Abstract
The widely used nonrelativistic QCD (NRQCD) factorization theory now encounters some notable difficulties in describing quarkonium production. This may be due to the inadequate treatment of soft hadrons emitted in the hadronization process, which causes bad convergence of velocity expansion in NRQCD. In this paper, starting from QCD we propose a rigorously defined factorization approach, soft gluon factorization (SGF), to better deal with the effects of soft hadrons. After a careful velocity expansion, the SGF can be as simple as the NRQCD factorization in phenomenological studies, but has a much better convergence. The SGF may provide a new insight to understand the mechanisms of quarkonium production and decay.
Highlights
As the simplest bound state of strong interactions, heavy quarkonium is a perfect system to study both perturbative and nonperturbative physics of QCD
X HnðPHÞhOHn i n þ where PH is the momentum of H, n denotes intermediate QQstates, whose quantum numbers are usually expressed in terms of spectroscopic notation 2Sþ1L1⁄2Jc, with c 1⁄4 1; 8 denoting color singlet or color octet of the pair, HnðPHÞ are perturbative calculable short-distance coefficients which can be expanded order by order in αs, hOHn i are gaugecompleted [25,26,27] nonperturbative long-distance matrix elements (LDMEs) which can be classified according to the power counting rules of v, and the ellipses denote other relativistic correction terms
We propose a soft gluon factorization (SGF) approach to describe quarkonium production and decay, which keeps the momentum difference between intermediate QQpair and physical quarkonium
Summary
As the simplest bound state of strong interactions, heavy quarkonium is a perfect system to study both perturbative and nonperturbative physics of QCD. With the color-octet mechanism, NRQCD can solve the infrared divergence problem encountered in the color-singlet model [11], explain the ψð2SÞ surplus [12], and describe the inclusive quarkonium production [13,14,15,16,17,18,19,20,21,22,23,24]. We show that the velocity expansion in the present NRQCD framework suffers from large high order relativistic corrections due to ignoring the momentum of soft hadrons, which are mainly soft gluons perturbatively, emitted in the hadronization process. The SGF has a much better convergence in the velocity expansion, and it may provide a new insight to fully understand the mechanisms of quarkonium production and decay.
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