Bound states of fermions should correspond to soliton solutions for the full effective Lagrangian for the fermions in the bound state. Such a soliton solution would include the effects of interactions of the fermions with virtual vacuum gauge bosons and fermion-antifermion pairs, but would be difficult to calculate in general. However, the interpretation of fermion bound states as solitons 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) for spacetime dimensions other than 4+1, there can be no pure gluon glueballs in the physical 3+1 dimensional spacetime of the effective field theory in the D4-D5-E6 model.
Since neutrinos are coupled only to the weak force, they do not produce easily observable bound states.
Since electrons, muons, and tauons interactions with the weak force and gravitation are too weak to produce easily observable bound states, and the only other force to which they are coupled is the electromagnetic force, their bound states can be approximated by the nonrelativistic quantum theory of Coulomb electromagnetic interactions.
Quark interactions with the weak force and gravitation are too weak to produce easily observable bound states. Quark interactions with the electromagnetic force are much weaker than interactions with the SU(3) color force, so the soliton bound states can be calculated for the SU(3) color force, with the electromagnetic effects to be considered as correction factors.
Such Soliton structures can be described in terms of Compton Radius 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 quarks is the proton.
Since only the proton, the lowest-energy bound state, forms an O(3) Model 3-Soliton, and it is useful to have a common reference frame in which to compare proton structure with the structure of other (qqq) Baryons,
To see the structure of the proton in those terms, begin with the QCD Lagrangian density, including quark mass term:
Fc/\*Fc + S(d - M)S.
where Fc is the color force curvature term, S is the spinor fermion term, d is the Dirac operator, M is the mass term, and some notation, 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 term Fc/\*Fc + SdS produces a 't Hooft-Polyakov monopole (see section 3.4 of Rajaraman.
To see this, coordinatize 3-dimensional space by the imaginary quaternions 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 quarks can be represented by the three components Ti, Tj, and Tk of a spherically symmetric scalar field T.
Spherical symmetry is obtained by allowing the red, blue, and green quarks to be rotated into each other, a process that can be represented by quaternion multiplication by the unit quaternions S3.
The approximation is that of considering only the S3 = SU(2) subgroup of color SU(3) to be effective in representing the lowest energy state. Then the soliton for the lowest energy state of a red-blue-green triple of quarks, or a proton, can be represented as the soliton in 3+1 dimensional space for a Yang-Mills SU(2) gauge field theory plus the three scalar fields Ti, Tj, and Tk, each corresponding to the red, blue, and green quark respectively.
As discussed in Rajaraman, the structure is that of a 't Hooft-Polyakov monopole. It has a simple static solution that is finite in size, with outer boundary radius R determined roughly by the rapidly decreasing function
exp(-r(M(proton)g^2 / 4¹)),
where r is the radius, M(proton) is the proton mass, and g is the color charge.
The proton mass comes from the mass term S(-M)S that is the sum of the constituent masses of the red, blue, and green quarks (two up and one 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 written as
Fc/\*Fc + S(d - M(crnt))S + S( - M)S ,
where M(crnt) is the current mass of the quarks in the general baryon, which is the excess of their constituent mass over the constituent mass of up and down quarks. In effect, the current mass of quarks is their net mass as quarks floating in a quark-antiquark sea made up primarily of up and down quarks.
Within a given baryon all quarks of a given type must have parallel spins, so that there can be no spin 1/2 baryons of the uuu or ddd type, only neutrons (udd) or protons (uud).
The proton is stable except with respect to quark to lepton decays which only occur by gravitation, so that the proton lifetime should be very long (I have seen estimates for the Hawking process proton lifetime 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 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 ...".
Friedberg-Lee Nontopological Soliton Model of (qqq) Baryons
(Wilets, particularly section 2.2):
Begin with the QCD Lagrangian density, including quark mass term:
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, the term
Fc/\*Fc + S d S
produces a 't Hooft-Polyakov monopole with mass due to the mass term
S(-M)S
that is the sum of the constituent masses of the red, blue, and green 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 written as
Fc/\*Fc + S(d - M(crnt))S - g(U)SS ,
where M(crnt) is the current mass of the quarks, which is the excess of the constituent mass over the constituent mass of up and down quarks.
A physical origin of the Friedberg-Lee phenomenological scalar field U has been established, as has the origin of the Friedberg-Lee bag as the 't Hooft-Polyakov monopole for the ground state proton.
At the center of the monopole bag, the color force gluons act just as the
Fc/\*Fc term of the Lagrangian, but the gluons are confined to the monopole bag.
Therefore the gluon term
Fc/\*Fc
should be multiplied by the Friedberg-Lee factor of k(u), where k(0) = 1 and k(u) = 0 for u > uv, where uv determines the boundary of 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 gluon term?
Following section 7.4 of Bhaduri, it should produce the quadratic and quartic terms of a Skyrme Lagrangian, interpretable physically as a soliton pion (Massless gluons are confined. Pions are the lightest unconfined hadrons.) cloud outside the monopole bag, in turn producing the remaining part (1/2)dudu - U(u) of the Friedberg-Lee Lagrangian, where U(u) is a quartic term, giving the full Friedberg-Lee Lagrangian:
k(u) Fc/\*Fc + S(d - M(crnt))S - g(u)SS + (1/2)dudu - U(u) .
REFERENCES:
Rajaraman, Solitons and Instantons, North-Holland 1982.
Feynman, The Feynman Lectures on Physics, Addison-Wesley 1963-4-5.
Wilets, Nontopological Solitons World 1989.
Bhaduri, Models of the Nucleon from Quarks to Soliton, Addison-Wesley 1988.
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