We want to find out what is the structure of the inner electric field of the electron, that is, the structure for a radius less than the electron classical radius of 2.8 * 10^-15 m.
High-energy Møller scattering is one way to find out.
Elastic Coulomb scattering of electrons in QED is apparently the same as for classical point charges where we only assume the Coulomb force between point masses, and do not take into account effects of the electric field of the electrons.
Bremsstrahlung: emission of a large photoni
E. Haug in 1975 calculated QED bremsstrahlung for Møller scattering.
Figure 7 in the paper plots the differential cross section for a single energetic photon emission per MeV for various collision energies. The plot is actually
cross section × energy of the photon in MeV.
It is kind of a power spectrum.
Let us look at a 10 + 10 MeV collision. A 5 MeV photon can be called a "large" photon. The differential cross section at 5 MeV is 6 mb / MeV. The total cross section for an emission of a > 5 MeV photon is roughly 20 mb.
For a 50 + 50 MeV collision, the total cross section for a > 25 MeV photon is roughly 40 mb.
Elastic scattering to a large angle without bremsstrahlung
What about elastic scattering to a large angle? Large angle scattering requires that the potential at the impact parameter b is comparable to half of the total energy of the colliding particle.
For a 10 MeV electron, b should be less than 2.8 * 10^-16 m, which implies a cross section of 3 * 10^-31 m^2, or 3 mb.
For a 50 MeV electron, the cross section for large angle elastic scattering is 0.1 mb.
We see that the cross section for a large photon emission is much bigger than the cross section of elastic scattering to a large angle.
Strange behavior: in elastic scattering the electron acts like it would have no electric field
Classically, according to the Larmor formula, if relativistic electrons pass at a distance < 1.4 * 10^-15 m, they will lose much of their energy in radiation. They cannot scatter elastically at all at short distances.
But in QED, electrons can also behave like point charges with no electric field: they can scatter elastically at very short distances, down to at least 10^-18 m. The elastic scattering is the simplest Feynman diagram.
What kind of a classical model could explain this peculiar behavior?
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