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SU(5) Grand Unification


SU(5) Unification:

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).

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

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:

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

the SU(5) GUT Magnetic Monopole:

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.

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. ...".

SU(5) Duality:

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

the monopoles of the SU(5) model are simply a different description of the fermions in the standard model.

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:

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. ...".


SU(5) Duality Overview:


Electron-Positron ------------------------- Planck Mass

0.001 GeV = 10^6 eV ---------------- 10^19 GeV = 10^28 eV

Pion -------------------------------------- SU(5) Monopole

0.1 GeV = 10^8 eV ------------------ 10^17 GeV = 10^26 eV

10 GeV = 10^10 eV -------------- 10^15 GeV = 10^24 eV

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

10^7 GeV = 10^16 eV ------- 10^9 GeV = 10^18 eV

10^8 GeV = 10^17 eV

It is interesting that:


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

J^2/M^4 = (J/M^2)^2 = 1 - (1/M)^2

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

J = sqrt(1 - (1/M)^2) M^2 = p_wesson M^2 = 137 M^2

so that

1 - (1/M)^2 = 137^2 and 1/M = sqrt(1 -137^2) = 137 i = 137 exp(pi/2)

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

Mwesson = Mplanck / 137 = 10^19 / 137 = 7.3 x 10^16 GeV

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

Wesson / Grav = ( 1 / (Mwesson)^2 ) / ( 1 / (Planck Mass)^2 ) = ( 1 / (0.01 Planck Mass)^2 ) / ( 1 / (Planck Mass)^2 ) = 10^4.
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.
The Wesson force and GUT SU(5) Monopoles may be related to the GraviPhoton force of the VoDou D4-D5-E6-E7 physics model.


Cosmic Ray Spectrum and SU(5) Duality

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:

... 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. ...".


Since the D4-D5-E6-E7-E8 VoDou Physics model is fundamentally a Planck Scale HyperDiamond Lattice Generalized Feynman Checkerboard model, it violates Lorentz Invariance at the Planck Scale.

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. ...


multi-TeV photons,

... 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:

E^2 - p^2 - m^2 = n E^2 ( E / Eplanck )^a = n p^2 ( E / Eplanck )^a

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

E^2 - p^2 - m^2 = - E^2 ( E / Eplanck ) = - p^2 ( E / Eplanck )

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

E1 + e = E2 + E3

p1 - q = p2 + p3

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

e = q

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

... This straightforwardly leads to the threshold equation

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

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

e = q + n q^(1+a) / 2 Eplanck^a

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

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

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

+ ( 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

e = q - q^2 / 2 Eplanck

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. ...".



F. W. Stecker has written an overview paper: astro-ph/0101072, The Curious Adventure of the UltraHigh Energy Cosmic Rays.


D5 Spin(1,9) = SL(2,O) and SU(5)

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:


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

SU(3) x U(2)

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

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

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:

Do the SU(3) and the U(2) inherit interrelationships from their common origin ?

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


the root vectors of SU(N) can be seen as half of the root vectors of Spin(2N).

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)


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


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)


the 4-complex-dimensional Physical SpaceTime:

U(2,2) root vectors




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.

This part of the D5 Lie algebra Spin(1,9) = SL(2,O) corresponds to Gravity, the Higgs Mechanism, and 4-dimensional Physical 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:

The 20 root vectors of this part of the D5 Lie algebra Spin(1,9) = SL(2,O) corresponds to the 20 root vectors of the A4 Lie algebra SU(5), which are the same as the root vectors of U(5) = SU(5)xU(1).

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).


Second, look at the remaining 20 root vectors (the other part of the 40 of D5) that correspond to

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


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.

This part of the D5 Lie algebra Spin(1,9) = SL(2,O) corresponds to the SU(3)xU(2) Standard Model and 4-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.

Which of the 20 root vectors of SU(5) correspond to the LeptoQuark X-bosons?

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 SU(5) Grand Unified LeptoQuark X-bosons correspond to the Special Conformal Transformations that are generators of Conformal GraviPhotons;


8 of the SU(5) Grand Unified LeptoQuark X-bosons correspond to 8-real-dimensional, or 4-complex-dimensional, Physical SpaceTime.

What are the properties of the LeptoQuark X-bosons?

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

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 decay has 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:

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 entitled Experimental 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. ...".


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