Bubble Wall Velocity at Strong Coupling

TL;DR

This work uses top-down holography to determine the steady-state velocity of expanding true-vacuum bubbles during chiral first-order transitions in strongly coupled QCD-like theories. By modeling the bubble wall as a trailing-brane configuration and computing the holographic drag force, the authors derive a universal zero-force condition that balances friction against the pressure difference between vacua, yielding a general velocity formula $v = C_d^{-1} \frac{T_c}{T_{boost}} \frac{p_t(T) - p_f(T_{boost})}{w_f(T_{boost})}$ valid across WSS and generic $Dp$-$Dq$ setups. The drag coefficient $C_d$ is connected to the gluonic equation of state via $C_d = 2\pi \frac{p_{glue}}{w_{glue}} \kappa_c$, with $\kappa_c$ a model-dependent constant, enabling a universal description of out-of-equilibrium bubble dynamics in strongly coupled plasmas. The results provide a non-perturbative benchmark for bubble dynamics and have potential implications for gravitational wave predictions from holographic phase transitions.

Abstract

Using the holographic correspondence as a tool, we determine the steady-state velocity of expanding vacuum bubbles nucleated within chiral finite temperature first-order phase transitions occurring in strongly-coupled large $N$ QCD-like models. We provide general formulae for the friction force exerted by the plasma on the bubbles and for the steady-state velocity. In the top-down holographic description, the phase transitions are related to changes in the embedding of $Dq$-${\bar Dq}$ flavor branes probing the black hole background sourced by a stack of $N$ $Dp$-branes. We first consider the Witten-Sakai-Sugimoto $D4$-$D8$-$\bar D8$ setup, compute the friction force and deduce the equilibrium velocity. Then we extend our analysis to more general setups and to different dimensions. Finally, we briefly compare our results, obtained within a fully non-perturbative framework, to other estimates of the bubble velocity in the literature.

Figures (7)

• Figure 1: Plot of the thin-wall bounce profile $x(y,\sigma)$ at the nucleation time, for $\tilde{L} = 0.62$ where $\tilde{L}\equiv 4\pi L T/3$ and $(x,\sigma)=(4\pi T/3)(x_4, \rho)$. The configuration smoothly interpolates between U-shaped profiles at $\sigma=0$ and disconnected branes at $\sigma \rightarrow \infty$.
• Figure 2: The trailing solution of eq. (\ref{['xiprime']}) for $v=0.5$ and $u_{J}=3u_T$. We have imposed $\xi(u_J)=0$. Notice that $\xi'(u_J)\neq0$ despite what one could guess from the picture.
• Figure 3: On the left, a cartoon of a possible steady-state solution, where $y=u/u_T$ and $(x,\sigma)=(4\pi T/3)(x_4, z)$. On the right a very schematic rotated zoom on the trailing wall and its boundaries.
• Figure 4: The bubble wall velocity as a function of the relative temperature $T/T_c$ from formula (\ref{['velhere']}). The constant line is the speed of sound of the flavor sector in the false vacuum, $c_s^2=dp_f/d\rho_f=1/6$.
• Figure 5: Solutions for the wall profile $\chi \equiv \xi/L$ for different values of the velocity.