# Proper TMD factorization for quarkonia production: $pp\to\eta_{c,b}$ as a study case

## TL;DR

This paper derives a proper factorization theorem for quarkonia production at low transverse momentum, revealing the entanglement of soft-gluon radiation and bound state formation, and introduces TMD shape functions as a new non-perturbative element.

## Contribution

It presents the first proper factorization theorem for $pp\to \eta_{c,b}$ at low transverse momentum, incorporating TMD shape functions to account for entangled soft mechanisms.

## Key findings

- Derived the factorization theorem for $pp\to \eta_{c,b}$.
- Identified the role of TMD shape functions in quarkonia production.
- Showed the entanglement of soft-gluon radiation and bound state formation.

## Abstract

Quarkonia production in different high-energy processes has recently been proposed in order to probe gluon transverse-momentum-dependent parton distribution and fragmentation functions (TMDs in general). However, no proper factorization theorems have been derived for the discussed processes, but rather just ansatzs, whose main assumption is the factorization of the two soft mechanisms present in the process: soft-gluon radiation and the formation of the bound state. In this paper it is pointed out that, at low transverse momentum, these mechanisms are entangled and thus encoded in a new kind of non-perturbative hadronic quantities beyond the TMDs: the TMD shape functions. This is illustrated by deriving the factorization theorem for the process $pp\to \eta_{c,b}$ at low transverse momentum.