Resonant electroweak clouds around Schwarzschild black holes

Abstract

We construct nonlinear electroweak clouds around Schwarzschild black holes in the bosonic Weinberg-Salam theory. These configurations are supported by harmonically time-dependent charged gauge fields, while their energy-momentum tensor remains static. Regularity at the horizon imposes a resonance condition relating the field frequency to the horizon electrostatic potential. Unlike linear clouds, these solutions exhibit a finite mass and electric charge gap, and exist only up to a maximal horizon size. We obtain both magnetic and electric clouds, including spherically symmetric and dipolar configurations, which reduce to electroweak balls in the vanishing-horizon limit. We also report a new family of dipolar electroweak balls, which can be energetically favoured over their spherical counterparts at fixed electric charge. Our results, obtained in the probe limit for electroweak couplings away from their measured values, establish a new realization of nonlinear resonant vector hair in a non-Abelian gauge-Higgs theory.