New partial resummation of the QED effective action

We explain a conjecture which states that the proper-time series expansion of the one-loop effective Lagrangian of quantum electrodynamics can be partially summed in all terms containing the field-strength invariants $\mathcal{F} = \frac{1}{4} F_{\mu\nu}F^{\mu\nu} (x)$, $\mathcal{G}= \frac{1}{4} \tilde F_{\mu\nu}F^{\mu\nu}(x)$. This summation is encapsulated in a factor with the same form as the (spacetime-dependent) Heisenberg-Euler Lagrangian density. We also discuss some implications and a possible extension in presence of gravity. We will focus on the scalar field case.