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

TL;DR

This paper examines the behavior of the invisible Higgs decay width in vector dark matter models, highlighting limitations of effective field theory and emphasizing the need for gauge-invariant UV completions.

Contribution

It clarifies the divergence issues in EFT approaches and demonstrates how UV completions yield finite decay widths, stressing the importance of gauge invariance in Higgs portal dark matter models.

Findings

EFT predicts divergent invisible Higgs decay width as VDM mass approaches zero.

UV completion with dark Higgs mechanism results in finite decay width.

Different limits in gauge coupling and dark Higgs VEV affect decay width behavior.

Abstract

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.