Bubble collisions in Abelian gauge theories and the geodesic rule

In an Abelian gauge symmetry, spontaneously broken at a first order phase transition, we investigate the evolution of two and three bubbles of the broken symmetry phase. The full field equations are evolved and we concentrate, in particular, on gauge-invariant quantities, such as the magnetic field and the integral around a closed loop of the phase gradient. An intriguing feature emerges, namely, the geodesic rule, commonly used in numerical simulations to determine the density of defects formed, is shown not to hold in a number of circumstances. It appears to be a function of the initial separation of the bubbles, and the coupling strength of the gauge field. The reason for the breakdown is that in the collision region the radial mode can be excited and it often oscillates about its symmetry-restoring value rather than settling to its broken symmetry value. This can lead to extra windings being induced in these regions and, hence, extra defects (antidefects) being formed.