  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.
