Investigating the structure of gluon fluctuations in the proton with incoherent diffraction at HERA

Impact parameter dependent dipole models are ideal tools for investigating the spatial structure of the proton. We investigate the incoherent $ep$ cross section in exclusive $J/\psi$ photoproduction as measured by HERA, and find that as $|t|$ increases, the models need several levels of the substructure of gluon density fluctuations in order to describe the measured data well. In lieu of a perturbative description, we add this substructure by hand. This substructure is modelled as hotspots within hotspots. This enables us to describe measurements for $|t|> 1$~GeV$^2$, which is necessary for describing any observable which integrates over the $t$-spectrum, such as the rapidity or $W_{\gamma p}$. We find that three levels of proton substructure are adequate for a good description of all available $ep$ data up to $|t|=30~$GeV$^2$. We note that the gluonic density fluctuation structure follows a scaling behaviour, such that the logarithms of the number of hotspots and their size fall on a line, effectively reducing the available parameter space of the model. Our findings systematically constrain and provide a benchmark for the development of a perturbative model of spatial gluon fluctuations in nucleons.