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Chiral bag plus skyrmion hybrid model for nucleons
Volume 167B, number 2
PHYSICS LETTERS
6 February 1986
C H I R A L BAG P L U S S K Y R M I O N H Y B R I D M O D E L F O R N U C L E O N S A. H O S A K A and H. T O K I Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA and Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan Received 27 August 1985; revised manuscript received 7 November 1985
The chiral bag plus skyrmion hybrid model is studied as a model for nucleons. The hybrid model is solved numerically, where great care has been taken to remove the divergence in the boundary condition at the chiral bag surface for the nonlinear differential equation of the skyrmion. The numerical results for the nucleon energy turn out to be finite for all bag radii, differing from the case of the linearized chiral model. With a suitable choice of the bag parameter (volume term), the nucleon mass, radius and the axial vector coupling constant gA come out to be very promising.
As a phenomenological model for nucleons in terms of the quark model, the MIT bag model has been successfully introduced to provide all the static properties of baryons and mesons [1 ]. The phenomenology for the quark confmement instead of solving the QCD lagrangian, however, has introduced an unfavorable aspect: the chiral symmetry is broken at the bag surface. The chiral bag model was then proposed by Chodos and Thorn [2] and by Brown and Rho [3] to restore the chiral symmetry by introducing pions outside of the M1T bag. Since then, several attempts have been made to solve the quark-pion coupled system either by linearization of the pion field [4] or by taking the classical approximation [5]. It was quickly realized, however, that the chiral bag becomes unstable as the bag radius is reduced [5,6]. About 25 years ago, Skyrme introduced a very interesting model for nucleons, where the nonlinear model provides a soliton solution with the "baryon number" = 1 by introducing a stabilizer (the so-called Skyrme term), which is identified as a nucleon [7]. The amazing feature of this model is that bosons create fermions by a topological twist. The static properties resemble remarkably those of nucleons [8]. The apparent success of the Skyrme model has motivated Mulders to construct a hybrid model of the above two models as a model for nucleons [9]. The appealing feature of the hybrid model is that the quark sector and the skyrmion sector (nonlinear pion 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
fields) share the baryon number, which is one at any bag radius [9,10]. The motivation of this study is therefore to work out the chiral bag plus skyrmion hybrid model (hereafter abbreviated as CSH) up to the end for nucleon static properties. One difficulty lying ahead of us is to remove the divergent behavior in the boundary condition for the skyrmion differential equation with great care. Independently from Mulders, Veptas et al. [12] and Brown et al. [13] have developed the similar hybrid model for nucleons. In the course of construction of the model, they have chosen to fix the strength of the Skyrme term by the requirement of the axial vector coupling constant gA having the correct value [ 13] This requirement is fulfilled only up to a certain bag radius R and therefore physical quantities like the mass have a kink as a function o f R . We prefer to choose the strength of the Skyrme term by the condition that the hybrid model reduces to the Skyrme model as R goes to 0 and then insist the strength e be constant while R is varied. We consider, hence, that the Skyrme term is inherent in the properties of the pion field. Let us start by writing the lagrangian of the hybrid model for clarification of the procedure of calculations, where the details are found in ref. [9]. The CSH model lagrangian consists of the quark, pion and the interaction parts: 153
Volume 167B, number 2 £ = £q + £~r + £int •
(1)
For simplicity, we consider the case of massless quark and pion. The quark lagrangian reads £q = (~i~v~U ~ - B) OB,
(2)
where B is a bag constant and 0B is one inside the bag and zero otherwise. The pion part is the Skyrme lagrangian, £~r =
{:}/2Tr i ) . US"U t
+ (1/32e 2) Tr[8~
UUt , 8 v UUt] 2} 0~.
with h = r/r,
(3)
(4)
and 0~ = 1 - OB. In eq. (3),/~ is the pion decay constant and e controls the strength of the Skyrme term. The third term, the interaction lagrangian, is £int = - ~ ~ exp(i~h0 75) ~b~B,
(5)
where the surface delta function 8 B is introduced such that the quark and the pion interact only at the bag surface. The variational method provides then a set of differential equations for ~ and 0 as given in eqs. (7), (8), (13), (14) and (71) in ref. [9]. In order to solve the equations of motion derived from the above lagrangian, we need to evaluate the quark axial current flow which appears in the equation of the conservation of the axial vector current, [f2 + (2 sin20)/e2r 2] O' - l 2(2 i [5, ixh~f5 exp(ixh~t50) ~k]) -
sgn(En) denotes the sign of the energy En. This expression is known to contain divergences which must be subtracted to perform a meaningful calculation. The strongest divergence, the order of which is l/r/, was found by Mulders using the multiple reflection technique [9] and the weaker logarithmic divergence was evaluated by Zahed et al. by comparing the axial current flow with the Casimir energy [11]. Writing the divergent terms explicitly, one has
C(0) = [ 1 / 1 6 m / - (In n)/30~r 2] sin 20 + Co(O),
Here U is the SU(2) unitary matrix for which we take the hedgehog ansatz with a chiral angle, U = exp(ixh0)
6 February 1986
PHYSICS LETTERS
where the finite term Co(O) is obtained by the direct numerical calculation of eq. (7). Now, if one insists that the hybrid model reduces to the Skyrme model whenR -~ 0, one needs to perform the subtraction with the condition dC(O)/d010_+0 = 0 ,
(9)
where C(0) denotes the subtracted axial current flow. Actually one finds the behavior of eq. (8) when 0 -~ 0 [9],
C(O) ~ [1/16rn7 - (In 7/)/301r2 + 3.75/30~r 2] sin 20 + 1.14(0/rr) 3 .
(I0)
We therefore drop the first term ofeq. (10) under the condition (9). This divergent term with the sin dependence might have appeared by treating the quark part quantum mechanically, which is to be cancelled out if one would calculate the quantum cor~ction from the pion sector [9]. Thus the subtracted C(O) is plotted in fig. 1. In this case we can determine the Skyrme pa-
(6) N
The expectation value on the right-hand side contains the valence and the vacuum contributions. The latter can be written in terms of the summation of the quark eigenmodes [9],
C(O)
r=R
N
C + Cv
0.05
O.I
0.05
d2S -Tr
x <01[~,ixh75 exp(ixh~/50) ~] 10) 1 -lim dEn 81r n-+0 ~n sgn(gn) -ffff-exp(-~ Ignl)
(7)
where En is the eigenenergy of the quark state n and 154
O.I
1
R =16~R~ f
_
(8)
-Tr/2 CHIRAL ANGLE /9 [red]
O
Fig. 1. The finite part C of the boundary condition for the chital angle as defined in eq. (9). For -n/2
Volume 167B, number 2
PHYSICS LETTERS
El 'Gev'/
rameter e from eq. (6) when R ~ 0. Using the numerical solution of the skyrmion equation, we find e =4.51
(12)
where EV is the volume energy and E~r is the classical energy of the pion. The quark energy consists of the valence energy and the Casimir energy. The Casirnir energy is obtained by integrating the relation [9,11,12] dEc/dO = 4~rC(O)/R ,
/
/
(11)
when the color number is Arc = 3. We shall comment here on an alternative method for fixing the Skyrme parameter e used by Brown et al. [13]. They use the continuity condition of the axial current at R [eq. ( 6 - 4 ) ] to determine e for each R so as to reproduce the axial vector coupling gA. Hence, e is a function of R and as a consequence the baryon mass comes out to have a kink around R 0.5 fm. Now we have only one value to fix in the model lagrangian, the bag constant B. For givenB we solve the equation of motion at various bag radii R and find a minimum of the total energy of the system. The total energy is written as E = Eq + E1r + E v ,
6 February 1986
I5
~
O
T
A
L
i
•
1.0 ~
i
QUARK
,
0.5
/
/ /i
,~"
".
', " "~Tr
i 1
"*°,
0.5
I~.0 R (fro)
i
1.5
Fig. 2. The energy of the CSH model ~/s a function of the bag radius R. The total energy E in eq. (12) is depicted by the solid line, where the bag constant of B 1/4 = 170 MeV is used. The quark energy Eq is shown by the dash-dotted line, where the Casimir energy is separately shown by the dotted line above R = 0.42 fm corresponding to 0 (r=R) = -~r]2. The pion part is depicted by the dashed line. The minimum corresponds to the stable configuration, where the total energy is E = 1260 MeV and the bag radius is R = 0.5 fm.
(13)
with the condition Ec(O = O) = O. In fig. 2 we show the numerical results of the energy as a function of the bag radius R for the bag constant B 1/4 = 170 MeV. The total energy starts from E ~ 1510 MeV a t R = 0 which corresponds to the skyrmion mass with parameter f~r = 93 MeV and e = 4.51 and decreases monotonically up t o R ~ 0.5 fm and then increases due to the volume term in eq. (12). The resulting minimum energy comes out to be ~1260 MeV at R ~ 0.5 fm. This energy is close to the average energy of delta and nucleon, Ear = 1173 MeV [5]. However, we have to work out the spin-isospin projection in order to make close comparison with the data. In fig. 2 we also show the quark and the pion contribution to the total energy. In the region -lr/2 < 0 < 0 or 0.42 fm < R < oo the quark energy is a sum of the valence part and the Casirnir part which is depicted by the dotted curve. F o r R < 0.42 fm the valence level dives into the negative sea and the energy comes only from the Casimir part. Another physical quantity we would like to discuss is the axial coupling constant gA. It is derived from the
volume integral of the axial vector current Aa(x). Using the conservation of the axial vector current, we can transform the volume integral to the asymptotic surface integral [8]. This shows that the axial coupling gA is determined by the asymptotic behavior of the Skyrme part which we express as #(r) ~ A / r 2. Using this A parameter we can obtain 8
2
gA = ~ rtf~rA
•
In fig. 3 we show gA as a function of the bag radius R. At R ~ 0.5 fm we obtain gA ~ 1.0. Zahed and Brown claim to multiply (No + 2)/Nc to the above equation in order to make comparison with datagA ~ 1.24 for the case of the pure skyrmion. We prefer here again that the spin-isospin projection is worked out before comparison with any nucleon data. In summary we have solved the chiral bag plus skyrmion hybrid model numerically. In order to set this model meaningful, we have carefully subtracted the divergent term and some finite contribution from the boundary condition for the chiral angle. This ad hoc subtraction has to be justified by explicitly calcu155
Volume 167B, number 2 I
PHYSICS LETTERS I
References
I
1.5 1.0 0"5t I
0.5
I
1.0
I
1.5 R [frn]
Fig. 3. The axial vector coupling gA as a function of the bag radius R.
lating the quantum correction o f the pion field. The resulting total energy and the axial coupling constant have been found to be smooth functions o f the bag, radius R and to be very reasonable. The CSH model seems to be a good candidate for the model o f hadrons. We are now strongly urged to work out the s p i n - i s o spin projection o f the CSH model. We acknowledge the financial support o f the National Science Foundation, which made our stay at Michigan State University possible, where the latter half o f this work was carried out. We thank M. Hosoda and the members o f the nuclear theory group at T o k y o Metropolitan University for enlightening discussions.
156
6 February 1986
[1] A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. D10 (1974) 2599; T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [2] A. Chodos and C.H. Thorn, Phys. Rev. D12 (1975) 2733. [3] G.E. Brown and M. Rho, Phys. Lett. 82B (1979) 177. [4] S. Theberg, A.W. Thomas and G.A. Miller, Phys. Rev. D22 (1980) 2838; F. Myhrer, G.E. Brown and Z. Xu, Nucl. Phys. A362 (1981) 317. [5] V. Vento, M. Rho, E.M. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413. [6] J. De Kam and H.J. Pirner, Nucl. Phys. A389 (1982) 640. [7] T.H.R. Skyrme, Prec. R. See. London A260 (1961) 127; Nucl. Phys. 31 (1962) 556. [8] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [9] P.J. Mulders, Phys. Rev. D30 (1984) 1073,in: Prec. Meeting on Electron and photon interactions at intermediate energies (Bad Hormef, Fed. Rep. Germany, 1984), eds. D. Menze, W. Pfefl and W.J. Schwille, p. 184. [10] J. Goldstone and R.L. Jaffe, Phys. Roy. Lett. 51 (1983) 1518. [11] I. Zahed, U.G. Meissner and A. Wirzba, Phys. Lett. 145B (1984) 117. [12] L. Vepstas, A.D. Jackson and A.S. Goldhaber, Phys. Lett. 140B (1984) 280. [13] G.E. Brown, A.D. Jackson, M. Rho and V. Vento, Phys. Lett. 140B (1984) 1285. [14] M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Lett. 51 (1983) 747. [15] I. Zahed and G.E. Brown, Lecture Note Los Alamos Summer School (1985).
PHYSICS LETTERS
6 February 1986
C H I R A L BAG P L U S S K Y R M I O N H Y B R I D M O D E L F O R N U C L E O N S A. H O S A K A and H. T O K I Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA and Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan Received 27 August 1985; revised manuscript received 7 November 1985
The chiral bag plus skyrmion hybrid model is studied as a model for nucleons. The hybrid model is solved numerically, where great care has been taken to remove the divergence in the boundary condition at the chiral bag surface for the nonlinear differential equation of the skyrmion. The numerical results for the nucleon energy turn out to be finite for all bag radii, differing from the case of the linearized chiral model. With a suitable choice of the bag parameter (volume term), the nucleon mass, radius and the axial vector coupling constant gA come out to be very promising.
As a phenomenological model for nucleons in terms of the quark model, the MIT bag model has been successfully introduced to provide all the static properties of baryons and mesons [1 ]. The phenomenology for the quark confmement instead of solving the QCD lagrangian, however, has introduced an unfavorable aspect: the chiral symmetry is broken at the bag surface. The chiral bag model was then proposed by Chodos and Thorn [2] and by Brown and Rho [3] to restore the chiral symmetry by introducing pions outside of the M1T bag. Since then, several attempts have been made to solve the quark-pion coupled system either by linearization of the pion field [4] or by taking the classical approximation [5]. It was quickly realized, however, that the chiral bag becomes unstable as the bag radius is reduced [5,6]. About 25 years ago, Skyrme introduced a very interesting model for nucleons, where the nonlinear model provides a soliton solution with the "baryon number" = 1 by introducing a stabilizer (the so-called Skyrme term), which is identified as a nucleon [7]. The amazing feature of this model is that bosons create fermions by a topological twist. The static properties resemble remarkably those of nucleons [8]. The apparent success of the Skyrme model has motivated Mulders to construct a hybrid model of the above two models as a model for nucleons [9]. The appealing feature of the hybrid model is that the quark sector and the skyrmion sector (nonlinear pion 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
fields) share the baryon number, which is one at any bag radius [9,10]. The motivation of this study is therefore to work out the chiral bag plus skyrmion hybrid model (hereafter abbreviated as CSH) up to the end for nucleon static properties. One difficulty lying ahead of us is to remove the divergent behavior in the boundary condition for the skyrmion differential equation with great care. Independently from Mulders, Veptas et al. [12] and Brown et al. [13] have developed the similar hybrid model for nucleons. In the course of construction of the model, they have chosen to fix the strength of the Skyrme term by the requirement of the axial vector coupling constant gA having the correct value [ 13] This requirement is fulfilled only up to a certain bag radius R and therefore physical quantities like the mass have a kink as a function o f R . We prefer to choose the strength of the Skyrme term by the condition that the hybrid model reduces to the Skyrme model as R goes to 0 and then insist the strength e be constant while R is varied. We consider, hence, that the Skyrme term is inherent in the properties of the pion field. Let us start by writing the lagrangian of the hybrid model for clarification of the procedure of calculations, where the details are found in ref. [9]. The CSH model lagrangian consists of the quark, pion and the interaction parts: 153
Volume 167B, number 2 £ = £q + £~r + £int •
(1)
For simplicity, we consider the case of massless quark and pion. The quark lagrangian reads £q = (~i~v~U ~ - B) OB,
(2)
where B is a bag constant and 0B is one inside the bag and zero otherwise. The pion part is the Skyrme lagrangian, £~r =
{:}/2Tr i ) . US"U t
+ (1/32e 2) Tr[8~
UUt , 8 v UUt] 2} 0~.
with h = r/r,
(3)
(4)
and 0~ = 1 - OB. In eq. (3),/~ is the pion decay constant and e controls the strength of the Skyrme term. The third term, the interaction lagrangian, is £int = - ~ ~ exp(i~h0 75) ~b~B,
(5)
where the surface delta function 8 B is introduced such that the quark and the pion interact only at the bag surface. The variational method provides then a set of differential equations for ~ and 0 as given in eqs. (7), (8), (13), (14) and (71) in ref. [9]. In order to solve the equations of motion derived from the above lagrangian, we need to evaluate the quark axial current flow which appears in the equation of the conservation of the axial vector current, [f2 + (2 sin20)/e2r 2] O' - l 2(2 i [5, ixh~f5 exp(ixh~t50) ~k]) -
sgn(En) denotes the sign of the energy En. This expression is known to contain divergences which must be subtracted to perform a meaningful calculation. The strongest divergence, the order of which is l/r/, was found by Mulders using the multiple reflection technique [9] and the weaker logarithmic divergence was evaluated by Zahed et al. by comparing the axial current flow with the Casimir energy [11]. Writing the divergent terms explicitly, one has
C(0) = [ 1 / 1 6 m / - (In n)/30~r 2] sin 20 + Co(O),
Here U is the SU(2) unitary matrix for which we take the hedgehog ansatz with a chiral angle, U = exp(ixh0)
6 February 1986
PHYSICS LETTERS
where the finite term Co(O) is obtained by the direct numerical calculation of eq. (7). Now, if one insists that the hybrid model reduces to the Skyrme model whenR -~ 0, one needs to perform the subtraction with the condition dC(O)/d010_+0 = 0 ,
(9)
where C(0) denotes the subtracted axial current flow. Actually one finds the behavior of eq. (8) when 0 -~ 0 [9],
C(O) ~ [1/16rn7 - (In 7/)/301r2 + 3.75/30~r 2] sin 20 + 1.14(0/rr) 3 .
(I0)
We therefore drop the first term ofeq. (10) under the condition (9). This divergent term with the sin dependence might have appeared by treating the quark part quantum mechanically, which is to be cancelled out if one would calculate the quantum cor~ction from the pion sector [9]. Thus the subtracted C(O) is plotted in fig. 1. In this case we can determine the Skyrme pa-
(6) N
The expectation value on the right-hand side contains the valence and the vacuum contributions. The latter can be written in terms of the summation of the quark eigenmodes [9],
C(O)
r=R
N
C + Cv
0.05
O.I
0.05
d2S -Tr
x <01[~,ixh75 exp(ixh~/50) ~] 10) 1 -lim dEn 81r n-+0 ~n sgn(gn) -ffff-exp(-~ Ignl)
(7)
where En is the eigenenergy of the quark state n and 154
O.I
1
R =16~R~ f
_
(8)
-Tr/2 CHIRAL ANGLE /9 [red]
O
Fig. 1. The finite part C of the boundary condition for the chital angle as defined in eq. (9). For -n/2
Volume 167B, number 2
PHYSICS LETTERS
El 'Gev'/
rameter e from eq. (6) when R ~ 0. Using the numerical solution of the skyrmion equation, we find e =4.51
(12)
where EV is the volume energy and E~r is the classical energy of the pion. The quark energy consists of the valence energy and the Casimir energy. The Casirnir energy is obtained by integrating the relation [9,11,12] dEc/dO = 4~rC(O)/R ,
/
/
(11)
when the color number is Arc = 3. We shall comment here on an alternative method for fixing the Skyrme parameter e used by Brown et al. [13]. They use the continuity condition of the axial current at R [eq. ( 6 - 4 ) ] to determine e for each R so as to reproduce the axial vector coupling gA. Hence, e is a function of R and as a consequence the baryon mass comes out to have a kink around R 0.5 fm. Now we have only one value to fix in the model lagrangian, the bag constant B. For givenB we solve the equation of motion at various bag radii R and find a minimum of the total energy of the system. The total energy is written as E = Eq + E1r + E v ,
6 February 1986
I5
~
O
T
A
L
i
•
1.0 ~
i
QUARK
,
0.5
/
/ /i
,~"
".
', " "~Tr
i 1
"*°,
0.5
I~.0 R (fro)
i
1.5
Fig. 2. The energy of the CSH model ~/s a function of the bag radius R. The total energy E in eq. (12) is depicted by the solid line, where the bag constant of B 1/4 = 170 MeV is used. The quark energy Eq is shown by the dash-dotted line, where the Casimir energy is separately shown by the dotted line above R = 0.42 fm corresponding to 0 (r=R) = -~r]2. The pion part is depicted by the dashed line. The minimum corresponds to the stable configuration, where the total energy is E = 1260 MeV and the bag radius is R = 0.5 fm.
(13)
with the condition Ec(O = O) = O. In fig. 2 we show the numerical results of the energy as a function of the bag radius R for the bag constant B 1/4 = 170 MeV. The total energy starts from E ~ 1510 MeV a t R = 0 which corresponds to the skyrmion mass with parameter f~r = 93 MeV and e = 4.51 and decreases monotonically up t o R ~ 0.5 fm and then increases due to the volume term in eq. (12). The resulting minimum energy comes out to be ~1260 MeV at R ~ 0.5 fm. This energy is close to the average energy of delta and nucleon, Ear = 1173 MeV [5]. However, we have to work out the spin-isospin projection in order to make close comparison with the data. In fig. 2 we also show the quark and the pion contribution to the total energy. In the region -lr/2 < 0 < 0 or 0.42 fm < R < oo the quark energy is a sum of the valence part and the Casirnir part which is depicted by the dotted curve. F o r R < 0.42 fm the valence level dives into the negative sea and the energy comes only from the Casimir part. Another physical quantity we would like to discuss is the axial coupling constant gA. It is derived from the
volume integral of the axial vector current Aa(x). Using the conservation of the axial vector current, we can transform the volume integral to the asymptotic surface integral [8]. This shows that the axial coupling gA is determined by the asymptotic behavior of the Skyrme part which we express as #(r) ~ A / r 2. Using this A parameter we can obtain 8
2
gA = ~ rtf~rA
•
In fig. 3 we show gA as a function of the bag radius R. At R ~ 0.5 fm we obtain gA ~ 1.0. Zahed and Brown claim to multiply (No + 2)/Nc to the above equation in order to make comparison with datagA ~ 1.24 for the case of the pure skyrmion. We prefer here again that the spin-isospin projection is worked out before comparison with any nucleon data. In summary we have solved the chiral bag plus skyrmion hybrid model numerically. In order to set this model meaningful, we have carefully subtracted the divergent term and some finite contribution from the boundary condition for the chiral angle. This ad hoc subtraction has to be justified by explicitly calcu155
Volume 167B, number 2 I
PHYSICS LETTERS I
References
I
1.5 1.0 0"5t I
0.5
I
1.0
I
1.5 R [frn]
Fig. 3. The axial vector coupling gA as a function of the bag radius R.
lating the quantum correction o f the pion field. The resulting total energy and the axial coupling constant have been found to be smooth functions o f the bag, radius R and to be very reasonable. The CSH model seems to be a good candidate for the model o f hadrons. We are now strongly urged to work out the s p i n - i s o spin projection o f the CSH model. We acknowledge the financial support o f the National Science Foundation, which made our stay at Michigan State University possible, where the latter half o f this work was carried out. We thank M. Hosoda and the members o f the nuclear theory group at T o k y o Metropolitan University for enlightening discussions.
156
6 February 1986
[1] A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. D10 (1974) 2599; T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [2] A. Chodos and C.H. Thorn, Phys. Rev. D12 (1975) 2733. [3] G.E. Brown and M. Rho, Phys. Lett. 82B (1979) 177. [4] S. Theberg, A.W. Thomas and G.A. Miller, Phys. Rev. D22 (1980) 2838; F. Myhrer, G.E. Brown and Z. Xu, Nucl. Phys. A362 (1981) 317. [5] V. Vento, M. Rho, E.M. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413. [6] J. De Kam and H.J. Pirner, Nucl. Phys. A389 (1982) 640. [7] T.H.R. Skyrme, Prec. R. See. London A260 (1961) 127; Nucl. Phys. 31 (1962) 556. [8] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [9] P.J. Mulders, Phys. Rev. D30 (1984) 1073,in: Prec. Meeting on Electron and photon interactions at intermediate energies (Bad Hormef, Fed. Rep. Germany, 1984), eds. D. Menze, W. Pfefl and W.J. Schwille, p. 184. [10] J. Goldstone and R.L. Jaffe, Phys. Roy. Lett. 51 (1983) 1518. [11] I. Zahed, U.G. Meissner and A. Wirzba, Phys. Lett. 145B (1984) 117. [12] L. Vepstas, A.D. Jackson and A.S. Goldhaber, Phys. Lett. 140B (1984) 280. [13] G.E. Brown, A.D. Jackson, M. Rho and V. Vento, Phys. Lett. 140B (1984) 1285. [14] M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Lett. 51 (1983) 747. [15] I. Zahed and G.E. Brown, Lecture Note Los Alamos Summer School (1985).