Let us continue the study of an electron e- scattering from a massive charge X+.
Overlapping probabilities in Feynman diagrams: we cannot assume no overlap
Consider the tree level elastic scattering diagram, and the tree level bremsstrahlung diagram.
It is clear that these cannot describe non-overlapping probabilities.
Elastic scattering, actually, never happens in the real world. The classical limit shows thst an infinite number of low-energy photons is always emitted, at least when |q| is small enough, so that we can describe the electron as a wave packet, passing X+ at a considerable distance.
Rule: Feynman diagrams do not necessarily describe non-overlapping probabilities. The infrared divergence in the bremsstrahlung diagram means that the electron emits an infinite number of photons. The probabilities for photons of various 4-momenta p overlap. Also, the probabilities in the tree level elastic scattering diagram overlap with those of the bremsstrahlung diagram.
In each individual case, we have to analyze which probabilities are disjoint, and which overlap.
The vertex function F1(q²): it is useless and should be ignored
k
~~~~~
p / \
e- ---------------------------------
| q
X+ ---------------------------------
We can imagine that the electron hits the electromagnetic field with a sharp hammer when it arrives close to X+. The Green's function creates virtual and real photons of various 4-momenta k.
If q would be 0, then the electron would absorb everything which it sent in the Green's function. But q disturbs this. A part of the wave from the hammer hit escapes as real photons, bremsstrahlung. That part is seen in the Feynman integral as the (negative) infrared divergence. Something is "missing" from the integral when q ≠ 0.
∫ d⁴k f(q²) - ∫ d⁴k f(0).
near k₀ near k₀
For 4-momenta close to some k₀, we define the "missing part" as the difference of the integral for q ≠ 0 from q = 0.
Large real photons cannot escape since the electron does not have enough kinetic energy to create them (though, classically, they would be able to escape). But the integral does have a "missing part" for them. The crucial question is how we should interpret the missing part?
e- ---------------------------------
| q
X+ ---------------------------------
Plane wave analysis of elastic scattering. Above we have the tree level diagram for elastic scattering. Let us analyze the tree diagram and the loopy diagram from the plane wave point of view.
1. We imagine that the plane wave describing the electron enters a cubic meter m³ where there is a time-independent electromagnetic wave q created by X+. An electron wave which is scattered by q absorbs the spatial momentum q.
2. A certain flux of the electron plane wave "absorbs" the momentum q and is scattered, according to the tree level diagram.
3. The loopy diagram means that a certain flux φ of the electron plane wave is scattered by a virtual photon k (which the electron itself sent).
4. That flux φ can be scattered by the q wave. Later, the flux is again scattered by k, this time absorbing back k. A part of the flux φ absorbed the momentum q.
5. The probability of the original electron plane wave scattering from q is the same as of the flux φ scattering from q.
6. It does not matter for the electron wave if it was the original planar wave, or the flux φ. The probability of absorbing q was the same.
7. We conclude that the probabilities described by the tree level diagram and the loopy diagram overlap completely. The loopy diagram does not contribute anything to the scattering probability.
8. Classically, reducing the electron mass makes the scattering probability larger, but that involves at least two momentum exchanges between the electron and X+. A Feynman diagram with just a simple q line should not be aware of this.
Empirical evidence. So far, we have not found data about scattering experients which would be accurate enough to reveal the numerical value of the vertex loop correction. The CERN LEP experiment probed vacuum polarization.
The electric vertex function F1(q²) is extremely small for small q². The paper at the link
claims that for q² << me², the electric form factor is
In the hydrogen atom, the kinetic energy of the electron is
Ekin = p² / (2 me),
and we can assume that q ≈ p. Then
q² / me² ≈ 2 Ekin / me
≈ 20 eV / 511 keV
≈ 4 * 10⁻⁵,
and
α / (3 π) * q² / me² ≈ 3 * 10⁻⁸.
The scattering probability changes very little from the (claimed) electric form factor. It is unlikely that such tiny changes can be measured.
The classical limit. It is nonsensical for the vertex function F₁. The fine structure constant is defined
α = 1 / (4 π ε₀) * e² / (ħ c).
Let us increase the mass of the electron, and its charge by some large ratio N >> 1. The value of the coupling constant
α(N) ~ N².
The formula for the form factor F₁(q²) would claim that the apparent charge of a macroscopic particle would greatly depend on q! This would badly break the classical limit of quantum mechanics. This observation strongly suggests that the formula for F₁(q²) is wrong. The vertex correction should be ignored.
k
~~~~~
/ \
e- ----------------------------------
| \
| ~~~~~~ real photon
| q
X+ -----------------------------------
Convergence when the loop radiates bremsstrahlung. On September 29, 2025 we remarked that the loop integral probably does not have an ultraviolet divergence if the electron inside the loop radiates a real photon. That is because the product gains one more electron propagator. Our analysis above suggests that the probability of this diagram overlaps with the tree level diagram, and we should not add the probability to the scattering of the electron. That is, we should ignore this diagram if we just look at the electron scattering.
If we are interested in bremsstrahlung, then we must analyze if the diagram calculates correctly the effect of the electron mass reduction.
Vacuum polarization
|
| e- ___
| q / \ q
| ~~~~~~ ~~~ ● X+ massive charge
| e+ \____/
|
| virtual pair
|
e-
^ t
|
Polarization P reduces the energy of the electric field, and thus makes the Coulomb force weaker between charges. Another way to measure polarization is the electric displacement D.
The relative permittivity εr ≥ 1. The Coulomb force is weaker in a medium because the electric field energy is reduced by polarization. Polarization happens because it takes the system to a lower energy state. Thus, it is trivial that polarization reduces field energy. By linear polarization we mean that εr is constant regardless of the electric field.
We define superlinear polarization as the case in which
εr(E)
the relative permittivity grows when the electric field |E| grows. Superlinear polarization further reduces the energy in the electric field. This makes the Coulomb force between opposite charges stronger. That is because we can further reduce the field energy by taking the charges closer.
Between charges of the same sign, superlinear polarization reduces the Coulomb force because it reduces the field energy when we take the charges closer to each other.
Superlinear polarization differs from the traditional interpretation of QED. In the traditional thinking, taking charges of the same sign closer to each other would increase the Coulomb repulsion because they would "see" the bare charge of each other.
Which is right: vacuum polarization increases the repulsive force or decreases it? If the electron and the positron were very massive, there would be no vacuum polarization. If we make them light, we increase the "freedom" of the system. Increasing the freedom should take the system to a lower energy state, which decreases Coulomb repulsion. It is very surprising if the traditional QED interpretation is right.
*** WORK IN PROGRESS ***
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