Ratio of \(J/\Psi\) and \(\Psi(2s)\) exclusive photoproduction cross-sections as an indicator for the presence of non-linear QCD evolution

We investigate the proposal that the rise with energy of the ratio of the exclusive photo-production cross-sections of vector mesons \(\Psi(2s)\) and \(J/\Psi\) can serve as an indicator for the presence of high gluon densities and associated non-linear high energy evolution; we study this proposal for both photoproduction on a proton and a lead nucleus. While previous studies were based on unintegrated gluon distributions subject to linear (Balitsky-Fadin-Kuraev-Lipatov) and non-linear (Balitsky-Kovchegov) evolution equations, the current study is based on the Golec-Biernat W\"usthoff (GBW) and Bartels Golec-Biernat Kowalski (BGK) models, which allow assessing more directly the relevance of non-linear corrections for the description of the energy dependence of the photoproduction cross-section. We find that the rise of the ratio is directly related to the presence of a node in the \(\Psi(2s)\) wave function and only manifests itself for the complete non-linear models, while it is absent for their linearized versions. We further provide predictions based on leading order collinear factorization and examine to which extent such an approach can mimic a ratio rising with energy. We also provide a description of recent ALICE data on the energy dependence of the photonuclear \(J/\Psi\) production cross-section and give predictions for the energy dependence of the ratio of \(\Psi(2s)\) and \(J/\Psi\) photoproduction cross-sections for both scattering on a proton and a lead nucleus.