CHK model simplified by means of the constant-cutoff method

NUCLEAR PHYSICS A

Nuclear Physics A554 ( 1993) 580-592 North-Holand

CHK model simplified by means of the constant-cutoff method Nils Dalarsson Royal Institute of Technology, S-100 44 Stockholm, Sweden

Received 7 July 1992 (Revised 26 October 1992)

Abstract: We suggest a quantum stabilization method for the SU(2) cr-model, based on the constant-cutoff

limit of the cutoff quantization method developed by Balakrishna, Sanyuk, Sehechter and Subbaraman, which avoids the difficulties with the usual soliton boundary conditions pointed out by Iwasaki and Ohyama. We investigate the baryon number n > 1 sector of the model and show that after the collective-coordinate quantization it admits a stable soliton solution which depends on a singledimensional arbitrary constant. We then show that the bound-state approach to the hyperons in the Skyrme model, developed by Caltan, Hornbostel and Klebanov (CHK), is considerably simplified by omitting the Skyrme stabilizing term and using the constant-cutoff stabilization method. We derive the results for spectra of strange and non-strange baryons and dibaryons and obtain the numerical results for the masses of some strange and non-strange baryons showing that for a specific value of the pion-decay constant we obtain very good agreement with the empirical values.

1. Introduction It was shown by Skyrme ‘) that baryons can be treated as solitons of a non-linear chiral theory. The original lagrangian of the chiral SU(2) a-model is 5?=$Trd,Ua*Ur,

(1.1)

where

U=$(cr+i7.Tr)

(1.2)

Tr

is a unitary operator ( UU+ = 1) and F, is the CT= tr(r) is a scalar-meson field and 1c= n(r) is The classical stability of the soliton solution requires the additional ad hoc term, proposed by &!C = &

Tr [ U’d,U,

pion-decay constant. In eq. (1.2) the pion isotriplet. to the chiral a-model lagrangian Skyrme ‘), to be added to eq. (l.l), U+&AFJ2

0375-9474/93/$~.00 @ 1993- Elsevier Science Publishers B.V. All rights reserved

(1.3)

N. Dalarsson / CHK model

581

with a dimensionless parameter e and where [A, B] = AB - BA. It was shown by several authors *)* that, after the collective quantization using the spherically symmetric ansatz V,(r) = exp [in. r,F( r)] ,

r,= r/r,

(1.4)

the chiral model, with both eqs. (1.1) and (1.3) included, gives a good agreement with the experiment for several important physical quantities. Thus it should be possible to derive the effective chiral lagrangian, obtained as a sum of eqs. (1.1) and (1.3), from a more fundamental theory like QCD. On the other hand it is not easy to generate a term like eq. (1.3) and give a clear physical meaning to the dimensionless constant e in eq. (1.3) using QCD. Mignaco and Wulck (MW) ‘) indicated therefore a possibility to build a stable single-baryon (n = 1) quantum state in the simple chiral theory, with the Skyrme stabilizing term (1.3) omitted. MW have shown that the chiral angle F(r) is in fact a function of a dimensionless variable s = &“(O)r, where x”(0) is an arbitrary dimensional parameter intimately connected to the usual stability argument against the soliton solution for the non-linear a-model lagrangian. Using the adiabatically rotated ansatz U(r, t) = A(t) U,(r)A+(t), where V,,(r) is given by eq. (1.4), MW obtained the total energy of the non-linear u-model soliton in the form E =&F;--

1 ,

x”(0)

(1.5)

where

(1.6) co

b=

I

ds $’

sin* (is),

0

(1.7)

and 9(s) is defined by F(r) = F(s)

= -m+@(s).

(1.8) The stable minimum of the function (1.5), with respect to the arbitrary dimensional scale parameter x”(O), is

(1.9) Despite the non-existence of the stable classical soliton solution to the non-linear u-model, it is possible, after the collective coordinate quantization, to build a stable chiral soliton at the quantum level, provided that there is a solution F = F(r) which satisfies the soliton boundary conditions, i.e. F(0) = -nw, F(m) = 0, such that the integrals (1.6) and (1.7) exist. * Including

an extensive

list of other references.

N. Dularsso~ / CHK model

582

However, as pointed out by Iwasaki and 0hyama4), the quantum stabilization method in the form proposed by MW ‘) is not correct since in the simple a-model the conditions F(0) = -mr and F(a) = 0 cannot be satisfied simultaneously. In other words if the condition F(O) = --rr is satisfied Iwasaki and Ohyama obtained numerically F(a)+ -&r, and the chiral phase F = F(r) with correct boundary conditions does not exist. Iwasaki and Ohyama also proved analytically that both boundary conditions F(0) = --nrr and F(E)) = 0 cannot be satisfied simultaneously. Introducing a new variable y = l/r into the differential equation for the chiral angle F = F(r) we obtain d2F 1 -=-sin2F. dy2 y2

(1.10)

There are two kinds of asymptotic solutions to eq. (1.10) around the point y = 0, which is called a regular singular point if sin 2F = 2F. These solutions are F(y) = ;rnn. + cy2

rn = even integer

(1.11)

F(y)=~m~+~cos~~~ln(cy)+~]

m = odd integer f

(1.12)

where c is an arbitrary constant and LYis a constant to be chosen adequately. When F(0) = -m then we want to know which of these two solutions are approached by F(y) when y +O (r+ cc)? In order to answer that question we multiply eq. (1.10) by y’F’(y), integrate with respect to y from y to cc and use F(0) = -nr. Thus we get a? 2y[F’(~)]~ dy = 1 - cos [2F(y)] . (1.13) y’F’(y)+ IY Since the left-hand side of eq. (1.13) is always positive, the value of F(y) is always limited to the interval nr - r < F(y) < nrr + v_ Taking the limit y + 0, eq. (1.13) is reduced to m

J

2y[ F’(y)]* dy = 1 - (-1)“)

(1.14)

0

where we used eqs. (1.11) and (1.12). Since the left-hand side of eq. (1.14) is strictly positive, we must choose an odd integer m. Thus the solution satisfying F(0) = -m approaches eq. (1.12) and we have F(m) + 0. The behaviour of the solution (1.11) in the asymptotic region y-+00 (r+O) is investigated by multiplying eq. (1.10) by F’(y), integrating from 0 to y and using eq. (1.11). The result is [F’(Y>12 =

2 sin*F(y) + y2

J

y 2 sin*F(y)

0

Y3

dy.

(1.15)

From eq. (1.15) we see that F’(y) + const as y + co, which means that F(P) = l/r for r+ 0. This solution has a singularity at the origin and cannot satisfy the usual boundary condition F(0) = -mr.

N. Dalarsson / CHK model

583

In ref. ‘) the present author suggested a method to resolve this difficulty by introducing a radial modification phase cp= q(r) in the ansatz (1.4), as follows: r,= r/r.

U(r)=exp[iT*@(r)+icp(r)],

(1.16)

Such a method provides a stable chiral quantum soliton but the resulting model is an entirely non-covariant chiral model, different from the original chiral a-model. In the present paper we use the constant-cutoff limit of the cutoff quantization method developed by Balakrishna, Sanyuk, Schechter and Subbaraman “) to construct a stable chiral quantum soliton within the original chiral a-model. Then we apply this method to the CHK model of strange baryons and derive the results for spectra of hyperons and strange dibaryons. The reason why the cutoff approach to the problem of chiral quantum soliton works is connected to the fact that the solution F = F(r) which satisfies the boundary condition F(a) = 0 is singular at r = 0. From the physical point of view the chiral quantum model is not applicable to the region about the origin, since in that region there is a quark-dominated bag of the soliton. However, as argued in ref. 6), when a cutoff E is introduced then the boundary conditions F(E) = -rm and F(W) = 0, can be satisfied. In ref. “) an interesting analogy with the damped pendulum has been discussed, showing clearly that as long as E > 0, there is a chiral phase F = F(r) satisfying the above boundary conditions. The asymptotic forms of such a solution are given by eq. (2.2) in ref. “). Fom these asymptotic solutions we immediately see that for E + 0 the chiral phase diverges at the lower limit. 2. Constant-cutoff

stabilization

In order to obtain a chiral soliton with an arbitrary baryon number n > 1, we use the variational ansatz discussed in ref. ‘). Compared to the usual n = 1 hedgehog ansatz (1.4), the variational ansatz has a different twist in the isovector field m rather than a different boundary condition for the chiral phase function, i.e. U(r)=exp[i~*ir,F(r)],

Gn=

(2.1)

where F = F(r) is the radial chiral-phase function satisfying the boundary conditions F(0) = -s- and F(m) = 0. Substituting eq. (2.1) into eq. (1.1) we obtain the static energy of the chiral baryon E. = &rF2, ~E~~jdr[r2(~)2+(n2+l)sin2F].

(2.2)

In eq. (2.2) we avoid the singularity of the profile function F = F(r) at the origin by introducing the cutoff e(t) at the lower boundary of the space interval r E [0, CO],

584

N. Ddarsson

f CHIC model

i.e. by working with the interval r E [E, co]. The cutoff itself is introduced following ref. “) as a dynamic time-dependent variable. From eq. (2.2) we obtain the following differential equation for the profile function F= F(r): =b(n*+l)sin2F

(2.3)

with the bounda~ conditions F(E) = -r and F(W) = 0, such that the correct soliton number is obtained. The profile function F = F[ r; s(t)] now depends implicitly on time t through E(t). Thus in the non-linear o-model lagrangian L=&F2,

Tr (8, U# U*) d3x,

(2.4)

we use the ansltze ~+(~,t)~A(t)~~(~,

u(r, t)= A(t)&tr, t)A+(t),

t)A+(t),

(251

where &(r, t)=exp{ir.r,,F[r;

E(t)]}.

(2.6)

The static part of the lagrangian (2.4), i.e. L=&FZ,

Tr (V U- V U’) d3x = - & ,

(2.7)

is equal to minus the energy & given by eq. (2.2). The kinetic part of the lagrangian is obtained using eq. (2.5) with eq. (2.6) and it is equal to L=&F2,

Tr (8, U&, U’) d3x = bx* Tr [&A&,A+] - c[i( t)]* ,

(2.8)

where m

b = $rF;

sin2Fy2 dy ,

c = $F:

~3my2(~)2y2cfy

(2.9)

with x(t)=[~(t)]~” and y = r/6. On the other hand the static energy functional (2.2) can be rewritten as E. = ux2j3 ,

+(n2+1)

sin*F

I

dy.

(2.10)

Thus the total lagrangian of the rotating soliton is given by L= d2-ax

2~3+2bx2&,&y,

(2.11)

N. Dalarsson / CHIC mtxfd

585

where Tr (~~A~~*) = 2&i’ and (Y,(v = 0, 1,2,3) are the collective coordinates defined as in ref. “). In the limit of a time-independent cutoff (a + 0) we can write (2.12) with ‘) y2'

-&J(l+l)+I(I+l& [

(I;‘)‘]

(2.13)

,

where (J2) = J(J + 1) is the eigenvalue of the square of the soliton laboratory angular momentum, (J2) = I(1 + 1) is the eigenvalue of the square of the soiiton laboratory isospin and Ii‘ is the isospin component relative to the body-fixed axes. A minimum of eq. (2.12) with respect to the parameter x is reached at E-1=

2 ab “4 -* [ 3Y2 I

(2.14)

The energy obtained by substituting eq. (2.14) into eq. (2.12) is given by E

4 3 a3 y2 ‘j4

=32‘7; [

I

(2.15)

,

and for n = 1 it becomes

1

l/4

(2.16)

*

This result is identical to the result obtained by Mignaco and Wulck which is easily seen if we rescale the integrals a and 6 in such a way that a -, $rF:a, b + $vFf,b and introduce fv = 2 -3’2F However, in the present approach, as shown in ref. 6), there is a profile function ; = F(y) with proper soliton boundary conditions F( 1) = --7f and F(m) = 0 and the integrals a, b and c in eqs. (2.9) and (2.10) exist and are shown in ref. “) to be a = 0.78 GeV2, b = 0.91 GeV2, c = 1.46 GeV’ for F, = 186 MeV. In the multiba~on case (n > 19, there is however one more point which deserves further attention. From eqs. (2.14) and (2.15) we see that the necessary condition for stability of a quantum chiral multiba~on is

1 ‘If2

Y2z-0,

J(J+l)+$t2+3)I(r+l)

,

(2.17)

since otherwise the results (2.14) and (2.15) are meaningless. The condition (2.17) is fortunately always satisfied since the third component of the body-fixed isospin is limited by the inequality

as argued in ref. ‘)_

586

N. Dalarsson

Using

eq. (2.16) we obtain

states as Mignaco for the integrals about

than the empirical

constant

agrees

and the nucleon.

a and b we obtain

25% higher

pion-decay

the same prediction

and Wulck ‘) which

ratio for the A-resonance

/ CHK model

equal

for the mass ratio of the lowest

rather

well with the empirical

Furthermore

the nucleon

using the calculated

mass M(N)

= 1167 MeV which is

value of 939 MeV. However,

to F, = 150 MeV we obtain

mass values

if we choose

a = 0.507 GeV’ and

the

b=

0.592 GeV2 giving the exact agreement with the empirical nucleon mass. Finally it is of interest to know how large the constant cutoffs are for the above values of the pion-decay constant in order to check if they are in the physically acceptable ball park. Using eq. (2.14) it is easily shown that for the nucleons (J = 4) the cutoffs are equal

to for F, = 186 MeV for F,, = 150 MeV .

(2.19)

From eq. (2.19) we see that the cutoffs are too small to agree with the size of the nucleon (0.72 fm), as we should expect since the cutoffs rather indicate a size of the quark-dominated bag in the center of the nucleon. Thus we find that the cutoffs are of the reasonable physical size. Since the cutoff is proportional to F,’ we see that the pion-decay constant must be less than 57 MeV in order to obtain a cutoff which exceeds the size of the nucleon. relevant to any physical phenomena.

Such values

of pion-decay

constant

are not

3. The simplified CHK model 3.1. THE

EFFECTIVE

INTERACTION

The lagrangian density for a bound model is given by [see ref. “)I

kaon-soliton

system

~==F’,Trd,Ua’“U++~F2,(mZ,+2rnZ,) +&& where

in the simplified

Tr (U-C U’-2)

FfJ m’, - m’,) Tr A,( U + U’) ,

m, and mK are pion and kaon masses, respectively, u = (V,)“2Uk(

CHK

&)I’2

(3.1) and

) (3.2)

In eq. (3.2) u, is the SU(2)-skyrmion field for which we choose the variational ansatz (2.1). The two-dimensional vector K in eq. (3.2) is the kaon doublet K=

K+=(K-

KO).

(3.3)

N. Dalarsson / CHK model

587

In addition to the simplified Skyrme-model action obtained using the lagrangian density (3.1), the Wess-Zumino action in the form d% ervolsr Tr [ U+d, UU+d,UU+d, UU+dp UU+a,U]

(3.4)

must be included into the total action of a kaon-soliton system. In eq. (3.4) N, is the number of colours in the underlying QCD. Substituting eq. (3.2) into the total action of the kaon-soliton system, expanding UK to second order in the kaon fields (3.3) and using the following ansatz for stationary antikaon states: K(r, t)=rmrr,k(r,

t)=e-ioZkp(r)r*~nx,

(3.5)

where x is a two-component spinor, we obtain the effective interaction lagrangian density for the kaon-soliton system in the lowest bound state [see ref. ‘)I, zre= l;‘i+-

dk+ dk dr -dr-tiA(r)(kfk-f’k)-k+[m~+V,,(r)]k,

(3.6)

where +n2+1 COS4(~F), r2

(3.7)

(3.8) The hamiltonian density corresponding

to the lagrangian density (3.6) is given by

~=lI+~+k+l7-6P,

(3.9)

where lI’=aZ/ak=k’+iA(r)k*,

(3.10)

~=~~/a~+=~-iA(r)k.

(3.11)

Using eqs. (3.6), (3.10) and (3.11) we obtain from eq. (3.9) %‘=I7+27--$ + kf[m$+

$--ih(r)(k’lT-Ki’k) V,,(r)]k+

k’A2(r)k.

(3.12)

From eq. (3.12) we obtain the radial wave equation for the lowest bound-state antikaon wave function u0 = rk,,( r) in the form d2U0 dr2

--ve~(r)uo+[02-m~+2wh(r)]u,=0,

(3.13)

588

A! Dalarsson

where w is the towest bound-state according to 87r

3.2. THE ROTATIONAL

/ CHIC nodal

energy, and the antikaon modes are normalized

00 drr2[w+A(r)]k;(r)kp(r)=l. IE

ENERGIES

OF THE STRANGE

(3.14)

MULTIBARYONS

In order to calculate the spectrum of the strange baryons we must take into account the rotational modes of the soliton ‘). We consider the spatial (R) and isospin (A) rotations, where the corresponding angular velocities & and fi with respect to the body-fixed axes are given by (R-‘doR),b = %&C,

(3.15)

A-‘&,A=$~ST,

(3.16)

where E& (a, 6, c = 1, 2, 3) is the totally antisymmetric kaon and the soliton fields are rotated according to

tensor with e123= 1. The

K + RAK = (RA~.ti&‘jAk,

(3.17)

uw+ RAu,A-’

(3.18)

.

Substituting eqs. (3.2) and (3.15)-(3.18) into eq, (3.1) and keeping in the expansion of the kaon held only the two eigenmodes in eq. (3.5) with coefficients a, and a2 for the up and down spinor x, respectively, we obtain the following expression for the total lagrangian of the kaon-soliton system: L= L(u,)+L(K)-N*P

+(1-c)[tN,B,+~,B*)-(~,~,+~*~*)~,,+N3(B3-n~,)l +fn[B:+~,‘+a(n”~3)(~~+~~) -2@,B,

+ 48&%*

+a

- n&)*1,

(3.19)

with +rF’, c=l-$0

cc’ dr r2 sinZF, ic 47 dr r2kz(r) cos*($F)k,(r) IE

(3.20)

,

(3.21)

where N is the spin of the antikaon modes given by N = $$‘r& and aAt(ii) are creation (annihilation)

operators of antikaon modes.

(3.22)

N. Dalarsson / CHK model

The quantization

589

of the rotational energy leads to

(3.23) where J is the total angular momentum of the kaon-soliton system, Z is the collective angular momentum of the rotating n > 1 soliton and T is the angular momentum of the kaons. 3.3. THE SPECTRUM

OF HYPERONS

In the special case when n = 1, i.e. for the spherically symmetric soliton, eq. (3.23) reduces to the well-known form ‘)

FZro,=~(Z+cT)2,

(3.24)

n=l.

The eigenvalue of the kaon angular momentum T is related to the strangeness as T = $571. Introducing the total angular momentum (3.25)

J=Z+T, we obtain the total energy of the n = 1 kaon-soliton

system

E=E,+w~S~+~[cJ(J+l)+(l-c)Z(Z+l)+~c(c-l)~S((~S~+2)],

(3.26)

where the static soliton energy E. is, for m, = 0, given by E, = $rF2, lemdr [ ~‘(~)‘+(n2+l)

sin'F]

(3.27)

,

and w is the bound-state energy eigenvalue for the actual kaon bound state. Using the constant-cutoff stabilization procedure we obtain the following spectrum of hyperons in the simplified CHK-model: 114

E=olSl+;

I

~~[d(~+1)+(1-c)Z(Z+l)+~c(c-l)~S~(~S~+2),

, I

(3.28) with a and 6 defined by eq. (2.10) and eq. (2.9), respectively. The numerical calculation for F,, = 186 MeV (a = 0.78 GeV2 and b = 0.91 GeV2) gives the mass of the A-particle, with J = f, Z = 0 and S = -1, equal to M(A) = 1120 MeV which is in excellent agreement with the empirical value of 1116 MeV. For F, = 150 MeV (a =0.507 GeV2 and 6 = 0.592 GeV’) we obtain M(A) = 1063 MeV, which is still in reasonable agreement with the empirical value. The spectrum of a few other hyperons is then presented in table 1.

590 TABLE

1

The hyperon spectrum in MeV E I

Hyperon

S

J (F,, = 186 MeV)

(F,., = 150 MeV)

M =P

A s

0 1

-1 -1

4 a

1120 1204 1648

1063 1559 1142

1116 1385 1195

E c”* R

f f 0

-2 -2 -3

: 5 t :

1284 1798 1947

1228 1711 1862

1315 1530 1672

3.4. THE ROTATIONAL

ENERGIES

OF STRANGE

DIBARYONS

The eigenfunctions of Z-Z,,,given by eq. (3.23) are product states of the spatial and isospin rotation matrices with the kaon eigenstates, i.e. (a, p, r, t[Z, z3; .Z,J3; Z;‘= Q- r,, .Z!‘= -nQ, Tj) = I-(21+ 1)(2&Z+1)]“2 - o~~,-no(a)D:,,o-=,(P)k7,(r, 87r*

(3.29)

r) ,

where we introduced the angular momenta Jbf and Zbf relative to body-fixed axes as dL

Jbf=__.-

fb’,%

and

(3.30)

ag%'

&k

such that the axial symmetry of the classical solution gives rise to the constraint .Z;r= --n(Z;f+ T3). For non-strange Q, Ji’= -nQ,

(3.31)

dibaryons ( T = $1S( = 0), the states of the form 1I, I3 ; J, J3; It’= are the eigenstates of H,,, and the rotational energy is given by

T3 = 0)

Q’] .

&J(J+l)+Z(Z+l)--&

(3.32)

For strange dibaryons (T =#Sl> 0), the eigenstates are obtained after the diagonalisation of Z-Z,,,. For dibaryons with strangeness S = -1 the isospin of the bound antikaon is T = i and the hamiltonian is a 2 x 2 matrix. If the basis states are /I, Z, ; J3 ; J3; Ztf = Qr f, Jt’= -nQ, T3 = *j), the rotational energy is given by Q2+c2

Q -f+$c

[Z(Z+l)-Q2+$]“*

1

[Z(Z+1)-Q2+$]1’2 -Q-i+&

1 *

(3.33)

h? ~aiaTsson / CYiK model

591

For dibaryons with strangeness S = -2 the isospin of the bound antikaons must be T = 1, since the antikaons are bosons, and the rotational hamiltonian has the form of a 3 x 3 matrix. If the basis states are chosen as IZ, Z, ; .Z,.Z3; Ztf = Q, .Zi’= -nQ, 7” = 0) and -nQ, T3 = f l), ordered according to increasing T3 IZ,z3;J,53;z~f=QIF1,J~f= values, the rotational energy is given by

--&(J+l)+Z(Z+l)2(Q-l)+e +ii

-$$

Q’+c’]

[21(f+l)-2Q(Q-1)]“2

[21(1+1)-2Q(Q-1)]“2 0

e [21(1+1)-ZQ(Q+l)]“’

0 [21(I+l)-2Q(Q+l)]“2 -2(Q+l)+c

a I

(3.34) In eqs. (3.32)-(3.34) the values of Z and .Z are subject to constraints Z 2 IZt’l and J > I.Zi’I.For n = 2 the parity of the states is P = (-)o and Q must be an integer. We now apply the constant-cutoff quantum stabilization method to the energy of the non-strange dibaryon (3.32). Thus we minimize the energy of the Skyrme soliton only, and not the energy of the kaon-soliton system as a whole. We then obtain the following expression for the moment of inertia (3.20) occurring in eqs. (3.32)-(3.34):

~=~;{;(~)2-&+J+l)+Z(Z+l)--$$Q2]~’4,

(3.35)

with a and b defined by eq. (2.10) and eq. (2.9), respectively. Thus the rotational energies of dibaryons in the simplified CHK approach are given by eqs. (3.32)-(3.34), where the moment of inertia R is given by eq. (3.35) with eqs. (3.9) and (3.10). 4. Conclusions The present paper shows a possibility to use the Skyrme model, or more precisely the CHK model, for calculation of the hyperon spectrum without use of the Skyrme stabilizing term, proportional to e-‘, which makes the practical calculations very lengthy and painful. The accuracy in the prediction of the nucleon mass for F, = 150 MeV is very good. The predicted hyperon masses, obtained using the same value of F,, are in good agreement with the experimental values for states with .Z=f. However, for states with .Z= 3 there is only a crude qualitative agreement. The reason for this insufficient accuracy is a considerable simplification of the employed model, where only the soliton energy (and not the energy of the kaon-

592

N. Dalarsson / CHK model

soliton system as a whole) is minimized and the dynamical term in the rotational pe~urbation is only evaluated to first order.

References 1) T.H.R. Skyrme, Proc. Roy. Sot. London A260 (1961) 127; Nucl. Phys. 31 (1962) 556 2) G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552; E. Witten, Nucl. Phys. 8160 (1979) 57; B223 (1983) 422,433; G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825; E.M. Nyman and D.O. Riska, Rep. Prog. Phys. 53 (1990) 1137 3) J.A. Mignaco and S. W&k, Phys. Rev. Lett. 62 (1989) 1449 4) M. Iwasaki and H. Ohyama, Phys. Rev. D40 (1989) 3125 5) N. Dalarsson, Mod. Phys. Lett. A6 (1991) 2345 6) B.S. Balakrishna, V. Sanyuk, J. Schechter and A. Subbaraman, Phys. Rev. D45 (1991) 344; P. Jam, J. Schechter and R. Sorkin, Phys. Rev. D39 (1989) 998 7) H. Weigel, B. Schwesinger and G. Holzwarth, Phys. Lett. B168 (1985) 321 8) R.K. Bhaduri, Models of the nucleon (Addison-Wesley, Reading, MA, 1988) 9) C.G. Callan and I. Klebanov, Nuct. Phys. B262 (1985) 365; C.G. Callan, K. Hornbostel and I. Klebanov, Phys. Lett. B202 (1988) 269; J. Kunz and P.J. Mulders, Phys. Lett. B215 (1988) 449 10) N. Dalarsson, Nucl. Phys. A536 (1992) 573 11) N. Dalarsson, Nucl. Phys. AS32 (1991) 708