Local and Covariant Flow Relations for OPE Coefficients in Curved Spacetime

The $n$-point functions of (perturbatively-renormalizable) quantum field theories are known to satisfy asymptotic relations called operator product expansions (OPEs) in the limit that all their spacetime points coincide. The coefficients of these expansions are state-independent and contain essential information about the quantum field theory itself. In (flat) Euclidean spacetime, Hollands et al. have derived novel ``flow equations'' which govern how OPE coefficients depend on the QFT's interaction parameters. Although proven to hold order-by-order in perturbation theory, these flow equations have been proposed as a potential avenue for defining OPE coefficients non-perturbatively. However, serious obstacles arise if one attempts to generalize the Hollands flow equations to curved Lorentzian spacetimes in a local and covariant manner. In this talk, I will describe these issues and sketch our resolutions for a solvable toy model: Klein-Gordon theory on curved spacetime with the (squared) mass, $m^2$, treated as an ``interaction parameter''. The strategies I describe for generalizing this toy model's flow relations in a local and covariant way are expected to be applicable, more generally, to QFTs with nonlinear interactions in curved Lorentzian spacetimes.