On the Question of the Point-Particle Nature of the Electron
  The electron and the positron treated as point particles in the Two Body Dirac equations of constraint dynamics for QED possess a new and as yet undiscovered peculiar ${}^1S_0$ bound-state which has a very large binding energy of about 300 keV, in addition to the usual ${}^1S_0$ positronium state with a binding energy of 6.8 eV. The production and detection of the peculiar ${}^1S_0$ state provide a test of the electron point-charge property. As the peculiar ${}^1S_0$ state lies lower than the usual ${}^1S_0$ state, the peculiar ${}^1S_0$ state can be produced by a two-photon decay of the usual $^{1}S_{0}$ state. We estimate the rate of the two-photon decay and show how it depends on the probability $P_{up}$ of the admixture of the peculiar component in the predominantly usual ${}^1S_0$ positronium. The produced peculiar ${}^1S_0$ state in turn annihilates into two photons with a total c.m. energy of about 723 keV. Thus the signature for this new peculiar ${}^1S_0$ positronium bound state would be the decay of the usual ${}^1S_0$ state into four photons, with two energies bunching around 150 and two around 360 keV. Such a four-photon decay of the usual ${}^1S_0$ state will not be present if the electron and positron are not point particles, or if the mixing probability $P_{up}$ is very small. 
Horace W. Crater, Cheuk-Yin Wong
hep-ph