PoS(CPOD 2013)038 http://pos.sissa.it/ ∗ [email protected] Speaker. System size dependence of hadroncleon Model production and properties the Statistical is Model discussedmulations. in within Similarities the and grand the differences canonical, Wounded between canonical Nu- predictions andof of micro-canonical conservation the for- laws models related are to exposed. thelike treatment A expansion with need conservation for laws models obeyed which in individual would collisions combine is a stressed. hydrodynamical- ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

8th International Workshop on Critical PointMarch and 11-15, Onset 2013 of Deconfinement Napa, California, USA Marek Gazdzicki Goethe-University Frankfurt, Frankfurt am Main, GermanyJan and Kochanowski University, Kielce, Poland E-mail: From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models PoS(CPOD 2013)038 ] in 3 . The main 1 Marek Gazdzicki ]). 5 ] in which production 2 , 2017/18/19 2017 2016 2015 2011/12/13 /142012 2009/10/11 1 ]. This is because they play a special 4 160 GeV] A 80 40 2 ]. It assumes that particle production in nucleon- 30 5 20 high stat. with new vertex detector high stat. with new beam energy [ detailed scan with existing detector 13 ] and the Statistical Model [ 3

p+p p+Pb

Xe+La Ar+Ca Be+Be Pb+Pb Pb+Pb system size system The NA61/SHINE data taking schedule for the ion program. The already registered reactions The paper is organized as follows. First the models are introduced, then the similarities and The Wounded Nucleon Model (WNM) was proposed by Bialas, Bleszynski and Czyz [ This work is motivated by the NA61/SHINE ion program [ 1976 as a late child of the S-matrix period [ nucleon and nucleus-nucleus collisions is an incoherent superposition of particle production from properties are studied as a functionand of plans size of of colliding the nuclei NA61/SHINE and data their taking collision within energy. The this status program are presented in Fig. role in of heavyapproaches, ion the collisions, hydrodynamical that and string-hadronic isas models. they the basic are Moreover, tools forefathers they to of interpret continuouslythem results the serve being on currently simple hadron and most production approximately in popular reproducingformulated high before several energy the basic collisions. QCD properties This era of and is the their duereview relation data. of to to multi-particle They QCD production were still in remains unclear high (for energy a collisions brief see historical Ref. [ differences in their predictionsremarks concerning conclude the the paper. system size dependence are discussed.2. Closing The Wounded Nucleon Model goals of this program arepoint the of study strongly of interacting theaffect matter. the onset system They of size can deconfinement dependence and be andview collision the of reached energy this search providing dependence I for ”trivial” are discuss the sufficiently phenomena hereof understood. first critical which conservation the In influence laws of on fluctuations thethe of Wounded system the Nucleon system size Model size, dependence. [ and second Two the models role are selected for this purpose: Figure 1: are indicated by green squares,program whereas are the shown in approved red future and data gray, taking respectively. and the proposed extension1. of the Introduction From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models PoS(CPOD 2013)038 ]. Its (2.1) 4 Marek Gazdzicki , where the ratio of ). In the SPS energy 2 4 / 1 NN s ≈ F ]( 7 in nucleus-nucleus collisions and A , 2 and 2 are mean numbers of wounded nu- / i NN W i h A h 3 = i W h / i A h is the mean multiplicity of hadron NN i A h GeV. At this energy the prediction of the WNM (see text) is strictly valid. and A i The ratio of mean pion multiplicity to the mean number of wounded nucleons for p+p interactions A ]. This is why the WNM and its string-hadronic successors are still in use. h 6 The Statistical Model of multi-particle production (SM) was initiated by Fermi in 1950 [ The most famous prediction of the model reads: 3. The Statistical Model where nucleon-nucleon interactions, respectively, whereas cleons in nucleus-nucleus and nucleon-nucleonfor collisions. the most This copiously prediction produced is ,mean approximately pions. pion valid This multiplicity is to demonstrated the inPb+Pb mean Fig. collisions number is of plotted wounded as nucleons(not a in shown function p+p in of interactions the collision and30% energy. plot) central [ energies From the the low ratio SPS for to p+p the and top Pb+Pb RHIC collisions differs ”only” by about basic assumption states that all possiblesion micro-states are equally of probable. the For macroscopic large systemproduced enough created systems hadron in (e.g. species a central colli- Pb+Pb (e.g. collisions)model, and pions, SM(GCE), abundantly kaons can be and used protons) for the particle inclusive grand spectra canonical and formulation mean of multiplicities. the This to Figure 2: and central Pb+Pb collisions plotted as a function of collision energy [ wounded nucleons (nucleons which interacted inelasticallythe and Glauber whose approach). number is Propertiesnuclei, calculated of e.g. using wounded they are nucleons the are same in independent p+p of and Pb+Pb the collisions size at of the same colliding collision energy per nucleon. From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models region shown in the plot, theat dependences about for 40 central Pb+Pb collisions and p+p interactions cross each other PoS(CPOD 2013)038 (3.1) ) and le ft Marek Gazdzicki . Two multiplicities B and are system volume, its tem- A E and T , , V T / E ]. In p+p interactions the spectra are approxi- 7 − e · , are measured to be approximately exponential 2 V 4 GeV [ m ∼ A + 3 2 T p dp GeV are presented. This is why the SM(GCE) and its / q n A 3 ) at 158 = d T right m ) at 158 denote particle transverse momentum and mass, respectively). This m right and T p , where examples of the transverse mass spectra in p+p interactions ( 3 is the particle momentum distribution and 3 dp / 2 GeV ( Examples of transverse mass spectra of the most abundant hadrons produced in p+p interactions n 3 ≤ d T In the WNM model hadrons are produced from ”decays” of independent wounded nucleons. ) and central Pb+Pb collisions ( m le ft is shown in Fig. Similarly in the SM(GCE)ments. model This they leads are to almost produced identicalproduction from predictions properties concerning ”decays” in the both of system models. size independent The dependence detailedusing volume of discussion as of ele- hadron an this conclusion example is multiplicities presented below of two different types of hadrons, for central Pb+Pb collisions ( hydrodynamical successors are in use today. 4. System size dependence: Similarities are needed in order to express theindependent model of predictions the on system the system size sizethe and dependence models its so themselves fluctuations. that as they well It are as obviously between simplifies the a latter comparison and between experimental data. Figure 3: ( mately exponential with the inverse slopethe parameter, which Statistical is Model. the The same spectra foris in different interpreted hadrons central as as Pb+Pb due predicted collisions to by deviate non-equilibrium somewhat effects from as the the prediction, collective which expansion of matter before freeze-out. the most famous prediction of the model, namely: perature and particle energy, respectively. Thisdata prediction if agrees spectra approximately in with the experimental direction transversein to the the transverse collision mass axis of are hadrons, considered. Namely, distributions From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models where PoS(CPOD 2013)038 , ] φ ) B , in , V ( A (4.2) (4.6) (4.7) (4.8) (4.9) (4.1) (4.3) (4.4) (4.5) V [ P (4.10) ∆ and and . ) W V W ( Marek Gazdzicki P and ) i W A and scaled variance, h calculated within the i , respectively, for the 2 + i ) V i i . B . AB X ) h are strongly intensive their h i ) ) ( and i i φ −h / W A V X W )) ( , , ]: i h , , and is independent of i − h ] ] 8 i − h B i − h ] ] ] 2 i 2 B ih B V V W W h ] = , B [ [ [ [ . V W , ( A h h h X A ω ω ω ω ( ( [ / . In the WNM they are proportional to / [ V W · · i i Σ i · i · i · i · 2 2 ]) B A i − h i i , h Var V V ] h B ] that properly constructed functions of the i ∼ W W i ∼ quantity is a reincarnation of the popular h h [ V h h 9 B W B B / / AB , / / ω h , Σ ih h i i h ih i i i and A 8 ( B B A [ B A B A 2 h h i h h 5 ∆ ih ih A . − A h − h A ] ] + ] + , h , B ] + ] + h ] B B A A B [ A [ [ + V [ [ or [ + W ∗ ∗ i ∗ ∗ ω i follows [ ω A i ω ω V ω ω i are quantities calculated for any fixed value of the system i ∼ h A W and the mixed second moment i ∼ B ∗ h h ∗ A h 4.8 i ∗ A in the WNM and the SM(GCE), respectively. Thus h ] = ] = - B ] = ] = i h B A ) ] + A B [ [ AB [ [ 4.6 V h A AB ] = ( ω ω [ ( distributions. The h ω ω and B P = ω stands for , B = i A i A i B [ X ]. As the quantities h and ∆ AB and h 10 AB ) , within the WNM and the SM(GCE), respectively. are probability (density) distributions of h and Eqs. A V ] = ( ) W ( B V , 4.5 P , where ( - i and A P [ X Σ h 4.3 quantities denoted by W / ] ]: and 8 X are strongly intensive quantities which measure fluctuations, i.e. they are sensitive ]. Such quantities should be used to study the system size dependence as they eliminate 4.8 [ ) 8 - ] W B ( , Var 4.3 P A [ From Eqs. First let us consider mean multiplicities, Second we consider multiplicity fluctuations characterized by second moments of multiplicity The scaled variances of Σ ] = X [ where Obviously the ratio of mean multiplicities is independent of the system size parameter, considered set of collisions. Quantities whichquantities [ have the latter property are called strongly intensive In Eqs. and are independent of and to second moments of and in the SM(GCE) they are proportional to the system volume: the WNM and the SM(GCE), respectively. Moreover, itthe is system easy size to show parameter that fluctuations. it Thus, is also the independent ratio of measure of fluctuations [ second moments, namely size parameter, the number of wounded nucleons: WNM read [ influence of usually poorly known distributions of the system size parameters, distributions. Two quantities of relevance are variance, From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models ω As may be expected they have a similar form in the SM(GCE) [ PoS(CPOD 2013)038 A denote p and π Marek Gazdzicki , K quantity in p+p and central ], where ) calculated within the SM(CE) φ p , K right ]. They concern three choices of 11 for the SM(GCE), the latter being propor- ] and [ z ) as a function of the mean multiplicity π , p le ft ( ], [ 4 π , 6 K ) and scaled variances ( ] = [ B le ft , A ] presents the results of calculations performed within the simplest 15 quantities are independent of the system size and its fluctuations. The number ]) and for second moments of multiplicity distributions since 2004 (see e.g. φ 14 , taken from Ref. [ ]). Three examples are presented below to illustrate the main results. and 4 13 16 Σ , The ratio of mean multiplicities ( , , 12 ∆ 15 hadron multiplicities, namely [ Figure In summary, predictions of the WNM and the SM(GCE) concerning the system size depen- Predictions of the Statistical Model concerning the volume dependence change qualitatively B from the SM(GCE), the latterwith being increasing proportional volume. to Thus the for system sufficiently large volume. systems The mean multiplicities ratio obtained approaches within one the Refs. [ model which allows to study thescaled effect variance of of the material the conservation multiplicity lawsand distribution. on negatively charged mean particles In multiplicity is and the assumed. modelthe The the SM(CE) ratio ideal of and the gas the mean of SM(GCE) multiplicities classical calculated is within positively plotted in Fig. and the SM(GCE) are plotted astional a to function the of system mean multiplicity volume.used The for ideal calculations. gas model of classical positively and negatively charged particles is values in p+p and Pb+Pb collisions areSM(GCE). equal. At this Of conference course, preliminary this experimental is results valid on only the within the WNM and the and Figure 4: From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models Pb+Pb collisions at the CERN SPS energies were presented [ dence are very similar.as In the particular, the models predict that the mean multiplicity ratio as well multiplicities of charged kaons,interactions pions are and close protons, to the respectively. ones for Noticeably, central the Pb+Pb results collisions. for p+p of wounded nucleons - the- system the size system parameter size in parameterpredictions the in for WNM the quantities SM(GCE) - which - is are is dependent discreet, on continous. whereas the the This system volume may size parameter distribution. to somewhat5. different System size dependence: Differences when material and/or motional conservationical laws are ensemble, introduced, the i.e. canonicalof instead of (CE) conservation the or laws grand micro-canonical has canon- Refs. (MCE) been [ extensively ensembles studied are for used. mean multiplicities The since effect 1980 (see e.g. PoS(CPOD 2013)038 ]. 16 Marek Gazdzicki ). The results for right ( 4 . For small energies of the 5 ] is shown in Fig. 17 7 Ratios of mean multiplicities of neutral hadrons calculated within the MCE and GCE formulations in the grand canonical formulation concerningcal. the system Differences size may dependence appear are only almost because(the identi- the number system size of parameter wounded is nucleons) discreet in and the continuous WNM in the SM(GCE) (the system volume). Finally, the comparison of mean hadron multiplicities calculated with the GCE and MCE 1. I demonstrated that the predictions of the Wounded Nucleon Model and the Statistical Model This is however not the case for the scaled variance as illustrated in Fig. system (in the GCE energysystem is energy and proportional volume. to The volume) observedproduction the peaks of and multiplicity various dips hadrons. ratios are strongly correlated depend with the on thresholds for the 6. Closing remarks Figure 5: of the hadron-resonance gas model are plottedto as a the function system of volume the system in energy, the the latter GCE. being proportional SM(GCE) can be used instead of mean multiplicities from the SM(CE) and the SM(MCE) [ formulations of the hadron resonance gas model [ the SM(CE) and the SM(GCE) approach eachthe scaled other variance when in the the volume SM(GCE) decreases is tofor one zero. the independent Of of scaled course, volume. variance Different in behaviourlarge is the volume observed SM(CE), approaches 0.5. it decreases with increasing volume and for a sufficiently From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models PoS(CPOD 2013)038 ]. 18 Marek Gazdzicki , 307 (2011) [arXiv:1006.1765 42 , 461 (1976). 111 , 014904 (2011) [arXiv:1101.4865 [nucl-th]]. , 127 (1992). 84 8 54 , 279 (1980). 97 , 791 (2012) [arXiv:1201.0485 [hep-ph]]. 43 [NA61/SHINE Collaboration], CPOD 2013, March 11-15, 2013, , 570 (1950). 5 et al. [NA61/SHINE Collaboration], CPOD 2013, March 11-15, 2013, Napa, [NA61/SHINE Collaboration], CERN-SPSC-2006-034. [NA61/SHINE Collaboration], CPOD 2013, March 11-15, 2013, Napa, California, et al. et al. et al. The similarity of predictions explain whyand these hydrodynamical models models) and can their successors co-exist. (string-hadronic collision. Within the Statistical Model theirmodel influence predictions obtained can using be different studied ensembles, bythe GCE, a predictions CE comparison are and of strongly MCE. dependent the The on result conservationmultiplicities is laws. that are For small affected systems whereas mostly mean the for GCE large formulation systems of the scaled Statisticalwhen variances Model both is are mean neither multiplicities modified. valid and for fluctuations small Thus are nor of for relevance. large systems els. In these models, similarly toages the over SM(GCE), many conservation collisions. laws are Thisurgent obeyed significantly to only limits develop for models their aver- which applicability. would Thus,servation combine laws it a obeyed in hydrodynamical-like seems individual expansion to collisions. with be Fortunately con- this effort has already started [ [hep-ph]]. US. California, US. Napa, California, US. I am thankful to Mark Gorenstein and Peter Seyboth for comments and corrections. This work 2. We all believe that material and motional conservation laws are obeyed in each high energy 3. Rich experimental data on hadron spectra in heavy ion collisions favor hydrodynamical mod- [7] S. Pulawski [8] M. I. Gorenstein and M. Gazdzicki, Phys. Rev. C [6] M. Gazdzicki, M. Gorenstein and P. Seyboth, Acta Phys. Polon. B [4] E. Fermi, Prog. Theor. Phys. [5] M. Gazdzicki, Acta Phys. Polon. B [9] M. Gazdzicki, M. I. Gorenstein and M. Mackowiak-Pawlowska, arXiv:1303.0871 [nucl-th]. [3] A. Bialas, M. Bleszynski and W. Czyz, Nucl. Phys. B [1] N. Antoniou [2] K. Grebieszkow [10] M. Gazdzicki and S. Mrowczynski, Z. Phys. C was supported by German Research Foundation (grants DFG GA 1480/2-1 and GAReferences 1480/2-2). Acknowledgments [11] M. Mackowiak-Pawlowska [12] J. Rafelski and M. Danos, Phys. Lett. B From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models PoS(CPOD 2013)038 , 1003 (2006) Marek Gazdzicki 32 , 054904 (2005) , 034901 (2004) 71 70 9 , 243 (2004) [hep-ph/0307061]. 35 , 201 (1980). 5 [nucl-th/0404056]. [nucl-th/0410044]. [nucl-th/0512070]. From p+p to Pb+Pb Collisions: Wounded Nucleon versus Statistical Models [13] K. Redlich and L. Turko, Z.[14] Phys. C F. Becattini and L. Ferroni, Eur. Phys. J. C [15] V. V. Begun, M. Gazdzicki, M. I. Gorenstein and O. S. Zozulya, Phys. Rev. C [16] V. V. Begun, M. I. Gorenstein, A. P. Kostyuk and O. S. Zozulya,[17] Phys. Rev. C V. V. Begun, L. Ferroni, M. I. Gorenstein, M. Gazdzicki and F. Becattini,[18] J. Phys.P. Huovinen G and H. Petersen, arXiv:1206.3371 [nucl-th].