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From Sets toQuarks:

Deriving the Standard Model plus Gravitation from Simple Operations on Finite Setsby Tony Smith Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras. Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction.                  Higgs Scalar, Gauge Bosons, and Fermions.                      Higgs Scalar Mass.                       Gauge Boson Masses.                           Weak Boson Masses.                                Parity Violation,                                 Effective Masses,                                 and Weinberg Angle.                      Fermion Masses.                           Renormalization.                                Perturbative QCD.                                Chiral Perturbation Theory.                                Lattice Gauge Theory.                           Truth Quark Mass.                           Other Fermion Masses.                                First Generation.                                Tauon Mass.                                Beauty Quark Mass.                                Muon Mass.                                 Strange Quark Mass.                                 Charm Quark Mass.                       K-M Parameters.                  Khokhlov: Nth Generation Fermions                            and N-Photon Processes (N=1,2,3).Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.

Mass = Amplitude to Change Direction.

  In the HyperDiamond Feynman Checkerboard modelthe mass parameter m is the amplitude for a particleto change its spacetime direction. Massless particlesdo not change direction, but continue on the samelightcone path.  In the D4-D6-E6-E7 Lagrangian continuum versionof this physics model, particle masses are calculated in termsof relative volumes of bounded complex homogeneous domains andtheir Shilov boundaries. The relationship betweenthe D4-D6-E6-E7 Lagrangian continuum approachandthe HyperDiamond Feynman Checkerboard discrete approachis that: the bounded complex homogeneous domains correspond toharmonic functions of generalized Laplaciansthat determine heat equations, or diffusion equations; while the amplitude to change directions in theHyperDiamond Feynman Checkerboard is a diffusion processin the HyperDiamond spacetime, also corresponding toa generalized Laplacian. Details of the D4-D6-E6-E7 Lagrangian continuum approachcan be found on the World Wide Web at URLs For the discrete HyperDiamond Feynman Checkerboardapproach of this paper, the only free mass parameteris the mass of the Higgs scalar. All other particlemasses are determined as ratios with respect to theHiggs scalar and each other. The Higgs mass is 145.789 GeV in theHyperDiamond Feynman Checkerboard model, since the Higgs Scalar field vacuum expectation value v is set at 252.514 GeV, a figure chosen so that the mass ratios of the model will give an electronmass of 0.5110 MeV.

Here is how the mass ratios work:

It is interesting that

the ratio of the sum of the masses of the weak bosons W+, W-,and W0 to the sum of the masses of the first generation fermions is259.031 GeV / 7.508 GeV = 34.5007

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.


Effectively, in the HyperDiamond FeynmanCheckerboard model, the electron mass is fixed at 0.5110 MeVand all other masses are determined from it by theratios calculated in the model.  

HiggsScalar, Gauge Bosons, and Fermions.

 Recall from Chapter 4 that 16x16 = 256-dimensional DCl(0,8)has graded structure

1 8 28 56 70 56 28 8 1

 that gives three types of particles for which mass ratioscan be calculated in the HyperDiamond Feynman Checkerboardmodel: the Higgs Scalar; the 28 bivector gauge bosons; and the 8 + 8 = 16 half-spinor fermions.  

Higgs ScalarMass.

 There is only one Higgs scalar. Its mass is 145.789 GeV.   

Gauge Boson Masses.

 After dimensional reduction to 4-dimensional spacetime,the 28 Spin(0,8) gauge bosons split into two groups: 12 Standard Model gauge bosons, which are 8 SU(3) gluons, 3 SU(2) weak bosons, and 1 U(1) photon,  which 12 gauge bosons see Internal Symmetry Spaceaccording to their group symmetry;  and 16 U(4) gauge bosons, which reduce to 15 SU(4) = Spin(6) gauge bosonsplus one U(1) phase for particle propagator amplitudes(this phase is what makes the sum-over-histories quantumtheory interferences work). The 15 SU(4) = Spin(6) gauge bosons further reduceto 10 Spin(5) deSitter gravitons that give physicalgravity by the MacDowell-Mansouri mechanism as described in with gravitons that see their Symmetry Spaceof Spacetime according to their group symmetry,    and 4 conformal generators and 1 scale generator. The 4 conformal generators couple to the Higgs scalarso that it becomes the mass-giver by the Higgs mechanismas described in and the 1 scale generatorrepresents the scale that we choseby setting the Higgs scalar field vacuum expectationvalue v to be 252.514 GeV. All these gauge bosons are massless at very high energies,but at energies comparable to the Higgs mass and below,the Higgs scalar couples to the SU(2) weak bosonsto give them mass. 

WeakBoson Masses.

 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 mW+ = mW- = 80.326  GeV,  and mW0 = 98.379  GeV.   

Parity Violation, Effective Masses,and Weinberg Angle.

 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:  mW+ = mW- = 80.326  GeV;  mZ0 = 80.326 + 11.536 = 91.862  GeV; and  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-levelHyperDiamond Feynman Checkerboard values somewhat.

According to hep-ex/0205080:

"... The NuTeV experiment has performed precision measurements of the ratio of neutral-current to charged-current cross-sections in high rate, high energy neutrino and anti-neutrino beams on a dense, primarily steel, target. The separate neutrino and anti-neutrino beams, high statistics, and improved control of other experimental systematics, allow the determination of electroweak parameters with significantly greater precision than past nu-N scattering experiments. Our null hypothesis test of the standard model prediction measures ...
sin(theta_W^(on-shell))^2 = 0.22773 +/- 0.00135 (stat) +/- 0.00093 (syst)

- 0.00022 x ( ( M_top^2 - ( 175 GeV/c^2 )^2 ) / ( 50 GeV/c^2 )^2 )

+ 0.00032 ln( M_Higgs / 150 GeV )

... A fit to the precision electroweak data, excluding neutrino measurements, predicts a value of 0.2227 +/- 0.00037 ... approximately 3 sigma from the NuTeV measurement. In the on-shell scheme ... sin(theta_W)^2 = 1 - ( M_W / M_Z )^2 , where M_W and M_Z are the physical gauge boson masses; therefore, this result implies M_W = 80.14 +/- 0.08 GeV ... The world-average of the direct measurements of M_W is 80.45 +/- 0.04 GeV ... A fit to precision data ... including NuTeV, has been performed by the LEPEWWG ... shown in Figure 1 ...

... This suggests that in the context of all the precision data, as compiled by the LEPEWWG, the NuTeV result is still a statistical anomaly suffcient to spoil the fit if the standard model is assumed. ....".


Note that there are several currently accepted ways to define theWeinberg angle. According to a19 November 2001 document of theParticle Data Group, "...



The D4-D5-E6-E7-E8 VoDou Physicsmodel definition of the Weinberg angle is fundamentally based onforce strengths (which, in turn, can be expressed in terms of thecalculated W and Z weak boson masses of M_W =80.326 GeV and M_Z = 91.862 GeV), so thatthe D4-D5-E6-E7-E8 VoDou Physicsmodel definition of the Weinberg angle is effectively theMSbar definition.

In the edited-quotes of the above images, c and cbar are functionsof the masses of the Higgs and the Truth Quark, and the above numbersmay be based on masses of around 100 GeV and 175 GeV, respectively.Even so, using those values of c and cbar, a calculation shows thatthe MSbar value of sin(theta_W)^2 of 0.235 of the D4-D5-E6-E7-E8VoDou Physics model corresponds to an on-shell tree levelvalue of sin(theta_W)^2 of 0.2267.


The hep-ex/0205080on-shell value for the NuTeV experimental result for sin(theta_W)^2 =0.2277 assumes that the Higgs MassM_Higgs = 150 GeV and that the Truth QuarkMass M_top = 175 GeV.

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, the Higgs Mass of 146 GeV isclose to 150 GeV, but the Truth Quark Mass of130 GeV is substantially lower than 175 GeV. If the value of 130 GeVis used for the Truth Quark Mass in the equation of hep-ex/0205080,then the value of sin(theta_W)^2 is increased by

0.00022 x ( 175^2 - 130^2 ) / 50^2 = 0.0012 GeV

giving for the on-shell value for the NuTeV experimental resultwith a 130 GeV Truth Quark Mass sin(theta_W^nu-N)^2 = 0.2289 withtotal statistical and systematic error bar +/- 0.00228, whichcompares (in my opinion) realistically with the D4-D5-E6-E7-E8VoDou Physics model on-shell tree level value of sin(theta_W)^2of 0.2267.

Therefore I do not agree with the conclusion of hep-ex/0205080that "... the NuTeV result is still a statistical anomaly sufficientto spoil the fit if the standard model is assumed ...".





Corresponding to the way that 24of the 28 Gauge Bosons of theD4-D5-E6-E7-E8 VoDou Physics model can be representedby the vertices of a 24-cell, thefirst-generation Fermion Particles and anti-Particles, andSpaceTime plus Internal SymmetrySpace, can be represented by the vertices of adual 24-cell:

In the dual 24-cell,  the 8 vertices

represent the first-generation electron; red, green, blue upquarks; red, green, blue down quarks, and neutrino. The 8 verticesrepresenting the first-generation antiparticles are denoted by thecyan, magenta, yellow, and square-black vertices.

These 8+8 = 16 vertices correspond to the 16 Complex Dimensions ofthe Bounded Domain corresponding to theSymmetric Space E6/ D5xU(1), and to the 16 Real Dimensions of its ShilovBoundary. The 32 Real Dimensions of that Complex Bounded Domaincorrespond to the 32 vertices of the5-dimensional HyperCube that make up 32 of the 72E6 root vectors.

In this image

only the 8 vertices representing the first-generation fermionparticles are marked.

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. In the HyperDiamond Feynman Checkerboard model,there are no higher generations of fermions than the Third. This can be seen algebraically as a consequence of thefact that the Lie algebra series E6, E7, and E8,has only 3 algebras, which in turn is a consequence ofnon-associativity of octonions, as described here and in geometrically as a consequence of the fact that, if 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 and target vertices are bothin the associative 4-dimensional physical spacetime,in which case the propagation is unchanged, and thefermion remains a FIRST generation fermion representedby a single octonion; 2 - the origin vertex is in the associative spacetime,and the target vertex in in the Internal Symmetry Space,in which case there must be a new link fromthe original target vertex in the Internal Symmetry Spaceto a new target vertex in the associative spacetime,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, and therefore as a SECOND generation fermion; 3 - the target vertex is in the associative spacetime,and the origin vertex in in the Internal Symmetry Space,in which case there must be a new link tothe original origin vertex in the Internal Symmetry Spacefrom a new origin vertex in the associative spacetime,so that a second octonion can be introduced at the originalorigin vertex in connection with the new link,so that the fermion can be regarded after dimensional reductionas a pair of octonions, and therefore as a SECOND generation fermion;and 4 - both the origin vertex and the target vertexare in the Internal Symmetry Space,in which case there must be a new link tothe original origin vertex in the Internal Symmetry Spacefrom a new origin vertex in the associative spacetime,and a second new link from the original target vertexin the Internal Symmetry Space to a new target vertexin the associative spacetime,so that a second octonion can be introduced at the originalorigin vertex 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, and therefore as a THIRD generation fermion. As there are no more possibilities, there are no more generations. 
 Particle masses and force strength constants are not really "constant" when you measure them, as the result of your measurement will depend on the energy at which you measure them.  Measurements at one energy level can be related to measurements at another by renormalization equations.  The particle masses calculated in the D4-D6-E6-E7-E7-E8 VoDou Physics model are, with respect to renormalization, each defined at the energy level of the calculated particle mass.  In the D4-D6-E6-E7-E7-E8 VoDou Physics model, Dilatation Scale Transformations of the Conformal Group provide a natural setting for the Renormalization Group Process.  For leptons, such as the electron, muon, and tauon, which carry no color charge, you can renormalize conventionally from that energy level to "translate" the result to another energy level, because those particles are not "confined" and so can be experimentally observed as "free particles"  ("free" means "not strongly bound to other particles, except for virtual particles of the active vacuum of spacetime").   For quarks, which are confined and cannot be experimentally observed as free particles, the situation is more complicated.  In the D4-D6-E6-E7-E8 VoDou Physics model, the calculated quark masses are considered to be constituent masses.   In hep-ph/9802425, Di Qing, Xiang-Song Chen, and Fan Wang, of Nanjing University, present a qualitative QCD analysis and a quantitative model calculation  to show that the constituent quark model [after mixing a small amount (15%) of sea quark components]remains a good approximation even taking into account the nucleon spin structure revealed in polarized deep inelastic scattering. The effectiveness of the NonRelativistic model of light-quark hadrons is explained by, and affords experimental Support for, the Quantum Theory of David Bohm. Consitituent particles are Pre-Quantum particles in the sense that their properties are calculated without using sum-over-histories Many-Worlds quantum theory.  ("Classical" is a commonly-used synonym for "Pre-Quantum".) Since experiments are quantum sum-over-histories processes, experimentally observed particles are Quantum particles.  The lightest experimentally observable particle containing quarks is the pion, which is a quark-antiquark pair made up of the lightest quarks, the up and down quarks. A quark-antiquark pair is the carrier of the strong force, and mathematically resembles a bivector gluon, which is the carrier of the color force.  The charactereistic energy level of pions is the square root of the sum of the squares of the masses of the two charged and one neutral pion. It is about 245 MeV (to more accuracy 241.4 MeV). The gluon-carried color force strength is renormalized to higher energies from about 245 MeV in the conventional way.  What about quarks, as opposed to gluons? Gluons are represented by quark-antiquark pairs, but a quark is a single quark. The lightest particle containing a quark that is not coupled to an antiquark is the proton,  which is a stable (except with respect to quantum gravity) 3-quark color neutral particle.  The characteristic energy level of the proton is about 1 GeV (to more accuracy 938.27 MeV).  Quark masses are renormalized to higher energies from about 1 GeV (or from their calculated mass, below which they do not exist except virtually) in the conventional way.  What about the 3 quarks (up, down, and strange) that have constituent masses less than 1 GeV? Below 1 GeV, they can only exist (if not bound to an antiquark) within a proton, so their masses are "flat", or do not "run", in the energy range below 1 GeV.  Since the 3 quarks, up, down, and strange, are the only ones lighter than a proton, they can be used as the basis for a useful low-energy theory, Chiral Perturbation Theory, that uses the group SU(3)xSU(3), or, if based only on the lighter up and down quarksthat uses the group SU(2)xSU(2).   A useful theory at high energies, much above 1 GeV, is Perturbative QCD, that treats the quarks and gluons as free, which they are asymptotically as energies become very high. However, Patrascioiu and Seiler indicate that Perturbative QCD may not be exactly 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. ...".

Perturbative QCD, Chiral Perturbation Theory, and LatticeGauge Theory:

Even though, as Patrascioiu and Seiler indicate,  Perturbative QCD may not be exactly physically accurate, Perturbative QCD can be useful in doing rough calculations and in understanding some aspects of QCD.  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 our 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.  Since the virtual sea is a quantum phenomenon, the current quarks of Perturbative QCD and Chiral Perturbation Theory are, in my view, Quantum particles.   The relation between current masses and constituent masses may be explained, at least in part, by the Quantum Theory of David Bohm. 
From a conventional point of view, in hep-ph/0006306, Szczepaniak and Swanson "... have shown how quark model phenomenology may be derived from a simple model of QCD through the careful use of a nonperturbative renormalization procedure coupled with a model of the vacuum which breaks chiral symmetry. ...".
 In our model:  a current quark is viewed as a composite combination of a fundamental constituent quark and Quantum virtual sea gluon, quarks, and antiquarks (compare the conventional picture of, for example, hep-ph/9708262, in which current quarks are Pre-Quantum and constituent quarks are Quantum composites); and   the input current quarks of Perturbative QCD and Chiral Perturbation Theory are Quantum, and not Pre-Quantum, so that we view Perturbative QCD and Chiral Perturbation Theory as effectively "second-order" Quantum theories (rather than fundamental theories) that are most useful in describing phenomena at high and low energy levels, respectively.  Therefore, in Perturbative QCD and Chiral Perturbation Theory,the up and down quarks roughly massless.  One result is that the current masses can then be used as input for the SU(3)xSU(3) Chiral Perturbation Theory that, although it is only approximate because the constituent mass of the strange quark is about 312 MeV, rather than nearly zero, can be useful in calculating meson properties.  Taekoon Lee, in hep-ph/0006349, points out that quark mass aligns the theta vacuum so that there is no strong CP violation, thus solving the QCD strong CP problem without an axion. This result can also be seen from the point of view of the D4-D5-E6-E7 physics model.  WHAT ARE THE REGIONS OF VALIDITY OF PERTURBATIVE QCD and CHIRAL PERTURBATION THEORY? Perturbative QCD is useful at high energies.  If Perturbative QCD is valid at energies above 4.5 GeV, then Yndurian in hep-ph/9708300 has shown that lower bounds for current quark masses are:  ms       at least 150 MeV   (compare D4-D6-E6-E7 312   MeV)  mu+md    at least  10 MeV   (compare D4-D6-E6-E7   6.5 MeV)  mu-md    at least   2 MeV   (compare D4-D6-E6-E7   2.1 MeV)  Yndurian bases his estimates on positivity, and uses MSbar masses defined at 1 Gev^2.   Chiral Perturbation Theory is useful at low energies. Lellouch, Rafael, and Taron in hep-ph/9707523 have shown roughly similar lower bounds using Chiral Perturbation Theory.    Lattice Gauge Theory calculations of Gough et al in hep-ph/9610223 give light quark current MSbar masses at 1 Gev^2 as:  ms       59    to  101   MeV  mu+md     4.6  to    7.8 MeV   Clearly, the strange quark current masses of Lattice Gauge Theory are a lot lighter than those calculated by Perturbative QCD and Chiral Perturbation Theory, as well as the D4-D6-E6-E7 model.   Further, recent observation of the decay of K+ to pion+ and a muon-antimuon pair gives a branching ratio with respect to the decay of K+ to pion+ and e+ e- that is 0.167, which is about 2 sigma below the Chiral Perturbation Theory prediction of 0.236.  These facts indicate that Perturbative QCD, Chiral Perturbation Theory, and Lattice Gauge Theory are approximations to fundamental theory, each useful in some energy regions, but not fully understood in all energy regions. 

Conventional Lattice Gauge Theory for fermions, such as quarks,has some fundamental problems:

The conventional lattice Dirac operator is afflicted with theFermion Doubling Problem, in which nearest-neighbor lattice sites areoccupied with (for 4-dim spacetime) 2^4 = 16 times too manyfermions;.

The conventional solutions to the Fermion Doubling Problem are toadd non-local terms that violate Chiral Symmetry on the lattice. Ifyou are trying to do Chiral Perturbation Theory on the lattice, thatseems to be a bad idea.

To solve the Fermion Doubling Problem without violating ChiralSymmetry, Bo Feng,Jianming Li, and Xingchang Song have proposed to modify theconventional lattice Dirac operator by adding a non-local term that(like an earlier approach of Drell et. al., Phys. Rev. D 14 (1976)1627) couples all lattice sites along a given direction instead ofcoupling only nearest-neighbor sites. Their modified lattice Diracoperator not only preserved Chiral Symmetry, it also gives theconventional D'Alembertian operator, and they are able to constructthe Weinberg-Salam Electro-Weak model on a lattice.

  Conventional Lattice Gauge Theory is formulated somewhat differently from  another approach to formulating physics models on lattices:Feynman Checkerboards,  

Truth Quark Mass.

 In the HyperDiamond Feynman Checkerboard model,the Higgs scalar couples most strongly with the particle-antiparticle pairmade of femions with the most charge in the highest generation. That means Third Generation fermions made of triples of octonions, carrying both color and electric charge, and therefore quarksrather than leptons, and carrying electric charge of magnitude 2/3 rather than 1/3,and therefore: the Higgs scalar couples most strongly with the Truth quark, whose tree-level constituent mass of 129.5155 GeV is somewhat lower than, but close to, the Higgs scalar mass of about 146 GeV.   The HyperDiamond Feynman Checkerboard model value ofabout 130 GeV is substantially different fromthe roughly 175 GeV figure advocated by FermiLab. I think that the FermiLab figure is incorrect. The Fermilab figure is based on analysis ofsemileptonic events. I think that the Fermilabsemileptonic analysis does not handle background correctly,and ignores signals in the data that are in roughagreement with the D4-D6-E6-E7 model tree level constituent massof about 130 GeV. Further, I think that dileptonic events are morereliable for Truth quark mass determination,even though there are fewer of them than semileptonic events. I disagree with the Fermilab D0 analysis of dileptonic events, which Fermilab says are in the range of 168.3 GeV. My analysis of those dileptonic eventsgives a Truth quark mass of about 136.7 GeV, in rough agreement with the D4-D6-E6-E7 model tree level Truth quark constituent massof about 130 GeV.   More details about these issues,including gif images of Fermilab data histogramsand other relevant experimental results,can be found on the World Wide Web at URLs I consider the mass of the Truth quark to bea good test of the D4-D6-E6-E7 model, as the model is falsifiable by experimental results.

The Truth Quark, through its stronginteraction with Higgs Vacua, may have two excited energy levels at225 GeV and 173 GeV, above a ground state at 130 GeV. The 173GeV excited state may exist due to appearance of a Planck-energyvaccum with < phi_vac2 > = 10^19 GeV in addition to thelow-energy Standard Model vacuum with < phi_vac1 > = 252GeV.

Other Fermion Masses.

 In the HyperDiamond Feynman Checkerboard model,the masses of the other fermions are calculated fromthe mass of the Truth quark, with the following results forindividual tree-level lepton masses and quark constituent masses:  me = 0.5110 MeV; mnue = mnumu = mnutau = 0 at 0-level but 1-level corrections exist; md = mu = 312.8 MeV (constituent quark mass); mmu = 104.8 MeV; ms = 625 MeV (constituent quark mass); mc = 2.09 GeV (constituent quark mass); mtau = 1.88 GeV; mb = 5.63 GeV (constituent quark mass); These results when added up,  with each quark having 3 color charge states, each fermion particle having an antiparticle, and each Dirac fermion having 2 helicity states,  give a total mass of first generation fermions: Sigmaf1 = 7.508  GeV   Here is how the individual fermion mass calculationsare done:
The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:
The Weyl fermion neutrino has at tree levelonly the left-handed state,whereas the Dirac fermion electron and quarks can haveboth left-handed and right-handed states,so that the total number of states correspondingto each of the half-spinor Spin(0,8) representations is 15.  In all generations, neutrinos are massless at tree level.  However, even though massless at tree level, neutrinos are spinors and therefore are acted upon by Gravity as shown by the Papapetrou Equations. Further, in Quantum Field Theory at Finite Temperature, the gravitational equivalence principle may be violated,causing mixing among neutrinos of different generations.     In the HyperDiamond Feynman Checkerboard model,  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.  In the HyperDiamond Feynman Checkerboard model, fermion masses are calculated as a product of four factors: V(Qfermion) x N(Graviton) x N(octonion) x Sym  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, md/me = 6V(S^7 x  RP^1) = 2 pi^5 = 612.03937  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 by assuming that the electron mass me = 0.5110 MeV.  Then, the constituent mass of the down quark ismd = 312.75 MeV, and the constituent mass for the up quark ismu = 312.75 MeV.  As the proton mass is taken to be the sum of the constituentmasses of its constituent quarks mproton = mu + mu + md = 938.25  MeV The D4-D6-E6-E7 model 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.  Therefore the tauon mass is 1.87704 GeV.  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 mb = 5.63111 GeV.  
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 D4-D6-E6-E7 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 D4-D6-E6-E7 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 D4-D6-E6-E7 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 D4-D6-E6-E7 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 mt = 129.5155 GeV  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 of third generation fermions: Sigmaf3 = 1,629 GeV   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.  Therefore, mmu = 104.76 MeV.

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 isms = 312.75  MeV + 312.75  MeV = 625.5  MeV  The 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 ismc = 312.75  MeV + 1,772.25  MeV = 2.085  GeV  The 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 D4-D6-E6-E7 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 fermions: Sigmaf2 = 32.9 GeV  

K-M Parameters.

 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. ...".

In hep-ph/0304132,Marco Battaglia says: "... This ... 330 page ... report containsthe results of the Workshop on the CKM Unitarity Triangle, held atCERN on 13-16 February 2002 to study the determination of the CKMmatrix from the available data of K, D, and B physics. This is acoherent document with chapters covering the determination of CKMelements from tree level decays and K and B meson mixing and theglobal fits of the unitarity triangle parameters. The impact offuture measurements is also discussed. ...".


Khokhlov- For N = 1, 2, 3:

NthGeneration Fermions and N-Photon Processes:


D. L. Khokhlov,in hep-ph/9809457, describes muon decay by 2-photon processes andtauon decay by 3-photon processes, a picture that is consistent withthe D4-D6-E6-E7 physics model whichdescribes

Since a muon would look like a pair of octonions, it should decayby a 2-photon process.

Khokhlov refers to Quantum Electrodynamics by Landau and Lifshitz(3rd ed, Nauka, 1989), for the 2-photon scattering cross sectionsigma_2, for the energy level hbar omega = m_e c^2 at which formationof real electron-positron pairs becomes possible,

sigma_2 = 0.031 alpha^2 r_cl^2

where alpha is the electromagnetic fine structure constant, r_cl =e^2 / m_e c^2 is the classical electron radius, omega is frequencey,hbar is Planck's constant h divided by 2 pi, c is the speed of light,and m_e is the electron mass.

As the effective radius r_2 of 2-photon scattering is given by r_2= sqrt( sigma_2 / pi )

r_2 = sqrt( 0.031 alpha^2 r_cl^2 / pi ) = ( 1 / 137 ) sqrt (0.031 / pi ) 2.8 x 10^(-13) cm =

= 2 x 10^(-16) cm

Since the Compton wavelength r_e of an electron is about 3.86 x10^(-11) cm, and the electron mass/energy m_e is about 0.511 MeV, theenergy level corresponding to muon 2-photon decay is 1.93 x 10^5 m_e= 98.6 GeV, which is the same order as the mass/energy of W-bosons(about 80 GeV) that mediate muon decay in the D4-D6-E6-E7physics model and the Standard Model



mu-neutrino + W


mu-neutrino + electron + e-antineutrino


Since 2-photon processes are characterized by the classicalelectron radius r_cl, the muon mass M_mu should be of the order

M_mu = hbar / c r_cl = ( r_e / r_cl ) 0.511 MeV =

= ( 3.86 x 10^(-11) / 2.8 x 10^(-13) ) 0.511 MeV = 70MeV

which is the same order as the experimental muon mass 105.66 MeVand the D4-D6-E6-E7 modelcalculated tree-level muon mass 104.8MeV.


Since 3-photon processes are characterized by the triplet state ofpositronium, and 2-photon processes are characterized by the singletstate of positronium, and, according to Sakurai (Advanced QuantumMechanics, Addison-Wesley 1967, page 227), for the n =1, s wave boundstates of positronium, the lifetime ratio T_singlet / T_triplet = 9pi / 4 ( pi^2 - 9 ) alpha = 1115, the tauon mass M_tau should be ofthe order

M_tau = M_mu sqrt( T_singlet / T_triplet ) = 2300MeV

which is the same order as the experimental tauon mass 1777 MeV,the D4-D6-E6-E7 modelcalculated tree-level tauon mass 1877 MeV,and the tauon mass estimate of Khokhlov of 2200 MeV.


From Sets to Quarks: Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras. Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.


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