A simple sum rule for the thermal gluon spectral function and applications

TL;DR

This work derives a simple finite-temperature sum-rule for the gluon spectral function, linking $f(z)$ to the difference of inverse propagator denominators at infinite and zero momentum, and enabling analytic progress in photon/dilepton rate calculations. By exploiting the sum-rule, the authors express the HTL-embedded quantities $J_{T,L}$ and $K_{T,L}$ in terms of a single function $F(x)$, derive precise asymptotic forms for $M_{\rm eff}$ relative to the gluon mass scale, and extend the framework beyond HTL by introducing generalized mass parameters ($m_P$, $m_D$, $m_{\rm mag}$) that govern the rates. They then apply this to the resummation of ladder diagrams and the LPM effect, obtaining a closed collision kernel $\mathcal{C}(\mathbf{l}_\perp)$ and reformulating the problem as a differential equation in impact-parameter space, which can be solved numerically via a shooting method. The results provide analytic and semi-analytic tools to compute photon and low-mass dilepton production in a quark–gluon plasma, with a clear separation from plasmon-mass dependence and applicability to realistic quasi-particle scenarios.

Abstract

In this paper, we derive a simple sum rule satisfied by the gluon spectral function at finite temperature. This sum rule is useful in order to calculate exactly some integrals that appear frequently in the photon or dilepton production rate by a quark gluon plasma. Using this sum rule, we rederive simply some known results and obtain some new results that would be extremely difficult to justify otherwise. In particular, we derive an exact expression for the collision integral that appears in the calculation of the Landau-Pomeranchuk-Migdal effect.

Figures (2)

• Figure 1: The two-loop diagrams contributing to photon and dilepton production.
• Figure 2: Important processes contained in the above 2-loop diagrams.