https://en.wikipedia.org/wiki/Møller_scattering
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If the energy of the collision is less than 1.022 MeV, then no new electron-positron pair can be produced.
If the energy is larger, then new pairs will be created.
Intuitively, the production of a pair should reduce the probability amplitude of an elastic collision where the electrons exchange a large amount p of 4-momentum.
But the possibility of pair production is not explicitly apparent in the Feynman formula which sums the t- and u-channels of elastic scattering.
Could it be that the Feynman formula for elastic scattering in some implicit way "knows" about the possibility of pair production? That is probably not true. We may imagine new physics where a lighter variant of electron exists and can produce a large number of new pairs. How could the Feynman formula for ordinary electrons be aware of such new physics?
Furthermore, the electron-electron collision may produce a photon. That is, the collision is not elastic. How could the elastic collision formula be aware of the (complex) process of photon radiation in the collision?
The simplest pair production diagram in Møller scattering is the following:
e- e+ e- e+
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The diagram contains four photon-electron vertices:
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while the elastic scattering only has two such vertices. If the integral formula for the pair production has a much smaller probability amplitude (or cross section) than the formula for elastic scattering, then in the first approximation we can ignore the effect which pair production has on the elastic probability amplitude.
We have not yet found the pair production cross section from literature.
A semiclassical treatment of pair production
Let us consider electrons and positrons as classical objects which obey special relativity.
For each classical trajectory of particles we associate an integral over a lagrangian density. The integral gives the phase, or the probability amplitude, of that history.
If an electron and a positron come to the distance 3 * 10^-15 m from each other, then their combined energy is zero, assuming that they have no kinetic energy.
We assume that we can add such a zero-energy pair to any history where the existing other particles bump into the particles in the pair, giving the pair a 4-momentum which makes them real, a 511 keV electron and positron.
That is, classical collisions can create new pairs by tearing apart an electron and a positron which exist as a zero-energy pair.
Our assumption is somewhat similar to the hole theory of Dirac. In Dirac's hole theory, an electron with an energy -511 keV gets excited to a state of an energy +511 keV, leaving behind a hole, which is the positron.
The zero-energy pair can be considered a zero-energy state of a positronium "atom". The atom gets excited by other particles, and goes into a state where the pair will annihilate again (= virtual pair), or goes into a state where the electron and the positron escape as real particles.
Is our model deterministic? Suppose that the electrons exchange more than 1.022 MeV of energy in a collision. What determines if a pair is produced, or if the collision is elastic?
Or should we make the model probabilistic?
The two electrons which enter the experiment can be considered as uncorrelated. Pair production can be seen as a positron moving backward in time, colliding with both electrons, and scattering forward in time.
But is the positron moving backward in time uncorrelated with the two electrons?
Suppose that the initial state of electrons is such that they would collide and exchange more than 1.022 keV of kinetic energy. Is there always some positron trajectory which will rob some of the energy and produce a real pair?
We believe that empirical tests show that elastic collisions are possible at large energies. Pairs are not always produced.
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