The Fierz–Pauli theory on curved spacetime at one-loop and its counterterms

We compute the first four heat-kernel coefficients required for the renormalization of Linearized Massive Gravity. We focus on the Fierz–Pauli theory in a curved spacetime, describing the propagation of a massive spin 2 field in a non-flat background. The background must be on-shell, i.e. must be an Einstein space, in order to ensure a consistent extension of the Fierz–Pauli theory beyond Minkowski space. Starting from the Stückelberg formulation with restored gauge invariance, we apply suitable gauge-fixing procedures to simplify the complicated non-minimal kinetic operators of the scalar, vector, and tensor sectors of the theory, reducing them into manageable minimal forms. Using standard techniques, we then compute the Seeley–DeWitt coefficients, with particular emphasis on the fourth coefficient, a 3 ( D ) a 3 ​ (D), in arbitrary spacetime dimensions. This result constitutes our main contribution, as it has not been previously reported in full generality in the literature.