Gluon fragmentation represents the dominant source of high energy prompt quarkonia at hadron colliders. Fragmentation approximations break down, however, when a quarkonium's transverse momentum becomes comparable to its mass. In this paper we identify a large class of color-octet diagrams that mediate quarkonia production at all energies and reduce to the dominant set of gluon fragmentation graphs in the high p1 limit. They contribute to quarkonia differential cross sections at the same order as color-singlet diagrams and bring theoretical predictions for U and Y production at the Fermilab Tevatron into agreement with experimental measurements. Using recent CDFdata we extract numerical values for bottomonia and charmonia color-octet matrix elements which are consistent with NRQCD scaling rules. We also find that quarkonia generated via the color-octet mechanism are strongly polarized at low as well as high energies. Spin alignment measurements can thus test the color-octet quarkonia production picture.
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