We show that the observed suppression of the $\Lambda(1520)/\Lambda$ ratio in
central Au + Au collisions at the Relativistic Heavy Ion Collider can be
naturally understood in the constituent quark recombination model.Comment: 3 pages, 1 figur

Berndt Müller

C. Markert

J. Zimanyi

Yoshiko Kanada-En'yo

English

arXiv

Kanada-En'yo, Y

Muller, B

Kyoto University Research Information Repository

Title Suppression of p-wave baryons in quark recombinationAuthor(s)Kanada-En'yo, Y; Muller, BCitationPHYSICAL REVIEW C (2006), 74(6)Issue Date2006-12URL http://hdl.handle.net/2433/50446RightCopyright 2006 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityRAPID COMMUNICATIONSPHYSICAL REVIEW C 74, 061901(R) (2006)Suppression of p-wave baryons in quark recombinationYoshiko Kanada-En’yoYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JapanBerndt Mu¨llerYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan andDepartment of Physics, Duke University, Durham, North Carolina 27708, USA(Received 4 August 2006; published 11 December 2006)We show that the observed suppression of the (1520)/ ratio in central Au+Au collisions at the BNLRelativistic Heavy Ion Collider can be naturally understood in the constituent quark recombination model.DOI: 10.1103/PhysRevC.74.061901 PACS number(s): 25.75.−q, 12.38.Mh, 12.39.JhRecent data from nuclear collisions observed at the BNLRelativistic Heavy Ion Collider (RHIC) show that the emissionof excited hadronic states is differentially suppressed incentral Au+Au collisions [1,2]. Similar results were alsofound in central collisions of Pb + Pb at the CERN SuperProton Synchrotron (SPS) [3]. The yields of some hadronsare suppressed in comparison with both those measured inp + p collisions and those predicted by statistical models.Curiously, not all excited hadrons are suppressed. For example,the ratio ∗/ shows no suppression at RHIC, and the ratioφ/K appears to be even slightly enhanced. The ratio K∗/Kis modestly suppressed, and the strongest suppression is seenin the ratio ∗/, where ∗ stands for the negative-paritypartner of the  hyperon with a mass of 1520 MeV.The differences of the relative yields of these strangehadrons compared with yields measured in p + p collisionsis not very surprising, because the production of hadronscontaining strange quarks is known to be strongly enhancedin nuclear collisions at RHIC energies. What is noteworthyis that the measured yield ratios deviate from the predictionsof the thermal statistical model, which generally explains theobserved hadron yields well [4–6]. The tentative explanationfor the observed deviation proposed by Adams et al. [1] is thatsome fraction of the short-lived excited hadrons decays beforethe medium has become so dilute that the decay fragments canescape unscattered. It is then possible to estimate the durationof the dense hadronic phase from the known lifetime of thehadronic state and the observed suppression [7]. The problemwith this explanation is that it yields vastly different estimatesfor the duration of the dense hadronic phase. For example, theK∗/K ratio demands a time span τ ≈ 2.5 fm/c, whereas the∗/ ratio would require a much longer duration of τ ≈9 fm/c. Such a long time span is difficult to reconcile with thevalues of the duration of the emission process derived fromtwo-pion intensity interferometry measurements, which givevalues for τ in the range 2–3 fm/c [8].The discrepancy can possibly be explained by includingthe effect of partial regeneration of excited hadronic states inthe evolving dense medium [9] if its lifetime is sufficientlylong. However, quantitative predictions of this effect dependon various unknown cross sections, and it is not clear whethera simple compelling picture emerges that can explain all themeasured ratios [1].In this article, we propose a simple explanation for theobserved strong suppression of the ∗/ ratio, compared withthe observed K∗/K ratio and the thermal model predictions.Our argument is based on the fact that the (1520) has adifferent internal quark structure than that of all other hadronsthat have been detected in the final state of relativistic heavyion reactions. In the framework of the constituent quark model,the (1520) is described as an orbital excitation of the ground-state  hyperon, where one quark is in a p-wave orbital [10].In the calculations, the (1520) is a mixture of flavor-SU(3)singlet and octet, which corresponds to a spatial wave functionwith the strange quark being predominantly in the p-wavestate, specifically,|(1520)〉 ≈ 0.91|1〉 + 0.40|8〉 ≈ −0.37|s〉 + 0.93|p〉, (1)where |1(8)〉 denotes the flavor-SU(3) state, and |s(p)〉 thesymmetry of the internal spatial wave function of the s quark.For simplicity, we shall neglect the minor component of thespatial wave function and approximate the ∗ state as a purep-wave excitation of the strange quark with the (ud) quarkpair being in a spin and isospin singlet state configuration.The internal structure of the emitted hadron is assumedto be of no relevance in the statistical (thermal) model ofparticle emission; only the mass and the quantum degeneracymatter. The assumption underlying the statistical model isthat all S-matrix elements for hadron production are ofroughly equal magnitude. This assumption may be justifiedfor the emission of ground-state hadrons, which all havesimilar spatial quark wave functions. The insensitivity to thehadronic wave function can be demonstrated in the frameworkof the sudden recombination model in the high-temperaturelimit, when the source is homogeneous over the size of thehadron [11]. In practice, however, the prevailing conditions athadronization do not correspond to this limit. The formationtemperature of hadrons, Tch ≈ 160 MeV, is comparable to theenergy scale of hadronic excitations, and the coherence lengthof quarks just before hadronization is also expected to besimilar to the size of a hadron. It is therefore plausible thatdetails of the spatial wave function of the emitted hadronshould have an effect on the production yield, leading todeviations from the predictions of the statistical model.0556-2813/2006/74(6)/061901(3) 061901-1 ©2006 The American Physical SocietyRAPID COMMUNICATIONSYOSHIKO KANADA-EN’YO AND BERNDT M ¨ULLER PHYSICAL REVIEW C 74, 061901(R) (2006)We show below in the framework of the nonrelativisticconstituent quark model that the internal p-wave structureof the (1520) hyperon leads to a significant suppression ofits creation by quark recombination, compared with s-wavebaryons. For reasonable choices of the parameters, we findthat the suppression is approximately one-half, similar to theexperimental observation. The amount of the suppression isquite insensitive to the parameters of our model. Our resultsuggests that the observed suppression of (1520) productionin central Au+Au collisions is not the consequence of final-state interactions, but of the production mechanism (quarkrecombination) itself. The insight that the internal p-wavestructure of a hadron can reduce its yield in nuclear reactionswas also reached in the framework of a quark-diquark modelof baryons [12]. Our work differs from this earlier study by ouruse of a more realistic internal wave function of the emittedbaryon, and it is based on the recombination model rather thanthe direct reaction model. These qualitative differences haveimportant quantitative consequences; in particular, the use ofa proper internal wave function of the baryon is essential forthe magnitude of the predicted suppression.Adopting the convention of Isgur and Karl [10], we denotethe positions of the u and d quark by x1 and x2, respectively,and the position of the s quark by x3. We assume that the uand d quarks have the same mass m, and the s quark has massms . The Jacobi coordinates are given byrc.m. = m(x1 + x2) + msx3M,r = (x1 − x2)/2, (2)r′ = (x1 + x2)/2 − x3,where M = 2m + ms denotes the total mass. We assume thatthe internal wave functions of the  and ∗ can be written as|〉 = 2−1/2s(bu/d, r)[χu × χd ](0)×s(bs, r′)χ (1/2)s |(ud − du)s〉, (3)|∗〉 = 2−1/2s(bu/d, r)[χu × χd ](0)× [p(bs, r′) × χs](3/2)|(ud − du)s〉, (4)where s and p are the lowest s- and p-wave eigenstates ofthe harmonic oscillator:s(b, r) = (πb2)−3/4e−r2/2b2 ,(5)p(b, r) = (πb2)−3/4√2rbe−r2/2b2 .We take the radius parameter of the constituent u, d-quarkwave functions as b ≈ 0.6 fm [13] and assume that theparameter scales as usual with the reduced mass: bu/d =b√2, bs = b√M/2ms .We further assume that the quarks just before hadronizationcan be described by Gaussian wave packets of the form	(R,D,k; x) = (πR2)−3/4 exp(− 12R2(r − D)2 + ik · r),(6)where D denotes the position and k the momentum of thecenter of the wave packet. The width parameter R describes thecoherence properties of the initial quarks, which is determinedby their mean free path in the hadronizing medium. SuchGaussian wave packets have been used, e.g., in the frameworkof quark molecular dynamics [14]. Because the magnitude ofR is not well known from phenomenological considerations,we shall explore a range of values for R in the vicinityof the confinement radius. We allow R2 to scale inverselywith the quark mass to ensure that quarks become localizedin the limit of infinite mass. We further assume that the initialquark momenta k are given by a thermal distribution withtemperature T = Tch. Finally, we assume that the hadrons areformed by recombination in a surface region, which we modelas a plane with a Gaussian thickness profile of half-width a.The ensemble distribution for the initial-state wave packet thushas the formF (D,k) =3∏i=1exp(−D2i,x2a2− k2i2miT). (7)With these model assumptions, the (sudden) transition matrixelements from an uncorrelated three-quark state to the hadronicstates (3) and (4) can be easily evaluated. The thermal averageand the integration over the hadronization zone can also beperformed analytically. We denote the averaged transitionprobabilities as W (s) and W (p),W (s) =∫ ∏idDidkiF (D,k) |〈|	i〉|2 , (8)and the analogous equation giving W (p) for ∗ formation. Thefinal result for the ratio of the transition probabilities into the and ∗ states, correcting for their spin(J, J∗ ) degeneracies,is(2J + 1)W (p)(2J∗ + 1)W (s) =2b23(R2 + b2)[1 +(2 + R2 + b2a2)−1+ 3R4(2b2R2 + R2 + b2mT)−1]. (9)We first note that the ratio tends to one in the limit R, T → ∞.This is not unexpected as it confirms the result obtained byFries et al. [11]. For finite values of R and T , the ratio isalways less than unity, i.e., the production of the p-wavehadron (the ∗) is suppressed. This is explored in Fig. 1,which shows the suppression factor (9) as a function of thehadronic size parameter b for several different choices ofthe quark coherence length R. Remarkably, in the range ofrealistic values for b ≈ 0.6 fm, the suppression is insensitiveto the precise value of R. This is gratifying, because R isby far the most uncertain parameter in our model. In thelarge-a limit for the surface thickness, which correspondsto a volume-dominated quark recombination, the suppressionfactor is found to be slightly (about 10%) enhanced.Figure 1 shows that the formation of p-wave baryonsis suppressed by a factor of 0.5–0.6 in the realistic rangeof parameters in our model. This is quite close to thesuppression, compared with the statistical model prediction,of the (1520)/ ratio in central Au+Au collisions [1]. Wedo not need to adjust the hadron size parameter b to fit the061901-2RAPID COMMUNICATIONSSUPPRESSION OF p-WAVE BARYONS IN QUARK . . . PHYSICAL REVIEW C 74, 061901(R) (2006) 0 0.2 0.4 0.6 0.8 10  0.2  0.4  0.6  0.8 1R/b=1.0R/b=1.5R/b=2.0R/b=10000Λ∗Λhadron size  b (fm)N    / NFIG. 1. Suppression factor for the formation of a p-wave baryonby recombination of three quarks from a thermal medium, asa function of the hadron size parameter b for four differentchoices of coherence length R of the initial-state quarks (R/b =1, 1.5, 2, 104). Temperature T = 160 MeV; ud-quark mass m =330 MeV; hadronization surface thickness a = 0.5 fm.observed suppression value, because we are using a realistichadron wave function. We emphasize that this suppressionis a formation effect, not an effect of final-state interactions.Within our model, it is predicted that the suppression should beindependent of the momentum of the baryon. If high-qualityspectra of  and ∗ hyperons can be obtained in futurehigh-statistics runs, this property could perhaps be used todistinguish between the suppression mechanism proposed hereand final-state suppression mechanisms. On general grounds,one expects that a suppression effect due to rescatteringwill diminish with increasing hadron momentum, becausethe particle spends less time in the medium. Of course, thisargument applies only to the momentum range in whichrecombination is thought to be the dominant hadronizationprocess (pT  4 GeV/c). Also, relativistic effects may modifyour prediction.In conclusion, we have shown that under the conditionsprevailing at hadronization of a quark-gluon plasma, therecombination model predicts that the yield of hadrons withan internal p-wave excitation is suppressed compared with theyield of ground-state hadrons. For the example of (1520)versus  formation, the suppression factor is calculated tobe in the range 0.5–0.6. This prediction is found to berather insensitive to the precise choice of model parameters.We suggest the mechanism pointed out here may be theorigin of the relative suppression, compared with statisticalmodel expectations, of (1520) production in central Au+Aucollisions at RHIC.The present work is based on the idea of the recombinationmodel. It extends previous work by taking into account theinternal structure of the emitted baryon and obtains the newinsight that an internal p wave may result in a reduced hadronyield. We note that the same mechanism affects the productionof all p-wave hadrons. It would be interesting to confirm thisexperimentally, but unfortunately it seems unlikely that thep-wave excitation of any other baryon or meson can beidentified in the final state of a heavy ion collision.B.M. would like to thank the members of the YITP,especially T. Kunihiro, for their hospitality during his stay inKyoto as YITP visiting professor. Y.K-E. thanks T. Kunihiroand H. Tsunetsugu for valuable discussions. This work wassupported in part by the Japan Society for the Promotion ofScience, a Grant-in-Aid for Scientific Research from the JapanMinistry of Education, Science, and Culture, and a grant fromthe U.S. Department of Energy (DE-FG02-05ER41367).[1] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 97,132301 (2006).[2] L. Gaudichet (STAR Collaboration), J. Phys. G: Nucl. Part. Phys.30, S549 (2004).[3] V. Friese (NA49 Collaboration), Nucl. Phys. A698, 487 (2002).[4] F. Becattini, Nucl. Phys. A702, 336 (2002).[5] J. Cleymans, B. Ka¨mpfer, M. Kaneta, S. Wheaton, and N. Xu,Phys. Rev. C 71, 054901 (2005).[6] A. Andronic, P. Braun-Munzinger, and J. Stachel, Nucl. Phys.A772, 167 (2006).[7] C. Markert, G. Torrieri, and J. Rafelski, in New States of Matter inHadronic Interactions: Pan-American Advanced Study Institute,AIP Conf. Proc. No. 631 (AIP, New York, 2002), p. 533.[8] J. Adams et al. (STAR Collaboration), Phys. Rev. C 71, 044906(2005).[9] M. Bleicher and J. Aichelin, Phys. Lett. B530, 81 (2002).[10] N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).[11] R. J. Fries, B. Muller, C. Nonaka, and S. A. Bass, Phys. Rev. C68, 044902 (2003).[12] J. Zimanyi and P. Levai, Acta Phys. Hung. A 27, 469 (2006).[13] N. Isgur and G. Karl, Phys. Rev. D 20, 1191 (1979).[14] T. Maruyama and T. Hatsuda, Phys. Rev. C 61, 062201 (2000).061901-3

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