Effective Potential in Curved Space and Cut-Off Regularizations

TL;DR

This paper compares two covariant cut-off regularization methods for deriving the effective potential of a scalar field in curved space, highlighting their differences in finite part details and extending analysis to fermion loops and renormalization group equations.

Contribution

It introduces and compares two covariant regularization schemes in curved space, providing detailed finite part analysis and extending to fermion contributions and renormalization insights.

Findings

Both regularization methods yield identical divergences.

The local momentum method provides more detailed finite parts.

Fermion loop contributions and RG equations are analyzed.

Abstract

We consider derivation of the effective potential for a scalar field in curved space-time within the physical regularization scheme, using two sorts of covariant cut-off regularizations. The first one is based on the local momentum representation and Riemann normal coordinates and the second is operatorial regularization, based on the Fock-Schwinger-DeWitt proper-time representation. We show, on the example of a self-interacting scalar field, that these two methods produce equal results for divergences, but the first one gives more detailed information about the finite part. Furthermore, we calculate the contribution from a massive fermion loop and discuss renormalization group equations and their interpretation for the multi-mass theories.