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