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

Abstract

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

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