Gluon fragmentation into polarized charmonium

Gluon fragmentation to ${\mathrm{\chi }}_{\mathit{cJ}}$(1P) followed by single photon emission represents the dominant source of prompt J/ψ’s at the Fermilab Tevatron for ${\mathit{p}}_{\mathrm{\perp }}$≳6 GeV. Since fragmenting gluons are approximately transverse, their products are significantly polarized. We find that gluon fragmentation populates the helicity levels of ${\mathrm{\chi }}_{\mathit{c}1}$, ${\mathrm{\chi }}_{\mathit{c}2}$, and J/ψ according to ${\mathit{D}}_{\mathrm{\chi }\mathit{c}1}^{(\mathit{h}=0)}$:${\mathit{D}}_{\mathrm{\chi }\mathit{c}1}^{(\mathrm{\Vert }\mathit{h}\mathrm{\Vert }=1)}$ ≃1:1, ${\mathit{D}}_{\mathrm{\chi }\mathit{c}2}^{(\mathit{h}=0)}$: ${\mathit{D}}_{\mathrm{\chi }\mathit{c}2}^{(\mathrm{\Vert }\mathit{h}\mathrm{\Vert }=1)}$:${\mathit{D}}_{\mathrm{\chi }\mathit{c}2}^{(\mathrm{\Vert }\mathit{h}\mathrm{\Vert }=2)}$ ≃1:2.8:6.0, and ${\mathit{D}}_{\mathit{J}/\mathrm{\psi }}^{(\mathit{h}=0)}$:${\mathit{D}}_{\mathit{J}/\mathrm{\psi }}^{(\mathrm{\Vert }\mathit{h}\mathrm{\Vert }=1)}$≃1:3.4. We also speculate that gluon fragmentation to the radially excited ${\mathrm{\chi }}_{\mathit{c}2}$(2P) state followed by radiative decay could represent a large source of ψ’(2S)’s. A measurement of these states’ polarizations would test this idea.