pi/K -> e nu branching ratios to O(e² p⁴) in Chiral Perturbation Theory

We calculate the ratios R_{e/mu}^{(P)} = Gamma(P -> e nu)/Gamma (P -> mu nu) (P=pi,K) in Chiral Perturbation Theory to order e^2 p^4. We complement the one- and two-loop effective theory results with a matching calculation of the local counterterm, performed within the large-$N_C$ expansion. We find R_{e/mu}^{(\pi)} = (1.2352 \pm 0.0001)*10^{-4} and R_{e/mu}^{(K)} = (2.477 \pm 0.001)*10^{-5}, with uncertainty induced by the matching procedure and chiral power counting. Given the sensitivity of upcoming new measurements, our results provide a clean baseline to detect or constrain effects from weak-scale new physics in these rare decays. As a by-product, we also update the theoretical analysis of the individual pi/K -> \ell nu modes.