Simple Gradient Flow Equation for the Bounce Solution

TL;DR

The paper tackles the challenge of computing the bounce solution, a saddle point governing false vacuum decay in Euclidean field theories. It builds a simple gradient flow that solves the CGM reduced problem by evolving ${\varphi}_i$ with $\frac{\partial}{\partial \tau} {\varphi}_i = \nabla^2 {\varphi}_i - \lambda[{\varphi}] \frac{\partial V}{\partial {\varphi}_i}$, where $\lambda[{\varphi}]$ keeps the potential energy ${\cal V}$ fixed while the kinetic energy ${\cal T}$ decreases, and obtains the bounce via the scale transformation $\phi_B(r) = \phi(\lambda^{-1/2} r)$. The method is shown to converge to a fixed point solving $\nabla^2 {\varphi}_i - \lambda \frac{\partial V}{\partial {\varphi}_i} = 0$, with the bounce arising as a scale-related degree of freedom, in agreement with CGM theory. Numerical tests on single- and multi-field potentials demonstrate robust convergence and excellent agreement with established codes like CosmoTransitions, validating the approach as a fast, reliable tool for computing bounce actions in models with multiple scalars. The preparation of a fast numerical package further enables practical deployment for decay-rate calculations in cosmology and particle physics.

Abstract

Motivated by the recent work of Chigusa, Moroi, and Shoji, we propose a new simple gradient flow equation to derive the bounce solution which contributes to the decay of the false vacuum. Our discussion utilizes the discussion of Coleman, Glaser, and Martin and we solve a minimization problem of the kinetic energy while fixing the potential energy. The bounce solution is derived as a scale-transformed of the solution of this problem. We also show that the convergence of our method is robust against a choice of the initial configuration.

Figures (7)

• Figure 1: A flow of the field configuration with the potential Eq. (\ref{['eq:single potential']}) with $d=4$ and the initial condition Eq. (\ref{['eq:single initial condition']}).
• Figure 2: Same as Fig. \ref{['fig:singlefield evolution']} except for the initial configuration.
• Figure 4: The black line is obtained from Eq. (\ref{['eq:bounce from gradient flow']}) in the limit of large $\tau$. The yellow dotted line is calculated by CosmoTransitions.
• Figure 5: The bounce solution in the $r$-$\phi$ plane is shown by solid lines. The dashed lines are results of CosmoTransitions. We take the potential Eq. (\ref{['eq:potential doublet scalar']}) with $c=2$ in $d=4$ space.
• Figure 6: The same bounce solution as Fig. \ref{['fig:doublescalar thickwall']} in $\phi_1$-$\phi_2$ plane.