psi' Polarization as a test of colour octet quarkonium production

We calculate the $\alpha_s$ corrections to the transverse polarization fraction of $\psi^\prime$'s produced at the Tevatron. If the `$\psi^\prime$-anomaly' is explained by gluon fragmentation into a colour octet Fock component of the $\psi^\prime$, the $\psi^\prime$'s should be $100\%$ transversely polarized at leading order in $\alpha_s$, up to spin symmetry breaking long-distance corrections. We find that the short-distance correction to the transverse polarization fraction is a few percent, so that a polarization measurement would provide a reliable test of the colour octet mechanism.