Muonic Event Reconstruction in Km3net-ORCA

MSc Physics and Astronomy Gravitation and Astroparticle Physics Amsterdam

Master Thesis

Muonic event reconstruction in KM3NeT-ORCA

written by

Max Merlijn Briel UvA ID: 10606513

September 2019 60 EC 2018-2019

Supervisor/Examiner: Second Examiner Dr. Ronald Bruijn Dr. Kazu Carvalho Akiba Abstract

The KM3NeT collaboration aims to solve the unknown neutrino mass ordering by measuring the direction and energy flux of atmospheric neutrinos. At a depth of over 2 kilometres in the Mediterranean Sea, it uses photomultiplier tubes (PMTs) to record Cherenkov radiation from the high-energy neutrino reaction products. A charged current interaction with a muon neutrino leaves a path, named a track, of photon hits through the detector. Using the hits and the properties of the PMTs, the direction of the track is approximated in a process called reconstruction and comprises two successive algorithms. The first generates a set of starting values for the second main algorithm, which uses a maximum likelihood method to find the best fitting track to the data. This research determines that the initial directional accuracy the main reconstruction has to be 10° or lower for the algorithm to achieve its full potential. Since the first algorithm only reaches this limit in 10% of the events, we implement new quality functions and likelihoods in the context of re-ranking and minimisation. They contain different parametrisations of the hit information, which have been developed for this work. We show that the ranking of the main algorithm is optimal in selecting the best track set, but a new likelihood improves their internal ranking. Minimisation with the new likelihoods increases the percentage below the limit up to 25%. A more complete likelihood is introduced with hit and no-hit information. Its likelihood space shows potential, and minimisation with the truth parameters leads to a significant improvement in the mean directional error from 1.92° to 0.81° compared to the current main reconstruction. 60% of the reconstructed tracks are now at a sub-degree accuracy. However, its likelihood contains many local minima and the pre-reconstruction is not optimised for this comprehensive likelihood putting it at a disadvantage. With better optimisation of the full reconstruction chain, the extended algorithm could provide better constraints on the direction and the neutrino mass ordering. Contents

1 Introduction 3

2 Neutrino Oscillations 6 2.1 The History of the Neutrino ...... 6 2.1.1 β decay ...... 6 2.1.2 Neutrino Discoveries: νe νµ ντ ...... 6 2.2 Neutrino Interactions ...... 7 2.3 Neutrino Oscillations ...... 8 2.4 Matter Oscillations ...... 11 2.4.1 Usage within KM3NeT-ORCA ...... 12 2.5 Current Parameter Constraints ...... 12

3 KM3NeT Infrastructure 14 3.1 Neutrino Telescopes: ARCA & ORCA ...... 14 3.1.1 Photomultiplier Tubes ...... 15 3.2 Detection Principle ...... 15 3.2.1 Cherenkov Emission ...... 15 3.2.2 Interaction Signatures ...... 16 3.3 Background Sources ...... 18 3.4 Data Acquisition ...... 18 3.5 Monte Carlo Simulation ...... 19

4 Current Muonic Event Reconstruction and PDFs 20 4.1 Track Parameters ...... 20 4.2 Maximum Likelihood Method ...... 21 4.3 PMT’s Probability Density Functions ...... 21 4.3.1 Light Emission ...... 22 4.3.2 Light Propagation ...... 23 4.3.3 Light Detection ...... 24 4.3.4 Direct and Indirect Light as PDFs ...... 25 4.4 The Pre-reconstruction: JPreﬁt ...... 27 4.5 The Main Reconstruction: JGandalf ...... 28 4.6 Other Algorithms: JStart & JEnergy ...... 28 4.7 Intrinsic Limits of the Reconstruction ...... 28

5 Evaluation of JGandalf’s Input 30 5.1 The Positions of JGandalf’s Minima ...... 30 5.2 Quantifying JGandalf’s Directional Input ...... 32 5.2.1 Deviates from the True Direction ...... 32 5.2.2 Deviates using JPreﬁt tracks ...... 33 5.3 JGandalf’s Input Limits ...... 36

1 5.4 Analysis of JPreﬁt’s Output ...... 36

6 Improvements to JGandalf’s Input 38 6.1 Optimising JPreﬁt’s Quality Function ...... 38 6.2 Introducing the New Likelihoods ...... 39 6.3 Parametrisation of np.e...... 41 6.4 Re-ranking of JPreﬁt tracks ...... 45 6.5 Minimisation ...... 47 6.6 Discussion of Input Improvements ...... 49

7 JGandalf Upgraded: JMerlin 52 7.1 JMerlin’s Likelihood Space ...... 52 7.2 Re-ranking of JPreﬁt and JGandalf tracks ...... 54 7.3 Minimisation ...... 54 7.4 JMerlin’s Input Requirements ...... 55

8 Discussion & Conclusion 59 8.1 Analysis of JPreﬁt and JGandalf ...... 59 8.2 Upgrading JPreﬁt’s output ...... 59 8.3 Extending JGandalf: JMerlin ...... 60 8.4 Future Improvements ...... 61 8.5 Conclusion ...... 62

Appendices 63

A Shower Emission Proﬁles 64

B JGandalf −LL Scans 66

C Global Minimum Scans 70

D JPreﬁt Quality Factor Exploration 71

E M-Estimator 73

F Parametrisation of PDFs 76 F.1 Individual Distance parametrisation ...... 76 F.2 Angle parametrisation ...... 78

G JGandalf vs JGandalfx vs JMerlin 92

2 Chapter 1

Introduction

The Standard Model is the best description of the subatomic world. It contains the 12 known particles, their antiparticles, three fundamental forces with their 4 gauge bosons, and generates mass using a single scalar boson [1–3]. The theory results from many years of theoretical and experimental research and has predicted the existence of new particles, such as the W ±/Z bosons [4–7], gluon [8–11], and the Higgs [12, 13]. The model, however, is incomplete and contains only three of the four fundamental forces. Gravity has been difficult to merge with the microscopic scale of particles. At this moment, its extreme weakness allows it to be safely ignored in experimental predictions [1]. More pressing issues with the Standard Model are at the small scales, such as neutrino masses. Early neutrino experiments, sensitive to only solar electron neutrinos, measured fewer interactions than predicted [14–20]. The most natural explanation was an in-flight flavour change of the neutrino [21]. For example, when an electron neutrino (νe) is created with energy Eν and travels a distance L, it has a probability to be detected as a muon neutrino (νµ). Through the transition νe → νµ, muon neutrinos appear in the detector, while electron neutrinos disappear, resulting in less than expected νe. Experiments measuring all three neutrino types were consistent with predictions and confirmed the neutrino mixing [22–30]. As a result, neutrinos are massive, which is in contrast with the massless Standard Model neutrinos. The mass states of the three neutrinos have to be unique and the flavour states are a linear combination of these mass eigenstates [31, 32]. The current neutrino oscillation framework comprises three flavours (νe, νµ, ντ ) and three mass eigenstates (ν1, ν2, ν3), although it is still an open question whether more non- interacting neutrinos exist [33]. Eν and L, and a set of constants determine the mixing probability. Most of them are contained within the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix) [31,32], which parametrises the mixing using three mixing angles and one CP violating phase, similar to the quark sector [34,35]. The other constants are the squared mass differences between the mass eigenstates. With three eigenstates, two mass differences are independent. So far, experiments are unable to measure the sign of one of the mass differences, which results in a degeneracy in the mass ordering. This is known as the Mass Hierarchy problem, also named the Mass Ordering problem. A promising method for determining the neutrino mass ordering is to exploit the effect of matter on the oscillation probability of neutrinos and antineutrinos [36], which alters the oscillation probability depending on energy, distance travelled, and the electron

3 density of the matter traversed. The ORCA (Oscillation Research with Cosmics in the Abyss) telescope of the KM3NeT (Cubic Kilometer Neutrino telescope) infrastructures plans to use this method with neutrinos and antineutrinos produced by cosmic rays. Cosmic rays are highly energetic charged particles that hit the atmosphere and create a shower of particles including νe/µ and ν¯e/µ. These atmospheric neutrinos oscillate through the Earth and create high-energy interaction products, such as muons, taus, and electrons. ORCA’s detector structure is optimised for the 3 to 20 GeV range, where matter effects have a strong presence. It is, however, unable to distinguish between neutrino flavour of the interaction directly. Instead, the detector can only separate two interaction signatures: showers and tracks. Neutrinos interacting with hadrons or electrons through the neutral current cause the former, while charged current interactions from electron neutrinos result in a similar signal. Tracks, on the other hand, are caused by muons that travel far through the detector. They originate from charged current muon neutrino interactions, but the decay of a tau can also generate muons. This thesis focuses on muonic events because of the excellent directional information provided by the long path of the muon through the detector. The muon’s direction links back to the amount of matter traversed by the neutrino. Together with the neutrino’s energy, they are essential in determining the mixing angles, mass differences, and mass ordering. To balance the extremely small neutrino interaction rate, the detector will encompass 5 megatons of sea water. At a depth of over 2 kilometres in the Mediterranean Sea, the ORCA detector uses photomultiplier tubes (PMTs) housed in pressure-resistant glass spheres called Digital Optical Modules (DOM) to record Cherenkov radiation from the high-energy reaction products. When a photon is registered on a PMT, it records a hit containing the time and duration of the measured signal. From a collection of such hits in combination with the direction and orientation of the PMTs (an event), we approximate the original path of the muon through the detector. This process is called reconstruction and consists of two successive algorithms: JPrefit and JGandalf. The former generates a set of 36 starting values for the latter. The JGandalf algorithm implements a maximum likelihood method to find the best fitting track to the data using these given initial guesses. However, it is sensitive to the directional accuracy of the tracks provided by JPrefit. Besides quantifying the sensitivity, this research implements new likelihood functions to improve the directional accuracy of the JPrefit tracks. The probability a PMT was hit during an event can provide additional information for a better directional reconstruction. In this research, different parametrisations of these probabilities are created from the analytical description of the emission, propagation, and detection of light from high energy charged particles. The new likelihoods with parametrisations are used to re-rank or minimise the JPrefit tracks to give better starting values for JGandalf. The absence of light on a PMT also provides information on the muon’s path through the detector. If a PMT is not hit during an event, it is known as a no-hit. Multiple no-hits can indicate a region where the track cannot have passed through due to the lack of hits. Using the full probability functions, a new more complete likelihood with hit and no-hit information is implemented: JMerlin.

4 This thesis starts with a description of the neutrino oscillation framework and the KM3NeT detector in Chapter 2 and 3, respectively. Chapter 5 evaluates JGandalf and the current tracks provided by the JPreﬁt algorithm. The tracks are improved upon in Chapter 6 using the likelihood with parametrisations. JMerlin is introduced and tested in Chapter 7. Chapter 8 ends the thesis with a discussion and conclusion about the new likelihoods and their implementation.

5 Chapter 2

Neutrino Oscillations

2.1 The History of the Neutrino

2.1.1 β decay The history of the neutrino starts at the end of the 19th century with the discovery of β− decay by Ernest Rutherford [37]. The nucleus changes charge from Z to Z + 1 alongside the emission of an electron depicted in 2.1. If we assume the interaction to be a two-body decay, as depicted, the electron’s energy spectrum is constant. Experiments have instead shown that the spectrum is continuous [38].

A(N,Z) → A(N − 1,Z + 1) + e− (2.1) One of the many proposed solutions came from Wolfgang Pauli. He suggested a 1 third light particle with a mass of 0.01mp, a spin 2 , and named it ”neutron”. After the discovery of the neutral nucleon, the neutron [39], the idea was forgotten until Fermi implemented the light particle in his theory for β-decay. He gave it the name we know today: neutrino; ”the light neutral one” [40].

2.1.2 Neutrino Discoveries: νe νµ ντ When Pauli included the particle in his theory, he worried it would be extremely difficult to detect [40]. Neutrino cross section calculations strengthened this idea, and the particle was expected to remain undetected [41]. With the rise of nuclear fission, intense neutrino and anti-neutrino sources became available, which vastly increased the chance of measuring a neutrino interaction. In the 40s and 50s, many experiments used nuclear reactors as their neutrino source [42–44]. In 1956, Reines and Cowan successfully measured a neutrino interaction using inverse beta decay (Eq. 2.2) for the first time [45]. An electron antineutrino from the nuclear reactor interacts with a proton and emits a positron and neutron. The positron immediately annihilates with electrons in the medium to photons, while a nucleus captures the neutron and emits delayed light allowing for background reduction [46].

ν¯ + p → e+ + n (2.2)

6 νl l ν¯l ¯l νl(¯νl) νl(¯νl)

W − W + Z0

f f 0 f f 0 f(f 0) f(f 0) (a) (b) (c)

Figure 2.1: Charged Current neutrino (a), Charged Current antineutrino (b), and Neutral Current (c) interactions, where l (¯l) is a (anti)lepton and f (f¯) is a (anti)fermion

After the discovery of the electron neutrino, it only took six years to find the muon ± neutrino [47]. Muons were observed to originate from π decays. Thus, a νµ had to be involved for a conserved interaction. The discovery of the tau lepton suggested a third neutrino flavour [48], which CERN experiments, such as DELPHI, indirectly confirmed through the Z0 decay width [3]. A direct measurement of a tau neutrino interaction took till 2000, when the DONUT collaboration imaged the interaction [49].

2.2 Neutrino Interactions

Neutrino interactions are complex and depend heavily on the target particles. As the literature thoroughly discusses many aspects of the interactions [46, 50, 51], this thesis will summarise those relevant for the KM3NeT-ORCA detector. The weak force mediates the interactions between the neutrino and ordinary matter [3], in which only left-handed flavour eigenstates of ν can partake. In the neutral current (NC) channel, the neutrino remains a neutrino, as seen in Figure 2.1c. Salam, Glashow and Weinberg postulated this channel [52–54] and the Gargamelle neutrino collaboration at CERN confirmed it in 1973 [55]. During the interaction, only some momentum of the neutrino is transferred to the other particle through a neutral Z boson. ν detection experiments used the charged current (CC) channel to confirm the neutrino’s existence. The neutrino transforms into a charged lepton of the same flavour (Figure 2.1a). It determines the flavour of the neutrino, while a W ± boson mediates the charge. In ORCA’s energy region from 3 to 20 GeV, the CC interaction cross section is 3 times larger than the NC channel [56]. For each interaction, an opposite one exists for the antimatter counterpart with one such example being depicted in Figure 2.1b. In the energy range of the KM3NeT-ORCA detector different types of interactions can take place between the neutrino and nucleons, such as (quasi-)elastic scattering, resonance production, and deep inelastic scattering [51]. Their combined cross section for neutrinos is twice as large as those for antineutrinos [56], as shown in Figure 2.2.

7 1.4 0.4 TOTAL / GeV) / GeV) 2 2 0.35 1.2 cm cm cm 0.3 -38 -38 1 (10 (10 TOTAL 0.25 0.8 0.2 DIS QE QE RES 0.6 0.15 DIS 0.4 0.1 RES 0.2 0.05 cross section / E cross section / E 0 0 10-1 1 10 102 10-1 1 10 102 E (GeV) E (GeV)

Figure 2.2: Nucleon interaction cross sections of the neutrino (left) and antineutrino (right) on an isoscalar target. The diﬀerent scattering processes are marked with RES for RESonance production, QE for Quasi-Elastic scattering, and Deep Inelastic Scattering as DIS. Figures taken from [51].

The KM3NeT’s target particle, however, is not a single proton or neutron. Their bound state in the water molecules increases the interaction rate with 1% through coherent scattering. Other eﬀects include the motion of the nucleon in the nucleus [57,58] and ﬁnal state interactions [59]. In the high energy regime, the relative contribution of the motion is negligible, and the KM3NeT infrastructure cannot distinguish individual particles, except the muon.

2.3 Neutrino Oscillations

As seen in Section 2.2, a neutrino takes part in an interaction with a specific flavour: electron (νe), muon (νµ), or tau (ντ ). These are eigenstates of the weak interaction, so the ν is created in one of these three flavours. A neutrino, however, does not have to stay in its original flavour state. The Homestake experiment measured the νe flux from the sun using a chlorine bath to capture neutrinos using inverse beta decay, which creates measurable radioactive argon, see Equation 2.3. Only around 1/3 of the expected electron neutrino flux was measured [14,17] and this deficit became known as the Solar Neutrino Problem.

ν + 37Cl → e− + 37Ar (2.3) The most natural solution was already proposed many years earlier as a reaction to a previous experiment from Davis using the same reaction 2.3 to measure reactor antineutrinos, which should be impossible. However, rumours spread that he had found such events and reached Bruno Pontecorvo, who came up with neutrino-antineutrino oscillations as an explanation [60, 61]. While the rumours turned out to be wrong, the idea of oscillations stayed. After the discovery of the second generation of neutrinos,

8 Maki, Nakagawa, and Sakata took the idea and proposed that the ”true neutrinos” are a linear combination of different neutrino flavours [32]. Pontecorvo extended this idea to include flavour oscillations between the different flavours, such as νe νµ [31]. These oscillations were the most natural solution to the Solar Neutrino Problem [21] and have major consequences. The neutrinos need to have distinct masses, and the flavour states are a linear combination of these mass eigenstates. X |να(t = 0)i = Uαj|νji with (α = e, µ, τ & j = 1, 2, 3), (2.4) j

in which Uαj, the Pontecorvo, Maki, Nakagawa, and Sakata (PMNS matrix), describes the mixing of the mass states in the ﬂavour eigenstates. 3 mixing angles and 6 phases parametrise this unitary matrix [3]. The number of phases that are real depends on the type of neutrino, Dirac or Majorana [34, 62]. In neutrino oscillation experiments, the diﬀerence is unimportant and the simplest form, the Dirac case, can be considered [63]. Five of the six phases are absorbed and one charge-parity violating phase (δCP ) remains. Together, with the three mixing angles, they parametrise the PMNS matrix, similar to the CKM matrix in the quark sector [3]. This is shown in Equation 2.5, where cij and sij are the cosine and sinus of the mixing angles: θ12, θ13, and θ23.

 −iδ c12c13 s12c13 s13e iδ iδ U = −s12c23 − c12s13s23e c12c23 − s12s13s23e c13s23  iδ iδ s12s23 − c12s13c23e −c12s23 − s12s13c23e c13c23    −iδ    1 0 0 c13 0 s13e CP c12 s12 0 = 0 c23 s23  0 1 0  −s12 c12 0 (2.5) iδ 0 −s23 c23 −s13e CP 0 c13 0 0 1 Neutrinos are always created with a specific flavour depending on the associated charged lepton, but each mass eigenstate propagates with its own phase factor: e−iEj t. The initial flavour state becomes a mixture of the other flavour states (Eq. 2.6).

X ? X ? −iEj t |νl(t)i = Uαj |vj(t)i = Uαj e |vj(t = 0)i j j

X ? −iEj t X = |νl(t)i = Uαj e Uβi|νβi (2.6) j β The probability of measuring a speciﬁc state is given by the probability amplitude of the interaction. For measuring a να as ﬂavour β with α 6= β the probability is:

2 X ? −iEj t P (να → νβ) = |hνβ|να(t)i| = UβjU e (2.7) αj j In the ultra-relativistic limit (E ≈ p), where the neutrinos of many experiments reside, Ej can be rewritten and Taylor expanded. In natural units, the time and distance travelled become equivalent (t ≈ L), and Ej simpliﬁes to:

9 ! ! q m2 m2 m2 m2 E = p2 + m2 = p 1 + j ≈ p 1 + j = p + j = E + j (2.8) j j p2 2p2 2p 2E

We can rewrite the oscillation probability to a real and imaginary part. The real part is charge-parity invariant, while the imaginary part is charge-parity violating and switches sign for the antineutrino.

2 ! X ∆mji P (ν → ν ) = δ − 4 Re U U ? U ? U sin2 L α β αβ αi βj αj βj 4E i

Equation 2.9 shows that the neutrino oscillation probability depends on elements of 2 the PMNS matrix, the squared mass difference between mass eigenstates (∆mji), the neutrino’s energy (E), and the distance travelled (L). To show the dependence on the PMNS matrix elements explicitly, we consider the two-flavour ν case. In this limit, the oscillation probability depends only on a single mass difference and a single mixing angle, as shown in Equations 2.10 and 2.11.

∆m2L P (ν → ν ) = 1 − sin2 (2θ) sin2 , (2.10) α α 4E

∆m2L P (ν → ν ) = sin2 (2θ) sin2 (2.11) α β 4E In the full neutrino flavour model, three mass differences are distinguished with two 2 2 being independent. The convention defines these as ∆m21 and ∆m31. Together they can 2 2 2 2 describe ∆m32 = ∆m31 − ∆m21, but a degeneracy remains. Since the sign of ∆m31 is unknown, it is impossible to determine the ordering of the neutrino mass eigenstates. 2 ∆m21 is defined as positive with m1 < m2 from experimental data. m3 can either be heavier than m2 or lighter than m1. This is the neutrino mass ordering or mass hierarchy problem. The first case, m1 < m2 < m3, is referred to as the normal ordering with 2 2 ∆m31 > 0. In the inverted ordering case, m3 < m1 < m2, ∆m31 is negative. Determining the exact order has become a focus of neutrino oscillation experiments, because it rules out or strengthens around half the proposed particle physics models [64]. It will help constraint result in leptonic CP violation, the absolute neutrino mass, and the Majorana-Dirac nature of the neutrino [65]

10 2.4 Matter Oscillations

0.08 Vacuum 0.07 Matter (Normal Ordering) − νe e 0.06 Matter (Inverted Ordering)

) NO e 0.05 ν

→ 0.04 Vac − µ ν

W (

P 0.03

0.02 IO 0.01 νe e− 0.00 2 4 6 8 10 Energy [GeV]

Figure 2.3: Electron and Figure 2.4: Electron neutrino appearance probability neutrino interaction called for a 1300 km baseline. Oscillation parameters are taken coherent forward scatter- from [66] and calculated using [67]. Vacuum oscillations ing resulting in the MSW for NO (dashed blue), matter oscillation for NO (blue) eﬀect. and IO (red) are shown.

The main idea to determine the neutrino mass ordering is to use the influence of the matter on the neutrinos. When travelling through a medium, the neutrino feels a potential caused by coherent forward scattering. The neutrino interacts with surrounding nucleons and electrons through the NC and CC interactions without disappearing [68]. Neutral current interaction influences all neutrino types equally, but CC interactions only take place between the electron (anti)neutrino and electron. This changes the effective mass and the oscillation probability of the neutrinos. In the two-flavour limit, we can rewrite this probability like their vacuum counterparts: ∆m2 L P (ν → ν ) = 1 − sin2 (2θ ) sin2 m (2.12) m α α m 4E ∆m2 L P (ν → ν ) = sin2 (2θ ) sin2 m (2.13) m α β m 4E

2 The matter alters the mixing angle θm and the mass diﬀerence ∆mm. But they still relate back to their vacuum values through Equations 2.14 and 2.15, where GF is Fermi’s constant, ne the electron density in the matter, and the sign depends on the involvement of a neutrino (+) or antineutrino (−).

√ 2 p 2 2 2 2 ∆mm = (∆m cos(2θ) − A) + (∆m sin 2θ) ,A = ±2 2EGF ne (2.14)

11 2 2 2 ∆m sin 2θ 1 A − ∆m cos 2θ sin θm = 2 = 1 + 2 (2.15) ∆mm 2 ∆mm From Equation 2.15,√ the amount of mixing between the two flavours can be maximised 2 when ∆m cos 2θ = 2 2GF neE. This resonance enhancement allows for mixing angles many times larger than in vacuum [68]. Whether the condition is satisfied depends on the neutrino mass ordering. With normal ordering, matter boosts the neutrino mixing, while with inverted ordering the antineutrino is boosted, as depicted in Figure 2.4. This is known as the Mikheyev-Smirnov-Wolfenstein effect, which fully explains the solar neutrino problem [69, 70]. Moreover, with a varying electron density, the mixing parameters themselves change, introducing an extra phase to the oscillations. The literature provides more details about this effect and other oscillation altering processes, such as parametric enhancement [71–73].

2.4.1 Usage within KM3NeT-ORCA To achieve the full eﬀectiveness of the resonance, a large mass is required. KM3NeT- ORCA employs the Earth for this purpose and measures atmospheric neutrino, which are products from cosmic charged particles interacting with the atmosphere. These so called cosmic rays are very energetic and create showers of particles including pions, which decay to νµ and νe. The neutrinos travel through the Earth towards the detector while oscillating. Since the matter boosts either the electron neutrinos or antineutrinos, their ﬂuxes provide information on the mass ordering. Details can be found in [36,74,75]. For the Earth, the resonance is largest in the region around 3 and 6 GeV, which is within ORCA’s sensitivity range [65].

2.5 Current Parameter Constraints

Three different research groups have constraint the parameters describing the neutrino oscillations using data from several experiments [66]. Table 2.1 shows their current values and limits. The leading mixing angles, θ12 and θ13 have been measured accurately, but θ23 still spreads a large part of the parameter space and δCP is almost unconstrained. At this moment, no preference for either mass ordering exists. The PMNS model has been shown to explain neutrino oscillations from experiments, but some anomalies remain. Gallium experiments have been measuring a lower rate from their radioactive sources than expected [76–83]. Furthermore, LSND [84, 85] and Micro- BooNE [86] have measured an excess in the low energy region of electron antineutrinos and is still unexplained [87]. Finally, a recalculation of the ν¯e flux showed that all short baseline reactor experiments have measured fewer events than expected [88–90]. More complete models for nuclear reactor decays have been suggested as a suitable solution for this reactor anomaly [91, 92]. For a full discussion of the neutrino anomalies, see [93]. Since most are only at a 2-3σ level, they might be a statistical fluke or the result of a

12 more complex neutrino theory [94]. For now, the 3ν scheme is the best current model in explaining neutrino oscillations.

Parameter Normal Ordering Inverted Ordering +0.78 +0.78 θ12 [deg] 33.82−0.76 33.82−0.76 +1.0 +1.0 θ23 [deg] 49.6−1.2 49.8−1.1 +0.13 +0.13 θ13 [deg] 8.61−0.13 8.65−0.13 +40 +27 δCP [deg] 215−29 284−29 2 −5 2 +0.21 +0.31 ∆m21[10 eV ] 7.39−0.20 7.39−0.20 2 −3 2 +0.034 +0.034 ∆m3l[10 eV ] 2.525−0.032 −2.512−0.032

Table 2.1: Current neutrino oscillation parameters for both mass orderings from [66] 2 2 2 without Super-Kamiokande data. ∆m3l is ∆m31 for normal ordering and ∆m32 for inverted ordering. Errors are 1σ deviations.

13 Chapter 3

KM3NeT Infrastructure

3.1 Neutrino Telescopes: ARCA & ORCA

For a better understanding of neutrinos and their astrophysical sources, two next generation neutrino telescopes are being built in the Mediterranean Sea as part of the KM3NeT infrastructure [75]. The ARCA telescope, Astroparticle Research with Cosmics in the Abyss, will look at the universe to find sources of high-energy neutrinos. These travel undisturbed from their source to the Earth and provide information about the sources and the particle acceleration mechanisms. Recently, a similar ice-based Cherenkov neutrino telescope at Antarctica, IceCube, has identified a blazar as the first high-energy astrophysical neutrino source [95]. The neutrino’s energy ranges from a few TeV to the PeV range. The second KM3NeT telescope, ORCA, will look at neutrinos with GeV energies to measure the neutrino oscillation parameters. Moreover, it will use the method described in Section 2.4 to determine the mass ordering of the neutrino mass states. Both detectors use the same 3-inch Photomultiplier Tubes (PMTs) housed in pressure- resistant glass spheres called Digital Optical Modules (DOMs) to record Cherenkov radiation from high-energy neutrino interaction products, such as muons, taus, and electrons. Each DOM contains 31 PMTs and other electronics, such as an accelerometer, compass, and many more components. Detailed information about all the contained electronics in a DOM can be found in the KM3NeT 2.0 Letter of Intent and the technical report [75, 96]. 18 DOMs are combined in a long vertical string, called the Detection Unit (DU). 115 DUs with all support infrastructure is called a building block. The separation between DUs and DOMs is optimised for cost and performance per detector based on the expected neutrino flux and the energy regime. ARCA will be build 100km offshore from Porto Palo di Capo Passero, Sicily, Italy at a depth of 3500m. 40km from Toulon, France in the Mediterranean Sea at a depth of 2450m the ORCA detector is under construction. The ARCA detector has DUs with 36m between each DOM and an average of 90m between each DU. ORCA’s design is denser to measure GeV neutrinos. Its DOMs are only separated by 9 meters, and DUs by 20 meters. The full ARCA detector will consist of 2 building blocks, adding up to a total volume of around 1 km3, while the ORCA detector will only be 1 building block with a volume 3 orders of magnitude smaller, around 5.4 × 106 m3.

14 3.1.1 Photomultiplier Tubes The PMTs are photosensitive devices with a 3-inch diameter capable of measuring single photons with wavelength from 280nm to 720nm [97]. When a photon hits a PMT, the photocathode layer releases an electron through the photoelectric effect. The efficiency of this process is the quantum efficiency and is wavelength dependent. Accelerated by an internal electric field, the released electron will hit several dynodes in quick succession, each releasing more electrons. If the charge pulse at the final dynode exceeds a set threshold, the charge pulse information is passed to the electronic within the DOM and a so called ”hit” is registered. Further steps are described in Section 3.4. KM3NeT’s 31 PMTs design provides several advantages over non-multi-PMT designs, such as ANTARES’s, IceCube’s, and Baikal Gigaton Volume Detector’s (GVD). The smaller PMT size reduces cost per photosensitive area. Furthermore, physical signals are easily recognised by requiring simultaneous hits on multiple PMTs of the same DOM. During extremely bright events, when normally PMTs are over-saturated, away-pointing PMTs are still able to provide valuable information on the event. And finally, the direction of the individual PMTs provides essential information about the direction and location of the event [98].

3.2 Detection Principle

3.2.1 Cherenkov Emission Neutrino telescopes do not measure the neutrino directly, but through the photons from reaction products. As described in Section 2.2, when a neutrino interacts through CC interactions, it can create neutral and charged particles with an very high energies. When the latter move through a dielectric medium, their surroundings are excited and emit electromagnetic waves spherically. With a suﬃciently high velocity of the charge particle, the emitted waves in the direction of movement constructively interfere to a wave front. This cone-like emission is known as Cherenkov radiation. Figure 3.1 depicts how the spherical emission interferes to create the cone-like Cherenkov emission. 1 cos (θ ) = (3.1) C βn

The light has a characteristic emission angle, θC given by Equation 3.1. For a relativistic particle (β = 1) in seawater (n = 1.35) the Cherenkov angle is approximately 42°. The amount of Cherenkov light emitted around this angle is described by the Frank-Tamm equation [3]:

d2N 2π 1 = 1 − (3.2) dxdλ αλ2 n2β2 It describes the number of photons per distance travelled (dx) per wavelength (dλ) and depends on the wavelength λ of the emission. A charged particle will continue to emit Cherenkov radiation until its velocity β drops below 1/n.

15 ϑ v=βc c

Figure 3.1: Visualisation of Cherenkov emission for charged particles moving through a dielectric medium to the right (red arrow). The particle’s movement causes the medium to emit electromagnetic waves spherically, which interfere to create a cone-like emission due to the high velocity of the charged particle. This is called Cherenkov radiation (blue arrows).

3.2.2 Interaction Signatures Depending on the neutrino ﬂavour and type of interaction between the neutrino and nucleons, the KM3NeT-ORCA detector distinguishes three main detection signatures. When the energy of the neutrino is large enough, it can recoil a quark and destroy the original nucleon, creating hadronic particles. They interact and decay to create more and more hadrons, resulting in a hadronic shower. This process is part of all neutrino interaction in the detector. Except for the NC channel, it is combined with other signatures. These come from charged current interactions with each neutrino ﬂavour having its own unique signature. Electrons from electron neutrino interactions cause a cascade of photons and electron-positron pairs: an electromagnetic shower. A muon, on the other hand, propagates through the detector with an energy dependent straight path length, and is, thus, referred to as a track. Besides emitting Cherenkov radiation, the charged particles lose energy through ionisation (δ-rays), Bremsstrahlung, and e−e+ pair production. In the low GeV regime ionisation is the most dominant energy loss process, but the contribution from Bremsstrahlung

16 and pair production increases linearly with energy and is dominant above 1 TeV, which Section 4.3.1 discusses in more detail. The emitted photons through these processes can cause small electromagnetic showers along the track of a muon through pair production or Compton scattering. Tracks, hadronic, and electromagnetic showers are also a signature of tau neutrino interactions, where the tau decays either into a muon, hadron, or electron. These have distinct signatures from the normal interaction due to the time delay in the tau decay, but can be indistinguishable because of the position resolution of the ORCA detector. All interactions and their signatures are summarised in Table 3.1.

Interaction Particle signature Detector signature

ν µ CC hadronic shower and µ track

track-like

hadronic shower and µ track (τ µ ν ν , 17% BR) ± → ± µ τ ∼

hadronic and EM shower ν CC τ (τ e ν ν , 18% BR) ± → ± e τ ∼

hadronic showers (τ hadrons, 65% BR) ± → ∼ point-like or shower-like

ν e CC hadronic and EM shower

ν NC hadronic shower

Table 3.1: Diﬀerent neutrino interactions with neutrinos (dashed black line), muons (orange), taus (green), hadronic showers (blue), and EM showers (red). The particle signature happens and the detector signature is how KM3NeT measured the event. Figure from [98].

17 3.3 Background Sources

The Mediterranean Sea is not a perfectly controlled experimental setup, and several background sources obscure the neutrino signal. 40K radioactive decay, bioluminescent organisms in the water, and the dark count rate result in the PMT registering ”hits” that do not originate from a photon from a muon. The average rate of these background hits is 8 kHz for a single PMT. Signal-like background comes from atmospheric muons from above the detector.

• K-40 decay The salt water contains radioactive isotopes that emit an electron in its beta decay or electron capture. A dominant source in seawater is the isotope Potassium-40, whose resulting electron has a maximum energy above the Cherenkov threshold. Other 40K decay products, such as 40Ar, emit photons of 1.46 MeV that can Compton scatter to Cherenkov emitting electrons. These processes provide a steady background rate for the detector.

• Dark count rate Even in the absence of light, thermal noise can release an electron from the photocathode layer. These are registered as hits and provide a signiﬁcant background.

• Bioluminescence Many organisms live within the sea, including a collection of luminescent creatures, such as the pyrosoma and siphonophores. Through a chemi- cal reaction they emit light in the optical range, which is visible to the KM3NeT PMTs. It is a slowly changing seasonal background rate, but large disturbances excites the organisms into emitting a bright burst of well above the background rate [99].

• Atmospheric muons Atmospheric showers from cosmic ray interactions create ν, µ and other particles of which ORCA only wishes to measure the neutrinos. The Earth shields the detector from the other shower particles in the up-going direction. In the down-going direction, a few kilometres of water shield the detector. If muons from above are suﬃciently energetic, they reach the detector. This atmospheric muon rate is a factor 105 larger than the neutrino rate from below. It is, therefore, important to reconstruct the direction of the tracks properly to reduce this atmospheric muon background.

3.4 Data Acquisition

When a pulse charge on a PMT is over threshold, its analogue signal is digitised into a time (t) and a time-over-threshold (T oT ). The combination of these two data values is known as an ”L0 hit”, also often referred to as just a ”hit”. Every 100 ms each DOM sends an identically sized time window containing all L0 hits to shore. To maintain time consistency between each DOM, a ﬁbre-optic network, an on-shore White Rabbit switch, and electronics embedded in the DOM work together to synchronise the complete detector

18 up to nanosecond precision [75, 100]. After the time slice arrives on-shore over Ethernet, the data stream has to be reduced. Without it, a full building block would result in a stream of 25 Gb/s; much of which is background noise. Therefore, many software triggers run over the timeslices in parallel to search for physics events and reduce the number of background events in the data stream. They look for ≥ N causally related hits from two diﬀerent PMTs on the same DOM within a 10ns time window (L1 hits). If found, all the L0 hits at the same time as the L1 hits, and those in a time window of approximately 10µs before and after the causally related hits are combined into a single event. This is written to disk and passed on to the reconstruction algorithms. A full overview and description of triggers can be found in [101].

3.5 Monte Carlo Simulation

Monte Carlo simulations are the only method to understand the detector response and develop new algorithms for analysis. They mimic the ORCA detector and its response to neutrino events. For this thesis, the neutrinos and their interactions are generated by gSeaGen [102]. Their energy is between 10 to 100 GeV to allow for better analysis of the reconstruction algorithms, which results in muons between 0 to 100 GeV. For the propagation of the charged particles and photons, a full simulation, KM3Sim, or a tabulated response, JSirene, can be used. The latter only propagates the initial lepton and uses probability density functions to calculate the expected amount and arrival time of light on PMTs [103, 104]. KM3Sim also propagates the initial charge particle, but continues to fully simulate the creation and propagation of photons and new charged particles using a Geant4 based algorithm [105]. The full simulation is used in this thesis. The final steps are to add background hits to the Monte Carlo events and simulate the detector and trigger response, which is done using the program, JTriggerEfficiency [106]. Version 5 of the Monte Carlo production using the above chain for a full ORCA building block with 20m horizontal and 9m vertical separation is used in the analysis of this thesis. Only a limited number of events can be analysed due to computational constraints, thus, in most cases the 1969 events from the file mcv5.0.gsg muon-CC 10-100GeV .km3sim.jte.1.root are used. When more statistics are required, additional Monte Carlo files are included.

19 Chapter 4

Current Muonic Event Reconstruction and Probability Density Functions

4.1 Track Parameters

After the trigger algorithms from Section 3.4 have found a potential event, the details of the ν-interaction have to be reconstructed. As discussed, the event contains L0 hits; each with the hit time and the time over threshold of the charge pulse. Together with the position and direction of the PMTs, this information is used to estimate the original lepton’s properties. This process is the event reconstruction. For which KM3NeT implements it as a multistage system that distinguishes between shower-like and track-like events. This chapter covers the reconstruction of track-like event, which consists of a pre-reconstruction, JPrefit, and a main-reconstruction phase, JGandalf. Their details are discussed in Sections 4.4 and 4.5. For the shower reconstruction see [75] and [107]. For a track-like event, the position of the lepton is described by three parameters: x, y, and t. No z position is required due to a degeneracy with the time parameter, t. Similarly, the direction of the lepton can be described a unit vector with two independent angles because of the cylindrical symmetry of the track. Instead of the actual angles, two parameters of the unit vector are used in the reconstruction: dx and dy. They are retrieved from the angle using simple geometry, shown in Equations 4.1, where θ and φ are the angles from the z-axis and around the z-axis, respectively. The directional parameters dx and dy can, therefore, only take on values between -1 and 1. The final property that is fitted is the muon energy E. All the track parameters are encompassed ~ in a vector θtrack for an easy of notation.

dz = cos θ dx = sin θ cos φ dy = sin θ sin φ (4.1)

20 4.2 Maximum Likelihood Method

The reconstruction chain uses a maximum likelihood method in which the probability that the data is generated by a speciﬁc statistical model is maximised [108]. Each data ~ point (xi) has a probability (P (xi|θ)) that it was generated under an assumed model with corresponding parameters (θ~). Their product gives a likelihood function: Y L(θ~|~x) = P (xi|θ~) (4.2) i When maximised, it gives the most probable model parameters, the maximum likelihood estimate. The likelihood can, of course, be comprised of more complex combinations of probabilities, as we will see in Section 4.5. The product of many small probabilities often runs into numerical issues. Thus, the likelihood is represented as a − log(L). This also makes it compatible with many minimisation algorithms. The optimum searching step is often the most complex due to the non-trivial landscape of the likelihood function and requires many initial starting values to cover a large parameters space.

4.3 PMT’s Probability Density Functions

~ ~ For KM3NeT, θtrack contains the model parameters, and the probability (P (x|θ)) is a description of the expected number of photo-electrons on a PMT as a function of light arrival time, also known as a probability density function (PDF). Besides depending on the details of the interaction, it is inﬂuenced by the water properties and geometry of the event with respect to the PMT. The PDFs are generated either by transforming data from a full Monte Carlo simulation into chance tables or by using a semi-analytical description. The latter only considers direct and single scattered photons, which is justiﬁed by the long scattering length in water. This thesis uses the analytical approach, because it shows the relation between the probability and physical parameters. Since they are an integral part of the reconstruction, we will discuss their origin and dependencies in three steps, based on an internal note by Maarten de Jong [109]. In this approach we will build a description of the expected number of photo-electrons per unit time on a PMT, which can be turned into a PDF by normalising the function with its integral. 1. Light Emission 2. Light Propagation 3. Light Detection For a simpler PDF, the topology of an event can be rotated and transformed, such that the track is the z-axis, as depicted in Figure 4.1. The PMT is rotated to be in the x − y plane with a closest distance R, zenith angle θ, and azimuth angle φ. This allows for easier storage of the probability values and a clear dependence on three parameters.

21 z0 z 6 J]J 6 J θ℘ J 0 J y ¨* R PP J ¨ φ PP ¨ ℘ (0, 0, 0) PJ -¨ x0

u u

(R, 0, 0)

θ0

(0, 0, z) u

Figure 4.1: Topology of a track event along the z-axis and the PMT in the x-y plane with a shortest distance R. θ℘ and φ℘ are the zenith and azimuth angles of the PMT. Image from [109].

4.3.1 Light Emission Cherenkov Emission Several of the light emission processes are already mentioned in Section 3.2. For minimum ionising particles, such as the low GeV muons, the main emission process is Cherenkov radiation. The amount of light emitted is described by Equation 3.2, but the number of photons at the PMT is diﬀerent. Using the cone hypothesis, the ﬂux of photons becomes Equation 4.3, with R the shortest distance between the PMT and track.

d2N 1 Φ0(R, λ) = (4.3) dxdλ 2πR sin θC

22 Showers Depending on the energy of the muon, different energy loss processes can cause small showers along the track. Due to their cascading nature, showers have a typical longitudinal profile of particles. After an initial steep increase in particles, their energy drops below the pair production limit and their total number decreases. Each particle above the Cherenkov limits emits their own radiation, which results in an effective emission angle of the shower around the normal Cherenkov angle in sea water. The longitudinal profile, dP 2 ( dz ) and angular emission profile (d P/[d cos θ0dφ0]) are parametrisations from Monte Carlo data. Appendix A provides more information on the the parametrisations. The longitudinal and angular profiles are combined into a photon flux at a PMT in Equation 4.4 and depend on the emission wavelength (λ) and angles (θ0, φ0). Moreover, dx/dE is the distance propagated by the shower per unit energy, which is around 4m per GeV in sea water.

dx d2N d2P Φ1(cos θ0, λ) = (4.4) dE dxdλ d cos θ0dφ0

Muon Energy Loss dE − = a(E) + b(E)E (4.5) dx The energy losses are classified ”constant” (a[E]) or linear with energy (b[E]). Ionisa- tion and δ-rays fall in the first category, while bremsstrahlung and e+e− pair production increase linearly and become the dominant energy loss process above 1 TeV, as depicted in Figure 4.2. Since they create small showers, their measurable flux becomes:

Φ2(cos θ0, E, λ) = b(E)EΦ1(cos θ0, λ) (4.6) The contribution from ionisation is minimal, but δ-rays are measurable. δ-rays are knocked-on electrons with a kinetic energy T . Their energy loss (T d2N/dT dx) is dependent on the kinetic energy, which decreases while travelling. By performing an integral over the allowed energies, the number of photons from δ-rays is found. Tmin and Tmax are constraint by kinematics of the interaction and can be found in [3]. Assuming that their emission is isotropic, the emitted number of photons is described by:

d2N 1 Z Tmax d2N Φ3(cos θ0, E, λ) = dT T (4.7) dEdλ 4π Tmin dT dx

4.3.2 Light Propagation Along the way to the PMT, the emitted photons are absorbed, scattered, or unaﬀected. This can be taken into account by using a full Monte Carlo simulation, a parametrisations, or a simpliﬁed analytical solution. The latter is used in this thesis to keep the relation between the PDFs and the physics clear.

23 Absorption is taken into account by adding an exponential factor (e−d/λabs ) to the expected number of photons, where d is the distance between the emission location and the PMT. λabs is the absorption length. The exact distance is wavelength dependent and shown in Figure 4.3. Only single scattering is taken into account for the semi-analytical solution. Other methods can go up to higher order, but all of them will reduce the number of photons hitting a PMT. For the case considered here, a similar exponential factor to the absorption is used with a scattering length (λscat), which is also wavelength dependent, as depicted in Figure 4.3. At the same time, light that would originally miss a PMT can now scatter into it. We model this indirect light by an eﬀective attenuation length (λatt) instead of the scattering and absorption coeﬃcients [109].

140 scattering length [m] 120 absorption

100

0 300 350 400 450 500 550 600 650 λ [nm]

Figure 4.2: Showing the energy loss per Figure 4.3: The scattering (red) and unit track from a muon. The ionisation absorption (black) distances as a function loss is almost constant over energy (dot- of wavelength for KM3NeT adapted dashed), while the bremsstrahlung (thin from [109]. solid line) and the pair production (dashed) increase linearly with energy. Nuclear interactions (dotted) will not be considered here. Image from [110].

4.3.3 Light Detection Two more obstacles stand in the way before a PMT measures a photon. First, the angle (θ ) between the incoming photon and the direction of the PMT determines whether it can actually see the photon. This is the angular acceptance () and is created from Monte Carlo simulations and measurements. It is tabulated to allow for easy access and interpolation between calculated values. Figure 4.4 shows the angular acceptance over

24 cos θ ’s range. The second obstacle is the quantum eﬃciency of the PMT. As described in Section 3.1.1, it is the eﬃciency per wavelength with which a photon releases an electron in the photocathode layer. Figure 4.5 shows that it further reduces the chance a PMT registers a hit.

∈ 0.3

70 QE

0.25 60

50 0.2

40 0.15

30 0.1 20

0.05 10

0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 650 700 θ cos ∅ λ [nm]

Figure 4.4: Angular acceptance of a PMT. Figure 4.5: Quantum eﬃciency. The ratio The cosine of the incoming photon angle of released electrons and incoming photons (cos θ ) determines if the PMT registers in the photocathode layer per unit wave- it. Image from [109] and originally from length. Image from [109]. Monte Carlo data.

4.3.4 Direct and Indirect Light as PDFs The emission, propagation, and detection parametrisations are combined to form a function to describe the expected number of photo-electrons with a specific arrival time on a PMT. This can be divided by its integral to give an actual probability density function for the expected arrival time. However, we will refer to the non-normalised functions as PDFs because of their implementation within the KM3NeT software. In the analytical case, a total of 6 function can be distinguished; two for each emission process. Their subtleties can be found in [109]. But here, we will consider a few of their important aspects. The direct light PDFs are the most straightforward with the three creation steps clearly visible; see Equations 4.8, 4.9, and 4.10. All three equations are dependent on the effective area of the PMT. The first has a direct dependence with the photocathode area (A), while the latter two are related to it by the solid angle of the PMT (dΩ). Furthermore, the first two direct light emission processes are wavelength dependent (∂t/∂λ). The arrival time from the muon energy loss, on the other hand, is dependent on the z position along the track (dt/dz). Indirect light, on the other hand, requires integration over several angles and distances, as seen in Equations 4.11, 4.12, and 4.13. These include the wavelength (λ), z position

25 (not for 4.12), zenith emission angle to the track (θ0), and for the last two PDFs the azimuth emission angle (φ0). Other important changes compared to the direct case are the factors 1/λs, dN/dx, and 1/2π. These, respectively, are the probability of scattering per unit length, the number of scatterings per track length, and a normalisation for the number of photons after the integration over angles. The latter is only required in the Cherenkov light PDF. As discussed in Section 4.3.2, the scattering and absorption have been replaced by a single attenuation factor (e−d/λatt ) and a scattering probability dPs/dΩs. Further details can be found in [109].

Direct Muon Cherenkov Light

−1 dnp.e. ∂t = Φ (R, λ) A (cos θ ) QE(λ) e−d/λabs e−d/λs (4.8) dt 0 ∂λ

Direct Shower Light

−1 dnp.e. ∂t = Φ (cos θ , λ) (cos θ ) QE(λ)e−d/λabs e−d/λs dΩ (4.9) dt 1 0 ∂λ

Direct Light from Muon Energy Loss Z −1 dnp.e. X dt = dλ Φ (cos θ , E, λ) dΩ (cos θ )QE(λ)e−e/λabs e−d/λs dt dz 2,3 0 z=z1,z2 (4.10)

Indirect Muon Cherenkov Light ZZZ −1 dnp.e. 1 dN 1 ∂t −d/λatt dPs = dλdzdφ0 (cos θ ) QE(λ)e dΩ (4.11) dt 2π dx λs ∂u dΩs Indirect Shower Light dn ZZZ −1 p.e. 1 ∂t −d/λatt dPs = dλdφ0d cos θ0Φ1(cos θ0, λ) (cos θ ) QE(λ)e dΩ dt λs ∂u dΩs (4.12)

Indirect Light from Muon Energy Loss ZZZZ −1 dnp.e. 1 ∂t = dλdzdφ0d cos θ0Φ2,3(cos θ0, E, λ) (4.13) dt λs ∂u

−d/λatt dPs (cos θ ) QE(λ)e dΩ dΩs

26 Tabulated PDFs Each time the probability for a hit on a PMT is required, all the integrals must be done for each wavelength. This is time consuming and ineﬃcient. To improve performance, the PDFs are integrated over wavelengths from 300 to 700 nm; the range where the PMT is most sensitive. The total probability is calculated for a grid of the following parameters:

• Distance between track and PMT (R)

• Zenith angle of the PMT (θ)

• Azimuth angle of the PMT (φ)

• Time diﬀerence from the Cherenkov light arrival time (dt) The calculated values are stored in a 4 dimensional table that allows for interpolation between the calculated grid values. These PDFs store the expected number of photo- electrons and calculate the probability using stored integrated values when calculating the likelihood.

4.4 The Pre-reconstruction: JPreﬁt

The pre-reconstruction phase, JPrefit, generates starting values for JGandalf, the main reconstruction algorithm. It does this by 3D clustering causally correlated L1 hits and finding their centre of weight. By performing a directional scan over the whole sky with 5° between guesses, around 800 tracks are created around the centre of hits. At this stage, the tracks length is assumed to be infinite. Per assumed direction a hit selection is performed using a 1D clustering algorithm that looks for causality between the track and the L0 hits within a 50m radius cylinder around the track. The radius of this cylinder is referred to as the road width. The algorithm uses the time information of the selected 2 hits to create the χ function is Equation 4.14, where ti and σi are the time of the hit and the time resolution of the PMT, while tC is the expected Cherenkov arrival time.

hits 2 X (ti − tC ) χ2 = (4.14) σ2 i i JPrefit searches for the optimal set of parameters that minimise the χ2. The set consist of the time parameter (t) and two positional parameters orthogonal to the direction of the track: x and y. The three parameters are combined with the direction to give a track guess. A quality is attributed to the track parameter, as defined in Equation 4.15, where NDF is the number of hits minus 3 for the fit parameters. The best 36 quality tracks are passed on to JGandalf. In some cases more tracks are selected based on whether downward pointing tracks are already included in the 36 selected tracks.

1 χ2 Q = NDF − (4.15) 4 NDF

27 4.5 The Main Reconstruction: JGandalf

The main reconstruction uses the maximum likelihood method to find the event hypothesis that best fits the observed data. It also performs a L0 hit selection within a road width of 50m. The time of the hits (D~ ) are compared to the probability of light to arrive ~ at that time (P (Di|θtrack)). Using these, the likelihood in Equation 4.16 is minimised for ~ all track parameters (θtrack), expect energy. Since the arrival time probability is required, the tabulated PDFs from Section 4.3 are used. JGandalf’s likelihood gives the quality of the track and its parameters. The process is repeated for all 36 input tracks from JPrefit and they are ranked according to their likelihood values. The highest ranked track should, in principle, provide the best possible track parameters, if the likelihood is properly defined.

hits ~ Y ~ L(θtrack|D~ ) = P (Di|θtrack) (4.16) i=1

4.6 Other Algorithms: JStart & JEnergy

After JGandalf the main reconstruction is over, but two more steps exist. The first, JStart, removes the assumption that the track is infinite and searches for the begin and end point of the track. This allows for a selection of hits consisting mostly of signal hits. This is required for the next step: JEnergy, which tries to reconstruct the energy of the muon and set a lower limit on it. This work does not consider these steps, as they no longer influence the final direction of the track.

4.7 Intrinsic Limits of the Reconstruction

The true interest of the reconstruction is to ﬁnd the original neutrino direction and energy, but even with the perfect reconstruction algorithm only the true muon direction and a lower limit for its energy can be found. Due to the scattering processes, the energy and direction of the muon will diﬀer from the neutrino. This introduces a fundamental limit for the reconstruction of the direction and the energy. The angle between the true neutrino and the muon direction shows the intrinsic limit of the reconstruction in Figure 4.6 (left). While the size of the limit is not necessarily a problem, it is the spread that introduces uncertainty in the neutrino’s direction. Luckily, higher neutrino energies lead to more forward boosting and push the median and the spread to lower values, as expected. Furthermore, the combined fraction of neutrino and antineutrino energy given to the muon decreases with higher energies, as Figure 4.6 (right) shows. The change from quasi-elastic and resonance scattering to deep inelastic scattering opens up the phase space for the muons energy with a larger component going to the shower. Another important element for the energy is whether the muon is contained within the detector. If it travels even a bit outside the detector volume, it becomes impossible to estimate

28 the original muon energy and only a lower limit can be set. These aspects will all restrict the measurements for the mass hierarchy, but a good directional reconstruction is still necessary to reduce background events.

45 1 true ν 0.9 [deg] 40 / E µ , ν true µ

θ 0.8

35 E 0.7 30 0.6 25 0.5 20 0.4 15 0.3 10 0.2 5 0.1 0 0 1 10 102 1 10 102 true true Eν [GeV] Eν [GeV]

Figure 4.6: The median of the intrinsic limit (blue) for the angle between the muon and (anti)neutrino (left) and the energy fraction from the (anti)neutrino going to the muon (right). Both neutrinos and antineutrinos are included. The 16% and 84% quantiles are marked in red. Retrieved from Monte Carlo simulations.

29 Chapter 5

Evaluation of JGandalf’s Input

Within the ORCA detector, the energy and direction are the most important parameters to reconstruct because of their sensitivity to the mass hierarchy. Their reconstruction principles are quite different, and in this thesis we will only consider the directional reconstruction of the muon. The accuracy of the measured direction is influenced by many parameters of the reconstruction, as introduced in Chapter 4, for example, the hit selection, the number of track guesses, and assumed coincidence window. These parameters are constantly optimised for the best results. Therefore, we focus on improving the reconstruction algorithms themselves. Section 6 and 7 discuss the improvements to JGandalf’s input and to JGandalf itself. It is important to first understand the current reconstruction framework and its limitations. Therefore, this section covers the input JGandalf requirements and whether JPrefit reaches these.

5.1 The Positions of JGandalf’s Minima

JGandalf uses a Levenberg-Marquardt minimisation algorithm, which is robust and finds solutions far from its starting value [111]. However, the KM3NeT collaboration knows that JGandalf does not perform well when starting values are far away from the true values. No quantitative limits for this exist and their nature has not yet been explored. For an event selected based on its high directional accuracy after the JGandalf algorithm, the -log likelihood space for the x parameter is shown in Figure 5.1. The other fit parameters are fixed to their truth values. The space has a minimum at the correct true value, indicated in red. However, the global minimum is not at the true parameter. Similar results can be seen in Appendix B for the other fit parameters and a collection of events from good to badly reconstructed events. The examples indicate a structural problem with the global minimum in the JGandalf likelihood. To explore this further, a scan over all parameters of 1969 events is performed. The global minimum is marked in the ranges -200m to 200m for the positional parameters and -1 and 1 for dx and dy. The global minimum positions from the X and Y parameters in Figure 5.2 show that most minima lie at the scan boundaries. Their combined count is an order of magnitude larger than at the true parameter. The same behaviour, although less extreme, shows in the directional fit parameters in Figure 5.3. The amount of tracks

30 at the boundaries is double those at the true parameter. JGandalf’s likelihood is not properly deﬁned. Minimisation to the global truth will lead to a wrong set of parameters. It is, therefore, the most probable cause for the initial value problem.

900

-Log LL 800 700 600 500 400 300 200 100

−150 −100 −50 0 50 100 150 Distance [m]

Figure 5.1: A negative log likelihood space scan for the X parameters done by JGandalf for a properly reconstructed event, where the Monte Carlo truth value is marked in red.

103 103 Counts Counts

102 102

10 10

1 1

−200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 X deviation Y deviation (a) (b)

Figure 5.2: The location of the global minimum for the X (a) and the Y (b) parameters. The true value is at 0.

31 3 10 103 Counts Counts

102 102

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 DX deviation DY deviation (a) (b)

Figure 5.3: The location of the global minimum for the DX (a) and the DY (b) parameters. The true value is at 0.

5.2 Quantifying JGandalf’s Directional Input

Let us quantify the influence of JGandalf’s initial value problem on the reconstruction. By searching till what deviation from the true direction JGandalf is still able to improve the directional information, a limit can be set. For this purpose, a directional accuracy is defined as the dot product between the fitted or initial track with the true direction. This is also known as the directional error.

~ ~ cos θARes = dinitial/ﬁt · dtrue (5.1)

5.2.1 Deviates from the True Direction The initial and output directional accuracy is compared by generating 639 tracks with 15 different deviations between 0 and 40 degrees from the Monte Carlo truth. Other fit parameters have been set to their corresponding truth value. After JGandalf minimises true the likelihood, the median of the final directional accuracy as a function of Eµ is calculated for each input track. The process is repeated for numerous events to reach enough statistics for Figure 5.4, where the relative change compared to the directional accuracy of the initial track is shown. JGandalf is successful when the final angle is smaller than the initial angle. This is the case for initial angles up to 10° above 20 GeV and up to 15° in the 10-20 GeV range. Below this energy range, the reconstruction is unable to properly reconstruct events due to the low number of signal photons measured. Tracks starting closest to their true value minimise to a worse directional accuracy, as the red in the lower left in Figure 5.4 indicates. Above 10° JGandalf is unable to reconstruct up to the same level of accuracy, but a slight improvement can still be seen for high energies and a large improvement around 10 GeV as can be seen in Figure 5.5. These effects decrease as the initial angle gets

32 larger. Above 90 GeV statistical effects are present, but with a sample of more high- energy neutrinos these should disappear. The best possible reconstruction achievable by JGandalf is indicated by the yellow line in Figure 5.5, because it starts at the true parameters. For initial directional deviations of 10° or less, the errors on the final tracks are similar to starting at the true values. For example, the 5.71429° deviate reconstructs, on average, to the same directional accuracy as the true tracks. Commonly, the reconstructed angles are plotted without the energy dependence as a count plot in log scale to show the sub-degree nature of the reconstruction. JGandalf’s best achievable directional accuracy is shown in Figure 5.6. Brute forcing many tracks would also result in this distribution if the likelihood is correctly defined. This, however, is computationally intensive and smarter methods, such as in KM3NeT, are required. This does result in an acceptableMedian loss in theof Final directional ARes accuracy. 1 30 0.8

25 0.6 0.4

20 0.2

0 15 −0.2

10 −0.4 Relative Directional Accuracy Change

Initial Directional Accuracy [deg] −0.6 5 −0.8

0 −1 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 5.4: The relative directional accuracy change by JGandalf as a function of initial directional accuracy and true muon energy. The initial tracks are directional deviates from the Monte Carlo truth. Above 90 GeV statistical eﬀects play a role.

5.2.2 Deviates using JPrefit tracks One drawback of the MC deviates is that it only takes into account change in directional parameters. In reality, the position also contains some deviations and will influence the reconstruction. For this purpose, we use JPrefit’s 36 track output to generate a sample of angles and positions. This does generate a biased set of tracks, since a selection and fit has been applied to the tracks. The relative change in directional accuracy in Figure 5.7 shows similar patterns as the MC deviates. However, the energy threshold has dropped to 10 GeV and the angle limit has increased to 15°. Figure 5.8 also shows that the improvements around 10 GeV

33 Median of Final ARes

30 Initial Dir.Acc. [deg] MC Truth; 0 5.71429 14.2857 25 20 28,5714

5 Median Final Directional Accuracy [deg]

0 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 5.5: Median final directional accuracy for five different initial directions. The yellow line is the reconstruction started from the Monte Carlo truth and is the best achievable directional accuracy for JGandalf. have been removed. Angles above the limit show much more improvements in the final median directional error than with the Monte Carlo deviates. The track selection by JPrefit is the most probable cause for the increase in these limits, because it only selects track from its collection of 800 tracks with a high quality factor.

34 90 Counts 80 70 60 50 40 30 20 10

0 − 10 3 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure 5.6: The best achievable directional accuracy for JGandalf by starting the reconstruction at the Monte Carlo truth parameters. fmedianARes 1 30 0.8

25 0.6 0.4

20 0.2

0 15 −0.2

10 −0.4 Relative Directional Accuracy Change

Initial Directional Accuracy [deg] −0.6 5 −0.8

0 −1 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 5.7: The relative directional accuracy change as a function of initial directional accuracy and true muon energy. The initial tracks are the 36 selected track from JPreﬁt. These have been run through JGandalf to give the ﬁtted tracks.

35 Median of Final ARes

Initial Dir.Acc. [deg] 22 0-2 2 20 3 2 2 - 5 1 18 3 3 8 - 10 2 16 3 16 - 18 2 14 3 21 1 - 24 12 3 10 8 6 4

Median Final Directional Accuracy [deg] 2 0 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 5.8: Median final directional accuracy for five different initial directional accuracy ranges. Initial tracks are JPrefit tracks and are run through JGandalf to get a fitted track.

5.3 JGandalf’s Input Limits

We conclude that the initial directional accuracy for the JGandalf algorithm has to be below 10° to 15° for energies above 20 GeV. The influence of the errors in the positional parameters on the directional reconstruction is still unclear, but the JPrefit tracks increase the range in initial directional accuracy up to 20° and the energy range goes down to 10 GeV. Within these limits, the JGandalf algorithm can still, on average, reconstruct the events to a similar accuracy as when starting at the true parameters. The increase in parameter space for JPrefit tracks is most likely caused by the pre-selection of tracks.

5.4 Analysis of JPreﬁt’s Output

The JPrefit algorithm increases the directional error JGandalf is able to handle, but the question remains whether the selected 36 tracks are within the directional input limits of JGandalf. JPrefit calculates 800 tracks, but only passes on the best 36. The normalised directional accuracy distributions for a large collection of events are differ significantly, as shown in Figure 5.9a, which are not corrected for their phase space. The best 36 tracks

36 and the top track have a much better distribution than all tracks combined. However, when combining the top 36 tracks for a collection of events, only 10% of these tracks have a directional accuracy below 10° and 21% below 15°. Considering that the best directional accuracy of the 800 tracks is always around 2.5° away from the truth, this is an extremely small fraction. However, in 65% of the cases the track with the best directional accuracy is included in the set of 36 tracks. There is signiﬁcant room for improvement here, which will be explored in Chapter 6. The position of a track has to be as close to the truth as possible to minimise its inﬂuence on the directional reconstruction. The majority (65%) of the 36 tracks is within 10 meters of the true position as depicted in Figure 5.9b, but in only 23% of the events the best position is included in the selected tracks. This is another clear avenue for improvements, but will be left for others to optimise.

Best Prefit Track Best JPrefit Track 0.16 0.16 Best 36 Prefit Tracks Best 36 JPrefit Tracks 0.14 All Prefit Tracks 0.14 All JPrefit Tracks

0.12 0.12

Normalised count 0.1 Normalised count 0.1

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0 0 0 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 60 70 80 90 100 θ true, reco [deg] Distancetrue, reco [m] (a) Normalised distributions of the directional (b) Normalised distributions of the position accuracy for three JPreﬁt cuts. for three JPreﬁt cuts.

Figure 5.9: JPreﬁt cuts: black is the best ranked track, blue all tracks, and the best 36 are marked in red.

37 Chapter 6

Improvements to JGandalf’s Input

The JPrefit algorithm does not always reach the directional accuracy requirements set in Chapter 5 for the JGandalf algorithm to achieve a mean error similar to minimisation starting from the true parameters. Better starting parameters for JGandalf can be achieved through different methods. In this chapter, we will discuss re-ranking and minimisation to improve the output from the JPrefit algorithm. Re-ranking finds a better set of 36 tracks from approximately 800 tracks created by JPrefit. This is achieved by reordering the tracks according to a new quality function. The new 36 tracks are scored by the fraction of events containing the track with the best directional accuracy from all JPrefit guesses: the containment fraction. A second quality check is performed by comparing the directional error of the top 36 tracks for overall a better set. Minimisation, on the other hand, updates the track parameters to be closer to the true track. For which we will utilise the maximum likelihood method. This requires new likelihood functions that will be introduced in Section 6.2.

6.1 Optimising JPreﬁt’s Quality Function

The simplest form of re-ranking is to alter the 1/4 factor in the current JPrefit quality in Equation 4.15. No internal documentation suggests that this is the best possible value to be used. Thus, we implement other factors than 0.25 and re-rank all 800 JPrefit tracks according to the new quality functions. After which, the containment fraction is calculated as a function of the top N tracks. A selection of new factors is shown in Figure 6.1 with the full range of factors in Appendix D. The current factor of 0.25 has the largest containment fraction of all the quality factors for most N, except when less than 10 tracks are selected. There, the larger quality factor outperforms the current factor. Moreover, it is especially good in finding the track with the best directional accuracy. The higher quality factor has this track as its first track in the ranking more often; an increase from 12% for a 0.25-factor to 16.5% for a 100-factor. When the number of tracks in a set increases, the performance of the larger quality factors starts to diminish. The smaller quality factors, on the other hand, start to perform better and eventually reach similar levels as the 0.25-factor. For a set of 36 tracks the 0.25-factor contains the track with the best directional accuracy most often, but the other factors might provide a better overall directional

38 accuracy for the set as a whole. Thus, we perform the second quality check and look at the directional error of the complete 36 track set for the 0.25 and 100 quality factors in Figure 6.2. With only minimal statistical deviations visible, it is clear that the factor does not inﬂuence the top 36 tracks. It does alter the internal ordering of those tracks, since the best track is found quicker by higher quality factors.

0.8

0.6

Quality Factor 0.4 0.10 0.25 0.2 100.00

Fraction Top N tracks containing the best D.A. 0 10 20 30 40 50 60 70 80 90 100 Top N tracks

Figure 6.1: The containment fraction over the top N ranked tracks with diﬀerent quality factors.

6.2 Introducing the New Likelihoods

Just changing the quality parameters has no to limited inﬂuence on the directional accuracy of the 36 tracks. Thus, we introduce more complex quality functions using likelihoods. Originally, the ORCA track reconstruction had a middle step between JPreﬁt and JGandalf: JSimplex. This did not lead to improvements in the directional accuracy after JGandalf. Consequently, it was left out of the reconstruction process. The likelihood used in the JSimplex algorithm compares the light arrival time with the Cherenkov cone hypothesis for which it used a M-Estimator Lorentz as a probability density function. For more information on the estimator, see Appendix E. The product of probabilities results in the following likelihood:

hits Y ~ L = PM-E(ti|θtrack) (6.1) i

39 ares_0.25

6000 Quality Factor 0.25 Counts 5000 100

4000

3000

2000

1000

0 − 10 3 10−2 10−1 1 10 102 103 Directional Accuracy [deg]

Figure 6.2: The directional accuracy of the 36 track set from two quality factors.

This likelihood will be used as a new quality parameter and will be expanded upon ~ with the probability a PMT was hit during an event (Phit(Di|θtrack)). It depends on the distance between the PMT and the track, the direction of the PMT, and duration of the event. This is also known as the hit and no-hit information. While both can be included, the no-hit information is not, because the algorithm would have to consider all non-hit PMT in the whole detector which is computationally intensive. Therefore, only the hit information is included in the likelihood:

hits Y ~ ~ L = PME(ti|θtrack)Phit(Di|θtrack) (6.2) i The hit probability is a simple counting experiment and is calculated using a Poisson distribution according to Equation 6.3. The Poisson distribution requires a description of the expected number of photo-electron (np.e.) on a PMT during the event. This information can be extracted from the PDFs described in Section 4.3.

~ ~ ~ Phit(Di|θtrack) = 1 − Pno hit(Di|θtrack) = 1 − P oisson(0|np.e.(Di, θtrack)) (6.3)

40 6.3 Parametrisation of np.e.

The expected number of photo-electron can be separated in signal and background hits (Equation 6.5). The latter is a product of the time of the event (∆t) and the background rate of the PMT (Ratebg), as seen in Equation 6.6. The expected number of signal photo-electron are calculated using the semi-analytical approach in Section 4.3.4. The JGandalf algorithm uses the full description of the expected arrival time on a PMT. A grid of calculated values are stored in tables as the expected number of photo-electrons arriving at a certain time with a dependence on the geometry of the event. The full tables are too memory and computationally intensive to use in a pre-reconstruction phase. Therefore, simplified parametrisations of these tables are introduced. We perform an integral over time to obtain the expected number of photo-electrons during an event (np.e.). Preferably, a parametrisation for the three remaining parameters is used. These are the two angles of the PMT with respect to the track (θ, φ) and the distance between the PMT and the track (R). As described in Section 4.3.4, six different light sources can be distinguished of which only four are dominant in the ORCA detector. Direct and indirect light from delta rays are left out of the reconstruction. Thus, they are not included in the parametrisation. Due to the complicated dependence of np.e. on R, θ, and φ, the tables are first simplified to only keep the distance dependence, which is achieved by averaging over θ and φ. This returns an average np.e., shown in Figure 6.3. Both direct light contributions have comparable shapes, but are shifted to different orders of magnitudes. Similarly, the indirect contributions look alike and have an order- of-magnitude shift. Each contribution can be individually fitted and then combined into a complete description, or the joint contributions can be fitted. The former has the benefit that it is modular. Individual contributions can be removed or added, but it will require significantly more parameters than the latter. As for the fit function, Figure 6.3 gives us the hint that at large distances an exponential factor plays a role due to the linear shape with the logarithmic y scale. Furthermore, the direct contributions have similar exponential factors, which most likely come from the e−d/λsc e−d/λabs factors in the light description from Equations 4.8 and 4.9. This also explains the different exponential factor for indirect light, which uses an effective attenuation length instead. Besides an exponential factor, it will require a combination of 1/Ra dependencies to account for the initial drop at short distances. There are many functions to choose from, but a rational function contains most possibilities. Combined with the exponential this gives the following function:

p + p R + p R2 + p R3 + p R4 −R/p1 2 3 4 5 6 hnp.e. signali = p0e 2 3 4 (6.4) p7 + p8R + p9R + p10R + p11R The density of data points in the tabulated PDF has been increased to to improve the accuracy of the fit. The fit is performed in two steps. First, the exponent is fitted to

41 > Total

p.e. 10 Direct Cherenkov

Figure 6.3: Time integrated and angle averaged expected number of photo-electrons for each contribution. the data points at the last 50 meters. After taking the ratio between fitted and data, we fit the rational part to it. In this section, the process of fitting the combined contributions is discussed, but the parameters for all types can be found in Appendix F. Figure 6.4a shows that the exponential fits properly in the 100 to 150 meter regime. Its residual ratio in Figure 6.4b also includes a great rational fit, and the full fit in Figure 6.4c looks good. Its maximum relative error is 1.1%. The shape of the residual is periodic and increases as the step sizes get larger with longer distances. Smaller steps might achieve better accuracy. However, for the implementation is this thesis, the current precision is adequate. Table 6.1 shows the best fit parameters, and the same process is repeated for the individual light contributions in Appendix F. The parametrisation of the combined fit is used, because it is the least computationally intensive of the two options. It might be possible to reduce the complexity of the individual fits, but this is a non-trivial task. The relation between the original equations in Section 4.3.4 and a ”simple” function is obscured by the integrals and other factors involved.

np.e. = np.e. bg + np.e. signal (6.5)

42 > fit p.e.

Data data Data

10−1

10 10−2

− 10 3

1 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Distance to track [m] Distance to track [m] (a) Exponential fit on the combined light con- (b) Fraction of the exponential fit on the com- tributions. bined light contributions and the rational fit to it. Exponential and rational fit

> 0.1 p.e.

data 0.08

Figure 6.4: Full ﬁt on the combined light contributions using an exponential and rational function.

np.e. bg = ∆t Ratebg (6.6) Furthermore, the average over the angles removes any directional information in the parametrisation, while the original PDF is extremely angle dependent. In Figure 6.5, the expected number of photo-electrons for two PMT directions has two orders of magnitude diﬀerence. Directional information has to be reintroduced. In Appendix F a parametrisation of the non-angle average PDFs is explored, but due to the complexity of the angle dependence, a simpler approximation is used. By multiplying hnp.e. signali with the PMT’s angular acceptance, some directionality is reintroduced.

np.e. signal(cos θ, R) = (cos θ)hnp.e. signal(R)i (6.7)

43 Parameter Value p 1.39072e-1 0 /dt Direction

p.e. −3 p1 2.92012e+1 10 SOUTH dn NORTH p 5.011141e00 − 2 10 4 p3 6.74735e-01 − 10 5 p4 8.90149e-3

− p5 5.83772e-5 10 6 p6 1.52862e-07 10−7 p7 1.30459e-3 −8 p8 4.71790e-2 10

− p9 8.65157e-3 10 9 p10 2.13413e-4 0 100 200 300 400 500 ∆t [ns] p11 -7.18605e-7

Table 6.1: Best ﬁt parame- Figure 6.5: The expected photo-electrons for PMTs ters. These are used in the pointing along the track in the direction of the parametrisation. muon’s propagation (NORTH) and in opposite direction (SOUTH).

Using the angle-averaged parametrisation, we can distinguish three new likelihoods and apply them in re-ranking and minimisation algorithms. Each can use L0 and L1 hits. We implement the former to include as much information as possible.

1. Time A likelihood with just a Lorentz distribution as an arrival time estimator. Similar to JSimplex but without any hit selection.

hits Y ~ L = P (ti|θtrack) (6.8) i

2. Time-Hit Expansion of Time using the hit information with only the distance parametrisation. hits Y ~ ~ L = P (ti|θtrack)Phit(Ri|θtrack) (6.9) i 3. Time-Hit-Angle Expansion of Time using the hit information with the distance parametrisation including the angular acceptance.

hits Y ~ ~ L = P (ti|θtrack)Phit(Ri, φi, θi|θtrack) (6.10) i

44 6.4 Re-ranking of JPreﬁt tracks

First, we will look at the re-ranking using the three new likelihoods. We perform the the same quality checks as with the quality factor analysis. For JPrefit the standard factor of 0.25 for the quality function is used. The containment fraction as a function of set size is shown in Figure 6.6. JPrefit finds the tracks with the best directional accuracy efficiently at large N. The Time-Hit likelihood performs the worst. This confirms the expectation that the orientation of the PMT is important to the expected number of photo-electrons. More interestingly, Time performs better than the more complex Time- Hit-Angle likelihood. Reasons for the underwhelming Time-Hit-Angle performance will be discussed in Section 6.6. 0.250000 1 0.9 0.8 0.7 0.6 0.5 Ranker 0.4 JPrefit Quality 0.3 Time LL Time Hit LL 0.2

Fraction of top N containing best D.A. Time Hit Angle LL 0.1 0 0 100 200 300 400 500 600 700 800 Top N Tracks

Figure 6.6: The fraction of events containing the track with the best directional accuracy in the top N tracks (containment fraction) for JPreﬁt and the three new likelihoods.

The reconstruction uses only the top 36 tracks; thus, we zoom in on that region. There we see that Time and Time-Hit-Angle perform better in the low top N tracks than JPrefit does. They both have the track with the best directional accuracy of the 800 as their first track in 19% and 22% of the cases, respectively. Time-Hit-Angle keeps outperforming JPrefit’s quality function up to the top 5 tracks, while Time continues until the top 10 tracks. The re-rankings also results in a new distribution of the directional error of the top 36 tracks, which is shown in Figure 6.8. First of all, the Time-Hit likelihood has the worst directional error of all four quality functions. But also Time and Time-Hit-Angle cannot select a better set of 36 tracks than JPrefit.

45 0.250000 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Fraction of top N containing best D.A. 0.1 0 0 5 10 15 20 25 30 35 40 Top N Tracks

Figure 6.7: Zoomed version of Figure 6.6 with the containment fraction. ares_0.25

6000 Quality Function JPrefit Time Counts 5000 TimeHit TimeHitAngle

4000

3000

2000

1000

0 10−1 1 10 102 Directional Accuracy [deg]

Figure 6.8: The directional accuracy of the best 36 tracks after re-ranking using the new quality functions.

46 With the selected 36 tracks, it is possible to re-rank those according to the new likelihood. This is depicted in Figure 6.9 and shows that the Time likelihood is able to rank track with the best directional accuracy higher than the JPreﬁt quality function. This could reduce the number of tracks necessary in future steps without loss of quality. The containment fraction does not add up to 1, because some events contain more track hypothesises, as explained in Section 4.4. 0.250000 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Fraction of top N containing best D.A. 0.1 0 0 5 10 15 20 25 30 35 Top N Tracks

Figure 6.9: The containment fraction for the tracks selected by JPreﬁt.

6.5 Minimisation

Although, Time-Hit-Angle ranks the track with the best directional accuracy most often as first, the set of 36 tracks shows no improvement at all. The other new likelihoods perform equally or even worse, but they can also be used to find better track parameters with a new and hopefully better minimum. To only analyse the likelihoods the minimisation is performed by MINUIT using the MiGrad method [112]. Using this algorithm, the first test of the likelihoods is to perform a fit starting at the Monte Carlo truth. Depicted in Figure 6.10, the Time-Hit likelihood has the worst directional accuracy. Counter-intuitively, the simpler Time likelihood is outperforming the more complex Time-Hit-Angle in terms of the directional error. Minimisation from the Monte Carlo truth shows that Time is the most likely to perform the best. The real question is whether or not they improve the directional accuracy of the tracks selection by JPrefit. As shown in Figure 6.11, all three likelihoods slightly improve the directional accuracy of the 36 tracks. A peak around 90° remains and Time-Hit increases this peak. Tracks with already a reasonable directional accuracy are brought below the 10° and 15° limits.

47 ares 200 Likelihoods 180

Counts Time 160 Time Hit 140 Time Hit Angle 120 100 80 60 40 20 0 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure 6.10: The directional accuracy of the three new likelihoods after performing a minimisation starting from the Monte Carlo truth.

ares

Likelihoods 10000 JPrefit Counts Time Time Hit 8000 Time Hit Angle

6000

4000

2000

0 10−2 10−1 1 10 102 103 Directional Accuracy [deg]

Figure 6.11: The directional accuracy of minimised tracks using the new likelihoods starting at the JPreﬁt’s 36 tracks.

48 Table 6.2 shows the percentage of tracks below the limits, which increases for all three likelihoods compared to JPreﬁt. The Time and Time-Hit-Angle likelihoods reach very similar results. However, the Time likelihood is fast and simple compared to Time- Hit-Angle due to it only requiring an estimator for the arrival time of the Cherenkov light.

Likelihood 10° limit 15° limit JPreﬁt 10.0% 21.5% Time 25.4% 34.5% Time Hit 19.6% 27.0% Time Hit Angle 25.5% 34.6%

Table 6.2: Percentages of tracks below the 10° and 15° limits for minimisation using JPreﬁt, Time, Time Hit, and Time Hit Angle likelihoods.

6.6 Discussion of Input Improvements

We have analysed a new quality factor and three new likelihoods in the context of re-ranking and minimisation. While changing the quality factor influences the ranking significantly, the current 0.25-factor remains the best for a set of 36 due to its high containment factor. The first track is not always the best track, but this is not a requirement at this stage of the reconstruction. Only the containment factor is of importance, since a set gets passed on to JGandalf. Time performs the best of the three, but is outperformed by JPrefit’s ranking in the 36 track regime. the Time-Hit-Angle and Time likelihoods are better in ranking the track with the best directional accuracy as their first, but it does not lead to an overall improvement of the directional error. The likelihood spaces of all three quality functions show minima at incorrect positions. Moreover, they are often asymmetric in their shape. As a result, one of two tracks at equal distance from the true parameter will be preferred. Or a track further away but at a steeper side of the likelihood is preferred, which results in a terrible track selection. This messes up the ranking and is the cause for the failing of the likelihoods in re-ranking. In the minimisation setting, the likelihoods perform significantly better. They improve the directional accuracy of the tracks with Time and Time-Hit-Angle taking the lead. They increase the percentage of events below JGandalf’s input requirement of 10° from 10% to around 25%. This will make it easier for JGandalf to improve the events. The fact that the Time-Hit likelihood does not perform well in either category is expected. The parametrisation of the average expected number of photo-electron on a PMT without directional information interferes with the reconstruction. For example, when a PMT is close but pointing away from a track, the parametrisation of the expected number of photo-electrons is similar to one pointing towards the track. In reality, these two numbers are different, thus driving the likelihood to places away from the true minimum, making Time-Hit a terrible likelihood.

49 THAnormalised

0.12 begin

L -L 0.1

0.08

0.06

0.04 Time Hit Angle

0.02 Time 0

0 5 10 15 20 25 30 35 40 45 50 Distance from MC Truth [m]

Figure 6.12: Likelihoods over distance from the Monte Carlo truth normalised by comparing it to its value at the Monte Carlo truth.

Time-Hit-Angle does use the directional information and closer resembles reality, but it performs equality if not worse than the likelihood with only the time information. The added parametrised hit information does not seem to provide extra information about the true muon direction. Several reasons for this behaviour have been identified. First, the Time-Hit-Angle likelihood is positional independent within 10 meters around the Monte Carlo truth for tracks coming from JPrefit. Figure 6.12 shows the likelihoods over distance for tracks from JPrefit and for the first 10 meters Time-Hit-Angle is flat. It even decreases compared to its starting position. Since most of JPrefit’s tracks are within a 10 meter radius from the Monte Carlo truth (see Figure 5.9b), Time-Hit-Angle is positional invariant in this region. This relates back to the directional resolution. Time Only Likelihood X=-10.00 Time Only Likelihood X=0.00 28200 28200

28000 28000 -Log LL -Log LL 27800 27800

27600 27600

27400 27400

27200 27200

27000 27000

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 DX deviation DX deviation (a) (b)

Figure 6.13: Time-Hit-Angle’s − log LL space scan for the DX with X at the MC truth (a) and DX with X 10 meters to the left (b).

For a cluster of hits with the optimal position directly below it, the optimal direction would be straight up. But if the position is oﬀ to the left, the direction vector has to

50 compensate to the right to best fit to the cluster of hits. Figure 6.13a shows the negative log likelihood space at a position at the Monte Carlo truth. Figure 6.13b shows the likelihood when the position is 10 meters off to the left. The directional minimum is now shifted to the right. The positional invariance leads to a directional invariance and Time-Hit-Angle is unable to distinguish between the tracks with the better directional accuracy. Moreover, Time increases in the first 10 meters and is better able to select the correct tracks. The Time likelihood is, therefore, the best minimiser to improve JGandalf’s input.

51 Chapter 7

JGandalf Upgraded: JMerlin

Besides improving the initial directional accuracy for JGandalf, it is also possible to improve JGandalf itself. In this chapter, we introduce a new likelihood as a full reconstruction step: JMerlin. This likelihood includes hit and no-hit information from L0 hits using the full PDFs instead of the parametrisations:

hits no hits Y Y L(θtrack|D~ ) = P (ti|θtrack) Phit(Di|θtrack) Pno-hit(Di|θtrack), (7.1) i i

where Phit and Pno-hit follow Equation 6.3 with all number of expected np.e. coming from the full PDF tables. By including the no-hits, the algorithm has to run over the complete detector structure, which is slow, but extra directional and positional information can be extracted from this data. For example, a track is pushed away from regions where no hits took place, making the likelihood space more centred around the best ﬁtting parameter. This reconstruction algorithm will not perform any hit or PMT selection with a road width or other criteria. Thus, a single PMT can have multiple hits in a single event, which are unlikely to all come from Cherenkov radiation. On the other hand, it could be the result of light scattering. While their inclusion adds extra computational time, the PDFs are able to handle these hits. Another algorithm exists, JGandalfx, which does implement a hit and PMT selection based on the road width. A complicated quality parameter is required to account for the diﬀerence in degrees of freedom between tracks. JMerlin’s quality function is just the likelihood, since all track guesses use the same number of hits and PMTs. A comparison between JGandalfx, JGandalf, and JMerlin can be found in Appendix G.

7.1 JMerlin’s Likelihood Space

The ﬁrst test for a correctly deﬁned likelihood is to inspect its likelihood space around the true parameters. Figure 7.1 shows JGandalf’s and JMerlin’s likelihood for the properly reconstructed event by JGandalf. It is immediately clear that JMerlin has a global minimum at the true value, while JGandalf does not, as already noted in Section 5. A parameter scan for many events is performed again and the position of the global minimum is shown in Figures 7.2 and 7.3. The global minima are closer to the truth than for JGandalf. Interestingly, most parameters are symmetric around the truth parameter,

52 Likelihood Comparison 1800 1600 1400 1200

Normalised LL JGandalf 1000 800 JMerlin 600 400 200 0 −200 −400 −200 −150 −100 −50 0 50 100 150 200 Rotated & translated Y position [m]

Figure 7.1: − log LL scan over the Y parameter for JGandalf and JMerlin with the other parameters fixed to the Monte Carlo truth. except the X parameter, which shows a bias towards one side. No obvious reason for this likelihood behaviour has been identified. The likelihood space, however, shows a complicated structure with many minima. They originate from the detector structure. When a track takes different positions and directions during the scan, it gets closer and further away from PMTs. If a tracks moves straight through a no-hit PMT, it becomes unlikely to be the correct track and the − log LL increases. The likelihood space is much harder to traverse, but its global minimum is properly defined. This chapter will explore the full potential of JMerlin in re-ranking and minimisation.

103 103 Counts Counts

102 102

10 10

1 1

−200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 X deviation Y deviation (a) (b)

Figure 7.2: Position of the global minimum for JMerlin’s likelihood for the X (a) and Y(b) parameters.

53 3 103 Counts 10 Counts

102 102

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 DX deviation DY deviation (a) (b)

Figure 7.3: Position of the global minimum for JMerlin’s likelihood for the DX (a) and DY (b) parameters.

7.2 Re-ranking of JPreﬁt and JGandalf tracks

At this stage in the reconstruction, the top-ranked track in the reconstructed tracks has to be as close to the truth as possible. No extra directional information will be added in the stages after JGandalf. First, we look at the containment fraction in Figure 7.4. JGandalf has a superior ranking over JMerlin. This is in contrast with the likelihoods for a good event seems to indicate. The steeper likelihood should result in a better discriminator of good and bad tracks, but this is not the case. Furthermore, JGandalf ranked track with the best directional accuracy ﬁrst in 9% of the events. JMerlin’s quality function only achieves this in 2.2% of the cases. This is at a similar level as randomly picking a track from the set. This indicates that JMerlin is not a viable re-ranker.

7.3 Minimisation

Once again, we start the minimisation at the Monte Carlo parameters, resulting in Figure 7.5. The extra information added by the hit and no-hit information improves the reconstruction significantly. JGandalf reaches a mean directional error of 1.92° and has some non-reconstructed events. These are visible as the blue peak at 102 in Figure 7.5. JMerlin removes this peak and improves the mean to 0.81°. Moreover, the sub-degree events go from 28% to 60%. This is a major improvement in the reconstruction of track-like events in KM3NeT-ORCA. Next, we perform the minimisation with JPrefit starting parameters to show if JMerlin can replace JGandalf. Figure 7.6 shows that this is not the case. JMerlin performs worse than JGandalf and the amount of sub-degree tracks decreases from 26% to 11%. Its most likely cause is the extreme likelihood space that JMerlin has due to the detector structure. The many local minima make it difficult for the minimisation to find the correct global minimum. JMerlin’s starting value problem might be worse than JGandalf’s. JMerlin’s

54 posbest

0.8

0.6

Ranking Algorithm

0.4 JMerlin

JGandalf

0.2 Fraction of events with best D.A. in top N 0 0 20 40 60 80 100 120 140 Top N tracks

Figure 7.4: Containment fraction of the track with the best directional accuracy for different top N tracks calculated with JGandalf’s and JMerlin’s quality function. input requirements are discussed in Section 7.4. The directional accuracy for minimisation of JGandalf’s output also gets worse, as Figure 7.7 shows. Many of the medium reconstructed events get a larger directional error after JGandalf. The number of sub-degree tracks goes from 26% to 18% and the mean doubles from 3.38° to 6.86°. The difference between the usage of the true parameters and the tracks selected by JPrefit is striking and might result from the optimisation of JPrefit for the JGandalf algorithm.

7.4 JMerlin’s Input Requirements

In the previous sections, we have seen that JMerlin performs well when it starts at the Monte Carlo truth, but fails when part of the reconstruction chain. Figure 7.1 shows that JMerlin’s likelihood space for a good event is complex with many local minima. Therefore, minimisation becomes a complicated endeavour if it starts far away from the truth. This became visible in the minimisation, where JMerlin shows great potential starting at the truth parameters, but shows no improvements with JPreﬁt or JGandalf tracks. These properties indicate an initial starting problem more pronounced than JGan- dalf’s. Therefore, we performed a directional scan using directional deviates, similar to JGandalf’s input analysis, which is shown in Figure 7.8. Due to the computational time of JMerlin the statistics are much lower than for JGandalf’s input analysis. Compared to JGandalf’s input scan, there are some striking diﬀerences. Tracks close to the true track with low energies no longer have an increasing directional error. They

55 ares

Algorithm 120 Count JMerlin JGandalf 100

0 − 10 3 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure 7.5: Directional accuracy comparison between JMerlin and JGandalf with starting parameters at the Monte Carlo truth. stay in the already found minimum. The change of improved tracks is much less than with JGandalf. JMerlin is only able to improve tracks below 5° from the true direction. The relative change of the improvement is also much lower. It is a clear sign that JMerlin gets stuck in a local minimum. Therefore, the initial directional accuracy has to be around 5° for JMerlin to improve the track. Taking into account the positional deviations using JPreﬁt tracks, we get Figure 7.9. Now, the tracks in the low energy regime and initially close to the truth decrease in directional accuracy. This is the expected behaviour. Compared to JGandalf, the angle limit is smaller. JMerlin’s initial directional error has to be lower than 10° to improve the track, which increases slightly for higher energies. Moreover, JMerlin only improves the tracks when the energy is above 30 GeV. It is much more sensitive to the initial values than JGandalf is. This explains the large improvements with the Monte Carlo truth starting values and the disappointing results on the JPreﬁt tracks. The initial guess has to be close to the truth; closer than JGandalf.

56 ares

90 Algorithm

Count 80 JMerlin 70 JGandalf

0 − 10 3 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure 7.6: The directional error of the best ranked track according to JGandalf and JMerlin after minimisation of the JPreﬁt tracks. ares

90 Algorithm

Count JMerlin 80 JGandalf 70 60 50 40 30 20 10

0 − 10 3 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure 7.7: The directional accuracy of the best ranked track according to JGandalf and JMerlin with JGandalf’s tracks as starting parameters for their minimisation.

57 Median of Final Direactional Accuracy 1 30 0.8

25 0.6 0.4

20 0.2

0 15 −0.2

10 −0.4 Relative Directional Accuracy Change

Initial Directional Accuracy [deg] −0.6 5 −0.8

0 −1 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 7.8: The relative directional accuracy change by JMerlin as a function of initial directional accuracy and true muon energy. The initial tracks are directional deviates from the Monte Carlo truth. Above 60 GeV the number of events is too low. fmedianARes 40 1

35 0.8 0.6 30 0.4

25 0.2

20 0 − 15 0.2 −

0.4 Relative Directional Accuracy Change 10 Initial Directional Accuracy [deg] −0.6

5 −0.8

0 −1 0 10 20 30 40 50 60 70 80 90 100 true Eµ

Figure 7.9: The relative directional accuracy change by JMerlin as a function of initial directional error and true muon energy. The initial tracks are the 36 selected track from JPrefit. These have been run through JGandalf to give the fitted track. Statistical effect are quite present above 90 GeV.

58 Chapter 8

Discussion & Conclusion

The approximation of the original muon direction within the KM3NeT-ORCA detector is a non-trivial task, which requires multiple reconstruction algorithms to transform measurements of light into a path through the detector. In this work, the two algorithms, JPreﬁt and JGandalf, have been analysed and improved upon in the context for track-like events. The likelihood space of the main reconstruction algorithm, JGandalf, shows a global minimum away from the true parameters. This is shown to be a structural problem and indicates a starting value problem; when the initial track direction is too far from the true parameters, the JGandalf algorithm cannot increase the directional accuracy.

8.1 Analysis of JPreﬁt and JGandalf

JGandalf is able to increase the directional accuracy for tracks with an initial directional accuracy below 10°, if the muon energy is above 20 GeV. In the 10 to 20 GeV range, the initial direction error can be larger; up to 15°. However, below 10 GeV the number of signal photons is limited and no improvement in the directional accuracy is achieved. The 36 tracks per event selected by the pre-reconstruction algorithm, JPreﬁt, are optimised for the JGandalf algorithm. Therefore, the directional accuracy of the initial tracks can be worse. The directional error on a track above 10 GeV can be up to 15°. On average, only 10% of the selected tracks by the JPreﬁt algorithm are below the 10° limit and 22% below 15°.

8.2 Upgrading JPreﬁt’s output

The low number of tracks below the initial directional limit of the JGandalf algorithm and the starting value problem of the algorithm are two areas for improvement in the reconstruction. The first is tackled by altering the current quality function and by introducing three new likelihoods. They incorporate an estimator for the expected arrival time of the Cherenkov light on a PMT and use different parametrisations of the expected number of photo-electrons during an event developed in the context of this work. The change in quality function of the JPrefit algorithm is achieved by changing the multiplication factor, and did not lead to an improvement in the directional accuracy.

59 The current factor of 0.25 is optimal for the selection of the best 36 tracks. The three new likelihoods are tested in two regimes: re-ranking and minimisation. Within the context of re-ranking, the three new likelihoods did not result in a better set of 36 tracks compared to the JPrefit ranking. In contrast, when looking at the 36 already selected track by JPrefit, the Time likelihood has the best ranking. This would allow fewer tracks to be passed on to the JGandalf algorithm without a loss of final directional accuracy. Minimisation results in a drastic improvement in the percentage of tracks below the 10° limit. The best performing likelihood, named Time-Hit-Angle, increases it to 25.5% with Time as a close second with 25.4%. Time-Hit performs sub-optimal and only increases the directional accuracy in the already low degree region, while also increasing the number of tracks in the high directional error region. Time and Time-Hit-Angle show similar results, indicating that the parametrisation of hit information does not add new information to the likelihood about the parameters of the track. Since the Time likelihood only uses an estimator for the expected photon arrival time, it is more computationally efficient than the Time-Hit-Angle likelihood. Moreover, Time-Hit-Angle’s likelihood space is flat within 10 meters from the true parameters. This leads to an indifference in the positional parameters within this regime. The Time likelihood, on the other hand, is not flat and increases correctly when further away from the true parameters. The preferred likelihood for minimisation is the Time likelihood. Future research should confirm that these improvements at this stage also lead to a better directional accuracy after the JGandalf algorithm.

8.3 Extending JGandalf: JMerlin

The second area for improvement is the likelihood space of the JGandalf algorithm. We proposed an extension to the current likelihood: JMerlin, which adds the hit and no-hit information of PMTs to the JGandalf likelihood. Its likelihood space has the global minimum near the true parameters for more events than JGandalf, but also contains many local minima that make the search for the correct minimum more difficult. The JMerlin algorithm is also tested in the context of re-ranking and minimisation. In the former, the likelihood cannot improve upon the JGandalf track ranking. Moreover, the track ranked first in the ranking is only the best track in 2.2% of the events. This is at a similar level as randomly picking a track from the set. A probable cause can be the abundance of minima in the likelihood landscape or a stronger dependence on the positional parameters of the JMerlin likelihood. A different method of track selection might improve JMerlin’s ranking. Using the true values as starting parameters for the minimisation of the JMerlin likelihood, it outperforms the JGandalf algorithm with an increase in directional accuracy from 1.92° to 0.81°. Moreover, the number of sub-degree events goes from 28% to 60%. This is a major improvement for the KM3NeT-ORCA reconstruction. However, when using tracks from JPrefit or JGandalf, the JMerlin algorithm does not converge to a solution closer to the true parameters. In Section 7.4, it is shown

60 that the initial directional accuracy for JMerlin is stricter at 5° instead of 10° for the JGandalf algorithm. Together with a stronger position dependence, this could lead to worse performance when using JPreﬁt tracks, especially since the JPreﬁt algorithm is optimised for JGandalf.

8.4 Future Improvements

This brings us to further improvements to the work presented here. The strict initial directional accuracy might be solved by looking at the positional dependence of the JMerlin likelihood and comparing it with the position distribution of the JPrefit and JGandalf tracks. The pre-reconstruction and its selection might introduce a bias, which does not favour the JMerlin algorithm. Furthermore, if only the direction is considered, JMerlin’s starting value problem might be solved by decreasing the directional error from 10° to 5°. One such solution would be to use the complete set of JPrefit tracks for the minimisation since the closest track will always be around 2.5° from the true direction. This is dependent on the assumption of no strong positional dependence of the likelihood. It is important to note that determining the initial directional accuracy constraints for JGandalf and especially JMerlin requires many events and is computationally intensive. While JGandalf’s limits could be determined with enough events in a reasonable amount of time, the no-hit component of the JMerlin algorithm increased the run time significantly. Therefore, the statistics for the JMerlin constraints are sub-optimal. This will have to be addressed in future work. Or a hit selection can be performed to reduce the number of considered PMTs and hits. Implementation of hit selection can speed up the reconstruction, but one has to be careful not to lose important information to the reconstruction. Not only can this be implemented in the JMerlin algorithm, the likelihoods for improving the initial directional accuracy of JGandalf can also benefit from a hit selection to reduce their computational intensity. Furthermore, the Time likelihood suggests that the expected arrival time of Cherenkov light contains essential information about the track. Its implemented estimator is a symmetric Lorentz distribution. The full probability density functions for the expected arrival time show that this distribution is non-symmetric. Therefore, the estimator can be replaced by a non-symmetric distribution that more closely resembles reality to improve the selection and minimisation results of this likelihood. Similarly, the JMerlin likelihood with future hit selections can also be upgraded. First, the remaining assumptions can be removed, such as the use of an infinite track length. In the KM3NeT-ORCA detector, the events are at relatively low energies and thus have a limited track length. The assumed infinite track length results in background being included as signal hits adding no information about the track parameters. Other improvements come from adding additional information to the likelihood. JMerlin already uses almost all available information. Only the time over threshold information can still be incorporated. This is a non-trivial task, because it requires a description of the time over threshold before it can be used in the likelihood

61 8.5 Conclusion

So concluding, the JGandalf algorithm has an initial value problem, such that the directional accuracy of the input tracks has to be below 10°. On average, the JPrefit algorithm only reaches this in 10% of the events. Minimisation using an estimator for the expected arrival time of Cherenkov light improves the fraction of events for which the limit is reached to 25.5%. Moreover, it also ranks the 36 selected tracks better than JPrefit according to their directional accuracy. The upgrade to JGandalf with hit and no-hit information, JMerlin, shows significant improvements when using the true values as starting parameters, but with tracks from JPrefit or JGandalf, the JMerlin algorithm does not improve the directional accuracy. The initial directional error constraints for JMerlin indicate a stricter starting value problem and a possible bias in the selected track introduced by the JPrefit algorithm. The results presented in this thesis provide an essential analysis of the current algorithms and show the potential for extending the reconstruction algorithms with more data. The increase in directional accuracy by the JMerlin algorithm leads to a better neutrino direction. A better directional accuracy of the KM3NeT-ORCA detector will lead to a faster determination of the neutrino mass ordering and, eventually, tighter constraints on the oscillation parameters, which brings us closer to a complete understanding of the sub-atomic particles.

62 Appendices

63 Appendix A

Shower Emission Proﬁles

Along the muon track, small showers are started by δ-ray emission, e−e+ pair production, and ionisation. Due to their cascading nature, showers have a typical energy deposition that initially increases and then slowly decreases till the energy of shower particles drops below the Cherenkov emission limit and the shower disappears. This can be parametrised using the following formula [3, 113]:

dP e−z/b = za−1 , (A.1) dz baΓ(a) where z is the distance from the interaction along the longitudinal direction of the dP shower and Γ a gamma function. a and b are optimised such that the integral of dz from 0 to ∞ is normalised to 1.

E a = 1.85 + 0.62 × log (A.2) [GeV ] b = 0.54 (A.3)

0.8 1 GeV 0.7 10 GeV 100 GeV 0.6 1 TeV y 10 TeV 0.5 100 TeV

probabili t 0.4

0.3 emissio n 0.2

0.1

0.0 0 2 4 6 8 10 longitudinal position [m]

Figure A.1: Longitudinal shower proﬁle for diﬀerent energies. Histogram are generated from Monte Carlo data and the parametrisation is overlaid. Figure from [113].

Besides the parametrisation for the longitudinal proﬁle, the angular emission is also parametrised. While the shower comprises many particles, each emitting their own

64 Cherenkov radiation, the eﬀective emission angle is around the Cherenkov angle in sea water. For energies above 1 GeV, the parametrisation is the following and nearly energy independent [109]:

2 d P a = c eb| cos θ0−cos θC | (A.4) d cos θ0dφ0 where a and b are the ﬁt parameters from [114] and [113]. c is used to normalise the probability to 1 over a full solid angle.

a = +0.35 (A.5)

b = −5.40 (A.6) 1 1 c = (A.7) 2π 0.06667

1 d φ θ P 1 2 10 d cos d

10 2

3 10 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 cos θ

Figure A.2: Parametrisation of the angular emission of an electromagnetic shower. Image taken from [109].

65 Appendix B

JGandalf −LL Scans

To determine the origin of the starting value problem of JGandalf, we looked at the likelihood space of three diﬀerent levels of reconstructed events. The events can be found in the mcv5.0.gsg muon-CC 10-100GeV.km3sim.jte.1.root ﬁle at the corresponding event number.

• Good Event an event reconstructed to 0.662559° by JGandalf (392)

• Medium Event reconstructed to 0.913204° by JGandalf (1965)

• Bad Event reconstructed to 59.9638° by JGandalf. (1948)

Because the global minimum for the time parameter is often at the Monte Carlo truth, it is left out of the likelihood scans. Below the likelihoods of the three events for all the other parameters will follow. All show a global minimum at a diﬀerent position than the true value. This indicates to a structural problem.

Good Event

900 900

-Log LL 800 -Log LL 800 700 700 600 600 500 500 400 400 300 300 200 200 100 100

−150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 Distance [m] Distance [m] (a) X Parameter (b) Y Parameter

66 900 900

-Log LL 800 -Log LL 800

700 700

600 600

500 500

400 400

300 300

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 DX Deviation DY deviation (c) DX Parameter (d) DY Parameter

Figure B.1: The likelihood space of JGandalf for the positional and directional ﬁt parameters for a good reconstruced event. The red line indicates the true value.

Medium Event

900 1000

-Log LL 800 -Log LL 800 700 600 600 500 400 400 300 200 200

100 0

−150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 Distance [m] Distance [m] (a) X Parameter (b) Y Parameter

67 900 1000 -Log LL -Log LL 800 700 800 600 600 500 400 400 300

200 200 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 DX Deviation DY deviation (c) DX Parameter (d) DY Parameter

Figure B.2: The likelihood space of JGandalf for the positional and directional ﬁt parameters for a medium reconstructed event. The red line indicates the true value.

Bad Event

200 1000 180 -Log LL -Log LL 160 800 140 120 600 100 80 400 60 40 200 20

−150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 Distance [m] Distance [m] (a) X Parameter (b) Y Parameter

68 180 1000 170 -Log LL -Log LL

160 800

150 600 140 400 130

120 200

110 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 DX Deviation DY deviation (c) DX Parameter (d) DY Parameter

Figure B.3: The likelihood space of JGandalf for the positional and directional ﬁt parameters for a badly reconstructed event. The red line indicates the true value.

69 Appendix C

Global Minimum Scans

The time parameter is properly defined. Therefore, the global minimum in its likelihood space is close to the truth value. There we show that this is indeed the case by performing a scan over the time parameter with the other track parameters fixed to their Monte Carlo truth value. Besides showing the global minima of JGandalf for the time parameter, the position of the global minima for JMerlin are also shown. The two distributions are similar with limited differences.

3 3

Counts 10 Counts 10

102 102

10 10

1 1

−100 −80 −60 −40 −20 0 20 40 60 80 100 −100 −80 −60 −40 −20 0 20 40 60 80 100 T deviation T deviation (a) JGandalf (b) JMerlin

Figure C.1: Positions of the global minima for the time parameter. The true values are at 0.

70 Appendix D

JPreﬁt Quality Factor Exploration

The influence of the quality factor on JPrefit’s ranking was as of the writing of this thesis unknown. Therefore, an exploration of this topic has been included in this thesis. The important points can be found in Section 6.1. Here we will show more quality factors that have been explored in the range from 0 to 100. Figure D.1 shows all their containment fractions in one figure. The outlier line is the 0-factor, which is a terrible ranker till around 250 when it catches up with the rest. Furthermore, the high quality factors in the redder colours all increase the fastest in the beginning. But lag in the higher track sets, where the lower quality factors take over as best fitters. Which quality factor is the best depends on the selected track size. Overall, the 0.25-factor quality function works reasonably well over the whole range of top N tracks, especially when considering the top 36 tracks. There it has the largest containment fraction of all tested factors with marginal differences from other factors. If, however, the track set size changes, the best quality factor will change with it.

Quality Factor Containment Quality Factor Containment 0.0 0.397149 1.0 0.629328 0.1 0.618126 1.5 0.626782 0.2 0.631365 2.0 0.623218 0.25 0.644603 3.0 0.626782 0.3 0.639511 4.0 0.626273 0.4 0.639002 5.0 0.630346 0.5 0.639511 10.0 0.626273 0.6 0.635947 15.0 0.622709 0.7 0.636965 20.0 0.62169 0.8 0.63391 30.0 0.617108 0.9 0.630855 100.0 0.60947

Table D.1: The containment fraction for diﬀerent quality factors in the JPreﬁt quality functions

71 1

0.8 Quality Factor 0.00 0.10 0.20 0.25 0.6 0.30 0.40 0.50 0.60 0.70 0.80 0.4 0.90 1.00 1.50 2.00 3.00 4.00 0.2 5.00 10.00 15.00 20.00 30.00 0 100.00 0 100 200 300 400 500 600 700 800 Fraction Top N tracks containing the best D.A. Top N tracks

Figure D.1

72 Appendix E

M-Estimator

hits Y L = P (ti|θtrack) (E.1) i JSimplex likelihood is the product of probabilities, see Equation E.1 These probabilities describe the expected arrival time of the Cherenkov cone. By comparing the actual arrival time of a hit, a probability can be calculated that states the chance that a hit occurred at that time. In JSimplex this probability is described by a Lorentz distribution to speed up the calculation. It is also known as a Cauchy distribution and follows the following formula:

γ2 f(x : xo, γ, I) = I 2 2 , (E.2) (x − x0) + γ where I is a normalisation factor and can be 1/πγ to normalise to 1. x0 is location of the peak of the distribution, γ determines the height and spread of the peak, and x is the input parameter. Figure E.1 shows a standard Lorentz distribution compared to a Normal distribution. Near the mean, the distributions are equal, but at the Lorentz the tails decrease slower than the Gaussian. This makes the Lorentz less sensitive to outliers. To show this, a normalised version of Equation E.1 is plotted for a properly reconstructed event with both a Lorentz and Normal distribution. Likelihood using the Gaussian distribution is much steeper and peaks at the wrong value. The Lorentz likelihood does peak at the right location.

73 1 Probability 0.8

0.6

0.4

0.2

0 −4 −2 0 2 4 x

Figure E.1: Gaussian (black) distribution with µ = 0 and σ = 1. The Lorentz (red) distribution has γ = 1 and I = 1.

600

500

400 Normalised -log LL

300

200 Time MEstimator

100 Time Gauss

× 6 0 10 52.61876 52.61878 52.6188 52.61882 time [ns] (a)

74 1000 1000

800 800

Normalised -log LL 600 Normalised -log LL 600

400 400 Time MEstimator Time MEstimator

200 Time Gauss 200 Time Gauss

0 0 −150 −100 −50 0 −100 −50 0 50 X position [m] Y position [m] (b) (c)

600 600

500 500

400 400 Normalised -log LL Normalised -log LL

300 300

200 Time MEstimator 200 Time MEstimator

Time Gauss Time Gauss 100 100

0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 DX direction DY direction (d) (e)

Figure E.2: Likelihood scans over all parameters for a good event with a Gaussian (red) and Lorentz distribution (black) as the time estimator. The vertical red line indicates the location of the MC truth.

75 Appendix F

Parametrisation of PDFs

JGandalf uses a full description of the expected arrival time on a PMT. These are stored in tables as the expected number of photo-electrons arriving at a certain time. The integral over time can be performed to obtain the expected number of photo-electrons during an event (np.e.). Preferably, a parametrisation for the three remaining parameters is used. These are the two angles of the PMT with respect to the track (θ, φ) and the distance between the PMT and the track (R). In Section 6.3 a parametrisation based on the time integrated and angle-averaged expected number of photo-electrons is introduced. Here, two different options are discussed. First, fits based on the individual light contributions instead of a fit on their combination. Secondly, the possibility for a parametrisation without angle-averaging is explored.

F.1 Individual Distance parametrisation

As described in Section 4.3.4, six different light sources can be distinguished. Because the contributions from δ-rays are minimal, they are left out of the parametrisation. For the distance parametrisation the PDF tables are integrated over time and averaged over θ, and φ. In Section 6.3, the fit process on the combined light contributions is covered. Here, the same process is repeated for individual light contributions. Their fits look great too, and some have even smaller relative errors than the combined fit. Only the direct Cherenkov contribution has trouble fitting the first meters. It is over-estimating the expected number of photo-electrons. In the end the parametrisation of the combined fit, because the individual fits are more computationally intensive. It might be possible to reduce the complexity of the fit functions for the individual contributions. This, however, is a non-trivial task. The relation between the original equations in Section 4.3.4 and a ”simple” function is obscured by the integrals and other factors involved.

76 Parameter Direct Cherenkov Indirect Cherenkov Direct EM Indirect EM p0 2.97185e-2 2.91034e-2 2.28628e-5 3.33769e-5 p1 1.52524e+1 2.74463e+1 1.76486e+1 2.92422e+1 p2 1.15877e+1 5.34003e00 1.28605e+1 5.70203e00 p3 2.76978e-2 3.04665e00 -3.75390e-2 2.77866e00 p4 1.38968e-3 3.10809e-3 8.42555e-4 6.46060e-3 p5 -6.80235e-6 1.63062e-4 -6.53198e-6 1.14147e-4 p6 1.36958e-8 -1.67102e-7 6.50376e-9 2.73494e-8 p7 -6.59486e-4 7.25288e-1 1.18694e-2 4.75299e-1 p8 1.16795e-1 6.89384e-1 7.65347e-2 4.96245e-1 p9 3.14282e-3 3.61473e-2 2.85991e-3 3.60447e-2 p10 -2.58075e-5 1.38979e-4 -3.32615e-5 1.21168e-4 p11 5.49159e-8 -8.44508e-7 8.54297e-8 -7.14626e-7

Table F.1: Best ﬁt parameters for the ﬁts on the individual light contributions.

Exponential and rational fit

> 0.05 10 p.e.

data 0.04

Exponential and rational fit

> 0.05 −1 p.e. 10

data 0.04

− 10 3 0 −0.01

− −0.02 10 4 −0.03 − − 0.04 10 5 −0.05 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Distance to track [m] Distance to track [m] (c) Full ﬁt on Indirect Cherenkov Light. (d) Residual of Full ﬁt on Indirect Cherenkov Light.

77 Exponential and rational fit

> 0.05

p.e. − 10 3

data 0.04

− 0.01 10 5 0 − 10 6 −0.01 −0.02 −7 10 −0.03 − − 0.04 10 8 −0.05 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Distance to track [m] Distance to track [m] (e) Full ﬁt on Direct Light from EM showers. (f) Residual of Full ﬁt on Direct Light from EM shower.

Exponential and rational fit

> 0.05 p.e.

data 0.04

Figure F.1: Fits on the individual light contributions.

F.2 Angle parametrisation

In the previous section, we have found a fit function that can accurately describe the individual and combined light contributions. These contributions, however, are averaged over the angles. The directional information is important, because different directions give different expected number of photo-electrons. In Section 6.3 this is reintroduced by implementing the angular acceptance of the PMT in the parametrisation for the expected number of photo-electrons. Another method is to not perform the averaging over the angles and perform a fit over R using Equation 6.4 for a range of values of θ and φ. The found fit parameters can then be fitted to reintroduce the directional information. Figure F.2 shows that for some directions the fit on the combined light contributions converges properly, while for others it does not. This is a problem that will need to be solved before a proper fit on the fit parameters can be performed. While different minimisation algorithms have been tried, the improvements were minimal and fits still failed to converge.

78 theta=0.00 phi=0.00 theta=2.39 phi=0.37

p.e. −1 p.e. 10

n 10 n

− 10 2 1

− −1 10 3 10

− −2 10 4 10

− −3 10 5 10

− − 10 6 10 4 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Distance to track [m] Distance to track [m] (a) θ = 0, φ = 0 (b) θ = 2.39, φ = 0.37

Figure F.2: Directional ﬁts for two diﬀerent directions.

For the individual light contributions, the situation is slightly different. The direct light seems to fit ”reasonably” well. There are still directions for which it does not minimise properly, but the p0 parameter shows a clear structure as shown in Figure F.3a. The peak is caused by the Cherenkov emission, and it is possible to fit a 2D Gaussian to this, although it is not a great fit. In the white area for p0 and p1 there is no light measured. This causes minimisation problems near its border. The other parameters show this problem region as either yellow or white. Although, sometimes it does achieve reasonable fits. Furthermore, the other parameters are mostly flat in the region where it is able to fit correctly. These parameters can be set to a fixed value without altering the p0 distribution too much. This shows potential to parameterise the direct Cherenkov Light. The indirect Cherenkov contribution does not have a clear Gaussian peak. It has a peak at the Cherenkov angle, but also a dip in the opposite direction for th p0 parameter. p1, on the other hand, looks different and has a peak and dip at locations perpendicular to the Cherenkov angle. The other fit parameters have quite some trouble properly minimising. Two distinct regions can be identified. The first is pointing towards the track, while the other it pointing away from the track. The scattering is the most probable origin for this division and the transition region is a difficult to minimise region. For the direct EM shower contribution, a similar pattern to the Direct light can be recognised, but the p0 maximum is less peaked and less Gaussian shaped. Moreover, the second parameter p1 now shows its own structure that does not look like a familiar pattern. The other parameters are relatively constant, but the minimisation has difficult to converge in many regions, mostly around θ = 1.5. Furthermore, the indirect EM shower light’s fit parameter space is even more complex. Regions similar to the direct EM shower contribution can be identified, but these transitions are more direct, which is difficult to parameterise. One would first have to overcome the minimisation issue and reduce the number of free parameters in the fit to capture most information in a single parameter. Another way to achieve a parametrisation could be to run a neural network on the data set and perfectly fit to it. That research will be left for others to pursue.

79 Fit Parameter Space

Direct Cherenkov Light

φ 3 φ 3 0.07 14

2.5 2.5 0.06 12

2 0.05 2 10

0.04 8 1.5 1.5

0.03 6 1 1 0.02 4

0.5 0.5 0.01 2

0 0 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

(a) p0 (b) p1

φ 20 φ 1 3 3

2.5 2.5 0.8 15

2 2 0.6

10 1.5 1.5 0.4

1 1 5 0.2 0.5 0.5

0 0 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

80 − ×10 3

φ 0.01 φ 3 3

0.2 2.5 0.008 2.5

0.15 2 0.006 2

0.1 1.5 0.004 1.5

1 1 0.05 0.002

0.5 0.5 0 0

0 0 −0.05 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

(e) p4 (f) p5

− ×10 6

φ 0.1 φ 3 3 0.0004

2.5 2.5 0.05 0.0002

2 2 0

0 −0.0002 1.5 1.5

−0.0004 1 −0.05 1 −0.0006

0.5 0.5 −0.0008 −0.1 0 0 −0.001 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

(g) p6 (h) p7

81 φ 1 φ 0.01 3 3

2.5 0.8 2.5 0.008

2 2 0.006 0.6

1.5 1.5 0.004 0.4 1 1 0.002 0.2 0.5 0.5 0

0 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

(i) p8 (j) p9

− ×10 6

φ 0.001 φ 1 3 3

0.5 2.5 0.0008 2.5

0 2 2 0.0006

−0.5 1.5 1.5 0.0004 −1 1 1

0.0002 −1.5 0.5 0.5

−2 0 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 θ θ

(k) p10 (l) p11

Figure F.3: The space of the fit parameters for a full fit on the direct Cherenkov contribution. The colour indicates the value of the parameter found by the minimisation. In white areas the fit was unable to be performed.

82 Indirect Cherenkov Light

φ 3 0.06 φ 3 32

2.5 0.05 2.5 31

30 2 0.04 2

29 1.5 0.03 1.5 28

1 0.02 1 27

0.5 0.5 0.01 26

25 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(a) p0 (b) p1

φ 10 φ 5 3 3 8

2.5 6 2.5 4

4 2 2 3 2

1.5 0 1.5 2 −2 1 1 −4 1 −6 0.5 0.5 −8 0 0 −10 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

83 φ 0.5 φ 0.001 3 3

0.4 0.0008 2.5 2.5

0.0006 2 0.3 2

0.0004 1.5 0.2 1.5

0.0002 1 0.1 1

0 0.5 0.5 0 −0.0002 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(e) p4 (f) p5

− ×10 6

φ 10 φ 1 3 3

8 2.5 2.5 0.8

6 2 2 0.6 4 1.5 1.5

2 0.4 1 1 0 0.2 0.5 0.5 −2

0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(g) p6 (h) p7

84 φ 5 φ 1 3 3

2.5 4 2.5 0.8

2 2 3 0.6

1.5 1.5

2 0.4 1 1

1 0.2 0.5 0.5

0 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(i) p8 (j) p9

− ×10 6

φ 0.01 φ 0 3 3 −1 0.008 2.5 2.5 −2

−3 0.006 2 2 −4

0.004 − 1.5 1.5 5

−6 0.002 1 1 −7

− 0 8 0.5 0.5 −9 −0.002 0 0 −10 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(k) p10 (l) p11

Figure F.4: The space of the fit parameters for a full fit on the indirect Cherenkov contribution. The colour indicates the value of the parameter found by the minimisation. In white areas the fit was unable to be performed.

85 Direct EM Shower

− ×10 3 φ

φ 0.1 20 3 3 0.09 18

2.5 0.08 2.5 16

0.07 14 2 2 0.06 12

1.5 0.05 1.5 10

0.04 8 1 1 0.03 6

0.02 4 0.5 0.5 0.01 2

0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(a) p0 (b) p1

φ 20 φ 10 3 3

15 2.5 2.5 8

10 2 2 6

1.5 5 1.5 4

1 0 1 2

0.5 −5 0.5 0

0 −10 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

86 φ 0.1 φ 0.001 3 3

2.5 0.08 2.5 0.0008

2 0.06 2 0.0006

1.5 1.5 0.04 0.0004

1 1 0.02 0.0002

0.5 0.5 0 0

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(e) p4 (f) p5

− ×10 6

φ 1 φ 0.02 3 3 0.8 0.015 2.5 0.6 2.5

0.4 0.01 2 2 0.2

1.5 0 1.5 0.005

−0.2 1 1 0 −0.4

−0.6 0.5 0.5 −0.005 −0.8

0 −1 0 −0.01 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(g) p6 (h) p7

87 φ 0.1 φ 0.05 3 3 0.09 0.045

2.5 0.08 2.5 0.04

0.07 0.035 2 2 0.06 0.03

1.5 0.05 1.5 0.025

0.04 0.02 1 1 0.03 0.015

0.02 0.01 0.5 0.5 0.01 0.005

0 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(i) p8 (j) p9

− ×10 6

φ 0.01 φ 3 3 0

2.5 0.008 2.5 −2

2 0.006 2

−4 1.5 1.5 0.004

−6 1 1 0.002

− 0.5 0.5 8 0

0 0 −10 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(k) p10 (l) p11

Figure F.5: The space of the fit parameters for a full fit on the direct electromagnetic shower contribution. The colour indicates the value of the parameter found by the minimisation. In white areas the fit was unable to be performed.

88 Indirect EM Shower

×10−3 φ 3 φ 3 32.5 0.07

32 2.5 0.06 2.5 31.5

0.05 2 2 31

0.04 30.5 1.5 1.5 30 0.03 1 1 29.5 0.02 29 0.5 0.5 0.01 28.5

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(a) p0 (b) p1

φ 8 φ 5 3 3 4.5 7 2.5 2.5 4 6 3.5 2 5 2 3

1.5 4 1.5 2.5

2 3 1 1 1.5 2 1 0.5 0.5 1 0.5

0 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

89 − ×10 3

φ φ 0.4 3 0.14 3 0.3 0.12 2.5 2.5 0.2 0.1

0.1 2 0.08 2

0.06 0 1.5 1.5 0.04 −0.1

0.02 − 1 1 0.2 0 −0.3 0.5 −0.02 0.5 −0.4 −0.04 0 0 −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(e) p4 (f) p5

− ×10 6

φ 10 φ 1 3 3

2.5 8 2.5 0.8

2 6 2 0.6

1.5 1.5 4 0.4

1 1 2 0.2

0.5 0.5 0 0

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(g) p6 (h) p7

90 φ 0.8 φ 0.05 3 3

0.7 0.045 2.5 2.5 0.04 0.6

2 2 0.035 0.5

1.5 1.5 0.03 0.4 0.025 1 1 0.3 0.02

0.5 0.5 0.2 0.015

0 0.1 0 0.01 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(i) p8 (j) p9

− ×10 6

φ 0.002 φ 3 3 0 0.0015 2.5 2.5 −2 0.001 2 2

−4 1.5 0.0005 1.5

−6 1 0 1

− 0.5 −0.0005 0.5 8

0 −0.001 0 −10 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θ θ

(k) p10 (l) p11

Figure F.6: The space of the fit parameters for a full fit on the indirect electromagnetic shower contribution. The colour indicates the value of the parameter found by the minimisation. In white areas the fit was unable to be performed.

91 Appendix G

JGandalf vs JGandalfx vs JMerlin

Within the KM3NeT reconstruction software, the JGandalfx reconstruction algorithm uses the hit and no-hit information. It is similar to JMerlin, but performs a hit and PMT selection, which consists of PMTs and hits within a cylinder around the track and ﬁrst hits on a PMT. This reduces the computational time, but signal hits might be lost. Furthermore, the likelihood calculations introduce a Lorentz function to smooth the hit probability. It also uses a likelihood ratio between the background and signal, since for each track diﬀerent number of hits and PMTs are selected. We have seen that JMerlin improves upon JGandalf’s reconstruction when starting from the Monte Carlo truth. For a comparison we have performed the same minimisation using JGandalfx, which is shown in Figure G.1. JGandalfx is run with the standard JGandalf parameters and has a similar distribution of the directional accuracy. It is slightly better, but nothing in comparison with the improvement JMerlin gives, suggesting that the hit and PMT selection for JGandalfx is too restrictive. In future research, larger road widths and JGandalfx’s usefulness in re-ranking should be explored. ares

Algorithm 140 JGandalf Count JGandalfx 120 JMerlin

100

0 − 10 3 10−2 10−1 1 10 102 Directional Accuracy [deg]

Figure G.1: The top ranked track after minimisation from the Monte Carlo truth for JGandalf, JGandalfx, and JMerlin.

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100 Acknowledgements

After a year of hard work, learning, and great experiences, it is ﬁnally over! Everyone always includes this little chapter and I don’t want to be the exception, because you never do research alone. In general, a big thank you to all the KM3NeT people at Nikhef and, of course, the DUNE peeps. Their help was sometimes appreciated, but always useful! In particular, I would like to thank Ronald Bruijn for being my supervisor and getting me involved in the KM3NeT collaboration and guiding my thesis. I want to thank Br´ıan for the nice discussions on JGandalf and the rest of JPP’s code base. Als laatst een stukje in het Nederlands, om mijn ouders, Robert en Marijke, te bedanken voor de steun en de mogelijkheid om te studeren waar ik van houd! Het moet niet altijd even makkelijk geweest zijn. En natuurlijk mijn zus, Robien, die altijd met super interessante vragen komt over de wereld. Thanks! Jullie zijn lief!

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