Implications of $B \to K \nu \bar{\nu}$ under Rank-One Flavor Violation hypothesis

We study the implications of the observed excess in $B^+ \to K^+ \nu \bar{\nu}$ 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 $K \to \pi \nu \bar{\nu}$, $B_s \to \mu^+ \mu^-$, 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 $p p \to \tau^+ \tau^-$. In all cases we find that a flavour alignment close to the third generation is generically preferred.