Towards excluding a light $Z^\prime$ explanation of $b\to s\ell^+\ell^-$

The discrepancies between $b\to s\ell^+\ell^-$ data and the corresponding Standard Model predictions constitute the most significant hints for new physics (at the TeV scale or below) currently available. In fact, many scenarios that can account for these anomalies have been proposed in the literature. However, only a single light new physics explanation, i.e. with a mass below the $B$ meson scale, is possible: a light $Z^\prime$ boson. In this article, we point out that improved limits on $B\to K^{(*)}\nu\nu$, including the experimental sensitivities required for a proper treatment of the necessarily sizable $Z^\prime$ width, together with the forthcoming Belle~II analyses of $e^+e^-\to\mu^+\mu^-+{\rm invisible}$, can rule out a $Z^\prime$ explanation of $b\to s\ell^+\ell^-$ data with a mass below $\approx4\,$GeV. Importantly, such a light $Z^\prime$ is the only viable single particle solution to the $b\to s\ell^+\ell^-$ anomalies predicting $R(K^{(*)})>0$ in high $q^2$ bins, therefore providing an essential consistency test of data.Comment: 8 pages, 4 figure