Probing lepton-flavour universality with \(K\rightarrow \pi \nu \bar{\nu }\) decays

We analyse the rare processes \(K\rightarrow \pi \nu \bar{\nu }\) 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, \(K\rightarrow \pi \nu \bar{\nu }\) 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 \(K\rightarrow \pi \nu \bar{\nu }\) branching ratios, closely correlated to similar effects in \(B\rightarrow K^{(*)}\nu \bar{\nu }\), are naturally expected. The correlation of \(\mathcal {B}(K \rightarrow \pi \nu \bar{\nu })\), \(\mathcal {B}(B\rightarrow K^{(*)}\nu \bar{\nu })\), and the LFU violations in B decays would provide a very valuable tool to shed more light on this interesting phenomenon.