We present a nonperturbative QCD calculation of elastic $J/psi$ meson production in photon-proton scattering at high energies. Using light cone wave functions of the photon and vector mesons, and the framework of the model of the stochastic QCD vacuum, we calculate the differential and integrated elastic cross sections for $gamma p goto J/psi p $ . With an energy dependence following the two-pomeron model we are able to give a consistent description of the integrated cross sections and the differential cross sections at low $|t|$ in the range from 20 GeV up to the highest HERA energies. We discuss different approaches to introduce saturation and find no specific effects up to energies presently available. We also calculate and compare to experiments the cross section for $upsilon$ photoproduction.
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