Worldline path integral for the massive graviton

We compute the counterterms necessary for the renormalization of the one-loop effective action of massive gravity from a worldline perspective. This is achieved by employing the recently proposed massive $\mathcal{N}=4$ spinning particle model to describe the propagation of the massive graviton on those backgrounds that solve the Einstein equations without cosmological constant, namely on Ricci-flat manifolds, in four dimensions. The model is extended to be consistent in $D$ spacetime dimensions by relaxing the gauging of the full SO(4) R-symmetry group to a parabolic subgroup, together with the inclusion of a suitable Chern-Simons term. Then, constructing the worldline path integral on the one-dimensional torus allows for the correct calculation of the one-loop divergencies in arbitrary $D$ dimensions. Our primary contribution is the determination of the Seleey-DeWitt coefficients up to the fourth coefficient $a_3(D)$, which to our knowledge has never been reported in the literature. Its calculation is generally laborious on the quantum field theory side, as a general formula for these coefficients is not available for operators that are non-minimal in the heat kernel sense. This work illustrates the computational efficiency of worldline methods in this regard. Heat kernel coefficients characterize linearized massive gravity in a gauge-independent manner due to the on-shell condition of the background on which the graviton propagates. They could serve as a benchmark for verifying alternative approaches to massive gravity, and, for this reason, their precise expression should be known explicitly.