Probing Lepton Flavour Universality with $K \to \pi \nu \bar\nu$ decays

We analyse the rare processes $K \to \pi\nu\bar\nu$ in view of the recent hints of violations of Lepton Flavour Universality (LFU) observed in B meson decays. If, as suggested by present data, the new interactions responsible for LFU violations couple mainly to the third generation of left-handed fermions, $K \to \pi\nu\bar\nu$ decays turn out to be particularly interesting: these are the only kaon decays with third-generation leptons (the $\tau$ neutrinos) in the final state. In order to relate B-physics anomalies and K decays we adopt an Effective Field Theory approach, assuming that the new interactions satisfy an approximate $U(2)_q\times U(2)_\ell$ flavour symmetry. In this framework we show that O(1) deviations from the Standard Model predictions in $K \to \pi\nu\bar\nu$ branching ratios, closely correlated to similar effects in $B \to K^{(*)}\nu\bar\nu$, are naturally expected. The correlation of $\mathcal{B}(K \to \pi\nu\bar\nu)$, $\mathcal{B}(B \to K^{(*)}\nu\bar\nu)$, and the LFU violations in B decays would provide a very valuable tool to shed more light on this interesting phenomenon.