Prediction for Maximum Supercooling in SU(N) Confinement Transition

TL;DR

The paper investigates the maximum supercooling in the confinement transition of SU($N$) Yang-Mills theory and its impact on gravitational wave signals. It combines lattice data for latent heat and domain-wall tension with analytically tractable softly broken $\mathcal{N}=1$ SYM on $\mathbb{R}^3\times S^1$ to derive an effective potential for the $N-1$ holonomies and compute the bounce action $S_b(\epsilon)$ near the critical temperature. The authors show that the metastable deconfined phase becomes unstable just below $T_{\rm cr}$, leading to a maximal supercooling $\epsilon_{\rm sc}^{YM}$ of order a few percent, and they argue this strongly suppresses the gravitational wave signal from the transition. The work provides testable lattice predictions and highlights the crucial role of multi-field holonomy dynamics in confining PTs, with potential implications for early-universe cosmology.

Abstract

The thermal confinement phase transition (PT) in $SU(N)$ Yang-Mills theory is first-order for $N\geq 3$, with bounce action scaling as $N^2$. Remarkably, lattice data for the action include a small coefficient whose presence likely strongly alters the PT dynamics. We give evidence, utilizing insights from softly-broken SUSY YM models, that the small coefficient originates from a deconfined phase instability just below the critical temperature. We predict the maximum achievable supercooling in $SU(N)$ theories to be a few percent, which can be tested on the lattice. We briefly discuss the potentially significant suppression of the associated cosmological gravitational wave signals.

Figures (6)

• Figure 1: Left: Potential of Eq. \ref{['eqn:approx_pot']} with $N=3$ as a function of $\vec{\phi}=(\phi_{1},\phi_{2})$ at $\epsilon=0$. The simplicity of this particular $N=3$ potential is misleading. Although the $\mathbb{Z}_{N}$ structure is present for $N>3$, the valleys connecting the vacua are no longer on straight paths in the field space. Right: Potential for $N=3$ as a function of $\vec{\phi}=(\phi_{1},0)$ for different values of $\epsilon$. For $\epsilon>\epsilon_{\rm sc}$ the deconfined phase is mechanically unstable.
• Figure 1: The bounce profile of nine holonomies $\vec{\phi}$ for $SU(10)$ at $\epsilon=0.037$ as function of $\rho$ that is the radial parameter of the $O(3)$-symmetric bounce.
• Figure 2: Maximum possible supercooling $\epsilon_{\rm sc}$ and superheating $\epsilon_{\rm sh}$, in the softly broken $\mathcal{N}=1$ SYM on $\mathbb{R}^3\times S^1$, fitted as a function of $N$ up to order $1/N^{2}$ .
• Figure 2: Projection of bounce trajectory for $SU(10)$ at $\epsilon=0.037$ on different planes. In the case of $SU(10)$, the field space has nine dimensions. Upper Left: $\phi_{1}-\phi_{3}$ plane, Upper Right: $\phi_{1}-\phi_{4}$ plane, Lower Left: $\phi_{1}-\phi_{7}$ plane, Lower Right: $\phi_{1}-\phi_{9}$ plane.
• Figure 3: The rescaled bounce action $\epsilon^{2}\widetilde{S}_{\rm b}$ as a function of $\epsilon$ for $SU(N)$ with $N=\{3,4,\ldots,15\}$. $\widetilde{S}_{\rm b}$ has two zeros, one at $\epsilon_{\rm sc}$ and the other at $\epsilon_{\rm sh}$, and a double pole at $\epsilon=0$ as expected. The results are seen to converge at large $N$. Numerical values are calculated using the FindBounce package Guada:2020xnz.