Abstract
In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $\eta_c$ are presented. As an application of those fragmentation functions, we study the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct nonrelativistic QCD approaches.
Highlights
Where ⊗ denotes a convolution in the momentum fraction z, the sum extends over all species of partons. dσ AþB→iþX indicates the partonic cross section that can be calculated in perturbation theory, while Di→H indicates the fragmentation function for the parton i into a hadron H. μF denotes the factorization scale which is introduced to separate the energy scales of the two parts
The factorization formula (1) was first derived by Collins and Soper for light hadron production [2]. This factorization formula can be applied to the heavy quarkonium production
Unlike the fragmentation functions for the production of the light hadrons which are nonperturbative in nature, the fragmentation functions for the heavy quarkonium production can be calculated through the nonrelativistic QCD (NRQCD) factorization [8]
Summary
According to QCD factorization theorem, the cross section for the inclusive production of a hadron H with high transverse momentum (pT) in a high-energy collision is dominated by the single parton fragmentation [1], i.e.,. The factorization formula (1) was first derived by Collins and Soper for light hadron production [2]. Fragmentation functions play an important role in the calculation of the cross sections under the LP factorization. Unlike the fragmentation functions for the production of the light hadrons which are nonperturbative in nature, the fragmentation functions for the heavy quarkonium production can be calculated through the nonrelativistic QCD (NRQCD) factorization [8]. In the early calculations of fragmentation functions for doubly heavy mesons [9,15], the fragmentation functions are determined through comparing the cross section calculated based on the NRQCD factorization with that calculated based on the factorization formula (1) for a process containing the doubly heavy meson being produced. We will calculate the fragmentation functions based on the operator definition suggested by Collins and Soper.
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