Addendum to "Invisible Higgs decay width versus dark matter direct detection cross section in Higgs portal dark matter models"

This article is an addendum to [S. Baek et al., Phys. Rev. D 90, 055014 (2014).]. Here, we discuss the invisible Higgs decay width Gamma(inv)(h) in the Higgs portal vector dark matter (VDM) model in the limit m(v) -> 0(+). In the effective field theory (EFT) approach where the VDM mass is attributed to the Stiickelberg mechanism, (Gamma(inv)(h))(EFT) is divergent, which is unphysical and puzzling. On the other hand (Gamma(inv)(h))(UV) becomes finite in a UV completion, where the VDM mass is generated by the dark Higgs mechanism. Then we can take the limit m(v) -> 0(+) by taking either (i) the dark gauge coupling g(X) -> 0(+) with a fixed dark Higgs vacuum expectation value v Phi, or (ii) v Phi -> 0(+) with a fixed g(X). Such a difference in the behavior of Gamma(inv)(h) in the massless VDM limit demonstrates another limitation of EFT for the Higgs portal VDM, and the importance of gauge-invariant and renormalizable models for the Higgs portal VDM.