Simplifying large spin bootstrap in Mellin space

TL;DR

This paper develops a simplified Mellin space approach to the large spin conformal bootstrap, providing explicit asymptotic formulas for anomalous dimensions and OPE coefficients, and clarifying their universal properties.

Contribution

It introduces a new Mellin space formulation using continuous Hahn polynomials, deriving explicit asymptotic expansions for large spin operators to all orders.

Findings

Derived explicit asymptotic expressions for anomalous dimensions.

Showed equivalence with Witten diagram approach at leading order.

Identified polynomial ambiguities and constraints at next order.

Abstract

We set up the conventional conformal bootstrap equations in Mellin space and analyse the anomalous dimensions and OPE coefficients of large spin double trace operators. By decomposing the equations in terms of continuous Hahn polynomials, we derive explicit expressions as an asymptotic expansion in inverse conformal spin to any order, reproducing the contribution of any primary operator and its descendants in the crossed channel. The expressions are in terms of known mathematical functions and involve generalized Bernoulli (Norlund) polynomials and the Mack polynomials and enable us to derive certain universal properties. Comparing with the recently introduced reformulated equations in terms of crossing symmetric tree level exchange Witten diagrams, we show that to leading order in anomalous dimension but to all orders in inverse conformal spin, the equations are the same as in the conventional formulation. At the next order, the polynomial ambiguity in the Witten diagram basis is needed for the equivalence and we derive the necessary constraints for the same.