Soft gluon emission process in the color-octet model for heavy quarkonium production

The color-octet model has been used successfully to analyze many problems in heavy quarkonium production. We examine some of the conceptual and practical problems of the soft gluon emission process in the color-octet model. We use a potential model to describe the initial and final states in the soft gluon emission process, as the emission occurs at a late stage, after the production of the heavy quark pair. It is found in this model that the soft gluon $M1\mathrm{}$ transition ${}^1{\overrightarrow{S_0^{\mathrm{}\left(8\right)\mathrm{}}}}^3{S}_1^{\mathrm{}\left(1\right)\mathrm{}}$ dominates over the $E1\mathrm{}$ transition ${}^3{\overrightarrow{P_J^{\mathrm{}\left(8\right)\mathrm{}}}}^3{S}_1^{\mathrm{}\left(1\right)\mathrm{}}$ for $J/\psi$ and ${\psi }^{\prime }$ production. Such a dominance may help resolve the questions of isotropic polarization and color-octet matrix element universality in the color-octet model.