- Unification
- Monopoles
- Dual Unification: Monopoles to Dyons to Fermions
- Duality Overview
- Cosmic Ray Spectrum and SU(5) Duality
- D5 of D4-D5-E6-E7-E8 VoDou Physics
- Experimental Verification of SU(5) GUT Proton Decay

The force strengths alpha_3 of the SU(3) Color Force and alpha_2 and alpha_1 of the U(2) = SU(2) x U(1) ElectroWeak Force have been measured at low energies (at and below around 90 GeV).

Those experimental measurements have been extrapolated to higher energies (up to around 8 x 10^16 GeV):

The left-hand graph (from Fig. 9-3 of Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, by Chris Quigg (Addison-Wesley 1983, 1997) shows that, although the effective Electromagnetic Fine Structure Constant alpha_EM remains smaller than 1/100 all the way up to about 8 x 10^16 GeV (so that I continue to use 1/137 for the rough calculations done on this web page, even though the calculations may be related to high energies);

the three force strengths alpha_3, alpha_2, and alpha_1 all tend to converge at an energy somewhere around 6 x 10^14 GeV.

The right-hand graph ( from CERN Courier, March 1991, page 1 ) is a larger-scale version of the region of convergence.

Although CERN said in that 1991 article: ".. several years ago ... extrpolations .. did not meet at a point ... suggesting that a Grand Unified Theory needed addtional physics input .. With the electroweak strengths ... now [1991] more accurately known ... and with measurements at the Z peak ... on the strong coupling ... it looks even less likely that extrapolations ... will converge at a point. ...",

in my opinion the facts:

- that the region of convergence is narrow ( 10^13 to 8 x 10^16 GeV ) relative to the 11 powers of 10 from 100 GeV to 10^13 GeV;
- that VoDou D4-D5-E6-E7 physics differs from the current (2000) consensus version of Minimal SU(5) Grand Unification in that Conformal GraviPhotons are present in VoDou D4-D5-E6-E7 physics and in that there is an apparent (to me, at least) incomplete understanding of some weak force phenomena ( some of which involved events that were initially misidentified as LeptoQuarks ) by the consensus; and
- that
**Standard Model SU(5) Grand Unification**predicts phenomena at both 6 x 10^14 GeV (**X-boson**mass and the Vacuum Expectation Value of**X-scalar Higgs**field ) and at 8 x 10^16 GeV (**Monopole**mass, about Mx / alpha_EM = 137 x Mx ) so that full convergence may not occur until the Monopole mass of about 8 x 10^16 GeV is reached;

give a pretty good indication that convergence may well be physically realistic, and could involve phenomena all the way up to the 8 x 10^16 GeV mass ( about Mplanck / 137 ) of

According to The Early Universe, by G. Borner (Springer-Verlag 1988), from which book's Fig. 6.21 the above SU(5) GUT illustration is taken, "... For GUT physics monopoles are extremely interesting objects: they have an onion-like structure ... which contains the whole world of grand unified theories.

- Near the center ( about 10^(-29) cm ) there is a GUT symmetric vacuum.
- At about 10^(-16) cm, out to the Yukawa tail ... exp( - Mw r ), the field is the electroweak colour field of the (3,2,1) standard model, and
- at ...[10^(-15) cm]... it is made up of photons and gluons, while
- at the edge [ 10^(-13) cm ] there are fermion-antifermion pairs.
- Far beyond nuclear distances it behaves as a magnetically-charged pole of the Dirac type.

This view of the GUT monopole raises the possibility that it may catalyze the decay of the proton ...".

According to The Early Universe, by Kolb and Turner (1994 paperback edition, Adddison-Wesley, page 241): "... The exponential expansion associated with inflation allows a small, sub-horizon sized region of space, within which the Higgs field is correlated, to encompass all of the presently observed Universe. The end result is less than one monopole in the entire observable Universe due to the Kibble mechanism. ...".

As to experimental results, the 2000 Review of Particle Physics says: "... a candidate event in a single superconducting loop in 1982 ... stimulated an enormous experimental effort to search for supermassive magnetic monopoles ... Monopole candidate events in single semiconductor loops have been detected [citing B. Cabrera, Phys. Rev. Lett. 48, 1378 (1982); A. D. Caplin et al., Nature 321, 402 (1986)] ... but no two-loop coincidence has been observed. ...".

Liu and Vachaspati in hep-th/9604138 say:

",,, The spectrum of monopoles is found to correspond to the spectrum of one family of standard model fermions and hence, is a starting point for constructing the dual standard model. ... there is an extra monopole state - the "diquark" monopole - with no corresponding standard model fermion. ...... It is worthwhile clarifying our use of the word "dual". What we have in mind is that

In other words, the standard model is an effective theory of fields that create and annihilate SU(5) monopoles. This is analogous to the sine-Gordon and Thirring model equivalence. ...".

Vachaspati in his paper hep-ph/9509271 asks "... why should the monopoles be fermions and not bosons?" and answers: "... isopin can lead to spin. The idea is that a bound state of a charged boson and a monopole forms a dyon that can have integer or half-integer spinif the isospin of the free boson is integer or half-integer respectively. Goldhaber has shown that dyons with half-integer spin also obey Fermi-Dirac statistics ...". (See also Vachaspati's paper hep-th/709149.)

Lepora in his paper at hep-ph/0008323 says "... the monopoles arising from gauge unification may actually be the familiar elementary particles. ... much of this work is completely in line with Vachaspati's original conjecture that the elementary particles originate from a magnetic SU(5) gauge unification. ... Rescaling the running gauge coupling ... then gives:

FIG. 1. Rescaled Running Gauge Couplings

which appears to unify at a thermal scale of a few GeV ... The picture of this dual unification is as follows:

- (i) Monopole Spectrum of SU(5) Theory: Vachaspati has identified the five stable monopoles of SU(5) unification as having magnetic charges that mirror the electric charges of one generation of standard model fermions. ...
- (ii) Dyon Spectrum of SU(5) Theory: A crucial observation is that although these monopoles are scalar, they all possess spin half dyonic excitations, with the correct fermionic statistics. All of these have both electric and magnetic charges ...
- (iii) Role of the Theta Vacuum: Another crucial observation is that SU(5) theory at vanishing theta angle is both parity and charge-parity symmetric, which is incapable of producing a particle spectrum consistent with the standard model. Importantly, a non-trivial theta vacuum violates both parity and charge-parity; it also ... is the main factor that determines the lowest energy dyonic excitations.
- (iv) Monopoles as Particles: Implicit in the monopoles as particles picture is that the unified SU(5) theory be strongly coupled: then the Compton wavelengths of the monopoles are greater than their core sizes; meaning that these magnetic charges are truly quantum mechanical excitations (particles). Consequently, at unification these magnetic interactions are weakly coupled and perturbative. Assuming these dyons become the principle states of the theory, they therefore have strongly coupled electric interactions.
- (v) Running Gauge Couplings: As the magnetic couplings are weak we assume they run like the electric gauge couplings in the standard model. This is motivated by the picture of electric-magnetic duality ... whose central theme is that there is little difference between electric and magnetic interactions; the behaviour of each sector being dictated by the strength of the gauge coupling. Because the electric sector is strongly coupled it should run according to Dirac's condition g g~ = 4 pi.
- (vi) Magnetic Colour Confinement: ... the picture of unification is ... a unification scale of a few GeV and gauge couplings consistent with the standard model. The magnetic colour coupling (dual to the standard model's electric sector) runs as in the standard model becoming confining at the QCD hadron scale of around a GeV.
- (vii) Electric Colour Breaking: By the electric-magnetic duality ... the electric colour sector is then broken to triviality, as is consistent with the dual superconductor model of confinement. Following Chan and Tsou we identify this broken colour symmetry with a breaking of horizontal family symmetry in the standard model. ... this splits the electrically colour charged dyons into three copies.
- (viii) Electric Isospin Confinement ... : At unification magnetic isospin is weakly coupled, implying that electric isospin is then strongly coupled and confikning. At present we do not know how to calculate this confinement scale. Therefore we assume its mass scale is large compared to the standard model, say over a hundred GeV. Although this appears higher than the unification scale suggested by ...[the above Fig. 1]... we interpret the monopoles being formed in a state where they are already confined. Consequently all electric isospin dyons are confined into very massive, dual isospin hadrons.
- (ix) Magnetic Isospin Breaking: By the electric-magnetic duality ... , magnetic isospin is therefore broken. ...

An important point emerging from this result is that the SU(5) gauge coupling appears to be strong. This is because the unified gauge coupling is around 0.7; ... A description of the physics of such non-Abelian dyons has been achieved by Chan and Tsou ... Their duality transformation is not expressed in terms of the usual field description, but in terms of non-local Polyakov loop variables. ... Above the unification scale there are no fermions, only the constituent bosons. Presumably creation of such a plasma would allow the occurance of many transitions disallowed in the standard model; for instance baryon or lepton number violation. ...".

10^3 GeV = 10^12 eV ------------ 10^13 GeV = 10^22 eV

Transparency to multi-TeV photons ------------- Highest Observed Cosmic Ray Energy

10^5 GeV = 10^14 eV ------------ 10^11 GeV = 10^20 eV

It is interesting that:

- The LeptoQuark X-boson energy scale Mx, the GUT monopole mass
Mmonopole, and the Planck mass Mplanck are related to the
Electromagnetic Fine Structure Constant alpha^EM roughly as
follows:
- Mplanck = 1.2 x 10^19 GeV
- alpha_EM x Mplanck = 8.8 x 10^16 GeV = Mmonopole = Mwesson, where Mwesson is a characteristic mass of relationships among Mass, Angular Momentum, and Magnetic Dipole Moment.
- alpha_EM x Mmonopole = 6.4 x 10^14 GeV = Mx

- The LeptoQuark X-boson energy scale is similar to some other
important energy scales:
- The High-Energy Singularity of the Higgs Mechanism model, Msing, beyond which the Higgs field vanishes, and the Maximum Energy Scale, Mmax, can be calculated in the Higgs Mechanism model. The fact that the Higgs field vanishes above Msing and Mmax may justify regarding the Higgs Mechanism model as a low energy effective theory, just as the D4-D5-E6-E7-E8 model is fundamentally a low (with respect to the Planck energy) energy effective theory. The values calculated by Ni, Lou, Lu, and Yang are Msing = 5.5 x 10^14 GeV and Mmax = 8.7 x 10^14 GeV; and
- the Reheating Energy of the Zizzi Inflationary Universe, about 10^14 GeV.

- The Maximum Cosmic Ray Energy is just below 10^13 Gev = 10^22 eV, which corresponds to the Cartan radius ( 10^(-26) cm ) of a nucleon.
- The Region above the Knee of the Cosmic Ray Energy Spectrum, about 10^16 eV to 10^18 ev, centered on about 10^17 eV, coincides with the Self-Dual Energy Region of SU(5) Duality.
- The Sum of the Masses of all 3x(8+8) Fermions in the D4-D5-E6-E7-E8 VoDou Physics model is about 2 x ( 0.001 + 3x0.3 + 3x0.3 + 0.1 + 3x0.6 + 3x2 + 2 + 3x5 + 3x130 ) = 2x(1.801 + 7.9 + 407 ) = 2x417 = 834 GeV.
- The Pion Mass is about 140 MeV.
- The Sum of the Electron and Positron Masses is about 1 MeV.

Consider a Compton Radius Vortex
Kerr-Newman Black Hole related to the **Wesson
force**. The equation (in units with G = c = hbar = 1) for a
Kerr-Newman Black Hole with coincident outer and inner event horizons
and with Q = 1

meaning that the Black Hole Core has UNIT amplitude to absorb or emit a gauge boson, in accord with Feynman's statement in his book QED (Princeton 1988): "... e - the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.0854... the inverse of its square: about 137.03... has been a mystery ... all good theoretical physicists put this number up on their wall ..."

is Q^2 + (J/M)^2 = 1 + (J/M)^2 = M^2

Dividing through by M^2, you get

For the Wesson force for which J = p_wesson M^2 with p_wesson = 1 / alpha_EM

so that

Then the magnitude | 1 / Mwesson | = 137 which (since the units are natural units with G = c = hbar = 1) implies that

which is consistent with the hypothesis that

**Mwesson**** = 7.3 x
10^16 GeV = Mmonopole**

Perhaps there is a **Wesson
force** that is carried by a Mwessson particle. If so, since
the strength of the gravitational force is ( 1 / (Planck Mass)^2 ) it
seems to me that Wesson may be seeing a force whose strength is ( 1 /
(Mwesson)^2 ) so that the ratio of the strength of the Wesson force
to gravitation is

Note that 0.01 = 1/100 is roughly the Electromagnetic Fine Structure Constant alpha_EM which is at low energies 1/137.03 ... but which is closer to 1/100 at high energy levels.

Cosmic Rays may be produced by Supernovae, Active Galactic Nuclei, and Gamma Ray Bursts. The Highest Observed Cosmic Ray Energy is the Fly's Eye Event, about 320 EeV = 3 x 10^20 GeV.

In astro-ph/0103477, The Origin of the Knee in the Cosmic-Ray Energy Spectrum, A. D. Erlykin and A. W. Wolfendale say: ".. .A sudden steepening of the cosmic-ray energy spectrum ( the knee ) is observed at an energy of about 3 PeV (1 PeV = 10^15 eV). The recent results on extensive air showers allow us to conclude that:

- a) the knee has an astrophysical origin;
- b) the `sharpness' and the fine structure of the knee rule out `Galactic Modulation' as the origin of the knee;
- c) most likely the knee is the result of the explosion of a single, recent, nearby supernova.

... the case when we are inside the shock ( or only just outside ) is preferable, compared with the opposite case when we are outside it ... this conclusion helps us to understand also the relatively small amplitude of the anisotropy in the knee region. ... The candidate for the source could not be too far from the solar system or too close to the moment of the explosion in order to have enough energy in cosmic rays and give the requred contribution to the cosmic-ray flux at the knee. ...

... The dotted line shows the typical propagation of the shock front. The dashed and dash-dotted lines are derived from the comparison of the energy contained in cosmic rays for our Single Source and the energy content of the cosmic rays accelerated by the supernova remnant ... The possible source candidates should lie inside the area delimited by these lines. ... At the moment, the sources which gave birth to Loop I and the Geminga pulsar are the most favorable contenders. ...".

The Region above the Knee
of the Cosmic
Ray Energy Spectrum, shown on the graph above taken from a graph
by P. Sokolsky on a
Goddard NASA web page, is about **5 x 10^15 eV** to **5 x
10^18 eV**, **centered on about 10^17 eV**, which coincides with
the Self-Dual Energy Region of SU(5)
Duality.

"... The high energy cosmic rays (CR) spectrum depicts a clear
**break at about 5 x 10^18 eV**. This break is accompanied by **a
transition in the CR composition from nuclei to protons**. ...",
according to astro-ph/0008107
by Giovanni Amelino-Camelia and Tsvi Piran.

Kalmakhelidze, Roinishvili, and Svanidzey, in hep-ex/0107011,
say: "... Classification of gamma-hadron families, registered by the
Pamir collaboration, on four groups of nuclei (P, He, middle and
heavy), responsible for their generation, is made, and fractions of
families in each of the groups are estimated. Results show, that
**below the knee of the energy spectrum the chemical composition of
primary cosmic rays** ... **remains the normal or is displaced
towards the light nuclei.** ...".

The GZK limit of about 5 x 10^19 eV is shown by the red line.

Fodor and Katz, in their paper hep-ph/0105348, Ultrahigh energy cosmic rays as a Grand Unification signal, say: "... We analyze the spectrum of the ultrahigh energy (above [about] 10^9 GeV) cosmic rays. With a maximum likelihood analysis we show that the observed spectrum is consistent with the decay of extragalactic GUT scale particles. The predicted mass for these superheavy particles is mX = 10^b GeV, where b = 14.6 +1.6 - 1.7 ... Altogether we study four different models: halo-SM, halo-MSSM, EG-SM and EG-MSSM. ... The UHECR data favors the EG-MSSM scenario. The goodnesses of the fits for the halo models are far worse. The SM and MSSM cases do not differ significantly. ...".

**Planck Scale violation of Lorentz
Invariance** is advocated by Giovanni Amelino-Camelia and Tsvi
Piran in astro-ph/0008107
to account for the facts that "... Significant evidence has
accumulated in recent years suggesting that ... the universe is more
transparent than what it was expected to be ... in two different
regimes,

**Ultra High Energy
Cosmic Rays** (UHECRs)

... UHECRs interact with the Cosmic Microwave Background Radiation (CMBR) and produce pions. ... These interactions should make observations of UHECRs with E > 5 x 10^19 eV (the GZK limit) ... unlikely. Still UHECRs above the GZK limit ... are observed. ...A sufficiently energetic CMBR photon, at the tail of the black body thermal distribution, is seen in the rest frame of an Ultra High Energy (UHE) proton with E > 5 x 10^19 eV as a > 140 MeV photon, above the threshold for pion production. UHE protons should loose energy due to photopion production and should slow down until their energy is below the GZK energy. The process stops when CMBR photons energetic enough to produce pions are not sufficiently abundant. The proton's mean free path in the CMBR decreasesexponentially with energy (down to a few Mpc) above the

GZK limit ( about 5 x 10^19 eV ). Yet more than 15 CRs have been observed with nominal energies at or above 10^20 +/- 30% eV. ...

and

... TeV photons interact with the Infra Red (IR) photons and produce electron-positron pairs. ... These interactions should make observations of ... gamma-rays with E > 20 TeV from distant sources unlikely. Still ... 20TeV photons from Mk 501 are observed. ...HEGRA has detected high-energy photons with a spectrum ranging up to 24 TeV from Markarian 501 (Mk 501), a BL Lac object at a redshift of 0.034 (about 157 Mpc). This observation indicates a second paradox of a similar nature. A high energy photon propagating in the intergalactic space can interact with an IR background photon and produce an electron-positron pair if the CM energy is above 2 me c^2 . The maximal wavelength of an IR photon that could create a pair with a 10 TeV photon is 40 microns. As the cross section for pair creation peaks at a center of mass energy of about 3 me c^2 , 10 TeV photons are most sensitive to 30 micron IR photon and the mean free path of these photons depends on the spectrum of the IR photons at the 15 to 40 micron range. ... we turn now to TeV photons from Mk 421 (another BL Lac object at a redshift of 0.031, corresponding to about 143 Mpc). It is not clear if the spectrum of this source extends high enough to pose a paradox comparable to the one indicated by Mk 501. ...

... In both cases low energy photons interact with high energy particles. The reactions should take place because when Lorentz transformed to the CM frame the low energy photon have sufficient energy to overcome an intrinsic threshold. In both cases the CM energies are rather modest (about 100 MeV for UHECRs and about 1 MeV for the TeV photons) and the physical processes involved are extremely well understood and measured in the laboratory. In both cases we observe particles above a seemingly robust threshold and the observations can be considered as a "threshold anomaly".

It is remarkable that in spite of these similarities at present there is only one mechanism that could resolve both paradoxes: a mechanism based on the single, however drastic, assumption of a violation of ordinary Lorentz invariance. ...

... We start by considering first, a class of dispersion relations ... which in the high-energy regime takes the form:

m, E and p denote the mass, the energy and the (3-component) momentum of the particle, Eplanck is the Planck energy scale (Eplanck = 10^22 MeV), while a and n are free parameters characterizing the deviation from ordinary Lorentz invariance ( in particular, a specifies how strongly the magnitude of the deformation is suppressed by Eplanck ). ...

Figure 1: The region of the a, n parameter space that provides a solution to both the UHECR and TeV threshold anomalies while satisfying the time-of-flight upper bound on LID. Only negative values of n are considered since this is necessary in order to have upward shifts of the threshold energies, as required by the present paradoxes. The solid thick line describes the time-of-flight upper bound. The [light blue] region above this line is excluded. The solid thin line and the dotted line describe the lower bound on LID obtained from the present UHECR (solid thin line [the hatched area below it is excluded]) and TeV (dotted line [the green area below it is excluded]) threshold anomalies. The anomalies disappear in the region above the lines. Within the narrow [white] region between the dotted line and the solid thick line the time of flight constraint is satisfied and both anomalies are resolved. The ... vertical ... [red line at a = 1 corresponds to a] ... favored quantum-gravity scenario ...

[A quadratic a = 2 scenario favored by L. Gonzalez-Mestres is not permitted by the TeV threshold anomaly, but is a solution of the UHECR anomaly and is consistent with the time-of-flight upper bound, lying on the red horizontal line in the green unhatched area.]

... The behaviour of the curves for upper and lower bounds on LID with respect to the bottom-left corner of the frame can be understood by noticing that at a fixed a ordinary Lorentz invariance can be reached taking the n -> infinity limit, while at fixed n this requires taking the a -> infinity ( i.e. the 1/a -> 0 ) limit. ... [The red lines intersect at a = 1, n = -1, leading to a dispersion relation

that solves both the UHECR and TeV-threshold anomalies.] ...

... Relevant for our phenomenological considerations is the process in which the head-on collision between a soft photon of energy e and momentum q and a high-energy particle of energy E1 and momentum p1 leads to the production of two particles with energies E2, E3 and momenta p2, p3. At threshold ( no energy available for transverse momenta ), energy conservation and momentum conservation imply

p1 - q = p2 + p3

moreover, using the ordinary Lorentz-invariant relation between energy and momentum, one also has the relations

Ei = sqrt( pi^2 + mi^2 ) = pi + mi^2 / 2 pi

... This straightforwardly leads to the threshold equation

This standard Lorentz-invariant analysis is modified by the deformations [ of Lorentz Invariance violation ] ... The key point is that ...[the equations for e and Ei] .. should be replaced by

Ei = pi + mi^2 / 2 pi + n pi^(1+a) / 2 Eplanck^a

Combining ...[equations]... one obtains a deformed equation describing the p1-threshold:

+ ( n p1,th^(2+a) / 4 e Eplanck^a ) ( ( (m2^(1+a) + m3^(1+a) ) / (m2 + m3)^(1+a) ) - 1 )

where we have included only the leading corrections (terms suppressed by both the smallness of Eplanck^(-1) and the smallness of e or m were neglected). ...

... in particular, if a = -n = 1 ...[the equations are

Ei = pi + mi^2 / 2 pi - pi^2 / 2 Eplanck

p1,th = ( (m2 + m3)^2 - m^2 ) / 4 e -

- ( p1,th^3 / 4 e Eplanck ) ( ( (m2^2 + m3^2 ) / (m2 + m3)^2 ) - 1 )

and] ... one would expect that the Universe be transparent to TeV photons. The corresponding result obtainable in the UHECRs context would imply that the GZK cutoff could be violated even for much smaller negative values of n ...

... it is quite remarkable that **the values expected from
quantum-gravity considerations (most notably the energy scale
characterizing the deformation being given by the Planck scale) are
in agreement with the strict limits we derive**. ...".

D5 Spin(1,9) has a 10-dimensional vector spacetime structure with signature (1,9) corresponding to the real Clifford algebra Cl(1,9) = M(32,R) = Cl(2,8) that at low energies (with respect to the Planck length) reduces to:

- a 6-dimensional (1,5) linear physical spacetime corresponding to the real Clifford algebra Cl(2,4) = M(4,Q) = Cl(1,5) = Cl(5,1) = Cl(6,0) of the Conformal group Spin(2,4) that linearizes the conformal transformations that are non-linear on 4-dimensional Minkowski physical spacetime; plus
- a 4-dimensional (0,4) internal symmetry space with geometry of CP2.

In the Weyl Group dimensional reduction mechanism of D4-D5-E6-E7-E8 VoDou physics, the Standard Model group can been written as

where the SU(3) and U(2) are independent parts of a Cartesian product.

However, both the SU(3) and the U(2) come from the 24-cell root vectors of the D4 Lie algebra Spin(8), so we can ask:

To see how this works, first recall some relevant facts:

- D(N) Lie algebras Spin(2N) have
- dimension 2N ( 2N - 1 ) / 2 = 2N^2 - N
- rank, or Cartan subalgebra dimension, N
- root vector number 2N^2 - N - N = 2 ( N^2 - N )

- U(N) Lie algebras have
- dimension N^2
- rank, or Cartan subalgebra dimension N
- root vector number N^2 - N

- A(N-1) Lie
algebras SU(N) have
- dimension N^2 - 1
- rank, or Cartan subalgebra dimension N - 1
- root vector number N^2 - 1 - ( N - 1 ) = N^2 - N

Therefore,

**
**

Now, **consider ****Spin(1,9) =
SL(2,O) and SU(5)**:

**Recall** that
the D4 is a subgroup of the D5 Lie
algebra Spin(1,9) = SL(2,O) whose 40 root vectors live in
5-dimensional space and correspond to

the 24 root vectors of the D4 Lie algebra Spin(8), physically corresponding to 24 of the 28 infinitesimal generators of Spin(8)

which reduces to 12 of the 16 dimensions of the Conformal U(2,2) of Gravity (the other 4 coming from the 4-dimensional Cartan Subalgebra of Spin(8))

and the Higgs Mechanism and the 12-dimensional Standard Model SU(3) x U(2)

plus

two copies of the 8-vertex 4-dimensional cross polytope, or hyperoctahedron, physically corresponding to

the real part of 8-complex-dimensional SpaceTime

which reduces to the real part of 4-complex-dimensional Physical SpaceTime and 4-complex-dimensional Internal Symmetry Space

and

the imaginary part of 8-complex-dimensional SpaceTime

which also reduces to the imaginary part of 4-complex-dimensional Physical SpaceTime and 4-complex-dimensional Internal Symmetry Space

**First, look at the 20 root vectors (half of the 40 of D5) that
correspond to **

**the 12 root vectors of U(2,2) = Spin(2,4) x U(1)**

**plus **

**the 4-complex-dimensional Physical SpaceTime**:

Since U(2,2) = SU(2,2) x U(1) ( where SU(2,2) = Spin(2,4) is the Conformal Group of Gravity and the Higgs Mechanism in D4-D5-E6-E7-E8 VoDou physics ) has rank 3+1 = 4, it uses 4 of the 5 elements of the Cartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O).

The 5th element of the Cartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O) is used to provide the U(1) symmetry of the complex 8-complex-dimensional space that includes, as a subspace, 4-complex-dimensional SpaceTime.

Since the U(2,2) root vectors correspond to the A3 = D3 cuboctahedron root vectors, and A3, with Lie algebra SU(4), has 4^2 - 1 - 3 = 4^2 - 4 = 12 root vectors.

A4, with Lie algebra SU(5), ( 4 + 1 )^2 - 1 - 4 = 4^2 + 2x4 + 1 - 1 - 4 = 20 root vectors.

Since the addtional 20 - 12 = 8 root vectors of A4 less A3 correspond to half of the addtional 8+8 = 16 root vectors of D5 less D4, which can be taken to be the 8 root vectors of the 4-complex-dimensional Physical SpaceTime:

**When the 5 Cartan subalgebra elements of this part of the D5
Lie algebra Spin(1,9) = SL(2,O) are included, you get 5+20 =
25-dimensional U(5)= SU(5)xU(1). **

**the 12-dimensional Standard Model SU(3) x U(2) **

**plus **

**the 4-complex-dimensional Internal Symmetry Space**:

Since SU(3)xU(2) of the Standard Model in D4-D5-E6-E7-E8 VoDou physics has rank 2+1+1 = 4 ( and since all 5 elements of the Cartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O) have already been used with respect to the U(2,2) of Gravity and the Higgs Mechanism ), 4 of the 12 root vectors corresponding to SU(3)xU(2) must be put into the Cartan Subalgebra of SU(3)xU(2), thus leaving these 8 root vectors of SU(3)xU(2):

The 5th element of the Cartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O) has been used to provide the U(1) symmetry of the complex 8-complex-dimensional space that includes, as a subspace, 4-complex-dimensional Internal Symmetry Space.

( Global Group Structure of the Standard Model is discussed here.)

Compare the 12 vertices corresponding to SU(3)xU(2) with the 12 vertices of the cuboctahedron of the A4 = SU(5) part of the D5 root vectors:

Now, remove the 4 Cartan Subalgebra vertices of SU(3)xU(2)

so that there is a correspondence between the 8 root vectors of SU(3)xU(2) and 8 of the 12 vertices of the cuboctahedron of the A4 = SU(5) part of the D5 root vectors

Now we have embedded the SU(3)xU(2) of the Standard Model in D4-D5-E6-E7-E8 VoDou physics in the SU(5) Lie algebra of the Georgi-Glashow SU(5) Grand Unified model.

By itself, the SU(3) can be represented by 3x3 matrices of the form

3 3 3 3 3 3 3 3 3

By itself, the U(2) can be represented by 2x2 matrices of the form

2 2 2 2

As interrelated parts of Grand Unified SU(5), the SU(3) and U(2) can be represented in 5x5 matrices of the form

3 3 3 X X 3 3 3 X X 3 3 3 X X X X X 2 2 X X X 2 2

where the 12 entries marked X correspond to Grand Unified LeptoQuark X-bosons that can mix quarks of SU(3) with leptons of U(2), thus causing phenomena such as proton decay.

4 of the LeptoQuark X-bosons corresponding to 4 of the cuboctahedral root vectors of the Conformal SU(2,2) = Spin(2,4)

plus the 8 corresponding to 4-complex-dimensional Physical SpaceTime

Physically, of the 12 root vectors of the Conformal SU(2,2) = Spin(2,4):

- 4 of the vertices of the cuboctahedron correspond to Rotations and Boosts ( with one Rotation and one Boost in the 3-dimensional Cartan Subalgebra of Spin(2,4) );
- 4 of the vertices of the cuboctahedron correspond to Translations; and
- 4 of the vertices of the cuboctahedron correspond to Special Conformal Transformations that are generators of Conformal GraviPhotons ( with the Conformal Scale Dilatation being the third element of the 3-dimensional Cartan Subalgebra of Spin(2,4) ).

Therefore:

and

From the SU(5) matrix pattern

3 3 3 X X 3 3 3 X X 3 3 3 X X X X X 2 2 X X X 2 2

it can be seen ( as in the April 1981 Scientific American article by Howard Georgi, reprinted in the Scientific American book Particle Physics in the Cosmos, ed. by Carrigan and Trower (Freeman 1989) ) that the LeptoQuark X-bosons have color charges

red red green green blue blue antired antigreen antiblue antired antigreen antiblue

and electric charges

-4/3 -1/3 -4/3 -1/3 -4/3 -1/3 +4/3 +4/3 +4/3 +1/3 +1/3 +1/3

**What about the LeptoQuark X-boson mass?**

The LeptoQuark X-bosons are related to **the Vacuum Expectation
Value of an X-scalar Higgs field**, analogous to the Vacuum
Expectation Value of the Higgs Scalar Field that gives mass to the
Dirac Leptons, Quarks, and Weak Bosons.

Since the LeptoQuark X-bosons correspond to GraviPhotons and Physical SpaceTime, and the Planck Energy is the characteristic energy of Gravity and Physical SpaceTime, the X Vacuum Expectation Value should be at the energy level corresponding to the Zizzi Decoherence Time of a field that begins at the Planck Energy scale, which time is about 10^(-34 ) seconds and which energy is about 10^14 GeV.

According to The Early Universe, by Kolb and Turner (1994 paperback edition, Adddison-Wesley, page 526): "... the full symmetry of the GUT cannot be manifest; if it were the proton would decay in 10^(-24) sec. The gauge group ... must be spontaneously broken to [ SU(3) x SU(2) x U(1) ]. For SU(5), this is accomplished by ... masses of the order of the unification scale for the twelve X ... gauge bosons. Thus, ... at energies below 10^14 GeV or so the processes mediated by X ... boson exchange can be treated as a four-fermion interaction with strength ... [proportional to 1 / M^2 ] ... where

M = 3 x 10^14 GeV is the unification scale. ... these new ... interactions are extremely weak at energies below 10^14 GeV. ...the proton lifetime must be ...[about]... 10^31 yr. ...".Although Kolb and Turner go on to say

"... The current limits to proton longevity are in excess of 10^32 yr, which seems to rule out the SU(5) GUT. ...",

Adarkar, Krishnaswamy, Menon, Sreekantan, Hayashi, Ito, Kawakami, Miyake, and Uchihori, authors of a paper entitled

Experimental evidence for G.U.T. Proton Decay, hep-ex/0008074, that was dated 30 August 2000 and appeared on the Los Alamos e-print arXiv on 31 August 2000, say:"...

in Kolar ... an experiment to detect proton decayhas been carried out since the end of 1980. Analysis of data yielded ...the life time of the proton is about 1 x 10^31 years...... it decays into wide spectrum of decay modes, p -> e+ pi0 , p -> nubar K+ and so on ...

... A number of other experiments have also looked ... The present consensus among these other experiments seems to be that they have not found any evidence for proton decay yet, and that the lower limit on the lifetime of proton is of the order of 10^33 years. ... The apparent contradiction between these conclusions does not mean a complete disagreement between the observations. ... there are a number of reports from other experiments, which have reached a general consensus among themselves that they have not found any conclusive evidence for proton decay event yet and that the life time must be as long as 10^33 years. This conclusion is in direct conflict with our results presented in this paper. In our opinion, there are many points of agreement between the observations in other experiments and ours, as mentioned below.

The Mont Blanc group ... have observed 9 events out of 21 events which are fully contained. This is consistent with our observations within statistical fluctuations. ...

The Frejus experiment has observed only one event of type p -> nubar K+ K+ -> mu+ nu ... their observed rate is in reasonable agreement with our result.

One of the results of Kamiokande ... shows two peaks ... The new background rate without these two bins is about half of their estimated value and the number of events after subtraction of the new background in these two bins are 5 and 6 respectively, in their observations with exposure factor 4.2 kty. Both of these rates are close to our observations in spite of different experimental techniques.

The IMB group ... have found 4 candidate events for e+ pi0 during the observation of about 4 kty. Out of these, two events have been rejected because of their association with muon decay signals. The other two events ... One of them has a concentrated Cerenkov light cone which may come from a slow proton and the other event has an extra light cone due to a lower energy particle. However, if these features are ascribed two fluctuations in cascade showers, these two events may remain as candidates for proton decay. Assuming such an interpretation, their observed rate of candidate event becomes close to our results. ...".

I have been told that some people have had difficulty downloading the source and postscript files from the e-print archive at hep-ex/0008074, so I have converted postscript to pdf files of the text file and 4 files of figures ( figa. 1-4, figs. 5-8, figs. 9-12, and figs. 13-14d ) and have put them on the web at these URLs:

- http://www.innerx.net/personal/tsmith/Kolartext.pdf
- http://www.innerx.net/personal/tsmith/Kolarfig1_4.pdf
- http://www.innerx.net/personal/tsmith/Kolarfig5_8.pdf
- http://www.innerx.net/personal/tsmith/Kolarfig9_12.pdf
- http://www.innerx.net/personal/tsmith/Kolarfig13_14d.pdf

I have also put a copy of my e-mail correspondence with one of the authors on the web as an html file at URL:

In my opinion, the Kolar results of Adarkar et al may well be correct, and

it is unfortunate that the possible correctness of SU(5) Grand Unification has been obscured by a "consensus" to the contrary that has been repeated as dogma by Kolb and Turner in their (otherwise) excellent book, and in many other articles and texbooks, even including the 1999 second edition of Lie Algebras in Particle Physics (Perseus Books) by Howard Georgi ( who, with Sheldon Glashow, invented SU(5) Grand Unification ), in which Howard Georgi said (at pages 235 and 236) "... Since SU(5) was first found theoretically, experimenters have looked for proton decay with more and more sensitive experiments, so far without success. In fact, the simplest version of the SU(5) unified theory is fairly convincingly ruled out by these experiments. ...".

For a further example, the **2000 Review of Particle
Properties** of the Particle Data
Group says, at page 686, "... p DECAY MODES ... See also the
"Note on Nucleon Decay" in our 1994 edition ... for a short review.
...". That
1994 Note says: "... There is as yet no compelling experimental
evidence for nucleon decay, despite the predictions. The observed
number of candidate events in each mode is roughly consistent with
the atmospheric neutrino background. **For the p -> e+ pi0
mode** ... No background contamination is as yet expected in the
current experiments ... **there are no candidate events** in the
three experiments ... from the three major detectors ( IMB,
Kamiokande, and Frejus ) ... Clearly, **the minimal SU(5) GUT has
already been ruled out**. ...".

In their paper entitledExperimental evidence for G.U.T. Proton Decay, hep-ex/0008074, Adarkar, Krishnaswamy, Menon, Sreekantan, Hayashi, Ito, Kawakami, Miyake, and Uchihori say: "....there are candidate events which nicely match with the decay scheme, p -> e+ pi0. ... The candidate events for this decay mode are listed below; [ Event No. 4268, Event No. 4910, Event No. 836-47 ] ... the background for this event seems to be negligible. ...".

......