Reconciling charmonium production and polarization data within the nonrelativistic QCD framework

A thorough study reveals that the only key parameter for $\psi$ ($J/\psi$, $\psi'$) polarization is the ratio $\langle O^\psi(^3S_1^{[8]})\rangle/\langle O^\psi(^3P_0^{[8]})\rangle$, if the velocity scaling rule holds. The fit of the yield data is incapable of the determination of this parameter. We provide a universal approach to fixing the long-distance matrix elements (LDMEs) for the $J/\psi$ and $\psi'$ production. Further, with the existing data, we implement this approach, and obtain a favorable set of the LDMEs, and manage to reconcile the charmonia production and polarization experiment. A quantitative analysis indicates that theoretical corrections to the short-distance coefficients change only the values of the LDMEs but not the phenomenological results.