Beam energy and system dependence of rapidity-even dipolar flow

New measurements of rapidity-even dipolar flow, v$^{even}_{1}$, are presented for several transverse momenta, $p_T$, and centrality intervals in Au+Au collisions at $\sqrt{s_{NN}}~=~200,~39$ and $19.6$ GeV, U+U collisions at $\sqrt{s_{NN}}~=~193$ GeV, and Cu+Au, Cu+Cu, d+Au and p+Au collisions at $\sqrt{s_{NN}}~=~200$~GeV. The v$^{even}_{1}$ shows characteristic dependencies on $p_{T}$, centrality, collision system and $\sqrt{s_{_{NN}}}$, consistent with the expectation from a hydrodynamic-like expansion to the dipolar fluctuation in the initial state. These measurements could serve as constraints to distinguish between different initial-state models, and aid a more reliable extraction of the specific viscosity $\eta/s$.


Introduction
Heavy-ion collisions (HIC) at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) are aimed at studying the properties of the strongly interacting quark-gluon plasma (QGP) created in such collisions. Recent studies have emphasized the use of anisotropic flow measurements to study the transport properties of the QGP [1][2][3][4][5][6][7]. A crucial question in these studies was the role of initial-state fluctuations and their influence on the uncertainties associated with the extraction of η/s for the QGP produced in HIC [8,9]. This work emphasizes new measurements for rapidity-even dipolar flow, v even 1 , which could aid a distinction between different initial-state models and facilitate the extraction of η/s with better constraints.
Anisotropic flow is characterized by the Fourier coefficients, v n , obtained from a Fourier expansion of the azimuthal angle (φ) distribution of the emitted particles [10]: where Ψ n represents the n th -order event plane, the coefficients v 1 , v 2 and v 3 are called directed, elliptic and triangular flow, respectively. The flow coefficients v n are related to the two-particle Fourier coefficients v n,n as: where p a T and p b T are the transvers momentum of particles (a) and (b), respectively, and δ NF is a so-called non-flow (NF) term, which includes possible contributions from resonance decays, Bose-Einstein correlations, jets, and global momentum conservation (GMC) [11][12][13][14][15]. The directed flow, v 1 , can be separated into an odd function of pseudorapidity (η) [16] which develops along the direction of the impact parameter, and a rapidity-even component [13,17] which results from the effects of initialstate fluctuations acting in concert with a hydrodynamic-like expansion; v 1 (η) = v even

Measurements
The correlation function technique was used to generate the two-particle ∆φ correlations: where (dN/d∆φ) same represent 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 a different event but with a similar classification for the vertex, centrality, etc. The pseudorapidity requirement |∆η| > 0.7 was also imposed on track pairs to minimize possible non-flow contributions associated with the short-range correlations from resonance decays, Bose-Einstein correlations and jets. The two-particle Fourier coefficients v n,n are obtained from the correlation function as: and then used to extract v even 1 via a simultaneous fit of v 1,1 as a function of p b T , for several selections of p a T with Eq. 2: Here, C ∝ 1/(�Mult��p 2 T �) takes into account the non-flow correlations induced by a global momentum conservation [14,15] and �Mult� is the mean multiplicity.
For a given centrality selection, the left hand side of Eq. 5 represents the N × N matrix which we fit with the right hand side using N + 1 parameters; N values of v even 1 (p T ) and one additional parameter C, accounting for momentum conservation [19]. Fig. 1

Results
Representative v even GeV are summarized in Figs. 2 and 3. The values of v even 1 (p T ) extacted for different centrality selections (0-10%, 10-20% and 20-30%) are shown in Fig. 2; the solid line in panel (a) shows the a hydrodynamic calculations with η/s = 0.16 [14], which in good agreement with our measurements, the inset shows the corresponding results for the associated momentum conservation coefficient, C, extracted for several centralities at √ s NN = 200 GeV. The v even 1 (p T ) 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 shifts with √ s NN . They also indicate that v even 1 increase as the centrality become more peripheral, as might be expected from the centrality dependence of ε 1 .
The extracted values of v even 1 (p T ), for different collision systems are compared in Fig. 3 for different values of �Mult�. Figs. 3(a), 3(b) and 3(c) indicate similar v even 1 (p T ) magnitudes for the systems specified at each �Mult�, as well as 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. This pattern confirms the predicted trends for rapidityeven dipolar flow [13,14,17] and further indicates that for the selected values of �Mult�, v even 1 (p T ) does not show a strong dependence on the collision system. This apparent system independence of v even 1 (p T ) for the indicated �Mult� values suggests that the fluctuations-driven initial-state eccentricity ε 1 , is similar for the six collision systems. It also suggests that the viscous effects that are related to η/s are comparable for the matter created in each of these collision systems.

Conclusion
In summary, we have used the two-particle correlation method to carry out new differential measurements of rapidity-even dipolar flow, v even 1 , in Au+Au collisions at different beam energies, and in U+U, Cu+Au, Cu+Cu, d+Au and p+Au collisions at √ s NN ≃ 200 GeV. The measurements confirm the characteristic patterns 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, expected when initial-state geometric fluctuations act in concert with the hydrodynamic-like expansion to generate rapidity-even dipolar flow. This measurements provide additional constraints which are important to discern between different initial-state models, and to aid precision extraction of the temperature dependence of the specific shear viscosity.