Dispersion Relation for CFT Four-Point Functions

TL;DR

This paper introduces a dispersion relation for conformal field theory four-point functions that allows reconstruction from spectral data without Regge assumptions, demonstrated explicitly in the Wilson-Fisher fixed point.

Contribution

It develops a new dispersion relation for CFT four-point functions that does not rely on Regge behavior and applies it to compute correlators in the Wilson-Fisher fixed point.

Findings

Derived a dispersion relation expressing four-point functions as integrals over discontinuities.

Showed the correlator depends only on the spectrum and low-twist OPE coefficients in perturbation theory.

Computed the four-point correlator in $ ext{phi}^4$ theory at the Wilson-Fisher fixed point to order $\e^2$.

Abstract

We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinning operators. As an application, the correlator $\langle \phi \phi \phi \phi \rangle$ in $\phi^4$ theory at the Wilson-Fisher fixed point is computed in closed form to order $\epsilon^2$ in the $\epsilon$ expansion.