Let me elaborate on what John wrote, in simple way.
Nothing new, only a pedagogical exercise.
Suppose protons of low energy, say 200 KeV, are
scattered by nearly stationary protons. What happens/
Those protons which pass far away are not scattered,
those which pass closer are scattered. The scattering
angle increase when so-called impact parameter
decreases. Backward scattering (180 degrees) takes
place in head-on collisions. There is an exact classical
theory of this; it predicts that the differential scattering
cross section of two point-like charges is inversely
proportional to the fourth power of sin(TET/2).
And experimental data are in perfect agreement with
this, but only at very low energies. At a sufficiently
high energy, such as 2000 keV, the predicted behavior
is confirmed at large angles only. The angle at which
the point-like theory starts to disagree with data was
used by E. Rutherford to estimate the radii of
scatterers (the nuclei of gold atoms).
The same approach can be used for any scatterer,
including hydrogen nuclei. That is how we learned
that the size of a proton is about one Fermi. John
was saying that for scattering of electrons on
electrons (I do not know about these experiments)
the predicted relation remains valid at all energies
we can produce and for all angles. In other words
even the GeV and TeV electrons in "supercolliders"
scatter according to predictions of a theory based
on point-like charges (?). That is why we say that
as far as we know electrons have no sizes.
But this may change when more powerful accelerators
are constructed. Neutrinos used to described as
mass-less particles; today we say that they may
have extremely small, but finite, masses. That is
still an open question, I think.
Ludwik Kowalski