Implications of $b\to s\mu\mu$ Anomalies for Future Measurements of $B \to K^{(*)} \nu \bar \nu$ and $K\to \pi \nu \bar \nu$

We investigate the consequences of deviations from the Standard Model observed in $b\to s\mu\mu$ transitions for flavour-changing neutral-current processes involving down-type quarks and neutrinos. We derive the relevant Wilson coefficients within an effective field theory approach respecting the SM gauge symmetry, including right-handed currents, a flavour structure based on approximate $U(2)$ symmetry, and assuming only SM-like light neutrinos. We discuss correlations among $B \to K^{(*)} \nu \bar \nu$ and $K\to \pi \nu \bar \nu$ branching ratios in the case of linear Minimal Flavour Violation and in a more general framework, highlighting in each case the role played by various New Physics scenarios proposed to explain $b\to s\mu\mu$ deviations.