Measurement of event-by-event transverse momentum and multiplicity fluctuations using strongly intensive measures Δ[ P T ,N] and Σ[ P T ,N] in nucleus-nucleus collisions at the CERN Super Proton Synchrotron
NA49 Collaboration, T. Anticic, B. Baatar, J. Bartke, H. Beck, L. Betev, H. Bialkowska, C. Blume, B. Boimska, J. Book, M. Botje, P. Buncic, P. Christakoglou, P. Chung, O. Chvala, J. Cramer, V. Eckardt, Z. Fodor, P. Foka, V. Friese, M. Gazdzicki, K. Grebieszkow, C.Hohne, K. Kadija, A. Karev, V. Kolesnikov, M. Kowalski, D. Kresan, A. Laszlo, R. Lacey, M. van Leeuwen, M. Mackowiak-Pawlowska, M. Makariev, A. Malakhov, G. Melkumov, M. Mitrovski, S. Mrowczynski, G. Palla, A. Panagiotou, J. Pluta, D. Prindle, F. Puhlhofer, R. Renfordt, C. Roland, G. Roland, M. Rybczynski, A. Rybicki, A. Sandoval, A. Rustamov, N. Schmitz, T. Schuster, P. Seyboth, F. Sikler, E. Skrzypczak, M. Slodkowski, G. Stefanek, R. Stock, H. Strobele, T. Susa, M. Szuba, D. Varga, M. Vassiliou, G. Veres, G. Vesztergombi, D. Vranic, Z. Wlodarczyk, A. Wojtaszek-Szwarc
MMeasurement of event-by-event transverse momentum and multiplicity fluctuationsusing strongly intensive measures ∆[ P T , N ] and Σ[ P T , N ] in nucleus-nucleus collisions atthe CERN Super Proton Synchrotron T. Anticic , B. Baatar , J. Bartke , H. Beck , L. Betev , H. Bia(cid:32)lkowska , C. Blume , B. Boimska , J. Book ,M. Botje , P. Bunˇci´c , P. Christakoglou , P. Chung , O. Chvala , J. Cramer , V. Eckardt , Z. Fodor , P. Foka ,V. Friese , M. Ga´zdzicki , , K. Grebieszkow , C. H¨ohne , K. Kadija , A. Karev , V. Kolesnikov , M. Kowalski ,D. Kresan , A. Laszlo , R. Lacey , M. van Leeuwen , M. Ma´ckowiak-Paw(cid:32)lowska , , M. Makariev , A. Malakhov ,G. Melkumov , M. Mitrovski , S. Mr´owczy´nski , G. P´alla , A. Panagiotou , J. Pluta , D. Prindle ,F. P¨uhlhofer , R. Renfordt , C. Roland , G. Roland , M. Rybczy´nski , A. Rybicki , A. Sandoval , A. Rustamov ,N. Schmitz , T. Schuster , P. Seyboth , F. Sikl´er , E. Skrzypczak , M. S(cid:32)lodkowski , G. Stefanek , R. Stock ,H. Str¨obele , T. Susa , M. Szuba , D. Varga , M. Vassiliou , G. Veres , G. Vesztergombi , D. Vrani´c ,Z. W(cid:32)lodarczyk , A. Wojtaszek-Szwarc (NA49 Collaboration) NIKHEF, Amsterdam, Netherlands. Department of Physics,University of Athens, Athens, Greece. E¨otv¨os Lor´ant University, Budapest, Hungary Wigner Research Center for Physics,Hungarian Academy of Sciences, Budapest, Hungary. MIT, Cambridge, Massachusetts, USA. H. Niewodnicza´nski Institute of Nuclear Physics,Polish Academy of Sciences, Cracow, Poland. GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,Darmstadt, Germany. Joint Institute for Nuclear Research, Dubna, Russia. Fachbereich Physik der Universit¨at, Frankfurt, Germany. CERN, Geneva, Switzerland. Institute of Physics,Jan Kochanowski University, Kielce, Poland. Fachbereich Physik der Universit¨at, Marburg, Germany. Max-Planck-Institut f¨ur Physik, Munich, Germany. Institute of Particle and Nuclear Physics,Charles University, Prague, Czech Republic. Nuclear Physics Laboratory,University of Washington, Seattle, Washington, USA. Institute for Nuclear Research and Nuclear Energy,BAS, Sofia, Bulgaria. Department of Chemistry,Stony Brook University (SUNYSB),Stony Brook, New York, USA. National Center for Nuclear Research, Warsaw, Poland. Institute for Experimental Physics,University of Warsaw, Warsaw, Poland. Faculty of Physics,Warsaw University of Technology, Warsaw, Poland. Rudjer Boskovic Institute, Zagreb, Croatia. (Dated: September 17, 2018)Results from the NA49 experiment at the CERN SPS are presented on event-by-event transversemomentum and multiplicity fluctuations of charged particles, produced at forward rapidities incentral Pb+Pb interactions at beam momenta 20 A , 30 A , 40 A , 80 A , and 158 A GeV/c, as well asin systems of different size ( p + p , C+C, Si+Si, and Pb+Pb) at 158 A GeV/c. This publicationextends the previous NA49 measurements of the strongly intensive measure Φ p T by a study ofthe recently proposed strongly intensive measures of fluctuations ∆[ P T , N ] and Σ[ P T , N ]. In theexplored kinematic region transverse momentum and multiplicity fluctuations show no significantenergy dependence in the SPS energy range. However, a remarkable system size dependence isobserved for both ∆[ P T , N ] and Σ[ P T , N ], with the largest values measured in peripheral Pb+Pbinteractions. The results are compared with NA61/SHINE measurements in p + p collisions, as wellas with predictions of the UrQMD and EPOS models. a r X i v : . [ nu c l - e x ] S e p PACS numbers: 25.75.-q, 25.75.Gz
I. INTRODUCTION AND MOTIVATION
Ultra-relativistic heavy ion collisions are studied mainly to understand the properties of strongly interacting matterunder extreme conditions of high energy densities when the creation of the quark-gluon plasma (QGP) is expected.The results obtained in a broad collision energy range by experiments at the Super Proton Synchrotron (SPS) atCERN, the Relativistic Heavy Ion Collider (RHIC) at BNL, and at the Large Hadron Collider (LHC) at CERNindeed suggest that in collisions of heavy nuclei such a state with sub-hadronic degrees of freedom appears when thesystem is sufficiently hot and dense.The phase diagram of strongly interacting matter is most often presented in terms of temperature ( T ) and bary-ochemical potential ( µ B ), which reflects net-baryon density. It is commonly believed that for large values of µ B thephase transition is of the first order and turns into a rapid but continuous transition (cross-over) for low µ B values.A critical point of second order (CP) separates these two regions. The phase diagram can be scanned by varying theenergy and the size of the colliding nuclei and the CP is believed to cause a maximum of fluctuations in the measuredfinal state particles. More specifically, the CP is expected to lead not only to non-Poissonian distributions of eventquantities like multiplicities or average transverse momentum [1, 2], but also to intermittent behavior of low-mass π + π − pair and proton production with power-law exponents calculable in QCD [3, 4].The NA49 experiment at the CERN SPS [5] pioneered the exploration of the phase diagram by an energy scan forcentral Pb+Pb collisions in the range 20 A to 158 A GeV ( √ s NN = 6.3–17.3 GeV), as well as a system size scan at thetop SPS energy of 158 A GeV. Evidence was found [6, 7] that quark/gluon deconfinement sets in at a beam energy ofabout 30 A GeV. Thus the SPS energy range is a region where the CP could be located. At present the search for thecritical point is vigorously pursued by the NA61/SHINE collaboration at the SPS [8] and by the beam energy scanprogram BES at RHIC [9].The NA49 experiment already measured multiplicity fluctuations in terms of the scaled variance ω of the distributionof event multiplicity N [10, 11] and event-by-event fluctuations of the transverse momentum of the particles employingthe strongly intensive measure Φ p T [12, 13]. The present paper reports a continuation of this NA49 study by analyzingtwo new strongly intensive measures of event-by-event transverse momentum and multiplicity fluctuations, ∆[ P T , N ]and Σ[ P T , N ] [14, 15]. These measures are dimensionless and have scales given by two reference values, namely theyare equal to zero in case of no fluctuations and one in case of independent particle production. Unlike Φ p T they allowto classify the strength of fluctuations on a common scale.This paper is organized as follows. In Sec. II the new strongly intensive measures of fluctuations ∆[ P T , N ] andΣ[ P T , N ] are introduced and briefly discussed. Data sets, acceptance used for this analysis, detector effects, andsystematic uncertainty estimates are discussed in Sec. III. The NA49 results on the energy and system size dependencesof transverse momentum and multiplicity fluctuations quantified by the new measures are presented and discussed inSec. IV. A summary closes the paper. II. STRONGLY INTENSIVE MEASURES OF TRANSVERSE MOMENTUM AND MULTIPLICITYFLUCTUATIONS
In thermodynamics extensive quantities are those which are proportional to the system volume. Examples ofextensive quantities in this case are the mean multiplicity or the variance of the multiplicity distribution. In contrast, intensive quantities are defined such that they do not depend on the volume of the system. It was shown [14] thatthe ratio of two extensive quantities is an intensive quantity, and therefore, the ratio of mean multiplicities, as wellas the commonly used scaled variance of the distribution of the multiplicity N , ω [ N ] = ( (cid:104) N (cid:105) − (cid:104) N (cid:105) ) / (cid:104) N (cid:105) , areintensive measures. Finally, one can define a class of strongly intensive quantities which depend neither on the volumeof the system nor on the volume fluctuations within the event ensemble. Such quantities can be truly attractive whenstudying heavy ion collisions, where the volume of the produced matter cannot be fixed and may change significantlyfrom one event to another. Examples of strongly intensive quantities are mean multiplicity ratios, the Φ measure offluctuations [16], and the recently introduced ∆ and Σ measures of fluctuations [14, 15]. In fact, it was shown [14]that there are at least two families of strongly intensive measures: ∆ and Σ. The previously introduced measure Φ isa member of the Σ-family.In nucleus-nucleus collisions the volume is expected to vary from event to event and these changes are impossibleto eliminate fully. Thus, the strongly intensive quantities allow, at least partly, to overcome the problem of volumefluctuations. Generally, the ∆ and Σ measures can be calculated for any two extensive quantities A and B . In thispaper B is taken to be the accepted particle multiplicity, N ( B ≡ N ), and A the sum of their transverse momenta P T ( A ≡ P T = (cid:80) Ni =1 p T i , the summation runs over the transverse momenta p T i of all accepted particles in a given event).Following Refs. [14, 15] the quantities ∆[ P T , N ] and Σ[ P T , N ] are defined as:∆[ P T , N ] = 1 (cid:104) N (cid:105) ω ( p T ) [ (cid:104) N (cid:105) ω [ P T ] − (cid:104) P T (cid:105) ω [ N ]] (1)and Σ[ P T , N ] = 1 (cid:104) N (cid:105) ω ( p T ) [ (cid:104) N (cid:105) ω [ P T ] + (cid:104) P T (cid:105) ω [ N ] − (cid:104) P T N (cid:105) − (cid:104) P T (cid:105)(cid:104) N (cid:105) )] , (2)where: ω [ P T ] = (cid:104) P T (cid:105) − (cid:104) P T (cid:105) (cid:104) P T (cid:105) (3)and ω [ N ] = (cid:104) N (cid:105) − (cid:104) N (cid:105) (cid:104) N (cid:105) (4)are the scaled variances of the two fluctuating extensive quantities P T and N , respectively. The brackets (cid:104) ... (cid:105) representaveraging over events. The quantity ω ( p T ) is the scaled variance of the inclusive p T distribution (all accepted particlesand events are used): ω ( p T ) = p T − p T p T (5)Equations (1) and (2) can be used only when assuming that ω ( p T ) is not equal to zero. There is an importantdifference between the ∆[ P T , N ] and Σ[ P T , N ] measures. Only the first two moments: (cid:104) P T (cid:105) , (cid:104) N (cid:105) , and (cid:104) P T (cid:105) , (cid:104) N (cid:105) are required to calculate ∆[ P T , N ], whereas Σ[ P T , N ] includes also the correlation term (cid:104) P T N (cid:105) − (cid:104) P T (cid:105)(cid:104) N (cid:105) . Therefore∆[ P T , N ] and Σ[ P T , N ] can be sensitive to various physics effects in different ways. In Ref. [14] all strongly intensivequantities containing the correlation term are named the Σ family, whereas those based only on mean values andvariances the ∆ family. As already mentioned, the previously studied [12, 13] measure Φ p T belongs to the Σ familyand obeys the relation: Φ p T = (cid:112) p T ω ( p T )[ (cid:112) Σ[ P T , N ] −
1] (6)With the normalization of ∆ and Σ proposed in Ref. [15] these quantities are dimensionless and have a common scalemaking possible a quantitative comparison of fluctuations of different, in general dimensional, extensive quantities.The basic properties of the ∆[ P T , N ] and Σ[ P T , N ] measures are the following:1. Absence of fluctuations . In the absence of event-by-event fluctuations ( N = const. , P T = const. ) the values of∆[ P T , N ] and Σ[ P T , N ] are equal to zero.2. Independent Particle Model (IPM). If the system consists of particles that are emitted independently from eachother (no inter-particle correlations) ∆[ P T , N ] and Σ[ P T , N ] are equal to one. For this case Φ p T vanishes.3. Model of Independent Sources (MIS). When particles are emitted by a number ( N S ) of identical sources, which areindependent of each other and P ( N S ) is the distribution of this number, then ∆[ P T , N ]( N S ) and Σ[ P T , N ]( N S )are independent of N S (intensive measures) and of its distribution P ( N S ) (strongly intensive measures). TheΦ p T measure has the same property. An example of MIS is the Wounded Nucleon Model (WNM) [17], where N S ≡ N W (number of wounded nucleons). Another example is a model where nucleus-nucleus ( A + A ) collisionsare an incoherent superposition of many independent nucleon-nucleon ( N + N ) interactions. For these cases allthree fluctuation measures, namely Φ p T , ∆[ P T , N ] and Σ[ P T , N ], are independent of the number of sources (andtherefore insensitive to the centrality of the collisions) and have the same values for A + A and N + N collisions.The measures ∆[ P T , N ] and Σ[ P T , N ] have similar advantages to ω [ N ]. Like ω [ N ] they have two reference values,namely ω [ N ] equals zero when the multiplicity is constant from event to event and one for a Poisson multiplicitydistribution. Therefore one can judge whether fluctuations are large ( > < ω [ N ] is not a strongly intensive quantity, and in the MIS one finds ω [ N ]( N s sources) = ω [ N ] (1 source) + (cid:104) n (cid:105) ω [ N S ], where (cid:104) n (cid:105) is the mean multiplicity of particles from a single source and ω [ N S ] representsfluctuations of N S .A comparison of the properties of ∆[ P T , N ], Σ[ P T , N ], and Φ p T is presented in Table I.The quantities ∆[ P T , N ] and Σ[ P T , N ] were studied in several models. The results of simulations of the IPM, theMIS, source-by-source temperature fluctuations (example of MIS), event-by-event (global) temperature fluctuations,and anti-correlation between P T /N and N were studied in Ref. [18]. Predictions from the UrQMD model on thesystem size and on the energy dependence of ∆[ P T , N ] and Σ[ P T , N ] are shown in Ref. [15]. Finally, the effectsof quantum statistics were discussed in Ref. [19]. The general conclusion is that ∆[ P T , N ] and Σ[ P T , N ] measuredeviations from the superposition model in different ways. Therefore, the interpretation of the experimental resultsmay benefit from a simultaneous measurement of both quantities. Unit No fluctuations Independent Particle Model Model of Independent SourcesΦ p T MeV/c Φ p T = − (cid:112) p T ω ( p T ) Φ p T = 0 independent of N S and P ( N S );Φ p T ( N S sources) = Φ p T (1 source)∆[ P T , N ] dimensionless ∆[ P T , N ] = 0 ∆[ P T , N ] = 1 independent of N S and P ( N S );∆[ P T , N ]( N S sources) = ∆[ P T , N ](1 source)Σ[ P T , N ] dimensionless Σ[ P T , N ] = 0 Σ[ P T , N ] = 1 independent of N S and P ( N S );Σ[ P T , N ]( N S sources) = Σ[ P T , N ](1 source)TABLE I: Properties of Φ p T , ∆[ P T , N ], and Σ[ P T , N ] in the absence of fluctuations ( N = const. , P T = const. ), in theIndependent Particle Model and in the Model of Independent Sources. III. DATA SELECTION AND ANALYSIS
The data used for the analysis, event and particle selection criteria, uncertainty estimates and corrections aredescribed in the previous publications of NA49 [12, 13] on the measure Φ p T . Here we only recall the key points.The analysis of the energy dependence of transverse momentum and multiplicity fluctuations uses samples of Pb+Pbcollisions at 20 A , 30 A , 40 A , 80 A and 158 A GeV/c beam momenta (center of mass energies from 6.3 to 17.3 GeV per N + N pair) for which the 7.2% most central reactions were selected. The analysis of the system size dependence isbased on samples of p + p , semi-central C+C, semi-central Si+Si, and minimum bias and central Pb+Pb collisionsat 158 A GeV/c beam momentum. Minimum bias Pb+Pb events were divided into six centrality bins (see Ref. [12]for details) but due to a trigger bias the most peripheral bin (6) is not used in the current analysis. For each bin ofcentrality the mean number of wounded nucleons (cid:104) N W (cid:105) was determined by use of the Glauber model and the VENUSevent generator [20] (see Ref. [12]).Tracks were restricted to the transverse momentum region 0 . < p T < . . < y ∗ π < . y ∗ π is the particle rapidity calculated in thecenter-of-mass reference system. For the study of system size dependence at 158 A GeV/c the rapidity was calculatedin the laboratory reference system and restricted to the region 4.0 < y π < . < y ∗ π < . y ∗ p calculated with the proton mass wasapplied ( y ∗ p < y ∗ beam − .
5) [13, 21]. This excludes the projectile rapidity domain where particles may be contaminatedby e.g. elastically scattered or diffractively produced protons.The acceptance of azimuthal angle φ was chosen differently for the study of energy and system size dependence(Fig. 1). For the energy scan a common region of azimuthal angle was selected for all five energies (only particleswithin the solid curves in Fig. 1 (left) were retained), whereas a wider range was used at 158 A GeV/c for the systemsize study (see Fig. 1 (right)). Together with the track quality criteria and rapidity cuts this results in using onlyabout 5 % respectively 20 % of all charged particles produced in the reactions.An additive correction for the limited two track resolution of the detector was applied to the values of ∆[ P T , N ]and Σ[ P T , N ]. The procedure to determine this correction was analogous to the one used to estimate the correctionsfor Φ p T in Refs. [12, 13]. Mixed events were prepared for each of the analyzed data sets and then processed by the [degrees] f -100 0 100 [ G e V / c ] T p
20A GeV/c [degrees] f -100 0 100 [ G e V / c ] T p
30A GeV/c [degrees] f -100 0 100 [ G e V / c ] T p
40A GeV/c [degrees] f -100 0 100 [ G e V / c ] T p
80A GeV/c [degrees] f -100 0 100 [ G e V / c ] T p [degrees] f -100 0 100 [ G e V / c ] T p < 1.4 p * [degrees] f -100 0 100 [ G e V / c ] T p < 2.6 p * FIG. 1: (Color online) Examples of NA49 ( φ, p T ) acceptances of charged particles with the azimuthal angle of negativelycharged particles reflected (see Ref. [13] for details). The solid lines represent the analytical parametrization of acceptance usedfor further analysis. Left: acceptance used for the energy scan of p T and N fluctuations, example for 2 . < y ∗ π < .
2. Right:acceptance used for the system size dependence of p T and N fluctuations, examples for 1 . < y ∗ π < . . < y ∗ π < . y ∗ p (see the text) not included. Figure reproduced from Refs. [12, 13] where parametrizations of the curvescan be found. [GeV] NN s , N ] ( TT R ) T [ P Dd -0.0500.05 all chargednegatively chargedpositively charged [GeV] NN s , N ] ( TT R ) T [ P Sd -0.04-0.0200.02 all chargednegatively chargedpositively charged FIG. 2: (Color online) Additive corrections δ ∆[ P T , N ] (left) and δ Σ[ P T , N ] (right) for limited two track resolution in the 7.2%most central Pb+Pb events at 20 A –158 A GeV/c. Estimates for positively charged, negatively charged and all charged particlesare distinguished by different markers (see legend).
NA49 simulation software. The resulting simulated raw data were reconstructed and the measures ∆[ P T , N ] andΣ[ P T , N ] were calculated using the same selection cuts as used for the real events. The additive two track resolutioncorrections δ ∆[ P T , N ] and δ Σ[ P T , N ]) were calculated as the difference between the values of ∆[ P T , N ] (or Σ[ P T , N ])after detector simulation and reconstruction and before this procedure. The resulting corrections for the data of theenergy scan are plotted in Fig. 2 and those for the data of the system size study in Fig. 3.The statistical uncertainties on ∆[ P T , N ] and Σ[ P T , N ] were obtained via the sub-sample method [12, 13]. Thesystematic uncertainties were estimated by varying event and track cut parameters (the procedures were identical tothose applied for Φ p T in Refs. [12, 13]). IV. RESULTS AND DISCUSSION
The results shown in this section refer to accepted particles, i.e., particles that are accepted by the detector andpass all kinematic cuts and track selection criteria as discussed in Sec. III. The data cover a broad range in p T (0 . < p T < . . < y ∗ π < . δ -rays is small. The selected azimuthal angle region is large and represents
A. Energy scan for central Pb+Pb interactions
Figure 4 presents for the 7.2% most central Pb+Pb interactions the energy dependence of the fluctuation measures∆[ P T , N ] and Σ[ P T , N ] calculated separately for all charged, negatively charged, and positively charged particles. Thesample of negatively charged particles is composed mainly of π − mesons, whereas the sample of positively chargedparticles is dominated by π + mesons and protons. Therefore, the measured values of ∆[ P T , N ] or Σ[ P T , N ] coulddiffer between both charges. Moreover, among all charged particles additional sources of correlations could exist thatare not present in positively or negatively charged particles separately. For all three charge selections the values of∆[ P T , N ] are smaller than one, the expectation for independent particle production. For Σ[ P T , N ] fluctuations for alland positively charged particles are close to the hypothesis of independent particle production (similar to the resultson Φ p T [13] which belongs to the same family of strongly intensive measures), whereas for negatively charged particlesΣ[ P T , N ] values are higher than one. It was suggested in Refs. [15, 19] that values of ∆[ P T , N ] < P T , N ] > p T > P T , N ] and Σ[ P T , N ] are compared to predictions of the UrQMD [24, 25] and EPOS [26,27] models in Fig. 4 (solid and dashed lines respectively). The models do not simulate a phase transition or thecritical point. However, resonance decays and effects of correlated particle production due to energy-momentum,charge and strangeness conservation laws are taken into account. The most central 7.2% interactions were selected forcomparison of the energy scan results, in accordance with the real NA49 events. The procedure of selecting the 7.2%most central events was the following: a sample of minimum bias Pb+Pb events was produced. Then the distributionof the impact parameter b was drawn and the value of b was determined below which 7.2% of the events remained.The resulting impact parameter range was 0 < b < .
35 fm in UrQMD and 0 < b < .
00 fm in EPOS. Finally, highstatistics samples of UrQMD and EPOS events were produced in these impact parameter ranges separately for eachenergy. , N ] T [ P D [GeV] NN s5 10 15 X N D all charged points - 7.2% Pb+Pbsolid lines - UrQMD 3.4dashed lines - EPOS 1.99 [GeV] NN s5 10 15 X N D neg. charged [GeV] NN s5 10 15 X N D pos. charged , N ] T [ P S [GeV] NN s5 10 15 X N S all charged [GeV] NN s5 10 15 X N S neg. charged points - 7.2% Pb+Pbsolid lines - UrQMD 3.4dashed lines - EPOS 1.99 [GeV] NN s5 10 15 X N S pos. charged FIG. 4: (Color online) Energy dependence of ∆[ P T , N ] (top) and Σ[ P T , N ] (bottom) for the 7.2% most central Pb+Pb interac-tions. Statistical uncertainties are denoted by lines, systematic ones by color boxes. Data (points) are compared to predictionsof the UrQMD 3.4 (solid lines) and EPOS 1.99 (dashed lines) models with acceptance restrictions as for the data. The measures ∆[ P T , N ] and Σ[ P T , N ] were calculated from charged particles, consistent with originating from themain vertex. This means that mostly pions, protons, kaons and their anti-particles from the primary interactionwere used because particles coming from the decays of K S , Λ, Σ, Ξ, Ω, etc. are suppressed by the track selectioncuts. Therefore, the analyses of UrQMD and EPOS events were also carried out by using primary charged pions,protons, and kaons and their anti-particles. The tracking time parameter in the UrQMD model was set to 100 fm/cand therefore the list of generated kaons, pions and (anti-)protons did not contain the products of weak decays. Inthe parameter settings of the EPOS model the decays of K S/L , Λ, Σ, Ξ, Ω, etc. particles were explicitly forbidden.Finally, in the analysis of the UrQMD and EPOS events the same kinematic restrictions were applied as for the NA49data.Figure 4 (top) shows that the energy dependence of ∆[ P T , N ] in the UrQMD model exhibits behavior similar tothat observed in the measurements. In both cases one finds ∆[ P T , N ] <
1, i.e. values below those for independentparticle production. As Bose-Einstein correlations are not implemented in the UrQMD model we conclude that in thismodel there must be another source(s) of correlation(s) leading to ∆[ P T , N ] <
1. The EPOS model shows ∆[ P T , N ]values which are significantly higher that those obtained from the NA49 data and UrQMD. The comparisons forΣ[ P T , N ] can be seen in Fig. 4 (bottom). Here the predictions of UrQMD lie above the measurements for all chargedand positively charged particles, whereas they are significantly below the results for negatively charged particles. Onthe other hand EPOS calculations for negatively charged particles are close to the data, but exceed the measurementseven more than the UrQMD predictions for all charged and positively charged particles.The measured energy dependences of ∆[ P T , N ] and Σ[ P T , N ] do not show any anomalies which might be attributedto approaching the phase boundary or the critical point. However, it should be noted that due to the limited acceptanceof NA49 and the additional restrictions used for this analysis the sensitivity for such fluctuations may be small if theunderlying range of correlations in momentum space is large. B. System size dependence at 158 A GeV/c
Figure 5 presents the dependence of ∆[ P T , N ] and Σ[ P T , N ] at 158 A GeV/c on the size of the colliding nuclei aswell as on the centrality of Pb+Pb interactions. The measured values for all accepted charged particles and also for , N ] T [ P D > W
90 in Pb+Pb collisionsis not reproduced by the UrQMD model. The EPOS model predictions for ∆[ P T , N ] are similar to the predictionsof UrQMD. Also the trends in Σ[ P T , N ] for negatively charged and positively charged particles are quite similar inUrQMD and EPOS. In contrast, for all charged particles the values of Σ[ P T , N ] are much higher in EPOS than inUrQMD and describe the NA49 results surprisingly well. C. Search for the critical point
When searching for possible indications of a critical point it is most appropriate to plot the strength of fluctuationsusing the standard phase diagram coordinates temperature T and baryochemical potential µ B . Moreover, centralcollisions of nuclei provide the cleanest interaction geometry. For such reactions fits of the hadron gas model (seee.g. Ref. [28]) were performed to determine the temperature T chem and baryochemical potential µ B of the producedparticle composition. These values are believed to be close to those of the hadronization along the transition line inthe phase diagram. The value of T chem was found to decrease somewhat for collisions of larger nuclei, whereas µ B decreases rapidly with collision energy.Results for ∆[ P T , N ] and Σ[ P T , N ] for inelastic p + p as well as central Pb+Pb collisions are shown in Fig. 6 versus µ B . The p + p results from NA61 [29, 30], plotted for comparison, were obtained using the NA49 acceptance cuts.One observes little dependence on µ B for both Pb+Pb or p + p collisions. In particular, there is no indication of amaximum that might be attributed to the critical point. A similar conclusion was reached from the µ B dependenceof Φ p T [13]. The measurements of ∆[ P T , N ] are consistent for the two reactions. The values of Σ[ P T , N ] are close tounity with the exception of the higher result in Pb+Pb for negatively charged particles.The dependence of ∆[ P T , N ], and Σ[ P T , N ] on T chem is shown in Fig. 7 at the beam momentum of 158 A GeV/cfor p + p , semi-central C+C, Si+Si and central Pb+Pb reactions. The results for p + p from NA49 (solid triangles)and NA61 (open triangles) are consistent. A maximum is observed for Si+Si interactions similar to the one foundpreviously for Φ p T in Ref. [12]. There it was interpreted as a possible effect of the critical point [31] consistent withQCD-based predictions of Ref. [1, 32]. Interestingly, for the same system, studies of intermittency in the productionof low mass π + π − pairs [33] and of protons [34] found indications of power-law behavior with exponents that wereconsistent with QCD predictions for a CP.Unfortunately, theoretical predictions, for fluctuations at CP, are not yet published for the new fluctuation measures∆[ P T , N ] and Σ[ P T , N ]. However, calculations for Si+Si collisions at 158 A GeV/c by using the Critical Monte Carlo(CMC) model [3, 35] are currently under study.
V. SUMMARY
This paper reports on the continuing search at the CERN SPS by the NA49 experiment for evidence of the criticalpoint of strongly interacting matter expected as a maximum of fluctuations. Results are presented on transversemomentum and multiplicity fluctuations of charged particles, produced at forward rapidities (1 . < y ∗ π < .
6) incentral Pb+Pb interactions at beam momenta 20 A , 30 A , 40 A , 80 A , and 158 A GeV/c, as well as in different systems( p + p , C+C, Si+Si, and Pb+Pb) at 158 A GeV/c. New strongly intensive measures of fluctuations, ∆[ P T , N ] andΣ[ P T , N ], were measured. This paper is an extension of previous NA49 studies [12, 13] where the strongly intensivemeasure Φ p T was used to determine transverse momentum fluctuations. The quantities ∆[ P T , N ] and Σ[ P T , N ] aredimensionless and have two reference values, namely they are equal to zero in case of no fluctuations ( P T = const. , N = const. ) and one in case of independent particle production. Therefore, ∆[ P T , N ] and Σ[ P T , N ] are preferableto Φ p T for which only one reference value is defined, i.e. Φ p T = 0 MeV/c for the model of independent particleproduction (IPM).The NA49 results show no indications of a maximum in the energy dependence of transverse momentum (see alsoRef. [31]) and previously measured multiplicity [31] fluctuations in central Pb+Pb collisions throughout the CERNSPS energy range (but finer steps in the scan would be desirable). At all energies the values of ∆[ P T , N ] are belowone, i.e. below the expectation from the IPM. The values of Σ[ P T , N ] are close to one for all charged and positivelycharged particles and slightly higher in case of negatively charged particles. The effect for negatively charged particlescan probably be explained as due to Bose-Einstein statistics. For positively charged particles the interpretation is lessclear because other sources of correlations (for example resonance decays) can contribute to the correlation measures.The system size dependence of ∆[ P T , N ] and Σ[ P T , N ] (and the related measure Φ p T [12, 31]) at 158 A GeV/cshows maxima for Si+Si and peripheral Pb+Pb interactions (for ∆[ P T , N ] this maximum is mostly seen for peripheralPb+Pb interactions). In central collisions values of ∆[ P T , N ] are lower than one, whereas values of Σ[ P T , N ] for allcharged and negatively charged particles are higher than one (weak anticorrelations). The maximum of Σ[ P T , N ] for0 , N ] T [ P D [MeV] B m
200 300 400 500 X N D all charged closed - 7.2% Pb+Pb (NA49)open - p+p (NA61 in NA49 acc.) [MeV] B m
200 300 400 500 X N D neg. charged [MeV] B m
200 300 400 500 X N D pos. chargedNA61 preliminary , N ] T [ P S [MeV] B m
200 300 400 500 X N S all charged closed - 7.2% Pb+Pb (NA49)open - p+p (NA61 in NA49 acc.) [MeV] B m
200 300 400 500 X N S neg. charged [MeV] B m
200 300 400 500 X N S pos. chargedNA61 preliminary FIG. 6: (Color online) Energy ( µ B ) dependence of ∆[ P T , N ] (top) and Σ[ P T , N ] (bottom) for the 7.2% most central Pb+Pbinteractions and a comparison to NA61 inelastic p + p interactions. µ B values for Pb+Pb collisions are taken from Ref. [28].NA49 data indicate [28] that at the top SPS energy µ B does not depend on the system size (C+C, Si+Si, Pb+Pb). Therefore,the µ B values for p + p are also displayed and assumed to be the same as for Pb+Pb. NA61 data were taken from Refs. [29, 30].For NA61 only statistical uncertainties are shown. all charged particles in C+C and Si+Si interactions is about 5% higher than the base line defined by the IPM. Alsopreviously studied multiplicity fluctuations for the most central A + A collisions were found to show a maximum forSi+Si reactions at 158 A GeV/c [31]. The excess of transverse momentum and multiplicity fluctuations is two timeshigher for all charged than for negatively charged particles as expected for the CP [1].The NA49 collaboration also searched for evidence of the critical point in an intermittency analysis of low-mass π + π − pair [33] and proton [34] production. Indications of power-law behavior consistent with that predicted for aCP were found in the same Si+Si interactions at 158 A GeV/c. The intriguing results strongly motivate the ongoingcritical point search by the successor experiment NA61/SHINE [8] which performs a systematic two-dimensional scan(SPS energies and system size ( p , Be, Ar, Xe, Pb)) of the phase diagram of strongly interacting matter. A maximumof several CP signatures, the so-called hill of fluctuations , would signal the existence of the CP. The RHIC BeamEnergy Scan [9] pursues a complementary program measuring higher order moments and cumulants of net-chargeand net-proton distributions in Au+Au collisions. So far no clear evidence for the CP was found [36, 37]. Thus thepossible existence of the CP remains an interesting and challenging question. Acknowledgments:
This work was supported bythe US Department of Energy Grant DE-FG03-97ER41020/A000,the Bundesministerium fur Bildung und Forschung (06F 137), Germany, the German Research Foundation (grant GA1480/2-2), the National Science Centre, Poland (grants DEC-2011/03/B/ST2/02617, DEC-2011/03/B/ST2/02634,and DEC-2014/14/E/ST2/00018), the Hungarian Scientific Research Foundation (Grants OTKA 68506, 71989, A08-77719 and A08-77815), the Bolyai Research Grant, the Bulgarian National Science Fund (Ph-09/05), the CroatianMinistry of Science, Education and Sport (Project 098-0982887-2878) and Stichting FOM, the Netherlands. [1] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D , 114028 (1999) [hep-ph/9903292]. , N ] T [ P D [MeV] chem T140 160 180 200 X N D all charged p+pC+CSi+SiPb+Pb all - + [MeV] chem T140 160 180 200 X N D neg. charged closed - (semi)centr. A+A (NA49)open - p+p (NA61 in NA49 acc.) [MeV] chem T140 160 180 200 X N D pos. chargedNA61 preliminary , N ] T [ P S [MeV] chem T140 160 180 200 X N S all charged [MeV] chem T140 160 180 200 X N S neg. charged closed - (semi)centr. A+A (NA49)open - p+p (NA61 in NA49 acc.) [MeV] chem T140 160 180 200 X N S pos. chargedNA61 preliminary p+pC+CSi+SiPb+Pb all - + FIG. 7: (Color online) System size ( T chem ) dependence of ∆[ P T , N ] (top) and Σ[ P T , N ] (bottom) for (semi)central A + A collisions at 158 A GeV/c and a comparison to NA61 inelastic p + p interactions. T chem values for p + p , C+C, Si+Si and mostcentral Pb+Pb collisions at 158 A GeV/c are taken from Ref. [28]. NA61 data were taken from Refs. [29, 30]. For NA61 onlystatistical uncertainties are shown.[2] C. Athanasiou, K. Rajagopal and M. Stephanov, Phys. Rev. D , 074008 (2010) [arXiv:1006.4636 [hep-ph]].[3] N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos and G. Mavromanolakis, Nucl. Phys. A , 149 (2005) [hep-ph/0505185].[4] N. G. Antoniou, F. K. Diakonos, A. S. Kapoyannis and K. S. Kousouris, Phys. Rev. Lett. , 032002 (2006) [hep-ph/0602051].[5] S. Afanasev et al. [NA49 Collaboration], Nucl. Instrum. Meth. A , 210 (1999).[6] S. V. Afanasiev et al. [NA49 Collaboration], Phys. Rev. C , 054902 (2002) [nucl-ex/0205002].[7] C. Alt et al. [NA49 Collaboration], Phys. Rev. C , 024903 (2008) [arXiv:0710.0118 [nucl-ex]].[8] see http://shine.web.cern.ch/[9] M. M. Aggarwal et al. [STAR Collaboration], arXiv:1007.2613 [nucl-ex].[10] C. Alt et al. [NA49 Collaboration], Phys. Rev. C , 064904 (2007) [nucl-ex/0612010].[11] C. Alt et al. [NA49 Collaboration], Phys. Rev. C , 034914 (2008) [arXiv:0712.3216 [nucl-ex]].[12] T. Anticic et al. [NA49 Collaboration], Phys. Rev. C , 034902 (2004) [hep-ex/0311009].[13] T. Anticic et al. [NA49 Collaboration], Phys. Rev. C , 044904 (2009) [arXiv:0810.5580 [nucl-ex]].[14] M. I. Gorenstein and M. Gazdzicki, Phys. Rev. C , 014904 (2011) [arXiv:1101.4865 [nucl-th]].[15] M. Gazdzicki, M. I. Gorenstein and M. Mackowiak-Pawlowska, Phys. Rev. C , no. 2, 024907 (2013) [arXiv:1303.0871[nucl-th]].[16] M. Gazdzicki and S. Mrowczynski, Z. Phys. C , 127 (1992).[17] A. Bialas, M. Bleszynski and W. Czyz, Nucl. Phys. B , 461 (1976).[18] M. I. Gorenstein and K. Grebieszkow, Phys. Rev. C , no. 3, 034903 (2014) [arXiv:1309.7878 [nucl-th]].[19] M. I. Gorenstein and M. Rybczynski, Phys. Lett. B , 70 (2014) [arXiv:1308.0752 [nucl-th]].[20] K. Werner, Phys. Rept. , 87 (1993).[21] K. Grebieszkow, Phys. Rev. C , 064908 (2007) [arXiv:0710.3594 [nucl-ex]].[22] S. Mrowczynski, Phys. Lett. B , 8 (1999) [nucl-th/9905021].[23] S. Mrowczynski, Phys. Lett. B , 6 (1998) [nucl-th/9806089].[24] S. A. Bass, M. Belkacem, M. Bleicher, M. Brandstetter, L. Bravina, C. Ernst, L. Gerland and M. Hofmann et al. , Prog.Part. Nucl. Phys. , 255 (1998) [Prog. Part. Nucl. Phys. , 225 (1998)] [nucl-th/9803035].[25] M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst, S. Soff, L. Bravina and M. Belkacem et al. , J. Phys. G , 1859(1999) [hep-ph/9909407].[26] K. Werner, Nucl. Phys. Proc. Suppl. , 81 (2008). [27] K. Werner, L. Karpenko, M. Bleicher and T. Pierog, EPJ Web Conf. , 05001 (2013).[28] F. Becattini, J. Manninen and M. Gazdzicki, Phys. Rev. C , 547C (2009) [arXiv:0907.4101 [nucl-ex]].[32] M. Stephanov, private communication.[33] T. Anticic et al. [NA49 Collaboration], Phys. Rev. C , 064907 (2010) [arXiv:0912.4198 [nucl-ex]].[34] T. Anticic et al. [NA49 Collaboration], arXiv:1208.5292 [nucl-ex].[35] N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos, A. I. Karanikas and C. N. Ktorides, Nucl. Phys. A , 799 (2001)[hep-ph/0012164].[36] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. Lett. , 092301 (2014) [arXiv:1402.1558 [nucl-ex]].[37] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. Lett.112