Worldline approach to vector and antisymmetric tensor fields II

TL;DR

The paper develops a massive extension of the worldline description for antisymmetric tensor fields, deriving the Seeley-DeWitt coefficients $a_0$, $a_1$, and $a_2$ by a simple shift $D\to D+1$ from the massless case and analyzing duality relations between massive forms. It provides a torus-based one-loop path integral representation for the effective action, analyzes topological mismatches in dualities, and computes the graviton self-energy contribution from both massless and massive antisymmetric tensors of arbitrary rank, reproducing known photon results in the massless and rank-1 limits. A Stückelberg interpretation is presented, showing how the Proca action for massive tensors can be embedded into a gauge-invariant framework with auxiliary fields and ghosts. Overall, the work demonstrates the power of the worldline formalism to yield general, tractable results for massive antisymmetric tensors in curved backgrounds, with potential applications to quantum gravity and string-inspired theories.

Abstract

We extend the worldline description of vector and antisymmetric tensor fields coupled to gravity to the massive case. In particular, we derive a worldline path integral representation for the one-loop effective action of a massive antisymmetric tensor field of rank p (a massive p-form) whose dynamics is dictated by a standard Proca-like lagrangian coupled to a background metric. This effective action can be computed in a proper time expansion to obtain the corresponding Seeley-DeWitt coefficients a0, a1, a2. The worldline approach immediately shows that these coefficients are derived from the massless ones by the simple shift D -> D+1, where D is the spacetime dimension. Also, the worldline representation makes it simple to derive exact duality relations. Finally, we use such a representation to calculate the one-loop contribution to the graviton self-energy due to both massless and massive antisymmetric tensor fields of arbitrary rank, generalizing results already known for the massless spin 1 field (the photon).