The simplest Coulomb scattering Feynman tree diagram diverges

Let us look at the very simplest Feynman diagram of elastic Coulomb scattering.

             p                                 k

       e- ------------------------------------

                            |  virtual

                            |  photon

                            |  q

       Z+ -----------------------------------

       momentum 0

                             

In the diagram, p, q, and k are spatial momenta. The spatial momentum of the nucleus Z+ is zero. We have

        k = p + q.

Since the scattering is elastic,

       |p| = |k|.

Possible values of k are on a surface of a sphere. The surface is 2-dimensional.

The propagator of the photon is

       ~ 1 / |q^2|.

Let us denote r = |q|. The Feynman integral for small r is of the form

       ∫ r / r^2  dr,

which diverges logarithmically for small r.

How to interpret this? An infinite number of electrons come out from the experiment where we send just one electron?

In the standard machinery of QED, we have to regularize the integral, that is, cut off the infrared infinity, and renormalize the value of the integral to 1, since we expect every electron in our experiment to be scattered, most of them just a little bit.

The original integral seems to describe a situation where a plane wave of electrons meets a single nucleus Z+. If the plane wave contains, say, one electron per square meter, then the integral does make sense, and say that an infinite number of electrons is scattered a little bit.

We cannot divide the probability distribution of a single electron evenly on R^2, but we can divide a countably infinite number of electrons evenly there.

In this simple case of infrared divergence, there is a perfectly logical mathematical explanation for regularization and renormalization.

The classical interpretation

Suppose that the experiment spans one square meter. A nucleus sits in the middle, and we shoot an electron from a random location in the square meter.

Classically, the electron will in all cases receive a momentum |q| which is larger than a small fixed number δ. We have a justification to cut off the integral at δ.

Hypothesis. All regularizations and renormalizations of infrared divergences have a classical justification. The momentum cannot be arbitrarily small in the experiment.