Perturbative Aspects of the Deconfinement Transition: Beyond the Faddeev-Popov Paradigm

Description

This book offers an original view of the color confinement/deconfinement transition that occurs in non-abelian gauge theories at high temperature and/or densities. It is grounded on the fact that the standard Faddeev-Popov gauge-fixing procedure in the Landau gauge is incomplete. The proper analysis of the low energy properties of non-abelian theories in this gauge requires, therefore, the extension of the gauge-fixing procedure, beyond the Faddeev-Popov recipe. The author reviews various applications of one such extension, based on the Curci-Ferrari model, with a special focus on the confinement/deconfinement transition, first in the case of pure Yang-Mills theory, and then, in a formal regime of Quantum Chromodynamics where all quarks are considered heavy. He shows that most qualitative aspects and also many quantitative features of the deconfinement transition can be accounted for within the model, with only one additional parameter. Moreover, these features emerge in a systematic and controlled perturbative expansion, as opposed to what would happen in a perturbative expansion within the Faddeev-Popov model. The book is also intended as a thorough and pedagogical introduction to background field gauge techniques at finite temperature and/or density. In particular, it offers a new and promising view on the way these techniques might be applied at finite temperature. The material aims at graduate students or researchers who wish to deepen their understanding of the confinement/deconfinement transition from an analytical perspective. Basic knowledge of gauge theories at finite temperature is required, although the text is designed in a self-contained manner, with most concepts and tools introduced when needed. At the end of each chapter, a series of exercises is proposed to master the subject. Preface Acknowledgements Contents Acronyms 1 Introduction: The Many Paths to QCD 2 Faddeev-Popov Gauge Fixing and the Curci-Ferrari Model 2.1 Standard Gauge Fixing 2.1.1 Gauge Invariance 2.1.2 Observables 2.1.3 Faddeev-Popov Procedure 2.1.4 Faddeev-Popov Action 2.1.5 BRST Symmetry 2.1.6 Gauge-Fixing Independence 2.2 Infrared Completion of the Gauge Fixing 2.2.1 Gribov Copies 2.2.2 Gribov-Zwanziger Approach 2.2.3 Serreau-Tissier Approach 2.2.4 Curci-Ferrari Approach 2.2.5 Connection to Other Approaches 2.3 Review of Results 2.3.1 Zero Temperature 2.3.2 Finite Temperature Appendix: BRST Transformations Under the Functional Integral Problems 3 Deconfinement Transition and Center Symmetry 3.1 The Polyakov Loop 3.1.1 Definition 3.1.2 Order Parameter Interpretation 3.2 Center Symmetry 3.2.1 The Role of Boundary Conditions 3.2.2 Relation to the Deconfinement Transition 3.2.3 The Center Symmetry Group 3.3 Center Symmetry and Gauge Fixing 3.3.1 Gauge-Fixed Measure 3.3.2 Constraints on Observables 3.3.3 Approximations and/or Modeling 3.3.4 Strategies to Be Followed Next 3.4 Effective Action 3.4.1 Definitions 3.4.2 Center Symmetry and Order Parameters 3.5 Lattice Implementations of the Gauge Fixing Problems 4 Background-Field Gauges: States and Symmetries 4.1 The Role of the Background Field 4.1.1 Center-Covariant Gauge-Fixed Measure 4.1.2 Background Gauge Fixing and the Polyakov Loop 4.1.3 Center-Covariant Effective Action 4.2 Self-Consistent Backgrounds 4.2.1 Background-Field Effective Action 4.2.2 Background Description of the States 4.2.3 Orbit Description of the States 4.2.4 Center-Symmetric States 4.3 Other Symmetries 4.3.1 Generalities 4.3.2 Charge Conjugation 4.4 Additional Remarks 4.4.1 Back to the Polyakov Loop 4.4.2 Hypothesis on the Gauge-Fixed Measure Convexity of W[J] Background Independence 4.4.3 Vanishing Background Problems 5 Background-Field Gauges: Weyl Chambers 5.1 Constant Temporal Backgrounds 5.1.1 Homogeneity and Isotropy 5.1.2 Restricted Twisted Gauge Transformations 5.1.3 Charge Conjugation 5.2 Winding and Weyl Transformations 5.2.1 Cartan–Weyl Bases 5.2.2 Winding Transformations 5.2.3 Weyl Transformations 5.2.4 Summary 5.3 Weyl Chambers and Symmetries 5.3.1 Periodic Winding Transformations 5.3.2 Weyl Chambers and Invariant States 5.3.3 Explicit Construction for SU(N) Appendix: Euclidean Spacetime Symmetries Problems 6 Yang–Mills Deconfinement Transition from the Curci–Ferrari Model at Leading Order 6.1 Landau–DeWitt Gauge 6.1.1 Faddeev–Popov Action 6.1.2 Curci–Ferrari Completion 6.1.3 Order Parameter Interpretation 6.2 Background-Field Effective Potential 6.2.1 Notational Convention 6.2.2 General One-Loop Expression 6.2.3 Checking the Symmetries 6.3 SU(2) and SU(3) Gauge Groups 6.3.1 Deconfinement Transition 6.3.2 Inversion of the Weiss Potential 6.3.3 Polyakov Loops 6.4 Thermodynamics 6.4.1 High- and Low-Temperature Behavior 6.4.2 Vicinity of the Transition Problems 7 Yang-Mills Deconfinement Transition from the Curci-Ferrari Model at Next-to-Leading Order 7.1 Feynman Rules and Color Conservation 7.1.1 Color Conservation 7.1.2 Feynman Rules 7.2 Two-Loop Effective Potential 7.2.1 Reduction to Scalar Sum-Integrals 7.2.2 Thermal Decomposition 7.2.3 Counterterm Contribution and Renormalization 7.2.4 UV-Finite Contributions 7.3 Next-to-Leading-Order Polyakov Loop 7.3.1 Setting Up the Expansion 7.3.2 Using the Weights 7.3.3 Completing the Calculation 7.4 Results 7.4.1 SU(2) and SU(3) Transitions 7.4.2 Polyakov Loops 7.4.3 Vicinity of the Transition 7.4.4 Low-Temperature Behavior Problems 8 Relation Between the Center Symmetry Group and the Deconfinement Transition 8.1 Polyakov Loops in Other Representations 8.1.1 N-Ality of a Representation 8.1.2 Center Symmetry Characterization 8.1.3 Fundamental Representations 8.2 SU(4) Weyl Chambers 8.2.1 Symmetries 8.2.2 Invariant States 8.3 One-Loop Results 8.3.1 Deconfinement Transition 8.3.2 Background-Dependent Polyakov Loops 8.4 Casimir Scaling Problems 9 Background-Field Gauges: Adding Quarks and Density 9.1 General Considerations 9.1.1 Polyakov Loops 9.1.2 Symmetries 9.1.3 Fermion Determinant 9.2 Continuum Sign Problem(s) 9.2.1 Imaginary Chemical Potential 9.2.2 Real Chemical Potential 9.3 Background-Field Gauges 9.3.1 Complex Self-Consistent Backgrounds 9.3.2 Background-Field Effective Potential 9.3.3 Background-Dependent Polyakov Loop 9.3.4 Other Approaches Problems 10 QCD Deconfinement Transition in the Heavy Quark Regime 10.1 Background Effective Potential 10.1.1 General One-Loop Expression 10.1.2 Real-Valuedness in the SU(3) Case 10.1.3 Polyakov Loop Potential 10.2 Phase Structure at μ=0 10.3 Phase Structure for μiR 10.3.1 Roberge-Weiss Symmetry 10.3.2 Results 10.4 Phase Structure for μR 10.4.1 Columbia Plot 10.4.2 T-Dependence of the Polyakov Loops 10.4.3 μ-Dependence of the Polyakov Loops Problems 11 A Novel Look at the Background-Field Method at Finite Temperature 11.1 Limitations of the Standard Approach 11.2 Center-Symmetric Landau Gauge 11.2.1 Constant Backgrounds 11.2.2 Twisted Boundary Conditions and Lattice Implementation 11.3 Implementation Within the Curci-Ferrari Model 11.3.1 One-Loop Potential 11.3.2 Practical Evaluation 11.3.3 Renormalization 11.3.4 Polyakov Loop 11.4 Results 11.4.1 Transition Temperatures 11.4.2 Polyakov Loop 11.4.3 Masses 11.5 Connection to the Self-Consistent Backgrounds 11.5.1 Case of an Ideal Gauge Fixing 11.5.2 In Practice Problems 12 Conclusions and Outlook A The SU(N) Lie Algebra A.1 Defining Weights of SU(N) A.2 Roots of SU(N) A.3 Relations Between Roots and Weights A.4 Complexified Algebra and Killing Form B Evaluating Matsubara Sums B.1 Basic Result B.2 Application to Sum-Integrals B.2.1 One-Loop Tadpole Sum-Integrals of Type I B.2.2 One-Loop Tadpole Sum-Integrals of Type II B.2.3 The Two-Loop Sunset Sum-Integral Solutions Problems of Chap. 2 Problems of Chap. 3 Problems of Chap. 4 Problems of Chap. 5 Problems of Chap. 6 Problems of Chap. 7 Problems of Chap. 8 Problems of Chap. 9 Problems of Chap. 10 Problems of Chap. 11 References Index