Exclusive diffractive processes, such as exclusive vector meson production, serve as excellent probes of hadron structure within the perturbative regime of quantum chromodynamics (QCD). The exclusive process involving light and heavy vector mesons, \(e p \rightarrow e V(V=J/\Psi ,\rho ,\phi )\) , has been investigated at the HERA accelerator facility. In this study, we focus on the theoretical prediction of exclusive \(J/\Psi \) and \(\rho ^0\) vector meson production. Employing the color dipole formalism for the study of the exclusive vector meson production in photon-proton interactions, we calculate both the differential cross-section and total cross-section of \(J/\Psi \) and \(\rho ^0\) vector mesons, employing an analytical solution of the Balitsky-Kovchegov (BK) equation. Furthermore, we present the ratio of the longitudinal to the transverse cross-section for \(J/\Psi \) and \(\rho ^0\) as a function of \(Q^2\) . Two well-known vector meson wave function models, boosted Gaussian (BG) and Gaus-LC (GLC), have been integrated into our analysis, showing slight sensitivity to the chosen vector meson wave functions. Our theoretical predictions agree well with the available experimental data for vector meson production. The analytical solution of the BK equation proves reliable for the theoretical prediction of exclusive vector mesons within a specific range of \(Q^2\) .
