Generation of magnetic field in a first-order electroweak phase transition

We study the generation of magnetic field during a first-order electroweak phase transition by evolving the classical equations of motion. The bubble production is done by a simple random nucleation scheme, controlled by a nucleation probability $p_b$, related to the quantum tunneling rate. The Higgs orientations of different bubbles are randomly and uniformly distributed on a 3-sphere. A damping term is introduced to damp the physical Higgs degree of freedom. We find magnetic field is generated during the phase transition, the strength of which is related to $p_b$ and the strength of the Higgs damping. Furthermore, we discuss the factors controlling the shape of the spectrum of the generated magnetic field.