Metastable vacuum decay and $θ$ dependence in gauge theory. Deformed QCD as a toy model

TL;DR

The paper analyzes metastable vacuum states and θ-dependence in gauge theories using a center-stabilized, deformed QCD toy model that remains confining and captures key topological features of QCD. It derives a dual 3D sine-Gordon description with $N$ monopole types and a θ-entry $\theta/N$, and identifies metastable vacua—populated by evenly spaced $\sigma_n$—with energy gaps $\epsilon\sim N^{-1}$ and spontaneous $\mathcal{P}$/$\mathcal{CP}$ violation via the topological density. The decay of these metastables is treated with the thin-wall Coleman picture, and the authors solve the bounce equations numerically, finding an exponentially small decay rate $\Gamma/V \sim \exp[-\mathcal{N}\,a N^{b}]$, with large-$N$ asymptotics aligning with recent analytic results. They further improve the large-$N$ analysis by adapting the domain size and initial guess, showing agreement with the Li 2014 asymptotic form for $N\gtrsim 35$, thereby validating the qualitative and semi-quantitative picture. The work speculates that metastable, CP-violating states could underlie event-by-event CP asymmetries in heavy ion collisions if such states are realized coherently during cooling, effectively inducing a spatially extended $\theta$ parameter and producing observable CME-like signals.

Abstract

We study a number of different ingredients related to the $θ$ dependence, metastable excited vacuum states and other related subjects using a simplified version of QCD, the so-called "deformed QCD". This model is a weakly coupled gauge theory, which however preserves all the relevant essential elements allowing us to study hard and nontrivial features which are known to be present in real strongly coupled QCD. Our main focus in this work is to test the ideas related to the metastable vacuum states (which are known to be present in strongly coupled QCD in large $N$ limit) in a theoretically controllable manner using the "deformed QCD" as a toy model. We explicitly show how the metastable states emerge in the system, why their life time is large, and why these metastable states must be present in the system for the self-consistency of the entire picture of the QCD vacuum. We also speculate on possible relevance of the metastable vacuum states in explanation of the violation of local $\cal{P}$ and $\cal{CP}$ symmetries in heavy ion collisions.

Figures (4)

• Figure 1: A plot of some simulation data for the one dimensional action (\ref{['1daction']}) as a function of the angle $\varphi$ between the boundary conditions done for $N = 7$.
• Figure 2: A plot of some simulation data for the $\sigma$ field configuration plotted across the domain wall done for $N = 7$.
• Figure 3: A plot of some simulation data for the decay exponent $F\left(N\right)$ plotted for $N$ in the range $15$ to $75$.
• Figure 4: A plot of the improved simulation data for the decay exponent $F\left(N\right)$ plotted for $N$ in the range $7$ to $75$.