[Paper Review] Phase Structure of Thermal QCD/QED:A Gauge Invariant Solution of the HTL Resummed Improved Ladder Dyson-Schwinger Equation

Hisao Nakkagawa, Hiroshi Yokota+1 more|ArXiv.org|Jul 6, 2007

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper proposes a gauge-invariant solution to the hard-thermal-loop (HTL) resummed improved ladder Dyson-Schwinger equation for fermion self-energy in thermal QED/QCD by enforcing the Ward-Takahashi identity through momentum-dependent gauge parameters. The method yields a critical temperature for chiral phase transition that is independent of gauge choice, confirming a second-order transition with reduced thermal fluctuation effects compared to previous Landau gauge analyses.

ABSTRACT

Based on the hard-thermal-loop resummed improved ladder Dyson-Schwinger quation for the fermion mass function, we propose a procedure how we can get the gauge invariant solution in the sense it satisfies the Ward-Takahashi identity. Results of the numerical analysis are shown and properties of the ``gauge-invariant'' solutions are discussed.

Motivation & Objective

• To resolve the gauge-parameter dependence of fermion mass solutions in finite-temperature Dyson-Schwinger equations.
• To enforce gauge invariance by satisfying the Ward-Takahashi identity in HTL-resummed, improved ladder approximations.
• To investigate the phase structure of thermal QED/QCD with a physically consistent, gauge-invariant solution.
• To determine whether chiral symmetry breaking/restoration is sensitive to the choice of gauge in finite-temperature field theory.
• To compare the strength of thermal fluctuations on chiral symmetry breaking with previous Landau gauge results.

Proposed method

• Solve the HTL-resummed improved ladder Dyson-Schwinger equation for fermion self-energy functions A(P), B(P), and C(P) in thermal QED/QCD.
• Use a momentum-dependent gauge parameter ξ to enforce the Ward-Takahashi identity, ensuring A(P) ≈ 1 for gauge invariance.
• Implement a nonlinear gauge with complex or real ξmn parameters to minimize |A(P) - 1|² numerically.
• Apply the instantaneous exchange approximation for the longitudinal photon propagator and use HTL-resummed gauge boson propagators.
• Enforce the condition A(P) ≈ 1 at p₀ = 0, p → 0 to ensure consistency with gauge invariance in the static limit.
• Compare results between real and complex ξ prescriptions to test gauge independence of the solution.

Experimental results

Research questions

• RQ1 Does the solution of the HTL-resummed improved ladder Dyson-Schwinger equation in thermal QED/QCD remain gauge-invariant when the gauge parameter ξ is allowed to depend on momentum?
• RQ2 What is the nature of the chiral phase transition in thermal QED/QCD when the Ward-Takahashi identity is enforced?
• RQ3 How does the critical temperature for chiral symmetry breaking depend on the choice of gauge parameter in the HTL-resummed framework?
• RQ4 To what extent do thermal fluctuations affect chiral symmetry breaking compared to previous Landau gauge analyses?
• RQ5 Is the fermion wave function renormalization constant A(P) stable and close to unity in the static limit under the proposed gauge-invariant procedure?

Key findings

• The fermion wave function renormalization constant A(P) is consistently close to unity across different gauge parameter prescriptions, confirming gauge invariance.
• The chiral phase transition in massless thermal QED/QCD is confirmed to be second-order, with a continuous generation of dynamical fermion mass at critical temperature or coupling.
• The critical temperature for chiral symmetry breaking is found to be independent of the gauge parameter when using momentum-dependent ξ, indicating gauge-invariant physical results.
• Thermal fluctuation effects on chiral symmetry breaking are smaller than previously estimated in Landau gauge analyses, as the symmetry-broken phase shrinks toward low temperatures and strong coupling.
• Solutions obtained with real and complex gauge parameters ξ show complete agreement, validating the robustness and gauge independence of the method.
• The invariant functions B(P) and C(P) exhibit weak momentum dependence near the static limit, supporting consistency with gauge invariance.