  This article is an addendum to Ref.~\cite{Baek:2014jga}. Here, we discuss the
invisible Higgs decay width $\Gamma_{h}^{\rm inv}$ in the Higgs portal vector
dark matter (VDM) model in the limit $m_V \rightarrow 0^+$. In the effective
field theory (EFT) approach where the VDM mass is attributed to the
St\"{u}ckelberg mechanism, $( \Gamma_{h}^{\rm inv} )_{\rm EFT}$ is divergent,
which is unphysical and puzzling. On the other hand $( \Gamma_{h}^{\rm inv}
)_{\rm UV}$ becomes finite in a UV completion, where the VDM mass is generated
by the dark Higgs mechanism. Then we can take the limit $m_V \rightarrow 0^+$
by taking either {\it (i)} the dark gauge coupling $g_X \rightarrow 0^+$ with a
fixed dark Higgs vacuum expectation value $v_\Phi$, or {\it (ii)} $v_\Phi \to
0^+$ with a fixed $g_X$. Such a difference in the behavior of $\Gamma_{h}^{\rm
inv}$ 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.
