Trackless ring fitting algorithmfor the RICH detector

Evelina Gersabeck, Gianluca Lamanna, Antonino Sergi, Soren Stamm

16.11.2011

Contents1 Introduction 2

2 Trackless ring finding algorithm 2

3 Monte Carlo generators 3

4 Results 4

A Appendix 8

1

1 IntroductionThe aim of the CERN NA62 experiment is to measure the branching ratio of thevery rare kaon decay K+ → π+νν with high precision. The theoretical predictionfor the branching ratio of this decay is (0.85 ± 0.07) · 10−10 and allows a test ofthe Standard Model by extracting the CKM parameter |Vtd|. The experiment usesa 400 GeV proton beam provided by the SPS in order to produce kaons with amomentum of 75 GeV. The difficult task in this fixed target experiment is to achievea signal to background ratio (S/B) of about 10:1 because of the large backgroundfrom all other kaon decays. This S/B ratio can be accomplished with using severalredundant measurements of the kinematics and detectors which are able to provideparticle identification.

This project is focused on the RICH detector placed downstream of the decayregion behind the spectrometer. The task of the RICH is to provide a separationbetween pions and muons in the momentum range from 15 – 35 GeV to reducebackground 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 of20 mirrors will be located at the end of the vessel in order to focus the cherenkovlight onto two spots. These spots consists of 2000 photomultipliers (PMT) whichcan detect the cherenkov light redirected by Winston cones (�18 mm) to the activeregions of the PMTs.

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

2 Trackless ring finding algorithmThe trackless ring finding algorithm for the RICH needs to fulfill certain criteria.Besides a good resolution for the reconstructed center and radius, the algorithmneeds 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 inputinformation 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 toselect 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 judgewhether the points belong to a circle or not.

Figure 1(b) shows an example of an event. The red points represent the PMT hitsin 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

2

A

B

C

D

(a) A cyclic quadrilateral

A

C

B

D

D

DD

(b) Work scheme of the algorithm, red pointsrepresent hits and the marked points (A-D) arerecognized by the algorithm. Any point in thebrown 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 ona ring, any fourth point is considered to be on a circle formed with the initial hitsif the difference is less than 300 mm2 1. After testing all hits with a given choice ofinitial start points, one will end up with a set of points that can be considered fora least squares fit. The results of the first fit define an acceptance region in whichpoints that are not yet accepted by the Ptolemy theorem are recollected and addedto the list of hits. All hits within a range of 15 mm are accepted or subtracted ifthey 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 thesample with the highest number of hits is chosen. If the event contains more thanone ring, the list of hits is cleaned and the algorithm runs again, trying to findanother ring using the remaining hits only.

3 Monte Carlo generatorsIn order to test this algorithm, two different Monte Carlo generators have beenused. On the one hand, a simple Toy Monte Carlo generator that uses only thePMT positions has been developed. All points are uniformly distributed over acircle and afterwards matched to the nearest PMT if the distance to this PMT issmaller than the radius of the Winston cone. Otherwise the point is dropped due tothe inefficiency of the PMT positioning. In order to keep this Monte Carlo generatorsimple, the points are only generated on one spot that is situated in the center of anarbitrary reference system. On the other hand, there is the Monte Carlo generatorusing a Geant4 simulation of the detector. Since there will be two PMT spots in thefinal experiment, these two spots are implemented in the Geant4 simulation. Forthe 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 thecombinations fulfill equation 1. (see appendix, figure 8)

3

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

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

Figure 2(a) shows a typical one ring event. Hits are marked with blue dotsand the fit result is represented by a red line. The comparison between true andfitted values for radius and center is satisfactory (cf. fig. 2(a)). In nearly all ofthe 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). Thestandard deviation is about 1.5 mm (Toy Monte Carlo) respectively 1.2 mm (Geant4Monte Carlo) at 25 GeV.

(a) Single ring event. Hits are represented byblue 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

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(b) Resolution for single ring events.Events generated by Geant4 Monte Carlosimulation 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 whereasthe Toy Monte Carlo does not show such a dependence2. This is due to the fact thatin the Toy Monte Carlo a fixed number of hits is used while the number of cherenkovphotons that are emitted by a charged particle transversing neon gas depends on theenergy of the particle. Figure 3 does not only show that the resolution depends onthe number of hits used for the fit, but also that Toy and Geant4 Monte Carlo arein 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 nexttest has been performed to determine the capability of the algorithm to find more

2compare appendix, fig 9

4

number of hits per ring5 10 15 20 25 30 35 40

σ

0

0.5

1

1.5

2

2.5 Toy Monte CarloGeant4 Monte Carlo

Figure 3: The reso-lution of the radiusdepends on the num-ber of hits. Fur-thermore, Geant4 andToy Monte Carlo arein good agreement de-spite the fact that inGeant4 Monte Carlo amean number of hitswas extracted by agaussian fit from thethe distribution of thenumber of hits.

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

Figure 4: Three ringevent, Toy Monte Carlo.Each color belongs toone paricle producingthose hits. The fit isrepresented by a redand the true circleby a black line. Fit-Results (MC truth):R1 = 159.6 (155.1),R2 = 130.4 (132.2),R3 = 139.6 (142.6),XC1 = −116.4 (−131.2),XC2 = −5.2 (−4.4),XC3 = −115.0 (−110.0),YC1 = 64.7 (−65.1),YC2 = −93.1 (−95.1),YC3 = 104.1 (103.5)(unit: mm)

Moreover, several checks have been implemented to extract the performance ofthe algorithm. The collection efficiency that is defined as follows

Number of Hits on ringNumber of possible Hits for this ring (2)

5

is around 77 %, respectively 78 %, for three ring events if the algorithm uses thePtolemy theorem only. This can be improved up to 85 % if one take the collectionstep into account. Figure 5 shows the collection efficiency at the different stepsof the algorithm and consistent shapes for Toy and Geant4 Monte Carlo. This

possible #hits on this ring#hits on ring

0 0.2 0.4 0.6 0.8 1

En

trie

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410Toy Monte Carlo, 0.77

Geant4 Monte Carlo, 0.78

efficiency of Ptolemy's theorem only

(a) excluding collection step

possible #hits on this ring#hits on ring

0 0.2 0.4 0.6 0.8 1

En

trie

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Toy Monte Carlo, 0.83

Geant4 Monte Carlo, 0.85

efficiency after collection step

(b) including collection step

Figure 5: Collection efficiency before and after collection step

is an import fact because it indicates that effects only depending on the specificgeometry and is mostly independent of other physics effects. In events with morethan one ring, it is possible to collect hits from other rings during the ring findingprocess. Consequently, the efficiency will be reduced in particular for the secondand third ring. Figure 6 shows on the one axis the efficiency and on the other axisthe difference between fitted radius and true Monte Carlo value. Consequently, thetrue radius can still be found only using down to 60 % of the total number of hits ofthis ring. On the other hand, there are a large number of events that contribute tothe background region (|∆R|>15). Those events might contain one ring that doesnot match a physical counterpart but represents an artificial combination of points.Nevertheless, the overall resolution shown in figures 7(a) (Toy Monte Carlo) and7(b) (Geant4 Monte Carlo) is still good and can be quoted to be around 2 mm. It isnot surprising that the resolution is not as good as the one ring resolution becausethe resolution depends on the number of hits that are used for the fit. The resolutionfor the center is around 2.6 – 3.2 mm for Geant4 Monte Carlo and about 2.4 mm forToy Monte Carlo. These values might as well depend on the number of hits for eachring.3

The time for processing per event is about 0.4 ms in the case of one ring only.For more than one ring this depends very much on the number of hits and a timebetween 1 ms and 1.8 ms quoted. There is no optimization done so far and the codemight be running faster if the trials are computed in parallel.

3compare figures 10 and 11

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)True - RFit

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Figure 6: Collection efficiency and comparison between true and fitted radius. Itis shown, that it is possible to find a good fit result even with a lower collectionefficiency.

/ ndf 2χ 51.6 / 44

const 8.1± 1229

µ 0.0139± 0.1664

σ 0.021± 1.907

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const 6.9± 951.3

µ 0.0126± 0.5497

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(b) Geant4 Monte Carlo

Figure 7: Resolution of the radius in the case of three ring events. The resolutiondepends on the number of hits and therefore it is slightly different for Toy andGeant4 Monte Carlo.

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A Appendix

Ptolemy Test Value0 500 1000 1500 2000

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Figure 8: Cut Valuefor the Ptolemy Theo-rem, the region abovethe red line includes90 % of all combina-tions, slightly dependson the energy.

ring radius [mm]100 120 140 160 180 200

σ

0

0.5

1

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2Toy Monte CarloGeant4 Monte Carlo

Figure 9: Resolutionand ring radius. Theresolution seem to bedependent on the en-ergy, respectively, onthe ring radius, butthe underlying effect isthe number of hits thatare produced per ring.Compare with fig. 3.

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/ ndf 2χ 24.52 / 16

const 14.5± 1486

µ 0.02167± 0.03579

σ 0.026± 2.363

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/ ndf 2χ 50.48 / 22

const 12.6± 1408

µ 0.02± -67.05

σ 0.021± 2.646

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const 12.6± 1408

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σ 0.021± 2.646

(b) Geant4 Monte Carlo

Figure 10: Resolution for the center position XC (test beam, 25 GeV) The differencebetween Toy and Geant4 Monte Carlo might depend on the number of hits as well.The offset in X results from the fact, that the RICH reference frame is different fromthe experiment reference frame.

/ ndf 2χ 6.957 / 16

const 14.5± 1494

µ 0.02156± 0.01116

σ 0.026± 2.359

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const 10.7± 1202

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const 10.7± 1202

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σ 0.028± 3.239

(b) Geant4 Monte Carlo

Figure 11: Resolution for the center position YC (test beam, 25 GeV). The differencebetween Toy and Geant4 Monte Carlo might depend on the number of hits as well.Since the global frame is only shifted in the x direction, there is no offset for YC .

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References[1] Ceccucci et al. Proposal to Measure the Rare Decay K+ → π+νν at the CERN

SPS. Technical Report CERN-SPSC-2005-013. SPSC-P-326, CERN, Geneva,Apr 2005.

[2] M. Lenti. The NA62 RICH detector. Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detectors and Associ-ated Equipment, 639(1):20 – 21, 2011. Proceedings of the Seventh InternationalWorkshop on Ring Imaging Cherenkov Detectors.

[3] Gianluca Lamanna. 2 rings trackless fitting for L1/L0 RICH trigger using GPU.TDAQ Meeting, May 2011.

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