Two-loop self-dual Euler-Heisenberg Lagrangians (II): Imaginary part and Borel analysis

TL;DR

This paper investigates the two-loop Euler-Heisenberg effective Lagrangian in QED with a self-dual background, revealing that the imaginary part's exponential prefactors have their own asymptotic expansions and validating Borel summation techniques for this complex structure.

Contribution

It provides a detailed analysis of the two-loop imaginary part structure and demonstrates the effectiveness of Borel summation in capturing the full prefactor.

Findings

The two-loop imaginary part includes exponential terms with asymptotic prefactors.

Borel dispersion relations accurately reproduce the leading imaginary contribution.

High-precision tests confirm the validity of Borel summation techniques.

Abstract

We analyze the structure of the imaginary part of the two-loop Euler-Heisenberg QED effective Lagrangian for a constant self-dual background. The novel feature of the two-loop result, compared to one-loop, is that the prefactor of each exponential (instanton) term in the imaginary part has itself an asymptotic expansion. We also perform a high-precision test of Borel summation techniques applied to the weak-field expansion, and find that the Borel dispersion relations reproduce the full prefactor of the leading imaginary contribution.