In an earlier blog post this week we observed that in annihilation, the virtual electron can be understood as a carrier of a "force".
e+ ---------------------------
| virtual
| photon
| p
e- ---------------------------
In Coulomb scattering, the pair exchanges a virtual photon. The propagator of the photon is
~ 1 / |p|^2,
which corresponds to the Coulomb potential 1 / r.
e+ ----------- ~~~~~~~~ photon
| virtual
| electron
| p
e- ----------- ~~~~~~~~ photon
Annihilation of a pair can be understood as scattering where the pair exchanges a virtual electron and changes to a pair of photons. The propagator of the virtual electron for large |p| is something like
~ 1 / |p|,
which would correspond to a 1 / r^2 potential. The force carrier is charged, in contrast to the Coulomb force.
There is a dilemma: why does the pair in the first reaction obey the Coulomb potential, but in the second reaction an unknown 1 / r^2 potential?
The pair certainly is under the Coulomb force in the second reaction, too.
It is as if there would be a 1 / r^3 force acting in the second reaction. We speculated that it might be the magnetic force between the spins, but that force between two dipoles is 1 / r^4. Also, the force between magnets is electromagnetic. It does not carry charge.
The "virtual electron force" acts between photons as well as pairs. That explains why the force affects the cross section from both directions in the second diagram.
The Coulomb force only acts between pairs. It would affect the reaction cross section only on the left side.
The photon as a rotating dipole would explain the 1 / r^3 force
Recall the "teleportation model" of the photon which we introduced in an earlier blog posting.
Let us assume that a circularly polarized photon is a rotating electron-positron dipole.
The strength of a dipole field on a single charge is ~ 1 / r^3. The potential is ~ 1 / r^2.
photon
e+ --------------------- e+
e- ------
\
---------- e- zero energy
---------- e+ photon
/
e+ -------
e- --------------------- e-
photon
In pair production, the dipole field of a photon succeeds in breaking apart the dipole of another photon. More precisely, both dipoles break apart. An electron and a positron fly away, and the remaining two particles form a zero energy photon.
Thus, pair production is a process where two photons collide and are broken apart into their constituent particles: the electron and the positron. Two particles fly away, the rest form a new photon of zero energy.
Pair annihilation is a process where an electron and a positron break apart a zero-energy photon and form two new photons.
If we have a lattice which contains charged particles, there photons can propagate as vibrations of the lattice. There pair production might really look like in the diagram above. The photon is a vibration which causes a surplus density of electrons at one location, and lack of electrons at another location. The collision of photons may be able to free one electron, leaving behind a hole.
Pair production is governed by a 1 / r^2 potential in the lattice and the cross section is described by the ~ 1 / |p| propagator.
To the other direction, the electron and the positron "roll down" the 1 / r^2 potential.
This would explain the β and 1 / β in the cross sections calculated from Feynman diagrams.
If the description is correct, an electron and a positron do not truly annihilate each other. They just regroup to form new photons.
Annihilation is not governed directly by the Coulomb force between the particles, but indirectly through the force that a dipole exerts on a single charge.
The Coulomb force does guide the opposite charges closer, but the annihilation reaction itself is between dipoles.
The "Coulomb correction", which is mentioned in literature about Delbrück scattering, probably tries to account for the focusing effect of the Coulomb attraction.
Does the new model solve the problem of extra degrees of freedom?
In the previous blog posting we discussed photon-photon scattering and the problem that the virtual pair stage has an extra degree of freedom, that is, |p| can have arbitrarily high values.
The problem arises from the assumption that the virtual pair can be born arbitrarily close to each other. If the particles are born very close, they must have huge kinetic energies and huge momenta.
In the diagram above, the e- and e+ which free themselves are not born at an arbitrary place. They are not born at all.
Rather, the electron and the positron are continuations of the two photons.
It may be possible to model the pair as a single particle which moves under the 1 / r^2 potential. Then there are no degrees of freedom at all. It is a classical deterministic path.
If an electron moves under the field of a nucleus and exchanges two virtual photons with the nucleus, then there is a loop in the Feynman diagram, and the rules would allow us to circulate any momentum p in the loop. But common sense says that such arbitrary momentum p cannot just pop up and join the electron. A single particle under a field does not have any degrees of freedom.
Yet another model: the semiconductor model of annihilation
Let us have a lattice of atoms. Let an electron escape from the lattice. What does the hole look like?
It probably polarizes the neighboring atoms. The electric field might look a bit like a dipole field?
That might explain the 1 / r^2 potential which governs annihilation.
When the hole and the electron are at a larger distance, polarization has no effect on the field of the hole far away.
Let us look at what models people have developed about semiconductor annihilation. LED lights work by doing this annihilation.
If a semiconductor is a perfect analogy of real electrons and positrons, then we can experimentally study pair production and photon-photon scattering without the problems of generating gamma quanta.
A brief Internet search does not reveal any potential associated with the electron-hole recombination in semiconductors.
Actually, polarization is always present in the lattice, whatever the distance of the electron and the hole.
The potential might be 1 / r far away from the hole, but steeper close to the hole if polarization for some reason is less effective close to the hole. In the case of atoms, once the distance of the electron is less than the grid spacing of the lattice, polarization might be zero. This would fit the picture that there is a cloud of polarization around an electron. The bare charge is larger but partially hidden by polarization.
The running of the coupling constant requires collisions of 100 GeV electrons and positrons to make a 8% difference. How could that explain the 1 / r^2 potential of annihilation at 1 MeV energies? It cannot.
Why the electron in the hydrogen atom does not "annihilate"?
Our classical calculation of the electromagnetic radiation says that when the electron approaches the proton, the acceleration is so large that the entire energy of the electron should radiate away when the distance is ~ 1.4 * 10^-15 m.
This is the classical problem about why electron orbits are stable in an atom.
In the s orbital, the angular momentum of the electron is zero. It swings through the proton every 5 * 10^-17 s.
The photograph model (or the uncertainty relation) says that to "draw" the electron at a position 10^-15 m from the proton, we would need spatial momentum > 1 GeV / c. Such an electron has energy > 1 GeV. The energy is not available, not even if we take into account the negative potential energy 1.5 MeV at that distance from the center of the proton.
Destructive interference in the photograph model wipes out all histories where energy is not conserved.
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