Strong-field regime within effective field theory

TL;DR

This work develops a Strong-Field Covariant Derivative Expansion (SF-CDE) to capture non-perturbative effects of strong background fields in scalar QED, deriving the full second-derivative corrections to the scalar Heisenberg–Euler action. The authors construct a momentum-space, worldline-based SF expansion around an operator that exactly includes field-strength terms, yielding a systematic, multiscale EFT with a consistent treatment of three scales: the mass $M$, the background field, and its derivatives. They obtain closed expressions for the $n=0$ Heisenberg–Euler action, compute the $n=1$ and $n=2$ derivative corrections, and present the full $oxed{ ext{L}}_{ m eff}^{\partial^2}$ in terms of symmetric form factors after diagonalizing the field strength. The derivative corrections preserve the transseries structure of Schwinger pair production while modifying coefficients, and the work includes cross-checks with weak-field limits and discussions of discrepancies with prior non-perturbative results, providing a practical framework for strong-field EFT in QED and a template for similar multiscale analyses in related theories.

Abstract

Building upon the Covariant Derivative Expansion, we develop a method to compute effective actions that is able to capture non-perturbative effects induced by strong background fields. We demonstrate the method in scalar QED, by deriving the full second-derivative corrections to the scalar Heisenberg--Euler effective action. The corresponding result is interpreted as an effective field theory with three characteristic scales, two of which are large (mass and field strength) in comparison with the remaining one (derivatives of the field). As an application, we show that, at this order, the transseries structure of the Schwinger pair production rate is preserved, even if the involved coefficients are modified. Our analysis also helps clarify recent disagreements concerning the coefficients of this effective action.