Structure of the ground state of the electroweak gauge theory in a strong magnetic field

The structure of the ground state of the Weinberg-Salam model in the presence of an external magnetic field is investigated. As the magnetic induction reaches a critical value $B_c$, $W$ pairs can be produced with zero energy. The equations for $B$, $Z$, $W$, and the Higgs field $\Phi$ are handled, near the transition point, by a perturbative method and solved exactly to first order in the parameter $e(B-B_c)$. Solutions with two types of boundary conditions are considered: (i) $B$ is produced in the interior of a large cylindrical solenoid by electric currents on the surface; (ii) a uniform background magnetic field exists throughout space, above $B_c$ a new phase with $W$ condensates emerges. The latter case admits solutions with lattice symmetry and an integer number of quanta of magnetic flux through each lattice cell. The $W$ wave function is expressed in terms of the Jacobi function ${\vartheta }_1\mathrm{}\left(z\right|\mathrm{}\tau )$ where $\tau$ is the lattice parameter. The average energy density $\overline{\mathcal{H}}$ as a function of $\tau$ is shown to be modular invariant. For $M_H>M_Z$, $\overline{\mathcal{H}}$ is minimal for a hexagonal lattice with a simple zero of $W$ at each center. Coherent quantum states are constructed for the $W$ pairs. The relation to type-II superconductivity is discussed.