Hadamard tails from flat-space perturbation theory

TL;DR

This paper introduces a perturbative approach to derive the Hadamard tail in curved spacetime, enabling analysis of short-distance singularities in quantum field theories, including interacting fields, using principles of flat-space perturbation theory.

Contribution

It presents an alternative derivation of the Hadamard tail leveraging perturbative field theory, extending applicability to interacting theories and arbitrary curved backgrounds.

Findings

Derived the Hadamard tail using perturbation theory methods.

Applied the approach to conformal field theories in curved space.

Enabled analysis of short-distance behavior in interacting quantum fields.

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.