The cosmological Higgstory of the vacuum instability

TL;DR

This work analyzes the cosmological implications of the SM Higgs vacuum instability, addressing gauge dependence with Nielsen identities and employing a canonical Higgs field to study long-wavelength fluctuations during inflation. It shows that AdS patches where the Higgs sits in the true vacuum pose a lethal threat unless the inflationary Hubble scale is sufficiently small, yielding bounds that depend on the reheating temperature and on the Higgs coupling to gravity or to the inflaton. Post-inflation dynamics, including pre-heating and reheating, can rescue some patches via thermal corrections, but the fate of AdS bubbles after inflation imposes strong constraints; a full GR treatment of bubble evolution clarifies under which conditions expanding bubbles threaten cosmology. The paper also explores a speculative quantum-gravity link that translates into precise Higgs-top mass correlations, which surprisingly align with current measurements, suggesting a possible deep connection between electroweak metastability and the ultimate fate of de Sitter space.

Abstract

The Standard Model Higgs potential becomes unstable at large field values. After clarifying the issue of gauge dependence of the effective potential, we study the cosmological evolution of the Higgs field in presence of this instability throughout inflation, reheating and the present epoch. We conclude that anti-de Sitter patches in which the Higgs field lies at its true vacuum are lethal for our universe. From this result, we derive upper bounds on the Hubble constant during inflation, which depend on the reheating temperature and on the Higgs coupling to the scalar curvature or to the inflaton. Finally we study how a speculative link between Higgs meta-stability and consistence of quantum gravity leads to a sharp prediction for the Higgs and top masses, which is consistent with measured values.

Figures (21)

• Figure 1: The dashed curves show the effective quartic coupling (left) and effective SM potential (right) computed at next-to-leading order in a generic Fermi $\xi$-gauge. The thick red dashed curve corresponds to the Landau gauge, $\xi=0$. The right handed panel shows that the height of the potential barrier is only approximately gauge-independent (a measure of the residual gauge dependence). The black continuous curves show the same potential expressed in terms of the canonical field $h_{\rm can}$: the gauge dependence in the potential gets compensated by the gauge-dependence of the kinetic term, such that the continuous curves nearly overlap.
• Figure 2: Random distribution of the Higgs field $\bar{h} = h/h_{\rm max}$ after $N=60$ e-folds of inflation with Hubble constant equal to the Higgs instability scale, $H = h_{\rm max}$. The blue dashed curve is the $V=0$ Gaussian approximation of eq. (\ref{['eq:hGauss']}). The red curve is the SM Higgs potential $\bar{V}(\bar{h})$, in arbitrary units.
• Figure 3: Minimal probability that, after $N=60$ e-folds of inflation, the Higgs fluctuated above the SM potential barrier (orange curve), or fall down to the true minimum (red curve). The continuous curves are the numerical results; the dashed curves are the analytical approximations presented in the text.
• Figure 4: Running of the Higgs coupling to gravity $\xi_H$ as a function of the renormalisation scale in the SM, for different initial conditions at the Planck scale. The dashed horizontal lines correspond to the special values $\xi_H=-1/6$ and $\xi_H=0$.
• Figure 5: Random distribution of the Higgs field $\bar{h} = h/h_{\rm max}$ after $N=60$ e-folds of inflation with Hubble constant equal to the Higgs instability scale, $H = h_{\rm max}$, and for $\xi_H = -0.01$. The blue dashed line is the Gaussian approximation of eq. (\ref{['eq:hGaussxi']}). The red curve is the Higgs potential $V_{\rm SM}(h) - 12\xi_H H^2 h^2/2$, in arbitrary units.