Smearing out contact terms in ghost-free infinite derivative quantum gravity

In the context of ghost-free infinite derivative gravity, we consider the single graviton exchange between two spinless particles, a spinless particle and a photon, or between a spinless particle and a spin-1/2 particle. To this end, we compute the gravitational potential for the three aforementioned cases and derive the $\mathcal{O}(G\hbar^2)$ correction that arises at the linearised level. In the local theory, it is well-known that such a correction appears in the form of a Dirac delta function. Here, we show that this correction is smeared out for the nonlocal theory and, in contrast to the local theory, takes on non-zero values for a non-zero separation between the two particles. In the case of the single graviton exchange between a spinless particle and a spin-1/2 particle, we also compute the $\mathcal{O}(G\hbar)$ correction that arises in the non-static case within the non-relativistic approximation and show that it is finite in the nonlocal theory.