A simple expression for calculating the classical potential concerning D-dimensional gravitational models is obtained through a method based on the generating functional. The prescription is then used as a mathematical tool to probe the conjecture that renormalizable higher-order gravity models -which are, of course, nonunitary -are endowed with a classical potential that is nonsingular at the origin. It is also shown that the converse of this statement is not true, which implies that the finiteness of the classical potential at the origin is a necessary but not a sufficient condition for the renormalizability of the model. The systems we have utilized to verify the conjecture were fourthand sixth-order gravity models in D-dimensions. A discussion about the polemic question related to the renormalizability of new massive gravity, which Oda claimed to be renormalizable in 2009 and three years late was shown to be nonrenormalizable by Muneyuki and Ohta, is considered. We remark that the solution of this issue is straightforward if the aforementioned conjecture is employed. We point out that our analysis is restricted to local models in which the propagator has simple and real poles.

Simple prescriptions for computing the D-dimensional classical potential related to electromagnetic and gravitational models, based on the functional generator, are built out. These recipes are employed afterward as a support for probing the premise that renormalizable higher-order systems have a finite classical potential at the origin. It is also shown that the opposite of the conjecture above is not true. In other words, if a higher-order model is renormalizable, it is necessarily endowed with a finite classical potential at the origin, but the reverse of this statement is untrue. The systems used to check the conjecture were D-dimensional fourth-order Lee–Wick electrodynamics, and the D-dimensional fourth- and sixth-order gravity models. A special attention is devoted to New Massive Gravity (NMG) since it was the analysis of this model that inspired our surmise. In particular, we made use of our premise to resolve trivially the issue of the renormalizability of NMG, which was initially considered to be renormalizable, but it was shown some years later to be non-renormalizable. We remark that our analysis is restricted to local models in which the propagator has simple and real poles.

Using a method based on the generating functional plus a kind of "correspondence principle" -which acts as a bridge between the electromagnetic and scalar fields -it is shown that the interparticle potential energy concerning a given D-dimensional electromagnetic model can be obtained in a simple way from that related to the corresponding scalar system. The D-dimensional electromagnetic potential for a general model containing higher derivatives is then found from the corresponding scalar one and the behavior of the former is analyzed at large as well as small distances. In addition, we investigate the presence of ghosts in the four-dimensional version of the potential associated with the model above and analyze the reason why the Coulomb singularity is absent from this system. The no-go theorem by Ostrogradski is demystified as well.