# Charged black holes in non-linear Q-clouds with O(3) symmetry

## TL;DR

This paper constructs novel non-linear charged scalar cloud solutions, called Q-clouds, around non-rotating charged black holes, revealing new insights into black hole hair and the weak-gravity conjecture.

## Contribution

It presents the first O(3) symmetric Q-cloud solutions around non-extremal, non-rotating charged black holes in Abelian gauge theory, extending the understanding of scalar hair.

## Key findings

- Existence of solutions with infinitely short hair at extremal black holes
- Q-clouds relate to the weak-gravity conjecture
- Discussion on why no-hair theorem does not apply to massive scalar fields

## Abstract

We construct charged soliton solutions around spherical charged black holes with no angular momentum in asymptotically flat spacetime. These solutions are non-linear generalizations of charged scalar clouds, dubbed Q-ball hair or Q-clouds, and they do not contradict the non-existence theorem for free (linear) scalar clouds around charged black holes. These solutions are the first examples of O(3) solutions for Q-clouds around a non-extremal and non-rotating BH in the Abelian gauge theory. We show that a solution exists with an infinitely short hair in the limit of extremal black holes. We discuss the evolution of scalar hair in a system with fixed total charge and describe how the existence of Q-clouds is related to the weak-gravity conjecture. The reason that the no-hair theorem by Mayo and Bekenstein cannot be applied to the massive scalar field is also discussed.