A positive definite formulation of vacuum decay with reduced symmetry

Author(s)

Espinosa, José R., Jinno, Ryusuke, Konstandin, Thomas, Matake, Shogo, Miyachi, Taiga

Abstract

The Euclidean bounce for vacuum decay enjoys an $O(4)$ symmetry that is lost in the presence of impurities than can catalyze the decay. We present a formulation for the calculation of the tunneling decay action, that is explicitly positive definite, for impurities whose effects are spherically symmetric so that the bounce symmetry is reduced to $O(3)$. The action constructed can be regarded as a generalization of the tunneling potential method, which implicitly assumed $O(4)$ symmetry. We show that the action obtained reduces to the tunneling potential for $O(4)$-symmetric cases and provide analytic examples with $O(3)$ symmetry and arbitrary wall thickness.

Figures

Caption

Potential $V(\phi,r)$ and tunneling potential $V_t(\phi,r)$ for the deformed thin-wall example of subsection \ref{subsec:thinwallO(3)} with $\alpha=1$ and $\phi_0$ chosen to give a bubble radius $R\simeq 6$.

Caption

Action density $s(\phi,r)$ for the deformed thin-wall example of subsection \ref{subsec:thinwallO(3)} with $\alpha=1$ and $\phi_0$ chosen to give a bubble radius $R\simeq 6$. The red line shows $\phi_B(r,0)$.

Caption

Action (normalized to the $O(4)$ value) for the deformed thin-wall example of subsection \ref{subsec:thinwallO(3)} with $\phi_0$ chosen to give a bubble radius $R\simeq 6$, $\alpha=\alpha_V$ in the potential and $\alpha=\alpha_V+\delta\alpha_\phi$ in the profile of the deformed bounce. The green line shows the case with same $\alpha$ value in both functions.