# An all-loop result for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian

## TL;DR

This paper derives an explicit all-loop expression for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian in QED, revealing that it depends solely on one-loop data and revising previous irreducible-based results.

## Contribution

It introduces a novel algebraic method to determine the all-loop strong magnetic field behavior using one-particle reducible contributions, simplifying previous approaches.

## Key findings

- Explicit all-loop expression for strong magnetic field limit.
- Leading behavior depends only on one-loop Lagrangian.
- Revises previous results based on irreducible diagrams.

## Abstract

We provide an explicit expression for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian for both scalar and spinor quantum electrodynamics. To this end, we show that the strong magnetic field behavior is fully determined by one-particle reducible contributions discovered only recently. The latter can efficiently be constructed in an essentially algebraic procedure from lower-order one-particle reducible diagrams. Remarkably, the leading strong magnetic field behavior of the all-loop Heisenberg-Euler effective Lagrangian only requires input from the one-loop Lagrangian. Our result revises previous findings based exclusively on one-particle irreducible contributions. In addition we briefly discuss the strong electric field limit and comment on external field QED in the large $N$ limit.