( contributed to SoutheastSection 2002 APS Meeting, October 31 - November 2, 2002, Auburn,Alabama )
For pdf format (450k), Click Here or go tophysics/0207095.
The material as presented to SESAPS 2002 is at physics/0207095and at the 2002 pdf version. This html pagehad been updated to include the August 2003material and further material (including some,identifiable by its original LaTeX format, from hep-ph/9708379v2) is in a 2003 pdf version HEREand on the web as EXT-2003-087( which supersedes earlier version EXT-2003-083) on CERN CDS. andv3 of physics/0207095. [Latermaterial is in [ ]. ]
Note that my theoretical model ( which Icall the D4-D5-E6-E7-E8VoDou Physics Model , because it is basedon the Liealgebras D4,D5,E6,E7,E8and on Cliffordalgebra periodicityrelated to IFA=VoDou ) meets Einstein'sCriterion:
"... a theorem which at present can not be based upon anythingmore than upon a faith in the simplicity, i.e., intelligibility, ofnature: there are no arbitrary constants ... that is to say,nature is so constituted that it is possible logically to lay downsuch strongly determined laws that within these laws onlyrationally completely determined constants occur (not constants,therefore, whose numerical value could be changed without destroyingthe theory). ...".
A theoretical model based on the D4Lie Algebra and HermitianSymmetric Spaces D5 /D4xU(1) and E6 /D5xU(1) and related ShilovBoundaries, along with combinatorial relations, allows thecalculation of ratios of particle masses:
and
as well as ratios of force strength constants:
The theoretical calculated electromagnetic fine structure constant= 1/137.03608 solves Feynman's mystery (QED, Princeton 1985,1988, at page 129): "... the inverse of ... about 137.03597... [the] square [of] ... the amplitude for a realelectron to emit or absorb a real photon ... has been amystery ever since it was discovered more than fifty years ago,and all good theoretical physicists put this number up on theirwall and worry about it. ...".
The Truth Quark constituent mass(tree-level) calculation of about 130 GeV had been made by February1984.
About 10 years later, in April 1994, Fermilab officially announcedobservation of the Truth Quark.
Fermilab's analysis of the events gives a T-quark mass of about170 GeV.
My independent analysis of the sameFermilab events gives a Truth Quark mass of about 130 GeV,consistent with the theoretical tree-level calculation.
The local Lagrangian of the theoretical model is based on thestructure of the real Cl(1,7) Clifford algebra which, through 8-foldperiodicity, might be used to construct a Generalized Hyperfinite II1von Neumann Algebra factor.
In April 1994, CDF at Fermilab (in FERMILAB-PUB-94/097-E) reporteda T-quark mass of 174 (+/-10)(+13/-12) GeV. The data analyzed by CDFincluded a 26-event histogram for Semileptonic events with W + (3 ormore) jets, without b-tags, which is Figure 65 of the report. In thehistogram, the green bars are in the140-150 GeV bin, close to the 130 GeV range that is deemedinsignificant by Fermilab's analysts, but considered by me torepresent the Truth quark; the cyan-bluebars represent bins in the 173 GeV range containing Semileptonicevents interpreted by Fermilab as Truth Quarks; and themagenta bars represent bins in the 225GeV range containing Semileptonic events not considered by Fermilab'sanalysts or by me as corresponding to either 130 GeV or 173 GeV TruthQuarks.
The peak of 8 events in the 140-150 GeV bin, shown ingreen, were excluded from the analysisby CDF on the grounds that (see page 140 of the report) "... the binwith masses between 140 and 150 GeV/c^2 has eight events.
We assume the mass combinations in the 140to 150 GeV/c^2 bin represent a statistical fluctuation since theirwidth is narrower than expected for a top signal. ...".
If the 140-150 GeV peak were only a statistical fluctuation seenby the CDF detector, one would not expect to find such a peakrepeated in the data seen by the D0 detector at Fermilab. However, inMarch 1997, D0 (in hep-ex/9703008) reported a T-quark mass of 173.3GeV (+/- 5.6 stat +/- 6.2 syst), based on data including a histogramsimilar to Figure 65 of the April 1994 CDF report which is Figure 3of the D0 report, to which I have added colors as describedabove:
Some of the D0 histogram events, shown incyan-blue, are are in the 150-190 GeVrange and do support the CDF analysis. However, similar to the140-150 GeV bin peak seen and thrown out by CDF,there is a peak of 5 events in the 130-140 GeVbin, shown in green, that wereexcluded from the analysis by D0. I did not see in the D0 report anexplicit discussion of the 5-event peak in the 130-140 GeV bin.
Tagged semileptonic events may be a more reliable measure ofT-quark mass, although there are fewer of them, so that statisticsare not as good.
CDF (in hep-ex/9801014, dated 30 September 1997) reported aT-quark mass of 175.9 +/- 4.8(stat.) +/- 4.9(syst.) GeV based onevents that were either SVX tagged, SVX double tagged, or untagged.However, CDF analysis of tagged semileptonicevents (14 of them) gave a T-quark Mass of 142 GeV (+33, -14),as shown in their Figure 2, which is a plot of events/10 GeV bin vs.Reconstructed Mass in GeV:
D0 (in hep-ex/9801025) also analyzed tagged semileptonic events,with the result shown in their figure 25:
The figure shows 3 events in the 130-150 GeVrange, one event in the 170-180 GeV bin, and one event in the200-210 GeV bin. According to footnote 10 of hep-ex/9801025,
One event which would have otherwisepassed the cuts, event (95653; 10822), wasremoved by D0 from its analysis because it was selected by thedilepton mass analysis. If this event is treated as a l + jetscandidate,
Dilepton events may be the most reliable measure of T-quark mass,although they are the least numerous type of event, so thatstatistics are not so good.
In October 1998 (in hep-ex/9810029) CDF analyzed 8 dilepton eventsand reported a T-quark mass of 167.4 +/- 10.3(stat) +/- 4.8(syst)GeV. Figure 2 of the report shows the 8 events:
I have colored green the events with T-quark mass less than 160GeV, and blue the events with T-quark mass greater than 160 GeV. Thehep-ex/9810029 CDF report stated that it "... supersedes ourpreviously reported result in the dilepton channel ...".
The superseded previous CDF dilepton report (hep-ex/9802017)analyzed 9 events out of a total of 11 events, which 11 events areshown on the following histogram:
I have colored green the events with T-quark mass less than 150GeV, and blue the events with T-quark mass greater than 150 GeV.
Note first, that in the earlier 11-event histogram 5 events areshown as greater than 150 GeV, but only 4 events are shown as greaterthan 160 GeV, while in the 8-event revised histogram 5 events areshown as greater than 160 GeV. This indicates to me that some changesin the analysis have shifted the event mass assignments upward byabout 10 GeV.
Note second, that the earlier 11-eventhistogram contains 3 events from 120-140 GeV that are omitted fromthe 8-event revised histogram.
D0 (in hep-ex/9706014 and hep-ex/9808029) has analyzed 6 dileptonevents, reporting a T-quark mass of about 168.4 GeV. The 1997 UCBerkeley PhD thesis of Erich Ward Varnes which can be found on theweb at http://wwwd0.fnal.gov/publications_talks/thesis/thesis.htmlcontains details of the events and the D0 analyses. Each of the 6events has its own characteristics. In this letter I will onlydiscuss one of them, Run 84676 Event 12814, an electron-muon dileptonevent. This figure
from page 159 of the Varnes thesis, shows a T-quark masslikelihood plot calculated by the neutrino weighting algorithm.
The solid line is the plot if all 3 jets are included, and thedashed line is the plot if only 2 of the jets are included byexcluding the third (lowest transverse energy) jet.
the 2-jet interpretation supports a130-140 GeV mass analysis that favors my calculated mass of about 130Gev.
Consider the Higgs mass - Truth Quark mass plane, based onFig. 3 of Froggatt's paper hep-ph/0307138:
The green dot corresponds to a 130GeV Truth Quark low-energy Standard Model one-vacuum ( < phi_vac1> = 252 GeV ) ground state that is well within the StabilityRegion below the Triviality Bound and above the Vacuum Stabilitybound for a Standard Model with a high-energy cut-off that goes allthe way to the Planck energy 10^19 GeV.
If accelerator-event collisions deposit up to 95 GeV of extraenergy into a Truth Quark, it will be pumped up along thered curve within the Stability Regionuntil it hits the Standard Model Critical Point at themagenta dot, where it will be a StandardModel Truth Quark excited state with mass-energy 225 GeV.
Since the 225 GeV Standard Model Truth Quark excited state isat a Critical Point, which is by definition on the Vacuum Stabilitycurve, the one low-energy Standard Model vacuum ( < phi_vac1> = 252 GeV ) is no longer stable, and a new vacuum phi_vac2forms.
In the theoretical model, the newvacuum phi_vac2 appears at the Planckenergy, where the low-energy Standard Model, Higgs, and Gravitywith 4-dimensional Physical Spacetime makes a transition to a moreunified structure with a Spin(1,7) gauge boson Lie algebra, fermionspinors from a Cl(1,7) Clifford algebra, an 8-dimensional Spacetime,and a corresponding high-energy vacuum with < phi_vac2 > =10^19 GeV = Planck energy.
When the second vacuum phi_vac2 appears, the structure of theStandard Model is altered ( as explained by Froggatt) so that the new Critical Point is at a 173 GeV Truth Quark mass, sothe magenta dot 225 GeV excited TruthQuark state decays, moving along the bluecurve along the Vacuum Stability bound to an intermediateexcited state at the cyan dot at a TruthQuark mass of 173 GeV.
If the region around the Truth Quark does not have enoughenergy-density to maintain the second Planck energy phi_vac2 vacuum (as is the case with present-day colliders that can do Truth Quarkexperiments ) then the cyan dot 173 GeVintermediate excited Truth Quark state decays along thegreen curve to the more stable TruthQuark low-energy Standard Model one-vacuum ground state atconstituent mass of 130 GeV, the greendot.
As a result:
There are some actual Fermilab events that seem to me to show thatprocess in action. They are dilepton events, for which I wouldnormally expect to see 2 jets in addition to the 2 leptons. However,some of the 6 D0 dilepton events describedin the 1997UC Berkeley PhD thesis of Erich Ward Varnes and in hep-ex/9808029have 3 jets, and I will here discuss the kinematics of two of thoseevents to illustrate the above-described Truth Quark - Higgs - Vacuaprocess. The kinematics of those two events are given in Appendix B.2of the 1997UC Berkeley PhD thesis of Erich Ward Varnes. (Similar kinematicdata are presented in D0August 1998, hep-ex/9808029.) In the Varnes Kinematics tables,there are two numbers for each jet: one is energy after CAFIXcorrections; and the second (in parentheses) is energy afterpost-CAFIX corrections.
The first dilepton event, Run 84395, Event 15530 ( mu mu ), asanalyzed using the neutrino weighting algorithm,
has, if all 3 jets are included ( the solid line in the graph ),energy around 200 GeV, corresponding to the Standard Model CriticalPoint Truth Quark excited state at the magentadot. If only the 2 highest energy jets are included ( thedashed line in the graph ), it has energy around 170 GeV,corresponding to the 2-vacuum intermediate excited Truth Quark stateat the cyan dot, and the energy of thethird jet would correspond to the decay down theblue curve along the Vacuum Stabilitybound. This same event, if analyzed usingthe matrix-element weightingalgorithm that, according to hep-ex/9808029,"... is an extension of the weight proposed in [R.H. Dalitz andG.R. Goldstein, Phys. Rev. D45, 1531 (1992)] ...",
indicates the eventual decay into the Truth Quark low-energyStandard Model one-vacuum ground state at constituent mass of 130 GeVat the green dot.
The other dilepton event that I will discuss is Run 84676, Event12814 ( e mu ), as analyzed using the neutrino weightingalgorithm:
It has, if all 3 jets are included ( the solid line in the graph), energy around 170 GeV, corresponding to the 2-vacuum intermediateexcited Truth Quark state at the cyandot. If only the 2 highest energy jets are included ( thedashed line in the graph ), it has energy around 130 GeV,corresponding to the Truth Quark low-energy Standard Model one-vacuumground state at constituent mass of 130 GeV at thegreen dot, and the energy of the thirdjet would correspond to the decay down thegreen curve.
Cl(1,7)Clifford Algebra,8-Periodicity,and aReal Hyperfinite von Neumann Algebrafactor.
Cl(2N;C) = Cl(2;C) x ...(N times tensorproduct)... x Cl(2;C)
Cl(2;C) = M2(C) = 2x2 complex matrices
spinor representation = 1x2 complex columnspinors
Hyperfinite II1 vonNeumann Algebra factor is the completionof the union of all the tensor products
Cl(2;C) x ...(N times tensor product)... xCl(2;C)
By looking at the spinor representation, you seethat "the hyperfinite II1 factor is the smallest von Neumann algebracontaining the creation and annihilation operators on a fermionicFock space of countably infinite dimension."
In other words, Complex Clifford Periodicity leadsto the complex hyperfinite II1 factor which represents Dirac'selectron-positron fermionic Fock space.
Now, generalize this to get arepresentation of ALL the particles and fields ofphysics.
Use RealClifford Periodicity to construct aReal HyperfiniteII1 factor as the completion of the unionof all the tensor products
Cl(1,7;R) x ...(N times tensor product)... xCl(1,7;R)
where the Real Clifford Periodicity is
Cl(N,7N;R) = Cl(1,7;R) x ...(N times tensorproduct)... x Cl(1,7;R)
The components of the Real Hyperfinite II1 factorare each
Cl(1,7;R)
[ my convention is (1,7) = (-+++++++) ]
Cl(1,7) is 2^8 = 16x16 =256-dimensional, and has graded structure
1 8 28 56 70 56 28 8 1
D4-D5-E6 Lagrangian Structure.
Construct the Standard Model plus Gravity Lagrangian of thetheoretical model based on the structure of the Cl(1,7) CliffordAlgebra. Cl(1,7) is 2^8 =16x16 = 256-dimensional,and has graded structure
1 8 28 56 70 56 28 8 1
There aretwo mirror imagehalf-spinors, each of the form of a real (1,7)column vector with octonionic structure.
The 1 represents:
the neutrino.
the electron;the red, blue, and green up quarks;
the red, blue, and green down quarks.
Second and third generation fermions come fromdimensional reduction of spacetime, so that
that reduces at lower energies to quaternionicstructures that are
There is a28-dimensonal bivectorrepresentation that corresponds to thegauge symmetry Lie algebraSpin(1,7)
that reduces at lower energiesto:
There is a 1-dimensional scalar representation forthe Higgsmechanism.
1 8 28 56 70 56 28 8 1 = (8+8)x(8+8)
the Integral over8-dimSpaceTime of
dd P' /\ * dd P +F /\*F +S' DS + GF+ GG
where d is the 8-dim covariant derivative S' and S are half-spinor fermion spaces
D is the 8-dim Dirac operator
GF is the gauge-fixing term
GG is the ghost term
the scalar part of the Lagrangian dd P' /\ * dd Pbecomes Fh8 /\ *Fh8
where Fh8 is an 8-dimensional Higgs curvature term.
After dimensional reduction to 4-dim SpaceTime, the scalar Fh8 /\*Fh8 term becomes the Integral over 4-dim Spacetime of
(Fh44 + Fh4I + FhII) /\ *(Fh44 + Fh4I + FhII)=
= Fh44 /\ *Fh44 + Fh4I /\ *Fh4I + FhII /\*FhII
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 SpaceTime of
Fh44 /\ *Fh44
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
FhII /\ *FhII
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.
In my notation (and that of Kane andBarger and Phillips), 2 M^2 is the squareMh^2 of the tree-level Higgs scalar particle mass.
In my notation (and that of Kane andBarger and Phillips), P is the Higgs scalarfield, and its tree-level vacuum expectation value is given by
The value of the fundamental mass scalevacuum expectation value v of the Higgs scalar field is set inthis model as the sum of the physical masses of the weak bosons, 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 be0.5110 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
In the theoretical model, thefundamental mass scale vacuum expectation value v of the Higgs scalarfield is the fundamental mass parameter that is to be set to defineall other masses by the mass ratio formulas of the model.
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 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 scale vacuum expectation value vof the Higgs scalar field is in the theoretical model set to be252.514 GeV so that the electron mass will turn out to be 0.5110MeV.
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,
FhII(X,Y) /\ *FhII(X,Y) = (1/4) M^2 H^2 - (1/16)M^2 v^2
The second term is the integral over 4-dim SpaceTime of theintegral over 4-dim Internal Symmetry Space of
Fh4I /\ *Fh4I
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 Doublet structure of P, the secondterm is the Integral over 4-dim SpaceTime 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,
Fh4I /\ *Fh4I = (1/8) dH dH
Combining the second and third terms, since the first term ismerged into the weak force part of the Lagrangian:
Fh4I /\ *Fh4I + FhII(X,Y) /\ *FhII(X,Y)=
= (1/8) dH dH + (1/4) M^2 H^2 - (1/16) M^2 v^2 =
= (1/8) ( dH dH + 2 M^2 H^2 - (1/2) M^2 v^2)
This is the form of the Higgs Lagrangian in Bargerand Phillips for a Higgs scalar particle of mass
Mh = M sqrt(2) = v / sqrt(3)= 145.789GeV
Three factors determine the probability for emission ofa gauge boson from an origin spacetime vertex to a target vertex: the part of the Internal Symmetry Spaceof the target spacetime vertex that is available for the gauge bosonto go to from the origin vertex; the volume of the spacetime link that is available for the gaugeboson to go through from the origin vertex to the target vertex; and an effective mass factor for forces(such as the Weak force and Gravity)that, in the low-energy ranges of our experiments,are carried effectively by gauge bosons that are notmassless high-energy. In this physics model, force strength probabilities are calculatedin terms of relative volumes of bounded complex homogeneous domains andtheir Shilov boundaries. The bounded complex homogeneous domains correspond toharmonic functions of generalized Laplaciansthat determine heat equations, or diffusion equations; while the amplitude to emit gauge bosons in theHyperDiamond Feynman Checkerboard is a process thatis similar to diffusion, andtherefore also corresponds to a generalized Laplacian. In this theoretical model, all force strengths are represented as ratios with respect to the geometric force strength of Gravity (that is, the force strength of Gravity without using the Effective Mass factor). Therefore, the only free charge, or force strength, parameteris the charge of the Spin(5) gravitons in theMacDowell-Mansouri formalism of Gravity. Note thatthese Spin(5) gravitons are NOT the ordinary spin-2gravitons of the low-energy region in which we live.The charge of the Spin(5) gravitons is taken to be unity, 1,so that its force strength is also unity, 1.All other force strengths are determined as ratioswith respect to the Spin(5) gravitons and each other. The force strength probability for a gauge boson tobe emitted from an origin spacetime HyperDiamond vertexand go to a target vertex is the product of three things: the volume Vol(MISforce) of the target Internal Symmetry Space,that is, the part of the Internal Symmetry Spaceof the target spacetime vertex that is available for the gauge bosonto go to from the origin vertex; the volume Vol(Qforce) / Vol(Dforce)^( 1 / mforce ) of the spacetime link to the target spacetime vertex from the origin vertex; and an effective mass factor 1 / Mforce^2 for forces(such as the Weak force and Gravity)that, in the low-energy ranges of our experiments,are carried effectively by gauge bosons that are notmassless high-energy SU(2) or Spin(5) gauge bosons,but are either massive Weak bosons due to the Higgs mechanismor effective spin-2 gravitons. For other forces, theeffective mass factor is taken to be unity, 1. Therefore, the force strength of a given force is alphaforce = (1 / Mforce^2 \) ( Vol(MISforce)) ( Vol(Qforce) / Vol(Dforce)^( 1 / mforce )) where: alphaforce represents the force strength; Mforce represents the effective mass; MISforce represents the part of the targetInternal Symmetry Space that is available for the gaugeboson to go to; Vol(MISforce) stands for volume of MISforce, and is sometimes also denoted by the shorter notation Vol(M); Qforce represents the link from the originto the target that is available for the gaugeboson to go through; Vol(Qforce) stands for volume of Qforce; Dforce represents the complex bounded homogeneous domainof which Qforce is the Shilov boundary; mforce is the dimensionality of Qforce,which is 4 for Gravity and the Color force,2 for the Weak force (which therefore is considered tohave two copies of QW for each spacetime HyperDiamond link),and 1 for Electromagnetism (which therefore is considered tohave four copies of QE for each spacetime HyperDiamond link) Vol(Dforce)^( 1 / mforce ) stands fora dimensional normalization factor (to reconcile the dimensionalityof the Internal Symmetry Space of the target vertexwith the dimensionality of the link from the origin to thetarget vertex). The Qforce, Hermitian symmetric space,and Dforce manifolds for the four forces are: Gauge Hermitian Type mforce Qforce Group Symmetric of Space Dforce Spin(5) Spin(7) / Spin(5)xU(1) IV5 4 RP^1xS^4 SU(3) SU(4) / SU(3)xU(1) B^6(ball) 4 S^5 SU(2) Spin(5) / SU(2)xU(1) IV3 2 RP^1xS^2 U(1) - - 1 - The geometric volumes needed for the calculations,mostly taken from Hua, are [with unit radius scale] Force M Vol(M) Q Vol(Q) D Vol(D) gravity S^4 8pi^2/3 RP^1xS^4 8pi^3/3 IV5 pi^5/2^4 5! color CP^2 8pi^2/3 S^5 4pi^3 B^6(ball) pi^3/6 weak S^2xS^2 2x4pi RP^1xS^2 4pi^2 IV3 pi^3/24 e-mag T^4 4x2pi - - - - Using these numbers, the results of thecalculations are the relative force strengthsat the characteristic energy level of thegeneralized Bohr radius of each force: Gauge Force Characteristic Geometric Total Group Energy Force Force Strength Strength Spin(5) gravity approx 10^19 GeV 1 GGmproton^2 approx 5 x 10^-39 SU(3) color approx 245 MeV 0.6286 0.6286 SU(2) weak approx 100 GeV 0.2535 GWmproton^2 approx 1.05 x 10^-5 U(1) e-mag approx 4 KeV 1/137.03608 1/137.03608 The force strengths are given at the characteristicenergy levels of their forces, because the forcestrengths run with changing energy levels. The effect is particularly pronounced with the colorforce. The color force strength was calculated using a simple perturbative QCD renormalization group equationat various energies, with the following results: Energy Level Color Force Strength 245 MeV 0.6286 5.3 GeV 0.166 34 GeV 0.121 91 GeV 0.106 Taking other effects, such as Nonperturbative QCD,into account, should give a Color Force Strength of about 0.125 at about 91 GeV
To calculate Weak Boson Masses and Weinberg Angle:
Denote the 3 SU(2) high-energy weak bosons (massless at energies higher than the electroweak unification) by W+, W-, and W0, corresponding to the massive physical weak bosons W+, W-, and Z0. The triplet { W+, W-, W0 } couples directly with the T - Tbar quark-antiquark pair, so that the total mass of the triplet { W+, W-, W0 } at the electroweak unification is equal to the total mass of a T - Tbar pair, 259.031 GeV. The triplet { W+, W-, Z0 } couples directly with the Higgs scalar, which carries the Higgs mechanism by which the W0 becomes the physical Z0, so that the total mass of the triplet { W+, W-, Z0 } is equal to the vacuum expectation value v ofthe Higgs scalar field, v = 252.514 GeV. What are individual masses of membersof the triplet { W+, W-, Z0 } ? First, look at the triplet { W+, W-, W0 } which can be represented by the 3-sphere S^3. The Hopf fibration of S^3 asS^1 --} S^3 --} S^2gives a decomposition of the W bosonsinto the neutral W0 corresponding to S^1 andthe charged pair W+ and W- correspondingto S^2. The mass ratio of the sum of the masses ofW+ and W- tothe mass of W0should be the volume ratio ofthe S^2 in S^3 tothe S^1 in S3. The unit sphere S^3 in R^4 isnormalized by 1 / 2. The unit sphere S^2 in R^3 isnormalized by 1 / sqrt3. The unit sphere S^1 in R^2 isnormalized by 1 / sqrt2. The ratio of the sum of the W+ and W- masses tothe W0 mass should then be(2 / sqrt3) V(S^2) / (2 / sqrt2) V(S^1) = 1.632993. Since the total mass of the triplet { W+, W-, W0 } is 259.031 GeV, the total mass of a T - Tbar pair, and the charged weak bosons have equal mass, we have and mW0 = 98.379 GeV.
The charged W+/- neutrino-electron interchange must be symmetric with the electron-neutrino interchange, so that the absence of right-handed neutrino particles requires that the charged W+/- SU(2)weak bosons act only on left-handed electrons. Each gauge boson must act consistentlyon the entire Dirac fermion particle sector,so that the charged W+/- SU(2) weak bosonsact only on left-handed fermions of all types. The neutral W0 weak boson does not interchange Weylneutrinos with Dirac fermions, and so is not restrictedto left-handed fermions,but also has a component that acts on both types of fermions,both left-handed and right-handed, conserving parity. However, the neutral W0 weak bosons are related tothe charged W+/- weak bosons by custodial SU(2)symmetry, so that the left-handed component of theneutral W0 must be equal to the left-handed (entire)component of the charged W+/-. Since the mass of the W0 is greater than the massof the W+/-, there remains for the W0 a componentacting on both types of fermions. Therefore the full W0 neutral weak boson interactionis proportional to(mW+/-^2 / mW0^2) acting on left-handed fermionsand (1 - (mW+/-^2 / mW0^2)) actingon both types of fermions. If (1 - (mW+/-2 / mW0^2)) is defined to besin(thetaw)^2 and denoted by K, and if the strength of the W+/- charged weak force(and of the custodial SU(2) symmetry) is denoted by T, then the W0 neutral weak interaction can be written as: W0L = T + K and W0LR = K. Since the W0 acts as W0L with respect to theparity violating SU(2) weak force and as W0LR with respect to the parity conserving U(1)electromagnetic force of the U(1) subgroup of SU(2), the W0 mass mW0 has two components: the parity violating SU(2) part mW0L that isequal to mW+/- ; and the parity conserving part mW0LR that acts like aheavy photon. As mW0 = 98.379 GeV = mW0L + mW0LR, and as mW0L = mW+/- = 80.326 GeV, we have mW0LR = 18.053 GeV. Denote by *alphaE = *e^2 the forcestrength of the weak parity conserving U(1)electromagnetic type force that acts through theU(1) subgroup of SU(2). The electromagnetic force strengthalphaE = e^2 = 1 / 137.03608 was calculatedin Chapter 8 usingthe volume V(S^1) of an S^1 in R^2,normalized by 1 / \qrt2. The *alphaE force is part of the SU(2) weakforce whose strength alphaW = w^2 was calculatedin Chapter 8 using the volume V(S^2) of an S^2 \subset R^3,normalized by 1 / sqrt3. Also, the electromagnetic force strength alphaE = e^2was calculated in Chapter 8 using a4-dimensional spacetime with global structure ofthe 4-torus T^4 made up of four S^1 1-spheres, while the SU(2) weak force strengthalphaW = w^2 was calculated in Chapter 8using two 2-spheres S^2 x S^2,each of which contains one 1-sphere ofthe *alphaE force. Therefore*alphaE = alphaE ( sqrt2 / sqrt3)(2 / 4) = alphaE / sqrt6, *e = e / (4th root of 6) = e / 1.565 , and the mass mW0LR must be reduced to an effective value mW0LReff = mW0LR / 1.565 = 18.053/1.565 = 11.536 GeV for the *alphaE force to act likean electromagnetic force in the 4-dimensionalspacetime HyperDiamond Feynman Checkerboard model: *e mW0LR = e (1/5.65) mW0LR = e mZ0, where the physical effective neutral weak boson isdenoted by Z0. Therefore, the correct HyperDiamond Feynman Checkerboard values forweak boson masses and the Weinberg angle thetaW are:
mZ0 = 80.326 + 11.536 =91.862 GeV;
Sin(thetaW)^2 = 1 - (mW+/- / mZ0)^2 =
= 1 - ( 6452.2663 / 8438.6270 ) =0.235.
Radiative corrections are not taken into account here,and may change these tree-level values somewhat.
Constituent Quark Masses:
To do calculations in theories such as Perturbative QCD and Chiral Perturbation Theory, you need to use effective quark masses that are called current masses. Current quark masses are different from the Pre-Quantum constituent quark masses of our model. The current mass of a quark is defined in this model as the difference between the constituent mass of the quark and the density of the lowest-energy sea of virtual gluons, quarks, and antiquarks, or 312.75 MeV.
First generation fermion particles are also representedby octonions as follows: Octonion Fermion Basis Element Particle 1 e-neutrino i red up quark j green up quark k blue up quark E electron I red down quark J green down quark K blue down quark First generation fermion antiparticles are representedby octonions in a similiar way. Second generation fermion particles and antiparticlesare represented by pairs of octonions. Third generation fermion particles and antiparticlesare represented by triples of octonions. There are no higher generations of fermions than the Third. This can be seen geometrically as a consequence of the fact thatif you reduce the original 8-dimensional spacetimeinto associative 4-dimensional physical spacetimeand coassociative 4-dimensional Internal Symmetry Space then if you look in the original 8-dimensional spacetimeat a fermion (First-generation represented by a single octonion)propagating from one vertex to another there are only 4 possibilities for the same propagationafter dimensional reduction: 1 - the origin o and target x vertices are bothin the associative 4-dimensional physical spacetime 4-dim Internal Symmetry Space -------------- 4-dim Physical SpaceTime ---o------x--- in which case the propagation is unchanged, and thefermion remains a FIRST generation fermion representedby a single octonion o 2 - the origin vertex o is in the associative spacetime and the target vertex * in in the Internal Symmetry Space 4-dim Internal Symmetry Space ----------*--- 4-dim Physical SpaceTime ---o---------- in which case there must be a new link fromthe original target vertex * in the Internal Symmetry Spaceto a new target vertex x in the associative spacetime 4-dim Internal Symmetry Space ----------*--- 4-dim Physical SpaceTime ---o------x--- and a second octonion can be introduced at the originaltarget vertex in connection with the new link so that the fermion can be regarded after dimensional reductionas a pair of octonions o and * and therefore as a SECOND generation fermion 3 - the target vertex x is in the associative spacetime and the origin vertex o in in the Internal Symmetry Space 4-dim Internal Symmetry Space ---o---------- 4-dim Physical SpaceTime ----------x--- in which case there must be a new link tothe original origin vertex o in the Internal Symmetry Spacefrom a new origin vertex * in the associative spacetime 4-dim Internal Symmetry Space ---o---------- 4-dim Physical SpaceTime ---O------x--- so that a second octonion can be introduced at the neworigin vertex O in connection with the new link so that the fermion can be regarded after dimensional reductionas a pair of octonions O and o and therefore as a SECOND generation fermion and 4 - both the origin vertex o and the target vertex * are in the Internal Symmetry Space, 4-dim Internal Symmetry Space ---o------*--- 4-dim Physical SpaceTime -------------- in which case there must be a new link tothe original origin vertex o in the Internal Symmetry Spacefrom a new origin vertex O in the associative spacetime,and a second new link from the original target vertex * in the Internal Symmetry Space to a new target vertex xin the associative spacetime 4-dim Internal Symmetry Space ---o------*--- 4-dim Physical SpaceTime ---O------x--- so that a second octonion can be introduced at the neworigin vertex O in connection with the first new link,and a third octonion can be introduced at the originaltarget vertex * in connection with the second new link,so that the fermion can be regarded after dimensional reductionas a triple of octonions O and o and * and therefore as a THIRD generation fermion. As there are no more possibilities, there are no more generations, and we have: The first generation fermionscorrespond to octonions O and second generation fermionscorrespond to pairs of octonions O x O and third generation fermionscorrespond to triples of octonions O x O x O To calculate the fermion masses in the model,the volume of a compact manifold representing thespinor fermions S8+ is used.It is the parallelizable manifold S^7 x RP^1. Also, since gravitation is coupled to mass,the infinitesimal generators of the MacDowell-Mansourigravitation group, Spin(0,5), are relevant. The calculated quark masses are constituent masses, not current masses.
Fermion masses are calculated as a product of four factors:
V(Qfermion) is the volume of the part of the half-spinor fermion particle manifold S^7 x RP^1 that is related to the fermion particle by photon, weak boson, and gluon interactions. N(Graviton) is the number of types of Spin(0,5) graviton related to the fermion. The 10 gravitons correspond to the 10 infinitesimal generators of Spin(0,5) = Sp(2). 2 of them are in the Cartan subalgebra. 6 of them carry color charge, and may therefore be considered as corresponding to quarks. The remaining 2 carry no color charge, but may carry electric charge and so may be considered as corresponding to electrons. One graviton takes the electron into itself, and the other can only take the first-generation electron into the massless electron neutrino. Therefore only one graviton should correspond to the massof the first-generation electron. The graviton number ratio of the down quark to thefirst-generation electron is therefore 6/1 = 6. N(octonion) is an octonion number factor relating up-type quarkmasses to down-type quark masses in each generation. Sym is an internal symmetry factor, relating 2nd and 3rdgeneration massive leptons to first generation fermions.It is not used in first-generation calculations.
The ratio of the down quark constituent mass to the electron massis then calculated as follows: Consider the electron, e. By photon, weak boson, and gluon interactions,e can only be taken into 1, the massless neutrino. The electron and neutrino, or their antiparticles,cannot be combined to produce any of themassive up or down quarks. The neutrino, being massless at tree level,does not add anything to the mass formula for the electron. Since the electron cannot be related to any other massive Diracfermion, its volume V(Qelectron) is taken to be 1. Next consider a red down quark I. By gluon interactions, I can be taken into J and K,the blue and green down quarks. By also using weak boson interactions, it can be taken into i, j, and k, the red, blue, and green up quarks. Given the up and down quarks, pions can be formed from quark-antiquark pairs, and the pions can decay to produce electrons and neutrinos. Therefore the red down quark (similarly, any down quark)is related to any part of S^7 x RP^1,the compact manifold corresponding to { 1, i, j, k, I, J, K, E } and therefore a down quark should have a spinor manifoldvolume factor V(Qdown quark) of the volume ofS^7 x RP^1. The ratio of the down quark spinor manifold volume factor tothe electron spinor manifold volume factor is just V(Qdown quark) / V(Qelectron) = V(S^7x RP^1)/1 = pi^5 / 3. Since the first generation graviton factor is 6, As the up quarks correspond to i, j, and k,which are the octonion transforms under E of I, J, and K of the down quarks, the up quarks and down quarkshave the same constituent mass mu = md. Antiparticles have the same mass as the correspondingparticles. Since the model only gives ratios of massses,the mass scale is fixed so that the electron mass
Then, the constituent mass of the down quark is
the constituent mass for the up quark is
These results when added up give a total mass offirst generation fermion particles:
As the proton mass is taken to be the sum of the constituentmasses of its constituent quarks
The theoretical calculation is close tothe experimental value of 938.27 MeV. The third generation fermion particles correspond to triples ofoctonions. There are 8^3 = 512 such triples. The triple { 1,1,1 } corresponds to the tau-neutrino. The other 7 triples involving only 1 and E correspondto the tauon:{ E, E, E }{ E, E, 1 }{ E, 1, E }{ 1, E, E }{ 1, 1, E }{ 1, E, 1 }{ E, 1, 1 } The symmetry of the 7 tauon triples is the same as the symmetry of the 3 down quarks, the 3 up quarks, and the electron,so the tauon mass should be the same as the sum of the masses of the first generation massive fermion particles. The calculated Tauon mass of 1.88 GeV is a sum of first generation fermion masses, all of which are valid at the energy level of about 1 GeV. However, as the Tauon mass is about 2 GeV, the effective Tauon mass should be renormalized from the energy level of 1 GeV (where the mass is 1.88 GeV) to the energy level of 2 GeV. Such a renormalization should reduce the mass. If the renormalization reduction were about 5 percent, the effective Tauon mass at 2 GeV would be about 1.78 GeV. The 1996 Particle Data Group Review of Particle Physics gives a Tauon mass of 1.777 GeV.
Note that all triples corresponding to thetau and the tau-neutrino are colorless. The beauty quark corresponds to 21 triples. They are triples of the same form as the 7 tauon triples,but for 1 and I, 1 and J, and 1 and K,which correspond to the red, green, and blue beauty quarks,respectively. The seven triples of the red beauty quark correspondto the seven triples of the tauon,except that the beauty quark interacts with 6 Spin(0,5)gravitons while the tauon interacts with only two. The beauty quark constituent mass should be the tauon mass times thethird generation graviton factor 6/2 = 3, so the B-quark mass is
The calculated Beauty Quark mass of 5.63 GeV is a consitituent mass, that is, it corresponds to the conventional pole mass plus 312.8 MeV. Therefore, the calculated Beauty Quark mass of 5.63 GeV corresponds to a conventional pole mass of 5.32 GeV. The 1996 Particle Data Group Review of Particle Physics gives a lattice gauge theory Beauty Quark pole mass as 5.0 GeV. The pole mass can be converted to an MSbar mass if the color force strength constant alpha_s is known. The conventional value of alpha_s at about 5 GeV is about 0.22. Using alpha_s (5 GeV) = 0.22, a pole mass of 5.0 GeV gives an MSbar 1-loop mass of 4.6 GeV, and an MSbar 1,2-loop mass of 4.3, evaluated at about 5 GeV. If the MSbar mass is run from 5 GeV up to 90 GeV, the MSbar mass decreases by about 1.3 GeV, giving an expected MSbar mass of about 3.0 GeV at 90 GeV. DELPHI at LEP has observed the Beauty Quark and found a 90 GeV MSbar mass of about 2.67 GeV, with error bars +/- 0.25 (stat) +/- 0.34 (frag) +/- 0.27 (theo). Note that the theoretical model calculated mass of 5.63 GeV corresponds to a pole mass of 5.32 GeV, which is somewhat higher than the conventional value of 5.0 GeV. However, the theoretical model calculated value of the color force strength constant alpha_s at about 5 GeV is about 0.166, while the conventional value of the color force strength constant alpha_s at about 5 GeV is about 0.216, and the theoretical model calculated value of the color force strength constant alpha_s at about 90 GeV is about 0.106, while the conventional value of the color force strength constant alpha_s at about 90 GeV is about 0.118. The theoretical model calculations gives a Beauty Quark pole mass (5.3 GeV) that is about 6 percent higher than the conventional Beauty Quark pole mass (5.0 GeV), and a color force strength alpha_s at 5 GeV (0.166) such that 1 + alpha_s = 1.166 is about 4 percent lower than the conventional value of 1 + alpha_s = 1.216 at 5 GeV.
Note particularly that triples of the type { 1, I, J },{ I, J, K }, etc.,do not correspond to the beauty quark, but to the truth quark. The truth quark corresponds to the remaining 483 triples, so theconstituent mass of the red truth quark is 161/7 = 23 times thered beauty quark mass, and the red T-quark mass is
The blue and green truth quarks are defined similarly. All other masses than the electron mass (which is the basis of the assumption of the value of the Higgs scalar field vacuum expectation value v = 252.514 GeV), including the Higgs scalar mass and Truth quark mass,are calculated (not assumed) masses in the HyperDiamond FeynmanCheckerboard model. The tree level T-quark constituent mass rounds off to 130 GeV. These results when added up give a total mass ofthird generation fermion particles:
The second generation fermion calculations are: The second generation fermion particles correspondto pairs of octonions. There are 8^2 = 64 such pairs. The pair { 1,1 } corresponds to the mu-neutrino. the pairs { 1, E }, { E, 1 }, and{ E, E } correspond to the muon. Compare the symmetries of the muon pairs to the symmetriesof the first generation fermion particles. The pair { E, E } should correspondto the E electron. The other two muon pairs have a symmetry group S2,which is 1/3 the size of the color symmetry group S3which gives the up and down quarks their mass of 312.75 MeV. Therefore the mass of the muon should be the sum ofthe { E, E } electron mass andthe { 1, E }, { E, 1 } symmetry mass,which is 1/3 of the up or down quark mass. According to the 1998 Review ofParticle Physics of the Particle Data Group, the experimentalmuon mass is about 105.66 MeV.
Note that all pairs corresponding tothe muon and the mu-neutrino are colorless. The red, blue and green strange quark each correspondsto the 3 pairs involving 1 and I, J, or K. The red strange quark is defined as the thrge pairs1 and I, because I is the red down quark. Its mass should be the sum of two parts:the { I, I } red down quark mass, 312.75 MeV, andthe product of the symmetry part of the muon mass, 104.25 MeV,times the graviton factor. Unlike the first generation situation,massive second and third generation leptons can be taken,by both of the colorless gravitons thatmay carry electric charge, into massive particles. Therefore the graviton factor for the second and third generations is 6/2 = 3. Therefore the symmetry part of the muon mass timesthe graviton factor 3 is 312.75 MeV, andthe red strange quark constituent mass isThe blue strange quarks correspond to thethree pairs involving J,the green strange quarks correspond to thethree pairs involving K,and their masses are determined similarly. The charm quark corresponds to the other 51 pairs.Therefore, the mass of the red charm quark shouldbe the sum of two parts: the { i, i }, red up quark mass, 312.75 MeV; and the product of the symmetry part of the strange quarkmass, 312.75 MeV, and the charm to strange octonion number factor 51/9,which product is 1,772.25 MeV. Therefore the red charm quark constituent mass isThe blue and green charm quarks are defined similarly,and their masses are calculated similarly. The calculated Charm Quark mass of 2.09 GeV is a consitituent mass, that is, it corresponds to the conventional pole mass plus 312.8 MeV. Therefore, the calculated Charm Quark mass of 2.09 GeV corresponds to a conventional pole mass of 1.78 GeV. The 1996 Particle Data Group Review of Particle Physics gives a range for the Charm Quark pole mass from 1.2 to 1.9 GeV. The pole mass can be converted to an MSbar mass if the color force strength constant alpha_s is known. The conventional value of alpha_s at about 2 GeV is about 0.39, which is somewhat lower than the teoretical model value. Using alpha_s (2 GeV) = 0.39, a pole mass of 1.9 GeV gives an MSbar 1-loop mass of 1.6 GeV, evaluated at about 2 GeV. These results when added up give a total mass ofsecond generation fermion particles:
The Kobayashi-Maskawa parameters are determined in terms of the sum of the masses of the 30 first-generation fermion particles and antiparticles, denoted by Smf1 = 7.508 GeV, and the similar sums for second-generation and third-generation fermions, denoted by Smf2 = 32.94504 GeV and Smf3 = 1,629.2675 GeV. The reason for using sums of all fermion masses (rather than sums of quark masses only) is that all fermions are in the same spinor representation of Spin(8), and the Spin(8) representations are considered to be fundamental.The following formulas use the above masses tocalculate Kobayashi-Maskawa parameters: phase angle d13 = 1 radian ( unit length on a phase circumference ) sin(alpha) = s12 = = [me+3md+3mu]/sqrt([me^2+3md^2+3mu^2]+[mmu^2+3ms^2+3mc^2]) = = 0.222198 sin(beta) = s13 = = [me+3md+3mu]/sqrt([me^2+3md^2+3mu^2]+[mtau^2+3mb^2+3mt^2])= = 0.004608 sin(*gamma) = = [mmu+3ms+3mc]/sqrt([mtau^2+3mb^2+3mt^2]+[mmu^2+3ms^2+3mc^2]) sin(gamma) = s23 = sin(*gamma) sqrt( Sigmaf2 / Sigmaf1 ) = = 0.04234886 The factor sqrt( Smf2 /Smf1 ) appears in s23 because an s23 transition is to the second generation and not all the way to the first generation, so that the end product of an s23 transition has a greater available energy than s12 or s13 transitions by a factor of Smf2 / Smf1 . Since the width of a transition is proportional to the square of the modulus of the relevant KM entry and the width of an s23 transition has greater available energy than the s12 or s13 transitions by a factor of Smf2 / Smf1 the effective magnitude of the s23 terms in the KM entries is increased by the factor sqrt( Smf2 /Smf1 ) . The Chau-Keung parameterization is used, as it allows the K-M matrix to be represented as the product of the following three 3x3 matrices:
1 0 0 0 cos(gamma) sin(gamma) 0 -sin(gamma) cos(gamma)
cos(beta) 0 sin(beta)exp(-i d13) 0 1 0 -sin(beta)exp(i d13) 0 cos(beta)
cos(alpha) sin(alpha) 0 -sin(alpha) cos(alpha) 0 0 0 1
The resulting Kobayashi-Maskawa parameters for W+ and W- charged weak boson processes, are: d s b u 0.975 0.222 0.00249 -0.00388i c -0.222 -0.000161i 0.974 -0.0000365i 0.0423 t 0.00698 -0.00378i -0.0418 -0.00086i 0.999 The matrix is labelled by either (u c t) input and (d s b) output, or, as above, (d s b) input and (u c t) output. For Z0 neutral weak boson processes, which are suppressed by the GIM mechanism of cancellation of virtual subprocesses, the matrix is labelled by either (u c t) input and (u'c't') output, or, as below, (d s b) input and (d's'b') output: d s b d' 0.975 0.222 0.00249 -0.00388i s' -0.222 -0.000161i 0.974 -0.0000365i 0.0423 b' 0.00698 -0.00378i -0.0418 -0.00086i 0.999 Since neutrinos of all three generations are massless at tree level, the lepton sector has no tree-level K-M mixing.
According to aReview on the KM mixing matrix by Gilman, Kleinknecht, and Renk inthe 2002 Review of Particle Physics:
"... Using the eight tree-level constraints discussed belowtogether with unitarity, and assuming only three generations, the 90%confidence limits on the magnitude of the elements of the completematrix are
d s b u 0.9741 to 0.9756 0.219 to 0.226 0.00425 to 0.0048 c 0.219 to 0.226 0.9732 to 0.9748 0.038 to 0.044 t 0.004 to 0.014 0.037 to 0.044 0.9990 to 0.9993
... The constraints of unitarity connect different elements, sochoosing a specific value for one element restricts the range ofothers. ... The phase d13 lies in the range 0 < d13 < 2 pi,with non-zero values generally breaking CP invariance for the weakinteractions. ... Using tree-level processes as constraints only, thematrix elements ...[ of the 90% confidence limit shown above]... correspond to values of the sines of the angles of s12 =0.2229 +/- 0.0022, s23 = 0.0412 +/- 0.0020, and s13 = 0.0036 +/-0.0007. If we use the loop-level processes discussed below asadditional constraints, the sines of the angles remain unaffected,and the CKM phase, sometimes referred to as the angle gamma = phi3 ofthe unitarity triangle ...
... is restricted to d13 = ( 1.02 +/- 0.22 ) radians = 59 +/- 13degrees. ... CP-violating amplitudes or differences of rates are allproportional to the product of CKM factors ... s12 s13 s23 c12 c13^2c23 sind13. This is just twice the area of the unitarity triangle.... All processes can be quantitatively understood by one value ofthe CKM phase d13 = 59 +/- 13 degrees. The value of beta = 24 +/- 4degrees from the overall fit is consistent with the value from theCP-asymmetry measurements of 26 +/- 4 degrees. The invariant measureof CP violation is J = ( 3.0 +/- 0.3) x 10^(-5). ... From a combinedfit using the direct measurements, B mixing, epsilon, and sin2beta,we obtain: Re Vtd = 0.0071 +/- 0.0008 , Im Vtd = -0.0032 +/- 0.0004... Constraints... on the position of the apex of the unitaritytriangle following from | Vub | , B mixing, epsilon, and sin2beta....
... A possible unitarity triangle is shown with the apex in thepreferred region. ...".
In hep-ph/0208080,Yosef Nir says: "... Within the Standard Model, the only source of CPviolation is the Kobayashi-Maskawa (KM) phase ... The study of CPviolation is, at last, experiment driven. ... The CKM matrix providesa consistent picture of all the measured flavor and CP violatingprocesses. ... There is no signal of new flavor physics. ... Verylikely, the KM mechanism is the dominant source of CP violation inflavor changing processes. ... The result is consistent with the SMpredictions. ...".
According to the 1986 CODATA Bulletin No. 63, the experimental value of the neutron mass is 939.56563(28) Mev, and the experimental value of the proton is 938.27231(28) Mev. The neutron-proton mass difference 1.3 Mev is due to the fact that the proton consists of two up quarks and one down quark, while the neutron consists of one up quark and two down quarks. The magnitude of the electromagnetic energy difference mN - mP is about 1 Mev, but the sign is wrong: mN - mP = -1 Mev, and the proton's electromagnetic mass is greater than the neutron's. The difference in energy between the bound states, neutron and proton, is not due to a difference between the Pre-Quantum constituent masses of the up quark and the down quark, calculated in the theory to be equal. It is due to the difference between the Quantum color force interactions of the up and down constituent valence quarks with the gluons and virtual sea quarks in the neutron and the proton. An up valence quark, constituent mass 313 Mev, does not often swap places with a 2.09 Gev charm sea quark, but a 313 Mev down valence quark can more often swap places with a 625 Mev strange sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (ms - md) (md/ms)^2 a(w) |Vds| = = 312 x 0.25 x 0.253 x 0.22 Mev = 4.3 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vds| = 0.22 is the magnitude of the K-M parameter mixing first generation down and second generation strange) so that the Quantum color force constituent mass Qmd of the down quark is Qmd = 312.75 + 4.3 = 317.05 MeV. Similarly, the up quark Quantum color force mass increase is about (mc - mu) (mu/mc)^2 a(w) |V(uc)| = = 1777 x 0.022 x 0.253 x 0.22 Mev = 2.2 Mev, (where |Vuc| = 0.22 is the magnitude of the K-M parameter mixing first generation up and second generation charm) so that the Quantum color force constituent mass Qmu of the up quark is Qmu = 312.75 + 2.2 = 314.95 MeV. The Quantum color force Neutron-Proton mass difference is mN - mP = Qmd - Qmu = 317.05 Mev - 314.95 Mev = 2.1 Mev. Since the electromagnetic Neutron-Proton mass difference is roughly mN - mP = -1 MeV the total theoretical Neutron-Proton mass difference is
an estimate that is fairly close to the experimental value of 1.3 Mev. Note that in the equation (ms - md) (md/ms)^2 a(w) |Vds| = 4.3 Mev Vds is a mixing of down and strange by a neutral Z0, compared to the more conventional Vus mixing by charged W. Although real neutral Z0 processes are suppressed by the GIM mechanism, which is a cancellation of virtual processes, the process of the equation is strictly a virtual process. Note also that the K-M mixing parameter |Vds| is linear. Mixing (such as between a down quark and a strange quark) is a two-step process, that goes approximately as the square of |Vds|: First the down quark changes to a virtual strange quark, producing one factor of |Vds|. Then, second,the virtual strange quark changes back to a down quark, producing a second factor of |Vsd|, which is approximately equal to |Vds|. Only the first step (one factor of |Vds|) appears in the Quantum mass formula used to determine the neutron mass. If you measure the mass of a neutron,that measurement includes a sum over a lot of historiesof the valence quarks inside the neutron.In some of those histories, in my view,you will "see" some of the two valence down quarksin a virtual transition state that is at a timeafter the first action, or change from down to strange,andbefore the second action, or change back.Therefore, you should take into accountthose histories in the sum in which you see a strange valence quark,and you get the linear factor |Vds| in the above equation. Note also that if there were no second generation fermions, or if the second generation quarks had equal masses, then the proton would be heavier than the neutron (due to the electromagnetic difference) and the hydrogen atom would decay into a neutron, and there would be no stable atoms in our world.
In this model, protons decay by virtual BlackHoles over 10^64 years, according to by Hawkingand his students who have studied thephysical consequences of creation of virtual pairs of Planck-energyBlack Holes.
UCC - DCC Baryon MassDifference:
According to a14 June 20002 article by Kurt Riesselmann in Fermi News: "... Thefour [ first and second generation ] flavors - up, down,strange, charm - allow for twenty different ways of putting quarkstogether to form baryons ... Protons, for example, consist of two upquarks and one down quark (u-u-d), and neutrons have a u-d-d quarkcontent. Some combinations exist in two different spinconfigurations, and the SELEX collaboration believes it hasidentified both spin levels of the u-c-c baryon. ... Physicistsexpect the mass difference between u-c-c and d-c-c baryons to becomparable to the difference in proton (u-u-d) and neutron (u-d-d)mass, since this particle pair is also related by the replacement ofan up by a down quark. The proton-neutron masssplitting, however, is sixty times smaller than the massdifference between the Xi_cc candidates observed by the SELEXcollaboration. ...
... Other questions, however, remain as well. The SELEXcollaboration is puzzled by the high rate of doubly charmed baryonsseen in their experiment. As a matter of fact, most scientistsbelieved that the SELEX collaboration wouldn't see any of theseparticles. ...".
An up valence quark, constituent mass 313 Mev, can swap places with a 2.09 Gev charm sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (mc - mu) a(w) |Vds| = = 1,777 x 0.253 x 0.22 Mev = 98.9 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vuc| = 0.22 is the magnitude of the K-M parameter mixing first generation up and second generation charm) so that the Quantum color force constituent mass Qmu of the up quark is Qmu = 312.75 + 98.9 = 411.65 MeV. A 313 Mev down valence quark can swap places with a 625 Mev strange sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (ms - md) a(w) |Vds| = = 312 x 0.253 x 0.22 Mev = 17.37 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vds| = 0.22 is the magnitude of the K-M parameter mixing first generation down and second generation strange) so that the Quantum color force constituent mass Qmd of the down quark is Qmd = 312.75 + 17.37 = 330.12 MeV.
Note that at the energy levels at which ucc and dcc live, theambient sea of quark-antiquark pairs has at least enough energy toproduce a charm quark, so that in the above equations there is nomass-ratio-squared suppression factor such as (mu/mc)^2 or (md/ms)^2,unlike the case of the calculation of theneutron-proton mass difference for which the ambient sea ofquark-antiquark pairs has very little energy since the proton isalmost stable and the neutron-proton mass difference is, according toexperiment, only about 1.3 MeV.
Note also that these rough calculations ignore the electromagneticforce mass differentials, as they are only on the order of 1 MeV orso, which for ucc - dcc mass difference is small, unlike the case forthe calculation of the neutron-proton massdifference.
The Quantum color force ucc - dcc mass difference is
Since the experimental value of the neutron-proton mass differenceis about 1.3 MeV, the theoretically calculated
which is consistent with the SELEX 2002 experimental result that:"... The proton-neutron mass splitting ... issixty times smaller than the mass difference between the Xi_cccandidates ...".
Root Vector Geometry ofFermions,Spacetime,GaugeBosons, andD4-D5-E6-E7-E8.
The 8 first-generation fermion particles can be represented as 8vertices of a 24-cell
The 8 dimensions of unreduced spacetime (which reduces to4-dimensional physical spacetime plus 4-dimensional internal SymmetrySpace) can be represented as another 8 vertices of the 24-cell
The third set of 8 vertices of the 24-cell then represents the 8first-generation fermion particles, so that the entire 24-cellrepresentation looks like
Note that the three sets of 8 vertices correspond to thetwo half-spinor and thevector representations of the D4 LieAlgebra are related by triality.
These relationships can also be viewed from the perspective of theCl(1,7) Clifford Algebra structures
1 8 28 56 70 56 28 8 1 = (8+8)x(8+8)
As to the 28-dimensional adjointrepresentation, 24 of the 28 gauge boson D4 generators can berepresented by the vertices of a dual 24-cell:
Note that the 24 + 24 = 48 vertices of the two dual 24-cells are48 of the 72 root vector vertices of the E6 Lie algebra, andcorrespond to the 48 root vector vertices of the F4 subalgebra ofE6:
The 24 adjoint gauge boson vertices correspond to the 24 root vector vertices of the D4 subalgebra of E6;When the 8 vector spacetime vertices are added, you get the 32 root vector vertices of the B4 subalgebra of E6;
When the 8+8 = 16 spinor vertices are added, you get the 48 root vector vertices of the F4 subalgebra of E6.
Here is how the 24 adjoint gauge bosonvertices break down after dimensional reduction to form U(2,2)for gravity plus SU(3)xSU(2)xU(1) for the Standard Model:
12 of the 24 vertices correspond to the 12 vertices of thecuboctahedron
that is the root vector polytope of the A3 = D3 Lie AlgebraSU(2,2) = Spin(2,4) of the 4-dimensional ConformalGroup. Then add the 4 D4 Cartan subalgebra generators to get the12+4 = 16-dimensional Lie Algebra SU(2,2)xU(1) = Spin(2,4)xU(1) =U(2,2) that, by a generalization of theMacDowell-Mansouri mechanism, produces Gravity and the Higgsmechanism, and
That leaves 24-12 = 12 remaining vertices
4 of which lie on a common line
and represent the generators of the 4-dimensional Lie Algebra U(2)= SU(2)xU(1).
The remaining 8 vertices
form a cube that can be labelled
tb----xb |\ | \ | zb----yb | | | | yr-|--zr | \| \| xr----tr
yb xb zb tb tr zr xr yr
with two central points surrounded by two interpenetratingtriangles, which is the root vector diagram of SU(3), Therefore:
There is a nice geometric way to see thestructures of D4-D5-E6-E7-E8LieAlgebras:
Floating above the Penrose-tiled planein the above image (adapted from Quasitiler)are, going from left to right:
E6 = g = g-2 + g-1 + g0 +g1 + g2
g0 = so(1,7) + R + iR
dim g-1 = 16
dim g-2 = 8
This gives real Shilov boundarygeometry of S1xS7 for (1,7)-dimensional high-energy spacetimerepresentation and for the first generation half-spinor fermionrepresentations, which is the local structure needed for a localLagrangian and calculation of ratios of particle masses and forcestrengths.
Geometric Structure of NonLocalQuantum Theory is given by E6,E7, and E8:
References:
The theoretical physics model described in this paper is not onlybased on the Lie Algebras D4, D5, E6, E7, and E8, but also on the256-dimensional Clifford Algebra Cl(1,7), whose 256 dimensionscorrespond to the 256 Odu of IFA, also knownas Vodou. Therefore, the name that I prefer for this theoreticalphysics model is
Many details and references can be found on myhome page on the web at URL
and on pages linked therefrom.
A few specific outside references are:
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
Quasitiler- The Penrose Tiling and 5-dim HyperCube in the image at the top ofthis page is from aweb page of Alex Feingold. Unfortunately, the link to Quasitilerat the Geometry Center of the University of Minnesota is no longergood, because not only was the URL http://freeabel.geom.umn.edu/changed to http://www.geom.umn.edu/ some time ago, but now theGeometry Center has gone away (NSF funds were cut off). Some of itspages live on in google cache, such ashttp://www.google.com/search?q=cache:s_WVB9Mv-4MC:www.geom.umn.edu/closed.html+&hl=en, but I don't know how to access all of them, and I feel that theloss of the Geometry Center is a loss to all of us who use theweb.
Encyclopedic Dictionary of Mathematics, second edition, MIT Press1993; Hua, Harmonic Analysis of Functions of Several ComplexVariables in the Classical Domains, Am. Math. Soc. 1979; Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic1978; Helgason, Groups and Geometric Analysis, Academic 1984; Besse,Einstein Manifolds, Springer-Verlag 1987; Rosenfeld, Geometry of LieGroups, Kluwer 1997; Gilmore, Lie Groups, Lie Algebras, and Some ofTheir Applications, John Wiley 1974; Edward Dunne's web site;Coquereaux and Jadczyk, Conformal Theories, Curved Phase SpacesRelativistic Wavelets and the Geometry of Complex Domains, Reviews inMathematical Physics, Volume 2, No 1 (1990) 1-44, which can bedownloaded from the web as a 1.98 MB pdf file.
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