More on polarization of the vacuum and quantum field theory: electroweak symmetry and mass
(Illustration above: by 'shell' I refer to the mean distance of the virtual positrons freed in the vacuum by the intense close in electric field, and the somewhat greater mean distance of the virtual electrons; obviously this is statistical and you not have every virtual electron at one radius and every virtual positron at another smaller radius from the real electron core.)
Quantum field theory implies the core of each real long-lived charge is surrounded by two concentric shells of charged vacuum particles: an inner shell with opposite charge to the core, and an outer one of similar charge to the core. The electric field arrow between the two shells points the other way to that for the charge from the core bare charge, so the latter is shielded. The shielding factor calculated in a previous post in this blog is approximately 137 or 1/alpha.
The physical reason why quarks have fractional charge can be explained very simply indeed. Electric charges are shielded by the polarized vacuum field they create at short distances. If you hypothetically put three electron charges close together so that they all share the same vacuum polarization cloud, the polarization in that cloud will be three times stronger. Hence, the shielding factor for electric charge will be three times greater. So the electric charge you would theoretically expect to get from each of the three electron-sized charges confined in close proximity is equal to: -1/3e. This is the electric charge of the downquark.
In the diagram above, the average distance of charged shells for any given species of vacuum loop charges is illustrated in a simple way. If the electron core couples to a particle in position A, then A will experience a charge 137 times stronger than a particle in position B. This is a simplified picture of the real situation (see recent posts on this blog for some issues with it).
However, it is mathematically correct if distance A corresponds to the upper collision energy cutoff corresponding to Standard Model fundamental force unification energy, and if distance B corresponds to the lower cutoff used in the quantum field theory.
Note that the electron core is not just a static electric charge but also has spin and a dipole magnetic field equal to 1 Bohr magneton in Dirac's quantum field theory. According to string theory, the core would be a closed string like an oscillating loop. However, string theory has no evidence and is not even wrong. If light contains oscillating positive to negative electric field energy, then you might split a gamma ray photon into two gravitationally trapped charges spinning around in small loops. Taking account of the Poynting vector for electromagnetic radiation, this predicts the spherically symmetrical radial electric field, dipole magnetic field, spin, mass and quantum field theory corrections to the dipole magnetic field of an electron.
In the diagram above, assume the real bare particle core can pair up with a transient virtual positron located in the shell position A. The increase in the magnetic moment which results for leptons is reduced by the 1/alpha or 137 factor due to shielding from the virtual positron's own polarization zone, and is also reduced by a factor of 2Pi because the two particles are aligned with opposite spins: the force gauge bosons being exchanged between them hit the spinning particles on the edges, which have a side-on length which is 2Pi times smaller than the full circumference of the particle. To give a real world example, it is well known that by merely spinning a missile about its axis you reduce the exposure of the skin of the missile to weapons by a factor of Pi. This is because the exposure is measured in energy deposit per unit area, and this exposed area is obviously decreased by a factor of Pi if the missile is spinning quickly. For an electron, the spin is half integer, so like a Mobius strip (paper loop with half a turn), you have to rotate 720 degrees (not 360) to complete a 'rotation' back to the starting point. Therefore the effective exposure reduction for a spinning electron is 2Pi, rather than Pi.
Hence by combining the polarization shielding factor with the spin coupling factor, we can account for the fact that the lepton magnetic moment increase due to this effect is approximately 1/(2.Pi x 137) = alpha/(2.Pi) added on to the 1 Bohr magneton of the bare real electron. This gives the 1.0116 Bohr magnetons result for leptons.
The Z boson of electroweak theory is unique because it has rest mass despite being an uncharged fundamental particle! You can easily see how charged particles acquire mass (by attracting a cloud of vacuum charges, which mire them, creating inertia and a response to the spacetime fabric background field which is gravity). But how does a non-composite neutral particle, the Z, acquire mass? This is essentially the question of electroweak symmetry breaking at low energy. The Z is related to the photon, but is different in that it has rest mass and therefore has a limited range, at least below electroweak symmetry breaking energy.
Z mass model: a vacuum particle with the mass of the electroweak neutral gauge boson (Z) semi-empirically predicts all the masses in the Standard Model. You use data for a few particles to formulate the model, but then it predicts everything else, including making many extra checkable predictions! Here's how. If the particle is at position A in the model above, it is inside the polarization range of the electron, but there is still its own polarization shell separating it from the real electron core. Because of the shielding of its own shell of vacuum polarization and from the spin of the electron core, the mass it gives the core is equal to Mz/(2.Pi x 137) =
Mzalpha/(2.Pi) ~ 105.7 MeV. Hence the muon!
Next, consider the lower energy state where the mass is at position B in the diagram above. In that case, the coupling between the central core charge and the mass at B is reduced by the additional distance (which empirically is a factor of ~1.5 reduction) and also the 137 or 1/alpha polarization attenuation factor. Hence
Mz(alpha)2/(1.5 x 2.Pi) ~ 0.51 MeV. Hence the electron!
Generalizing, for n real charge cores (such as a bare lepton or 2-3 bare quarks), and N masses of Z boson mass at position A (a high energy, high mass state), the formula for predicting the observable mass of the elementary particle is: Men(N+1)/(2.alpha)
This does make predictions! It is based on known facts of polarization in the vacuum, the details of which have evidence from many experiments. It has more physics and checkable tests going for it than the periodic table of chemistry had first proposed from sheer empirical association by Newlands and Mendeleev, which today would doubtless be suppressed sneeringly as 'numerology' (the reason for the periodic table had to await the discovery of quantum mechanics). It produces a kind of periodic table of elementary particle masses which are directly comparable to measured data. This fact-guided methodology of doing physics is in stark contrast to stringy, abject, useless extra dimensional speculation.
The stringy model doesn't predict the force mechanisms, strengths, particle masses, and cosmological results which it suppressed with censorship.
But for a comparison of the above heuristic ideas with those quantum field theory between the Tomonaga-Feynman-Schwinger quantum field theory renormalized calculation of the magnetic field increase for an electron due to the vacuum (which Dirac and Feynman, as well as others like Dr Chris Oakley, raise objects to on mathematical grounds, as being incomplete/fiddled) see:
'This paper is based on the elementary remark that the extraction of gauge invariant results from a formally gauge invariant theory is ensured if one employs methods of solution that involve only gauge covariant quantities. We illustrate this statement in connection with the problem of vacuum polarization by a prescribed electromagnetic field. The vacuum current of a charged Dirac field, which can be expressed in terms of the Green's function of that field, implies an addition to the action integral of the electromagnetic field. Now these quantities can be related to the dynamical properties of a "particle" with space-time coordinates that depend upon a proper-time parameter. The proper-time equations of motion involve only electromagnetic field strengths, and provide a suitable gauge invariant basis for treating problems. Rigorous solutions of the equations of motion can be obtained for a constant field, and for a plane wave field. A renormalization of field strength and charge, applied to the modified lagrange function for constant fields, yields a finite, gauge invariant result which implies nonlinear properties for the electromagnetic field in the vacuum. The contribution of a zero spin charged field is also stated. After the same field strength renormalization, the modified physical quantities describing a plane wave in the vacuum reduce to just those of the maxwell field; there are no nonlinear phenomena for a single plane wave, of arbitrary strength and spectral composition. The results obtained for constant (that is, slowly varying fields), are then applied to treat the two-photon disintegration of a spin zero neutral meson arising from the polarization of the proton vacuum. We obtain approximate, gauge invariant expressions for the effective interaction between the meson and the electromagnetic field, in which the nuclear coupling may be scalar, pseudoscalar, or pseudovector in nature. The direct verification of equivalence between the pseudoscalar and pseudovector interactions only requires a proper statement of the limiting processes involved. For arbitrarily varying fields, perturbation methods can be applied to the equations of motion, as discussed in Appendix A, or one can employ an expansion in powers of the potential vector. The latter automatically yields gauge invariant results, provided only that the proper-time integration is reserved to the last. This indicates that the significant aspect of the proper-time method is its isolation of divergences in integrals with respect to the proper-time parameter, which is independent of the coordinate system and of the gauge. The connection between the proper-time method and the technique of "invariant regularization" is discussed. Incidentally, the probability of actual pair creation is obtained from the imaginary part of the electromagnetic field action integral. Finally, as an application of the Green's function for a constant field, we construct the mass operator of an electron in a weak, homogeneous external field, and derive the additional spin magnetic moment of α/2π magnetons by means of a perturbation calculation in which proper-mass plays the customary role of energy.'