Centrality determination with a forward detector in the RHIC Beam Energy Scan
CCentrality determination with a forward detector in the RHIC Beam Energy Scan
Skipper Kagamaster and Rosi Reed
Lehigh University, Bethlehem PA
Michael Lisa
The Ohio State University, Columbus, OH (Dated: September 4, 2020)Recently, Chatterjee et al [1] used a hadronic transport model to estimate the resolution withwhich various experimental quantities select the impact parameter of relativistic heavy ion collisionsat collision energies relevant to the Beam Energy Scan (BES) program at the Relativistic HeavyIon Collider (RHIC). Measures based on particle multiplicity at forward rapidity were found to besignificantly worse than those based on midrapidity multiplicity. Using the same model, we showthat a slightly more sophisticated measure greatly improves the resolution based on forward rapidityparticles; this improvement persists even when the model is filtered through a realistic simulationof a recent upgrade detector to the STAR experiment. These results highlight the importanceof optimizing centrality measures based on particles detected at forward rapidity, especially forexperimental studies that search for a critical point in the QCD phase diagram. Such measurementsusually focus on proton multiplicity fluctuations at midrapidity, hence selecting events based onmultiplicity at midrapidity raises the possibility of nontrivial autocorrelations.
I. INTRODUCTION
A central goal of relativistic heavy ion collision physicsis to probe the nonperturbative regime of quantum chro-modynamics (QCD) by exploring the phase diagram ofpartonic matter as a function of thermodynamic vari-ables. At very high temperatures achievable at theLarge Hadron Collider or Relativistic Heavy Ion Col-lider (RHIC), the quark-gluon plasma (QGP) is formed,a phase in which colored partons are the dynamical de-grees of freedom [2–5]. Collisions at progressively lowerenergies form matter with lower temperature and higherbaryochemical potential, eventually reaching conditionsat which the system is in the hadronic gas phase, in whichpartons are confined. Understanding the nature of thetransition between QGP and hadronic matter has been acentral goal of the Beam Energy Scan (BES) program atRHIC [6, 7]. Of particular interest is a possible criticalpoint (CP) [8–20] at the terminus of the line character-izing a first-order phase transition on the QCD phasediagram, sketched in figure 1.In an infinite system in equilibrium, measured mo-ments of conserved quantities such as charge or baryonnumber are directly sensitive to the correlation lengthof the system. A non-monotonic energy dependenceof higher-order moments (e.g. the kurtosis) of thenet baryon distribution may reveal the a QCD criti-cal point [22, 23]. The STAR Collaboration has mea-sured [24] the first four moments of the net-proton (astand-in for net-baryon) distribution in heavy ion col-lisions at collision energies √ s NN = 7 . −
200 GeV,recently reporting non-monotonic behavior at the 3 σ level [25]. This may suggest that the discovery of oneof the long-sought features of bulk QCD is within reach.However, great care must be taken in this analysis ofsmall effects, to avoid nontrivial autocorrelations associ-ated with event selection. In particular, the protons and FIG. 1. The QCD phase diagram, showing various states ofQCD matter as functions of baryon chemical potential ( µ B )and temperature (T). The CP location (and existence) is atthis time not confirmed. [21] antiprotons used to construct the net-proton distributionare measured in the rapidity range | y | < .
5. A criticalfeature of the observation is that the interesting behavioronly occurs for the most central collisions, where central-ity is estimated by the multiplicity of charged particles– except for protons and antiprotons – with pseudorapidity | η | <
1. In other words, one looks at (possibly correlated)fluctuations in the multiplicities of protons and antipro-tons as a function of charged particle (except proton andantiproton) multiplicity. Exclusion of the protons andantiprotons from the centrality measure does not triv-ially remove autocorrelations (by which we mean mea-surement of an event-wise quantity while selecting eventswith a measure directly related to that quantity), sincevery few (anti)protons are “direct.” Almost all are de-cay products of higher-mass baryons (e.g. ∆ ++ or N ∗ )whose sibling decay products are pions or other chargedparticles which are counted in the centrality measure. a r X i v : . [ nu c l - e x ] S e p A recent study by Chatterjee, et al. [1] based on ahadronic transport model concludes that the effect ofthese autocorrelations on the higher-order moments anal-ysis is small. However, the model employed in the studydoes not itself include critical fluctuations, so it is un-clear how general this theoretical conclusion is. In anyevent, it would be useful to check the robustness of theexperimental observation by using a centrality estimatorbased on particles not emitted at midrapidity.The STAR experiment has recently commissioned theEvent Plane Detector (EPD) [26] to upgrade its capa-bilities in the BES program. The EPD provides highly-segmented coverage for charged particles emitted in awide pseudorapidity range (2 . < | η | < .
1) and full az-imuth. An analysis that used the EPD for a centralityestimate would be free of potential autocorrelations dis-cussed above.Chatterjee and collaborators [1] report that thesummed multiplicity in the EPD acceptance would bea poor estimator of event centrality, because participantand spectator contributions are anticorrelated. In whatfollows, we extend the study of Chatterjee and demon-strate that a slightly more sophisticated treatment of thesignal in the EPD region improves the resolution substan-tially. We show that this improvement persists even whenthe model is filtered through a realistic simulation of theSTAR EPD, and that the detector can provide a measureof centrality with only slightly lower resolution than thatfrom the STAR Time Projection Chamber (TPC, withcoverage in the pseudorapidity range | η | < II. METHODS
The Ultra-Relativistic Quantum Molecular Dynamic(UrQMD) model is a microscopic transport model usedto simulate relativistic heavy ion collisions in the sameenergy ranges as the BES data [27]. For this study, weused 55k events per center of mass energy for Au+Aucollisions at √ s NN = 7.7, 11.5, 14.5, and 100k events for19.6 GeV. - - - - h h / d c h d N = (GeV) NN sUrQMD 7.711.514.519.6EPD FIG. 2. Charged particle pseudorapidity distributions forBES energies of √ s NN = 19.6 (magenta triangle), 14.5 (bluetriangle), 11.5 (red square), 7.7 (black circle) GeV. There isno centrality selection for these distributions. The EPD ac-ceptance is shown as a green box. A. Calculation of global centrality estimators
Our aim is to explore the connection between experi-mental global observables, generically designated X , withthe impact parameter of the collision. Naturally, thisconnection depends on the collision model. Glauber mod-els [28] have been successfully used to make associateparticle multiplicities at midrapidity in high-energy col-lisions. However, at lower energies (relevant to the RHICBES program), the spectator-participant paradigm be-comes less justified. Furthermore, the longitudinal dy-namics of baryon stopping, which drives the physics ofthe forward direction covered by upgrade detectors [26],requires a dynamical transport model.We use the UrQMD transport model [27, 29], as wasused by Chatterjee [1]. For a given impact parameter b ,the model begins with realistic non-smooth initial condi-tions and a Monte-Carlo approach; subsequent evolutionof the collision is based on string dynamics and hadronicrescattering to produce final-state particles that wouldbe measured in a detector.In the mid-rapidity region, we compute two quanti-ties calculated by Chatterjee for reference. The first is X RM1 , defined as the charged-particle multiplicity in thepseudorapidity range | η | < .
5. The second is X RM3 ,the charged particle multiplicity with | η | < .
0, exclud-ing protons and antiprotons. These centrality estimatorshave been used in several analyses at RHIC [24, 30, 31].Within the EPD acceptance, the natural analog tothese measures is X FWD , the charged-particle multiplic-ity in the range 2 . < | η | < .
1. For completeness and F W D X = 19.6 GeV NN SUrQMD 00.0050.010.0150.020.0250.030 2 4 6 8 10 12 14b (fm)0100200300400500600700 F W D X = 11.5 GeV NN SUrQMD 00.0050.010.0150.020.0250.030 2 4 6 8 10 12 14b (fm)0100200300400500600700 F W D X = 14.5 GeV NN SUrQMD 00.0050.010.0150.020.0250.030 2 4 6 8 10 12 14b (fm)0100200300400500600700 F W D X = 7.7 GeV NN SUrQMD
FIG. 3. The sum of all charged particle yields in the EPD ac-ceptance ( X FWD ) vs impact parameter b for the four collisionenergies considered in this paper. to compare to Chatterjee, we calculate this quantity aswell, but two problems with X FWD must be considered:it cannot be unambiguously measured in the EPD as theEPD is essentially a segmented calorimeter [26]; whichwe discuss this in more detail in section II B, and mostimportantly, the nontrivial dynamics of stopping drivesyields in the forward direction. These dynamics dependon the collision energy as the spectator region rangeswithin the EPD acceptance (see Figure 2). They alsodepend on centrality, which varies the relative contribu-tions of participants and spectators. This can be seen inFigure 3, which is the total charged particle yield in theEPD acceptance ( X F W D ) versus the impact parameterfor a variety of energies. In this figure we see that thecorrelation between forward multiplicity and impact pa-rameter decreases with decreasing collision energy due tothe contribution from spectators and participants withinthe EPD acceptance. However, due to the segmentationof the EPD in η , it is possible to separate the differentcontributions, as can be seen in Figure 4. In section II C,we present a novel approach to use this nontrivial behav-ior to craft an observable centrality estimator with strongcorrelation to impact parameter at forward rapidity. B. EPD Simulation
The STAR TPC, which covers mid-rapidity in theexperiment, measures individual charged particles with90% efficiency [32], so a centrality measure based onmultiplicity is appropriate. This is not the case forthe STAR EPD, which consists of 744 tiles of 1.2-cm-thick scintillator [26]. When a relativistic charged parti-cle passes through a tile, it deposits a small amount ofenergy, dE , probabilistically according to the Landaudistribution ρ ( dE ). This distribution is characterized FIG. 4. The sum of charged particle yields for a given EPDring using simulated UrQMD events at √ s NN = 7.7 GeV.Ring 1 is the ring closest to the beam-pipe and Ring 16 is theclosest to mid-rapdity. The precise acceptance boundaries foreach ring are listed in Table I.EPD Ring | η l | | η h | EPD Ring | η l | | η h | | η l | ) and high ( | η h | ) values for the | η | ranges of each EPD ring [26]. by two parameters: the most probable value ( dE MPV )for energy loss and a width ( dE WID ); both the energyscale and the relative width ( dE WID /dE
MPV ) are deter-mined by the thickness of the scintillator. For the EPD, dE WID /dE
MPV ≈ . ζ ≡ dEdE MPV . (1)The single-particle distribution dP dζ then peaks at unityand has Landau width ∼ .
2. Figure 5 shows the simu-lated distribution for a tile that is dominated by one ortwo particles crossing per collision event.Depending on the collision energy and centrality,
N >
FIG. 5. A low-flux tile which is dominated by either oneor two particles passing per collision events. The colouredpeaks show the 1 (grey) and 2 (red) particle Landau distri-butions, whereas the spectrum (black circles) is a sum of theconvoluted distributions (green line). dP N dζ ( ζ ) = (cid:90) ∞ dζ dP dζ ( ζ ) dP N − dζ ( ζ − ζ ) . (2)The total energy deposited into each tile is recorded; forevery collision event, the distribution is a weighted sumof dP N dζ , seen for N ≤ N = 1 , , , , ζ will be.It can seem natural to construct EPD-based centralitymeasures analogous to those used in the TPC by sub-stituting ζ for multiplicity. However, the most probablevalue for dP N dζ is not found at ζ = N , and there is clearlyconsiderable overlap between dP N dζ distributions. Indeed,for a low-flux tile (c.f. Figure 5), it is most accurate toassume N = 1 in any collision event for which ζ > N = ζ (which is not even correct “on average”as the Landau distribution does not have a well-definedmean) in this case only builds in unwanted noise for anevent-wise centrality measure. Even in the high-flux tileof Figure 6, a ζ = 4 event is most likely caused by threeparticles having passed through the tile.How these effects influence the correlation between acentrality estimator and the true impact parameter de-pends on the collision model and the detector geometry.Using the UrQMD model, the collisions are assumed to FIG. 6. A high-flux tile which experiences a range of particlemultiplicities, resulting in a spectrum (black circles) that isa sum of convoluted Landau distributions (green line). Thecoloured peaks show the 1 (grey), 2 (red), 3 (blue), 4 (purple)and 5 (dark grey) particle Landau distributions. take place at the center of the STAR experiment (forsimplicity) and, upon emission, charged particles are as-sumed to propagate in a straight line until they strikethe EPD. A precise geometric model of the active ele-ments of the EPD [26] is used to register the passage ofcharged particles through each tile. Each charged parti-cle deposits energy according to the Landau distribution,as discussed above, with the net signal from a tile beingthe sum of all deposited energy.[33]Figure 7 shows the (anti-)correlation between the im-pact parameter from the model and the sum of the sig-nals ( X ζ ≡ (cid:80) i ζ i ) from the 744 EPD tiles for 19.6 GeVAu+Au collisions. The “noise” effect of Landau fluctu-ations is clear in the extended tail of the distribution atlarge X ζ , reducing the correlation.The effect of these fluctuations may be reduced by“truncating” the signal from each tile, replacing a tile’ssignal with: ζ (cid:48) ≡ (cid:40) ζ, if ζ < MxMx , otherwise (3)We chose the value of Mx = 3 for all energies and central-ities for this paper, though this could be tuned based onan analysis of the most probable value of the number ofparticles that will pass through a given tile. The resultof applying this methodology can be seen in Figure 8,which plots X ζ (cid:48) ≡ (cid:80) i ζ (cid:48) i versus impact parameter. ThePearson coefficient is about 0.98, as compared to only0.39 for the correlation in Figure 7. z X NN SUrQMD
FIG. 7. X ζ value versus impact parameter for √ S NN = 19.6GeV UrQMD simulations. The extended tail of the distribu-tion is due to Landau fluctuations. ' z X NN SUrQMD
FIG. 8. X ζ (cid:48) value versus impact parameter for √ s NN = 19.6GeV UrQMD simulations. The truncation of ζ (from Equa-tion 3) reduces the noise from Landau fluctuations seen in thecorrelation between X ζ and b in Figure 7. C. A new centrality estimator using ring weights
As we discuss in detail in section III, the 16 rings ofthe EPD (corresponding to different | η | ranges listed inTable I) can be affected quite differently as the impactparameter of the collision is varied. Indeed, dependingon the collision energy, the signal in some rings may in-crease as b is increased, while the signal in others may decrease . Thus their contributions to the simple sum X ζ (cid:48) discussed may partly cancel, reducing the sensitivity ofthis measure to collision centrality.The differential response can be accounted for– indeed,even be exploited to increase sensitivity– by constructinga new simple measure that weights each ring’s “contribu-tion” differently. Below, we consider two ways to quantifythis contribution.
1. Weighted sum of ζ (cid:48) First, we define a ring’s contribution to be the sum ofthe truncated signals in each tile in the ring: C r ≡ (cid:88) tile j inring r ζ (cid:48) j (energy-loss based) (4)where we have indicated explicitly that the ring’s con-tribution is based on energy loss (in the next section,we will discuss an analogous, particle based approach).Since we consider symmetric collisions occurring mid-waybetween the EPD wheels, ring r on the wheel positionedat z = −
375 cm is summed with ring r on the wheel at z = +375 cm.Based on these contributions, we define our centralitymeasure as a weighted sum: X W,ζ (cid:48) ≡ (cid:88) r =1 W r C r + W (5)where W i are parameters determined below.We wish to maximize the correlation between X ζ (cid:48) andsome global quantity G (for the moment, G is the im-pact parameter, b , but we generalize the discussion insection IV). A figure of merit may be the squared resid-ual, summed over events: χ = N events (cid:88) j =1 ( X W,ζ (cid:48) ,j − G j ) , (6)where X W,ζ (cid:48) ,j and G j are respectively the values of theestimator and global quantity (e.g. impact parameter)for event j .Maximizing χ yields 17 linear equations: (cid:88) q =1 A q,t W q = B t (7)where: A q,t = N events (cid:88) j =1 C q,j C t,j for q, t = 1 ...
16 (8) A ,t = N events (cid:88) j =1 C t,j for t = 1 ...
16 (9) A , = N events (10) B t = N events (cid:88) j =1 G j C t,j for t = 1 ...
16 (11) B = N events (cid:88) j =1 G j (12)Since A is a symmetric, real, 17 ×
17 matrix it can beeasily inverted to find the unique best parameters W t .
2. Weighted sum of particles
As discussed above, an EPD tile measures in a givenevent the energy deposited in the tile, and not the numberof particles that actually passed through the tile. How-ever, it is natural to ask whether the sensitivity would beimproved if a ring’s contribution would be the number ofcharged particles passing through tiles in that ring, i.e.if C r in equation 4 would be redefined: C r ≡ (cid:88) tile j inring r N j (particle based) (13)Here, N j is the number of particles that passed throughtile j in the event.Analogous to X W,ζ (cid:48) a weighted-sum centrality mea-sure X W,F W D may then be constructed from the contri-butions of equation 13, with the weights determined byequations 6-12.
III. RESULTS
With the mathematical formalism described in SectionII C, we can examine the performance of the EPD as itpertains to relating signals within the detector to theimpact parameter using UrQMD.
A. Correlations between yields and b In order to apply the formalism described in the previ-ous section, we will compare the two mid-rapidity observ-ables ( X RM and X RM ) with four forward rapidity ob-servables ( X ζ (cid:48) , X W,ζ (cid:48) , X
F W D , and X W,F W D ), where onlythe observables designated by W use the linear weightmethod formalized above. The weights determined by FIG. 9. The weights for both forward η particles and ζ (cid:48) whenusing the linear weight scheme from Section II C, by EPD ring.The sign change in ring weights is motivated by spectatorproton intrusion into the EPD’s acceptance window, as canbe seen in Figure 2. this method versus EPD ring number can be seen in Fig-ure 9, with the acceptance of the EPD rings detailedin Table I. The sign of the weights changes when mov-ing from a distribution that is dominated by spectatorsversus participants. In Figure 9, we can see that theEPD ring this occurs in changes as the collision energychanges due to the change in beam rapidity (see Fig-ure 2). The weights for EPD distributions with ζ (cid:48) arevery similar to those required by the raw particle counts.These weights were then used to determine the corre-lation between { X W,F W D , X
W,ζ (cid:48) } and b , the results ofwhich can be seen in Figure 10.Figure 3 is in agreement with the conclusions of theChatterjee analysis [1], and indicates that a summationof the yield over the EPD acceptance is a poor observable.However, when we apply the linear weighting techniquedescribed in Section II C, we recover a much more usablecorrelation between forward η particle yields and b . Thisis summarised in Figure 10. For all the energies underconsideration, X RM (middle row) is well correlated withthe impact parameter, whereas X ζ (cid:48) from the EPD (bot-tom row) shows a decreasing correlation with decreasingenergy. However, the linear weighted X W,ζ (cid:48) (top row)shows that the correlation is restored.Experimentally, the centrality is not determined by amodel dependent relationship between a global observ-able and the impact parameter, but rather by consider-ing the global observable’s distribution quantiles (thoughit should be noted that real world analyses include aGlauber model as the efficiency of recording an eventonly approaches 100% for the most central collisions [28]). ' z = 19.6 GeV NN S R M X = 19.6 GeV NN S ' z W , X = 19.6 GeV NN S ' z = 14.5 GeV NN S R M X = 14.5 GeV NN S ' z W , X = 14.5 GeV NN S ' z = 11.5 GeV NN S R M X = 11.5 GeV NN S ' z W , X = 11.5 GeV NN S ' z = 7.7 GeV NN S R M X = 7.7 GeV NN S ' z W , X = 7.7 GeV NN S FIG. 10. Correlation between the impact parameter, b , andthree observables: X W,ζ (cid:48) (top), X RM (middle), X ζ (cid:48) (bot-tom), for four different collision energies. This indicates that, as long as a global observable is rea-sonably correlated with the impact parameter, the cen-trality distribution based on this selection criteria willalso be reasonable.In Figure 11, the impact parameter distributions thatare determined by using the appropriate quantiles for theglobal variables are shown along with distributions thatresult from directly using the quantiles of the impact pa-rameter distribution. For √ s NN = 19.6 GeV, all meth-ods performed similarly (quantified in Section III B). Themid-rapidity observables, X RM and X RM , have b dis-tributions which peak within the b distribution slices anddo not change drastically as the collision energy decreasesfrom 19.6 GeV to 7.7 GeV. The forward observables with-out any weighting, X F W D and X ζ (cid:48) , have distributionswhich no longer lie under the b distribution slices at thelower energies; this suggests a poor centrality resolutionfor these observables in the lower energy ranges of theBES program. This potential loss in resolution, how-ever, can be compensated for by applying the weightingscheme discussed in section II. We see that the distribu-tions for X W,F W D and X W,ζ (cid:48) are under the b distributionslices for all collision energies under consideration. B. Centrality Resolution
From the b distributions in Figure 11, we determinedthe centrality resolution for all X . As in [1], we employthe centrality resolution metric Φ: - - - d P / db = 7.7 GeV NN S RM1 X RM3
X 0 2 4 6 8 10 12 14 16b (fm) - - - d P / db = 11.5 GeV NN S ' z X ' z W, X - - - d P / db = 14.5 GeV NN S FWD X W,FWD X - - - d P / db = 19.6 GeV NN Sb FIG. 11. The impact parameter distributions for centrality se-lections 0 - 5%, 20 - 30% and 90 - 100%. The black histogramsare the b impact parameter distributions if the centrality se-lection is determined from the impact parameter directly. Thegreen circles are determined using X RM (closed) and X RM (open), the triangles are determined using X ζ (cid:48) (open blue)and X W,ζ (cid:48) (closed red), and the squares are determined using X FWD (open blue) and X W,FWD (closed red). t o
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10 0 t o Centrality110 s / x s = F RM1 X RM3 X ' z X ' z W, X FWD X W,FWD X FIG. 12. Centrality resolution for collision energies √ s NN = 19.6, 14.5, 11.5, and 7.7 GeV. The observables based onmid-rapidity multiplicity are X RM (green closed circles) and X RM (dark green open circles). The observables based onunweighted distributions from the forward region are X FWD (blue open square) for particle yield and X ζ (cid:48) (blue open tri-angle) for truncated energy loss in the EPD. The observablesbased on linear weights are X W,FWD (closed dark red square)and X W,ζ (cid:48) (closed red triangle). Φ i ≡ σ b X,i σ b i (14)where σ b i is the variance from the impact parameter dis-tribution for a given centrality range i when centrality isdetermined by b , and σ b X,i is the variance from the im-pact parameter distribution for the same centrality range i where centrality is determined by X .Results for Φ can be seen in Figure 12. At thetop RHIC energies, all centrality methods perform sim-ilarly. The resolution at lowest energies is poorest forunweighted distributions in the forward region ( X F W D for particle yield, and X ζ (cid:48) for truncated energy lossin the EPD), but this resolution is clearly recoveredwhen we apply the linear weights detailed in SectionII C ( X W,F W D and X W,ζ (cid:48) , respectively). Both X RM and X RM are based on mid-rapidity multiplicities. Theanomalous, upward point at the 5-10% range is due tothe smaller centrality bins used for our two most centralselections, which increases the standard deviation of thedistributions.The poor performance of X F W D and X ζ (cid:48) , which agreeswith the conclusions in [1], is due to spectator proton in-trusion into the EPD’s acceptance. If we do not weightthe EPD rings which are dominated by the spectator pro-tons yield, which have positive correlation with b , with adifferent sign than the EPD rings where we are dominateby participants, which have a negative correlation, theywill cancel out. This is the entire purpose of the method-ology in section II C; thus the correlation weights foundfor rings with spectators will have inverted signs com-pared with those rings with only participants (Figures 2and 9). The linear weight method recovers the centralityresolution lost by the simple sum method of X F W D and X ζ (cid:48) IV. SUMMARY AND DISCUSSION
Observables in heavy-ion experiments can suffer fromauto-correlation effects if the particles used to constructthe observable are from the same acceptance as the eventcentrality is defined (e.g. κ and S for net-proton mul-tiplicity being studied as part of the STAR BES pro-gram) [34]. For observables analysed at mid-rapidity,such as those measured using the TPC in the STAR ex-periment, it would thus be preferable to select centralityusing a forward detector. The EPD is the prime candi-date for a forward centrality selection detector at STAR,but at the lower collision energies of the BES there is sig-nificant spectator proton intrusion into the η acceptancewindow of the EPD. It had been suggested that this spec-tator intrusion could potentially degrade the centralityresolution of the EPD due to the inverse correlation effecton particle yields with b in those η regions where the EPDand spectator protons coincide [1]. However we showed that by treating the particle yield correlations from theEPD rings individually, instead of simply summing theyield from the total EPD acceptance, we can account forcorrelations with both participants and spectators in anevent. This treatment leverages the spectator protons inthe EPD acceptance range as a relevant marker for globalquantities (such as b ).In this paper, we outlined a method of applying a lin-ear weight to the rings (Equation 5) as one example ofa procedure that improves the centrality resolution byproperly weighting the contribution from each EPD ringbased on minimizing the residual between a global ob-servable and the EPD rings. Results from this methodwere shown for correlations with b , but the method issufficiently general that weighting may be found for anyglobal quantity G ; for instance, the method could also beemployed to weigh the EPD ring contributions as theycorrelate with X RM or V.More sophisticated methods than the simplest one wedescribed in section II C are certainly possible and areunder development.Using this method we recovered the centrality reso-lution in the forward η region that is lost in the lowerenergy ranges of STAR BES when simply considering asum of the yields over the entire EPD η range. Further,EPD simulation shows no marked degradation of central-ity resolution when comparing centrality using the linearweighted sum of particles in the EPD acceptance range( X W,F wd ) versus simulated energy deposition in the EPDitself ( X W,ζ (cid:48) ). This strongly suggests the EPD can beused as a reliable centrality detector in STAR BES en-ergy ranges of √ s NN = 7.7, 11.5, 14.5, 19.6, and 200GeV, which would greatly reduce the possibility of auto-correlations in analyses of observables at mid-rapidity.We conclude by briefly considering the implication ofthis study in an experimental analysis, in which the trueimpact parameter is unknown. Any estimate of the im-pact parameter resolution of any measurable estimator isthen completely model-dependent; X ζ (cid:48) may in fact be thebest estimator in reality, despite being the worst in theUrQMD calculations (c.f. figure 12). In such a case– es-pecially if even a years-long program of high-quality dataresults in a subtle wiggle in fluctuations at only the 3 σ level [25]– nontrivial effects of autocorrelations must beruled out in a model-independent way, not relying solelyon transport calculations [1]. The experimental proce-dure must be to (1) perform the analysis, using a com-mon estimator (e.g. X RM ) to select on centrality; (2)in a kinematic region far from the fluctuation measure-ment, construct a weighted estimator by following theprocedure of section II C 1, where the common estimatoris used as the global quantity G in equations 11-12; (3)repeat the analysis, using the new estimator (e.g. X W,ζ (cid:48) )to select on centrality. Persistence of the signal when us-ing the new estimator would lead to greater confidencethat autocorrelations are not influencing the signal itself.Such confidence would be most welcome for the subtlestsignals of fundamental physics of QCD.
ACKNOWLEDGEMENTS
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