# Effective field theory for black holes with induced scalar charges

## TL;DR

This paper develops an effective field theory framework to describe how black holes can acquire scalar charges when interacting with time-dependent scalar fields, extending previous models to include massive scalars and astrophysical scenarios.

## Contribution

It introduces a novel EFT approach for black holes with induced scalar charges, incorporating both conservative and dissipative effects via the in-in formalism.

## Key findings

- Reproduces known results for massless scalars
- Generalizes to massive scalar fields
- Analyzes black hole binaries in dark matter halos

## Abstract

While no-hair theorems forbid isolated black holes from possessing permanent moments beyond their mass, electric charge, and angular momentum, research over the past two decades has demonstrated that a black hole interacting with a time-dependent background scalar field will gain an induced scalar charge. In this paper, we study this phenomenon from an effective field theory (EFT) perspective. We employ a novel approach to constructing the effective point-particle action for the black hole by integrating out a set of composite operators localized on its worldline. This procedure, carried out using the in-in formalism, enables a systematic accounting of both conservative and dissipative effects associated with the black hole's horizon at the level of the action. We show that the induced scalar charge is inextricably linked to accretion of the background environment, as both effects stem from the same parent term in the effective action. The charge, in turn, implies that a black hole can radiate scalar waves and will also experience a "fifth force." Our EFT correctly reproduces known results in the literature for massless scalars, but now also generalizes to massive real scalar fields, allowing us to consider a wider range of scenarios of astrophysical interest. As an example, we use our EFT to study the early inspiral of a black hole binary embedded in a fuzzy dark matter halo.