Probing valence quark width of the proton in deeply virtual Compton scattering at high energies

We use the refined hot spot model to study the valence quark shape of the proton with the deeply virtual Compton scattering at high energies in the framework of Color Glass Condensate. To investigate the individual valence quark shape, a novel treatment of the valence quark width is employed. We calculate the cross-sections for the coherent and incoherent deeply virtual Compton scattering, for the first time, by using different widths ($\mathrm{B_u}$ and $\mathrm{B_d}$) for the profile density distributions of the up and down quarks instead of using the same width as in the literature. We find that the cross-sections calculated with $\mathrm{B_u \geq B_d}$ at each collision energy are consistent with each other themselves, which is in agreement with the theoretical expectations. While the ones computed with $\mathrm{B_u < B_d}$ show some discrepancies. This outcome implies that the up quark might emit more gluons than the down quark leading to $\mathrm{B_u \geq B_d}$ at high energy. The energy impact of the outcome is estimated. Our results show that as the collision energy increases, the aforementioned discrepancies are not only significantly broadened, but also shift to relatively smaller momentum transfer range in the future Electron-Ion Collider (EIC) and Large Hadron Electron Collider (LHeC) energies, which indicates that the EIC and LHeC can provide unprecedented chance to access the shape of the valence quark of the proton.