Eliminating experimental bias in anisotropic-flow measurements of high-energy nuclear collisions

We argue that the traditional event-plane method, which is still widely used to analyze anisotropic flow in ultrarelativistic heavy-ion collisions, should be abandoned because flow fluctuations introduce an uncontrolled bias in the measurement. Instead, one should use an alternative, such as the scalar-product method or cumulant method, which always measures an unambiguous property of the underlying anisotropic flow and therefore eliminates this bias, and does so without any disadvantages. It is known that this correction is important for precision comparisons of traditional $v_n$ measurements requiring better than a few-percent accuracy. However, we show that it is absolutely essential for correlations between different harmonics, such as those that have been recently measured by the ATLAS Collaboration, which can differ from the nominally measured quantity by a factor two or more. We also describe how, using the corrected analysis method, the information from different subevents can be combined in order to optimize the precision of analyses.

Figure 1

Ratio of rms $v_n$ corresponding to the low-resolution limit of Eq. (11) integrated over $p_T$, to the mean $v_n$, corresponding to the high-resolution limit, as a function of the number of participant nucleons (binned in 5$\%$ centrality intervals), for a Pb-Pb collision at 2.76 TeV per nucleon pair. We model flow fluctuations by assuming $v_n\propto {\varepsilon }_n$ in each event, where ${\varepsilon }_n$ is the anisotropy of the initial distribution in the corresponding harmonic, defined as in Ref. [30]. The value of ${\varepsilon }_n$ in each event is obtained from the Phobos Monte Carlo Glauber model [31]. For $v_3$, the ratio is close to $2/\sqrt{\pi }\simeq 1.13$, corresponding to Gaussian fluctuations [32].

Figure 2

(top, left) Low- and high-resolution limits of the correlation between ${\Phi }_2$ and ${\Phi }_4$, defined by Eqs. (28) and (29), versus number of participants. (top, right) Low- and high-resolution limits for integrated $v_4\left\{{\Psi }_2\right\}$, defined by Eqs. (30), (31), and (32). (bottom panels) Ratios of low-resolution to high-resolution results. The nominally measured values correspond to the high-resolution limit, which can differ from what is actually measured (usually closer to the low-resolution limit) by up to a factor of two.