O(N) Invariance of the Multi Field Bounce

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 that year by Coleman, Glaser, and Martin, who further extended the proof to N Euclidean dimensions. Their proof holds for N>2 and was again restricted non-trivially to the case of 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. In the talk, I will provide such proof. More precisely, I will show that if a non-trivial minimum action solution of the Euclidean field equations exists, then it is O(N) symmetric. This talk is based on arXiv:1611.04570.