Renormalizability, van Dam-Veltman-Zakharov discontinuity, and Newtonian singularity in higher-derivative gravity

It was proposed that if a higher-derivative gravity is renormalizable it implies necessarily a finite Newtonian potential at the origin, but the reverse of this statement is not true. Here, we show that the reverse is true when taking into account the van Dam-Veltman-Zakharov discontinuity, which states that the theory obtained from the massive one by taking a zero mass limit is not equivalent to the theory obtained in the zero mass case. The surviving degree of freedom in the zero mass limit is an extra scalar that does not affect the light bending angle but affects the Newtonian potential. This asserts that in order to make the singularity cancellation the number of massive ghost and healthy tensors matches with that of massive ghost and healthy scalars.