Complementarity of $B\to K^{(\ast )}\mu \overline{\mu }$ and $B\to K^{(\ast )}+\text{inv}$ for searches of GeV-scale Higgs-like scalars

The rare decays $B^{+}\to K^{+}\mu \overline{\mu }$ and $B^0\to K^{\ast 0}\mu \overline{\mu }$ provide the strongest constraints on the mixing of a light scalar with the Higgs boson for GeV-scale masses. The constraints sensitively depend on the branching ratio to muons. Additional decay channels like an invisible partial width may substantially weaken the constraints. This scenario will be probed at Belle II in $B\to K^{(\ast )}+$inv. We illustrate the complementarity of scalar decays to muons and invisible decays using the currently available results of LHCb and BaBar. We provide two simple model realisations providing a sizeable invisible scalar width, one based on a real scalar and one based on a $U(1{)}_{B-L}$ gauge symmetry. In both examples the scalar decays into heavy neutral leptons which can be motivated by the seesaw mechanism for neutrino masses.