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

Wenliang Li

doi:10.1007/jhep01(2020)055

Abstract

In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions of generic scalar primaries. Based on the nonperturbative pole structure in spin of Lorentzian inversion, we propose the natural basis functions for cross-channel conformal blocks. In this new basis, we find a closed-form expression for crossed conformal blocks in terms of the Kampé de Fériet function, which applies to intermediate operators of arbitrary spin in general dimensions. We derive the general Lorentzian inversion for the case of identical external scaling dimensions. Our results for the lightcone limit also shed light on the complete analytic structure of conformal blocks in the lightcone expansion.

Highlights

In the conformal bootstrap method, the nontrivial equation from crossing symmetry schematically reads: direct conformal block = crossed conformal block
We find a closed-form expression for crossed conformal blocks in terms of the Kampe de Feriet function, which applies to intermediate operators of arbitrary spin in general dimensions
G(z, z) = Pi Gτi, i(z, z), i where Pi is the product of two operator product expansions (OPEs) coefficients associated with the intermediate operator Oi, Gτi, i(z, z) denotes the direct-channel conformal blocks, τ = ∆ − is known as twist, and ∆i, i are the scaling dimension and spin of Oi

Summary

Nonperturbative poles in spin

To find the natural basis functions, we will study the pole structure of the Lorentzian inversion of crossed conformal blocks. As a quadratic differential equation for the crossed blocks is not yet available, we will instead study the singularity structure of a nontrivial integral transform, the Lorentzian inversion.. The standard expansion around z = 1 uses (1 − z)p as the basis functions. The pole structure seems to depend on the choice of basis functions, but the nonperturbative results should be independent of our choice. As spurious poles will cancel out and nonperturbative poles will appear in the final results, it is reasonable to consider basis functions that are manifestly consistent with the nonperturbative pole structure, which should lead to a more natural formulation of conformal blocks. We will discuss the series expansion in terms of (2.6)

Lightcone limit of crossed conformal blocks

Identical external scaling dimensions

Conclusion