Back to Cover Page

3.6. Renormalization of Force Strengths.

 The lowest order renormalization group equations for the force strengths and Yukawa couplings in the F4 model can be studied by using Mathematica NDSolve:


The variables c (and cc), w, and e (and ee) are the color, weak, and electromagnetic charges, using the unconventional convention that a = g^2 rather than

a = g^2/4 for charge g.

The conventional renormalization equation is of the form

MdC/dM = - 7C^3/16^2, with C^2/4 = ac, so that C = (ac)(2ù).

If c^2 = ac, then C = c 2ù so that Md(2ùc)/dM = - 7 x 8ùc^3/16^2, so the lowest order renormalization equations are then:

Mdc/dM = -7c^3/4 ; Mdw/dM = -3.3w^3/4; and Mde/dM = +4e^3/4 .

Since Md /dM = d /dlnM = d / 2.3logM , there is a conversion factor for plotting c, w, and e against logM.

The U(1) electromagnetic charge e takes the values 0.085 at 0.1 GeV, 0.086 at 100 GeV, and 0.095 at 10^19 GeV.

The F4 model predicts that e = 0.085 = 1/137.03608 at the low characteristic energies for QED.

The SU(2) weak charge w takes the values 0.50 at 100 GeV and 0.20 at 10^19 GeV.

The F4 model predicts that w = 0.50 = 0.2534577 at the characteristic energy for the SU(2) weak force, the mass-energy range of the weak bosons, 100 GeV.


LEP has recently measured c^2 = as at the neutral weak boson mass 91 GeV to be29:

0.106 (+0.005;-0.004) DELPHI energy-energy correlation;

0.121 (+0.010;-0.008) L-3 energy-energy correlation;

0.115 (+0.009;-0.008) L-3 energy-energy correlation asymmetry;

0.131 (±0.009) OPAL energy-energy correlation;

0.117 (±0.009) OPAL energy-energy correlation asymmetry;

0.121 (+0.013;-0.014) ALEPH global event shape variables;

0.117 (+0.006;-0.009) ALEPH clustered energy-energy correlation.

The 1990 Review of Particle Properties30 gives

c2 = as = 0.179 ±0.009 at the b-quark mass, about 5.3 GeV, and

c2 = as = 0.132 ±0.016 at 34 GeV.

In the F4 model, the fundamental color force energy LQCD is

LQCD = (m+^2 + m0^2 + m-^2) 0.242 GeV. That is the energy below which the color force is completely confined. Since 0.242 GeV is close to the s-quark current mass of (0.625 - 0.312) GeV = 0.313 GeV, the number of quarks Nf at that energy is considered to be Nf = 3.

At the c-quark current mass of 1.78 GeV, Nf = 4.

At the b-quark current mass of 5.32 GeV, Nf = 5.

At the t-quark current mass of 130 GeV, Nf = 6.

In the F4 model, LQCD is not varied as Nf increases, but the increase of Nf is taken into acccount in the modified Macintosh Pascal program, which starts by setting c = 0.79 at 0.245 GeV.

The F4 model predicts the value c = 0.79 = 0.6286 at the characteristic energy LQCD for QCD. As shown by the above figure, the SU(3) color charge c is set at 0.79 at 0.245 GeV, and it evolves to

c2 = as = 0.166539 at 5.3 GeV (0.179±0.009)

c2 = as = 0.121178 at 34 GeV (0.132±0.016

c2 = as = 0.105704 at 91 GeV (from 0.106 (DELPHI) to 0.131 (OPAL))

c = 0.14 at 1019 GeV,

with the experimental values29,30 given in parentheses.

Unlike grand unified theories, there is no requirement in the F4 model that all force strengths converge to a single value at a high unification energy. In the F4 model, the breakdown into 4 separate forces occurs at the Planck energy and is based on the Weyl group structure rather than the simple group structure of the high-energy unified gauge group.



Back to Cover Page