Where do the momenta disappear in "classical" annihilation of a pair?

Let us imagine that a classical electron and a classical positron collide head-on. Let the initial velocity of the particles be small.

                   -------   electromagnetic wave

                    -----

                     ---

         e- ------>  <------ e+

                     ---

                    -----

                   -------   electromagnetic wave

We calculated using the Larmor formula that at the distance 1.4 * 10^-15 m, the pair emits all its energy, 1.022 MeV, in a very sharp "wrinkle" in the electric field of each particle.

The emitted wave looks like a dipole wave and does not carry much horizontal momentum in the diagram.

What happens to the substantial horizontal momenta of the particles at the moment of annihilation?

In the Feynman diagram, a mysterious virtual electron carries the momentum between the annihilating particles.

The spins of the electron and the positron are assumed to be opposite at annihilation. Both particles are magnets whose north pole points to the same direction.

We calculated in an earlier blog posting about the Pauli exclusion pronciple that the magnetic repulsion exceeds the electric attraction already at the distance 3 * 10^-13 m if the magnets are side-by-side.

Maybe the magnets are positioned linearly? Then the magnetic force would strengthen the attraction.

There is a lot of horizontal momentum in the electric field of each particle at the moment of collision. But the electric fields almost cancel each other at that point. The neutralization of momentum may actually happen in the electromagnetic field.

The collision of static electric fields would eventually cancel the big opposite momenta of the particles. The mass-energy of the particles themselves might be already zero at that point.

In the drum skin model, we let the two pressing fingers approach each other, and at the moment of "annihilation" suddenly remove the fingers. If most of the energy escapes to the normal direction from the "collision", then the momenta in the original static fields (depressions) of the fingers has to get neutralized in some way.

Comparison of the output electromagnetic wave in a classical model versus a Feynman diagram

Let us have |p| very small in the propagator G_F(p - q) of the previous blog posting. Then the value of the propagator is almost the same for all directions of q.

The Feynman diagram predicts that the two photons will be emitted to a random direction from the annihilation (they have opposite momenta, but the direction of q is random).

The classical model gives approximately the same prediction: in most cases, the electron and the positron will orbit a little around each other. The direction of the dipole in the final annihilation may be almost random.

What about the spin? If the particles have opposite spins, their north poles point to the same direction. We suggested that they will be aligned as a line at the annihilation. The photons will be emitted mostly normal to the direction of the spin.

Let us check what literature says. A brief Internet search did not return any data about the direction of the emitted photons versus the direction of the spins of the electron and the positron.