Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect

TL;DR

This work addresses the electroweak vacuum instability in the Standard Model for Higgs masses in the LHC-relevant range by introducing a heavy scalar singlet with a nonzero vev that couples to the Higgs via a Higgs-portal. The key idea is a tree-level threshold correction δλ = $λ_{HS}^2/λ_S$ which, when integrating out the singlet at scale $M_S$, yields an effective Higgs quartic $λ = λ_H - δλ$ that can stabilize the potential; the effect is robust and non-decoupling. If $λ_{HS}>0$, the threshold condition $λ_H(μ) > δλ$ above $M_S$ is softened at larger scales and RG contributions further boost stability, potentially raising the instability scale up to very high values. If $λ_{HS}<0$, stabilization relies on RG effects above $M_S$, with a modified running for the effective coupling $λ$, and larger couplings are typically required. The mechanism is illustrated in three UV-m motivated contexts—see-saw neutrino masses, invisible axion, and unitarized Higgs inflation—showing the Higgs mass windows in which stabilization is feasible. Overall, a minimal addition of one singlet scalar can render the electroweak vacuum absolutely stable across plausible parameter ranges, offering a simple, UV-complete path to resolve a potential cosmological and phenomenological tension.

Abstract

We show how a heavy scalar singlet with a large vacuum expectation value can evade the potential instability of the Standard Model electroweak vacuum. The quartic interaction between the heavy scalar singlet and the Higgs doublet leads to a positive tree-level threshold correction for the Higgs quartic coupling, which is very effective in stabilizing the potential. We provide examples, such as the see-saw, invisible axion and unitarized Higgs inflation, where the proposed mechanism is automatically implemented in well-defined ranges of Higgs masses.

Figures (5)

• Figure 1: Feynman diagram for the tree-level threshold correction to the Higgs quartic coupling.
• Figure 2: Running of the Higgs quartic coupling in the SM and in the model with a scalar singlet, here assumed to have the mass $M_S =10^8\,{\rm GeV}$. Left: if $\lambda_{HS}>0$, thanks to the tree level shift at the singlet mass, the coupling never enters into the instability region, even assuming that singlet contributions to the RG equations are negligible. Right: if $\lambda_{HS}<0$ the instability can be shifted away or avoided only by singlet contributions to the RG equations.
• Figure 3: For $m_h=125$ GeV and $\lambda_{HS}>0$, bands of the modified instability scale $\Lambda_I$ versus the threshold correction $\delta\lambda$ to the Higgs quartic coupling due to a scalar singlet with mass $M_S=10^4, 10^6, 10^8, 10^{10}$GeV (from left to right). For a fixed $M_S$ value the lowest boundary of the band corresponds to small $\lambda_S, \lambda_{HS}$ and the highest boundary to $\lambda_S(M_{\rm Pl})=4\pi$.
• Figure 4: For $m_h=125$ GeV and $\lambda_{HS}<0$, the modified instability scale $\Lambda_I$ versus the threshold correction $\delta\lambda$ to the Higgs quartic coupling due to a scalar singlet with mass $M_S= 10^8$GeV and $\lambda_S(M_S)=0.01, 0.1, 0.2$, as indicated.
• Figure 5: The SM instability scale $\Lambda_I$ increasing as a function of the Higgs mass. The central line corresponds to $M_t=173.2$ GeV and $\alpha_{\rm s}(M_Z)=0.1184$ and the side-bands to 1 sigma deviations as indicated (with the larger deviation for the top mass uncertainty). The horizontal lines and band mark several values or ranges of interest for $\Lambda_I$. The three lowest lines are relevant for the see-saw case and correspond to lower limits on the mass $M_1$ of the lightest right-handed neutrino $N_1$ coming from thermal leptogenesis. The bound depends on the initial density $\rho_{N_1}$: $M_1> 2\times 10^7$GeV for $\rho_{N_1}\sim 0$; $M_1> 5\times 10^8$GeV for thermal $\rho_{N_1}$ and $M_1>2\times 10^9$GeV for $\rho_{N_1}$ dominating the universe; the shaded band shows the range of singlet masses $10^9-10^{12}$GeV relevant for the axion case; and the upper line is the singlet mass $10^{13}$GeV relevant for the unitarized Higgs inflation case.