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Transcript of Trackless ring fitting algorithm for the RICH · PDF file Trackless ring fitting algorithm...

  • Trackless ring fitting algorithm for the RICH detector

    Evelina Gersabeck, Gianluca Lamanna, Antonino Sergi, Sören Stamm


    Contents 1 Introduction 2

    2 Trackless ring finding algorithm 2

    3 Monte Carlo generators 3

    4 Results 4

    A Appendix 8


  • 1 Introduction The aim of the CERN NA62 experiment is to measure the branching ratio of the very rare kaon decay K+ → π+νν̄ with high precision. The theoretical prediction for the branching ratio of this decay is (0.85 ± 0.07) · 10−10 and allows a test of the Standard Model by extracting the CKM parameter |Vtd|. The experiment uses a 400 GeV proton beam provided by the SPS in order to produce kaons with a momentum of 75 GeV. The difficult task in this fixed target experiment is to achieve a signal to background ratio (S/B) of about 10:1 because of the large background from all other kaon decays. This S/B ratio can be accomplished with using several redundant measurements of the kinematics and detectors which are able to provide particle identification.

    This project is focused on the RICH detector placed downstream of the decay region behind the spectrometer. The task of the RICH is to provide a separation between pions and muons in the momentum range from 15 – 35 GeV to reduce background events (e.g. from the decay K+ → µ+ν). It consists of a 17 m long, 4 m wide cylindrical vessel, filled with neon at a atmospheric pressure. A total of 20 mirrors will be located at the end of the vessel in order to focus the cherenkov light onto two spots. These spots consists of 2000 photomultipliers (PMT) which can detect the cherenkov light redirected by Winston cones (�18 mm) to the active regions of the PMTs.

    The goal is to test an algorithm for trackless reconstruction of cherenkov rings in the detector. The advantage of a trackless reconstruction is that it does not rely on other detectors and the reconstructed track parameters can be used for a cross check with other detectors. The disadvantage is that there is no initial seed for the algorithm. [1, 2]

    2 Trackless ring finding algorithm The trackless ring finding algorithm for the RICH needs to fulfill certain criteria. Besides a good resolution for the reconstructed center and radius, the algorithm needs to be capable to work for a multi ring event and if it is used at trigger level, the time for processing one event is limited to a few microseconds. Different input information for the ring fitting is available at trigger or event level.

    Since the aim of the project was to develop a trackless ring finding algorithm, only the PMT coordinates can be used. It was proposed to use Ptolemy theorem to select a set of hits that can be considered for an algebraic fit. [3]

    Ptolemy theorem. Four points belong to a cyclic quadrilateral if and only if the relation

    AB · CD +BC · AD = AC ·BD (1)

    is valid.

    The difference between both sides of equation 1 provides a decision value to judge whether the points belong to a circle or not.

    Figure 1(b) shows an example of an event. The red points represent the PMT hits in the spots. With a given set of three initial hits (A, B, C), any other fourth hit (D) can be tested if it belongs to a cyclic quadrilateral and therefore can be considered


  • A




    (a) A cyclic quadrilateral






    D D

    (b) Work scheme of the algorithm, red points represent hits and the marked points (A-D) are recognized by the algorithm. Any point in the brown band will be found in the recollection step.

    Figure 1: Ptolemy’s Theorem and the algorithm’s working principle

    to belong to a ring. Due to the finite size of the PMTs, the hits are not perfectly on a ring, any fourth point is considered to be on a circle formed with the initial hits if the difference is less than 300 mm2 1. After testing all hits with a given choice of initial start points, one will end up with a set of points that can be considered for a least squares fit. The results of the first fit define an acceptance region in which points that are not yet accepted by the Ptolemy theorem are recollected and added to the list of hits. All hits within a range of 15 mm are accepted or subtracted if they are too far away.

    For each event there will be several combinations of initial points to be tested. These trials will result in different sets of hits that might represent a ring and the sample with the highest number of hits is chosen. If the event contains more than one ring, the list of hits is cleaned and the algorithm runs again, trying to find another ring using the remaining hits only.

    3 Monte Carlo generators In order to test this algorithm, two different Monte Carlo generators have been used. On the one hand, a simple Toy Monte Carlo generator that uses only the PMT positions has been developed. All points are uniformly distributed over a circle and afterwards matched to the nearest PMT if the distance to this PMT is smaller than the radius of the Winston cone. Otherwise the point is dropped due to the inefficiency of the PMT positioning. In order to keep this Monte Carlo generator simple, the points are only generated on one spot that is situated in the center of an arbitrary reference system. On the other hand, there is the Monte Carlo generator using a Geant4 simulation of the detector. Since there will be two PMT spots in the final experiment, these two spots are implemented in the Geant4 simulation. For the purpose of using the same algorithm for both Monte Carlo generators, the hits

    1The cut value is chosen on the basis that in an event with only one ring at least 90 % of the combinations fulfill equation 1. (see appendix, figure 8)


  • from both spots are merged into one spot. For a later stage of the algorithm the spots should be considered independent and the algorithm should try to find a ring in each spot first, and only merge the spots at the latest stage.

    4 Results The first tests of the algorithm have been performed on events that only contain one ring. Therefore all hits should belong to the same ring and there are no misplaced hits expected.

    Figure 2(a) shows a typical one ring event. Hits are marked with blue dots and the fit result is represented by a red line. The comparison between true and fitted values for radius and center is satisfactory (cf. fig. 2(a)). In nearly all of the single ring events it is possible to collect all of the hits belonging to that ring. Furthermore, the resolution of the radius is quite good as shown in fig. 2(b). The standard deviation is about 1.5 mm (Toy Monte Carlo) respectively 1.2 mm (Geant4 Monte Carlo) at 25 GeV.

    (a) Single ring event. Hits are represented by blue points, the fit with a red and the true cir- cle with a black line. Fit-Results (MC truth): R = 175.6 (175.8), XC = −61.2 (−60.5), YC = 75.2 (72.3) (unit: mm)

    / ndf 2χ 65.81 / 44

    const 7.0± 794.6

    µ 0.0089± 0.2627

    σ 0.008± 1.147

    ) [mm]True - RFit R (R∆ -10 -5 0 5 10

    E n

    tr ie











    / ndf 2χ 65.81 / 44

    const 7.0± 794.6

    µ 0.0089± 0.2627

    σ 0.008± 1.147

    (b) Resolution for single ring events. Events generated by Geant4 Monte Carlo simulation with pion test beam, 25 GeV

    Figure 2: Single Ring Event and resolution of the radius

    The resolution of the Geant4 Monte Carlo seems to depend on the energy whereas the Toy Monte Carlo does not show such a dependence2. This is due to the fact that in the Toy Monte Carlo a fixed number of hits is used while the number of cherenkov photons that are emitted by a charged particle transversing neon gas depends on the energy of the particle. Figure 3 does not only show that the resolution depends on the number of hits used for the fit, but also that Toy and Geant4 Monte Carlo are in good agreement. Therefore, the resolution varies between 2.5 mm and 1.0 mm.

    After determining the resolution of the one ring track finding algorithm, the next test has been performed to determine the capability of the algorithm to find more

    2compare appendix, fig 9


  • number of hits per ring 5 10 15 20 25 30 35 40







    2.5 Toy Monte Carlo Geant4 Monte Carlo

    Figure 3: The reso- lution of the radius depends on the num- ber of hits. Fur- thermore, Geant4 and Toy Monte Carlo are in good agreement de- spite the fact that in Geant4 Monte Carlo a mean number of hits was extracted by a gaussian fit from the the distribution of the number of hits.

    than two rings in the same event. Therefore the algorithm was extended as described in section 2. A typical three ring event can be seen in fig. 4. The different colors of the hits belong to different particles that produced those hits. Furthermore, the three red circles are the results of the fit and in addition to those, the black rings represent the true generated ring positions. The agreement between generated and fitted rings is satisfactory and fig. 4 shows that the algorithm is able to distinguish between two rings positioned close to each other. On the other hand, this event shows the limitations of the algorithm as well, because in the crossing region the algorithm cannot distinguish between the two circles.

    Figure 4: Three ring event, Toy Monte Carlo. Each color belongs to one paricle producing those hits. The fit is represented by a red and the true circle by a black line. F