NLO fragmentation functions of heavy quarks into heavy quarkonia

Abstract

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