δV(r)
for the low-momentum vacuum polarization case, such that the Fourier transform of the potential is the vacuum polarization correction to the photon propagator (section 7.5, page 254).
The potential goes to zero exponentially fast,
δV(r) ~ exp(-2 m r),
when r grows larger than 1/2 of the reduced Compton wavelength of the electron.
Energy non-conservation
We complained in our August 7, 2021 post that a running coupling constant breaks the classical limit and energy conservation. If a force depends on the exchanged momentum q, then the force depends on the state of the motion of the particle, for example, its speed v. Such a force usually is not conservative. Think about drag in water, for example.
A conservative force is often a result from a potential which only depends on the position of the particle.
The Fourier transform of most potentials does not describe the scattering cross sections of the potential
The vacuum polarization correction
Π_μν(q)
to the photon propagator depends on the exchanged momentum q. The correction is used to calculate scattering cross sections.
But Peskin and Schroeder define a radial (Uehling) potential, which produces a conservative force field. What is going on?
The Fourier transform of the Coulomb potential 1 / r describes perfectly the classical scattering caused by the potential. The photon propagator gives the exact right classical cross sections. The 1 / r potential is a special case.
Other potentials do not necessarily behave that nicely. For example, the Fourier transform of a Dirac delta potential is a constant function. The Fourier transform does not describe the classical scattering caused by the deep and narrow potential well. The scattering cross section is infinitesimal, but that is not reflected by the Fourier transform.
According to the vacuum polarization matrix
Π_μν(q),
the corrected photon propagator for small |q| is
1 / q² + constant,
or
1 / q² * (1 + q² * constant).
The correction may be significant even for quite small |q|.
However, the correction potential δV(r) goes to zero exponentially fast with r. For a small momentum exchange|q|, it is essentially zero. The corrected potential does not classically reproduce the scattering cross sections described by the corrected photon propagator.
The Schrödinger equation with the corrected potential does not reproduce the cross sections described by the corrected propagator, either.
Does the Feynman diagram calculate scattering cross sections or does it calculate the Fourier transform of a corrected potential?
We have been believing that a Feynman diagram calculates probability amplitudes for scattering.
But literature thinks that in the case of vacuum polarization, it calculates the Fourier transform of a corrected potential, which is a very different thing, as we explained above. Energy conservation and the classical limit are restored if we interpret that it calculates a potential, not scattering cross sections.
Energy conservation is very important. It makes sense to respect it.
Question. What is the corrected potential in the high-momentum q case? Does that potential reproduce (approximately) the scattering which is described by Π_μν(q)?
Question. How to interpret Feynman diagrams in the cases where they do not calculate probability amplitudes for scattering? Should we start using non-perturbative ideas? The momentum exchange q should be split into many smaller parts?
Question. If we correct the potential in such a way that it really reproduces the scattering amplitudes of the Feynman vacuum polarization diagram, does that solve the proton radius puzzle? The change might be around 1 / 1,000.
A possible solution to the energy conservation problem: the extra force is a dynamic process which cannot be reduced to a correction term in the Coulomb potential
If we enforce energy conservation separately for each encounter of the charged particles, then there is no energy conservation problem. Energy conservation is enforced in Feynman diagrams.
Then it is possible that the extra force does depend on |q| and not directly on the separation r of the particles. Then the muon in muonic hydrogen feels a different extra force at a distance r than the electron in ordinary hydrogen at the same distance.
In the standard calculation of the vacuum polarization contribution to the Lamb shift in ordinary hydrogen, it is assumed that the correction to the potential is a Dirac delta function. No assumption is made about the detailed form of the potential. It is just a pit very close to the proton.
In muonic hydrogen, the exact form of the potential is relevant because the Bohr radius of muonic hydrogen is only 2.6 * 10⁻¹³ m, which is just over half of the reduced Compton wavelength of the electron.
Thus, it might be that the Feynman diagram does calculate scattering probability amplitudes, and it is an error to interpret it as a way to calculate the Fourier transform of a static corrected Coulomb potential which is the same for all different cases and particles. It is a dynamic process when an electron or a muon encounters a proton. The process does conserve energy, but it does not define a single static force field in space.
We solved the proton radius puzzle?
The observation of the previous section may solve the proton radius puzzle. The vacuum polarization contribution in muonic hydrogen was incorrectly calculated. That is the reason why it gives a wrong value for the proton radius.
The correction potential, which goes exponentially to zero for large r, is a wrong approximation for large r. It should be replaced with a potential which approaches zero slower for large r.
We still need to check if other methods of measuring the proton radius assume anything about a fixed, corrected Coulomb potential, or do they estimate the radius using scattering amplitudes.
A very crude calculation shows that using a potential which better reflects scattering amplitudes may affect the vacuum polarization correction up to 10%. We only need to explain a difference of 0.15%.
In the link is the radial probability density for the 2s and 2p orbitals. If we just fix the potential at > 5 Bohr radii to reflect scattering amplitudes, then the effect is to increase the Lamb shift by about
1,000 eV * 1 / 100,000 * 0.05 = 0.5 meV,
while we should increase it by 0.3 meV to explain the measurement by Pohl et al.
The classical limit problem
If we keep an electron close to a proton for a long time, then the exchanged momentum q may be very large over a long period of time. We do not believe that the correction for the Coulomb force should depend on q in such a case. Should we split q in smaller pieces, and if yes, how?
A Feynman diagram calculates the dynamics of a short encounter, a collision of particles. How should we calculate long lasting or static events?
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