Hadamard tails from flat-space perturbation theory

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.