Cancellation of UV divergences in ghost-free infinite derivative gravity

We consider the most general covariant gravity action up to terms that are quadratic in curvature. These can be endowed with generic form factors, which are functions of the d'Alembert operator. If they are chosen in a specific way as an exponent of an entire function, the theory becomes ghost-free and renormalizable at the price of non-locality. Furthermore, according to power-counting arguments, if these functions grow sufficiently fast along the real axis, divergences may only appear at the first order in loop expansion. Using the heat kernel technique, we compute the one-loop logarithmic divergences in the ultraviolet limit and determine the conditions under which they vanish completely, apart from the Gauss--Bonnet term and a surface term, both of which can be neglected on a four-dimensional manifold without a boundary. We identify form factors both within the Tomboulis class and beyond it that lead to vanishing logarithmic divergences. The general expression for the one-loop beta functions of the dimensionless couplings in quadratic gravity with asymptotically monomial form factors is given.