Study of Azimuthal Anisotropy of High-p_T Charged Particles in Au Au Collisions at √s_NN = 200 GeV with RHIC-PHENIX ^†

Abstract

We study the path length dependence of energy-loss in the Quark Gluon Plasma (QGP) by measuring the azimuthal anisotropy coefficient and transverse momentum ($p_{\mathrm{T}}$) spectra for charged hadrons in Au + Au at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$ at the RHIC-PHENIX experiment. To estimate the strength of the energy-loss as a function of $p_{\mathrm{T}}$, we use the $\Delta p_{\mathrm{T}}$ which is the difference of $p_{\mathrm{T}}$ which provide the same yields at in-plane and out-of-plane directions. The results indicate that there are different structures between low-$p_{\mathrm{T}}$ and high-$p_{\mathrm{T}}$ regions. At high-$p_{\mathrm{T}}$, the size of $\Delta p_{\mathrm{T}}$ increases as the centrality goes up. We also calculate the difference of the path length of in-plane and out-of-plane directions for each centrality. The difference of the path length increases along with the centrality and the tendency is the same with the $\Delta p_{\mathrm{T}}$ results.

1. Introduction

In high energy heavy ion collisions, hard scattered partons can loose their energy because of the interaction with QGP. From the previous results of the nuclear modification factor $R_{AA}$, it is suggested that the energy-loss plays an important role for the suppression of the yields in QGP relative to nucleon scattering. The previous study in PHENIX by using Au + Au collisions and proton + proton (p + p) collsions [1] has been focused on understanding the strength of energy loss. It compares the strength of the energy loss as a function of transverse momentum ($p_{\mathrm{T}}$) in Au + Au collision from the central collision to the peripheral to that in p + p. The study indicates that the amount of the energy loss at all centralities tends to be independent of the $p_{\mathrm{T}}$. In this research, we intend to clarify the path-length dependence of the QGP energy-loss. The hard scattered partons have different QGP path-lengths depending on the azimuthal angle of the particle emission. The yield difference at the different azimuthal angle for high-$p_{\mathrm{T}}$ particles in the momentum space can be seen as a result of the different amount of energy-loss in the QGP since the original emission angle should be isotropic, azimuthally. In this analysis, we use the azimuthal anisotropy coefficient ($v_2$) to estimate the azimuthal-angle dependence of the particle yield. The analysis using $v_2$ is unique and has advantages that cancel the systematic errors comparing to the previous method [1], since this method uses only the Au + Au collision system. The strength of energy loss can be investigated by measuring the $v_2$ at high $p_T$, and we can calculate it more accurately.

2. Analysis Methods

We assume that the azimuthal distribution follows the Equation (1) since we consider only $v_2$ component in this analysis.

$$dN/d\varphi \propto 1+2v_{2}cos\left(2\varphi \right)$$

(1)

We use two previous results, inclusive $p_{\mathrm{T}}$ spectra and the azimuthal anisotropy $v_2$ for charged hadrons, to obtain the “in-plane yield” and the “out-of-plane yield”. The “in-plane” means the plane parallel to the reaction plane direction while the “out-of-plane” is the one perpendicular to that. For this study, we use preliminary results of the azimuthal anisotropy $v_2$ measured by the PHENIX experiment in Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$ in 2014 data shown in Figure 1 [2,3].

Figure 1. Azimuthal anisotropy coefficient $v_2$ as a function of $p_{\mathrm{T}}$ in Au + Au at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$ ( PHENIX preliminary results [3], for different region of the centrality from 0–10% to 40–50%. The results are shown by different symbols as explained in the legend.). Bars indicate the statistical errors and boxes indicate the systematic errors.

The inclusive $p_{\mathrm{T}}$ spectrum in Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$ shown as black points in Figure 2, is taken from [4]. For the given $p_{\mathrm{T}}$ range, one can get the azimuthal distribution, Equation (1), illustrated by the black line in the left panel of Figure 3. Here, we define the in-plane yield (out-of-plane yield) as the yield where the azimuthal distribution is assumed to be flat and has a constant value of 1 + 2$v_2$ (1 − 2$v_2$) which is the value at $\varphi =0$$(\varphi =\pi /2)$. The integral value of the black line is equal to the that of the yellow flat line indicated by “inclusive”. The right panel in Figure 3 shows a cartoon of the inclusive, in-plane and out-of-plane yields as a function of $p_{\mathrm{T}}$. These three lines can be obtained from the corresponding distributions for a given $p_{\mathrm{T}}$ in the left panel.

Figure 2. Inclusive, in-plane and out-of-plane yields as a function of $p_{\mathrm{T}}$ for the centrality 20–30% in the Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$. Bars indicate the statistical errors.

Figure 3. Left: A typical inclusive azimuthal anisotropic distribution. Right: Cartoon of the inclusive, in-plane and out-of-plane yields as a function of $p_{\mathrm{T}}$.

Figure 2 shows the differential yield as a function of the $p_{\mathrm{T}}$ in the case of centrality 20 to 30% in Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$. The black points are the inclusive yield, while the red and the blue points show the particle yields in the in-plane and the out-of-plane, respectively. We fit these yields by a function, f($p_{\mathrm{T}}$), given in Equation (2), where $P_0$, $P_1$, $P_2$, $P_3$, $P_4$ are parameters to be determined by a fit.

$$f\left(p_{\mathrm{T}}\right)=P_0(p_{\mathrm{T}}/e^{P_1{p}_{\mathrm{T}}})+P_2{(1.0+p_{\mathrm{T}}/P_3)}^{P_4}$$

(2)

We determine the values of these parameters, separately, by fitting the inclusive in-plane and out-of-plane yield. By using the fitting results, one can obtain the values of $p_{\mathrm{T}}$s, $p_{\mathrm{T},\mathrm{in}}$ and $p_{\mathrm{T},\mathrm{out}}$, that give the same in-plane and out-of-plane yields, respectively ($f\left(p_{\mathrm{T},\mathrm{in}}\right)=f^{\prime }(p_{\mathrm{T},\mathrm{out}}$)). We define the difference $\Delta p_{\mathrm{T}}$ = $p_{\mathrm{T},\mathrm{in}}$ - $p_{\mathrm{T},\mathrm{out}}$ as the estimator of the energy-loss within QGP for given $p_{\mathrm{T}}$.

3. Results

The obtained values of $\Delta p_{\mathrm{T}}$ are shown in Figure 4 as a function of $p_{\mathrm{T}}$ for the various centrality regions from 0 to 50% in 10% steps. In each figure, the vertical axis is $\Delta p_{\mathrm{T}}$ and the horizontal is the in-plane $p_{\mathrm{T}}$. For low $p_{\mathrm{T}}$ the $\Delta p_{\mathrm{T}}$ increases as $p_{\mathrm{T}}$ increases. On the other hand, at high $p_{\mathrm{T}}$, $\Delta p_{\mathrm{T}}$ is almost constant, i.e., $\Delta p_{\mathrm{T}}$ does not depend on its own $p_{\mathrm{T}}$. The results indicate that the mechanisms causing the $\Delta p_{\mathrm{T}}$ seem to be different between low $p_{\mathrm{T}}$ and high $p_{\mathrm{T}}$. This is consistent with the previous pictures that the yield difference between in and out of plane at low $p_{\mathrm{T}}$ is due to the elliptic flow [5] and that at high $p_{\mathrm{T}}$ is due to the parton energy loss described in the introduction. The results also indicate that although the shapes are similar for each centrality, for 0–30% centrality, it tends to increase $\Delta p_T$ as centrality goes up, and for 30–50% centrality it increases more gently.

Figure 4. $\Delta p_{\mathrm{T}}$ vs. $p_{\mathrm{T}}$ of in-plane in Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$ for the centrality region from 0% to 50% by a 10% step. In this proceeding, we are using an arbitrary scale for the vertical axis. Error bars indicate statistical errors.

In order to study the relation between the $\Delta p_{\mathrm{T}}$ and the parton path length within the QGP, we calculate the distance from the center of the collision to the collision surface in-plane direction ($L_{\mathrm{in}}$) and the out-of-plane direction ($L_{\mathrm{out}}$) as the simplest case. We calculate $L_{\mathrm{in}}$ and $L_{\mathrm{out}}$ geometrically from the relationship between the centrality and the impact parameter of gold nuclei, and take the difference $(dL=L_{\mathrm{out}}-L_{\mathrm{in}})$ between them as shown in the left panel of Figure 5. The radius of the gold nuclei is taken to be $7.27\times {10}^{-15}\mathrm{m}$ . The right panel of Figure 5 shows the calculated dL as a function of the centrality from 0 to 50%. One can clearly see that dL increases with the centrality up to 30%. This behavior is in line with the result for $\Delta p_{\mathrm{T}}$ at higher $p_{\mathrm{T}}$, supporting the interpretation based on a path length dependent energy loss.

Figure 5. Left: Definition of $L_{\mathrm{in}}$, $L_{\mathrm{out}}$ and dL. Right: The value of the path-length difference dL as a function of the centrality in Au + Au collisions at $\sqrt{s_{NN}}$ = 200 $\mathrm{GeV}$. Error bars indicate the statistical errors.

4. Conclusions

We obtain the in-plane and out-of-plane yields from the inclusive $p_{\mathrm{T}}$ spectra and the $v_2$ measurement using our previous results. From these yields, we estimate the transverse momentum loss, $\Delta p_{\mathrm{T}}$, as a function of $p_{\mathrm{T}}$ (in-plane) for the centrality 0 to 50%. The $\Delta p_{\mathrm{T}}$ seems to be independent of its $p_{\mathrm{T}}$ at high $p_{\mathrm{T}}$. The dL increases along with the centrality and the tendency is the same as for the $\Delta p_{\mathrm{T}}$ results.