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NUCLEAR MATTER ASPECTS OF SKYRMIONS∗

arXiv:hep-ph/9211295v1 23 Nov 1992NUCLEAR MATTER ASPECTS OF SKYRMIONS∗ANDREAS WIRZBAInstitut f¨ur Kernphysik, Technische Hochschule Darmstadt, Schloßgartenstr. 9D-W-6100 Darmstadt, GermanyABSTRACTAs an alternative approach to the infinite-array description of dense matter in theSkyrme model, we report about the properties of a single skyrmion on a compact3-sphere of finite radius.

The density of this matter can be increased by decreasingthe hypersphere radius. As in the array calculations one encounters a transition toa distinct high density phase characterized by a delocalization in energy and baryoncharge and by increased symmetries.

We will argue that the high density phase hasto be interpreted as chirally restored one. The arguments are based on the formationof complete chiral multiplets and the vanishing of the pionic massless Goldstonemodes in the high-density fluctuation spectrum.

We show that the restoration ofchiral symmetry is common to any chirally invariant extension of the usual Skyrmemodel - whether via higher-order contact terms or via the introduction of stabilizingvector mesons which act over a finite range.1. IntroductionIn this lecture we will present some studies about the high-density behavior ofbaryonic configurations in the Skyrme model 1 and its variants 2, 3.

These modelsbelong to a class of effective models which treat the baryon stabilization (and hencethe baryon structure) and the interaction between baryons on the same footing. Thereis no principal difference between the stabilization and the interaction mechanism,the models just act in different topological sectors: in the baryon (=winding) numberB = 1 sector for the structure physics 4, 3 and in the B > 1 sector for the interac-tion physics 5.

For B →∞one has a new parameter (the density) at ones disposalin order to tune the interplay between the stabilization and the interaction mecha-nisms. Because of these features the Skyrme model and its variants have the inherentpossibility to allow for radical changes when baryonic matter is compressed to highdensities: the baryons which at low densities are well-separated and clearly definedobjects might completely loose their identity and the baryon matter can become uni-form.

There are at least two distinct ways for investigating the high-density behaviorof Skyrme-type models.The first approach which was pioneered by Igor Klebanov 6 uses infinite periodicarrays of skyrmions with baryon number (winding number) of one per unit cell in realspace-time. The details of the lattice structure and the periodic boundary conditionsto which the mesonic fields are subject can be chosen as to avoid nearest neighborfrustration at low densities.

The aim is to find the classical static field configura-∗Talk at the workshop on “Baryons as Skyrme Solitons”, Siegen, Germany, Sept.27-30, 1992.TH-Darmstadt-Preprint IKDA 92/37 — hep-ph/92112951

tion which minimizes the energy per cell volume for a fixed baryon charge per arraycell. The equations of motion (which are non-linear partial differential equations) aresolved numerically on a lattice grid subject just to two constraints: the choice of thecrystal and the form of the twisted periodic boundary conditions which guarantee theperiodic structure and minimize the frustrations in the field gradients (and thus theenergy) between neighboring cells in the asymptotic low density region.

The bound-ary conditions are then extrapolated without alteration to high densities as well. Atlow densities the skyrmions in the periodic arrays are well-separated, ensembled in aphase of weakly interacting baryons.

With increased density, however, they grow insize as measured by their r.m.s. radius until at a critical density - as first observedby W¨ust, Brown and Jackson 7 - they “melt” to a distinct high-density phase wherethe skyrmions completely lose their identity.

The new phase is characterized by theapproximate uniformity in the baryon as well as the energy density, by the fact thatthe averages of the σ- and π-fields vanish over the cell volume and that there appearsan additional symmetry at the critical density: the “half-skyrmion symmetry” aofGoldhaber and Manton.8The short-comings of these calculations are: they are by fiat of numerical nature,the rotational and translational symmetries are explicitly broken down to discrete onesby the artificial crystalline grid structure. Finally the calculations depend on the formof the crystal and the twisted periodic boundary conditions which are extracted inthe asymptotic low density regime, but nevertheless applied at high densities.

Forinstance, the order of the phase transition can depend on these choices: For thesame simple cubic lattice it is first order for Klebanov’s choice of boundary terms,while it is second order for the modified rectangular boundary terms of Jackson andVerbaarschot 9 where a preferred direction in space is singled out. But the fact thatthere are phase transitions are common for all these calculations - and even the criticaldensities are approximately the same.

Klebanov’s crystal is not of minimal energy, aface-centered cubic arrangement of skyrmions (at low densities) is preferable. Again,as the density increases there is a second-order phase transition and the half-skyrmionsymmetry emerges.10 The crystal of minimal energy is in this higher density phaseand has energy per baryon only of 3.8% above the topological lower bound.

Furthergeneralizations are the inclusion of temperature and the derivation of the equationof state for the Skyrme matter in the case of the simple cubic lattice 11 as well asthe fcc structure 12 by T.S. Walhout.

He has also studied periodic arrays using anω-stabilized variant of the Skyrme model.13Nevertheless, the question might arise whether these phase transitions are lat-tice artifacts or not.Fortunately, there is an alternative approach to study thesame phenomena by replacing the periodic arrays in flat space R3 by few-skyrmion-systems (or even a single skyrmion) on the compact manifold S3(L) pioneered by NickManton.14, 15 The finite baryon number on S3(L) corresponds to a finite baryon den-sity in R3(with infinite baryon number), so that one obtains a model for skyrmionaThis symmetry is characterized by families of planes on which the scalar field, σ is zero. The pionicfields have a reflection symmetry about these planes while the scalar field is reflection antisymmetric.2

matter which is appreciably simpler to study than any lattice model. The density ofthis matter can be increased by decreasing the hypersphere radius L, the radius ofthe 3-sphere in four dimensions.

In the limit L →∞a skyrmion localized on thehypersphere has basically the same properties as an isolated skyrmion in flat spaceor in a periodic array for infinite separation. For a large hypersphere it is still welllocalized and dominated by its stabilization mechanism, but because of the curvatureand finite size effects of the hypersphere it acts as if the tail of another skyrmion (hereof course nothing else than itself seen via the opposite pole of the 3-sphere) is present.By shrinking the hypersphere radius this interaction mechanism will become strongerand stronger.

Finally it can become so strong that the stabilizing mechanism cannotlocalize the skyrmion any longer: the skyrmion will be smeared (=delocalized) overthe whole 3-sphere. The connection between both approaches can be made by iden-tifying the averaged baryon densities: in the periodic array calculation the baryondensity is averaged over the cell volume, on the hypersphere it is given by the ratioB/(2π2L3dim) of the baryon charge B and the (surface-) volume of the 3-sphere interms of the dimensioned radius Ldim.

The S3(L) approach has the advantage that itis technically far simpler than the period array one band that it allows for mathemat-ically rigorous results for the properties of the “high-density” (or “small L”) phaseof a single skyrmion on S3(L). Furthermore neither the (continuous) rotational northe translational symmetry are broken.

The disadvantage is obvious: The physicalspace is not a hypersphere - at least not a small one. Both approaches supplementeach other in a complementary way: the first is realized in normal space-time and forinfinite systems (a precondition for a phase transition), but is technically complicatedand plagued by its crystalline nature, by the ambiguities in choosing the array itselfand the boundary terms; the other is simple and exact, but is set up in an unphysicalworld and limited to finite systems which therefore should be not identified with aninfinite periodic array, but just with one array cell.The talk is organized as follows.

In sect. 2 we will give Manton’s general descriptionof the Skyrme model on the 3-sphere and review some of the B = 1 properties.

Insect. 3 we present arguments why the phase transition has to be interpreted as chiralrestoration.

In sect. 4 we report on the generalization to variants of the Skyrme-modelwhich involve higher-order contact terms or which are stabilized by vector mesons.Sect.

5 contains a discussion about the order of the phase transition. We end the talkwith a short discussion section.2.

The Skyrme Model on the HypersphereAs mentioned there is considerable interest in studying the Skyrme modelL2,4 = fπ24 Tr∂µU†∂µU+ ǫ424 TrhU†∂µU, U†∂νUi2 ,(1)on a 3-dimensional hypersphere of radius L, S3(L).14, 15 We will report here espe-bIn fact most of the calculations can be done analytically or involve at most the numerical task ofsolving ordinary non-linear differential equations.3

cially on the findings of refs.15, 16, 17. In order to get parameterization-independentresults it is useful to relate the quaternion representation U of the Skyrme modelto a cartesian representation {Φα} = (Φ0, Φz, Φx, Φy) with the imposed constraintΦαΦα = 1 as U = Φ0 + i⃗τ · ⃗Φ and to introduce the strain tensor 15Kij≡∂iΦa∂jΦa(with{Φα} = (σ, πz, πx, πy))=−14Tr{U†∂iU, U†∂jU}+,(2)where i and j label the space coordinates.

This is a symmetric 3 × 3 matrix withthree positive eigenvalues which we will denote as λ2a, λ2b, λ2c. Manton 15 has shownthat the λi have a simple geometrical interpretation.

They correspond to the lengthchanges of the images of any orthonormal system in a given space manifold (hereR3 or S3(L)) under the conformal map, U, onto the group manifold, here SU(2) ∼=S3. Therefore the name strain tensor which refers to such a general “rubber-sheet”geometry.

Among the invariants of Kij, three are fundamental cand have a simplegeometric meaning:Tr(K)=λ2a + λ2b + λ2c=P length212n(Tr(K))2 −Tr(K2)o=λ2aλ2b + λ2bλ2c + λ2cλ2a=P area2det(K)=λ2aλ2bλ2c=volume2 . (3)The first one measures the sum of the squared length changes of the mapped or-thonormal frame, the second one the sum of the squared area changes and thethird the squared volume change.With the help of Eq.

(2) it is easy to see thatthese invariants are (modulo normalization factors) the static energy densities of thesecond-order non-linear σ model, (f 2π/2)Tr(∂iU∂iU†), the fourth-order Skyrme term,−(ǫ24/4)Tr[U†∂iU, U†∂jU], and the sixth-order term proportional to the square of thebaryon density (see Eq. (24) in section 4).

The static energy of a general skyrmionconfiguration on the hypersphere has in this language the form dE=ZS3(L) dV (λ2a + λ2b + λ2c + λ2bλ2c + λ2cλ2a + λ2aλ2b)=ZS3(L) dV (λa −λbλc)2 + (cycl. perm.′s) + 6ZS3(L) dV λaλbλc.

(4)The last term is just 12π2B with the baryon (winding) number B, since λaλbλc/(12π2)is the Jacobian of the map S3(L)→S3(1)∼= SU(2) - in other words the baryon (wind-ing) number density. Because of the positive definiteness of the terms in the lastexpression of (4) it is obvious that any skyrmion configuration has to respect thetopological boundE ≥12π2B.

(5)cAll other invariants can be constructed from them.dWe have adopted a dimensionless form of the Skyrme lagrangian.To obtain the dimensionedquantities, one divides the λi’s and multiplies L by 2√2ǫ4/fπ and multiplies E by√2ǫ4fπ.4

Furthermore, one can immediately see that there is exactly one possibility to satisfythe topological bound: The insertion of the identity map, λa = λb = λc, with λi = 1which corresponds to an isometric mapping from the spacial manifold into the targetmanifold and which has baryon (winding) number B = 1. Specifying to the case athand where the target manifold is SU(2) ∼= S3(1), the unit 3-sphere, we see thatthe only way for saturating the topological bound (the absolute minimum of anyskyrmion configuration with a non-zero winding number) is the isometric mapping ofthe spatial S3(L) with L = 1 onto the target iso-spin sphere S3(1).

We can thereforeconclude that the topological bound can nether be saturated by any B > 1 system orby any skyrmion configuration in flat space, R3.eAfter minimizing E[λa, λb, λc] underthe constraintZS3(L) dV λaλbλc = 2π2B(6)Manton 15 and Loss 16 found that for L < 1 there is even a stronger bound on anyB = 1 skyrmion configuration,Eidentity = 6π2L + 1L,(7)which is still satisfied by the identity map λa = λb = λc = 1/L independently of theparameterization of the B = 1 configuration. In other words for L ≤1 the identitymap is the absolute minimum of any B = 1 configuration.Finally, it is now simple to show in full generality 15, 16 that for L <√2 theidentity map is stable for all allowed (δB = 0) small-amplitude perturbations λi →(1/L) + δi(x), whereas it becomes a saddle for L >√2.fUnder the small perturbationgiven above the energy is given to second-order asE[λa, λb, λc] = Eidentity + 2L + 4L3 ZI1 +1 + 2L2 ZI2 + 4L2ZI3 + O(δ3)(8)with I1 = δa + δb + δc, I2 = δ2a + δ2b + δ2c and I3 = δaδb + δbδc + δcδa.

Using theB-constraint (6) we have up to third order corrections the relation1L2ZS3(L) dV I1 = −1LZS3(L) dV I3 + O(δ3).After inserting this relation in Eq. (8) and using the fact that the δi’s are local vari-ations, one can finally find that the identity map is stable against small-amplitudefluctuations, i.e.

E[λa, λb, λc] > Eidentity, as long as the following local inequalityholds1 + 2L2(δ2a + δ2b + δ2c) −2(δaδb + δbδc + δcδa) > 0,(9)eOf course, we cannot exclude the possibility that such a configuration might come infinitesimallyclose to the topological bound.fThat means that the identity map is at least a local minimum for 1 < L <√2.5

in other words as long as L <√2. This concludes Manton’s general proof - indepen-dent of any parameterization - that the identity map is at least a local minimum upto Lc =√2 and becomes a saddle for L >√2.So far we have not specified any parameterization.

In the following we will spe-cialize to the familiar hedgehog parameterization. It has the following properties:(a) according to a general theorem by Palais 18 about reduced variational equationsfor symmetric fields a variational solution of hedgehog form is a solution of the fullset of Euler-Lagrange equations of the Skyrme model, (b) the hedgehog solution isthe energetically lowest solution known so far in the B = 1 sector in R3 and (c)it is numerically proven that the B = 1 hedgehog solution is at least a local min-imum.

Hedgehog configurations on the hypersphere are conveniently described interms of the conventional ‘polar’ coordinates µ, θ, φ on S3(L) with 0 ≤µ, θ ≤π and0 ≤φ ≤2π, so that a typical point on S3(L) has Cartesian coordinates(L cos µ, L sin µ cos θ, L sin µ sin θ cos φ, L sin µ sin θ sin φ). (10)In these coordinates the metric is ds2 = L2(dµ2 + sin2 µ dθ2 + sin2 µ sin2 θ dφ2) andthe volume element isdV = L3 sin2 µ dµ sin θ dθ dφ .

(11)The field components of a hedgehog of baryon number B on S3(L) have then theformΦ0=cos f(µ)Φz=sin f(µ) cos θΦx=sin f(µ) sin θ cos φΦy=sin f(µ) sin θ sin φ,(12)where the “radial” profile function f(µ) is subject to the boundary condition f(0) = 0and f(π) = Bπ, B integer. The topological winding-number B is of course the baryoncharge of the corresponding field configuration.

The field components, fπΦα (fπ isthe pion-decay constant) should be identified with the familiar σ, πz, πx, πy fields.In the normal quaternion representation the hedgehog ansatz readsU ≡Φ0 + i⃗τ · ⃗Φ = exp(i⃗τ · ˆr(θ, φ) f(µ))(13)where ˆr(θ, φ) is the usual radial S2 unit vector. Note that for the hedgehog con-figuration the eigenvalues of the strain tensor are simply λ2a = f ′2/L2, λ2b = λ2c =sin2 f/(L2 sin2 µ) where f ′ stands for dfdµ.The energy of the hedgehog field configuration (13) on S3(L) is gE = 4πLZ π0 sin2 µdµ f ′2 + 2sin2 fsin2 µ!+4π 1LZ π0 sin2 µdµ sin2 fsin2 µ"2f ′2 + sin2 fsin2 µ#!.

(14)gNote in order to obtain the dimensioned quantities, one multiplies L in (14) by 2√2ǫ4/fπ and Eby√2ǫ4fπ.6

Variation of (14) with respect to the radial profile function f(µ) leads us to the Eulerequationd2fdµ2 + sin 2µsin2 µdfdµ −sin 2fsin2 µ+ 2L2sin2 fsin2 µd2fdµ2 +sin 2fL2 sin2 µ dfdµ!2−sin2 fsin2 µ=0 . (15)Again because of Palais’ general theorem 18, solutions of (15) yield solutions of thefull set of Euler-Lagrange equations of the Skyrme model on S3(L) when substitutedinto (12) or (13).

It is very simple to check that one solution of Eq. (15) (in thecase B = 1) is the map f(µ) = µ which corresponds to the uniform mapping ofS3(L) onto the isospin manifold S3(1) ∼= SU(2).

One can easily see that for theuniform hedgehog map, f(µ) = µ, the eigenvalues of the strain tensor (2) are simplyλ2a = λ2b = λ2c = 1/L2. In other words it is a special parameterization of the identitymap.

The uniform map, f(µ) = µ, is a solution for all L, but has no finite L →∞limit. The energy associated with the uniform map is of course given by (7).

Asmentioned for 1 < L <√2, the uniform map is still at least a local minimum. ForL >√2 there exist a lower energy configuration which represents a skyrmion localizedabout one point, i.e.

f(µ) ̸= µ in the hedgehog parameterization (12).15, 17 In factin this parameterization the identity map bifurcates into two solutions of the sameenergy concentrated around the north or south pole of the hypersphere which arerelated by 17fN(µ) = π −fS(π −µ). (16)In the limit L →∞, one recovers the usual flat space skyrmion.17 The identitymap has O(4) symmetry (see Eq.

(12)) and is therefore completely uniform in energyand baryon density. For L >√2 the symmetry of the energetically most favorablesolution is broken to O(3) indicating that the solution is localized in energy andbaryon distribution.

The bifurcation at L =√2 is of the standard “pitchfork” typecorresponding to a second order phase transition. In Landau’s scheme the bifurcationis characterized by a symmetric quartic polynomial whose quadratic term changessign.In order to discriminate the delocalized (high-density) phase from the localized(low-density) phase in a way which can be generalized to few-skyrmion systems onthe hypersphere or to flat space array-calculations, several candidates for an orderparameter were studied in ref.

17: One candidate is the integrated squared deviation ofthe local energy/baryon density from the averaged density normalized by the squaredaveraged density times the hypersphere volume. It converges to the value 1 for thelocalized solution in the limit L →∞, for smaller L it decreases and shows a square-root bifurcation at L =√2 to the value zero belonging to the identity map.

Theinterpretation would be that this order parameter signalled deconfinement in theenergy and baryon distribution. This is, however, a special property of the B = 1system on S3(L).

For few-skyrmion systems 17, 19 and flat space arrays 9, 10 this7

parameter never becomes zero in the high-density phase, although there is still asizable decrease in its value from the low-density to the high-density phase.Thesecond suggestion of ref. 17 on the other hand works also in these cases.

It is thesquared chiral expectation value⟨σ/fπ⟩2 + ⟨⃗π/fπ⟩2,(17)where the σ and pion fields are averaged over the hypersphere volume (or the cellvolume for periodic arrays). Because of the residual O(3) symmetry for the hedgehog(or discrete symmetries in the array calculations) which cause ⟨⃗π⟩to be zero it hasthe same content as the parameter ⟨σ⟩alone.

For the identity map it is obvious(see Eq. (12)) that the parameter (17) vanishes.

In the case of the localized solutionit approaches the value 1 with increasing L, since in most of the space the U fieldof the localized skyrmion is approximately close to the vacuum value U0 = 1 andonly markedly deviates from this in the small region where the skyrmion is located.The interpretation is that a zero in this order parameter signals chiral symmetryrestoration or at least “chiral democracy”, since there is no escape from the “chiral-circle” constraint σ2 + ⃗π2 = f 2π. But nevertheless as shown for few-skyrmion systemson the hypersphere 17, 19 and for flat space arrays 9, 10 the σ and pion profiles ofthe high-density phase are so equally distributed about the chiral circle that theiraverages vanish.Let me summarize the B = 1 results on S3(L): There is a second-order transitionat a baryon density 1/(2π2Lc3) for Lc =√2 from a localized and chirally broken low-density solution to a “chirally restored” high-density solution.

Restoring dimensionfulparameters into the Skyrme lagrangian (by using the empirical value of the pion decayconstant fπ = 93 MeV and an ǫ4 = 0.0743 guaranteeing a reasonable value for gA)one findsρc = fπ2√2ǫ4!31√2312π2 ≈0.20fm−3,(18)which is far too low.h(remember that the nuclear matter density is ρnm = 0.16fm−3).The important result, however, is that the S3(L) value for ρc is more or less thesame than the one found in the periodic array calculations 7, 9, 10 under the sameinput parameters, approximately ρc = 0.17fm−3.Furthermore, the correspondingtransition densities ρc in the few-skyrmion calculation 17, 19 interpolate between thesetwo numbers. Thus the identification of the hypersphere formalism with the periodicarray calculation seems to be justified even quantitatively.hThis might improve if the stabilization term is replaced by more realistic ones.

Furthermore allcontributions from kinetic terms are neglected. Finally, there is the problem of the missing centralattraction 5 which is probably linked to loop corrections 20 or missing 1/Nc corrections 21.

Anyadditional attractive term tends to localize the skyrmion and will therefore move the transition pointup to higher densities.8

3. Indications for a Chiral Symmetry RestorationIn this section I will present some arguments (see refs.

22, 23 for more details)why the transition reviewed in the preceeding section should be interpreted as chiralsymmetry restoration.As mentioned the complete uniformity of the high-densityB = 1 solution on S3(L) is linked to the strong condition f(µ) = µ and does notgeneralize to few-skyrmion systems 17, 19 on S3(L) or periodic arrays 7, 9, 10 in flatspace. However, the “chiral democracy” as discussed at the end of the preceedingsection can even be achieved via a weaker conditionf(µ) = π −f(π −µ)(19)which indicates that in the high-density phase there is a symmetry about the hyper-sphere equator between the “northern” and the “southern” hemisphere.

In fact thisis just the half-skyrmion symmetry 8 which is a common signal of the high-densityphase for few-skyrmion systems on the hypersphere 17, 19 as well as for periodic ar-rays in flat space 8, 24.So the chiral symmetry restoration seems to be prior tothe delocalization. In fact this can be tested by adding a (pion mass) term whichexplicitly breaks chiral symmetry:22LSB = mπ2f 2π4Tr(U + U† −2).

(20)When this term is added, the phase transition is not a sharp one any longer, butit is smeared out. The chiral order parameter (17) still approaches zero for higherand higher densities, but never actually becomes exactly zero.

Of course, this is justa consequence of the explicit breaking of the chiral symmetry. A second effect ofadding the term (20) is a small shift of the now approximate phase transition tohigher densities, since the term induces an attractive force.In ref.

22 it is furthermore argued that the vanishing of the order parameter ⟨σ⟩(for the Skyrme model without explicit symmetry breaking of course) implies thevanishing of the matrix element for pion decay ⟨0|Aaµ(x)|πb⟩in case the axial currentAaµ is calculated in the mean-field approximation in the frame-work of a σ-model,and therefore the vanishing of the “effective” pion decay constant and the quarkcondensate.The most convincing arguments why the phase transitions in the Skyrme modelought to be identified with a chiral symmetry restoration are the following two: Inthe B = 1 case on S3(L) it can be shown that the transition from the low-densityphase to the high-density phase is accompanied(i) by the formation of complete chiral multiplets in the fluctuation spectrum(ii) and by the vanishing of the three pionic Goldstone modes from the spectrum.These two points are treated at length in ref. 23 where the local stability of theuniform and the localized skyrmion solution on S3(L) and thus their small-amplitudenormal modes are investigated.

If all normal modes have positive energy, the solution9

is locally stable. If some normal modes have negative energy, the solution is a saddlepoint.

The study of the small-amplitude fluctuations about the B = 1 solutions onthe hypersphere revealed the following scenario:(i) In general, the modes about a hedgehog solution have just the usual O(3) de-generacies, i.e. a degeneracy 2N + 1 in terms of the principal quantum numberN ≥1 of the modes.

(ii) The modes about the identity map on the other hand can be classified accordingto the following representations of the group O(4): the symmetric tensor repre-sentation (N, 0) which has (N + 1)2 degenerate states and as static eigenvalues iλN,0 = L {N(N + 2) −4} + 2 {N(N + 2) −2} /L,(21)and the (N, 1) representation with a 2((N + 1)2 −1) degeneracy and the staticeigenvaluesλN,1 = {N(N + 2) −3} (L + 1/L). (22)Thus their degeneracy exceeds by far the one of the modes about localizedhedgehog solutions.

Furthermore for each mode about the identity map with agiven parity, there exists at least one degenerate mode of opposite parity. Usingthe covering group SU(2)L × SU(2)R of SO(4) it can be shown (see ref.

23)that the (N, 0) modes belong to a trajectory based on the four-fold degenerate(1/2, 1/2) representations of SU(2)L × SU(2)R, while the (N, 1) modes belongto a trajectory based on the six-fold degenerate (1, 0) + (0, 1) representation.Thus the modes form complete multiplets of chiral symmetry either by theadditional (σ,⃗π) degeneracy of the (N, 0) modes (21) or by the parity doublingof the (N, 1) modes (22). Note that in ref.

23 the (N, 0) modes were identified aspurely “electric” grand-spin modes (but with a degeneracy between e.g. grandspin K = 0+ and K = 1−modes), while the (N, 1) modes were identified asdegenerate “magnetic” and “electric” modes of the same grand spin K, butopposite parity.

The (N = 1, 1) modes are the sixth zero modes about theidentity map. (iii) At high densities (L <√2) all normal modes about the identity map havepositive energies, the identity map is stable.

There are four degenerate low-lying modes in the spectrum, the four (N = 1, 0) modes, one with σ quantumnumbers and three with pion ones indicating that the σ and all three π-fieldsare degenerate and treated on the same footing. They have the static eigenvalueλN=1,0 = −L + 2/L.

(23)(iv) The four (N = 1, 0) modes cross zero at L =√2 and become negative forL >√2 signalling there the instability of the identity map.iWe use the phrase “static eigenvalue” as a short-hand notation for the change of the static energyunder the small perturbation of the fluctuation-mode.10

(v) At L =√2 there is the bifurcation into the normal localized hedgehog solutionwith the reduced O(3) symmetry. All modes about the localized solution arestable for L ≥√2.

The modes with N ≥1 have (with the exception of someaccidental degeneracies for special values of L) only the usual O(3) degeneracy2N + 1. However, there exist 9 zero modes.jThe three additional zero modes(compared with the number of zero modes for the identity map) correspond tothe three unstable pionic (N = 1, 0) modes of the identity map.

Their fourthpartner, the σ-like mode, corresponds to a positive stable excitation about thelocalized hedgehog (to an infinitesimally small conformal variation in the profilefunction f(µ) →f(µ) + δ(µ)). The breakdown of the continuous symmetryO(4) of the identity map at L =√2 to the O(3) symmetry of the localized forL >√2 energetically favorable hedgehog solution is therefore closely linked tothe appearance of three new zero modes.

(vi) These three zero modes should be in fact interpreted as the three pionic Gold-stone modes linked to the spontaneous breakdown of chiral symmetry.Thearguments for this identification are the following: Whereas six of the nine zeromodes stay normalizable even in the limit L →∞, three have no finite L →∞limit. This is consistent with the fact that the flat space hedgehog has onlysix zero modes.

Furthermore when the chiral symmetry breaking term (20) isadded to the Skyrme lagrangian, the above mentioned normalizable zero modesstay zero modes, whereas the three additional zero modes are shifted in energyby the pion mass. These are the three pionic Goldstone modes which becomemassive when an explicitly chiral symmetry breaking term is added and whichshow up as non-normalizable plain wave excitations of the vacuum in the flatspace limit.Since the three Goldstone modes disappear for the high-density delocalized phaseand the high-density modes form complete chiral multiplets, the chiral symmetryrestoration is established.4.

Generalizations4.1. Contact TermsIn the following we will show that the form of the Skyrme model (1) and especiallythe stabilization by the fourth-order Skyrme term is not a necessary preconditionfor the very existence of the chiral phase transition.Let us for instance replacethe fourth-order stabilizing term in (1) by a stabilizing term of sixth-order (see e.g.ref.

25)L2,6 = fπ24 Tr∂µU†∂µU−c6BµBµ(24)jNote that a hedgehog in flat space has only 6 zero modes: 3 rotational (which cannot be separatedfrom iso-rotational ones because of the hedgehog symmetry) and 3 translation ones.11

(note c6 > 0) where Bµ is the topological baryon currentBµ = εµναβ24π2 Tr(U†∂νU) (U†∂αU) (U†∂βU). (25)As reported in section 2 the fourth-order Skyrme term as well as the sixth-order termand the second-order non-linear sigma model term have a geometrical meaning whenexpressed in the strain tensor language.

One can apply therefore Manton’s generalmachinery 15 to show that the sixth-order term allows for the identity map as asolution and to construct the critical value Lc where the identity map of the model(24) becomes unstable and bifurcates into a localized solution. In fact all qualitativephenomena show up as before.

Naturally the value of Lmin for the minimum of thestatic energy is in general not the same for both models and more importantly thecritical hypersphere radius Lc where the chiral restoration occurs isLc = (3)14Lmin(26)for the model (24) instead of Lc =√2Lmin as it was the case for the usual Skyrmemodel (1).17As shown in ref. 26 any lagrangian which (a) is a polynomial of the fundamentalinvariants (3) of the strain tensor, (b) which has the usual vacuum properties and(c) which allows for (locally) stable B = 1 Skyrmion solutions, leads to a chirallyrestored phase at high densities.

In addition the position of the phase transition canbe uniquely derived just from the lagrangian without solving any equation of motion.Let us discuss for this purpose a lagrangian of the analytic formLG ≡−G(−L2, L4, L6) = −∞Xj1, j2, j3=0Aj1j2j3 (−L2)j1(−L4)j2(−L6)j3(27)where j1, j2, j3 are integer indices and Aj1j2j3 Taylor coefficients. The arguments ofthe analytic function G and the signs are chosen in such a way, that the correspondingstatic energy density has the simple formE ≡G(E2, E4, E6) =∞Xj1, j2, j3=0Aj1j2j3 E2j1E4j2E6j3.

(28)The function G should be of course positive definite to ensure a positive energy densityand G(0, 0, 0) ≡0 (e.g., A000 = 0) to ensure triviality in the B = 0 sector. By insertingf = 0 or f ′ = 0 one furthermore learns that G(x, 0, 0) ≥0 and G(x, x2/4, 0) ≥0 forany real x ≥0.

The first inequality signals that a possible negative term expressedsolely by the second-order term can nether be compensated by adding fourth- orsixth-order terms in any combination to the static energy density. Therefore Skyrme-type models which at fourth-order are not positive definite cannot be saved by theaddition of a sixth-order term proportional to BµBµ or any power of this term.Note that the energy density (28) is the most general symmetric function of theeigenvalues, λ2i , of the strain tensor Kij (2) and includes all forms which can come12

form lagrangians which involve arbitrary combinations of powers of first derivativesof the fields to even order.In the chirally restored (uniform) regime on the hypersphere the energy densitiesof the second-, fourth- and sixth-order term, E2, E4 and E6, have the simple form(modulo a prefactor which can be incorporated into the Taylor coefficients A(j1j2j3))E2i = 3L2i ,i = 1, 2, 3. (29)In ref.

26 the eigenvalues of all possible small amplitude (static) perturbations aroundthe identity map were obtained as the following analytical result:λGN,a =Xi=1,2,3λi,N,a∂∂E2i+ λ3,N,a2L23 ∂∂E2+ 2L2∂∂E4+ 3L4∂∂E6!2G(E2, E4, E6)(30)withλi,N,a = iN(N + 2) + a2 −4 −i −12{a(N + 1)}2 −4L3−2i. (31)where the indices N ≥1 and a = 0, 1 characterize the allowed O(4) representations:k(N, a = 0), the symmetric tensor representations which have (N + 1)2 degeneratestates and the (N, a = 1) representations which have a 2{(N + 1)2 −1} degeneracy(see refs.

23, 26 for further details). Note that in the case a = 1 the expression forλGN,a=1 simplifies toλGN,a=1 = L{N(N + 2) −3} ∂∂E2+ 1L2∂∂E4!G(E2, E4, E6).

(32)A negative value for any eigenvalue λN,1 in (32) would mean that infinitely many ofthe (N, 1) modes would pass through zero energy simultaneously. This is probablyindicative of a non-perturbative configuration of lower dimension, e.g., a string-likeconfiguration.

Fortunately, one can exclude this pathological instability, since it vio-lates the constraint that the model (27) should have the same vacuum properties asthe nonlinear σ model (see ref.26 for a discussion about this point). So the (N, 1) fluc-tuations cannot lead to an instability.

In case the identity map is (locally) stable inthe high-density region (i.e. all λGN,a ≥0), the monotonic increase of the eigenvaluesλGN,a of the small-amplitude normal modes with increasing N is guaranteed.

Then,the critical Lc where the identity map becomes unstable against small-amplitude per-turbations is given by the value of L where the first λGN,0 in Eq. (30) becomes negative.Finally, letG(E2, E4, E6) →E2for E2i →0(33)andG(E2, E4, E6) →C1E22 + C2E4for E2i →∞(34)kSee section 3 for the corresponding results for the usual Skyrme model (1).13

where i runs from 1 to 3 and C1 ≥0 and C2 > 0 are fixed positive coefficients. Thecondition (33) enforces that asymptotically for low densities (large L) the lagrangian(27) is dominated by the non-linear sigma model lagrangian, whereas for high densities(small L) it scales as a free Fermi gas (i.e.

EG ∝1/L4). The low density behavioris of course indisputable, the high density constraint on the other hand requires theadditional input that such effective models can be applied even for densities wherethe underlying theory, QCD, becomes to leading order a free Fermi gas.Back to the stability analysis: Note that the signs of the static eigenvalues λGN,0 andtherefore the stability of the uniform regime follow from the signs of the derivativesacting on G and from the signs of the coefficients λi,N,0 (31).

From all possible λi,N,awith i = 1, 2, 3, N ≥1 and a = 0, 1 only the term λi=1,N=1,a=0 is negative, all theother coefficients are guaranteed to be larger than zero or at most equal zero. (To thelatter category belong the six O(4) zero modes with N = 1 and a = 1.

Furthermoreall the terms λi=3,N,1 are zero indicating that the sixth-order term itself has infinitelymany zero modes.) Now taking the high-density (34) and the low-density behavior(33) into account, we can conclude that at high densities the uniform B = 1 skyrmionsolution on the hypersphere is bound to be stable whereas at sufficiently low densities,where the non-linear σ model term becomes the dominant one, the uniform solutionis unstable since λGN=1,0 becomes negative eventually.

Furthermore we have to takeinto account that in the stable regime - even infinitesimally close to the instability -all λGN,0 have to increase monotonically with N, that the λGN,1 modes have to be stableby fiat and that the eigenvalues of the modes have to be smooth functions of L. Thenthe existence of a critical value of L is guaranteed where λG1,0 becomes negative suchthat the uniform high density phase becomes unstable. There is a bifurcation fromthe uniform solution which is chirally restored to the usual, localized and chirallybroken B = 1 hedgehog solution.

So even for the most general geometric lagrangian(27) built form first order field derivatives the existence of the chiral phase transition(as discussed at length in the preceeding chapters for the normal Skyrme model)is guaranteed under the assumption of a few reasonable constraints (the vacuumstructure, Eqs. (33) and (34)) on the form of the lagrangian.4.2.

Vector Meson StabilizationWe have discussed so far lagrangians which ensured the stabilization of skyrmionsby higher-order (contact) terms in the field derivatives. Naturally the question ariseswhether the existence of the chiral phase transition is linked to the stabilization bycontact terms or whether vector mesons can be added which act over a finite range,as e.g.

ω (and ρ) mesons. Let us take as an example a Skyrme-type variant withω meson stabilization.

(See ref.13 for a periodic array study of this model.) Thecorresponding lagrangian has the structure 27L2,ω = f 2π4 Tr(∂µU†∂µU) −14ωµνωµν + m2ω2 ωµωµ + gωωµBµ(35)where the first part is the usual non-linear σ model term, the second one the ω kineticterm expressed through the ω field-strength tensor ωµν = ∂µων −∂νωµ, the third part14

is the ω mass term and the last one is the coupling of the ω field to the topologicalbaryon current (25). As shown by Adami 28 the lagrangian (35) has stable B = 1solitons.By putting an ω-stabilized B = 1 hedgehog on the hypersphere one can easily findthat one solution of the equations of motion is always the uniform solution f(µ) = µfor the hedgehog profile andω0 = −gωm2ωB0 = −gωm2ω12π2L3(36)for the ω0-component.

The ω0-component is in this case spatially constant and de-pends only on the hypersphere radius L. (As in flat space the spatial componentsωi are identically zero because of the static hedgehog form of the soliton profile. )When the uniform solutions f(µ) = µ and (36) are reinserted into the correspondingenergy density of the lagrangian (35), one recovers the same structure as for a modelstabilized by a sixth-order term with L6 = −(g2ω/2m2ω)BµBµ.

This should not comeas a surprise, since in the limit mω, gω →∞, under a constant ratio gω/mω, the la-grangian (35) converges to the lagrangian (24) with c6 = (g2ω/2m2ω). Both lagrangianshave the same high-density phase in the static sector.

The physical reason for theidentical behavior of the static solutions of both lagrangians is that at high densi-ties it is energetically favorable that the ω0-component becomes spatially constant,otherwise the “costs” in energy from a finite spatial gradient term in this componentbecomes higher and higher with increasing density. This together with the equationof motion for the ω0-component necessarily lead to Eq.

(36) which in turn guaranteesthe identical high-density behavior of both lagrangians. However, the ω-stabilizedmodel bifurcates at a smaller Lc and the low-density behavior of both lagrangians isdifferent.

Under increasing parameters gω and mω with the ratio gω/mω kept constantthe critical density and the low density phase of the ω stabilized model will approachthe corresponding quantities of the model (24). So the inclusion of finite range vectormesons leads to an increase in the transition density in comparison to the correspond-ing contact-term model, see also ref.13.

In summary, also the ω-stabilized Skyrmemodel has a critical hypersphere radius Lc where the uniform (chirally restored) phaseon the hypersphere becomes unstable and bifurcates into a localized (chirally broken)phase which has the usual B = 1 hedgehog as flat space limit.A similar behavior follows when the L2,ω model is extended by the introductionof ρ mesons. This can be done in different ways.

For our purposes the only essen-tial precondition is that the total lagrangian is still explicitly chiral symmetric. Forsimplicity let us consider in the following the hidden ρ meson coupling `a la Bando etal.

29 in a minimal way 30, i.e.L2,ω,ρ=f 2π4 Tr(∂µU†∂µU) −af 2π4 Tr(ξ†∂µξ −ξ∂µξ† −2igρµ)2 −14ρµνρµν−14ωµνωµν + m2ω2 ωµωµ + gωωµBµ + (ωρπ coupling terms)(37)15

with ξ ≡√U, and ρµν = ∂µρν −∂νρµ −ig[ρµ, ρν], the non-abelian field-strengthtensor. The second term in (37) is responsible for the generation of the ρ mass andthe ρππ coupling with a coupling constant g. (The standard choice for the parametera is a = 2, see ref.

29.) Note that under the static hedgehog ansatz only the spatialρi components are excited.

Again it is easy to show that the equations of motion ofthis model are satisfied by the uniform profilesf(µ)=µ(38)ω0=−gωm2ωB0(39)ρi=12ig(ξ†∂iξ −ξ∂iξ†)with ξ = exp(iτ · ˆrµ/2). (40)The last equation guarantees that the second term in (37) vanishes and that the ρkinetic term behaves as fourth-order Skyrme term with a coefficient ǫ24 = 1/(8g2).

Thelagrangian (37) (without extra ωρπ couplings) has therefore the same high densitybehavior as the simple contact lagrangianL2,4,6 = f 2π4 Tr(∂µU†∂µU) +132g2Tr[U†∂µU, U†∂νU]2 −12 gωmω2BµBµ. (41)This behavior presented here for the simplest ωρπ-model is generic – just the co-efficient of the sixth-order term in (41) may change when ωρπ-coupling terms areintroduced.

For all models with vector-meson stabilization which have been studiedso far 2, 31, 32 the following is true: It can be numerically shown that(i) the only existing B = 1 solution at high densities is the uniform (delocalized)hedgehog one, the static solutions are identical to those of contact-term la-grangians with suitable coefficients and the static vector meson fields are equalto their driving pionic currents (see e.g. (36) or (40)),(ii) there exists a critical model-dependent hypersphere-radius Lc where the uniformsolution becomes unstable and bifurcates into the usual localized configurationand(iii) at low densities the solutions on the hypersphere S3(L) converge asymptoticallyto the corresponding solutions in the flat space.The essential conditions on these models are two-fold: they should be explicitly chiralsymmetric and they should allow for locally stable B = 1 hedgehog solitons.Incase these two conditions are met, there will be necessarily a chiral phase transitionat high densities with the same features as already discussed.

Still, under realisticvalues for the mesonic input parameters (e.g. fπ, ǫ4, c6, see ref.

25, or gω and g, seerefs. 30, 32, 2)) the transition densities ρc come out systematically too low, ρc ≈ρnm(0.16 fm−3).

Apparently these models lack terms which generate additional attraction.Note that the hedgehog mass under such realistic parameters is systematically toohigh (∼1.6 GeV) whereas there is not sufficient central attraction in the B = 216

system.5 If one would tune the stabilizing parameters – while keeping fπ = 93MeVfixed – such that the classical hedgehog mass would be 0.87 GeV (consistent with anucleon mass of 0.94 GeV), then the transition density ρc would be approximatelythree times the nuclear matter density.5. Second- Versus First-OrderWhereas in the periodic array calculations second as well as first order phasetransitions were found depending on the choice of the lattice and the twisted boundaryconditions, we have so far encountered only second order transitions between thedelocalized and the localized phase on the 3-sphere.

This does not need to be thecase in general: All the Skyrme-type lagrangians presented are explicitly not scaleinvariant and do not “know” (yet) about the trace anomaly in QCD. Let us thereforetry to incorporate the same scaling behavior as in QCD into these lagrangians (seerefs.

33, 34, 35), e.g. consider the lagrangianL2,4,χ=χ2χ20f 2π4 Tr(∂µU†∂µU) + ǫ244 Tr[U†∂µU, U†∂νU]2 + 12(∂µχ)2−BB1 + χχ0!4log χ4eχ04!(42)as simplest extension of the usual Skyrme lagrangian.

The scalar field χ with thevacuum expectation value χ0 = ⟨0|χ|0⟩is introduced with the purpose of making thefirst term in (42) scale invariant (the Skyrme term and the χ-kinetic term are alreadyscale invariant), whereas the last term in (42), the χ-potential term, is adjusted tofit the trace anomaly of QCD. BB is the “bag constant” which can be expressedin terms of the gluon condensate as BB = (9/32)⟨0|(αs/π)G2|0⟩.

For values of theparameters χ0 and BB see ref. 34.

(Note that the fluctuations of the χ-field correspondto glueball-excitations.) In ref.

36 it was numerically shown that the lagrangian (42)again possesses a phase transition to a uniform delocalized solution at high densitieswith f(µ) = µ and a constant χ profile which – depending on the choice of parameters– can be even χ = 0. Because of the insufficient accuracy in the numerics, the authorsof ref.

36 missed the fact that the phase transition is of first order. The latter is moreor less obvious from the form of the χ-potential term which is adjusted to be minimalfor χ = χ0.

Thus it is impossible to find a smooth transition between the constantχ < χ0 (high-density) profile and the localized (low density) χ profile (which at thesouth pole on the 3-sphere is exactly χ0) without violating the χ equation of motion:χ′′ + 2cos µsin µ χ′ −χχ20f 2πL2 f ′2 + 2sin2 fsin2 µ!−4BBχ3χ40log χ4χ40!= 0. (43)For a non-vanishing nonlinear sigma term (as in the present case since f(µ) ̸= 0)the χ field cannot both be constant and equal to χ0.

So the χ field can only do thetransition from the low- to the high-density phase and vice versa by a jump. Thus by17

incorporating a scalar field `a la Schechter 33 in the Skyrme model (or its extensions)the second order phase transition can be easily changed to a first order one. See alsoref.

22 for a different mechanism to achieve the same thing.It is rather easy to extend the lagrangian (27) to the most general with the QCDtrace anomaly consistent geometrical form involving only first derivatives to evenorder:LG,χ=− χχ0!4G −(χ0χ )2L2, −(χ0χ )4L4, −(χ0χ )6L6!+ 12(∂µχ)2 −BB1 + χχ0!4log χ4eχ04!. (44)Note that the restoration of scale invariance, χ = 0, is only consistent with thelagrangian (44) if G has the asymptotics (34), i.e.

χχ0!4G −(χ0χ )2L2, −(χ0χ )4L4, −(χ0χ )6L6!→C1(−L2)2 + C2(−L4)if χ →0. (45)We know from what was said before that one can expect in this case a first-order phasetransition to a chirally restored phase.

However, the χ-profile in the high-densityphase does not need to become zero, it can equally well be just a constant χ < χ0.Scale invariance implies chiral restoration, but not vice versa. Nevertheless, just thepossibility to have a scale invariant limit at high densities leads to strong constraintson the form of the effective model.

If a lagrangian is considered which includes anyterm higher than fourth-order in the derivatives of the pion fields, terms of all ordershave to be included as well both to satisfy the high-density asymptotic behavior (45)and to ensure the stability of the chiral symmetric high-density phase.26 The necessityfor infinitely many terms is consistent with the large Nc philosophy.37, 38 Because ofasymptotic freedom, the effective model which follows from QCD to leading order inthe 1/Nc expansion must involve infinitely many mesons. The hope is – as shownfor the ω and ρ meson case – that the high density phase of this infinite tower ofmeson resonances can be approximated by infinitely many suitable contact terms.This might lead to “dreams” of Regge trajectories.6.

DiscussionEven effective models of the Skyrme class (which do not possess any quark degreesof freedom) can have a phase transitions to a high-density phase with restored chiralsymmetry. There are essentially just two conditions on such a model: It must have therelevant symmetries (especially the lagrangian must be explicitly chirally invariant)and it must treat the baryon structure and interaction on the same footing.

Normally,the phase transitions are of second order, but one can easily change them to first-order ones by incorporating the QCD trace anomaly for instance. This allows also -depending on the input parameters - for a scale invariant high density limit.

Scaleinvariance implies chiral symmetry restoration, but not vice versa.18

Under the same parameter input (from the meson sector) the periodic array cal-culations as well as the 3-sphere calculations give approximately the same values forthe transition densities. The predictions, however, are for both systematically toolow, ρc ≈ρnm.

This problem may be connected with two other deficiencies whichplague the Skyrme-like models: the missing central attraction in the B = 2 channeland the predicted high value for the hedgehog mass. Quantum (higher loop or higher1/Nc) corrections may have a chance to cure all three problems at the same time byproviding extra non-local terms which generate attraction.AcknowledgementsThe author would like to thank A.D. Jackson, L. Castillejo and C. Weiss for usefuldiscussions.References[1] T.H.R.

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