Regularized conformal electrodynamics: Novel $C$ metric in $2+1$ dimensions

Conformal electrodynamics is a particularly interesting example of power Maxwell nonlinear electrodynamics, designed to possess conformal symmetry in all dimensions. In this paper, we propose a regularized version of conformal electrodynamics, minimally regularizing the field of a point charge at the origin by breaking the conformal invariance of the theory with a dimensionful "Born-Infeld-like" parameter. In four dimensions the new theory reduces to the recently studied regularized Maxwell electrodynamics, distinguished by its "Maxwell-like" solutions for accelerated and slowly rotating black hole spacetimes. Focusing on three dimensions, we show that the new theory shares many of the properties of its four-dimensional cousin, including the existence of the charged $C$-metric solution (currently unknown in the Maxwell theory).