Renormalization Proof for Massive phi_4^4 Theory on Riemannian Manifolds

TL;DR

This paper proves the renormalizability of massive phi^4_4 theory on Riemannian manifolds using Wilson flow equations, establishing bounds on Schwinger functions and heat kernel behavior, and showing the minimal form of the bare action.

Contribution

It provides the first inductive renormalizability proof for phi^4_4 theory on curved manifolds using Wilson flow equations, extending flat space results.

Findings

Renormalizability established for phi^4_4 on Riemannian manifolds.

Derived bounds on Schwinger functions indicating decay properties.

Identified the form of counter terms including a curvature-dependent divergence.

Abstract

In this paper we present an inductive renormalizability proof for massive $\vp_4^4$ theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite are precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show, that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counter terms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.