Worldline formalism in a gravitational background
Fiorenzo Bastianelli, Andrea Zirotti
TL;DR
This paper develops the worldline representation for a scalar particle in a gravitational background and demonstrates that, with worldline dimensional regularization and proper zero-mode handling, the one-loop effective action ${\Gamma}[g]$ reproduces standard graviton tadpole and self-energy results. It analyzes two zero-mode schemes (DBC and PBC), showing that PBC introduces noncovariant total-derivative terms in the effective action, while DBC with DR preserves covariance and yields reliable correlation-function results. Ward identities arising from general coordinate invariance are explicitly verified, and a compact tensor decomposition using $R_i$, $S_1$, and $S_2$ ensures consistency with Feynman diagram calculations. The work suggests practical routes for extending the approach to other particle loops and connecting with open/closed string amplitude techniques, while clarifying the trade-offs between boundary-condition choices for correlation functions versus the effective action.
Abstract
We analyze the worldline formalism in the presence of a gravitational background. In the worldline formalism a path integral is used to quantize the worldline coordinates of the particles. Contrary to the simpler cases of scalar and vector backgrounds, external gravity requires a precise definition of the ultraviolet regularization of the path integral. Taking into account the UV regularization, we describe the first quantized representation of the one-loop effective action for a scalar particle. We compute explicitly the contribution to the graviton tadpole and self-energy to test the validity of the method. The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way. Finally, we comment on the technical problems related to the factorization of the zero mode from the path integral on the circle.