# Correct Definition of Color Singlet P-Wave Non-Perturbative Matrix Element of Heavy Quarkonium Production

## TL;DR

This paper addresses the issue of non-canceling infrared divergences in color singlet P-wave heavy quarkonium production by emphasizing the importance of gauge links in the non-perturbative matrix element to ensure gauge invariance and consistent factorization.

## Contribution

It introduces the necessity of gauge links in the non-perturbative matrix element for P-wave quarkonium to handle infrared divergences at all orders.

## Key findings

- Gauge links are essential for gauge invariance of P-wave matrix elements.
- Proper definition resolves non-canceling infrared divergences.
- Ensures consistent factorization in heavy quarkonium production.

## Abstract

Recently we have proved factorization of infrared divergences in NRQCD S-wave heavy quarkonium production at high energy colliders at all orders in coupling constant. One of the problem which still exists in the higher order pQCD calculation of color singlet P-wave heavy quarkonium production/anihillation is the appearance of non-canceling infrared divergences due to real soft gluons exchange, although no such infrared divergences are present in the color singlet S-wave heavy quarkonium. In this paper we find that since the non-perturbative matrix element of the color singlet P-wave heavy quarkonium production contains derivative operators, the gauge links are necessary to make it gauge invariant and be consistent with the factorization of such non-canceling infrared divergences at all orders in coupling constant.