Collectivity in small systems measured with PHENIX at RHIC

TL;DR

This paper demonstrates that azimuthal particle correlations in small collision systems at RHIC are linked to initial geometry and are well described by hydrodynamical models including QGP formation.

Contribution

It provides the first definitive evidence linking flow coefficients to initial geometry in small systems and validates hydrodynamical models for these conditions.

Findings

$v_2$ and $v_3$ are correlated to initial geometry.

Hydrodynamical models with QGP describe flow data.

Flow signals are present in small systems.

Abstract

In this paper we show azimuthal particle correlations in three different small-system collisions with different intrinsic initial geometries. The simultaneous constraints of $v_2$ and $v_3$ in $p/d/^3$He$+$Au collisions definitively demonstrate that the $v_n$'s are correlated to the initial geometry. In addition, we find that hydrodynamical models which include QGP formation describe simultaneouly the elliptic and triangular flow data in a statistically acceptable manner in all three systems.

Equations

$$v_{2}^{\mbox{$p$$

+

$$Au}}<v_{2}^{\mbox{$$

d

$$+$Au}}\approx v_{2}^{\mbox{${}^{3}$He$+$Au}},\hskip 50.0ptv_{3}^{\mbox{$p$$

+

$$Au}}\approx v_{3}^{\mbox{$$

d

$$+$Au}}<v_{3}^{\mbox{${}^{3}$He$+$Au}}.$$

$$v_{n}^{\mbox{$p$$

+

$$Au}}>v_{n}^{\mbox{$$

d

$$+$Au}}>v_{n}^{\mbox{${}^{3}$He$+$Au}},$$

Full text

1 Introduction

One of the key discoveries at RHIC is the identification of quark-gluon plasma (QGP) and its characterization as a near-perfect fluid via its collective flow [1, 2, 3, 4, 5]. One of the first observations of collective longitudinal and radial flow and their hydrodynamical coupling in the invariant momentum distribution and Bose-Einstein correlations was made by the EHS/NA22 experiment  [6] in $h$+$p$ collisions at CERN SPS at the beam momentum of 250 GeV/c, corresponding to $\sqrt{s}\approx 22$ GeV. As one of the first results of the d+Au beam energy scan at RHIC, PHENIX observed collective hydrodynamical behaviour of elliptic flow in d+Au collisions [7, 8], providing evidence for collectivity in d+Au collisions from $\sqrt{s_{NN}}=20$ GeV to 200 GeV. The LHC experiments observed similar features in small-system collisions [9, 10, 11, 12]. These results not only broaden the domain of the applicability of the hydrodynamical paradigm to a previously unexpected domain, but also raise several fundamental questions as well. Is it due to the appereance of sQGP (i.e. a strongly coupled fluid)? If yes, how much time is spent in the QGP phase? What is the origin of final state collectivity? Is it due to initial geometry and hydrodynamics? Is the initial state geometry the primary driver of final state momentum correlations in small systems?

In order to test and answer these questions RHIC performed not only beam energy scan but also geometry scan measurements which allows for the investigation of the phase diagram of QCD matter by varying the beam energy in the region where the change from crossover to first order phase transition is suggested to occur. The beam-energy-scan program found real-valued $v_{2}$ in $d+$Au at all collision energies, providing evidence for collectivity in $d+$Au at all energies. Applying the unique capabilities of RHIC a projectile geometry scan [13] was utilized in order to discriminate between hydrodynamical models that couple to the initial geometry and initial-state momentum correlation models that do not.

To characterize the fluidity of QGP, the azimuthal distribution of each event’s final-state particles, $\frac{dN}{d\phi}$, is decomposed into a Fourier series as follows: $\frac{dN}{d\phi}\propto 1+\sum_{n}2v_{n}(p_{T})\cos(n(\phi-\psi_{n})),$ where $p_{t}$ and $\phi$ are the transverse momentum and the azimuthal angle of a particle relative to the beam direction, respectively, and $\psi_{n}$ is the orientation of the $n^{\rm th}$ order symmetry plane of the produced particles. The second ($v_{2}$) and third ($v_{3}$) Fourier coefficients represent the amplitude of elliptic and triangular flow, respectively.

Varying the collision system from p$$+Au, to d$$+Au, to 3He+Au changes the initial geometry from dominantly circular, to elliptical, and to triangular configurations, as characterized by the 2nd and 3rd order spatial eccentricities, which correspond to ellipticity and triangularity, respectively. The mean $\varepsilon_{2}$ and $\varepsilon_{3}$ values for small impact parameter $p/d/^{3}$He+Au collisions are shown in Fig. 1a. The definition of the $n^{\rm th}$ order spatial eccentricity of the system, $\varepsilon_{n}$, is $\varepsilon_{n}=\frac{\sqrt{\langle r^{n}\cos(n\phi)\rangle^{2}+\langle r^{n}\sin(n\phi)\rangle^{2}}}{\langle r^{n}\rangle},$ where $r$ and $\phi$ are the polar coordinates of participating nucleons [14]. Based on the calculation from a MC Glauber model, the average second and third order spatial eccentricities ($\epsilon_{2}$ and $\epsilon_{3}$) are shown as columns in Fig. 1a. The second and third order spatial eccentricities are called ellipticity and triangularity, respectively.

Hydrodynamical models begin with an initial spatial energy-density distribution with a given temperature that evolves in time following the laws of relativistic viscous hydrodynamics using an equation of state determined from lattice QCD [15]. Examples of this temperature evolution are shown for $p/d/^{3}$He+Au collisions in Fig. 1b using the hydrodynamical model SONIC [16]. Based on haydrodynamical models a clear prediction for the ordering of the experimentally accessible $v_{2}$ and $v_{3}$ can be given, namely

[TABLE]

This ordering assumes that hydrodynamics can efficiently translate the initial geometric $\varepsilon_{n}$ into dynamical $v_{n}$, which is indeed seen in hydrodynamical simulations with small values of specific shear viscosity, as indicated on Fig. 1.

There exist a class of alternative explanations where $v_{n}$ is not generated via flow, but rather is created at the earliest time in the collision process as described by so-called color glass condensate or initial momentum space correlation models [17]. The expectation from models based on initial-state momentum domain correlations for the ordering of the magnitude of the $v_{2}$ and $v_{3}$ coefficients is:

[TABLE]

while the MSTV model in which gluons from the Au target do not resolve the individual color domains in the projectile $p/d/^{3}$He does not follow Eq. (2).111Please see the Note Added in Proof at the end of this manuscript for an important update regarding the MSTV calculation.

2 Models vs. data

Fig. 2 summarizes the results of elliptic and triangular flow measurements in the RHIC $p/d/^{3}$He+Au geometry scan. The data points follow a geometrical ordering in a qualitative agreement with expectations from hydrodynamics.

Fig. 3 compares quantitatively the PHENIX elliptic and triangular flow measurements for $p/d/^{3}$He+Au collisions with the results of numerical simulations. Two of these, SONIC and iEBE-VISHNU indicate predictions from numerical solutions of 2d+1 relativistic hydrodynamics with lattice QCD equation of state. The third model MSTV is on the other hand is based on initial state correlations and a color glass condensate initial state. Hydrodynamical models are consistent with the $v_{n}$ data in all three systems, however, they tend to diverge at higher $p_{T}$ in case of $v_{3}$, which may be more sensitive to the hadronic scattering. Focusing on the MSTV, Fig. 3 shows that this model does a fair job in case of $v_{2}$, but fails in case of $v_{3}$.

In order to distinguish these models, a statistical significance test was made and provided a $p$-value for the MSTV calculations of $v_{2}$ and $v_{3}$ for the three collision systems of effectively zero, in contradiction to the robust values found for the hydrodynamical models.

The MSTV paper made a clear prediction that the $v_{2}$ will be identical between systems when selecting on the same event multiplicity. Shown in Fig. 4 are the previously published d$$+Au(20-40%) and p$$+Au(0-5%) $v_{2}$ where the measured mean charged particle multiplicities ($dN_{\mathrm{ch}}/d\eta$) match [18]. Our results contradict to this MSTV prediction, as they indicate clear differences between the $v_{2}$ of $d$+Au and $p$+Au collisions even if they are measured in the same multiplicity class, as indicated by Fig. 4. The results are however in a reasonable qualitative agreement with hydrodynamical predictions.

Note Added in Proof

Subsequent to the preparation of this manuscript we were made aware that there is an issue in the MSTV calculation and that the calculation no longer agrees with the PHENIX data when the issue is corrected. For details see http://www.int.washington.edu/talks/WorkShops/int_19_1b/People/Mace_M/Mace.pdf .

Acknowledgments

The author is grateful for the support of EFOP 3.6.1-16-2016-0001, and NKFIH grant FK 123842 - 123959 (Hungary), as well as to the full list of PHENIX funding agencies.