New Rosenbluth formula including the Coulomb correction in the second Born approximation

The Coulomb correction factor is derived for $M1$ scattering on a pointlike nucleus in the second Born approximation. The Coulomb correction for $M1$ scattering off the proton is larger than that for $C0$ scattering by 2%. The difference affects the determination of the proton radius where an identical Coulomb correction has been applied for both components so far. A new Rosenbluth formula, that includes the Coulomb corrections, is applied to electron scattering on the proton with low-energy electron beams. The Coulomb corrections are essential to discuss the electric and magnetic radii of the proton.