Higgs portal to dark matter and $B\rightarrow K^{(*)}$ decays

We consider a Higgs portal model in which the 125-GeV Higgs boson mixes with a light singlet mediator h$_{2}$ coupling to particles of a Dark Sector and study potential b→sh$_{2}$ decays in the Belle II experiment. Multiplying the gauge-dependent off-shell Standard-Model b-s-Higgs vertex with the sine of the Higgs mixing angle does not give the correct b-s-h$_{2}$ vertex. We clarify this issue by calculating the b-s-h$_{Zahl}$ vertex in an arbitrary R$_{ξ}$ gauge and demonstrate how the ξ dependence cancels from physical decay rates involving an on-shell or off-shell h$_{2}$. Then we revisit the b→sh$_{2}$ phenomenology and point out that a simultaneous study of B→K∗h$_{2}$ and B→Kh$_{2}$ helps to discriminate between the Higgs portal and alternative models of the Dark Sector. We further advocate for the use of the h$_{2}$ lifetime information contained in displaced-vertex data with h$_{2}$ decaying back to Standard-Model particles to better constrain the h$_{2}$ mass or to reveal additional h$_{2}$ decay modes into long-lived particles.