Tony Smith's Home Page

SU(5) GrandUnification


 

SU(5) Unification:

The force strengths alpha_3 of the SU(3) Color Force and alpha_2and alpha_1 of the U(2) = SU(2) x U(1) ElectroWeak Force have beenmeasured at low energies (at and below around 90 GeV).

( Global Group Structure of the Standard Model is discussedhere.)

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

The left-hand graph (from Fig. 9-3 of Gauge Theories of theStrong, Weak, and Electromagnetic Interactions, by Chris Quigg(Addison-Wesley 1983, 1997) shows that, although the effectiveElectromagnetic Fine Structure Constant alpha_EM remains smaller than1/100 all the way up to about 8 x 10^16 GeV (so that I continue touse 1/137 for the rough calculations done on this web page, eventhough the calculations may be related to high energies);

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

The right-hand graph ( from CERN Courier, March 1991, page 1 ) isa 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 GrandUnified Theory needed addtional physics input .. With the electroweakstrengths ... now [1991] more accurately known ... and withmeasurements at the Z peak ... on the strong coupling ... it lookseven less likely that extrapolations ... will converge at a point....",

in my opinion the facts:

give a pretty good indication that convergence may well bephysically realistic, and could involve phenomena all the way up tothe 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-Verlag1988), from which book's Fig. 6.21 the above SU(5) GUT illustrationis taken, "... For GUT physics monopoles are extremely interestingobjects: they have an onion-like structure ... which contains thewhole world of grand unified theories.

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

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

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

SU(5) Duality:

Liu and Vachaspati in hep-th/9604138say:

",,, The spectrum of monopoles is found to correspond to thespectrum of one family of standard model fermions and hence, is astarting point for constructing the dual standard model. ... there isan extra monopole state - the "diquark" monopole - with nocorresponding standard model fermion. ...... It is worthwhileclarifying our use of the word "dual". What we have in mind isthat

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

In other words, the standard model is an effective theory offields that create and annihilate SU(5) monopoles. This is analogousto the sine-Gordon and Thirring modelequivalence. ...".

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

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

FIG. 1. Rescaled Running Gauge Couplings

which appears to unify at a thermal scale of a few GeV ... Thepicture 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 unifiedgauge coupling is around 0.7; ... A description of the physics ofsuch non-Abelian dyons has been achieved by Chan and Tsou ... Theirduality transformation is not expressed in terms of the usual fielddescription, but in terms of non-local Polyakov loop variables. ...Above the unification scale there are no fermions, only theconstituent bosons. Presumably creation of such a plasma would allowthe occurance of many transitions disallowed in the standard model;for instance baryon or lepton number violation. ...".

 



SU(5) Duality Overview:

 

Electron-Positron ------------------------- PlanckMass

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


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

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


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


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

Transparency to multi-TeV photons------------- Highest Observed Cosmic RayEnergy

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


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

10^8 GeV = 10^17 eV


It is interesting that:

 

Consider a Compton Radius VortexKerr-Newman Black Hole related to the Wessonforce. The equation (in units with G = c = hbar = 1) for aKerr-Newman Black Hole with coincident outer and inner event horizonsand 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 = 137exp(pi/2)

Then the magnitude | 1 / Mwesson | = 137 which (since the unitsare 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 x10^16 GeV = Mmonopole

Perhaps there is a Wessonforce that is carried by a Mwessson particle. If so, sincethe strength of the gravitational force is ( 1 / (Planck Mass)^2 ) itseems 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 forceto gravitation is

Wesson / Grav = ( 1/ (Mwesson)^2 ) / ( 1 / (Planck Mass)^2 ) = ( 1 / (0.01 PlanckMass)^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 forceand GUT SU(5) Monopoles may be relatedto the GraviPhotonforce of the VoDou D4-D5-E6-E7physics model.

  


Cosmic Ray Spectrum and SU(5)Duality

CosmicRays may be produced by Supernovae,Active Galactic Nuclei, and GammaRay Bursts. The HighestObserved Cosmic Ray Energy is theFly'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 thecosmic-ray energy spectrum ( the knee ) is observed at an energy ofabout 3 PeV (1 PeV = 10^15 eV). The recent results on extensive airshowers 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 smallamplitude of the anisotropy in the knee region. ... The candidate forthe source could not be too far from the solar system or too close tothe moment of the explosion in order to have enough energy in cosmicrays and give the requred contribution to the cosmic-ray flux at theknee. ...

... The dotted line shows the typical propagation of the shockfront. The dashed and dash-dotted lines are derived from thecomparison of the energy contained in cosmic rays for our SingleSource and the energy content of the cosmic rays accelerated by thesupernova remnant ... The possible source candidates should lieinside the area delimited by these lines. ... At the moment, thesources which gave birth to Loop I and theGeminga pulsar are the most favorable contenders. ...".

The Region above the Kneeof the CosmicRay Energy Spectrum, shown on the graph above taken from a graphby P. Sokolsky on aGoddard NASA web page, is about 5 x 10^15 eV to 5 x10^18 eV, centered on about 10^17 eV, which coincides withthe Self-Dual Energy Region of SU(5)Duality.

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

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

The GZK limit of about 5 x 10^19 eV isshown 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 likelihoodanalysis we show that the observed spectrum is consistent with thedecay of extragalactic GUT scale particles. The predicted mass forthese 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-MSSMscenario. The goodnesses of the fits for the halo models are farworse. The SM and MSSM cases do not differ significantly. ...".

 

Since the D4-D5-E6-E7-E8VoDou Physics model is fundamentally a Planck ScaleHyperDiamond LatticeGeneralized FeynmanCheckerboard model, it violates LorentzInvariance at the Planck Scale.

Planck Scale violation of LorentzInvariance is advocated by Giovanni Amelino-Camelia and TsviPiran in astro-ph/0008107to account for the facts that "... Significant evidence hasaccumulated in recent years suggesting that ... the universe is moretransparent than what it was expected to be ... in two differentregimes,

Ultra High EnergyCosmic 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

multi-TeVphotons,

... 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 energyparticles. The reactions should take place because when Lorentztransformed to the CM frame the low energy photon have sufficientenergy to overcome an intrinsic threshold. In both cases the CMenergies are rather modest (about 100 MeV for UHECRs and about 1 MeVfor the TeV photons) and the physical processes involved areextremely well understood and measured in the laboratory. In bothcases we observe particles above a seemingly robust threshold and theobservations can be considered as a "threshold anomaly".

It is remarkable that in spite of these similarities at presentthere is only one mechanism that could resolve both paradoxes: amechanism based on the single, however drastic, assumption of aviolation 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 thedeviation from ordinary Lorentz invariance ( in particular, aspecifies how strongly the magnitude of the deformation is suppressedby Eplanck ). ...

Figure 1: The region of the a, n parameter space that provides asolution to both the UHECR and TeV threshold anomalies whilesatisfying the time-of-flight upper bound on LID. Only negativevalues of n are considered since this is necessary in order to haveupward shifts of the threshold energies, as required by the presentparadoxes. The solid thick line describes the time-of-flight upperbound. The [light blue] regionabove this line is excluded. The solid thin line and the dotted linedescribe the lower bound on LID obtained from the present UHECR(solid thin line [the hatched area below it is excluded]) andTeV (dotted line [the green areabelow it is excluded]) threshold anomalies. The anomaliesdisappear in the region above the lines. Within the narrow[white] region between the dotted line and the solid thickline the time of flight constraint is satisfied and both anomaliesare resolved. The ... vertical ...[red line at a = 1 corresponds toa] ... 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 LIDwith respect to the bottom-left corner of the frame can be understoodby noticing that at a fixed a ordinary Lorentz invariance can bereached taking the n -> infinity limit, while at fixed n thisrequires 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 theprocess in which the head-on collision between a soft photon ofenergy e and momentum q and a high-energy particle of energy E1 andmomentum p1 leads to the production of two particles with energiesE2, E3 and momenta p2, p3. At threshold ( no energy available fortransverse momenta ), energy conservation and momentum conservationimply

E1 + e = E2 + E3

p1 - q = p2 + p3

moreover, using the ordinary Lorentz-invariant relation betweenenergy 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 thedeformations [ of Lorentz Invariance violation ] ... The keypoint is that ...[the equations for e and Ei] .. should bereplaced 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 deformedequation 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 (termssuppressed by both the smallness of Eplanck^(-1) and the smallness ofe 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 toTeV photons. The corresponding result obtainable in the UHECRscontext would imply that the GZK cutoff could be violated even formuch smaller negative values of n ...

... it is quite remarkable that the values expected fromquantum-gravity considerations (most notably the energy scalecharacterizing the deformation being given by the Planck scale) arein 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) andSU(5)

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

 

In the Weyl Group dimensional reductionmechanism of D4-D5-E6-E7-E8 VoDouphysics, 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 Cartesianproduct.

( Global Group Structure of the Standard Model is discussedhere.)

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

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

To see how this works, first recall somerelevant facts:

Therefore,

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


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

Recall thatthe D4 is a subgroup of the D5 Liealgebra Spin(1,9) = SL(2,O) whose 40 root vectors live in5-dimensional space and correspond to

the 24 root vectors of the D4 Lie algebra Spin(8), physicallycorresponding 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 CartanSubalgebra of Spin(8))

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

plus

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

the real part of 8-complex-dimensional SpaceTime

which reduces to the real part of 4-complex-dimensional PhysicalSpaceTime 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-dimensionalPhysical SpaceTime and 4-complex-dimensional Internal SymmetrySpace


First, look at the 20 root vectors (half of the 40 of D5) thatcorrespond to

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

plus

the 4-complex-dimensional Physical SpaceTime:

U(2,2) root vectors

real

imaginary

 

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 HiggsMechanism in D4-D5-E6-E7-E8 VoDouphysics ) has rank 3+1 = 4, it uses 4 of the 5 elements of theCartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O).

The 5th element of the Cartan Subalgebra of the D5 Lie algebraSpin(1,9) = SL(2,O) is used to provide the U(1) symmetry of thecomplex 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-dimensionalPhysical SpaceTime.

Since the U(2,2) root vectors correspond to the A3 = D3cuboctahedron 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 A3correspond to half of the addtional 8+8 = 16 root vectors of D5 lessD4, which can be taken to be the 8 root vectors of the4-complex-dimensional Physical SpaceTime:

The 20 root vectors of this part of the D5 Lie algebraSpin(1,9) = SL(2,O) corresponds to the 20 root vectors of the A4 Liealgebra 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 D5Lie 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 otherpart of the 40 of D5) that correspond to

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

plus

the 4-complex-dimensional Internal Symmetry Space:

SU(3)xU(2)

real

imaginary

 

Since SU(3)xU(2) of the Standard Model in D4-D5-E6-E7-E8VoDou physics has rank 2+1+1 = 4 ( and since all 5 elements ofthe Cartan Subalgebra of the D5 Lie algebra Spin(1,9) = SL(2,O) havealready been used with respect to the U(2,2) of Gravity and the HiggsMechanism ), 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 leavingthese 8 root vectors of SU(3)xU(2):

The 5th element of the Cartan Subalgebra of the D5 Lie algebraSpin(1,9) = SL(2,O) has been used to provide the U(1) symmetry of thecomplex 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-dimensionalInternal Symmetry Space.

( Global Group Structure of the Standard Model is discussedhere.)

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

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

so that there is a correspondence between the 8 root vectors ofSU(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 inD4-D5-E6-E7-E8 VoDou physics in theSU(5) Lie algebra of the Georgi-Glashow SU(5) Grand Unifiedmodel.

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

3 3 3 3 3 3 3 3 3

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

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 UnifiedLeptoQuark X-bosons that can mix quarks of SU(3) with leptons ofU(2), thus causing phenomena such as proton decay.

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

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

plus the 8 corresponding to 4-complex-dimensional PhysicalSpaceTime

real

imaginary

 

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

Therefore:

4 of the SU(5) Grand Unified LeptoQuark X-bosons correspondto the Special Conformal Transformationsthat are generators of Conformal GraviPhotons;

and

8 of the SU(5) Grand Unified LeptoQuark X-bosons correspondto 8-real-dimensional, or 4-complex-dimensional, PhysicalSpaceTime.

What are the properties of the LeptoQuarkX-bosons?

( Global Group Structure of the Standard Model is discussedhere.)

From the SU(5) matrix pattern

3 3 3 X X3 3 3 X X3 3 3 X XX X X 2 2X X X 2 2

it can be seen ( as in the April 1981 Scientific American articleby Howard Georgi, reprinted in the Scientific American book ParticlePhysics 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 ExpectationValue of an X-scalar Higgs field, analogous to the VacuumExpectation Value of the Higgs Scalar Field that gives mass to theDirac Leptons, Quarks, and Weak Bosons.

Since the LeptoQuark X-bosons correspond to GraviPhotons andPhysical SpaceTime, and the Planck Energy is the characteristicenergy of Gravity and Physical SpaceTime, the X Vacuum ExpectationValue should be at the energy level corresponding to theZizzi Decoherence Time of a field that begins at the Planck Energyscale, which time is about 10^(-34 ) secondsand 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 downloadingthe source and postscript files from the e-print archive at hep-ex/0008074,so I have converted postscript to pdf files of the textfile and 4 files of figures ( figa.1-4, figs. 5-8, figs.9-12, and figs. 13-14d ) andhave put them on the web at these URLs:

I have also put a copy of my e-mailcorrespondence with one of the authors on the web as an html fileat URL:

 

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

it is unfortunate that the possiblecorrectness of SU(5) Grand Unification has been obscured by a"consensus" to the contrary that has been repeated as dogma byKolb and Turner in their (otherwise) excellent book, and in manyother articles and texbooks, even including the 1999 second editionof Lie Algebras in Particle Physics (Perseus Books) by Howard Georgi( who, with Sheldon Glashow, invented SU(5) Grand Unification ), inwhich Howard Georgi said (at pages 235 and 236) "... Since SU(5) wasfirst found theoretically, experimenters have looked for proton decaywith more and more sensitive experiments, so far without success. Infact, the simplest version of the SU(5) unified theory is fairlyconvincingly ruled out by these experiments. ...".

For a further example, the 2000 Review of ParticleProperties of the Particle DataGroup says, at page 686, "... p DECAY MODES ... See also the"Note on Nucleon Decay" in our 1994 edition ... for a short review....". That1994 Note says: "... There is as yet no compelling experimentalevidence for nucleon decay, despite the predictions. The observednumber of candidate events in each mode is roughly consistent withthe atmospheric neutrino background. For the p -> e+ pi0mode ... No background contamination is as yet expected in thecurrent experiments ... there are no candidate events in thethree experiments ... from the three major detectors ( IMB,Kamiokande, and Frejus ) ... Clearly, the minimal SU(5) GUT hasalready 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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