Conformal Partial Waves and the Operator Product Expansion

TL;DR

This paper derives explicit formulas for conformal partial waves in four and six dimensions using differential equations and hypergeometric functions, enhancing understanding of operator contributions in conformal field theories.

Contribution

It provides new, succinct expressions for conformal partial waves in multiple dimensions, connecting differential equations, hypergeometric functions, and Jack polynomials.

Findings

Explicit conformal partial wave formulas in 4D and 6D.

Unified expression for any dimension using Jack polynomials.

Connections between conformal invariants and hypergeometric functions.

Abstract

By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for $O(d,2)$ succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension $\Delta$ and spin $\ell$ together with its descendants to conformal four point functions for $d=4$, recovering old results, and also for $d=6$. The results are expressed in terms of ordinary hypergeometric functions of variables $x,z$ which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves.