Investigation of nonlinear dipole cross sections

The nonlinear corrections to the Golec-Biernat Wusthoff (GBW) and Bartels- Golec- Kowalski (BGK) models, as discussed by Peredo-Hentschinski [M.A.Peredo and M.Hentschinski, Phys.Rev.D{109}, 014032 (2024)], are analyzed in terms of the gluon density${}^{,}$s nonlinear behavior. We incorporate these nonlinear corrections into the low $x$ gluon distribution of the modified BGK model for electron-ion colliders. By verifying that these results allow for a more direct assessment of the relevance of nonlinear corrections in describing the dipole cross section of heavy nuclei, we determine the dipole cross section at a fixed $\sqrt{s}$ and $Q^2$ to the minimum value of $x$ given by $Q^2\mathrm{/}s$ of the center-of-mass energy of electron-ion colliders. Additionally, we examine the nonlinear gluon density in the photoproduction cross sections of vector mesons $\mathrm{\Psi }(2s)$ and $J\mathrm{/}\mathrm{\Psi }$ on a lead nucleus.