Revisiting the $\Gamma(K \to e \nu)/\Gamma(K \to \mu \nu)$ ratio in supersymmetric unified models

20 pages, 6 Figures in Encapsulated PostscriptInternational audienceIt has been pointed out that supersymmetric extensions of the Standard Model can induce significant changes to the theoretical prediction of the ratio $\Gamma(K \to e\nu)/\Gamma(K \to \mu\nu)= R_K$, through lepton flavor violating couplings. In this work we carry out a full computation of all one-loop corrections to the relevant $\nu \ell H^+$ vertex, and discuss the new contributions to $R_K$ arising in the context of different constrained (minimal supergravity inspired) models which succeed to account for neutrino data, further considering the possibility of accommodating a near future observation of a $\mu \to e \gamma$ transition. We also re-evaluate the prospects for $R_K$ in the framework of unconstrained supersymmetric models. In all cases, we address the question of whether it is possible to saturate the current experimental sensitivity on $R_K$ while in agreement with the recent limits on BR($B_{(s)} \to \mu \mu$). Our findings reveal that in view of the recent bounds, and even when enhanced by effective sources of flavour violation in the right-handed $\tilde e-\tilde \tau$ sector, constrained supersymmetric (seesaw) models typically provide excessively small contributions to $R_K$. While unconstrained models can indeed account for sizable contributions, the associated regions in parameter space are in general disfavoured due to the new $B_{s}\to \mu \mu$ bound