Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes

Abstract

For Euclidean quantum field theories, Holland and Hollands have shown operator product expansion (OPE) coefficients satisfy "flow equations": For interaction 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. In this paper, we generalize these results for flat Euclidean space to curved Lorentzian spacetimes in the context of the solvable "toy model" of massive Klein-Gordon scalar field theory, with $m^2$ viewed as the "self-interaction parameter". Even in Minkowski spacetime, a serious difficulty arises from the fact that all integrals must be taken over a compact spacetime region to ensure convergence but any integration cutoff necessarily breaks Lorentz covariance. We show how covariant flow relations can be obtained by adding compensating "counterterms" in a manner similar to that of the Epstein-Glaser renormalization scheme. We also show how to eliminate dependence on the "infrared-cutoff scale" $L$, thereby yielding flow relations compatible with almost homogeneous scaling of the fields. In curved spacetime, the spacetime integration will cause the OPE coefficients to depend non-locally on the spacetime metric, in violation of the requirement that quantum fields should depend locally and covariantly on the metric. We show how this potentially serious difficulty can be overcome by replacing the metric with a suitable local polynomial approximation about the OPE expansion point. We thereby obtain local and covariant flow relations for the OPE coefficients of Klein-Gordon theory in curved Lorentzian spacetimes. In an appendix, we develop an algorithm for constructing local and covariant flow relations beyond our "toy model" based on the associativity properties of OPE coefficients, and we apply our method to $λϕ^4$-theory.