MARCH 11, 2005: On a new page, I have added a brief PDF file containing a first attempt to do A Gordon-Like Magnetic Moment Decomposition for Fermion Mass, which I believe will ultimately be necessary to obtain an exact fit with the experimental mass data.  I was motivated in this direction by a couple of folks who have been playing around with Fermion masses and magnetic moments on http://www.physicsforums.com/showthread.php?t=46055&page=7&pp=15.  In particular, they found that certain lepton magnetic moments came very close to the ratio of certain lepton masses divided by certain electroweak bosons, but were worried that their mass formulas did not have any perturbative expansion as is expected for the magnetic moments.  They observed that the draft result in this author’s Fermion Mass paper restored the missing perturbation loops to their theory.  And their work suggested to me that any exact prediction of the Fermion masses is likely to require consideration of generating some part of the mass through magnetic moments, and that magnetic moment may in fact be what is needed to generate a small, non-zero mass for the neutrino which otherwise cannot acquire mass – at least through what has been laid out so far – because it lacks an electric charge, i.e., its Q=0.  

 

I would like to see if this decomposition can help these folks complete their efforts to characterize the magnetic moment, and whether it can be used to help me complete the Fermion mass fit being attempted here.

 

FEBRUARY 21, 2005:  I’ve believe it may be possible to characterize the masses in each lepton generation to within 1% based on strictly theoretical parameters, based on the first four loops of a perturbative expansion.  More importantly, this appears to confirm the hypothesis that each generation is defined by the loops that appear between the observer and the vacuum, i.e., by how the particle is “dressed” by the scalar bosons with which it interacts.  DETAILS HERE:

 

FEBRUARY 18, 2005: The paper on electrodynamics and the spacetime vacuum is updated.  The third rank object *jiτσν has many features of a baryon.  I believe now, that the proper interpretation of this object is as a baryon.  Details are on the other page. 

 

Also, I have thought quite a bit about it, and I am abandoning the neutrino mass prediction.  Others have opined, and I agree, that the perturbation loop interpretation is the reason for the e4 term, not anything having to do with lepton number.  The electron results are unchanged, and the latest information is on this page.  I am now in a collaboration to update the paper and prepare it for publication.

Fermion Mass Revelation in Electroweak Interactions (Full Paper PDF)

ABSTRACT: It is demonstrated how to reveal non-zero Fermion masses during electroweak symmetry breaking.  The sum over three generations of the observed electron masses at low probe energy is related to the weak mixing angle according to

Mass (electron) + Mass (muon) + Mass (tau) = v·[e2 + e4]·[1+e·sinθW]1 See possible updated formula for running e^2

where v= 246.220 GeV is the Fermi vacuum expectation, e2=1/137.036 is the running electromagnetic coupling at low probe energy, and Sin2θW  is the weak mixing angle.  Observed electron masses over the range of uncertainty in Mass (electron) + Mass (muon) + Mass (tau) =  1883.16  MeVhttp://pdg.lbl.gov/2004/tables/lxxx.pdf, are used to predict a value for the weak mixing angle of:

 .223 149 <  Sin2θW < .226 523,

which does not appear to be ruled out by experimental observations of Sin2θW.

 

1 Alternatively,  =

 

2 The e2 couples through electrostatic charge Qelectron=-1; Qneutrino=0.  The e4 couples through lepton number L=1 for both electron and neutrino.  The e·sinθW arises from the Higgs scalar.

 

CONTENTS

Motivations, Results, and Overview

1.  A Review of “Vector Minus Axial” Weak Interactions

2.  “Scalar Plus Pseudoscalar” Weak Fermion Masses

3.  Scalar Currents and Bosons in the Weak Fermion Mass Term

4.  Electroweak Step One: Constructing a Parity-Conserving Electromagnetic Interaction

5.  Electroweak Step Two: Charged and Neutral Current Mixing

6.  Interlude: Reconciling μ2>0 with μ2<0 for the Term μ2(φ†φ)

7.  Electroweak Step Three: Spontaneous Symmetry Breaking

8.  Revealing Fermion Mass by Restoring a Parity-Conserving Mass Term

9.  Prediction of Electron Masses to Within 4.58%, Based on the Ground State of the Vacuum

10.  Prediction of Electron Masses to within 0.75%, As Well as the Higgs Mass and Couplings, Based Perturbations of the Vacuum

11.  Prediction of Electron and Neutrino Masses to Within Experimental Error, Based on Lepton Number

 

THEORETICAL OVERVIEW

 

            The mass sum Mass (electron) + Mass (muon) + Mass (tau) in the above formulae is employed, rather than any individual mass, because a careful analysis of bi-unitary transformations on the three-generation mass matrix reveals (see section 9) that one can construct these mass matrixes to leave these mass sums invariant, irrespective of the chosen magnitudes of the three real mixing angles and the one phase that come into play for three generations of Fermion.  In short, these mass sums are mass-matrix-mixing invariants.

 

            The theoretical foundation begins with parity violation in the weak interaction, and in particular, with the proposition that the weak interaction violates parity not only in the current / vector boson interactions, but in the mass term as well.  Now, one would expect that having γ5 Dirac matrices in the mass term would wreak havoc, but that is exactly the point.  In exactly the same way that the standard electroweak model constructs a parity-conserving electromagnetic current and a massless mediating photon out of parity-violating weak and hypercharge interactions, it is demonstrated here how to construct a parity-conserving electroweak mass term out of parity-violating weak and hypercharge mass terms.  For the mass term, the symmetry constraints are even tighter than in the standard model.  At the end of the day in the standard model, the W±μ and the Zμ still mediate parity-violating interactions and it is only the photon Aμ which mediates a parity-conserving interaction.  For the mass term, we allow no γ5 in the end, and this is a very tight symmetry constraint.  After restoring parity to electromagnetic current, mixing neutral currents via sinθW, and breaking symmetry following the usual standard model prescription without deviation, and additionally, after imposing parity conservation on the mass term, the total m(e) + m(mu) + m(tau) above is revealed to be v·e2, which falls just under 5% short of the observed mass and is the dominant term in the electron mass formula recited above.

 

            Then, the question becomes how to find the final 5% of this mass total Mass (electron) + Mass (muon) + Mass (tau).  Now, the first 95% of the mass, v·e2, arises not from Higgs perturbations, but from the ground state of the vacuum.  The scalar particle involved is a massless scalar photon, not a massive Higgs.  When we turn to examine perturbations from the ground state, and start to look at the Higgs, it turns out that a second electroweak scalar boson, which is associated with the I3 weak isospin generator, has a wavefunction which already been normalized through eliminating the γ5 from the mass, to a value e·sinθW.  Coupling of the mass to this particle – which we then associate with the Higgs – turns v·e2 into v·e2[1 + e·sinθW].  This extra factor brings m(e) Mass (electron) + Mass (muon) + Mass (tau) to within 0.75% of what is observed.

 

            For the remaining mass, we note that when the ground state mass term v·e2 is revealed, the couplings for the parity violating weak and hypercharge interactions are forced out of the mass term.  That is, only parity-conserving charges (no γ5) contribute to ground state mass.  So, we need to look for another parity-conserving charge to give this final 0.75% of the mass to the electron.  QCD color is out of the question, because the leptons don't interact strongly.  So, we turn to lepton number and hypothesize that this is a parity-conserving charge with its own coupling gL and massless vector boson(s).  We then ask, what is the magnitude of the gL lepton coupling needed to yield the last 0.75% of the electron mass total Mass (electron) + Mass (muon) + Mass (tau)?  It turns out that gL = e2 = 1/137.036 is precisely the magnitude of gL required to bring the mass prediction within experimental error.  This adds a gL2 = e4 factor to the mass formula, yielding the final result Mass (electron) + Mass (muon) + Mass (tau) =  v·[e2 + e4]·[1+e·sinθW] recited at the outset.  Other than the very desirable result that gL ends up related to the known coupling e2 and is not an independent coupling, it is not clear theoretically why gL turns out to be so close to 1/137.036, though this is likely a clue to something deeper.   THIS FINAL RESULT REGARDING LEPTON NUMBER AND THE NEUTRINO IS NOW SUPERSEDED BY WHAT FOLLOWS:

 

Back to top

 

WHAT THE GENERATIONS REALLY MEAN (REV 3.0) (ALWAYS WORK IN PROGRESS)

 

It appears possible to fit the electron masses very closely to the loops, along the following lines.  It is to be emphasized that this work is still in development, but the closeness of the numbers set forth below, based on about a dozen terms which are all generated by the theoretical foundation in Fermion Mass Revelation in Electroweak Interactions, and which rely only on the Fermi vacuum expectation, the electromagnetic running coupling, and the weak mixing angle, does suggest the possibility with further development of obtaining an exact fit with experiment.  Further, the chances that so few terms would come so close to all three lepton masses, within 1% for each, merely by coincidence seems to be of very low probability:

 

The electron mass is only observed through a “chain” of scalar bosons.  Each time the “a” scalar gets between the electron and the observer, it attenuates the observed mass by a factor of e.  Each time the Higgs gets between the electron and the observer, it attenuates the observed mass by a factor of e sin θW.  These factors result from normalizing the scalar bosons to restore parity conservation to the mass term, see equation (9.38) in section 9 of the paper, as well as (9.41) which shows the underlying form of the mass term Lagrangian for a single scalar photon, with e2 attenuation.

 

Referring to the Feynman diagram below: 

Each generation of electron is characterized by the particular combination of scalar bosons that come between the observer and the “bare” particle.  The leading terms for the electron mass, for example, are all based on four loops between the observer and the bare particle.  The larger-mass tau and muon are observed through less loops in their leading order, and so the bare mass is less attenuated and these are observed to have larger mass.  But, these are all the same underlying particle, merely viewed through different sets of loops.

 

Following is a chart of all perturbation terms for zero to five loops between the mass and the observer, along with their calculated values based on v = 246.22046 GeV, e2= 7.29735256824 x 10-3 and Sin2θW = .23120. http://pdg.lbl.gov/2004/reviews/consrpp.pdf  All value are shown in MeV:

 

# of scalar bosons attenuating mass observation:

 

0                        1                       2                        3                         4                         5                       # of Higgs:   # scalar photons

 

ve                    ve2                     ve3                    ve4                      ve5                        ve6                             0                    5

21033.270            1796.756                153.487                 13.112                     1.120                         .096

 

                             ve2 ·SinθW         ve3 ·SinθW         ve4 ·SinθW            ve5  ·SinθW           ve6 ·SinθW                 1                    4

                          863.940                   73.802                   6.304                       .539                          .0460

 

                                                          ve3 ·Sin2θW        ve4 ·Sin2θW           ve5  ·Sin2θW          ve6 ·Sin2θW               2                    3

                                                    35.486               3.031                  .259                      .022

 

                                                                                    ve4 ·Sin3θW          ve5  ·Sin3θW          ve6 ·Sin3θW               3                    2

                                                                                             1.458                       .125                           .011

 

                                                                                                               ve5  ·Sin4θW          ve6 ·Sin4θW               4                    1

                                                                                                                             .0599                         .005

 

                                                                                                                                    ve6 ·Sin5θW              5                    0

                                                                                                                                     .002

 

 

Now, we will let the data and what we know about the masses determine how we group these terms, and try to see is an infinite series can be set up which satisfactorily fits the electron mass data out to five loops.  We can use each term only once, and we will compare with experimental mass data: http://pdg.lbl.gov/2004/tables/lxxx.pdf

 

In particular, the following assignment comes very close with only the first four loops:

 

ve2 - ve4 - ve4 ·SinθW = 1777.34 MeV       (Observed tau mass is 1776.99 + 0.29 / - 0.26 MeV)

 

ve3 ·SinθW + ve3 ·Sin2θW -ve4 ·Sin2θW -ve4 ·Sin3θW = 104.799 MeV  (Observed muon mass is 105.65839 ± 0.000009 MeV)

 

ve5 - ve5  ·SinθW - ve5  ·Sin2θW +ve5  ·Sin3θW +ve5  ·Sin4θW (e) = .507 MeV (Observed electron mass is .51099892 ± 0.00000004 MeV)

 

Now, the reader may be concerned that we have put some “minus” signs into the above in what appears to be an ad-hoc manner, and that some other choice would lead to different masses.  Yet, we know in perturbation theory that some loops are added and some loops are subtracted, so all we have really done is use the observed experimental masses to guide us in understanding which loops are positive and which are negative.

 

Now, we rewrite the chart in light of the above, that is, we rewrite this to show which loops are positive and which are negative based on the experimental mass data.:

 

# of scalar bosons attenuating mass observation:

 

0                        1                       2                        3                         4                         5                       # of Higgs:   # scalar photons

 

ve                    +ve2 (τ)                 ve3                  -ve4 (τ)                  ve5 (e)                   ve6                       0                    5

21033.270            +1796.756                153.487                 -13.112                     1.120                         .096

 

                             ve2·SinθW            ve3·SinθW (μ)     -ve4·SinθW (τ)      -ve5·SinθW (e)       ve6·SinθW              1                    4

                          863.940                   73.802                   -6.304                       -.539                          .0460

 

                                                          ve3·Sin2θW(μ)      -ve4 ·Sin2θW (μ)   -ve5·Sin2θW (e)      ve6·Sin2θW               2                    3

                                                    35.486               -3.031                  .-259                     .022

 

                                                                                     -ve4 ·Sin3θW (μ)    +ve5·Sin3θW (e)    ve6·Sin3θW               3                    2

                                                                                             -1.458                       +.125                        .011

 

                                                                                                                 +ve5·Sin4θW (e)    ve6·Sin4θW               4                    1

                                                                                                                              +.060                         .005

 

                                                                                                                                      ve6·Sin5θW              5                    0

                                                                                                                                     .002

 

 

Each of the masses can be predicted to within 1% using the loop combinations shown, out to four loops.  Next to each term, we show the particle it is associated with, as well as whether it is added or subtracted.  We have used all terms except ve = 21033.270 MeV, ve2·SinθW = 863.940 MeV, and ve3 = 153.487 MeV.  Now, let us see if these results are amenable to being fitted to a perturbative function involving products of the e and the e·SinθW.  That is, is there a perturbative series which actually yields not only the magnitudes of these loops. but also their signs as needed to fit the data.  If we cannot find a series, then this might be seen as an ad hoc grouping of terms, But, if these is a series which fits, then that gives credibility to an interpretation of each generation as being characterized by the particular set of mass-attenuating loops between the observer and the bare mass.

 

It turns out that the above can indeed be fitted to a series. The expansion which fits this data is written as:

 

v·[+ e - e2 - e3 + e4 + e5 - e6 - e7 + e8. . .]·[1- e·sinθW + e2·sin2θW - e3·sin3θW+ e4·sin4θW + . . .]

 

From this expansion, let us again write out the above chart, but with the signs as given by this expansion:

 

0                        1                       2                        3                            4                         5                       # of Higgs:   # scalar photons

 

ve (γ)               -ve2 (τ)                -ve3 (γ)              +ve4 (τ)                 +ve5 (e)                 -ve6                        0                    5

21033.270            -1796.756                -153.487                 +13.112                     +1.120                        -.096

 

                             -ve2·SinθW (γ)     +ve3·SinθW (μ)    +ve4·SinθW (τ)      -ve5·SinθW (e)      -ve6·SinθW              1                    4

                         -863.940                 +73.802                   +6.304                       -.539                          -.0460

 

                                                           +ve3·Sin2θW(μ)     -ve4 ·Sin2θW (μ)   -ve5·Sin2θW (e)      +ve6·Sin2θW           2                    3

                                                   35.486                 -3.031                  .-259                     +.022

 

                                                                                       -ve4 ·Sin3θW (μ)    +ve5·Sin3θW (e)    +ve6·Sin3θW           3                    2

                                                                                               -1.458                       +.125                        +.011

 

                                                                                                                   +ve5·Sin4θW (e)    -ve6·Sin4θW             4                   1

                                                                                                                                +.060                         -.005

 

                                                                                                                                      -ve6·Sin5θW             5                    0

                                                                                                                                       -.002

 

In the above, we now have:

 

-ve2 + ve4 + ve4 ·SinθW = - 1777.34 MeV       (observed -1776.99 - 0.29 / + 0.26 MeV))

 

ve3·SinθW + ve3 ·Sin2θW -ve4 ·Sin2θW -ve4 ·Sin3θW = 104.799 MeV  (The positive muon mass fits the expansion, observed 105.65839 ± 0.000009 MeV)

 

ve5 - ve5  ·SinθW - ve5  ·Sin2θW +ve5  ·Sin3θW +ve5  ·Sin4θW (e) = .507 MeV (The positive electron mass fits the expansion, observed .51099892 ± 0.00000004 MeV)

 

The tau mass enters this series as a negative number.  This does not mean the mass is negative which would be contrary to antiparticles having positive mass.  It means only that the tau mass is subtracted from the other two masses in this perturbation series.

 

Also, the 5-loop term is given by:

 

-ve6 -ve6·SinθW +ve6·Sin2θW +ve6·Sin3θW -ve6·Sin4θW -ve6·Sin5θW = -.116 MeV

 

If all of the terms in the above were to go into a single particle, that particle would have its mass reduced by .116 MeV.  It is not clear which particle these terms might go into; this would require going to higher precision to fit the data which we shall not do here.

 

Finally, we can deduce the series out to infinite order using 1/1-x = 1 + x + x2 + x3 + x4 . . .  This is given by:

 

 = 18,343.690 MeV

 

and is valid for running e and SinθW out to infinite order.

 

Now, let’s briefly see what we can make of these results.  Referring to the earlier Feynman diagram, apparently, leading loops for the tau electron involve a single scalar photon loop ve2, the three-scalar photon loop ve4, and a two-scalar photon one-Higgs loop ve4·SinθW.  For the muon, the leading loops involve one scalar photon and one Higgs ve3·SinθW, two Higgs ve3 ·Sin2θW, one scalar photon and two Higgs ve4 ·Sin2θW, and three Higgs ve4 ·Sin3θW.

 

The electron’s leading loops are all four-scalar boson loops, and it is interesting that the electron seems to use all five of the four-loop terms, that is, four scalar photons and no Higgs, all the way to no scalar photons and fours Higgs.

 

Nevertheless, through 4 loops, this appears to be the closest fit to date of the electron data to date.  This is based solely on theoretical parameters, and there are many loops still remaining to be carefully considered for an even tighter fit.  The full series above, out to infinite order, subtracts a total of 1.333 MeV from the masses given by the first four loop order; that is, the loop terms of fifth order and higher sum up mover all orders to – 1.333 MeV.

 

Comparing further with experimental data:

 

Most importantly, it appears that the generations may be understood on the basis of shielding by a scalar photon and the Higgs.

 

If you have read the pertinent parts of the paper, it is clear why the terms e and the e·SinθW come into play in determining mass.  I will, however, expand this discussion in the near future to connect these results clearly to the physics underlying the main paper.

 

A key open question, which needs study, is why particular loops dress the electron, the muon, and the tauon.  That is, the experimental values of the masses and the terms developed here seems to suggest a heuristic understanding of the generation based on the loops which dress each lepton.  But the question of how and why particular loops come to be associated with particular particles is an open question for now.  It is planned in an updated draft to explore several possibilities in this regard.