Instability of electroweak homogeneous vacua in strong magnetic fields

We consider the classical vacua of the Weinberg-Salam (WS) model of electroweak forces. These are no-particle, static solutions to the WS equations minimizing the WS energy locally. We study the WS vacuum solutions exhibiting a non-vanishing average magnetic field of strength $b$, and prove that (i) there is a magnetic field threshold $b_*$ such that for $b < b_*$, the vacua are translationally invariant (and the magnetic field is constant), while for $b > b_*$ they are not, (ii) for $b > b_*$, there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plan transversal to $b$, and (iii) the lattice minimizing the energy per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold $b_*$. In the absence of particles, the Weinberg-Salam model reduces to the Yang-Mills-Higgs (YMH) equations for the gauge group $U(2)$. Thus our results can be rephrased as the corresponding statements about the $U(2)$-YMH equations.