Endorsing black holes with beyond Horndeski primary hair: An exact solution framework for scalarizing in every dimension

This work outlines a straightforward mechanism for endorsing primary hair into Schwarzschild black holes, resulting in a unique modification within the framework of a special scalar-tensor theory, the so-called beyond Horndeski type. The derived solutions are exact, showcase primary hair with an everywhere regular scalar field profile, and continuously connect with the vacuum geometry. Initially devised to introduce primary hair in spherically symmetric solutions within General Relativity in any dimension, our investigation also explores the conditions under which spherically symmetric black holes in alternative gravitational theories become amenable to the endorsement of primary hair through a similar pattern. As a preliminary exploration, we embark on the process of endorsing primary hair to the Reissner-Nordstr\"om black hole. Subsequently, we extend our analysis to encompass spherically symmetric solutions within Lovelock and cubic quasitopological gravity theories.