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Cluster Production in Ultrarelativistic Heavy Ion Collisions Ru¨diger Scheibl and Ulrich Heinz D-93040 Universität Regensburg ...mayproveaninterestingtoolto extractphysicalquan- V approximately known from HBT analysis for central eff tities of the fireball, once the formation process has been Pb+Pb collisions at the SPS and 0.7 [3], 2 3 hC i ≈ hC i ≈ understood. We assume an emission function of the form we obtain for these reactions B 8 10−4GeV2 and 2 ≈ · [1] with gaussian source parametrization and local ther- B 4 10−7GeV2 in reasonable agreement with ex- 3 ≈ · mal equilibrium distributions for nucleons in the hot, cen- periments [4, 5]. We further consider the ratio SAA′ of tral reaction region. Using a Wigner function coalescence the invariant momentum spectra (1) of different cluster mechanism [2], we obtain a formula for the invariant mo- species at zero particle momentum P~A=P~A′ =~0 (in the mentum spectra of clusters of A nucleons [3]: rest frame of the fireball). These particles are likely to 8 comefromthecentralcollisionzone. Frommeasuredanti- d3N 2J +1 µ M 99 EdP3 =M (2π)3 exp(cid:18) AT− (cid:19) hCAiVe(ffA) particle/particle ratios SA¯A the fugacity µA/T can be de- termined. Since is only weakly dependent on flow n 1 ×exp(cid:18)−MtT−M − 2A(∆Yη2)2(cid:19) , (1) aonfdittteomexpterraactturteh,ehCwfrAeeieczaen-ouusteteamnpaepraptruorxeimTaoteftkhneofiwrleebdaglel a ∗ from the ratio of different cluster spectra: J where the first factors are the cluster mass M = Am (m 9is the nucleon mass), the cluster’s spin degeneracy over (A′ A)m T = − . (6) v1 pµhAasoefspthaecec,launstdert.heTfhuegaccoiatyleswcietnhcethceorcrheecmtiiocnal pAotenctoina-l ln(SAA′)−lnAA′((22JJAA′++11)) −ln(κ)+ µA′T−µA hC i 4tains the quantum mechanis of coalescence and basically The variable κ contains the ratio of the A-depending ef- 1relatestheclusterwavefunctiontosysteminhomogeneities fective volumes and the coalescence corrections: 0 due to flow (for nucleons, 1.0). Flow also causes a 1 hC1i ≡ A′ 3/2 strongcorrelationbetween momentumand spatialcoordi- κ= hCAi . (7) 0 8nates of a particle, which makes the effective volume Veff (cid:18)A(cid:19) hCA′i 9of the source of particles with certain momentum smaller Inapurelythermalmodelforclusteremission,κ=1. The /than the overallfireballvolume. In our model, V is only h eff later was used by NA52 to extract a freeze-out tempera- tslightlydependentontransversemomentumandnearlyin- ture from their data on minimum bias Pb+Pb collisions - ldependent of longitudinal rapidity Y, but scales with the at 158GeV per nucleon and resulted in substantially dif- c clustermass. Further,itisrelatedtotheHBT-parameters u ferent temperatures fromdifferent particle ratios[5]. This nRk and R⊥ in the YKP-parametrization[1]: contradiction no longer exists in our analysis: : iv Ve(ffA) = A−32 Ve(ff1) , (2) SAA′ T [MeV] X V(1) mt R (m )R2(m ) . (3) thermal model [5] our model r eff ∼ m k t ⊥ t p/d 120.7 1.8 a p¯/d¯ 117.1±3.1 144 Lastly, ∆η is the width of the spacetime rapidity distri- ± d/t 135.8 9.3 bution and T∗ the effective temperature of the transverse d¯/t 139.5±4.9 146 momentum distribution. T∗ is calculated from the freeze- ± out temperature T and the transverse flow rapidity ηf: We usedslightlydifferentchemicalpotentials forneutrons and protons (known from p¯/p and d¯/d), which make the T =T +(M/A)η2 . (4) ∗ f temperaturesfromp/dandp¯/d¯(p/tandd¯/t,respectively) Note that T is independent of cluster mass: through naturallycoincide. Thiscanbejustifiedbytheinitialneu- ∗ M/A = m only the nucleon mass enters the formula. tronexcesss in the cold Pbion, a remainderof which may Thisisincontradictionwithexperiments,wheredeuterons still be found in the central collision region at freeze-out. show muchhigher effective temperatures thanprotons (at least in the mid-rapidity region of central collisions, z.B. References [4]). This can be cured by a box profile for the transverse [1] U. Heinz, Rio de Janeiro Workshop 97, fireballgeometryinsteadofagaussian,andleadstoanon- nucl-th/9710065 analytic formula instead of (1) and to a different scaling in (2), e.g. A−21. A more substantial study is on its way [2] R. Scheibl and U. Heinz, GSI Yearly Report 96, nucl-th/9701053 [3]. From (1) follows a momentum-independent invariant [3] R. Scheibl, PhD thesis at University Regensburg, in coalescence factor B : A preparation 2J +1 (2π)3 1 A−1 [4] M. Murray for the NA44 Collaboration, Jaipur Con- A B = . (5) A A ference 97, nucl-ex/9706007 hC i √A2A (cid:18) m Veff(cid:19) [5] The Newmass (NA52) Collaboration: Preprint Bern Aboxprofileforthetransversefireballgeometrywillresult BUHE-97-05,submitted to Phys. Lett. B in a rise of B with larger transverse momentum. With A