Conformal partial waves in momentum space

TL;DR

This paper derives a closed-form expression for conformal partial waves in Minkowski momentum space for scalar operators across arbitrary dimensions, facilitating analysis of 4-point functions in conformal field theory.

Contribution

It provides the first explicit formula for conformal partial waves in momentum space valid in any dimension, expressing them as sums over spin partial waves with factorized coefficients.

Findings

Closed-form conformal partial waves in arbitrary dimensions.

Decomposition of partial waves into spin partial waves with factorized coefficients.

Application to scalar box integral in 4D demonstrating practical utility.

Abstract

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension $d \geq 3$ (including non-integer $d$). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in $d = 4$ dimensions.