Dense skyrmion systems

Nuclear Physics A501 (1989) Sol-812 North-Holland. Amsterdam

DENSE

SKYRMION

L. CASTILLEJO

SYSTEMS

and P.S.J. JONES UK

A.D. JACKSON

and J.J.M.

VERBAARSCHOT

A. JACKSON

Received 28 November 1988 (Revised 13 April 1989) Abstract: The Skyrme model is used to investigate dense baryonic matter, Solutions of static close-packed skyrmions in arrays with fee symmetry as well as bee and intermediate symmetry are considered as a function of the density, and the phase transition into regular arrays of half-skyrmions is investigated. The fee array of skyrmions has lowest energy in the condensed half-skyrmion phase hut not in the uncondensed phase, in which the energy depends very little on the precise symmetry. The lowest energy, just 3.8?& above the theoretical lower bound, occurs in the condensed halfskyrmion phase corresponding to the fee array.

1. introduction The topological very successful in nucleon-nucleon skyrmion contains

model for nucleons introduced by Skyrme in 1960 ‘) has been describing the qualitative features of single nucleons and of the interaction 7-4). The classical spherically symmetric solution or nucleons, deltas and higher resonances, and to obtain properties

of these particles it is necessary to project out the relevant spin and isospin state. However, at very high densities nuclear matter not only contains nucleons, but also coherent delta amplitudes, and it is therefore interesting to see what the Skyrme lagrangian predicts for dense skyrmionic matter with no attempt at projecting out the spin and isospin content. The first calculations of dense skyrmion matter were by Klebanov ‘) who studied a simple cubic array of skyrmions with neighbouring skyrmions relatively isospin rotated by 7~ so as to maximise their attraction at large separation. These calculations were repeated over a wider range of densities I”), and it was seen that, as the density was increased, there was a transition from a phase of isolated skyrmions to a phase in which the energy was much more uniformly distributed and in which (v), the mean value of the CTfield, vanished. This phase transition was better understood following the work of Manton “) who considered tl3:~-9474/gYiSor,sO 0 i North-Holland Physics

Elsevier Science Publishers Publishing Division)

R.V.

L. Ca.stillejo et al. / Dense Skyrmion

802

systems

skyrmions on a sphere S3 rather than on flat space; there the phase transition shows up as the skyrmion envelops the sphere uniformly below a critical radius. Goldhaber and Manton

‘) were able to identify

of an additional of half-skyrmions

symmetry

the phase

in which Klebanov’s

with alternating

transition

in R, as the appearance

array of skyrmions

signs of v. Klebanov’s

become

an array

choice of cubic lattice was

mainly guided by symmetry and maximising nearest neighbour interaction, but in dense neutron matter the long-range tensor force attraction between neutrons leads to an alternative spin-isospin symmetry 14). The analogue of this symmetry for skyrmions was investigated by Jackson and Verbaarschot “) and it produced a lower energy both in the uncondensed phase and in the condensed half-skyrmion phase. Though the long-range forces may dictate the optimal array at low densities the strong repulsion between skyrmions enforced by their topological properties would indicate that close-packed arrays could be energetically favoured at high density. In particular a fee cubic arrangement would enable the long-range forces to be effective and at the same time avoid too close an approach. This symmetry for skyrmion matter has been proposed independently in refs. x,y,‘O); Cook and Dallacasa 15) have even proposed it for nuclei. Kugler and Shtrikman I”) have performed variational calculations of the minimum energy configuration with halfskyrmion symmetry. In the remainder of this paper we shall investigate fee and bee and a range of interpolating symmetries as a function of density. 2. Skyrmion

arrays

Skyrme ‘) and Klebanov ‘) have shown that at large separation, energy between two hedgehog skyrmions with a relative isospin an axis a may be expressed as: V=2c(3(r.a)‘--r’a’)/r’,

r, the interaction rotation 0 about

(1)

where c is a constant and the magnitude of a is sin :O. This interaction can be summed over all the neighbours of any given skyrmion to provide an asymptotic formula for the energy for the dilute phase. Consider thus the following arrays: (i) The Klebanov cubic lattice ‘) in which translation by a lattice spacing a in the x direction involves isospin rotation by r about the z axis, and cyclicly. This is best expressed in terms of U=(a+i7-mT)/f,

(2)

U(x+a,y,z)=TZU(X,y,z)T;,

(3)

and cyclicly. Note that the six nearest neighbours are attractive but the twelve second nearest are repulsive for large separation. (ii) The Jackson and Verbaarschot cubic lattice “) consists of alternate layers parallel to the xy plane, one layer with skyrmions alternately unrotated and rotated

L. Castillejo

by rr about about

et al. / Den.se

Slqwnion

ry~tems

the z axis, and the next layer with skyrmions

the x and y axes. This can be written U(x+a,y,

Z)’ U(x,y,

U(x,y-ta, z+a)=

803

alternately

rotated

by rr

as z)=7,U(x,y,

Z)T,,

T,U(X,J’, z)r,..

(4) (5)

Both nearest and next nearest neighbours are now attractive at large a. (iii) The fee array in which the skyrmions at the corners of the cube are unrotated and those on the faces are rotated by r about the normal to the corresponding face.

U(x+~a, ?‘+$a, z) = T,U(X, y, Z)T,

(6)

and cyclicly. There are now twelve nearest neighbours with maximal long range attraction but the six next nearest have zero attraction. (iv) The bee array we consider can be obtained from the fee case above by contracting the z-axis by a factor a, the fee nearest neighbours in the xy plane now becoming the corners of the bee cube and the skyrmions on the centres of the xz and yz faces becoming the ones at the centre of the bee cube. Alternatively in terms of the bee variables U(x~a,y,z)=U(.u,y~u,z)=T,U(X,y,Z)T~,

(7)

U(x~~u,~~+~u,z+~u)=~(7,.f~,)U(.~,y,2)(7,.~F7,).

(8)

The eight nearest and four of the six next nearest neighbours are maximally attractive, and the remaining two neighbours produce no long-range attraction. (v) A regular array which interpolates between fee and bee by changing the unit aspect ratio of the fee cube of side a into a rectangular region of aspect ratio y3 with lattice displacements in the x and y directions of ru and lattice displacements in the z directions of u/r’. Otherwise, the boundary unaltered. We define p = r- l/r to measure the deviation Hence, p = 0.23 ( r3 = v’?) describes a bee array, p s dimensional columns of closely packed skyrmions, and planes of square arrays of skyrmions. Performing next nearest

the sums neighbours

conditions of eq. (6) are from fee symmetry (p = 0). 1 describes separate onep < -1 describes separate

of the asymptotic interaction in eq. (1) over nearest and as well as over a large number of neighbours gives the

results shown in table 1. The contribution of the distant skyrmions is of course only meaningful in the limit of zero pion mass, which is the case considered here. The binding energy is given at the same densities for all the cases listed above. The table shows that the bee is the most bound while the fee and Jackson and Verbaarschot cubic are next and about equal. The dependence on p of this asymptotic formula indicates that the fee energy is a local maximum dropping away indefinitely for negative p since the separated planes contain alternating unrotated and rotated skyrmions which attract each other. For positive p the limit of separate columns of skyrmions is noninteracting and of zero energy so there is a minimum in the

804

L. ~a.~~~liej~ et al. / Lkrzsr S~~rlni~/i~ .sysremr

asymptotic energy near p = 1.3. But these arguments fail as soon as the short-range repulsion comes into play. Since the fee array is close packed it is quite likely to have a lower energy than the others at high density.

Furthermore,

the limits p + +a

of the asymptotic formula at fixed density are unphysical since at some value of p the effect of the repulsive cores of the nearby skyrmions can no longer be neglected. This indicates the existence in the minimum at positive

of another p.

minimum

3. Numerical

at negative

p and a major change

calculations

Numerical calculations of the minimum energy of the Skyrme lagrangian as a function of a have been carried out for the fee and bee configurations and for a range ofcrystal arrays interpolating between them with -0.35

(K)]+-$[(Tr

(K))‘-Tr

(K’)]

with ,

.

itX,

ii,X;

.i;=z!ad!L

fl0)

and the boundary conditions as given above for the fee array. The minimisation of the energy was performed in a single octant of the face centered cube for the dilute phase, and in an octant of the half-skyrmion cube in the condensed phase. Exactly the same procedure was used for the bee and intermediate cases by trivially changing the aspect ratio of the cube into a rectangular region, with lattice displacements in the x and J’ or z directions becoming ra or a/r’ respectively, but otherwise keeping the same boundary conditions. The numerical procedure was to discretize Hamilton’s equations on a rectangular mesh and then set the momenta to zero after each ‘time’ step, thus ensuring that the motion is always downhill in energy. The mesh size was typically 18” mesh points to the octant of the cube in the dilute phase and the same number or fewer points in the octant of the half-skyrmion in the condensed phase providing substantially greater accuracy. Convergence was assumed when the extrapolated energy did not vary significantly over several hundred iterations during which the baryon number was stable to 0.01%. The results so obtained were first extrapolated to an infinite number of iterations as in ref. “) and then extrapolated to zero mesh size. This was easy in the condensed phase, where the energy and baryon density are spread fairly uniformly in space, so that the energy varies linearly with l/r?‘: where n3 is the number of mesh points; but in the dilute phase the number of points covering each skyrmion reduces as the size ofthe cube increases and the ext~polation

L. Cavtilkjo

is less clean. numerical

The baryon

calculation

et nl. /

density

Lkmr

Sk~wnion

also reflects

of the baryon

number

due to finite mesh size and this provided The value of 1 -h is proportional to l/i?’

.sy.strms

this feature b from

805

so the deviation

1 is also a measure

an alternative extrapolation for large n in the condensed

of the of errors

procedure. phase, but

the slope varies with a. The extrapolations to zero mesh size was therefore performed by linear extrapolation of the energy with respect to 1 -b, the slope determined by varying n for some typical cases. This procedure is very accurate for the condensed phase leading to errors of a fraction of an MeV, but in the dilute phase typical errors are two or three MeV. To ensure continuity at the second-order phase transition points the energies in the dilute phase were adjusted (within their errors) to agree with the more accurate values determined with the condensed phase boundary conditions. 4. Results The solutions show some very general features which hold for all values of the aspect ratio p so far investigated. Consider first the fee array, p = 0. At low density the skyrmions are well localised around their lattice positions (0,0, 0), (0, ia, :a), (;a, 0, lu), (ia, la, 0), each with a nearly spherical surface cr = 0 separating the inner half-skyrmion, u < 0, from the outer half, a> 0. This latter region extends to neighbouring skyrmions, is connected and fills most of space. The space average (a) of the CTfield is thus close to 1, while (rr) = 0 due to the cubic symmetry. As the density is increased the energy per skyrmion drops smoothly, as does ((7). The skyrmions spread out a little and the cr=O surfaces get more cubical. Then a second-order phase transition is reached, the energy still dropping smoothly, but the u = 0 surfaces now become perfect cubes of side iu which touch along the edges leaving the u > 0 region divided up into exactly similar cubes. The u = 0 surfaces are orthogonal planes with either x, y or z equal to :u or :a, the half-skyrmions rr < 0 and ~7> 0 now have identical n distributions, but there is no longer a unique way of associating any c < 0 half-skyrmion with a corresponding cr> 0 halfskyrmion. If the density were to be decreased from this symmetrical phase the skyrmion density could either reconcentrate around the original points where (T = -1, or alternatively on the points where c = 1. Above the phase transition, a = a,., we find (a)- (a - a,)‘, see fig. 1. This is the signature of such a second-order phase transition and the linear plot of (u)~ against a, as also shown in fig. 1, provides the most accurate determination of a,. The extra symmetry acquired at the phase transition can be written: cr(xi~u,y,z)=-a(x,?~,=), 7T,(.X+tu,?,,z)=-~,(x,~,z), rr,.(x-+lu,

_tJ,Z) = frr,.(x,

~,(x+lu, y, z)

_r, ‘7) ,

= +7r,(x, y, z)

(11)

806

L. Castillejo

0.7 -

(CT)

et al. / Dense Skvrmion

systems

-+-

lUY --•-_ 0.6 0.5 -

0.1 + 0.0 2.b

2.5

2.6

2.7

L (F,-,,) Fig 1. Plots of (0) and (a)’ versus

L with L = (density)m””

and cyclicly. Exactly the same second-order phase transition, with corresponding scaling of x, y and z coordinates occurs for all values of p in the range -0.35

807

50 -

L (F,)

’ ‘; (a) fee array, (b) hcc array, (c) Klebanovs cubic array ‘). Fig. 2. Plots of energy versus L = (density) The curve for the Jackson cubic array “1 has not been plotted as it lies just above the bee array. Solid lines have half-skyrmion symmetry and dashed do not. are deceptive since p = 0.23 corresponds to a factor 1.4 in aspect ratio while the same contour can be reached by increasing L by 18%. The distributions of the g and rr fields at the minimum are extremely well approximated by the formulae o = cos (5) cos (7) cos (0, 7r, = sin ([)Jl

- lsin’(n)-$sin’(l)++sin”(n)sin’(i),

(12) (13)

and cyclicly for nTT,. and nTT,,where .$ = 2_rrxla, 17 = 2n-y/a and 5 = 2vz/a. This formula is a three-dimensional analogue of the exact two-dimensional solution for the non-linear tr model described in ref. “) in terms of the Jacobi elliptic functions sn and cn which replace sines and cosines. An extension of formulae (12) and (13), including variational parameters which modify the arguments 6, 7, 5 and the factors 4 and $, can be found to fit the whole range of densities and deformations considered here for the half-skyrmion condensed phase “); and conformal transformations then provide variational solutions in the dilute phase as in ref. I”).

L. Castillejo

808

et al. /

systems

Deme Skyrmion

807

0.2 P I 0.0

t

\

-041

1.0

’

1,

’

’

’

15

’

’

’

’

’

2.0

\

,

I

2.5

L(F,) Fig. 3. Contour plots of equal energy in the (L, p) plane where L = (density) ‘W and p measures the deviation from fee symmetry (see text). Also shown are the curves for minimum energy with respect to L at fixed p and of the phase transition points separating the condensed half-skyrmion phase from the dilute phase.

Near the minimum we might expect that small changes in L and p would produce predominantly a stretching of the fields in the appropriate directions without essentially changing their shape. The two components of energy E, and Ed, quadratic and quartic in derivatives, would then scale as: E2=+Eo

?+r4 ( r2

L

-=$E,((l+p’)+&.‘+.

) L

E,=$E,,

from fee symmetry both E, and E, are p = 0. The total energy E is a minimum numerical solutions do not support such small correction, a better approximation

. .I+,

0

(15) separately stationary with respect to p at E, when L = L,, and Ez = E, = IE,. But the a simple scaling. For p = 0 there is only a being

(16) with F = 0.0515. But the dependence on p is more interesting. E, does indeed have a minimum at p = 0 with roughly the expected coefficient for p2, though it falls more slowly with L, but E2 has a maximum rather than a minimum with a negative coefficient for p’ which increases linearly with L. At L, the p’ contribution of E,

L. Chrtilkjo to

E

is negative

and smaller

cl

809

al. / Dfnse S~~,r?~ifJr7Yi_Ftems

than the positive

E4 contribution

but it increases

with

L until it gets larger, so that the curvature of E in p varies smoothly from a positive value at Lo through zero at a critical density corresponding to L = Ld = 2.43 fm to negative curvature above this second critical point. There are therefore at least two independent soft modes which go negative at L, and L,,, the mode which breaks the half-skyrmion symmetry but keeps fee symmetry and the mode which breaks fee symmetry but maintains the half-skyrmion symmetry. Though L,. and L,, are very close they appear not to be equal, but the two phase transitions may nonetheless be related, and since the second is triply degenerate there may be a whole class of modes which go negative at this criticai density signalling perhaps a solid-liquid phase transition. The above picture explains the salient features of fig. 3, the near independence of both the critical point L, and the energy on p near the critical density. It enables us to express the energy as a smooth function of L, p and (u) such that for each L and p minimising with respect to (a) gives the calculated energies and values of {a). The phase transition which breaks the half-skyrmion symmetry then arises from E( L, p, (v) = 0) changing from minimum to maximum in (a) as L increases through L,, and similarly fee symmetry is broken at Ld as E( L,p = 0,(a))changes from a minimum to a maximum in p. From symmetry the energy must be an even function of (u} and for small (CT) it may be parametrised in terms of (rr)’ and (~r)~ as:

where the coefficients

are given by

ru(L~=0.649-0.487~+0.0*~~, 0

~(L~=O.30O+O.o~~-f--0.11Y~, ‘0

The first four coefficients are known rather accurately since they can be calculated from solutions for the condensed half-skyrmion phase and are chosen to fit the p dependence of the energy right through the region L,, < L < 2.8 for -0.35 < p < 0.35, but the last two coefficients describe the start of the dilute phase and would be less accurately determined from solutions in the dilute phase. But here also one can use the solutions in the condensed phase to determine 9(L) and u(L) below L, by including a pion mass term E, = f mJI,)‘(l

-(cr))L”

(19)

i.. Ca.sfillejo et al. / Dense Skyrmion

810

.sptems

for each skyrmion to drive the solutions away from (g>=O and half-skyrmion symmetry, the pion mass acting like a chemical potential for (v). This enables us to determine the coethcients in (17) much more accurately. The first factor in q(L) was chosen to change sign at the critical length L = L, = 2.45, a point

which

is best determined

from the extrapolation

of (a)’ in fig. 1, and the

magnitudes and forms of q(L) and v(L) are then adjusted to fit the values of (o) as a function of L which arise from including the pion term above for various values of mw. These predict results consistent with the values of (m) in fig. 1 and the difference in energy between the condensed and dilute phases above L, from fig. 2. We have not included p’ (c>* terms in E, though there is a small effect, nor have we included the slight p dependence of L, in the parametrization. The coefficient a(L) changes sign at L= LCi= 2.43 and the coeflicient y(L) also changes sign at nearly the same L. Thus for fixed 15, I. < Ld and small IpI the energy has a minimum at p = 0 and a maximum for p < 0 which merge at L= Ldr the maximum is at p = -0.175 for L = 2.25 and at p = -0.46 for L = 2.0. For L just above Ld there is a maximum at p = 0 and a minimum at positive p, at a distance from p = 0 which depends mostly on the p3 coefficient /3(t) and which merges with the maximum at Ld. There is presumably another minimum at larger negative p for L > 2 which is outside the range of the present calculation and does not show up in fig. 3. It is hard to calculate the field distributions for densities lower than those shown in figs. 2 and 3, or for large Ipi, but the asymptotic form of the energy arising from eq. (I) (see table 1) predicts a maximum for fee symmetry, p = 0, with energy falling continuously for negative p. Due to short range repulsion one would expect that there would be a minimum for fixed density at some negative p, but, as shown above, this is beyond the range of our calculations. For positive p the minimum which appears in the asymptotic formula starts at the phase transition Ld at p = 0, and moves to larger p as L increases. But there is really no good reason to believe that at low densities the cubic or rectangular structures studied so far should be minima. It is worth noting that the ratios LJ Lo = 1.47 and L,,/ L, = 1.46 are only 4% and 3% above the critical ratio of fi which Manton “) derived for skyrmions

The asymptotic

P

many

potentials

from eq.

(1) for a single skyrmion

divided

by the skyrmion

density

L’(potential)

Klebanov

Jackson

fee

bee

near neighbours many neighhours

-949 -931

-1640 -1485

-2075 -1593

-199s - 1634

--0.3 -1776

-0.2 -1663

-0. I -1614

0.0 -1593

0.1 -1604

0.2 -1614

0.3 -1651

0.6 -1697

2.0 -1619

Near neighbours includes nearest and next nearest and many neighbours includes about 160 for Klebanov ‘) and Jackson ‘) arrays and all neighbours within a distance of 20a for fee, bee and intermediate p type arrays.

L. Castillejjo et al. / Dense Skyrmion

on S, where the lower bound

can be achieved

in R, is only 3.8% above the lower bound, an approximate but the argument

way to explain

exactly.

Manton’s

.q~stems

811

Since the energy argument

for the fee

can be modified

in

why L, differs by only a few per cent from fiL,,,

for L,, lying so close is less clear.

5. Conclusion There are several rather general conclusions to be drawn from all these detailed calculations. The salient feature is that there is a robust phase transition as the density is increased from a system of isolated skyrmions with no strongly preferred symmetry to a regular lattice of half-skyrmions. The transition is in general second order and the condensed system has lowest energy with fee symmetry. The phase transition and condensed phase look remarkably similar to the solution of Manton “) on a 3-sphere. The energy minimum occurs at a density of 0.217 fin-’ and the phase transition at 0.068 fin-‘. These should not be compared directly to nuclear matter density 0.17 fin-‘. The Klebanov ‘) choice of parameters sets energy and length scales determined by ,f, which is set at the unrealistic value of 64.5 MeV. Also skyrmion matter contains both nucleons and d and the latter have not been projected out. At high density this may be a reasonable approximation but not at low densities. Furthermore, the present calculations contain only potential terms and no kinetic energy contributions. The effects of including these to one loop order can be crudely estimated from the work of Zahed et al. “) for a single skyrmion on RJ and Generalis and Williams “) for SJ. One loop contributions appear to lower the skyrmion mass by about 20%, in the dilute phase or near the phase transition, with most of the contribution occurring effectively in the E4 term rather than in Ez. Even larger corrections may occur near the minimum energy of the condensed phase. Such changes in the relative strength of E, and E2 would increase the above densities by a significant factor to bring them more in line with predictions for nuclear matter. In most of the above calculations we have concentrated on the zero pion mass case, using a finite mass pion only to explore the energy surface as a function of (CT).This seems a good procedure for several reasons. Our main interest is to explore the effects of the transition to half-skyrmion symmetry and the pion term destroys this symmetry explicitly. The pion mass is important for well separated skyrmions because over large regions of space (far from the center of the skyrmions) it is the dominant term in the action. But in our dense system this occurs nowhere and so the pion and (T distributions will not be altered much. There is also the question whether it is correct to use the free field values of the pion mass and ,f, in such dense systems. According to Forkel et al. I’)) the half-skyrmion transition corresponds to chiral symmetry restoration, and they argue one should consider that the ,f, in eq. (19) really corresponds to (a), and would thus give no contribution in the condensed phase. The mass term in (19) ensures that skyrmions will prefer to concentrate around points with u = -1 rather than CJ= 1. The dominant effect of

812

L. Cartilkjo

cf al. / Lhm

Skjwnion

is to introduce

.~~3rem.s

the pion

mass on the energy

occupied increasing

by the skyrmion; this shifts the minimum, decreasing L,, by 2.2% and its energy by 42 MeV; it also slightly increases the energy difference

between the bee and fee minima. The pion (c). For L< L,, (m) is small and dominated

a term proportional

to the volume

mass term always leads to a nonzero by the balance between the ((7) term

in E, and the (g)’ term in (17), it is proportional to mi/q(L) so gets larger as L approaches L,. At the minimum (q) = 0.12. Near L, the quartic term in (a) in (17) becomes important and this has the effect of smoothing the abrupt increase of ((7) which occurs at the phase transition as seen in fig. 1. Above L, again the shift in (c) is dominantly proportional to mf, but involves both q(L) and u(L). Essentially in view of the smallness of the pion mass it merely leads to a smooth transition from the low to the high density phase. Of course, (g) will never be rigorously equal to zero, and in a strict sense there will be no phase transition for nonzero pion mass. Similar conclusions hold on S, [ref. “‘)]. This work was supported in part by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract Nos. DE-AC03-76F00098 and DE-FG02-88ER40388; also by SERC under grants CR/E/2085 and GR/E/37514. One of the authors (L.C.) would like to thank the Aspen Institute for Physics for support.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) 1.5) 16) 17) 18)

T.H.R. Skyrme, Nucl. Phys. 31 (1962) 5.56 A.D. Jackson and M. Rho, Phys. Rev. Lett. 51 (1983) 7.51 A. Jackson and A.D. Jackson, Nucl. Phys. A457 (1986) 6X7 G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 52 I. Klehanov, Nucl. Phys. B262 (1985) 133 N.S. Manton, Common. Math. Phys. III (1987) 469 AS. Goldhaber and N.S. Manton, Phys. Lett. BlY8 (1987) 231 A.D. Jackson and J.J.M. Verbaarschot, Nucl. Phys. A484 (1988) 419 D.I. Dyakonov and A.D. Mirlin, preprint 1327, Leningrad Nuclear Physics Institute (19X7) M. Kugler and S. Shtrikman, Phys. Lett. 8208 (19X8) 491 I. Zahed, A. Wirzha and U.-G. Meisaner, Phys. Rev. D33 (1986) 830 S. Generalis and G. Williams, Nucl. Phys. A484 (1988) 620, and to he published A.D. Jackson, J.J.M. Verhaarschot, 1. Zahed and L. Castillejo, Stony Brook preprint R.A. Smith and V.J. Pandharipande, Nucl. Phys. A237 (1975) 407 N.D. Cook and V. Dallacasa, Phys. Rev. C35 (1987) 18X3 G.E. Brown, A.D. Jackson and E. Worst, Nucl. Phys. A468 (1985) 450 A.D. Jackson and C. Weiss, private communication. A.D. Jackson, C. Weiss, A. Wirzba and A. Lande, Accurate variational forms for multi-skyrmion configurations, Stony Brook preprint (198X) 19) H. Forkel, A.D. Jackson, M. Rho, C. Weiss and A. Wirzba, Chiral symmetry restoration and the Skyrme model, Stony Brook preprint (1989 1 20) A. Wirzba, private communication and Skyrmion Workshop, Nordita, (1988)