# Renormalization Proof for Massive $\phi_4^4$ Theory on Riemannian Manifolds

September 29, 2006

In this paper we present an inductive renormalizability proof for massive
$\phi_4^4$ 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.

Keywords:

renormalization