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

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