Multidimensional Phase Space and Sunset Diagrams

TL;DR

This paper derives general expressions for the phase-space of particle decay into multiple particles across any number of dimensions, and computes the associated sunset diagram imaginary parts using hypergeometric functions.

Contribution

It provides a unified, dimension-independent formulation of phase-space and sunset diagrams, expressed through hypergeometric functions, applicable to arbitrary dimensions.

Findings

Explicit formulas for phase-space in any dimension.

Closed-form expressions for sunset diagram imaginary parts.

Hypergeometric series representations that simplify threshold analysis.

Abstract

We derive expressions for the phase-space of a particle of momentum $p$ decaying into $N$ particles, that are valid for any number of dimensions. These are the imaginary parts of so-called `sunset' diagrams, which we also obtain. The results are given as a series of hypergeometric functions, which terminate for odd dimensions and are also well-suited for deriving the threshold behaviour.