e- -----------------------------------
| virtual photon q
|
k O -k + q virtual loop
|
| virtual photon q
------------------------------------
proton
Above q is spatial momentum and k is arbitrary 4-momentum.
What is the sign in the vertex of the loop and the virtual photon? The vertex of the photon and the electron naturally carries a coefficient -e, and the vertex of the photon and the proton +e. But what is the coefficient when the photon couples to the loop? Does it couple to the electron (like in the Dirac sea model), or to the positron? In the dipole model, the force between the dipole and the charge is always attractive.
The Feynman parametrization is a relevant procedure in the calculation. The loop integral diverges quadratically. The Feynman parametrization affects the terms in the integral. Then terms which have an odd power of the arbitrary 4-momentum k, are dropped. That is, the terms in the new formula reveal symmetric integrals which are zero if the integration area is symmetric. Mathematically, this is dubious. Why the symmetries in this particular written form of the integral are relevant? Why not some other form?
The Ward identity is strange. In the integral, Hagen Kleinert claims (Formula (12.447)) that a particular part of the integral (which actually seems to diverge) "should be" zero based on a Ward identity. The identity claims that the loop matrix
Π_μν(q)
represents the "polarization" of the photon q, and it must be normal to q because the polarization of a (real) photon is always normal to the momentum of the photon.
We do not understand this. The virtual photon q is a direct momentum exchange from the Coulomb force. It does not have any defined polarization. Moreover, if we define the polarization as the direction of the electric field, the photon q is longitudinally polarized.
A semiclassical "bump into" model. Think about a model where the electron "bumps into" a virtual pair and gives it the momentum q. Then it may happen that the pair bumps into the proton and the proton absorbs the momentum q. Does this process make the Coulomb force appear stronger? If the electron would get the momentum -q toward the proton, then classically, the pair will move away from the proton, and it is improbable that it will bump into the proton. Momentum conservation bans a process where the pair does not give up its momentum. Thus, classically, the process makes the attraction appear weaker. The momentum q points toward the proton.
But what about an electron and an antiproton? In this case, the process would make the repulsion appear stronger. The model is not consistent with vacuum polarization.
What about phase shifts? What would the phase be for the electron which bumped into the virtual pair? Suppose that a mildly relativistic electron scatters to a large angle from the proton. The phase shift when the electron is roughly at the distance of the electron classical radius from the proton is around 1/1,000 cycles. The total phase shift, if the electron is sent from a distance 1 meter, might be around 15/1,000 cycles. The phase shift from bumping into a virtual pair might be of a similar magnitude. We conclude that phase shifts are almost negligible.
Cross sections in this classical model would be larger than for plain Coulomb scattering. The electron could get the momentum exchange q in two ways: through the Coulomb force or through bumping.
What would the propagator be? Maybe something like
1 / q² * 1 / q² * 1 / m²
for small q. But that becomes infinite for very small q. We believe that the lifetime of the virtual pair is very short. The formula fails to take that into account and is nonsensical. For large q, the value is too small.
Why does the traditional calculation of vacuum polarization give empirically correct values even though there are so many ad hoc tricks in the calculation? Above we mentioned some fuzzy parts of the calculation. On top of that there is regularization. Even if the motivation for the traditional calculation is shaky, it does give the correct vacuum polarization contribution to the Lamb shift at least with a precision 1 / 1,000.
For high-momentum scattering, measurements at the LEP have an accuracy 10% or worse. Thus, we do not know if the traditional calculation is very accurate in the high-momentum case.
The measurement of the Lamb shift of muonic hydrogen (Randolf Pohl et al., 2010) gives a proton charge radius which is 4% less than the radius measured from electron scattering. This suggests that there might be something wrong with traditional vacuum polarization calculations when the momentum exchange is of an intermediate size. The measured Lamb shift in muonic hydrogen is 206.3 meV, and the discrepancy, which gives the strange proton radius, is 0.3 meV.
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