Proton and Lambda-Hyperon Production in Nucleus-Nucleus Collisions
aa r X i v : . [ h e p - ph ] M a y PROTON AND Λ -HYPERON PRODUCTIONIN NUCLEUS-NUCLEUS COLLISIONSG.H. Arakelyan , C. Merino , and Yu.M. Shabelski A.Alikhanyan National Scientific Laboratory(Yerevan Physics Institut)Yerevan, 0036, ArmeniaE-mail: [email protected] Departamento de F´ısica de Part´ıculas, Facultade de F´ısicaand Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE)Universidade de Santiago de Compostela15782 Santiago de CompostelaGaliza-SpainE-mail: [email protected] Petersburg Nuclear Physics InstituteNCR Kurchatov InstituteGatchina, St.Petersburg 188350, RussiaE-mail: [email protected]
The experimental data on net proton and net Λ-hyperon spectra obtained by theNA35 Collaboration, as well as the inclusive densities of Λ and ¯Λ obtained by NA49,NA57, and STAR collaborations, are compared with the predictions of the Quark-Gluon String Model. The contributions of String Junction diffusion, interactions withnuclear clusters, and the inelastic screening corrections are accounted for. The levelof numerical agreement of the calculations with the experimental data is of about20 − Introduction
The Quark-Gluon String Model (QGSM) [1] is based on the Dual Topological Unita-rization, Regge phenomenology, and nonperturbative notions of QCD. This model issuccessfully used for the description of multiparticle production processes in hadron-hadron [2, 3, 4, 5], hadron-nucleus [6, 7], and nucleus-nucleus [8, 9, 10] collisions. Inparticular, the rapidity dependence of inclusive densities of different secondaries ( π ± , K ± , p , and ¯ p ) produced in Pb-Pb collisions at 158 GeV/c per nucleon were reasonablydescribed in ref. [10]. In the present paper we consider the yields of p and ¯ p , as wellas Λ and ¯Λ, produced in the collisions of different nuclei at CERN SpS and RHICenergies.In the QGSM high energy interactions are considered as proceeding via the exchangeof one or several Pomerons, and all elastic and inelastic processes result from cuttingthrough or between Pomerons [11]. Inclusive spectra of hadrons are related to thecorresponding fragmentation functions of quarks and diquarks, which are constructedusing the Reggeon counting rules [12].In the case of interaction with nuclear target, the Multiple Scattering Theory(Gribov-Glauber Theory) is used. It allows to consider the interaction with nucleus asthe superposition of interactions with different numbers of target nucleons [13, 14, 15,16].Also in the case of nucleus-nucleus collisions the Multiple Scattering Theory allowsto consider the interaction as the superposition of separate nucleon-nucleon interac-tions. Though in this case the analytical summation of all the diagrams is impos-sible [17], the signiificant classes of diagrams can be analytically summed up in theso-called rigid target approximation [18] which is used in the present paper.The significant differences in the yields of baryons and antibaryons in the central(midrapidity) region are present even at high energies. This effect can be explained[4, 5, 19, 20, 21, 22, 23, 24] in QGSM by the special structure of baryons consisting ofthree valence quarks together with a special configuration of gluon field, called StringJunction [25, 26, 27, 28].One additional contribution comes from the coherent interaction of a projectile withmultiquark clusters inside the nuclei. The existance of these interactions is confirmedby the presence, with not such a small probability, of a cumulative effect [29].These contributions were incorporated into the QGSM in [30, 31], and they allowto describe a number of experimental facts.At very high energies the contribution of the enhancement Reggeon diagrams be-2omes important, leading to a new phenomenological effect, the supression of the in-clusive density of secondaries [32] in the central (midrapidity) region.In this paper we present the description of p , ¯ p , Λ, and ¯Λ production on nucleartargets at CERN SpS and RHIC energies. The QGSM [1, 2, 3] allows one to make quantitative predictions for different featuresof multiparticle production, in particular, for the inclusive densities of different secon-daries, both in the central and in the beam fragmentation regions.In QGSM, each exchanged Pomeron corresponds to a cylindrical diagram, and thus,when cutting one Pomeron, two showers of secondaries are produced (see Fig. 1 a,b).The inclusive spectrum of a secondary hadron h is then determined by the con-volution of the diquark, valence quark, and sea quark distributions, u ( x, n ), in theincident particles, with the fragmentation functions, G h ( z ), of quarks and diquarksinto the secondary hadron h . Both the distributions and the fragmentation functionsare constructed using the Reggeon counting rules.In particular, in the case of n >
1, i.e. in the case of multipomeron exchange, thedistributions of valence quarks and diquarks are softened due to the appearance of asea quark contribution. There is some freedom [6] in how to account for this effect.In principle, the valence and sea quarks can depend on n in a different manner, forexample: u uu ( x, n ) = C uu · x α R − α B +1 · (1 − x ) − α R + m u ud ( x, n ) = C ud · x α R − α B · (1 − x ) − α R + m u u ( x, n ) = C u · x − α R · (1 − x ) α R − α B + m u d ( x, n ) = C d · x − α R · (1 − x ) α R − α B +1+ m u s ( x, n ) = C d · x − α R · (1 − x ) α R − α B +( n − d , (1)where every distribution u i ( x, n ) is normalized to unity, d is a parameter, and thevalues of m and m can be found from the conditions: h x uu i + h x d i + 2( n − · h x s i = 13igure 1: (a) Cylindrical diagram representing a Pomeron exchange within the DTU classification(quarks are shown by solid lines); (b) One cut of the cylindrical diagram corresponding to the single-Pomeron exchange contribution in inelastic pp scattering; (c) One of the diagrams for the inelasticinteraction of one incident proton with two target nucleons N and N in a pA collision. h x ud i + h x u i + 2( n − · h x s i = 1 . (2)The details of the model are presented in [1, 2, 3, 19]. The averaged number ofexchanged Pomerons h n i pp slowly increase with the energy. The Pomeron parametershave been taken from [3].For a nucleon target, the inclusive rapidity, y , or Feynman- x , x F , spectrum of asecondary hadron h has the form [1]: dndy = x E σ inel · dσdx F = ∞ X n =1 w n · φ hn ( x ) + w D · φ hD ( x ) , (3)where the functions φ hn ( x ) determine the contribution of diagrams with n cut Pomerons, w n is the relative weight of this diagram, and the term w D · φ hD ( x ) accounts for thecontribution of diffraction dissociation processes.4n the case of pp collisions: φ hn ( x ) = f hqq ( x + , n ) · f hq ( x − , n )+ f hq ( x + , n ) · f hqq ( x − , n )+2( n − · f hs ( x + , n ) · f hs ( x − , n ) , (4) x ± = 12 [ q m T /s + x ± x ] , (5)where f qq , f q , and f s correspond to the contributions of diquarks, valence quarks, andsea quarks, respectively.These contributions are determined by the convolution of the diquark and quarkdistributions with the fragmentation functions, e.g., f hq ( x + , n ) = Z x + u q ( x , n ) · G hq ( x + /x ) dx . (6)In the calculation of the inclusive spectra of secondaries produced in pA collisionswe should consider the possibility of one or several Pomeron cuts in each of the ν blobsof proton-nucleon inelastic interactions. For example, in Fig. 1c it is shown one of thediagrams contributing to the inelastic interaction of a beam proton with two targetnucleons. In the blob of the proton-nucleon1 interaction one Pomeron is cut, while inthe blob of the proton-nucleon2 interaction two Pomerons are cut. The contributionof the diagram in Fig. 1c to the inclusive spectrum is x E σ pAprod · dσdx F = 2 · W pA (2) · w pN · w pN · n f hqq ( x + , · f hq ( x − ,
1) ++ f hq ( x + , · f hqq ( x − ,
1) + f hs ( x + , · [ f hqq ( x − ,
2) + f hq ( x − ,
2) ++ 2 · f hs ( x − , } , (7)where W pA (2) is the probability of interaction with namely two target nucleons.It is essential to take into account all digrams with every possible Pomeron config-uration and its corresponding permutations. The diquark and quark distributions andthe fragmentation functions are the same as in the case of pN interaction.The total number of exchanged Pomerons becomes as large as h n i pA ∼ h ν i pA · h n i pN , (8)where h ν i pA is the average number of inelastic collisions inside the nucleus (about 4 forheavy nuclei at SpS energies). 5he process shown in Fig. 1c satisfies [13, 14, 15, 16] the condition that the absorp-tive parts of the hadron-nucleus amplitude are determined by the combination of theabsorptive parts of the hadron-nucleon amplitudes.In the case of a nucleus-nucleus collision, in the fragmentation region of projectilewe use the approach [8, 9, 10], where the beam of independent nucleons of the projectileinteract with the target nucleus, what corresponds to the rigid target approximation[18] of Glauber Theory. In the target fragmentation region, on the contrary, the beamof independent target nucleons interact with the projectile nucleus, these two resultscoinciding in the central region. The corrections for energy conservation play here avery important role if the initial energy is not very high. This approach was used in[10] for the succsessful description of π ± , K ± , p , and ¯ p produced in Pb-Pb collisions at158 GeV per nucleon.In the present paper we consider the spectra of secondary baryons and antibaryons,as well as their differences, i.e. the net baryon spectra. At low energies, the net baryonspectra coincide with the spectra of baryon, while at asymptotically high energiesthey are negligible due to the even signature of the Pomeron trajectory. The energydependence of the net baryon spectra between these two limits strongly depends onthe production mechanism. In the string models, baryons are considered as configurations consisting of three con-nected strings (related to three valence quarks), called String Junction (SJ) [25, 26, 27,28], this picture leading to some quite general phenomenological predictions.The production of a baryon-antibaryon pair in the central region usually occurs via SJ - SJ pair production (SJ has upper color indices, whereas anti-SJ ( SJ ) has lowerindices), which then combines with sea quarks and sea antiquarks into a B ¯ B pair[27, 33], as it is shown in Fig. 2a.However, in the processes with incident baryons there exists another possibility toproduce a secondary baryon in the central region, called SJ diffusion. The quantitativedescription of the baryon number transfer due to SJ diffusion in rapidity space wasobtained in [19] and following papers [4, 5, 20, 21, 22, 24].In the QGSM the differences in the spectra of secondary baryons and antibaryonsappear for processes which present SJ diffusion in rapidity space. These differencesonly vanish rather slowly when the energy increases.To obtain the net baryon charge, and according to ref. [19], we consider three6igure 2: QGSM diagrams describing secondary baryon production: (a) usual B ¯ B central productionwith production of new SJ pair; (b) initial SJ together with two valence quarks and one sea quark;(c) initial SJ together with one valence quark and two sea quarks; (d) initial SJ together with threesea quarks. different possibilities. The first one is the fragmentation of the diquark giving rise toa leading baryon (Fig. 2b). A second possibility is to produce a leading meson in thefirst break-up of the string and a baryon in a subsequent break-up [12, 34] (Fig. 2c).In these two first cases the baryon number transfer is possible only for short distancesin rapidity. In the third case, shown in Fig. 2d, both initial valence quarks recombinewith sea antiquarks into mesons, M , while a secondary baryon is formed by the SJtogether with three sea quarks [5, 19, 35].The fragmentation functions for the secondary baryon B production correspondingto the three processes shown in Fig. 2b, 2c, and 2d can be written as follows (see [19]for more details): G Bqq ( z ) = a N · v Bqq · z . , (9) G Bqs ( z ) = a N · v Bqs · z · (1 − z ) , (10) G Bss ( z ) = a N · ε · v Bss · z − α SJ · (1 − z ) , (11)where a N is the normalization parameter, and v Bqq , v Bqs , v Bss are the relative probabilitiesfor different baryons production that can be found by simple quark combinatorics[36, 37]. These probabilities depend on the strangeness suppression factor
S/L , andwe use
S/L = 0 .
32 following [38]. 7he contribution of the graph in Fig. 2d has in QGSM a coefficient ε which deter-mines the small probability for such a baryon number transfer.The fraction z of the incident baryon energy carried by the secondary baryon de-creases from Fig. 2b to Fig. 2d. Only the processes in Fig. 2d can contribute to theinclusive spectra in the central region at high energies if the value of the intercept of theSJ exchange Regge-trajectory, α SJ , is large enough. The analysis in [24] gives a value of α SJ = 0 . ± .
1, that is in agreement with the ALICE Collaboration result, α SJ ∼ . α SJ = 0 . ε = 0 . . (12) In the case of interaction with a nuclear target some secondaries can be produced inthe kinematical region forbidden for the interaction with a free nucleon. Such processesare called the cumulative ones, the simplest example being the production of secondarynucleons in the backward hemisphere in the laboratory frame.Usually, the cumulative processes are considered as a result of the coherent inter-action of a projectile with a multiquark cluster, i.e. with a group of several nucleonswhich are at short distances from each other that appears as as a fluctuation of thenuclear matter [29, 40].The inclusive spectra of the secondary hadron h in the central region is determinedat high energies by double-Pomeron diagrams [41]. The case of pp collision is shownin Fig. 3a. In the case of proton-nucleus collisions two different possibilities exist, theinteractions with individual target nucleons (Fig. 3b) and the secondary production oncluster (Fig. 3c). For nucleus-nucleus collisions there are four possibilities (Figs. 3d,3e, 3f, and 3g). The one in Fig. 3g corresponds to a new process where a secondaryhadron is produced by the interaction of two clusters.It was shown in refs. [30, 31] that in the case of secondary production from thecluster fragmentation, the inclusive spectra can be calculated in the framework of theQGSM with the same quark and diquark fragmentation functions. The only differencecomes from the quark and diquark distributions, u clq ( x, n, k ) and u clqq ( x, n, k ), where k is the number of nucleons in the cluster. The distributions u cli ( x, n, k ) can also becalculated by using the Reggeon counting rules, and they take the form: u cluu ( x, n ) = C uu · x α R − α B +1 · ( k − x ) − α R N + m u clud ( x, n ) = C ud · x α R − α B · ( k − x ) − α R + N + m Reggeon diagrams for the different possibilities corresponding to the inclusive spectra ofa secondary hadron h produced in the central region in (a) pp , (b,c) p A, and (d − g) AB collisions.Pomerons are shown by wavy lines. u clu ( x, n, k ) = C u · x − α R · ( k − x ) α R − α B + N + m u cld ( x, n ) = C d · x − α R · ( k − x ) α R − α B +1+ N + m u cls ( x, n ) = C d · x − α R · (1 − x ) α R − α B +( n − d , (13)where N = 2 · (1 − α B ) · ( k − . (14)The function u cls ( x, n ) does not depend on k . All these functions u cli ( x, n, k ) are nor-malized to unity and the values m and m can be found from conditions in Eq. (2).This approach was succesfully used in [30, 31] for the description of cumulativeparticles produced in hA collisions. In the present paper we use it for describing theenhancement of strangeness production on nuclear targets in the central region.The probability to find a proton in the backward hemisphere in high energy pA collisions reach values up to 10%. Keeping in mind that it is only a part of cumulativeprocesses, in the numerical calculations and for every hadron-nucleon interaction wetake the probability to interact with a cluster V cl , and the probability to actuallyinteract with a nucleon 1 − V cl . We use the numerical values: V cl = 0 . , V cl = 0 . , (15)as the maximal reasonable values of V cl . 9 .4 Inelastic screening (percolation) effects The QGSM gives a reasonable description [5, 6, 10, 42] of the inclusive spectra ofdifferent secondaries produced both in hadron-nucleus and in nucleus-nucleus collisionsat energies √ s NN = 14 −
30 GeV.At RHIC energies the situation drastically changes. The spectra of secondariesproduced in pp collisions are described rather well [5], but the RHIC experimental datafor Au+Au collisions [43, 44] give clear evidence of the inclusive density suppressioneffects which reduce by a factor ∼ (a) Multiperipheral ladder corresponding to the inclusive cross section of diagram (b), and(c) fusion of several ladders corresponding to the inclusive cross section of diagram (d). At energies √ s NN ≤ −
40 GeV, the inelastic processes are determined by theproduction of one (Fig. 4a) or several (Fig. 4c) multiperipheral ladders, and the corre-sponding inclusive cross sections are described by the diagrams of Fig. 4b and Fig. 4d.In accordance with the Parton Model [48, 49], the fusion of multiperipheral laddersshown in Fig. 4c becomes more and more important with the increase of the energy,resulting in the reduction of the inclusive density of secondaries. Such processes cor-respond to the enhancement Reggeon diagrams of the type of Fig. 4d, and to evenmore complicate ones. All these diagrams are proportional to the squared longitudinalform factors of both colliding nuclei [32], so their contribution becomes negligible whenthe energy decreases. Following the estimations presented in reference [32], the RHICenergies are just of the order of magnitude needed to observe this effect.However, all quantitative estimations are model dependent. The numerical weight10f the contribution of the multipomeron diagrams is rather unclear due to the many un-known vertices in these diagrams. The number of unknown parameters can be reducedin some models, and, as an example, in reference [32] the Schwimmer model [50] wasused for the numerical estimations. Also, in [51] the phenomenological multipomeronvertices of eikonal type were introduced for the summation of the enhancement dia-gram.Another possibility to estimate the contribution of the diagrams with Pomeron in-teraction comes [52, 53, 54, 55, 56] from Percolation Theory. The percolation approachand its previous version, the String Fusion Model [57, 58, 59], predicted the multiplicitysuppression seen at RHIC energies, long before any RHIC data were measured.New calculations of inclusive densities and multiplicities in percolation theory bothin pp [60, 61], and in heavy ion collisions [61, 62], are in good agreement with theexperimental data in a wide energy region.In order to account for the percolation (inelastic screening) effects in the QGSM,it is technically more simple [23] to consider the maximal number of Pomerons n max emitted by one nucleon in the central region that can be cut. These cut Pomerons leadto the different final states. Then the contributions of all diagrams with n ≤ n max areaccounted for as at lower energies. The larger number of Pomerons n > n max can alsobe emitted obeying the unitarity constraint, but due to the fusion in the final state (atthe quark-gluon string stage), the cut of n > n max Pomerons results in the same finalstate as the cut of n max Pomerons.By doing this, all model calculations become very similar to the percolation ap-proach. The QGSM fragmentation formalism allows one to calculate the integratedover p T spectra of different secondaries as the functions of rapidity and x F .In this frame, we obtain a reasonable agreement with the experimental data onthe inclusive spectra of secondaries produced in d+Au collisions at RHIC energy [23]with a value n max = 13, and in p+Pb collisions at LHC energy [63] with the value n max = 23. It has been shown in [64] that the number of strings that can be used forthe secondary production should increase with the initial energy.11 Numerical results
One example of the QGSM description of the x F -spectra of secondary protons andantiprotons measured in pp collisions at 158 GeV/c by NA49 Collaboration [65] ispresented in Fig. 5. -3 -2 -1 Figure 5:
QGSM x F -spectra of secondary protons and antiprotons produced in pp collisions at 158GeV/c compared to the experimental data by the NA49 Collaboration [65]. The QGSM results for net proton ( p − ¯ p ) and net Λ-hyperon (Λ − ¯Λ) productionsat 200 GeV per nucleon are compared to the experimental data by the NA35 Collab-oration [66] in Figs. 6 and 7.The data for net baryon production by proton beam interaction with S and
Aunuclear targets are presented in Fig. 6 in the central and beam fragmentation regions asfunction of rapidity in the laboratory system. The absolute normalization of dn/dy inall cases is determined by the data of proton and antiproton production in pp collisionsat similar energies.The results of the QGSM calculations without SJ and cluster contributions are12 -2 -1 -2 -1 -3 -2 -1 -3 -2 -1 Figure 6:
Net proton p − ¯ p (upper panels) and net Λ-hyperon Λ − ¯Λ (lower panels) production in p − S (left panels) and p − Au (right panels) collisions at 200 GeV per nucleon. Solid curves showthe QGSM calculations with both SJ and cluster contributions, dashed curves with SJ contributionsbut without the cluster ones, and dotted curves without both SJ and cluster contributions. shown in Fig. 6 by dotted lines. Dashed lines show the same calculations with SJcontributions but without the cluster ones, and solid lines show the results with both SJand cluster contributions. The corrections for very high energy interactions describedin Subsection 2.4 are negligible at this energy.In the case of net proton production in p − S collisions, all three curves are closeto each other and they are in reasonable agreement with the experimental data. In thecase of p − Au collisions the number of net protons is too small at small rapidities,what can be explained by the influence of the target fragmentation region. The nuclearcluster contribution, which is important mainly in the beam fragmentation region,seems to be too large.In the case of net Λ-hyperon production the experimental error bars are rather large,13 -1 -1 Figure 7:
Net proton p − ¯ p (upper panels) and net Λ-hyperon Λ − ¯Λ (lower panels) production in S - S (left panels) and in S -
Au (right panels) collisions at 200 GeV per nucleon. Solid curves showthe QGSM calculations with both SJ and cluster contributions, dashed curves with SJ contributionsbut without the cluster ones, and dotted curves without both SJ and cluster contributions. and one can talk of general semiquantitative agreement of the QGSM calculations withthe data in the lower panels of Fig. 6.For the sulphur S beam shown in Fig. 7, the theoretical calculations are in rea-sonable agreement with the data for net proton production, whereas the data on netΛ-hyperon production are systematicaly higher than all calculated curves. In any case,though, all disagreements are not large than ∼ .2 Λ and ¯Λ production in midrapidity region The NA49 Collaboration obtained experimental data [68] for yields of Λ and ¯Λ hyperonsin midrapidity region ( | y | < .
4) in the central C+C, Si+Si, and Pb+Pb collisions (5%centrality), at 158 GeV per nucleon. These results are presented in Table 1, togetherwith the QGSM results obtained for the same rapidities and centralities. √ s (GeV) Reaction QGSM Experiment dn/dy17.2 C +C → Λ 0.237 0 . ± . ± .
04, [68]C + C → ¯Λ 0.064 0 . ± . ± .
010 [68]17.2 Si + Si → Λ 0.69 0 . ± . ± .
13, [68]Si + Si → ¯Λ 0.17 0 . ± . ± . → Λ 9.4 12 . ± . ± . → ¯Λ 2.05 1 . ± . ± . → Λ 0.034 0 . ± . ± .
003 [69](m.b.) p + Be → ¯Λ 0.010 0 . ± . ± .
001 [69]17.2 p + Pb → Λ 0.074 0 . ± . ± .
006 [69](m.b.) p + Pb → ¯Λ 0.0019 0 . ± . ± .
002 [69]17.2 Pb + Pb → Λ 9.4 18 . ± . ± . → ¯Λ 2.05 2 . ± . ± .
24 [69]62.4 Au + Au → Λ 11.1 15 . ± . ± . → ¯Λ 8.2 8 . ± . ± . → Λ 3.82 4 . ± .
45 [71]Cu + Cu → ¯Λ 3.34 3 . ± .
37 [71]200 Au + Au → Λ 14.2 14 . ± . → ¯Λ 12.1 11 . ± . → Λ 36.2 -Pb+Pb → ¯Λ 35.6 -Table 1. Experimental NA49 [68], NA57 [69], and STAR [70, 71] data for Λ and¯Λ production at 158GeV per nucleon, and at STAR energies, and the correspondingdescription by the QGSM.On the other hand, the NA57 Collaboration obtained the experimental data [69] forΛ and ¯Λ yields in midrapidity region | y | < . p +Be and p +Pbinteractions, and in central (5% centrality) Pb+Pb collisions at 158 GeV per nucleon.Unfortunately, the data by the NA49 and NA57 Collaborations are not compatible,as one can see from Table 1, where the values of dn/dy for different hyperons measured15y one collaboration are far outside the error bars of the corresponding values publishedby the other collaboration for the same centrality. This is probably due to differentexperimental event selection. .Here again one can see that the calculated yields of Λ and ¯Λ are in agreement withexperimental data on the level of 20 −
30% accuracy.Hyperon production at higher energies in midrapidity region was also measuredat RHIC. The data by the STAR Collaboration [70, 71] for Au + Au and Cu+Cucollisions at √ s NN = 62.4 GeV and 200 Gev are presented in Table 1. We also give inTable 1 the QGSM predictions for central Pb+Pb collisions at the LHC energy √ s NN = 3 TeV. The QGSM provides a reasonable description of nucleon and Λ, as well as their an-tiparticles, production in nucleon-nucleus and nucleus-nucleus collisions at high ener-gies. The level of numerical accuracy is of about 20 − + hyperons are reasonably reproduced for the cases of p+Beand p+Pb collisions [69], but they are several times underestimated in the case ofcentral Pb+Pb interactions [69]. For Ω + production in central Pb+Pb collisions thedisagreement is larger than one order of magnitude. The physical reasons for theseobserved disagreements will be discussed in a separate paper. Acknowledgements
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