Skyrmion approach to finite density and temperature
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Chapter 1Skyrmion Approach to finite density and temperature
Byung-Yoon Park and Vicente Vento Department of Physics, Chungnam National UniversityDaejon 305-764, [email protected] Departament de Fisica Te`orica and Institut de F´ısica CorpuscularUniversitat de Val`encia and Consejo Superior de Investigaciones Cient´ıficasE-46100 Burjassot (Val`encia), [email protected]
We review an approach, developed over the past few years, to describe hadronicmatter at finite density and temperature, whose underlying theoretical frameworkis the Skyrme model, an effective low energy theory rooted in large N c QCD. Inthis approach matter is described by various crystal structures of skyrmions,classical topological solitons carrying baryon number, from which conventionalbaryons appear by quantization. Chiral and scale symmetries play a crucial rolein the dynamics as described by pion, dilaton and vector meson degrees of free-dom. When compressed or heated skyrmion matter describes a rich phase dia-gram which has strong connections with the confinement/deconfinement phasetransition.
An important issue at present is to understand the properties of hadronic matterunder extreme conditions, e.g., at high temperature as in relativistic heavy-ionphysics and/or at high density as in compact stars. The phase diagram of hadronicmatter turns out richer than what has been predicted by perturbative QuantumChromodynamics (QCD). Two approaches have been developed thus far to discussthis issue : on the one hand, Lattice QCD which deals directly with quark and gluondegrees of freedom, and on the other, effective field theories which are described interms of hadronic fields. We shall describe in here a formalism for the secondapproach based on the topological soliton description of hadronic matter firstlyintroduced by Skyrme.
Lattice QCD, the main computational tool accessible to highly nonperturba-tive QCD, has provided much information on the the finite temperature transition,such as the value of the critical temperature, the type of equation of state, etc. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
However, due to a notorious ‘sign problem’, lattice QCD could not be applied tostudy dense matter. Only in the last few years, it has become possible to simu-late QCD with small baryon density. Chiral symmetry is a flavor symmetry ofQCD which plays an essential role in hadronic physics. At low temperatures anddensities it is spontaneously broken leading to the existence of the pion. Latticestudies seem to imply that chiral symmetry is restored in the high temperatureand/or high baryon density phases and that it may go hand-in-hand with the con-finement/deconfinement transition. The quark condensate h ¯ qq i of QCD is an orderparameter of this symmetry and decreases to zero when the symmetry is restored.The Skyrme model, is an effective low energy theory rooted in large N c QCD, which we have applied to dense and hot matter studies.
The model does not haveexplicit quark and gluon degrees of freedom, and therefore one can not investigatethe confinement/deconfinement transition directly, but we may study the chiralsymmetry restoration transition which occurs close by. The schemes which aimat approaching the phase transition from the hadronic side are labelled ‘bottomup’ schemes. The main ingredient associated with chiral symmetry is the pion,the Goldstone boson associated with the spontaneously broken phase. The variouspatterns in which the symmetry is realized in QCD will be directly reflected inthe in-medium properties of the pion and consequently in the properties of theskyrmions made of it.The most essential ingredients of the Skyrme model are the pions, Goldstonebosons associated with the spontaneous breakdown of chiral symmetry. Baryonsarise as topological solitons of the meson Lagrangian. The pion Lagrangian canbe realized non-linearly as U = exp( i~τ · ~π/f π ), which transforms as U → g L U g † R under the global chiral transformations SU L ( N f ) × SU L ( N f ); g L ∈ SU L ( N f ) and g R ∈ SU R ( N f ). Hereafter, we will restrict our consideration to N f = 2. In the caseof N f = 2, the meson field π represents three pions as ~τ · ~π = (cid:18) π √ π + √ π − − π (cid:19) . (1.1)The Lagrangian for their dynamics can be expanded in powers of the right andleft invariant currents R µ = U ∂ µ U † and L µ = U † ∂ µ U , which transforms as R µ → g L R µ g † L and L µ → g R L µ g † R . The lowest order term is L σ = f π ∂ µ U † ∂ µ U ) . (1.2)Here, f π = 93 MeV is the pion decay constant.Throughout this paper, we take the following convention for the indices: (i) a, b, · · · = 1 , , i, j, · · · = 1 , , µ, ν, · · · =0 , , , α, β, · · · = 0 , , , ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature In the next order, one may find three independent terms consistent with Lorentzinvariance, parity and G -parity as L = α tr[ L µ , L ν ] + β tr { L µ , L ν } + + γ tr( ∂ µ L ν ) . (1.3)In his original work, Skyrme introduced only the first term to be denoted as L sk = 132 e tr[ L µ , L ν ] , (1.4)which it is still second order in the time derivatives. The value of the “Skyrmeparameter” may be evaluated by using ππ data. In the Skyrme model, it is alsodetermined, for example, as e = 5 . to fit the nucleon-Delta masses, or as e = 4 . to fit the axial coupling constant of nucleon.One may build up higher order terms with more and more phenomenologicalparameters. However, this naive derivative expansion leads to a Lagrangian whichhas an excessive symmetry; that is, it is invariant under U ↔ U † , which is not agenuine symmetry of QCD. To break it, we need the Wess-Zumino-Witten term. The corresponding action can be written locally as S W ZW = − iN c π Z d xε µνλρσ tr( L µ L ν · · · L σ ) , (1.5)i.e. in a five dimensional space whose boundary is the ordinary space and time. For N f = 2 this action vanishes trivially, but for N f = 3 it provides a hypothesizedprocess KK → π + π π − . When the action is U (1) gauged for the pions to interactwith photons, this term plays a nontrivial role even with two flavors.Chiral symmetry is explicitly broken by the quark masses, which provides themasses to the Goldstone bosons. The mass term can be incorporated in the sameway as chiral symmetry is broken in QCD; that is, L m = f π m π U + U † − ∼ − h ¯ qq i M ( U + U † − , (1.6)where M = (cid:18) m u m d (cid:19) . (1.7)We neglect the u- and d-quark mass difference.The approach has been generalized to more sophisticated meson Lagrangianswhich are constructed by implementing the symmetries of QCD. The scale dilatonhas been incorporated into the effective scheme to describe in hadronic languagethe scale anomaly.
The vector mesons ρ and ω with masses m ρ,ω ∼
780 MeVcan be incorporated into the Lagrangian by using the hidden local symmetry (HLS) and guided by the matching of this framework to QCD in what is called ‘vectormanifestation’ (VM). We shall discuss these generalizations, when required in thediscussion of skyrmion matter, later on.The classical nature of skyrmions enables us to construct a dense system quiteconveniently by putting more and more skyrmions into a given volume. Then, ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento skyrmions shape and arrange themselves to minimize the energy of the system.The ground state configuration of skyrmion matter are crystals. At low densityit is made of well-localized single skyrmions. At a critical density, the systemundergoes a structural phase transition to a new kind of crystal. It is made of‘half-skyrmions’ which are still well-localized but carry only half winding number.In the half-skyrmion phase, the system develops an additional symmetry whichleads to a vanishing average value of σ = T r ( U ), the normalized trace of the U field. In the studies of the late 80’s, the vanishing of this average value h σ i wasinterpreted as chiral symmetry restoration by assuming that h σ i is related to theQCD order parameter h ¯ qq i . However, in Ref. 7, it was shown that the vanishingof h σ i cannot be an indication of a genuine chiral symmetry restoration, becausethe decay constant of the pion fluctuating in such a half-skyrmion matter does notvanish. This was interpreted as a signal of the appearance of a pseudogap phasesimilar to what happens in high T c superconductors. The puzzle was solved in Ref. 10 by incorporating a suitable degree of freedom,the dilaton field χ , associated to the scale anomaly of QCD. The dilaton field takesover the role of the order parameter for chiral symmetry restoration. As the densityof skyrmion matter increases, both h σ i and h χ i vanish (not necessarily at the samecritical density). The effective decay constant of the pion fluctuation vanishes onlywhen h χ i becomes zero. It is thus the dilaton field which provides the mechanismfor chiral symmetry restoration.Contrary to lattice QCD, there are few studies on the temperature dependenceof skyrmion matter. Skyrmion matter has been heated up to melt the crystal into aliquid to investigate the crystal-liquid phase transition a phenomenon which isirrelevant to the restoration of chiral symmetry. We have studied skyrmion matterat finite density and temperature and have obtained the phase diagram describingthe realization of the chiral symmetry. The contents of this review are as follows. Section 1.2 deals with the history ofskyrmion matter and how our work follows from previous investigations. We alsostudy of the pion properties inside skyrmion matter at finite density. To confront theresults with reality, in Sec. 1.3 we show that the scale dilaton has to be incorporatedand we discuss how the properties of the pion change thereafter. Section 1.4 isdevoted to the study of the temperature dependence and the description of thephase diagram. In Sec. 1.5 we incorporate vector mesons to the scheme and discussthe problem that arises due to the coupling of the ω meson and our solution to it.Finally the last section is devoted to a summary of our main results and to someconclusions we can draw from our study. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature Skyrmion matter
The Skyrme model describes baryons, with arbitrary baryon number, as static soli-ton solutions of an effective Lagrangian for pions.
The model has been used todescribe not only single baryon properties, but also has served to derive thenucleon-nucleon interaction, the pion-nucleon interaction, properties of lightnuclei and of nuclear matter. In the case of nuclear matter, most of the develop-ments done in late 80’s involve a crystal of skyrmions. m =0 single skyrmion FCC half skyrmion CC E / B L F single skyrmion FCC m =140 MeV inhomogeneous phase Fig. 1.1. Energy per single skyrmion as a function of the size parameter L . The solid circles showthe results for massless pions and the open circles are those for massive pions. Note the rapidphase transition around L ∼ . The first attempt to understand the dense skyrmion matter was made byKutchera et al. These authors proceeded by introducing a single skyrmion intoa spherical Wigner-Seitz cell without incorporating explicit information on the in-teraction. The presently considered conventional approaches were developed later.In them one assumes that the skyrmions form a crystal with a specific symmetryand then performs numerical simulations using this symmetry as a constraint. Thefirst guess at this symmetry was made by Klebanov. He considered a systemwhere the skyrmions are located in the lattice site of a cubic crystal (CC) and haverelative orientations in such a way that the pair of nearest neighbors attract max-imally. Goldhaber and Manton suggested that contrary to Klebanov’s findings,the high density phase of skyrmion matter is to be described by a body-centeredcrystal (BCC) of half skyrmions. This suggestion was confirmed by numerical cal-culations. Kugler and Shtrikman, using a variational method, investigated theground state of the skyrmion crystal including not only the single skyrmion CC andhalf-skyrmion BCC but also the single skyrmion face-centered-cubic crystal (FCC)and half-skyrmion CC. In their calculation a phase transition from the single FCC ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento to half-skyrmion CC takes place and the ground state is found in the half-skyrmionCC configuration. Castillejo et al. obtained similar conclusions.In Fig. 1.1 we show the energy per baryon E/B as a function of the FCC boxsize parameter L a . Each point in the figure denotes a minimum of the energy for theclassical field configuration associated with the Lagrangians (1.2), (1.4) and (1.6)for a given value of L . The solid circles correspond to the zero pion mass calculationand reproduce the results of Kugler and Shtrikman. The quantities L and E/B ,appearing in the figure, are given in units of ( ef π ) − ( ∼ . f π = 93 MeVand e = 4 .
75) and
E/B in units of (6 π f π ) /e ( ∼ E/B easily with its Bogolmoln’ybound for the skyrmion in the chiral limit, which can be expressed as
E/B = 1 inthis convention.In the chiral limit, as we squeeze the system from L = 6 to around L = 3 .
8, onesees that the skyrmion system undergoes a phase transition from the FCC singleskyrmion configuration to the CC half-skyrmion configuration. The system reachesa minimum energy configuration at L = L min ∼ . E/B ∼ . L > L min with the constrainedsymmetry may not be the genuine low energy configuration of the system for thatgiven L . Note that the pressure P ≡ ∂E/∂V is negative, which implies that thesystem in that configuration is unstable. Some of the skyrmions may condenseto form dense lumps in the phase leaving large empty spaces forming a stableinhomogeneous as seen in Fig. 1.1 for L = L min . Only the phase to the leftof the minimum, L < L min , may be referred to as “ homogeneous ” and there thebackground field is described by a crystal configuration. m =140 MeV m =0 L F single skyrmion FCChalf skyrmion CC Fig. 1.2. h σ i as a function of the size parameter L . The notation is the same as in Fig. 1.1 a A single FCC is a cube with a sidelength 2 L , so that there are 4 single skyrmions in a volume of8 L , that is, the baryon number density is related to L as ρ B = 1 / L . ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature The open circles are the solutions found with a nonvanishing pion mass, m π =140 MeV. b Comparing to the skyrmion system for massless pions, the energy perbaryon is somewhat higher. Furthermore, there is no first order phase transition atlow densities.
Fig. 1.3. Local baryon number densities at L = 3 . L = 2 . L = 2 . In Fig. 1.2, we show h σ i , i.e. the space average value of σ as a function of L . In the chiral limit, h σ i rapidly drops as the system shrinks and reaches zero at L ∼ .
8, where the system goes to the half-skyrmion phase. This phase transitionwas interpreted as a signal for chiral symmetry restoration. However, as we sallsee in the next section, this is not the expected transition. In the case of massivepions, the transition in h σ i is soft. Its value decreases monotonically and reacheszero asymptotically, as the density increases. Furthermore, as we can see in Fig.1.3, where the local baryon number density is shown, for L = 2 (left) and L = 3 . z = 0 plane, the system becomes a half-skyrmion crystal at highdensity.Another scheme used to study multi-skyrmion systems is the procedure based onthe Atiyah-Manton Ansatz. In this scheme, skyrmions of baryon number N areobtained by calculating the holonomy of Yang-Mills instantons of charge N . ThisAnsatz has been successful in describing few-nucleon systems. This procedurehas been also applied to nuclear matter with the instanton solution on a four torus. The energy per baryon was found to be (
E/B ) min = 1 .
058 at L min = 2 .
47, which b Incorporating the pion mass into the problem introduces a new scale in the analysis and thereforewe are forced to give specific values to the parameters of the chiral effective Lagrangian, the piondecay constant and the Skyrme parameter, a feature which we have avoided in the chiral limit.In order to proceed, we simply take their empirical values, that is, f π = 93 MeV and e = 4 . ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento V ( d ) d L F =5.0 L F =10.0 Fig. 1.4. The energy cost to shift a single skyrmion from its stable position by an amount d inthe direction of the z -axis. is comparable to the variational result of Kugler and Shtrikman. In Ref. 43, the Atiyah-Maton Ansatz is employed to get skyrmion matter fromthe ’t Hooft’s multi-instanton solution, which is modified to incorporate dynamicalvariables such as the positions and relative orientations of the single skyrmions. Thisdescription provides information on the dynamics of a single skyrmion in skyrmionmatter. Shown in Fig. 1.4 is the energy change of the system when a single skyrmionis shifted from its FCC lattice site by an amount d in the direction of the z-axis.Two extreme cases are shown. In the case of a dense system ( L F ≡ L = 5 . d , it is almost quadratic in d . It implies that thedense system is in a crystal phase. On the other hand, in the case of a dilute system( L F = 10 . d , whichimplies that the system is in a gas (or liquid) phase. If we let all the variables varyfreely, the system will prefer to change to a disordered or inhomogeneous phase inwhich some skyrmions will form clusters, as we have discussed before. Pions in Skyrmion matter
The Skyrme model also provides the most convenient framework to study the pionproperties in dense matter. The basic strategy is to take the static configuration U ( ~x ) discussed in Sec. 1.2.1 as the background fields and to look into the propertiesof the pion fluctuating on top of it. This is the conventional procedure used to findsingle particle excitations when one has solitons in a field theory. The fluctuating time-dependent pion fields can be incorporated on top of thestatic fields through the Ansatz U ( ~x, t ) = p U π U ( ~x ) p U π , (1.8) ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature where U π = exp (cid:18) i~τ · ~φ ( x ) /f π (cid:19) , (1.9)with ~φ describing the fluctuating pions.When U ( ~r ) = 1( ρ B = 0), the expansion in power of φ ’s leads us to L ( φ ) = 12 ∂ µ φ a ∂ µ φ a + 12 m π σ ( ~x ) φ a φ a + · · · , (1.10)which is just a Lagrangian for the self-interacting pion fields without any interactionswith baryons. Here, we have written only the kinetic and mass terms relavant forfurther discussions. With a non trivial U ( ~r ) describing dense skyrmion matter, theLagrangian becomes, L = 12 G ab ( ~x ) ∂ µ φ a ∂ µ φ b + 12 m π σ ( ~x ) φ a φ a + · · · , (1.11)with G ab ( ~x ) = σ δ ab + π a π b . (1.12)The structure of our Lagrangian is similar to that of chiral perturbation theoryEq. (1.13) of Refs. 46,47. These authors start with a Lagrangian containing all thedegrees of freedom, including nucleon fields, and free parameters. They integrateout the nucleons in and out of an `a priori assumed Fermi sea and in the processthey get a Lagrangian density describing the pion in the medium. Their resultcorresponds to the above Skyrme Lagrangian except that the quadratic (currentalgebra) and the mass terms pick up a density dependence of the form − f π (cid:18) g µν + D µν ρf π (cid:19) Tr( U † ∂ µ U U † ∂ ν U ) + f π m π (cid:18) − Σ πN f π m π ρ (cid:19) Tr( U + U † − , (1.13)where ρ is the density of the nuclear matter and D µν and σ are physical quantitiesobtained from the pion-nucleon interactions. Note that in this scheme, nuclearmatter is assumed ab initio to be a Fermi sea devoid of the intrinsic dependencementioned above.We proceed via a mean field approximation consisting in averaging the back-ground modifications G ab ( ~x ) and σ ( ~x ) appearing in the Lagrangian which are re-duced to constants, h G ab i = Gδ ab and h σ i . Then, the Lagrangian can be rewrittenas L ( φ ∗ ) = 12 ∂ µ φ ∗ a ∂ µ φ ∗ a + 12 m ∗ π φ ∗ a φ ∗ a + · · · , (1.14)where we have carried out a wavefunction renormalization, φ ∗ a = √ Gφ a , which leadsto a medium modified pion decay constant and mass as f ∗ π f π = √ G, (1.15) m ∗ π m π = h σ i√ G . (1.16) ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento m / m & f * /f < > < > m * /mf * /f Fig. 1.5. Estimates of f ∗ π /f π and m ∗ π /m π as functions of the baryon number density of skyrmionmatter. In Fig. 1.5 we show the estimates of f ∗ π /f π and m ∗ π /m π as a function of thedensity. As the density increases, f ∗ π decreases only to ∼ . f π and then it remainsconstant at that value. Our result is different from what was the general believe: the vanishing of h σ i is not an indication of chiral symmetry restoration since thepion decay constant does not vanish. Note that h σ i has the same slope at low densities, which leads to m ∗ π /m π ∼ G becomes a constant, m ∗ π /m π decreaseslike h σ i / with a factor which is greater than 1. As the density increases, higherorder terms in ρ come to play important roles and m ∗ π /m π decreases. A morerigorous derivation of these quantities can be obtained using perturbation theory. The slope of h σ i at low density is approximately 1/3. If we expand h σ i about ρ = 0 and compare it with Eq.(1.13), we obtain h σ i ∼ − ρρ + · · · ∼ − Σ πN f π m π ρ + · · · , (1.17)which yields Σ πN ∼ m π f π / (3 ρ ) ∼
42 MeV, which is comparable with the experi-mental value 45 MeV c . This comparison is fully justified from the point of view ofthe N expansion since both approaches should produce the same result to leadingorder in this expansion. The liner term is O (1).The length scale is strongly dependent on our choice of the parameters f π and e .Thus one should be aware that the ρ scale in Fig. 1.5 could change quantitativelyconsiderably if one chooses another parameter set, however the qualitative behaviorwill remain unchanged. c While this value is widely quoted, there is a considerable controversy on the precise value of thissigma term. In fact it can even be considerably higher than this. See Ref. 48 for a more recentdiscussion. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx
Skyrmion Approach to finite density and temperature Note that the density dependence of the background is taken into account toall orders. No low-density approximation, whose validity is in doubt except at verylow density, is ever made in the calculation. The power of our approach is that thedynamics of the background and its excitations can be treated in a unified way onthe same footing with a single Lagrangian.
Dilaton dynamics
The dynamics introduced in Sec. 1.1 as an effective theory for the hadronic inter-actions is probably incomplete. In fact, it is not clear that the intrinsic densitydependence required by the matching to QCD is fully implemented in the model.One puzzling feature is that the Wigner phase represented by the half-skyrmionmatter with h σ i = 0 supports a non-vanishing pion decay constant. This may beinterpreted as a possible signal for a pseudogap phase. However, at some point,the chiral symmetry should be restored and there the pion decay constant shouldvanish.This difficulty can be circumvented in our framework by incorporating in thestandard skyrmion dynamics the trace anomaly of QCD in an effective manner. The end result is the skyrmion Lagrangian introduced by Ellis and Lanik andemployed by Brown and Rho for nuclear physics which contains an additionalscalar field, the so called scale dilaton.The classical QCD action of scale dimension 4 in the chiral limit is invariantunder the scale transformation x → λ x = λ − x, λ ≥ , (1.18)under which the quark field and the gluon fields transform with the scale dimension3/2 and 1, respectively. The quark mass term of scale dimension 3 breaks scaleinvariance. At the quantum level, scale invariance is also broken by dimensionaltransmutation even for massless quarks, as signaled by the non-vanishing of the traceof the energy-momentum tensor. Equivalently, this phenomenon can be formulatedby the non-vanishing divergence of the dilatation current D µ , the so called traceanomaly, ∂ µ D µ = θ µµ = X q m q ¯ qq − β ( g ) g Tr G µν G µν , (1.19)where β ( g ) is the beta function of QCD.Broken scale invariance can be implemented into large N c physics by modifying ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento the standard skyrmion Lagrangian, introduced in Sec. 1.1, to L = f π (cid:18) χf χ (cid:19) Tr( ∂ µ U † ∂ µ U ) + 132 e Tr([ U † ∂ µ U, U † ∂ ν U ]) + f π m π (cid:18) χf χ (cid:19) Tr( U + U † − ∂ µ χ∂ µ χ − m χ f χ (cid:20) χ (cid:18) ln( χ/f χ ) − (cid:19) + 14 (cid:21) . (1.20)We have denoted the non vanishing vacuum expectation value of χ as f χ , a constantwhich describes the decay of the scalar into pions. The second term of the traceanomaly (1.19) can be reproduced by the potential energy V ( χ ), which is adjustedin the Lagrangian (1.20) so that V = dV /dχ = 0 and d V /dχ = m χ at χ = f χ . The vacuum state of the Lagrangian at zero baryon number density is definedby U = 1 and χ = f χ . The fluctuations of the pion and the scalar fields about thisvacuum, defined through U = exp( i~τ · ~φ/f π ) , and χ = f χ + ˜ χ (1.21)give physical meaning to the model parameters: f π as the pion decay constant, m π as the pion mass, f χ as the scalar decay constant, and m χ as the scalar mass. Forthe pions, we use their empirical values as f π = 93MeV and m π = 140MeV. We fixthe Skyrme parameter e to 4.75 from the axial-vector coupling constant g A as in Ref.50. However, for the scalar field χ , no experimental values for the correspondingparameters are available.In Ref. 51, the scalar field is incorporated into a relativistic hadronic model fornuclear matter not only to account for the anomalous scaling behavior but also toprovide the mid-range nucleon-nucleon attraction. Then, the parameters f χ and m χ are adjusted so that the model fits finite nuclei. One of the parameter setsis m χ = 550 MeV and f χ = 240 MeV (Set A). On the other hand, Song et al. obtain the “best” values for the parameters of the effective chiral Lagrangian withthe “soft” scalar fields so that the results are consistent with “Brown-Rho” scaling, explicitly, m χ = 720 MeV and f χ = 240 MeV (Set B). For completeness, we consideralso a parameter set of m χ = 1 GeV and f χ = 240 MeV (Set C) corresponding toa mass scale comparable to that of chiral symmetry Λ χ ∼ πf π . Dynamics of the single skyrmion
The procedure one has to follow can be found in Ref. 10 and is similar to the onediscussed in Sec. 1.2.1. The first step is to find the solution for the single skyrmionwhich includes the dilaton dynamics. The skyrmion with the baryon number B = 1can be found by generalizing the spherical hedgehog Ansatz of the original Skyrmemodel as U ( ~r ) = exp( i~τ · ˆ rF ( r )) , and χ ( ~r ) = f χ C ( r ) , (1.22) ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature with two radial functions F ( r ) and C ( r ). Minimization of the mass equation leadsto a coupled set of equations of motion for these functions. In order for the solutionto carry a baryon number, U has the value − F ( x = 0) = π ,while there is no such topological constraint for C ( x = 0). All that is required isthat it be a positive number below 1. At infinity, the fields U ( ~r ) and χ ( ~r ) shouldreach their vacuum values. set C : M B=1sol =1466 MeV, < r > =0.43 fmset B : M B=1sol =1389 MeV, < r > =0.44 fm P r o f il e F un c t i on s rF(r)/C(r) set A : M B=1sol =1364 MeV, < r > =0.44 fm Fig. 1.6. Profile functions F ( x ) and C ( x ) as a function of x . Shown in Fig. 1.6 are profile functions as a function of x (= ef π r ). F ( r ) andconsequently the root mean square radius of the baryon charge show little depen-dence on m χ . On the other hand, the changes in C ( r ) and the skyrmion mass arerecognizable. Inside the skyrmion, especially at the center, C ( r ) deviates from itsvaccum value 1. Note that this changes in C ( r ) is multiplied by f π in the currentalgebra term of the Lagrangian. Thus, C ( r ) ≤ effective f π inside thesingle skyrmion, which implies a partial restoration of the chiral symmetry there.The reduction in the effective pion decay constant is reflected in the single skyrmionmass.The larger the scalar mass is, the smaller its coupling to the pionic field andthe less its effect on the single skyrmion. In the limit of m χ → ∞ , the scalar fieldis completely decoupled from the pions and the model returns back to the originalone, where C ( r ) = 1, M sk = 1479 MeV and h r i / = 0 .
43 fm.
Dense skyrmion matter and chiral symmetry restoration
The second step is to construct a crystal configuration made up of skyrmions witha minimal energy for a given density.Referring to Refs. 10,11 for the full details, we emphasize here the role thedilaton field in the phase transition scenario for skyrmion matter. Let the dilaton ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento field χ ( ~r ) be a constant throughout the whole space as χ/f χ = X. (1.23)Then the energy per baryon number of the system for a given density can be cal-culated and conveniently expressed as E/B ( X, L ) = X ( E /B ) + ( E /B ) + X ( E m /B ) + (2 L ) (cid:0) X (ln X − ) + (cid:1) , (1.24)where E , E and E m are, respectively, the contributions from the current algebraterm, the Skyrme term and the pion mass term of the Lagrangian to the energy ofthe skyrmion system, described in Sec. 1.1, and (2 L ) is the volume occupied by asingle skyrmionThe quantity E/B ( X, L ) can be taken as an in medium effective potential for X , modified by the coupling of the scalar to the background matter. Using theparameter values of Ref. 11 for the Skyrme model without the scalar field, theeffective potential E/B ( X ) for a few values of L behaves as shown in Fig. 1.7(a).At low density (large L ), the minimum of the effective potential is located closeto X = 1. As the density increases, the quadratic term in the effective potential E/B ( X ) develops another minimum at X = 0 which is an unstable extremum of thepotential V ( X ) in free space. At L ∼ X ∼
1. At higher density, the minimum shifts to X = 0 wherethe system stabilizes. L=1.0 fmL=1.1 fmL=0.8 fm E / B ( M e V ) XL=1.4 fm
X=0phase E / B ( M e V ) L (fm) X=0phase \ X (a) (b) Fig. 1.7. (a) Energy per single skyrmion as a function of the scalar field X for a given L . Theresults are obtained with the ( E /B ), ( E /B ), and ( E m /B ) of Ref. 11 and with the parametersets B, (b) Energy per single skyrmion as a function of L . In Fig. 1.7(b), we plot
E/B ( X min , L ) as a function of L , which is obtainedby minimizing E/B ( X, L ) with respect to X for each L . The figure in the smallbox is the corresponding value of X min as function of L . There we see the explicitmanifestation of a first-order phase transition. Although the present discussion isbased on a simplified analysis, it essentially encodes the same physics as in the more ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature rigorous treatment of χ given in Ref. 10 ρ < σ > & <χ / f χ > <σ> <χ/ f χ >χ SBphase pseudogap chiral symmetryrestored phase(A) (B) (C)half−skyrmion CCphase π σσ<σ>π chiralcircle π σ (A)(B)(C)
Fig. 1.8. Average values of σ = Tr( U ) and χ/f χ of the lowest energy crystal configuration at agiven baryon number density. We show in Fig. 1.8 the average values h σ i and h χ/f χ i over space for the mini-mum energy crystal configurations obtained by the complete numerical calculationwithout any approximation for χ . These data show that a ‘structural’ phase transi-tion takes place, characterized by h σ i = 0, at lower density then the genuine chiralphase transition which occurs when h χ i = 0. The value of h σ i becomes 0 whenthe structure of the skyrmion crystal undergoes a change from the single skyrmionFCC to the half-skyrmion CC. Thus, the pseudogap phase persists in an interme-diate density region, where the h χ/f χ i does not vanish while h σ i does. A similarpseudogap structure has been also proposed in hot QCD. The two step phase transition is schematically illustrated in Fig. 1.8. Let ρ p and ρ c be the density at which h σ i and h χ i vanish, repectively.(A) At low density ( ρ < ρ p ), matter slightly reduces the vacuum value of the dilatonfield from that of the baryon free vacuum. This implies a shrinking of the radiusof the chiral circle by the same ratio. Since the skyrmion takes all the valueson the chiral circle, the expectation value of σ is not located on the circle butinside the circle. Skyrmion matter at this density is in the chiral symmetrybroken phase.(B) At some intermediate densities ( ρ p < ρ < ρ c ), the expectation value of σ vanishes while that of the dilaton field is still nonzero. The skyrmion crystalis in a CC configuration made of half skyrmions localized at the points where σ = ±
1. Since the average value of the dilaton field does not vanish, the radiusof the chiral circle is still finite. Here, h σ i = 0 does not mean that chiralsymmetry is completely restored. We interpret this as a pseudogap phase.(C) At higher density ( ρ > ρ c ), the phase characterized by h χ/f χ i = 0 becomesenergetically favorable. Then, the chiral circle, describing the fluctuating piondynamics, shrinks to a point. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
The density range for the ocurrence of a pseudogap phase strongly depends on theparameter choice of m χ . For small m χ below 700 MeV, the pseudogap has almostzero size.In the case of massive pions, the chiral circle is tilted by the explicit (mass)symmetry breaking term. Thus, the exact half-skyrmion CC, which requires asymmetric solution for points with value σ = +1 and those with σ = − h σ i = 0 does not existfor any density. Thus no pseudogap phase arises. However, h σ i is always inside thechiral circle and its value drops much faster than that of h χ/f χ i . Therefore, only ifthe pion mass is small a pseudogap phase can appear in the model. Pions in a dense medium with dilaton dynamics
Since we have achieved, via dilaton dynamics, a reasonable scenario for chiral sym-metry restoration, it is time to revisit the properties of pions in a dense medium. Aswas explained in Sec. 1.2.2 and in Ref. 7, we proceed to incorporate the fluctuationson top of the static skyrmion crystal. (We refer to Refs. 10,11 for details.)Using a mean field approximation we calculate the in-medium pion mass m ∗ π and decay constant f ∗ π obtaining, Z π = *(cid:18) χ ( ~x ) f χ (cid:19) (1 − π ( ~x )) + ≡ (cid:18) f ∗ π f π (cid:19) , (1.25) m ∗ π Z π = *(cid:18) χ ( ~x ) f χ (cid:19) σ ( ~x ) m π + . (1.26)The wave function renormalization constant Z π gives the ratio of the in-mediumpion decay constant f ∗ π to the free one, and the above expression arises from thecurrent algebra term in the Lagrangian. The explicit calculation of m ∗ χ is given inRef. 11.In Fig. 1.9 we show the (exact) ratios of the in-medium parameters relativeto their free-space values. Only the results obtained with the parameter set B areshown. The parameter set A yields similar results while set C shows a two stepstructure with an intermediate pseudogap phase. Not only the average value of χ over the space but also χ ( ~r ) itself vanishes at any point in space. This is the reasonfor the vanishing of m ∗ π and f ∗ π . That is, f ∗ π really vanishes when ρ < ρ c in theSkyrme model with dilaton dynamics.At low matter density, the ratio f ∗ π /f π can be fitted to a linear function f ∗ π f π ∼ − . ρ/ρ ) + · · · (1.27)At ρ = ρ , this yields f ∗ π /f π = 0 .
76, which is comparable to the other predictions.In Ref. 11, the in medium modification of the χ decay into two pions is alsostudied using the mean field approximation. Gathering the terms with a fluctuating ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature r a t i o / c < > < /f > f * /f m * /m m * /m m * / m / c m * / m / c * / m * / c (a) (b) Fig. 1.9. (a) The ratios of the in-medium parameters to the free space parameters. The graph in asmall box shows the masses of the pion and the scalar, (b) the in-medium decay width Γ ∗ ( χ → ππ )as a function of ρ . scalar field and two fluctuating pion fields, we get the Lagrangian density for theprocess χ → ππ L M,χπ = χ f χ ( δ ab + g ab ) χ∂ µ φ a ∂ µ φ b , (1.28)where only the term from L σ is used. Averaging the space dependence ofthe background field configuration modifies the coupling constant by a factor h ( χ /f χ )(1 + g ) i = h ( χ /f χ )(1 − π ) i . Taking into account the appropriatewave function renormalization factors, Z π , and the change in the scalar mass, weobtain the in-medium decay width asΓ ∗ ( χ → ππ ) = 3 m ∗ χ πf χ (cid:12)(cid:12)(cid:12)(cid:12) h ( χ /f χ )(1 − π ) ih ( χ /f χ ) (1 − π ) i (cid:12)(cid:12)(cid:12)(cid:12) ≈ m ∗ χ πf ∗ χ . (1.29)We show in Fig. 1.9 the in-medium decay width predicted with the parameter setB. In the region ρ ≥ ρ pt where χ = 0, Γ ∗ cannot be defined to this order. Near thecritical point, the scalar becomes an extremely narrow-width excitation, a featurewhich has been discussed in the literature as a signal for chiral restoration. Another interesting change in the properties of the pion in the medium is asso-ciated with the in medium pion dispersion relation. This relation requires, besidesthe mass, the so-called in medium pion velocity, v π . This property allows us togain more insight into the real time properties of the system under extreme con-ditions and enables us to analyze how the phase transition from normal matterto deconfined QCD takes place from the hadronic side , the so called ‘bottom up’approach.At nonzero temperature and/or density, the Lorentz symmetry is broken by themedium. In the dispersion relation for the pion modes (in the chiral limit) p = v π | ~p | , (1.30) ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento the velocity v π which is 1 in free-space must depart from 1. This may be studiedreliably, at least at low temperatures and at low densities, via chiral perturbationtheory. The in-medium pion velocity can be expressed in terms of the time com-ponent of the pion decay constant, f tπ and the space component, f sπ , h | A a | π b ( p ) i in-medium = if tπ δ ab p , h | A ia | π b ( p ) i in-medium = if sπ δ ab p i . (1.31)The conservation of the axial vector current leads to the dispersion relation (1.30)with the pion velocity given by v π = f sπ /f tπ . (1.32)In Ref. 60 two decay constants, f t and f s , are defined differently from those ofEq.(1.31)), through the effective Lagrangian, L eff = f t ∂ U † ∂ U ) − f s ∂ i U † ∂ i U ) + · · · , (1.33)where U is an SU(2)-valued chiral field whose phase describes the in-medium pion.In terms of these constants, the pion velocity is defined by v π = f s /f t . (1.34)In Ref. 11, it is shown that local interactions with background skyrmion matterlead to a breakdown of Lorentz symmetry in the dense medium and to an effectiveLagrangian for pion dynamics in the form of Eq.(1.33). The results are shown inFigs. 1.10. Both of the pion decay constants change significantly as a function ofdensity and vanish – in the chiral limit – when chiral symmetry is restored. However,the second-order contributions to the f s and f π , which break Lorentz symmetry,turn out to be rather small, and thus their ratio, the pion velocity, stays v π ∼ ∼ .
9. Note, however, the drastic change in its behaviorat two different densities. At the lower density, where skyrmion matter is in thechiral symmetry broken phase, the pion velocity decreases and has the minimumat ρ = ρ p . If one worked only at low density in a perturbative scheme, one wouldconclude that the pion velocity decreases all the way to zero. However, the presenceof the pseudogap phase transition changes this behavior. In the pseudogap phase,the pion velocity not only stops decreasing but starts increasing with increasingdensity. In the chiral symmetry restored phase both f t and f s vanish. Thus theirratio makes no sense.In Refs. 12,13, the in-medium modification of the neutral pion decay process intotwo gammas and neutrino-anti-neutrino pair are studied in the same manner. The π → γγ process is shown to be strongly suppressed in dense medium, while theprocess π → ν ¯ ν forbidden in free space becomes possible by the Lorentz symmetrybreaking effect of the medium. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature ρ r a t i o f t /f π χ S phase χ SB pseudogap phasephase υ π =f s /f t chiral phasef π∗ / f π =0.78 at ρ =0.6 ρ ~0.004m ρ ~0.01m χ transition Fig. 1.10. In-medium pion decay constants and their ratio, the pion velocity.
There are many studies of lattice QCD at finite temperature. The situation is com-pletely different for skyrmion matter where the number of studies is limited. Forexample, skyrmion matter has been heated up to melt the crystal into a liquid tostudy the crystal-liquid phase transition.
However this phenomenon is irrele-vant to study the restoration of chiral symmetry, which interests us here for thereasons discussed in previos sections.What happens if we heat up the system? Naively, as the temperature increases,the kinetic energy of the skyrmions increases and the skyrmion crystal begins tomelt. The kinetic energy associated with the translations, vibrations and rotationsof the skyrmions is proportional to T . This mechanism leads to a solid-liquid-gasphase transition of the skyrmion system. However, we are interested in the chiralsymmetry restoration transition, which is not related to the melting. Therefore, anew mechanism must be incorporated to describe chiral symmetry restoration. Weshow in what follows that the thermal excitation of the pions in the medium is theappropriate mechanism, since this phenomenon is proportional to T and thereforedominates the absorption of heat.The pressure of non-interacting pions is given by P = π T , (1.35)where we have taken into account the contributions from three species of pion, π + , π , π − . This term contributes to the energy per single skyrmion volume as3 P V ( χ/f χ ) . The kinetic energy of the pions arises from L σ (1.2), and thereforescale symmetry implies that it should carry a factor χ . The factor 3 comes fromthe fact that our pions are massless. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
To estimate the properties of skyrmion matter at finite temperature let us take χ as a constant field as we did in Sec. 1.3.3. After including thermal pions, Eq.(1.24) can be rewritten as E/B ( ρ, T, X ) = (cid:18) E /B )( ρ ) + π T V (cid:19) X +( E /B )( ρ )+ X (ln X − )+ ) , (1.36)where we have dropped the pion mass term.As in Sec. 1.3.3, chiral restoration will occur when the value of X min thatminimizes E/B vanishes. By minimizing
E/B with respect to X , we observe thatthe phase transition from a non-vanishing X = e − / to X = 0. Thus, the natureof the phase transition is of the first order.After a straightforward calculation we obtain, ρ c ( E /B ) + π T c = f χ m χ e / . (1.37)which leads to T c = π f χ m χ e / − ρ c ( E /B )( ρ c ) !! / (1.38)For ρ = 0 (zero density), our estimate for the critical temperature is T c = π f χ m χ e / ! / ∼
205 MeV , (1.39)where we have used the following values for the parameters. f χ = 210 MeV and m χ = 720 MeV. It is remarkable that our model leads to T c ∼
200 MeV, which isclose to that obtained by lattice QCD and in agreement with the data. To usthis is a confirmation that the mechanism chosen for the absorption of heat plays afundamental role in the hadronic phase.The numerical results on E /B that minimize the energy of the system for agiven ρ B can be approximated by E /B = (cid:26) f π /ρ / , ρ > ρ f π /e sk , ρ < ρ , (1.40)where ρ = ( e sk f π / . .Using Eq.(1.40) for E /B , we obtain the critical density for chiral symmetryrestoration at zero temperature as ρ c ( T = 0) = f χ m χ e / f π ! / ∼ .
37 fm − . (1.41)Since ρ = 0 .
24 fm − < ρ c ( T = 0) our result is consistent with the high densityformula for E /B used.The resulting critical density ρ c ( T = 0) ∼ .
37 fm − is only twice normal nuclearmatter density and it is low with respect to the expected values. This result does ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature (f m /f ) T c ~205(1-3.1 ) Chiral Symmetry Broken Phase T ( M e V ) (fm -3 ) Chiral Symmetry Restored PhaseT c ~205(1-1.9 ) (f m ) Fig. 1.11. The chiral phase transition. The solid line shows the exact calculation, while the graylines two approximate estimates. not represent a problem since ρ c ( T = 0) scales with ( f χ m χ /f π ) and T ρ =0 c with( f χ m χ ) / and small changes in the parameters lead to larger values for the criticaldensity without changing the critical temperature too much.For a finite density smaller than ρ c ( T = 0), we obtain the corresponding criticaltemperature by substituting the asymptotic formulas (1.40) for E /B , T c = (cid:26) T c ( ρ = 0) (1 − . ρ c ) / for ρ < ρ ,T c ( ρ = 0) (1 − . ρ / c ) / for ρ > ρ , (1.42)where the density is measured in fm − The gray lines in Fig. 1.11 show these twocurves. The results from the exact calculations obtaned by minimization of theenergy (1.36) are shown by black dots connected by black line in Fig. 1.11. Theresulting phase diagram has the same shape but the values of the temperatures anddensities are generally smaller than in the approximate estimates.
In our effort to approach the theory of the hadronic interactions and inspired byWeinberg’s theorem we proceed to incorporate to the model the lowest-lyingvector mesons, namely the ρ and the ω . In this way we also do away with the adhoc Skyrme quartic term. It is known that these vector mesons play a crucial rolein stabilizing the single nucleon system as well as in the saturation of normalnuclear matter. We consider a skyrmion-type Lagrangian with vector mesons possessing hiddenlocal gauge symmetry, spontaneously broken chiral symmetry and scale symme-try. Such a theory might be considered as a better approximation to reality ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento than the extreme large N c approximation to QCD represented by the Skyrme model.Specifically, the model Lagrangian, which we investigate, is given by L = f π (cid:18) χf χ (cid:19) Tr( ∂ µ U † ∂ µ U ) + f π m π (cid:18) χf χ (cid:19) Tr( U + U † − − f π a (cid:18) χf χ (cid:19) Tr[ ℓ µ + r µ + i ( g/ ~τ · ~ρ µ + ω µ )] − ~ρ µν · ~ρ µν − ω µν ω µν + gω µ B µ + ∂ µ χ∂ µ χ − m χ f χ (cid:2) ( χ/f χ ) (ln( χ/f χ ) − ) + (cid:3) , (1.43)where, U = exp( i~τ · ~π/f π ) ≡ ξ , ℓ µ = ξ † ∂ µ ξ , r µ = ξ∂ µ ξ † , ~ρ µν = ∂ µ ~ρ ν − ∂ ν ~ρ µ + g~ρ µ × ~ρ ν , ω µν = ∂ µ ω ν − ∂ ν ω µ , and B µ = π ε µναβ Tr( U † ∂ ν U U † ∂ α U U † ∂ β U ). Notethat the Skyrme quartic term is not present. The vector mesons, ρ and ω , areincorporated as dynamical gauge bosons for the local hidden gauge symmetry ofthe non-linear sigma model Lagrangian and the dilaton field χ is introduced so thatthe Lagrangian has the same scaling behavior as QCD. The physical parametersappearing in the Lagrangian are summarized in Table 1.1 . Table 1.1. Parameters of the model Lagrangiannotation physical meaning value f π pion decay constant 93 MeV f χ χ decay constant 210 MeV g ρππ coupling constant 5.85 ∗ m π pion mass 140 MeV m χ χ meass 720 MeV m V vector meson masses 770 MeV † a vector meson dominance 2 ∗ obtained by using the KSFR relation m V = m ρ = m ω = af π g with a = 2. cf. g ρππ = 6 .
11 from the decay width of ρ → ππ . † experimentally measured values are m ρ =768 MeV and m ω =782 MeV. Dynamics of the single skyrmion
The spherically symmetric hedgehog Ansatz for the B = 1 soliton solution of thestandard Skyrme model can be generalized to U B =1 = exp( i~τ · ˆ rF ( r )) , (1.44) ρ a,B =1 µ = i = ε ika ˆ r k G ( r ) gr , ρ a,B =1 µ =0 = 0 , (1.45) ω B =1 µ = i = 0 , ω B =1 µ =0 = f π W ( r ) , (1.46) χ B =1 = f χ C ( r ) . (1.47) ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature F ( r ) r (fm) model model model model model model model model G ( r ) r (fm) W ( r ) r (fm) model model C ( r ) r (fm) model model Fig. 1.12. Profile functions - F ( r ), G ( r ), W ( r ) and C ( r ). The boundary conditions that the profile functions satisfy at infinity are F ( ∞ ) = G ( ∞ ) = W ( ∞ ) = 0 , C ( ∞ ) = 1 , (1.48)and at the center ( r = 0) are F (0) = π, G (0) = − , W ′ (0) = C ′ (0) = 0 . (1.49)The profile functions are obtained numerically by minimizing the soliton masswith the boundary conditions (see Ref. 8 for the technical details). The resultsare summarized in Table 1.2 and the corresponding profile functions are given inFig. 1.12. The role of the ω meson that provides a strong repulsion is prominent.Comparing the πρ model with the πρω model, the presence of the ω increases themass by more than 415 MeV and the size, i.e. h r i , by more than .28 fm .How does the dilaton affect this calculation? The πρ model with much smallerskyrmion has a larger baryon density near the origin and this affects the dilaton,significantly changing its mean-field value from its vacuum one. The net effect ofthe dilaton mean field on the mass is a reduction of ∼
150 MeV, whereas for the πρω model it is only of 50 MeV. The details can be seen in Table 1.2. The effect onthe soliton size is, however, different: while the dilaton in the πρ model produces anadditional localization of the baryon charge and hence reduces h r i from .21 fm to.19 fm , in the πρω model, on the contrary, the dilaton produces a delocalization and increases h r i from .49 fm to .51 fm . We will see, however, that this strongrepulsion provided by ω causes a somewhat serious problem in the chiral restorationof the skyrmion matter at higher density. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
Table 1.2. Single skyrmion mass and various contributions to it.Model h r i E B =1 E B =1 π E B =1 πρ E B =1 ρ E B =1 ω E B =1 WZ E B =1 χ πρ -model 0.27 1054.6 400.2 + 9.2 110.4 534.9 0.0 0.0 0.0 πρχ -model 0.19 906.5 103.1 + 1.4 155.1 504.1 0.0 0.0 142.8 πρω -model 0.49 1469.0 767.6 + 39.9 33.2 370.7 -257.6 515.1 0.0 πρωχ -model 0.51 1408.3 646.0 + 29.2 34.9 355.7 -278.3 556.7 64.2 model model E / B L (fm) < > model < > model <> & <> L (fm)< > model E / B L (fm) model model < > model< > model< > model <> & <> ) L (fm) Fig. 1.13.
E/B , h χ i and h σ i as a function of L in the models (a) without the ω and (b) withomega. Skyrmion Matter : an FCC skyrmion crystal
Again, the lowest-energy configuration is obtained when one of the skyrmions isrotated in isospin space with respect to the other by an angle π about an axisperpendicular to the line joining the two. If we generalize this Ansatz to many-skyrmion matter, we obtain that the configuration at the classical level for a givenbaryon number density is an FCC crystal where the nearest neighbour skyrmionsare arranged to have the attractive relative orientations. Kugler’s Fourier seriesexpansion method can be generalized to incorporate the vector mesons, althoughsome subtelties associated with the vector fields have to be implemented. Thedetails can be found in Ref. 8.Figs. 1.13 are the the numerical results of the energy per baryon E/B , h χ i and h σ i in various models as a function of the FCC lattice parameter L . In the πρχ ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature model, as the density of the system increases ( L decreases), E/B changes little. It isclose to the energy of a B = 1 skyrmion up to a density greater than ρ ( L ∼ . ω , the dilaton field plays a dramatic role. A skyrmion mat-ter undergoes an abrupt phase transition at high density at which the expectationvalue of the dilaton field vanishes h χ i = 0. (In general, h χ i = 0 does not necessarilyrequire h χ i = 0. However, since χ ≥ h χ i = 0 always accompanies χ = 0 in thewhole space.) The ρ meson on the other hand is basically a spectator at the classi-cal level, producing little change with respect to our previously studied πχ modelexcept that at high densities, once the ρ starts to overlap, the energy of nuclearmatter increases due to its the repulsive effect at short distances. The densitieshave to be quite high since these skyrmions are very small. Since χ vanishes at thephase transition, we recover the standard behavior, namely, f ∗ π = 0 and m ∗ ρ = 0.In the πρωχ model, the situation changes dramatically. The reason is thatthe ω provides not only a strong repulsion among the skyrmions, but somewhatsurprisingly, also an intermediate range attraction. Note the different mass scalesbetween Figs. 1.13(a) and 1.13(b). In both the πρω and the πρωχ models, at highdensity, the interaction reduces E/B to 85% of the B = 1 skyrmion mass. Thisvalue should be compared with 94% in the πρ model. In the πρχ -model, E/B goesdown to 74% of the B = 1 skyrmion mass, but in this case it is due to the dramaticbehavior of the dilaton field.In the πρωχ model the role of the dilaton field is suppressed. It provides a only asmall attraction at intermediate densities. Moreover, the phase transition towardsits vanishing expectation value, h χ i = 0, does not take place. Instead, its valuegrows at high density!The problem involved is associated with the Lagrangian (1.43) which includesan anomalous part known as Wess-Zumino term, namely the coupling of the ω tothe Baryon current, B µ . To see that this term is the one causing the problem,consider the energy per baryon contributed by this term. (cid:18) EB (cid:19) W Z = 14 ( 3 g Z Box d x Z d x ′ B ( ~x ) exp( − m ∗ ω | ~x − ~x ′ | )4 π | ~x − ~x ′ | B ( ~x ′ ) (1.50)where “Box” corresponds to a single FCC cell. Note that while the integral over ~x is defined in a single FCC cell, that over ~x ′ is not. Thus, unless it is screened, theperiodic source B filling infinite space will produce an infinite potential w whichleads to an infinite ( E/B ) W Z . The screening is done by the omega mass, m ∗ ω . Thusthe effective ω mass cannot vanish. Our numerical results reflect this fact: at highdensity the B - B interaction becomes large compared to any other contribution.In order to reduce it, χ has to increase, and thereby the effective screening mass m ∗ ω ∼ m ω h χ i becomes larger. In this way we run into a phase transition where theexpectation value of χ does not vanish and therefore f π does not vanish but instead ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento m =720 MeV P r o f il e F un c t i on s r(fm)F(r)W(r) C(r)G(r) f L F (fm) m =3000 MeV P r o f il e F un c t i on s r(fm)F(r)W(r) C(r)G(r) f L F (fm) Fig. 1.14. A small and large skyrmion obtained with m χ = 720MeV (left) and m χ = 3000 MeV.Shown in small boxes are h χ i and h σ i as a function of the FCC lattice size L F . increases. A Resolution of the ω problem Assuming that there is nothing wrong with (1.43), we focus on the Wess-Zuminoterm in the Lagrangian. Our objective is to find an alternative to (1.43) thatleads to a behavior consistent with the expected behavior. In the absence of anyreliable clue, we try the simplest, admittedly ad hoc , modification of the Lagrangian(1.43) that allows a reasonable and appealing way-out. Given our ignorance as tohow spontaneously broken scale invariance manifests in matter, we shall simplyforego the requirement that the anomalous term be scale invariant and multiply theanomalous ω · B term by ( χ/f χ ) n for n ≥
2. We have verified that it matters littlewhether we pick n = 2 or n = 3. We therefore take n = 3: L ′ an = g ( χ/f χ ) ω µ B µ (1.51)This additional factor has two virtues:i) It leaves meson dynamics in free space (i.e. χ/f χ = 1) unaffected, since chiralsymmetry is realized `a la sigma model as required by QCD.ii) It plays the role of an effective density-dependent coupling constant so that athigh density, when scale symmetry is restored and χ/f χ →
0, there will beno coupling between the ω and the baryon density as required by hidden localsymmetry with the vector manfiestation.The properties of this Lagrangian for the meson ( B = 0) sector are the same asin our old description. The parameters of the Lagrangian are determined by mesonphysics as given in Table 1.1Fig. 1.14 summarizes the consequences of the modification. Depending on thedilaton mass, the properties of a single skyrmion show distinguished characters ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature and consequently undergoes different phase transition. A small dilaton mass, say m χ ∼ . ω − B coupling, which leads us to such a smallsized skyrmion. Since these small skyrmions are already chiral-symmetry-restoredobjects, simply filling the space with them restores the symmetry. As shown inthe small box, chiral symmetry is restored simultaneously when h σ i vanishes. Incase of having a large mass, the dilaton does not play any significant role in thestructure of a single skyrmion. This scenario leads, as the density of skyrmionmatter increases, first to a pseudogap phase transition where h σ i = 0 and thereafter,at higher density, to a genuine chiral symmetry restoration phase transition where h χ/f χ i = 0. Anyway, whether the dilaton is light or heavy, we finally have areasonable phase transition scenario that at some critical density chiral symmetryrestoration occurs where h χ/f χ i vanishes.Under the same mean field approximation, this skyrmion approach to the densematter leads us to the scaling behaviors of the vector mesons m ∗ ρ m ρ = m ∗ ω m ω = s h (cid:18) χf χ (cid:19) i , (1.52)while that of the pion decay constant is f ∗ π f π = s h (cid:18) χf χ (cid:19) (1 + ( a − π ) i . (1.53)With a = 1, a remarkably simple BR scaling law is obtained. These scaling lawsimply that as the density of the matter increases the effecive quantities in mediumscale down.We have shown how a slight modification of the Lagrangian resolves the ω prob-lem. However, in modifying the Lagrangian we have taken into account only thephenomenological side of the problem. Multiplying the Wess-Zumino term by thefactor ( χ/f χ ) n has no sound theoretical support. It breaks explicitly the scale in-variance of the Lagrangian. Recall that the dilaton field was introduced into themodel to respect scale symmetry. Furthermore, we don’t have any special rea-son for choosing n = 3, except that it works well. Recently, a more fundamentalexplanation for the behavior in Eq.(1.51) has been found. d In trying to understand what happens to hadrons under extreme conditions, it isnecessary that the theory adopted for the description be consistent with QCD. Interms of effective theories this means that they should match to QCD at a scale close d Private communication by M. Rho on work in progress by H.K. Lee and M. Rho. ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento to the chiral scale Λ χ ∼ πf π ∼ which provides a theoretical support for alow-energy effective field theory for hadrons and which gives, in the chiral limit, anelegant and unambiguous prediction of the behavior of light-quark hadrons at hightemperature and/or at high density. Following the indications of the HLS theory,we have described a Skyrme model in which the dilaton field χ , whose role in densematter was first pointed out by Brown and Rho, and the vector meson fields ρ and ω were incorporated into the Skyrme Lagrangian to construct dense skyrmionmatter.We have presented an approach to hadronic physics based on Skyrme’s philoso-phy, namely that baryons are solitons of a theory described in terms of meson fields,which can be justified from QCD in the large N c expansion. We have adopted thebasic principles of effective field theory. Given a certain energy domain we describethe dynamics by a Lagrangian defined in terms of the mesonic degrees of freedomactive in that domain, we thereafter implement the symmetries of QCD and VM,and describe the baryonic sectors as topological winding number sectors and solvein these sectors the equations derived from the Lagrangian with the appropriateboundary conditions for the sector. In this way one can get all of Nuclear Physicsout of a single Lagrangian. We have studied the B=1 sector to obtain the proper-ties of the single skyrmion, the B=2 sector to understand the interaction betweenskyrmions, and our main effort has been the study skyrmion matter, as a model forhadronic matter, investigating its behavior at finite density and temperature andthe description of meson properties in that dense medium.Skyrme models have been proven successful in describing nuclei, the nucleon-nucleon interaction and pion-nucleon interactions. It turns out that Skyrme modelsalso represent a nice tool for understanding low density cold hadronic matter andthe behavior of the mesons in particular the pion inside matter. We have shownin here that when hadronic matter is compressed and/or heated Skyrme modelsprovide useful information on the chiral phase transitions. Skyrmion matter isrealized as a crystal and we have seen that at low densities it is an FCC crystalmade of skyrmions. The phase transition occurs when the FCC crystal transformsinto a half skyrmion CC one. In our study we have discovered the crucial role of thescale dilaton in describing the expected phase transition towards a chiral symmetryrestored phase. We have also noticed the peculiar behavior of the ω associated toits direct coupling to the baryon number current and we have resolved the problemby naturally scaling the coupling constant using the scale dilaton.Another aspect of our review has been the study of the properties of elementarymesons in the medium, in particular those involved in the model, the pion and thedilaton. Moreover we have described how their properties change when we movefrom one phase to another.A description of the chiral restoration phase transition in the temperature- ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx Skyrmion Approach to finite density and temperature density plane has been presented, whose main ingredient is that the dominantscenario is the absorption of heat by the fluctuating pions in the background ofcrystal skyrmion matter. This description leads to a phase transition whose dy-namical structure is parameter independent and whose shape resembles much theconventional confinement/deconfiment phase transition. We obtain, for parametervalues close the conventional ones, the expected critical temperatures and densities.For clarity, the presentation has been linear, in the sense, that given the La-grangian we have described its phenomenology, and have made no effort to interpretthe mechanisms involved and the results obtained from QCD. In this way we havetaken a ‘bottom up’ approach: the effective theory represents confined QCD and itshould explain the hadronic phenomenology in its domain of validity.The main result of our calculation is the realization that phase transition sce-nario is not as simple as initially thought but contains many features which makeinteresting and phenomenologically appealing. It is now time to try to collect ideasbased on fundamental developments and see how our effective theory and the prin-ciples that guide it realize these ideas. In this line of thought, it is exciting to haveunveiled scenarios near the phase transition of unexpected interesting phenomenol-ogy in line with recent proposals. Acknowledgements
We would like to thank our long time collaborators Dong-Pil Min and Hee-JungLee whose work is reflected in these pages and who have contributed greatly to theeffort. We owe inspiration and gratitude to Mannque Rho, who during many yearshas been a motivating force behind our research. Skyrmion physics had a boom inthe late 80’s and thereafter only a few groups have maintained this activity obtain-ing very beautiful results, which however, have hardly influenced the community.We hope that this book contributes to make skyrmion physics more widely appre-ciated. Byung-Yoon Park thanks the members of Departamento de F´ısica Te´oricaof the University of Valencia for their hospitality. Byung-Yoon Park and VicenteVento were supported by grant FPA2007-65748-C02-01 from Ministerio de Cienciae Innovaci´on.
References
1. J.-e. Alam, S. Chattopadhyay, T. Nayak, B. Sinha, and Y. P. Viyogi, Quark Matter2008,
J. Phys.
G35 , 100301, (2008).2. T. H. R. Skyrme, A Nonlinear field theory,
Proc. Roy. Soc. Lond.
A260 , 127, (1961).3. T. H. R. Skyrme, A Unified Field Theory of Mesons and Baryons,
Nucl. Phys. ,556, (1962).4. F. Karsch, Recent lattice results on finite temerature and density QCD, part II, PoS . LAT2007 , 015, (2007).5. M. Fromm and P. de Forcrand, Revisiting strong coupling QCD at finite temperatureand baryon density. (2008). ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
6. G. ’t Hooft, A Planar Diagram Theory For Strong Interactions,
Nucl. Phys.
B72 ,461, (1974).7. H.-J. Lee, B.-Y. Park, D.-P. Min, M. Rho, and V. Vento, A unified approach to highdensity: Pion fluctuations in skyrmion matter,
Nucl. Phys.
A723 , 427, (2003).8. B.-Y. Park, M. Rho, and V. Vento, Vector mesons and dense skyrmion matter,
Nucl.Phys.
A736 , 129, (2004).9. B.-Y. Park, M. Rho, and V. Vento, The Role of the Dilaton in Dense Skyrmion Matter,
Nucl. Phys.
A807 , 28, (2008).10. H.-J. Lee, B.-Y. Park, M. Rho, and V. Vento, Sliding vacua in dense skyrmion matter,
Nucl. Phys.
A726 , 69, (2003).11. H.-J. Lee, B.-Y. Park, M. Rho, and V. Vento, The Pion Velocity in Dense SkyrmionMatter,
Nucl. Phys.
A741 , 161, (2004).12. A. C. Kalloniatis, J. D. Carroll, and B.-Y. Park, Neutral pion decay into nu anti-nuin dense skyrmion matter,
Phys. Rev.
D71 , 114001, (2005).13. A. C. Kalloniatis and B.-Y. Park, Neutral pion decay in dense skyrmion matter,
Phys.Rev.
D71 , 034010, (2005).14. B.-Y. Park, H.-J. Lee, and V. Vento, Skyrmions at finite density and temperature:the chiral phase transition. (2008).15. E. Witten, Current Algebra, Baryons, and Quark Confinement,
Nucl. Phys.
B223 ,433, (1983).16. G. S. Adkins, C. R. Nappi, and E. Witten, Static Properties of Nucleons in the SkyrmeModel,
Nucl. Phys.
B228 , 552, (1983).17. A. D. Jackson and M. Rho, Baryons as Chiral Solitons,
Phys. Rev. Lett. , 751,(1983).18. E. Witten, Global Aspects of Current Algebra, Nucl. Phys.
B223 , 422, (1983).19. S. Weinberg, Phenomenological Lagrangians,
Physica . A96 , 327, (1979).20. A. A. Migdal and M. A. Shifman, Dilaton Effective Lagrangian in Gluodynamics,
Phys. Lett.
B114 , 445, (1982).21. J. R. Ellis and J. Lanik, Is scalar gluonium observable?,
Phys. Lett.
B150 , 289, (1985).22. M. Bando, T. Kugo, and K. Yamawaki, Nonlinear Realization and Hidden LocalSymmetries,
Phys. Rept. , 217, (1988).23. M. Harada and K. Yamawaki, Hidden local symmetry at loop: A new perspective ofcomposite gauge boson and chiral phase transition,
Phys. Rept. , 1, (2003).24. I. R. Klebanov, Nuclear matter in the skyrme model,
Nucl. Phys.
B262 , 133, (1985).25. A. S. Goldhaber and N. S. Manton, Maximal symmetry of the skyrme crystal,
Phys.Lett.
B198 , 231, (1987).26. A. D. Jackson and J. J. M. Verbaarschot, Phase structure of the skyrme model,
Nucl.Phys.
A484 , 419, (1988).27. Z. Tesanovic, O. Vafek, and M. Franz, Chiral symmetry breaking and phase fluctua-tions: A QED-3 theory of the pseudogap state in cuprate superconductors,
Phys. Rev.
B65 , 180511, (2002).28. G. Kaelbermann, Nuclei as skyrmion fluids,
Nucl. Phys.
A633 , 331, (1998).29. O. Schwindt and N. R. Walet, Soliton systems at finite temperatures and finite den-sities. (2002).30. I. Zahed and G. E. Brown, The Skyrme Model,
Phys. Rept. , 1, (1986).31. A. Jackson, A. D. Jackson, and V. Pasquier, The Skyrmion-Skyrmion Interaction,
Nucl. Phys.
A432 , 567, (1985).32. B. Schwesinger, H. Weigel, G. Holzwarth, and A. Hayashi, The skyrme soliton in pion,vector and scalar meson fields: pi n scattering and photoproduction,
Phys. Rept. ,173, (1989). ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx
Skyrmion Approach to finite density and temperature
33. L. Castillejo, P. S. J. Jones, A. D. Jackson, J. J. M. Verbaarschot, and A. Jackson,Dense Skyrmion Systems,
Nucl. Phys.
A501 , 801, (1989).34. M. Kugler and S. Shtrikman, A new skyrmion crystal,
Phys. Lett.
B208 , 491, (1988).35. N. S. Manton and P. M. Sutcliffe, Skyrme crystal from a twisted instanton on a fourtorus,
Phys. Lett.
B342 , 196, (1995).36. M. Kutschera, C. J. Pethick, and D. G. Ravenhall, Dense matter in the chiral solitonmodel,
Phys. Rev. Lett. , 1041, (1984).37. M. Kugler and S. Shtrikman, Skyrmion crystals and their symmetries, Phys. Rev.
D40 , 3421, (1989).38. H. Forkel et al., Chiral symmetry restoration and the skyrme model,
Nucl. Phys.
A504 , 818, (1989).39. M. F. Atiyah and N. S. Manton, Skyrmions from Instantons,
Phys. Lett.
B222 , 438,(1989).40. M. F. Atiyah and N. S. Manton, Geometry and kinematics of two skyrmions,
Commun.Math. Phys. , 391, (1993).41. R. A. Leese and N. S. Manton, Stable instanton generated Skyrme fields with baryonnumbers three and four,
Nucl. Phys.
A572 , 575, (1994).42. N. R. Walet, Quantising the B=2 and B=3 Skyrmion systems,
Nucl. Phys.
A606 ,429, (1996).43. B.-Y. Park, D.-P. Min, M. Rho, and V. Vento, Atiyah-Manton approach to Skyrmionmatter,
Nucl. Phys.
A707 , 381, (2002).44. R. Jackiw, Quantum meaning of classical field theory,
Rev. Mod. Phys. (3), 681.45. S. Saito, T. Otofuji, and M. Yasino, Pion Fluctuations about the Skyrmion, Prog.Theor. Phys . , 68, (1986).46. H. Yabu, F. Myhrer, and K. Kubodera, Meson condensation in dense matter revisited, Phys. Rev.
D50 , 3549, (1994).47. V. Thorsson and A. Wirzba, S-wave Meson-Nucleon Interactions and the Meson Massin Nuclear Matter from Chiral Effective Lagrangians,
Nucl. Phys.
A589 , 633, (1995).48. W. R. Gibbs and W. B. Kaufmann, The Contribution of the Quark Condensate tothe pi N Sigma Term, nucl-th/0301095 . (2003).49. G. E. Brown and M. Rho, Scaling effective Lagrangians in a dense medium,
Phys.Rev. Lett. , 2720, (1991).50. G. E. Brown, A. D. Jackson, M. Rho, and V. Vento, The nucleon as a topologicalchiral soliton, Phys. Lett.
B140 , 285, (1984).51. R. J. Furnstahl, H.-B. Tang, and B. D. Serot, Vacuum contributions in a chiral effectiveLagrangian for nuclei,
Phys. Rev.
C52 , 1368, (1995).52. C. Song, G. E. Brown, D.-P. Min, and M. Rho, Fluctuations in ’Brown-Rho scaled’chiral Lagrangians,
Phys. Rev.
C56 , 2244, (1997).53. H. Reinhardt and B. V. Dang, Modified Skyrme Model with correct QCD scalingbehavior on S3,
Phys. Rev.
D38 , 2881, (1988).54. K. Zarembo, Possible pseudogap phase in qcd,
JETP Lett. , 59, (2002).55. T. Hatsuda and T. Kunihiro, The sigma-meson and pi pi correlation in hot/densemedium: Soft modes for chiral transition in QCD. (2001).56. H. Fujii, Scalar density fluctuation at critical end point in NJL model, Phys. Rev.
D67 , 094018, (2003).57. R. D. Pisarski and M. Tytgat, Propagation of Cool Pions,
Phys. Rev.
D54 , 2989,(1996).58. H. Leutwyler, Nonrelativistic effective Lagrangians,
Phys. Rev.
D49 , 3033, (1994).59. M. Kirchbach and A. Wirzba, In-medium chiral perturbation theory and pion weakdecay in the presence of background matter,
Nucl. Phys.
A616 , 648, (1997). ctober 24, 2018 17:2 World Scientific Review Volume - 9.75in x 6.5in sm-xxx B.-Y. Park and V. Vento
60. D. T. Son and M. A. Stephanov, Real-time pion propagation in finite-temperatureQCD,
Phys. Rev.
D66 , 076011, (2002).61. A. Bochkarev and J. I. Kapusta, Chiral symmetry at finite temperature: linear vsnonlinear σ -models, Phys. Rev.
D54 , 4066, (1996).62. I. Arsene et al., Quark-gluon plasma and color glass condensate at RHIC? The per-spective from the BRAHMS experiment,
Nuclear Physics A . (1-2), 1.63. U. G. Meissner, Low-energy hadron physics from effective chiral lagrangians withvector mesons, Phys. Rept. , 213, (1988).64. B. D. Serot and J. D. Walecka, The relativistic nuclear many body problem,
Adv.Nucl. Phys. , 1, (1986).65. U.-G. Meissner, A. Rakhimov, and U. T. Yakhshiev, The nucleon nucleon interactionand properties of the nucleon in a pi rho omega soliton model including a dilaton fieldwith anomalous dimension, Phys. Lett.
B473 , 200, (2000).66. L. McLerran, Quarkyonic Matter and the Phase Diagram of QCD. (2008).67. L. McLerran and R. D. Pisarski, Phases of Cold, Dense Quarks at Large N c , Nucl.Phys.