Azimuthal harmonics in small and large collision systems at RHIC top energies

TL;DR

This study measures azimuthal harmonic coefficients in various collision systems at RHIC energies, revealing system-independent behavior for some harmonics and scaling laws that shed light on initial geometry and viscous effects.

Contribution

It provides the first comprehensive comparison of azimuthal harmonics across small and large collision systems at RHIC, highlighting scaling behaviors and initial-geometry effects.

Findings

$v_1^{ ext{even}}$ and $v_3$ are system-independent at fixed multiplicity.

$v_2$ scales with initial eccentricity $ ext{and}$ multiplicity.

$ ext{ln}(v_2/ ext{eccentricity})$ scales linearly with $ ext{N}_{ch}^{-1/3}$.

Abstract

The first ($v_1^{\text{even}}$), second ($v_2$) and third ($v_3$) harmonic coefficients of the azimuthal particle distribution at mid-rapidity, are extracted for charged hadrons and studied as a function of transverse momentum ($p_T$) and mean charged particle multiplicity density $\langle \mathrm{N_{ch}} \rangle$ in U+U ($\roots =193$~GeV), Au+Au, Cu+Au, Cu+Cu, $d$+Au and $p$+Au collisions at $\roots = 200$~GeV with the STAR Detector. For the same $\langle \mathrm{N_{ch}} \rangle$, the $v_1^{\text{even}}$ and $v_3$ coefficients are observed to be independent of collision system, while $v_2$ exhibits such a scaling only when normalized by the initial-state eccentricity ($\varepsilon_2$). The data also show that $\ln(v_2/\varepsilon_2)$ scales linearly with $\langle \mathrm{N_{ch}} \rangle^{-1/3}$. These measurements provide insight into initial-geometry fluctuations and the role of viscous hydrodynamic attenuation on $v_n$ from small to large collision systems.

Equations

$$V_n\equiv v_n{e}^{in{\Psi }_n}=\{e^{in\varphi }{\}}_j,$$

$$\frac{d{N}^{pairs}}{d\Delta \varphi }\propto 1+2\sum _{n=1}^{\infty }{v}_{nn}\cos (n\Delta \varphi ),$$

$$v_{nn}(p_{\mathrm{T}}^a,p_{\mathrm{T}}^b)=v_n(p_{\mathrm{T}}^a)v_n(p_{\mathrm{T}}^b)+{\delta }_{NF},$$

$${\mathcal{E}}_n\equiv {\epsilon }_n{e}^{in{\Phi }_n}=-\frac{\int d^2{r}_{\perp }{r}^m{e}^{in\phi }{\rho }_e(r,\phi )}{\int d^2{r}_{\perp }{r}^m{\rho }_e(r,\phi )},$$

$$\ln (v_n/{\epsilon }_n)\propto -n^2⟨\frac{\eta }{s}(T)⟩⟨N_{ch}{⟩}^{-1/3},$$

$$C_{\text{r}}(\Delta \varphi ,\Delta \eta )=\frac{(dN/d\Delta \varphi {)}_{\text{same}}}{(dN/d\Delta \varphi {)}_{\text{mixed}}},$$

$$c_2\{4\}=⟨⟨4⟩⟩-2⟨⟨2⟩{⟩}^2,$$

$$v_{nn}$$

$$v_{nn}(p_{\mathrm{T}}^a,p_{\mathrm{T}}^b)=v_n(p_{\mathrm{T}}^a)v_n(p_{\mathrm{T}}^b),$$

$$v_{11}(p_{\mathrm{T}}^a,p_{\mathrm{T}}^b)=v_1^{even}(p_{\mathrm{T}}^a)v_1^{even}(p_{\mathrm{T}}^b)-K{p}_{\mathrm{T}}^a{p}_{\mathrm{T}}^b,$$

Full text

An important goal of the experimental program at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL), is to provide quantitative experimental data which can (i) give insight on the dynamical evolution of the quark-gluon plasma (QGP) created in heavy ion collisions, and (ii) serve as important constraints for the extraction of the associated transport coefficients.” The azimuthal anisotropy of particle emission in the transverse plane, known as anisotropic flow, is a key observable because it reflects the viscous hydrodynamic response to the initial spatial distribution in energy density (both from intrinsic geometry and fluctuations), produced in the early stages of the collision Danielewicz et al. (1998); Ackermann et al. (2001); Adcox et al. (2002); Heinz and Kolb (2002); Hirano et al. (2006); Huovinen et al. (2001); Hirano and Tsuda (2002); Romatschke and Romatschke (2007); Luzum (2011); Song et al. (2011); Qian et al. (2016); Schenke et al. (2011a); Teaney and Yan (2012); Gardim et al. (2012); Lacey et al. (2016a).

Experimentally, anisotropic flow manifests as an azimuthal asymmetry of the measured single-particle distribution, quantified by the complex flow coefficients Bilandzic et al. (2011); Luzum (2011); Teaney and Yan (2012):

[TABLE]

where $\phi$ denotes the azimuthal angle around the beam direction, of a particle emitted in the collision and $\{\}_{j}$ denotes the average over all particles emitted in the event. The weighted average ${{\left<\left|V_{n}\right|^{2}\right>}^{1/2}=v_{n}}$ (which accounts for multiplicity variations) and ${\Psi_{n}}$ denote the magnitude and azimuthal direction of the $n^{\text{th}}$-order harmonic flow vector that fluctuates from event to event. The first three coefficients, ${v_{1}}$, ${v_{2}}$, and ${v_{3}}$, are termed directed, elliptic, and triangular flow, respectively. The fluctuations-driven component of ${v_{1}}$, termed ${v^{\text{even}}_{1}}$, is proportional to the dipole asymmetry of the collision system Teaney and Yan (2011); Luzum and Ollitrault (2011).

The ${v_{n}}$ coefficients are also related to the Fourier coefficients ${v_{nn}}$ which characterize the amplitude of the two-particle correlations in relative azimuthal angle $\Delta\phi=\phi_{\mathrm{a}}-\phi_{\mathrm{b}}$ Poskanzer and Voloshin (1998); Lacey (2006) for the particles $\mathrm{a}$ and $\mathrm{b}$, which comprise the pairs:

[TABLE]

where $\delta_{\text{NF}}$ signify the contributions of short-range non-flow correlations due to resonance decays, Bose-Einstein correlations and jet-like decays, as well as long-range contributions, which result from global momentum conservation Lacey (2006); Luzum and Ollitrault (2011); Retinskaya et al. (2012); Aad et al. (2012).

The initial anisotropic density profile $\rho_{e}(r,\varphi)$ in the transverse ($\perp$) plane, which drives anisotropic flow, can be similarly characterized by complex eccentricity coefficients Alver et al. (2010); Petersen et al. (2010); Lacey et al. (2011a); Teaney and Yan (2011); Qiu and Heinz (2011):

[TABLE]

where ${\Phi_{n}}$ is the angle of the so-called ${n^{th}}$-order participant plane; ${m={n}}$ for ${n{\geq\,}2}$ and ${m=3}$ for ${n=1}$ Teaney and Yan (2011). Theoretical investigations show that ${v_{n}\propto\varepsilon_{n}}$ for elliptic and triangular flow (${n=2,3}$) Qiu and Heinz (2011); Fu (2015); Niemi et al. (2016); Noronha-Hostler et al. (2016), and the temperature ($T$) dependent specific shear viscosity $\frac{\eta}{s}(T)$, of the created medium, reduces the ratio ${v_{n}/\varepsilon_{n}}$. Thus, the comparison of viscous hydrodynamical model calculations to this ratio is commonly employed to estimate $\frac{\eta}{s}(T)$ and its average $\left<\frac{\eta}{s}(T)\right>$, over the system’s evolution Hirano et al. (2006); Romatschke and Romatschke (2007); Song et al. (2011); Schenke et al. (2011b); Bozek et al. (2010); Qiu and Heinz (2011); Schenke et al. (2011a); Niemi et al. (2012); Gardim et al. (2012); McDonald et al. (2017); Bernhard et al. (2016). The viscous attenuation of ${v_{n}/\varepsilon_{n}}$ can also be understood within an acoustic model framework, akin to that for viscous relativistic hydrodynamics Liu and Lacey (2018a, b); Lacey et al. (2013); Magdy (2018, 2017); Shuryak and Zahed (2013); Lacey et al. (2011b):

[TABLE]

where $\mathrm{\langle N_{ch}\rangle}$ is the charged particle multiplicity density and $\mathrm{\langle N_{ch}\rangle^{-1/3}}$ is a proxy for the dimensionless size of the system Liu and Lacey (2018a, b); Lacey et al. (2016b).

Recent measurements at both RHIC and the Large Hadron Collider (LHC), have indicated sizable ${v_{2}}$ and ${v_{3}}$ values in high multiplicity $p+p$ Khachatryan et al. (2016); Aad et al. (2016), $d$+Au Adare et al. (2015); Adamczyk et al. (2015a) and $p$+Pb collisions Chatrchyan et al. (2013); Abelev et al. (2013); Aad et al. (2013), reminiscent of those observed in peripheral A+A collisions. These measurements have generated considerable debate on whether the final-state collective effects, which dominate the mechanism for anisotropic flow in A+A collisions, also drive the anisotropy measured in high-multiplicity $p+p$ and $p$+A ($d$+A) collisions Dusling and Venugopalan (2013); Bozek and Broniowski (2013); Dusling et al. (2016). The related question of whether the properties of the medium produced in the small $p+p$, $p$+A and $d$+A systems are similar to those produced in the larger A+A systems is also not fully settled.

In this letter we present and compare a comprehensive set of ${v^{\text{even}}_{1}}$, ${v_{2}}$ and ${v_{3}}$ measurements for U+U ($\sqrt{s_{{}_{NN}}}=193$ GeV), Au+Au, Cu+Cu, Cu+Au, $d$+Au, and $p$+Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV, that should prove invaluable for the interpretation of collectivity in small systems, and in ongoing efforts to constrain theoretical models and obtain a robust extraction of $\frac{\eta}{s}(T)$.

The data for the six colliding systems presented in this work, were collected with the STAR detector at RHIC using a minimum-bias trigger. Charged-particle tracks, measured in the full azimuth and pseudorapidity range ($|\eta|<1.0$) of the Time Projection Chamber (TPC) Anderson et al. (2003), were used to reconstruct the collision vertices. Events were selected with vertex positions $\pm 30$ cm from the nominal center of the TPC (in the beam direction).

Collision centrality and the associated $\langle\mathrm{N_{ch}}\rangle$ was determined from the measured event-by-event multiplicity with the aid of a tuned Monte Carlo Glauber calculation Adamczyk et al. (2012). Analyzed tracks were required to have a distance of closest approach to the primary vertex of less than 3 cm, and have at least 15 TPC space points used in their reconstruction. To remove split tracks, the ratio of the number of fit points to a maximum possible number of TPC space points was required to be larger than 0.52. Analyzed tracks were restricted to $0.2<p_{\mathrm{T}}<4$ GeV/$c$.

Two-particle $\Delta\phi$ correlation functions ($C_{\text{r}}$) were generated to extract the flow coefficients:

[TABLE]

where $(dN/d\Delta\phi)_{\text{same}}$ represents the distribution of track pairs, in relative azimuthal angle $\Delta\phi$, taken from the same event. $(dN/d\Delta\phi)_{\text{mixed}}$ represents the $\Delta\phi$ distribution for track pairs in which each member is selected from different events in the same $\langle\mathrm{N_{ch}}\rangle$ and vertex position classes. The pseudorapidity requirement $|\Delta\eta|>0.7$ was imposed for all track pairs to suppress short-range non-flow contributions Adam et al. (2018). A further check for the dominance of flow correlations was obtained by measuring the second-order four-particle cumulant $c_{2}\{4\}$:

[TABLE]

where $\langle\langle\rangle\rangle$ represents the averaging first over particles in an event and then over all events within a given event class. The three sub-events method Jia et al. (2017) was used for these evaluations with sub-events for $\eta_{1}<-0.35$, $|\eta_{2}|<0.35$ and $\eta_{3}>0.35$.

Figures 1(a-e) show the correlation functions obtained for U+U, Au+Au, Cu+Au, Cu+Cu, $d$+Au and $p$+Au collisions for $\langle\mathrm{N_{ch}}\rangle=21\pm 3$. They indicate patently similar correlation patterns with a visible enhancement of near-side ($\Delta\phi\sim 0$) pairs, reminiscent of the so-called “ridge” observed in high multiplicity $p+p$ Khachatryan et al. (2016); Aad et al. (2016), $d$+Au Adare et al. (2015); Adamczyk et al. (2015a) and $p$+Pb collisions Aad et al. (2013); Chatrchyan et al. (2013). The corresponding values for $c_{2}\{4\}$ vs. $\langle\mathrm{N_{ch}}\rangle$, shown in Fig. 1 (g), indicate negative values which suggests the absence of significant short-range non-flow contributions, and the dominance of flow correlations to $C_{\text{r}}$ Borghini et al. (2001); Zhao et al. (2018). Note that the paucity of central $p$+Au events precluded the extraction of $c_{2}\{4\}$ from these events.

Similar sets of correlation functions were generated as a function of ${p_{\mathrm{T}}}$ and $\langle\mathrm{N_{ch}}\rangle$ to allow a study of ${v^{\text{even}}_{1}}$, ${v_{2}}$ and ${v_{3}}$ (for each collision system) for different dimensionless sizes and eccentricities. Monte Carlo quark Glauber (MC-qGlauber) calculations Liu and Lacey (2018a) were used to compute ${\varepsilon_{n}}$ as a function of collision centrality or $\langle\mathrm{N_{ch}}\rangle$ for all collision systems, from the two-dimensional profile of the density of quark participants in the transverse plane (c.f Eq. 3). The model takes account of the finite size of the nucleon, the wounding profile of the nucleon, the distribution of quarks inside the nucleon, and quark cross sections which reproduce the NN inelastic cross section at $\sqrt{s_{NN}}$ = 200 GeV.

The ${v_{nn}}$ coefficients were obtained from the correlation function as:

[TABLE]

and then used to extract ${v_{n}}$ for ${n>1}$,

[TABLE]

and the ${v^{even}_{1}}$ component of $v_{1}$,

[TABLE]

where $K\propto 1/(\langle\mathrm{N_{ch}}\rangle\langle p_{\mathrm{T}}^{2}\rangle)$ takes account of the long-range non-flow correlations induced by global momentum conservation Retinskaya et al. (2012); Aad et al. (2012); Adam et al. (2018). A simultaneous fit of ${v_{11}(p_{\mathrm{T}}^{b})}$ for several selections of ${p_{\mathrm{T}}^{a}}$ (c.f. Eq. 9) was used to facilitate the extraction of ${v^{\text{even}}_{1}}$ Adam et al. (2018).

The systematic uncertainties associated with the ${v_{n}}$ extractions were estimated through studies of the influence of the choice of the cuts for z-vertex position, track selection, efficiency correction, $\Delta\eta$ and the fitting procedure. The uncertainty associated with $\Delta\eta$ dominate for the $d$+Au and $p$+Au systems. The respective uncertainties, ranging from 2% to 10%, were added in quadrature to obtain an overall systematic uncertainty for the respective measurements.

The extracted values of ${v^{\text{\text{even}}}_{1}(p_{\mathrm{T}})}$, ${v_{2}(p_{\mathrm{T}})}$ and ${v_{3}(p_{\mathrm{T}})}$ for the collision systems are compared in Fig. 2 for different values of $\langle\mathrm{N_{ch}}\rangle$. Figures 2(a) - 2(c) indicate similar ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ magnitudes for the systems specified at each $\langle\mathrm{N_{ch}}\rangle$, as well as the characteristic pattern of a change from negative ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ at low-${p_{\mathrm{T}}}$ to positive ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ for ${p_{\mathrm{T}}\gtrsim 1}$ GeV/$c$. This pattern confirms the predicted trends for dipolar flow Teaney and Yan (2011); Luzum and Ollitrault (2011); Retinskaya et al. (2012); Adam et al. (2018) and further indicates that for the selected values of $\langle\mathrm{N_{ch}}\rangle$, ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ is essentially independent of collision system. Figures 2(d) - 2(f), show similar system-independent patterns for ${v_{3}(p_{\mathrm{T}})}$, but with magnitudes and trends that differ from those for ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$. The system independence of ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ and ${v_{3}(p_{\mathrm{T}})}$ for the indicated $\langle\mathrm{N_{ch}}\rangle$ values suggests that the fluctuations-driven initial-state eccentricities $\varepsilon_{1}$ and $\varepsilon_{3}$, and the subsequent final-state interactions are similar for the indicated collision systems.

The ${v_{2}(p_{\mathrm{T}})}$ values shown in Figs. 2(g) - 2(i) contrasts with those for ${v^{\text{even}}_{1}(p_{\mathrm{T}})}$ and ${v_{3}(p_{\mathrm{T}})}$. That is, the trends for a given $\langle\mathrm{N_{ch}}\rangle$ are independent of the collision system, but the magnitudes are not system-independent, albeit with differences that grow with $\langle\mathrm{N_{ch}}\rangle$. The system dependent differences, apparent for $\langle\mathrm{N_{ch}}\rangle$ = 140 and 70 (Figs. 2(g) and 2(h)), can be attributed to the system-dependent $\varepsilon_{2}$ values for each $\langle\mathrm{N_{ch}}\rangle$. For $\langle\mathrm{N_{ch}}\rangle\sim 21$ (Fig. 2(i)), the MC-qGlauber eccentricities for the different systems, do not vary strongly.

Figures 2(j) and 2(k) confirm the influence of the system-dependent $\varepsilon_{2}$ values for $\langle\mathrm{N_{ch}}\rangle$ = 140 and 70 . That is, they show data collapse onto a single curve for ${v_{2}/\varepsilon_{2}}$ vs. $p_{\mathrm{T}}$ for U+U, Au+Au, Cu+Au and Cu+Cu systems. Fig. 2(l) also indicates an approximate collapse of the scaled results for $p$+Au and $d$+Au onto the curve for the eccentricity-scaled A+A data. This pattern is suggestive of a dominant collective flow contribution to the measured anisotropy in high multiplicity $p$+A($d$+A) collisions Liu and Lacey (2018b). However, a quantitative estimate of a possible long-range non-flow contribution is required to fully establish the degree of this apparent scaling.

The $\langle\mathrm{N_{ch}}\rangle$ dependence of ${v^{\text{even}}_{1}}$, ${v_{2}}$, and ${v_{3}}$ are compared for all six collision systems in Figs. 3(a) - 3(c); they are in good agreement with the $v_{2}$ data reported for U+U and Au+Au collisions in Ref. Adamczyk et al. (2015b). The inset in Fig. 3(a) compares the associated values of $K$ vs. $\langle\mathrm{N_{ch}}\rangle^{-1}$ (c.f. Eq. 9) for each system.

For $\langle\mathrm{N_{ch}}\rangle\gtrsim 170$, the ${v_{n}}$ values all show a decrease with increasing values of $\langle\mathrm{N_{ch}}\rangle$, consistent with the expected decrease of ${\varepsilon_{n}}$ as collisions become more central. The apparent decrease in the values of ${v_{2}}$ for $\langle\mathrm{N_{ch}}\rangle\lesssim 170$ corroborate the dominant role of size-driven viscous attenuation of the flow harmonics for these multiplicities. Note that $\varepsilon_{2}$ increases for \langle\mathrm{N_{ch}}\rangle\mbox{<}170. Figures 3(a) and 3(b) indicate system-independent magnitudes and trends for ${v^{\text{even}}_{1}}$ and ${v_{3}}$ analogous to the ${p_{\mathrm{T}}}$-dependent results shown in Fig. 2.

The ${v_{2}}$ comparisons shown in Fig. 3(c), accentuate the system-dependent patterns observed in Figs. 2(g), 2(h) and 2(i). Here, the sizable uncertainties for the $p$+Au and $d$+Au data points for $\langle\mathrm{N_{ch}}\rangle\sim 21$ reflect the systematic uncertainty estimates for residual non-flow contributions which are smaller for these $p_{\mathrm{T}}$-integrated measurements. The striking system-dependent patterns shown in Fig. 3(c) can be attributed to the strong dependence of $\varepsilon_{2}$ on system size for a fixed value of $\langle\mathrm{N_{ch}}\rangle$. This shape dependence, which weakens for low $\langle\mathrm{N_{ch}}\rangle$, is confirmed via the plot of ${v_{2}/\varepsilon_{2}}$ vs. $\langle\mathrm{N_{ch}}\rangle^{-1/3}$ shown in Fig. 4. A similar plot, reflecting the $n^{2}$ dependence of viscous attenuation Liu and Lacey (2018a, b), was obtained for ${v_{3}/\varepsilon_{3}}$ vs. $\langle\mathrm{N_{ch}}\rangle^{-1/3}$. The inset in Fig. 4 indicates a marked similarity between the slopes of the eccentricity-scaled ${v_{2}}$ for U+U, Au+Au, Cu+Au and Cu+Cu collisions. The eccentricity-scaled results for $d$+Au and $p$+Au also follow the data trend for these heavier collision species with larger systematic uncertainty. Hydrodynamic simulations for Au+Au collisions Alba et al. (2018) exhibit similar scaling trends within the same range of $\langle\mathrm{N_{ch}}\rangle$.

In summary, we have used the two-particle correlation method to carry out a comprehensive set of measurements of ${v^{\text{even}}_{1}}$, ${v_{2}}$, and ${v_{3}}$ as a function of ${p_{\mathrm{T}}}$ and $\langle\mathrm{N_{ch}}\rangle$ in U+U ($\sqrt{s_{{}_{NN}}}=193$ GeV) and Au+Au, Cu+Au, Cu+Cu, $d$+Au, and $p$+Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. The detailed comparisons of the measurements highlight the sensitivity of ${v_{n}}$ to the magnitude of the initial-state eccentricity, system size and the final-state interactions in the expanding matter. The wealth of the A+A measurements lead to data collapse of ${\ln(v_{n}/\varepsilon_{n})}$ vs. $\langle\mathrm{N_{ch}}\rangle^{-1/3}$ onto a single curve. Similarly scaled results for $d$+Au and $p$+Au (for $\langle\mathrm{N_{ch}}\rangle\sim 21$) are also observed with larger uncertainty. The combined measurements and their scaling properties provide a new set of constraints which could prove invaluable for the interpretation of collectivity in small systems and for detailed theoretical extraction of the temperature-dependent $\frac{\eta}{s}$.

Acknowledgments

Acknowledgements.

We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. National Science Foundation, the Ministry of Education and Science of the Russian Federation, National Natural Science Foundation of China, Chinese Academy of Science, the Ministry of Science 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.