The final theory... (an incomplete post)

http://arxiv.org/abs/hep-th/0510040, Luis Alvarez-Gaume, Miguel A. Vazquez-Mozo, Introductory Lectures on Quantum Field Theory, equation 7.17, p70, states:

[observable electric charge of electron] = [charge observed at low energy, ~1/137 in dimensionless QFT charge units].[ 1 + {log_e(M/m)²(1/137)/(12Pi²)}] ~ 1/128,

for electron mass m = 0.511 MeV, Z_o boson mass M = 92,000 MeV (for non-mathematical readers, such dimensionless ratios lime (M/m)ⁿ can use any units you like, because so long as the same units throughout, any unit conversion factors will cancel out and you get the same answer; hence you can use kg, J, eV, MeV, GeV, TeV, whatever for the rest mass-energy so long as you use it throughout the ratio, obviously don't use 0.511 MeV for electron and 92 GeV for Z_o in these mixed units).

The equation above is false! It gives 1/136.9, a mere 0.07% increase in charge of the electron for collisions at an energy equal to the rest mass energy of the Z_o, 92 GeV. It states it gives 1/128, a 7% increase. What's wrong?

http://www.math.uab.edu/hainzl/vacuum.pdf on page 6 (at the top) ["Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation" by Christian Hainzl, Mathieu Lewin & Eric Sere}, gives the formula:

[observable electric charge of electron] = [charge observed at low energy, ~1/137 in dimensionless QFT charge units].[ 1 + {log_e(M/m)²(1/137)/(3Pi/2)}] ~ 1/132,

where M/m is ratio of (energy of interest)/(low (infrared cutoff) energy), gives a 3.7% increase in electric charge at 92 GeV. Still not the required 7% increase that QFT is supposed to give to predict Levine's experimental results of 1997, but much closer than the other paper. I don't doubt that QFT can give the right equation, I'm just commenting that the pages and pages of technically trivial drivel on the internet on this subject - both at arXiv.org and other academic places - doesn't give consistent, useful results.

Now why is there a difference? Why has the geometric part of the multiplication factor for the logarithmic term changed from 1/(12Pi²) to 1/(3Pi/2)? Could it be that the quantum field theorists are complete crackpots?

I like the conceptual basis of heuristic QFT:

'If the forces really are unified at some high energy scale, then we would expect the electromagnetic, weak and strong forces to have the same strength at this unification scale.' - http://hep-www.colorado.edu/~nlc/SUSY_Wagner/susy/node1.html
‘All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs. The virtual fermions with charges opposite to the bare charge will be, on average, closer to the bare charge than those virtual particles of like sign. Thus, at large distances, we observe a reduced bare charge due to this screening effect.’ – I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424.

'... we find that the electromagnetic coupling grows with energy. This can be explained heuristically by remembering that the effect of the polarization of the vacuum ... amounts to the creation of a plethora of [virtual] electron-positron pairs around the location of the charge. These virtual pairs behave as dipoles that, as in a dielectric medium, tend to screen this [bare core] charge, decreasing its value at long distances (i.e. lower energies).' - arxiv hep-th/0510040, p 71.
Equation 7.17 on page 70 of the paper above gives the basic law for the effect of polarization for collisions at different energies. The topic is connected to SUSY (supersymmetry) because electric charge increases 7% from 1/137 to 1/128 as energy of collision increases to 92 GeV.

Over the same range, the color charge of a quark falls in strength, and at higher energies this leads to asymptotic freedom of quarks at short ranges (inside nucleons for instance). It is a good guess that the short-range increase in strong force charge is associated with the fall of electromagnetic charge at the same range (suggesting a conservation of total field energy of gauge bosons).
At extremely high energy SUSY theories predict that you can completely break through the vacuum polarization shield and the same charge strength value.
There are lots of ways of approaching this speculative problem without SUSY. The electric charge rises by 7% from 1/137 to 1/128.5 when going from low energy to 92 GeV. This is due to vacuum polarisation being penetrated. Charge is determined by exchange radiation, so where is the extra gauge boson energy coming from? Is gauge boson energy conserved so it simply gets shared between the different force fields?
That would produce unification because the total sum of field energies would be constant, and the increase in electromagnetic observed charge would be compensated for by a fall in the observed charges of other fields until they unified when the polarisation shield was completely broken down.

According to http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html, the strong force strength given by quantum chromodynamics varies with energy as

alpha (strong) = 12Pi/{(33 - 2n_f)ln[(E/L)²]},

note that strong force charge strength, alpha, at low energy is usually stated to be ~1 (i.e., ~137 times the electromagnetic force at low energy, which has alpha = 1/137),

where n_f is the number of quarks active in pair production (up to 6), and L is the experimentally determined cutoff of about 200 MeV, required to give the right answers.

Cutoffs are extremely important in renormalisation in all quantum field theories, electrodynamics and nuclear forces (weak and strong). If you integrate a function proportional to 1/r over all distances r from a given value out to infinity, the result is proportional to the logarithm of infinity, which equals infinity. Any number multiplied by infinity is not a physical result of a useful kind. The fall off as 1/r is characteristic of the field energy potential associated with inverse square law forces(unlike the force inverse square law of the field, proportional to 1/r²).

To make physics work, you have to fiddle with the logarithmic result of the integration using a 'cutoff' which is literally that. You cut off the integration at a value chosen to make the calculation give the right answer. Basically this amounts to fiddling about with the theory to make it work. However the problem is real, not a mathematical invention. So the cutoff is real too. What is interesting is physically interpreting what the cutoff energy signifies for the underlying energy.

Two types of cutoff are referred to commonly, minimum energy ('infrared cutoff') and maximum energy ('ultraviolet cutoff'). When you integrate the potential of a field 1/r over distance r, you get a result proportional to: (log B) - (log A) = log(B/A), where A and B are the infrared and ultraviolet cutoff distances (distances can easily be related to energy for particle scattering in physics). Short distances correspond to high energies, and long distances to low energies, in collisions (i.e., in particle accelerators).

Wikipedia states:

'In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy approaching infinity, or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a finite answer is a potential problem.

'The ultraviolet (UV) divergences are often unphysical effects that can be removed by regularization and renormalization. If they cannot be removed, they imply that the theory is not perturbatively well-defined at very short distances.

'The classic example of an ultraviolet divergence, and the scenario from which the name arises, occurs when one attempts to calculate the amount of radiation emitted by a black body using classical mechanics. As the wavelengths become shorter, there are more possible modes for the object to vibrate in. The calculation results in the object supposedly emitting infinite amounts of energy. This problem, which was known as the ultraviolet catastrophe, is addressed by quantum mechanics, which limits the amount of radiation emitted at short wavelengths by requiring that short-wavelength light exist in larger energy packets.

'The dependence of physical quantities on the chosen cutoffs (especially the ultraviolet cutoffs) is the main focus of the theory of renormalization group.'

Wikipedia also has an entry on the short ranged asymptotic freedom of quarks within nucleons, which claims the mechanism is: 'The Landau pole behavior of QED is a consequence of screening by virtual charged particle-antiparticle pairs, such as electron-positron pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases.

'In QCD, the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Roughly speaking, each gluon carries both a color charge and an anti-color charge. The net effect of polarization of virtual gluons in the vacuum is not to screen the field, but to augment it and affect its color. This is sometimes called antiscreening. Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance.

'Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors.'

This is not as helpful as http://arxiv.org/abs/hep-th/0510040, which states:

'... for real QCD (N_C = 3, N_f = 6) ... we have ... for a theory that is weakly coupled at an energy scale mu(o) the coupling constant decreases as the energy increases mu -> infinity. This explains the apparent freedom of quarks inside the hadrons [example: nucleons]: when the quarks are very close together their effective color charge tend to zero. This phenomenon is called asymptotic freedom.

'Asymptotic free theories displac a behavior that is opposite to that found above in QED. At high energies their coupling constant approaches zero whereas at low energies they become strongly coupled (infrared slavery). These features are at the heart of the success of QCD as a theory of strong interactions, since this is exactly the type of behavior found in quarks...'