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, Phys. Rev. D 90, 055014 (2014).PRVDAQ1550-799810.1103/PhysRevD.90.055014]. Here, we discuss the invisible Higgs decay width Γhinv in the Higgs portal vector dark matter (VDM) model in the limit mV→0+. In the effective field theory (EFT) approach where the VDM mass is attributed to the Stückelberg mechanism, (Γhinv)EFT is divergent, which is unphysical and puzzling. On the other hand (Γhinv)UV becomes finite in a UV completion, where the VDM mass is generated by the dark Higgs mechanism. Then we can take the limit mV→0+ by taking either (i) the dark gauge coupling gX→0+ with a fixed dark Higgs vacuum expectation value vφ, or (ii) vφ→0+ with a fixed gX. Such a difference in the behavior of Γhinv 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.