Bounds on nonlinear effective field theories via resurgent relative entropy

We study nonlinear effective field theories (EFTs) with factorially growing perturbative expansions, focusing on a class in which the relative entropy encodes an infinite tower of higher-dimensional operators. Using the resummed relative entropy, we derive bounds on EFT coefficients: the non-negativity of the resummed relative entropy fixes the sign of their asymptotic growth, while its violation signals instabilities. In fermionic QED, analytic continuation from Euclidean to Minkowski spacetime yields a concrete example: the Schwinger effect, a nonperturbative instability captured by the resummed relative entropy.