First extraction of the proton mass radius and scattering length $\left|\alpha_{\rho^0 p}\right| $ from $\rho^0$ photoproduction

As it involves the lightest physical states excited from the vacuum by the vector quark current, near-threshold $\rho^0$ photoproduction is considered a possible way to research the proton radius and the absolute value of the scattering lengths of the $\rho^0$--proton interaction. In this work, under the assumption of a scalar form factor of dipole form, the value of the proton mass radius is calculated as $0.85\pm 0.06 \text{ fm }$ by fitting the differential cross section of the $\gamma p \rightarrow \rho^0 p$ reaction at near-threshold energy. For light vector meson photoproduction, because the exchange of a scalar quark--antiquark pair is not suppressed and should dominate the scalar gluon exchange, the radius we extract from $\rho^0$ photoproduction is likely to represent the quark radius of the proton. This fact may explain why the value obtained in this work is very near the proton charge radius. Moreover, the absolute value of the $\rho^0$--proton scattering length $|\alpha_{\rho^0 p}|= 0.31 \pm 0.06 \text{ fm}$ is obtained for the first time within the vector meson dominance model. This result disobeys the rule that the absolute value of the vector meson and proton scattering length $|\alpha_{V p}|$ increases with the meson's mass, which can be attributed to treating the $\rho^0$ meson as a point in the analysis. These results provide useful theoretical information for an in-depth understanding of proton structure and the proton--vector meson interaction.