Vacuum Stability in the Standard Model and Beyond

Abstract

We revisit the stability of the Standard Model vacuum, and investigate its quantum effective potential using the highest available orders in perturbation theory and the most accurate determination of input parameters to date. We observe that the stability of the electroweak vacuum centrally depends on the values of the top mass and the strong coupling constant. We estimate that reducing their uncertainties by a factor of two to three is sufficient to establish or refute SM vacuum stability at the $5σ$ level. We further investigate vacuum stability for a variety of singlet scalar field extensions with and without flavor using the Higgs portal mechanism. We identify the BSM parameter spaces for stability and find sizable room for new physics. We further study the phenomenology of Planck-safe models at colliders, and determine the impact on the Higgs trilinear, the Higgs-to-electroweak-boson, and the Higgs quartic couplings, some of which can be significant. The former two can be probed at the HL-LHC, the latter requires a future collider with sufficient energy and precision such as the FCC-hh.

Figures (13)

• Figure 1: Shown is the coupling $\alpha_{\lambda,\text{eff}}$ characterizing the quantum effective potential at the field value $h$ where the top mass is given by the PDG 2024 central value $M_t^{\sigma}$ (thick blue line) as well as three standard deviations in either direction (thin lines). For comparison, we also show the resummed tree-level potential with the central value $M_t^{\sigma}$ as the top mass (thick dashed line). The gray band indicates the Planck scale. Effective potentials for which $\alpha_{\lambda,\text{eff}}$ stays positive for all field values $h$ are stable.
• Figure 2: Regions of stability for the SM Higgs potential as a function of the top mass $M_t$ and the strong coupling constant $\alpha_s^{(5)}(M_Z)$. Color-coding indicates stability ($\alpha_{\lambda,\text{eff}}(\mu)\ge 0$, green) or otherwise (red/gray). Uncertainties are derived by combining the $1\sigma-5\sigma$ intervals of both observables (dashed or solid rings). Upper panel: 2024 PDG values with top mass $M_t^\sigma$ (gray background) and uncorrelated uncertainties as well as CMS analysis CMS:2019esx with correlated uncertainties. Lower panel: PDG 2024 central value for the top mass $M_t^\text{MC}$ (black dot) and uncertainties added in quadrature. For comparison, we also show $M_t^\sigma$ (blue dot) and its $1\sigma$ uncertainty range (blue crosshair).
• Figure 3: BSM critical surface for the $O(N_S)$ model with $N_S=1,10,100$ and $10^4$ generations of real scalar singlets spanned by the pure BSM quartic $\alpha_v(M_s)$ and the Higgs portal $\alpha_\delta(M_s)$. Red areas correspond to a RG evolution featuring Landau poles below the Planck scale. Brown indicates instabilities in the Higgs potential ($\min \alpha_\lambda(\mu) \leq -10^{-4}$). Gray regions feature an unstable BSM scalar potential i.e. violation of \ref{['eq:Vstab1']} at some scale $\mu$ before $M_\text{Pl}$. Yellow regions correspond to a SM-like RG evolution with $-10^{-4} \leq \min \alpha_\lambda(\mu),\,\alpha_\lambda(M_\text{Pl}) \leq 0$ hinting a metastable Higgs. Dark (light) blue areas correspond to a strictly (softly) Planck-safe RG evolution with a stable potential all the way up to (at) $M_\text{Pl}$, see App. \ref{['sec:PS']}. Hatched black areas violate tree-level perturbative unitarity \ref{['eq:unitarity']}, while gray, lightly hatched ones are experimentally excluded by \ref{['eq:betabound']} on the scalar mixing angle, see Sec. \ref{['sec:Pheno']} for details.
• Figure 4: Two-loop renormalization group flow in the SM (dashed lines) and the scalar $O(N_S)$ model (solid lines) for $N_S=1$, $M_s=1\text{ TeV}$, $(\alpha_\delta,\,\alpha_v)|_{M_s}= (10^{-3}, \,10^{-2})$ (top) and $(\alpha_\delta,\,\alpha_v)|_{M_s}= (10^{-3}, \,10^{-4})$ (bottom). In the upper plot quartic couplings are trapped in a walking regime before the Planck scale, whereas in the lower plot the running to $M_\text{Pl}$ occurs within a weakly coupled regime without walking.
• Figure 5: BSM critical surface in the $M_s-\alpha_\delta(M_s)$-plane in a minimal BSM model featuring $N_S=1$ real BSM scalars and $O(N_S)$ symmetry for fixed $\alpha_v(M_s)=10^{-2}$ (top) and $\alpha_v(M_s)=10^{-4}$ (bottom). Same color coding as Fig. \ref{['fig:surface-O(N)']}.