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 $V(\varphi )=\frac{m^2}{2}{\varphi }^2-\frac{\lambda }{4}{\varphi }^4+\frac{g}{6}{\varphi }^6$, we construct a nonradial solution explicitly: two bubble-tubes of opposite sign wrapping orthogonal rings, invariant under $\mathrm{O}(2)\times \mathrm{O}(2)$ 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.

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