Tree-level lepton universality violation in the presence of sterile neutrinos: impact for $R_K$ and $R_{\pi }$

We consider a tree-level enhancement to the violation of lepton flavour universality in light meson decays arising from modified $W\mathrm{\ell }\nu$ couplings in the standard model minimally extended by sterile neutrinos. Due to the presence of additional mixings between the active (left-handed) neutrinos and the new sterile states, the deviation from unitarity of the leptonic mixing matrix intervening in charged currents might lead to a tree-level enhancement of $R_P=\mathrm{\Gamma }(P\to e\nu )\mathrm{/}\mathrm{\Gamma }(P\to \mu \nu )$, with $P=K,\pi$. We illustrate these enhancements in the case of the inverse seesaw model, showing that one can saturate the current experimental bounds on $\mathrm{\Delta }{r}_K$ (and $\mathrm{\Delta }{r}_{\pi }$), while in agreement with the different experimental and observational constraints.