In the F4 model prior to dimensional reduction, the fermion particles are all massless. The neutrinos obey the Weyl equation and must remain massless and left-handed after dimensional reduction. The electrons and quarks obey the Dirac equation and acquire mass after dimensional reduction. After dimensional reduction, the charged W± of the SU(2) weak force can interchange Weyl fermion neutrinos with Dirac fermion electrons. It is required (as an ansatz or part of the F4 model) that the charged W± neutrino-electron interchange must be symmetric with the electron-neutrino interchange, so that the absence of right-handed neutrino particles requires that the charged W± SU(2) weak bosons act only on left-handed electrons. It is also required (as an ansatz or part of the F4 model) that each gauge boson must act consistently on the entire Dirac fermion particle sector, so that the charged W± SU(2) weak bosons act only on left-handed fermions of all types. Therefore, for the charged W± SU(2) weak bosons, the 4-dimensional spinor fields S8± contain only left-handed particles and right-handed antiparticles. So, for the charged W± SU(2) weak bosons, S8± can be denoted S8±L.
The neutral W0 weak bosons do not interchange Weyl neutrinos with Dirac fermions, and so may not entirely be restricted to left-handed spinor particle fields S8±L, but may have a component that acts on the full right-handed and left-handed spinor particle fields S8± = S8±L + S8±R.
However, the neutral W0 weak bosons are related to the charged W± weak bosons by custodial SU(2) symmetry, so that the left-handed component of the neutral W0 must be equal to the left-handed (entire) component of the charged W±. Since the mass of the W0 is greater than the mass of the W±, there remains for the W0 a component acting on the full S8± = S8±L + S8±R spinor particle fields.
Therefore the full W0 neutral weak boson interaction is proportional to (Mw±^2 / Mw0^2) acting on S8±L and
(1 - (Mw±^2 / Mw0^2)) acting on S8± = S8±L + S8±R.
If (1 - (Mw±^2 / Mw0^2)) is defined to be sin^2qw and denoted by x , and if the strength of the W± charged weak force (and of the custodial SU(2) symmetry) is denoted by T, then the W0 neutral weak interaction can be written as: W0L ~ T + x and W0R ~ x.
The F4 model allows calculation of the Weinberg angle qw,
by mw+ = mw- = mw0 cosqw . The Hopf fibration of S3 as
S1 -> S3 -> S2 gives a decomposition of the W bosons into the neutral W0 corresponding to S1 and the charged pair W+ and W- corresponding to S2. The mass ratio of the sum of the masses of W+ and W- to the mass of W0 should be the volume ratio of the S2 in S3 to the S1 in S3.
The unit sphere S3 in R4 is normalized by 1 / 2;
the unit sphere S2 in R3 is normalized by 1 / Ã3; and
the unit sphere S1 in R2 is normalized by 1 / Ã2.
The ratio of the sum of the W+ and W- masses to the W0 mass should then be (2 /Ã3) V(S2) / (2 / Ã2) V(S1) = 1.632993.
The sum SmW = mw+ + mw- + mw0 has been calculated to be
v Ãaw = 517.798 Ã0.2534577 = 260.774 GeV.
Therefore, cos^2qw = mw±^2 / mw0^2 = (1.632993/2)2 = 0.667 , and sin2qw = 0.333, so mw+ = mw- = 80.9 GeV, and mw0 = 98.9 GeV. However, these values must be corrected for the fact that only part of the w0 acts through the parity-violating SU(2) weak force and the rest acts through a parity-conserving U(1) electromagnetic-type force.
In the F4 model, the weak parity-conserving U(1) electromagnetic-type force acts through the U(1) subgroup of SU(2), which is not exactly like the F4 electromagnetic U(1) with force strength aE = 1 / 137.03608 = e^2 .
The W0 mass mw0 has two parts: the parity-violating SU(2) part mw0± that is equal to mw± ; and the parity-conserving part mw0 that acts like a heavy photon. As mw0 = 98.9 GeV = mw0± + mw00, and
as mw0± = mw± = 80.9 GeV, we have mw00 = 18 GeV.
Denote by a`E = `e^2 the force strength of the weak parity-conserving U(1) electromagnetic-type force that acts through the U(1) subgroup of SU(2).
The F4 electromagnetic force strength aE = e^2 = 1 / 137.03608 was calculated using the volume V(S1) of an S1 in R2 , normalized by 1 / Ã2.
The a`E force is part of the SU(2) weak force whose strength
aw = w2 was calculated using the volume V(S2) of an S2 in R3 , normalized by 1 / Ã3.
Also, the F4 electromagnetic force strength aE = e^2 was calculated using four 1-spheres S1, while the SU(2) weak force strength aw = w^2 was calculated using two 2-spheres S2, each of which contains one 1-sphere of the a`E force.
Therefore a`E = aE (Ã2 / Ã3)(2/4) = aE / Ã6 ,
`E = e / 4Ã6 = e / 1.565 , and the mass mw00 must be reduced to an effective value mw00eff = mw00 / 1.565 = 18/1.565 = 11.5 GeV for the a`E force to act like an electromagnetic force in the 4-dimensional spacetime of the F4 model: `E mw00 = e (1/5.65) mw00 = e mz0 , where the physical effective neutral weak boson is denoted by Z rather than W0.
Therefore, the correct F4 values for weak boson masses and the Weinberg angle are:
mw+ = mw- = 80.9 GeV; mz = 80.9+11.5 = 92.4 GeV; and
sin^2qw = 1 - (mw±/mz)^52 = 1 - 6544.81/8537.76 = 0.233.
Radiative corrections are not taken into account here, and may change the F4 value somewhat.
If g¶w1± is the Dirac operator corresponding to the covariant derivative for the charged weak force,
g¶w10± is the Dirac operator corresponding to the covariant derivative for the parity violating part of the neutral weak force,
g¶w100 is the Dirac operator corresponding to the covariant derivative for the parity conserving part of the neutral weak force, and
Y is the Yukawa coupling, then
the result of all this is that the full Weak Force Lagrangian after action of the Higgs mechanism is:
º [ - Fw14*Fw14 + mw+^2 W+W+ + mw-^2 W-W- + mz^2 ZZ +
+ (1/2) [ aw W+ÝW+ + aw W-ÝW- + aw ZÝZ ] [ H^2 + 2vH ] +
+ (1/2)(¶H)^2 - (1/2)(µ^2/2)H^2 +
+ (1/16)µ^2v^2[1 - 4H^3/v^3 - H^4/v^4] ] +
+ `S8±L (g¶w1± - Yv) S8±L +
+ `S8±L (g¶w10± - Yv) S8±L + + `S8± (g¶w100 - Yv) S8± +
- `S8± YH S8±
The F4 Higgs-weak force Lagrangian is similar to the standard Higgs-weak force Lagrangian set out in, e.g., Barger and Phillips33.