Conformal Bootstrap in Mellin Space

TL;DR

This paper introduces a novel Mellin space bootstrap approach for Conformal Field Theories, successfully reproducing known results and providing new higher-order OPE coefficients, enhancing analytical understanding of critical models.

Contribution

It combines Polyakov's bootstrap with Mellin representation, using crossing-symmetric Witten diagrams to derive operator data in CFTs analytically.

Findings

Reproduces anomalous dimensions in epsilon expansion.

Obtains higher-order OPE coefficients than existing methods.

Improves agreement with numerical results in the 3d Ising model.

Abstract

We propose a new approach towards analytically solving for the dynamical content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the epsilon expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in epsilon than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement of certain observables in the 3d Ising model, with the precise numerical values that have been recently obtained.