Divergence in the vertex correction in QED

NOTE December 22, 2020: 

https://www.mpi-hd.mpg.de/personalhomes/harman/HCI-WS2013-2014/Lecture-12.pdf

Zoltan Harman (2014) writes that the ultraviolet divergence is only in the case where the momentum q of the incoming photon is zero.

There is no ultraviolet divergence if q != 0.

The case q = 0, of course, can be reduced to the electron self-energy case, which we handled by banning self-energy loops altogether (the center of mass argument).

There is an infrared divergence. That is handled by setting a little mass m to the photon carrying the momentum k.

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In the past two weeks we have argued that the Feynman integral in QED for vacuum polarization converges (but not absolutely), and the electron self-energy loop can be removed altogether.

The third divergence in QED is in the vertex correction:

                     photon

                     4-momentum k

                     ~~~~~

                   /               \

          e- ------------------------

                        /

             ~~~~

          photon

          4-momentum q

The divergence is logarithmic.

For example, the electron may scatter from the field of a nucleus Z+. Then the incoming photon carries spatial momentum q.

Our rubber plate model of the electric field of the electron says that the plate will bend when the electron accelerates toward the nucleus. The plate resists the acceleration.

When the electron recedes from the nucleus, the rubber plate tries to keep it going. The electron emitted a photon which contains spatial momentum k, to the rubber plate, and then absorbed the same photon.

Thus, in the classical world, the electron does send an off-shell photon to itself in the encounter with the nucleus.

If the electron exchanges momentum q with the nucleus, we can calculate the rough geometry of the encounter. 

We have the principle that virtual photons only live for about 0.1 times their (Compton) wavelength. From the geometry of the rendezvous we can then set a cutoff for possible momenta k. The divergence is removed.

Alternatively, we may use destructive interference. If the approaching electron would send a short-wavelenth off-shell photon (momentum k) to itself, there would be lots of destructive interference at the point where it absorbs the photon. We would get an extra coefficient 1 / |k| to the Feynman integrand, which would remove the divergence.

The famous anomalous magnetic moment of the electron is calculated using many diagrams with the vertex correction. We need to check if our cutoff procedure would somehow alter the calculation. The calculation agrees with the experimental value at a precision 10^-10. We do not want to change that.

Note that in the vertex correction, we could have very heavy particles with large charges doing the rendezvous. The calculation has to agree with the classical limit. We need to verify that is the case.

In classical electrodynamics, the magnetic field resists changes in the current. A single electron can be regarded as a current. The rubber plate in this case seems to do the work of the magnetic field: it resists changes in the velocity vector of the electron.

Our rubber plate model tries to solve the long-standing problem of the interaction of a charge with its own field. The vertex correction is a prime example of such self-interaction.