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

TL;DR

This paper extends no-hair theorems for black holes to include charged scalar fields in the Abelian Higgs model, relevant for holographic superconductors, across various dimensions and horizon topologies.

Contribution

It generalizes existing no-hair theorems to charged scalar fields and establishes bounds for scalar hair development on extremal black holes.

Findings

No-hair theorems for neutral scalar fields are extended.

Bounds on scalar field mass and charge for scalar hair are derived.

Results apply to black holes with different horizon topologies.

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.