Confinement in Holographic Theories at Finite Theta

Abstract

A strongly coupled confining gauge theory with a non-zero vacuum angle undergoing a deconfinement to confinement phase transition is studied in the holographic gravitational description. A simplified five-dimensional setup is constructed where a bulk scalar models the effect of the vacuum angle, and the suitable boundary conditions on the ultra-violet and the infra-red boundaries are identified. In this five-dimensional dual geometry, and in the limit of small back-reaction in the infra-red, the critical temperature for the phase transition is shown to reduce quadratically with the vacuum angle, matching lattice results. The rate for the phase transition is estimated and is seen to be enhanced (reduced) when the field theory has a relevant (irrelevant) deformation at high-energies. Crucially, for the irrelevant case, the confined phase can get destabilized for a range of parameters. In the context of early universe dynamics, if the vacuum angle is time-dependent, the transition history changes strongly: the deconfined phase can last till much lower temperatures than naively expected, and one can trigger a transition to the confined phase by a change in the vacuum angle, thus providing a controlled way to generate supercooling. As a phenomenological application, the peak frequency and the power of resulting gravitational wave signal from bubble collisions changes, affecting their visibility in detectors. Possible generalizations of the scenario are discussed.

Figures (5)

• Figure 1: Radion potential for the three cases discussed: type $\textbf{A}$ has a barrier at smaller $\varphi$, while type $\textbf{B, C}$ do not.
• Figure 2: Effect of $\theta$ on the radion potential of type $\textbf{A}$.
• Figure 3: Location of minimum as a function of $\kappa_\theta$ for type $\mathbf{A, B, C}$ type potentials in black, red, green respectively. Dashed gray line shows the maximum that merges with the minimum for type $\mathbf{A}$ type potential.
• Figure 4: The bounce action $S_b$ for $N=3$ as a function of $T/T_c(0)$, for $\theta = 0^\circ$ (blue solid) and $\theta = 36^\circ$ (red solid). Left panel shows for type $\mathbf{B}$ type radion potential where $\kappa < 0$ originally, and right panel shows for type $\mathbf{C}$ radion potential where $\kappa>0$ originally. Dotted lines shows $S_b^\text{max}$, which must be larger than the corresponding $S_b$ for the phase transition to proceed. For type $\mathbf{B(C)}$, we have taken $\kappa = -1/2(1/2), \epsilon = 1/10(-1/10)$ which gives $\kappa_\theta = 0.14$ for $N=3$ and $T_c(\theta)/T_c = 0.03(0.08)$. For both these cases, the other parameters of the potential are chosen to have the minimum at $\theta = 0$, $\left<\varphi\right>\sim 10^{-15}$. We have used the functional form of $\widetilde{S}_b$ from eq. \ref{['eq:Sb-function-thick-wall']} which gives $a_0 = 146.3, a_1 = -154.2, a_2 = 11.3, b_1 = 0.02, b_2 = 0.07$ for type $\mathbf{B}$ and $a_0 = 199.0, a_1 = -195.7, a_2 = 0, b_1 = 0.005, b_2 = 1.515$ for type $\mathbf{C}$.
• Figure 5: A cartoon of an early universe scenario, where the angle $\theta$ transitions from $\sim 30^\circ$ at early times to $\sim 0^\circ$ at late times. At early times, the bounce action traces the red curve as a function of the red-shifting temperature and the action is too large for the phase transition to proceed. Once the $\theta$ transition happens, the bounce action starts to trace the blue curve, and is now low enough for the phase transition to proceed. For some choice of parameters, the change in $\theta$ can trigger the deconfinement to confinement phase transition.