Contents -- Preface -- 1. Introduction -- 2. General Background from Statistical Physics -- 2.1 Generalities -- 2.1.1 Phase transitions in statistical systems -- 2.1.1.1 First- and second-order transitions in the infinite volume limit -- 2.1.1.2 Landau's free energy -- 2.2 Generating functional n-point correlations and effective potentials -- 2.3 The molecular-mean field approximation -- 2.3.1 Self-consistent equation of state for a ferromagnet -- 2.3.1.1 Critical exponents in the molecular-mean field approximation -- 2.3.2 Variational estimates for the free energy of a spin system -- 2.3.3 Molecular-mean field approximation for an N-component scalar field theory in D dimensions -- 2.3.3.1 Solutions of the mean-field equations -- 2.3.3.2 Critical exponents in the symmetric phase -- 2.3.3.3 Critical exponents in the broken phase -- 2.3.3.4 First-order transitions within the molecular-mean field approximation -- 2.3.3.5 Tricritical behavior -- 2.3.4 Variational estimates for the SU(2) Higgs model -- 2.3.4.1 Solutions of the mean-field equations of the SU(2) Higgs model -- 2.3.5 Improved variational estimates for the SU(2) Higgs model -- 2.3.6 Summary -- 2.4 Renormalization group -- 2.4.1 Generalities -- 2.4.2 Block-spin transformations -- 2.4.3 Iteration of the block-spin transformation -- 2.4.4 Field renormalization -- 2.4.5 Linearized renormalization-group transformation and universality -- 2.4.6 Scaling-sum rules -- 2.4.6.1 Anomalous dimension -- 2.4.6.2 Correlation length -- 2.4.6.3 Two-point susceptibility -- 2.4.6.4 Vacuum expectation value -- 2.4.6.5 Specific heat -- 2.4.6.6 Summary -- 2.4.7 Violation of scaling-sum rules for critical exponents -- 2.5 Finite-size scaling analysis for second-order phase transitions -- 2.5.1 Shift in the pseudo-critical parameter.

2.5.2 T-like and h-like scaling fields -- 2.6 Finite-size scaling analysis for first-order phase transitions -- 2.6.1 Predictions for rounding and shifting of singularities in first-order transitions -- 2.6.2 Finite-size scaling with linked cluster expansions -- 2.6.3 Summary of criteria -- 3. Field Theoretical Framework for Models in Particle Physics -- 3.1 The standard model in limiting cases I: Spin models as a guideline for the phase structure of QCD -- 3.1.1 Renormalization-group analysis in the chiral limit -- 3.1.1.1 Renormalization-group equation scaling behavior of the Green functions and zeros of the B-function -- 3.1.1.2 Computation of the B-function for an O(N)-symmetric scalar field theory -- 3.1.1.3 Generalization to the SU(N1) x SU(N1)- symmetric model -- 3.1.1.4 Including nonzero bare masses -- 3.1.2 Limit of the pure SU(Nc) gauge theory -- 3.1.2.1 Effective versus physical temperatures -- 3.1.2.2 The phase structure of Z(N)-spin models -- 3.1.2.3 Influence of dynamical quarks -- 3.1.2.4 Finite quark masses and external fields -- 3.1.2.5 Summary -- 3.1.2.6 The Columbia plot -- 3.2 The standard model in limiting cases II: Phase transitions in the electroweak part of the standard model -- 3.2.1 The electroweak phase transition in cosmology -- 3.2.2 Perturbative approach in four dimensions in the continuum -- 3.2.3 The lattice approach in four dimensions -- 3.2.3.1 Gross phase structure with deconfinement and Higgs transition -- 3.2.3.2 The Higgs transition in four dimensions -- 3.2.3.3 Effect of the gauge fields -- 3.2.4 The lattice approach in three dimensions -- 3.2.4.1 Measurements of the surface tension -- 3.2.4.2 Localization of the critical endpoint -- 3.2.4.3 Universality class of the critical endpoint -- 3.2.5 Summary of results and open questions -- 3.3 A primer to lattice gauge theory.

3.3.1 The QCD Lagarngian -- 3.3.2 Introducing the lattice cutoff -- 3.3.3 Scalar field theories on the lattice -- 3.3.4 Gauge field theories on the lattice -- 3.3.4.1 Functional measure and the gauge orbit -- 3.3.5 Fermions on the lattice -- 3.3.5.1 The Wilson-fermion action and hopping parameters -- 3.3.5.2 Staggered fermions and flavor symmetries -- 3.3.5.3 Sources of errors -- 3.3.6 Translating lattice results to continuum physics -- 3.3.6.1 The critical temperature Tc -- 3.3.6.2 Choosing the appropriate extension in time direction -- 3.3.6.3 A test of asymptotic scaling -- 3.3.6.4 Translation to physical units -- 3.3.7 Summary and outlook -- 3.3.8 Transfer matrix and Polyakov loops -- 3.3.8.1 Path integral formulation of finite temperature field theory -- 3.3.8.2 Quantum mechanics of gluons -- 3.3.8.3 Finite-temperature partition function -- 3.3.8.4 Representation as path integral -- 4. Analytic Methods on the Lattice and in the Continuum -- 4.1 Convergent versus asymptotic expansions -- 4.1.1 Asymptotic expansions -- 4.1.2 Borel resummations -- 4.1.3 Polymer expansions -- 4.1.4 Strong coupling expansions -- 4.1.5 A theorem by Osterwalder and Seiler -- 4.1.6 Linked cluster expansions as convergent expansions -- 4.1.7 Convergent power series and the critical region: radius of convergence and physical singularity -- 4.2 Linked cluster expansions in more detail -- 4.2.1 Historical remarks -- 4.2.2 Introduction to linked cluster expansions -- 4.2.3 Classification of graphs -- 4.2.4 Towards a computer implementation of graphs -- 4.2.5 Application to phase transitions and critical phenomena -- 4.2.6 Some results -- 4.3 Renormalization perturbation theory and universality at zero temperature - the continuum limit -- 4.3.1 Generalities -- 4.3.2 Renormalization -- 4.3.3 Perturbative renormalization.

4.3.4 The counterterm approach -- 4.3.5 Power counting in the continuum -- 4.3.6 Power counting for Feynman integrals in momentum space: UV-divergence for rational functions and integrals -- 4.3.7 Power counting for Feynman diagrams: UV-divergence degrees of propagators vertices fields and diagrams -- 4.3.8 Renormalization or: Removing UV-divergencies under the integral sign -- 4.3.9 Renormalization or: Removing UV-divergencies in the counterterm approach -- 4.3.10 Power counting in lattice field theory -- 4.3.11 Preliminaries of lattice-power counting and lattice UV-divergence degrees -- 4.3.12 Power-counting theorem on the lattice -- 4.3.13 Renormalization on the lattice -- 4.3.14 Massless fields and IR-power counting -- 4.3.15 Gauge theories -- 4.4 Weak coupling expansion at finite temperature -- 4.4.1 Motivation and problems -- 4.4.2 Finite-temperature Feynman rules and renormalization -- 4.4.3 IR-divergencies and resummation techniques -- 4.4.4 Resummation leading to a reasonable weak coupling expansion -- 4.4.5 Gauge theories and the magnetic-mass problem -- 4.4.6 Hard-thermal loop resummation -- 4.4.6.1 Example of a hard-thermal loop -- 4.4.6.2 Effective HTL-action -- 4.5 Constraint effective potential and gap equations -- 4.5.1 Generalities -- 4.5.2 Scalar field theories -- 4.5.2.1 Mass resummation and gap equations -- 4.5.3 The order of the phase transition -- 4.5.4 The weak-electroweak phase transition -- 4.6 Dimensional reduction at high temperature -- 4.6.1 Generalities -- 4.6.2 Matsubara decomposition and the Appelquist-Carazzone decoupling theorem -- 4.6.3 General outline of dimensional reduction -- 4.6.4 Dimensional reduction of a O4-theory from four to three dimensions -- 4.6.4.1 Large-cutoff and high-temperature expansions of one-loop integrals.

4.6.4.2 The steps after dimensional reduction -- 4.6.5 Pure gauge theories and QCD -- 4.6.5.1 Pure gauge theories -- 4.6.5.2 QCD with fermions -- 4.6.5.3 Dimensional reduction for pure SU(Nc) gauge theories -- 4.6.5.4 Renormalization and zero-momentum projection -- 4.6.5.5 Nonperturbative verification of dimensional reduction and determination of the screening masses -- 4.6.5.6 Dimensional reduction in QCD with dynamical fermions -- 4.6.6 The Gross-Neveu model in three dimensions -- 4.6.6.1 Large N -- 4.6.6.2 Phase structure at infinite N -- 4.6.6.3 Dimensional reduction for finite N -- 4.6.6.4 Phase structure at finite N -- 4.6.6.5 The strong-coupling limit -- 4.6.6.6 Finite couplings and LCE-expansions -- 4.6.6.7 Summary -- 4.6.7 Dimensional reduction in the SU(2) Higgs model -- 4.6.7.1 Guidelines for an alternative form of dimensional reduction -- 4.6.7.2 Performing the integration step -- 4.6.7.3 Integrating upon the superheavy modes -- 4.6.7.4 Examples for the matching procedure to integrate upon the superheavy modes -- 4.6.7.5 Integrating upon the heavy modes -- 4.7 Flow equations of Polchinski -- 4.7.1 Generalities -- 4.7.2 Flow equations for effective interactions -- 4.7.3 Effective average action -- 4.7.4 High-temperature phase transition of O(N) models -- 4.7.5 Appendix: Perturbative renormalization -- 5. Numerical Methods in Lattice Field Theories -- 5.1 Algorithms for numerical simulations in lattice field theories -- 5.2 Pitfalls on the lattice -- 5.3 Pure gauge theory: The order of the SU(3)-deconfinement transition -- 5.3.1 The order of limits -- 5.3.2 Correlation lengths mass gaps and tunneling events -- 5.3.3 Correlation functions in the pure SU(3) gauge theory -- 5.4 Including dynamical fermions -- 5.4.1 Finite-size scaling analysis -- 5.4.2 Finite-mass scaling analysis -- 5.4.3 Bulk transitions.

5.4.4 Results for two and three flavors: The physical mass point.