Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime

TL;DR

This paper explores the connection between zeta function regularization and spectral cutoff methods in curved spacetime, providing a rigorous foundation and extending the approach to higher loops with explicit calculations.

Contribution

It introduces a generalized zeta function framework for higher-loop regularization in curved spacetime, linking Feynman diagrams to meromorphic zeta functions.

Findings

Established a rigorous justification for zeta function regularization at one loop

Extended the method to higher loops with generalized zeta functions

Performed explicit one-loop and two-loop calculations, including conformal anomaly evaluation

Abstract

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.