Relating centrality to impact parameter in nucleus-nucleus collisions

In ultrarelativistic heavy-ion experiments, one estimates the centrality of a collision by using a single observable, say $n$, typically given by the transverse energy or the number of tracks observed in a dedicated detector. The correlation between $n$ and the impact parameter $b$ of the collision is then inferred by fitting a specific model of the collision dynamics, such as the Glauber model, to experimental data. The goal of this paper is to assess precisely which information about $b$ can be extracted from data without any specific model of the collision. Under the sole assumption that the probability distribution of $n$ for a fixed $b$ is Gaussian, we show that the probability distribution of the impact parameter in a narrow centrality bin can be accurately reconstructed up to $5\%$ centrality. We apply our methodology to data from the Relativistic Heavy Ion Collider and the Large Hadron Collider. We propose a simple measure of the precision of the centrality determination, which can be used to compare different experiments.

Figure 1

Histograms of the probability distribution of $n$ measured by different experiments. Dashed line is STAR data on $\text{Au}+\text{Au}$ collisions at $\sqrt[]{s}=130\mathrm{GeV}$ [16]. The other curves show LHC data on $\text{Pb}+\text{Pb}$ collisions at $\sqrt[]{s}=2.76\mathrm{TeV}$. Circles are ALICE data [11]. Dot-dashed line is ATLAS data [13]. Solid line is CMS data, extracted from Fig. 2 of Ref. [18]. The horizontal axis of each histogram has been rescaled by the value of $n$ at the knee (see text).

Figure 2

Shaded areas correspond to histograms of the probability distribution of $n$ for fixed impact parameter in $\text{Pb}+\text{Pb}$ collisions at $\sqrt[]{s}=2.76\mathrm{TeV}$ in the $\mathrm{T}R\mathrm{ENTo}$ model [17]. The values $b=0$, 5, 8, 12 fm were used, which correspond to $c_b=0$, 10%, 26%, 58%, respectively. We generated $5\times {10}^5$ events for each value of $b$. Solid lines are Gaussian fits. The quantity $n$ is in arbitrary units.

Figure 3

Circles show $P\left(n\right)$ obtained from the $\mathrm{T}R\mathrm{ENTo}$ model of initial conditions. Dashed line shows fit of $P\left(n\right)$ using Eqs. (3) and (6) in scenario (A). The vertical line indicates the position of the knee (see Table 1). The quantity $n$ is in arbitrary units.

Figure 4

Mean value of $n$ versus $b$ centrality. Circles show result calculated directly by binning $\mathrm{T}R\mathrm{ENTo}$ results in $c_b$. Dashed line shows result reconstructed from the fit of $P\left(n\right)$ alone. The inset shows a zoom of the most-central collisions, where we compare $\overline{n}\left(c_b\right)$ to $n\left(c\right)$ (dotted line).

Figure 5

Probability distribution of $b$ centrality in a very narrow centrality bin. Histograms correspond to direct calculation using $\mathrm{T}R\mathrm{ENTo}$. Lines show probability distributions of $c_b$ reconstructed through Bayes' theorem, Eq. (9). Solid and dashed lines correspond to scenarios (A) and (B), respectively. The second panel corresponds to $n=n_{\mathrm{knee}}$. Note that, for $n\ge n_{\mathrm{knee}}$ (two uppermost panels), the most probable value of $c_b$ is $c_b=0\%$. Left of the knee, on the other hand, the most probable value of $c_b$ is $c$ [7], as is clearly visible in the two lowermost panels. All the curves displayed in this figure have area normalized to unity.

Figure 6

(a) Empty symbols show distribution of the VZERO amplitude (denoted by $n$), used by the ALICE collaboration to determine the collision centrality [11]. Full symbols show the Glauber model used by the ALICE collaboration to fit the measured distribution. Line shows fit of data provided by Eq. (6). The inset is a zoom of the central part of the histogram. The quantity $n$ is in arbitrary units. (b) Symbols show STAR data [9]. Line shows fit using Eq. (6). The vertical line in both panels indicates the position of the knee.

Figure 7

Probability distributions of $b$ centrality for selected values of the centrality reconstructed from STAR (dashed lines) and ALICE (solid lines) data by using Bayes' theorem. All the curves shown in this figure have area normalized to unity.

Figure 8

Probability distribution of $c_b/c_{\mathrm{knee}}$ extracted from STAR (solid lines) and ALICE (dashed lines) data for three selected values of $c/c_{\mathrm{knee}}$.