For Euclidean-signature quantum field theories with renormalizable self-interactions, Holland and Hollands have shown the operator product expansion (OPE) coefficients satisfy "flow equations": For a (renormalized) self-coupling parameter λ, the partial derivative of any OPE coefficient with respect to λ is given by an integral over Euclidean space of a sum of products of other OPE coefficients evaluated at λ. These Euclidean flow equations were proven to hold order-by-order in perturbation theory, but they are well defined non-perturbatively and thus provide a possible route towards giving a non-perturbative construction of the interacting field theory. The purpose of this paper is to generalize the Holland and Hollands results for flat Euclidean space to curved Lorentzian spacetimes in the context of the solvable "toy model" of massive Klein-Gordon scalar field theory on globally-hyperbolic curved spacetimes, with the squared mass, m 2 , viewed as the "self-interaction parameter". There are a number of difficulties that must be overcome to carry out this program. Even in Minkowski spacetime, a serious difficulty arises from the fact that all integrals must be done over a compact region of spacetime to ensure convergence. However, there does not exist any Lorentz-invariant function of compact support, so any flow relations that involve only integration over a compact region cannot be Lorentz covariant. We show how covariant flow relations can be obtained by the addition of "counterterms" that cancel the non-covariant dependence on the cutoff function in a manner similar to that used in the Epstein-Glaser renormalization scheme. The necessity of integration over a finite region also effectively introduces an "infrared cutoff scale" L into the flow relations, *

No abstract