General Form of $s$, $t$, $u$ Symmetric Polynomial and Heavy Quarkonium physics

TL;DR

This paper derives a general form for symmetric polynomials in Mandelstam variables, simplifying heavy quarkonium production calculations and proving proportionality of cross sections at various orders in velocity expansion.

Contribution

It introduces a manifestly symmetric polynomial form for Mandelstam variables and applies it to simplify and analyze heavy quarkonium production cross sections.

Findings

Simplifies expressions by reducing their length to one-sixth.

Reproduces exact differential cross section for $J/$ production at leading order.

Proves proportionality of cross sections at all orders in velocity expansion.

Abstract

Induced by three gluons symmetry, Mandelstam variables $s$, $t$, $u$ symmetric expressions are widely involved in collider physics, especially in heavy quarkonium physics. In this work we study general form of $s$, $t$, $u$ symmetric polynomials, and find that they can be expressed as polynomials where the symmetry is manifest. The general form is then used to simplify expressions which asymptotically reduces the length of original expression to one-sixth. Based on the general form, we reproduce the exact differential cross section of $J/\psi$ hadron production at leading order in $v^2$ up to four unknown constant numbers by simple analysis. Furthermore, we prove that differential cross section at higher order in $v^2$ is proportional to that at leading order. This proof explains the proportion relation at next-to-leading order in $v^2$ found in previous work and generalizes it to all order.