Towards Completing the Standard Model: Vacuum Stability, EWSB and Dark Matter

TL;DR

The SM, viewed up to the $U(1)_Y$ Landau pole, suffers from a false vacuum due to dimensional transmutation that places a global minimum near $10^{26}$ GeV, making it phenomenologically untenable. The authors propose the minimal, classically scale-invariant addition of a complex scalar singlet $S$, whose real part induces EWSB via dimensional transmutation and through the Higgs portal stabilizes the Higgs potential, while the CP-odd component serves as a stable dark matter candidate; the CP-even component can also drive inflation under severe tuning. They demonstrate that a wide region of parameter space yields the correct DM relic density and remains compatible with Higgs phenomenology and DM direct detection limits, with the scalar sector perturbativity persisting up to scales near Planck, though the Landau pole remains below the SM $U(1)_Y$ pole. The framework provides a compact, testable route to address vacuum stability, EW-scale generation, DM, and inflation within a perturbative, gravity-agnostic extension, with concrete predictions for future DM searches and precision Higgs measurements.

Abstract

We study the standard model (SM) in its full perturbative validity range between $Λ_QCD$ and the $U(1)_Y$ Landau pole, assuming that a yet unknown gravitational theory in the UV does not introduce additional particle thresholds, as suggested by the tiny cosmological constant and the absence of new stabilising physics at the EW scale. We find that, due to dimensional transmutation, the SM Higgs potential has a global minimum at 10^26 GeV, invalidating the SM as a phenomenologically acceptable model in this energy range. We show that extending the classically scale invariant SM with one complex singlet scalar S allows us to: (i) stabilise the SM Higgs potential; (ii) induce a scale in the singlet sector via dimensional transmutation that generates the negative SM Higgs mass term via the Higgs portal; (iii) provide a stable CP-odd singlet as the thermal relic dark matter due to CP-conservation of the scalar potential; (iv) provide a degree of freedom that can act as an inflaton in the form of the CP-even singlet. The logarithmic behaviour of dimensional transmutation allows one to accommodate the large hierarchy between the electroweak scale and the Landau pole, while understanding the latter requires a new non-perturbative view on the SM.

Figures (9)

• Figure 1: Running of the gauge couplings, the top Yukawa and the Higgs self-coupling in the standard model. The Higgs quartic coupling is evaluated at 1-loop, the top Yukawa and the gauge couplings at 2-loop order.
• Figure 2: The SM Higgs effective potential as a function of the Higgs field strength, $V(h)=-\mu^2 h^2 + \lambda_H(h) \, h^4$. The Higgs mass parameter is approximated as a constant and the running quartic coupling is evaluated at 1-loop level, where the scale is set by the field strength $h$. The global minimum at $\sim 10^{26}$ GeV is generated by $\lambda_H$ running positive (from low to high energy) at this scale. The local minimum at the electroweak scale is caused by the negative Higgs mass parameter.
• Figure 3: The renormalization group running of the scalar couplings, for the initial values $\lambda_{RI} = 0.3$, $\lambda_R = -1.2 \times 10^{-3}$, $\lambda_{HI} = 0.35$, $\lambda_I=0.01$, $\lambda_{RH}=-10^{-4}$, $\lambda_H=0.12879$ and $m_t= 173.1$ GeV at the top mass scale. Notice that the Higgs self coupling remains positive in the whole range, while $\lambda_R$ runs negative around $10^4$ GeV, shown in the inset, generating a VEV at that scale.
• Figure 4: The perturbative range of validity of the model, i.e. the position of the Landau pole of the scalar couplings, as a function of the initial value of $\lambda_{IH}$ at the EW scale. $\lambda_{RI}\simeq 0$ and $\lambda_{R}\simeq 0$ (solid black line), $\lambda_{RI}=0.5$ and $\lambda_{R}\simeq 0$ (solid red line) and $\lambda_{RI}=0.5$ and $\lambda_{R}=0.1$ (dashed red line). The gray horizontal line is the Planck scale.
• Figure 5: Isocurves of $\lambda_H$ as functions of $\lambda_{RI}$ and $\lambda_{RH}$. The color scale represents the deviation $\delta\lambda_H$ from the SM value $\lambda_H\approx 0.13$. The black region corresponds to $|\delta\lambda_H|<0.001$, the darkest grey region to $|\delta\lambda_H|<0.002$ and so on, with the lightest grey region corresponding to $|\delta\lambda_H|<0.005$. The white region corresponds to large mixing, $\sin\theta_{SH}>0.3$, which we do not consider in this work. To the left of/below the white region the CP even scalar is lighter than the Higgs $m_s<m_h$. To the right/above the white region, $m_s>m_h$.