Higgs portal to dark matter and B→ K(*) decays

We consider a Higgs portal model in which the 125-GeV Higgs boson mixes with a light singlet mediator h₂ h2 coupling to particles of a Dark Sector and study potential b→ s h₂ b→sh2 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 - h₂ h2 vertex. We clarify this issue by calculating the b - s - h₂ h2 vertex in an arbitrary R_ξ Rξ gauge and demonstrate how the ξ ξ dependence cancels from physical decay rates involving an on-shell or off-shell h₂ h2 . Then we revisit the b→ s h₂ b→sh2 phenomenology and point out that a simultaneous study of B→ K^* h₂ B→K∗h2 and B→ K h₂ B→Kh2 helps to discriminate between the Higgs portal and alternative models of the Dark Sector. We further advocate for the use of the h₂ h2 lifetime information contained in displaced-vertex data with h₂ h2 decaying back to Standard-Model particles to better constrain the h₂ h2 mass or to reveal additional h₂ h2 decay modes into long-lived particles.