Leading low-temperature correction to the Heisenberg-Euler Lagrangian

TL;DR

This paper presents an efficient method to derive the leading low-temperature correction to the Heisenberg-Euler Lagrangian at two loops from its one-loop zero-temperature form, using real-time quantum field theory.

Contribution

It introduces a simplified approach to extract low-temperature two-loop corrections from one-loop data and resums these contributions across all loop orders.

Findings

Leading low-temperature correction obtained from one-loop zero-temperature Lagrangian.

Method effectively dresses low-temperature two-loop contributions with tadpole structures.

Resummation of low-temperature contributions to all loop orders achieved.

Abstract

In this note, we show that the well-known leading low-temperature correction to the Heisenberg-Euler Lagrangian in a constant electromagnetic field arising at two loops can be efficiently extracted from its one-loop zero-temperature analogue. Resorting to the real-time formalism of equilibrium quantum field theory that explicitly separates out the zero-temperature contribution from the finite-temperature corrections the determination becomes essentially trivial. In essence, it only requires taking derivatives of the Heisenberg-Euler Lagrangian at one loop and zero temperature for the field strength. As a bonus, we then effectively dress the low-temperature contribution at two loops by one-particle reducible tadpole structures. This generates a subset of higher-loop contributions to the Heisenberg-Euler Lagrangian in the limit of low temperatures. We extract their leading strong-field behavior at a given loop order, and finally resum these to all loop orders.