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_* b ∗ such that for b<b_* b < b ∗ , the vacua are translationally invariant (and the magnetic field is constant), while, for b>b_* b > b ∗ , they are not, (ii) for b>b_* 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 plane 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_* 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.