A brief Internet search did not return any measured data for low-energy annihilation cross sections. It looks like other processes dominate over the simple annihilation process.
There is some data for positronium formation, as well as for complicated processes when a positron meets a molecule.
The formula which is derived from the Feynman diagram suggests that the cross section is
σ = 1 / β π r_0^2,
where the speed of the particles is β c, and r_0 is the classical electron radius.
The cross section using a classical particle model is very different. The pair quickly radiates away all their energy when their distance is about r_0 / 2.
Let us assume that the maximum possible angular momentum of the pair is J_a at the moment of annihilation, or just before it.
The linear momentum of a particle is initially
p = m_e β c.
If the impact factor is b, then the initial angular momentum is
J = b m_e β c,
and the maximum value for b is
b = J_a / (m_e β c).
The classical cross section is proportional to 1 / β^2, not to 1 / β as in the Feynman formula.
For a 2 eV electron, β = 0.003 and for a 2 keV electron, β = 0.1.
The difference in predicted cross sections is very large.
The classical limit
Let us increase the masses of the particles by a large factor M, and their charges by a factor sqrt(M).
The particles will follow the same classical trajectory as a classical electron and a positron.
We can monitor the trajectory of the heavy particles accurately, because now they really are classical particles. If they lose energy in radiation as in the Larmor formula, they have lost all their energy when they are roughly 1.4 * 10^-15 m from each other. We may assume that they are annihilated by then.
The heavy particles send many photons whose wavelength is in the range 10^-15 m. The process is very different from the Feynman diagram for electron-positron annihilation.
The classical formula for the cross section gives the right answer in the classical limit, the Feynman formula a wrong answer.
Who is right?
The Feynman model assumes a very short interaction of particles, which otherwise fly freely. Could this be the reason for the discrepancy?
The electron and the positron according to the Schrödinger equation
The electron and the positron probably obey the Schrödinger equation. They are low-energy particles like the orbital electron of the hydrogen atom.
If we just look at the position difference vector r of the electron and the positron, we can model the system as a single particle in a central Coulomb potential.
For a free electron, the Schrödinger equation probably calculates roughly the same trajectory as classical physics.
Thus, the cross section for a close encounter really is ~ 1 / β^2, also according to the Schrödinger equation. If there is a close encounter, why would annihilation probability be proportional to 1 / β, like the Feynman formula claims?
The pair annihilation cross section once again
In the February 6, 2021 blog posting we calculated that the ratio of the Feynman integrals to opposite directions of the reaction is 1 / β^2 for a massive Klein-Gordon "pseudoelectron".
If we calculate the probability of annihilation of the "pseudoelectron" by integrating over all possible photon momenta q, and |p| is small, then the Feynman integral value is essentially independent of p.
That would mean that the annihilation probability does not depend on p at all for small p, which would be very strange.
To the other direction, the integral over possible p is ~ β^2. That sounds more sensible.
p q
e- --------- ~~~~~~~~ photon
| p - q
| virtual
| electron
e+ --------- ~~~~~~~~ photon
-p -q
Let us calculate the integrals again, this time using the spinor field propagator. That is, we look at the real electron instead of the Klein-Gordon "pseudoelectron".
The Feynman propagator for a spinor electron is
i (k-slash - m I) / (k^2 - m^2 + i ε).
There, k is the 4-momentum of the virtual electron. The energy in the virtual electron is zero in annihilation, because of symmetry. I is the unit 4 x 4 matrix.
Let |p| be small and p constant. Then |q| is a little larger than m.
We have to integrate
i (-p-slash + q-slash + m I)
/ (q^2 + m^2),
where we used the fact that k^2 is roughly -q^2.
Let us vary q. The denominator is constant. What do the slash terms give?
Each spatial component of q varies symmetrically around zero.
The calculation is complicated.
In the link, S. Bragin and A. Di Piazza (2020) perform the steps and get the same result as V. B. Beretstetskii et al. (1982) in the book Quantum Electrodynamics.
At this stage we have to believe that Feynman formulas really give the results known in literature. The cross section is ~ 1 / β, which for an unknown reason differs from the classical formula ~ 1 / β^2.
The virtual electron can be understood as the carrier of a 1 / r^2 potential "force"?
The cross section for pair annihilation is essentially constant for small β if we calculate the Feynman diagram using the propagator for the massive Klein-Gordon equation (the "pseudoelectron"). The propagator for large |p| is roughly 1 / |p|^2.
The propagator for the Dirac equation for large |p| is something like ~ 1 / |p|.
We may interpret the Dirac propagator as a carrier of a force whose potential is 1 / r^2.
When the photons scatter from this 1 / r^2 Dirac "potential", they are converted into a pair.
The Coulomb 1 / r potential has the property that if we shoot a particle horizontally from the surface of a sphere, the particle will go to infinity if its kinetic energy is larger than the negative potential energy.
But if the potential is like close to the event horizon of a black hole, then a horizontally shot particle will crash into the surface of the sphere. The particle has to be shot at an angle to make it climb up the potential.
According to the Feynman formula, the cross section for pair production is ~ 1 / β. That suggests that the potential, which the particle has to roll up, is steeper than 1 / r.
The classical cross section for annihilation in a Coulomb 1 / r potential is ~ 1 / β^2.
If the potential is steeper, the cross section may be, e.g., ~ 1 / β.
The magnetic force of the electron and positron spins may make the potential steeper than 1 / r.
The Dirac equation is kind of a "square root" of the massive Klein-Gordon equation. The propagator clearly reflects this fact. It might be that the potential 1 / r^2 and the cross section ~ 1 / β somehow are a result of being a "square root".
We need to check the form of the magnetic force between the spins, and also figure out why the classical limit suggests that the cross section should be like for the 1 / r potential.
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