A Tale of Invisibility: Constraints on New Physics in $b\to s\nu\nu$

The Belle II experiment will measure the rare decays $B\to K\nu\nu$ and $B\to K^* \nu\nu$ with increased sensitivity which can hence be expected to serve as a very efficient probe of new physics. We calculate the relevant branching ratios in low-energy effective field theory (LEFT) including an arbitrary number of massive sterile neutrinos and discuss the expected sensitivity to the different operators. We also take into account the longitudinal polarisation fraction $F_L$ and the inclusive decay rate $B\to X_s\nu\nu$. In our investigation we consider new physics dominantly contributing to one and two operators both for massless and massive (sterile) neutrinos. Our results show a powerful interplay of the exclusive decay rates $B\to K\nu\nu$ and $B\to K^*\nu\nu$, and a surprisingly large sensitivity of the inclusive decay mode to vector operators even under conservative assumptions about its uncertainty. Furthermore, the sensitivity of $F_L$ is competitive with the branching ratio of $B\to K^* \nu\nu$ in the search for new physics contributing to scalar operators and thus also complementary to $B\to K\nu\nu$ and $B\to X_s\nu\nu$.