Light New Physics in $B\to K^{(*)}\nu\bar\nu$?

The study of the rare decays $B\to K^{(*)} \nu \bar\nu$ offers a window into the dynamics operating at the electroweak scale, allowing studies of the Standard Model and searches for heavy new physics. However, the analysis of these decays is also potentially sensitive to the on-shell production of new light bosons $X$ through the process $B\to K^{(*)} X$. In particular, Belle~II has recently measured $B^+\to K^+\nu\bar\nu$, finding a $2.8\sigma$ excess under the assumption of heavy new physics. Since this excess is rather localized in the kaon energy, a fit that includes the decay mode $B^+\to K^+ X$ to the kinematic distributions prefers $m_X\approx2\,$GeV with branching fraction Br$[B\to KX]=(8.8\pm2.5)\times 10^{-6}$ and a significance of $\approx3.6\sigma$. However, no excess was found in the BaBar measurements of $B\to K^{(*)} \nu \bar\nu$, and a global analysis of the Belle II and BaBar data leads to Br$[B\to KX]=(5.1\pm2.1)\times 10^{-6}$ with a reduced significance of $\approx2.4\sigma$. We then study various simplified dark-flavoured models and present a possible UV completion based on a gauged $B_3-L_3$ symmetry, highlighting the discovery potential of dedicated searches for $B\to K^{(*)}X$ at Belle II.