Particle Flow Calorimetry and the PandoraPFA Algorithm
aa r X i v : . [ phy s i c s . i n s - d e t ] J u l Abstract
Key words: [] Preprint submitted to Nuclear Physics B July 21, 2009 r X i v : . [ phy s i c s . i n s - d e t ] J u l Abstract
Key words: [1]
Preprint submitted to Nuclear Physics B July 21, 2009 r X i v : . [ phy s i c s . i n s - d e t ] J u l CU-HEP-09 / Particle Flow Calorimetry and the PandoraPFA Algorithm
M. A. Thomson
Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom.
Abstract
The Particle Flow (PFlow) approach to calorimetry promises to deliver unprecedented jet energyresolution for experiments at future high energy colliders such as the proposed InternationalLinear Collider (ILC). This paper describes the PandoraPFA particle flow algorithm which is thenused to perform the first systematic study of the potential of high granularity PFlow calorimetry.For simulated events in the ILD detector concept, a jet energy resolution of σ E / E . . −
400 GeV jets. This result, which demonstrates that high granularity PFlowcalorimetry can meet the challenging ILC jet energy resolution goals, does not depend stronglyon the details of the Monte Carlo modelling of hadronic showers. The PandoraPFA algorithm isalso used to investigate the general features of a collider detector optimised for high granularityPFlow calorimetry. Finally, a first study of the potential of high granularity PFlow calorimetry ata multi-TeV lepton collider, such as CLIC, is presented.
Key words:
Particle Flow Calorimetry, Calorimetry, ILC
PACS: + c
1. Introduction
In recent years the concept of high granularity Particle Flow calorimetry [1] has been devel-oped in the context of the proposed International Linear Collider (ILC). Many of the interestingphysics processes at the ILC [2] will be characterised by multi-jet final states, often accompa-nied by charged leptons and / or missing transverse energy associated with neutrinos or the lightestsuper-symmetric particles. The reconstruction of the invariant masses of two or more jets willprovide a powerful tool for event reconstruction and event identification. Unlike at LEP, wherekinematic fitting [3] enabled precise invariant mass reconstruction, at the ILC di-jet mass recon-struction will rely on the jet energy resolution of the detector. The goal for jet energy resolutionat the ILC is that it is su ffi cient to cleanly separate W and Z hadronic decays. An invariant massresolution comparable to the gauge boson widths, i.e. σ m / m = . ≈ Γ W / m W ≈ Γ Z / m Z , leadsto an e ff ective 3 . σ separation of the W → q ′ q and Z → qq mass peaks, i.e. the optimal invariantmass cut corresponds to + . σ ( − . σ ) in the reconstructed W (Z) mass distributions. Email address: [email protected] (M. A. Thomson)
Preprint submitted to Nuclear Instruments and Methods July 21, 2009 n the traditional calorimetric approach, the jet energy is obtained from the sum of the ener-gies deposited in the electromagnetic and hadronic calorimeters (ECAL and HCAL). This typi-cally results in a jet energy resolution of the form σ E E = α √ E(GeV) ⊕ β. (1)The stochastic term, α , is usually greater than ∼
60 %. The constant term, β , which encom-passes a number of e ff ects, is typically a few per cent. For high energy jets there also will bea contribution from the non-containment of the hadronic showers. The stochastic term in thejet energy resolution results in a contribution to the di-jet mass resolution of σ m / m ≈ α/ p E j j ,where E j j is the energy of the di-jet system in GeV. At the ILC, operating at centre-of-massenergies √ s = . − . −
350 GeV. Hence to achieve the ILC goal of σ m / m = . .
30 % / √ E(GeV). This is unlikely to be achievable with a traditional approach tocalorimetry.
Measurements of jet fragmentation at LEP have provided detailed information on the particlecomposition of jets ( e.g. [4, 5]). On average, after the decay of short-lived particles, roughly 62%of the jet energy is carried by charged particles (mainly hadrons), around 27% by photons, about10% by long-lived neutral hadrons ( e.g. n , n and K L ), and around 1.5 % by neutrinos. Hence,approximately 72 % of the jet energy is measured in the HCAL and the jet energy resolution islimited by the relatively poor HCAL energy resolution, typically &
55 % / √ E(GeV). The LEPcollaborations, most notably ALEPH, and other collider experiments ( e.g.
H1, D0 and CMS)have obtained improved jet energy resolution using the Energy Flow [6] approach, wherebyenergy deposits in the calorimeters are removed according to the momentum of the chargedparticle tracks. Using this method, ALEPH achieved a jet energy resolution (for √ s = M Z )equivalent to σ E / E ≈
65 % / √ E (GeV) [6]. This is the best jet energy resolution of the four LEPexperiments, but is roughly a factor two worse than required for the ILC.It is widely believed that the most promising strategy for achieving the ILC jet energy goalis the Particle Flow (PFlow) approach to calorimetry. This extends the concept of Energy Flow toa highly granular detector. In contrast to a purely calorimetric measurement, PFlow calorimetryrequires the reconstruction of the four-vectors of all visible particles in an event. The recon-structed jet energy is the sum of the energies of the individual particles. The momenta of chargedparticles are measured in the tracking detectors, while the energy measurements for photonsand neutral hadrons are obtained from the calorimeters. In this manner, the HCAL is used tomeasure only ∼
10 % of the energy in the jet. If one were to assume calorimeter resolutionsof σ E / E = . / √ E(GeV) for photons and σ E / E = . √ E(GeV) for hadrons, a jet energyresolution of 0 . / √ E(GeV) would be obtained with the contributions from tracks, photons andneutral hadrons as given in Table 1. In practice, this level of performance can not be achievedas it is not possible to perfectly associate all energy deposits with the correct particles. For ex-ample, if the calorimeter hits from a photon are not resolved from a charged hadron shower,the photon energy is not accounted for. Similarly, if part of charged hadron shower is identifiedas a separate cluster, the energy is e ff ectively double-counted as it is already accounted for by The only alternative proposed to date is that of Dual Readout calorimetry as studied by the DREAM collaboration [7]. confusion rather than calorimetric performance is the limiting factorin PFlow calorimetry. Thus, the crucial aspect of PFlow calorimetry is the ability to correctlyassign calorimeter energy deposits to the correct reconstructed particles. This places stringentrequirements on the granularity of the ECAL and HCAL. From the point of view of event recon-struction, the sum of calorimeter energies is replaced by a complex pattern recognition problem,namely the Particle Flow reconstruction Algorithm (PFA). The jet energy resolution obtained isa combination of the intrinsic detector performance and the performance of the PFA software.
Component Detector Energy Fract. Energy Res. Jet Energy Res.
Charged Particles ( X ± ) Tracker ∼ . E j − E X ± < . × − E j Photons ( γ ) ECAL ∼ . E j . p E γ . p E j Neutral Hadrons ( h ) HCAL ∼ . E j . √ E h . p E j Table 1: Contributions from the di ff erent particle components to the jet-energy resolution (all energies in GeV). The tablelists the approximate fractions of charged particles, photons and neutral hadrons in a jet of energy, E j , and the assumedsingle particle energy resolution. The PandoraPFA algorithm was developed to study PFlow calorimetry at the ILC. PandoraPFAis a C ++ implementation of a PFA running in the MARLIN [8] reconstruction framework. It wasdeveloped and optimised using simulated physics events generated with the MOKKA [9] pro-gram, which provides a detailed Geant4 [10] simulation of potential detector concepts for theILC. In particular, PandoraPFA was developed using the MOKKA simulation of the LDC [11]detector concept and, more recently, the ILD [12] detector concept. The algorithm is designed tobe su ffi ciently flexible to allow studies of PFlow for di ff erent detector designs. Whilst a numberof PFAs [13, 14, 15] have been developed for the ILC, PandoraPFA is the most sophisticatedand best performing algorithm. In this paper PandoraPFA is described in detail. It is then usedto study the potential at a future high energy lepton collider of PFlow calorimetry with a highlygranular detector, in this case the ILD detector concept.
2. Overview of the ILD Detector Model
The ILD detector concept [12], shown in Figure 1, consists of a vertex detector, trackingdetectors, ECAL, HCAL and muon chambers. It represents a possible configuration of a de-tector suitable for PFlow calorimetry. Specifically, for the ECAL and HCAL the emphasis ison granularity, both longitudinal and transverse, rather than solely energy resolution. Suitablecandidate technologies are being studied by the CALICE (calorimetry for the ILC) collabora-tion [16]. Amongst these are the Silicon-Tungsten ECAL and Steel-Scintillator HCAL designsassumed for the baseline ILD detector simulation.Both the ECAL and HCAL are located inside a solenoid which is taken to produce the 3.5 Tmagnetic field. The main tracking detector is simulated as a time projection chamber (TPC) withan active gas volume of half-length 2.25 m and inner and outer radii of 0.39 m and 1.74 m re-spectively. The vertex detector consists of 6 layers of Silicon with an inner radius of 15 mm fromthe interaction point (IP). The tracking is complemented by two barrel Silicon strip detectorsbetween the vertex detector and the TPC and seven Silicon forward tracking disks. The ECAL issimulated as a Silicon-Tungsten sampling calorimeter consisting of 29 layers. The first 20 layers3ave 2.1 mm thick Tungsten and the last 9 layers have 4.2 mm thick Tungsten. The high resis-tivity Silicon is segmented into 5 × pixels. At normal incidence, the ECAL correspondsto 23 radiation lengths ( X ) and 0.8 nuclear interaction lengths ( λ I ). The HCAL is simulated asa Steel-Scintillator sampling calorimeter comprising 48 layers of 20 mm thick Steel and 5 mmthick 3 × plastic scintillator tiles. At normal incidence the HCAL is 6 λ I thick.The ECAL and HCAL in the ILD concept are well matched to the requirements of PFlowcalorimetry. Tungsten is the ideal absorber material for the ECAL; it has a short radiation lengthand small Moli`ere radius (see Table 2) which leads to compact electromagnetic (EM) showers. Italso has a large ratio of interaction length to radiation length which means that hadronic showerswill tend to be longitudinally well separated from EM showers. The 5 × transversesegmentation takes full advantage of the small Moli`ere radius. Steel is chosen as the HCALabsorber, primarily for its structural properties. The 3 × HCAL transverse segmentation isbelieved to be well matched to the requirements of PFlow calorimetry (see Section 9.5).
Material λ I / cm X / cm ρ M / cm λ I / X Fe 16.8 1.76 1.69 9.5Cu 15.1 1.43 1.52 10.6W 9.6 0.35 0.93 27.4Pb 17.1 0.56 1.00 30.5
Table 2: Comparison of interaction length, λ I , radiation length, X , and Moli`ere radius, ρ M , for Iron, Copper, Tungstenand Lead. Also given is the ratio of λ I / X .
3. Reconstruction Framework
The performance of PFlow calorimetry depends strongly on the reconstruction software. Forthe results obtained to be meaningful, it is essential that both the detector simulation and the re-construction chain are as realistic as possible. For this reason no Monte Carlo (MC) informationis used at any stage in the reconstruction as this is likely to lead to an overly-optimistic evaluationof the potential performance of PFlow calorimetry.PandoraPFA runs in the MARLIN [8] C ++ framework developed for the LDC and ILD de-tector concepts. The input to PandoraPFA (in LCIO [17] format) is a list of digitised hits inthe calorimeters and a list of reconstructed tracks. Tracks in the TPC are reconstructed us-ing a MARLIN processor, LEPTrackingProcessor, adapted from the TPC pattern recognitionsoftware [18] based on that used by ALEPH and track fitting software used by DELPHI [19].Reconstruction of tracks in the inner Silicon detectors is performed by a custom processor,SiliconTracking [21]. TPC and Silicon track segments are combined in a final tracking pro-cessor, FullLDCTracking [21].PandoraPFA combines the tracking information with hits in the high granularity calorimetersto reconstruct the individual particles in the event. As an example of the information used inthe reconstruction, Figure 2 shows a photon, a charged hadron ( π + ) and a neutral hadron ( K L ) assimulated in the ILD detector concept. 4 . The PandoraPFA Particle Flow Algorithm The PandoraPFA algorithm performs calorimeter clustering and PFlow reconstruction ineight main stages:
1) Track Selection / Topology: track topologies such as kinks and decays ofneutral particles in the detector volume ( e.g K S → π + π − ) are identified.
2) Calorimeter Hit Se-lection and Ordering: isolated hits, defined on the basis of proximity to other hits, are removedfrom the initial clustering stage. The remaining hits are ordered into pseudo-layers and informa-tion related to the geometry and the surrounding hits are stored for use in the reconstruction. the main clustering algorithm is a cone-based forward projective method [22] work-ing from innermost to outermost pseudo-layer. The algorithm starts by seeding clusters usingthe projections of reconstructed tracks onto the front face of the ECAL.
PandoraPFA can be run in a mode where the above clustering algorithm is performed intwo stages. In the first stage, only ECAL hits are considered with the aim of identifying energydeposits from photons. In the second stage the clustering algorithm is applied to the remaininghits.
4) Topological Cluster Merging: by design the initial clustering stage errs on the side ofsplitting up true clusters rather than merging energy deposits from more than one particle intoa single cluster. Clusters are then combined on the basis of clear topological signatures in thehigh granularity calorimeters. The topological cluster merging algorithms are only applied toclusters which have not been identified as photons.
5) Statistical Re-clustering:
The previousfour stages of the algorithm are found to perform well for jets with energies of less than 50 GeV.For higher energy jets the performance degrades due to the increasing overlap between hadronicshowers from di ff erent particles. Clusters which are likely to have been created from the mergingof hits in showers from more than one particle are identified on the basis of the compatibility ofthe cluster energy, E C , and the associated track momentum, p . In the case of an inconsistentenergy-momentum match, attempts are made to re-cluster the hits by re-applying the clusteringalgorithm with di ff erent parameters, until the cluster splits to give a cluster energy consistentwith the momentum of the associated track.
6) Photon Recovery and Identification:
A moresophisticated, shower-profile based, photon-identification algorithm is then applied to the clus-ters, improving the tagging of photons. It is also used to recover cases where a primary photon ismerged with a hadronic shower from a charged particle.
7) Fragment Removal: “neutral clusters”which are fragments of charged particle hadronic showers are identified.
8) Formation of Parti-cle Flow Objects:
The final stage of the algorithm is to create the list of reconstructed particles,Particle Flow Objects (PFOs), and associated four-momenta.The essential features of each of the above stages are described in more detail below. Thedescription includes the main configuration parameters which determine the behaviour of thealgorithms. These can be defined at runtime. The default values, which are optimised for theILD concept, are given. / topology Tracks are projected onto the front face of the ECAL using a helical fit to the last 50 hitson the reconstructed track (no account is taken for energy loss along the trajectory in the TPCgas). Tracks are then classified according to their likely origin. For example, neutral particledecays resulting in two charged particle tracks ( V s) are identified by searching for pairs of trackswhich are consistent with coming from a single point displaced from the IP. Charged particledecays to a single charged particle and any number of neutral particles (kinks) are identifiedon the basis of the distance of closest approach of the parent and daughter tracks. Similarly,interactions in the tracking volume (prongs) are identified. This information, along with the5riginal track parameters and the projection of the track onto the front face of the ECAL, isstored in ExtendedTrack objects for use in the subsequent event reconstruction.
In addition to the reconstructed tracks, the input to PandoraPFA is a list of digitised calorime-ter hits. For each hit, the position ( x , y , z ), the energy deposition, and the physical layer in theECAL / HCAL are specified. Based on this information,
ExtendedCaloHit objects are formed.These hits are self-describing and incorporate information relating to both the geometry of thedetector (accessed from the GEAR[23] geometry description) and information related to thedensity of calorimeter hits in the neighbouring region. The five main steps in the calorimeter hitprocessing (calibration, geometry, isolation, MIP identification, ordering) are described below.
The energy of each calorimeter hit is converted to a minimum ionising particle (MIP) equiva-lent (at normal incidence) using a calibration factor
CALMIPcalibration . Di ff erent calibrationfactors are used for ECAL and HCAL hits. Hits are only retained if they are above a MIP-equivalent threshold of CalMIPThreshold (with default values of [0.5] and [0.3] for ECALand HCAL respectively). The MIP equivalent energy deposit is then converted into calorimetricmeasurement using MIP to GeV calibration factors,
CalMIPToGeV , for the ECAL and HCAL.In general, the calorimeters will not be compensating, and separate energy measurements arecalculated for the hypotheses that the hit is either part of an EM or hadronic shower. The finalchoice of which energy to use depends on the whether the shower to which a hit is associatedis ultimately identified as being EM in nature. To allow for calorimeters with di ff erent absorberthicknesses as a function of depth, the calibration factor applied is proportional to the absorberthickness of the layer in front of the hit. Initial values for the calibration factors are determinedfrom MC samples of single muons, photons and K L s. The muon sample is used to determine theMIP calibration, the photon sample is used to determine the ECAL calibrations and the K L s areused to determine the initial HCAL calibration. Since the neutral hadrons in jets are a mixtureof K L s, neutrons and anti-neutrons, the initial HCAL calibration is modified (typically by ∼ The PandoraPFA reconstruction is designed to minimise the dependence on the detectorgeometry to enable comparisons of di ff erent detector designs. For this reason, information isadded to the digitised calorimeter hits such that they become self describing. For example, the ExtendedCaloHit objects store the size of the corresponding detector pixel. To reduce the de-pendency of the clustering algorithms on the detector geometry, hits are ordered in increasingdepth in the calorimeter. This is achieved by defining “pseudo-layers” which follow the gen-eral layer structure of the calorimeters. This is necessary for calorimeter layouts such as in theECAL stave-like structure being studied by the CALICE collaboration, shown schematically inFigure 3. Here there are regions where the first layer in a calorimeter stave can be deep in theoverall calorimeter structure. 6 .2.3. Isolation Requirements
Low energy neutrons produced in hadronic showers can travel a significant distance from thepoint of production and thus produce isolated energy deposits. For PFlow calorimetry, these en-ergy deposits are of little use as it is impossible to unambiguously associate them with a particularhadronic shower. For this reason, and to improve the performance of the clustering algorithms,isolated hits are identified and excluded from the initial cluster finding. Isolated hits are definedusing one of two possible criteria: i) less than a minimum number of calorimeter hits within apre-defined distance from the hit in question; or ii) a cut on the local weighted hit number density, ρ i , defined by: ρ i = X j w i j = X j r ⊥ i j ) n where r ⊥ i j = r i × ( r i − r j ) | r i | . Here r i is the position of the hit in question, the sum over j is for all hits within a certain numberof pseudo-layers of hit i , and the default value for n is 2. By default, method i) is used. Hits which are consistent with having originated from a minimum ionising particle (MIP) areflagged based on energy deposition and the surrounding hits in the same calorimeter layer. Fora hit to be tagged as MIP-like: a) the energy deposition must be no more than
MipLikeMipCut[5.0] times the mean expected MIP signal, and b) of the adjacent (usually 8) pixels in the samelayer, no more than
MipMaxCellsHit [1] should have hits above threshold. This informationis used in the identification of minimum ionising tracks within the calorimeter.
Prior to applying the clustering algorithm, hits within each pseudo-layer are ordered eitherby energy (the default) or by local hit density, ρ i , defined above. The latter option is intendedprimarily to be used for the case of digital calorimetry, where a simple hit count replaces theanalogue energy information. The main clustering algorithm of PandoraPFA is a cone-based forward projective methodworking from innermost to outermost pseudo-layer. In this manner hits are either added to exist-ing clusters or they are used to seed new clusters. Throughout the algorithm clusters are assigneda direction (or potentially directions) in which they are propagating. This allows the clusteringalgorithm to follow tracks in the calorimeters. The input to the clustering algorithm is a vector ofhits (
ExtendedCaloHit s) ordered by pseudo-layer and energy (or local hit density) and a vectorof tracks (
ExtendedTrack s).The algorithm starts by seeding clusters using the projections of reconstructed tracks onto thefront face of the ECAL. The initial direction of a track-seeded cluster is obtained from the trackdirection at the ECAL front face. The hits in each subsequent pseudo-layer are then looped over.Each hit, i , is compared to each clustered hit, j , in the previous layer. The vector displacement, r ij ,is used to calculate the parallel and perpendicular displacement of the hit with respect to the unitvector(s) ˆu describing the cluster propagation direction(s), d k = r ij . ˆu and d ⊥ = | r ij × ˆu | . Associ-ations are made using a cone-cut, d ⊥ < d k tan A + bD pad , where A is the cone half-angle, D pad is7he size of a sensor pixel in the layer being considered, and b is the number of pixels added to thecone radius. Di ff erent values of A and b are used for the ECAL and HCAL with the default valuesset to { tan A E = . , b E = . } and { tan A H = . , b H = . } respectively. The values can be mod-ified using the steering parameters ClusterFormationAngle and
ClusterFormationPads .For hits in layer k , associations are first searched for in layer k −
1. If no association is made,possible associations with clustered hits in layers k − k − SameLayerPadCut = [2.8] ([1.8]) for pixels in the ECAL (HCAL). If a hit remainsunassociated, it is used to seed a new cluster. Clusters seeded with calorimeter hits are assignedan initial direction corresponding to radial propagation from the IP. This procedure is repeatedsequentially for the hits in each pseudo-layer working outward from ECAL front-face.
Clusters which are consistent with being from EM showers from photons are identified. Forreasons of speed, simple cut based criteria are used. The fast photon identification requirements are: no associated track; the cluster must start within 10 X of the front face of the ECAL; thecluster direction (obtained from a linear fit to the energy-weighted centroids of the hits in eachpseudo-layer) must point to within 20 ◦ of the IP; the rms deviation of the hits in the cluster aroundthe linear fit to the centroids in each calorimeter layer must be less than 40 cm; and the fractionof hits classified as MIP-like must be less than 30 %. In addition, weak cuts on the longitudinaldevelopment of the shower are imposed. Photon clusters are essentially frozen at this stage inthe PandoraPFA algorithm; they are not used in the subsequent topological cluster merging orreclustering algorithms. Rather than attempting to cluster all calorimeter hits in a single pass, PandoraPFA can be runin a mode (
PhotonClustering >
0) where the clustering algorithm described above is first ap-plied solely to the ECAL hits to identify photons as the first stage of PFlow reconstruction. Theclustering algorithm parameters are chosen to reflect the narrowness of EM showers. Recon-structed clusters which are consistent with the expected EM transverse and longitudinal showerprofiles (see Section 4.6) are stored and the associated calorimeter hits are not considered in thesecond pass of the clustering algorithm. The identified photon clusters are added back to theevent just prior to the formation of the PFOs. For the results presented in this paper, photonclustering is run prior to the main clustering algorithm.
By design the initial clustering algorithm errs on the side of splitting up true clusters ratherthan merging energy deposits from more than one particle into a single cluster. Hence, the nextstage in the PandoraPFA algorithm is to merge clusters which are not already associated to tracks(termed “neutral clusters”) with clusters which have an associated track (termed “charged clus-ters”). The merging algorithms are based on the clear topological signatures shown schematicallyin Figure 4. The exact cut values depend on the cluster energy and the values below are those given in the text are the defaultvalues for a 10 GeV cluster. r + κ ˆd , is used to project forwards or backwards in the calorimeter. Similarly, theentire cluster may be classified as track-like. The main topological rules for cluster associationare:(i) Looping tracks: Because of the forward projective nature of the primary clustering algo-rithm, tracks which turn back in the calorimeter due to the high magnetic field are oftenreconstructed as two track-like clusters. The track-like segments at the ends of the clustersare projected forwards and the clusters are combined if the distance of closest approach ofthe two forward-going track projections is less than LooperClosestApproachCutECAL[5 cm] .(ii) Broken tracks: Non-continuous tracks in the calorimeters can arise when particles crossboundaries between physical sub-detectors or cross dead regions of the calorimeters. Suchinstances are identified using track-like segments in the last six layers of a charged clusterand the first six layers in a neutral cluster. The clusters may be merged if the distance ofclosest approach of the forward-going and backward-going track-like segment projectionsis less than
TrackMergeCutEcal [2.5 cm] .(iii) Tracks pointing to showers: If, when projected forward, a track-like charged cluster pointsto within
TrackMergeCutEcal [2.5 cm] of the start of a cluster deeper in the calorimeter,the clusters may be merged.(iv) Track-like clusters pointing back to hadronic interactions: If the start of a neutral cluster is atrack-like segment and it points to within
TrackBackMergeCut [3.0 cm] of the identifiedfirst hadronic interaction of charged cluster, the clusters may be merged.(v) Back-scattered tracks: Hadronic interactions can produce tracks in the calorimeter whichpropagate backwards in the calorimeter. Due to the forward projective nature of the cluster-ing algorithm, these often will be reconstructed as separate clusters. Back-scattered tracksare identified as track-like clusters which point to within
TrackBackMergeCut [3.0 cm] of the identified hadronic interaction of a charged cluster.(vi) Hadronic interactions pointing to neutral clusters: If a charged-cluster has track-like seg-ment prior to the identified interaction point, and it points to within
TrackForwardMergeCut[5.0 cm] of the start of a cluster deeper in the calorimeter, the clusters may be merged.(vii) Proximity-based merging: The minimum distance between a charged cluster, of energy E C ,and a neutral cluster, of energy E N , is defined as the smallest distance between any of thehits in the two clusters. If this distance is less than ProximityCutDistance [5 cm] thenthe clusters maybe merged if there is additional evidence that the two clusters originatefrom a single hadronic shower. To suppress false matches the χ consistency between theoriginal and merged cluster energies and the associated track momentum, p , is used. Themerged cluster energy, E ′ = E C + E N , must be consistent with the track momentum, χ ′ = ( E ′ − p ) /σ E ′ < EnergyChi2ForCrudeMerging [2.5] , where σ E ′ is the uncertainty on themerged cluster energy assuming that it is a hadronic shower. In addition, the χ consistencymust not be significantly worse than that for the original cluster, ∆ χ = ( χ ′ ) − χ < EnergyDeltaChi2ForCrudeMerging [1.0] , where χ = ( E C − p ) /σ E C .9viii) Cone-base merging: Starting from the identified hadronic interaction point of each chargedcluster, a cone of half-angle CosineConeAngle [0.9] is defined in the direction of thetrack-like segment of the cluster. Neutral clusters deeper in the calorimeter with morethan 50 % of the energy of lying within this cone may be merged providing the above χ consistency requirements are satisfied. If there is no track-like segment at the start of thecharged cluster, the track direction is used.(ix) Photon recovery: In dense jets minimum ionising particles may pass through the EMshower from a photon, resulting in a single reconstructed cluster. Cases where the hadroninteracts a significant distance after the end of the EM shower are identified and photonsoverlapping with charged clusters are recovered. The previous four stages of the PandoraPFA algorithm are found to perform well for jets withenergy less than about 50 GeV. At higher energies the jet energy resolution degrades due to theincreasing overlap between the hadronic showers from di ff erent particles. It is possible to detectsuch reconstruction failures by comparing the charged cluster energy, E C , with the momentumof the associated track, p . A possible reconstruction failure is identified if | ( E C − p ) /σ E C | > ChiToAttemptReclustering [3.0] . In this case the PandoraPFA algorithm attempts to finda more self-consistent clustering of the calorimeter hits. If, for example, a 10 GeV track isassociated with a 20 GeV calorimeter cluster, shown schematically in Figure 5a), a potentialreconstruction failure is identified. One possible approach would be to simply remove hits fromthe cluster until the cluster energy matched the track momentum. However, this does not use thefull information in the event. Instead, the clustering algorithm is modified iteratively with thehope that a more correct clustering of the hits will be found. This is implemented by passingthe hits in the cluster and the associated track(s) to the main clustering algorithm described inSections 4.3 and 4.4. The algorithm is applied repeatedly, using successively smaller values ofthe parameters A and b , with the aim of splitting the original cluster so that the track momentumand associated cluster energy are compatible, as indicated in Figure 5a). In principle, completelydi ff erent clustering algorithms could be tried. In cases where no significant improvement in the χ compatibility of the track and associated cluster is found, the original cluster is retained.In steps vii) and viii) of the topological clustering, described in Section 4.4, the case wheretoo little energy is associated with the track is addressed. However, in a dense jet environment,the neutral cluster which should be associated with a charged cluster may itself be merged withanother neutral cluster, as indicated in Figure 5b). In such cases the reclustering procedure acts onthe combination of hits in the charged cluster associated to the track and nearby neutral clusters. A relatively sophisticated photon identification algorithm is applied to the reconstructed clus-ters. The longitudinal profile of the energy deposition, ∆ E obs , as a function of number of radiationlengths from the shower start, t , is compared to that expected [24] for an EM shower: ∆ E EM ≈ E ( t / a − e − t / Γ ( a ) t , where a = . +
12 ln E E c , is the shower energy and E c is the critical energy, which is chosen to give the appropriateaverage MC shower profile in the ECAL. The level of agreement is parameterised by the sumover samplings in radiation length of the fractional deviation of the cluster profile compared theexpectation for an EM shower: δ = E X i | ∆ E iobs − ∆ E iEM | . This approach was preferred to a χ -based metric as it is less sensitive to large local deviationswhich might arise from energy deposits from other nearby particles. The quantity δ is minimisedas a function of the assumed starting point of the shower, t . Hence the output of the showershape algorithm is a measure of the consistency with the expected EM shower profile, δ , and thestarting depth of the shower in the ECAL, t (in radiation lengths). These variables are used asthe basis for identifying clusters as photons. Transverse information is not used as this wouldmake the photon identification algorithm more sensitive to over-lapping EM showers from veryclose photons. The compact nature of EM showers is utilised in an attempt to identify photons which mayhave been merged into the cluster associated with a hadronic shower. The transverse energydistribution (ECAL only) of the reconstructed clusters is determined assuming that the clusteroriginates from the IP. A peak finding algorithm attempts to identify localised energy depositionswhich are displaced from the associated track. If the longitudinal energy profile in these regionsis consistent with being an EM shower, the relevant hits are removed from the cluster and used toform a new cluster (assumed to be a photon). Cases where removing the candidate photon wouldresult in the remaining cluster energy being inconsistent with the associated track momentum arevetoed.
At this late stage in PandoraPFA there are still a significant number of “neutral clusters”(not identified as photons) which are fragments of charged particle hadronic showers. An at-tempt is made to identify these clusters and merge them with the appropriate parent chargedcluster. All non-photon neutral clusters, i , are compared to all charged clusters, j . For eachpair of clusters a quantity, e i j , is defined which encapsulates the evidence that cluster i is afragment from cluster j using the following information: the number of calorimeter layers inwhich the minimum distance between the hits in the two clusters are separated by less than FragmentRemovalContactCut [2] pixels; the fractions of the energy of cluster i within threenarrow cones defined by the first hadronic interaction in cluster j ; the minimum distance of thecentroid within a layer of cluster i to the fitted helix describing the track associated to cluster j ;and the minimum distance between any of the hits in the two clusters. The requirement, R i j , forthe clusters to be merged, i.e. the cut on e i j , depends on the location of the depth of the neutralcluster in the calorimeter and the change in the χ for the track − cluster energy consistency thatwould occur if the clusters were merged, ∆ χ = ( p − E j ) /σ E j − ( p − E j − E i ) /σ E ij . If e i j > R i j the clusters are merged. This ad hoc procedure gives extra weight to cases wherethe consistency of the track momentum and associated cluster energy improves as a result ofmerging the neutral cluster with the charged cluster.11 .8. Formation of Particle Flow Objects The final stage of PandoraPFA is to create Particle Flow Objects (PFOs) from the results ofthe clustering. Tracks are matched to clusters on the basis of the distance closest approach ofthe track projection into the first 10 layers of the calorimeter. If a hit is found within a distance
TrackClusterAssociationDistance [10 mm] of the track extrapolation, an association ismade. If an identified kink is consistent with being from a K ± → µ ± ν or π ± → µ ± ν decay theparent track is used to form the PFO, otherwise the daughter track is used. Relatively primitiveparticle identification is applied and the reconstructed PFOs, including four-momenta, are writtenout in LCIO [17] format. Figure 6a) shows an example of a PandoraPFA reconstruction of a100 GeV jet from a Z → uu decay at √ s =
200 GeV. The ability to track particles in the highgranularity calorimeter in the ILD detector concept can be seen clearly.
5. Parameterising Particle Flow Performance: rms Figure 7 shows the distribution of PFA reconstructed energy for simulated (Z /γ ) ∗ → qqevents (light quarks only, i.e. q = u,d,s) generated at √ s =
200 GeV with the Z decaying at rest,termed “Z → uds” events. A cut on the polar angle of the generated qq system, θ qq , is chosento avoid the barrel / endcap overlap region, | cos θ qq | < .
7. Only light quark decays are consid-ered as, currently, PandoraPFA does not include specific reconstruction algorithms to attempt torecover missing energy from semi-leptonic decays of heavy quarks. The reconstructed energydistribution of Figure 7 is not Gaussian. This is not surprising; one might expect a Gaussiancore for perfectly reconstructed events, and tails corresponding to the population of events whereconfusion is significant. Quoting the rms, in this case 5.8 GeV, as a measure of the jet energyresolution over-emphasises the importance of these tails. In this paper, performance is quoted interms of rms , which is defined as the rms in the smallest range of reconstructed energy whichcontains 90 % of the events. For the data shown in Figure 7, rms = . is that it is robust and isrelatively insensitive to the tails of the distribution; it parameterises the resolution for the bulk ofthe data. One possible criticism of this performance measure is that for a true Gaussian distribu-tion, rms would be 21 % smaller than the true rms. However, for the non-Gaussian distributionfrom PFlow reconstruction, this is not a fair comparison. For example, the central region of thereconstructed energy distribution is 15 % narrower than the equivalent Gaussian of σ = rms as shown in Figure 7. To determine the equivalent Gaussian statistical power, a MC study wasperformed assuming a signal with the shape of the PFA reconstructed energy distribution centredon x and a flat background. A fit to determine the value of x was performed using the shape of thePFA distribution as a resolution function (fitting template). The process was repeated assuminga signal with same number of events but now with a Gaussian energy distribution. The width ofthe Gaussian (for both the signal and the fitting function) was chosen to give the same statisticalprecision on x as obtained with the PFA resolution function. From a fit to signal and backgroundcomponents the same fitted uncertainty, σ x , is obtained for a Gaussian with standard deviationof 1 . × rms . On this basis it is concluded that the statistical power for PFlow reconstructionwith PandoraPFA yielding rms is equivalent to a Gaussian resolution with σ = . × rms .This conclusion does not depend strongly on the assumed relative normalisation of the signal andbackground or the total energy of the generated events. Here the best fit Gaussian to the region 196 −
205 GeV has an rms of 3.5 GeV . Particle Flow Performance The performance of the PandoraPFA algorithm with the ILD detector concept is studiedusing MC samples of approximately 10000 Z → uds generated with the Z decaying at restwith E Z = √ s = . − . E Z =
750 GeV and 1 TeV. Jet fragmentation and hadronisation wasperformed using the PYTHIA [25] program tuned to the fragmentation data from the OPALexperiment [26]. The events were passed through the MOKKA simulation of the ILD detectorconcept which is described in detail in [12]. The
LCPHYS [27] Geant4 physics list was used for themodelling of hadronic showers. For each set of events, the total energy is reconstructed and the jetenergy resolution is obtained by dividing the total energy resolution by √
2. Figure 8 shows thejet energy resolution as a function of the polar angle of the quarks in Z → qq events. The energyresolution does not vary significantly in the region | cos θ | < . . < | cos θ | < .
8. In addition,there is a small degradation in performance at cos θ ≈ Jet Energy rms rms ( E j j ) rms ( E j j ) / p E j j rms ( E j ) / E j
45 GeV 3.4 GeV 2.4 GeV 25.2 % (3 . ± .
05) %100 GeV 5.8 GeV 4.1 GeV 29.2 % (2 . ± .
04) %180 GeV 11.6 GeV 7.6 GeV 40.3 % (3 . ± .
04) %250 GeV 16.4 GeV 11.0 GeV 49.3 % (3 . ± .
05) %375 GeV 29.1 GeV 19.2 GeV 81.4 % (3 . ± .
05) %500 GeV 43.3 GeV 28.6 GeV 91.6 % (4 . ± .
07) %
Table 3: Jet energy resolution for Z → uds events with | cos θ qq | < .
7, expressed as: i) the rms of the reconstructed di-jetenergy distribution, E jj ; ii) rms for E jj ; iii) the e ff ective constant α in rms ( E jj ) / E jj = α ( E jj ) / p E jj (GeV); and iv)the fractional jet energy resolution for a single jet where rms ( E j ) = rms ( E jj ) / √ Table 3 summarises the current performance of the PandoraPFA algorithm applied to ILDdetector simulation. For the typical ILC jet energy range, 45 −
250 GeV, the energy resolution issignificantly better than the best resolution achieved at LEP, σ E / E ≈ . / √ E (GeV). Table 3also lists the single jet energy resolution. For jet energies in the range 45 −
375 GeV this isbetter than 3.8 %, which is necessary to resolve hadronic decays of W and Z bosons. These re-sults clearly demonstrate the potential of PFlow calorimetry at the ILC; the jet energy resolutionobtained is approximately a factor two better than might be achievable with a traditional calori-metric approach. Furthermore, it is expected that the performance of PandoraPFA will improvewith future refinements to the algorithm.It is worth noting, that for perfect PFlow reconstruction, the energy resolution would bedescribed by σ E / E ≈ α/ √ E(GeV), where α is a constant. The fact that this does not applyis not surprising; as the particle density increases it becomes harder to correctly associate thecalorimetric energy deposits to the particles and the confusion term increases. Also it should benoted that in a physics analysis involving multi-jet final states, the resolution may be degradedby imperfect jet finding. 13 . Understanding Particle Flow Performance PandoraPFA is a fairly complex algorithm, consisting of over 10,000 lines of C ++ . It has anumber of distinct stages which interact with each other in the sense that reconstruction failures inone part of the software can be corrected at a later stage. The relative importance of the di ff erentstages in the reconstruction is investigated by turning o ff parts of the PandoraPFA algorithm.Table 4 compares the full PandoraPFA reconstruction with the algorithm run: a) without thetopological cluster merging phase; b) without the reclustering phase; c) without running thephoton clustering stage prior to the running the full clustering; d) without fragment removal; ande) the case where tracks from V s and kinks are not used in the event reconstruction. There area number of notable features. The topological clustering and fragment removal algorithms areimportant at all energies. For low energy jets, the reclustering stage is not particularly important.This is because the primary clustering and topological clustering algorithms are su ffi cient inthe relatively low particle density environment. With increasing jet energy, the reclustering stagebecomes more important. For high energy jets ( E >
100 GeV) it is the single most important stepin the reconstruction after the initial clustering. Running the dedicated photon clustering stagebefore the main clustering algorithm is advantageous for higher energy jets. The V / kink findingdoes not significantly improve the resolution, although it is an important part in the identificationof the final reconstructed particles. Algorithm Jet Energy Resolution rms ( E j ) / E j [%] E j =
45 GeV E j =
100 GeV E j =
180 GeV E j =
250 GeVFull PandoraPFA 3 . ± .
05 2 . ± .
04 3 . ± .
04 3 . ± . . ± .
05 3 . ± .
04 3 . ± .
05 3 . ± . . ± .
05 3 . ± .
04 3 . ± .
05 4 . ± . . ± .
05 2 . ± .
04 3 . ± .
04 3 . ± . . ± .
05 3 . ± .
04 3 . ± .
04 3 . ± . V / Kink Tracks 3 . ± .
05 2 . ± .
04 3 . ± .
04 3 . ± . Table 4: Jet energy resolutions (rms / E ) for the full PandoraPFA reconstruction compared to that obtained: a) withoutthe topological cluster merging phase; b) without the reclustering phase; c) without running the photon clustering stageprior to the running the full clustering; d) without fragment removal; and e) the case where tracks from V s and kinksare not used in the event reconstruction. The contributions to the jet energy resolution have been estimated by replacing di ff erent stepsin PandoraPFA with algorithms which use MC information to perform: a) perfect reconstructionof photons as the first phase of the algorithm; b) perfect reconstruction of neutral hadrons; andc) perfect identification of fragments from charged hadrons. The jet energy resolutions obtainedusing these “perfect” algorithms enable the contributions from confusion to be estimated. In ad-dition, studies using a deep HCAL enable the contribution from leakage to be estimated. Finally,MC information can be used to perform ideal track pattern recognition enabling the impact ofimperfect track finding code to be assessed. Table 5 lists the estimated breakdown of the total jetenergy into its components, including the contributions from calorimetric energy resolution ( i.e. the energy resolution for photons and neutral hadrons). For the current PandoraPFA algorithm,the contribution from the calorimetric energy resolution, ≈
21 % / √ E , dominates the jet energyresolution for 45 GeV jets. For higher energy jets, the confusion term dominates. This behaviouris summarised in Figure 9. The contributions from resolution and confusion are roughly equal for1420 GeV jets. From Table 5 it can be seen that the most important contribution for high energyjets is confusion due to neutral hadrons being lost within charged hadron showers. For all jetenergies considered, fragments from charged hadrons, which tend to be relatively low in energy,do not contribute significantly to the jet energy resolution. Contribution Jet Energy Resolution rms ( E j ) / E j E j =
45 GeV E j =
100 GeV E j =
180 GeV E j =
250 GeVTotal 3.7 % 2.9 % 3.0 % 3.1 %Resolution 3.0 % 2.0 % 1.6 % 1.3 %Tracking 1.2 % 0.7 % 0.8 % 0.8 %Leakage 0.1 % 0.5 % 0.8 % 1.0 %Other 0.6 % 0.5 % 0.9 % 1.0 %Confusion 1.7 % 1.8 % 2.1 % 2.3 %i) Confusion (photons) 0.8 % 1.0 % 1.1 % 1.3 %ii) Confusion (neutral hadrons) 0.9 % 1.3 % 1.7 % 1.8 %iii) Confusion (charged hadrons) 1.2 % 0.7 % 0.5 % 0.2 %
Table 5: The PFlow jet energy resolution obtained with PandoraPFA broken down into contributions from: intrinsiccalorimeter resolution, imperfect tracking, leakage and confusion. The di ff erent confusion terms correspond to: i) hitsfrom photons which are lost in charged hadrons; ii) hits from neutral hadrons that are lost in charged hadron clusters; andiii) hits from charged hadrons that are reconstructed as a neutral hadron cluster. The numbers in Table 5 can be used to obtain an semi-empirical parameterisation of the jetenergy resolution: rms E = √ E ⊕ . ⊕ . E ⊕ . (cid:18) E (cid:19) . % , where E is the jet energy in GeV. The four terms in the expression respectively represent: theintrinsic calorimetric resolution; imperfect tracking; leakage and confusion. This functional formis shown in Figure 10. It is worth noting that the predicted jet energy resolutions for 375 GeVand 500 GeV jets are in good agreement with those found for MC events (see Table 3); these datawere not used in the determination of the parameterisation of the jet energy resolution.The ILC jet energy goal of σ E / E < . −
420 GeV. Figure 10 also shows a parameterisation of the jet energy resolution (rms ) obtainedfrom a simple sum of the total calorimetric energy deposited in the ILD detector concept. It isworth noting that even for the highest energies jets considered, PFlow reconstruction significantlyimproves the resolution. The performance of PFlow calorimetry is compared to 60 % / √ E (GeV) ⊕ . indication of the resolution which might be achieved using atraditional calorimetric approach. For a significant range of the jet energies relevant for the ILC,PFlow results in a jet energy resolution which is roughly a factor two better than the best at LEP.
8. Dependence on Hadron Shower Modelling
The results of the above studies rely on the accuracy of the MC simulation in describingEM and hadronic showers. The Geant4 MC provides a good description of EM showers as hasbeen demonstrated in a series of test-beam experiments [28] using a Silicon-Tungsten ECAL of15he type assumed for the ILD detector model. However, the uncertainties in the development ofhadronic showers are much larger [29]. There are a number of possible e ff ects which could a ff ectPFlow performance: the hadronic energy resolution; the transverse development of hadronicshowers which will a ff ect the performance for higher energy jets where confusion is important;and the longitudinal development of the shower which will a ff ect both the separation of hadronicand EM showers and the amount of leakage through the rear of the HCAL.To assess the sensitivity of PFlow reconstruction to hadronic shower modelling uncertainties,five Geant4 physics lists are compared: • QGSP BERT , Quark-Gluon String model[30] with the addition of the Precompound modelof nuclear evaporation[31] (QGSP) for high energy interactions, and the Bertini (BERT)cascade model[32] for intermediate energy interactions; • QGS BIC , Quark-Gluon String (QGS) for high energy interactions and the Binary cascade(BIC) model[33] for intermediate and low energies; • FTFP BERT , the Fritiof (FTF) string-based model[34] with Precompound[31] for high en-ergy interactions and the Bertini cascade model for intermediate energies; • LHEP , based on the Low and High Energy Parameterised modes (LEP and HEP) of theGHEISHA package[35] used in Geant3; • LCPhys [27], which uses a combination of the QGSP, LEP and BERT models.These physics lists represent a wide range of models and result in significantly di ff erent predic-tions for total energy deposition, and the longitudinal and transverse shower profiles. For eachPhysics list, the calibration constants in PandoraPFA are re-tuned, but no attempt to re-optimisethe algorithm is made. The jet energy resolutions obtained are given in Table 6. Whilst non-statistical di ff erences are seen, the rms variations are relatively small, less than 4.2 %. Whilstthis might seem surprising, it should be noted that the e ff ect on the jet energy resolution of thehadronic modelling is likely to be predominantly from the neutral hadron confusion term. Thistends to dilute the sensitivity to the modelling of hadronic showers. For example, from Table 4 itcan be seen that if the neutral hadron confusion term for 250 GeV jets is increased by 25 %, whenadded in quadrature to the other terms, the overall jet energy resolution would only increase by10 %. Physics List Jet Energy Resolution r = rms ( E j ) / E j
45 GeV 100 GeV 180 GeV 250 GeV
LCPhys (3 . ± .
05) % (2 . ± .
04) % (3 . ± .
04) % (3 . ± .
05) %
QGSP BERT (3 . ± .
06) % (2 . ± .
06) % (2 . ± .
06) % (3 . ± .
07) %
QGS BIC (3 . ± .
06) % (2 . ± .
05) % (3 . ± .
07) % (3 . ± .
07) %
FTFP BERT (3 . ± .
08) % (3 . ± .
06) % (3 . ± .
06) % (3 . ± .
08) %
LHEP (3 . ± .
07) % (3 . ± .
06) % (3 . ± .
06) % (3 . ± .
06) % χ (4 d.o.f) 23.3 17.8 16.0 6.3rms / mean ( σ r / r ) 4.2 % 3.9 % 3.5 % 2.5 % Table 6: Comparison of the jet energy resolution obtained using di ff erent hadronic shower physics lists. The χ consis-tency of the di ff erent models for each jet energy are given as are the rms variations between the five models. −
250 GeV jets, the jet energy resolutionobtained from PFlow calorimetry as implemented in PandoraPFA does not depend strongly onthe hadronic shower model; the observed di ff erences are less than 5 %. This is an importantstatement; it argues strongly against the need for a test beam based demonstration of PFlowcalorimetry (the design of such an experiment would be challenging). From test beam data theperformance of the ECAL and HCAL systems can be demonstrated using single particles andthe MC can be validated. Once the single particle performance is demonstrated, the uncertaintiesin extrapolating to the full PFlow performance for jets, which arise from the detailed modellingof hadronic showers, are likely to be less than 5 %.
9. Detector Design for Particle Flow Calorimetry
PFlow calorimetry requires the full reconstruction of the individual particles from the in-teraction. The optimisation of a detector designed for PFlow calorimetry extends beyond thecalorimeters as tracking information plays a major role. This section presents a study of thegeneral features of a detector designed for high granularity PFlow reconstruction.
PFlow calorimetry requires the e ffi cient separation of showers from charged hadrons, pho-tons and neutral hadrons. This implies high granularity calorimeters with both the ECAL andHCAL inside the detector solenoid. For high energy jets, failures in the ability to e ffi ciently sep-arate energy deposits from di ff erent particles, the confusion term, will dominate the jet energyresolution. The physical separation of calorimetric energy deposits from di ff erent particles willbe greater in a large detector, scaling as the inner radius of the ECAL, R , in the barrel region andthe detector length, L , in the endcap region. There are also arguments favouring a high magneticfield, as this will tend to deflect charged particles away from the core of a jet. The scaling lawhere is less clear. The separation between a charged particle and an initially collinear neutral par-ticle will scale as BR . However, there is no reason to believe that this will hold on average fora jet of non-collinear neutral and charged particles. The true dependence of PFlow performanceon the global detector parameters, B and R has to be evaluated empirically. The dependence of the PFlow performance on the main detector parameters has been inves-tigated using PandoraPFA. The studies are based on full reconstruction of the tracking and thecalorimetric information. The results presented here use the Geant4 simulation of the LDC de-tector concept [11] which, from the point of view of PFlow, is essentially the same as the ILDdetector concept described in Section 2. The starting point for the optimisation studies is the LD-CPrime detector model with a 3.5 T magnetic field, an ECAL inner radius of 1820 mm and a 48layer (6 λ I ) HCAL. The ECAL and HCAL transverse segmentations are 5 × and 3 × respectively. The jet resolution is investigated as a function of a number of parameters. For good PFlow performance both the ECAL and HCAL need to be within the detectorsolenoid. Consequently, in addition to the cost of the HCAL, the HCAL thickness impacts thecost of the overall detector through the radius of the superconducting solenoid. The thickness of17he HCAL determines the average fraction of the jet energy that is contained within the calorime-ter system. The impact of the HCAL thickness on PFlow performance is assessed by changingthe number of HCAL layers in the LDCPrime model from 32 to 63. This corresponds to avariation of 4 . − . λ I in the HCAL (4 . − . λ I in the ECAL + HCAL combined).The study of the optimal HCAL thickness depends on the possible use of the instrumentedreturn yoke (the muon system) to correct for leakage of high energy showers out of the rear ofthe HCAL. The e ff ectiveness of this approach is limited by the fact that, for much of the polarangle, the muon system is behind the relatively thick solenoid (2 λ I in the MOKKA simulationof the detector). Nevertheless, to assess the possible impact of using the muon detector as a“tail-catcher”, the energy depositions in the muon detectors were included in the PandoraPFAreconstruction. Whilst the treatment could be improved upon, it provides an indication of howmuch of the degradation in jet energy resolution due to leakage can be recovered in this way.The results are summarised in Figure 11 which shows the jet energy resolution obtained fromPandoraPFA as a function of the HCAL thickness. The e ff ect of leakage is clearly visible, withabout half of the degradation in resolution being recovered when including the muon detectorinformation. For jet energies of 100 GeV or less, leakage is not a major contributor to the jetenergy resolution provided the HCAL is approximately 4 . λ I thick (38 layers). However, for180 −
250 GeV jets this is not su ffi cient; for leakage not to contribute significantly to the jetenergy resolution at √ s = . − . λ I for an ILC detector. The LDCPrime model assumes a magnetic field of 3.5 T and an ECAL inner radius of1820 mm. A number of variations on these parameters were studied: i) variations in the ECALinner radius from 1280 − B = . B from 2 . − . R = B and R . In total thirteen sets of parameters were con-sidered spanning a wide range of B and R . The parameters include those considered by the LDC,GLD [36], and SiD [37] detector concept groups for the ILC. In each case PFlow performancewas evaluated for 45, 100, 180, and 500 GeV jets.Figure 12 shows the dependence of the jet energy resolution as a function of: a) magnetic field(fixed R ) and b) ECAL inner radius (fixed B ). For 45 GeV jets, the dependence of the jet energyresolution on B and R is rather weak because, for these energies, it is the intrinsic calorimetricenergy resolution rather than the confusion term that dominates. For higher energy jets, where theconfusion term dominates the resolution, the jet energy resolution shows a stronger dependenceon R than B .The jet energy resolutions are reasonably well described by the function:rms E = √ E ⊕ . ⊕ . E ⊕ . (cid:18) R (cid:19) − . (cid:18) B . (cid:19) − . (cid:18) E (cid:19) . % , where E is measured in GeV, B in Tesla, and R in mm. This is the quadrature sum of four terms: i)the estimated contribution to the jet energy resolution from the intrinsic calorimetric resolution;ii) the contribution from track reconstruction; iii) the contribution from leakage; and iv) thecontribution from the confusion term obtained empirically from a fit to the data of Figure 12 andseveral models where both B and R are varied [12]. In fitting the confusion term, a power-law18orm, κ B α R β E γ , is assumed. This functional form provides a reasonable parameterisation of thedata; the majority of the data points lie within 2 σ of the parameterisation.From the perspective of the optimisation of a detector for PFlow, these studies show that forthe PandoraPFA algorithm, the confusion term scales as approximately B . R , i.e. for good PFlowperformance a large detector radius is significantly more important than a very high magneticfield. The dependence of PFlow performance on the transverse segmentation of the ECAL wasstudied using modified versions of the LDCPrime model. The jet energy resolution is determinedfor di ff erent ECAL Silicon pixel sizes; 5 × , 10 ×
10 mm , 20 ×
20 mm , and 30 ×
30 mm .The two main clustering parameters in the PandoraPFA algorithm were re-optimised for eachECAL granularity. The PFlow performance results are summarised in Figure 13a. For 45 GeVjets, the dependence is relatively weak since the confusion term is not the dominant contributionto the resolution. For higher energy jets, a significant degradation in performance is observedwith increasing pixel size. Within the context of the current reconstruction, the ECAL transversesegmentations have to be at least as fine as 10 ×
10 mm to meet the ILC jet energy requirementof σ E / E < . √ s = × being preferred.A similar study was performed for the HCAL. The jet energy resolution obtained fromPandoraPFA was investigated for HCAL scintillator tile sizes of 1 × , 3 × , 5 × and 10 ×
10 cm . The PFlow performance results are summarised in Figure 13b. From thisstudy, it is concluded that the ILC jet energy resolution goals can be achieved an HCAL trans-verse segmentation of 5 × . For higher energy jets going to 3 × leads to a significantimprovement in resolution. From this study there appears to be no significant motivation for1 × granularity over 3 × . The results quoted here are for an analogue scintillator tilecalorimeter. The conclusions for a digital, e.g. RPC-based, HCAL might be di ff erent. Based on the above studies, the general features of a detector designed for high granularityPFlow calorimetry are: • ECAL and HCAL should be inside the solenoid. • The detector radius should be as large as possible, the confusion term scales approximatelywith the ECAL inner radius as R − . • To fully exploit the potential of PFlow calorimetry the ECAL transverse segmentationshould be at least as fine as 5 × . • For the HCAL longitudinal segmentation considered here, there is little advantage in trans-verse segmentation finer than 3 × . • The argument for a very high magnetic field is relatively weak as the confusion term scalesas B − . .These studies, based on the PandoraPFA algorithm, motivated the design of the ILD detectorconcept for the ILC as is discussed in more detail in Chapter 2 of [12].19
0. Particle Flow for Multi-TeV Colliders
In this section the potential of PFlow Calorimetry at a multi-TeV e + e − collider, such asCLIC [38], is considered. Before the results from the LHC are known it is di ffi cult to fully definethe jet energy requirements for a CLIC detector. However, if CLIC is built, it is likely that theconstruction will be phased with initial operation at ILC-like energies followed by high energyoperation at √ s ∼ a priori clear that PFlow calorimetry is suitable for higher energies. This questions needs to be consid-ered in the context of the possible physics measurements where jet energy resolution is likely tobe important at √ s ∼ + e − → qqevents is unlikely to be interest. Assuming the main physics processes of interest consist of finalstates with between six and eight fermions, the likely relevant jet energies will be in the range375 −
500 GeV. To study the potential of the PFlow calorimetry for these jet energies the ILDconcept, which is optimised for ILC energies, was modified; the HCAL thickness was increasedfrom 6 λ I to 8 λ I and the magnetic field was increased from 3.5 T to 4.0 T. The jet energy reso-lution obtained for jets from Z → uu , dd , ss decays at rest are listed in Table 7. For high energyjets, the e ff ect of the increased HCAL thickness (the dominant e ff ect) and increased magneticfield is significant. Despite the increased particle densities, the jet energy resolution (rms ) for500 GeV jets obtained from PFlow is 3.5 %. This is equivalent to 78 % / √ E (GeV). This is likelyto be at least competitive with a traditional calorimetric approach, particularly when the constantterm in Equation 1 and the contribution from non-containment are accounted for. Furthermore,it should be remembered that PandoraPFA has not been optimised for such high energy jets andimprovements can be expected. It is also worth noting that the purely calorimetric energy reso-lution (rms ) for 500 GeV jets with the modified ILD concept is equivalent to 115 % / √ E (GeV)and, thus, the gain from PFlow reconstruction is still significant. Jet Energy rms ( E j j ) / p E j j rms ( E j ) / E j λ I λ I λ I λ I
45 GeV 25.2 % 25.2 % (3 . ± .
05) % (3 . ± .
05) %100 GeV 29.2 % 28.7 % (2 . ± .
04) % (2 . ± .
04) %180 GeV 40.3 % 37.5 % (3 . ± .
04) % (2 . ± .
04) %250 GeV 49.3 % 44.7 % (3 . ± .
05) % (2 . ± .
05) %375 GeV 81.4 % 71.7 % (3 . ± .
05) % (3 . ± .
05) %500 GeV 91.6 % 78.0 % (4 . ± .
07) % (3 . ± .
07) %
Table 7: Comparisons of jet energy resolutions for two sets of detector parameters. This jet energy resolution shownis for (Z /γ ) ∗ → uds events with | cos θ qq | < .
7. It is expressed as: i) the e ff ective constant α in rms ( E jj ) / E jj = α ( E jj ) / p E jj (GeV), where E jj is the total reconstructed energy; and ii) the fractional jet energy resolution for a singlejet where rms ( E j ) = rms ( E jj ) / √ A requirement for a detector at a future linear collider is the ability to separate hadronic Wand Z decays. It was on this basis that the ILC jet energy resolution goal of σ E / E . . / Z decays will not be at rest and the di-jet system will be boosted. At a multi-TeVlepton collider the boost may be significant as the energies of the gauge bosons are potentially inthe range 500 GeV − ff ects associated withhighly boosted jets: • The jet particle multiplicities are lower than those for jets of the same energy producedfrom decays at rest. This increases the average energy of the particles in the jet and,consequently, will result in less containment of the hadronic showers (greater leakage); • The energies of the jets in the di-jet system will, in general, not be equal. Where one ofthe jets is much higher in energy than the other PFlow performance will tend to degrade. • The high jet boost decreases the average separation of the particles in the jet. This willtend to increase the confusion term. • The two jets from the decay of a highly boosted gauge boson will tend to overlap to forma “mono-jet”, as shown in Figure 14. The overlapping of jets has the potential to increasethe confusion term.Due to the likely increased confusion term, reconstructing the invariant mass of high energygauge bosons presents a challenge for PFlow calorimetry. However, it should be noted that itmay be even more challenging for a traditional calorimetric approach as it is now necessary toreconstruct the invariant mass of a single system of nearby particles which will not be well-resolved in the calorimeters.The PFlow reconstruction of boosted gauge bosons has been investigated by generating MCsamples of ZZ → dd νν and W + W − → ud µ − ν µ events at √ s = ff ectsdescribed above. However, the mass resolution (rms ) of 2.8 GeV obtained from decays ofgauge bosons with E =
125 GeV is compatible with that expected from the jet energy resolutionof Table 7 after accounting for the gauge boson width.For the ILC operating at √ s = . − . E W / Z = −
250 GeV. Here the reconstructed W andZ mass peaks are well resolved. The statistical separation, which is quantified in Table 8, isapproximately 2 . σ , i.e. the separation between the two peaks is approximately 2.5 times greaterthe e ff ective mass resolution.For CLIC operating at √ s = . − . . σ ) betweenthe W and Z peaks. Even for 1 TeV W / Z decays, where the events mostly appear as a singleenergetic mono-jet, the mass resolution achieved by the current version of PandoraPFA allowsseparation between W and Z decays at the 1 . σ level. It should be remembered that PandoraPFA21 W / Z rms ( m ) σ m / m W / Z sep ǫ
125 GeV 2.8 GeV 2.9 % 2 . σ
91 %250 GeV 3.0 GeV 3.5 % 2 . σ
89 %500 GeV 3.9 GeV 5.1 % 2 . σ
84 %1000 GeV 6.4 GeV 7.0 % 1 . σ
78 %
Table 8: Invariant mass resolutions for the hadronic system in simulated ZZ → dd νν and W + W − → ud µ − ν µ events inthe ILD detector concept. The W / Z separation numbers, which take into account the tails, are defined such that a 2 σ separation means that the optimal cut in the invariant mass distribution results in 15.8 % of events being mis-identified.The equivalent W / Z identification e ffi ciencies, ǫ , are given in the final column. Even with infinitely good mass resolution,the best that can be achieved is 94 % due to the tails of the Breit-Wigner distribution and, thus, the possible range for ǫ is50 −
94 %; has not been optimised for such high energy jets, and these results represent a lower bound onwhat can be achieved. From this result it is concluded that PFlow calorimetry is certainly notruled out for a multi-TeV lepton collider.
11. Conclusions
A sophisticated particle flow reconstruction algorithm, PandoraPFA, has been developed tostudy the potential of high granularity Particle Flow calorimetry at a future linear collider. Thealgorithm incorporates a number of techniques, e.g. topological clustering and statistical reclus-tering, which take advantage of the highly segmented calorimeters being considered for the ILCand beyond.PandoraPFA has been applied to the reconstruction of simulated events in the ILD detectorconcept for the ILC. The results presented in this paper provide the first conclusive demonstrationthat Particle Flow Calorimetry can meet the ILC requirements for jet energy resolution. For jetsin the energy range 40 −
400 GeV, the jet energy resolution, σ E / E , is better than 3.8 %. For the jetenergies relevant at the ILC, the jet energy resolution is approximately a factor of two better thanthe best achieved at LEP. The conclusions do not depend strongly on the details of the modellingof hadronic showers.PandoraPFA has been used to investigate the factors limiting the performance of Particle Flowcalorimetry. For jet energies below approximately 100 GeV, the intrinsic calorimetric resolutiondominates the jet energy resolution. For higher energy jets, the confusion term ( i.e. imperfectreconstruction) dominates. The largest single contribution to the confusion term arises from themis-assignment of energy from neutral hadrons.PandoraPFA has been used to study design of a detector optimised for high granularity Parti-cle Flow calorimetry demonstrating the importance of high transverse segmentation in the elec-tromagnetic and hadron calorimeters. The confusion term, which dominates the jet energy reso-lution for high energy jets, scales as approximately B − . R − , where B is the solenoidal magneticfield strength and R is the inner radius of the electromagnetic calorimeter.In addition, PandoraPFA has been used to perform a preliminary study of the potential ofParticle Flow calorimetry at a multi-TeV collider such as CLIC. For decays at rest, a jet energyresolution below 3 . / Z energiesof up to approximately 1 TeV. 22n conclusion, the studies described in this paper provide the first proof of principle of ParticleFlow calorimetry at a future lepton collider. For ILC energies, √ s = . − .
12. Acknowledgements
I would like to acknowledge: the UK Science and Technology Facilities Council (STFC) forthe continued support of this work; my colleagues on the ILD detector concept for providingthe high quality simulation and software frameworks used for these studies; Vasily Morgunovfor many interesting and useful discussions on Particle Flow reconstruction; and David Ward forreading the near final draft of this paper.
References [1] J.-C. Brient and H. Videau, “The calorimetry at a future e + e − linear collider”, arXiv:hep-ex / et al. , ILC-REPORT-2007-001 (2007).[3] M. A. Thomson, Proc. of EPS-HEP 2003, Aachen. Topical Vol. of Eur. Phys. J. C Direct (2004).[4] I. G. Knowles and G. D. La ff erty, J. Phys. G23 (1997) 731.[5] M. G. Green, S. L. Lloyd, P. N. Rato ff and D. R. Ward, “Electron-Positron Physics at the Z”, IoP Publishing (1998).[6] ALEPH Collaboration, D. Buskulic et al., Nucl. Instr. and Meth. A360 (1995) 481.[7] N. Akchurin, et al. , Nucl. Instr. and Meth.
A537 (2005) 29.[8] F. Gaede, Nucl. Instr. and Meth.
A559 (2006) 177.[9] P. de Freitas, et al. , http: // polzope.in2p3.fr:8081 / MOKKA.[10] GEANT4 collaboration, S. Agostinelli et al. , Nucl. Instr. and Meth.
A506 (2003) 3;GEANT4 collaboration, J. Allison et al. , IEEE Trans. Nucl. Sci. 53 (2006) 1.[11] Detector Outline Document for a Large Detector Concept, D. Kisielewska et al. , (2006).http: // / documents / dod / [12] ILD Letter of Intent, http: // / [13] V. L. Mogunov, “Energy-flow Method for Multi-jet E ff ective Mass Reconstruction in the Highly Granular TESLACalorimeter”, proceedings of the Snowmass Summer Study on the Future of Particle Physics, Snowmass, U.S.A.(2001).[14] A. Raspereza, arXiv:physics / et al. , “CALICE Report to the R&D Review Panel”, ILC-DET-2007-024 (2007); arXiv:0707.1245.[17] F. Gaede, T. Behnke, N. Graf, and T. Johnson, “LCIO: A persistency framework for linear collider simulationstudies”, Proceedings of CHEP 03, La Jolla, California, March 2003;arXiv-physics:0306114 (2003),[18] T. Behnke, et al. , “Track Reconstruction for a Detector at TESLA”, LC-DET-2001-029 (2001), and referencestherein.[19] DELPHI Collaboration, DELPHI Data Analysis Program (DELANA) User’s Guide, DELPHI 89-44 PROG 137(1989).[20] TESLA Technical Design Report, DESY 2001-011, ECFA 2001-2009 (2001).[21] A. Raspiereza, “LDC Tracking Package”, Proceedings of LCWS2007, DESY, Hamburg, June 2007.[22] M. A. Thomson, “Progress with Particle Flow Calorimetry”, Proceedings of LCWS2007, DESY, Hamburg, June2007.arXiv:0709.1360.[23] ”GEAR - a geometry description toolkit for ILC reconstruction software”,http: // ilcsoft.desy.de / portal / software packages / gear”.[24] Particle Data Group, Phys. Lett. B667 (2008) 1.[25] T. Sj¨ostrand, Comp. Phys. Comm. (2001) 238.[26] OPAL Collaboration, G. Alexander et al. , Z. Phys.
C69 (1996) 543.[27] “Linear Collider Physics List”,http: // / comp / physics / geant / slac physics lists / ilc / ilc physics list.html
28] CALICE Collaboration, C. Adlo ff et al. , accepted for publication by NIMA (2009); arXiv:0811.2354.[29] R. Wigmans, Proc. of HSW06, Fermilab, 2006, AIP Conf. Proc. 896 (2007) 123.[30] G. Folger and J.-P. Wellisch, “String Parton Models in Geant4”, CHEP03, La Jolla, California, March 2003, nucl-th / et al. , Phys. Rev. Lett. (1991) 1523;L. V. Bravina et al. , Phys. Rev. Lett. (1995) 49.[31] Geant4 Physics Reference Manual, Section IV, Chapter 28.[32] M. P. Guthrie, R. G. Alsmiller and H. W. Bertini, Nucl. Instr. and Meth. A66 (1968) 29.[33] G. Folger, V. N. Ivanchenko and J.-P. Wellisch, Eur. Phys. J.
A21 (2004) 407.[34] B. Anderson, G. Gustafson and B. Nielsson-Almqvist, Nucl. Phys. (1987) 289.[35] H. Fesefeld, “Simulation of hadronic showers, physics and applications”, Technical Report PITHA 85-02, Aachen,Germany, September 1985.[36] GLD Concept Study, arXiv:physics / // siliconddetector.org / SiD / LOI.[38] G. Guignard (ed.), et al. , “A 3 TeV Linear Collider Based on CLIC Technology”, CERN-2000-008 (2000). Figure 1: A quadrant of the ILD detector concept showing the main dimensions and layout of the sub-detector compo-nents.
800 -600 -400 -200 0 200 400 600 800120014001600180020002200240026002800300032003400 γ TPCECALHCAL a) -800 -600 -400 -200 0 200 400 600 800120014001600180020002200240026002800300032003400 + π b) -800 -600 -400 -200 0 200 400 600 800120014001600180020002200240026002800300032003400 K c) Figure 2: Example simulated single particle interactions in the ILD detector concept: a) a 10 GeV photon; b) a 10 GeV π + and c) a 10 GeV K L . Hits in the TPC, ECAL and HCAL are shown. For the ECAL (HCAL) all hits with energydepositions > . .
3) minimum ionising particle equivalent are displayed. Simulated TPC hits are digitised assuming227 radial rows of readout pads. a)y x y z b)IP
Figure 3: Schematic showing the definition of the pseudo-layer assignment for calorimeter hits. The solid lines indicatethe positions of the physical ECAL layers and the dashed lines show the definition of the virtual pseudo-layers. a) The xy -view showing the CALICE stave structure for the ECAL. Here hits in the first layer of the stave can be deep in theoverall calorimeter. b) The xz -view showing a possible layout for the ECAL barrel / endcap overlap region. Here thepseudo-layers are defined using the projection back to the IP such that the pseudo-layer is closely related to the depth inthe calorimeter. igure 4: The main topological rules for cluster merging: i) looping track segments; ii) track segments with gaps; iii)track segments pointing to hadronic showers; iv) track-like neutral clusters pointing back to a hadronic shower; v) back-scattered tracks from hadronic showers; vi) neutral clusters which are close to a charged cluster; vii) a neutral cluster nearto a charged cluster; viii) cone association; and ix) recovery of photons which overlap with a track segment. In each casethe arrow indicates the track, the filled points represent the hits in the associated cluster and the open points represent thehits in the neutral cluster. igure 5: Schematic examples of the main reclustering strategies used in PandoraPFA. The arrows indicates the track,the filled points represent the hits in the associated charged cluster and the open points represent the hits in the neutralcluster. a) Here the charged cluster energy is initially significantly greater than the associated track momentum. The hitsare reclustered using modified parameters for the clustering algorithm in the hope that a more consistent solution can befound. b) Here the cluster energy is significantly less than the associated track momentum. The topological associationalgorithms vii) and viii) have not added the neutral cluster as his would have resulted in a charged cluster with too muchenergy for the track momentum. The hits are reclustered in the hope that the neutral cluster naturally splits in such a waythat the topological association algorithm will now make the correct association. /mm
500 1000 1500 2000 2500 y / mm -1000-5000 Figure 6: PandoraPFA reconstruction of a 100 GeV jet in the MOKKA simulation of the ILD detector. The di ff erentPFOs are shown by colour / grey-shade according to energy. econstucted Energy/GeV
180 190 200 210 220 E v en t s PandoraPFAG(rms) ) G(rms
Figure 7: The total reconstructed energy from reconstructed PFOs in 200 GeV Z → uds events for initial quark directionswithin the polar angle acceptance | cos θ qq | < .
7. The dotted line shows the best fit Gaussian distribution with an rms of5.8 GeV. The solid line shows a Gaussian distribution, normalised to the same number of events, with standard deviationequal to rms ( i.e. σ = . θ |cos E / G e V / r m s
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
Figure 8: The jet energy resolution, defined as the α in σ E / E = α/ √ E(GeV), plotted versus cos θ qq for four di ff erentvalues of √ s . The plot shows the resolution obtained from (Z /γ ) ∗ → qq events (q = u,d,s) generated at rest. GeV
JET E [ % ] j e t / E r m s TotalResolutionConfusion
OtherLeakage
Figure 9: The contributions to the PFlow jet energy resolution obtained with PandoraPFA as a function of energy. Thetotal is (approximately) the quadrature sum of the components. GeV jet E [ % ] j e t / E r m s Particle Flow (ILD+PandoraPFA)Particle Flow (confusion term)Calorimeter Only (ILD) 2.0 % ⊕ E(GeV)60 % /
Figure 10: The empirical functional form of the jet energy resolution obtained from PFlow calorimetry (PandoraPFAand the ILD concept). The estimated contribution from the confusion term only is shown (dotted). The dot-dashedcurve shows a parameterisation of the jet energy resolution obtained from the total calorimetric energy deposition in theILD detector. In addition, the dashed curve, 60 % / √ E (GeV) ⊕ . (HCAL) I λ E / G e V / r m s
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
Figure 11: Jet energy resolutions (rms ) for the LDCPrime as a function of the thickness (normal incidence) of theHCAL. In addition, the ECAL contributes 0 . λ I . Results are shown with (solid markers) and without (open markers)taking into account energy depositions in the muon chambers. All results are based on Z → uu , dd , ss with generatedpolar angle in the barrel region of the detector, | cos θ qq | < . Field/Tesla [ % ] j e t / E r m s = 1825 mm ECAL ra)
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
ECAL Inner Radius/mm [ % ] j e t / E r m s B = 3.5 Tesla b)
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
Figure 12: a) the dependence of the jet energy resolution (rms ) on the magnetic field for a fixed ECAL inner radius. b) the dependence of the jet energy resolution (rms ) on the ECAL inner radius a fixed value of the magnetic field. Theresolutions are obtained from Z → uu , dd , ss decays at rest. The errors shown are statistical only. ECAL Cell Size/cm [ % ] j e t / E r m s a)
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
HCAL Cell Size/cm [ % ] j e t / E r m s b)
45 GeV Jets100 GeV Jets180 GeV Jets250 GeV Jets
Figure 13: a) the dependence of the jet energy resolution (rms ) on the ECAL transverse segmentation (Silicon pixelsize) in the LDCPrime model. b) the dependence of the jet energy resolution (rms ) on the HCAL transverse segmenta-tion (scintillator tile size) in the LDCPrime model. The resolutions are obtained from Z → uu , dd , ss decays at rest. Theerrors shown are statistical only. igure 14: An example of a Z → dd decay with E Z = + e − → ZZ → νν dd interaction inthe ILD detector concept. GeV qq M
60 80 100 120 E v en t s / G e V W/Z
Ea) /GeV qq M
60 80 100 120 E v en t s / G e V W/Z
Eb) /GeV qq M
60 80 100 120 E v en t s / G e V W/Z
Ec) /GeV qq M
60 80 100 120 E v en t s / G e V W/Z
Ed)
Figure 15: Reconstructed invariant mass distributions for the hadronic system in simulated ZZ → dd νν and W + W − → ud µ − ν µ events as simulated in the modified ILD detector model.events as simulated in the modified ILD detector model.