Holographic phase transitions via thermally-assisted tunneling

TL;DR

The paper addresses vacuum decay in holographic, strongly coupled gauge theories and studies the regime where thermally-assisted tunneling dominates. It introduces and constructs a thermal bounce, an $O(3)$-symmetric time-dependent Euclidean solution that connects the zero-temperature $O(4)$ bubble to the high-temperature critical bubble, enabling exact bounce actions $B(T)$ at low $T$. The main results show that the thermal bounce can dominate vacuum decay in certain parameter regions, shifting the nucleation temperature $T_n$ and yielding revised gravitational-wave predictions for critical temperatures $T_c$ spanning from the TeV scale up to $10^{12}$ GeV. This work thereby extends the phenomenological reach of holographic phase transitions, revealing observable stochastic gravitational waves for supercooled transitions in future detectors, while noting limitations of the 4D effective theory and the need for a full 5D analysis.

Abstract

We construct the thermal bounce solution in holographic models that describes first-order phase transitions between the deconfined and confined phases in strongly-coupled gauge theories. This new, periodic Euclidean solution represents transitions that occur via thermally-assisted tunneling and interpolates between the $O(4)$-symmetric vacuum bubble at zero temperature and the high temperature $O(3)$-symmetric critical bubble associated with classical thermal fluctuations. The exact thermal bounce solution can be used to obtain the bounce action at low temperatures which allows for a more accurate determination of vacuum decay rates, significantly improving previous estimates in holographic models. In particular, provided the phase transition is sufficiently supercooled, new predictions are obtained for the gravitational wave signal strength for critical temperatures ranging from the TeV scale up to $10^{12}$ GeV, some of which are within reach of future gravitational wave detectors.

Figures (10)

• Figure 1: Left: The radion potential (\ref{['eq:rad_pot_approx']}) rescaled to a form which is independent of the model parameters. Right: The magnitude of the curvature of the potential scaled by $\varphi_{\rm min}^2$, assuming $m_\varphi^2=0.1$ (red dashed line) and $m_\varphi^2=0.9$ (blue solid line). The radion mass is determined from the value of the curvature at $\varphi = \varphi_{\rm min}$, shown as the dot-dashed vertical line. The kink at $\varphi/\varphi_{\rm min}\approx 0.7$ corresponds to an inflection point, where the curvature changes sign from negative to positive as $\varphi$ increases.
• Figure 2: The static critical bubble (top row) and thermal bounce (bottom row) solutions, ${\bar{\varphi} = \varphi_b(\bar{\tau},\bar{r})/T}$, in the 4D effective radion theory (\ref{['eq:Seff']}), in the quantum regime (${T<T_q}$), and as a function of the dimensionless coordinates $\bar{\tau}=\tau T$, $\bar{r}=rT$. We assume $\kappa^4=0.1$, $\epsilon_2=-0.04$, $\epsilon_3 = -0.12, \,\alpha_{\rm IR} = 20$, $v_{\rm UV} = 0.35$ and $N_c = 3$ corresponding to $m_\varphi^2=0.9$ and $T_c=1$ TeV. (These solutions have explicit dependence only on the two parameters: $T/T_c$ and $m_\varphi^2$.) For comparison, both solutions are shown for a single period of $\bar{\tau}$ and identical ranges of the $\bar{\varphi}, \bar{r}$ axes. The boundary where $\varphi_b(\bar{\tau},\bar{r}) = 0$, is shown as a red curve in the $\bar{\tau} \bar{r}$-plane.
• Figure 3: Left: The quantum-to-classical transition temperature $T_q$ as a function of the critical temperature $T_c$ for $m_\varphi^2 = 0.1$ (dashed red line) and $m_\varphi^2 = 0.9$ (solid blue line). For $10^3\,{\rm GeV}\leq T_c \leq 10^{12}\,{\rm GeV}$, the ratio $T_q/T_c$ has the constant value of $7.4\times 10^{-3}$ and $0.13$ for $m_\varphi^2 = 0.1$ and $0.9$, respectively. The shaded region where $T_q>T_c$ is not physically relevant. Right: The ratio $T_q/T_c$ as a function of the radion mass-squared $m_\varphi^2$ depicting the two benchmark values (solid points) used in our analysis. Note that as $m_\varphi^2$ increases, there is more parameter space where the thermal bounce is important.
• Figure 4: The action $B$ (scaled by $1/N_c^2$) of the dominant false vacuum decay solution (solid blue line) as a function of the temperature $T$ for the large radion mass $m_\varphi^2=0.9$. Below the temperature $T_q$, depicted by the vertical, dot-dashed gray line, the thermal bounce dominates over the critical bubble solution. For $T<T_q$, the critical bubble action is shown by the blue dashed line for comparison.
• Figure 5: The nucleation temperature $T_n$ (scaled by 1/$T_c$) as a function of the critical temperature $T_c$ for $N_c = 3$ ($6$) shown as solid blue (red) lines, assuming $m_\varphi^2 = 0.1$ (left plot) and $0.9$ (right plot). For comparison, the nucleation temperature determined from the critical bubble solution is shown as a dashed blue (red) line for $N_c = 3$ ($6$). The horizontal gray dot-dashed line indicates the classical-to-quantum transition temperature, $T_q$ (again scaled by $1/T_c$) which separates the classical and quantum regimes. The values of the ratio where $T_n/T_c>1$ are not physically relevant. Note that for each value of $N_c$, the curves end at a maximum value of $T_c$, above which the transition never takes place $(i.e.~\Gamma < H^4)$.