Beam energy dependence of rapidity-even dipolar flow in Au+Au collisions

New measurements of directed flow for charged hadrons, characterized by the Fourier coefficient \vone, are presented for transverse momenta $\mathrm{p_T}$, and centrality intervals in Au+Au collisions recorded by the STAR experiment for the center-of-mass energy range $\mathrm{\sqrt{s_{_{NN}}}} = 7.7 - 200$ GeV. The measurements underscore the importance of momentum conservation and the characteristic dependencies on $\mathrm{\sqrt{s_{_{NN}}}}$, centrality and $\mathrm{p_T}$ are consistent with the expectations of geometric fluctuations generated in the initial stages of the collision, acting in concert with a hydrodynamic-like expansion. The centrality and $\mathrm{p_T}$ dependencies of $\mathrm{v^{even}_{1}}$, as well as an observed similarity between its excitation function and that for $\mathrm{v_3}$, could serve as constraints for initial-state models. The $\mathrm{v^{even}_{1}}$ excitation function could also provide an important supplement to the flow measurements employed for precision extraction of the temperature dependence of the specific shear viscosity.

High-energy nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) can result in the creation of a plasma composed of strongly coupled quarks and gluons (QGP). Full characterization of this hot and dense matter is a major goal of present-day high-energy physics research. Recent studies have emphasized the use of anisotropic flow measurements to study the transport properties of this matter [1,2,3,4,5,6,7,8,9]. A current focus is centered on delineating the role of initial-state fluctuations, as well as reducing their influence on the uncertainties associated with the extraction of the temperature dependent specific shear viscosity (i.e. the ratio of shear viscosity to entropy density η s (T)) of the QGP produced in these collisions [4,5,6,7,8,9,10,11,12,13,14].
The v n coefficients used to characterize anisotropic flow, are normally obtained from a Fourier expansion of the azimuthal angle (φ) distribution of the particles produced orthogonal to the beam direction [15,16]: where Ψ n represents the n th order event plane, i.e., e inφ = v n e inΨ n and the brackets indicate averaging over particles and events. The coefficient v 1 is commonly termed directed flow, v 2 is the elliptic flow, v 3 is the triangular flow etc. For flow dominated distributions, the v n coefficients are related to the Fourier coefficients v nn used to characterize two-particle correlations in relative azimuthal angle ∆φ = φ a − φ b for particle pairs a, b [17]: However, so-called non-flow (NF) correlations can also contribute to the two-particle correlations [17,18,19,20,21]: where δ NF includes possible contributions from resonance decays, Bose-Einstein correlations, jets, and global momentum conservation (GMC).
The magnitude of v odd 1 (η) can be made negligible via a symmetric pseudorapidity selection, to give a straightforward measurement of v even 1 (η). The rapidity-even v 1 is proportional to the fluctuationsdriven dipole asymmetry ε 1 of the system [19,23,24]; v even 1 ∝ ε 1 , where ε 1 ≡ |r 3 e iφ | / r 3 and averaging is taken over the initial energy density after re-centering the coordinate system, i.e., |r 3 e iφ | = 0. Hydrodynamical model calculations [20] indicate that the magnitude of v even 1 is sensitive to η/s, albeit with less sensitivity than for the higher order harmonics, n ≥ 2. It has not been experimentally established whether this sensitivity depends on the temperature T, baryon chemical potential µ B or both. Similarly is has not been established whether this sensitivity could reflect the influence of a possible critical end point (CEP) in the phase diagram for nuclear matter [25]. Therefore, differential v even 1 measurements that span a broad range of √ s NN (T and µ B ), could potentially provide (i) unique supplemental constraints to discern between different initialstate models, (ii) aid precision extraction of η/s and study its possible dependence on T and µ B , and (iii) give insight on the CEP. It is noteworthy that the paucity of v even 1 measurements at RHIC energies precludes their current use as constraints.
The present work employs two-particle correlation functions to extract v 11 = cos ∆φ values as a function of p T a , p T b and centrality for a broad selection of beam energies. In turn the GMC ansatz [18,26] is used in conjunction with the two-component fitting procedure outlined in Refs. [20,21] and discussed below, to extract v even 1 as a function of p T and centrality for each value of √ s NN . The measurements indicate the characteristic p T -dependent directed flow patterns associated with rapidity-even dipolar flow [19,23,24], as well as striking centrality and √ s NN dependencies which could serve as constraints for initial-and final-state model inputs.
The data reported in this analysis are from Au+Au collisions spanning the full range of energies, √ s NN = 7.7 − 200 GeV, in beam energy scan I (BES-I), collected with the STAR detector using a minimum bias trigger. The collision vertices were reconstructed using charged-particle tracks measured in the Time Projection Chamber (TPC) [27]. The TPC covers the full azimuth and has a pseudorapidity range of |η| 1−2 cm with respect to the beam axis. Note that the distribution of the vertex positions broadens (in the beam direction) as the beam energy is lowered. The centrality of each collision was determined by measuring event-by-event multiplicity and interpreting the measurement with a tuned Monte Carlo Glauber calculation [28,29]. Analyzed tracks were required to have a distance of closest approach to the primary vertex to be less than 3 cm, and to have at least 15 TPC space points used in their reconstruction. Furthermore, the ratio of the number of fit points to the maximum possible number of TPC space points was required to be larger than 0.52 to remove split tracks. The p T of tracks was limited to the range 0.2 < p T < 4 GeV/c.
The correlation function technique [17] was used to generate the two-particle ∆φ correlations, where ∆η = η a − η b is the pseudorapidity separation between the particle pairs a, b, (dN/d∆φ) same represents the normalized azimuthal distribution of particle pairs from the same event and (dN/d∆φ) mixed represents the normalized azimuthal distribution for particle pairs in which each member is selected from different events but with a similar classification for the vertex, and centrality. The pseudorapidity requirement |∆η| > 0.7 was also imposed on track pairs to minimize possible nonflow contributions associated with the short-range correlations from resonance decays, Bose-Einstein correlations and jets. The two-particle Fourier coefficients v nn are obtained from the correlation function as: where the ∆φ bin width was chosen to optimize statistical significance. The v nn values were then used to extract v even 1 via a simultaneous fit of v 11 as a function of p T b for several selections of p T a with Eq. (3), Here, K ∝ 1/( N ch p 2 T ) takes into account the nonflow correlations induced by global momentum conservation [20,21]; N ch is the mean multiplicity and p 2 T is proportional to the variance of the transverse momentum over the full phase space. The charged particle multiplicity measured in the TPC acceptance is used as a proxy for N ch . For a given centrality selection, the left hand side of Eq. (6) represents a N-by-M v 11 matrix (i.e., N values for p T b for each of the M p T a selections) which we fit with the right hand side of Eq. (6) using N + 1 parameters: N values of v even 1 (p T ) and one additional parameter K, the coefficient of momentum conservation [30]. Figure 1 illustrates the efficacy of the fitting procedure for 0-5% central Au+Au collisions at √ s NN = 200 GeV. The solid curve (obtained with Eq. (6)) in each panel illustrates the effectiveness of the simultaneous fits, as well as the constraining power of the data. That is, v 11 (p T b ) evolves from purely negative to negative and positive values as the selection range for p T a is increased.
The v even 1 extractions, were carried out for several centrality intervals at each beam energy, depending on the available statistics. The associated systematic uncertainties were estimated from variations in the extracted values after (i) varying all of the analysis cuts by a chosen range about the standard values, (ii) crosschecks to determine the uncertainty associated with the expectation that p T v even 1 (p T ) ∼ 0 and (iii) varying the number of data points used in the fits. The resulting relative uncertainties, which range from ∼ 2% to ∼ 10%, were added in quadrature to assign an overall systematic uncertainty for each measurement. The overall uncertainty for each measurement ranges from ∼ 4% at √ s NN = 200 GeV and grows to ∼ 20% at √ s NN = 7.7 GeV.
The resulting extracted values of v even 1 (p T ) for 0-10% central Au+Au collisions are shown for the full span of BES-I energies in Fig. 2. These values indicate the characteristic pattern of a change from negative v even 1 (p T ) at low p T , to positive v even 1 (p T ) for p T 1 GeV/c, with a crossing point that only very slowly shifts with √ s NN .
This predicted pattern for rapidity-even dipolar flow [19,23] is also indicated by the solid line in panel (a), which shows the result of a hydrodynamic model calculation [20]. It stems from the requirement that the net transverse momentum of the system is zero, i.e., p T v even 1 (p T ) = 0, which implies that the hydrodynamic flow direction of low-p T particles is opposite to those for high-p T particles. Crosschecks made with a large sample of the data, confirmed that p T v even 1 (p T ) ∼ 0, within systematic uncertainties. The crossing point is also expected to shift with √ s NN since the p T and p T 2 values change with √ s NN [30]. For these data, there is little, if any, shift due to the weak dependence of the p T on √ s NN for the indicated centrality selection. It is noteworthy that the low statistical significance of the data for √ s NN <19.6 GeV, precluded similar centrality dependent plots for these beam energies.
The centrality dependencies of the p T -weighted |v even 1 | and K are shown in Fig. 3 for several √ s NN values as indicated, and for 0.4<p T <0.7 GeV/c; this p T range was selected to minimize the associated statistical uncertainties. The increase in the magnitude of |v even 1 | as collisions become more peripheral (Fig. 3(a)), is expected since v even 1 is driven by fluctuations which become more important for smaller systems, i.e., for more peripheral collisions. For each value of √ s NN , Fig. 3(b) indicates a linear dependence of K on N ch −1 with slopes that decrease with increasing √ s NN . This is to be expected since K ∝ 1/( N ch p T 2 ) and the values for p T 2 increase with √ s NN for most of the centrality range.  data are reflected about zero to facilitate a comparison of the magnitudes. The v 3 data, which are obtained from the present analysis, are in good agreement with the data reported in Ref. [31] for the same centrality and p T cuts. The comparison indicates strikingly similar magnitudes and trends for |v even 1 | and v 3 , suggesting a much larger viscous attenuation of v 3 . Note that while ε 1 and ε 3 are both fluctuations-driven, ε 3 ∼ 2ε 1 for 0-10% central Au+Au collisions [23,32] over the √ s NN range of interest. A similar pattern was observed for comparisons made at higher p T , albeit with lower statistical significance. These excitation functions are expected to provide important experimental input to ongoing theoretical attempts to pin down initial state mod-els and make precision extractions of the specific shear viscosity.
In summary, we have employed two-particle correlation functions to carry out new measurements of the p T and centrality dependence of the anisotropic flow coefficient v even 1 in Au+Au collisions spanning the beam energy range √ s NN = 7.7 − 200 GeV. The results show the expected patterns for momentum conservation and the characteristic pattern of an evolution from negative v even 1 (p T ) for p T 1 GeV/c, to positive v even 1 (p T ) for p T 1 GeV/c. That is, the trends expected when initial-state geometric fluctuations act in concert with hydrodynamiclike expansion to generate rapidity-even dipolar flow. The measured dependencies on √ s NN , centrality and p T , as well as the similarity in magnitude and trend of the excitation functions for v even 1 and v 3 , constitute a new set of experimental constraints. These new constraints could prove invaluable to future theoretical attempts to discern between different initial-state models, as well as for precision extraction of the temperature dependence of the specific shear viscosity. and Technology of China and the Chinese Ministry of Education, the National Research Foundation of Korea, GA and MSMT of the Czech Republic, Department of Atomic Energy and Department of Science and Technology of the Government of India; the National Science Centre of Poland, National Research Foundation, the Ministry of Science, Education and Sports of the Republic of Croatia, RosAtom of Russia and German Bundesministerium fur Bildung, Wissenschaft, Forschung and Technologie (BMBF) and the Helmholtz Association.