Invisible Higgs Decay Width vs. Dark Matter Direct Detection Cross Section in Higgs Portal Dark Matter Models

The correlation between the invisible Higgs branching ratio ($B_h^{\rm inv}$) vs. dark matter (DM) direct detection ($\sigma_p^{\rm SI}$) in Higgs portal DM models is usually presented in the effective field theory (EFT) framework. This is fine for singlet scalar DM, but not in the singlet fermion DM (SFDM) or vector DM (VDM) models. In this paper, we derive the explicit expressions for this correlation within UV completions of SFDM and VDM models with Higgs portals, and discuss the limitation of the EFT approach. We show that there are at least two additional hidden parameter in $\sigma_p^{\rm SI}$ in the UV completions: the singlet-like scalar mass $m_2$ and its mixing angle $\alpha$ with the SM Higgs boson ($h$). In particular, if the singlet-like scalar is lighter than the SM Higgs boson ($m_2 < m_h \cos \alpha / \sqrt{1 + \cos^2 \alpha}$), the collider bound becomes weaker than the one based on EFT.