Proca theory of four-dimensional regularized Gauss-Bonnet gravity and black holes with primary hair

Abstract: We introduce a novel, well-defined four-dimensional regularized Gauss-Bonnet theory of gravity by applying a dimensional regularization procedure. The resulting theory is a vector-tensor theory within the generalized Proca class. We then consider the static spherically symmetric solutions of this theory and find black hole solutions that acquire primary hair. Notably, one of the integration constants associated with the Proca field is not manifest in the original metric, but under a disformal transformation of the seed solution, it emerges as a second, independent primary hair. This additional hair acts as an effective cosmological constant in the disformed geometry, even in the absence of a bare cosmological constant term. We further generalize these black hole solutions to include electromagnetic charges and effects related to the scalar-tensor counterparts of the regularized Gauss-Bonnet theory. We discuss the implications of our findings to observations.