# The NLO fragmentation functions of heavy quarks into heavy quarkonia

## TL;DR

This paper derives the next-to-leading order fragmentation functions for heavy quarks into heavy quarkonia, providing a tool for precise predictions of quarkonium production at high-energy colliders.

## Contribution

It presents the first NLO calculation of heavy quark to quarkonium fragmentation functions, including operator renormalization and initial scale determination.

## Key findings

- Fragmentation functions are obtained at NLO accuracy.
- The functions can be evolved to any scale using DGLAP equations.
- Application to $J/ ext{}/ ext{Psi}$ and $ ext{Upsilon}$ production at $Z$-pole colliders.

## Abstract

In the paper, we derive the next-to-leading order (NLO) fragmentation function for a heavy quark, either charm or bottom, into a heavy quarkonium $J/\Psi$ or $\Upsilon$. The ultra-violet divergences in the real corrections are removed through the operator renormalization, which is performed under the modified minimal subtraction scheme. We then obtain the NLO fragmentation function at an initial factorization scale, e.g. $\mu_{F}=3 m_c$ for $c\to J/\Psi$ and $\mu_{F}=3m_b$ for $b\to \Upsilon$, which can be evolved to any scale via the use of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation. As an initial application of those fragmentation functions, we study the $J/\Psi$ ($\Upsilon$) production at a high luminosity $e^+e^-$ collider which runs at the energy around the $Z$ pole and could be a suitable platform for testing the fragmentation function.