Centrality determination with a forward detector in the RHIC Beam Energy Scan

Our aim is to explore the connection between experimental global observables, generically designated $X$, with the impact parameter of the collision. Naturally, this connection depends on the collision model. Glauber models Miller et al. (2007) have been successfully used to make associate particle multiplicities at midrapidity in high-energy collisions. However, at lower energies (relevant to the RHIC BES program), the spectator-participant paradigm becomes less justified. Furthermore, the longitudinal dynamics of baryon stopping, which drives the physics of the forward direction covered by upgrade detectors Adams et al. (2020), requires a dynamical transport model.

We use the UrQMD transport model Bass et al. (1998); Bleicher et al. (1999), as was used by Chatterjee Chatterjee et al. (2020). For a given impact parameter $b$, the model begins with realistic non-smooth initial conditions and a Monte-Carlo approach; subsequent evolution of the collision is based on string dynamics and hadronic rescattering to produce final-state particles that would be measured in a detector.

In the mid-rapidity region, we compute two quantities calculated by Chatterjee for reference. The first is $X_{\rm RM1}$, defined as the charged-particle multiplicity in the pseudorapidity range $|\eta|<0.5$. The second is $X_{\rm RM3}$, the charged particle multiplicity with $|\eta|<1.0$, excluding protons and antiprotons. These centrality estimators have been used in several analyses at RHIC Adamczyk et al. (2015, 2013, 2014).

Within the EPD acceptance, the natural analog to these measures is $X_{\rm FWD}$, the charged-particle multiplicity in the range $2.1<|\eta|<5.1$. For completeness and to compare to Chatterjee, we calculate this quantity as well, but two problems with $X_{\rm FWD}$ must be considered: it cannot be unambiguously measured in the EPD as the EPD is essentially a segmented calorimeter Adams et al. (2020); which we discuss this in more detail in section II.2, and most importantly, the nontrivial dynamics of stopping drives yields in the forward direction. These dynamics depend on the collision energy as the spectator region ranges within the EPD acceptance (see Figure 2). They also depend on centrality, which varies the relative contributions of participants and spectators. This can be seen in Figure 3, which is the total charged particle yield in the EPD acceptance ($X_{FWD}$) versus the impact parameter for a variety of energies. In this figure we see that the correlation between forward multiplicity and impact parameter decreases with decreasing collision energy due to the contribution from spectators and participants within the EPD acceptance. However, due to the segmentation of the EPD in $\eta$, it is possible to separate the different contributions, as can be seen in Figure 4. In section II.3, we present a novel approach to use this nontrivial behavior to craft an observable centrality estimator with strong correlation to impact parameter at forward rapidity.

The STAR TPC, which covers mid-rapidity in the experiment, measures individual charged particles with 90% efficiency Anderson et al. (2003), so a centrality measure based on multiplicity is appropriate. This is not the case for the STAR EPD, which consists of 744 tiles of 1.2-cm-thick scintillator Adams et al. (2020). When a relativistic charged particle passes through a tile, it deposits a small amount of energy, $dE_{1}$, probabilistically according to the Landau distribution $\rho(dE_{1})$. This distribution is characterized by two parameters: the most probable value ($dE_{\rm MPV}$) for energy loss and a width ($dE_{\rm WID}$); both the energy scale and the relative width ($dE_{\rm WID}/dE_{\rm MPV}$) are determined by the thickness of the scintillator. For the EPD, $dE_{\rm WID}/dE_{\rm MPV}\approx 0.2$ Adams et al. (2020).

As is the practice experimentally, we will quantify the signal in a tile by normalizing to the peak in the single-particle distribution:

The single-particle distribution $\frac{dP_{1}}{d\zeta}$ then peaks at unity and has Landau width $\sim 0.2$. Figure 5 shows the simulated distribution for a tile that is dominated by one or two particles crossing per collision event.

Depending on the collision energy and centrality, $N>1$ charged particles may pass through a tile in a given collision event, each leaving energy independently. The total energy in such events follows a distribution:

The total energy deposited into each tile is recorded; for every collision event, the distribution is a weighted sum of $\frac{dP_{N}}{d\zeta}$, seen for $N\leq 5$ as the histogram in Figure 6 (distributions for $N=1,2,3,4,5$ are shown as shaded curves). This distribution has been shown to reproduce the EPD experimental measurements closely Adams et al. (2020).

In general, the more particles that pass through a tile in a collision event, the larger the measured $\zeta$ will be. It can seem natural to construct EPD-based centrality measures analogous to those used in the TPC by substituting $\zeta$ for multiplicity. However, the most probable value for $\frac{dP_{N}}{d\zeta}$ is not found at $\zeta=N$, and there is clearly considerable overlap between $\frac{dP_{N}}{d\zeta}$ distributions. Indeed, for a low-flux tile (c.f. Figure 5), it is most accurate to assume $N=1$ in any collision event for which $\zeta>0$. Assuming $N=\zeta$ (which is not even correct “on average” as the Landau distribution does not have a well-defined mean) in this case only builds in unwanted noise for an event-wise centrality measure. Even in the high-flux tile of Figure 6, a $\zeta=4$ event is most likely caused by three particles having passed through the tile.

How these effects influence the correlation between a centrality estimator and the true impact parameter depends on the collision model and the detector geometry. Using the UrQMD model, the collisions are assumed to take place at the center of the STAR experiment (for simplicity) and, upon emission, charged particles are assumed to propagate in a straight line until they strike the EPD. A precise geometric model of the active elements of the EPD Adams et al. (2020) is used to register the passage of charged particles through each tile. Each charged particle deposits energy according to the Landau distribution, as discussed above, with the net signal from a tile being the sum of all deposited energy.¹¹1Our simulation does not include the effects of the magnetic field in the STAR experiment; however, at these rapidities, this is relatively unimportant for measuring multiplicity as a function of $\eta$. Secondary scattering and production in surrounding material (magnet iron, beam pipe, etc) is a larger effect that is not accounted for, in our simulation. Exploring corrections due to these effects is outside the scope of the present study.

Figure 7 shows the (anti-)correlation between the impact parameter from the model and the sum of the signals ($X_{\zeta}\equiv\sum_{i}\zeta_{i}$) from the 744 EPD tiles for 19.6 GeV Au+Au collisions. The “noise” effect of Landau fluctuations is clear in the extended tail of the distribution at large $X_{\zeta}$, reducing the correlation.

The effect of these fluctuations may be reduced by “truncating” the signal from each tile, replacing a tile’s signal with:

We chose the value of ${\rm Mx}=3$ for all energies and centralities for this paper, though this could be tuned based on an analysis of the most probable value of the number of particles that will pass through a given tile. The result of applying this methodology can be seen in Figure 8, which plots $X_{\zeta^{\prime}}\equiv\sum_{i}\zeta^{\prime}_{i}$ versus impact parameter. The Pearson coefficient is about 0.98, as compared to only 0.39 for the correlation in Figure 7.

As we discuss in detail in section III, the 16 rings of the EPD (corresponding to different $|\eta|$ ranges listed in Table 1) can be affected quite differently as the impact parameter of the collision is varied. Indeed, depending on the collision energy, the signal in some rings may increase as $b$ is increased, while the signal in others may decrease. Thus their contributions to the simple sum $X_{\zeta^{\prime}}$ discussed may partly cancel, reducing the sensitivity of this measure to collision centrality.

The differential response can be accounted for– indeed, even be exploited to increase sensitivity– by constructing a new simple measure that weights each ring’s “contribution” differently. Below, we consider two ways to quantify this contribution.

In order to apply the formalism described in the previous section, we will compare the two mid-rapidity observables ($X_{RM1}$ and $X_{RM3}$) with four forward rapidity observables ($X_{\zeta^{\prime}},X_{W,\zeta^{\prime}},X_{FWD}$, and $X_{W,FWD}$), where only the observables designated by $W$ use the linear weight method formalized above. The weights determined by this method versus EPD ring number can be seen in Figure 9, with the acceptance of the EPD rings detailed in Table 1. The sign of the weights changes when moving from a distribution that is dominated by spectators versus participants. In Figure 9, we can see that the EPD ring this occurs in changes as the collision energy changes due to the change in beam rapidity (see Figure 2). The weights for EPD distributions with $\zeta^{\prime}$ are very similar to those required by the raw particle counts. These weights were then used to determine the correlation between $\{X_{W,FWD},X_{W,\zeta^{\prime}}\}$ and $b$, the results of which can be seen in Figure 10.

Figure 3 is in agreement with the conclusions of the Chatterjee analysis Chatterjee et al. (2020), and indicates that a summation of the yield over the EPD acceptance is a poor observable. However, when we apply the linear weighting technique described in Section II.3, we recover a much more usable correlation between forward $\eta$ particle yields and $b$. This is summarised in Figure 10. For all the energies under consideration, $X_{RM3}$ (middle row) is well correlated with the impact parameter, whereas $X_{\zeta^{\prime}}$ from the EPD (bottom row) shows a decreasing correlation with decreasing energy. However, the linear weighted $X_{W,\zeta^{\prime}}$ (top row) shows that the correlation is restored.

Experimentally, the centrality is not determined by a model dependent relationship between a global observable and the impact parameter, but rather by considering the global observable’s distribution quantiles (though it should be noted that real world analyses include a Glauber model as the efficiency of recording an event only approaches 100% for the most central collisions Miller et al. (2007)). This indicates that, as long as a global observable is reasonably correlated with the impact parameter, the centrality distribution based on this selection criteria will also be reasonable.

In Figure 11, the impact parameter distributions that are determined by using the appropriate quantiles for the global variables are shown along with distributions that result from directly using the quantiles of the impact parameter distribution. For $\sqrt{s_{NN}}$= 19.6 GeV, all methods performed similarly (quantified in Section III.2). The mid-rapidity observables, $X_{RM1}$ and $X_{RM3}$, have $b$ distributions which peak within the $b$ distribution slices and do not change drastically as the collision energy decreases from 19.6 GeV to 7.7 GeV. The forward observables without any weighting, $X_{FWD}$ and $X_{\zeta^{\prime}}$, have distributions which no longer lie under the $b$ distribution slices at the lower energies; this suggests a poor centrality resolution for these observables in the lower energy ranges of the BES program. This potential loss in resolution, however, can be compensated for by applying the weighting scheme discussed in section II. We see that the distributions for $X_{W,FWD}$ and $X_{W,\zeta^{\prime}}$ are under the $b$ distribution slices for all collision energies under consideration.

From the $b$ distributions in Figure 11, we determined the centrality resolution for all $X$. As in Chatterjee et al. (2020), we employ the centrality resolution metric $\Phi$:

where $\sigma_{b_{i}}^{2}$ is the variance from the impact parameter distribution for a given centrality range $i$ when centrality is determined by $b$, and $\sigma_{b_{X,i}}^{2}$ is the variance from the impact parameter distribution for the same centrality range $i$ where centrality is determined by $X$.

Results for $\Phi$ can be seen in Figure 12. At the top RHIC energies, all centrality methods perform similarly. The resolution at lowest energies is poorest for unweighted distributions in the forward region ($X_{FWD}$ for particle yield, and $X_{\zeta^{\prime}}$ for truncated energy loss in the EPD), but this resolution is clearly recovered when we apply the linear weights detailed in Section II.3 ($X_{W,FWD}$ and $X_{W,\zeta^{\prime}}$, respectively). Both $X_{RM1}$ and $X_{RM3}$ are based on mid-rapidity multiplicities. The anomalous, upward point at the 5-10% range is due to the smaller centrality bins used for our two most central selections, which increases the standard deviation of the distributions.

The poor performance of $X_{FWD}$ and $X_{\zeta^{\prime}}$, which agrees with the conclusions in Chatterjee et al. (2020), is due to spectator proton intrusion into the EPD’s acceptance. If we do not weight the EPD rings which are dominated by the spectator protons yield, which have positive correlation with $b$, with a different sign than the EPD rings where we are dominate by participants, which have a negative correlation, they will cancel out. This is the entire purpose of the methodology in section II.3; thus the correlation weights found for rings with spectators will have inverted signs compared with those rings with only participants (Figures 2 and 9). The linear weight method recovers the centrality resolution lost by the simple sum method of $X_{FWD}$ and $X_{\zeta^{\prime}}$