Nonidentical protons

We have calculated the proton charge radius by assuming that the real proton radius is not unique and the radii are randomly distributed in a certain range. This is performed by averaging the elastic electron-proton differential cross section over the form factor cut-off. By using a dipole form factor and fitting the middle value of the cut-off to the low $Q^2$ Mainz data, we found the lowest $\chi^2/N$ for a cut-off $\Lambda=0.8203\pm 0.0003$ GeV, which corresponds to a proton charge radius $r_E=0.8333\pm 0.0004$ fm. The result is compatible with the recent precision measurement of the Lamb shift in muonic hydrogen as well as recent calculations using more sophisticated techniques. Our result indicates that the relative variation of the form factor cut-off should be around 21.5%. Based on this result we have investigated effects of the nucleon radius variation on the symmetric nuclear matter (SNM) and the neutron star matter (NSM) by considering the excluded volume effect in our calculation. The mass-radius relation of neutron star is found to be sensitive to this variation. The nucleon effective mass in the SNM as well as the equation of state of both the SNM and the NSM exhibit a similar sensitivity.