A Mellin space approach to the conformal bootstrap

TL;DR

This paper introduces a Mellin space approach to the conformal bootstrap, enabling analytic and numerical analysis of conformal field theories, including the Wilson-Fisher fixed point and large spin expansions.

Contribution

The paper develops a Mellin space framework for the conformal bootstrap, providing new analytic predictions for operator dimensions and OPE coefficients, and extends existing methods to arbitrary dimensions.

Findings

Reproduces Feynman diagram results for operator dimensions to O(ε^3)

Provides new analytic predictions for OPE coefficients at the Wilson-Fisher fixed point

Extends bootstrap methods to large spin expansions in any dimension

Abstract

We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of $CFT_d$ four point functions and expands them in terms of crossing symmetric combinations of $AdS_{d+1}$ Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about $d=4$. We reproduce Feynman diagram results for operator dimensions to $O(\epsilon^3)$ rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at $\epsilon =1$). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.