Magnetic field generation in first order phase transition bubble collisions

We consider the formation of a ring-like magnetic field in collisions of bubbles of broken phase in an Abelian Higgs model. Particular attention is paid to multiple collisions. The small collision velocity limit, appropriate to the electroweak phase transition, is discussed. Assuming the relevant features of the Abelian Higgs model are appropriate also to the electroweak case, we argue that after the completion of the electroweak phase transition, when averaged over nucleation center distances, there exists a mean magnetic field $B\simeq 2.0\times {10}^{20}\mathrm{G}$ with a coherence length $9.1\times {10}^3{\mathrm{GeV}}^{\mathrm{-}1}$ (for $m_H=68\mathrm{}\mathrm{GeV}$). Because of the ring-like nature of $B,$ the volume average behaves as $B\sim 1/L.$ Taking into account the turbulent enhancement of the field by inverse cascade, we estimate that colliding electroweak bubbles would give rise to a mean field $B_{\mathrm{rms}}\simeq {10}^{-21}\mathrm{G}$ at 10 Mpc comoving scale today.