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Fermion particle masses, where quark masses are constituent masses, can be calculated in the F4 model in which the first generaton fermion particles (two helicity states for Dirac electron and quarks, one for Weyl neutrino, for a total of 15 states) correspond to the 8-dimensional half-spinor representation S8+ of Spin(8) and the 15 fermion antiparticle states correspond to the mirror image 8-dimensional half-spinor representation S8- of Spin(8).

In the F4 model, the Kobayashi-Maskawa parameters are determined in terms of the sum of the masses of the 30 first-generation fermion particles and antiparticles, denoted by Smf1 = 7.508 GeV, and

the similar sums for second-generation and third-generation fermions, denoted by Smf2 = 32.94504 GeV and Smf3 = 1,629.2675 GeV.

Since both leptons and quarks come from the same 8-dimensional half-spinor Spin(8) representations, the F4 model requires that leptons and quarks have the same mixing angles. Lepton mixing is suppressed by the fact that neutrinos are massless at tree level.

Since the fermions of the F4 model are based on the two mirror image 8-dimensional half-spinor representations of Spin(8), and therefore on the Spin(8) Clifford algebra, the neutrinos as Weyl fermions can exist as either real Majorana-Weyl neutrinos or pure imaginary pseudo-Majorana-Weyl neutrinos7.

At tree level, all neutrinos in the F4 model are Weyl fermions and are massless. If the real Majorana-Weyl spinors are taken to correspond to left-handed particles (and right-handed antiparticles), then the pure imaginary pseudo-Majorana-Weyl spinors could correspond to right-handed particles (and left-handed antiparticles).

Then a radiative process that could take a left-handed real Majorana-Weyl spinor (such as the electron neutrino ñel) and transform it into a right-handed pure imaginary pseudo-Majorana-Weyl spinor, denoted by ner. The interaction Leff = l (1/M) (ne)(ne) (H)(H)

(see eq. 3.11.1 of the second edition of Mohapatra41, and Langacker40)

involving two Higgs bosons can be considered to be a nonrenormalizable effective four-particle interaction that is fundamentally an exchange of a massive particle M that interacts with neutrinos, as shown in Fig. 16.

In the F4 model, M should be a Planck-mass black hole with gravitational interaction that couples to the neutrino spin 1/2.

The resulting pseudo-Majorana-Weyl mass, and the corresponding Majorana-Weyl mass, is (eq. 3.11.1 of Mohapatra41, second edition)

mne = 4 mw2 l / g2 M = 4 x 81 x 81 x 0.25 / 0.25 x 1.2 x 1019 GeV =

= 2.187 x 10-6 eV.

If the second and third generation muon and tauon neutrinos get mass in the same ratio as the muon and tauon to the electron, then the effective masses for all three generations of neutrinos are

mne = 2.187 x 10-6 eV

mnµ = 205.5 x 5.7 x 10-6 eV = 4.49 x 10-4 eV

mnt = 3686.3 x 2.187 x 10-6 eV = 8.06 x 10-3 eV .

If the mixing angles for leptons are the same as in the Kobayashi-Maskawa matrix, as they should be in the F4 model since quarks and leptons all come from the same Spin(8) spinor representation, then the lepton Cabibbo angle q12 = a is given by sina = 0.221975, and

sin2(q12) = 0.04927. For q13 = b, sinb = 0.004608.

For q23 = g , assuming that the same width factor applies for leptons,

sing = 0.0423.

The MSW effect is described by Bethe and Bahcall42. The F4 model mass square differences and mixing angles for the MSW effect are:

Dm122 = 1.998 x 10-7 eV2 sin2(q12) = 0.04927

Dm132 = 6.496 x 10-5 eV2 sin2(q13) = 2.123 x 10-5

Dm232 = 5.79 x 10-5 eV2 sin2(q23) = 1.789 x 10-3.

According to Bethe and Bahcall42, the 37Cl results and the

Kamiokande II results, taken together, give Dm122 sin2(q12)= 1 x 10-8.

As Dm122 sin2(q12) = 0.9844 x 10-8, the experimental results, with the neutrino masses and mixing angles as derived above, are consistent with the (non-adiabatic) MSW effect for ne ´ nµ mixing and the standard solar model without any cooling of the solar interior by WIMPs.

According to Fig. 1 of Bethe and Bahcall42, the permitted region of the parameter space Dm122 plotted versus sin2(q12) defined by

Dm122 sin2(q12)= 1 x 10-8 requires that 71Ga experiments should see about 20 ± 20 SNU (including neutrinos that go through the Earth and arrive at the counter at night) if Dm122 = 1.998 x 10-7 eV2 and sin2(q12) = 0.04927.

The SAGE gallium experiment reported 20 +15 -20 ±32 SNU in 1991. GALLEX reported in 83 ± 19 ± 8 SNU in 1992. The standard solar model predicts 125-132 SNU.

SAGE preliminary results in 1992 range from 0 to 100 SNU.

GALLEX had a problem with 68Ge formed when the gallium was stored above ground for a year while the laboratory was being built inside the mountain at Gran Sasso.

It may be necessary to wait for results from the heavy water Sudbury Neutrino Detector to get the situation cleared up.

The other neutrino masses and mixing angles have not yet been tested experimentally, and would not produce the MSW effect in the sun.

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