With nonrelativistic QCD factorization, we present the first complete next-to-leading order study on the polarization of prompt J/ψ hadroproduction by including feeddown from χ(c)((3)P(J)(1),(3)S(1)(8)) and ψ(2s) which turn out to be very important parts. By using the color-octet long-distance matrix elements obtained from a combined fit of the measurements at the Tevatron and LHC for J/ψ, ψ(2s) and χ(c), the prompt J/ψ polarization predictions are presented, and the results are in agreement with the CDF run I data (except two points), but in conflict with the CDF run II data, while they are close to the ALICE data (inclusive J/ψ). The measurements at the LHC are expected to clarify the situation.

Next-to-leading-order QCD corrections to J= plus c production in e e ÿ annihilation at s p 10:6 GeV are calculated in this paper, and an analytic result is obtained. By choosing proper physical parameters, a K factor (ratio of next-to-leading order to LO) of about 2, which is in agreement with the result in Y.-J. Zhang, Y.-j. Gao, and K.-T. Chao, Phys. Rev. Lett. 96, 092001 (2006), is obtained. The plot of the K factor vs the center-of-mass energy s p shows that it is more difficult to obtain a convergent result from the perturbative QCD without resummation of lns=m 2 c terms as the s p becomes larger.