Fate of three-dimensional black holes coupled to a scalar field and the Bekenstein-Hawking entropy

Three-dimensional black holes coupled to a self-interacting scalar field is considered. It is known that its statistical entropy a la Strominger does not agree with the Bekenstein-Hawking (BH) entropy. However, I show that, by a careful treatment of the vacuum state in the canonical ensemble with a fixed temperature, which is the same as that of the BTZ black hole without the scalar field, the BH entropy may be exactly produced by the Cardy's formula. I discuss its several implications, including the fate of black holes, no-scalar-hair theorems, stability, mirror black holes, and higher-order corrections to the entropy