# Renormalizability, vDVZ discontinuity and Newtonian singularity in higher-derivative gravity

## TL;DR

This paper explores the relationship between renormalizability, vDVZ discontinuity, and Newtonian singularity in higher-derivative gravity, revealing conditions for singularity cancellation involving degrees of freedom.

## Contribution

It demonstrates that considering the vDVZ discontinuity clarifies the link between renormalizability and finite Newtonian potential at the origin.

## Key findings

- The vDVZ discontinuity affects the Newtonian potential.
- Matching degrees of freedom cancels singularities.
- Renormalizability implies a finite potential under certain conditions.

## Abstract

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 vDVZ discontinuity which states that the theory obtained from massive one by taking 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 which 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.