Chiral symmetry restoration and the skyrme model

Nuclear Physics A504 (1989) 818-828 North-Holland, Amsterdam

CHIRAL SYMMETRY RESTORATION

A N D THE SKYRME M O D E L * H. FORKEL~, A.D. JACKSON, Mannque RHO2 and C. WEISS Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA

A. WIRZBA NORDITA, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark

H. BANG The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark

Received 5 April 1989 Abstract: Applications of the Skyrme model to dense baryonic matter reveal a transition from a low-density phase of localized skyrmions to a high-density phase possessing an additional halfskyrmion symmetry. We suggest that this transition should be regarded as the restoration of chiral symmetry at high densities. This is based on (i) the vanishing of the matrix element for pion decay and (ii) the appearance of parity doublets in both the strange and non-strange sectors at high density and (iii) the appearance of a triplet of massless "Goldstone bosons" at low density.

The Skyrme m o d e l 1) is k n o w n to provide a qualitatively sensible description of low-energy ~-zr scattering, the properties o f single b a r y o n s 2) a n d the n a t u r e of the n u c l e o n - n u c l e o n i n t e r a c t i o n 3). F r o m the p o i n t of view of n u c l e a r physics, applications of this m o d e l ( a n d its m a n y variants) to e x t e n d e d systems with a finite b a r y o n density are o f great interest. The fact that these effective l a g r a n g i a n models provide a s i m u l t a n e o u s d e s c r i p t i o n of b a r y o n structure a n d b a r y o n i c interactions raises the possibility that such c a l c u l a t i o n s may indicate f u n d a m e n t a l changes in the structure of b a r y o n i c matter at high densities. A variety of n u m e r i c a l studies suggest that this is, i n d e e d , the case. These calculations fall into two b r o a d classes. The first class involves the study of periodic arrays o f s k y r m i o n s in flat space (R3). O n e seeks classical field configurations which m i n i m i z e the energy for a fixed b a r y o n density. I n such c a l c u l a t i o n s the tr a n d rt fields are m i n i m i z e d n u m e r i c a l l y w i t h o u t c o n s t r a i n t except for the i m p o s i t i o n of one of a variety of "twisted" p e r i o d i c b o u n d a r y c o n d i t i o n s a i m e d at m a i n t a i n i n g the average b a r y o n density a n d a v o i d i n g nearestn e i g h b o r frustration 4-7). I n all cases, one finds u n a m b i g u o u s evidence of a phase * Work supported in part by the US Department of Energy under Grant No. DE-FG02-88ER40388. Supported by a research fellowship from the scientific council of NATO administered by the DAAD, 2 Permanent address: Service de Physique Th6orique, I.R.F., CEN Saclay, 91191 Gif-sur-Yvette, France. 0375-9474/89/$03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

t-1. Forkel et al. / Chiral symmetry restoration

819

transition. At low baryon density, both the baryon number and the energy are well-localized. The fields describe an ensemble of interacting baryons. At (and above) some critical density, the field configurations change dramatically, and an additional half-skyrmion symmetry emerges*. As a consequence, the baryon number and energy are distributed equally in all octants of the unit cell, and it becomes impossible to provide a unique identification for individual baryons. Further, the unit cell average of the scalar field, which is the de facto order parameter for this transition, vanishes. (Evidently, (or) must approach f~ in the limit of zero baryon density 8).) For the energetically favorable boundary conditions of refs. 6) and 7), this transition is second order with (or)- pv/-~c-p in the vicinity of pc. Fig. 1 shows (0-) as a function of density for a face-centered cubic array of skyrmions. Similar high-density field configurations are found for a variety of twisted boundary conditions and for a variety of chiral symmetric modifications of the basic Skyrme lagrangian (either through replacement of the stabilizing fourth-order term or the introduction of explicit vector mesons s)). Indeed, Manton and Goldhaber 9) have offered general arguments that the lowest energy field configuration at sufficiently high baryon density will necessarily possess the half-skyrmion symmetry encountered in numerical calculations. Calculations of R 3 arrays of skyrmions are technically complicated and require the numerical solution of Hamilton's equations on an uncomfortably dense mesh of points. Thus, Manton 10) was led to consider a far simpler calculation in which the manifold R 3 is replaced by the compactified manifold S 3 ( L ) . The introduction of a single baryon into a compactified space can be regarded as approximating baryonic matter with an average density of 1/2rr2L 3. In the limit of large L this replacement is passive, and the usual isolated fiat-space skyrmion emerges. At higher densities this change results in significant technical simplifications due to the fact that the target manifold of the (o-, ~r) fields is naturally S3(f~) due to the constraints 1.0

0.8

i

i



0.6 0.4

0.2 0

_~

s ~ r n ~ ._tj

i 1

.0

L

i 2

5

Fig. 1. The solid line shows the order parameter, (o-) for a single skyrmion on a hypersphere as a function of L and indicates that chiral symmetry is restored for L<~x/2. The dotted line shows (o-) when chiral symmetry is broken explicitly through a pion mass term and indicates the expected result that chiral symmetry is not restored at any finite density in this case. * The fields reveal families of planes on which the scalar field, o', is zero. The pionic fields are reflection symmetric about these planes while the scalar field is reflection antisymmetric.

820

H. Forkel et al. / Chiral symmetry restoration

of the non-linear sigma model. Such calculations have been shown to provide a remarkably faithful reproduction of the results obtained for flat-space arrays of skyrmions H). In effect, curvature effects simulate the interactions with other baryons. In particular, the phase transition noted above is again observed. To see how this emerges, we introduce the hedgehog approximation for the fields: tr-= c o s f ( t z ) ,

or-- sinf(/~)~

(1)

with the boundary conditions f ( 0 ) = 0 and f ( T r ) = 7r. For large L, the skyrmion profile, f ( / x ) , is concentrated near one of the poles of the hypersphere. (Specifically, the baryon density is concentrated in an angular range o f / x - 1 / L corresponding to a fixed size in flat space.) A solution of equal energy can be obtained with the related profile f(~) = rr-f(rr-

~).

(2)

The replacement of f by f merely interchanges the poles of the hypersphere. As L decreases, the skyrmion becomes less localized i n / , until some Lc is reached. For all L ~< Lc, the profile becomes the identity map, f ( ~ ) = ~, and the profiles f and f become identical. The equivalence o f f and f is the half-skyrmion symmetry noted above in flat space. As a consequence of this symmetry alone, one finds equal baryon number and equal energy in the two hemispheres of S3(L) and finds that the average value of o- over the hypersphere is necessarily zero. The fact that the actual profile is the identity map and that the full SO(4) symmetry of the problem is restored at high densities is a feature of the S3(L) calculations not shared by flat-space arrays of skyrmions. As in flat space, this transition is the rule rather than the exception; it has been seen with a wide variety of chiral symmetric effective lagrangians. Given the robustness of this phase transition, it is of interest to provide a physical interpretation. That is the purpose of the present manuscript. To be specific, we claim that this phase transition describes the restoration of chiral symmetry at high baryon density. Our illustrations will focus on the technically simpler results obtained on a hypersphere, and we shall find that many (but not all) of the arguments depend only on the existence of a half-skyrmion symmetry. Their extension to flat-space arrays of skyrmions seems guaranteed. The results of fig. 1 suggest that (o-) is a good candidate for an order parameter. Our aim is to make plausible the belief that (tr) = 0 indicates the restoration of chiral symmetry. First, consider the chiral symmetry breaking term in an SU(3) generalization of the Skyrme model. Current algebra indicates that this term should have the form ~sa = - ~ Tr ( U M + M * U* - M - M * ) .

(3)

The quantity # is to be identified with the quark condensates in the (possibly non-trivial) vacuum: 2# = (~u) = (dd) ~ (gs).

(4)

H. Forkel et al. / Chiral symmetry restoration

Here, M is the mass matrix, diag (mu, md, determined by the physical meson masses. 2

2,Y ~

2

ms). In 2

821

the zero-density vacuum t~ is 2

m ~.f~

m Kf~-

m u -b m d

m s

(5)

wheref~ is the pion decay constant in free space. However, in a baryonic environment the quark condensates change, and eq. (5) must be modified. To do this we note that the term Tr ( U M + M t U*) in eq. (3) plays precisely the role of the usual mass term, ~lMq. Also, the unitary matrix, U, and the quark bilinear, ~1(1+ Ts)q, have the same transformation properties under SU(Ny) x SU(NI). It seems reasonable to equate these quantities (up to some dimensional constant) ~2). Thus, the d appearing in eqs. (3)-(5) is proportional to the spatial average of the scalar field, (tr), arising in Skyrme model calculations. (This identification with the spatial average of tr is an additional but natural assumption.) This suggests a simple interpretation for the results of fig. 1: At sufficiently high baryon density the quark condensates, (clq), are precisely 0 and chiral symmetry is restored. This identification has important consequences. If eq. (3) is to be used in a description of dense matter, it is necessary that tY is not a constant. Equivalently, one can allow the meson masses to run with density. Thus, in ref. 13), we were led to parameterize the meson masses as Mi = m,(1 + ~//i(o-))

(6)

where i = u, d for pions and i = s for kaons. The parameters Yi are chosen to give the physical pion and kaon masses in the limit of zero baryon density. We emphasize that it is necessary to impose such running meson masses. This physically important feature is not built in by the use of eq. (3) with a constant value of ~. It is at least as important to emphasize that it is neither necessary nor appropriate for f~ to run in a similar manner. As we shall now indicate, the identification of ~ with (~r) is sufficient to ensure the vanishing of the matrix element for pion decay at high baryon density. Pion decay in a baryonic medium is governed by the matrix element of the axial a current, A , ( x ) , M~a b (x) = (0[A~(x)[w b)

(7)

where a and b are isospin indices. Here, [0) is a general vacuum on which the pion is excited as a Goldstone mode. In free space and for a pion of four-momentum q~, M~ b is given in full generality by the form M~b(O) = t8 • ~b[f~q~

(8)

where f,~ is a dimension 1 constant identified as the pion decay constant. Let us express this in the chiral model employing the usual representation of the fields U ( x ) = 1 [tr(x) + i~r- ~ ( x ) ]

(9)

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H. F o r k e l et al. / C h i r a l s y m m e t r y restoration

U*(x) U(x) = 1. The unitarity of U(x) can also be expressed in terms of the vacuum expectation values (cr2(X))o + (~2(X))o =f~.

(10)

We make a mean-field approximation in the scalar sector and regard the pionic excitation as a fluctuation about this mean field. Thus, M~b(O) = (Ol~(O)lO) (O[a~° (0)[~ b)

(11)

from which it follows that

(01o-(0)10) -- (o"(0))o =f~

(O]asr" (O)]'n "b) =

is"bq..

(12)

We wish to extend this result to a non-trivial vacuum such as we encounter in the case of a single baryon o n S3(L) or arrays of skyrmions in R 3. The argument is not rigorous. We assume that eq. (11) remains valid and that the relevant quantity is (o-(0))~ (where 0 denotes the non-trivial vacuum). Precisely as in eq. (3), this term can be expressed in a quark language as a ?:lqtadpole which carries zero momentum. It is a constant in free space. In a non-trivial vacuum it is not. Following the above arguments, it is again reasonable that the correct extension is the average value of or. Thus, pion decay is governed by -a M ,ab (x)=(cr)~(01O,zr ( x ) [ l rb) .

(13)

On S3(L),

(cr)o = f . ~

d/.~ sin"/z c o s f ( / z ) .

(14)

Fig. 1 shows that (o-)~- 0 at the phase transition. By parity, (~)~ is identically 0. Thus, after the phase transition (i.e., in the homogeneous phase) we find (,~)~ + (~)~ -- 0

(15)

which is to be compared with the free space result 2 (or)o+ (~)o2 = f ~2.

(16)

The difference between eqs. (15) and (16) is the effect of interactions (as summarized partially by the curvature of the hypersphere.) These results, particularly eq. (15), should be contrasted with eq. (10) which is always valid. An identical result is obtained for arrays of skyrmions in R 3. The vanishing of the pion decay matrix element is a traditional indication of the restoration of chiral symmetry. The above arguments indicate that a term (~r) enters naturally so that the pion decay matrix element vanishes at high baryon density even for a fixed value off~. We note that (or) is non-zero at all finite baryon densities if chiral symmetry is broken explicitly through the introduction of a pion mass term using eq. (3). Specifically, to leading order in the pion mass, (or) - -m~L4/(2 - L 2)

14. Forkel et al. / Chiral symmetry restoration

823

and vanishes only for L = 0. It is interesting to note the general result that this perturbative estimate, given for L < v/2, breaks down at the phase transition (when L = x/2). The order parameter for non-zero m~, obtained from a complete calculation (i.e., not using this perturbative estimate), is also shown in fig. 1. It is straightforward but complicated to study the energy of small-amplitude fluctuations in (tr, ~t) fields about the S3(L) hedgehog background as a function of the baryon density ~4). Manton studied the softest mode in this problem for L < x/2 where f ( / z ) = / z [ref. ~o)]. In this region, the lowest-energy normal m o d e is the infinitesimal conformal m a p for which 8 f = s i n / z . The energy of this mode is 12rr2(-L + 2/L). The fact that this energy vanishes at L = v/2 indicates that there is a second-order phase transition. As a consequence of the SO(4) symmetry of the identity map, there is no reason to regard o- as special. Thus, when the skyrmion profile is given by the identity map, a similar conformal transformation can be applied to each of the three pion fields and three additional independent modes of equal energy must a p p e a r for each L ~). These (and all other) modes can be characterized by the (x, y, z) parity of tS~. These E1 modes have a parity opposite to that of the original E0 mode. (Our notation for the modes follows Walliser and Eckart ~5).) A similar parity doubling is found for all normal modes. For L > ~ the skyrmion profile is no longer given by the identity map. Constructing the normal modes about the now-localized skyrmion, we find that the E0 mode has positive energy while the triplet of E1 modes remains at energy 0 for all L > ~ . Thus, we find a parity doubling everywhere in the spectrum of non-strange excitations when chiral symmetry is restored. To be more specific, there are two distinct classes of modes ~4). One set of modes has energies proportional to (L + 1/L). The lowest modes of this class are degenerate E1 and M ! modes while the K t h modes are degenerate E1 • • • E K and M1 • • • M K modes. The second class of modes has energies in which the dependences on L and I/L are different. The lowest modes of this class are degenerate E0 and E1 modes the modes discussed in greater detail above. The K t h modes in this set are degenerate E 0 . . . E K modes. Thus, all states have degenerate partners of opposite parity. The behaviour of the lowest E0 mode (the infinitesimal conformal map) and its E1 partners is shown in fig. 2. (These modes have been constructed with the usual assumption of sinusoidal time dependence. In the limit of large L theseeigenvalues can be related to the phase shifts of ref. ~5). There is also a straightforward but non-trivial relation to the static eigenvalue problem considered in ref. ~4).) It is again interesting to see what happens if chiral symmetry is broken explicitly. As noted above, there is no longer a phase transition. The energy of these modes is never zero. At high densities, the soft E0 and E1 modes are split by a term proportional to rn,~ which vanishes in the limit as L-* 0. There is no parity doubling when chiral symmetry is explicitly broken. In order to understand the symmetry properties of this model it is necessary to determine the symmetry groups of the hamiltonian and its B = 1 ground state in the

824

H. Forkel et aL / Chiral symmetry restoration (~L) 2

/El

i

i

i

i

"

1

2

Fig. 2. The energies of the "soft" E0 and (three-fold degenerate) El modes for a single skyrmion on a hypersphere as a function of L with m~ = 0. For L<~x/2 these modes are degenerate. For L~>x/2 the three E1 modes have zero energy and result in the appearance of three Goldstone modes. two phases. Internal and space symmetries of the hamiltonian are summarized by the twelve-dimensional group G = SO(4)x® SO(4)sp~ce- the direct product of the chiral group and the group of transformations of S3 onto itself. The SO(4)space consists of the SO(3) subgroup of rotations around a given fixed point (generated by the usual angular momentum operators, L~) and the transformations by the remaining three generators, Li, spanning the coset SO(4)/SO(3), which we will call translations. The chiral group can be decomposed similarly in the isospin SO(3) subgroup with generators Ii and the coset spanned by the axial generators, L. As in free flat space, the localized hedgehog solution breaks G down to the "grand spin" SO(3) generated by K, = L~ + I,.

(17)

As a result, the nine broken generators give rise to nine zero modes of the hedgehog. These include the three "grooming" M1 modes (generated by Li - Ii) as well as the "translational" E1 modes (generated by ~ ) and the "axial" E1 modes (generated by L)*. The excitation spectrum shows the usual degenerate grand-spin multiplets. The situation changes dramatically with the phase transition. The o--component of the chiral four-vector loses its preferred role, and the three additional generators

/L=E,+L

(18)

now leave the hedgehog invariant. Consequently, there remain only six zero modes corresponding to "grooming" and the generators, (L~-/~). The remaining "pionic" modes acquire a finite energy and are no longer zero modes for all L ~x/2) signalling the appearance o f Goldstone bosons. In the homogeneous phase the characteristic connection by the hedgehog of internal and space symmetries is extended to a coupling of axial transformations * In free space the three modes correspondingto the axial generators are frozen because they change the asymptotic form of the skyrmion for r--,oo (i.e., the B = 0 vacuum). They appear, however, on the compact hypersphere is).

H. Forkel et al. / Chiral symmetry restoration

825

and translations. The phase transition restores the full diagonal SO(4) subgroup of G which contains the grand spin subgroup in the same way that the chiral group contains the isospin. The enlarged symmetry of the ground state is reflected in the appearance of degenerate SO(4) multiplets in the excitation spectrum. After projecting to the physical quantum numbers of the nucleon by quantizing the spurious translational zero modes** as collective coordinates, chiral symmetry will emerge from the diagonal SO(4). This is the precise analogue of the usual restoration of isospin from grand spin through the quantization of rotational modes. Again, the introduction of a pion mass removes the additional degeneracy in the spectrum in the homogenous phase. In the localized phase the energy of the three lowest axial modes is no longer zero but approaches m~ like 1 / L 3. It is to be expected that the appearance of Goldstone bosons at low densities is related to the breaking of the continuous SO(4) symmetry and not to the breaking of the discrete half-skyrmion symmetry. This can easily be demonstrated to be the case by introducing a softer symmetry-breaking. As noted, the usual symmetrybreaking term, proportional to Tr ( U - 1), destroys both the half-skyrmion symmetry and the SO(4) symmetry for all non-zero L. No phase transitions are encountered. However, the introduction of a term of the form Tr ( U 2 - 1), which is consistent with QCD symmetry, does lead to a phase transition and a high-density profile which retains the half-skyrmion symmetry but which does not have the full SO(4) symmetry. The related second-order problem again reveals a "soft" E0 mode, as in fig. 2, which has zero energy at the (second-order) phase transition. However, the associated E1 mode is not degenerate with this E0 mode above the phase transition and does not have zero energy below the phase transition in spite of the breaking of the half-skyrmion symmetry. This suggests the following observations. A soft symmetry breaking can induce the restoration of the half-skyrmion symmetry and give rise to a vanishing (o-) without restoring chiral symmetry. This means that (cr) = 0 is not always the relevant order parameter of chiral phase transitions. Therefore, within the hypersphere framework, it is a necessary condition but not sufficient for the chiral phase transition. On the other hand, the identity map implies a half-skyrmion symmetry (the inverse is not true) and, hence, makes {o-) a relevant order parameter. We believe that the definition of the pion decay constant, eq. (13), in terms of (or) is also subject to this non-uniqueness. A similar parity doubling is found in the SU(3) extension of the Skyrme model on S3(L) [ref. 13)] which follows the work of Callan and Klebanov 16). (In this case we adopt a technically simpler variant of the Skyrme model in which stabilization is provided by a finite-mass to-meson 17).) Fig. 3 shows the lowest energy (p-wave) A and (s-wave) A* excitations around the usual SU(2) skyrmion. We also show (o-)/f,~. In obtaining these results f~ = 62 MeV, m~ = 0, a strange quark mass, ms, ** Note that, even in the phase with homogeneousbaryon and energy density, the solition still breaks translational invariance.

826

H . F o r k e l et al. / C h i r a l s y m m e t r y .

400

.

.

.

1

.

t.o ( MeV )

.

.

restoration .

f

-

/

,o~,

soo

<(3"> IO0

0

i

,

1

2

Fig. 3, The energies of the A and A* modes for a single skyrmion on a hypersphere as a function of L calculated with m~ =0. The order parameter, (o-}, for this variant of the SU(3) Skyrme model is also shown (in arbitrary units). When (o')= 0, the A and A* modes are degenerate.

of 175 MeV and a "running" kaon mass of m K -~ ms(1 + 3/(0-)) were adopted with 3/= 0.0295 MeV -1. At those densities for which (tr) = 0, parity doubling is also found in the strange sector. Several comments are in order. First, the vanishing of the energy of these strangeness excitations is an indication of the onset of kaon condensation. Due to small amplitude approximations for the kaon fields made in both r e f s . |3,16), these calculations cannot be followed into the kaon-condensed region. Second, a similar parity doubling is expected for all strange excitations. As in the case of the non-strange excitations, the origin of this degeneracy lies in the half-skyrmion symmetry and not in the full SO(4) of the identity map: The forms of the A and A* fluctuations transform into each other under interchange of the poles of S 3. When the skyrmion profile transforms into itself under this interchange, the energy of these excitations will be the same. We have set rn~ = 0 to allow for exact chiral symmetry restoration and the consequent A/A* degeneracy. As an unwanted by-product of this choice, the critical density for the transition to the homogeneous phase is lowered considerably, and kaon condensation occurs at a significantly higher density than the chiral phase transition. For the physical pion mass, used in ref. 13), kaons condense in the region of (the now approximate) chiral symmetry restoration. The use of a finite pion mass breaks the A/A* degeneracy by a term proportional to m r2. Our various indicators of chiral symmetry restoration on the hypersphere are all consistent; there are no internal debates as to whether chiral symmetry is broken or restored. This is due in large measure to the important role played by half-skyrmion symmetry. Since it is precisely the half-skyrmion symmetry which is common to the high-density phases in R 3 arrays and single skyrmions o n S 3 ( L ) , many of the present hypersphere results provide an accurate reflection of what happens in real dense matter. It remains to be seen how parity doubling of the non-strange excitations at high density and the appearance of Goldstone bosons at low densities will emerge in calculations with flat space arrays of skyrmions where continuous symmetry equivalent to the SO(4) symmetry in the hypersphere is lacking. Nonetheless, it is our opinion that the existence of Goldstone bosons (i.e., massless pions) in such

H. Forkel et al. / Chiral symmetry restoration

827

systems is virtually inevitable and that (o-) will continue to be an order parameter for chiral symmetry in fiat space arrays. The phase transition described here is of second order. The existence of a transition to a homogeneous phase at high densities is a robust prediction which is largely independent of the details of the model. The order of the transition is far more model dependent. For example, following the arguments given above, one might choose to break chiral symmetry (in the SU(2) sector) through the addition of a term proportional to m~(tr) Tr ( U - 1),

(19)

where rn= is the pion mass in free space and (o-) must now be determined through the solution of a (trivial) integro-differential equation. This self-consistent solution does, indeed, show a transition to the identity map. In this case, however, the transition is of first-order. It is tempting to view the uniform distribution of baryon and energy densities - the loss of identity of individual baryons - as anecdotal evidence that this transition might be related to deconfinement. One should resist this temptation. There are no grounds for identifying the homogeneous phase with the quark-gluon plasma. Indeed, following the arguments of Atick and Witten J9) there is every reason to believe that the mesons of our effective theory (Riemann surfaces in the terminology of string theories) are not the correct degrees of freedom for the description of the q u a r k - g l u o n plasma. Such effective theories must break down at the critical point. Nevertheless, they are in principle valid up to this point. Finally, we note that m a n y models of dense hadronic matter and its phase transitions currently discussed in connection with heavy-ion physics neglect precisely the kinds of medium-dependent effects which a r e included here. By ignoring the consequences of running masses and the degeneracies which emerge naturally on the way to chiral symmetry restoration, many people have been led to significant underestimates of the entropy of " n o r m a l " hadronic matter at high density. This has important consequences, e.g., for estimates of the lifetime of the quark-gluon plasma. One of the implications of chirai symmetry restoration is that a large latent heat is unlikely. We are grateful for discussions with G.E. Brown on questions related to "running" masses in dense matter and with N.S. Manton on questions related to zero-modes. Comments, discussions and criticism for L. Castillejo, H.B. Nielsen, T.H. Hansson, N. Kaiser and I. Zahed are also gratefully noted. References 1) T.H.R. Skyrme, Nucl. Phys. 31 (1962) 556 2) For the original work see: G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552; for a review and further references see: 1. Zahed and G.E. Brown, Phys. Reports 142 (1986) 1;

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3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

18) 19)

H. Forkel et al. / Chiral symmetry restoration

U.-G. Meissner and I. Zahed, Adv. Nucl. Phys. 17 (1986) 143; G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825; G.S. Adkins, in: Chiral solitons, ed. K.F. Liu (World Scientific, Singapore, 1987); U.-G. Meissner, Phys. Reports 161 (1988) 213 For the original work see: A. Jackson, A.D. Jackson and V. Pasquier, Nucl. Phys. A432 (1985) 567; for a review and further references see the report articles under ref. 2) I. Klebanov, Nucl. Phys. B262 (1985) 133 G.E. Brown, A.D. Jackson and E. Wrist, Nucl. Phys. A468 (1987) 137 A.D. Jackson and J.J.M. Verbaarschot, Nucl. Phys. A484 (1988) 419 A. Jackson, A.D. Jackson, J.J.M. Verbaarschot, P.S. Jones and L. Castillejo, Nucl. Phys., to be published A.D. Jackson, A. Wirzba and L. Castillejo, Nucl. Phys. A486 (1988) 634 A.S. Goldhaber and N.S. Manton, Phys. Lett. B198 (1987) 231 N.S. Manton, Commun. Math. Phys. 111 (1987) 469 A.D. Jackson, C. Weiss, A. Wirzba and A. Lande, Nucl. Phys. A494 (1989) 523 J.F. Donoghue and C.R. Nappi, Phys. Lett. B168 (1986) 105 M. Rho, H. Forkel, N.N. Scoccola and A.D. Jackson, to be published A.D. Jackson, N.S. Manton and A. Wirzba, Nucl. Phys. A495 (1989) 499 H. Walliser and G. Eckart, Nucl. Phys. A429 (1984) 514 C.G. Callan and 1. Klebanov, Nucl. Phys. B262 (1985) 365; C.G. Callan, K. Hornbostel and 1. Klebanov, Phys. Lett. B202 (1988) 269 N.N. Scoccola, H. Nadeau, M.A. Nowak and M. Rho, Phys. Lett. B201 (1988) 425; The results of this simple model are corroborated by the more realistic calculations of N.N. Scoccola, D.P. Min, H. Nadeau and M. Rho, Stony Brook preprint N.S. Manton, private communication J.J. Atick and E. Witten, Nucl. Phys. B310 (1988) 291