Soft Gluon Approach for Diffractive Photoproduction of $J/\psi$

We study diffractive photoproduction of $J/\psi$ by taking the charm quark as a heavy quark. A description of nonperturbative effect related to $J/\psi$ can be made by using NRQCD. In the forward region of the kinematics, the interaction between the $c\bar c$-pair and the initial hadron is due to exchange of soft gluons. The effect of the exchange can be studied by using the expansion in the inverse of the quark mass $m_c$. At the leading order we find that the nonperturbative effect related to the initial hadron is represented by a matrix element of field strength operators, which are separated in the moving direction of $J/\psi$ in the space-time. The S-matrix element is then obtained without using perturbative QCD and the results are not based on any model. Corrections to the results can be systematically added. With a reasonable assumption we find that in the large energy limit the S-matrix element is related to the asymptotic behavior of the gluon distribution for $x\to 0$. Our approach is different than previous approaches and our results are also different than those in these approaches. The differences are discussed in detail. A comparison with experiment is also made and a qualitative agreement is found