False Vacuum Decay Rate From Thin To Thick Walls

TL;DR

This paper derives explicit formulas for the false vacuum decay rate in scalar field theories, providing accurate analytical and numerical results across thin and thick wall regimes in 3 and 4 dimensions.

Contribution

It introduces a unified approach to compute the decay rate, including bounce action and one-loop determinants, valid from thin to thick wall regimes in arbitrary dimensions.

Findings

Analytical bounce action expansion in thin wall parameter

Numerical functional determinant calculations in 3 and 4 dimensions

Compact formula for decay rate valid across regimes

Abstract

We consider a single real scalar field in flat spacetime with a polynomial potential up to $\phi^4$, that has a local minimum, the false vacuum, and a deeper global minimum, the true vacuum. When the vacua are almost degenerate we are in the thin wall regime, while as their difference in potential energy increases, we approach the thick wall regime. We give explicit simple formulae for the decay rate of the false vacuum in 3 and 4 spacetime dimensions. Our results include a careful treatment both of the bounce action, which enters at the exponent of the decay rate, and of the functional determinant at one loop, which determines the prefactor. The bounce action is computed analytically as an expansion in the thin wall parameter in generic $D$ dimensions. We find that truncating such an expansion at second order we obtain a remarkably accurate bounce action also deep into thick wall regimes. We calculate the functional determinant numerically in 3 and 4 dimensions and fit the results with simple polynomials of the same thin wall parameter. This allows us to write the complete one-loop decay rate as a compact expression, which works accurately from thin to thick wall regimes.