Electromagnetic Generalized Quasi-topological gravities in $(2 + 1)$ dimensions

The construction of Quasi-topological gravities in three-dimensions requires coupling a scalar field to the metric. As shown in arXiv:2104.10172, the resulting "Electromagnetic" Quasi-topological (EQT) theories admit charged black hole solutions characterized by a single-function for the metric, $-g_{tt}=g^{-1}_{rr}\equiv f(r)$, and a simple azimuthal form for the scalar. Such black holes, whose metric can be determined fully analytically, generalize the BTZ solution in various ways, including singularity-free black holes without any fine-tuning of couplings or parameters. In this paper we extend the family of EQT theories to general curvature orders. We show that, beyond linear order, $f(r)$ satisfies a second-order differential equation rather than an algebraic one, making the corresponding theories belong to the Electromagnetic Generalized Quasi-topological (EGQT) class. We prove that at each curvature order, the most general EGQT density is given by a single term which contributes nontrivially to the equation of $f(r)$ plus densities which do not contribute at all to such equation. The proof relies on the counting of the exact number of independent order-$n$ densities of the form $\mathcal{L}(R_{ab},\partial_a \phi)$, which we carry out. We study some general aspects of the new families of EGQT black-hole solutions, including their thermodynamic properties and the fulfillment of the first law, and explicitly construct a few of them numerically.