Mass radius and mechanical properties of the proton via strange $\phi$ meson photoproduction

In this work, the cross sections of $\phi$ photoproduction are systematically investigated with the two gluon exchange model and the pomeron model. The results obtained show that the theoretical values of the two models agree well with the experimental data. Since the mass radius and mechanical properties of the proton are encoded in the scalar gravitational form factor of the momentum energy tensor, based on the differential cross sections of $\phi$ photoproduction at near-threshold predicted by the two gluon exchange model, the average mass radius of the proton is derived as $\sqrt{\left\langle r^{2}\right\rangle_\mathrm{m} }=0.78 \pm 0.06$ fm. As a comparison, we directly extract the proton mass radius from the experimental data of $\phi$ photoproduction to obtain an average value of $0.80\pm0.05$ fm, which is very close to the result given by the two gluon exchange model. Taking a similar approach, we extracted the average value of $m_{D}$ to obtain the distribution of the pressure and shear force contributed within the proton, and compared the results with other groups. The results of this paper may provide necessary theoretical information for the subsequent in-depth study of the internal structure and properties of proton.