# Classification of shift-symmetric Horndeski theories and hairy black holes

## TL;DR

This paper classifies shift-symmetric Horndeski theories based on their scalar configurations and explores the conditions under which black holes can have scalar hair, revealing that certain couplings lead to secondary hair determined by horizon properties.

## Contribution

It provides a comprehensive classification of shift-symmetric Horndeski theories regarding scalar configurations and analyzes scalar hair around black holes in these theories.

## Key findings

- Theories without linear Gauss-Bonnet coupling admit all GR solutions.
- Scalar hair in theories with linear Gauss-Bonnet coupling is secondary and horizon-determined.
- Local Lorentz symmetry influences the admissibility of trivial scalar configurations.

## Abstract

No-hair theorems for scalar-tensor theories imply that the trivial scalar field configuration is the unique configuration around stationary black hole spacetimes. The most basic assumption in these theorems is that a constant scalar configuration is actually admissible. In this paper, we classify shift-symmetric Horndeski theories according to whether or not they admit the trivial scalar configuration as a solution and under which conditions. Local Lorentz symmetry and the presence of a linear coupling between the scalar field and Gauss-Bonnet invariant plays feature prominently in this classification. We then use the classification to show that any theory without linear Gauss-Bonnet coupling that respects Local Lorentz symmetry admits all GR solutions. We also study the scalar hair configuration around black hole spacetimes in theories where the linear Gauss-Bonnet coupling is present. We show that the scalar hair of the configuration is secondary, fixed by the regularity of the horizon, and is determined by the black hole horizon properties.