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Christoph Kopper, Volkhard F. M\"uller
Renormalization Proof for Massive $\vp_4^4$ Theory
on Riemannian Manifolds
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.