System-size dependence of the pion freeze-out volume as a potential signature for the phase transition to a Quark Gluon Plasma
aa r X i v : . [ nu c l - t h ] S e p System-size dependence of the pion freeze-out volume as apotential signature for the phase transition to a Quark GluonPlasma
Qingfeng Li ∗ , Caiwan Shen , and Marcus Bleicher ,
1) School of Science,Huzhou Teachers College, Huzhou 313000,People’s Republic of China2) Frankfurt Institute for Advanced Studies (FIAS),Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany3) Institut f¨ur Theoretische Physik,Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany
Abstract
Hanburry-Brown-Twiss (HBT) correlation functions and radii of negatively charged pions fromC+C, Si+Si, Cu+Cu, and In+In at lower RHIC/SPS energies are calculated with the UrQMDtransport model and the CRAB analyzing program. We find a minimum in the excitation functionof the pion freeze-out volume at low transverse momenta and around E lab ∼ − A GeV whichcan be related to the transition from hadronic to string matter (which might be interpreted asa pre-cursor of the QGP). The existence of the minimum is explained by the competition of twomechanisms of the particle production, resonance decays and string formation/fragmentation.
PACS numbers: 25.75.Gz,25.75.Dw,24.10.Lx ∗ E-mail address: [email protected] R O , R S ) is related to the emission durationof the source. Thus, a ratio larger than unity is expected when the system crosses the first-order phase transition [32] and remains for a long time in the mixed phase. However, analmost unity value has been shown by experiments throughout the beam energies from AGSto RHIC [30]. This seems to imply a strong space-momentum correlation [33] which has tobe taken into account in the interpretation of the experimental data. In recent years boththe non-monotonous energy dependence of the freeze-out volume and the small R O /R S -ratiopresent in the AGS, SPS, and RHIC energy regions have been explained fairly well by bothanalyzing non-Gaussian effects and by considering a stronger early pressure which comesfrom the contribution of mean-field potentials for both formed and preformed hadrons [33–36]. Recently, it was pointed out [37] that a sufficient spatial size of the fire ball is essentialto locate the critical point, the crossover and the first-order phase transition in the phasediagram.On the transport model side, benchmark test have been performed, which provide asolid basis for the present and future investigations. The results obtained previously arein line with the experimental facts that the extracted HBT radii are always rather smalland change smoothly for the large span of explored beam energies. I.e. no unexpectedobservations about the freeze-out volume of two particle correlations is found at low SPSenergies where the energy threshold for the onset of deconfinement might be reached. Thisimplies that the expected sensitivity of the pion freeze-out volume to the possible phasetransition is relatively weak and needs to be analyzed carefully. On the experimental sidea detailed exploration of different collision systems is currently underway with the NA61experiment taking data at beam energies from 10A GeV to 158A GeV for systems from p+pto Au+Au [4, 12]. Finally, we suggest to conduct a careful transverse-momentum analysisfor the excitation function of the pion freeze-out volume in order to find out any althoughweak but unusual phenomenon for the phase transition.In this work, we provide a baseline calculation for the system-size and the transverse-momentum dependence of the excitation function of the HBT pion freeze-out volume withina relativistic transport model. In line with our previous findings, we explore two scenarios,the standard cascade calculations and a modified version of the model which includes mean3eld potentials of both formed and preformed hadrons to increase the pressure in the earlystage of the reaction. As a tool we employ a modified version of the UrQMD model which -apart from other changes - includes a repulsive interaction for the early stage of the reactionto mimic the explosive expansion of the source encountered at the highest energies (fordetails of the implementation, the reader is referred to [33–35]).For the present analysis we simulate central collisions ( σ/σ T otal < σ being the crosssection) for four mass-symmetric reactions, C+C, Si+Si, Cu+Cu, and In+In, with beamenergies from 2A GeV to 80A GeV (for detailed energy points, please see the figures below).Firstly we extract the inverse slope parameter T (“apparent temperature” or “temperature”)from the transverse mass ( m t = ( m + p t ) / ) spectra of negatively charged pions at mid-rapidity ( | y cm | < .
5, where y cm = log E + p k E − p k , E and p k are the energy and longitudinalmomentum of the pion meson in the center-of-mass system) according to the expression1 m t dN dm t dy cm = f ( y cm ) exp( − m t T ) (1)where f ( y cm ) = const . Fig. 1 shows the excitation function of the extracted T values atAGS and SPS energies (up to 80A GeV). Calculations with (“SM-EoS”) and without (“Cas-cade”) mean-field potentials of both formed and preformed hadrons from C+C and In+Inreactions are compared with each other. In order to extract the temperature parameter T a transverse mass upper limit is chosen to avoid potential problems with deviations from asingle exponential spectrum. Here we use the range | m t − m π | < . /c for the fittingprocess, in line with the transverse momentum range for the HBT analysis later on.As shown in previous UrQMD calculations and the experimental results [38], the exci-tation function of the extracted temperature from pion spectra is weakly dependent on thesize of the system, which is shown in Fig. 1. At AGS a rapid increase of the T from about0 .
11 GeV to 0 .
14 GeV with increasing beam energies is seen. With the further increase ofbeam energies, the T becomes flat and shows no obvious beam-energy dependence especiallyfor the light system and in cascade calculations. From the results shown in Fig. 1 one alsofinds that the mean-field potential contribution to the modification of T is also of minorimportance and disappears completely for the light C+C system. Note that the mean-fieldpotential for pre-formed hadrons (string fragments) is also considered in the present calcu-lations. Therefore, the weak dependence of the apparent temperature on different equationsof state in such a large beam-energy region implies that most of the previously predicted4ignatures to the confinement-deconfinement phase transition would not be that bright inthe real dynamic transport process [39, 40].Let us now turn to the correlation’s study. The program Correlation After Burner(CRAB) (version 3.0 β ) [41, 42] is used to analyze the two-particle interferometry. Thecorrelator C of two particles is decomposed in Pratt’s (so-called longitudinal co-movingsystem LCMS or “Out-Side-Long”) three-dimensional convention (Pratt-radii). The three-dimensional correlation function is fit with the standard Gaussian form (using ROOT [43]and minimizing the χ -squared) C ( q L , q O , q S ) = 1 + λe − R L q L − R O q O − R S q S − R OL q O q L . (2)In Eq. (2), λ is usually referred to as an incoherence factor. However, it might also beaffected by many other factors, such as the contaminations from long-lived resonances orthe details of the Coulomb modification in final state interactions (FSI). Here we regard itas a free parameter and do not assign a specific physical meaning to it. R L , R O , and R S are the Pratt-radii in longitudinal, outward, and sideward directions, while the cross-term R OL plays a role at large rapidities. q i is the pair relative momentum q ( q = p − p ) inthe i direction. Furthermore, the correlator is also k -dependent (the transverse componentis preferred under a rapidity cut), where k = ( p + p ) / negatively charged pion pairs are calculated at mid-rapiditywith the two correlated particles at | Y ππ | < . Y ππ = log( E + E + p k + p k E + E − p k − p k ) is the pairrapidity with pion energies E and E and longitudinal momenta p k and p k in the center ofmass system) in each CRAB analyzing run. Fig. 2 depicts excitation functions of the HBTparameters λ (upper-left), R L (upper-right), R O (bottom-left), and R S (bottom-right) of π − − π − pairs from central C+C collisions at two k T (= ( p T + p T ) /
2) bins 0 −
100 MeV /c and 100 −
200 MeV /c . Calculations with cascade mode are shown with scattered symbolswhile calculations with SM-EoS are shown by different lines.Similarly, Fig. 3 gives the results of In+In collisions. A beam-energy range 10A-80AGeV is selected and the number of calculated energy points is large enough in order togive clearer information about the energy dependence of these HBT parameters. The λ parameter is less than unity for both systems at AGS and SPS energies and decreases withincreasing beam energy. This behaviour is related to the increasing importance of long5
20 40 60 800.100.120.140.160.180.20 T ( G e V ) E lab (A GeV) In+In,Cascade In+In,SM-EoS C+C,Cascade C+C,SM-EoS - T <5 %|y cm |<0.5|m t -m |<0.65 GeV/c FIG. 1: Excitation function of the extracted temperature parameter T of negatively charged pionsat mid-rapidity from central C+C and In+In reactions at AGS and SPS energies. The comparisonof calculations between with and without mean-field potentials is shown for each colliding system. lived resonances and substantial rescattering. It was shown that the excitation functionof the experimental λ values from Au+Au collisions at AGS energies can be reproducedwell by the RQMD model [18]. Although the effect of the mean-field potential on λ isweak in both reactions at all beam energies as shown in [18], it indeed becomes visiblein the heavier In+In system especially at high SPS energies. We further find that at thelow k T bin the λ value is driven up slightly when potentials are considered and vice versafor the high k T bin. This is easy to understand because the attractive potential leads tomore rescattering process for pions with small momenta, which implies a larger incoherence.While the repulsive part of the potential leads to less rescattering of the resonances beforefreeze-out, which implies thus a larger coherence. If we turn to look into the beam-energydependence of the HBT radii, surprisingly, we find some non-monotonous behaviour at lowSPS energies, especially in the longitudinal direction. In C+C collisions and within the k T bin 0 −
100 MeV /c , a minimum R L is seen at ∼ A GeV for calculations both with andwithout potentials. While the appearance of the minimum value of R L from In+In collisionsdepends heavily on both the consideration of the potential and the k T interval selected - the6 .50.60.70.80.91.0 C+C, / T <5%,|y |<0.5 SM-EoS, k T < 100 MeV/c SM-EoS, 100 20 40 60 80123 R S ( f m ) R L ( f m ) R O ( f m ) E lab (A GeV) 20 40 60 80123 FIG. 2: Excitation functions of the HBT parameters λ (upper-left), R L (upper-right), R O (bottom-left), and R S (bottom-right) of π − − π − pairs from central C+C collisions at two k T bins 0 − /c and 100 − 200 MeV /c . A pion-pair rapidity cut | Y ππ | < . minimum appears at ∼ A GeV only from the calculation with potentials and at the larger k T bin 100 − 200 MeV /c . Furthermore, a stronger beam-energy dependence of the HBTradii with mean-field potentials is also seen especially in the longitudinal direction of pionpairs from In+In collisions which is certainly due to the density dependence of the Skyrme-like terms in SM-EoS [33]. Therefore, it is seen clearly that for light system, the minimumpoint appears at both k T bins, while for heavy system, the appearance of the minimumdepends on the consideration of both the certain k T bin and the mean-field potentials. Thisis interesting phenomenon which can be addressed by the system size and energy scan ofthe NA61 experiment.Fig. 4 shows the excitation function of the pion source volume V f (= (2 π ) / R L R S ) atfreeze-out (calculated as [28]) from central C+C (in plot (a) ), Si+Si ( (b) ), Cu+Cu ((c) ), and In+In ( (d) ) collisions. First of all, for the calculations in the cascade mode,the excitation function of V f shows a minimum only for light systems (especially for the7 .50.60.70.80.91.0 In+In, / T <5%,|y |<0.5 SM-EoS, k T < 100 MeV/c SM-EoS, 100 20 40 60 80345 R S ( f m ) R L ( f m ) R O ( f m ) E lab (A GeV) 20 40 60 80345 FIG. 3: Excitation functions of the HBT parameters λ (upper-left), R L (upper-right), R O (bottom-left), and R S (bottom-right) of π − − π − pairs from central In+In collisions at two k T bins 0 − /c and 100 − 200 MeV /c . A pion-pair rapidity cut | Y ππ | < . middle-sized Si+Si system). In the In+In case, the minimum is not seen in any k T bins.It implies that the occurrence of the minimum V f is system-size dependent if the cascademode is adopted in calculations. Secondly, let us explore the results with the inclusion ofmean-field potentials (lines), in C+C case, the excitation function of V f shows a minimum atlow SPS energies for the calculation at lower k T bin (the V f value decreases about 7% fromthe beam energy 10A Gev to 20A GeV). This minimum keeps for heavier systems Si+Si,Cu+Cu, and In+In, but, the valley within the lower k T becomes shallow and disappearwhile that in the higher k T bin starts to become important (the amplitude of the decreaseof V f from 10A GeV to 35A GeV is about 9%). Therefore, the minimum seems alwaysto be present (although in different k T bins) in the excitation function of the pion sourcevolume V f of all systems for all calculations with the mean-field potentials. It implies thatthe occurrence of the minimum V f is k T dependent but not system-size dependent if themean-field potentials are considered in calculations. Finally, the minimum, if it is present,8 d)c)b) In+InCu+CuSi+Si V f ( f m ) C+C a) 20 40 60 80200220240260280300320 20 40 60 80350400450500550600650700750 20 40 60 80800900100011001200130014001500 E lab (A GeV) FIG. 4: Excitation functions of the pion source volume V f from central C+C (in plot (a) ), Si+Si( (b) ), Cu+Cu ( (c) ), and In+In ( (d) ) collisions at two k T bins 0 − 100 MeV /c (solid lines andsquares) and 100 − 200 MeV /c (dashed lines and circles). A pion-pair rapidity cut | Y ππ | < . lies always around the beam energy 30 A GeV. This is because the dominant mechanism ofthe particle production starts to change from the decay of resonances to the fragmentationof strings [44]. Although the transport of free quark degree of freedom is not considered incurrent version of the UrQMD, the mean-field potentials for the pre-formed hadrons maypartly compensate the lack of pressure from the (missing) partonic stage. The decrease ofthe pion source volume before the minimum is due to the fact that more pions starts tobe produced from the excitation and fragmentation of strings which happens at an earlierstage of the collision, i.e. at a smaller freeze-out size. With the further increase of the beamenergy, the increase of the pion source volume is due to the increase of pion yields at highSPS energies. However this effect is washed out in heavier system since both the productionof pions from the decay of resonance and the rescattering at the late stage of collisions playimportant roles. 9n summary, we have calculated the HBT correlation functions of negatively chargedpions for several systems from the light C+C, middle-sized Si+Si and Cu+Cu, to the heavyIn+In system with the UrQMD transport model and the correlation after-burner CRAB 3.0.We found that the non-monotonous energy dependence of the pion freeze-out volume mightsignal a change in the degrees of freedom from hadronic to partonic matter around 30 A GeV. The existence of the minimum was explained by the combination and competition ofthe resonance decay with the string excitation and fragmentation (which we interpreted asa pre-cursor to a QGP). The results presented here, especially for the middle-sized systems,are in reach of the NA61/SHINE experiment at the CERN-SPS. Acknowledgements We would like to thank S. Pratt for providing the CRAB program and acknowledgesupport by the computing server C3S2 in Huzhou Teachers College. The work is supportedin part by the key project of the Ministry of Education of China (No. 209053), the NationalNatural Science Foundation of China (Nos. 10905021,10979023, 10979024), the ZhejiangProvincial Natural Science Foundation of China (No. Y6090210), and the Qian-Jiang TalentsProject of Zhejiang Province (No. 2010R10102). This work was supported by the HessianLOEWE initiative through the Helmholtz International Center for FAIR (HIC for FAIR). [1] T. Matsui and H. Satz, Phys. Lett. B , 416 (1986).[2] S. Soff, S. A. Bass, M. Bleicher, L. Bravina, E. Zabrodin, H. St¨ocker and W. Greiner, Phys.Lett. B , 89 (1999)[3] A. Dumitru and R. D. Pisarski, Phys. Lett. B , 95 (2002)[4] M. Gazdzicki [NA49-future Collaboration], arXiv:nucl-ex/0612007.[5] U. Heinz and G. Kestin, arXiv:nucl-th/0612105.[6] K. Grebieszkow [NA49 Collaboration and NA61 Collaboration], Acta Phys. 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