Simplifying large spin bootstrap in Mellin space

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 (Nørlund) 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.

Bootstrapping inflationary correlators in Mellin space

“…The success of this revival came from combining this observation with the Mellin representation of Witten diagrams in position space [12], which has led to important advances in analytic approaches to the Conformal Bootstrap (see e.g. [63,124,[140][141][142][143][144]…”

A crossing-symmetric OPE inversion formula

“…We will refer to these hypothetical functions as the Polyakov blocks. It has been proposed that P ∆,J (z,z) exist and are equal to the sum of the s-, t-and u-channel Witten diagrams, supplemented by appropriate contact terms [34,35,50]. However, no universal prescription for fixing the contact terms or a general proof of consistency of this approach has been presented.…”

Closed-form expression for cross-channel conformal blocks near the lightcone

“…The Lorentzian inversion integral should commute with the α-derivative when no singularity is encountered. According to (20), the Lorentzian inversion of a conserved-current block has a simpler expression. For efficient numerical evaluation, one can write the well-poised 7 F 6 hypergeometric series as a sum of two 1-balanced 4 F 3 hypergeometric series, and replace ∂ α f (α) by [f (α) − f (0)]/α, where α is set to a small number, such as 10 −n .…”

“…The analytic conformal bootstrap can be formulated as an algebraic problem [13][14][15]. More recently, significant advances towards the nonperturbative regime have been made by upgrading the analytical toolkit from asymptotic expansion at large spin [13][14][15][16][17][18][19][20][21][22] to convergent Lorentzian inversion at finite spin [23,24]. (See also [25] for convergent results from a different method.)…”