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 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 \(b_*\). 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.

我们考虑弱电力的温伯格-萨拉姆 (WS) 模型的经典真空。这些是 WS 方程的无粒子静态解，可局部最小化 WS 能量。我们研究了具有强度b的非零平均磁场的 WS 真空解，并证明 (i) 存在磁场阈值\(b_*\)，使得对于\(b<b_*\)，真空为平移不变（并且磁场是恒定的），而对于\(b>b_*\)，它们不是，(ii) 对于\(b>b_*\)，存在具有较低能量的非平移不变解每单位体积的二维晶格在垂直于b 的平面上具有离散平移对称性，并且 (iii) 当磁场强度接近阈值\(b_*\)时，使每单位体积的能量最小化的晶格接近六边形。。在没有粒子的情况下，Weinberg-Salam 模型简化为规范组U (2) 的 Yang-Mills-Higgs (YMH) 方程。因此，我们的结果可以改写为关于U (2)-YMH 方程的相应陈述。