$B\to K^* M_X$ vs $B\to K M_X$ as a probe of a scalar-mediator dark matter scenario

Recently, Belle II reported the observation of the decay $B\to K M_X$, $M_X$the missing mass, with the branching ratio much exceeding ${\cal B}(B\to K\nu\bar\nu)$ which is the only Standard Model (SM) process contributing to thisreaction. If confirmed, this might be an indication of new nonSM particlesproduced in this decay. One of the possible explanations of the observed effectcould be light dark-matter (DM) particles produced via a scalar mediator field.We give simple arguments, that a combined analysis of the $B\to K M_X$ and$B\to K^* M_X$ reactions would be a clean probe of the scalar mediatorscenario: (i) making use of an observed value ${\cal B}(B\to K M_X)\simeq 5.4\,{\cal B}(B\to K \nu\bar\nu)_{\rm SM}$ and (ii) assuming that the effect is dueto the light dark matter coupling to the top quark via a {\it scalar} mediatorfield, one finds an upper limit ${\cal B}(B\to K^* M_X) < 2.8 \, {\cal B}(B\toK^* \nu\bar\nu)_{\rm SM}$. Within the discussed scenario, this upper limit doesnot depend on the mass of the scalar mediator nor on the specific details ofthe unobserved dark-matter particles in the final state.