A few statistical problems faced by the event reconstruction in ALICE experiment at CERN are discussed in this paper. We outline several ad-hoc extensions of traditional Kalman- lter track nding which seem to increase the quality of tracks reconstructed in high multiplicity events anticipated for Pb Pb collisions at LHC. These extensions, however, need a stricter formulation and justi cation from the theoretical side. The particle identi cation in ALICE is done by combining the information from different detecting systems using a Bayesian method. Having many clear advantages, this approach introduces into the analysis additional complications which are also discussed here

Belikov, Iouri

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ALICE Statistical Wish-listIouri Belikov for the ALICE collaborationCERN, Geneva, SwitzerlandAbstractA few statistical problems faced by the event reconstruction in ALICE exper-iment at CERN are discussed in this paper. We outline several ad-hoc ex-tensions of traditional Kalman-lter track nding which seem to increase thequality of tracks reconstructed in high multiplicity events anticipated for PbPb collisions at LHC. These extensions, however, need a stricter formulationand justication from the theoretical side. The particle identication in ALICEis done by combining the information from different detecting systems usinga Bayesian method. Having many clear advantages, this approach introducesinto the analysis additional complications which are also discussed here.1 IntroductionA Large Ion Collider Experiment (ALICE) [1] at CERN is a general-purpose heavy-ion experiment de-signed to study the physics of strongly interacting matter and the Quark-Gluon Plasma in nucleus-nucleuscollisions at the LHC. In addition to heavy systems, the ALICE Collaboration will study collisions oflower-mass ions, in order to vary the energy density, and protons (both pp and pA), which primarilyprovide reference data for the nucleusnucleus collisions. The pp data will also allow for a number ofgenuine pp physics studies.The detector consists of a central part (see Fig. 1), which, event-by-event, measures hadrons, elec-trons and photons, and of a forward spectrometer to measure muons. The central part, which coverspolar angles from 45◦ to 135◦ over the full azimuth, is embedded in the large L3 solenoidal mag-net. It consists of an Inner Tracking System (ITS) of high-resolution silicon detectors; a cylindricalTime-Projection Chamber (TPC); three particle identication arrays, a Time-Of-Flight (TOF) detector, aTransition-Radiation Detector (TRD) and a single-arm ring imaging Cherenkov detector (HMPID) and asingle-arm electromagnetic calorimeter (PHOS). The forward muon spectrometer (covering polar angles180◦−θ = 2◦−9◦) consists of a complex arrangement of absorbers, a large dipole magnet, and fourteenplanes of tracking and triggering chambers. Several smaller detectors for global event characterizationand triggering are located at forward angles.The detector is optimized for charged-particle density dNch/dy = 4000 and its performance ischecked in detailed simulations up to dNch/dy = 8000. The track reconstruction efciency in theacceptance of the TPC is about 80% down to transverse momentum of pt ∼ 0.2 GeV/c and about 90%for tracks with pt > 1 GeV/c. It is limited only by the particle decays and small dead zones betweenthe TPC sectors. Typical momentum resolution obtained with the magnetic eld of 0.5 T is ∼ 1% atpt ∼ 1 GeV/c and ∼ 4% at pt ∼ 100 GeV/c. The secondary vertices can be reconstructed with theprecision better then 100 µm.The detector has excellent particle identication (PID) capabilities. From p ∼ 0.1 GeV/c to afew GeV/c the charged particles are identied by combining the PID information provided by ITS, TPC,TRD, TOF and HMPID. Statistically, the charged particles can be identied up to a few tens GeV/c usingthe relativistic rise of dE/dx in the TPC. Electrons above 1 GeV/c are identied by the TRD, and muonsare registered by the muon spectrometer.To achieve the benchmarks described above, the ALICE reconstruction has to cope with a fewstatistical problems. We will discuss some of them in this paper (the details can be found in Chapter 5 ofthe ALICE Physics Performance Report [2]).83Fig. 1: Schematic layout of the ALICE detector.2 Statistical problems with track finding in ITSThe track reconstruction in ALICE starts in the TPC, and then the tracks have to be prolonged in the ITS.This is difcult, because the distance between the inner wall of the TPC and the outer layer of the ITS israther large and the track density inside the ITS is so high that there are always many ITS clusters foundwithin the prolongation ‘window’ dened, mainly, by the multiple scattering in the material. The sameoften happens between the ITS layers as well (see Fig. 2). All this leads to a non-negligible probabilityof assigning to tracks many wrong clusters, if we use just the criterion of minimal χ2 at each layer.Therefore we have to nd the ways to improve the classical Kalman lter track-nding procedure [3].For each event, we do two reconstruction passes over the set of clusters in the ITS: rst, witha ‘primary vertex constraint’ (see below) and then without the constraint. In the both cases, we try toassign to a track, one by one, all the hits within the predicted window that have a χ2 below a given limit,and not only the one with minimal χ2. This way, for each track from TPC, we build a whole ‘tree’ ofall possible prolongations in ITS. To speed up building the tree, the branches are sorted after each layeraccording to χ2 and only a restricted number of acceptable branches are propagated further. Finally, wechoose the most probable track candidate (i.e. the path along the tree) taking into account the quality ofthe whole path (sum of χ2s at the layers, total number of assigned clusters and a few other criteria).Because most of tracks are expected to be primary, the rst reconstruction pass is done applying anad-hoc ‘primary vertex constraint’ (see Fig. 3). Since the primary vertex in ALICE can be reconstructedin advance sufciently well, the idea is to use this additional information during the track nding. Whengoing over the clusters within the ‘window’, we take into account not only the positions of clusters andthe track intersection point with the layer, but also the direction towards the primary vertex. Technically,this is done by extending the vector of measurement mmT = {y, z} → {y, z, sin(φ), tan(λ)},where {y, z} are the coordinates of the cluster position and the angles {φ, λ} dene the direction to theprimary vertex and are calculated using the current value of the track curvature. The elements of thecovariance matrix of the extended measurement vector that correspond to the two angles are evaluatedconsidering the material which this track would cross on its way to the primary vertex. The subsequentevaluation of the χ2 and update of the track parameters become thus 4-dimensional problem.I. BELIKOV84Fig. 2: The problem of track finding in ITS in high multiplicity events: Several clusters are found within theprolongation ‘window’ from one layer to another.Fig. 3: Taking into account the information about the primary vertex position by applying a ‘vertex constraint’(see the text).ALICE STATISTICAL WISH-LIST85Detailed Monte-Carlo studies performed with ALICE ofine simulation and reconstruction frame-work AliRoot [1] show that the outlined ad-hoc ‘vertex constraint’ signicantly reduces the probabilityof wrong cluster assignment, and so the quality of reconstructed tracks improves. Unfortunately, theprocedure is not free of aws. For example, for each of the tracks, it uses several times (even thoughwith different ‘weights’) the same information about the primary vertex position. This is done as manytimes per a track as there are detector layers. Consequently, one of the undesirable features is that theresulting covariance matrix of the track parameters becomes underestimated (which can be overcome byan additional retting step, however).In future, we would like to incorporate the vertex constraint into the Kalman-lter track ndingin a stricter way. A possible solution can probably be found by introducing the information about theprimary vertex position in the form of Bayesian priors.3 Statistical problems with particle identificationThe ALICE experiment is able to identify particles with momenta from 0.1 GeV/c and up-to a few tensGeV/c (statistically, on the relativistic rise of dE/dx in TPC). This can be achieved by combining severaldetecting systems that are efcient in some narrower and complementary momentum sub-ranges. Thesituation is complicated by the amount of data to be processed (about 107 events with about 104 tracksin each). Thus, the particle identication (PID) procedure should satisfy the following requirements:1. It should be as much as possible automatic.2. It should be able to combine PID signals of different nature (e.g. dE/dx and time-of-ight mea-surements).3. When several detectors contribute to the PID, as it is shown in Fig. 4, the procedure must protfrom this situation by providing an improved PID.4. When some of the detectors can not separate the particle species, the signals from the other detec-tors must not affect the combined PID.5. It should take into account the fact that, due to different event and track selection, the PID dependson the kind of analysis.The method described here is similar to that in Ref. [4]. Let r(s|i) be a conditional probabilitydensity function to observe in some detector a PID signal s if a particle of type i (i = e, µ, pi,K,p, ...) isdetected. The probability to be a particle of type i if the signal s is observed, w(i|s), depends not only onr(s|i), but also on how often this type of particles is registered in the experiment (a priori probabilitiesCi to nd a particle of i-type in the detector). The corresponding relation is given by Bayes’s formula:w(i|s) = r(s|i)Ci∑k=e,µ,pi,... r(s|k)Ck. (1)If Ci and r(s|i) are not strongly correlated we can rely on the following approximation: The functions r(s|i) reect only properties of the detector (‘detector response functions’) and donot depend on other external conditions like event and track selections. On the contrary, the quantities Ci (‘relative concentrations’ of particles of type i) do not dependon the detector properties, but reect the external conditions, selections etc.In the case of several detectors, the signal s is replaced by a vector of the PID measurements s¯ inthe detectors. The response function r(s|i) becomes some ‘combined response function’ R(s¯|i) of thewhole system of the detectors involved (in the simplest case, this is the product of the single-detectorPID response functions). The PID procedure is then done in steps:I. BELIKOV86 First, the detector response functions are obtained (theoretically, or in beam tests). This can bedone ‘once and forever’ before the reconstruction starts as a part of detector calibration. Second, for each track, a value R(s¯|i) is calculated using the PID signals measured for this track.This is done during the event reconstruction. Third, the relative concentrations of particle species Ci are estimated for the subset of events andtracks selected for a specic physics analysis. For obtaining better results, the particle concentra-tions Ci can be considered as functions of momentum. Finally, for each track within the selected subset, the array of probabilities w(i|s¯) is calculatedusing the formula (1). This steps, as well as the previous one, can be done only during the physicsanalysis of the data.Doing the particle identication in this way, we naturally satisfy all the requirements mentioned at thebeginning of this paragraph. However there are two problems which we are still working on.Fig. 4: The particle identification for the shown track is done by combining PID signals from five detectors: ITS,TPC, TRD, TOF and HMPID.Since the results of such a PID procedure depend explicitly on the choice of a priori probabilitiesCi (and, in fact, this kind of dependence is unavoidable in any approach), the question of stability ofthe results with respect to the choice of Ci becomes important. This problem seems to be related to the‘Punzi effect’ discussed in Ref. [5]. At lower momenta, there is always some momentum region wherethe single-detector response functions for different particle types of at least one of the detectors do notsignicantly overlap, and so the stability is guaranteed. The nal PID weights w(i|s¯) are dened bythe detector response functions. The more detectors enter the combined PID procedure, the wider thismomentum region becomes and the results are more stable. But, nally, as the momentum goes up, allthe detectors lose separation power, and the PID decision is given by the bare priors Ci (which we cannot in this case estimate independently). The question is: can we somehow quantify the ‘contribution ofpriors’ to the nal PID weights so that if they become dominant, relative to the ‘contribution of detectorresponses’, we know when to stop trying to identify particles of higher momenta?ALICE STATISTICAL WISH-LIST87The second problem is of a different nature. The formula (1) fundamentally assumes that allthe components of vector s¯ are the results of PID measurements done for the same particle. In otherwords, the procedure of assigning clusters to tracks has to be ideal, which is not the case in reality. Theconsequences are seen, for example, in Fig. 5, where the PID efciency and contamination in ALICETOF detector are shown. In spite of the fact that separation of the particle species by the time of ightmethod improves greatly with decreasing momentum (Fig. 5, upper pad), the actual situation with thePID becomes worse, especially for the particles below 0.5 GeV/c (Fig. 5, lower pad). This is becausethe low-momentum particles decay, suffer from scattering in material and so have a higher probabilityof being assigned to the wrong cluster in the TOF detector. This mismatching effect is not taken intoaccount by the formula (1), and so the combined PID result becomes biased at low momenta.The effect of mismatching can be corrected by excluding from the vector s¯ the components thatdeviate too much from ‘reasonable expectation’. This is possible, for example, in the case of the ALICETOF detector, because we calculate the expected time of ight during the track nding in ITS, TPC andTRD. However, in general case, we may not know what the ‘reasonable expectation’ is. Also, applyingsharp cuts in an otherwise smooth procedure based on formula (1) may cause additional difcultieswith nding the best values for the cuts. Thus, a better solution for the problem of dealing with themismatching is still to be found.p (GeV/c)0.5 1 1.5 2 2.5 3<beta>0.30.40.50.60.70.80.911.1Kaonsp (GeV/c)0.5 1 1.5 2 2.5 3Efficiency (Contamination)0.20.30.40.50.60.70.80.91Kaon identification: TOF only  PID efficiency  PID contanimationFig. 5: PID efficiency and contamination for charged kaon identification with ALICE TOF detector. The deterio-ration of PID at low momenta is the consequence of the mismatching effect (see the text).I. BELIKOV884 The wish-listThe statistical problems arising in event reconstruction in ALICE and discussed above represent theALICE reconstruction statistical wish-list. In short, we would like to nd better, theory supported, waysfor1. introducing constraints in the standard Kalman lter (needed for improving the track nding inhigh-multiplicity events in the ALICE ITS);2. quantifying the relative importance of prior information and the results of actual measurementswhen making Bayesian decision (needed to dene the highest momentum up-to which the Bayesianparticle identication still makes sense);3. taking into account mismeasurements in Bayesian combination of information (needed for im-proving the low-momentum particle identication in ALICE).References[1] F. Carminati et al. [ALICE Collaboration], J. Phys. G 30, (2004) 1517.[2] B. Alessandro et al. [ALICE Collaboration], J. Phys. G 32 (2006) 1295.[3] P. Billoir, Nucl. Instrum. Meth. A 225 (1984) 352.[4] M. Aguilar-Benitez et al., Z. Phys. C 50 (1991) 405.[5] P. Catastini and G. Punzi, PHYSTATO5: Statistical Problems in Particle Physics, Astrophysics andCosmology, Oxford, England, United Kingdom, 12-15 Sep 2005.ALICE STATISTICAL WISH-LIST89

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