The process AB ->-quarkonium +7 -h X ^ for a heavy quarkonium state such as J/xj; or T, is considered. The ratio of the cross sections for this process in pp and pp collisions is shown to be a very sensitive probe of models of quarkonium formation.PACS numbers: 13.85.Ni, 13.85.Qk, 14.40.Gx Over the years, there has been considerable interest in the study of heavy quarkonia because the leptonic decay channels of these resonances provide the cleanest signals of heavy quark formation. The production of quarkonia takes place through subprocesses that are dominated by gluons. These processes have, therefore, been used to study gluon distributions in the hadron.The cross section for the production of a QQ bound state factorizes into two parts: a short-distance part, which corresponds to the production of a heavy quark pair from a hard collision of the incident particles, while the other part specifies how the QQ pair produced in the collision forms a quarkonium bound state. The shortdistance part is, of course, computable in perturbative QCD but the formation of the bound state from the QQ pair can be specified only in the context of some phenomenological model of hadronization. In particular, it is not known whether the hard scattering subprocess is sensitive to the quantum numbers of the bound state, or whether this information is completely screened by the subsequent hadronization.For several years now, two models of quarkonium formation have been used to describe J/T/? and T production in high energy collisions: the color-singlet model [1] and the semilocal duality model [2]. The latter model has also been referred to as the color-evaporation model in the literature. In the color-singlet model, one starts with the full QQ production amplitude and then projects out the state with the proper spin, parity, and chargeconjugation assignments to describe a '^^'^^Lj quarkonium state. This state is required to be a color singlet, which is achieved by the radiation of a hard parton in the final state. The matrix element so obtained is then convoluted with the modulus squared of the wave function at the origin, |i^oP {ox its derivative, in the case of P-wave quarkonia), which finally yields the matrix element for the quarkonium production process. The wave function and its derivative are fixed from the leptonic or hadronic decay widths of the quarkonia. In this model, because the amplitude for the production of a state with definite J^^ are computed, it is possible to predict the production cross section for different resonances in a given family of quarkonia. It is also important to note that the hard scattering vertex is governed by selection rules that involve the quantum numbers of the quarkonium. For example, in leptoproduction of charmonia, the ^^i state (the J/ip) is produced in the subprocess 7^ -> ^Sig, but the production of the corresponding P states (the x's) is disallowed.In the semilocal duality model, the hard scattering vertex is completely blind to the quantum numbers of the quarkonium resonance. The quant...