Electroweak Vacuum Instability and Renormalized Higgs Field Vacuum Fluctuations in the Inflationary Universe

TL;DR

This paper examines the electroweak vacuum instability during inflation, emphasizing the importance of renormalized Higgs field fluctuations in curved spacetime and their impact on vacuum stability.

Contribution

It provides a quantum field theory analysis of Higgs vacuum fluctuations in curved spacetime, incorporating renormalization to assess stability during inflation.

Findings

Vacuum fluctuations grow with Hubble scale, risking vacuum decay.

Renormalization is essential to accurately evaluate physical effects.

Vacuum stability depends on the Hubble scale being below a critical value.

Abstract

In this work, we investigated the electroweak vacuum instability during or after inflation. In the inflationary Universe, i.e., de Sitter space, the vacuum field fluctuations $\left< {\delta \phi }^{ 2 } \right>$ enlarge in proportion to the Hubble scale $H^{2}$. Therefore, the large inflationary vacuum fluctuations of the Higgs field $\left< {\delta \phi }^{ 2 } \right>$ are potentially catastrophic to trigger the vacuum transition to the negative-energy Planck-scale vacuum state and cause an immediate collapse of the Universe. However, the vacuum field fluctuations $\left< {\delta \phi }^{ 2 } \right>$, i.e., the vacuum expectation values have an ultraviolet divergence, and therefore a renormalization is necessary to estimate the physical effects of the vacuum transition. Thus, in this paper, we revisit the electroweak vacuum instability from the perspective of quantum field theory (QFT) in curved space-time, and discuss the dynamical behavior of the homogeneous Higgs field $\phi$ determined by the effective potential ${ V }_{\rm eff}\left( \phi \right)$ in curved space-time and the renormalized vacuum fluctuations $\left< {\delta \phi }^{ 2 } \right>_{\rm ren}$ via adiabatic regularization and point-splitting regularization. We simply suppose that the Higgs field only couples the gravity via the non-minimal Higgs-gravity coupling $\xi(\mu)$. In this scenario, the electroweak vacuum stability is inevitably threatened by the dynamical behavior of the homogeneous Higgs field $\phi$, or the formations of AdS domains or bubbles unless the Hubble scale is small enough $H< \Lambda_{I} $.