N-Point Vertex Functions, Ward-Takahashi Identities and Dyson-Schwinger Equations in Thermal QCD/QED in the Real Time Hard-Thermal-Loop Approximation

Abstract

In this paper we calculated the n-point hard-thermal-loop (HTL) vertex functions in QCD/QED for n= 2, 3 and 4 in the physical representation in the real time formalism (RTF). The result showed that the n-point HTL vertex functions can be classified into two groups, a) those with odd numbers of external retarded indices, and b) the others with even numbers of external retarded indices. The n-point HTL vertex functions with one retarded index, which obviously belong to the first group a), are nothing but the HTL vertex functions that appear in the imaginary time formalism (ITF), and vise versa. All the HTL vertex functions belonging to the first group a) are of O(g 2 T 2 ) , and satisfy among them the simple QED-type Ward-Takahashi identities, as in the ITF. Those vertex functions belonging to the second group b) never appear in the ITF, namely their existence is characteristic of the RTF, and their HTL's have the high temperature behavior of O(g 2 T 3 ), one-power of T higher than usual. Despite this difference we could verify that those HTL vertex functions belonging to the second group b) also satisfy among themselves the QED-type Ward-Takahashi identities, thus guaranteeing the gauge invariance of the HTL's in the real time thermal QCD/QED.

Gluon Energy at Next-to-Leading Order in Hot QCD Using the Real-Time Formalism

“…The real-time formalism [2,[7][8][9] is used instead of the imaginary-time one. It also gives expressions for the vertex functions using the Feynman parameterization.…”

NLO dispersion laws for slow-moving quarks in HTL QCD

“…Finally, as we mentioned early in this appendix, the notation we use here for the vertex functions is close to that used in [75]. However, the notation we use in the main text is close to the one used in [68].…”

“…We give there an explicit expression of Σ (1) in terms of the three and four HTL-dressed vertex functions. These functions are derived ab initio as discrepancies between different results in the literature are found [74,75]. Then, in section four, we introduce a Feynman parametrization to help perform the solid-angle integrals present in the vertex hard thermal loops.…”

“…(3.6) have been written in [53] using a different notation. The HTL-dressed vertex functions Γ are derived in the literature [74,75,77] . However, there are discrepancies in these results [74,75], which led us to rederive all three and fourpoint HTL-dressed vertex functions ab initio in the CTP formalism.…”

Gluons dispersion relations in hot QCD using the real time formalism

“…In the Keldysh representation [2,10,11], the retarded/advanced (ra/ar) gluon propagators are given by: (1) and the {rr}-component given by:…”