O($N$) Invariance of the Multi-Field Bounce

TL;DR

This paper proves that the bounce solution in multi-field vacuum decay problems is invariant under O(N) rotations, extending previous results limited to single scalar fields and addressing a gap in the theoretical understanding.

Contribution

It provides a proof that the O(N) invariance of the bounce holds for multiple scalar fields, generalizing earlier proofs for single fields and higher dimensions.

Findings

O(N) invariance of the bounce for multiple scalar fields established

Reduced problem approach links kinetic energy minimization to bounce symmetry

Supports the assumption of symmetry in phenomenological tunneling studies

Abstract

In his 1977 paper on vacuum decay in field theory: The Fate of the False Vacuum, Coleman considered the problem of a single scalar field and assumed that the minimum action tunnelling field configuration, the bounce, is invariant under O(4) rotations in Euclidean space. A proof of the O(4) invariance of the bounce was provided later by Coleman, Glaser, and Martin (CGM), who extended the proof to $N>2$ Euclidean dimensions but, again, restricted non-trivially to a single scalar field. As far as we know a proof of O($N$) invariance of the bounce for the tunnelling problem with multiple scalar fields has not been reported, even though it was assumed in many works since, being of phenomenological interest. We make progress towards closing this gap. Following CGM we define the reduced problem of finding a field configuration minimizing the kinetic energy at fixed potential energy. Given a solution of the reduced problem, the minimum action bounce can always be obtained from it by means of a scale transformation. We show that if a solution of the reduced problem exists, then it and the minimum action bounce derived from it are indeed O($N$) symmetric.