Heat kernel approach to the one-loop effective action for nonlinear electrodynamics

TL;DR

This paper introduces a heat kernel method to compute the one-loop effective action in nonlinear electrodynamics, addressing non-minimal operators and focusing on the weak-field regime and conformal theories.

Contribution

It develops a novel heat kernel approach for nonlinear electrodynamics, including calculations of divergences and contributions for non-minimal operators in the weak-field limit.

Findings

Calculated the $a_0$, $a_1$, and $a_2$ heat kernel coefficients in the weak-field regime.

Computed the $a_0$ contribution to all orders for conformal NLED theories.

Commented on the role of causality in the convergence of heat kernel contributions.

Abstract

We develop a heat kernel method to compute the one-loop effective action for a general class of nonlinear electrodynamic (NLED) theories in four dimensional Minkowski spacetime. Working in the background field formalism, we extract the logarithmically divergent part of the effective action, the so-called induced action, corresponding to the DeWitt $a_2$ coefficient of the heat kernel. In NLED, quantisation yields non-minimal differential operators, for which standard heat kernel techniques are not immediately applicable. Considering the weak-field regime, we calculate the $a_0$, $a_1$ and $a_2$ contributions to leading order in the background electromagnetic field strength. Finally, we consider conformal NLED theories and compute the $a_0$ contribution to all orders. For this class, we comment on the role of causality being necessary and sufficient for the convergence of the exact $a_1$ and $a_2$ contributions.