Conformal Four-Point Correlation Functions from the Operator Product Expansion

TL;DR

This paper develops a method to compute conformal blocks for operators in various Lorentz representations using group theory and Gegenbauer polynomials, providing explicit examples and simplifying the calculation process.

Contribution

It introduces a systematic approach to derive conformal blocks for arbitrary Lorentz representations, reducing the problem to group theory and substitution rules, with explicit examples.

Findings

Explicit formulas for conformal blocks in various Lorentz representations

Simplified computation method using Gegenbauer polynomial substitutions

Demonstrated calculations for scalar, tensor, and mixed operators

Abstract

We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in arXiv:1905.00036 and arXiv:1905.00434, and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group.