Deriving the Standard Model plus Gravitation from Simple Operations on Finite Sets by Tony Smith Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras. Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. 3-Quark Protons. Quark-AntiQuark Pions. Spin-2 Physical Gravitons. Planck Mass. Appendix A - Errata for Earlier Papers. References.

In his 1994 Georgia Tech Ph. D. thesis under David Finkelstein, Spacetime as a Quantum Graph, Michael Gibbs describes some 4-dimensional HyperDiamond lattice structures, that he considers likely candidates to represent physical particles. The structures developed by Michael Gibbs in his thesis are useful not only for his model but also for the HyperDiamond Feynman Checkerboard model. Since his model is somewhat different from the HyperDiamond Feynman Checkerboard in some respects, I use a different terminology here. However, I want to make it clear that I have borrowed, in some cases in a modified form, structures from his thesis. Three structures that are useful for the HyperDiamond Feynman Checkerboard model are: 3-link Rotating Propagator, useful for building a proton out of 3 quarks; 2-link Exchange Propagator, useful for building a pion out of a quark and an anti-quark; and 4-link Propagator, useful for building a physical spin-2 physical graviton out of Spin(5) Gauge bosons. In the 2-dimensional Feynman Checkerboard, there is only one massive particle, the electron. What about the D4-D5-E6 model, or any other model that has different particles with different masses? In the context of Feynman Checkerboards, mass is just the amplitude for a particle to have a change of direction in its path. More massive particles will change direction more often. In the HyperDiamond Feynman Checkerboard model the fundamental path segment length epsilon of any particle is the Planck length LPL. However, in the sum over paths for a particle of mass m, it is a useful approximation to consider the path segment length to be the Compton wavelength Lm of the mass m, Lm = h/mc That is because the distances between direction changes in the vast bulk of the paths will be at least Lm, and those distances will be approximately integral multiples of Lm, so that Lm can be used as the effective path segment length. This is an important approximation because the Planck length LPL is about 10^-33 cm, while the effective length L100GeV for a particle of mass 100 GeV is about 10^-16 cm. In this section, the HyperDiamond lattice is given quaternionic coordinates. The origin 0 designates the beginning of the path. The 4 future lightcone links from the origin are given the coordinates 1+i+j+k, 1+i-j-k, 1-i+j-k, 1-i-j+k The path of a particle at rest in space, moving 7 steps in time, is denoted by Particle 0 1 2 3 4 5 6 7 Note that since the HyperDiamond speed of light is sqrt3, the path length is 7 sqrt3. The path of a particle moving along a lightcone path in the 1+i+j+k direction for 7 steps w ith no change of direction is Particle 0 1+ i+ j+ k 2+2i+2j+2k 3+3i+3j+3k 4+4i+4j+4k 5+5i+5j+5k 6+6i+6j+6k 7+7i+7j+7k At each step in either path, the future lightcone can be represented by a Square Diagram of lines connecting the future ends of the 4 future lightcone links leading from the vertex at which the step begins.

In the following subsections, protons, pions, and physical gravitons will be represented by multiparticle paths. The multiple particles representing protons, pions, and physical gravitons will be shown on sequences of such Square Diagrams, as well as by a sequence of coordinates. The coordinate sequences will be given only for a representative sequence of timelike steps, with no space movement, because the notation for a timelike sequence is clearer and it is easy to transform a sequence of timelike steps into a sequence of lightcone link steps, as shown above. Only in the case of gravitons will it be useful to explicitly discuss a path that moves in space as well as time.

Since particle masses can only be observed experimentally for particles that can exist in a free state ("free" means "not strongly bound to other particles, except for virtual particles of the active vacuum of spacetime"), and since quarks do not exist in free states, the quark masses that I calculate are interpreted as constituent masses (not current masses). The relation between current masses and constituent masses may be explained, at least in part, by the Quantum Theory of David Bohm. In hep-ph/9802425, Di Qing, Xiang-Song Chen, and Fan Wang, of Nanjing University, present a qualitative QCD analysis and a quantitative model calculation to show that the constituent quark model [after mixing a small amount (15%) of sea quark components] remains a good approximation even taking into account the nucleon spin structure revealed in polarized deep inelastic scattering. The effectiveness of the NonRelativistic model of light-quark hadrons is explained by, and affords experimental Support for, the Quantum Theory of David Bohm. Consitituent particles are Pre-Quantum particles in the sense that their properties are calculated without using sum-over-histories Many-Worlds quantum theory. ("Classical" is a commonly-used synonym for "Pre-Quantum".) Since experiments are quantum sum-over-histories processes, experimentally observed particles are Quantum particles. Consider the experimentally observed proton. A proton is a Quantum particle containing 3 constituent quarks: two up quarks and one down quark; one Red, one Green, and one Blue. The 3 Pre-Quantum constituent quarks are called "valence" quarks. They are bound to each other by SU(3) QCD. The constituent quarks "feel" the effects of QCD by "sharing" virtual gluons and virtual quark-antiquark pairs that come from the vacuum in sum-over-histories quantum theory. Since the 3 valence constituent quarks within the proton are constantly surrounded by the shared virtual gluons and virtual quark-antiquark pairs, the 3 valence constituent quarks can be said to "swim" in a "sea" of virtual gluons and quark-antiquark pairs, which are called "sea" gluons, quarks, and antiquarks.

This internal structure of the Proton can be described in terms of a Compton Radius Vortex.

In the model, the proton is the most stable bound state of 3 quarks, so that the virtual sea within the proton is at the lowest energy level that is experimentally observable. The virtual sea gluons are massless SU(3) gauge bosons. Since the lightest quarks are up and down quarks, the virtual sea quark-antiquark pairs that most often appear from the vacuum are up or down pairs, each of which have the same constituent mass, 312.75 MeV. If you stay below the threshold energy of the strange quark, whose constituent mass is about 625 MeV, the low energy sea within the proton contains only the lightest (up and down) sea quarks and antiquarks, so that the Quantum proton lowest-energy background sea has a density of 312.75 MeV. (In the model, "density" is mass/energy per unit volume, where the unit volume is Planck-length in size.) Experiments that observe the proton as a whole do not "see" the proton's internal virtual sea, because the paths of the virtual sea gluon, quarks, and antiquarks begin and end within the proton itself. Therefore, the experimentally observed mass of the proton is the sum of the 3 valence quarks, 3 x 32.75 MeV, or 938.25 MeV which is very close to the experimental value of about 938.27 MeV.

To study the internal structure of hadrons, mesons, etc., you should use sum-over-histories quantum theory of the SU(3) color force SU(3). Since that is computationally very difficult (For instance, in my view, the internal structure of a proton looks like a nonperturbative QCD soliton. See WWW URL http://www.innerx.net/personal/tsmith/SolProton.html ) you can use approximate theories that correspond to your experimental energy range. For high energy experiments, such as Deep Inelastic Scattering, you can use Perturbative QCD. For low energies, you can use Chiral Perturbation Theory. Renormalization equations are conventionally used to relate experimental observations that are made at different energy levels.

The HyperDiamond structure used to approximate the proton is the 3-link Rotating Propagator, in which 3 quarks orbit their center somewhat like the 3 balls of an Argentine bola. Here is a coordinate sequence representation of the approximate HyperDiamond Feynman Checkerboard path of a proton: R-Quark G-Quark B-Quark 1-i-j+k 1-i+j-k 1+i-j-k 2-i+j-k 2+i-j-k 2-i-j+k 3+i-j-k 3-i-j+k 3-i+j-k 4-i-j+k 4-i+j-k 4+i-j-k Here is a Square Diagram representation of the approximate HyperDiamond Feynman Checkerboard path of a proton:

time = 1: time = 2: time = 3: time = 4:

In the HyperDiamond Feynman Checkerboard model, where the proton is represented by two up quarks and one down quark, and quark masses are constituent masses: the spins of the quarks should be in the lowest energy state, with one spin anti-parallel to the other two, so that the spin of the proton is + 1/2 + 1/2 - 1/2 = + 1/2; the color charge of the proton is + red + blue + green = 0, so that the pion is color-neutral; the electric charge of the proton is + 2/3 + 2/3 - (- 1/3) = +1; the theoretical tree-level mass is 3 x 312.75 MeV = 938.25 MeV while the experimental mass is 938.27 MeV; the proton is stable with respect to decay by the color, weak, and electromagnetic forces, while decay by the gravitational force is so slow that it cannot be observed with present technology. WHAT ABOUT THE NEUTRON MASS? According to the 1986 CODATA Bulletin No. 63, the experimental value of the neutron mass is 939.56563(28) Mev, and the experimental value of the proton is 938.27231(28) Mev. The neutron-proton mass difference 1.3 Mev is due to the fact that the proton consists of two up quarks and one down quark, while the neutron consists of one up quark and two down quarks. The magnitude of the electromagnetic energy difference mN - mP is about 1 Mev, but the sign is wrong: mN - mP = -1 Mev, and the proton's electromagnetic mass is greater than the neutron's. The difference in energy between the bound states, neutron and proton, is not due to a difference between the Pre-Quantum constituent masses of the up quark and the down quark, calculated in the theory to be equal. It is due to the difference between the Quantum color force interactions of the up and down constituent valence quarks with the gluons and virtual sea quarks in the neutron and the proton. An up valence quark, constituent mass 313 Mev, does not often swap places with a 2.09 Gev charm sea quark, but a 313 Mev down valence quark can more often swap places with a 625 Mev strange sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (ms - md) (md/ms)^2 a(w) |Vds| = = 312 x 0.25 x 0.253 x 0.22 Mev = 4.3 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vds| = 0.22 is the magnitude of the K-M parameter mixing first generation down and second generation strange) so that the Quantum color force constituent mass Qmd of the down quark is Qmd = 312.75 + 4.3 = 317.05 MeV. Similarly, the up quark Quantum color force mass increase is about (mc - mu) (mu/mc)^2 a(w) |V(uc)| = = 1777 x 0.022 x 0.253 x 0.22 Mev = 2.2 Mev, (where |Vuc| = 0.22 is the magnitude of the K-M parameter mixing first generation up and second generation charm) so that the Quantum color force constituent mass Qmu of the up quark is Qmu = 312.75 + 2.2 = 314.95 MeV. The Quantum color force Neutron-Proton mass difference is mN - mP = Qmd - Qmu = 317.05 Mev - 314.95 Mev = 2.1 Mev. Since the electromagnetic Neutron-Proton mass difference is roughly mN - mP = -1 MeV the total theoretical Neutron-Proton mass difference is mN - mP = 2.1 Mev - 1 Mev = 1.1 Mev, an estimate that is fairly close to the experimental value of 1.3 Mev. Note that in the equation (ms - md) (md/ms)^2 a(w) |Vds| = 4.3 Mev Vds is a mixing of down and strange by a neutral Z0, compared to the more conventional Vus mixing by charged W. Although real neutral Z0 processes are suppressed by the GIM mechanism, which is a cancellation of virtual processes, the process of the equation is strictly a virtual process. Note also that the K-M mixing parameter |Vds| is linear. Mixing (such as between a down quark and a strange quark) is a two-step process, that goes approximately as the square of |Vds|: First the down quark changes to a virtual strange quark, producing one factor of |Vds|. Then, second, the virtual strange quark changes back to a down quark, producing a second factor of |Vsd|, which is approximately equal to |Vds|. Only the first step (one factor of |Vds|) appears in the Quantum mass formula used to determine the neutron mass. If you measure the mass of a neutron, that measurement includes a sum over a lot of histories of the valence quarks inside the neutron. In some of those histories, in my view, you will "see" some of the two valence down quarks in a virtual transition state that is at a time after the first action, or change from down to strange, and before the second action, or change back. Therefore, you should take into account those histories in the sum in which you see a strange valence quark, and you get the linear factor |Vds| in the above equation. Note also that if there were no second generation fermions, or if the second generation quarks had equal masses, then the proton would be heavier than the neutron (due to the electromagnetic difference) and the hydrogen atom would decay into a neutron, and there would be no stable atoms in our world.

In the D4-D5-E6-E7-E8 VoDou Physics model, protons decay by virtual Black Holes over 10^64 years, according to by Hawking and his students who have studied the physical consequences of creation of virtual pairs of Planck-energy Black Holes.

According to a
14 June 20002 article by Kurt Riesselmann in Fermi News: "... The
four [ first and second generation ] flavors - up, down,
strange, charm - allow for twenty different ways of putting quarks
together to form baryons ... Protons, for example, consist of two up
quarks and one down quark (u-u-d), and neutrons have a u-d-d quark
content. Some combinations exist in two different spin
configurations, and the SELEX collaboration believes it has
identified both spin levels of the u-c-c baryon. ... Physicists
expect the mass difference between u-c-c and d-c-c baryons to be
comparable to the difference in proton (u-u-d) and neutron (u-d-d)
mass, since this particle pair is also related by the replacement of
an up by a down quark. **The
proton-neutron mass splitting**, however, **is sixty times
smaller than the mass difference between the Xi_cc candidates**
observed by the SELEX collaboration. ...

... Other questions, however, remain as well. The SELEX collaboration is puzzled by the high rate of doubly charmed baryons seen in their experiment. As a matter of fact, most scientists believed that the SELEX collaboration wouldn't see any of these particles. ...".

An up valence quark, constituent mass 313 Mev, can swap places with a 2.09 Gev charm sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (mc - mu) a(w) |Vds| = = 1,777 x 0.253 x 0.22 Mev = 98.9 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vuc| = 0.22 is the magnitude of the K-M parameter mixing first generation up and second generation charm) so that the Quantum color force constituent mass Qmu of the up quark is Qmu = 312.75 + 98.9 = 411.65 MeV. A 313 Mev down valence quark can swap places with a 625 Mev strange sea quark. Therefore the Quantum color force constituent mass of the down valence quark is heavier by about (ms - md) a(w) |Vds| = = 312 x 0.253 x 0.22 Mev = 17.37 Mev, (where a(w) = 0.253 is the geometric part of the weak force strength and |Vds| = 0.22 is the magnitude of the K-M parameter mixing first generation down and second generation strange) so that the Quantum color force constituent mass Qmd of the down quark is Qmd = 312.75 + 17.37 = 330.12 MeV.

Note that at the energy levels at which ucc and dcc live, the ambient sea of quark-antiquark pairs has at least enough energy to produce a charm quark, so that in the above equations there is no mass-ratio-squared suppression factor such as (mu/mc)^2 or (md/ms)^2, unlike the case of the calculation of the neutron-proton mass difference for which the ambient sea of quark-antiquark pairs has very little energy since the proton is almost stable and the neutron-proton mass difference is, according to experiment, only about 1.3 MeV.Note also that these rough calculations ignore the electromagnetic force mass differentials, as they are only on the order of 1 MeV or so, which for ucc - dcc mass difference is small, unlike the case for the calculation of the neutron-proton mass difference.

The Quantum color force ucc - dcc mass difference is mucc - mdcc = Qmu - Qmd = 411.65 MeV - 330.12 MeV = 81.53 MeV

Since the experimental value of the neutron-proton mass difference is about 1.3 MeV, the ucc - dcc mass difference calculated by D4-D5-E6-E7-E8 VoDou Physics is about

81.53 / 1.3 = 62.7 times the experimental value of the neutron-proton mass difference,

which is consistent with the SELEX 2002 experimental result that: "... The proton-neutron mass splitting ... is sixty times smaller than the mass difference between the Xi_cc candidates ...".

In this HyperDiamond Feynman Checkerboard version of the D4-D5-E6 model, pions are made up of first generation valence quark-antiquark pairs. The pion bound state of valence quark-antiquark pairs has a soliton structure that would, projected onto a 2-dimensional spacetime, be a Sine-Gordon breather. See WWW URL http://www.innerx.net/personal/tsmith/SnGdnPion.html Marin, Eilbeck, and Russell, in their paper Localized Moving Breathers in a 2-D Hexagonal Lattice, show "...that highly localized in-plane breathers can propagate in specific directions with minimal lateral spreading ... This one-dimensional behavior in a two-dimensional lattice was called quasi-one-dimensional (QOD) ..." (In their paper, dimensionality refers to spatial dimensionality.) They use QOD behavior to describe phenomena in muscovite mica crytstals. The D4-D5-E6 model uses similar QOD behavior to describe the pion. Take the pion as the fundamental quark-antiquark structure and assume that the quark masses are constituent masses that include the effects of the complicated structure of sea quarks and binding gluons within the pion at the energy level of about 313 MeV that corresponds to the Compton wavelength of the first generation quark constituent mass. The sea quarks and binding gluons within the pion constitute the dressing of the proton valence quarks with the quark and gluon sea in which the valence quarks swim within the pion. The quark and gluon sea has characteristic energy of roughly the constituent quark masses of 312.8 MeV so that the valence quarks float freely within the pion sea.

This internal structure of the Pion can be described in terms of a Compton Radius Vortex.

To express the assumption of free-floating quarks more mathematically, define the relationship between the calculated constituent quark masses, denoted by mq, and QCD Lagrangian current quark masses, denoted by Mq, by Mq = mq - mu = mq - md = mq - 312.8 MeV. This makes, for the QCD Lagrangian, the up and down quarks roughly massless. One result is that the current masses can then be used as input for the SU(3)xSU(3) chiral theory that, although it is only approximate because the constituent mass of the strange quark is about 312 MeV, rather than nearly zero, can be useful in calculating meson properties. In hep-ph/9802425, Di Qing, Xiang-Song Chen, and Fan Wang, of Nanjing University, present a qualitative QCD analysis and a quantitative model calculation to show that the constituent quark model [after mixing a small amount (15%) of sea quark components] remains a good approximation even taking into account the nucleon spin structure revealed in polarized deep inelastic scattering. The effectiveness of the NonRelativistic model of light-quark hadrons is explained by, and affords experimental Support for, the Quantum Theory of David Bohm.

The HyperDiamond structure used to approximate the pion is the 2-link Exchange Propagator. Here is a coordinate sequence representation of the approximate HyperDiamond Feynman Checkerboard path of a pion, in coordinates where 1+i+j+k and 1-i+j-k represent time and longitudinal space, and 1-i-j+k and 1+i-j-k represent transverse space. Quark AntiQuark 0 0 1+i+j+k 1-i+j-k 2 2 3-i+j-k 3+i+j+k 4 4 5+i+j+k 5-i+j-k 6 6 Here is a Square Diagram representation of the approximate HyperDiamond Feynman Checkerboard path of a pion:

Q and Qbar both at the spatial origin at time 0. time = 1: Q and Qbar both at the spatial origin at time 2. time = 3: Q and Qbar both at the spatial origin at time 4. time = 5: Q and Qbar both at the spatial origin at time 6.

The pion can be thought of as a bound quark-antiquark pair oscillating with respect to each other in the 2-dimensional subspace of 1+i+j+k and 1-i+j-k representing time and longitudinal space on the HyperDiamond Lattice. As the pion is represented in the HyperDiamond Feynman Checkerboard model on a 2-dimensional subspace of time and longitudinal space, the pion can be thought of as a Sine-Gordon doublet breather solution to the Sine-Gordon equation L = (1/2)(df)^2 - mo^2(m^2/j)(cos(sqrt j /m)f - 1). The doublet breather solution fv(mx,mt) = = 4(m/sqrt(j))arctan(sin(vmt/sqrt(1+v^2))/vcosh(mx/sqrt(1+v^2))). is confined in space (y-axis) but periodic in time (x-axis).

This stereo image of a doublet breather was generated by the program 3D-Filmstrip for Macintosh by Richard Palais. You can see the stereo with red-green or red-cyan 3D glasses. The program is on the WWW at http://rsp.math.brandeis.edu/3D-Filmstrip. The doublet can be thought of as a bound soliton-antisoliton pair oscillating with respect to each other. In the doublet's rest frame, its field is confined within the envelope ±4(m/sqrt(j))arctan((1/v)sech(mx/sqrt(1+v^2))) . If Msol is taken to be the mass of the soliton (or antisoliton) that is oscillating with respect to the antisoliton (or soliton), and if m is taken to be the mass of the doublet bound state, called the meson mass, then in the quantum theory, to leading order, Msol = 8m^3/ j - m /pi , where the term 8m^3/ j is the classical mass and the term -m /pi is the leading order quantum correction. At the classical level the parameter j can be scaled away but at the quantum level the parameter j determines the energy levels of the quantum theoretical Sine-Gordon equation, as described by Rajaraman in his book Solitons and Instantons (North-Holland 1987). In the HyperDiamond Feynman Checkerboard model the fundamental quantum energy level is j = pi m^2 (the area of a circle of radius m), so that Msol = (8m - m)/pi = (7/pi) m. Then, if Msol is interpreted as the constituent mass of a first-generation quark or antiquark and m as the pion mass, the HyperDiamond Feynman Checkerboard model gives m(pion) = (pi/7) Msol(quark) = 312.75 pi / 7 = 140.4 Mev. Experimentally, m(pion) = 139.57 Mev for charged pions. Charged pions, with quark-antiquark pair containing one up quark and one down quark, are represented as quark-antiquark pairs, whereas neutral pions are a bit more complicated, being a superposition of a pair of two up quarks and a pair of two down quarks. To see the other properties of the pion from the HyperDiamond Feynman Checkerboard picture, take for definiteness the quark to be a red up quark and the antiquark to be an anti-red down quark. Then in the HyperDiamond Feynman Checkerboard model: the spins of the quarks in a pion should be in the lowest energy state, with spins anti-parallel, so that the spin of the pion is + 1/2 - 1/2 = 0; the color charge of the pion is + red - red = 0, so that the pion is color-neutral; the electric charge of the pion is + 2/3 - (- 1/3) = +1; the theoretical tree-level mass is 140.4 MeV, while the experimental mass is 139.57 MeV; the dominant decay path should be by annihilation of the quark-antiquark pair, producing the next less massive pair of particles, the anti-muon (104.8 MeV) of electric charge +1 plus a massless neutral muon neutrino.

WHAT ABOUT THE NEUTRAL PION? The quark content of the charged pion is u_d or d_u , both of which are consistent with the sine-Gordon picture. Experimentally, its mass is 139.57 Mev. The neutral pion has quark content (u_u + d_d)/sqrt(2) with two components, somewhat different from the sine-Gordon picture, and a mass of 134.96 Mev. The effective constituent mass of a down valence quark increases (by swapping places with a strange sea quark) by about DcMdquark = (Ms - Md) (Md/Ms)2 aw V12 = = 312x0.25x0.253x0.22 Mev = 4.3 Mev. Similarly, the up quark color force mass increase is about DcMuquark = (Mc - Mu) (Mu/Mc)2 aw V12 = = 1777x0.022x0.253x0.22 Mev = 2.2 Mev. The color force increase for the charged pion DcMpion± = 6.5 Mev. Since the mass Mpion± = 140.4 Mev is calculated from a color force sine-Gordon soliton state, the mass 140.4 Mev already takes DcMpion± into account. For pion0 = (u_u + d_d)/ sqrt 2 , the d and _d of the the d_d pair do not swap places with strange sea quarks very often because it is energetically preferential for them both to become a u_u pair. Therefore, from the point of view of calculating DcMpion0, the pion0 should be considered to be only u_u , and DcMpion0 = 2.2+2.2 = 4.4 Mev. If, as in the nucleon, DeM(pion0-pion±) = -1 Mev, the theoretical estimate is DM(pion0-pion±) = DcM(pion0-pion±) + DeM(pion0-pion±) = = 4.4 - 6.5 -1 = -3.1 Mev, roughly consistent with the experimental value of -4.6 Mev.

WHAT ABOUT SU(3)xSU(3) CHIRAL THEORY OF SPIN-0 MESONS? To see how chiral theory might be useful, define the relationship between the calculated constituent quark masses, denoted by mq, and QCD Lagrangian current quark masses, denoted by Mq, by Mq = mq - mu = mq - md = mq - 312.8 MeV. The relation between current masses and constituent masses may be explained, at least in part, by the Quantum Theory of David Bohm. This makes, for the QCD Lagrangian, the up and down quarks roughly massless. Therefore the current masses can be used as input for the SU(3)xSU(3) chiral theory that, although it is only approximate because the constituent mass of the strange quark is about 312 MeV, rather than nearly zero, can be useful. SU(3)xSU(3) chiral theory can be used to get a rough estimate of the pion mass, and other meson masses. While I think that SU(3)xSU(3) chiral theory is not as accurate as the sine-Gordon soliton approach for mesons, the SU(3)xSU(3) chiral theory is more conventional, more highly developed, and more widely understood, so I will describe it here: To get the pion mass by the SU(3)xSU(3) chiral theory, recall that the 3 quarks of the SU(3) are the up quark and down quark, with roughly zero current mass for Mu and Md, and the strange quark, for which the calculated constituent mass ms = 625 MeV, so that the current Ms = ms - 313 = 312 MeV. This is somewhat higher than the 100 to 200 MeV usually used for the current strange mass Ms, but I will use it for SU(3)xSU(3) chiral theory to get the Gell-Mann-Okubo relation 3 eta^2 + pion^2 = 4 kaon^2 (where eta, pion, and kaon denote masses of those mesons) Then assume: 1 - there are two types of spin-0 mesons, pion-type, made up of a quark-antiquark pair that "virtually cancel each other out part of the time", with maximal cancellation for the pion made of up and down quarks, so that for pion-type mesons the total mass is the pion mass plus the current masses (or excess of constituent mass over up or down) of any quarks other than up or down (in the chiral theory, the pion mass is considered unknown until it is calculated using the Gell-Mann-Okubo formula); and eta-type, made up of a quark-antiquark pair that do not "virtually cancel each other out part of the time" but "float together on a pion-type sea" so that meson mass is the sum of the constituent masses of the quark and anti-quark less the pion mass; 2 - the kaon is pion-type, in fact just like a pion except that one of the quark-antiquark pair is strange rather than down, so the kaon mass is the pion mass plus the excess of the strange constituent mass over the down constitutent mass, or, from another point of view, the current mass of the strange quark, so that kaon = pion + 312 MeV; 3 - the eta is eta-type, so that the eta mass is twice the up or down quark constituent mass, less the pion mass because the quark and antiquark "float in a pion sea" so that eta = 2 x 313 - pion = 626 MeV - pion. Then, from Gell-Mann-Okubo: 3 eta^2 + pion^2 = 4 kaon^2 3(391,876 - 1248 pion + pion^2) + pion^2 = = 4(97,344 + 624 pion + pion^2) 3(391,876 - 1248 pion + pion^2) + pion^2 = = 4(97,344 + 624 pion + pion^2) 1,175,628 - 3,744 pion + 3 pion^2 + pion^2 = = 389,376 + 2,496 pion + 4 pion^2 786,252 MeV = 6,240 pion pion = 126 MeV kaon = 126 + 312 MeV = 438 MeV eta = 626 - 126 MeV = 500 MeV Experimentally, the results are: pion = 134 MeV (for neutral pion) kaon = 497 MeV (for neutral kaon) eta = 547 MeV If continue, you can exhaust the possibilities of up-down-strange quarks combined in pion-type or eta-type ways, to make: eta' as pion-type with two strange parts, for mass eta' = pion + 312 + 312 MeV = 750 MeV (experimentally eta' = 958 MeV) eta(1295) as eta-type with one strange part, for mass eta(1295) = 625 + 313 - pion = 812 MeV (experimentally eta(1295) = 1,295 MeV) eta(1440) as eta-type with two strange parts, for mass eta(1440) = 625 + 625 - pion = 1,124 MeV (experimentally eta(1440) = 1,440 MeV)

CHARGED AND NEUTRAL KAONS: Just as there is a mass difference between charged and neutral pions, there is also a mass difference between charged and neutral kaons: K0 mesons, quark content (s_d or _sd), have mass about 497.7 Mev, and K± mesons (s_u or _su) have mass about 493.6 Mev, which is somewhat greater than the sum of the pion mass (= 135 Mev) and the excess of the strange quark mass over the up or down quark mass (= 313 Mev). In the K0, the effective constituent mass of a down valence quark increases (by swapping places with a strange sea quark) by about (Ms - Md) (Md/Ms)2 aw V12 = 312x0.25x0.253x0.22 Mev = 4.3 Mev. If the strange quark swaps places with a down sea quark, it just gives the down sea quark "kinetic" energy in the amount of the mass excess of a strange quark over a down quark, so the effective constituent mass of the strange quark is unchanged, and DcMK0 = 4.3 Mev. However, in the K±, the up quark color force mass increase is (Mc - Mu) (Mu/Mc)2 aw V12 = 1777x0.022x0.253x0.22 Mev = 2.2 Mev, so DcM(K0-K±) = 4.3 - 2.2 = 2.1 Mev. If DeM(K0-K±) = -1 Mev, then DM(K0-K±) = -1 + 2.1 = 1.1 Mev, roughly consistent with the experimental value of 4.1 Mev.

WHAT ABOUT SPIN-1 MESONS? The rho vector mesons have the same quark content as pions, but have parallel spins, and so are spin-1 instead of spin-0. Therefore they do not form sine-Gordon doublet type solitons. The rho mass, about 770 Mev, is approximately the sum of the constituent masses of an up quark and a down antiquark plus the mass of a pion binding them (312.75 + 312.75 + 139.187 = 764.687 Mev).

WHAT ABOUT MESONS MADE OF HEAVY QUARKS? For mesons made up of quarks substantially more massive than the up and down quarks of the pion, the sine-Gordon soliton picture and the chiral Lagrangian picture are relatively less important than the constituent quark picture in which the meson mass is roughly the sum of the valence quark constituent masses. D0 mesons, quark content (c_u o _cu), have mass about 1864.5 Mev, and D± mesons (c_d or _cd) have mass about 1869.3 Mev, which is somewhat less than the sum of the pion mass (=135 Mev) and the excess of the charm quark mass over the u-d quark mass (=1,770 Mev). In the D±, the effective constituent mass of a down valence quark increases by about 4.3 Mev. If the charm quark swaps places with an up sea quark, it just gives the up sea quark "kinetic" energy in the amount of the mass excess of a charm quark over a up quark, so the effective constituent mass of the charm quark is unchanged, and DcMD± = 4.3 Mev. However, in the D0, the up quark color force mass increase is about 2.2 Mev, so DcM(D0-D±) = 2.2 - 4.3 = -2.1 Mev. If DeM(D0-D±) = -1 Mev, then DM(D0-D±) = -1 - 2.1 = -3.1 Mev, roughly consistent with the experimental value of -4.8 Mev. Ds mesons, quark content (c,s), have mass about 1,970 Mev, which is somewhat less than the sum of the charm quark mass (2,085 Mev) and the strange quark mass (625 Mev). etac mesons, quark content (c,_c), have mass about 2,980 Mev, which is somewhat less than twice the charm quark mass (4,170 Mev). upsilon mesons, quark content (b,`b), have mass about 9,460 Mev, which is somewhat less than twice the beauty quark mass (11,260 Mev). B0 mesons, quark content (b_d or _bd), have mass about 5279.4 Mev, and B± mesons (b_u or _bu)have mass about 5277.6 Mev, which is somewhat less than the sum of the pion mass (135 Mev) and the excess of the beauty quark mass over the u-d quark mass (5,317 Mev). In the B0, the effective constituent mass of a down valence quark increases by about 4.3 Mev. If the beauty quark swaps places with an down or strange sea quark, it just gives the down or strange sea quark "kinetic" energy in the amount of the mass excess of a beauty quark over a down or strange quark, so the effective constituent mass of the beauty quark is unchanged, and DcMB0 = 4.3 Mev. However, in the B±, the up quark color force mass increase is about 2.2 Mev, so DcM(B0-B±) = 4.3 - 2.2 = 2.1 Mev. If DeM(B0-B±)= -1 Mev, then DM(B0-B±) = -1 + 2.1 = 1.1 Mev, roughly consistent with the experimental value of 1.8 Mev.

In this HyperDiamond Feynman Checkerboard version of the D4-D5-E6 model, spin-2 physical gravitons are made up of the spin-1 gauge bosons of the 10-dimensional Spin(5) de Sitter subgroup of the 15-dimensional Spin(6) Conformal group used to construct Einstein-Hilbert gravity in the D4-D5-E6 model described in URLs http://xxx.lanl.gov/abs/hep-ph/9501252 http://xxx.lanl.gov/abs/quant-ph/9503009 The action of Gravity on Spinors is given by the Papapetrou Equations. The spin-2 physical gravitons are massless, but they can carry energy up to and including the Planck mass. Unlike the pions and protons, which are made up of fermion quarks that live on vertices of the HyperDiamond Lattice, the gravitons are gauge bosons that live on the links of the HyperDiamond Lattice. The Planck energy spin-2 physical gravitons are fundamental structures with HyperDiamond Feynman Checkerboard path length LPlanck. Here is a coordinate sequence representation of the HyperDiamond Feynman Checkerboard path of a fundamental Planck-mass spin-2 physical graviton, where T, X, Y, Z represent infinitesimal generators of the Spin(5) de Sitter group: T X Y Z 1+i+j+k 1+i-j-k 1-i+j-k 1-i-j+k 2+i+j+k 2+i-j-k 2-i+j-k 2-i-j+k 3+i+j+k 3+i-j-k 3-i+j-k 3-i-j+k 4+i+j+k 4+i-j-k 4-i+j-k 4-i-j+k Here is a Square Diagram representation of the HyperDiamond Feynman Checkerboard path of a fundamental Planck-mass spin-2 physical graviton:

time = 1,2,3,4:

The representation above is for a timelike path at rest in space. With respect to gravitons, we can let the path move in space as well. Let T and Y represent time and longitudinal space, and X and Z represent transverse space. Then, as discussed in Feynman's Lectures on Gravitation, section 3.4, the Square Diagram representation shows that our spin-2 physical graviton is indeed a spin-2 particle.

Spin-2 physical gravitons of energy less than the Planck mass are more complicated composite gauge boson structures with approximate HyperDiamond Feynman Checkerboard path length Lgraviton energy. They can be deformed from a square shape, but retain their spin-2 nature as described by Feynman:

Calculation of the Planck mass is at WWW URL http://www.innerx.net/personal/tsmith/Planck.html Here is a summary of a combinatorial calculation: Consider an isolated single point, or vertex in the lattice picture of spacetime. In the HyperDiamond Feynman Checkerboard model, fermions live on vertices, and only first-generation fermions can live on a single vertex. (The second-generation fermions live on two vertices that act at our energy levels very much like one, and the third-generation fermions live on three vertices that act at our energy levels very much like one.) At a single spacetime vertex, a Planck-mass black hole is the Many-Worlds quantum sum of all possible virtual first-generation particle-antiparticle fermion pairs permitted by the Pauli exclusion principle to live on that vertex. The Planck mass in 4-dimensional spacetime is the sum of masses of all possible virtual first-generation particle-antiparticle fermion pairs permitted by the Pauli exclusion principle. There are 8 fermion particles and 8 fermion antiparticles for a total of 64 particle-antiparticle pairs. A typical combination should have several quarks, several antiquarks, a few colorless quark-antiquark pairs that would be equivalent to pions, and some leptons and antileptons. Due to the Pauli exclusion principle, no fermion lepton or quark could be present at the vertex more than twice unless they are in the form of boson pions, colorless first-generation quark-antiquark pairs that are not subject to the Pauli exclusion principle. Of the 64 particle-antiparticle pairs, 12 are pions. A typical combination should have about 6 pions. If all the pions are independent, the typical combination should have a mass of .14x6 GeV = 0.84 GeV. However, just as the pion mass of .14 GeV is less than the sum of the masses of a quark and an antiquark, pairs of oppositely charged pions may form a bound state of less mass than the sum of two pion masses. If such a bound state of oppositely charged pions has a mass as small as .1 GeV, and if the typical combination has one such pair and 4 other pions, then the typical combination should have a mass in the range of 0.66 GeV. Summing over all 2^64 combinations, the total mass of a one-vertex universe should give: mPlanck = 1.217-1.550 x 10^19 GeV. There is also a quaternionic calculation of about 1.3 x 10^19 GeV.

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

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