Double gluon fragmentation to $J/\psi$ pairs at the Tevatron

It has been proposed that the large cross sections for prompt $\psi$, $\psi'$, and $\chi_c$ production at the Fermilab Tevatron $p\bar p$ collider can be explained by a dominant color-octet term in the fragmentation function for a gluon to split into quarkonium. We show that this mechanism makes testable predictions for double-quarkonium $\psi\psi$, $\psi\psi'$, $\psi\chi_c$, $\psi\Upsilon$ and $\psi\chi_b$ production, as well as for $W \psi$ production, using color-octet matrix elements previously determined from charmonium production data. The $\psi\psi$ signal would already be measurable at the Tevatron, while the $\psi\chi_c$ and $W\psi$ signals would be on the edge of present detectability.