Pion freeze-out as seen through HBT correlations in heavy ion collisions from FAIR/AGS to RHIC energies
aa r X i v : . [ nu c l - t h ] J u l Pion freeze-out as seen through HBT correlations in heavy ioncollisions from FAIR/AGS to RHIC energies
Qingfeng Li, ∗ Marcus Bleicher, and Horst St¨ocker ,
1) Frankfurt Institute for Advanced Studies (FIAS),Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany2) Institut f¨ur Theoretische Physik,Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany
Abstract
We perform a systematic analysis of several HBT parameters in heavy ion collisions from E beam =2 AGeV to √ s NN = 200 GeV within the UrQMD transport approach and compare the results toexperimental data where available. We find that the ’lifetime’ of the emission source as calculatedfrom τ ∼ q R O − R S , is larger than the experimentally observed values at all investigated energies.The calculated volume of the pion source ( V f ) is found to increase monotonously with increasingbeam energy and the experimentally observed decrease of the measured V f at AGS is not seen.Finally, we calculate the mean free path λ f = 0 . − PACS numbers: 25.75.Gz,25.75.Dw,24.10.Lx ∗ E-mail address: liqf@fias.uni-frankfurt.de
1n order to create the theoretically predicted deconfined phase of Quantum Chromo-dynamics (QCD) heavy ions have been collided with energies from less than √ s ∼ . . −
20 GeV (FAIR/AGS and SPS) up to 20 −
200 GeV (RHIC).Indeed, it seems that some nontrivial signals - such as charmonium suppression, relativestrangeness enhancement, etc. - of the (phase) transition to the deconfined phase havebeen observed in heavy ion collisions (HICs) at SPS energies [1, 2, 3, 4, 5, 6]. Additionalinformation about the matter created in such collisions can be obtained from the investiga-tion of the space-time structure of the particle emission source (the region of homogeneity).The established tool to extract this information is known as Femtoscopy [7] or originallyas Hanbury-Brown-Twiss interferometry (HBT) [8, 9, 10]. Experimentally this technique isquite often used to extract the information on the spatio-temporal evolution of the particlesource, which has been scanned thoroughly by several separate experimental collaborationsover the whole discussed energy region [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 35]. However,so far the measured excitation function of HBT parameters shows no obvious discontinuitieswithin the large span of explored beam energies [7].A comprehensive theoretical investigation on the excitation function of the HBT parame-ters is thus highly required but still absent so far [7]. Recently, based on the Ultra-relativisticQuantum Molecular Dynamics (UrQMD, v2.2) transport model (employing hadronic andstring degrees of freedom) (for details, the reader is referred to Refs. [23, 24, 25, 26]) and theprogram CRAB (v3.0 β ) [27, 28, 29], we have investigated the transverse momentum, system-size, centrality, and rapidity dependence of the HBT parameters R L , R O , R S (dubbed asHBT radii or Pratt radii), and the cross term R OL of pion source at AGS [30], SPS [31] andRHIC [32] energies, respectively. In general, the calculations are satisfying and well in linewith the experimental data although discrepancies are not negligible. Such as, I), the calcu-lated R L and R S values for Au+Au collisions at low AGS energies are visibly smaller thanthe data if the default UrQMD version 2.2 (cascade mode) is adopted. II), the HBT-’puzzle’with respect to the ’duration time’ of the pion source, is present at all energies. In orderto understand the origin of this HBT-’puzzle’, some efforts were made, however, a completeunderstanding of this effect is still lacking.In this paper, we present pion interferometry results on the ”duration-time” related quan-tity q R O − R S , the freeze-out volume V f , and derive the mean free path λ f of pions atfreeze-out. The analysis is based on the comprehensive comparison of the excitation func-2ion of calculated HBT radii R L , R O , and R S at small transverse momenta with data, in theFAIR/AGS, SPS and RHIC energy regime. The standard UrQMD v2.2 in cascade mode isemployed throughout this paper to serve as a benchmark for further discussions[47].To calculate the two-particle correlator, the CRAB program is based on the formula: C ( k , q ) = R d x d x g ( x , p ) g ( x , p ) | φ ( q , r ) | R d x g ( x , p ) R d x g ( x , p ) . (1)Here g ( x, p ) is the probability for emitting a particle with momentum p from the space-time point x = ( r , t ). φ ( q , r ) is the relative two-particle wave function with r being theirrelative position. q = p − p and k = ( p + p ) / q for bosons and can be fitted approximatelyby a Gaussian form. Using Pratt’s three-dimensional convention (the LCMS system), thestandard parametrisation of the correlation function in Gaussian form reads C ( q O , q S , q L ) = 1 + λ exp( − R L q L − R O q O − R S q S − R OL q O q L ) . (2)Here q i and R i are the components of the pair momentum difference q and the homogeneitylength (HBT radii) in the i direction, respectively. The pre-factor λ is the incoherenceparameter and lies between 0 (complete coherence) and 1 (complete incoherence) in realisticHICs. The term R OL is called cross-term and vanishes at mid-rapidity for symmetric systems,while it deviates from zero at large rapidities [31, 33, 34].We compare our calculations of the Pratt parameters of the pion source with experimentaldata for the following central collisions of heavy nuclei:1. Au+Au at the AGS beam energies E b = 2, 4, 6, and 8A GeV ( <
11% of the totalcross section σ T ), a rapidity cut | Y cm | < . Y cm = log( E cm + p k E cm − p k ), E cm and p k are theenergy and longitudinal momentum of the pion meson in the center-of-mass system)is employed. The experimental (E895) data are taken from [12].2. Au+Au at the AGS beam energy 11 .
6A GeV (the <
5% most central collisions), arapidity cut | Y cm | < . E b = 20, 30, 40, 80, and 160A GeV ( < . σ T ofmost central collisions), a pion-pair rapidity cut | Y ππ | < . Y ππ = log( E + E + p k + p k E + E − p k − p k )3s the pair rapidity with pion energies E and E and longitudinal momenta p k and p k in the center of mass system) is employed. The experimental (NA49) data aretaken from [14, 15].4. Pb+Au at the SPS beam energies E b = 40, 80, and 160A GeV (the <
5% most centralcollisions), the pion-pair rapidity cut Y ππ = − . ∼ . − . ∼
0, and − . ∼ − . √ s NN = 30 ( < σ T ),62 . < σ T ), 130 ( < σ T ), and 200 GeV ( < σ T ). Here a pseudo-rapiditycut | η cm | < . η cm = log( p + p k p − p k ), ( p is the momentum of the pion) is employed. Theexperimental (PHOBOS, STAR, and PHENIX) data are taken from [17, 18, 19, 20, 21].Fig. 1 shows the excitation function of the calculated HBT radii R L [in (a)], R O [(b)], R S [(c)], and the duration-time related quantity q R O − R S [(d)] at k T = 100 MeV (fulllines, black) and 200 MeV (dotted lines, red). The experimental data within this transversemomentum region are shown for comparison. Since the experimental data from NA49 [14, 15]and from CERES [16] collaborations overlap at beam energies 40, 80, and 160A GeV, weshow the calculations and data with respect to CERES energies separately as dashed-dottedlines and open symbols.The calculated R L (Fig. 1(a)) increases faster than R O and R S with increasing energies,which is also observed in data. Meanwhile, with increasing beam energies, the splitting of R L with different k T becomes stronger, which can be attributed to the flow-dominated freeze-outscenario. This observation is also in line with previous results [32], where it was found thatthe decrease of R L with increasing k T at RHIC energies is stronger for central reactions thanfor peripheral ones. It should also be noted that the small k T behaviour of the correlationfunction is also affected by some other factors, such as the decay of resonances, potentialinteractions and/or the treatment of resonance life-times and widths in the medium.Fig. 1 (b) shows the calculations on R O in comparison to the experimental data. Here wefind that model calculation in data agree fairly well at AGS and SPS-NA49 but deviationsfrom data are observable at RHIC energies. The comparison to the CERES data (with theappropriate cuts) however shows a rather large discrepancy between calculation and data.This deviations was also report in a previous study [31] and might hint to systematic differ-ences (apart from different experimental centralities and cuts) between the NA49 analysis4 R L ( f m ) Exp. UrQMD(v2.2) Energy Region k T =100 MeV/c AGS/SPS(NA49)/RHIC k T =200 MeV/c k T =100 MeV/c SPS(CERES) k T =200 MeV/c (b)(a) R O ( f m ) (c) R S ( f m ) E b (A GeV) (d) ( R O - R S ) / ( f m ) FIG. 1: (Color Online) Excitation function of HBT radii R L [in (a)], R O [(b)], R S [(c)], andthe quantity q R O − R S [(d)]. The calculations are shown at k T = 100 ±
50 MeV (full line)and 200 ±
50 MeV /c (dotted line), respectively. The gray areas between the k T = 100 MeVand k T = 200 MeV lines are shown for better visibility. The data are at k T ∼ /c forreactions at E b = 2 , , , A GeV (AGS-E895) [12] and 20 , , , , A GeV (SPS-NA49) [15],at k T ∼ /c for reaction at √ s NN = 130 GeV [19, 21], at k T ∼ /c for reactionsat E b = 11 .
6A GeV (AGS-E802) [13], E b = 40 , , A GeV (SPS-CERES) [16], and √ s NN =62 . ,
200 GeV [17, 18, 20]. K T dependence of theHBT radii (with K T up to 1 . /c ) in Pb+Pb collisions at 40A GeV, and it was foundthat the NA57-data, especially the R O , are in line with the NA49-data [35].In Fig. 1 (c), we notice that the calculated sideward radii R S are in qualitative agreementwith the data but seem to be 15% smaller for almost all energies. Furthermore, the increaseof the measured R S at low AGS energies can not be reproduced [30] and might be due tothe omission of potential interactions that gain importance at low beam energies. Detailedinvestigations will be presented in a future publication. The excitation function of the HBTradii from several systems (from light to heavy) inspired by the NA49-future collaboration[4] might also provide new insights into this problem, and predictions are in progress.The HBT duration time ”puzzle”, i.e. the fact of the theoretical quantity q R O − R S being larger than extracted from the data, is present at all investigated energies (see Fig.1 (d)): The calculated values of q R O − R S are about 3 . ∼ . ∼ q R O − R S , as well as the elliptic flow [41], over the whole energy range.Fig. 2 (a) shows the excitation function of the pion source volume V f at freeze-out,calculated as [42] V f = (2 π ) R L R S . (3)Note that the radius R O is not considered to calculate the pion freeze-out volume since itcontains the contribution of the temporal extent of the pion source. Fig. 2 (a) shows clearlythat the UrQMD cascade calculations do provide a reasonable freeze-out volume for the pionsource at RHIC energies. At SPS energies, the agreement is fine with CERES data while itslightly underpredicts those of NA49. Towards even lower energies, the model underpredictsthe measured freeze-out volume due to the omission of the strong interaction potential andother in-medium effects. E.g. at E b = 2 A GeV, the measured V f is about 2 −
10 100 1000 100000.00.51.01.5
Exp. UrQMD(v2.2) Energy Region k T =100 MeV/c AGS/SPS(NA49)/RHIC k T =200 MeV/c k T =100 MeV/c SPS(CERES) k T =200 MeV/c f ( f m ) E b (A GeV) (b)(a) V f ( f m ) FIG. 2: (Color Online) (a): Excitation function of the pion freeze-out volume V f (according to Eq.3) at transverse momenta between k T = 100 MeV and 200 MeV (gray area), compared with data inthis k T -region. (b): Excitation function of the mean free path λ f of pions at freeze-out (accordingto Eq. 4) at the same transverse momenta. The experimental value for λ f at √ s NN = 200 GeVis obtained with the help of recent dN/dy data in [43], at all other energies the λ f data are takenfrom [42]. for an improvement of the HBT-radii at small k T and hence reproduce the data better.From the discussions above, it is clear that the major part of this difference is related to thesideward radius R S .We are now ready to estimate the mean free path λ f of the pions at freeze-out from thefollowing expression [42] λ f = V f N σ = V f N N σ πN + N π σ ππ . (4)7ith the averaged pion-nucleon cross section σ Nπ = 72 mb and the averaged pion-pion crosssection σ ππ = 13 mb (Note that within the present model calculations these values areslightly energy dependent. However, here we have adopted the explicit numbers from Ref.[42] to compare to the results presented there). The nucleon and pion multiplicities N N and N π are calculated as N N = y th · √ π · dN nucleons dy | y mid , (5)and N π = y th · √ π · dN pions dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y mid . (6)using the assumption of a thermal equilibrated system at freeze-out with a temperature T f = 120 MeV. Here, y th is the estimated thermal homogeneity scale in rapidity at a certain k T and T f , and is given by the expressions: y th = arctanh( h β th i ), with h β th i = q h γ i / h γ i and h γ i = 1+1 / K ( m T /T f ) /K ( m T /T f ) − T f /m T . Here K n ( z ) is the modified Besselfunction of order n and m T = q m π + k T . At k T = 100 MeV and 200 MeV, the calculatedhomogeneity lengths in rapidity y th are 0.98 and 0.81, respectively. dN/dy | y mid is the rapiditydensity of pion (nucleons) at mid-rapidity[48]. Recent calculations using the present UrQMDtransport model [25], have shown that the calculated pion and nucleon yields are reasonablyin agreement with data.Fig. 2 (b) shows the excitation function of λ f of pions at freeze-out. It is seen that thetheoretical λ f value increases gradually from ∼ . ∼ k T . The experimental values of λ f are also between0 . − λ f below the AGS energy deviates from the data due to thesmaller calculated sideward radii (resulting in a too small freeze-out volume).Certainly, the assumption of a constant kinetic freeze-out temperature T f = 120 MeV inthe calculations of y th over the whole energy range is not justified, especially for the collisionsat low beam energies. By inserting an energy dependent temperature (assuming a kineticfreeze-out temperature of 70% of the chemical freeze-out temperature shown in [44]) intoEqs. 5 and 6, one finds that at lower AGS energies, the number of nucleons and pions inthe pion source volume is visibly reduced due to the decreased T f . As a result λ f increasesand a flatter distribution of λ f as a function of the beam energy is obtained in the presentcalculation. 8he observation (both experimentally and theoretically) of a nearly energy independentmean free path on the order of 0 . R L , R O , and R S ,the quantity q R O − R S , the volume V f , and the mean free path λ f of pions at freeze-outfor heavy systems with energies ranging from lowest AGS to the highest RHIC energies.Generally, the model calculations with UrQMD v2.2 (cascade mode) are in line with thedata over the whole inspected energy range. Although discrepancies especially in the lowerAGS energy region are found and have to be resolved. Especially the HBT parameter R S andthe HBT duration-time related ”puzzle” deserve further attention. Finally we re-interpretthe measured and calculated radii in terms of a mean free path for pions and kinetic freeze-out. Here we find a nearly constant mean free path for pions on the order of λ f = 0 . Acknowledgements
We would like to thank S. Pratt for providing the CRAB program and acknowledgesupport by the Frankfurt Center for Scientific Computing (CSC). Q. Li thanks the FrankfurtInstitute for Advanced Studies (FIAS) for financial support. This work is partly supportedby GSI, BMBF, DFG and Volkswagenstiftung.9
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