Regular black holes from pure gravity in four dimensions

We derive static spherically symmetric regular black holes as vacuum solutions to purely gravitational theories in four dimensions. To that end, we construct four-dimensional nonpolynomial gravities starting from subclasses of two-dimensional Horndeski actions. By construction, these theories possess second-order equations of motion on spherically symmetric backgrounds. We show that a subset of these nonpolynomial gravities, referred to as nonpolynomial quasitopological gravities, admit single-function static spherically symmetric solutions whereby the metric function is determined by an algebraic equation. Solutions to these theories include the Hayward regular black-hole spacetime, for which a corresponding gravitational action is stated explicitly.