Analysis of the Phase Structure of Thermal QED/QCD through the HTL Improved Ladder Dyson-Schwinger Equation --On the Gauge Dependence of the Solution--

TL;DR

This paper numerically investigates the gauge dependence of fermion mass generation in thermal QED/QCD using the HTL improved ladder Dyson-Schwinger equation, revealing significant gauge dependence and violations of Ward-Takahashi identity.

Contribution

It demonstrates that solutions to the Dyson-Schwinger equation in this context are highly gauge-dependent and violate the Ward-Takahashi identity, highlighting the need for gauge fixing procedures.

Findings

Fermion wave function renormalization deviates from unity.

Solutions depend strongly on gauge parameters.

No solutions consistent with Ward-Takahashi identity were found.

Abstract

We solved with a numerical procedure the HTL improved ladder DS equation for the retarded fermion self-energy function $\Sigma_R$ to study the spontaneous generation of fermion mass in thermal QCD/QED, and studied the gauge-dependence of the solution within a general covariant gauge where the gauge parameter $\xi$ is any constant number. With the numerical solutions thus obtained, we found the followings; i) The fermion wave function renormalization function $A(P)$ always deviates largely from unity even at the momentum where the mass is defined, thus the corresponding solutions explicitly contradict with the Ward-Takahashi identity. ii) As a result, the obtained solutions strongly depend on the choice of gauge parameters: the critical temperatures and the critical coupling constants significantly change gauge by gauge. In all gauges we studied in the present analysis, we could not find any solution, having a possibility to be consistent with the Ward-Takahashi identity. Thus we are forced to investigate the procedure to find a gauge which enables us to get a solution being consistent with the Ward-Takahashi identity, otherwise we can not obtain any physically sensible conclusions through the analysis of the point-vertex ladder DS equation no matter how the gauge propagator gets improved.