A Generic Renormalization Method in Curved Spaces and at Finite Temperature

TL;DR

This paper introduces a universal renormalization method applicable to scalar field theories in curved spaces and at finite temperature, simplifying calculations by relying on coordinate space principles and propagator properties.

Contribution

It develops a generic coordinate space renormalization procedure that works across various backgrounds, including curved geometries and thermal settings, using minimal inputs.

Findings

Derived closed-form renormalized amplitudes at 1- and 2-loop orders.

Established a method preserving diffeomorphism invariance in curved backgrounds.

Validated the approach with explicit calculations at finite temperature.

Abstract

Based only on simple principles of renormalization in coordinate space, we derive closed renormalized amplitudes and renormalization group constants at 1- and 2-loop orders for scalar field theories in general backgrounds. This is achieved through a generic renormalization procedure we develop exploiting the central idea behind differential renormalization, which needs as only inputs the propagator and the appropriate laplacian for the backgrounds in question. We work out this generic coordinate space renormalization in some detail, and subsequently back it up with specific calculations for scalar theories both on curved backgrounds, manifestly preserving diffeomorphism invariance, and at finite temperature.