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6.4. Third Generation Fermion Masses.

The t-quark mass calculation is an important prediction of the F4 model, so details of the third generation calculation are given here.

The third generation fermion particles correspond to triples of octonions. There are 8^3 = 512 such triples.

The triple (1,1,1) corresponds to the tau-neutrino.

The other 7 triples involving only 1 and e correspond to the tauon:

(e,e,e)

(e,e,1) (e,1,e) (1,e,e)

(1,1,e) (1,e,1) (e,1,1) .

The symmetry of the 7 tauon triples is the same as the symmetry of the 3 down quarks, the 3 up quarks, and the electron, so the tauon mass should be the same as the sum of the masses of the first generation massive fermion particles.

Therefore the tauon mass is mt = 1.87704 GeV.

Note that all triples corresponding to the tau and the tau-neutrino are colorless.

The beauty quark corresponds to 21 triples. They are triples of the form (1,1,ie), (1,ie,1), (ie,1,1), (ie,ie,1), (ie,1,ie), (1,ie,ie), and (ie,ie,ie), and the similar triples for 1 and je and for 1 and ke.

Note particularly that triples of the type (1,ie,je), (ie,je,ke), etc., do not correspond to the beauty quark, but to the truth quark, because the octonionic products of at least some pairs within such octonionic triples as (1,ie,je), (ie,je,ke), etc., are of the type -k, or -k, -i, or -j, etc., that correspond to the truth quark.

The red beauty quark is defined as the seven triples (1,1,ie), (1,ie,1), (ie,1,1), (ie,ie,1), (ie,1,ie), (1,ie,ie), and (ie,ie,ie), because ie is the red down quark. The seven triples of the red beauty quark correspond to the seven triples of the tauon, except that the beauty quark interacts with 6 Spin(5) gravitons while the tauon interacts with only two. The beauty quark constituent mass should be the tauon mass times the third generation graviton factor 6/2 = 3, so mb = 5.63111 GeV.

The blue beauty quarks correspond to the seven triples involving je, and the green beauty quarks correspond to the seven triples involving ke.

The truth quark corresponds to the remaining 483 triples, so the constituent mass of the red truth quark is 483/21 = 69/3 = 23 times the red beauty quark mass, or mt = 129.5155 GeV.

The blue and green truth quarks are defined similarly, and their

masses are calculated similarly. Therefore

Smf3 = 4(mt + 3mb + 3mt) =

= 4(1.87704 + 3&endash;5.63111 + 3&endash;129.5155) GeV = 1,629.2675 GeV.

In detail, the red truth quark is defined as the 161 triples:

(e,j,ke), (e,ke,j), (e,i,e), (e,e,i), (e,k,je), (e,je,k), (e,1,ie), (e,ie,1),

(ke,ie,k), (ke,k,ie), (ke,ke,i), (ke,i,ke), (ke,j,e), (ke,e,j), (ke,1,je), (ke,je,1),

(je,ie,j), (je,j,ie), (je,je,i), (je,i,je), (je,k,e), (je,e,k), (je,1,ke), (je,ke,1),

(ie,ke,j), (ie,j,ke), (ie,ie,i), (ie,i,ie),(ie,je,k), (ie,k,je), (ie,1,e), (ie,e,1),

(k,ke,ie), (k,ie,ke), (k,k,i), (k,i,k), (k,je,e), (k,e,je), (k,1,j), (k,j,1),

(j,ie,je), (j,je,ie), (j,j,i), (j,i,j), (j,ke,e), (j,e,ke), (j,1,k), (j,k,1),

(i,e,e), (i,ke,ke), (i,je,je), (i,ie,ie), (i,k,k), (i,j,j), (i,1,1), (i,i,i)

(1,e,ie), (1,ie,e), (1,ke,je), (1,je,ke), (1,k,j), (1,j,k), (1,i,1), (1,1,i)

because the octonion product of the elements of each such triple is ±i, the red up quark;

(e,je,ke), (e,ke,je), (e,ie,e), (e,e,ie), (e,k,j), (e,j,k), (e,1,i), (e,i,1),

(ke,ie,ke), (ke,ke,ie), (ke,k,i), (ke,i,k), (ke,je,e), (ke,e,je), (ke,1,j), (ke,j,1),

(je,ie,je), (je,je,ie), (je,j,i), (je,i,j), (je,ke,e), (je,e,ke), (je,1,k), (je,k,1),

(ie,e,e), (ie,ke,ke), (ie,je,je), (ie,k,k), (ie,j,j), (ie,i,i), (ie,1,1), (ie,ie,ie)

(k,ke,i), (k,i,ke), (k,k,ie), (k,ie,k), (k,j,e), (k,e,j), (k,1,je), (k,je,1),

(j,i,je), (j,je,i), (j,j,ie), (j,ie,j), (j,k,e), (j,e,k), (j,1,ke), (j,ke,1),

(i,ke,k), (i,k,ke), (i,je,j), (i,j,je), (i,ie,i), (i,i,ie), (i,1,e), (i,e,1),

(1,e,i), (1,i,e), (1,ke,j), (1,j,ke), (1,k,je), (1,je,k), (1,ie,1), (1,1,ie)

because the octonion product of the elements of each such triple is ±ie, the red down quark - the outline [ shown here as cyan ] type face elements are omitted because they belong to the red beauty quark;

(ie,ke,je), (ie,je,ke), (ie,j,k), (ie,k,j), (ie,ie,e), (ie,e,ie), (ie,1,i), (ie,i,1)

(i,i,e), (i,e,i), (i,1,ie), (i,ie,1), (i,ke,j), (i,j,ke), (i,je,k), (i,ke,j)

(e,ie,ie), (e,i,i) (1,ie,i), (1,i,ie)

because although the octonion product of the elements of each such triple is colorless ±e, the first element not 1 or e is ie or i; and

(e,ie,i), (e,i,ie) (1,i,i), (1,ie,ie)

(ie,i,e), (ie,e,i), (ie,ie,1), (ie,1,ie), (ie,ke,j), (ie,j,ke), (ie,je,k), (ie,k,je)

(i,ke,je), (i,je,ke), (i,j,k), (i,k,j), (i,ie,e), (i,e,ie), (i,i,1), (i,1,i)

because although the octonion product of the elements of each such triple is colorless ±1, the first element not 1 or e is ie or i

- the outline [ shown here as cyan ] type face elements are omitted because they belong to the red beauty quark.

It is interesting that in the F4 model the mass of the t`t bound state

m(t`t ) Å 2mt Å 260 GeV Å SmW = mW+ + mW- + mW0 = mHiggs.

This is a coincidence in the F4 model, arising because 69 Å 1 / 2aE,

aW Å 1 / 4, and mt = (69/3) mb = (69/3)(3) mt = 69 (Smf1 / 4) =

= 69 (aE / aW) SmW / 4 , and because mHiggs = mW+ + mW- + mW0.

[ In 1998, after reading quant-ph/9806009 by Guang-jiong Ni, I realized that I made some errors in my 1992 calculation of the Higgs Vacuum Expectation Value and the Higgs mass, and I now calculate

the Higgs Vacuum Expectation Value as about 260 GeV

and the Higgs mass as about 145 GeV. However, I don't think that the general geometric/structural ideas were incorrect. ]

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