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5.2. Spinor Fermion part of the Lagrangian.

 Consider the spinor fermion term `S8± gw S8± .

For each of the S2's in S2 x S2, the part of Q8+ on which Spin(4) acts locally is Q3+ = RP1 x S2, the Silov boundary of the bounded domain D3+ that is isomorphic to the symmetric space D3+ = Spin(5)/SU(2) x U(1).

Due to the reducibility of Spin(4), the Dirac operator gw decomposes as gw = gw1 + gw2 , where gw1 is the Dirac operator corresponding to the surviving SU(2) and gw2 is the Dirac operator corresponding to the other SU(2). Then the spinor term is:

[ `S8± gw1 S8± + `S8± gw2 S8± ] .

The Dirac operator term gw2 in the Higgs S3 = SU(2) has dimension of mass. After integration S3 `S8± gw2 S8± over the Higgs S3, gw2 becomes the real scalar Higgs scalar field Y = (v + H) that comes from the complex SU(2) doublet F after action of the Higgs mechanism.

If integration over the Higgs S3 involving two fermion terms

`S8± S8± is taken to change the sign by i2 = -1,then, by the Higgs mechanism, `S8± gw2 S8± -> (S3 `S8± gw2 S8± ) ->

-> - `S8± YY S8± = - `S8± Y(v+H) S8± ,

where: H is the real physical Higgs scalar, mH = (lv2/2) =261 GeV.

v is the vacuum expectation value of the scalar field Y, the free parameter in the theory that sets the mass scale. Denote the sum of the three weak boson masses by SmW. v = SmW ((2)/aw) =

= 260.774 &endash; 2 / 0.5034458 = 732.53 GeV, a value chosen so that the electron mass will be 0.5110 MeV.

Y is the Yukawa coupling between fermions and the Higgs field. Y acts on all 28 elements (2 helicity states for each of the 7 Dirac particles and 7 Dirac antiparticles) of the Dirac fermions in a given generation, because all of them are in the same Spin(8) spinor representation. Denote the sum of the first generation Dirac fermion masses by Smf1.

Then Y = (2) Smf1/v , just as (aw) = (2) SmW / v.

Y should be the product of two factors:

e^2, the square of the electromagnetic charge e = aE , because in the term S3 `S8± gw2 S8± -> `S8± Y(v+H) S8± each of the Dirac fermions S8± carries electromagnetic charge proportional to e ; and

1/g, the reciprocal of the weak charge g = aW , because an SU(2) force, the Higgs SU(2), couples the scalar field to the fermions.

Therefore Smf1 = Y v / 2 = (e^2/g) v / 2 = 7.508 GeV and

Smf1 / SmW = (e^2/g)v / gv = e^2 / g^2 = aE / aW.

The Higgs term -`S8± Y(v+H) S8± = -`S8± Yv S8± - `S8± YH S8± = = - `S8± ((2) Smf1) S8± - `S8± ((2) Smf1/v) S8± .

The resulting spinor term is of the form

[ `S8± (gw1 - Yv) S8± - `S8± YH S8± ] ,

where (gw1 - Y) is a massive Dirac operator. How much of the total mass Smf1 = Y v / 2 is allocated to each of the first generation Dirac fermions is determined by calculating the individual fermion masses in the F4 model, and those calculations also give the values of Smf2, Smf3, and individual second and third generation fermion masses.

 

 


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