N-point vertex functions, Ward-Takahashi identities and Dyson-Schwinger equations in thermal QCD/QED in the real time hard-thermal-loop approximation

In this paper we calculate the n-point hard-thermal-loop (HTL) vertex functions in QCD/QED for n=2, 3 and 4 in the physical representation using the real time formalism (RTF). The result reveals that the n-point HTL vertex functions can be classified into two groups, a) those with odd numbers of external retarded indices, and b) those with even numbers of external retarded indices. The n-point HTL vertex functions with one retarded index, which obviously belong to group a), are identical to the HTL vertex functions that appear in the imaginary time formalism (ITF). All the HTL vertex functions belonging to group a) are O(g2T2) and satisfy among them the simple QED-type Ward-Takahashi identities, as in the ITF. Those vertex functions belonging to group b) never appear in the ITF, their existence being characteristic of the RTF, and their HTLs exhibit high temperature behavior that is O(g2T3), one power of T higher than usual. Despite this difference, we were able to verify that those HTL vertex functions belonging to group b) also satisfy among themselves the QED-type Ward-Takahashi identities, thus guaranteeing the gauge invariance of the HTLs in the real time thermal QCD/QED. As an application, we explicitly derive the HTL resummed Dyson-Schwinger equations for the physical fermion mass function and for the gauge boson polarization tensor in thermal QCD/QED. (author)