Sterile neutrinos and $R_K$

Abstract: We study the violation of lepton flavour universality in light meson decays due to the presence of non-zero mixings between the active neutrinos with new sterile states. The modified $W \ell \nu$ vertices, arising from a non-unitarity leptonic mixing matrix intervening in charged currents, might lead to a tree-level enhancement of $R_{P} = \Gamma (P \to e \nu) / \Gamma (P \to \mu \nu)$, with $P=K, \pi$. These enhancements are illustrated in the case of the inverse seesaw, showing that one can saturate the current experimental bounds on $\Delta r_{K}$ (and $\Delta r_{\pi}$), while in agreement with the different experimental and observational constraints.