Hadamard tails from flat-space perturbation theory

TL;DR

This work develops a perturbative, equivalence-principle–based framework for quantum field theory in curved spacetime to extract the universal short-distance, non-analytic structure of two-point functions. By expanding the metric as a small perturbation around flat space and relating curved-space correlators to flat-space correlators with insertions of the stress-energy tensor, it reproduces the Hadamard tail and extends the analysis to interacting theories and to two-point functions of scalar primaries in generic CFTs. The authors derive explicit results for a free scalar with arbitrary non-minimal coupling, showing how curvature and metric derivatives enter the leading short-distance singularities via geometric invariants such as the geodesic distance $s^2$ and the van Vleck determinant $\mathcal{D}$, with a universal tail controlled by $R$. They further show that for conformal primaries the leading curvature correction in a curved background is fixed by conformal data and is non-analytic in the geodesic distance, providing EFT predictions for gravitational corrections to OPE data in curved spacetimes. The framework offers a path to explore quantum fields in curved backgrounds beyond free theories, with potential applications to black-hole and cosmological settings and connections to quasinormal modes through retarded Green’s function poles.

Abstract

The short-distance singular structure of the two-point function of a free scalar field in curved spacetime has a universal behavior that characterizes well-behaved states (called Hadamard states). This includes a non-analytic term proportional to the Ricci scalar curvature known as the Hadamard tail. This is usually derived by solving a differential equation for the Green's function of a Klein-Gordon field in curved spacetime. We present an alternative derivation which leverages the equivalence principles and makes use of perturbative field theory methods. This allows for the computation of the short-distance singular behavior of correlators of QFTs in curved space, including for interacting field theories, where the traditional Green's function strategy cannot be easily generalized. As an example, we apply these ideas to the two-point function of two scalar primary operators of an arbitrary Conformal Field Theory placed in an arbitrary curved background.