21 September 2000

Ž .Physics Letters B 489 2000 273–279www.elsevier.nlrlocaternpe

Correlation between the charged kaon ratioand the baryon phase-space density in heavy-ion collisions

Fuqiang Wang 1

Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received 28 April 2000; received in revised form 10 July 2000; accepted 11 August 2000Editor: J.P. Schiffer

Abstract

It is found that the average baryon phase-space density obtained from the ratio of deuteron to proton yields is nearlyconstant over centrality in AuqAu collisions at the AGS. The finding offers an explanation for the puzzling centralityindependence of the ratio of charged kaon total yields. The correlation between the charged kaon ratio and the averagebaryon phase-space density is studied for central heavy-ion collisions of various systems over a wide range of beam energy.It is found that the charged kaon ratio and the average baryon phase-space density both increase with decreasing beamenergy, and are strongly correlated. Such study may provide a new approach to search for medium effects on the kaon mass.q 2000 Elsevier Science B.V. All rights reserved.

PACS: 25.75.-q; 25.75.Dw

Strangeness production has been extensively stud-ied in heavy-ion collisions because enhancedstrangeness production may signal the formation ofQuark-Gluon Plasma. It has been observed that kaonproduction per participant nucleon increases withcentrality in SiqAl, SiqAu, and AuqAu colli-

Ž .sions at the Alternating Gradient Synchrotron AGS ,reaching a factor of 3–4 enhancement in central

w xcollisions with respect to pqp interactions 1 .However, the ratio of charged kaon total yieldsŽ q y.K rK varies little with the collision centralityw x1 . Similar results have been also observed in Niq

Ž .Ni collisions at the Schwer Ionen Synchrotron SIS

1 Current address: Department of Physics, Purdue University,1396 Physics Building, West Lafayette, IN 47907, USA

w x2 and PbqPb collisions at the Super Proton Syn-Ž . w x q ychrotron SPS 3 . The nearly constant K rK is

puzzling because Kq and Ky are thought to beproduced by different mechanisms: Ky ’s are pro-duced by pair production together with a Kq, whileKq ’s can be produced, in addition, by associateproduction together with a hyperon. These differentproduction mechanisms lead to different rapidity dis-tributions which are observed, namely, the Kq ra-

y w xpidity distribution is broader than K ’s 1 . Onenaively expects that in heavy-ion collisions the rela-tive contribution of associate over pair productionincreases with centrality, because the associate pro-duction threshold for kaons is lower than the pairproduction threshold, and there are more particle

Ž .re-interactions at lower than the full energy incentral than peripheral collisions. Therefore, one ex-pects an increasing KqrKy with centrality.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00975-8

( )F. WangrPhysics Letters B 489 2000 273–279274

On the other hand, the constituent quark modelw x4,5 has been successfully applied to describe parti-

w xcle ratios in heavy-ion collisions 6,7 . In the con-q ystituent quark model, K sus and K sus. Hence,

q y ŽK rK depends on the baryon baryonyanti-.baryon phase-space density established in the colli-

Žsion zone at chemical freeze-out when particlescease to interact inelastically hence particle abun-

. w xdances and ratios are fixed 8,9 . In this picture, theobserved constant KqrKy implies a constant baryonphase-space density over the collision centrality. Theextraction of the baryon phase-space density atchemical freeze-out heavily relies on chemical equi-librium models; such model studies have been so farlimited to central collisions. On the other hand, thebaryon phase-space density at final kinetic freeze-outŽwhen particles cease to interact both inelastically

.and elastically can be readily extracted fromdeuteron coalescence data. This is because deuterons,due to its small binding energy, are mostly formedby a proton and a neutron overlapping in phase-space,and cannot subsequently survive collisions with otherparticles. In this Letter, we study the kinetic freeze-

Ž² :Kin.out average baryon phase-space density f as aB

function of centrality, especially for AuqAu colli-² :Kinsions at the AGS. We show that the f value inB

AuqAu collisions is nearly constant over centrality.The result may indicate that the average baryonphase-space densities at kinetic and chemical freeze-

Ž² :Ch .out f are connected in a way that is indepen-B

dent of centrality. We further study the correlationq y ² :Kinbetween K rK and f in central heavy-ionB

collisions and demonstrate that KqrKy increases² :Kin Žwith f from SPS, to AGS, to SIS decreasingB

.beam energy .However, there is a physics origin that may result

in an opposite behavior, a decreasing KqrKy with² :Chincreasing f , and that is mass modification inB

nuclear medium. It is predicted that the Ky effectivemass is lower in nuclear medium than in free space,

q w xand the K effective mass slightly higher 10,11 . Ay Ž q.decreasing K andror an increasing K effective

mass would yield a reduced KqrKy ratio. In fact,the KaoS Collaboration has studied KqrKy in NiqNi collisions at the SIS and inferred a significant

y w xdrop in the K effective mass 2 . On the theoryside, however, different conclusions can be reached

w xdepending on model assumptions 12,13 . In this

Letter, we take a different approach, i.e., studyingq y ² :KinK rK as a function of f . If the mass modifi-B

cation is large, one may observe a spectacular phe-nomenon: KqrKy increases and then decreases with² :Kinf . Such phenomenon is not observed, however,B

we argue that this new approach may provide a moredirect way to search for kaon medium modifications.

The Letter is organized as follows. First we pre-sent the rapidity distributions of the average nucleonphase-space density at the AGS for collisions of

² :Kinvarious centralities. Then we extract f for eachB

² :Kincentrality and show that the value of f is nearlyB

constant. Finally we demonstrate that KqrKy is² :Kinstrongly correlated with f using the centralB

collision data and discuss implications of the resultson medium effects to the kaon mass.

In the coalescence model, the neutron phase-spacedensity is related to the ratio of deuteron to proton

Ž . w x w xyields drp 14–16 . Following Ref. 16 , but notassuming identical neutron and proton yields, weobtain the spatial average nucleon phase-space den-sity at kinetic freeze-out as

1qprn 1 dNrdyŽ . dKin² :f y s , 1Ž . Ž .N2 3 dNrdyŽ . p

Ž . Ž .w here pr n s dN r dy r dN r dy , andp nŽ .dNrdy are the rapidity densities of proton,p,n,d

neutron, and deuteron, respectively. The first termarises from isospin asymmetry and will be referredto as

a s 1qprn r2. 2Ž . Ž .I

The value of prn depends on the combination of theprojectile and target species, the collision centralityand nucleon rapidity. The value of prn may varylittle with nucleon transverse momentum because ofthe isospin symmetry of strong interaction.

The deuteron and proton data from SiqAl andw xSiqAu at 14.6 AGeVrc 17 and AuqAu at 11.6

w x ² Ž .:KinAGeVrc 18 are used to extract f y . WeN

assume prns1 for the nearly isospin symmetricSiqAl collisions. For SiqAu and AuqAu, thevalues of prn are estimated using the Relativistic

Ž . w xQuantum Molecular Dynamics RQMD model 19 .For the rapidity ranges covered by the data, themodel indicates 0.8-prn-1, hence 0.9-a -1I

for both systems.

( )F. WangrPhysics Letters B 489 2000 273–279 275

² Ž .:KinFig. 1 shows the obtained f y as a functionN

² Ž .:Kinof the proton rapidity. f y varies little withN

rapidity in central SiqAl and AuqAu collisions.In peripheral SiqAl and AuqAu and both central

² Ž .:Kinand peripheral SiqAu collisions, f y is largerN

at target rapidity than mid-rapidity. We note that attarget rapidity deuterons may be produced by mecha-

Žnisms other than coalescence e.g. target fragmenta-. w x ² Ž .:Kintion 18 , resulting in an overestimate of f y N

Ž .by Eq. 1 . The magnitude of this possible effect isnot investigated in the present work.

The average baryon phase-space density is therelevant quantity. Hyperons are the most significantcontributors to the baryons besides nucleons. Theabundance of antibaryons is negligible at AGS ener-gies, and is small at SPS energies. Generally, theaverage baryon phase-space density is obtained from

Kin Kin² : ² :f s f 1yNrNŽ .B N

= 1q YyY r NyN , 3Ž . Ž . Ž .² :Kinwhere f is the rapidity averaged nucleonN

Ž .phase-space density see below , and N, N,Y, and Yare the nucleon, antinucleon, hyperon, and antihy-

Fig. 1. The spatial average nucleon phase-space density at kinetic² Ž .:Kinfreeze-out, f y , as a function of the proton center-of-massN

Ž .rapidity y in central and peripheral heavy-ion collisions at theAGS. Collision centralities are indicated by the cross section

Ž .ranges. The data points at backward rapidities y-0 are mea-Ž .sured, and are reflected about mid-rapidity ys0 . Errors shown

are statistical only. The experimental systematic effects largelycancel in the drp ratio. The solid lines connecting the data pointsare to guide the eye. The dotted lines are an extrapolation to theunmeasured regions.

peron total yields, respectively. It has been assumedthat the phase-space distributions of hyperons and

w xantibaryons are the same as of nucleons 20 . Thisassumption does not introduce significant errors on² :Kinf as the contributions of hyperons and an-B

Ž .tibaryons are generally small see Table 1 .We denote

a s1yNrNf1yprp 4Ž .N

where NrN has been approximated by the antipro-Ž .ton to proton ratio prp , and

a s1q YyY r NyN . 5Ž . Ž . Ž .Y

Hence,

² :Kin ² :Kinf s f Pa Pa . 6Ž .B N N Y

The conservations of global baryon number andstrangeness give, respectively,

NyNsN y YyY 7Ž . Ž .part

whereN is the total number of participant nucle-part

ons, and

q yYyYfKyKs 1qa P K yK 8Ž . Ž . Ž .K

0 0 q yŽ . Ž .where a s K y K r K yK . Therefore, Eq.KŽ .5 becomes

y1q ya s 1y 1qa P K yK rN . 9Ž . Ž . Ž .Y K part

The value of a is not one in isospin asymmetricK

collisions, however, we will use a f1 because theKw xeffect of a non-unitary value 21 is reduced by

Ž q y.K yK rN which is a small quantity for colli-part

sions we consider. It should be noted that only oneunit of strangeness per multi-strange hyperon is

Ž .counted in Eq. 8 . This is safe because the produc-tion of multi-strange hyperons is relatively small

w xeven at the SPS energies 22 .The AGS E859rE866 proton and deuteron mea-

surements cover a broad rapidity range exploiting theŽ .near symmetry of the SiqAl and AuqAu sys-

w x Ž .tems 23 . We take the dNrdy weighted averagep² Ž .:Kin Ž .of f y in Eq. 1 over this rapidity range toN

² :Kinobtain f . The unmeasured drp ratio in theN

mid-rapidity region is approximated by the dottedŽlines in Fig. 1 connecting the last point i.e., closest

.to mid-rapidity with the reflected one. For SiqAuwe assume that the unmeasured drp ratio at the

()

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Table1q y ² :Kin Ž .The K rK ratio and the average baryon phase-space density at kinetic freeze-out f extracted from drp ratios in central heavy-ion collisions. The factors a , a , andB I N

Ž . q y ² :Kina take into account, respectively, the effects of isospin asymmetry, antinucleon and anti hyperon yields. For K rK and f the first error is statistical. The systematicBY

Ž .errors in percentage are introduced by the extrapolation in rapidity, the uncertainties in a , a , and a , and the preliminary nature of some of the data. The experimentalI N Y

systematic effects largely cancel in the KqrKy and drp ratios.Kinq y ² :Collision Beam energy Centrality K rK a a a f BI N Y

Ž .system per nucleon mbbw x w x w xS q S 200 GeV 160 1.58"0.09"5% 26 1 0.89 27 1.32 0.0039"0.0007"10% 28

w x w x w x w xPbqPb 158 GeV 340 1.83"0.02"5% 3 0.97 19 0.93 3 1.32 0.0069"0.0010"10% 28,29w x w xSiqAl 14.6 GeVrc 100 4.45"0.32 1 1 1 1.11 0.0221"0.0017"5% 17w x w x w xSiqAu 14.6 GeVrc 260 5.25"0.42 1 0.95 19 1 1.13 0.0269"0.0006"10% 17

a a w x w x w xAuqAu 11.1, 11.6 GeVrc 420, 470 6.35"0.48 1 0.96 19 1 1.12 0.0297"0.0009"5% 18a w x w xNiqNi 1.8, 1.93 GeV 100 21.1"3.4"5% 2 1 1 1 0.0879"0.0021"5% 30

a The first number is for the KqrKy data; the second is for the drp data. Quoted for NiqNi are the kinetic beam energies.b The PbqPb a value is used because the total yield Kqy Ky is not available for SqS.Y

( )F. WangrPhysics Letters B 489 2000 273–279 277

more forward rapidities is the same as the last data² :Kinpoint. We assign a 8% systematic error on f N

for SiqAu due to this extrapolation by examining asmoothly dropping drp ratio from the low rapidity

² :Kin ² :Kindata points. From f , we extract f usingN B

Ž . Ž . Ž .Eqs. 6 , 4 , and 9 . For all three systems a f1N

and a ranges from 1.03 in peripheral to 1.13 inY

central collisions. For SiqAl and AuqAu, we² :Kinassign a 5% systematic error on f due to theB

² :Kinassumptions made to extract f and errors onB

the correction factors. For SiqAu, we assign a 10%² :Kinsystematic error on f .B

q y w x ² :KinFig. 2 shows K rK 1 and f at the AGSB

as a function of N . Both quantities are nearlypart

constant. It should be noted that the constant valuesdo not necessarily demonstrate that KqrKy and² :Kinf are correlated. However, it will be shown thatB

they are correlated in central collisions at differentenergies. Therefore, it is fair to argue that they arecorrelated also in the same collision system and areindependent of the collision centrality. BecauseKqrKy is fixed at chemical freeze-out, the results

² :Kinmay imply that the f values at kinetic freeze-B

out, which presumably happens later than chemicalfreeze-out in heavy-ion collisions, are connected to

q y Ž .Fig. 2. The K rK ratio filled symbols, left scale and the² :Kinaverage baryon phase-space density at kinetic freeze-out f B

Ž .open symbols, right scale as a function of the total number ofŽ .participant nucleons N at the AGS. Errors shown are statisti-part

cal for KqrKy and statistical and systematic errors added in² :Kinquadrature for f . The experimental systematic effects largelyB

cancel in the KqrKy and drp ratios. Note that both quantitiesare nearly constant over centrality. The dashed curve shows the

Žaverage nucleon phase-space density at kinetic freeze-out right. ² :Kinscale ; the errors are comparable to the f ones.B

² :Chthe chemical freeze-out f values in a way thatB

is independent of centrality.Ž qSince the difference between the kaon yields K

y. ² :KinyK enters into f as a correction, one mayB

be concerned about auto-correlation. However, suchŽ .concern is unnecessary because 1 no correlation is

q y q y Žpresent between K rK and K yK which in-w x. Ž .creases with N more than linearly 1 , and 2 thepart

q y ² :Kin ŽK yK correction to f is small at mostB

. q y13% . In fact, one may correlate K rK with theaverage nucleon phase-space density which is shownas the dashed curve in Fig. 2.

It should be noted that the correlation exists only² :Kinbetween the rapidity integrated values of f andB

q y ² :KinŽ .K rK ; locally in rapidity, f y andBq yŽ . ŽK rK y are not or much less correlated espe-

.cially in central AuqAu collisions . This may bedue to longitudinal flow which can be different for

q y w xK , K , and nucleon 24 . Similar problems areencountered in equilibrium model studies of particleratios: thermodynamic relations between various par-ticle ratios are only valid for ratios of integratedyields; differentially they do not hold because of

w xlongitudinal and transverse flow 25 .Now we study the correlation between KqrKy

² :Kinand f in central collisions at different beamBq y ² :Kinenergies. The values of K rK and f ex-B

tracted from experimental data are tabulated in Tableq y w x1. The K rK values for SqS 26 and NiqNi

w x ² :Kin2 are at mid-rapidity. The f values for SqSB

w x w x28 and PbqPb 28,29 are determined from drp atmid-rapidity. These values are approximated as forthe total yield ratios with the systematic errors quoted

q y ² :Kinin Table 1. The K rK and f values forB

w x w xAuqAu 1,18 and NiqNi 2,30 are obtained frommeasurements at slightly different beam energies.For NiqNi we have assumed prns1.

q y ² :KinFig. 3 shows K rK as a function of f .B

Both quantities decrease with beam energy. The datafrom various collision systems at different beamenergies follow a similar dependence. Motivated byequilibrium models, we fit the data to power-lawfunction, yielding

1.20"0.10Kinq ² :K f Bs1q . 10Ž .y ž /K 0.0076"0.0009

The fit result is plotted in Fig. 3 as the dashed curve.

( )F. WangrPhysics Letters B 489 2000 273–279278

Fig. 3. The KqrKy ratio as a function of the average baryon² :Kinphase-space density at kinetic freeze-out f in central heavy-B

ion collisions. Errors shown are statistical and systematic errorsadded in quadrature. The data are tabulated in Table 1. Thedashed curve is a fit to power-law function. The dotted curve is asimilar fit but excluding the NiqNi data point.

The KaoS Collaboration observed a KqrKy ra-tio in NiqNi collisions at the same equivalentŽ .sub-threshold beam energy that is about seven timessmaller than in pqp near threshold. The KaoSCollaboration contributed this difference to a drop inthe Ky mass in NiqNi collisions, and inferred amass drop of 270q55 MeVrc2 by using the mea-y90

w x ² :Kinsured kaon excitation function 2 . Since f is aB

measure of matter density, we argue that studyingq y ² :KinK rK against f may provide a different andB

probably more direct approach to search for mediumeffect on kaon mass: If the Ky mass is modified by

Ž .medium at low energy but not or differently at highenergy, then the low energy KqrKy should deviatefrom extrapolation of the high energy data. ThisLetter is a first attempt of this new approach: The fitresult in Fig. 3 seems to imply no difference betweenthe SIS and AGSrSPS data; fitting the data exclud-ing the NiqNi point to power-law function, yield-ing the result shown in the dotted curve, seems togive the same implication.

Moreover, as noted, the average baryon phase-space densities at chemical and kinetic freeze-outmay be different; the more suitable variable to corre-late with KqrKy is the chemical freeze-out one. It

w xhas been found in Ref. 16 , under the assumption of

chemical and thermal equilibrium, that the chemical² :Chfreeze-out f is twice the kinetic freeze-outB

² :Kinf at both the AGS and SPS. Using the thermalB

model parameters obtained from the NiqNi dataw x ² :Ch31 , the f value is found to be the same as theB

² :Kin ² :Chf value. In Fig. 3, if the f values wereB B

plotted, then the NiqNi data point would lie abovea sensible extrapolation from the AGS and SPS databy about a factor of two. This is even more contra-dictory to Ky mass drop, as Ky mass drop wouldgive a reduced KqrKy ratio.

However, one should be careful to conclude fromthe above results that there is no Ky mass drop inthe NiqNi data. This is because:

1. Although the power law parameterization is moti-vated by equilibrium models, it is not a priori for

q y ² :Kinthe relation between K rK and f . In fact,B

w xdata from CqC collisions at the SIS 32 do notfollow the parameterization shown in Fig. 3: therea large KqrKy ratio of 39"6 was observed; nodrp data is available, but it can be conjectured

² :Kinthat the f value is lower in CqC than inB

NiqNi. It is possible that the CqC data followa different systematic consistent with theAGSrSPS data, while the NiqNi data lie belowthis systematic. This would be an interesting re-sult because it implies a Ky mass drop in NiqNicollisions.

2. Since the leverage of the AGS and SPS data issmall, the systematic errors on these data arecritically important to the extrapolation to the SISdata.

In order to address these concerns, more data areneeded to fill in the gap between AGS and SIS.These data are underway from the E895 Collabora-

w xtion 33 . New data from the SPS at 40 AGeV wouldbe also valuable as they fill in the gap between thefull-energy AGS and SPS data.

In summary, we have extracted the average baryonphase-space density at kinetic freeze-out from thedeuteron to proton ratio in SiqAl, SiqAu and

w xAuqAu collisions at the AGS 17,18 . The averagebaryon phase-space density at kinetic freeze-out isfound to be nearly constant over the collision central-ity, which offers an explanation for the puzzling

( )F. WangrPhysics Letters B 489 2000 273–279 279

centrality independence of the charged kaon totalw xyield ratio 1 . The correlation between the charged

kaon ratio and the average baryon phase-space den-sity is studied for central heavy-ion collisions ofvarious systems over a wide range of beam energy. Itis found that the charged kaon ratio increases stronglywith the average baryon phase-space density, andboth variables increase with decreasing beam energy.It is argued that studying the correlation betweenthese variables may provide a different and moredirect approach to search for medium effect on kaonmass. Although no conclusion can be drawn from thepresent study regarding medium modification to the

w xkaon mass in NiqNi collisions at the SIS 2 , newdata from the AGS low energy collisions shouldshed light on this question.

Acknowledgements

The author gratefully acknowledges discussionswith Drs. C. Chasman, B.A. Cole, E. Garcia, F.Laue, G. Odyniec, G. Rai, H.G. Ritter, M.J. Tannen-baum, and N. Xu. He also thanks G. Cooper for thecritical reading of the manuscript. This work wassupported by the US Department of Energy undercontract DE-AC03-76SF00098.

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