The O(4)-breaking bubble

Abstract

False vacuum decay in field theory is thought to be dominated by Coleman's O(4)-symmetric bounce, the minimum action nontrivial solution of the imaginary time equations of motion. Beyond the bounce, non-constructive existence proofs of O(4)-breaking solutions are available in the mathematics literature, but the solutions themselves, and their physics, have remained unknown. Considering the simple, bounded-below, scalar field potential $V(ϕ)=\frac{m^2}{2}ϕ^2-\fracλ{4}ϕ^4+\frac{g}{6}ϕ^6$, we construct a nonradial solution explicitly: two bubble-tubes of opposite sign wrapping orthogonal rings, invariant under ${\rm O}(2)\times{\rm O}(2)$ rotations combined with a parity that exchanges the rings. The solution admits valid Cauchy data for real time evolution from a $t=0$ slice, and supports an odd number of unstable deformation modes.