Measurement of the elliptic anisotropy of charged particles produced in PbPb collisions at $\sqrt{s_{NN}}=2.76$ TeV

The anisotropy of the azimuthal distributions of charged particles produced in $\sqrt{s_{NN}}=2.76$ TeV PbPb collisions is studied with the CMS experiment at the LHC. The elliptic anisotropy parameter, $v_2$, defined as the second coefficient in a Fourier expansion of the particle invariant yields, is extracted using the event-plane method, two- and four-particle cumulants, and Lee-Yang zeros. The anisotropy is presented as a function of transverse momentum ($p_{\mathrm{T}}$), pseudorapidity ($\eta$) over a broad kinematic range, $0.3<p_{\mathrm{T}}<20$ GeV/c, $\left|\eta \right|<2.4$, and in 12 classes of collision centrality from 0 to $80\%$. The results are compared to those obtained at lower center-of-mass energies, and various scaling behaviors are examined. When scaled by the geometric eccentricity of the collision zone, the elliptic anisotropy is found to obey a universal scaling with the transverse particle density for different collision systems and center-of-mass energies.

Figure 1

A schematic diagram of a noncentral nucleus-nucleus collision viewed in the plane orthogonal to the beam. The azimuthal angle $\phi$, the impact parameter vector b, and the reaction-plane angle ${\Psi }_{\text{R}}$ are shown. The event-plane angle $\Psi$, with respect to which the flow is measured, fluctuates around the reaction-plane angle.

Figure 2

A schematic view of a PbPb collision with an impact parameter $b=6$ fm, as obtained from the Glauber model. The nucleons that participate in inelastic interactions are marked with solid circles. The $x$ and $y$ coordinates represent the laboratory frame, while $x^{\prime }$ and $y^{\prime }$ represent the frame that is aligned with the axes of the ellipse in the participant zone. The participant eccentricity ${\epsilon }_{\text{part}}$ and the standard deviations of the participant spatial distribution ${\sigma }_{y^{\prime }}$ and ${\sigma }_{x^{\prime }}$ from which the transverse overlap area of the two nuclei is calculated are also shown. The angle ${\Psi }_{\text{R}}$ denotes the orientation of the reaction plane.

Figure 3

Efficiency (top), fake rate (middle), and momentum resolution (bottom) of charged tracks obtained from hydjet simulated events in four pseudorapidity regions: $\left|\eta \right|<0.8$, $0.8<\left|\eta \right|<1.6$, $1.6<\left|\eta \right|<2.0$, and $2.0<\left|\eta \right|<2.4$ displayed from left to right, and for the five centrality classes given in the legend.

Figure 4

Event-plane resolution correction factors as a function of centrality for the two event planes (HF$-$ and HF+) used in determining the elliptic anisotropy parameter $v_2$. The corrections determined with the three-subevent method used in the analysis are shown as open squares and star symbols. The results from a two-subevent method used in evaluating the systematic uncertainties are shown as solid circles, though they overlap the other points in all but the most peripheral bin.

Figure 5

An example of the modulus of the second harmonic LYZ generating function $G^{\theta }\left(ir\right)$ as a function of the imaginary axis coordinate $r$ for $\theta =0$. Both the sum and the product generating functions are shown, calculated from events with centrality 15–$20\%$, $\left|\eta \right|<0.8$, and $0.3<p_{\mathrm{T}}<12$ GeV/c. An enlargement of $|G^{\theta }\left(ir\right)|$ around its first minimum is shown in the inset.

Figure 6

Results from the event-plane (EP) method for $v_2$ as a function of $p_{\mathrm{T}}$ at midrapidity $\left|\eta \right|<0.8$ for the 12 centrality classes given in the legend. The error bars show the statistical uncertainties only.

Figure 7

Second-order cumulant results for $v_2$ as a function of $p_{\mathrm{T}}$ at midrapidity $\left|\eta \right|<0.8$ for the 12 centrality classes given in the legend. The error bars show the statistical uncertainties only.

Figure 8

Fourth-order cumulant results for $v_2$ as a function of $p_{\mathrm{T}}$ at midrapidity $\left|\eta \right|<0.8$ for the ten centrality classes given in the legend. The error bars show the statistical uncertainties only.

Figure 9

Lee-Yang zeros results for $v_2$ as a function of $p_{\mathrm{T}}$ at midrapidity $\left|\eta \right|<0.8$ for the eight centrality classes given in the legend. The error bars show the statistical uncertainties only.

Figure 10

Comparison of the four different methods for determining $v_2$ as a function of $p_{\mathrm{T}}$ at midrapidity ($\left|\eta \right|<0.8$) for the 12 centrality classes given in the figures. The error bars show the statistical uncertainties only.

Figure 11

(Top) Integrated $v_2$ as a function of centrality at midrapidity $\left|\eta \right|<0.8$ for the four methods. The boxes represent the systematic uncertainties. The magnitudes of the statistical uncertainties are smaller than the size of the symbols. (Bottom) The values from three of the methods are divided by the results from the event-plane method. The boxes represent the systematic uncertainties excluding the sources that are common to all methods. The magnitudes of the statistical uncertainties are smaller than the size of the symbols.

Figure 12

(Left) Centrality dependence of the integrated $v_2$ divided by the participant eccentricity, ${\epsilon }_{\mathrm{part}}$, obtained at midrapidity $\left|\eta \right|<0.8$ for the event-plane and two- and four-particle cumulant methods. (Right) The measurements of $v_2$ as a function of centrality are the same as in the left panel, but here the two-particle and four-particle cumulant results are divided by their corresponding moments of the participant eccentricity, $\epsilon \left\{2\right\}$ and $\epsilon \left\{4\right\}$. In both panels, the error bars show the sum in quadrature of the statistical and systematic uncertainties in the $v_2$ measurement, and the lines represent the systematic uncertainties in the eccentricity determination.

Figure 13

Pseudorapidity dependence of $v_2$ for $0.3<p_{\mathrm{T}}<3$ GeV/c with all four methods in 12 centrality classes. The boxes give the systematic uncertainties. The magnitudes of the statistical uncertainties are smaller than the size of the symbols.

Figure 14

Results from the event-plane method for $v_2\left(p_{\mathrm{T}}\right)$ in three pseudorapidity regions and in 12 centrality classes. The error bars show the statistical uncertainties that in most cases have magnitudes smaller than the size of the symbol.

Figure 15

Inclusive charged-particle spectra at midrapidity (left) and forward rapidity (right), for the 12 centrality classes given in the legend. The distributions are offset by arbitrary factors given in the legend for clarity. The shaded bands represent the statistical and systematic uncertainties added in quadrature, including the overall normalization uncertainties from the trigger efficiency estimation.

Figure 16

Mean transverse momentum of the charged-particle spectra as a function of $N_{\mathrm{part}}$ in three pseudorapidity intervals marked in the figure. The error bars represent the quadratic sum of the statistical and systematic uncertainties.

Figure 17

The values of $v_2\left\{2\right\}$ and $v_2\left\{4\right\}$ obtained with the cumulant method, as a function of $p_{\mathrm{T}}$ from CMS (solid symbols) and ALICE (open symbols) [13], measured in the range $\left|\eta \right|<0.8$ for the 40–$50\%$ centrality class. The error bars show the statistical uncertainties, and the boxes give the systematic uncertainties in the CMS measurement.

Figure 18

Comparison of results for $v_2\left(p_{\mathrm{T}}\right)$ obtained with the event-plane method from CMS (solid symbols) and ATLAS (open symbols) for the centrality classes marked in the figure. The error bars show the statistical and systematic uncertainties added in quadrature.

Figure 19

(Top panels) Comparison of $v_2\left(p_{\mathrm{T}}\right)$ using the event-plane method as measured by CMS (solid circles) at $\sqrt{s_{NN}}=2.76$ TeV, and PHENIX [73] (open diamonds) at $\sqrt{s_{NN}}=200$ GeV for midrapidity ($\left|\eta \right|<0.8$ and $\left|\eta \right|<0.35$, respectively). The error bars represent the statistical uncertainties. The solid line is a fit to the CMS data. (Bottom panels) Ratios of the CMS fit to the PHENIX data (open diamonds) and to the CMS data (open circles). The error bars show the statistical uncertainties, while the shaded boxes give the quadrature sum of the CMS and PHENIX systematic uncertainties.

Figure 20

(Top panels) Comparison of $v_2\left(p_{\mathrm{T}}\right)$ using the two-particle (left) and the four-particle (right) cumulant method as measured by CMS (solid circles) at $\sqrt{s_{NN}}=2.76$ TeV, and STAR [74] (open stars) at $\sqrt{s_{NN}}=200$ GeV at midrapidity ($\left|\eta \right|<0.8$ and $\left|\eta \right|<1.3$, respectively). The error bars represent the statistical uncertainties. The line is a fit to the CMS data. (Bottom panels) Ratios of the CMS fit values to the STAR data (open diamonds) and to the CMS data (open circles). The error bars show the statistical uncertainties, while the shaded boxes give the quadrature sum of the CMS and STAR systematic uncertainties.

Figure 21

The CMS integrated $v_2$ values for 20–$30\%$ centrality from the range $\left|\eta \right|<0.8$ and $0<p_{\mathrm{T}}<3\mathrm{GeV}/\mathrm{c}$ obtained using the event-plane method is compared as a function of $\sqrt{s_{NN}}$ to results at midrapidity and similar centrality from ALICE [13], STAR [76], PHENIX [23], PHOBOS [49, 77, 78], NA49 [79], E877 [80], and CERES [81]. The error bars for the lower-energy results represent statistical uncertainties; for the CMS and ALICE measurements the statistical and systematic uncertainties are added in quadrature.

Figure 22

Mean transverse momentum of the charged-particle spectra as a function of $N_{\mathrm{part}}$ measured by CMS in PbPb collisions at $\sqrt{s_{NN}}=2.76$ TeV (solid circles) and by STAR [82] in AuAu collisions at $\sqrt{s_{NN}}=200$ GeV (open circles). The error bars represent the quadratic sum of statistical and systematic uncertainties.

Figure 23

The CMS integrated $v_2$ values from the event-plane method divided by the participant eccentricity as a function of $N_{\mathrm{part}}$ with $\left|\eta \right|<0.8$ and $0<p_{\mathrm{T}}<3\mathrm{GeV}/\mathrm{c}$. These results are compared with those from PHOBOS [34] for different nuclear species and collision energies. The PHOBOS $v_2$ values are divided by the cumulant eccentricity $\epsilon \left\{2\right\}$ (see text). The error bars give the statistical and systematic uncertainties in the $v_2$ measurements added in quadrature. The dashed lines represent the systematic uncertainties in the eccentricity determination.

Figure 24

Eccentricity-scaled $v_2$ as a function of the transverse charged-particle density from CMS and PHOBOS [34]. The error bars include both statistical and systematic uncertainties in $v_2$. The dashed lines represent the systematic uncertainties in the eccentricity determination.

Figure 25

Measurements of $v_2$ as a function of the pseudorapidity of particles in the rest frame of the colliding nuclei $\eta \pm y_{\mathrm{beam}}$ from CMS (solid symbols) and PHOBOS (open symbols) [77] in three centrality intervals. The error bars show the statistical uncertainties and the boxes give the systematic ones.