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7.1 Calculation of Kobayashi-Maskawa Parameters.

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.

Calculation of all fermion masses permits calculation of Kobayashi-Maskawa parameters in terms of sums of masses (counting each helicity state of Dirac fermions separately and counting both particles and antiparticles) of all fermions in each of the three generations, denoted by Ímfn . The mixing matrix between the up, charm, and truth quarks and the down, strange, and beauty quarks is due to the action on them of the weak force, and can be written as follows:

d s b

u Vud Vus Vub

c Vcd Vcs Vcb

t Vtd Vts Vtb

 

In the Chau-Keung36 parameterization, the 3¥3 Kobayashi-Maskawa matrix is given by the product of three 3¥3 matrices

with parameters a, b, e, and g :

1 0 0 cb 0 sb e-ie ca sa 0

0 cg sg 0 1 0 -sa ca 0

0 -sg cg -sb e+ie 0 cb 0 0 1

where sa = sin(a) = Smf1 / (Smf1^2 + Smf2^2) = 0.222197704;

sb = sin(b) = Smf1 / (Smf1^2 + Smf3^2) = 0.00460816;

sg = sin(g) = [Smf2 / (Smf2^2+Smf3^2)] [(Smf2 / Smf1)] =

= 0.04234886385;

ca = cos(a); cb = cos(b); and cg = cos(g);

Smf1 = 7.508 GeV, Smf2 = 32.94504 GeV, and Smf3 = 1629.2675 GeV,

and -e is the CP-violating phase angle, which is -e = -/2 because it arises from the dimensional reduction projection defined by

: O=HƒHe -> Hƒ(He)(-e) = HƒH &emdash;diagonal map-> H ,

where a basis for H is {1,i,j,k}, a basis for He is {e,ie,je,ke}, and

e is the imaginary octonion e in the basis {1, i, j, k, e, ie, je, ke} of O.

The factor [(Smf2 / Smf1)] appears in sg because an sg transition is to the second generation and not all the way to the first generation, so that the end product of an sg transition has a greater available energy than sa or sb transitions by a factor of Smf2 / Smf1. Since the width G of a transition is proportional to the square of the modulus of the relevant KM entry, i.e., G [proportional to] |V|^2, and the width Gg of an sg transition has greater available energy than the sa or sb transitions by a factor of Smf2 / Smf1 ,

the effective magnitude of the sg terms in the KM entries is increased by a factor (Sm2 / Smf1).

The reason for using sums of all fermion masses (rather than sums of quark masses only) is that all fermions are in the same spinor representation of Spin(8), and the Spin(8) representations are taken to be fundamental in the F4 model.

 

The result, calculated by Mathematica Notebook,

 

is the following effective K-M matrix (with K-M phase e = / 2):

d s   b u 0.975 0.222 -0.00461 i c -0.222    0.974 0.0423 -0.000191 i -0.0000434 i t 0.00941 -0.0413 0.999 -0.00449 i -0.00102 i

 


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