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Bound states of fermions should correspond to soliton solutionsfor the full effective Lagrangian for the fermions in the boundstate. Such a soliton solution would include the effects ofinteractions of the fermions with virtual vacuum gauge bosons andfermion-antifermion pairs, but would be difficult to calculate ingeneral. However, the interpretation of fermion bound states assolitons leads to calculable results in some cases:
Since, by Deser's theorem, there are no solitons for pure SU(n)Yang-Mills theories (without fermion or scalar source terms) forspacetime dimensions other than 4+1, there can be no pure gluonglueballs in the physical 3+1 dimensional spacetime of the effectivefield theory in the D4-D5-E6 model.
Since neutrinos are coupled only to the weak force, they do notproduce easily observable bound states.
Since electrons, muons, and tauons interactions with the weakforce and gravitation are too weak to produce easily observable boundstates, and the only other force to which they are coupled is theelectromagnetic force, their bound states can be approximated by thenonrelativistic quantum theory of Coulomb electromagneticinteractions.
Quark interactions with the weak force and gravitation are tooweak to produce easily observable bound states. Quark interactionswith the electromagnetic force are much weaker than interactions withthe SU(3) color force, so the soliton bound states can be calculatedfor the SU(3) color force, with the electromagnetic effects to beconsidered as correction factors.
Such Soliton structures can be described in terms of ComptonRadius Vortex structures.
There are two ways a color-neutral soliton can be formed:
a red-blue-green triple of quarks; or
a quark-antiquark pair.
This paper deals with red-blue-green (qqq) triples of quarks.
The lowest energy bound state of a red-blue-green triple of quarksis the proton.
Since only the proton, the lowest-energy bound state, forms anO(3) Model 3-Soliton, and it is useful to have a common referenceframe in which to compare proton structure with the structure ofother (qqq) Baryons,
To see the structure of the proton in those terms, begin with theQCD Lagrangian density, including quark mass term:
Fc/\*Fc + S(d - M)S.
where Fc is the color force curvature term, S is the spinorfermion term, d is the Dirac operator, M is the mass term, and somenotation, as conjugation of an S, is omitted.
Separate the mass term:
Fc/\*Fc + SdS + S(-M)S.
Then, for the lowest energy baryon case of the proton, the termFc/\*Fc + SdS produces a 't Hooft-Polyakov monopole (see section 3.4of Rajaraman.
To see this, coordinatize 3-dimensional space by the imaginaryquaternions i, j, and k; and
take the red, blue, and green quarks are to correspond to the i,j, and k axes respectively.
Then for the lowest energy state the red, blue, and green quarkscan be represented by the three components Ti, Tj, and Tk of aspherically symmetric scalar field T.
Spherical symmetry is obtained by allowing the red, blue, andgreen quarks to be rotated into each other, a process that can berepresented by quaternion multiplication by the unit quaternionsS3.
The approximation is that of considering only the S3 = SU(2)subgroup of color SU(3) to be effective in representing the lowestenergy state. Then the soliton for the lowest energy state of ared-blue-green triple of quarks, or a proton, can be represented asthe soliton in 3+1 dimensional space for a Yang-Mills SU(2) gaugefield theory plus the three scalar fields Ti, Tj, and Tk, eachcorresponding to the red, blue, and green quark respectively.
As discussed in Rajaraman, the structure is that of a 'tHooft-Polyakov monopole. It has a simple static solution that isfinite in size, with outer boundary radius R determined roughly bythe rapidly decreasing function
exp(-r(M(proton)g^2 / 4¹)),
where r is the radius, M(proton) is the proton mass, and g is thecolor charge.
The proton mass comes from the mass term S(-M)S that is the sum ofthe constituent masses of the red, blue, and green quarks (two up andone down) in the proton, and M(proton) = 938.25 Mev.
The experimental value, according to the 1986 CODATA Bulletin No.63, is 938.27231(28) Mev.
The Lagrangian for a general (qqq) baryon can now be writtenas
Fc/\*Fc + S(d - M(crnt))S + S( - M)S ,
where M(crnt) is the current mass of the quarks in the generalbaryon, which is the excess of their constituent mass over theconstituent mass of up and down quarks. In effect, the current massof quarks is their net mass as quarks floating in a quark-antiquarksea made up primarily of up and down quarks.
Within a given baryon all quarks of a given type must haveparallel spins, so that there can be no spin 1/2 baryons of the uuuor ddd type, only neutrons (udd) or protons (uud).
The proton is stable except with respect to quark to lepton decayswhich only occur by gravitation, so that the proton lifetime shouldbe very long (I have seen estimates for the Hawking process protonlifetime ranging from 10^50 years to 10^122 years).
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 diffence 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) V12 == 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 V12 = 0.22 is the K-M parameter mixing generations 1 and 2) 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(12) == 1777 x 0.022 x 0.253 x 0.22 Mev = 2.2 Mev, 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 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.
According to a14 June 20002 article by Kurt Riesselmann in Fermi News: "... Thefour [ first and second generation ] flavors - up, down,strange, charm - allow for twenty different ways of putting quarkstogether to form baryons ... Protons, for example, consist of two upquarks and one down quark (u-u-d), and neutrons have a u-d-d quarkcontent. Some combinations exist in two different spinconfigurations, and the SELEX collaboration believes it hasidentified both spin levels of the u-c-c baryon. ... Physicistsexpect the mass difference between u-c-c and d-c-c baryons to becomparable to the difference in proton (u-u-d) and neutron (u-d-d)mass, since this particle pair is also related by the replacement ofan up by a down quark. Theproton-neutron mass splitting, however, is sixty timessmaller than the mass difference between the Xi_cc candidatesobserved by the SELEX collaboration. ...
... Other questions, however, remain as well. The SELEXcollaboration is puzzled by the high rate of doubly charmed baryonsseen in their experiment. As a matter of fact, most scientistsbelieved that the SELEX collaboration wouldn't see any of theseparticles. ...".
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 differenceis about 1.3 MeV, the ucc - dcc mass difference calculated byD4-D5-E6-E7-E8 VoDou Physics isabout
81.53 / 1.3 = 62.7 times the experimental value of theneutron-proton mass difference,
which is consistent with the SELEX 2002 experimental result that:"... The proton-neutron mass splitting ... is sixtytimes smaller than the mass difference between the Xi_cc candidates...".
Friedberg-Lee Nontopological Soliton Model of (qqq)Baryons
(Wilets, particularly section 2.2):
Begin with the QCD Lagrangian density, including quark massterm:
Fc/\*Fc + S(d-M)S .
Separate the mass term:
Fc/\*Fc + S d S + S(-M)S .
Then, for the lowest energy baryon case of the proton, theterm
Fc/\*Fc + S d S
produces a 't Hooft-Polyakov monopole with mass due to the massterm
that is the sum of the constituent masses of the red, blue, andgreen quarks (two up and one down) in the proton.
Now identify the mass term S(-M)S
with the fermion-scalar interaction term -g(u) SS
of the Friedberg-Lee non-topological soliton model,
where u is the Friedberg-Lee phenomenological scalar field.
The Lagrangian for a general (qqq) baryon can now be writtenas
Fc/\*Fc + S(d - M(crnt))S - g(U)SS ,
where M(crnt) is the current mass of the quarks, which is theexcess of the constituent mass over the constituent mass of up anddown quarks.
A physical origin of the Friedberg-Lee phenomenological scalarfield U has been established, as has the origin of the Friedberg-Leebag as the 't Hooft-Polyakov monopole for the ground stateproton.
At the center of the monopole bag, the color force gluons act justas the
Fc/\*Fc term of the Lagrangian, but the gluons are confined to themonopole bag.
Therefore the gluon term
should be multiplied by the Friedberg-Lee factor of k(u), wherek(0) = 1 and k(u) = 0 for u > uv, where uv determines the boundaryof the monopole bag, to get:
k(u) Fc/\*Fc + S(d - M(crnt))S - g(u)SS .
What happens to the "outside" part (1- k(u)) Fc/\*Fc of the gluonterm?
Following section 7.4 of Bhaduri, it should produce the quadraticand quartic terms of a Skyrme Lagrangian, interpretable physically asa soliton pion (Massless gluons are confined. Pions are the lightestunconfined hadrons.) cloud outside the monopole bag, in turnproducing the remaining part (1/2)dudu - U(u) of the Friedberg-LeeLagrangian, where U(u) is a quartic term, giving the fullFriedberg-Lee Lagrangian:
k(u) Fc/\*Fc + S(d - M(crnt))S - g(u)SS + (1/2)dudu - U(u) .
Rajaraman, Solitons and Instantons, North-Holland 1982.
Feynman, The Feynman Lectures on Physics, Addison-Wesley1963-4-5.
Wilets, Nontopological Solitons World 1989.
Bhaduri, Models of the Nucleon from Quarks to Soliton,Addison-Wesley 1988.
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