Probing lepton-flavour universality with $K → πν\overlineν$ decays

We analyse the rare processes $K → πν\overlineν$ 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, $K → πν\overlineν$ decays turn out to be particularly interesting: these are the only kaon decays with third-generation leptons (the τ 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 $U(2)_q×U(2)_l$ flavour symmetry. In this framework we show that O(1) deviations from the Standard Model predictions in $K → πν\overlineν$ branching ratios, closely correlated to similar effects in $B→K^{(∗)}ν\overline{v}$, are naturally expected. The correlation of $\mathcal{B}(K → πν\overline{v})$, $\mathcal{B}(B→K^{(∗)}ν\overline{v})$, and the LFU violations in B decays would provide a very valuable tool to shed more light on this interesting phenomenon.