Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

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viii contents 4.6 Algebraic formulation 78 4.6.1 Magnetic flux 82 4.6.2 Allowed magnetic charges 83 4.7 The punchline 84 5 fractional electric charge and quark confinement 87 5.1 Center symmetry breaking by quarks 88 5.2 A hidden global symmetry 90 5.2.1 Historical notes 92 5.2.2 The proposal 96 5.3 Explorations in a 2-color world 98 5.3.1 Model setup 98 5.3.2 Dynamical restoration of symmetry 100 5.3.3 Physical scales 102 5.3.4 Back to the hopping expansion 106 5.3.5 Tuning the Z 2 disorder 111 5.4 Fundamental Higgs with fractional electric charge 112 5.5 Discussion 115 6 concluding remarks 119 a twist away from criticality 121 a.0.1 Magnetic twist and spatial string screening 121 a.0.2 Center vortex string formation at high temperature 123 a.0.3 Electric string formation at zero temperature 125 b more on c-periodic boundary conditions 129 b.1 Classification of C-periodic twists 129 b.2 Mixed C-periodic boundary conditions 130 b.2.1 x direction C-periodic, y, z directions periodic 130 b.2.2 y, z directions C-periodic, x direction periodic 133 c low-energy representations from a lattice gut 137 bibliography 139
I N T R O D U C T I O N 1 The marriage of Quantum Chromodynamics (QCD) and the electroweak theory of Weinberg, Glashow and Salam [1, 2, 3] is resilient. It has withstood more than thirty years of collider experiments and accommodated multiple generations of fermions. This Standard Model of particle physics must be extended to account for neutrino oscillations [4, 5], dark matter, gravity, and other theoretical puzzles, but this does not inhibit its potency as a low energy effective theory. Even the recent discovery of a 125 GeV particle at the Large Hadron Collider (LHC) is so far consistent with the textbook electroweak paradigm [6, 7]. Yet our theoretical understanding of the Standard Model remains incomplete. This is especially true of QCD, the theory of conspiring quarks and gluons that contribute the vast majority of the mass of ordinary matter. On the one hand, asymptotic freedom indicates that QCD is weakly interacting at high energies [8, 9], allowing one to identify the partons of deep inelastic electron-proton scattering experiments with the quanta of quark and gluon fields. On the other hand, isolated quarks and gluons have never been observed. What’s more, the masses of protons and neutrons are much larger than the sum of the current masses of a handful of ‘up’ and ‘down’ quarks [10]. ‘Where are the quarks? Why are we so heavy?’ These questions should be explained by the non-linear nature of QCD at low energies. The interactions of quarks and gluons are not mere perturbations to the kinetic terms of the Lagrangian. They are dominant, allowing for the long-range, infra-red (IR) physics behind quark confinement and dynamical mass generation. The pursuit of an understanding of IR QCD dynamics forms a noteworthy contrast with beyond Standard Model endeavors that include extra symmetries or dimensions. They introduce exotic new physics that become relevant under exotic conditions. These conditions may be out of reach of collider experiments. Meanwhile, QCD is challenging us at room temperature. The complexity of QCD is not diminished as pressure and temperature are increased to the regimes of colliding nuclei or compact stars. In spite of naive expectations from asymptotic freedom, the soup of quarks and gluons produced by the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) does not allow for a simplistic interpretation as a phase of weakly interacting, deconfined particles [11]. Indeed, this ‘quark-gluon plasma’ (QGP) remains strongly coupled. I will refrain from presenting a sketch of the QCD phase diagram at non-zero pressure (chemical potential) and temperature that is commonly inferred from model studies and theoretical arguments. Such a sketch implies that we have a more complete understanding of strong dynamics than we actually do, especially in regimes of large chemical chemical potential where experimental probes and ab initio calculations are lacking. 1
Page 1 and 2: T O P O L O G Y, S Y M M E T RY, A
Page 3: To Angela.
Page 6 and 7: Z U S A M M E N FA S S U N G Wir un
Page 10 and 11: 2 introduction While waiting on exp
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Page 15 and 16: 2.2 quantization 7 The non-abelian
Page 17 and 18: 2.3 lattice field theory 9 Figure 2
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50 universal aspects of confinement
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’ T H O O F T M O N O P O L E S I
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4.1 magnetic charges in the continu
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4.2 magnetic charge on the lattice
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4.4 twisted c-periodic boundary con
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4.5 intuitive picture 73 Therefore
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4.5 intuitive picture 75 The constr
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4.5 intuitive picture 77 This illus
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4.6 algebraic formulation 79 It fol
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4.6 algebraic formulation 81 Figure
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4.6 algebraic formulation 83 Applyi
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4.7 the punchline 85 this restricti
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88 fractional electric charge and q
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100 fractional electric charge and
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C O N C L U D I N G R E M A R K S 6
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T W I S T AWAY F R O M C R I T I C
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twist away from criticality 123 0.1
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130 more on c-periodic boundary con
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L O W- E N E R G Y R E P R E S E N
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B I B L I O G R A P H Y [1] S. Glas
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ibliography 141 [30] L. von Smekal,
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ibliography 143 [60] J. Engels, E.
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ibliography 145 [89] A. S. Kronfeld
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ibliography 147 [120] Y. Aoki, G. E
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ibliography 149 [151] M. P. Nightin
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ibliography 151 [179] C. Korthals A
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C U R R I C U L U M Name: Samuel Ry
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