Scaling of deuteron production in ultrarelativistic nucleus-nucleus collisions

27 July 1995

PHYSICS LETTERS B Physics Letters B 355 (1995) 27-31

ELSEVIER

Scaling of deuteron production in ultrarelativistic nucleus-nucleus collisions H. Sorge a,1, J.L. Nagle b, B.S. Kumarb a hash’tatfiir theoretische Physik, J.W Goethe Universitiit Frankfurt, Frankfurt, Germany b A.W Wright Nuclear Structure Laboratory? Yale University, New Haven, CC USA

Received 23 May 1995; revised manuscript received 13 June 1995 Editor: G.F. Bertsch

Abstract Deuteron production in S- and Pb-induced collisions at ultrarelativistic energies is studied in a transport model, treating the deuteron production by phase space coalescence. The production rate of deuterons as a function of rapidity, dNd/dy, is approximately found to satisfy a simple scaling behavior in terms of the proton and the pion rapidity distributions, dN,/dy and dN,/dy, namely dNd/dy N (dh$/dy)‘/( dN,/dy). The scaling is a consequence of the near equality of the freeze-out volumes the nucleons and mesons. Violation of the scaling is attributed to differences between these freeze-out volumes.

1. Introduction High energy nucleus-nucleus collisions are under intense study as as a means of creating a new state of matter [ 1,2]. Observables which are directly related to the space-time structure of the reactions are of particular interest, because they may provide a final ‘snapshot’ of the hadron-emitting source. The deuteron production rate is in this category; only nucleons with similar positions and momenta fit into the deuteron wave function, and the formation probability for a deuteron approximately proportional to the square the nucleon phase space density. The deuteron production rates do not change once entropy production has ceased in the collision [ 31. Hence, deuterons are sensitive to the state of the system at much earlier times than when they are formed. In this paper, we suggest an observable that is sensitive to the relative sizes of ’ E-mail: [email protected]. 0370-2693/95/$X)9.50 @ 1995 Elsevier Science B.V. All rights reserved XDZ0370-2693(95)00763-6

proton and pion sources in AA collisions at energies above 100 A GeV. The final source sizes depends on the interaction cross sections of the particles in the medium [4,5] as well as the production mechanisms for the particles [ 61. The transverse size of the baryon source is expected to be relatively smaller in symmetric AA collisions, because projectile stopping is a function of the target thickness. 2 Since nucleons have a larger cross section than pions in hadronic matter having a small baryon fraction, they decouple from the medium at lower density and their freeze-out volume is larger.

‘On average a nucleon has to propagate through considerable nuclear material-on the order of 12 fm at nuclear ground state density-to end up in the central rapidity region at CERN energy (200 A GeV). The longitudinal velocity has to be degraded by three units of rapidity. A single inelastic NN collision shifts a nucleon only by about 0.7 units. The effective shift in pA collisions

is even smaller, approximately 0.5 per collision [ 71.

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H. Serge et al. /Physics Letters B 3.55 (1995) 27-31

2. Scaling laws It was suggested a long time ago that the invariant yields of light nuclei with mass A scale with the A-th power of the proton yields d3NA EA= dp:

(1)

The scale factor BA is determined by the nuclear parameters of the ingoing projectile and target [ 81. This scaling relation was very successful in describing the yields of light nuclei in nucleus-nucleus interactions at Bevalac and SIS energies around 1 A GeV [ 31. However, scaling is violated by a factor of up to 40 in nuclear collisions at the beam energy of lo-15 A GeV [ 91. Production of additional particles that are neglected in Eq. ( 1) may affect cluster production at the higher beam energy. Since additional particles can interact with the original nucleons as well, the system must expand to a lower density before it can break up. In particular, the formation and decay of resonances play an important role in the space-time evolution of the heavy ion collisions, according to calculations with the transport model RQMD (Relativistic Quantum Molecular Dynamics) [ 10,111. Recently, the spectra of deuterons and protons produced in Si collisions with a heavy target at 14.5 A GeV were measured [9,12] and compared to calculations with RQMD [ 131. The data gave evidence for considerable transverse expansion beyond the volume of geometric overlap between the colliding nuclei. The transverse expansion is most pronounced for central collisions. Of course, this is expected if the interactions between original nucleons and secondary particles are of importance for cluster formation. In this paper we propose a new scaling relation for deuteron yields which may be appropriate for ultrarelativistic energies. We suggest that the deuteron yields scale inversely with the rapidity density of the mesons produced in AA collisions at high beam energies, that is Bd = c . (dN-/dy)-I, with c a universal implicitly depends

(2) constant. Under this scaling Bd on the source size of the reac-

tion, because the meson rapidity density 3 is proportional to their freeze-out volume, assuming hadrons freeze out at constant density. This last assumption is given support by the observed scaling of radii extracted from Bose-Einstein correlations [ 141 , R N (dN’/-/dy) ‘13. Eq. (2) should be applied only for collisions which are dominated by meson production, i.e. with many more pions than baryons in the final state. We shall test the scaling relation for nucleusnucleus reactions at energies 160-200 A GeV, which are currently under experimental study at the CERNSPS. We determine phase space distributions using the RQMD model (version 1.09) [6,15,16] and calculate the cluster coalescence with the Wigner function method [ 17-191. RQMD is a cascade model in which the secondary particles are produced as fragmentation products of decaying color strings, ropes and resonances. Afterwards they may interact with each other and the nucleons present. Nuclear cluster formation is not included dynamically in RQMD. We add it after the strong interactions have ceased and the particles are streaming freely. The deuteron yields are calculated from the product of proton and neutron source functions (gp and g,) at freeze-out and the coalescence factor, integrated over phase space: Nd=/d4nld4x2/

dpl dp2 g,(xl,pl)gn(.ap2)

x Pd(1,2). The coalescence Pd =

3/g.

(3) factor pd is taken as

f~hm&-~s)

.

(4)

Here f$ denotes the Wigner transform of the deuteron wave function, with center-of-mass motion removed. Its arguments are the distance between the nucleons at the larger of the two freeze-out times and the relative momentum, all values evaluated in the deuteron 3We have chosen - somewhat arbitrarily -the negative particles for the scaling law. Note that mesons are populating the three charge states in approximately equal amounts at ultrarelativistic energies. We have included feeding from weak decays (except A and KS) in the negatives, because it is experimentally much easier to subtract hyperon feed-down from measured proton yields than to control the amount of meson decays, e.g. of 7 and r)‘. Our conclusions in this paper are not affected by this procedure.

29

H. Serge et al. /Physics Letters B 355 (1995) 27-31

5

F

s+s

-._.

__..

-7

:-

:__

z1

P Tl

;.:

I.

,__:

‘..

:_,~?-‘-‘“‘....i

v+

?,-“’

.._.,

:..

..: . ..._,,,; a__...1

..1 :..._ __._ 1

i-i._.,

i....

z...,

-1

:-.-.-.,,_._._.

(\I - lo-+

.

r

\

i

! -.-v-.-.

?..y._., :.....

d p

Pb+PI:

Fig. 1. Final rapidity distributions of negatively charged hadrons, primordial protons and deuterons calculated from RQMD (version 1.09). The RQMD calculations have been carried out for three different systems, S on S (impact parameter b < 1 tin), S on W (b < 4 fm), both at a projectile energy of 200 A GeV, and Pb on Pb (b < 1 fm), with beam energy 160 A GeV. The distributions are dispIayed as histograms, straight line for protons, dashed for deuterons (multiplied with 50) and dotted for negatives (multiplied with 0.1). The rapidity is calculated in the equal-speed-system of projectile and target. The generated events for the symmetric systems have been reflected z + -z in order to improve statistics.

center of mass frame where pt + p2 = 0. This is parameterized with a harmonic oscillator wave function as in Ref. [ 181. The prefactor in Eq. (4) comes from statistical isospin and spin projection of a np pair onto the corresponding deuteron quantum numbers. In the RQMD the final particle energies are set on shell. The calculated rapidity distributions of deuterons and protons neglecting feeding from weak decays are displayed in Fig. 1. The RQMD calculations have been carried out for three different systems, S on S (impact parameterbc lfm),SonW(b<4fm),bothata beam energy of 200 A GeV, and Pb on Pb (b < 1 fm) , with beam energy 160 A GeV. The different shapes

0

’

0.2

’

0.4

’

0.5

’

0.8

’

’

1

Am, (GeV/c*) Fig. 2. Transverse mass spectra of primordial protons and deuterons in the rapidity window ymid - 0.5 < y < v,,,id + 0.5. The spectra have been calculated for S+S and Pb+Pb using the RQMD model under the same conditions as explained in caption to Fig. 1. They are represented as histograms, protons from PbtPb (straight line), deuterons from Pb+Pb (dashed line), protons from S+S (dotted line), and deuterons from S+S (dashed-dotted line). The calculated spectra arc compared to Boltzmann distributions N rnt . exp( -mr/T) with slope parameters fitted to the large rnr tail of the distributions, with T parameters (from top to bottom) 232, 180,300 and 180 MeV.

of the baryon distributions, discussed in [ 151, reflect the dependence of the stopping power on target and projectile mass number. The calculated d/p ratios are less than one percent in the central rapidity region, much smaller than the data measured at the lower beam energies of lo-15 A GeV [ 121. The main motivation apply a scaling relation. like Eqs. ( 1) , (2) to the deuteron yields is that it accounts for the fact that more particles need a larger freezeout volume. Other collective dynamical effects in ultrarelativistic collisions such as collective flow may lead to scaling violations. Indeed, an immediate consequence of Eq. ( 1) is that the transverse mass spectrum of the nucleon clusters will have the same slope as the primordial nucleon spectra. This is only consistent with Eq. (3) if the freeze-out distribution gN factorizes into a product of separate functions of position and momentum, thus precluding collective flow. Transverse mass spectra calculated from RQMD are shown in Fig. 2 (histograms), compared to fits with a Boltzmann distribution (symbols). The primordial

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H. Serge et al. /Physics

Letters B 355 (199.5) 27-31

s+s

Pb+Pb

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Y Fig. 3. The scaled deuteron rapidity density [dNd/dy/ ( dNp/dy )‘I . dN- jdy as a function of rapidity. The RQMD calculations have been carried out for the same three systems as in Fig. 1, S on S (straight line), S on W (dashed histogram) and Pb on Pb (dotted histogram).

proton spectra are well described by the thermal distributions with ‘apparent temperature’ (slope parameter) 180 MeV for S+S and 232 MeV for Pb+Pb collisions. At large m, behavior of the deuteron distribution fits with same slope parameter only in the S+S case; for Pb+Pb collisions the slope parameter is 300 MeV. Furthermore, the calculated deuteron distributions differ markedly from a thermal shape at low ml values, exhibiting a convex curvature which is characteristic of transverse flow. We may therefore conclude that the coalescence formula Eq. ( 1) is inapplicable to the calculated momentum distributions. We next consider the validity of a coalescence formula with the particle yields integrated over transverse momentum, integration over transverse momentum

dNd& = Bd. W7p/dy)2,

(5)

which should be less sensitive to flow effects. The scaled deuteron rapidity density [ dNd/dy/ (dN, / dy)‘] . dN-/dy will be constant if Eqs. (2), (5) are satisfied. The ratio obtained from RQMD is displayed in Fig. 3 as a function of rapidity for the three collid-

;a transverse

distance

(fm)

Fig. 4. Distribution of transverse distances (to the collision center) 111-t . dN/dr, at freeze-out for different particle species in S on S (top), S on W (middle) and Pb on Pb (bottom) collisions. Only particles around central rapidity ( y& f 1) are taken into account, primordial protons (straight line), pions (dashed histogram), and neutral (anti-)kaons KO+F?O (dotted histogram). Note that the pion and kaon distributions have been renomralized that the integral gives the same yield as for the protons.

ing systems under consideration. Indeed, the scaled deuteron rapidity density turns out to be remarkably independent of rapidity over a range of about three units 4 . The S-induced reactions show no target dependence going from a S to a W target. While scaling holds for the Pb+Pb collisions as a function of rapidity, it is clear from Fig. 3 that the scaled deuteron yields are smaller in this case than in the S-fA reactions, by about 15%. This points towards a relative increase of the nucleon source in comparison to the meson source. The relative increase of the nucleon source in central Pb+Pb collisions can be checked directly by look“It is quite naturai that scaling breaks down in the projectile and target fragmentation region, because the basic assumptions, in particular meson dominance, are not fulfilled here.

H. Sorge et al. /Physics Letters B 355 (1995) 27-31

ing at the RQMD freeze-out distributions (see Fig. 4). Fig. 4 shows transverse freeze-out distributions for pions, protons and kaons at midrapidity ( yfid - 1 < y < ymid + 1) . The relative size of the proton source grows relatively to the meson source by going from S+S to Pb+Pb. While the ratio of proton to pion transverse RMS radius (at midrapidity) is 1.05 in S+S reactions (5.4 fm/5.1 fm); it increases to 1.19 in central Pb+Pb collisions ( 10.3 fm/8.6 fm). The system has expanded considerably before freeze-out, because the initial transverse RMS sizes of S (Pb) have values 2.4 (4.5) fm only.

Acknowledgements H. S. is grateful to M. Leltchouk for discussions about collective flow. He also thanks J. Simon-Gillo and R. Mattiello for stimulating discussions. This work was supported in part by GSI and DFG, and the U.S. DOE, grant number DE-FG02-9 lER-40609.

References [l]

A. Poskanzer et al., eds., Quark Matter ‘95, Proceedings of the 11th International Conference on Quark Matter (Monterey, CA, 1995), Nucl. Phys. A (1995), in print.

31

[2] H.-A. Gustafsson et al., eds., Quark Matter ‘93, Proceedings of the 10th international conference on Quark Matter (Borlange, Sweden, 1993), Nucl. Phys. A 566 (1994). [3] L.F? Csemai and J.I. Kapusta, Phys. Rep. 131 (1986) 223. [4] K. Haglin and S. Pratt, Phys. Lett. B 328 (1994) 255. [5] J. Cleymans, K. Redlich, H. Satz and E. Suhonen, Z. f. Phys.

C 58 (1993) 347.

161 H. Serge, Phys. Lett. B 344 (1995) 35. [71 W. Busza and R. Ledoux,

Ann. Rev. Nucl. Part. Sci. 38 (1988) 119. [81 S.T. Butler and C.A. Pearson, Phys. Rev. 129 (1963) 836. [91 ES14 Collaboration, J. Barrette et al., Phys. Rev. C 50 (1994) 1077. IlO1 H. Sorge, R. Mattiello, H. Stocker and W. Greiner, Phys. Lett. B 271 (1991) 37. [Ill M. Hoffmann, R. Mattiello, H. Sorge, H. Stocker and W. Greiner, preprint UFTP-374 (1994), Phys. Rev. C (1995), in print. T. Abbott et al., E802 Collaboration, Phys. Rev. C 50 ( 1994) 1024. J. Nagle et al., Phys. Rev. Lett. 73 (1994) 1219. R. Stock, Ann. Phys. (Leipzig) 48 (1991) 195. H. Sorge, UFTP 376-94 (preprint), Z. f. Phys. C (1995) in print. M. Berenguer, H. Sorge, and W. Greiner, Phys. Lett. B 332 (1994) 15. E.A. Remler, Ann. Phys. (NY) 136 (1981) 293, and references therein. M. Gyulassy, K. Frankel and E.A. Remler, Nucl. Phys. A 402 (1983) 596. R. Mattiello, A. Jahns, H. Stocker, W. Greiner, and H. Serge, Phys. Rev. Lett. 74 (1995) 2180. PEPEPE