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

TL;DR

This work develops a heat-kernel, Volterra-series framework to quantise nonlinear electrodynamics with non-minimal differential operators, enabling computation of the one-loop effective action in four dimensions. By performing a background-field split and gauge fixing, the authors derive a non-minimal operator and construct the heat kernel expansion to extract the DeWitt coefficients $a_0$, $a_1$, and $a_2$, first in a weak-field regime and then exactly for conformal NLED (for $a_0$). They provide explicit quartic-order results for each coefficient, and, in the conformal case, establish that causality conditions are necessary and sufficient for the convergence of the exact $a_1$ and $a_2$ contributions; ModMax serves as a concrete example. The findings illuminate how nonlinear electromagnetic backgrounds influence quantum corrections and pave the way for extending the approach to curved spacetimes, with potential implications for effective actions in string theory and beyond.

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.