A way to get a well-defined derivative expansion of real-time thermal effective actions

TL;DR

This paper develops a method to systematically derive a well-defined derivative expansion of real-time thermal effective actions for scalar fields, ensuring analyticity and addressing initial time dependence.

Contribution

It introduces a generalized real-time formalism that allows for a consistent derivative expansion of thermal effective actions with clear analytic properties.

Findings

Derived the quadratic thermal effective action for scalar fields.

Established the analyticity of the result at zero external momenta.

Discussed initial time dependence in the context of equilibrium expansions.

Abstract

We compute the quadratic part of the thermal effective action for real scalar fields which are initially in thermal equilibrium and vary slowly in time using a generalised real-time formalism proposed by Le Bellac and Mabilat \cite{belmab}. We derive both Real Time and Imaginary Time Formalisms and find that the result is analytic at the limits of zero external four-momenta when using our full time contour. We expand the fields in time up to the second derivative and discuss the initial time dependence of our result before and after the expansion in terms of equilibrium.