Bounce configuration from gradient flow

Abstract

Based on the gradient flow, we propose a new method to determine the bounce configuration for false vacuum decay. Our method is applicable to a large class of models with multiple fields. Since the bounce configuration is a saddle point of an action, a naive gradient flow method does not work. We point out that a simple modification of the flow equation can make the bounce configuration its stable fixed point while the false vacuum configuration an unstable one. Consequently, the bounce configuration can be obtained simply by following the flow without a careful choice of an initial configuration. With numerical analysis, we confirm the validity of our claim, checking that the flow equation we propose indeed has solutions that flow into the bounce configuration.Introduction: Study of false vacua (and metastable states) has been important in various fields, like particle physics, cosmology, nuclear physics, condensed matter physics, and so on. For example, in the field of particle physics and cosmology, the stability of the electroweak vacuum has been attracted much attention. In particular, taking the best-fit values of the observed top-quark and Higgs-boson masses, and assuming that the standard model of particle physics is valid up to a very high scale (like the Planck scale), the electroweak vacuum is metastable [1,2]. It is because the Higgs quartic coupling constant becomes negative at a high energy scale due to the renormalization group effects. Performing precise calculation based on relativistic quantum field theory, the decay rate of the electroweak vacuum per unit volume is known to be ∼ 10 −582 Gyr −1 Gpc −3 [3-5], with which the stability of our universe looks plausible for the present cosmic time scale. However, this conclusion may be altered with the introduction of new physics beyond the standard model. The studies of the stability of the electroweak vacuum in such new physics models remain important.In relativistic quantum field theory, the decay of the false vacuum is mainly induced by the field configuration called "bounce" [6][7][8]. Bounce is a configuration obeying the classical equation of motion (EOM) derived from the Euclidean action. With the bounce, which we denote as φ, the decay rate of the false vacuum per unit volume is given in the following form:

Classical production of ’t Hooft–Polyakov monopoles from magnetic fields

“…This is a considerably harder computational task than finding a minimum-energy solution. We achieve this using a modified gradient flow algorithm proposed by Chigusa et al [23]. This converges on saddle points by including a term in each flow iteration that lifts the negative mode, and is summarized briefly below; for full details consult the referenced paper.…”

Electroweak sphaleron in a strong magnetic field

“…Here we give a brief overview of the modified gradient descent method used in our calculations to find saddle point solutions in lattice electroweak theory. For full details, the reader is invited to consult the original paper by Chigusa et al introducing the method [24].…”

“…[22]: we employed a modified gradient flow first proposed in Ref. [24] to converge on saddle points of the discretized energy. For full numerical details, readers are invited to consult these papers, but we include a brief summary of the method in Appendix.…”

“…After smoothing, using standard gradient flow, the modified gradient flow of Ref. [24] could be used to find a saddle-point solution of the fields that tends to the Ambjørn-Olesen vortex lattice at large distances. After the first sphaleron solution against an Ambjørn-Olesen background was found, the magnitude of the external field was increased further by changing the scalar field VEV.…”

SimpleBounce: A simple package for the false vacuum decay

“…Our package utilizes a gradient flow equation. This framework is recently proposed by Chigusa, Moroi, and Shoji [19]. Our package solves a flow equation which is proposed by the author [1].…”