Principles of Phase Structures in Particle Physics

Hildegard Meyer-Ortmanns, Thomas Reisz9789810234416, 981-02-3441-4

The phase structure of particle physics shows up in matter at extremely high densities and/or temperatures as they were reached in the early universe, shortly after the big bang, or in heavy-ion collisions, as they are performed nowadays in laboratory experiments. In contrast to phase transitions of condensed matter physics, the underlying fundamental theories are better known than their macroscopic manifestations in phase transitions. These theories are quantum chromodynamics for the strong interaction part and the electroweak part of the Standard Model for the electroweak interaction. It is their non-Abelian gauge structure that makes it a big challenge to predict the type of phase conversion between phases of different symmetries and different particle contents. The book is about a variety of analytical and numerical tools that are needed to study the phase structure of particle physics. To these belong convergent and asymptotic expansions in strong and weak couplings, dimensional reduction, renormalization group studies, gap equations, Monte Carlo simulations with and without fermions, finite-size and finite-mass scaling analyses, and the approach of effective actions as supplement to first-principle calculations.

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Table of contents :
Contents……Page 12
Preface……Page 8
1. Introduction……Page 20
2.1.1.1 First- and second-order transitions in the infinite volume limit……Page 36
2.1.1.2 Landau’s free energy……Page 40
2.2 Generating functional n-point correlations and effective potentials……Page 45
2.3.1 Self-consistent equation of state for a ferromagnet……Page 52
2.3.1.1 Critical exponents in the molecular-mean field approximation……Page 54
2.3.2 Variational estimates for the free energy of a spin system……Page 60
2.3.3 Molecular-mean field approximation for an N-component scalar field theory in D dimensions……Page 64
2.3.3.1 Solutions of the mean-field equations……Page 69
2.3.3.2 Critical exponents in the symmetric phase……Page 74
2.3.3.3 Critical exponents in the broken phase……Page 77
2.3.3.5 Tricritical behavior……Page 80
2.3.4 Variational estimates for the SU(2) Higgs model……Page 81
2.3.4.1 Solutions of the mean-field equations of the SU(2) Higgs model……Page 90
2.3.5 Improved variational estimates for the SU(2) Higgs model……Page 93
2.3.6 Summary……Page 98
2.4.1 Generalities……Page 99
2.4.2 Block-spin transformations……Page 101
2.4.3 Iteration of the block-spin transformation……Page 111
2.4.4 Field renormalization……Page 114
2.4.5 Linearized renormalization-group transformation and universality……Page 120
2.4.6 Scaling-sum rules……Page 123
2.4.6.2 Correlation length……Page 125
2.4.6.4 Vacuum expectation value……Page 127
2.4.6.5 Specific heat……Page 128
2.4.7 Violation of scaling-sum rules for critical exponents……Page 129
2.5 Finite-size scaling analysis for second-order phase transitions……Page 132
2.5.1 Shift in the pseudo-critical parameter……Page 137
2.5.2 T-like and h-like scaling fields……Page 139
2.6 Finite-size scaling analysis for first-order phase transitions……Page 141
2.6.1 Predictions for rounding and shifting of singularities in first-order transitions……Page 148
2.6.2 Finite-size scaling with linked cluster expansions……Page 160
2.6.3 Summary of criteria……Page 166
3.1 The standard model in limiting cases I: Spin models as a guideline for the phase structure of QCD……Page 172
3.1.1 Renormalization-group analysis in the chiral limit……Page 174
3.1.1.1 Renormalization-group equation scaling behavior of the Green functions and zeros of the B-function……Page 177
3.1.1.2 Computation of the B-function for an O(N)-symmetric scalar field theory……Page 182
3.1.1.3 Generalization to the SU(N1) x SU(N1)- symmetric model……Page 185
3.1.1.4 Including nonzero bare masses……Page 188
3.1.2 Limit of the pure SU(Nc) gauge theory……Page 189
3.1.2.1 Effective versus physical temperatures……Page 192
3.1.2.2 The phase structure of Z(N)-spin models……Page 193
3.1.2.3 Influence of dynamical quarks……Page 194
3.1.2.4 Finite quark masses and external fields……Page 195
3.1.2.5 Summary……Page 197
3.1.2.6 The Columbia plot……Page 199
3.2.1 The electroweak phase transition in cosmology……Page 202
3.2.2 Perturbative approach in four dimensions in the continuum……Page 204
3.2.3 The lattice approach in four dimensions……Page 206
3.2.3.1 Gross phase structure with deconfinement and Higgs transition……Page 207
3.2.3.2 The Higgs transition in four dimensions……Page 209
3.2.3.3 Effect of the gauge fields……Page 211
3.2.4 The lattice approach in three dimensions……Page 213
3.2.4.1 Measurements of the surface tension……Page 216
3.2.4.2 Localization of the critical endpoint……Page 218
3.2.4.3 Universality class of the critical endpoint……Page 220
3.2.5 Summary of results and open questions……Page 221
3.3.1 The QCD Lagarngian……Page 223
3.3.2 Introducing the lattice cutoff……Page 225
3.3.3 Scalar field theories on the lattice……Page 226
3.3.4 Gauge field theories on the lattice……Page 231
3.3.4.1 Functional measure and the gauge orbit……Page 233
3.3.5 Fermions on the lattice……Page 235
3.3.5.1 The Wilson-fermion action and hopping parameters……Page 238
3.3.5.2 Staggered fermions and flavor symmetries……Page 240
3.3.5.3 Sources of errors……Page 242
3.3.6 Translating lattice results to continuum physics……Page 243
3.3.6.1 The critical temperature Tc……Page 245
3.3.6.3 A test of asymptotic scaling……Page 246
3.3.6.4 Translation to physical units……Page 247
3.3.7 Summary and outlook……Page 248
3.3.8.1 Path integral formulation of finite temperature field theory……Page 249
3.3.8.2 Quantum mechanics of gluons……Page 252
3.3.8.3 Finite-temperature partition function……Page 258
3.3.8.4 Representation as path integral……Page 259
4.1 Convergent versus asymptotic expansions……Page 264
4.1.1 Asymptotic expansions……Page 266
4.1.2 Borel resummations……Page 270
4.1.3 Polymer expansions……Page 271
4.1.4 Strong coupling expansions……Page 280
4.1.5 A theorem by Osterwalder and Seiler……Page 285
4.1.6 Linked cluster expansions as convergent expansions……Page 292
4.1.7 Convergent power series and the critical region: radius of convergence and physical singularity……Page 297
4.2.1 Historical remarks……Page 303
4.2.2 Introduction to linked cluster expansions……Page 306
4.2.3 Classification of graphs……Page 314
4.2.4 Towards a computer implementation of graphs……Page 319
4.2.5 Application to phase transitions and critical phenomena……Page 322
4.3.1 Generalities……Page 327
4.3.2 Renormalization……Page 329
4.3.3 Perturbative renormalization……Page 334
4.3.4 The counterterm approach……Page 336
4.3.5 Power counting in the continuum……Page 340
4.3.6 Power counting for Feynman integrals in momentum space: UV-divergence for rational functions and integrals……Page 341
4.3.7 Power counting for Feynman diagrams: UV-divergence degrees of propagators vertices fields and diagrams……Page 343
4.3.8 Renormalization or: Removing UV-divergencies under the integral sign……Page 345
4.3.9 Renormalization or: Removing UV-divergencies in the counterterm approach……Page 347
4.3.10 Power counting in lattice field theory……Page 348
4.3.11 Preliminaries of lattice-power counting and lattice UV-divergence degrees……Page 350
4.3.12 Power-counting theorem on the lattice……Page 353
4.3.13 Renormalization on the lattice……Page 355
4.3.14 Massless fields and IR-power counting……Page 358
4.3.15 Gauge theories……Page 366
4.4.1 Motivation and problems……Page 373
4.4.2 Finite-temperature Feynman rules and renormalization……Page 377
4.4.3 IR-divergencies and resummation techniques……Page 383
4.4.4 Resummation leading to a reasonable weak coupling expansion……Page 390
4.4.5 Gauge theories and the magnetic-mass problem……Page 392
4.4.6 Hard-thermal loop resummation……Page 397
4.4.6.1 Example of a hard-thermal loop……Page 400
4.4.6.2 Effective HTL-action……Page 405
4.5.1 Generalities……Page 407
4.5.2 Scalar field theories……Page 409
4.5.2.1 Mass resummation and gap equations……Page 416
4.5.3 The order of the phase transition……Page 423
4.5.4 The weak-electroweak phase transition……Page 427
4.6.1 Generalities……Page 436
4.6.2 Matsubara decomposition and the Appelquist-Carazzone decoupling theorem……Page 440
4.6.3 General outline of dimensional reduction……Page 448
4.6.4 Dimensional reduction of a O4-theory from four to three dimensions……Page 454
4.6.4.1 Large-cutoff and high-temperature expansions of one-loop integrals……Page 461
4.6.4.2 The steps after dimensional reduction……Page 464
4.6.5 Pure gauge theories and QCD……Page 467
4.6.5.1 Pure gauge theories……Page 468
4.6.5.2 QCD with fermions……Page 469
4.6.5.3 Dimensional reduction for pure SU(Nc) gauge theories……Page 473
4.6.5.4 Renormalization and zero-momentum projection……Page 476
4.6.5.5 Nonperturbative verification of dimensional reduction and determination of the screening masses……Page 484
4.6.5.6 Dimensional reduction in QCD with dynamical fermions……Page 488
4.6.6 The Gross-Neveu model in three dimensions……Page 496
4.6.6.1 Large N……Page 501
4.6.6.2 Phase structure at infinite N……Page 510
4.6.6.3 Dimensional reduction for finite N……Page 512
4.6.6.4 Phase structure at finite N……Page 519
4.6.6.5 The strong-coupling limit……Page 522
4.6.6.6 Finite couplings and LCE-expansions……Page 523
4.6.6.7 Summary……Page 525
4.6.7 Dimensional reduction in the SU(2) Higgs model……Page 527
4.6.7.1 Guidelines for an alternative form of dimensional reduction……Page 528
4.6.7.2 Performing the integration step……Page 529
4.6.7.3 Integrating upon the superheavy modes……Page 530
4.6.7.4 Examples for the matching procedure to integrate upon the superheavy modes……Page 533
4.6.7.5 Integrating upon the heavy modes……Page 534
4.7.1 Generalities……Page 536
4.7.2 Flow equations for effective interactions……Page 538
4.7.3 Effective average action……Page 545
4.7.4 High-temperature phase transition of O(N) models……Page 548
4.7.5 Appendix: Perturbative renormalization……Page 563
5.1 Algorithms for numerical simulations in lattice field theories……Page 568
5.2 Pitfalls on the lattice……Page 573
5.3.1 The order of limits……Page 576
5.3.2 Correlation lengths mass gaps and tunneling events……Page 577
5.3.3 Correlation functions in the pure SU(3) gauge theory……Page 579
5.4 Including dynamical fermions……Page 580
5.4.1 Finite-size scaling analysis……Page 581
5.4.2 Finite-mass scaling analysis……Page 582
5.4.3 Bulk transitions……Page 587
5.4.4 Results for two and three flavors: The physical mass point……Page 590
5.5.1 Thermodynamics for the pure gauge theory……Page 595
5.5.2 Including dynamical fermions in thermodynamic observables……Page 600
5.6 Interface tensions……Page 601
5.6.1 The two-coupling method……Page 603
5.6.2 The histogram method……Page 604
5.6.3 The vacuum-tunneling correlation length……Page 607
5.6.4 No relics of the type of phase conversion in the early universe?……Page 608
5.7.1 Simulations with Wilson fermions……Page 609
5.7.2 Improved actions……Page 610
5.7.3 QCD at finite baryon density or nonzero chemical potential……Page 611
6.1 Postulating effective actions for QCD……Page 614
6.2 QCD and dysprosium……Page 615
6.3 The chiral transition in chiral perturbation theory……Page 621
6.4 Mass sensitivity of the chiral transition……Page 624
6.5 A network of gluonic strings……Page 630
6.6 Further reading……Page 637
7.1.1 Scales and observables……Page 638
7.1.2 The hydrodynamic framework……Page 645
7.1.3 The bag-model equation of state……Page 650
7.2.1 Thermodynamic observables……Page 654
7.2.2 Dileptons……Page 658
7.2.3 Strangeness production……Page 667
7.2.4 Pion interferometry……Page 669
7.2.5 Intermittency analysis……Page 671
7.2.6 Outlook……Page 675
References……Page 678
Index……Page 692