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.
