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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 for F4 physics.

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.

There are two ways a color-neutral soliton can be formed:

a red-blue-green triple of quarks; or

a quark-antiquark pair.

Protons and red-blue-green (qqq) triples of quarks:

The lowest energy bound state of a red-blue-green triple of quarks is the proton.

To see the structure of the proton, begin with the QCD Lagrangian density, including quark mass term:

Fc4/\*Fc4 + `j4(g¶c - M)j4 .

Separate the mass term:

Fc4/\*Fc4 + `j4 g¶c j4 + `j4( - M)j4 .

Then, for the lowest energy baryon case of the proton, the term Fc4/\*Fc4 + `j4 g¶c j4 produces a 't Hooft-Polyakov monopole

(§3.4 of Rajaraman62).

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 ji, jj, and jk of a spherically symmetric scalar field j.

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 ji, jj, and jk, each corresponding to the red, blue, and green quark respectively.

As discussed in §3.4 of Rajaraman62, 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(Mprotong2/4¹)), where r is the radius,

Mproton is the proton mass, and g is the color charge.

The proton mass comes from the mass term `j4( - M)j4 that is the sum of the constituent masses of the red, blue, and green quarks (two up and one down) in the proton, and Mproton = 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

Fc4/\*Fc4 + `j4(g¶c - mcrnt)j4 + `j4( - M)j4 , where mcrnt 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 quarkülepton 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 from 1050 years to 10122 years).

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 |DeM(n-p)| is about 1 Mev, but the sign is wrong: DeM(n-p) = -1 Mev, and the proton's electromagnetic mass is greater than the neutron's (Feynman63, Vol. II, p. 28-11).

The difference in energy between the bound states, neutron and proton, is not due to a difference between the elementary 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 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 effective constituent mass of the down valence quark is heavier by about

(Ms - Md) (Md/Ms)^2 aw V12 = 312 x 0.25 x 0.253 x 0.22 Mev Å 4.3 Mev. Similarly, the up quark color force mass increase is about

(Mc - Mu) (Mu/Mc)^2 aw V12 = 1777 x 0.022 x 0.253 x 0.22 Mev Å

Å 2.2 Mev, and

the color mass difference is DcM(n-p) = 2DcM(d-u) - DcM(u-d) =

= DcM(d-u) Å 4.3 Mev - 2.2 Mev = 2.1 Mev.

Therefore, the total neutron-proton mass difference is

DM(n-p) = DcM(n-p) + DeM(n-p) Å 2.1 Mev -1 Mev = 1.1Mev,

an estimate that is fairly close to the experimental value of 1.3 Mev.

Pions and quark-antiquark (q/\q) pairs:

From the point of view of the SU(3) color force, and ignoring the electromagnetic force, the charged pion is the lowest energy bound quark-antiquark pair state, either u`d or `u d . It should be spherically symmetric in space, and therefore representable by a soliton solution in 1+1 dimensions if the 3+1 dimensional solution is of the form (1/r2)j(r,t) so that the corresponding 1+1 dimensional solution is of the form j(r,t) and if all pertinent operators are similarly transformed.

If the quark is taken to correspond to a soliton and the antiquark to an antisoliton, the natural representation is by a doublet or breather solution to the sine-Gordon equation

L = (1/2)(¶µj)2 - m02(m2/l)(cos(÷l /m)j - 1).

At the classical level, the sine-Gordon Lagrangian can be scaled to be independent of the parameter l /m2, but the quantum energy levels depend on l /m2 as is discussed by Rajaraman62.

According to Rajaraman62, classical solutions to the sine-Gordon equation include:

the soliton js(mx) = 4(m / ÷l) arctan(exp(mx));

the antisoliton ja(mx) = - 4(m / ÷l) arctan(exp(mx)) = -fs(mx); and the doublet, or breather

jv(mx,mt) = 4(m / ÷l) arctan(sin(vmt / Ã(1+v2))/vcosh(mx / Ã(1+v2))) =

= 4(m / ÷l) arctan( Ã(`t2-1) sin(mt /`t )/cosh(mx Ã(`t2-1) /`t).

In the doublet, v is a parameter related to Lorentz-transforming the solution away from its rest frame. The doublet is a periodic solution with period t = (2¹Ã(1+v2)/v , and can be thought of as a bound soliton-antisoliton pair oscillating with respect to each other with period t.

`t is defined to be mt/2¹.

In the doublet's rest frame, its field is confined within the envelope

±4 (m / ÷l) arctan( (1/v) sech(mx / Ã(1+v2)) ) .

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 = 8m3/ l - m /¹ , where the term 8m3/ l is the classical mass and the term -m /¹ is the leading order quantum correction.

Assuming that the quantum correction term should also be of the form m3/ l , the leading order formula for Msol requires that l/m2 = ¹. In that case, Msol = (8m - m)/¹ = (7/¹) m.

Then, if Msol is interpreted as the constituent mass of a first-generation quark or antiquark and m as the pion mass, the F4 model gives

m(pion) = (¹/7) Msol(quark) = 312.75¹/7 = 140.4 Mev.

Experimentally, m(pion) = 139.57 Mev, which is in pretty good agreement with the F4 model leading order calculation.

For the sine-Gordon doublet, an approach similar to the WKB formula for the hydrogen atom, called by Coleman64 the DHN formula for its originators Dashen, Hasslacher, and Neveu, may be an exact formula for the doublet energy levels.

Assuming that l /m^2 = ¹, the DHN formula gives

m(pion) = 2 Msol(quark) sin((l /m2)/16(1-(l /m2)/8¹)) =

= 2 Msol(quark) sin(¹/14).

Then, since in the F4 model Msol(quark) = 312.8 Mev,

m(pion) = 625.6 sin(¹/14) = 139.2 Mev.

That is closer to the experimental value m(pion) = 139.57 Mev.

Sine-Gordon and Massive Thirring equations:

In §7.3, Rajaraman62 uses the approach of Mandelstam65 to start with the sine-Gordon operator j to construct the operator y of the massive Thirring equation

L = i`y gµ¶µ y - mF`yy - (1/2)g(`y gµ y ) (`y gµ y ),

where m02(m2/ l)cos((÷l/m)f) = - mF`yy , l/4¹m2 = 1/(1 + g/¹), and

-(÷l/2¹m)emn¶nf = `y gn y .

As discussed by Coleman64 as well as Rajaraman62, the massive Thirring model in y describes a massive fermion-antifermion pair in one space dimension and is equivalent to the sine-Gordon equation in j.

Given j , the two-component Fermi field y in 1+1 dimensions is defined by y1(x) = C1 exp(A1(x)) and y2(x) = C2 exp(A2(x)) , where

Then there is the commutation relation

[j(y),y(x)] = (2¹m / ÷l) q(x-y) y(x) , where q(x-y) is the step function.

The operator y(x) raises the value of the field j by 2¹m / ÷l to the left of x and leaves it unchanged to the right of x.

The operator y(x) reduces the soliton state

[j(°) - j(-°)] = 2¹m / ÷l to the vacuum state [j(°) - j(-°)] = 0 .

If, according to the main assumption of this section, l/m2 = ¹, then the soliton state j(°) = - j(-°) = Ã¹ ,

For l/m2 = ¹ the first coefficient of A1 and A2 is -i2Ã¹ and the second coefficient is ±(iÃ¹)/2 . Then A1(°) - A2(°) = -i¹.

Mesons other than the charged 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)/Ã2 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

(Ms - Md) (Md/Ms)^2 aw V12 = 312 x 0.25 x 0.253 x 0.22 Mev Å 4.3 Mev. Similarly, the up quark color force mass increase is about

(Mc - Mu) (Mu/Mc)^2 aw V12 = 1777 x 0.022 x 0.253 x 0.22 Mev Å 2.2 Mev.

The color force increase for the charged pion DcM¹± = 6.5 Mev. Since the mass M¹± = 139.57 Mev is calculated from a color force sine-Gordon soliton state, the mass 139.57 Mev already takes DcM¹± into account.

For ¹0 = (u`u + d`d)/Ã2 , 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 DcM¹0, the ¹0 should be considered to be only u`u , and

DcM¹0 Å 2.2+2.2 = 4.4 Mev.

If, as in the nucleon, DeM(¹0-¹±) Å -1 Mev, the theoretical estimate is DM(¹0-¹±) = DcM(¹0-¹±) + DeM(¹0-¹±) Å 4.4 - 6.5 -1 = -3.1 Mev, roughly consistent with the experimental value of -4.6 Mev.

The r vector mesons have the same quark content as pions, but have parallel spins. Therefore they do not form sine-Gordon doublet type solitons. The r 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).

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 (Å135 Mev) and the excess of the strange quark mass over the u-d quark mass (Å313 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 = 312 x 0.25 x 0.253 x 0.22 Mev Å 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 Å 4.3 Mev.

However, in the K±, the up quark color force mass increase is

(Mc - Mu) (Mu/Mc)2 aw V12 = 1777 x 0.022 x 0.253 x 0.22 Mev Å 2.2 Mev, so DcM(K0-K±) = 4.3 - 2.2 = 2.1 Mev.

IfDeM(K0-K±) Å -1 Mev, then DM(K0-K±) Å -1 + 2.1 = 1.1 Mev, roughly consistent with the experimental value of 4.1 Mev.

D0 mesons, quark content (c`u or`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 (Å135 Mev) and the excess of the charm quark mass over the u-d quark mass (Å1,770 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± Å 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±) Å -1 Mev, then DM(D0-D±) Å -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

(Å2,085 Mev) and the strange quark mass (Å 625 Mev).

h' mesons, quark content (s,`s), have mass about 960 Mev, which is somewhat less than twice the strange quark mass (Å1,250 Mev).

hc mesons, quark content (c,`c), have mass about 2,980 Mev, which is somewhat less than twice the charm quark mass (Å 4,170 Mev).

° mesons, quark content (b,`b), have mass about 9,460 Mev, which is somewhat less than twice the beauty quark mass (Å11,260 Mev).

B0 mesons, quark content (b/\d or /\bd)(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 (Å135 Mev) and the excess of the beauty quark mass over the u-d quark mass (Å5,317 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 Å 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±)Å -1 Mev, then DM(B0-B±) Å -1 + 2.1 = 1.1 Mev, roughly consistent with the experimental value of 1.8 Mev.

For mesons made up of quarks substantially more massive than the up and down quarks of the pion, the sine-Gordon soliton picture is less important than the constituent quark picture.

Friedberg-Lee Nontopological Soliton Model

of (qqq) Baryons (Wilets66, particularly §2.2):

Begin with the QCD Lagrangian density, including quark mass term:

Fc4/\*Fc4 + `j4(g¶c - M)j4 .

Separate the mass term:

Fc4/\*Fc4 + `j4 g¶c j4 + `j4( - M)j4 .

Then, for the lowest energy baryon case of the proton, the term Fc4/\*Fc4 + `j4 g¶c j4 produces a 't Hooft-Polyakov monopole with mass due to the mass term `j4( - M)j4 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 `j4( - M)j4 with the fermion-scalar interaction term -g(s)`jj of the Friedberg-Lee non-topological soliton model, where s is the Friedberg-Lee phenomenological scalar field.

The Lagrangian for a general (qqq) baryon can now be written as

Fc4/\*Fc4 + `j4(g¶c - mcrnt)j4 - g(s)`jj ,

where mcrnt 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 s 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 Fc4/\*Fc4 term of the Lagrangian, but the gluons are confined to the monopole bag. Therefore the gluon term Fc4/\*Fc4 should be multiplied by the Friedberg-Lee factor of k(s), where k(0) = 1 and k(s) = 0 for

s > sv, where sv determines the boundary of the monopole bag, to get:

k(s) Fc4/\*Fc4 + `j4(g¶c - mcrnt)j4 - g(s)`jj .

What happens to the "outside" part (1- k(s)) Fc4/\*Fc4 of the gluon term? Following §7.4 of Bhaduri67, 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)¶ms¶ms - U(s) of the Friedberg-Lee Lagrangian, where U(s) is a quartic term, giving the full Friedberg-Lee Lagrangian:

k(s) Fc4/\*Fc4 + `j4(g¶c - mcrnt)j4 - g(s)`jj +

+ (1/2)¶ms¶ms - U(s) .

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