Extraction of the proton mass radius from the vector meson photoproductions near thresholds

We present an extraction of the proton mass radius by studying the $t$-dependence of the differential photoproduction cross sections of the vector mesons near thresholds. At low energy, the photoproduction of heavy quarkonium off the proton is connected to the gravitational scalar form factor of the proton and its Fourier transform -- the mass distribution inside the proton. Under an assumption of the dipole form of the gravitational form factor, the mass radius of the proton are determined via the near-threshold photoproduction data of J/$\psi$, $\phi$ and $\omega$ vector mesons. The overall proton mass radius from the combined analysis is estimated to be $\sqrt{\left } = 0.66\pm0.03$ fm, with the dipole parameter of $m_{grav.}=1.06\pm0.05$ GeV. We also provide a statistical projection of the proton mass radius for a suggested measurement of $\Upsilon$ electroproduction at the future EicC.