Dispersion Relation for CFT Four-Point Functions

TL;DR

The paper derives a dispersion relation for conformal field theory four-point functions, expressing the correlator as an integral over its single discontinuity and exploiting OPE analyticity and crossing symmetry. In perturbative settings, the correlator becomes fully determined by the spectrum and low-twist OPE data, enabling unambiguous reconstruction without assuming Regge behavior. The framework is demonstrated via two key applications: the Wilson-Fisher fixed point, where the four-point function is computed to order $\varepsilon^2$ in $d=4-\varepsilon$, and holographic supergravity in AdS, where leading and subleading OPE data are extracted from the single discontinuity. The approach complements the Lorentzian inversion formula and offers a direct, potentially nonperturbative route to CFT data, with promising links to extremal functionals and Polyakov-Mellin bootstrap techniques.

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 $\langle φφφφ\rangle$ in $φ^4$ theory at the Wilson-Fisher fixed point is computed in closed form to order $ε^2$ in the $ε$ expansion.

Figures (2)

• Figure 1: Analytic structure of $g^{(d)}_{\Delta,\ell}(z,\overline{z})$ and factors contributing to branch cuts.
• Figure 2: Contour deformation determining the dispersion relation.