Triangular flow of negative pions emitted in PbAu collisions at $\sqrt{s_{NN}} = $ 17.3 GeV

Differential triangular flow, $v_3(p_T)$, of negative pions is measured at $\sqrt{s_{NN}}$= 17.3 GeV around midrapidity by the CERES/NA45 experiment at CERN in central PbAu collisions in the range 0-30\% with a mean centrality of 5.5\%. This is the first measurement as a function of transverse momentum of the triangular flow at SPS energies. The $p_T$ range extends from about 0.05 GeV/c to more than 2 GeV/c. The triangular flow magnitude, corrected for the HBT effects, is smaller by a factor of about 2 than the one measured by the PHENIX experiment at RHIC and the ALICE experiment at the LHC. Within the analyzed range of central collisions no significant centrality dependence is observed. The data are found to be well described by a viscous hydrodynamic calculation combined with an UrQMD cascade model for the late stages.


Introduction
The azimuthal anisotropy of particles emitted in heavy-ion collisions is used to study properties of hot and dense systems created in such collisions. The almond shape of the overlapping region in a non-central collision manifests itself in the appearance of the elliptic flow anisotropy (1) driven by strong interactions among constituents of the expanding medium. By these interactions the geometrical anisotropy of the overlap zone evolves, following the pressure gradients, into the momentum space anisotropy that is measured by the second harmonic coefficient v 2 . But due to fluctuating positions of the colliding nucleons, the event plane derived from the elliptic anisotropy is not a strict plane of symmetry, and higher-order anisotropies may appear (2). In fact, among the prominent results from collider experiments are observations of significant triangular flow, at the Relativistic Heavy Ion Collider (RHIC) at nucleon-nucleon center-of-mass energy up to √ s NN = 200 GeV (3,4,5), and at the Large Hadron Collider (LHC) at √ s NN = 2.76 TeV (7,8,9), both in central and non-central collisions.
The large v 2 values of collective flow agree well with predictions of relativistic hydrodynamics (10) without dissipation. This suggests that elliptic flow is developed in the early phase of a locally equilibrated, strongly interacting Quark Gluon Plasma (QGP). The QGP behaves as a nearly perfect liquid with a very small ratio η/s of shear viscosity to entropy density, close to its string-theoretical limit of 1/4π (11,12).
The average elliptic flow magnitude, v 2 , is about 20% larger at the LHC compared to RHIC (13,14,15). This increase is mainly due to the harder p T spectrum at LHC energies. The measured v 2 is in agreement with hydrodynamical extrapolations from RHIC data using the same η/s value (16,17) and also in agreement with a hybrid calculation treating the QGP by ideal hydrodynamics and the late stages by a hadronic cascade model (18). Contrary to the elliptic flow, the triangular flow is nearly independent of centrality. The triangular flow can be described using viscous hydrodynamics and transport models. Triangular flow is found to be a sensitive probe of initial geometry fluctuations and viscosity (19).
The elliptic flow magnitude v 2 measured at the Super Proton Synchrotron (SPS) energy, √ s NN = 17.3 GeV, is about 30% lower than those at the top RHIC energy of √ s NN = 200 GeV (3). Except for the most central collisions (20), the differential flow data v 2 (p T ) at SPS (21,22,23), although very similar in shape to the RHIC and LHC data, stay below calculations of ideal hydrodynamics (24). This failure of ideal hydrodynamics at the top SPS energy has been ascribed to insufficient number densities at very early collision stages (25) and strong dissipative effects at the late hadronic stages (11,12,26,27).
In this paper, we present the first measurement of triangular flow at SPS energy. Experimental results comprise differential triangular flow v 3 (p T ) of negative pions emitted from central 158 AGeV PbAu collisions. The results are compared with the measurement of the triangular flow performed by the PHENIX collaboration at RHIC and the ALICE Collaboration at LHC and also with a hydrodynamics calculation coupled with a UrQMD cascade model (28) to describe the late stages. These findings might shed some light on the late stage of collective expansion characterized by rescattering in the 'hadronic corona' (29).

Experiment and data sample
A sample of 30 · 10 6 central PbAu events was collected with the upgraded CERES/NA45 spectrometer during the heavy-ion run at the top SPS energy of 158 AGeV. Within the polar angle acceptance of 7.7 • < ϑ < 14.7 • , which corresponds to a pseudorapidity range 2.05 < η < 2.70 near midrapidity (y mid = 2.91), the CERES spectrometer has axial symmetry around the beam direction. As it covers the full azimuth φ, it is very suitable for studies of azimuthal anisotropy. A detailed description of the CERES experiment is given in (30).
A precise momentum determination is provided by the radial-drift Time Projection Chamber (TPC) (31) which is operated inside an axially symmetric magnetic field with a radial component providing deflection in r φ . Negative pions are identified using the differential energy loss dE/dx along their tracks in the TPC. For vertex reconstruction and tracking outside the magnetic field, two radial Silicon Drift Detectors (SDD) (32) are placed at 10 and 13 cm downstream of a segmented Au target. Negative pions are reconstructed by matching track segments in the SDD doublet and in the TPC using a momentum-dependent matching window. Depending on pion momentum, the relative momentum resolution varies between 2% and 8%.
A mix of three triggers designed to enhance central events has been used for data collection in the range 0 -30% of σ/σ geo with an average centrality of 5.5% in the data sample. The trackmultiplicity distribution for all triggers combined ('all triggers') is shown in Fig. 1   The mix of all triggers, with a resulting mean centrality of 5.5 %, is labeled 'all triggers' and displayed by black circles. The vertical axis represents the differential cross-section expressed in barns (b). The σ/σ geo axis on top applies to minimum-bias data only. symbols. At low multiplicities, it strongly deviates from the minimum-bias distribution labeled (a). Beside minimum-bias data, which contribute 0.5%, a semi-central trigger, labeled (b), contributes 8.3 % to the total. The biggest share of data, 91.2%, is collected with a most central trigger labeled (c) in Fig. 1. Note that the data will be presented here, besides 'all triggers', for 'top-central' and 'mid-central' triggers by selecting N track > 159 or ≤ 159, with weighted mean centralities of 2.4% and 9.8%, respectively. We remark that because of the unconventional shape of the 'all triggers' and 'mid-central' distributions, we have supplied the actual distributions in digitized form for theory comparisons.

Analysis and results
Among the higher-order harmonics, the triangular collective flow is of particular interest. It is quantified by v 3 , the third-order harmonic coefficient of the azimuthal particle distribution measured with respect to Ψ 3 , the azimuthal angle of the 3rd-order participant event-plane. The angle Ψ 3 is determined as: Here, φ i is the azimuthal angle of the i-th particle out of n used for event-plane reconstruction, and w i (p T i ) are weights used to optimize the event-plane resolution. In the same way as in (20) cos[3( proach is used to avoid the trivial autocorrelation effect, and some contribution from short-range correlations. In order to correct for local detector inefficiency, a shifting and flattening procedure has been applied (for more details see (33)) to ensure an azimuthally isotropic event-plane distribution. The azimuthal anisotropy of particle tracks is then measured with respect to the 3rd-order event plane reconstructed by employing tracks from non-adjacent slices only. The finite resolution of the event plane orientation is obtained from the differences between the event planes reconstructed from two sliced subevents, a and b. The corresponding correction factor is calculated as 3 )] ) −1/2 , and used to compensate the raw v 3 for finite event-plane resolution. As the latter depends on multiplicity, the correction factor is calculated for different centralities. Both, the event multiplicity and the v 3 magnitude influence the event-plane resolution. Fig. 2 shows that a decrease of the dispersion in event-plane orientation with increasing multiplicity is weaker than the decrease in anisotropy. In order to reduce statistical errors, the v 3 results, presented in this paper, are obtained by merging the results obtained in six narrower multiplicity bins.
Since almost all particles accepted for analysis are negative pions, subsamples become partially correlated due to the Hanbury Brown & Twiss effect (HBT) of identical bosons. This effect produces a space-momentum correlation between two pions of the same charge if the product of their momentum difference and the source radius R is below the uncertainty limit, i.e., | p 2 − p 1 | ≤ /R. In the rather central collisions under study here, R is typically 7 fm, and consequently /R ≈ 30 MeV/c, much smaller than the mean pion momentum p T ≈ 400 MeV/c. Moreover, the HBT correlation is short range also in azimuth, and it is significant only if |φ 1 − φ 2 | ≤ /Rp T 0.1. As we deal with bosons, the correlation is positive like flow itself, and therefore applying the flow analysis to the HBT correlations would result in a spurious flow.
In order to subtract the non-flow HBT contribution we follow (34,35) and use the standard Bertsch-Pratt parametrization in the comoving system. The corresponding parameters R side , R out , R long describe the dimensions of the source and the so-called chaoticity parameter λ their degree of non-coherence. λ is allowed to be varied by generous ±50% to account for different track resolution in TPC and the SDD doublet; the former is used for the determination of the source parameters, the latter for the event plane determination. The numerical values R side , R out , R long and λ are obtained from the CERES HBT data (36,37,38) by averaging over k t ≤ 0.6 GeV/c.
As expected, the size of the corrections applied to the data displayed in Fig. 3 is quite large at low p T , but decreases rapidly with increasing p T . The systematic uncertainty in the HBT contribution is derived by calculating the correction varying all source parameters by ±1 σ together and independently, and then taking the error of the mean of the resulting distribution to represent the systematic uncertainty in each bin. The systematic uncertainties in the corrected v 3 have significant size (up to 0.4%) just in the p T region where the HBT effect is greatest. They become negligible for p T approaching 0.8 GeV/c. In order to stabilize the final HBT-corrected value of the triangular flow, several iterations of the correction procedure described in (34) have been performed until the difference between the final HBT-corrected v 3 value and the one before it became smaller than 10 −4 . The triangular flow values corrected this way increase about linearly, starting from zero at transverse momenta close to zero up to 0.04 at p T around 2 GeV/c. Fig. 4 compares our triangular flow results with those from the PHENIX and ALICE Collaborations at √ s NN = 200 GeV and √ s NN = 2.76 TeV (3,39), respectively, in the limited p T range accessible to CERES and at comparable centrality. By inspection of Fig. 4 we conclude that the magnitudes of triangular flow at RHIC and at LHC energy are nearly equal (40). In contrast, the magnitude at the top SPS energy reaches only about one half of the corresponding value at LHC energy. The transverse momentum range of the analyzed SPS data is small with respect to that covered by ALICE data (39). In this restricted p T range the data suggest a linear v 3 (p T ) dependence starting from zero.
We like to remark that ALICE uses large gaps in pseudo-rapidity between tracks used for event-plane reconstruction and tracks to measure v 3 ; this way non-flow contributions from jets and mini-jets might have been effectively suppressed. Although jet-like correlations have been observed at SPS energy (22) at the much lower √ s NN compared to LHC, the minijet density is strongly reduced. In Ref (22)  the total jet yield is about 0.02 per event which is more than an order of magnitude smaller with respect to the corresponding yield at the LHC energy, while the charged particle pseudo-rapidity density is only 4 times smaller (41). This is quite fortunate, since to employ a pseudo-rapidity gap is no option for the limited acceptance in CERES. On Fig. 3 in (5) is shown of the p T -integrated two-particle Fourier coefficients, i.e. of the squared v 3 magnitude as a function of √ s NN energy with a shallow minimum between 10 and 20 GeV. The integration has been performed for p T > 0.2 GeV/c. The ratio between the v 3 at 19.6 GeV, which is quite close to the top SPS energy of 17.3 GeV/c, with respect to the v 3 measured at 200 GeV is about 0.63. In Fig. 5 are depicted corresponding ratios for 17.3, 19.6, 200 and 2760 GeV/c where the p T integration has been done within the range 0.3 < p T < 2.1 GeV/c. The v 3 ratio between the top SPS and the top RHIC energy is about 0.66 which is quite close to the one found in (5). This is alo in a rather good agreement with an AMPT predictions from (6) for the ratio of about 0.6. In contrast to elliptic flow which reflects the initial anisotropy of the fireball and thus depends strongly on centrality (see Fig. 24 in (20) and Fig. 6.22 in (33)), triangular flow arises entirely from fluctuations of the initial shape, and we see from Fig. 6 that its magnitude is not significantly different for mid-central and top-central collisions, with mean averaged centralities of 2.4% and 9.8%, respectively. A rather weak centrality dependence has also been reported by ALICE (see Fig. 1 in (39, 40)) where a very slight increase of v 3 with centrality has been observed. The different centrality behaviour of elliptic and triangular flow can also be observed from the corresponding p T -dependencies displayed in Fig. 7. The systematic errors of the v 3 data shown in Fig. 6   Approaches combining the relativistic hydrodynamics with transport models (so-called hybrid models) have been applied to describe the expansion stage of heavy-ion collisions at ultrarelativistic energies. In such models viscous or ideal fluid dynamics is used to describe the evo-lution of the hot and dense quark-gluon plasma, and hadron transport to describe the evolution of the late sparse hadron gas. The calculations shown here are done using a hybrid model (28) combining the vHLLE viscous hydrosolver (42) with UrQMD hadron cascade (43). In this model both kinetic and chemical freeze-outs are described dynamically by the UrQMD hadron cascade, and thus there are no clear freeze-out temperatures. With this approach, particle yields, in particular for strange mesons and baryons, are not well described. However, since we deal here with pions only, this may not be a serious shortcoming. The switch from fluid to cascade, the 'particlization' (28), is set to take place on a constant energy density surface where = 0.5 GeV/fm 3 . Since the net baryon density is not uniform on such a surface, this density does not correspond to a single temperature. Within the chiral model of the Equation of State (EoS) used in this hydrodynamics description, the value of the switching density sw corresponds to T ≈175 MeV at µ B =0. The remaining parameters of the model are the two Gaussian radii for the initial distribution of energy, and the starting time for the hydrodynamic phase. Their values, together with the value of the switching density, sw = 0.5 GeV/fm 3 , are based on reproduction of the data in collisions at RHIC energies, and are kept unchanged at the SPS for simplicity. In Ref. (28)  In Fig. 8, the comparison between the predictions by this hydrosolver+UrQMD model and our v 3 (p T ) measurements of negative pions in PbAu collisions is shown. The model predictions are calculated for hadrons within 0.2 < p T < 2.0 GeV/c and −1 < η < 1, which is very close to the experimental acceptance. Also, the centrality samples which roughly correspond to the experimental ones are simulated. Comparing the presented distributions, one can conclude that the model predictions are in a rather good agreement with the experimental results, except in the p T region between 0.3 and 0.7 GeV/c where the model slightly underpredicts the experimental data.

Summary
The triangular flow appears as a hydrodynamic response of the system created in heavy-ion collision to the fluctuation of the positions of the overlapping nucleons at the moment of impact. In this paper, for the first time, results on the differential triangular flow v 3 (p T ) are presented measured at the top SPS energy. The magnitudes of v 3 are found to be about one half of the ones measured at the top RHIC and LHC energies. The v 3 measured by CERES at SPS energy of √ s NN = 17.3 GeV is similar to the one measured by STAR at RHIC energy of √ s NN = 19.6 GeV.
The hydrosolver+UrQMD model is able to reproduce the experimental data rather well. This comparison could shed some light on the dynamics of the system created in heavy-ion collisions at top SPS energy.