Role of charged gauge fields in generating magnetic seed fields in bubble collisions during the cosmological electroweak phase transition

We calculate the magnetic field generated during bubble collisions in a first-order electroweak phase transition that may occur for some choices of parameters in the minimal supersymmetric standard model. We derive equations of motion from the electroweak Lagrangian that couple the Higgs field and the gauge fields of the standard model sector. We show that for sufficiently gentle collisions, where the Higgs field is relatively unperturbed in the bubble overlap region, the equations of motion can be linearized so that in the absence of fermions the charged $W^{\pm }$ fields are the source of the electromagnetic current for generating the seed fields. Solutions of the equations of motion for the charged gauge fields and Maxwell’s equations for the magnetic field in $O(1,2)$ space-time symmetry are expressed in closed form by applying boundary conditions at the time of collision. Our results indicate that the magnetic fields generated by charged $W^{\pm }$ fields in the collision are comparable to those found in previous work. The magnetic fields so produced could seed galactic and extra-galactic magnetic fields observed today.

Figure 1

Two bubbles colliding in the Coleman model as explained in the text. The distance scale is in arbitrary units.

Figure 2

A vertical slice through the colliding bubbles shown in Fig. 1 as explained in the text. The distance scale is in arbitrary units.

Figure 3

Magnitude of the azimuthal magnetic field calculated in the theory of the present paper. The field is shown as a function of distance $\rho$ from the axis of collision in the symmetry plane at time $\delta t=5$, 10, 15, and 20.

Figure 4

Magnitude of the azimuthal magnetic field calculated in the Abelian Higgs model. The field is shown as a function of distance $\rho$ from the axis of collision in the symmetry plane at time $\delta t=5$, 10, 15, and 20.

Figure 5

Magnitude of the azimuthal magnetic field calculated in the theory of the present paper. The field is shown as a function of distance $z$ along the axis of collision at a distance $\rho =1$ from the axis of collision for times $\delta t=5$, 10, 15, and 20.

Figure 6

Magnitude of the azimuthal magnetic field calculated in the Abelian Higgs model. The field is shown as a function of distance $z$ along the axis of collision at a distance $\rho =1$ from the axis of collision for times $\delta t=5$, 10, 15, and 20.

Figure 7

Spectral limit.