  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.
