Pair annihilation / production: a generator of pairs
e- e+
| | | |
| | | ----- axis of rotation
| | | |
photon ---> <--- photon
In the previous blog post we suggested that mainly the spins of the incoming electron and positron are responsible for producing the outgoing to two photons in annihilation.
If we place two rotating charges as above, with opposite spins, they will together act as an approximate (linear) electric dipole transmitter.
The reverse process is pair production. Suppose that we let two extremely strong laser beams collide as above. They might create a continuous flow of pairs.
The production of pairs would then be a non-perturbative classical process.
There are two "cranks" in this case. Turning them with incoming laser beams can be interpreted as a "generator" of pairs, or alternatively, the cranks form a "transmitter" of pairs. The process would be classical.
The classical description of the electron and positron streams would be a classical wave. Individual particles are then a quantum phenomenon, just like a photon is a quantum of a classical electromagnetic wave.
However, we have a problem: how to explain the collision of two very high-frequency photons? How do they give the kinetic energies to the created electron and positron?
Pair annihilation cross section
In our February 8, 2021 blog post we noted that the cross section for the annihilation of slow electron - positron pairs is
σ ~ 1 / β²
in a simple classical model, while it is
σ ~ 1 / β
in Feynman diagrams. Here, the speed of the colliding electron and the positron is
v = β c,
where c is the speed of light.
The disrepancy suggests that we cannot treat a slow electron as a "scalar" point particle, like we did in our simple classical model. That makes sense, since the electron does have a spin.
What about large energies?
Douglas M. Gingrich (2004) gives the cross section for fast pairs:
σ ~ 1 / E * ln(2 E / m),
where m is the electron mass and E is the kinetic energy of the the electron and the positron. The particles approach from opposite directions.
The result differs from a classical calculation where we try to approximate the production of the photons with the Larmor formula. The cross section is too large. It looks like the electron is not like a classical particle even with large energies.
A. Hartin (2007) gives the cross section for the Breit-Wheeler process of pair production from the collision of two gamma ray photons.
If the energies of the photons are ~ 1 MeV, then the cross section for annihilation, pair production, and Compton scattering is of the order
σ ~ π r₀²,
where r₀ is the electron classical radius. Does this offer us any insight?
We have earlier calculated that in elastic scattering of electrons and positrons, we can use the classical approximation of charged point particles.
How can we reconcile all this? If we want to build a (semi)classical model of the electron and the photon, it should be able to explain these properties.
Conclusions
The crank model does not seem to help us.
Let us once again look at the "length scale problem" of bremsstrahlung, which we have discussed several times in the past three years.
The problem is the following. If we model a 1 MeV electron as a classical point particle which zooms past a proton, then the it should pass the proton at a distance of
~ 10⁻¹⁵ m
to shed most of its kinetic energy.
But the wavelength of the bremsstrahlung photon is roughly 1,000 times larger. How can the encounter produce such a photon?
In pair annihilation we have the length scale problem if we assume that the electron is a point particle.
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