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We solved with a numerical procedure the HTL improved ladder DS equation forthe retarded fermion self-energy function $\Sigma_R$ to study the spontaneousgeneration of fermion mass in thermal QCD/QED, and studied the gauge-dependenceof 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) Thefermion wave function renormalization function $A(P)$ always deviates largelyfrom unity even at the momentum where the mass is defined, thus thecorresponding solutions explicitly contradict with the Ward-Takahashi identity.ii) As a result, the obtained solutions strongly depend on the choice of gaugeparameters: the critical temperatures and the critical coupling constantssignificantly change gauge by gauge. In all gauges we studied in the presentanalysis, we could not find any solution, having a possibility to be consistentwith the Ward-Takahashi identity. Thus we are forced to investigate theprocedure to find a gauge which enables us to get a solution being consistentwith the Ward-Takahashi identity, otherwise we can not obtain any physicallysensible conclusions through the analysis of the point-vertex ladder DSequation no matter how the gauge propagator gets improved.

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