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? No, there exists a classical vertex function
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: is it nonsensical for the vertex function F₁(q²)? The fine structure constant is defined
α = 1 / (4 π ε₀) * e² / (ħ c)
≈ 1/137.
However, in natural units, the fine structure constant is simply e².
Let us increase the charge of the electron by some large factor N, and its mass by a factor N², so that it becomes a macroscopic particle. Then we can track its path in a classical fashion.
We assume that the electron passes X+ at some fixed distance R. In the vertex correction,
α q² / m²
stays constant, since α grows by a factor N², q by N, and m by N².
The formula for the form factor F₁(q²) would claim that the apparent charge of a macroscopic particle would significantly (about 0.1% * q² / me²) depend on the momentum q it absorbs from a large, macroscopic charge X+. Is this nonsensical? Classically, the far electric field of the electron does not have time to react as the electron passes X+. The electron will have somewhat reduced mass, which causes it to go closer to X+ and receive more momentum. But is the effect so large that it could be 0.1% * q² / m²?
If we increase the mass of the electron, then its Compton wavelength will become smaller than its classical radius. Maybe there is a law of nature which prohibits this?
If we take the classical limit by decreasing h, then the vertex correction claims that the apparent charge of an electron varies very much depending on the momentum q it absorbs. Actually, h is set 1 in the formulae. We cannot change it.
The assumption of a fixed distance R when we grow the mass and charge may not be realistic. If the distance grows with the mass, then the correction does go to zero. This could be called a classical limit.
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.
The classical vertex correction
Let us calculate an order of magnitude estimate for the electron in the hydrogen atom. The frequency of the orbit is
6.6 * 10¹⁵ Hz.
The far electric field which does not have time to take part in the scattering (= orbit) is
1.5 * 10⁻¹⁶ s * c
= 4.5 * 10⁻⁸ m
= r
away. The ratio
Δ = re / r ≈ 5 * 10⁻⁸
tells us how much the mass of the electron is reduced.
^
● / proton
e- • --------
Let us denote by R = 1 the Bohr radius. As the electron passes past the proton, it receives an impulse which accelerates it the distance ~ 1 up in the diagram.
If the mass of the electron is reduced by some small fraction Δ, then the electron passes slightly closer to the proton, say,
Δ / 4.
The cross section of the scattering grows because of this, by a factor Δ / 2:
Δ / 2 ~ 2.5 * 10⁻⁸.
The order of magnitude is the same as in the QED vertex correction.
Let us study how the classical vertex correction depends on q. If we make R = 2, then q is halved. The mass reduction Δ is halved because the time to go past the proton is double.
The upward force in the diagram is 1/4 but the time to go past the proton is double. The acceleration upward still moves the electron the distance 1 upward.
The effect on the scattering is 1/4, because Δ is halved and R is doubled. This agrees with the QED vertex correction.
Hypothesis. The "missing part" of the vertex correction integral, which cannot escape as bremsstrahlung, is "detached" from the electron during the scattering, and reduces the effective mass of the electron. This, in the classical way, increases the scattering amplitude of the electron.
Question. How can this classical effect in the Feynman integral depend on the Planck constant h?
If literature always sets h = 1 in the calculations, then the formulae above contain a hidden factor h. Then the value does not depend on h, after all.
The QED vacuum polarization for small |q|has a roughly similar magnitude as the vertex correction. Why?
Hypothesis 2. The QED vertex correction is the classical effect. We have been suspecting this in our blog for many years.
If Hypothesis 2 is true, then the renormalization in the vertex correction is what is needed to make the calculation correct and classical. It is not ad hoc, but is mandatory.
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.
The classical limit. Peskin and Schroeder (1995) give:
Recall that in the metric signature (+ - - -), q² < 0. Let us then grow e by a large factor N and m by a factor N². For small |q|,
Π₂(q²) ~ e² q² / m²,
which is
~ e⁴ / m²
if the electron passes at some fixed distance R from X+. The value of Π₂ does not change. The correction will stay reasonably large.
For small momenta |q|, the vacuum polarization correction is equivalent to the Coulomb potential correction term above. The term is called the Uehling potential, and it makes the potential pit deeper.
For large momenta |q|, the coupling constant grows by the formula above. There, A = exp(5/3).
If |q²| << m², then the integral for Π₂(q²) - Π₂(0) looks much like the vertex correction, and probably does not contain h as a factor. That is, vacuum polarization might be a "classical" effect for small |q|. But what classical effect is it?
If the electric field tries to hit the Dirac field in order to create a pair and reduce the energy of the electric field, this does not need to depend on the Planck constant h. The energy to create the pair is 2 me c², which does not contain the Planck constant.
The Planck constant is involved in the energy and wavelength of real particles. A transient hit to a field does not create real particles, and it might be that we do not need to bother about the value of the Planck constant.
In our favorite model, the rubber membrane and the sharp hammer model, the hit produces various transient waves. If some of them would escape, then in the quantum description, we would need to worry about the fact that the energy is h f. But transient waves may have a lot of freedom to be whatever they like. That would explain the absence of h.
In this blog we have remarked that momentum transfers are not quantized. It may be that most transient phenomena are not quantized.
Hypothesis 3. Vacuum polarization for small |q| is a phenomenon of the "classical" Dirac field and the electromagnetic field.
Conclusions
We once again stressed that Feynman diagrams may calculate overlapping classical probabilities. The infrared divergence of bremsstrahlung is a prime case: the electron always sends an infinite number of real photons.
We observed that the classical vertex correction, which is due to the far field of the electron not following instantaneously the electron, may be the correct electric form factor in QED. In QED we see various formulae for the electric form factor F₁(q²), but they always depend on the "photon mass", which is used to cut off the infrared divergence. Thus, we do not know what researchers suggest that F₁(q²) should be. Anyway, the classical vertex correction is the best bet, and satisfies the classical limit.



















