Natural Science Forum / Physics / Particle Physics / February 2005

For a number of years, I have been doing independent research into the
question of why the Fermions have the masses they have.  During the past two
months I believe I have made a breakthrough with respect to revealing and
predicting the lepton masses based squarely on the standard electroweak
model.  I will try in this Email to very briefly summarize my results.
Please let me know if you would like me to Email you two papers further
detailing these results.  I start with the experimental predictions and work
back to the underlying theory.

If you take the sum of the electron masses over the three observed
generations, this sum turns out to be given within experimental error by
m(e) + m(mu) + m(tau) = v[e^2 + e^4][1 + e sin thetaW], where e^2 =
1/137.036 is the electromagnetic coupling at low probe energy, v = 246,220
MeV is the Fermi vacuum, thetaW is the weak mixing angle, and m(e) + m(mu) +
m(tau) = 1883.16 +0.29 / -0.26 MeV is the experimentally-observed mass total
for the three electrons.  The Fermi vacuum v and electromagnetic coupling e
are known with fair precision; m(tau) which is the error-controlling term in
the mass and sin thetaW are less-accurately known.  If you plug this mass
range into this formula (get out a calculator and try it), it is simple to
see that this mass range predicts a mixing angle of .223 149 < sin^2 thetaW
< .226 523, which seems to be right in the middle of what people have found
and actually straddles the two extremes of this NuTeV anomaly that has
received some attention these days.

If one takes the sum of the neutrino masses over the three generations --  
which are found in my research to be non-zero -- this sum turns out to be
predicted by m(e-neutrino) + m(mu-neutrino) + m(tau-neutrino) = v[e^4][1 + e
sin thetaW].  This is the same formula as for the electrons, but with the
e^2 removed.  This is because e^2 couples to the charge Q=-1 of the
electron, and the neutrinos have no charge.  The e^4 couples to lepton
number, which is L=1 for both the electron and the neutrino, so this is in
both formulae.  The range of 223 149 < sin^2 thetaW < .226 523 set out in
the last paragraph leads to the prediction that m(e-neutrino) +
m(mu-neutrino) + m(tau-neutrino) = 13.64 MeV, and the particle databases
indicate that the upper bound on the tau neutrino mass is m(tau-neutrino) <
18.2 MeV, so this prediction is also within experimental uncertainty.
Because the mu-neutrino is experimentally limited to be < 0.19 MeV and the
electron neutrino is negligible, this leads to a second prediction that
13.45 MeV < m(tau-neutrino) < 13.65 MeV.  It is my hope that this result can
aid in the hunt for a neutrino mass, by setting out a narrow band in which
to look for this mass.  And, once m(tau-neutrino) is pinned down, we will
know where to look for m(mu-neutrino), and after that, for m(e-neutrino).

The theoretical foundation for this begins with parity violation in the weak
interaction.  I start with what at first seems an odd idea, that the weak
interaction violates parity not only in the current / vector boson
interactions, but in the mass term as well.  Now, you will immediately
realize that having gamma-5 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, I am able to construct a parity conserving
electroweak mass term out of parity-violating weak and hypercharge mass
terms.  But for the mass term, the symmetry constraints are even tighter.
At the end of the day in the standard model, the W+/- and the Z still
mediate parity-violating interactions and it is only the photon A which
mediates a parity-conserving interaction.  For the mass term, getting rid of
all the gamma-5 ends up very tightly constraining the mass term.  After
restoring parity to electromagnetic current, mixing neutral currents via sin
theta-W, and breaking symmetry following the usual standard model
prescription without deviation, and additionally imposing parity
conservation on the mass term, the total m(e) + m(mu) + m(tau) above is
revealed to be ve^2, which falls just under 5% short of the observed mass,
as you can readily calculate. This term, ve^2, you will note, is dominant
term in the electron formula recited above.

Then, the question becomes how to find the final 5% of this mass total.
Now, this first 95% 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, which is a w3 associated with the I3 weak isospin
generator, has a wavefunction which already been normalized through symmetry
breaking and eliminating the gamma-5 to a value e sin thetaW.  Coupling of
the mass to this particle -- which we then associate with the Higgs, turns
ve^2 into v[e^2][1 + e sin thetaW].  This extra factor brings m(e) + m(mu) +
m(tau) to within 0.75% of what is observed, which again, is easy to
calculate.

For the remaining mass, looking back on how we got to this point, we note
that when the ground state mass term e^2 was revealed, the couplings for the
parity violating weak and hypercharge interactions were forced out of the
mass term.  That is, only parity-conserving charges (no gamma-5) 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 m(e) + m(mu) + m(tau)?  It turns out that
g-L = e^2 = 1/137.036 is precisely the magnitude of g-L required to bring
the mass prediction within experimental error.  This adds a gl^2 = e^4
factor to the mass formula, yielding the final result m(e) + m(mu) + m(tau)
= v[e^2 + e^4][1 + e sin thetaW] recited at the outset.

Now, in all candor, I have no idea at this point, from a theoretical
viewpoint, why g-L = e^2 = 1/137.036 turns out to be exactly what is needed
to predict the mass total within experimental error, though it is clearly
highly desirable that g-L turns out to be related to a known coupling rather
than an independent coupling.  I suspect there is a deep clue in this of
some sort wider unification that at the very least will incorporate lepton
number formally and explicitly into the electroweak scheme, but I have no
idea right now how to decipher it.

In any event, the addition of the e^4 factor for the electron completes the
formula I recited at the outset, and brings the electron mass total into the
range of experimental uncertainty.  This e^4 also implies that the neutrino
mass total is v[e^4][1 + e sin thetaW], also recited above.  From this, we
arrive at the 13.45 MeV < m(tau-neutrino) < 13.65 MeV prediction.

A few other items are worth noting: because the Higgs becomes associated
with a w3 scalar wavefunction, the Higgs mass is the same as that of this
w3, and this w3 has the same mass as W+/- in the standard theory,  So, mass
(Higgs) = mass (W+/+) is another prediction.  I know the Higgs predictions
have been upwardly-revised recently, but I also know that the Higgs is often
thought to shoulder the whole burden of generating mass and my results
suggests that for the leptons, the scalar photon carries just over 95% of
this burden (ve^2) and the Higgs only about 4% (e sin thetaW).  So, the
Higgs probably does not need to be as heavy as some folks believe.

You may ask why I am using the sums m(e) + m(mu) + m(tau) and m(e-neutrino)
+ m(mu-neutrino) + m(tau-neutrino) in the formulas I have listed.  This is
because, a careful analysis of bi-unitary transformations on the
three-generation mass matrix reveals 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 electron and neutrino.  In short, these
low-energy-probe mass totals are mass-matrix-mixing invariants and thus give
us good targets for prediction.

You may also ask why the focus on leptons and not quarks.  That is simple: I
am restricting myself to the standard electroweak model to try to predict
masses.  Quarks have color, electroweak interactions can't account for
color, so we are certain leave something out if we aim to predict quark
masses with electroweak theory.  Leptons have no color, so it is reasonable
to expect that electroweak theory may contain most or all the tools needed
to predict the lepton masses.  I believe my results bear this out.

While electroweak theory does not provide the tools to predict the quark
masses, these results do make the very important point that only the
couplings for interactions which conserve parity are revealed in the vacuum
ground state mass.  For leptons, this gets us over 95% of the way to an
accurate mass prediction, and so we can preview that for quarks, the strong
coupling gS^2 versus e^2 for leptons will be the dominant predictor of mass.
This gives a rough quark-to-lepton mass ratio on the order of 16 to 1, which
is not at all out of the ballpark for what is actually observed in the first
generation.  The top quark, of course, is off the charts, but if one looks
at the mass of the top quark times the mass of the bottom quark -- which is
part of the invariant amplitude for interactions between the top and the
bottom, and takes the square root, this approximate 16 to 1 ratio makes
sense there too.  Again the formal aspects of my research are focused on
lepton masses, but do lead to at least a qualitative understanding of what
will need to go into quark masses.

Finally, I believe more than ever following this work that it is vital to
construct the strong interaction as a parity-conserving interaction which
sits across two (or more) interactions which violate parity.  This may
include baryon number, this may include weak isospin and hypercharge, this
may include something we don't yet know about.  But I have learned that
non-zero rest masses are only revealed when you start out with a mass term
which violates parity, and then mix together two or more parity-violating
interactions to construct a parity-conserving interaction.  Whether this can
be done for the strong interaction apart from electroweak interactions, or
whether this is central to merging these interaction together, I do not
know.  But, I am convinced that we must find a way to sit SU(3) QCD across
two or more parity-violating interactions just as is done in the electroweak
model, construct SU(3) color as parity conserving, mix the neutral currents
(which may use a unitary matrix with more than one mixing angle, and may
include complex phases), break the symmetry to make the gluons massless, and
then eliminate all the gamma-5 from the mass term.

If you are interested in more details about this work, I have just completed
a twelve-page paper summarizing my results in a little more detail, and
within a few days expect to complete the finishing touches on a 115 page
paper which I have prepared during the last four weeks that lays everything
out in very thorough detail.

Please let me know if you would like the twelve page summary which is now
fully prepared, and / or the full exposition when it is fully prepared.

Very truly yours,

Jay R. Yablon
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com