No-hair theorems for black holes in the Abelian Higgs model

TL;DR

The authors analyze no-hair properties for black holes in the Abelian Higgs model across dimensions and horizon topologies, using horizon-adapted coordinates and a crucial first-integral of the field equations. They prove a no-hair theorem for neutral scalars in AdS with negative cosmological constant and positive potentials, and identify constraints on extremal hairy solutions by examining near-horizon geometries. For extremal horizons with $k=0,1$, they show the near-horizon region must be $AdS_2\times \Sigma$ with the scalar vanishing there, and derive a bound $m^2_{ ext{eff}}(r_+)\ge 4|\Lambda|$ that excludes regular extremal black branes with $m^2-2q^2<4|\Lambda|$. These results constrain the phase structure of holographic superconductors and the space of AdS gravity solutions with charged scalar hair.

Abstract

Motivated by the study of holographic superconductors, we generalize no-hair theorems for minimally coupled scalar fields charged under an Abelian gauge field, in arbitrary dimensions and with arbitrary horizon topology. We first present a straightforward generalization of no-hair theorems for neutral scalar hair. We then consider the existence of extremal black holes with scalar hair, and in the case of horizons with zero or positive curvature, provide a bound on the mass and charge of the scalar field that are necessary for the scalar hair to develop.