Feynman parametrization and Mellin summation at finite temperature

TL;DR

This paper demonstrates that Mellin summation technique (MST) effectively computes finite-temperature loop integrals in thermal field theory, resolving issues with Feynman parametrization by properly handling periodicity and domain changes.

Contribution

It clarifies that problems with Feynman parametrization at finite temperature are due to domain and periodicity mishandling, and shows MST as a robust solution.

Findings

MST provides a well-defined method for finite-temperature loop integrals.

Proper implementation of periodicity avoids issues with Feynman parametrization.

Application to magnetic field effects demonstrates the method's practical utility.

Abstract

We show that the Mellin summation technique (MST) is a well defined and useful tool to compute loop integrals at finite temperature in the imaginary-time formulation of thermal field theory, especially when interested in the infrared limit of such integrals. The method makes use of the Feynman parametrization which has been claimed to have problems when the analytical continuation from discrete to arbitrary complex values of the Matsubara frequency is performed. We show that without the use of the MST, such problems are not intrinsic to the Feynman parametrization but instead, they arise as a result of (a) not implementing the periodicity brought about by the possible values taken by the discrete Matsubara frequencies before the analytical continuation is made and (b) to the changing of the original domain of the Feynman parameter integration, which seemingly simplifies the expression but in practice introduces a spurious endpoint singularity. Using the MST, there are no problems related to the implementation of the periodicity but instead, care has to be taken when the sum of denominators of the original amplitude vanishes. We apply the method to the computation of loop integrals appearing when the effects of external weak magnetic fields on the propagation of scalar particles is considered.