Abstract
We study the fragmentation function of the gluon to color-octet 3S1 heavy quark-antiquark pair using the soft gluon factorization (SGF) approach, which expresses the fragmentation function in a form of perturbative short-distance hard part convoluted with one-dimensional color-octet 3S1 soft gluon distribution (SGD). The short distance hard part is calculated to the next-to-leading order in αs and all orders in velocity expansion. By deriving and solving the renormalization group equation of the SGD, threshold logarithms are resummed to all orders in perturbation theory. The comparison with gluon fragmentation function calculated in NRQCD factorization approach indicates that the SGF formula resums a series of velocity corrections in NRQCD which are important for phenomenological study.
Highlights
The velocity expansion suffers from large high order relativistic corrections due to ignoring the effects of soft hadrons emitted in the hadronization process
We study the fragmentation function of the gluon to color-octet 3S1 heavy quark-antiquark pair using the soft gluon factorization (SGF) approach, which expresses the fragmentation function in a form of perturbative short-distance hard part convoluted with one-dimensional color-octet 3S1 soft gluon distribution (SGD)
The comparison with gluon fragmentation function calculated in NRQCD factorization approach indicates that the SGF formula resums a series of velocity corrections in NRQCD which are important for phenomenological study
Summary
Before studying the gluon fragmentation function, we first briefly review the SGF formula for quarkonium production cross section. In the above formula, dσ[nn ](PH /z, mQ, μf ) are the short-distance hard parts which, roughly speaking, produce a QQpair with momentum PH /z and quantum number n in the amplitude and n in the complex conjugate. As the short distance hard parts dσnn (PH /z, mQ, μf ) do not depend on nonperturbative physics, they can be perturbatively calculated. To this end, we replace the quarkonium H in eq (2.1) by an on-shell QQpair with certain quantum number m in the amplitude and m in the complex-conjugate amplitude, which results in dσQQ[mm ] =. On, which express short-distance hard parts in terms of perturbative calculated dσQQ[nn ] and F[nn ]→QQ[mm ]. We will confirm this conclusion at one-loop level in this paper
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