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6.2. First Generation Fermion Masses.

Since only the two factors V(Q) and N(Graviton) are involved, the ratio of the down quark constituent mass to the electron mass can now be calculated as follows:

Consider the electron, e. By photon, weak boson, and gluon interactions, e can only be taken into 1, the massless neutrino. The electron and neutrino, or their antiparticles, cannot be combined to produce any of the massive up or down quarks. The neutrino, being massless, does not add anything to the mass formula for the electron. Since the electron cannot be related to any other massive Dirac fermion, its volume V(Q) is taken to be 1.

Next consider a red down quark ie. By gluon interactions, ie can be taken into je and ke, the blue and green down quarks. By weak boson interactions, it can be taken into i, j, and k, the red, blue, and green up quarks. Given the up and down quarks, pions can be formed from quark-antiquark pairs, and the pions can decay to produce electrons and neutrinos. Therefore the red down quark (similarly, any down quark) is related to any part of Q8+ = RP1 x S7, the continuum limit compact manifold corresponding to

{1, i, j, k, e, ie, je, ke}, and therefore a down quark should have a spinor manifold volume factor of the volume of Q8+ = V(Q8+).

The ratio of the down quark spinor manifold volume factor to the electron spinor manifold volume factor is just

V(Q8+) / 1 = V(Q8+) = ¹5/3 (see Hua11).

Since the first generation graviton factor is 6,

md / me = 6V(Q8+) = 2¹5 = 612.03937.

As the up quarks correspond to i, j, and k, which are isomorphic to

ie, je, and ke of the down quarks, the up quarks and down quarks have the same constituent mass mu = md.

Antiparticles have the same mass as the corresponding particles.

The sum of the first generation fermion masses Smf1 is given by

Smf1 = 4(me + 3md + 3mu) = (4 + 24 x 612.03937)me.

As me = 0.5110 MeV is assumed to fix the mass scale because the F4 model only gives ratios of masses, Smf1 = 7.508 GeV and the electron to quark mass ratio then gives a constituent mass of the down quark

md = 312.75 MeV, and the same constituent mass for the up quark

mu = 312.75 MeV.

As the proton mass is taken to be the sum of the constituent masses of its constituent quarks, m(proton) = mu + mu + md = 938.25 MeV,

the F4 model is close to the experimental value of 938.27 MeV.

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