## From Sets to Quarks:

Deriving the Standard Model plus Gravitation
from Simple Operations on Finite Sets
by Tony Smith

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 model
the mass parameter m is the amplitude for a particle
to change its spacetime direction. Massless particles
do not change direction, but continue on the same
lightcone path.

In the D4-D6-E6-E7 Lagrangian continuum version
of this physics model, particle masses are calculated in terms
of relative volumes of bounded complex homogeneous domains and
their Shilov boundaries.

The relationship between
the D4-D6-E6-E7 Lagrangian continuum approach
and
the HyperDiamond Feynman Checkerboard discrete approach
is that:

the bounded complex homogeneous domains correspond to
harmonic functions of generalized Laplacians
that determine heat equations, or diffusion equations;

while the amplitude to change directions in the
HyperDiamond Feynman Checkerboard is a diffusion process
in the HyperDiamond spacetime, also corresponding to
a generalized Laplacian.

Details of the D4-D6-E6-E7 Lagrangian continuum approach
can be found on the World Wide Web at URLs

http://xxx.lanl.gov/abs/hep-ph/9501252

For the discrete HyperDiamond Feynman Checkerboard
approach of this paper, the only free mass parameter
is the mass of the Higgs scalar. All other particle
masses are determined as ratios with respect to the
Higgs scalar and each other.

The Higgs mass is 145.789 GeV in the
HyperDiamond 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 electron
mass 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 is 259.031 GeV / 7.508 GeV = 34.5007

which is very close to

the ratio of the geometric part of the Weak Force Strength to the Electromagnetic Fine Structure Constant is 0.253477 / ( 1 / 137.03608 ) = 34.7355.

Effectively, in the HyperDiamond Feynman
Checkerboard model, the electron mass is fixed at 0.5110 MeV
and all other masses are determined from it by the
ratios calculated in the model.

### Higgs Scalar, Gauge Bosons, and Fermions.

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

### 1 8 28 56 70 56 28 8 1

that gives three types of particles for which mass ratios
can be calculated in the HyperDiamond Feynman Checkerboard
model:

the Higgs Scalar;

the 28 bivector gauge bosons;

and the 8 + 8 = 16 half-spinor fermions.

### Higgs Scalar Mass.

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 Space
according to their group symmetry;

and

16 U(4) gauge bosons,

which reduce to 15 SU(4) = Spin(6) gauge bosons
plus one U(1) phase for particle propagator amplitudes
(this phase is what makes the sum-over-histories quantum
theory interferences work).

The 15 SU(4) = Spin(6) gauge bosons further reduce
to 10 Spin(5) deSitter gravitons that give physical
gravity by the MacDowell-Mansouri mechanism as described in

http://xxx.lanl.gov/abs/hep-ph/9501252

with gravitons that see their Symmetry Space
of Spacetime according to their group symmetry,

and 4 conformal generators and 1 scale generator.

The 4 conformal generators couple to the Higgs scalar
so that it becomes the mass-giver by the Higgs mechanism
as described in

http://xxx.lanl.gov/abs/hep-ph/9501252

and the 1 scale generator
represents the scale that we chose
by setting the Higgs scalar field vacuum expectation
value 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 bosons
to give them mass.

### Weak Boson 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 of
the Higgs scalar field, v = 252.514 GeV.

What are individual masses of members
of 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 as
S^1 --} S^3 --} S^2
gives a decomposition of the W bosons
into the neutral W0 corresponding to S^1 and
the charged pair W+ and W- corresponding
to S^2.

The mass ratio of the sum of the masses of
W+ and W- to
the mass of W0
should be the volume ratio of
the S^2 in S^3 to
the S^1 in S3.

The unit sphere S^3 in R^4 is
normalized by 1 / 2.

The unit sphere S^2 in R^3 is
normalized by 1 / sqrt3.

The unit sphere S^1 in R^2 is
normalized by 1 / sqrt2.

The ratio of the sum of the W+ and W- masses to
the 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 consistently
on the entire Dirac fermion particle sector,
so that the charged W+/- SU(2) weak bosons
act only on left-handed fermions of all types.

The neutral W0 weak boson does not interchange Weyl
neutrinos with Dirac fermions, and so is not restricted
to 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 to
the charged W+/- weak bosons by custodial SU(2)
symmetry, so that the left-handed component of the
neutral W0 must be equal to the left-handed (entire)
component of the charged W+/-.

Since the mass of the W0 is greater than the mass
of the W+/-, there remains for the W0 a component
acting on both types of fermions.

Therefore the full W0 neutral weak boson interaction
is proportional to
(mW+/-^2 / mW0^2) acting on left-handed fermions
and

(1 - (mW+/-^2 / mW0^2)) acting
on both types of fermions.

If (1 - (mW+/-2 / mW0^2)) is defined to be
sin(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 the
parity 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 is
equal to mW+/- ; and

the parity conserving part mW0LR that acts like a
heavy 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 force
strength of the weak parity conserving U(1)
electromagnetic type force that acts through the
U(1) subgroup of SU(2).

The electromagnetic force strength
alphaE = e^2 = 1 / 137.03608 was calculated
in Chapter 8 using
the volume V(S^1) of an S^1 in R^2,
normalized by 1 / \qrt2.

The *alphaE force is part of the SU(2) weak
force whose strength alphaW = w^2 was calculated
in 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^2
was calculated in Chapter 8 using a
4-dimensional spacetime with global structure of
the 4-torus T^4 made up of four S^1 1-spheres,

while the SU(2) weak force strength
alphaW = w^2 was calculated in Chapter 8
using two 2-spheres S^2 x S^2,
each of which contains one 1-sphere of
the *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 like
an electromagnetic force in the 4-dimensional
spacetime HyperDiamond Feynman Checkerboard model:

*e mW0LR = e (1/5.65) mW0LR = e mZ0,

where the physical effective neutral weak boson is
denoted by Z0.

Therefore, the correct HyperDiamond Feynman Checkerboard values for
weak 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-level
HyperDiamond 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 the Weinberg angle. According to a 19 November 2001 document of the Particle Data Group, "...

...

...".

The D4-D5-E6-E7-E8 VoDou Physics model definition of the Weinberg angle is fundamentally based on force strengths (which, in turn, can be expressed in terms of the calculated W and Z weak boson masses of M_W = 80.326 GeV and M_Z = 91.862 GeV), so that the D4-D5-E6-E7-E8 VoDou Physics model definition of the Weinberg angle is effectively the MSbar definition.

In the edited-quotes of the above images, c and cbar are functions of the masses of the Higgs and the Truth Quark, and the above numbers may be based on masses of around 100 GeV and 175 GeV, respectively. Even so, using those values of c and cbar, a calculation shows that the MSbar value of sin(theta_W)^2 of 0.235 of the D4-D5-E6-E7-E8 VoDou Physics model corresponds to an on-shell tree level value of sin(theta_W)^2 of 0.2267.

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

In the D4-D5-E6-E7-E8 VoDou Physics model, the Higgs Mass of 146 GeV is close to 150 GeV, but the Truth Quark Mass of 130 GeV is substantially lower than 175 GeV. If the value of 130 GeV is 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 result with a 130 GeV Truth Quark Mass sin(theta_W^nu-N)^2 = 0.2289 with total statistical and systematic error bar +/- 0.00228, which compares (in my opinion) realistically with the D4-D5-E6-E7-E8 VoDou Physics model on-shell tree level value of sin(theta_W)^2 of 0.2267.

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

### Fermion Masses.

Corresponding to the way that 24 of the 28 Gauge Bosons of the D4-D5-E6-E7-E8 VoDou Physics model can be represented by the vertices of a 24-cell, the first-generation Fermion Particles and anti-Particles, and SpaceTime plus Internal Symmetry Space, can be represented by the vertices of a dual 24-cell:

In the dual 24-cell,  the 8 vertices

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

These 8+8 = 16 vertices correspond to the 16 Complex Dimensions of the Bounded Domain corresponding to the Symmetric Space E6 / D5xU(1), and to the 16 Real Dimensions of its Shilov Boundary. The 32 Real Dimensions of that Complex Bounded Domain correspond to the 32 vertices of the 5-dimensional HyperCube that make up 32 of the 72 E6 root vectors.

In this image

only the 8 vertices representing the first-generation fermion particles are marked.

First generation fermion particles are also represented
by 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 represented
by octonions in a similiar way.

Second generation fermion particles and antiparticles
are represented by pairs of octonions.

Third generation fermion particles and antiparticles
are 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 the
fact that the Lie algebra series E6, E7, and E8,
has only 3 algebras, which in turn is a consequence of
non-associativity of octonions, as described here and in

http://www.innerx.net/personal/tsmith/E678.html

or geometrically as a consequence of the fact that,

if you reduce the original 8-dimensional spacetime
into associative 4-dimensional physical spacetime
and coassociative 4-dimensional Internal Symmetry Space,

then, if you look in the original 8-dimensional spacetime
at a fermion (First-generation represented by a single octonion)
propagating from one vertex to another,
there are only 4 possibilities for the same propagation
after dimensional reduction:

1 - the origin and target vertices are both
in the associative 4-dimensional physical spacetime,
in which case the propagation is unchanged, and the
fermion remains a FIRST generation fermion represented
by 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 from
the original target vertex in the Internal Symmetry Space
to a new target vertex in the associative spacetime,
and a second octonion can be introduced at the original
target vertex in connection with the new link,
so that the fermion can be regarded after dimensional reduction
as 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 to
the original origin vertex in the Internal Symmetry Space
from a new origin vertex in the associative spacetime,
so that a second octonion can be introduced at the original
origin vertex in connection with the new link,
so that the fermion can be regarded after dimensional reduction
as a pair of octonions, and therefore as a SECOND generation fermion;
and

4 - both the origin vertex and the target vertex
are in the Internal Symmetry Space,
in which case there must be a new link to
the original origin vertex in the Internal Symmetry Space
from a new origin vertex in the associative spacetime,
and a second new link from the original target vertex
in the Internal Symmetry Space to a new target vertex
in the associative spacetime,
so that a second octonion can be introduced at the original
origin vertex in connection with the first new link,
and a third octonion can be introduced at the original
target vertex in connection with the second new link,
so that the fermion can be regarded after dimensional reduction
as a triple of octonions, and therefore as a THIRD generation fermion.

As there are no more possibilities, there are no more generations.

RENORMALIZATION OF PARTICLE MASSES AND FORCE STRENGTHS:

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 quarks
that 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.

Further,

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

### Perturbative QCD, Chiral Perturbation Theory, and Lattice Gauge 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 the Fermion Doubling Problem, in which nearest-neighbor lattice sites are occupied with (for 4-dim spacetime) 2^4 = 16 times too many fermions;.

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

To solve the Fermion Doubling Problem without violating Chiral Symmetry, Bo Feng, Jianming Li, and Xingchang Song have proposed to modify the conventional 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 of coupling only nearest-neighbor sites. Their modified lattice Dirac operator not only preserved Chiral Symmetry, it also gives the conventional D'Alembertian operator, and they are able to construct the 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 pair
made 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 quarks
rather 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 of
about 130 GeV is substantially different from
the roughly 175 GeV figure advocated by FermiLab.

I think that the FermiLab figure is incorrect.

The Fermilab figure is based on analysis of
semileptonic events. I think that the Fermilab
semileptonic analysis does not handle background correctly,
and ignores signals in the data that are in rough
agreement with the D4-D6-E6-E7 model tree level constituent mass

Further, I think that dileptonic events are more
reliable 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 events
gives a Truth quark mass of about 136.7 GeV,
in rough agreement with
the D4-D6-E6-E7 model tree level Truth quark constituent mass

including gif images of Fermilab data histograms
and other relevant experimental results,
can be found on the World Wide Web at URLs

http://www.innerx.net/personal/tsmith/TCZ.html

I consider the mass of the Truth quark to be
a good test of the D4-D6-E6-E7 model,
as the model is falsifiable by experimental results.

The Truth Quark, through its strong interaction with Higgs Vacua, may have two excited energy levels at 225 GeV and 173 GeV, above a ground state at 130 GeV. The 173 GeV excited state may exist due to appearance of a Planck-energy vaccum with < phi_vac2 > = 10^19 GeV in addition to the low-energy Standard Model vacuum with < phi_vac1 > = 252 GeV.

### Other Fermion Masses.

In the HyperDiamond Feynman Checkerboard model,
the masses of the other fermions are calculated from
the mass of the Truth quark, with the following results for
individual 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);

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 calculations
are done:
The Discrete HyperDiamond Generalized Feynman Checkerboard and Continuous Manifolds are related by Quantum Superposition:
The Weyl fermion neutrino has at tree level
only the left-handed state,
whereas the Dirac fermion electron and quarks can have
both left-handed and right-handed states,
so that the total number of states corresponding
to 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 fermions
correspond to octonions   O

and second generation fermions
correspond to pairs of octonions   O x  O

and third generation fermions
correspond to triples of octonions   O x  O x  O.

To calculate the fermion masses in the model,
the volume of a compact manifold representing the
spinor 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-Mansouri
gravitation 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 mass
of the first-generation electron.

The graviton number ratio of the down quark to the
first-generation electron is therefore 6/1 = 6.

N(octonion) is an octonion number factor relating up-type quark
masses to down-type quark masses in each generation.

Sym is an internal symmetry factor, relating 2nd and 3rd
generation 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 mass
is 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 the
massive 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 Dirac
fermion, 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 manifold
volume factor V(Qdown quark) of the volume of
S^7 x RP^1.

The ratio of the down quark spinor manifold volume factor to
the 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 quarks
have the same constituent mass mu = md.

Antiparticles have the same mass as the corresponding
particles.

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 is
md = 312.75 MeV, and

the constituent mass for the up quark is
mu = 312.75 MeV.

As the proton mass is taken to be the sum of the constituent
masses of its constituent quarks

mproton = mu + mu + md = 938.25  MeV

The D4-D6-E6-E7 model calculation is close to
the experimental value of 938.27 MeV.

The third generation fermion particles correspond to triples of
octonions. 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 correspond
to 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 the tau 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 correspond to 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 the third generation graviton factor 6/2 = 3, so the B-quark mass is   mb = 5.63111 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 the constituent mass of the red truth quark is 161/7 = 23 times the red 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 Feynman Checkerboard 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 correspond to 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 symmetries of the first generation fermion particles.   The pair { E, E } should correspond to the E electron.   The other two muon pairs have a symmetry group S2, which is 1/3 the size of the color symmetry group S3 which gives the up and down quarks their mass of 312.75 MeV.     Therefore the mass of the muon should be the sum of the { E, E } electron mass and the { 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 of Particle Physics of the Particle Data Group, the experimental muon mass is about 105.66 MeV.

Note that all pairs corresponding to
the muon and the mu-neutrino are colorless.

The red, blue and green strange quark each corresponds
to the 3 pairs involving 1 and I, J, or K.

The red strange quark is defined as the thrge pairs
1 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, and
the 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 that
may 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 times
the graviton factor 3 is 312.75 MeV,
and
the red strange quark constituent mass
is
ms = 312.75  MeV + 312.75  MeV = 625.5  MeV

The blue strange quarks correspond to the
three pairs involving J,
the green strange quarks correspond to the
three 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 should
be 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 quark
mass, 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
is
mc = 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,

These results when added up give a total mass of second 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 to

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.

"... Using the eight tree-level constraints discussed below together with unitarity, and assuming only three generations, the 90% confidence limits on the magnitude of the elements of the complete matrix 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, so choosing a specific value for one element restricts the range of others. ... The phase d13 lies in the range 0 < d13 < 2 pi, with non-zero values generally breaking CP invariance for the weak interactions. ... Using tree-level processes as constraints only, the matrix 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 as additional constraints, the sines of the angles remain unaffected, and the CKM phase, sometimes referred to as the angle gamma = phi3 of the unitarity triangle ...

... is restricted to d13 = ( 1.02 +/- 0.22 ) radians = 59 +/- 13 degrees. ... CP-violating amplitudes or differences of rates are all proportional to the product of CKM factors ... s12 s13 s23 c12 c13^2 c23 sind13. This is just twice the area of the unitarity triangle. ... All processes can be quantitatively understood by one value of the CKM phase d13 = 59 +/- 13 degrees. The value of beta = 24 +/- 4 degrees from the overall fit is consistent with the value from the CP-asymmetry measurements of 26 +/- 4 degrees. The invariant measure of CP violation is J = ( 3.0 +/- 0.3) x 10^(-5). ... From a combined fit 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 unitarity triangle following from | Vub | , B mixing, epsilon, and sin2beta. ...

... A possible unitarity triangle is shown with the apex in the preferred region. ...".

In hep-ph/0208080, Yosef Nir says: "... Within the Standard Model, the only source of CP violation is the Kobayashi-Maskawa (KM) phase ... The study of CP violation is, at last, experiment driven. ... The CKM matrix provides a consistent picture of all the measured flavor and CP violating processes. ... There is no signal of new flavor physics. ... Very likely, the KM mechanism is the dominant source of CP violation in flavor changing processes. ... The result is consistent with the SM predictions. ...".

In hep-ph/0304132, Marco Battaglia says: "... This ... 330 page ... report contains the results of the Workshop on the CKM Unitarity Triangle, held at CERN on 13-16 February 2002 to study the determination of the CKM matrix from the available data of K, D, and B physics. This is a coherent document with chapters covering the determination of CKM elements from tree level decays and K and B meson mixing and the global fits of the unitarity triangle parameters. The impact of future measurements is also discussed. ...".

## Nth Generation Fermions and N-Photon Processes:

D. L. Khokhlov, in hep-ph/9809457, describes muon decay by 2-photon processes and tauon decay by 3-photon processes, a picture that is consistent with the D4-D6-E6-E7 physics model which describes

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

Khokhlov refers to Quantum Electrodynamics by Landau and Lifshitz (3rd ed, Nauka, 1989), for the 2-photon scattering cross section sigma_2, for the energy level hbar omega = m_e c^2 at which formation of 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 x 10^(-11) cm, and the electron mass/energy m_e is about 0.511 MeV, the energy 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-E7 physics model and the Standard Model

muon

to

mu-neutrino + W

to

mu-neutrino + electron + e-antineutrino

Since 2-photon processes are characterized by the classical electron 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 = 70 MeV

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

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

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

which is the same order as the experimental tauon mass 1777 MeV, the D4-D6-E6-E7 model calculated 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.