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[ 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. ]

The F4 model of physics is an 8-dimensional theory that breaks down to a 4-dimensional theory at the Planck mass. Below the Planck mass, the F4 model weak force has gauge group Spin(4) = SU(2) x SU(2) = S3 x S3. Spin(4) acts naturally on the quaternionic manifold S2 x S2 = [Spin(3)/Spin(2)] &endash; [Spin(3)/Spin(2)], giving the 4-dimensional Lagrangian

where Fw4 is the weak force Spin(4) curvature and g¶w is the Dirac operator.

Each SU(2) in Spin(4) is 3-dimensional and corresponds three gauge bosons, W+, W-, and W0, so that if the two SU(2)'s were independent over each of their corresponding S2's in S2 x S2, the resulting gauge boson over S2 x S2 might be a W+ on one of the S2's and a W- on the other S2, a physically unrealistic result. Therefore the two SU(2)'s in Spin(4) must agree, so that only one SU(2) effectively survives to be the weak force SU(2). The other SU(2) is the Higgs SU(2).

Consider the gauge boson term º - Fw4 /\ *Fw4 .

The curvature Fw4 has two components, each of which can be in either the weak force SU(2) or the Higgs SU(2).

Therefore Fw4 = Fw14 + Fw124 + Fw24 , where Fw14 has both components in the weak force SU(2), Fw24 has both components in the Higgs SU(2), and the mixed term Fw124 has one component in each of the SU(2)'s. Then, using an approach similar to that of Mayer23:

º - Fw4 /\ *Fw4 =

= º (- Fw14 /\ *Fw14 - 2 Fw124 /\ *F1w24 - Fw24 /\*Fw24 ).

As all possible paths should be taken into account in the sum over histories path integral picture of quantum field theory, the terms involving the Higgs SU(2) should be integrated over the Higgs SU(2)=S3. Integrating over the Higgs S3 gives

º ( - Fw14 /\*Fw14 + 2 º S£ -Fw124 /\*F1w24 + º S£-Fw4/\*Fw4 ).

The first term is just º - Fw14 /\*Fw14 .

The third term, º ºS£ - Fw24 /\*Fw24 , after integration over the Higgs S£, produces terms of the form - l (FÝF)2 + m2 FÝF by a process similar to the Mayer mechanism.

Proposition 11.4 of chapter II of volume 1 of Kobayashi and Nomizu32 states that

where L takes values in the SU(2) Lie algebra. If the action of the Hodge dual * on L is such that *L = -L and *[L,L] = [L,L], then

and if integration of L over the Higgs S3 is ºS3 L µ F = (F+, F0), then

= (1/4) [ - l (FÝF)^2 + m^2 FÝF ] ,

where ¬ is the strength of the scalar field self-interaction and m2 is the other constant in the Higgs potential, and where F is a 0-form taking values in the SU(2) Lie algebra. The SU(2) values of F are represented by complex SU(2)=Spin(3) doublets F = (F+, F0).

In real terms, F+ = (F1 + iF2)/Ã2 and F0 = (F3 + iF4)/Ã2, so F has 4 real degrees of freedom.

In terms of real components, FÝF = (F12 + F22 + F32 + F42)/2 . The nonzero vacuum expectation value of the - l(FÝF)^2 + m^2FÝF term is v = m / Ãl , and <F0> = <F3> = v / Ã2.

In the unitary gauge, F1 = F2 = F4 = 0, and

F = (F+, F0) = (1/Ã2)(F1 + iF2, F3 + iF4) = (1/Ã2)(0, v + H) , where

F3 = (v + H)/Ã2 , v is the Higgs potential vacuum expectation value and

H is the real surviving Higgs scalar field.

Since l = µ2 / v2 and F = (v + H) / Ã2 ,

(1/4)[- l(FÝF)^2 + m^2 FÝF] =

= -(1/16)(µ^2/v^2)( v + H )4 + (1/8)µ^2( v + H )^2 =

= (1/16) [ -µ^2v^2 -4µ^2vH -6µ^2H^2 -4µ^2H^3/v -µ^2H^4/v^2 +

+2µ^2v^2 +4µ^2vH +2µ^2H^2 ] =

= - (1/4) µ^2H^2 + (1/16) µ^2v^2 [ 1 -4H^3/v^3 -H^4/v^4 ] .

The second term, º 2ºS£ (- Fw124 /\ *Fw124), gives º ¶wFÝ¶wF, where ¶w is the weak force covariant derivative, by a process similar to the Mayer mechanism.

From Proposition 11.4 of chapter II of volume 1 of Kobayashi and Nomizu32:

where L takes values in the SU(2) Lie algebra.

If the X component of Fw124(X,Y) is in the surviving weak force SU(2) space, and the Y component of Fw124(X,Y) is in the Higgs

S3 = SU(2) space, then, since the SU(2)'s in Spin(4) = SU(2) x SU(2) are independent, the Lie bracket product [X,Y] = 0 so that L([X,Y]) = 0 and

Fw124(X,Y) = (1/2) [L(X),L(Y)] . Since the X direction is in the surviving spacetime, orthogonal to the Higgs S3 of the Y direction, the Lie bracket [L(X),L(Y)] is the covariant derivative of L(Y) in the X direction, and this case contributes Fw124(X,Y) =

= (1/2) [L(X),L(Y)] = (1/2) ¶XL(Y) to the total value of Fw124(X,Y) .

By symmetry, the Y component of Fw124(X,Y) could be in the surviving weak force SU(2) space and the X component of Fw124(X,Y) be in the Higgs S3 = SU(2) space, and these symmetric cases would also contribute Fw124(X,Y) = (1/2) [L(X),L(Y)] = (1/2) ¶wXL(Y) to the total value of Fw124(X,Y).

To the extent that X and Y are in the same S3 = SU(2),

[L(X),L(Y)] - L([X,Y]) would cancel out, because [L(X),L(Y)] would be the same SU(2) term as L([X,Y]).

The total value of Fw124(X,Y) is then Fw124(X,Y) = ¶wXL(Y) .

Integration of L over the Higgs S3 gives ºY in S3 ¶wXL(Y) = ¶wXF,

where F is a 0-form taking values in the SU(2) Lie algebra.

The SU(2) values of F are represented by complex SU(2)=Spin(3) doublets F = (F+, F0).

In real terms, F+ = (F1 + iF2) / Ã2 and F0 = (F3 + iF4) / Ã2,

so F has 4 real degrees of freedom.

As discussed above, in the unitary gauge, F1 = F2 = F4 = 0, and

F = (F+, F0) = (1/Ã2)(F1 + iF2, F3 + iF4) = (1/Ã2)(0, v + H) ,

where F3 = (v + H) / Ã2 , v is the Higgs potential vacuum expectation value, and H is the real surviving Higgs scalar field.

The second term is then:

º (2 ºS3 - Fw124 /\ *Fw124) =

= º (ºS3 (-1/2) [L(X),LY)] /\ * [L(X),L(Y)] ) = º ¶wFÝ¶wF

where the weak-Higgs covariant derivative ¶w is

¶w = ¶ + Ãaw (W+ + W-) + Ã(aw / cos2qw) W0 , where qw is the Weinberg angle. Then ¶wF = ¶w(v + H)/Ã2 =

= [ ¶H + Ã(aw)W+(v+H) + Ã(aw)W-(v+H) + Ã(aw)W0(v+H) ]/Ã2

In the F4 model the W+, W-, W0, and H terms are considered to be linearly independent. v = v+ + v- + v0 has linearly independent components v+, v-, and v0 for W+, W-, and W0. H is the Higgs component. ¶wFÝ¶wF is the sum of the squares of the individual terms. Integration over the Higgs S3 involving two derivatives ¶wX¶wX is taken to change the sign by i^2 = -1. Then:

¶wFÝ¶wF = (1/2) (¶H)^2 +

+ (1/2) [ aw v+^2 W+ÝW+ + aw v-^2 W-ÝW- + aw v0^2 W0ÝW0 ] +

+ (1/2) [ aw W+ÝW+ + aw W-ÝW- + aw W0ÝW0 ] [ H^2 + 2vH ] .

Then the full curvature term of the weak-Higgs Lagrangian,

º - Fw14/\*Fw14 + ¶FÝ¶F - l (FÝF)2 + m^2 FÝF , is, by the Higgs mechanism:

º [ - Fw14/\*Fw14 +

+ (1/2) [ aw v+^2 W+ÝW+ + aw v-^2 W-ÝW- + aw v0^2 W0ÝW0 ] +

+ (1/2) [ aw W+ÝW+ + aw W-ÝW- + aw W0ÝW0 ] [ H^2 + 2vH ] +

+ (1/2) (¶H)2 - (1/4)m^2H^2 +

+ (1/16) m^2v^2 [ 1 -4H^3/v^3 -H^4/v^4 ] ] .

The weak boson Higgs mechanism masses, in terms of

v = v+ + v- + v0, are: (aw/2) v+^2 = mw+^2 ; (aw/2) v-^2 = mw-^2 ; and (aw/2) v0^2 = mw0^2, with

( v = v+ + v- + v0 ) = ((Ã2)/Ãaw) ( mw+ + mw- + mw0 ) . Then:

º [ - Fw14/\*Fw14 + mw+^2 W+W+ + mw-^2 W-W- + mw0^2 W0W0 +

+ (1/2) [ aw W+ÝW+ + aw W-ÝW- + aw W0ÝW0 ] [ H^2 + 2vH ] +

+ (1/2)(¶H)2 - (1/2)(m^2/2)H^2 +

+ (1/16m^2v^2[1 - 4H^3/v^3 - H^4/v^4] ] .

The Higgs vacuum expectation value v = ( v+ + v- + v0 ) is the only particle mass free parameter. In the F4 model, v is set so that the electron mass me = 0.5110 MeV. Therefore,

((Ãaw)/Ã2) v = mw+ + mw- + mw0 = 260.774 GeV, the value chosen so that the electron mass (which is to be determined from it) will be 0.5110 MeV. In the F4 model, aW is calculated to be aW= 0.2534577,

so ÃaW = 0.5034458 and v = 732.53 GeV.

The Higgs mass mH is given by the term

(1/2)(¶H)2 - (1/2)(m^2 / 2)H^2 = (1/2) [ (¶H)^2 - (m™/2) H^2 ]

to be mH^2 = m^2/2 = l v^2/2, so that mH = Ã(m^2/2) = Ã(l v^2/2) .

l is the scalar self-interaction strength. It should be the product of the "weak charges" of two scalars coming from the Higgs SU(2) in Spin(4), which should be the same as the weak charge of the surviving weak force SU(2) and therefore just the square of the SU(2) weak charge, Ã(aw)^2 = aw , where aw is the SU(2) geometric force strength.

Therefore l = aw = 0.2534576 , Ãl = 0.5034458, and

v = 732.53 GeV, so that the mass of the Higgs scalar is

mH= vÃ(l/2) = 260.774 GeV.

[ 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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