Direct fragmentation of $ \eta_{c}^{}$ η c (1S) and $ \eta_{b}^{}$ η b (1S) considering the transverse momentum effect of constituents

Abstract

In this work, we calculate the fragmentation function (FF) of pseudoscalar S -wave heavy quarkonia from gluon splitting. We also study, for the first time, the Fermi motion effect of constituent quarks on the fragmentation production of the $ \eta_c(1S)$ and $ \eta_b(1S)$ states. In all previous studies of heavy quarkonium FFs, by ignoring the Fermi motion of constituents a Dirac-delta functional form was approximated for the wave function of bound state. Here, to study the Fermi motion of quarkonium constituents we consider a typical wave function which is different from the delta form and is the nonrelativistic limit of the solution of the Bethe-Salpeter equation with the QCD kernel. We shall present our results for the fragmentation of $ g\rightarrow (\eta_c, \eta_b)$ and show how the Fermi motion effect changes the results.