Lifetimes of (near) eternal false vacua

TL;DR

The paper demonstrates that long-lived or eternal false vacua can arise in quantum field theories when a $(d-1)$-form global symmetry protects degenerate vacua (universes). It shows how modifying the instanton sum to restrict topological charge induces such higher-form symmetries, with lifetimes set by $(d-2)$-brane tensions, and analyzes concrete realizations in the 2d Schwinger model and 4d QCD, where a 3-form ${f Z}_p^{(3)}$ symmetry mixes with a 0-form symmetry to form a nontrivial 4-group. In 2d, universes appear as degenerate vacua labeled by discrete shifts, with domain walls effectively non-dynamical; in 4d, a similar construction yields multiple universes connected through a higher-group structure, and eternal false vacua persist unless the higher-form symmetry is explicitly broken. The results reveal that decay rates for false vacua can be parametrically suppressed by UV-scale tensions, and they offer a framework for exploring global structure ambiguities and their phenomenological implications in QFT and potentially the Standard Model. All mathematical structures are encoded via higher-form symmetries and their anomalies, providing a unified perspective on universes across dimensions.

Abstract

We consider examples of long-lived false vacua in quantum field theory that arise from so-called `universes'. These false vacua are protected by a $(d-1)$-form global symmetry, where $d$ is the dimension of spacetime. The lifetimes of the false vacua are set by UV data: the tension of $(d-2)$-branes charged under a $(d-2)$-form gauge symmetry. The lifetimes can be made parametrically long even when the difference in energy density between the false and true vacua is large compared to the natural scales of the field theory. We study examples of near-eternal false vacua in abelian gauge theories in two dimensions and in four-dimensional QCD. In both cases, it is possible to view the $(d-1)$-form symmetries as arising from a modification of the sum over instantons. We find that the modification of the instanton sum in 4d QCD leads to a higher-group symmetry structure involving the 3-form and conventional 0-form global symmetries.

Figures (8)

• Figure 1: Effective potential $V_{\text{eff}}(x)$ for $p=3$, for different realizations of the $({\mathbb Z}_p)_x$ and $({\mathbb Z}_p)_z$ symmetries. The potential on the right is obtained by numerically diagonalizing the Hamiltonian \ref{['eq:Hqm2']} with $\mu = .02\, g$. The curvature of the potential near $x=0$ is roughly $m_{\text{eff}} \approx .02 \, g < \sqrt{p}\, \mu$, so the Born-Oppenheimer approximation is justified.
• Figure 2: Multi-branched structure of the effective potential for massless Schwinger models at $\theta =0$ with charge $p=1$ and $p=3$. For $p=1$, all branches correspond to the same universe. For $p=3$, there are three distinct universes.
• Figure 3: Multi-branched structure of the effective potential for $p=1,\theta =\pi$. The metastable vacuum (red dot) in the $k=0$ branch unstable. It decays to the minimum marked by the blue dot (on the same $k=0$ branch). By a shift of $2\pi$, this is equivalent to one of the true global minima, marked by the black dot.
• Figure 4: Multi-branched structure of the effective potential for $p=3,\theta =0$.
• Figure 5: Effective potential for bosonized scalars $\varphi,\eta$ with $p=3, m =0, M>0$. The four curves coincide with the four branches in Figure \ref{['fig:p=3a']}.