$J/\psi$ meson production in SIDIS: matching high and low transverse momentum

We consider the transverse momentum spectrum and the $\cos 2\phi$ azimuthal distribution of $J/\psi$ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering, where the electron and the proton are unpolarized. At low transverse momentum, we propose factorized expressions in terms of transverse momentum dependent gluon distributions and shape functions. We show that our formulae correctly match with the collinear factorization results at high transverse momentum, obtained in the framework of collinear, nonrelativistic QCD (NRQCD). We conclude that the shape functions at high transverse momentum are independent of the quantum numbers of the produced quarkonium state, except for the overall magnitude given by the appropriate NRQCD long distance matrix element.