(Zoltan Harman, 2014)
The mathematical limit of the vacuum polarization loop integral is not well-defined if we integrate over all momenta p. The contributions of various values of p may cancel each other out: the limit may depend on the order of integration.
Or the limit may be infinite for all integration orders.
By the integration order we mean the order in which the contributions of various p are summed to the integral.
The integral can be seen as an infinite series where the terms have various complex values. Some values cancel each other out. The limit of the series may depend on the ordering of the terms. The limit may be infinite for some or all orderings.
The basic idea of Pauli-Villars:
1. We guess that unknown laws of nature suppress the contribution of high |p|. Nature imposes a smooth cutoff for high |p|.
2. We guess that the smooth cutoff is like the one of Pauli-Villars, for some large Λ. Or if the integral converges for arbitrarily large Λ, then we guess that the "right" integration order is the order given by Pauli-Villars.
Kenneth Wilson's scale hypothesis
According to a hypothesis of Kenneth Wilson, microscopic behavior of Nature imposes a cutoff on the ill-defined integral.
There exists some correct microscopic theory T of nature. Vacuum polarization masks the low-level theory T. In a laboratory we measure the sum of the "true" electric potential of an electron (in T) and the masking effect caused by vacuum polarization.
What we measure is the familiar Coulomb potential. One can then speculate what the "true" electric potential of an electron might be in T, behind the mask of vacuum polarization.
One hypothesis is that the coupling constant is stronger for high momenta in the vacuum polarization loop. This is called a running coupling constant. With the running, we can get the vacuum polarization loop integral to stay constant for different large cutoffs Λ' and Λ.
Assume that the unmasked electric potential is much stronger than the measured potential. There is an unknown large Λ, up to which we have to integrate the vacuum loop so that we get the measured Coulomb potential.
If we just integrate the loop over Λ' < |p| < Λ, does the limited integral carry some meaning?
Could the limited integral give us some intermediate low-level theory for an "energy scale" Λ'?
If we assume the running coupling constant hypothesis, then the integrals from 0 up to Λ' and Λ have the same value. The limited integral from Λ' to Λ is zero.
To get concreteness to all this, we should be able to observe scattering so that only virtual pairs whose momentum |p| is in a certain range, need to be summed over the low-level theory T.
But in a Feynman diagram, we need to sum over all |p|.
Suppose that in a collider with very high energies, scattering fails to conform to the simplest Feynman diagram. There might be more scattering than predicted. Then one may speculate that the coupling constant is larger for large energies.
Let us find out what experimental evidence there is about a running coupling constant in QED.
Classically, a surprisingly large cross section for Coulomb scattering with high energies implies that the electric potential is steeper than 1 / r.
The OPAL Collaboration (2006) measured the coupling constant at 1.81 GeV^2 and 6.07 GeV^2.
In literature, from the LEP there is robust evidence that the cross section really is larger than expected if we would assume that the coupling constant stays constant for large energies.
In the link, (Michiel Botje, 2013), the running is explained by dividing the diverging integral into one over 1 / z and a converging part which depends on the momentum q transferred in Bhabha scattering. The converging part does change when |q| is raised. The integral over 1 / z is made finite by imposing a cutoff M for large momenta.
We need to think more about this. What kind of vacuum polarization processes can conserve the speed of the center of mass? Can those processes explain the running of the coupling constant in QED?
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