Application of analytic functionals to mean field theory and Wilson-Fisher fixed point

TL;DR

The paper addresses the analytic bootstrap problem by deploying a complete basis of analytic functionals dual to double-trace blocks to extract CFT data for mean-field theory and the Wilson-Fisher fixed point. By relating the action of these functionals to coefficients in exchange Witten diagrams and Polyakov blocks, it derives OPE coefficients to first order in the $4-\varepsilon$ expansion, reproducing known MFT results and obtaining explicit $O(\varepsilon)$ corrections for the Wilson-Fisher case. The key contributions include a concrete recursion-based link between functional data and Witten-diagram coefficients, yielding $b_{0,\ell}$ in four dimensions and $O(\varepsilon)$ shifts $(-1)^\ell \lambda^{(1)}_{n,\ell}$ with harmonic-number structures. This work validates the analytic functional approach as a powerful, analytic route to perturbative CFT data and suggests clear avenues for higher-order expansion and extension to more complex theories.

Abstract

We consider application of the analytic functional approach to the conformal field theories associated with mean field theory and Wilson-Fisher fixed point. We study the constraints imposed by the crossing symmetry on the coefficients of the operator product expansion. Making use of these constraints along with a few other additional inputs, we obtain expressions of the coefficients of the operator product expansion up to first order of $ε$.