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Renormalizationof Color, Weak, and Electromagnetic Forces

SU(5) Grand Unifiedstructure produces

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


Earlier, I did some much cruder calculations my self inMathematica, which produced the following chart, which should not beconsidered 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 wascalculated from the lowest order renormalization group equations forthe force strengths and Yukawa couplings in the D4-D5-E6-E7model. Since, in the D4-D5-E6-E7model, the forces remain separate below the Planck energy, wherea sharp transition to unification occurs, that model does not requirethe force strengths to smoothly converge to a single value (although,if you include higher order processes, that might indeed occur). Inthe D4-D5-E6-E7 model, DilatationScale Transformations of the Conformal Group provide a naturalsetting for the Renormalization Group Process.

The calculations were done using Mathematica NDSolve (where { isless 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]]*			((3.3)/(4*Pi))*w[M]^3,		e'[M] == N[Log[10]]*			(4/(4*Pi))*e[M]^3,		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 +			1.5*e[M]^2))/(16*Pi^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 -			0.25*t[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},			{c,w,e,t,b,lt,l},			{M,2,19}] 

The variables c (and cc), w, and e (and ee) are the color, weak,and electromagnetic charges, using the unconventional convention thata = g^2 rather than a = g^2/4pi for charge g.

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

The D4-D5-E6model predicts that e = 0.085 = sqrt 1/137.03608 at the lowcharacteristic energies for QED.

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

The D4-D5-E6 model predicts that w = 0.50 = sqrt 0.2534577 at thecharacteristic energy for the SU(2) weak force, the mass-energy rangeof 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 completelyconfined.

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 isconsidered 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, butthe increase of Nf is taken into acccount in the renormalizationcomputations by setting c = 0.79 at 0.245 GeV.

The D4-D5-E6 model predicts the value c = 0.79 = sqrt 0.6286 atthe characteristic energy LQCD for QCD.

As shown by the above figure, the SU(3) color charge c is set at0.79 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 hasnoted 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 colorforce strength at the Z peak of about 0.11 with Lambda_QCD = 200 MeV+/-.

The low energy results and lattice calculations are closer to thetree 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'spaper indicates that there may be problems with Standard Modelglobal fit methods used at CERN, which could explain their high(relative to the D4-D5-E6 model) values of thetruth quark mass.

Patrascioiu and Seiler propose to resolve the discrepancy between high energy observation, giving  ALPHA_s = 0.123, and low energy observations that extend by the renormalization group to ALPHA_s = 0.113 and are closer to the D4-D5-E6 model valuethat 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 thatI used to calculate ALPHA_s is not valid, so that ALPHA_s = 0.106 isnot the true value for 91 GeV under the D4-D5-E6-E7model.

The true 91 GeV value for the D4-D5-E6-E7model ALPHA_s may be higher, closer to the high-energyexperimental 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 theirpaper, Can Renormalization Group Flow End in a Big Mess?,hep-th/0304178, say: "... The field theoretical renormalizationgroup equations have many common features with the equations ofdynamical systems. In particular, the manner how Callan-Symanzikequation ensures the independence of a theory from its subtractionpoint is reminiscent of self-similarity in autonomous flows towardsattractors. Motivated by such analogies we propose that besidesisolated fixed points, the couplings in a renormalizable field theorymay also flow towards more general, even fractal attractors. Thiscould lead to Big Mess scenarios in applications tomultiphase systems, from spin-glasses and neural networks tofundamental ... theory. We argue that ... such chaotic flows ... poseno obvious contradictions with the known properties of effectiveactions, the existence of dissipative Lyapunov functions, and eventhe strong version of the c-theorem. We also explain the difficultiesencountered when constructing effective actions with chaoticrenormalization group flows and observe that they have many commonvirtues with realistic field theory effective actions. We concludethat if chaotic renormalization group flows are to be excluded,conceptually novel no-go theorems must be developed. ... in theclassical Yang-Mills theory chaotic behaviour has already been wellestablished ... Consequently such chaotic behaviour will not beconsidered here. Obviously, a chaotic RG flow also necessitates theconsideration of field (string) theories with at least threecouplings. In the present article we shall be interested in thepossibility of chaotic RG flows in the IR limits of quantum field andstring theories. ... we consider limit cycles from the point of viewof RG flows, and inspect vorticity as a RG scheme independent toolfor describing multicoupling flows. ... we explain how to constructmodel effective actions from the beta-function flows. In particular,we explain how the construction fails in case of chaotic flows andsuggests this parallels the problems encountered in constructingactual field theory effective actions. This also explains why it isvery hard to construct actual field theory models with chaotic RGflow. ...".


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