Energy dependence of the proton geometry in exclusive vector meson production

The gluon radius of the proton is expected to increase at small gluon momentum fractions $x$, an effect which has hitherto not been considered in the dipole model framework. We investigate the energy dependence of exclusive $J/\psi$, $\phi$, and $\rho$ production by introducing three models for $x$ dependence of the gluon thickness function. We allow the transverse width of the proton to increase as $x$ decreases, using novel parametrisations in the spherical proton and the hotspot model. We compare these with a model where the number of hotspots increases as $x$ decreases and confront the models with HERA data. The models exhibit clear differences in the slope of the $t$-spectra and in the cross section ratios between coherent and incoherent events. Comparisons to $t$-slopes and $W_{\gamma p}$ measurements show a preference for models where the proton's size increases as $x$ decreases.