Probing the nuclear equation of state by $K^+$ production in heavy ion collisions

5 Pages·0.12 MB·English

by  C. Fuchs

arXiv:nucl-th/0011102v2 17 Jan 2001
1
Probing the nuclear equation of state by K+ production in heavy ion collisions
C. Fuchs, Amand Faessler, E. Zabrodin
Institut fu¨r Theoretische Physik der Universit¨at Tu¨bingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
Yu-Ming Zheng
China Institute of Atomic Energy, P.O. Box 275 (18), Beijing 102413, China
peron,respectively)andprocessesπB YK+ induced
ThedependenceofK+productiononthenuclearequation −→
by pion absorption. In the early studies on subthreshold
of stateis investigated in heavyion collisions. Anincrease of K+ production only the baryon induced channels have
theexcitationfunctionofK+ multiplicitiesobtainedinheavy
been considered [12,13]. As shown in [14–16] the pionic
(Au+Au) over light (C+C) systems when going far below
channelplays animportantrole,particularinheavysys-
thresholdwhichhasbeenobservedbytheKaoSCollaboration
tems. Taking this fact into account the kaon yield could
strongly favours a soft equation of state. This observation
beexplainedadoptingrealisticmomentumdependentnu-
holds despite of the influence of an in-medium kaon poten-
clear interactions [9,11,15,16]. However, the dependence
tialpredicted byeffectivechiralmodelswhich isnecessary to
reproduce theexperimental K+ yields. of the kaon production on the nuclear EOS turned now
out to be too small for definite conclusions.
The aim of the present work is to study the question
if in the meantime decisive information on the nuclear
EOS can be extracted from subthreshold kaon produc-
From the very beginning kaons have been considered tion in heavy ion collisions. There are several reasons
as one ofthe best probes to study dense and hot nuclear why it appears worthwhile to do this: Firstly, there has
matter formed in relativistic heavy ion collisions [1]. In been significant progress in the recent years towards a
particular at incident energies below the corresponding more precise determination of the elementary kaon pro-
production thresholds in free space K+ mesons are cre- duction cross sections [17,18], based also on new data
atedintheearlyandhighdensityphaseofsuchreactions pointsformtheCOSY-11forthereactionspp pK+X
−→
and – due to strangeness conservation – are not reab- very close to threshold [19]. Secondly, the KaoS Collab-
sorbed by the nuclear environment. Furthermore, there oration has performed systematic measurements of the
existstrongevidencesthatkaonschangetheirproperties K+ production far below threshold in heavy (Au+Au)
insidethenuclearmediumaspredictedbyeffectivechiral and light (C +C) systems [20]. Looking at the ratios
models[2,3]. Theinvestigationofapartialrestorationof built from heavy and light systems possible uncertain-
chiral symmetry in dense matter probed by K mesons ties whichmightstill existinthe theoreticalcalculations
has strongly stimulated both experimental and theoreti- should cancel out to a large extent which allows to draw
cal efforts in the recent years [4–11]. reliable conclusions. Furthermore, far below threshold
The original motivation to study the kaon production the kaon production is a highly collective process and a
in heavy ion reactions at intermediate energies, namely particular sensitivity to the compression of the partici-
to extract information on the nuclear equation of state pant matter is expected.
(EOS)athighdensitiesisamatterofcurrentdebate. Al- The present investigations are based on the Quantum
ready in the first theoretical investigations by transport Molecular Dynamics (QMD) transport model [21]. For
models it was noticed that the K+ yield reacts sensi- the nuclear EOS we adopt soft and hard Skyrme forces
tive on the nuclear equation of state [1,12–14], i.e. it correspondingto a compressionmodulus of K=200MeV
was found to be about a factor 2–3 larger when a soft and380MeV,respectively,andwithamomentumdepen-
EOSwasapplied comparedtoa hardEOS.At thattime dence adjusted to the empirical optical nucleon-nucleus
the available data [5] already favoureda soft equationof potential [21]. The saturation point of nuclear matter is
state. However, calculations as well as the experimen- thereby fixed at E = 16 MeV and ρ = 0.17 fm−3
B sat
tal data were still burdened with large uncertainties. It [21]. The calculations−include ∆ and N∗(1440) reso-
was further noticed [12] that the influence of the repul- nances with [22]. The QMD approach with Skyrme in-
sivemomentumdependentpartofthenuclearinteraction teractionsiswelltested,containsacontrolledmomentum
leadstoastrongsuppressionofthekaonabundances. An dependence and provides a reliable descriptionof the re-
underprediction of K+ yields using more realistic mo- action dynamics in the SIS energy range, expressed e.g.
mentum dependent forces was due to the fact that the by collective nucleon flow observables as well as particle
production mechanism is twofold: baryon induced pro- production. Also the EOS predicted by microscopic ap-
cesses BB BYK+ where the kaon is created via proaches (G-matrix) [23] is similar to the soft version of
−→
binary baryon–baryon collisions (B stands either for a the Skyrme interaction for densities up to 2ρ .
sat
nucleon or a ∆–resonance and Y for a Λ or a Σ hy- We further consider the influence of an in-medium
2
22
kaonpotentialbasedoneffectivechiralmodels[2–4,9,25]. 1100
The K+ mean field consists of a repulsive vector part
Vµ = 3/8fπ∗2jµ and an attractive scalar part ΣS = Au+Au*10−1
m m∗ = m m2 ΣKNρ +V Vµ. Here j is
K − K K −q K− fπ2 s µ µ 110011
the baryon vector current and ρ the scalar baryon den-
s
sity and ΣKN = 450 MeV. Following [25] in the vector b]b]
field the piondecay constant in the medium f∗2 =0.6f2 mm
π π
is used. However, the enhancement of the scalar part ) [) [ 110000 C+C
using f∗2 is compensated by higher order contributions ++
π KK
in the chiral expansion [25], and therefore here the bare ((
σσ
value is used, i.e. Σ ρ /f2. Compared to other chi-
KN s π
ral approaches [3,4] the resulting kaon dispersion rela- 1100−−11 soft EOS, w/o pot
hard EOS, w/o pot
tion shows a relatively strong density dependence. The
soft EOS, with pot
increase of the in-medium K+ mass m˜K, Eq. (2), with hard EOS, with pot
this parameterisation is still consistent with the em- KaoS (exp)
pirical knowledge of kaon-nucleus scattering and allows 1100−−22
to explore in-medium effects on the production mecha- 00..55 11..00 11..55 22..00
nism arising from zero temperature kaon potentials. For EE [[GGeeVV]]
the kaon production via pion absorption πB YK+ llaabb
−→
the elementary cross section of [24] are used. For the FIG. 1. Excitation function of the K+ production cross
NN BYK+ channels we apply the cross sections section in Au+Au (scaled by 10−1) and C +C reactions.
−→
of Ref. [17] which give a good fit to the COSY-data The calculations are performed with in-medium kaon poten-
close to threshold. For the case of N∆ BYK+ and tial and using a hard/soft nuclear EOS and are compared to
∆∆ BYK+ reactions experimental−a→re rare. Thus data from the KaoS Collaboration [8,20]. For C +C also
−→
we rely on the model calculation of ref. [18]. In the case calculations without kaon potential are shown.
∗
that a N resonance is involved in the reaction we used
the same cross section as for nucleons. In the presence In Fig. 1 the K+ excitation function for Au + Au
of scalar and vector fields the kaon optical potential in and C +C reactions starting from 0.8 AGeV which is
·
nuclearmatterhasthesamestructureasthecorrespond- far below threshold (Ethr = 1.58 GeV) are shown. The
ingSchroedingerequivalentopticalpotentialfornucleons calculations are performed for a soft/hard EOS includ-
[23] ing the in-medium kaon potential. For both systems the
agreement with the KaoS data [8,20] is very good when
1 Σ2 V2 a soft EOS is used. In the large system there is a visi-
U (ρ,k)= Σ + k Vµ+ S − µ . (1)
opt S µ ble EOS effect which is absent in the light system. To
− m 2m
K K
estimate the influence of the in-medium kaon potential
and leads to a shift of the thresholds conditions inside for C+C also calculations without potential are shown.
the medium. To fulfil energy-momentum conservation Already in the light system the K+ yield is reduced by
the optical potential is absorbed into an newly defined about 50% by the influence of the potential which is es-
effective mass sential to reproduce the data [8].
To extractmoreclear informationonthe nuclearEOS
m˜ (ρ,k)= m2 +2m U (ρ,k) (2) next we consider the ratio R of the kaon multiplicities
K q K K opt
obtainedinAu+AuoverC+C reactions,normalisedto
whichisaLorentzscalarandsetsthecanonicalmomenta the correspondingmass numbers. In Fig. 2central(b=0
on the mass-shell 0 = k2 m˜2. Thus, e.g., the thresh- fm) collisions are analysed. Going far below threshold
µ− K
oldconditionforK+ productioninbaryoninducedreac- R stronglyincreaseswhen a soft EOSis appliedwhereas
tions reads √s m˜ +m˜ +m˜ with √s the centre– the increase of R is much less pronounced using the stiff
B Y K
of–mass energy≥of the colliding baryons. For a consis- EOS and R even slightly drops at the lowest energy.
tent treatment of the thresholds the scalar and vector Hence, this ratio reflects the higher compression
baryon mean fields entering into eq. (2) are determined achieved in the heavy system. The C +C system, on
from two versions of the non-linear Walecka model with the other hand, is too small to develop a significantly
K=200/380MeV, respectively [13]. The hyperon field is larger compression in the case of a soft EOS compared
therebyscaledby2/3whichyieldsalsoagooddescription tothehardEOS.Moreover,inthelattercasetheslightly
oftheΛflow[26]. Sincetheparameterisationschosenfor moreenergeticbinarycollisionsleadeventoahigherK+
the non-linearWaleckamodelyieldthe sameEOSasthe yield in the case of a hard EOS at 0.8 AGeV. Remark-
·
Skyrme ones, the overall energy is conserved. The kaon ably, this behaviour is seen despite of the presence of an
production is treated perturbatively and does generally in-mediumkaonpotentialwhichactsoppositetotheEOS
not affect the reaction dynamics [6]. effect: a higher compressionincreases the kaonyield but
3
alsothe value ofthe in-mediumkaonmasswhich,onthe createdandP isthecorrespondingproductionprobabil-
i
other hand, tends to lowerthe yield again. However,the ity. Forthecomparisonofthetwosystemsthecurvesare
increase of the in-medium mass goes linear with density normalised to the corresponding mass numbers. Fig.3
whereas the collision rate per volume increases approxi- illustrates several features: Only in the case of a soft
mately with ρ2. E.g. in centralAu+Au reactions at0.8 EOSthemeandensitiesatwhichkaonsarecreateddiffer
AGeV the average density < ρ > at kaon production is significantly for the two different reaction systems, i.e.
·
enhanced from 1.47 to 1.57 ρ switching from the hard <ρ/ρ >=1.46/1.40(C+C) and 1.47/1.57(Au+Au)
sat sat
to the soft EOS. This leads to an averageshift of the in- using the hard/soft EOS. Generally, in C +C densities
mediummass(2)comparedtothevacuumvalueof55/61 above 2ρ are rarely reached whereas in Au+Au the
sat
MeV using the hard/soft EOS, i.e. a relative shift of 6 kaonsarecreatedatdensitiesuptothreetimessaturation
MeV between soft and hard. However, collective effects density. Furthermore,forC+C thedensitydistributions
like the accumulation of energy by multiple scattering areweaklydependentonthe nuclearEOS.Thesituation
show a higher sensitivity on the compression resulting changes completely in Au+Au. Here the densities pro-
in an enhancement of the available energy < √s >= 90 fileshowsapronouncedEOSdependence[13]. Moreover,
MeV applying the soft EOS. For C + C this effect is the excess of kaons obtained with the soft EOS origi-
reverse (< √s >= 45 MeV) since the system is too nates almost exclusively from high density matter which
−
small to develop a significant difference in compression demonstrates that compression effects are probed.
and more repulsive collisions enhance the K+ yield at
low energies. This effect disappears above 1.0 AGeV.
·
ThereexiststhusavisuableEOSdependenceofthekaon
multiplicities.
Au+Au soft EOS
hard EOS
5.0 3m] 10−4
soft EOS, w/o kaon pot [f
4.5 hard EOS, w/o kaon pot ρ
C soft EOS, with kaon pot d
)C+ 4.0 hard EOS, with kaon pot A)/
A /
+
/K+ 3.5 MK
M
(
d
(
/ u 3.0 C+C
A
+
u
A 2.5
)
A
/
K+ 2.0 −5
M 10
0 1 2 3 4
(
1.5 ρ/ρ
sat
1.0
FIG.3. Kaon multiplicities (normalised to the mass num-
0.5 1.0 1.5 2.0
bers of the colliding nuclei) as a function of the baryon den-
E [GeV] sityatthespace-timecoordinateswheretheK+ mesonshave
lab
beencreated. Central(b=0fm)Au+AuandC+C reactions
FIG. 2. Excitation function of the ratio of K+ multiplic- at0.8A·GeVareconsidered. Thecalculations areperformed
ities obtained in central (b=0 fm) Au + Au over C + C within-mediumkaonpotentialandusingahard/soft nuclear
reactions. The calculations are performed with/without EOS.
in-mediumkaonpotentialandusingahard/softnuclearEOS.
The comparison to the KaoS data [20] is finally made
Toobtainaquantitativepictureoftheexploreddensity
in Fig. 4. Here only calculations including the kaon po-
effects in Fig. 3 the baryondensities are shownat which
tential are shown since it is already clear from Fig. 1
the kaons are created. The energy is chosen most below
that without the potential one is not able to reproduce
threshold, i.e. at 0.8 AGeV and only central collisions
the experimental yields. The calculations are performed
·
are consideredwhere the effects are maximal. dM /dρ
K+ under minimal bias conditions with bmax = 11 fm for
is defined as
Au +Au and b = 5 fm for C +C and normalised
max
to the experimental reaction cross sections [8,20]. Both
NK+ dPi calculations show an increase of R with decreasing inci-
dM /dρ= (3)
K+ X dρB(xi,ti) dent energy down to 1.0 AGeV. As already seen for the
i ·
central collisions this increase is much less pronounced
where ρ is the baryon density at which the kaon i was
B
4
using the stiff EOS. In the latter case R even drops for calculationsindicatethattheK+ productiongetshardly
0.8 AGeV whereas the soft EOS leads to an unrelieved affected by compressional effects in C +C but is highly
·
increase of R. At 1.5 AGeV which is already very close sensitivetothehighdensitymatter(1 ρ/ρ 3)cre-
sat
tothresholdthedifferen·cesbetweenthetwomodelstend atedinAu+Aureactions. Resultsfort≤heK+ ex≤citation
to disappear. The overall behaviour of R is found to function in Au+Au over C +C reactions as measured
be quite independent of the various production channels bytheKaoSCollaboration,stronglysupportthescenario
with initial states i = NN,πN,N∆,π∆,∆∆. Ratios withasoftEOS.Thisstatementisalsovalidwhenanen-
R built separately for the individual channels show in hancement of the in-medium kaon mass as predicted by
i
both cases (soft or hard) a similar energy dependence as chiral models is taken into account.
the total R (except of R which tends to remain large The authors would like to acknowledge valuable dis-
π∆
alsoathighenergies). Thetransportcalculationsfurther cussions with J. Aichelin, H. Oeschler, P. Senger and C.
demonstratethattheincreaseofRisnotisnotcausedby Sturm. ThisworkwassupportedinpartbytheDeutsche
a trivial, i.e. EOS independent limitation of phase space Forschungsgemeinschaft (DFG) under grant 446 CHV-
at low energy in the small system. This is supported by 113/91/1,and the National Natural Science Foundation
the fact that the number of collisions which the involved ofChina(NSFC)undergrantno19775068and19975074.
particles encountered prior to the production of a kaon
and which is a measure of the collectivity provided by
the systemdoes notreacha sharplimit for C+C atlow
energies. ThestrongincreaseofRcanbedirectlyrelated
to higher compressible nuclear matter. The comparison
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5