Hadronic $ψ$ production calculated in the NRQCD factorization formalism

The NRQCD factorization formalism of Bodwin, Braaten, and Lepage prescribes how to write quarkonium production rates as a sum of products of short-distance coefficients times non-perturbative long-distance NRQCD matrix elements. We present, in the true spirit of the factorization formalism, a detailed calculation of the inclusive cross section for hadronic $\psi$ production. We find that in addition to the well known {\it color-singlet} production mechanisms, there are equally important mechanisms in which the $c\bar{c}$ pair that forms the $\psi$ is initially produced in a {\it color-octet} state, in either a ${}^3S_1$, ${}^1S_0$, ${}^3P_0$ or ${}^3P_2$ angular-momentum configuration. In our presentation, we emphasize the ``matching'' procedure, which %is the method that allows us to determine the short-distance coefficients appearing in the factorization formula. We also point out how one may systematically include relativistic corrections in these calculations.