The isovector equation-of-state in heavy ion collisions and neutron stars

Progress in Particle and Nuclear Physics 59 (2007) 365–373www.elsevier.com/locate/ppnp

Review

The isovector equation-of-state in heavy ion collisionsand neutron stars

H.H. Wolter∗

Department of Physics, University of Munich, Munich, Germany

Abstract

We discuss the investigation of the symmetry energy as a function of density in heavy ion collisions,which is still rather uncertain from a theoretical point of view, but of importance in nuclear structure physicsand also in astrophysics. We briefly review transport theory and non-equilibrium and fluctuation effects. Wethen report on investigations of both the low density EOS in the Fermi energy regime and the high densityEOS in intermediate energy collisions. At low density we discuss, in particular, the role and treatment offluctuations and the dynamical fragment formation. At the high density we make connections with neutronstars.c© 2007 Elsevier B.V. All rights reserved.

Keywords: Nuclear equation-of-state; Symmetry energy; Heavy ion collisions; Non-equilibrium; Fragmentation; Kaonproduction; Neutron stars

1. Introduction

A primary motivation for the study of heavy ion collisions has been the investigation of thephase diagram of strongly interacting matter. In this contribution we are interested in the hadronicsector of the phase diagram, where we study the equation-of-state (EOS) of nuclear matter as afunction of density and temperature. However, in recent years another coordinate in this phasediagram, that of the proton to neutron ratio, or the asymmetry, has become of major interest.The asymmetry dependence of the EOS is of relevance today in connection with radioactive

∗ Tel.: +49 89 2891 4009; fax: +49 89 2891 4008.E-mail address: [email protected].

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366 H.H. Wolter / Progress in Particle and Nuclear Physics 59 (2007) 365–373

beam physics and the structure of exotic nuclei, and because of its astrophysical implications.Indeed, the nuclear EOS, and in particular its asymmetry dependence, is crucial in the physicsof supernova explosions and the structure of neutron stars, e.g. in the question, whether neutronstars may be exotic objects in their interior.

The nuclear phase diagram can be explored in the laboratory by heavy ion collisions. Inparticular, it will be of great interest to fully exploit the asymmetry degree of freedom incollisions of very exotic nuclei. A heavy ion collision can be thought of as exploring a path inthe phase diagram depending on the bombarding energy. It proceeds from initial normal nuclei atmore or less asymmetry, to high density and heated nuclear matter at maximum compression todilute nuclear matter in the expansion stage with the possible observation of a phase transition.However, a heavy ion collision is a transient and very non-equilibrated process. One usually doesnot “see” directly the nuclear EOS – a static concept – in a heavy ion collision, but rather has toinfer back to it via theoretical tools, in particular transport theory. While semiclassical transporttheory has been used very successfully in the last few decades to correlate many heavy ion data,there are still open questions about its use in non-equilibrated matter and for fragmentation.

In Section 2 we will discuss the modelization of the EOS, in particular, of its asymmetrydependence. I will briefly discuss transport theory and its extensions and problems. In thefollowing section we will address the low density behavior, and fragmentation reactions witha particular view on the symmetry energy. I will then present some investigations of the highdensity behavior of the EOS with respect to flow variables and kaon production, as well as theconsistency with neutron star observables.

2. The nuclear equation-of-state

Nuclear matter is an idealized concept. Approximately, it exists in the interior of heavy nuclei,and its properties at saturation are extracted from finite nuclei and their low energy excitations.Very asymmetric nuclear matter can most likely be found in the outer core of neutron stars;however, their inner structure is far from certain. Thus theory has to try to give guidance to thebehavior of nuclear matter for high densities, temperatures, and asymmetries.

In recent years a relativistic formulation for nuclear matter has become increasingly popular,because it naturally leads to a much more elegant description of the strong spin–orbit force innuclei. In a field theoretical approach one starts with the hadronic Lagrangian density [1]

L(ψ; σ, ω, ρ, δ, . . .) = ψ[γµDµ− (M − Γσσ − Γδ Eτ · Eδ )]ψ + L0

mes (1)

Dµ= (i∂µ − Γωωµ − Γρ Eρµ · Eτ). (2)

Here, ψ is the nucleon field, and σ, ω, ρ, and δ are the various meson fields, which are coupled tothe nucleon field in a minimal way via coupling constants Γi . L0

mes is the free meson Lagrangian,possibly with self interactions of the mesons. The big success of this formulation is that, withthe coupling fixed from two-body data, i.e. in a “realistic” way, one obtains a good descriptionof nuclear matter in Brueckner theory.

A compilation and assessment of theoretical predictions of the nuclear EOS for symmetricand asymmetric nuclear matter has recently been given in an article for the so-called WCI groupby Fuchs and Wolter [6]. The uncertainty in the symmetric EOS today has been considerablyreduced due to heavy ion experiments, some of which will be referred to later on. The symmetryenergy, on the other hand, is shown in Fig. 1 for several models. These include non-relativisticSkyrme models (SKM∗, SkLya), Brueckner–HF (DBHF), variational calculations with three-body forces (var AV18), RMF with non-local, density-dependent and ρ mesons (NL3, DD-TW,

H.H. Wolter / Progress in Particle and Nuclear Physics 59 (2007) 365–373 367

Fig. 1. Symmetry energy as a function of density as predicted by different models (see text). The left hand panel enlargesthe low density region.

DD-ρδ) and chiral perturbation theories (ChPT) (detailed references are found in Ref. [6]). Whiledifferent theories agree fairly well at about saturation energy, the density dependence is largelyunconstrained both for densities below and above saturation. It is one of the aims of heavy ionphysics today to determine this behavior. This directly translates into an uncertainty of variousother important quantities, like the difference of the effective masses for protons and neutrons orthe isospin-dependent optical potential, or Lane potential [6].

In finite nuclei calculations are done in the Hartree (usually called Relativistic Mean Field,RMF), or sometimes in the Hartree–Fock approximation. Then the coupling coefficients are notthe “realistic” ones from the two-body data, but they are fixed in this approximation to theproperties of nuclear matter, and/or to a number of finite nuclei. However, a connection canbe established between the many-body DBHF calculation and the RMF approach via density-dependent coupling coefficients Γi (ρ), to obtain a density-dependent hadronic field theory [3].The functional dependence can be taken directly from DB calculations [2], or can be adjustedby a fit to data of nuclear matter and some spherical nuclei [4,5]. Parametrizations of this typeprovide highly precise descriptions of nuclear ground states and collective features of nuclei.This approach can be thought of as a density functional approach, where the vertex functionseffectively contain correlations beyond the mean field approach.

In the Lagrangian in Eq. (2) we have included also a δ-meson exchange, or rather a δ-likefield. This is an isovector–scalar field, the analog of the σ field in the isovector sector. It shouldnot necessarily be thought of as the exchange of a physical δ meson, which, in fact, is rathermassive. Rather it takes care of the full isospin nature of the interaction. It is seen e.g. inRef. [2], that it is naturally predicted in a microscopic DB approach, even without introducingexplicitly a δ-meson there. Such a field has usually not been necessary in empirical RMFparametrizations [4]. This is understood since the symmetry energy in this parametrization isproportional to ( fρ − fδ(m∗/EF )

2)ρB , with fi = Γi/mi . Thus the contributions of the ρ and δfields interfere destructively just as this occurs in the isoscalar sector with the σ and ω fields. Theδ field has a density-dependent factor, which originates from the coupling to the scalar (isovector)density. At a fixed density, like approximately in finite nuclei, one may parametrize the symmetryenergy with a ρ field, only. However, a difference appears for the density dependence of thesymmetry energy, because the contribution of the δ field is weakened at higher density. Thus, aρδ parametrization of the isovector part of the EOS predicts a stiffer density dependence.


3. Transport descriptions of heavy ion collisions

Heavy ion collisions represent a strong non-equilibrium process, which is extremely difficultto describe fully in a quantum-mechanical way. For many purposes such a description is alsonot needed, since, because of the complicated final state, ensemble averages are appropriateto describe the data. While for certain questions the use of thermal or hydrodynamical modelsmay be sufficient, for a complete evolution from the initial to the final state, one has to resortto transport approaches, which describe the evolution of the one-body phase space distributionf (x, p) in a semiclassical approximation, with a collision term of the Boltzmann type. This is theBoltzmann–Uehling–Uhlenbeck (BUU) or Boltzmann–Nordheim–Vlasov (BNV) approach [8],which has been used extensively and altogether successfully in the last decades to interpret heavyion collisions.

In a relativistic form the transport equation reads[p∗µ∂x

µ +(

p∗ν Fµν + m∗∂µx m∗

)∂ p∗

µ

]f (x, p∗) = IC, (3)

where p∗µ = pµ − Σµ is the kinetic momentum, m∗

= m − Σs the effective (Dirac) mass, andFµν = EpµΣ ν

− EpνΣµ is the field strength tensor. The lhs is the Vlasov equation of the evolutionof the phase space distribution in a mean field, namely the scalar and vector self energies Σs,Σµ.The dissipation effects due to two-body collisions, on the average, are included in the collisionterm IC on the rhs of Eq. (3)

IC =12

∫d4 p2

E∗p2(2π)3

d4 p3

E∗p3(2π)3

d4 p4

E∗p4(2π)3

(p∗+ p∗

2)2 dσ

dΩ inmedδ4(p + p2 − p3 − p4)

×[

f (x,p3) f (x,p4) f (x,p) f ((x,p2))− f (x,p) f (x,p2) f (x,p3) f ((x,p4))].

(4)

Here dσdΩ inmed is the in-medium NN cross section, and the last terms are the occupation and

blocking factors f = (1 − f ), where the latter represent the action of the Pauli principle. Thistransport equation is a non-linear partial integro-differential equation, which has not been solveddirectly as such. Rather, the so-called test particle method is used to simulate it [8]. The collisionterm is usually evaluated stochastically by using the criterion, that two test particles perform anelastic collision, if they approach within a distance depending on the in-medium cross sectiond =

√σinmed/Nπ (N , number of test particles/nucleon).

Since we are interested here especially in effects of asymmetry, then one has to treat protonsand neutrons separately, since their mean fields and also their isospin-dependent cross sectionsare different. Thus one has coupled transport equations for protons and neutrons. At higherenergies new particles can be produced through inelastic NN collisions, e.g. above about 300MeV cm energy ∆ resonances are produced, which decay into nucleons and pions. At higherenergies also strangeness is produced in form of hyperons and K mesons. Again many differentchannels have to be considered and many inelastic cross sections are needed to describe theseprocesses which only partially can be taken from experiment.

However, in a non-equilibrium approach the particles are actually quasiparticles with aspectral function of finite width, i.e. with a finite life time, either due to collision broadening orbecause of decay processes. This is expressed in the quantum transport theory of Kadanof–Baymtheory [9], but not yet fully explored [10]. It also links the mean field terms in the Vlasovpart to the in-medium cross section in the collision term, which are connected through the


G-Matrix [12]. The G-Matrix depends on the density, or more generally on the phase spacedistribution, since the intermediate propagator depends on this, and one would have to obtain theG-Matrix for the non-equilibrium phase space configuration of the heavy ion collision, whichis clearly not possible. There have been attempts in our group to take this into account in anapproximate way [13].

4. Low density EOS and dynamical fragmentation

The low density behavior of the EOS can be investigated in heavy ion collisions in the finalexpansion stage of a central collision or in the decay of the spectator in a peripheral collision.In particular, one may enter the region of thermodynamical instability, i.e. the region of theliquid–gas type phase transition, which manifests itself in fragmentation processes, i.e. in aseparation of the dilute system into heavier fragments (“liquid”) embedded in free nucleons orlight nuclei (“gas”). Fragmentation is in fact the dominant final state in heavy ion reactions inthis regime, as demonstrated for central collisions by the INDRA and for spectator decay by theALADIN collaborations. An extensive review of the work of the Catania group in this domainhas been given by Baran et al. [18].

Transport theories of the BUU type treats the evolution of the one-body density under theinfluence of a mean field and the average dissipative effects of two-body collisions. Dissipationand fluctuations are intimately connected, as expressed by the fluctuation–dissipation theorem.Fragmentation, on the other hand, is intrinsically a many-body correlation, which is not containedin a one-body description. However, in a thermodynamically instable region it is triggered byfluctuations of the one-body density, which act as the seeds of fragment formation and areexponentially enhanced by mean field dynamics. Transport theory with fluctuations is describedby the Boltzmann–Langevin (BL) equation, which in addition to the collision term also includesa fluctuation term. The BL equation has been studied numerically in model systems [11] butis too complicated for realistic applications. Various approximation schemes to such a fulltreatment have been proposed. Because of the fluctuations the distribution function can besplit as f (r, p, t) = f (r, p, t) + δ f , i.e. into a mean field part f , which is described by thedissipative BUU equation, and a fluctuation part δ f . We have argued [21] that the fluctuationsare given locally in space and time by the statistical fluctuations of an equilibrated system ofthe corresponding density and temperature. Then the equilibrium variance of the distributionfunction is given as σ 2

equil = 〈( f − f )2〉 = f (1 − f ). In the Stochastic Mean Field (SMF)model [19] we have implemented such a scheme, where the fluctuations are inserted into thephase space distribution at appropriate times during the instability phases. With a fragmentrecognition (coalescence) algorithm the fragments are, finally, identified. Thus fragments aregenerated dynamically. It was shown that this procedure is consistent in schematic models withthe BL equation [20].

Fragmentation processes have also be used to investigate the isovector EOS. The initialasymmetry of the system (here 124Sn +

124Sn, I = (N − Z)/A = .19) can be compared with theasymmetry of the produced IMF’s. This is shown in Fig. 2, where the mean asymmetry of theIMF is shown as a function of the charge of the fragment for central (left) and peripheral (right)collisions. The initial asymmetry is indicated. It is seen that in the central collision the asymmetryof the fragments is lower than the initial asymmetry. Thus a charge fractionation has taken place,and the dilute system splits into more symmetric fragments (“liquid”) and into neutrons andneutron-rich light particles (“gas”). The fragment asymmetry, however, is different for the neckfragmentation, also shown in the figure. Here the neck IMF are actually more asymmetric than


Fig. 2. Fragmentation observables in 124Sn +124Sn reactions for central (left, b = 2 fm) and semiperipheral (right,

b = 6 fm) collisions. In each case the following quantities are shown: mean asymmetry I = (N − Z)/A of fragments asa function of fragment charge at the time of freeze-out.

the original nuclei. This is due to a neutron flow into the dilute neck region, which has beencalled isospin migration. Both these phenomena can be well interpreted from the behavior ofthe chemical potential for protons and neutrons as a function of density. Since these chemicalpotentials depend on the isovector EOS, the fractionation and migration phenomena describedabove also depend on the isovector EOS. Indeed, this is what is seen in actual calculation [21].It is thus a possible way to obtain information on the symmetry energy at low densities. Inpractice, however, the secondary evaporation of the fragments tends to reduce the sensitivity tothe isovector EOS. It has been suggested that fragment–fragment correlations may be a way toobtain more information here [22].

5. The EOS at high density and neutron stars

The high density EOS can be investigated with relativistic collision energies, where up tothree times the saturation density is reached. In particular, the high density symmetry energycan be studied. This question touches intimately with the question of the structure of neutronstars. The primary observable sensitive to the pressure at the maximum density is the nucleon (orparticle) flow, i.e. the momentum distributions of the hadrons in the final state. The momentumdistribution is usually expanded in terms of a Fourier series for the azimuthal distribution of theyield N (Θ, y, pt ; b) = N0(1 + v1(y, pt ) cos(Θ) + v2(y, pt ) cos(2Θ) + · · ·) as a function ofthe rapidity y and transverse momentum pt for a given impact parameter b, with v1 the sidewardand v2 the elliptic flow. Systematic investigations have been performed by Danielewicz et al.in a non-relativistic momentum-dependent parametrization of the EOS [14]. From this one mayobtain limits of the EOS, which are compatible with the data, and it is seen that several modelsfall outside of this region. One may also see that the neutron EOS is much less constrained, whichpoints to the need to obtain more information experimentally. In Ref. [7] we have systematicallyinvestigated flow in the above relativistic formulation. We describe the flow from low SIS to AGSenergies with microscopic interactions, provided non-equilibrium effects, as mentioned above,are taken into account.

Probably the most stringent determination of the EOS has been obtained from kaon productionnear threshold energies. Pions and ∆s are the most copious particles in intermediate energycollisions. For strangeness production at energies below the kaon threshold the primary source


Fig. 3. Ratios of π−/π+ (left) and K +/K 0 (right) yields as a function of the incident energy for central Au + Aucollisions for different assumptions on the symmetry energy: N L (no isovector dependence, black disks), N Lρ (asysoft,red squares), N Lρδ (asystiff, green diamonds). In addition we show for Ebeam = 1 A GeV results with a densitydependent ρ-coupling fρ (ρB ) (blue triangles) and for the N L model including kaon potentials (purple crosses). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

are secondary reactions, of which the most important ones are associated strangeness productionN∆ → NΛK and the strangeness exchange Nπ → ΛK . In Ref. [15] the ratio of kaonproduction in a heavy system (Au + Au) with a high compression is compared to the productionin a light system (C + C). It is seen that this clearly favors a soft EOS.

The sensitivity seen in the investigation in Ref. [15] arises from two effects: from the closenessof the collision energy to the production threshold and to the fact that K + mesons interact weaklywith nuclear matter, and thus carry information from the dense phase of the collision. There are,however, different species of kaons with different charges, which have slightly different massesand also different mean field potentials, and thus different thresholds. Therefore the ratio ofdifferently charged kaons, i.e. the ratio K +/K 0, should be an interesting observable to investigatethe high density symmetry energy. This is demonstrated in Fig. 3 from Ref. [16], where the ratiosof π−/π+ and K 0/K + are calculated for isovector EOS’s of different stiffness (exemplified hereby the RMF EOS discussed above without isovector dependence, and with isovector energy withand without a δ-meson). Pions interact strongly with nuclear matter and thus carry informationabout the whole evolution of the collision and not only the high density phase, and thus theirsensitivity is low. In contrast, the sensitivity of the kaon ratio is appreciable, and it should bea suitable variable to investigate the symmetry energy at high density. We also show that theeffect depends essentially on the presence of the δ-meson and is robust against assumptions onthe ρ-coupling and on a kaon mean field potential.

The high density isovector EOS is also of direct relevance to the structure of neutron stars(NS). The solution of the Tolman–Oppenheimer–Volkov equation yields the NS mass for a givenEOS for a given starting central density. On the other hand, the proton fraction x = Z/N fromβ-equilibrium and charge neutrality is a direct function of the symmetry energy. In Fig. 4 weshow this dependence for the NS mass and the proton fraction for a number of different EOS’s,which represent a fair selection of recent theoretical models [17]. The masses obtained have tobe compared to observed NS masses, which are typically in the range of 1–1.5 solar masses.All EOS’s obtain such masses. Recently a very heavy neutron star of 2.1 solar masses wasdiscovered, and some of the EOS’s begin to have problems, explaining such a heavy NS.


Fig. 4. (Left panel) Neutron star masses as a function of the central density for different EOS (see text). Shown are limitsfor typical neutron star masses and those for the newly discovered heavy NS with one and two σ errors. (Right panel)Proton fraction x = Z/A for the same EOS’s as a function of central density. Indicated also is the region of the onset ofthe direct URCA process. This point is indicated on the curves of the left hand panel as a dot.

In addition one has to consider the cooling behavior of a NS. Above a proton fraction of about11% the direct URCA process, i.e. the β-decay of the proton, can take place, which leads toemission of neutrinos and thus to a rapid cooling of the NS, such that it is not observable. Thusone has to require, that the URCA limit is not reached for stars with masses of the heavy NS, andmuch less for typical NS masses. The point where the URCA limit is reached in the right handpanel is marked as a dot in the curves on the left hand panel. It is seen that several of the EOS’sreach this limit even in the range of typical NS masses, and are thus not compatible with thecooling curves. These considerations are somewhat qualitative, because additional considerationsfor the heat conduction through the crust are important, but this does not change the qualitativeresult.

Other checks have been performed in Ref. [17] against further observables of NS, such asmass–radius relations, gravitational vs. baryonic mass, etc., but also for the results of heavy ioncollisions, such as whether the EOS falls into an acceptable region of Ref. [14], and for kaonproduction. It was found that none of all the EOS’s considered satisfies all checks. This maybe interpreted in several ways: it shows the usefulness of such a simultaneous comparison toNS and heavy ion data, and stresses the need to find even better EOS’s. If, however, one wouldnot succeed to find a satisfactory hadronic EOS, then this could be an indication that exoticphenomena have to be considered more seriously in NS.

6. Summary

In this lecture we have reviewed how heavy ion collisions can be used to investigate variousaspects of the nuclear equation-of-state, such as the high and low density behavior, the isospinpart of the EOS, and phase transitions. Transport theory is used to describe the very non-equilibrated collision process. Microscopic input can be used in such transport descriptions,making a link to nuclear many-body theory, as well as to phenomenological approaches in thevein of density functional theory. The density dependence of the symmetry energy is discussedin particular. At high density we propagate kaon production as a sensitive probe, and we discussconnections to neutron star physics. Fragmentation is a process that appears at low density andwhich can be seen as a consequence of a liquid–gas phase transition, and which is sensitive to theisovector EOS. The role of fluctuations is essential to describe fragmentation, and it is discussedhow these can be treated in a transport approach. Many of the effects discussed obviously scale


with the asymmetry of the system, and thus the availability of very asymmetric, exotic beams, isvery desirable in order to obtain more constraints on the symmetry energy.

Acknowledgements

This account of the work on the isovector equation of state has been done together with manyco-workers, which are too many to list them all as authors. I want to recognize, in particular,M. Colonna, M. Di Toro, G. Ferini, T. Gaitanos, and C. Fuchs.

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The isovector equation-of-state in heavy ion collisions and neutron stars