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Here is how the mass ratios work:
It is interesting that
which is very close to
the ratio of the geometric part of the Weak Force Strength to theElectromagnetic Fine Structure Constant is 0.253477 / ( 1 / 137.03608) = 34.7355.
Before I read quant-ph/9806009,
by Guang-jiong Ni of the Department of Physics, Fudan University,Shanghai 200433, P. R. China, and the related paper hep-ph/9801264by Guang-jiong Ni, Sen-yue Lou, Wen-fa Lu, and Ji-feng Yang,
I did not correctly understand the Higgs mechanism. Therefore, inmy earlier papers I had wrongly stated that the D4-D5-E6-E7-E7-E8VoDou Physics model gives a Higgs scalar mass of about 260 GeV and aHiggs scalar field vacuum expectation value of about 732 GeV. Inow see that my earlier values were wrong, and that the correctvalues under the D4-D5-E6-E7-E8 VoDou Physics model are a Higgsscalar mass of about 146 GeV and a a Higgs scalar field vacuumexpectation value of about 252 GeV. The fault was not with theD4-D5-E6-E7-E8 VoDou Physics model itself, but with my incorrectunderstanding of it with respect to the Higgs mechanism.
Guang-jiong Ni, Sen-yue Lou, Wen-fa Lu, and Ji-feng Yang,in hep-ph/9801264,used a new Regularization-Renormalization (R-R) method to calculatethe Higgs mass in the Standard Model to be about 140 GeV. Guang-jiongNi has further described the R-R method in quant-ph/9806009.
When the R-R method of Ni is applied to the D4-D5-E6-E7-E8VoDou Physics model, it is seen that, at tree level,
To see how the R-R method of Ni, Lou, Lu, and Yang works,consider Integrals over 4-dim SpaceTime or its dual 4-dim MomentumSpace of factors (ignoring factors like 2 and pi) like
for which the Integral I diverges as the Integral of dK^4 /K^4
or as definite integral from 0 to R of dK / K in the limit of Rgoing to infinity, which is a logarithmic divergence of the most weaktype. Other Integrals encountered in physics give linear, quadratic,and other higher divergences.
Their new R-R method deals with the divergence (for example, ofthe Integral I) by:
1 - taking the partial derivative with respect to M^2 of thedivergent Integral of
to get the convergent Integral d I / d M^2 of
whose value is (ignoring factors like 2 and pi) d I / d M^2 = 1 /M^2.
2 - after working with the convergent Integral, then going back tothe original Integral by Integrating the convergent expression d I /d M^2 = 1 / M^2 with respect to M^2, getting
where C1 = - ln M1^2 is a Constant of Integration and M1 has thedimension of mass.
3 - Use the Chain Approximation to derive a renormalized mass MR =M + dM.
then dM = ( Alpha / 4 Pi ) M ( 5 - 3 ln M^2 / M1^2 )
where Alpha = e^2 / 4 Pi is the Electromagnetic Fine StructureConstant.
4 - Fix the constant M1 so that the parameter M in the originalLagrangian is interpreted as the observed renormalized mass MR. Thiscannot be done by Perturbative Quantum Field Theory, so M1 should befixed by experiment. The condition dM = 0 gives
5 - When the particle is off the mass-shell so that p^2 =/= M^2,combine the above with other Feynman Diagram Integrals of QuantumElectroDynamics to calculate such things as the Lamb shift of the2S(1/2) state with respect to the 2P(1/2) state in Hydrogen, whichcan then be understood as a mass modification of the electron boundin the Hydrogen atom.
As Ni says in quant-ph/9806009,"... by means of perturbative QFT, we can evaluate the massmodification but never the mass generation. ... we re replace thedivergence by arbitrary constant M1, then the latter is fixed by themass M measured in the experiment. ... the crucial point lies in thefact that we should act from the beginning, act before thecounterterm in introduced, act until the bottom is reached. That is,to take the derivative of integral I with respect to M^2 (or to aparameter Sigma added by hand, say M^2 -- goes to -- M^2 + Sigma)enough times until it becomes convergent, then perform the same timesof integration with respect to M^2 ( Sigma then setting Sigma -- goesto -- 0 again) for going back to I. Now instead of divergence,weobtain some arbitrary constant Ci ... to be fixed ... Each Ci has itsunique meaning and role ..."
The difference between the D4-D5-E6-E7-E8 VoDou Physics modelHiggs mass of 146 GeV and the Ni, Lou, Lu, and Yang calculated valueof 138 GeV is due to these facts:
where d is the 8-dim covariant derivativeP is the scalar field F is the Spin(8) curvatureS' and S are half-spinor fermion spaces D is the 8-dim Dirac operatorGF is the gauge-fixing termGG is the ghost term
After dimensional reduction to 4-dim SpaceTime, the scalar Fh8 /\*Fh8 term becomes the Integral over 4-dim Spacetime of
where cross-terms are eliminated by antisymmetry of the wedge /\ product and 4 denotes 4-dim SpaceTime and I denotes 4-dim Internal Symmetry Space
The first term is the integral over 4-dim SpaceTimeof
Since they are both SU(2) gauge group terms, this term merges intothe SU(2) weak force term that is the integral over 4-dim SpaceTimeof Fw /\ *Fw (where w denotes Weak Force).
The third term is the integral over 4-dim SpaceTime of theintegral over 4-dim Internal Symmetry Space of
The third term after integration over the 4-dim Internal SymmetrySpace, produces, by a process similar to the MayerMechanism developed by MeinhardMayer, terms of the form
where L is the Lambda term, P is the Phi scalar complex doubletterm, and M is the Mu term in the wrong-sign Lamba Phi^4 theorypotential term, which describes the Higgs Mechanism. The M and L arewritten above in the notation used by Kaneand Barger and Phillips. Ni,and Ni, Lou, Lu, and Yang, use a differentnotation
so that the L that I use (following Kane and Barger and Phillips)is different from the Ln of Ni, andNi, Lou, Lu, and Yang, and the P that I useis different from Pn, and the 2 M^2 that I use is ( 1 / 2 )Sigma.
Proposition 11.4 of chapter II of volume1 of Kobayashi and Nomizu states that
where P takes values in the SU(2) Lie algebra. If the action ofthe Hodge dual * on P is such that *P = -P and *[P,P] =[P,P], then
If integration of P over the Internal Symmetry Space gives P =(P+, P0), where P+ and P0 are the two components of the complexdoublet scalar field, then
which is the Higgs Mechanism potential term.
The value of the fundamental mass scalevacuum expectation value v of the Higgs scalar field is set inthe D4-D5-E6-E7 model as the sum of the physical masses of the weakbosons, W+, W-, and Z0, whose tree-level masses will be 80.326 GeV,80.326 GeV, and 91.862 GeV, respectively, and so that the electronmass will be 0.5110 MeV.
The resulting equations, in my notation (and that of Kaneand Barger and Phillips), are:
In their notation, Ni, Lou, Lu, and Yanghave 2M^2 = (1/2) Sigma and P^2 = 6 Sigma / Ln, and for thetree-level value of the Higgs scalar particle mass Mh they have Mh^2/ Pn^2 = Ln / 3.
By combining the non-perturbative Gaussian Effective Potential(GEP) approach with their Regularization-Renormalization (R-R)method, Ni, Lou, Lu, and Yang findthat:
Mh and Pn are the two fundamental mass scales of the Higgsmechanism, and
the fundamental Higgs scalar field mass scale Pn of Ni,Lou, Lu, and Yang is equivalent to the vacuum expectation value vof the Higgs scalar field in my notation and that of Kaneand Barger and Phillips, and
Ln (and the corresponding L) can not only be interpreted as theHiggs scalar field self-coupling constant, but also can beinterpreted as determining the invariant ratio between the masssquares of the Higgs mechanism fundamental mass scales, Mh^2 and Pn^2= v^2. Since the tree-level value of Ln is Ln = 1, and since Ln / 3 =Mh^2 / Pn^2 = Mh^2 / v^2 = 2 L, the tree-level value of L is L = Ln /6 = 1 / 6, so that, at tree-level
so that it is equal to the sum of the physical masses of the weakbosons, W+, W-, and Z0, whose tree-level masses will be 80.326 GeV,80.326 GeV, and 91.862 GeV, respectively, and
so that the electron mass will be 0.5110 MeV.
Then, the tree-level mass Mh of the Higgs scalar particle is givenby
The Critical Mass (or Energy, or Temperature) Mssb for restorationof the Spontaneous Symmetry Breaking (SSB) symmetry, which is Mssb =Mh sqrt(12/Ln), so that, for the tree-level value Ln = 1,
The High-Energy Singularity of the Higgs Mechanism model, Msing,beyond which the Higgs field vanishes, and the Maximum Energy Scale,Mmax, can be calculated in the Higgs Mechanism model. The fact thatthe Higgs field vanishes above Msing and Mmax may justify regardingthe Higgs Mechanism model as a low energy effective theory, just asthe D4-D5-E6-E7-E8 VoDou Physics model isfundamentally a low (with respect to the Planck energy) energyeffective theory. The values calculated by Ni,Lou, Lu, and Yang are
The Planck energy is
The Complex Doublet P = ( P+, P0) = (1/sqrt(2)) ( P1 + iP2, P3 +iP4 ) = (1/sqrt(2)) ( 0, v + H ), so that
where v is the vacuum expectation value and H is the realsurviving Higgs field.
The value of the fundamental mass scalevacuum expectation value v of the Higgs scalar field is in theD4-D5-E6-E7 physics model set to be 252.514 GeV so that the electronmass will turn out to be 0.5110 MeV.
Now, to interpret the term
in terms of v and H, note that L = M^2 / v^2 and that P =(1/sqrt(2)) ( v + H ), so that
= (1/16) ((M^2 / v^2) ( v + H )^4 - (1/8) M^2 ( v + H )^2 =
= (1/4) M^2 H^2 - (1/16) M^2 v^2 ( 1 - 4 H^3 / v^3 - H^4 / v^4)
Disregarding some terms in v and H,
The second term after integration over the 4-dim Internal SymmetrySpace, produces, by a process similar to the MayerMechanism, terms of the form
where P is the Phi scalar complex doublet term and d is thecovariant derivative.
Proposition 11.4 of chapter II of volume1 of Kobayashi and Nomizu states that
where P(X) takes values in the SU(2) Lie algebra. If the Xcomponent of Fh4I(X,Y) is in the surviving 4-dim SpaceTime and the Ycomponent of Fh4I(X,Y) is in the 4-dim Internal Symmetry Space, thenthe Lie bracket product [X,Y] = 0 so that P([X,Y]) =0 and therefore
Integration over Internal Symmetry Space of (1/2) dx P(Y) gives(1/2) dx P, where now P denotes the scalar Higgs field and dx denotescovariant derivative in the X direction.
Taking into account the Complex Doubletstructure of P, the second term is the Integral over 4-dimSpaceTime of
= (1/4) (1/2) d ( v + H ) /\ *d ( v + H ) = (1/8) dH dH + (someterms in v and H)
Disregarding some terms in v and H,
Combining the second and third terms, since the first term ismerged into the weak force part of the Lagrangian:
In the D4-D5-E6-E7-E8 VoDou Physicsmodel, the geometric part of the weak force strength, and thegeometric weak charge that is its square root, are:
In more customary particle physics notation, such as that found inKane, there are two weak charges, g1 andg2, such that their squares are weak force strengths. In theD4-D5-E6-E7 model, the geometric weak charge is the average of thecustomary two weak charges g1 and g2:
Combining some aspects of the D4-D5-E6-E7 model and some aspectsof the customary picture gives tree-level estimate results that areoff by a few percent. Estimated Weak Boson masses areapproximately
Mz = sqrt( g1^2 + g2^2 ) v / 2 = 93.98 GeV
Some other relations given by Kane, and the results of using inthem some D4-D5-E6-E7 model values, and the estimates immediatelyabove, are
from the Weinberg angle ThetaW: g2 = e / sin(ThetaW) = 0.6247
g1 = e / cos(ThetaW) = 0.3463
the Fermi constant, GF = sqrt(2) g2^2 / 8 Mw^2 = 1.11 x 10^(-5)GeV^(-2)
For comparison, the D4-D5-E6-E7 model value of the Fermi constantis
= sqrt(80.326^2 + 80.326^2 + 92.862^2) = 146.09298 GeV.
The tree level mass of a Higgs scalar, about 146 GeV, is somewhat higher than, but roughly similar to, the tree level Truth quark mass of about 130 GeV.
the sum of the tree level masses mW+ + mW- + mZ0of the 3 weak bosons W+, W-, and Z0, that is, the physical weak bosons below the Higgs mass scale, is the fundamental energy level vacuum expectation value vof the Higgs scalar field. To give the tree-level particle masses mW+ = mW- = 80.326 GeV and mZ0 = 91.862 GeV (as well as the other calculatedparticle masses and force strength constants of the D4-D5-E6-E7 model), P is set equal to 252.514 GeV. The D4-D5-E6-E7 model assumed value v = about 80+80+92 = 252 GeV.
The Higgs Vacuum Expectation Value of 252 GeV is roughly theCompressibility of theAether and the SuperpositionSeparation of an entire single Tubulin in the Brain,and is close to the tree level mass of the TruthQuark T-T(bar) Meson of about 260 GeV. It is also ofthe same order of magnitude as the geometric mean (about 650 GeV) ofthe Planck Energy (10^19 GeV) and theHydrogen Lamb Shift Energy (4.3x 10^(-5) eV = 4.3 x 10^(-14) GeV (see Weinberg, The QuantumTheory of Fields, Vol. I, p. 593).
Mw, the square root of the sum of the squares of the tree level masses of the 3 weak bosons W+, W-, and Z0, that is, the physical weak bosons below the Higgs mass scale, is 146.09298 GeV, which is very close to the mass of the tree level Higgs scalar mass of 145.789 GeV. The tree level mass of a pair of Higgs scalars, about 292 GeV, is somewhat higher than, but roughly similar to, the fundamental energy level vacuum expectation value vof the Higgs scalar field, about 252 GeV, and the truth quark T-T(bar) meson mass of about 260 GeV.
Each quark in the T-Tbar can decay very rapidly to W + b + X (where X is just an indication for other miscellaneous stuff).
Note that , roughly,
In hep-ph/0102137,12 Februay 2001, G. Degrassi says: "... The last months of the year2000 ... seems a good moment to try to review what we (do not)know about the Higgs.
... Given these two pieces of information it is often said thatone of the greatest achievement of LEP has been to have pin down theHiggs mass between 113 (from the 95% C.L. lower bound of thedirect searches) and 170 GeV (from the 95% C.L. upper bound ofthe global fit to electroweak data). ...
Formulae ...[ that are used for indirect ] Higgs massinference from precision measurements ... following a Bayesianapproach ... to construct f( mH | ind. ), the p.d.f. of the Higgsmass conditioned by this indirect information under the assumption ofthe validity of the S.M.... are very accurate for 75 < MH < 350GeV with the other parameters in the ranges 170 < Mt < 181 GeV... 0.113 > ALPHAs(MZ) > 0.123 ... The experimental inputs Iuse to construct f( mH | ind. ) are: s2eff = 0.23146 +/- 0.00017, MW= 80.419 +/- 0.038 GeV, ... Mt = 174.3 +/- 5.9 GeV, ALPHAs(MZ) =0.119 +/- 0.003. ... Table 1 summarizes the results ...
... As expected, the inclusion of the direct search informationdrifts the p.d.f. towards higher values of MH by cutting regionsbelow 110 GeV. ...".
In my opinion, the fact that the indirect prediction of a 90 to105 GeV Higgs mass is shown to be wrong by failure to detect a Higgswith a direct search up to 110 GeV indicates that either
In the paper hep-ph/0204345
Marko B. Popovic says:
The range of Higgs masses in the vicinity of 140 GeV satisfies all theoretical requirements for healthy physics up to the Plank scale
... the theoretical and logical necessity for the existence of ... Supersymmetry, technicolor, extra dimensions and other exotic scenarios around the TeV scale ... is strongly questioned here ... experimental data and theoretical knowledge strongly point towards the SM as the best candidate for the physics that will be observed in near future experiments. ...
... The vacuum energy problem is ... cited as a serious obstacle to the SM. However, the vacuum energy problem only guides toward a conflict between the classical general theory of relativity and its, yet to be understood, "quantum-compatible" version.....
.... The well-known non-zero vacuum expectation value (vev) of the scalar Higgs field, vEW = 2.462 x 10^2 GeV, sets the scale of the electroweak interactions. Another well-known energy scale is the Planck scale, /\planck = 1.2 x 10^19 GeV, probably setting the scale of the unified theory incorporating gravity. Obviously, many orders of magnitude separate these numbers. Traditionally, this fact is considered a serious obstacle for the SM. The problem termed hierarchy problem expresses doubts that the SM alone can provide a good physical description over such a broad range of energies. ...
... Only when united with the fine-tuning problem can the hierarchy look persuasive as an obstacle to the SM at high energies.... To characterize the fine-tuning problem it is useful to introduce the dimensionless mass parameter ... mu = mH(/\)^2 / /\^2 . The parameter mH(/\) represents a renormalized Higgs mass at the cut-off energy scale /\ ...
... Figure 1 ...
... Illustration of the fine-tuning problem in traditional sense (top sketch) and in the more realistic SM case (bottom sketch). The purpose of these drawings is to show the qualitative distinctions between the functional forms of the two cases; by no means should the two plots be quantitatively compared or separately analyzed. The gauge and Yukawa couplings are labeled with gi and gf respectively while mu_a and mu_b identify the two possible evolutions of dimensionless mass parameter. ...
... from fine-tuning ... in traditional sense ... it may be concluded that the SM can be a good description only up to very small energies, maybe ten or so times larger than a physical Higgs mass! ...
... However, there is no fundamental physical or logical basis for the claim that the loop corrections need to be smaller than the tree level value!... the logic behind fine-tuning problem ... in traditional sense ... fails. And the reason is rather simple - the coupling constants g^2 and lambda run logarithmically, and moreover, the parameter mu intersects with g^2 . Therefore, the more the theory is finely tuned, the smaller a hierarchy (see Fig. 1.b). And the fine-tuning is shown to be benign. ...
... The simple truth is that the SM already gives the best explanation! ...
... one should be ambitious and try to explain the hierarchy up to the Planck (or GUT) scale with the SM structure alone. ...
... In Fig. 4 the dimensionless mass parameter mu is shown over the range, 140-230 GeV, of Higgs masses. Due to the stability and perturbativity constraints, only the 140-170 GeV curves are accepted up to the Planck scale. 5 An exciting feature is easily observed; the mu's in the vicinity of 140 GeV are negative over a whole range of energies, and they very slowly approach zero in the high energy limit. The negativity of dimensionless parameter mu guarantees the good behavior of the scalar propagator at high energies. In other words, the negativity of the mass squared suggests that the mass parameter in the propagator at high energies should be set to zero and that the theory is surely not breaking down below the Planck scale.
The stability curve roughly traces the energy scale at which quartic coupling lambda goes to zero (more pedantically the scale at which deeper minima form) as a function of physical Higgs mass. Therefore, in Fig. ... 4 it is seen that the lambda's for Higgses in the vicinity of 140 GeV are positive while mu's are negative over the whole energy range. Therefore, the minimum of the effective potential can not be formed in this range. ... That is the SM explanation of the fictitious hierarchy problem! ...
... It should be noted that at the Planckian energies both dimensionless quantities characterizing scalar potential, i.e. lambda and mu, happens to be suspiciously close to zero! ...
... Interestingly, the Higgs masses in the vicinity of 140 GeV may happen to be those preferred by the fine-tuning quantities presented in Figs. 2 and 3 as well. ...
... mu for Higgses in the vicinty of 140 GeV runs almost logarithmically (i.e. as expected) all the way up to the Planck scale. The slowly varying parameter alpha in Fig. 3 is tightly confined to the region near the value of 0.05.
Why does experimental data and theoretical knowledge strongly point towards the SM as the best candidate for the physics that will be observed in near future experiments? The answer is drawn from the following facts:
- -The SM is the correct "theory" at low energies.
- -The hierarchy and fine-tuning are not serious problems of the SM.
- -The Planck scale, as an important gravity related dimensional quantity, may be addressed by the current theoretical framework. The range of Higgs masses in the vicinity of 140 GeV satisfies all theoretical requirements for healthy physics up to the Plank scale:
- perturbativity and
- negativity of the dimensionless mass parameter mu.
- -The range of Higgs masses in the vicinity of 140 GeV falls inside 1 sigma of the ... electroweak precision data (ewpd) ... analysis preferred range (or 2 sigma with Z -> bbar b forward backward asymmetries excluded.
- -Current new physics models at the 1 TeV scale are neither minimal nor required by some burning theoretical necessity. ...
... If Higgs happens to [weigh] around 140 GeV ... Nature has indeed chosen the large SM high-energy desert in front of us ...".
Such a desert is consistent with the possibility that theTruth Quark, through its strong interaction with Higgs Vacua, mayhave two excited energy levels at 225 GeV and 173 GeV, above aground state at 130 GeV. The 173 GeV excited state may existdue to appearance of a Planck-energy vaccum with < phi_vac2 > =10^19 GeV in addition to the low-energy Standard Model vacuum with< phi_vac1 > = 252 GeV.
Weinberg, The Quantum Theory of Fields (2 Vols.), Cambridge1995,1996.
Barger and Phillips, Collider Physics, updated edition, AddisonWesley 1997.
Gockeler and Schucker, Differential Geometry, Gauge Theories, andGravity , Cambridge 1987.
Kane, Modern Elementary Particle Physics, updated edition, AddisonWesley 1993.
Kobayashi and Nomizu, Foundations of Differential Geometry, vol.1, John Wiley 1963.
Kobayashi and Nomizu, Foundations of Differential Geometry, vol.2, John Wiley 1969.
Mayer,Hadronic Journal 4 (1981) 108-152, and also articles in NewDevelopments in Mathematical Physics, 20th Universitatswochen furKernphysik in Schladming in February 1981 (ed. by Mitter andPittner), Springer-Verlag 1981, which articles are:
Ni, To Enjoy the Morning Flower in the Evening - What does theAppearance of Infinity in Physics Imply?, quant-ph/9806009.
Ni, Lou, Lu, and Yang, hep-ph/9801264.
Particle Properties from theParticle Data Group at LBL
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