pseudoreal irreps. Unlike in the 1-2 case, the coupling c associated with the anomaly-implementing term is a dimensionless four-point

~~2 coupling. In this case, βvr will be corrected by a term proportional c , so we cannot simply solve~~

~~βvr = 0 and remove vr from the problem as we do elsewhere in the calculation. Instead, we will~~

~~simply include βvr and solve for vr as we do with the other couplings; with only ﬁve β functions for each of these Lagrangians, the added complexity is not an issue.~~

~~For the 1-C3 Lagrangian, we ﬁnd~~

~~2 2 βu1 = −u1 + 5u1 + 9w~~

~~2 2 2 2 βur = −ur + 13ur + 12urvr + 3vr + w + 4c~~

~~2 2 (6.62) βvr = −vr + 6urvr + 6vr − 4c~~

~~2 2 βw = −w + 2u1w + 10urw + 6vrw + 2w + 8c~~

~~βc = −c + 3urc − 3vrc + 3wc; for the 1-R3 Lagrangian, we ﬁnd~~

~~2 2 βu1 = −u1 + 5u1 + 6w~~

~~2 3 2 2 2 βur = −ur + 10ur + 8urvr + 2 vr + w + 4c~~

~~11 2 2 (6.63) βvr = −vr + 6urvr + 2 vr − 4c~~

~~2 2 βw = −w + 2u1w + 7urw + 4vrw + 2w + 6c~~

~~3 βc = −c + 3urc − 2 vrc + 3wc; 141 and for the 1-P3 Lagrangian, we ﬁnd~~

~~2 2 βu1 = −u1 + 5u1 + 15w 3 β = −u + 19u2 + 10u v + v2 + w2 − 1 c2 ur r r r r 2 r 2 7 2 2 (6.64) βvr = −vr + 6urvr + 2 vr + c~~

~~2 3 2 βw = −w + 2u1w + 16urw + 5vrw + 2w − 2 c~~

~~βc = −c + 3urc − 3vrc + 3wc where couplings subscripted with 1 are for the nr = 1 irrep and with r are for the nr = 3 irrep.~~

All coefficients in these β functions are numerical, so the fixed points are straightforward to solve for numerically. The values of the couplings at the nontrivial real-valued fixed points for each

Lagrangian are listed in Table 6.5. In principle, we could have found up to 25 = 32 fixed points, but in practice we find only 6 with real-valued couplings, as with all other two-rep theories that we have examined. Although in principle we could have found vr 6= 0, we find vr = 0 for all fixed points. We also find c = 0 for all fixed points. It follows that the values of the remaining couplings at each fixed point are the same as those in the corresponding two-irrep Lagrangians without an anomaly-implementing term.

3 The 1-1 Lagrangian lacks a vr term, so we treat it separately. The anomaly-implementing term is c δLAI = [φ φ3 + (c.c)] (6.65) 3! 1 2 where c has been redeﬁned to absorb combinatoric factors. For its β functions, we ﬁnd

~~2 2 βu1 = −u1 + 5u1 + w~~

~~2 2 2 βu2 = −u2 + 5u2 + w + 2c (6.66) 2 2 βw = −w + 2u1w + 2u2w + 2w + 2c~~

~~βc = −c + 3u2c + 3wc.~~

The values of the couplings at the nontrivial real-valued ﬁxed points for this Lagrangian are listed in Table 6.6. Unlike in the other 1-3 cases, this theory admits two ﬁxed points where c 6= 0. 142

~~Computing the stability matrices, plugging in each ﬁxed point, and diagonalizing, we obtain the number of relevant directions (i.e. negative eigenvalues) about each ﬁxed point. For the 1-C3,~~

~~1-R3, and 1-P3 Lagrangians we find at least two negative eigenvalues for each fixed point, so none of the fixed points is stable.~~

For the 1-13 Lagrangian, we find a single fixed point with no negative eigenvalues, but two zero eigenvalues, indicating exactly (at the order of this calculation) marginal directions. Whether this fixed point is stable or not will depend on NNLO contributions beyond the scope of this calculation. Thus, the results are ambiguous for these theories: we cannot exclude the existence of a stable fixed point, but cannot conclusively determine that one exists.

~~6.7.4 2-2 anomaly~~

~~For the C2-R2 Lagrangian, the anomaly-implementing term in the Lagrangian is~~

~~AI δL = c Det φC Det φR (6.67) and for the P2-R2 Lagrangian, it is~~

~~AI δL = c Pf φP Det φR. (6.68)~~

~~As with the 1-3 Lagrangians above, the c coupling associated with the anomaly-implementing terms~~

~~2 here is dimensionless and βvr receives a correction proportional to c . Thus, we leave βvr in the analysis and solve for vr as with all other couplings. For the C2-R2 Lagrangian, we ﬁnd~~

~~β = −u + 8u2 + 8u v + 3v2 + 3w2 + 6c2 uC C C C C C~~

~~β = −v + 6u v + 4v2 − 6c2 vC C C C C~~

~~β = −u + 7u2 + 6u v + 3 v2 + 4w2 + 8c2 uR R R R R 2 R (6.69) β = −v + 6u v + 9 v2 − 8c2 vR R R R 2 R~~

~~2 2 βw = −w + 5uCw + 4vCw + 4uRw + 3vRw + 2w + 4c~~

~~1 βc = −c + uCc − vCc + uRc − 2 vRc + 4wc. 143~~

Theory u1 ur w Nλ<0 1-C3 0.2 0 0 4 0 0.0769231 0 4 0.2 0.0769231 0 2 0.0714286 0.0714286 0.0714286 3 0.175975 0.0744978 -0.0484642 3 1-R3 0.2 0 0 4 0 0.1 0 4 0.2 0.1 0 2 0.0909091 0.0909091 0.0909091 3 0.19154 0.0986309 -0.0367477 3 1-P3 0.2 0 0 4 0 0.0526316 0 4 0.2 0.0526316 0 2 0.05 0.05 0.05 3 0.15 0.05 -0.05 3

Table 6.5: Nontrivial fixed points for the β functions Eqs. 6.62, 6.63, and 6.64. For all fixed points, vr = 0 and c = 0. The value of each coupling is provided to six digits of precision. Nλ<0 indicates the number (out of five) of negative eigenvalues of the stability matrix at this fixed point.

~~Theory u1 u2 w c Nλ<0 1-13 0.2 0 0 0 3 0 0.2 0 0 3 0.2 0.2 0 0 2 0.166667 0.166667 0.166667 0 0∗ 0.155312 0.147047 0.186287 ±0.045991 2~~

Table 6.6: Nontrivial ﬁxed points for the β functions Eq. 6.66. The value of each coupling is provided to six digits of precision. Nλ<0 indicates the number (out of four) of negative eigenvalues of the stability matrix at this ﬁxed point; 0∗ indicates that two of the eigenvalues are exactly zero. 144

~~For the P2-R2 Lagrangian, we ﬁnd~~

~~β = −u + 10u2 + 6u v + 3 v2 + 3w2 + 3 c2 uP P P P P 2 P 2~~

~~β = −v + 6u v + 3 v2 − 3c2 vP P P P 2 P~~

~~β = −u + 7u2 + 6u v + 3 v2 + 6w2 + 3c2 uR R R R R 2 R (6.70) β = −v + 6u v + 9 v2 − 3c2 vR R R R 2 R~~

~~2 2 βw = −w + 7uPw + 3vPw + 4uRw + 3vRw + 2w + c~~

~~1 βc = −c + uPc − vPc + uRc − 2 vRc + 4wc.~~

~~2 The 1 -C2, etc., Lagrangians lack one of the vr couplings, so we treat them separately. For~~

~~2 2 the 1 -C2 and 1 -R2 Lagrangians, the correction is c δLAI = δLAI = (φ )2 Det φ + (c.c.); (6.71) 12-C2 12-R2 2! 1 2~~

~~2 and for 1 -P2 Lagrangian, it is c δLAI = (φ )2 Pf φ + (c.c.) (6.72) 12-P2 2! 1 2~~

~~2 where c has been redeﬁned to absorb combinatoric factors. The β functions for the 1 -C2 case are~~

~~2 2 2 βu1 = −u1 + 5u1 + 4w + 4c~~

~~β = −u + 8u2 + 8u v + 3v2 + w2 + 2c2 uC C C C C C~~

~~β = −v + 6u v + 4v2 − 2c2 (6.73) vC C C C C~~

~~2 2 βw = −w + 2u1w + 5uCw + 4vCw + 2w + 4c~~

~~βc = −c + u1c + uCc − vCc + 4wc.~~

~~2 For the 1 -R2 case, we ﬁnd~~

~~2 2 2 βu1 = −u1 + 5u1 + 3w + 3c~~

~~β = −u + 7u2 + 6u v + 3 v2 + w2 + 2c2 uR R R R R 2 R~~

~~β = −v + 6u v + 9 v2 − 2c2 (6.74) vR R R R 2 R~~

~~2 2 βw = −w + 2u1w + 4uRw + 3vRw + 2w + 4c~~

1 βc = −c + u1c + uRc − 2 vRc + 4wc. 145 C2-R2 P2-R2 uC uR w Nλ<0 uP uR w Nλ<0 0.125 0 0 5 0.1 0 0 5 0 0.142857 0 5 0 0.142857 0 5 0.125 0.142857 0 3 0.1 0.142857 0 3 0.0909091 0.0909091 0.0909091 4 0.0769231 0.0769231 0.0769231 4 0.116175 0.130925 -0.0522872 4 0.0896469 0.120929 -0.0556215 4

Table 6.7: Nontrivial fixed points for the β functions Eqs. 6.69 and 6.70. For all fixed points, vr = 0 and c = 0. The value of each coupling is provided to six digits of precision. Nλ<0 indicates the number (out of six) of negative eigenvalues of the stability matrix at this fixed point.

~~2 For the 1 -P2 case, we ﬁnd~~

~~2 2 3 2 βu1 = −u1 + 5u1 + 6w + 2 c~~

~~β = −u + 10u2 + 6u v + 3 v2 + w2 + 1 c2 uP P P P P 2 P 2~~

~~β = −v + 6u v + 3 v2 − c2 (6.75) vP P P P 2 P~~

~~2 2 βw = −w + 2u1w + 7uPw + 3vPw + 2w + c~~

~~βc = −c + u1c + uPc − vPc + 4wc.~~

As with the 1-3 Lagrangians, all coefficients in the β functions are numerical so we may solve for the zeros numerically. The values of the couplings at the nontrivial real-valued fixed points for each Lagrangian are listed in Tables 6.7 and 6.8. We again find 6 real-valued fixed points for each set of β functions, again with vr = 0 and c = 0 such that the values of the remaining couplings at each fixed point are the same as those in the corresponding two-irrep Lagrangian without an anomaly-implementing term. For each fixed point, the stability matrix has at least two negative eigenvalues, so we again find no stable fixed points. 146

Theory u1 ur w Nλ<0 2 1 -C2 0.2 0 0 4 0 0.125 0 4 0.2 0.125 0 2 0.111111 0.111111 0.111111 3 0.199458 0.124865 -0.0116204 3 2 1 -R2 0.2 0 0 4 0 0.142857 0 4 0.2 0.142857 0 3 0.125 0.125 0.125 3 0.199278 0.142617 0.0154886 2 2 1 -P2 0.2 0 0 4 0 0.1 0 4 0.2 0.1 0 2 0.0909091 0.0909091 0.0909091 3 0.19154 0.0986309 -0.0367477 3

Table 6.8: Nontrivial fixed points for the β functions Eqs. 6.73, 6.74, and 6.75. For all fixed points, vr = 0 and c = 0. The value of each coupling is provided to six digits of precision. Nλ<0 indicates the number (out of five) of negative eigenvalues of the stability matrix at this fixed point. 147

~~6.8 Discussion~~

~~Our analysis ﬁnds stable ﬁxed points in the three 1-1 theories listed in Table 6.1. In the three~~

1-13 theories, it finds a fixed point with no relevant directions but two exactly marginal ones (at leading nontrivial order); its stability thus depends on NNLO corrections and is ambiguous at this order. Otherwise, our analysis finds a complete absence of stable fixed points in the full infinite class of multirep theories with vectorlike fermion content that we enumerated in Chapter2. Our analysis thus indicates that the three 1-1 theories and three 1-13 theories are the only multirep theories that may not exhibit first-order phase transitions, and that first-order phase transitions are an almost completely generic feature of multirep theories.

~~This ﬁnding applies to both speciﬁc cases of Ferretti’s model and the lattice-deformed Ferretti model. Our lattice studies indicate that the lattice-deformed Ferretti model indicated a strongly~~

first-order transition. This calculation provides independent analytic support for that finding. In addition, this calculation makes the same prediction for both of these models, providing support for the idea that our lattice result will apply for the flavor content of Ferretti’s model as well.

~~It is worth reiterating the limitations of the calculation.~~

Our calculation assumed simultaneous chiral symmetry breaking in all irreps. As discussed above, the phase transitions in the ten theories where we find stable fixed points are inherently simultaneous phase transitions, because the spontaneously broken symmetry that defines the system is one involving both of the irreps present. However, if a theory is found to exhibit multiple distinct chiral transitions, this calculation will not apply.

~~The validity of these results depend on whether working to one loop and leading order in the~~

~~ expansion (i.e., lowest nontrivial order) is suﬃcient to exclude the existence of stable ﬁxed points.~~

The original application of this method applies to single-irrep theories where the fermions are in a complex irrep (e.g., fundamentals in QCD) [130], and predicted that the chiral transition must be first order for Nf ≥ 2 (without anomaly-implementing terms). More recently, however, a more sophisticated version of this calculation predicted a stable fixed point in the NF = 2 theory that is 148 missed by this leading-order treatment Nf > 2 [23, 24, 39]. This may permit a stable decoupled- product fixed point in, for example, some theory with two distinct complex irreps with n1 = n2 = 2.

Further, we cannot argue that such stable ﬁxed points will not appear generically if the calculation done here is taken to higher orders. However, in favor of the validity of this calculation, our numerical study of the lattice-deformed Ferretti model suggests a ﬁrst order transition consistent with the predictions of this calculation.

~~6.A Constraints due to vacuum stability and χSB pattern~~

The relative values of the bare couplings in each Lagrangian must be constrained to ensure that the vacuum is stable, to give the desired chiral symmetry breaking pattern, and to ensure that all irreps of the theory are chirally condensed at zero temperature. When rr < 0 the solution for the φr ﬁeld associated with the correct χSB pattern when r is a complex or real irrep is

~~φ = φ0I, (6.76) where I is the nr × nr identity matrix. For pseudoreal irreps, the desired solution is~~

~~P φ = φ0J , (6.77) where J P is the nr × nr antisymmetric matrix Eq. 4.11. In practice, this diﬀerent form for the~~

~~2 necessary pseudoreal solution will not matter, as we will only work with |φ0| ∼ I in the analysis below.~~

For real reps, in cases where nr is even, it is possible to arrange the Weyl fermions in to Dirac fermions and demand that the theory respect a U(nr/2)V vector symmetry. The Vafa-Witten theorem states that this symmetry will not be broken by a QCD-like theory [161]. Accommodating this condition would require us to make θ ∝ J R, where J R = |J P| is the symmetric equivalent of the matrix J P (Eq. 4.11)[23]. It is not possible to construct this matrix when nr is odd, which reﬂects that it is not possible to deﬁne SU(nr/2)V with an odd number of Weyl degrees of freedom

~~2 2 [62]. Because (J R) = I and all results below are in terms of |θ| ∝ I, using J R in cases where nr is even would not change anything. 149~~

~~There is another family of solutions that corresponds to a diﬀerent, physically-irrelevant SSB pattern,~~

~~φ = φ0Ik (6.78)~~

where Ik is a matrix which is all zeroes except for k many 1s on the diagonal. In practice, it is suﬃcient to consider the extremal case I1 to develop constraints [23]. These undesired solutions provide additional channels through which the vacuum may destabilize, so they must also be taken into account. Even when the vacuum is stable, further constraints are required to guarantee they are not the minimum of the potential, which would give the wrong chiral symmetry breaking pattern.

In the discussion that follows we will mostly specialize to the case of two distinct irreps and discuss how each result generalizes to the R > 2 case. Following the lattice-deformed Ferretti model, we name these F and A2, for which we will name the ﬁelds φ and θ, respectively. What the two irreps actually are has no eﬀect on the analysis.

Enumerating over the the desired vacuum ∼ I and the undesired vacuum ∼ I1 for each of R irreps, there are 2R possible ground states of the overall potential to consider. In what follows, the couplings always appear in characteristic combinations when expressions are evaluated for a given ground state. For notational clarity and to avoid repeatedly enumerating lengthy expressions for each of the four ground states, we will write expressions below in terms of general couplings whose deﬁnition depends on the ground state of interest. The couplings from the single-irrep sectors are unaware of the other sector, so we deﬁne

~~RF ≡ NF rF ,UF ≡ NF (NF uF + vF ) when φ ∝ I (6.79)~~

~~RF ≡ rF ,UF ≡ uF + vF when φ ∝ I1 and w w w RA2 ≡ NA2 rA2 ,UA2 ≡ NA2 (NA2 uA2 + vA2 ) when θ ∝ I (6.80)~~

~~RA2 ≡ rA2 ,UA2 ≡ uA2 + vA2 when θ ∝ I1. 150~~

~~The irrep-coupling sector is aware of the ground state of both single-irrep sectors, so we deﬁne~~

~~w W = NF NA2 w, when (φ, θ) ∝ (I, I)~~

~~W = NF w, when (φ, θ) ∝ (I, I1) (6.81) w W = NA2 w when (φ, θ) ∝ (I1, I)~~

~~W = w when (φ, θ) ∝ (I1, I1).~~

~~In the discussion that follows, one need only plug in the appropriate deﬁnitions to recover the expressions for each ground state.~~

~~Plugging in the non-trivial ground states, we ﬁnd that the potential for our theory is~~

~~2 1 4 2 1 4 1 2 2 V (φ0, θ0) = RF |φ0| + 4 UF |φ0| + RA2 |θ0| + 4 UA2 |θ0| + 2 W |φ0| |θ0| . (6.82)~~

~~At large values of the ﬁeld, only the quartic part of the potential~~

~~2 2 1 4 1 4 1 2 2 V4(|φ0| , |θ0| ) = 4 UF |φ0| + 4 UA2 |θ0| + 2 W |φ0| |θ0| (6.83) is pertinent to vacuum stability. For each of the four ground states, we require that~~

~~2 2 lim lim V4 |φ0| , |θ0| > 0 (6.84) 2 2 |φ0| →∞ |θ0| →∞ where the inequality is strict or the full potential Eq. 6.82 will be unbounded from below when~~

~~either of RF < 0 or RA2 < 0. The condition Eq. 6.84 must be satisﬁed regardless of how the two limits are taken. To explore this condition, we demand~~

~~2 2 lim V4 a|x| , b|x| > 0 2 |x| →∞ (6.85) 1 2 1 2 1 ⇔ 4 UF a + 4 UA2 b + 2 W ab > 0 for all a ≥ 0, b ≥ 0, a + b > 0, and simultaneously for all four ground states.~~

~~Taking a = 0 we ﬁnd UF > 0, and taking b = 0 we ﬁnd UA2 > 0: the single-irrep subsectors~~

~~must be independently stable. Plugging back in for UF and UA2 , we recover the stability conditions familiar from analyses of the single-rep subsectors of the multirep theory [130, 23, 39],~~

~~w NF uF + vF > 0,NA2 uA2 + vA2 > 0, (6.86)~~

~~uF + vF > 0, uA2 + vA2 > 0. 151~~

Note that the Nu + v conditions do not subsume the u + v conditions, as u may be positive or negative. These single-irrep conditions generalize trivially to the case of R irreps. When these conditions are satisﬁed, it is obvious from the form of Eq. 6.85 that a positive W cannot destabilize the potential. To bound negative W s, consider a continuation of the condition Eq. 6.85 where we allow a and b to range over all real numbers including negatives. When both of a < 0 and b < 0, the signs of all terms in Eq. 6.85 are unchanged. However, if only one of a or b is negative, the sign of the W term is ﬂipped. Thus, allowing negative a and b and demanding stability amounts to simultaneously requiring

~~1 2 1 2 1 4 UF a + 4 UA2 b + 2 |W |ab > 0 (6.87) 1 2 1 2 1 4 UF a + 4 UA2 b − 2 |W |ab > 0 for all a ≥ 0, b ≥ 0, a + b > 0. Assuming the single-irrep subsectors are independently stable~~

~~(U > 0), the +|W | subcondition will always be satisﬁed and the condition thus bounds only~~

W < 0. With a and b allowed to range over all reals, condition Eq. 6.85 is equivalent to the requirement for a positive-deﬁnite quadratic form in (a, b). A positive-deﬁnite quadratic form has positive eigenvalues; demanding that this is true for the LHS of Eq. 6.85 yields the condition that

~~2 UF UA2 > W . Because this is only required when W < 0, we obtain~~

~~p W > − UF UA2 . (6.88)~~

~~This reduces to~~

w > −uF uA2 (6.89) in the usual case where v = 0. Generalizing, we must have stability in any two-irrep subtheory of an R-irrep theory to have stability in the overall potential; thus, this condition applies separately to all w-type couplings in any multirep theory.

To ensure that the correct χSB pattern is realized, we must constrain the couplings such that the φ ∝ I and θ ∝ I solutions minimize the potential. When φ and θ are extrema of the potential, the value of the potential can be expressed as [23]

~~1 2 1 2 1 2 1 2 Vsoln. = 2 rF |φF | + 2 rA2 |θ| = 2 RF |φ0| + 2 RA|θ0| . (6.90) 152~~

~~There is obviously a disordered phase where (φ, θ) = (0, 0) and thus V = 0. When RF < 0, there~~

2 2 exists a phase where only φ is ordered. In this phase |φ0| = −2RF /UF and so V = −RF /UF . Similarly, |θ |2 = −2R /U and V = −R2 /U when R < 0 and only θ is ordered. The 0 A2 A2 A2 A2 A solution when both φ and θ are ordered is

~~2 WRA2 − RF UA2 2 WRF − RA2 UF |φ0| = 2 2 , |θ0| = 2 2 (6.91) UF UA2 − W UF UA2 − W for which the value of the potential is~~

~~2WR R − R2 U − R2 U F A2 F A2 A2 F V = 2 . (6.92) UF UA2 − W~~

~~To obtain conditions on the couplings, we demand that no ground state with φ ∝ I1 and/or~~

θ ∝ I1 is the minimum of the potential anywhere. For brevity, we henceforth refer to the ground state where (φ, θ) ∝ (I, I) as the (I, I) ground state and the potential for this ground state as V (I, I), etc. For the case where only a single sector is ordered, demanding that V (I, 0) < V (I1, 0) yields the condition vF ≥ 0 and similarly demanding that V (0, I) < V (0, I1) yields the condition vA2 ≥ 0. This generalizes obviously to the case of R irreps. For the case where both φ and θ are ordered, we similarly find that vF ≥ 0 ensures that both V (I, I) < V (I1, I) and V (I, I1) < V (I1, I1) hold; and that vA2 ≥ 0 ensures that both V (I, I) < V (I, I1) and V (I1, I) < V (I1, I1) hold. Applying transitivity, if vF ≥ 0 and vA2 ≥ 0, then the (I, I) ground state minimizes the potential when both φ and θ are ordered. We find no additional conditions on either v-type coupling from the two-rep condition, which we will assume generalizes to the case of R irreps. These conditions are the same as what is found in analyses of the single-irrep subsectors of the potential [130, 39, 23], so irrep coupling does not seem to affect which χSB pattern is realized. Bibliography

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~~Autocorrelations in MCMC Simulations~~

For a sequence of measurements of any given observable Ot in MCMC time t, the autocorrelation function is deﬁned as 1 PN−τ (O(t) − hOi)(O(t + τ) − hOi) C(τ) = N−τ t=1 (A.1) hO2i − hOi2 where h· · · i is the usual MCMC expectation value taken by averaging over all t (cf. Eq. 3.30).

The autocorrelation function is normalized such that C(0) = 1. For long τ, C(τ) ∼ exp[−τ/τAC ], where τAC is the “autocorrelation time”. It is often stated in the MCMC literature that there is a unique longest autocorrelation time in any given simulation of a system, and that all observables are sensitive to all autocorrelation times in the system. Thus, at long τ, we should observe the same τAC in all observables. However, the author has not observed this to be the case in lattice simulations.

As an extremely pathological example, the autocorrelation time in the topological charge (which remains static between tunneling events where the system moves to a diﬀerent topological sector) is typically dramatically longer than τAC as measured for any other observable. Less pathologically, the author has observed that long-distance observables like Wilson-ﬂowed operators and Polyakov loops typically have longer autocorrelation times than short-distance observables like the plaquette.

~~If C(τ) = exp[−τ/τAC ] is exact, then Z ∞ dτC(t) = τAC . (A.2) 0 This inspires the “integrated autocorrelation time”, ∞ 1 X τ ≡ + C(τ). (A.3) I 2 τ=1 166~~

~~In practice, C(τ) becomes very noisy at long times due to limited statistics (autocorrelation functions are notoriously noisy), so summing out to τ = ∞ does not produce the desired quantity.~~

~~Instead, one can truncate the sum in τI self-consistently like~~

~~(I) 10τAC (I) 1 X τ ≡ + C(τ). (A.4) AC 2 τ=1~~

This quantity requires no fits to compute and serves as a practical means of estimating τAC (given sufficient statistics), which is notoriously difficult to estimate. τI is typically less than τAC [136].

~~This self-consistent truncation procedure is sometimes called the “Madras-Sokal” procedure [117].~~

Autocorrelations impede the convergence of observables computed with MCMC. Suppose we generate N configurations using MCMC and measure some observable O on them. If our measurements Ot are autocorrelated, we have less statistical information about O than if we had N uncorrelated measurements. This is obviously true in the pathological case where the autocorrelation time diverges and all N configurations are identical. In this case, we have only one independent measurement of O and thus infinite uncertainty about its value. Clearly, we must account for autocorrelations to compute the uncertainty in O. We may do so analytically, where if we are able to estimate τ accurately we can write

~~N δO2 = g δ = g h(O − hOi)2i (A.5) O O,naive O N − 1 where~~

~~O O g = 1 + 2τAC ≈ 2τI . (A.6)~~

~~For data without autocorrelations, τAC → 0 and so g → 1 and we recover the uncorrected form for~~

~~2 2 δO (cf. Eq. 3.29). Conversely, if the autocorrelation time diverges so does δO, indicating that our estimate lacks statistical validity with only one statistically-independent measurement.~~

In practice, one more frequently corrects for autocorrelations using “blocking” or “binning”, which can be applied without attempting to measure τAC . Autocorrelations are an issue when the separation ∆t between measurements in Monte Carlo time is not signiﬁcantly greater than the autocorrelation time τAC . To increase the separation between measurements, we can simply thin 167 the data, discarding all but every Bth measurement and thus increasing the separation between measurements to B∆t. However, thinning throws away statistical information: if τ is ﬁnite and

~~C(t) 6= 1, then any set of measurements contains more information than an individual measurement.~~

~~The idea of blocking is to use all N autocorrelated measurements to estimate a smaller set of N/B measurements without autocorrelations. To block a sequence of N measurements of some observable~~

~~Ot where t ∈ [1,N], group the measurements into N/B blocks of B consecutive measurements.~~

~~Compute the average of Oi within each bin, i.e., for bin index b ∈ [1, N/B],~~

~~bB 1 X O = O . (A.7) b B i i=(b−1)B~~

Finally, replace your N measurements Oi with the resulting set of N/B measurements Ob separated by B∆t. Determining a suﬃciently long block size B requires some art; in practice, one usually tries increasingly large values of B until the error in the measurement stops growing larger.

An often-overlooked subtlety in blocking is how to correctly measure correlated observables from autocorrelated data. Suppose we have two observables, X and Y , which are correlated such that hXY i= 6 hXihY i. Generate N autocorrelated conﬁgurations and measure Xi and Yi on each.

~~To estimate hXY i correctly, we must compute the composite observable (XY )i = XiYi for each P conﬁguration, block the result (XY )i → (XY )b, then ﬁnally compute hXY i = 1/(N/B) b(XY )b.~~

The incorrect alternative is to separately block Xi → Xb and Yi → Yb, then attempt to estimate P hXY i as 1/(N/B) b XbYb. This procedure is simply inconsistent. Heuristically, we know X and Y will ﬂuctuate in a correlated way; blocking damps ﬂuctuations, so if we average blocks of X and

Y independently we will wash away the correlations in their ﬂuctuations. We may also consider the limiting case where the binsize B → N; in this limit our uncertainty diverges, but the value of the P P single bin b(XY )b → (XY )1 = hXY iB=1 while b XbYb → X1Y1 = hXiB=1hY iB=1, where the h· · ·iB=1 are computed over the unbinned dataset. Finally, consider a pathological case where X is uniformly distributed in [0, 2] and Y = 1 if A > 1 and 0 otherwise such that hXY i = 0.75. Our results should remain correct as long as B N. However, taking B τAC , we will have Ab → 1 P and Bb → 0.5 such that ( b AbBb)/N/B → 0.5. Appendix B

~~Group Theory~~

~~B.1 Diagnosing complexity classes of irreps of SU(N)~~

~~For any complex irrep R, there exists a conjugate irrep R (whose Dynkin labels are the reﬂection of R’s). If the Dynkin labels ai are invariant under reversal, then the irrep is self-conjugate~~

~~(i.e., R = R). More precisely, given the Dynkin labels ai for a given irrep, a self-conjugate irrep is deﬁned as an irrep where~~

~~ai = aN−1−i ∀a. (B.1)~~

If an irrep is not self-conjugate, it is complex; self-conjugate irreps are either real or pseudoreal. To diﬀerentiate between real and pseudoreal irreps, we require a second diagnostic. Deﬁne the vector

I(Nc) = (1Nc, 2(Nc − 1), 3(Nc − 2),...Nc1), (B.2) and from that “irrep height” N −1 Xc I = aiI(Nc)i (B.3) i (we note that this quantity is standardly denoted T , but we want to avoid confusion with the trace of an irrep). If a self-conjugate irrep has an even height, it is real; if a self-conjugate irrep has an odd height, it is pseudoreal [150].

~~B.2 Algebraic methods for linear sigma models~~

In the calculation described in Chapter6, we found that the form of the fields in the linear sigma model Lagrangian depend on the complexity class of the associated irrep of fermion. Specif- 169 ically, the fields are arbitrary complex matrices for complex irreps, symmetric complex matrices for real irreps, and antisymmetric complex matrices for pseudoreal irreps. One may obtain these forms simply by asking what fields are invariant under the appropriate chiral symmetry group, as described in Section 4.2. However, we may obtain the forms of the fields through a more physical argument: the fields in a linear sigma model may be thought of as parametrizing the scalar and pseudoscalar modes of the theory of interest.

~~To see this, note that the scalar order parameter (chiral condensate) ﬁeld φ may be expressed in terms of the coset of broken generators τ i associated with the χSB pattern of the relevant irrep.~~

~~The axial anomaly is accounted for by symmetry-breaking terms in the Lagrangian so we include~~

U(1)A → 1 in determining the coset of broken generators; this leads to the set of generators for the cosets associated with real and pseudoreal irreps including (in any basis with one non- traceless element) an element proportional to the identity. The broken coset for complex irreps is U(N) × U(N)/U(N) ≈ U(N), for which the generators τ span the fundamental irrep of u(N).

For real irreps, the broken coset is U(N)/O(N), for which we choose τ that span the fundamental irrep of u(N)/o(N). To construct a basis for this coset, notice that the generators of O(N) are all antisymmetric; we thus construct a basis for u(N) wherein generators are either symmetric or antisymmetric, and remove the antisymmetric ones. The broken coset for pseudoreal irreps is

~~SU(2N)/Sp(2N), which satisfy J PX = XT J P = −(J PX)T for some antisymmetric orthogoanl J P.~~

~~By picking some basis for 2N × 2N antisymmetric matrices, then multiplying by an appropriate choice of J P, we obtain a basis for the broken coset.~~

Consider the case of a complex irrep. We may decompose φ like φ = SP where S ≡ sjτ j, with sj real, is a Hermitian matrix describing the scalar modes; and P ≡ exp[ipjτ j], with pj real, is a unitary matrix describing the pseudoscalar modes. The product SP is an arbitrary N D × N D complex matrix, with (2N D)2 real degrees of freedom parameterized by the sj and pj. In this form, it is straightforward to recover the ﬁrst (only non-irrelevant) term in the chiral Lagrangian by tuning couplings to decouple the scalar modes and anomalous axial pseudoscalar mode. A similar procedure works for real and pseudoreal irreps, up to complications due to additional group theory 170

~~T 1 T 1 structure: to get the correct symmetry φ = ±φ , one must represent φ = (P 2 ) JSP 2 where~~

~~J 2 = ±1 for real/pseudoreal irreps.~~

We may obtain a convenient basis for calculation by manipulating these physically-motivated decompositions. By expanding the exponential in P , reducing products of multiple τs to sums of single τs, and gathering coefficients, we find that φ may instead be parameterized as a sum over the broken generators with complex coefficient fields. Specifically, for complex irreps, the field φ may be expanded like

~~a i i a φb = ϕ (τ )b (B.4) where τ are as described above for each irrep and the φi are complex scalar ﬁelds. For real and pseudoreal irreps, the expansion is~~

a i i a φb = ϕ (Jτ )b (B.5) where J 2 = ±J 2 for real (+) or pseudoreal (−) irreps. With complex coefficients, (J times) the cosets associated with complex, pseudoreal, and real irreps provide bases for arbitrary, antisymmetric, and symmetric complex matrices, respectively. For pseudoreal irreps, the complexity of the coefficient field ϕ allows the choice of the set of generators iJτ for a basis; with the additional factor of i, these form a basis for so(N).

In these bases, the Feynman rules are simply those for coupled complex |φ|4 theories with additional ﬂavor group structure multiplying each vertex. Computing the group-theoretic weights associated with each diagram reduces to an exercise in generator algebra. For the coset U(N), the usual u(N) algebra identities are available. Meanwhile, the set of generators of U(N)/O(N) is not closed under commutation, so they do not form a Lie algebra and only a reduced set of generator

I I identities is available. Taking in to account that the generators are symmetric τAB = τBA, where I is an adjoint index and A and B are fundamental indices, we ﬁnd a suﬃcient set of identities to 171 perform the computation is:

~~I J IJ IJ Tr τ τ = TF δ = δ~~

~~I I 1 (τ )AB(τ )CD = 2 (δAC δBD + δADδBC ) (B.6) II U(N) O(N) 1 δ = dG − dG = 2 N(N + 1)~~

~~I I U(N) O(N) 1 (τ )AB(τ )BC = CF − CF δAC = 2 (N + 1)δAC where TF = 1 is the trace of the fundamental representations of U(N) and O(N), set to be consistent~~

~~U(N) with the conventional normalization of the kinetic term for complex scalar ﬁelds, and CF and O(N) CF are the quadratic Casimirs of the fundamental representations of U(N) and O(N). As~~

~~i discussed above, for the coset SU(2N)/Sp(2N) one may expand φ = φiτ with τi the generators of~~

SO(2N), so the usual generator identities for so(2N) generators are available. Summing all one- loop diagrams contributing to a process and using coset generator identities to reduce the flavor group structure, the contribution to each counterterm can be found as the coefficient of the flavor group structure associated with the corresponding coupling. Appendix C

~~Gradient Flow Phase Diagnostics~~

~~C.1 Flowed anisotropy~~

~~As discussed in Section 3.6, the ﬂowed observable ht2E(t)i is commonly used to compute scales~~

~~2 like t0/a on zero-temperature lattices. When measured on ﬁnite temperature lattices, spatial- temporal anisotropy in this same observable can be employed to diagnose the phase [55, 54, 166].~~

µν Usually, one computes hEi = hFµνF i with F as the clover terms for each site, with each clover’s orientation being deﬁned by the two directions it extends in. This observable can be decomposed as

2 2 2 ht E(t)i = ht Ess(t)i + ht Est(t)i (C.1) where Ess is the contribution to E from the three space-space (xy, xz, yz) clovers and Est is the contribution from the three space-time (xt, yt, zt) clovers. Previous applications of this diagnostic have used the quantity

~~2 ∆(t) = t hEss(t) − Est(t)i. (C.2)~~

~~In this work we will use a related quantity, the ﬂowed anisotropy RE, deﬁned as~~

~~RE(t) ≡ hEss(t)/Est(t)i, (C.3)~~

~~(cf. Ref. [37]). This observable probes spontaneous symmetry breaking of hypercubic symmetry~~

~~(which is exact in hypercubic volumes L4 without dynamical fermions, but approximate otherwise).~~

~~In the low-temperature conﬁned and chirally broken phase, the gauge ﬁelds are roughly isotropic. 173~~

2 2 Figure C.1: Evolution under Wilson flow of the split flow observables t Ess and t Est , shown 3 for three different 12 × 6 ensembles on a slice at constant (β, κ6) = (7.4, 0.1285) in the lattice- deformed Ferretti model (see Chapter5). Also shown is the behavior of RE − 1, where RE ≡ hEss(t)i / hEst(t)i. The left panel shows typical confined behavior: the split flow observables are degenerate and RE = 1, indicating isotropy. The central panel shows some hints of impending deconfinement: the split flow observables separate slightly, and RE − 1 becomes nonzero. The right panel shows typical deconfined behavior: the split flow observables break apart, and RE − 1 grows rapidly in flow time, indicating anisotropy. 174

In this case, spacelike and timelike clovers are related by (approximate) hypercubic symmetry and the observable RE(t) ≈ 1 for any reasonable flow time t. In the high-temperature deconfined and chirally restored phase, hypercubic symmetry is broken strongly: the temporal center symmetry is broken, while the spatial center symmetries are still (approximately) preserved. In such anisotropic phases, RE(t) departs from unity even at small flow times. In this paper, we always measure RE(t) at flow time t/a2 = 1. Thus, at finite t, this quantity provides a sharp diagnostic of phase, as seen in Fig. C.1.

~~C.2 Polyakov loops at long ﬂow times~~

The behavior of Polyakov loops at long flow times provides a sharp diagnostic of confinement. √ For a lattice with temporal extent Nt, the flow time ratio ct = 8t/(aNt) is a rough measure of the Wilson-flow smearing in the temporal direction. Defining “long flow time” as ct > 1, we find that flowed Polyakov loops exhibit strongly divergent behavior at long flow times depending on

~~3 the phase. For the 12 × 6 lattices that form the bulk of the numerics shown in this work, ct = 1 corresponds to t ∼ 5.~~

As found in Chapter5, the lattice-deformed Ferretti model exhibits a strongly first-order phase transition. For this system, the behavior of Polyakov loops at long flow time is virtually binary. Figure C.2 shows the behavior of Polyakov loops for individual configurations under flow for three typical ensembles: one confined, one confined but near the phase boundary, and one deconfined. Fig. C.3 shows the ensemble average for the same data. On deconfined configurations, volume-averaged Polyakov loops rapidly reach their maximal values max |PR| = d(R), where d(R) is the dimension of the representation R (cf. right panels, Figs. C.2 and C.3). On confined lattices, volume-averaged Polyakov loops wander or move only very slowly towards their maximal values

(cf. left panels, Figs. C.2 and C.3). The diﬀerence in behavior is very sharp. Even ensembles near the phase transition, like the one shown in the central panels of Figs. C.2 and C.3, will not order at even extremely long ﬂow times (t > 100).

~~For continuous transitions and ﬁrst-order transitions smoothed into crossovers, this diagnostic 175~~

Figure C.2: Fundamental Polyakov loops under Wilson flow, depicted for three different 123 × 6 ensembles on a slice of constant (β, κ6) = (7.4, 0.1285) in the lattice-deformed Ferretti model (see Chapter5). Each panel depicts the complex plane. Each line is the evolution of PF under Wilson flow on a configuration in the ensemble (i.e., each line is the complex function PF (t) where t is the flow time). In the two left panels, we see typical confined behavior. In the right panel, we see typical deconfined behavior. The ensemble shown in the central panel sits almost directly on top of the confinement transition.

Figure C.3: Ensemble-averaged evolution of PF under Wilson flow, depicted for three different 3 12 × 6 ensembles on a slice of constant (β, κ6) = (7.4, 0.1285) in the lattice-deformed Ferretti model (see Chapter5). In the left panel and center panels, we see typical confined behavior: the Polyakov loop either does not increase in magnitude or increases very slowly. In the right panel, we see typical deconfined behavior: the magnitude of Polyakov loop rapidly approaches its maximum value (max PF = d(F ) = 4). 176 appears to be less effective. Consider for example QCD-like NF = 2 SU(Nc) gauge theories, which appear to exhibit crossovers (as expected for analogues of QCD with approximately strange-mass quarks) rather than an obvious first-order phase transition [63]. In these cases, the Polyakov loop is no longer binary under flow. The first-order transition seen in the multirep lattice theory thus appears to sharpen the long-flow time diagnostic. As visible in Fig. C.4, at low β the Polyakov loop remains closer to zero while at high β it approaches its maximum value as t/a2 increases; it interpolates smoothly across intermediate βs. This is convenient for automation because it is true independent of bare parameters and Nt. Thus, rather than having to specify bare-parameter- dependent cuts on hP i, we may simply make a global definition of phase.

It is a concern that this procedure could wash out an exotic partially conﬁned phase, hiding it from discovery. However, as will be discussed in Chapter5 we do not observe such a phase in the lattice-deformed Ferretti model, and so we have been unable to test whether this is the case. However, if the mechanism behind such a phase is a partial breaking of center symmetry (for

SU(4) theories like the lattice-deformed Ferretti model, this would go as Z4 → Z2 → Z1 cooling from infinite temperature), then it is sufficient for the gradient flow to preserve center symmetry to assure that exotic phases will not be washed out. In fact, this is the case: the application of gradient flow commutes with center symmetry. Under the Wilson flow, each gauge link evolves as dictated by the differential equation Eq. 3.54. Equation 3.54 involves only topologically trivial

~~Wilson loops, so the Wilson ﬂow equations are invariant under center transformations.~~

~~This is not the ﬁrst exploration of the Polyakov loop under smearing [156], including in the context of improving signal in the Polyakov loop, for which Ref. [142] used RG-blocking. Wilson~~

ﬂow has also previously been applied to the Polyakov loop to remove lattice artifacts, obtain renormalized Polyakov loops, and amplify signal [54, 143, 128]. However, all of these studies have stopped short of the extremal “long ﬂow time case” that we discuss here. 177

~~1.0 t/a2 = 0.01 t/a2 = 0.25 t/a2 = 0.5 2 0.8 t/a = 1.0 t/a2 = 2.0 t/a2 = 4.0 t/a2 = 8.0 0.6 c N / | P | 0.4~~

~~0.2~~

~~0.0~~

~~5.10 5.15 5.20 5.25 5.30 5.35 5.40~~

Figure C.4: The evolution of the Polyakov loop under flow along a slice through bare parameter D 3 space varying β at constant κ = 0.128 for NF = 2 SU(3) on 12 ×6. Each line is the same slice, but at different flow times t/a2. As t/a2 increases, Polyakov loops for deconfined ensembles saturate to their maximum value of Nc, while Polyakov loops for confined ensembles remain small. The dashed lines indicate the boundaries used in this study to define confined, ambiguous, and deconfined, for t/a2 = 2.0.