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6.1. Fermion Mass Formula.

The F4 model puts the 8 first generation fermion particles in one 8-dimensional half-spinor representation S8+ of Spin(8), with the 8 antiparticles being in the mirror image 8-dimensional half-spinor representation S8- of Spin(8).

Given a basis {1, i, j, k, ke, je, ke} for O, where 1 is the basis element for the real axis and, as to the seven imaginaries, i, j, and k are just the three imaginary quaternions, and e, ie, je, and ke are constructed from the four quaternionic basis elements 1, i, j, and k by introducing an octonionic imaginary e. The octonionic basis for S8+ corresponds to fermion particles as follows:

1 - electron neutrino

i, j, k - red, blue, and green up quarks

e - electron

ie, je, ke - red, blue, and green down quarks.

The antiparticle correpsondence for S8- is similar.

In the F4 model, the Weyl fermion neutrino has at tree level only the left-handed state, whereas the Dirac fermion electron and quarks can have both left-handed and right-handed states, so that the total number of states corresponding to each of the half-spinor Spin(8) representations Spin(8) is 15.

In the F4 model, the first generation fermions correspond to octonions O, while second generation fermions correspond to pairs of octonions O x O and third generation fermions correspond to triples of octonions O x O x O. This structure is a result of the dimensional reduction mechanism from an 8-dimensional theory to a 4-dimensional theory at the Planck energy, as explained in Section 2.6.

To calculate the fermion masses in the F4 model, the volume of a manifold related to the spinor fermions, Q8+ = RP1 x S7, is used.

It is a parallelizable manifold whose structure is described in detail in Section 2.2.

Also, since gravitation is coupled to mass, the infinitesimal generators of the F4 model gravitation gauge group, Spin(5), are used in the fermion mass calculations. The construction of gravity from Spin(5) is described in detail in Chapter 4.

The fundamental F4 quark masses are constituent masses, not current masses.

In the F4 model, fermion masses are calculated as a product of four factors:

V(Q) x N(Graviton) x N(octonion) x Sym

V(Q) is the volume of the part of the half-spinor fermion particle manifold Q8+ = RP1 x S7 that is related to the fermion particle by photon, weak boson, and gluon interactions.

N(Graviton) is the number of types of graviton related to the fermion. The 10 gravitons correspond to the 10 infinitesimal generators of

Spin(5) = Sp(2). 2 of them are in the Cartan subalgebra.

6 of them carry color charge, and may therefore be considered as corresponding to quarks.

The remaining 2 carry no color charge, but may carry electric charge and so may be considered as corresponding to electrons.

One takes the electron into itself, and the other can only take the first-generation electron into the massless electron neutrino.

Therefore only one graviton should correspond to the mass of the first-generation electron.

The graviton number ratio of the down quark to the first-generation electron is therefore 6/1 = 6.

For second and third generations, massive leptons may be of the form (1,e) or (1,e,e), and can be taken into other massive lepton forms such as (e,1) or (e,1,1) by the second electically charge graviton. Therefore the graviton number ratio of quarks to massive leptons for the second and third generations is 6/2 =3.

N(octonion) is an octonion number factor relating up-type quark masses to down-type quark masses in each generation, since 2nd and 3rd generation fermions can be considered to correspond to pairs or triples of 1st generation fermions, which can be considered to be octonions.

Sym is an internal symmetry factor, relating 2nd and 3rd generation massive leptons to first generation fermions.




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