Implications of \(B \rightarrow K \nu {\bar{\nu }}\) under rank-one flavor violation hypothesis

We study the implications of the observed excess in \(B^+ \rightarrow K^+ \nu \bar{\nu }\) under the assumption of Rank-One Flavour Violation, i.e. that New Physics couples to a single specific direction in flavour space. By varying this direction we perform analyses at the level of the low-energy EFT, the SMEFT, and with explicit mediators such as leptoquarks and colorless vectors (\(Z^\prime \) and \(V^\prime \)). We study correlations with other flavour, electroweak and collider observables, finding that the most interesting ones are with \(K \rightarrow \pi \nu \bar{\nu }\), \(B_s \rightarrow \mu ^+ \mu ^-\), meson mixing and the LHC searches in \(\tau ^+ \tau ^-\) high-energy tails. Among the various mediators, the scalar leptoquarks \(\tilde{R}_2\) and \(S_1\) offer the best fits of the Belle-II excess, while being consistent with the other bounds. On the other hand, colorless vectors are strongly constrained by meson mixing and resonance searches in \(p p \rightarrow \tau ^+ \tau ^-\). In all cases we find that a flavour alignment close to the third generation is generically preferred.