The Reconstruction Software for the MICE Scintillating Fibre Trackers

Transcription

1 Preprint typeset in JINST style - HYPER VERSION The Reconstruction Software for the MICE Scintillating Fibre Trackers A. Dobbs a, C. Hunt a, K. Long a, E. Santos a and C. Heidt b a Imperial College London, London, SW7 2AZ, U.K. b University of California Riverside, 900 University Ave, Riverside, CA 92521, U.S.A. adobbs@imperial.ac.uk ABSTRACT: A report on the reconstruction software of the scintillating fibre trackers of the Muon Ionization Cooling Experiment (MICE). MICE will demonstrate the principle of muon beam emittance reduction via ionisation cooling for the first time, for application in future muon acceleratorbased facilities, such as a Neutrino Factory or Muon Collider. The beam emittance before and after cooling is required to be measured to high precision, which is to be achieved with two scintillating fibre trackers, one upstream and one downstream of the cooling channel. This paper describes the reconstruction software of the fibre trackers, including the GEANT4 based simulation, geometry and configuration implementations, digitisation, spacepoint reconstruction, pattern recognition and the final track fit based on a Kalman filter. The performance of the software is evaluated by means of MC studies using momentum residuals, single particle analyses and various other methods, and the precision of the final reconstructed emittance is evaluated. KEYWORDS: Pattern recognition, cluster finding, calibration and fitting methods; Performance of High Energy Physics detectors; Particle tracking detectors. Corresponding author.

2 Contents 1. The MICE Experiment Overview The Scintillating Fibre Trackers 3 2. Coordinate systems and Reference Surfaces Channels and Digits Planes and Clusters Stations and Spacepoints Trackers and Tracks 5 3. The MAUS Framework 6 4. Data Structure General MAUS, MC and DAQ data structures Tracker reconstruction data structure Cross links and the MC - reconstruction bridge 8 5. Geometry 9 6. Simulation Noise Reconstruction Digitization Clustering Spacepoint Reconstruction Cluster selection Pattern Recognition Straight Line Pattern Recognition Helical Pattern Recognition Track Fit Track Model and Measurement Equation Stages of the Kalman Filter Goodness of fit Performance Conclusion 25 1

3 A. Appendix 1: Least Squares Fitting 26 A.1 Straight Line Fit 28 A.2 Circle Fit The MICE Experiment 1.1 Overview The Muon Ionization Cooling Experiment (MICE) is designed to demonstrate the feasibility of ionization cooling for the first time. Ionization cooling is a neccessary technology for any future facility based on high intensity muon beams, such as a Neutrino Factory [1], the ultimate tool to study leptonic CP symmetry violation, or Muon Collider [2], a potential route to multi-tev lepton - anti-lepton collisions. Any such facility will require beam cooling, that is, a reduction in the emittance of the beam, so that after generation of the muon beam via pion decay the phase space may be shrunk so that it falls within the acceptance of downstream acceleration components. Without cooling much of the muon beam is lost, leading to a severe reduction in useful particle rates. The short muon lifetime neccessitates the use of a fast beam cooling technique, which traditional cooling techiniques are unable to provide. To counter this, ionisation cooling was proposed in the early 1970s [3, 4], but has yet to be demonstrated. Ionization cooling reduces beam emittance by first passing a beam through some suitable, low atomic number material, such a hydrogen. This leads to reduction in beam momentum in all directions due to ionisation energy losses. After the absorber, momentum is restored in the longitudinal direction only by means of RF cavities. The sequence is repeated leading to an overall reduction in the transverse phase space of the beam. Absorber Focus Coil Module 1 Absorber Focus Coil Module 2 Downstream PID: TOF2, KL, EMR 4 T Spectrometer Solenoid and Tracker MHz 4 Cavity RF Coupling Coil Module 4 T Spectrometer Solenoid and Tracker 2 Figure 1. The Step V cooling channel, representing one half-cell of a Neutrino Factory Feasibility Study II cooling channel. SS refers to Spectrometer Solenoids, used to created helical particle paths for tracking. 2

4 ISIS Target LM CKOV = Cerenkov detector GVA1 = Scintillator counter D = Dipole Magnet KL = KLOE Light Detector DS = Decay Solenoid LM = Luminosity Monitor DSA = Decay Solenoid Area Q = Quadrupole magnet EMR = Electron Muon Ranger TOF = Time of Flight Q1p3 D2 Q4p6 Q7p9 TOF2 KL DS D1 DSA MICE Hall GVA1 TOFv CKOVadb TOF1 EMR Figure 2. The MICE Muon Beamline. A pion production target intercepts the protons of the circulating ISIS beam. Some of the subsequent pions are then captured by quadrupoles and guided down the beamline, passing through various PID dectors along the way, before delivery to the cooling channel itself. Emittance is measured immediately before and after the cooling channel using the scintillating fibre trackers. MICE, when complete, will represent the first practical demonstration of sustainable muon cooling. A schematic of an ionisation cooling channel known as Step V is shown in Fig. 1. The final MICE cooling channel will follow this design or a variant thereof. MICE is based at Rutherford Appleton Laboratory, U.K., and is a staged experiment, built and run in discrete steps. A schematic of the MICE Step I beamline [5] is shown in Fig. 2. The lab s ISIS 800 MeV synchrotron serves as a proton driver, with a titanium target being inserted into the circulating beam to generate a pion shower. A fraction of this shower is captured by a quadrupole triplet, and directed into the MICE experimental hall using a pair of dipole magnets. A 5 T superconducting decay solenoid is used to increase the pion path length, allowing time for decay into muons. These muons are then transported via a further series of quadrupoles to the cooling channel. MICE recently completed Step I of its programme, consisting of the muon beamline with particle identification (PID). Step IV, which will introduce the trackers and the first absorber module, is due to begin taking data in The Scintillating Fibre Trackers In order to meet the stringent requirements of the emittance measurement, MICE is equipped with two high precision scintillating fibre ( scifi ) trackers. Each tracker is housed in a 4 T superconducting solenoid to allow momentum measurement, completing the spectrometer design. Tracker 1 is located upstream of the cooling channel, Tracker 2 downstream. Tracker 1 is rotated by 180 with respect to Tracker 2 in the beamline, with station 5 being most upstream in Tracker 1, and station 1 being most upstream in Tracker 2. Each tracker consists of 5 detector stations, labelled 1 to 5, spaced at increasing separations along the beam axis, as illustrated in Fig. 3. Tracker 1 is orientated such that Station 5 sees the 3

5 yt xt Station 1 Station 2 Station 3 zt Station 4 Station 5 Figure 3. Left: A schematic of the tracker carbon fibre frame, showing the detector station positions. The fibre planes are glued on to the upstream edge (lower zt position) of the carbon fibre station frames (shown in green). Right: A photograph of a tracker. The orange tint is due to the special lighting needed to protect the fibres. The intersecting lines visible on the station faces indicates the direction of the fibres in each plane. beam first, while Tracker 2 is rotated by 180 such that Station 1 sees the beam first. Thus in both trackers Station 1 is always nearest the cooling channel, which sits between them. Each station is in formed of three planes of 350 µm scintilating fibres, orientated at 120 degrees to each other. The fibres in each plane are arranged in a doublet layer structure, in order to give 100% coverage of the plane area, as illustrated in Fig. 4, and are backed by a sheet of mylar plastic. The planes, also known as views, are labelled U, V and W. Plane U is attached to the station frame directly, plane W on to plane U, and plane V on to plane W. The fibre plane orientations are illustrated in fig. 5. In Tracker 1 as seen by the incoming beam, the U plane fibres are orientated such that they run approximately parallel to the ground, the W plane fibres run from 4 o clock to 10 o clock, and the V plane fibres from 2 o clock to 8 o clock. In Tracker 2 as seen by the incoming beam, the U plane fibres are orientated such that they run approximately parallel to the ground, the W plane fibres run from 2 o clock to 8 o clock, and the V plane fibres from 4 o clock to 10 o clock. The fibres produce scintillation light when ionising radiation passes through them, which is guided to visible light photon counters (VLPCs) in a cryostat for readout. 2. Coordinate systems and Reference Surfaces 2.1 Channels and Digits The V and W planes consist of 214 channels, labelled 0 to 213, while the U plane has 212 channels, labelled 0 to 211. The channel number increases from left to right if a plane is placed fibre side down, mylar side up, with the fibre readout pointing downwards, as illustrated in Fig Planes and Clusters As stated, the planes are labelled V, U and W. They are also assigned numbers, plane V being assigned 0, plane U is assigned 1 and plane W is assigned 2. The plane reference surface is defined to be the flat plane that is tangential to the outer surface of the mylar plane. The measured position perpendicular to the direction of the fibres in each plane is labelled with α (v, u, w), defined to increase in the same direction as the channel number. The 4

6 Mylar (a) W U V (b) Figure 4. (a) Arrangement of the doublet layers in the scintillating-fibre stations. The outer circle shows the solenoid bore while the inner circle shows the limit of the active area of the tracker. The grey, blue, and green regions indicate the direction that the individual 350 µm fibres run (moving outward from the centre) in the U, V, and W planes respectively. (b) Detail of the arrangement of the scintillating fibres in a doublet layer. The fibre spacing and the fibre pitch are indicated on the right-hand end of the figure in µm. The pattern of seven fibres ganged for readout as a single channel, via a single clear-fibre light-guide, is shown in red. The sheet of mylar plastic glued to the doublet layer is indicated. U W V... V W U Tracker 1 Station 5 Tracker 2 Station 1 Figure 5. The orientation of the fibres in each plane, as seen be the incoming beam, for both trackers. The green object is station frame. The order of the planes is reversed between trackers as Tracker 1 is rotated by 180 with respect to Tracker 2 in the beamline. z axis of the plane coordinate system is defined to be perpendicular to the plane reference surface and points in the direction from the mylar towards the fibres. The direction in which the fibres run defines the final plane coordinate, β, with the direction defined to complete a right-handed coordinate system (β plays the role of x, α corresponds to y, and z p to z). The origin of the (β,α) coordinate system is taken to be at the centre of the circular active area of the plane. 2.3 Stations and Spacepoints The stations are labelled 1 to 5. The station reference surface is defined to coincide with the reference surface of the V doublet layer. The station coordinate system is defined such that the y s axis is coincident with the v axis (that is, perpendicular to the run of the fibres in the V plane), the z s axis is coincident with the z p axis of the V layer and the x s axis completes a right-handed coordinate system. 2.4 Trackers and Tracks The upstream tracker is labelled Tracker 1, the downstream Tracker 2. The tracker reference surface is defined to coincide with the reference surface of Station 1. The tracker coordinate system is defined such that the z t axis coincides with the nominal axis of cylindrical symmetry of the tracker 5

7 as shown in figure 3. The tracker z t coordinate increases from Station 1 to Station 5. The tracker y t axis is defined to coincide with the y s axis of Station 1 and the tracker x t axis completes a right-handed coordinate system. (a) β Doublet layer (mylar side up) Z p (b) y s x s α Channel 1 Central Fibre Channel 212 or 214 Fibre run to the station optical connnectors towards the bottom of the figure v w u z s Figure 6. (a) The channel numbering within a plane, and the (β,α,z p ) plane coordinate system (a righthanded system). (b) The fibre plane ordering with respect to the station body and the station coordinate frame (a right-handed system). In Tracker 1 the beam approaches from the right, in Tracker 2 from the left. Note that z s is by definition equivalent to z p of the V plane. 3. The MAUS Framework The tracker software exists within the context of the MICE software framework, known as MAUS (MICE Analysis User Software). MAUS is used to perform Monte Carlo (MC) simulation and both online and offline data reconstruction for MICE. It is built using a combination of C++ (primarily in the backend) and Python (primarily in the frontend). Simulation is supported by GEANT4 [6], and analysis with ROOT [7]. The input and output data formats can be either ROOT or JavaScript Object Notation (JSON) files (with ROOT being the standard). MAUS also reads in the custom binary format written by the MICE data acquisition system (DAQ). MAUS is controlled by the user using a top level Python script, together with a configuration file (also written in Python). The structure at the top level is set out in a funcional coding manner, with different modules being able to be swapped in and out depending on the task at hand. The modules come in four types: Input, Output, Map and Reduce. Input modules provide the initial data to MAUS, such from a ROOT or JSON file, or via the DAQ. Maps perform most of the simulation and analysis work, including calling the simulation routines and performing the reconstruction. Maps may be processed in parallel across multiple nodes. Reducers are used to display output, such as for online reconstruction plots, and are capable of accumulating data sent from maps over multiple spills. Lastly, output modules provide data persistency such as saving to a ROOT or JSON file. The tracker software is called using, at present, four maps and one reducer. The maps cover digitisation of MC data, digitisation of real DAQ data, the addition of noise to MC data and recon- 6

8 struction. The reducer provides various event plots and run information. The modules contain little code themselves but instead tend to call backend C++ classes. 4. Data Structure 4.1 General MAUS, MC and DAQ data structures A simplified schematic of the tracker data structure, with the relevant entries from the more general MAUS data structure, is shown in Fig. 7. All the objects listed represent container classes for different parts of the simulation, raw data and reconstruction. The top level object is known as the spill, which contains the data associated with the burst of particles which flows through the MICE beamline after one dip of the pion production target. Within the spill the data is split into three branches: real data from the DAQ, MC data generated by simulation, and reconstruction data, which is formed from data in either the real or MC branches. Two keys principles apply to how the different branches of the data structure should interact: 1. The reconstruction should proceed indentically whether the data originates from the MC or real branches; 2. The reconstruction branch should contain no direct references to the MC data. This is to preserve the honesty of any studies performed using MC data which are intended to evaluate the reconstruction performance (as well as making for a cleaner structure). The real DAQ data is held in an object known as TrackerDAQ. This is then further split depending on where the raw data originated; if from the full MICE DAQ then it is stored in a VLSB object, or if from the cosmic data test run DAQ then in a VLSB_C object. The MC event holds data on scifi hits produced by tracks passing through sensitive volumes (the tracker fibre planes) and any associted noise hits originating in those planes. The scifi hit is implemented as a class based on the generic Hit class template from which all the different MICE MC detector hit classes are derived. Other relevant data held in the MC event though not part of the tracker data structure includes simulated track objects which hold truth information on position, momentum and particle types. Such data is used to evaluate the reconstruction performance against the known truth data (see section 8). 4.2 Tracker reconstruction data structure The reconstruction data itself for tracker is all held within a class known as SciFiEvent. This then contains C++ standard library vectors of the following container classes, representing the higher level reconstructed tracker data: Digits, representing the digitisation of a detector channel response to an incident track; Clusters, which represent groups of neighbouring digits arising from the same particle crossing multiple channels; Spacepoints, which group clusters from adjacent detector planes to give a real space position in terms of (x,y,z); 7

9 Spill MC Event Recon Event Tracker DAQ SciFi Hit SciFi Noise Hit SciFi Event VLSB VLSB_C Digit Cluster Spacepoint SPR Track HPR Track Kalman Track Contains pointers to (owns memory) Contains pointers to (does not own memory) Linked by a Lookup Trackpoint Figure 7. The tracker software data structure, and relevant MAUS data structure. The spill is the top level object below which data is split into real data, MC data and reconstruction sides. When an object owns the memory of a set of other objects, these are held as standard vectors of pointers. When an object contains cross links to another set of objects, without owning their memory, these are held as a ROOT TRefArray of pointers. SPR = Straight Pattern Recognition, HPR = Helical Pattern Recognition. Straight pattern recognition tracks, which group together spacepoints from different tracker stations according to the particle track which generated them, in the case that the originating track is straight (i.e. neutrals or all tracks when the solenoidal field is off). The track parameters given in eqn are also stored; Helical pattern recognition tracks, which group together spacepoints from different tracker stations with the particle track which generated them, in the case that the originating track is helical (i.e. charged particle tracks when the solenoidal field is on). The track parameters given in eqn are also stored; Scifi tracks, holding the final Kalman fit parameters of the particle track. Scifi tracks also contain trackpoints, which hold the fit parameters at each detector plane, including the optimal momentum and position of the track. 4.3 Cross links and the MC - reconstruction bridge Each higher level object also contains cross links in the form of pointers back to the objects within the SciFiEvent which were used to create it; thus clusters contain pointers to digits, spacepoints pointers to clusters, and pattern recognition tracks to spacepoints. The scifi tracks are slightly different, with the trackpoints within the scifi track containing a link back to the cluster which produced them. In this manner all higher level objects can be traced back fully to the original digits. 8

10 Tracker 1 offsets in mm Station 1 Station 2 Station 3 Station 4 Station 5 X Y Z Tracker 2 offsets in mm Station 1 Station 2 Station 3 Station 4 Station 5 X Y Z Table 1. surface. Position of tracker stations from CMM measurements, positions taken with respect to reference In the case of a MC run (as opposed to real data) the digits themselves are linked via an ID number and lookup table back to the MC hits used to produce them. The ID is defined as tspccc, where t is the tracker number, s is the station number, p is the plane number, and c is the channel number. The bridge is created in this manner so that the reconstruction side of the datastructure is kept free of any direct reference to the MC data, preserving the impartiality of the reconstruction to either MC or real data. 5. Geometry The absolute position of each tracker station was determined by use of a coordinate measuring machine (CMM) at Imperial College London. The measurements were made by supporting the trackers upon blocks on the CMM without clamping the stations to any support structure. This was done to avoid introducing inaccuracies that may arise from the extra stresses put on the frame from securing the structure. Four measurements were then made along the plane of each of the stations (see table 1). Tracker station positions are stored in the MICE configuration database (CDB) and made available for the whole experiment. Through the CDB the MICE tracker configuration is integrated into the larger MICE configuration, and any alterations needed can be applied and monitored. The CDB is a bitemporal database, alterations are tracked by date and run number, ensuring the proper geometry is used in analysis of historic data. Configurations are stored in CDB as a collection of XML files which are translated into the native MAUS format "MICE modules," when pulled from the CDB and called by GEANT4 at run-time. The MICE modules are human readable, nested, text documents that, when taken as a collective, contain all the needed information to simulate the various MICE systems and detectors. MAUS is designed such that the same detector geometry descriptions are used for both MC and real data processing. The only differences present in the MC and real geometries relate to nonactive portions of the experiment and field mapping. Beyond the detector description, included in the MC geometry only is the epoxy resin, which was used in securing the individual scintillating 9

11 Figure 8. A visualisation produced by GEANT4 of the Step IV simulation geometry, for a 20 primary muon track spill. fibres to the tracker station body, and the thin films of mylar placed between each scintillating fibre plane. The carbon fibre body of the trackers have not been included in either the real or MC geometries as their exposure to the MICE beam and the effects from multiple scattering upon the beam through the body is expected to be minimal. The ISIS beam is scheduled to restart in March of 2015, which will allow for many new measurements to be made of the final Step IV position of the trackers. At the time, beam will be used to finalize a measurement of the tracking stations relative to each other, align the trackers with one another, and determine the alignment of the trackers with respect to the axis of the solenoidal field. 6. Simulation The tracker MC forms part of the general MAUS MC, which is built upon the GEANT4 standard physics libraries. A screenshot showing a GEANT4 visualisation of a simulation accumulated over 20 tracks is shown in Fig. The trackers are simulated on a per fibre basis and layered into the fibre planes as shown in Fig. 4. Simulated particles incident upon a tracker fibre are stepped through that fibre. The GEANT4 library determines if the particle interacts with the fibre at each step, and if so how much energy is deposited into the fibre. The tracker MC applies a series of conversion factors to the GEANT4 output to determine the corresponding number of photo electrons (NPE) that would have been read out by the tracker VLPCs. At the present a single calibration is used for each of the VLPCs. Step IV commissioning of the trackers will provide information on individual VLPC calibration constants. A channel map is used to determine which channel on the VLPC the active fibre is read into. The channel number along with the NPE and timing information from GEANT4 are used to create a SciFiDigit record, which is then ready to be used for reconstruction. In order to follow the principles laid out in section 4.1, that the reconstruction should be agostic as to whether the input 10

12 Events Magnitude of Noise in NPE Entries Mean Constant ± 0.03 Slope ± Number of Recorded Photo Electrons (NPE) Events Number of Channels With Noise per Particle Event 2400 Entries Mean Number of Channels Recording Noise Figure 9. (a) The magnitude of each background event recorded in NPE. The left side of this plot has been truncated at one NPE due to reconstruction only recording signal greater than a single PE. Data is fit to an exponetial of the form y = e c+mx (b) Distribution of background noise events for each recorded particle event in the SST. data orginates from MC or real data, every effort as been taken to insure the output digit from MC is identical to that from the DAQ. 6.1 Noise In May 2012 a single station test (SST) was carried out. This test consisted of a single tracker station placed in the MICE beam. Analysis of the low level reconstruction, on the level of digits, from the SST is the basis for the tracker MC noise simulation. Two cuts are made on the pool of digits from the SST. Firstly all confirmed signals, defined as any digits that were used for spacepoint reconstruction, were removed from the pool. Second, any digit from a channel with an anomalous signal strength was also removed, as sources of anomalous signal could be mis-calibrated channels or problems with the scintillating fibers. The cutoff for anomalous channels was 300% of the minimum signal of 1.0 PE. The results of the SST noise analysis are shown in Fig. 9. Fig. 9a shows the number of events as a function of the NPE produced by noise. The plot is fitted with an exponential function which is then used as the model of the magnitude of the MC noise. Fig 9b shows a histogram of the number of events for which noise occured in a given number of channels. This distribution is used to model how many channels in the MC should have noise added to them, with the specific channel numbers then being chosen at random. Taken together this then provides the complete description of the scifi MC noise. 7. Reconstruction The reconstruction begins either with real data from the DAQ, or with MC hit data. The data is then digitised, creating the most primitive reconstruction object, the SciFiDigit, representing a hit in a tracker plane channel. After this point, the subsequent reconstruction proceeds in a manner independent of whether the digits originating from MC or real data, in accordance with the principles of section 4.1. This is illustrated in Fig

13 Raw Data (bin) Unpacker InputCppDAQOfflineData MapCppTrackerRecon Raw Data (json) SciFi Space Pt Pattern Recognition SciFi PR Track Mapping + Calibration MapCppTrackerDigits Space Point Recon SciFi Digit Cluster Recon SciFi Cluster Track Fitting SciFi Track MC Recon MapCppTrackerMCDigitisation Real Data SciFi Hit MapCppSimulation MC SciFi Noise Monte Carlo Reconstruction MC Particle MapPyBeamMaker Figure 10. The reconstruction data flow. Data originates either from MC or real DAQ data, the two branches meeting after digitisation, after which the reconstruction proceeds independently of which branch the data came from. The relevant MAUS modules for each step are indicated. 7.1 Digitization For real data the electronic signals produced by the VLPCs are digitised using analogue-to-digital converters (ADCs). The DAQ system records the pulse height in ADC counts together with the DAQ channel number. The pulse height is then calibrated to the number of photoelectrons produced by the VLPC associated with the channel hit, and the DAQ channel number converted back to the tracker channel number. This information is then used to form a SciFiDigit. The analagous process for MC data is described in section Clustering When a particle traverses a station plane it will generate a pulse usually at most in one or two channels. Higher numbers of channels are possible due to noise or showering, but this is less likely. Clustering takes hits in adjacent channels in a plane and groups them together, producing a cluster associated with a single particle. Even if no adjacent channel hits are found, a single channel hit is still used to produce a cluster object prior to moving on the next level of reconstruction. The clustering algorithm is straightforward, looping over every combination of pairs of digits in a scifi event, and then combining any which occur in neighbouring channels in the same detector plane into a cluster. A value for α is then calculated by subtracting the central fibre channel number from the digit channel number. In the case of multi-digit cluster, the average channel value is calculated prior to subtracting the central fibre number: α = 1 N N i=0 a i a CF (7.1) where N is the number of digits in the cluster, a i is the channel number of the i th digit in the cluster, and a CF is the central fibre channel number. 12

14 Once alpha has been determined the position from the central fibre may be translated in to millimeters using the following formula: d[mm] = α p q (7.2) where p is the distance separating the centres of two neighboring fibres, known as the pitch, having a value of mm, and q is the number of fibres separating the centres of two adjacent channels, which is 3.5 (see Fig. 4). 7.3 Spacepoint Reconstruction Spacepoints are formed from clusters, to give the position a track traversed a station. Each spacepoint is formed from either three clusters (a triplet spacepoint) or two clusters (a doublet spacepoint). The clusters must be from the same detector station, but from different planes within that station. Once the clusters for a spacepoint have been selected, the crossing point of the channels, represented by the clusters from the different planes, is calculated to provide the (x,y) coordinates of the spacepoint Cluster selection In order to determine which clusters from each plane originate from the same track, and so may be used to correctly form a spacepoint, we follow a principle known as Kuno s conjecture, which states that for a given triplet spacepoint, the sum of the channel numbers of each cluster will be a constant. To see how this comes about, consider the coordinate system defined by the u, v and w axes. The u, v and w coordinates my be written in terms of the polar coordinates (r,φ) as follows: u = r cos[φ] (7.3) [ ] 2π v = r cos 3 φ (7.4) [ ] 4π w = r cos 3 φ (7.5) The sum u + v + w may now be written: { [ ] [ ]} 2π 4π u + v + w = r cosφ + cos 3 φ + cos 3 φ (7.6) { [ ( ) ( ) ] 2π 2π = r cosφ + cos cosφ + sin sinφ+ + (7.7) 3 3 [ ( cos 2π ) ( cosφ + sin 2π ) ]} sinφ+ (7.8) 3 3 { ( ) } 2π = r cosφ + 2cos cosφ (7.9) 3 = r {cosφ + [ cosφ]} (7.10) = 0 (7.11) If the sum is performed using the fibre numbers for the channels hit, the sum of the three views will equal the sum of the central-fibre numbers, i.e. if the central fibre numbers of each of the U, V and W doublet-layers is 106.5, then the sum of channel numbers will be =

15 The three clusters, one each from planes, that make up a space point must therefore satisfy: n u + n v + n w = n u 0 + n v 0 + n w 0 ; (7.12) where n u, n v and n w are the fibre numbers of the clusters in the u, v and w views respectively and n u 0, nv 0 and nw 0 are the respective central-fibre numbers. A triplet space point is then selected if: (n u + n v + n w ) (n u 0 + n v 0 + n w 0 ) < K. (7.13) Once all triplet space-points have been found, doublet space-points are created from pairs of remaining clusters from different views. In this case, the only condition for acceptance is that the crossing point position of the pair is within an acceptable radius based on the tracker fiducial volume, currently set at 160 mm. 7.4 Pattern Recognition Pattern Recognition refers to the process of grouping spacepoints according to the particle tracks which created them and performing initial track fits. The algorithm is based on looping over different combinations of spacepoints and performing a fit, based on a simple linear least squares technique. The algorithm treats helical and straight tracks separately, though much of the code is shared. Presently helical track fitting is attempted first, with straight then following using any spacepoints that remain in the SciFiEvent. Each tracker is treated separately and identically Straight Line Pattern Recognition In the absence of a magnetic field, the tracks passing through the tracker may be described using a straight line in three dimensions. Taking the z coordinate as the independent parameter, the track parameters may be taken to be: x 0 v sl y 0 = t x ; (7.14) t y where, x 0 and y 0 are the position at which the track crosses the tracker reference surface, t x = dx dz and t y = dy dz. The track model may then be written: x = x 0 + zt x (7.15) y = y 0 + zt y (7.16) Pattern recognition then proceeds as follows. First an attempt is made to find all the tracks containing 5 spacepoints present in the event. After this, all four point then all three point tracks are searched for in turn. To begin, a spacepoint is chosen in each of two stations, i and j where i and j label two different stations and j > i. Ideally, i = 1 and j = 5 (which is always the case for a five point track). However, a search of all combinations of pairs for which j i > 1 is made, taking the pairs in the 14

16 order of decreasing separation in z; i.e. in order of decreasing z ji = z j z i. Initial values for the track parameters, are then calculated as follows: v sl Init = x Init 0 y Init 0 t Init x t Init y, (7.17) t Init x = x j x i z j z i ; (7.18) x Init 0 = x i z i t Init x ; (7.19) t Init y = y j y i z j z i ; and (7.20) y Init 0 = y i z i t Init y ; (7.21) where (x i,y i,z i ) are the coordinates of space-point i, etc. A search is then made for spacepoints in each of the intermediate stations, k, between station i and station j. The distance between the x and y coordinates of the space-points in the stations k; j < k < i and the line defined by the initial track parameters is then calculated as follows: Points are accepted as part of a trial track if: δx k = x k (x Init 0 + z k t Init x ) and (7.22) δy k = y k (y Init 0 + z k t Init y ). (7.23) δx k < and (7.24) δy k <. (7.25) where is known as the road cut. If at least one spacepoint satisfies this selection for each station, a trial track is formed consisting of the space-points selected in stations i, k,... and j. For each trial track, a straight-line fit is performed to calculate the fit χ 2 in both x and y. The trial track is accepted, and the spacepoints are labelled as used, if the x and y fit χ 2 satisfy: χ 2 x N 2 < χ2 cut, (7.26) χ 2 y N 2 < χ2 cut, (7.27) where N is the number of spacepoints used in the fit. This proceeds until either all the spacepoint combinations have been tried, or there no longer remain enough unused spacepoints to form a track. 15

17 7.4.2 Helical Pattern Recognition Helical pattern recognition proceeds by selecting a spacepoint from each station, attempting a circle fit in (x,y), calculating the true turning angle to determine the values of s, and finally attempting a straight line fit in (z,s). If each step succeeds, the spacepoints are accepted and associated with a helical track object. If any step fails, a different combination of spacepoints is tried. Helical Track Parameters In the presence of a magnetic field, the tracks passing through the tracker may be described using a helix. In the tracker coordinate frame, the tracks form circles in the (x,y) plane. Defining s to be the length of the arc swept out by the track in the (x,y) plane, a track may be described using a straight line in the (s,z) plane. Five parameters are required to uniquely determine a helix in three dimensions: three from the circle, one to define the compactness and one to describe the phase at which it crosses a reference surface. Taking the z coordinate as the independent variable, the track parameters for pattern recognition may be taken to be: v hlx = x c y c r s 0 t s ; (7.28) where (x c,y c ) are the coordinates of the centre of the circle, r is the circle radius, s 0 the intercept of the line in the (z,s) and t s = ds dz, which represents the compactness of the helix (this serves the same purpose as the dip angle parameter often used when describing helices elsewhere). Note that φ 0, the turning angle of the most upstream spacepoint (defined to increase in an anticlockwise sense in the (x,y) plane), could be used instead of s 0, and is also stored in track object. Helical Track Model Having defined the helix parameters, the next step is to describe a track model, that is, a means of relating the measured (x,y,z) coordinates of the spacepoints to the helix parameters. A helix may be described by the parametric equations: x = ρ cos(φ) + x c (7.29) y = ρ sin(φ) + y c (7.30) z = ts 1 s s 0 = ts 1 ρφ s 0 (7.31) where the definition of the radian has been used to transform s into phi: s = ρφ (7.32) A diagram illustrating these relations is shown in Fig. 11. The turning angle plays the role of the parameter in these equations. It can be determine from the measured positions using: ( ) y φ = tan 1 yc (7.33) x x c 16

18 Here φ refers specially to the raw or observed value of the turning angle. The true value of the turning angle may vary from this by extra 2nπ, corresponding to the number of rotations that a particle makes when travelling between two tracker stations: φ = φ + 2nπ, 0 < φ < 2π (7.34) where n is an integer, representing the number of times the particle track has passed through the origin of φ coordinate system between two given stations. Determining the value of n requires some work, and is described below in the discussion on fitting in the (z,s) plane. Figure 11. The helical track geometry and track model. Helix (z, x) and (z, y) projections While not necessary for the procedure for determining the track parameters, it is also useful to note the projections of the helix in the (z,x) and (z,y) planes. Starting from eqns. 7.29, 7.31 and 7.32 it can be seen that: ( ) zts s 0 x = ρ cos + x c (7.35) ρ Similarily, starting from eqns. 7.30, 7.31 and 7.32, it can again be seen: ( ) zts s 0 y = ρ sin + y c (7.36) ρ Circle Fitting in (x,y) The circle formed by the track in the (x,y) plane is required to be parameterised, so that it may be readily solved using the least squares method. The standard equation for a circle is given by: (x X 0 ) 2 + (y Y 0 ) 2 = ρ 2 (7.37) 17

19 where (X 0,Y 0 ) is the position of the centre of the circle and ρ is its radius. Expanding: which implies: (x 2 + y 2 ) 2X 0 x 2Y 0 y = ρ 2 (X 2 0 +Y 2 0 ) (7.38) (x 2 + y 2 ) ρ 2 (X 2 0 +Y 2 0 ) 2X 0 x ρ 2 (X 2 0 +Y 2 0 ) 2Y 0 y ρ 2 (X 2 0 +Y 2 0 ) = 1 (7.39) This form then indicates a second system which may used to parameterise the circle: where: α(x 2 + y 2 ) + βx + γy + κ = 0 (7.40) α = These equations are readily inverted to yield: 1 ρ 2 (X 2 0 +Y 2 0 ) (7.41) β = 2X 0 α (7.42) γ = 2Y 0 α (7.43) κ = 1 (7.44) X 0 = β 2α Y 0 = γ 2α β ρ = 2 + γ 2 4α 2 κ α (7.45) (7.46) (7.47) The least squares fit is performed using the paramterisation given in eqn. 7.40, then translated using the above relations into the standard form for a circle given by eqn The least squares fit itself is described in section A. Prior to performing the fits, spacepoints must selected. As in the straight line case, the helical algorithm first attempts to find tracks with spacepoints in all five tracker stations then in any four stations (but not in any three stations, as a circle may always be fitted to three points). In contrast to the straight fit however no trial circle is formed, followed by use of road cuts to select spacepoints. Instead the algorithm loops over all the combinations of spacepoints in all the stations, and then moves directly to attempt a circle fit. If the χ 2 of the circle fit is sufficiently small, then the reconstruction proceeds to performing a straight line in the (z,s) plane. If the fit fails the chi 2 cut, the the algorithm continues looping over unused spacepoints in the stations, until all possibilities have been exhausted. 18

20 Straight Line Fitting in (z,s) The projection of a helical track in the (z,s) plane is that of a straight line. Before a fit can be attempted however, it is necessary to calculate s for each spacepoint. As mentioned earlier however, a difficultly arises in determining the true value of the turning angle, φ, as it is possible that a particle track may execute a number of rotations between tracker stations. This is not apparent, of course, when looking only at the circle formed in the (x,y) plane. The ambiguity can be resolved by making use of the varying longitudinal separations of the tracker stations. Consider the turning angle of the most upstream spacepoint, referred to as φ 0. As this is the first turning angle it is by definition the true turning angle: φ 0 φ 0 (7.48) Next, consider the angles turned through as the track propogates from station i to station j ( φ ji ) and from station j to station k ( φ k j ): φ ji = φ j φ i (7.49) φ k j = φ k φ j (7.50) φ ki = φ k φ i (7.51) It is known that, neglecting the effects of energy loss and multiple scattering, the ratio of the difference in turning angles between any two stations, and the difference in longitudinal positions between those stations, is a constant for any given track: Re-stating in terms of the observed turning angles: φ k j + 2n k jπ z k j φ ji z ji = C (7.52) φ k j z k j = φ ji z ji = φ ki z ki (7.53) = φ ji + 2n jiπ z ji = φ ki + 2n kiπ z ki (7.54) where n ji is the number of times the track passes through the origin of the φ coordinate system between stations i and j. It then follows that: n ki = n k j + n ji (7.55) There are a number of ways forward which could be taken from this point to find the different n ji and determine the true values of the turning angles. Firstly, let all the separations be with calculated with respect to the most upstream spacepoint, with turning angle φ 0 and position z 0 : φ j0 + 2n j0π z j0 = φ i0 + 2n i0π z i0 (7.56) φ j0 + 2n j0π φ i0 + 2n i0π = z j0 (7.57) z i0 19

21 Consider next a candidate value for the number of turns which occur between the first and last stations under consideration. As 0 refers to the first station under consideration, let f label the final station under consideration. This candidate number of turns may then be referred to as n c f 0. Using this value it is possible to predict what the true value of each turning angle should be for each intermediate station, as the separations in z are always known: φ c i0 = ( φ f 0 + 2n c f 0π ) z i0 z f 0 (7.58) where φ c i0 is the predicted true turning angle of the ith station with respect to the first station. If the candidate value of the total number of turns is correct then this candidate turning angle will be correct. If the turning angle calculated in this manner is correct, then dividing this angle by 2π should result in a remainder of the original observer turning angle for that station: φ i0 = φ c i0 mod 2π if φ c i0 = φ i0 (7.59) This then provides a test for the correctness of n c f 0. If for each itermediate station the candidate total number of turns between stations produces predicted turning angles whose remainders when divided by 2π are equal to the original observered turing angle, within some small tolerance, then candidate total number of turns between stations is accepted as correct. The values calculated for the intermediate turning angles are then refined; as it is known the difference between the observed turning angle and the true turning angle is an integer number of 2π, the closest such number to the predicted turning angle found earlier is calculated, and this value then assigned as the true turning angle. For example, if an observed turning angle is π and the predicted true turning angle is calculated to be 3π + 0.1, then the true turning angle for that station is taken to be 3π. The removes any small error introduced by extrapolating the turning angle using equation With the true turning angles found, the values of s for each spacepoint may be calculated using equation Once the values of s have been determined a fit in the (z,s) plane is performed, employing exactly the same algorithm used in the straight track fit. If the χ 2 of this fit is also sufficiently small then the track is accepted, the spacepoints marked as used, and the loop repeats looking for more tracks. 7.5 Track Fit After a collection of points hypothesised to belong to a particle s trajectory are put together by the pattern recognition routines, a Kalman Filter [8, 9] is used to find the optimal estimation of the track parameters associated to each track. A Kalman Filter is a recursive algorithm in which the track parameters are free to change, to some degree, at each point the particle being tracked finds some material. This is in contrast with global fit methods in which an unique set of parameters is fit to the whole particle trajectory. The necessity of an algorithm such as the Kalman Filter is a consequence of the multiple Coulomb scattering (MCS) and energy loss which add noise to the otherwise deterministic track model which defines the trajectory of the particle. In the absence of process noise, the Kalman Filter is equivalent to the global least squares fit. 20

22 The ingredients for the track fitting are: a track model which describes the path of the particle in the detector, a measurement equation which relates the (x, y) coordinate in the detector volume with the corresponding detector measurement (α), the resolution of the measurements (σ α ) and, finally, an accurate description of the geometry of the detector. Besides grouping the measurements that are likely to form the same track, the pattern recongition also provides also an initial estimate of the track parameters. The Global Least Squares Method is a suitable option for this purpose. In the following stage, the track fitting algorithm, employing the Kalman Filter, produces the optimal estimate of the track parameters. In this setup, the pattern recognition must have a broad acceptance and the track candidate constructed is ultimately kept or rejected by a goodness-of-fit test performed by the track fitting routine. Summarising, the goals of the track fitting routine are to compute the best estimate of the track parameters and to produce a confidence test which confirms the hypothesis that the measurements put together represent a particle s trajectory. In order to compute this test statistic, a covariance matrix must be associated to each fitted point. In the following sections, it will be shown how these goals are met by a Kalman Filter, while keeping tracking routines numerically stable and robust against error estimates. The final track parameters are optimal at the plane nearest to the cooling channel, to be used in emittance studies and also optimal at the opposite end of the detector, for extrapolation to other MICE detectors Track Model and Measurement Equation Each track is characterised by a set of parameters which are allowed to change along the propagation through the detector. This parametrisation contains information about the position (x,y,z) and momentum (p x, p y, p z ) of the particle. For the track fit, the parametrisation chosen consists of the 5-dimensional state vector: a = x p x y p y κ Q/p z ; (7.60) where Q is the charge of the particle. Whilst this is a different paramterisation from that used at the pattern recogntion stage (given by eqn. 7.28) the conversion is straight forward. The rule to extrapolate the state vector along the detector length is a fundamental ingredient in the track fitting routine and it is usually referred to as the track model. Typically, the state vector is chosen such that the correlation between elements is minimised. The track fitting set up is prepared to handle both helical and straight trajectories. For the straight track case, the parametric equations of motion are trivial: the track gradients m x = p x /p z and m y = p y /p z are constant and used to update the transverse position coordinates, x = x + m x + z and y = y + m y + z. For helical tracks, the 21

23 coordinates are updated according to: x = x + p x p t Rsin θ p y p t R(1 cos θ) (7.61) y = y + p y p t Rsin θ + p x p t R(1 cos θ) (7.62) z = z + z (7.63) p x = p x cos θ p y sin θ (7.64) p y = p y cos θ + p x sin θ (7.65) p z = p z ; (7.66) where θ = cbq z/p z, with c 0.299MeV/cT 1 mm 1, Q the charge of the particle and B is the magnitude of the magnetic field inside the tracker volume. Each (x, y) point in a detector plane corresponds to a fibre channel measurement (α), i.e. a scifi cluster. In the track fitting formalism, the measurement is represented by a vector, m = [α], which in this case is unidimensional. The measurement equation transforms the track-parameter vector into the measurement vector: h(a) = m = [α]; (7.67) or, in matrix form: Ha = m = [α]. (7.68) The error associated with the measurement is stored in the measurement covariance matrix: V = [σ 2 α]; (7.69) where σ α = w/ 12 is the variance of the channel measurement, given that the probability of measuring α is uniform over the channel width w = 1. This uniformity is in conflict with the assumption of Gaussian errors Stages of the Kalman Filter Being a recursive algorithm, the Kalman filter requires an estimate of the initial values to be fitted. At the initialisation stage, an initial state vector (a) and covariance matrix (C) are defined for the first measurement plane. The covariance matrix is constructed under the assumption that the initial errors are uncorrelated, so it is assumed to be diagonal, with randomly large elements (C ii = 1000) so that the final result isn t biased towards the initial values. The initial position estimate, (x,y), is taken from the spacepoint formed in that station. The other components of the state vector are as found by the pattern recognition. Some caution is necessary at the initialisation stage: if the covariance matrix is initialised with values too large, the early steps of the calculation might diverge. It can be shown that for estimation error remains bounded if the initial estimation error and the measurement error are small enough. The fitting is done in a set of steps: first, the track parameters are extrapolated according to the track model: a k 1 k = f (a k 1 ); (7.70) 22