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Renormalization of Color, Weak, and Electromagnetic Forces

SU(5) Grand Unified structure produces

coupling constant unification of the color and electroweak forces at about 10^14 GeV.


Earlier, I did some much cruder calculations my self in Mathematica, which produced the following chart, which should not be considered as accurate at high energies as SU(5) Grand Unified calculations.

The chart of strengths of color charge (red) weak charge (blue), and electromagnetic charge (green) at energies up to 10^19 GeV was calculated from the lowest order renormalization group equations for the force strengths and Yukawa couplings in the D4-D5-E6-E7 model. Since, in the D4-D5-E6-E7 model, the forces remain separate below the Planck energy, where a sharp transition to unification occurs, that model does not require the force strengths to smoothly converge to a single value (although, if you include higher order processes, that might indeed occur). In the D4-D5-E6-E7 model, Dilatation Scale Transformations of the Conformal Group provide a natural setting for the Renormalization Group Process.

The calculations were done using Mathematica NDSolve (where { is less than and } is greater than):

F[M_]:= Which[-0.61 {= M { -0.20 , 2,
				-0.20 {= M { 0.25, 3,
				0.25 {= M { 0.73 , 4,
				0.73 {= M { 2.11 , 5,
			  	2.11 {= M {= 19 , 6, True, 0]
NDSolve[{c'[M] == -N[Log[10]]*
			((33 - 2*F[M])/(12*Pi))*c[M]^3,
		w'[M] == -N[Log[10]]*
		e'[M] == N[Log[10]]*
		t'[M] == N[Log[10]]*
			(t[M]^3 - t[M]*(2.67*c[M]^2 +
			1.5*w[M]^2 + 0.72*e[M]^2))/(16*Pi^2),
		b'[M] == N[Log[10]]*
			(b[M]*t[M]^2 - b[M]*(2.67*c[M]^2 +
			1.5*w[M]^2 + 0.39*e[M]^2))/(16*Pi^2),
		lt'[M] == N[Log[10]]*
			(lt[M]*t[M]^2 - lt[M]*(w[M]^2 +
		l'[M] == N[Log[10]]*(12/(16*Pi^2))*
			(l[M]^2 + 0.5*l[M]*t[M]^2 -
			0.375*l[M]*w[M]^2 + 0.046*w[M]^4 -
			c[2] ==  0.32,
			w[2] ==  0.5,
			e[2] ==  0.086,
			t[2] ==  1,
			b[2] ==  0.043,
			lt[2] ==  0.015,
			l[2] == 0.25},

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^2/4pi for charge g.

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 D4-D5-E6 model predicts that e = 0.085 = sqrt 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 D4-D5-E6 model predicts that w = 0.50 = sqrt 0.2534577 at the characteristic energy for the SU(2) weak force, the mass-energy range of the weak bosons, =100 GeV.

In the D4-D5-E6 model, the fundamental color force energy /\QCD is /\QCD = sqrt(M(pi+)^2 + M(pi0)^2 + M(pi-)^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 D4-D5-E6 model, /\QCD is not varied as Nf increases, but the increase of Nf is taken into acccount in the renormalization computations by setting c = 0.79 at 0.245 GeV.

The D4-D5-E6 model predicts the value c = 0.79 = sqrt 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

c2 = as = 0.121178 at 34 GeV

c2 = as = 0.105704 at 91 GeV

c = 0.14 at 10^19 GeV.

Shifman has noted that Standard Model global fits at the Z peak, about 91 GeV, give a color force strength of about 0.125 with Lambda_QCD = 500 MeV +/-,

whereas low energy results and lattice calculations give a color force strength at the Z peak of about 0.11 with Lambda_QCD = 200 MeV +/-.

The low energy results and lattice calculations are closer to the tree level D4-D5-E6 model value at 91 GeV of 0.106.

Also, the D4-D5-E6 model has Lambda_QCD = 245 MeV.

(For the pion mass, upon which the Lambda_QCD calculation depends, see this part of my home page.)

Shifman's paper indicates that there may be problems with Standard Model global fit methods used at CERN, which could explain their high (relative to the D4-D5-E6 model) values of the truth quark mass.

Patrascioiu and Seiler propose 
to resolve the discrepancy between 
high energy observation, giving  ALPHA_s = 0.123, 
low energy observations 
that extend by the renormalization group to ALPHA_s = 0.113 
and are closer to the D4-D5-E6 model value
that extends by the renormalization group to ALPHA_s = 0.106  
by using nonperturbative lattice LQCD 
instead of perturbative PQCD.  
They contend that PQCD and asymptotic freedom are NOT true 
properties of physical QCD, 
and that LQCD suggests a nonzero fixed point for ALPHA_s. 
They point out that the slower running of ALPHA_s under LQCD 
"...may also be expressed by saying that Lambda_QCD is not constant 
but an increasing function of the momentum Q."  
If they are correct, the renormalization group equation that I used to calculate ALPHA_s is not valid, so that ALPHA_s = 0.106 is not the true value for 91 GeV under the D4-D5-E6-E7 model.

The true 91 GeV value for the D4-D5-E6-E7 model ALPHA_s may be higher, closer to the high-energy experimental ALPHA_s = 0.123.

Leader and Stamenov have proposed that a QCD fixed point 
(such as proposed by Patrascioiu and Seiler) 
may explain the observation of CDF at Fermilab, in hep-ex/9601008, 
that 19.5 pb^(-1) of data indicate that 
"The cross section for jets with ET greater than 200 GeV 
is significantly higher than current predictions based on 
third-order alpha_s perturbative QCD calculations." 
so that Perturbative QCD may not be physically accurate. 


Alexei Morozov and Antti J. Niemi, in their paper, Can Renormalization Group Flow End in a Big Mess?, hep-th/0304178, say: "... The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan-Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental ... theory. We argue that ... such chaotic flows ... pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed. ... in the classical Yang-Mills theory chaotic behaviour has already been well established ... Consequently such chaotic behaviour will not be considered here. Obviously, a chaotic RG flow also necessitates the consideration of field (string) theories with at least three couplings. In the present article we shall be interested in the possibility of chaotic RG flows in the IR limits of quantum field and string theories. ... we consider limit cycles from the point of view of RG flows, and inspect vorticity as a RG scheme independent tool for describing multicoupling flows. ... we explain how to construct model effective actions from the beta-function flows. In particular, we explain how the construction fails in case of chaotic flows and suggests this parallels the problems encountered in constructing actual field theory effective actions. This also explains why it is very hard to construct actual field theory models with chaotic RG flow. ...".


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