Let me give you the crazy idea that the radius of an electron and a proton is fixed but complex, where the real part is the mean and the imaginary part is the standard deviation. Then the classical radius of an electron and a proton determines the mean value, and the root-mean-square value is variable in its meaning. The electron radius is pointwise at high energies, when relativistic corrections are applied, and the scattering cross section is proportional to the square off classical electron radius.
The formula for the scattering cross section of a photon by an electron does not need to be regularized and determines the scattering cross section $$Re\sigma=\sigma(0)-\sigma(\infty)=\frac{8}{3}\pi r_e^2;\sigma(x)=\sigma(\frac{\hbar \omega}{mc^2})$$ In this case, the radius in complex form is $$R_e=r_e(1\pm\sqrt{(Re\sigma-\pi r_e^2)/\pi}i)=r_e(1\pm 1.29i)$$ its modulus determines the scattering cross section $$|R_e|=r_e|1\pm1.29i|=1.63r_e=\sqrt{\frac{8}{3}}r_e$$ The formulas for the cross section of the scattering of an electron by an electron and the annihilation of an electron and a positron with the formation of two photons require regularization. The regularization parameter must be chosen so that the size of the electron coincides with the size of the electron when a photon is scattered by an electron. It turns out that the three formulas equally determine the size of the electron.
There is no unambiguous value for the size of elementary particles. Elementary particles do not have a finite size and it is impossible to determine an unambiguous final size by their charge. 795
For an electron, there are scattering cross-sections of various reactions, and with their help I was able to determine the complex size of an electron. The complex size of an electron is determined up to the imaginary part. For a proton, this cannot be done, since there are no formulas describing the cross-sectional area of ​​reactions. Nuclear forces are not described by the perturbation theory, therefore only measurements are made and there are no theoretical formulas. The classical radius of the electron is greater than the classical radius of the proton. But this does not mean anything, the size of the proton is unknown. 969