"Appendix 2.4" of "Measurement of the higher-order anisotropic flow coefficients for identified hadrons in Au$+$Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV"

Measurements of the anisotropic flow coefficients $v_2\{\Psi_2\}$, $v_3\{\Psi_3\}$, $v_4\{\Psi_4\}$, and $v_4\{\Psi_2\}$ for identified particles ($\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$) at midrapidity, obtained relative to the event planes $\Psi_m$ at forward rapidities in Au$+$Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV, are presented as a function of collision centrality and particle transverse momenta $p_T$. The $v_n$ coefficients show characteristic patterns consistent with hydrodynamical expansion of the matter produced in the collisions. For each harmonic $n$, a modified valence quark number $N_q$ scaling (plotting $v_n\{\Psi_m\}/(N_q)^{n/2}$ versus ${\rm KE}_T/N_q$) is observed to yield a single curve for all the measured particle species for a broad range of transverse kinetic energies ${\rm KE}_T$. A simultaneous blast-wave model fit to the observed $v_n\{\Psi_m\}(p_T)$ coefficients and published particle spectra identifies radial flow anisotropies $\rho_n\{\Psi_m\}$ and spatial eccentricities $s_n\{\Psi_m\}$ at freeze-out. These are generally smaller than the initial-state participant-plane (PP) geometric eccentricities $\varepsilon_n\{\Psi_m^{\rm PP}\}$, as also observed in the final eccentricity from quantum interferometry measurements with respect to the event plane.

本文报道了质心系能量$sqrt{s_{_{NN}}}=200 mathrm{GeV}$的金金（Au+Au）碰撞中，基于前向快度区域事件平面$Psi_m$测得的快度中段鉴别粒子（$pi^pm$、$K^pm$及$p+ar{p}$）的各向异性流系数$v_2{Psi_2}$、$v_3{Psi_3}$、$v_4{Psi_4}$与$v_4{Psi_2}$随碰撞中心度及粒子横向动量$p_T$的变化关系。上述$v_n$系数呈现出与碰撞产生物质的流体动力学膨胀相符的特征模式。针对每一级谐波$n$，修正价夸克数标度方案（即以$mathrm{KE}_T/N_q$为横轴，绘制$v_n{Psi_m}/(N_q)^{n/2}$随横轴的分布）被观测到可将所有被测粒子种类在宽泛的横向动能$mathrm{KE}_T$范围内统一为单一条曲线。同时结合爆轰波模型（blast-wave model）对观测到的$v_n{Psi_m}(p_T)$系数与已发表的粒子谱进行联合拟合，可提取出冻出阶段的径向流各向异性$
ho_n{Psi_m}$与空间偏心率$s_n{Psi_m}$。这类偏心率总体上小于初始态参与子平面（participant-plane, PP）几何偏心率$varepsilon_n{Psi_m^{
m PP}}$，该结论也与基于事件平面的量子干涉测量得到的末态偏心率观测结果一致。