A Mellin space approach to the conformal bootstrap

TL;DR

This work develops a Mellin-space conformal bootstrap built from crossing-symmetric AdS$_{d+1}$ Witten exchange blocks, enabling analytic constraints without a fixed gravity dual. By enforcing the vanishing of spurious Mellin poles, the authors derive systematic equations that determine operator dimensions and OPE coefficients, first demonstrating the ε-expansion about $d=4$ for the Wilson–Fisher fixed point and then extracting large-spin data in general dimensions. The results reproduce known Feynman-diagram dimensions to $O(ε^3)$ and yield new OPE data, with consistency checks against 3d Ising-model numerics showing improved agreement when higher-order terms are included. The approach also yields universal large-spin formulas and discusses numerical prospects, suggesting a complementary route to conventional bootstrap analyses and potential connections to AdS/CFT in critical phenomena.

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 $O(ε^3)$ 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 $ε=1$). 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.

Figures (2)

• Figure 1: Witten Diagrams in $AdS_{d+1}$ for identical scalars with exchange of a field corresponding to an operator of dimension $\Delta$ and spin $\ell$.
• Figure 2: Plot of $\frac{c_T}{c_{free}}$ against $\epsilon$, showing its variation in $2<d<4$. The dots indicate numerical bootstrap results epsnum. The dashed red line indicates the analytic result with only the $O(\epsilon^2)$ term. The smooth green line is the one including the $O(\epsilon^3)$ and gives a more closer match.