Vacuum Stability, Perturbativity, and Scalar Singlet Dark Matter

TL;DR

This work analyzes one-loop vacuum stability and perturbativity bounds in a Z2-symmetric real scalar singlet extension of the SM (Z2xSM). The authors show that the Higgs–singlet coupling $a_2$ modifies the RG running of the Higgs quartic coupling, relaxing the SM Higgs mass lower bound and lowering the perturbativity upper bound on $M_h$ for a given cutoff $\Lambda$, while the singlet self-coupling $b_4$ and relic-density constraints further constrain the parameter space. They derive the RGEs for $\lambda$, $a_2$, and $b_4$ from the one-loop effective potential and enforce absolute vacuum stability up to $\Lambda$, using two perturbativity criteria to avoid spurious Landau poles; the results show that strong $a_2$ can shrink or even eliminate the viable Higgs window at high $\Lambda$, and that relic-density considerations impose lower bounds on the singlet mass $M_S$ tied to $M_h$ and $b_4$. The analysis highlights that a light scalar DM discovery could provide indirect information on the singlet self-coupling and emphasizes the sensitivity of the theory to the assumed perturbativity bounds and cutoff scale, with implications for direct-detection searches and the interpretation of future Higgs/dark-matter data.

Abstract

We analyze the one-loop vacuum stability and perturbativity bounds on a singlet extension of the Standard Model (SM) scalar sector containing a scalar dark matter candidate. We show that the presence of the singlet-doublet quartic interaction relaxes the vacuum stability lower bound on the SM Higgs mass as a function of the cutoff and lowers the corresponding upper bound based on perturbativity considerations. We also find that vacuum stability requirements may place a lower bound on the singlet dark matter mass for given singlet quartic self coupling, leading to restrictions on the parameter space consistent with the observed relic density. We argue that discovery of a light singlet scalar dark matter particle could provide indirect information on the singlet quartic self-coupling.

Figures (10)

• Figure 1: The change in the Higgs mass upper and lower bounds as $a_2\left(M_Z\right)$ is increased while requiring stability of the potential and perturbativity of the quartic coupling constants up to cutoff scales $\Lambda=\left(10^{19},\, 10^9,\, 10^6,\, 10^3\right)\,\text{GeV}$ (the purple [dark gray], blue [light gray], green [medium gray], and orange [light gray] regions, respectively). Here, $b_4\left(M_Z\right)=0.001$ and $b_2\left( v_0\right) = 0$. Perturbativity of the potential using the constraints of Eq. $\left(\ref{['perturbativity_more']}\right)$ is indicated by the solid-colored regions; the combined solid-colored and striped regions indicate perturbativity of the potential using Eq. $\left(\ref{['perturbativity_less']}\right)$.
• Figure 2: One-loop vacuum stability and perturbativity bounds on the Standard Model Higgs mass $M_h$ as a function of the cutoff $\Lambda$ for a set of values of $a_2\left(M_Z\right)$. We show $a_2\left(M_Z\right)=\left(0.005, 0.4, 0.8\right)$ using green [medium gray], orange [light gray], and purple [dark gray], respectively. The upper bounds show the variation between the less restrictive constraints of Eq. $\left(\ref{['perturbativity_less']}\right)$ and the more restrictive constraints of Eq. $\left(\ref{['perturbativity_more']}\right)$.
• Figure 3: The analogous plot to Fig. \ref{['fig:mhva2']} for the case $b_4\left(M_Z\right)=0.4$. Here there is no region of Higgs masses and $S$-$H$ couplings $a_2$ that allow for perturbativity of the potential up to $10^{19}\,\text{GeV}$. The color and grayscale schemes are the same as Fig. \ref{['fig:mhva2']}.
• Figure 4: The analogous plot to Fig. \ref{['fig:funnelall']} for the case $b_4\left(M_Z\right)=0.4$, demonstrating that stability and perturbativity of the potential cannot be maintained to arbitrarily large ( i.e.,$\mathcal{O}\left(M_{Pl}\right)$) scales when $b_4$ is increased. The color and grayscale schemes are the same as Fig. \ref{['fig:funnelall']}.
• Figure 5: Values of $a_2\left(M_Z\right)$ and $M_S$ consistent with WMAP relic density measurements (blue [dark gray] band) and vacuum stability/perturbativity (unshaded regions) for fixed $M_h=200\,\text{GeV}$ and $b_4\left(M_Z\right)=0.4$. The unshaded region below the red [medium gray] curve has stable $\left(h=v_0,\, S=0\right)$ minima up to a $\Lambda=1\,\text{TeV}$ cutoff scale. The shaded region above the red curve is excluded because of the occurrence of deeper minima along the $S$ direction of the effective potential. Additionally, we indicate the region excluded by CDMS-II (brown [dark gray] arrow) and the region of projected sensitivity for SuperCDMS (green [light gray] arrow).