Instability of electroweak homogeneous vacua in strong magnetic fields

TL;DR

This paper investigates the stability and structure of classical electroweak vacua under strong magnetic fields, revealing a transition from uniform to lattice-structured solutions with lower energy, approaching a hexagonal pattern near a critical magnetic field strength.

Contribution

It demonstrates a magnetic field threshold where electroweak vacua transition from translationally invariant to lattice-structured solutions with lower energy, revealing a new phase structure in the Weinberg-Salam model.

Findings

Existence of a magnetic field threshold $b_*$ for vacuum stability.

Below $b_*$, vacua are translationally invariant.

Above $b_*$, non-invariant solutions form a 2D lattice with lower energy.

Abstract

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.