Track reconstruction of cosmic muons in the XENON1T muon veto

MSc Physics and Astronomy

Gravitation, Astro-, and Particle

Physics

Master Thesis

Track reconstruction of cosmic muons in the XENON1T

muon veto

by

Davey Oogjes

11909714

November 2019

60 ECTS

November 2018 – November 2019

Supervisor/Examiner:

Examiner:

dhr. prof. dr. M.P. Decowski

mevr. D.F.E. Samtleben

Abstract

The XENON1T experiment is one of the most sensitive dark matter detectors in the world. One of the ways this high sensitivity is reached is through the use of a Muon Veto. This work presents a first analysis of events in the Muon Veto. For this, a simulation algorithm was developed, as well as two reconstruction methods, one that uses timing information and one that uses amplitude information.

Using this simulation, these two reconstruction methods are tested and com-pared. After comparison, the time fitting method is determined to poorly recon-struct tracks, while the amplitude fitting method performs well at reconrecon-structing simulated tracks.

The simulation is also used to design cuts on the Muon Veto data, resulting in 97.8% of events being removed. The time fitting algorithm seems to perform well at reconstructing the remaining tracks, but this is determined to be due to too much noise being present in the data after cuts. Likewise, the amplitude fitting algorithm does not reconstruct real tracks well due to the presence of noise. Although this first analysis is unsuccesful at reconstructing muon tracks, methods are presented through which future analysis could succeed.

Contents

1 Introduction 1

2 Particles and relevant interactions 3

2.1 Muons . . . 3 2.1.1 Particle showers . . . 3 2.1.2 Cosmic Rays . . . 4 2.1.3 Cherenkov radiation . . . 5 2.2 Neutrons in XENON1T . . . 7 2.2.1 Production mechanisms . . . 7

3 The XENON1T Experiment 10 3.1 Photomultiplier Tubes . . . 10

3.2 XENON1T . . . 11

3.2.1 Neutron shielding . . . 13

3.3 The XENON1T Muon Veto . . . 13

3.3.1 Design of the Muon Veto . . . 13

3.3.2 Readout and triggers . . . 14

3.3.3 Calibration . . . 15

4 Muon track simulation and reconstruction 17 4.1 Muon track principles . . . 17

4.2 The simulation algorithm . . . 19

4.3 Track reconstruction algorithm . . . 24

4.3.1 Fit performance . . . 26

5 Data selection and cuts 35 6 Reconstructed muon tracks 39 6.1 Timeline fit . . . 39

6.2 Amplitude fit . . . 41

7 Conclusions and Discussion 46

References I

Chapter 1

Introduction

One of the biggest mysteries in modern physics is the nature of dark matter. Multiple observations have shown that ordinary, luminous matter can only account for 15% of the matter content of the universe, with the remaining 85% of matter being some invisible or dark matter, seemingly only interacting with other matter through gravitational interactions. Many theories exist to explain dark matter, but one of the best motivated theories is that of the WIMP, a Weakly Interacting Massive Particle. WIMPs interact with ordinary matter through the electroweak interaction and have a mass in the GeV range, and fit nicely into popular theories of physics beyond the Standard Model like Super Symmetry.[1]

However, recent experiments aimed at discovering dark matter have not been able to find the WIMP, so instead limits are being placed on its mass and cross section with ordinary matter. Some of the strongest of these limits were set by the XENON collaboration with the XENON1T experiment[2], which uses a so-called dual phase time projection chamber(TPC).

As WIMPs have only a small chance to interact with the detector target, it is vital to shield the target material from as much background radiation as possible. For this reason, the TPC is placed inside of a water tank, protecting the TPC with ∼4 m of water on all sides. This water tank serves three purposes: the hydrogen atoms have masses similar to neutrons and thus slow them down quickly, the water shields the experiment from electromagnetic radiation, and the tank is outfitted with 84 PMTs, which can detect light emitted by charged particles travelling through the water by the Cherenkov radiation they emit. In most cases, these charged particles are cosmic muons, produced by cosmic rays which interact with our atmosphere. As these muons are quite energetic and do not lose much of their energy as they travel through the atmosphere and the Gran Sasso mountain, they are able to reach the water tank with ease.

While shielding and providing a veto when a muon veto event coincides with a TPC events are the main functions of the muon veto, it is possible to reconstruct the muon tracks using the timing and charge information gathered by the PMTs. This makes the water tank not only a muon veto, but also a muon tracking detec-tor. In this thesis, a first analysis of the muon data will be presented using two reconstruction methods, one that uses timing information and one that uses signal amplitude information.

2 Introduction

This work is structured as follows. In Chapter 2, muons and their interactions with other matter that are relevant to this work will be discussed, as well as one of the products of muons, the neutron. After that, Chapter 3 discusses the work-ing principle of XENON1T, followed by a description of the workwork-ing principle and technical design of the Muon Veto. Next, Chapter 4 treats the methods used to simulate and reconstruct muon tracks, as well as the results of using these recon-struction methods on simulated events. Following, Chapter 5 discusses the data used and the cuts that were designed using the simulated events. Lastly, Chapter 6 presents the results of using the reconstruction algorithms on the XENON1T Muon Veto data.

Chapter 2

Particles and relevant interactions

Dark matter has only a very slim chance of interacting with normal matter, which means that background signals caused by other types of particles need to be well understood. While some of these background events originate in radioactive decays in the detector material or the surrounding rock, there is also a cosmic source of background noise, which is the muon. In this section, the muon will be discussed together with the various interactions it can have with the detector. In the second part of this chapter, the neutron will be discussed, as it is a very dangerous source of background due to leaving a WIMP-like signal.

2.1

Muons

2.1.1

Particle showers

When a high-energy particle collides with matter, it can produce a number of new particles at lower energies, which can again interact, producing more particles until the energies are too low and the particles are stopped by the medium. This is called a particle cascade or particle shower. Within particle showers, there are two categories, depending on the interaction that governs the shower:

• Electromagnetic showers

Electromagnetic showers are started by either a high-energy electron/positron or a photon interacting primarily through the electromagnetic interaction. When their energy is above a few MeV, electrons and positrons will emit pho-tons through bremsstrahlung when in the vicinity of an atomic nucleus. These photons will then interact with nuclei primarily through electron-positron pair production, and these electrons and positrons can again interact through bremsstrahlung. This will continue until other energy loss processes like the photoelectric effect and Compton scattering start to dominate for the elec-trons/positrons, and the photon energies fall below the threshold for pair pro-duction.

• Hadronic showers

When the parent particle is a hadron, there is a chance for spallation to oc-cur. In this process, the parent particle interacts with the nucleons of the target, starting a cascade in which more hadrons are produced. The so-called secondary particles escape from the interacting atom, while other particles

4 Particles and relevant interactions

deposit their entire energy in the nucleus by exciting it. When the nucleus de-cays to a lower energy level, it expels low energy particles, primarily neutrons. Meanwhile the secondary particles might spallate other nuclei producing more particles, or decay into lower mass particles, which can start electromagnetic showers if their energy is above the threshold for bremsstrahlung or pair pro-duction. The entire collection of particles involved in this event is called a hadronic shower.

2.1.2

Cosmic Rays

When high-energy cosmic particles, e.g. protons, enter the upper atmosphere, they collide with atmospheric nuclei. In this collision, often a hadronic shower is started, primarily producing pions and kaons as secondary particles. An example of this type of event is shown in fig. 2.1, where an incoming proton interacts with an atmospheric nucleus. Some of these produced pions and kaons particles will interact with other nuclei through the strong interaction, producing more particles until these secondary particles reach energies below the threshold for pion production. Others will decay before any collision. In the case of neutral pions, the products of its decays are photons, which can initiate an electromagnetic shower.

The charged mesons however are not allowed to decay into photons due to con-servation of charge, and will instead mostly decay into muons:

π+→ µ+_{+ ν}

µ π− → µ−+ νµ

K+→ µ+_{+ ν}

µ K− → µ−+ νµ.

While these muons only have a mean lifetime of 2.2 µs, their highly relativistic velocities allow them to reach the surface of the Earth. As they typically lose only

about 2 MeV per g/cm2 _{to ionization as they pass through matter, they have an}

average energy of ∼4 GeV when they reach the surface of the Earth. The flux is

around 1 muon cm−2s−1 at sea level.[3]

As muons lose little energy as they traverse matter, they are able to reach un-derground labs like LNGS, which is the site of the XENON1T experiment. Some experiments, including Borexino[5], GERDA[6], LVD[7] and MACRO[8], have pub-lished results on muons detected in the underground lab, including their energies, angular distribution and flux. In Hall B of LNGS, the muon flux is (3.31 ± 0.03 ×

10−8) cm−2s−1, with an average energy of ∼270 GeV.[2] The muon angular

distri-bution, in both the inclination and azimuthal angle for the Borexino experiment in Hall A, is shown in fig. 2.2. In the left figure an inclination cos(θ) of 1 means that a muon travels straight down while a value of 0 means it travels horizontally. As expected the distribution shows that most muons travel vertically. At ground level the azimuthal distribution is expected to be isotropic. However, fig. 2.2 shows two peaks and two valleys instead. This observation demonstrates the effect of the varying rock density and the shape of the Gran Sasso mountain on the muon flux.

2.1 Muons 5

Figure 2.1: A high-energy proton colliding with an atmospheric nucleus, producing a particle shower. In this shower many particles are produced, including a large number of muons. From [4].

(a) (b)

Figure 2.2: Angular distribution of muons in Hall A of the LNGS underground laboratory, as measured by the Borexino experiment [9].

2.1.3

Cherenkov radiation

When a charged particle moves through a polarizeable medium, it excites the molecules in the medium to higher energy levels. As they relax to their ground state, photons are emitted in the form of electromagnetic waves, which move outward spherically,

as is shown in fig. 2.3. When v < cm, where cm is the phase-velocity of light in

the medium and v the velocity of the particle, these waves interfere destructively.

However, when v > cm, they interfere constructively, resulting in coherent radiation

6 Particles and relevant interactions

is called the Cherenkov angle, and can be calculated through

cos(θc) =

1

βn, (2.1)

where β = v

c, c is the speed of light in vacuum and n is the refractive index of

the medium. When ultra-relativistic, meaning β ≈ 1, muons travel through water

which has a refractive index nwater = 1.333, the Cherenkov photons are emitted at

a constant angle of θc = 41.25◦. The amount of photons emitted per unit length

travelled by the charged particle and per unit wavelength can then be calculated through d2_N dxdλ = 2παz2 λ2 1 − 1 β2_n2 ! , (2.2)

where α = ₁₃₇1 is the fine structure constant and z = 1 the charge number.[3]

The 1

λ2 dependence means that photon production at higher energies is suppressed.

Therefore Cherenkov radiation is mostly in the ultraviolet range.

Figure 2.3: Electromagnetic waves produced by charged particle travelling through a polarizable medium for v<c and v>c. When v > c, Cherenkov radiation is produced,

travelling in the direction θC with respect to the direction of the particle. From [10].

In the following chapters, eq. (2.2) will be used to simulate and reconstruct muon tracks, by integrating the equation over certain wavelength ranges. First however, one of the products of cosmic muons will be discussed, namely neutrons.

2.2 Neutrons in XENON1T 7

2.2

Neutrons in XENON1T

2.2.1

Production mechanisms

Neutrons are an important source of background radiation for XENON1T, as they leave a WIMP-like signal. They are produced through a variety of mechanisms, either radioactive in origin or cosmic in origin:

• Alpha decay

An alpha-particle consists of two protons and two neutrons bound together, similar to a helium-4 ion. In unstable nuclei an alpha particle experiences Coulomb repulsion between it and the rest of the nucleus. This is balanced by the weak nuclear force, but there is a possibility for the α-particle to escape the nucleus through quantum tunneling. This ejected α-particle can then spallate a nucleus in the surrounding material, knocking out a neutron in the process. As an example, there is the α-decay of Radium-226:

226 88 Ra → 222 86 Rn + 4 2α,

which could then lead to a reaction like:

4 2α + 9 4Be → 12 6 C + 1 0n, releasing a neutron.

• Beta-delayed neutron emission

In a beta-delayed neutron emission (denoted βn), a proton experiences a β-decay which turns the atom into a different element and leaves it in an unstable excited state. In the case of βn, a neutron is emitted as the excited nucleus decays into its ground state. As an example, there is the beta-delayed neutron

emission decay of 17 7 N: 17 7 N → 17 8 O ∗ + β−+ νe 17 8 O ∗ _→16 8 O + n.

In neutron rich nuclei typically βn occurs, as the nucleus lacks the binding energy to hold on to the additional neutron, but likewise beta-delayed alpha or proton emission can occur in other nuclei.

• Spontaneous fission

The nuclear binding energy reaches its maximum at mass numbers of about A = 56. This means that very heavy particles are unstable, and have a chance to spontaneously fission. When this occurs, the nucleus splits up into two daugh-ter particles that are relatively close in mass, which can lose a few neutrons or other particles if they lack the binding energy.

• Cosmogenic neutrons

Cosmogenic neutrons on the other hand are not produced in radioactive decays, but in the interaction of cosmic radiation, generally muons, with the material

8 Particles and relevant interactions

surrounding the experiment. As the muons travel through the rock, they can spallate nuclei within the rock, knocking out neutrons and other particles. The products of this spallation can then either directly influence the experiment, or start a particle shower, resulting in a large number of neutrons if the nature of the shower is hadronic.

To protect the XENON1T experiment from radiation, it was placed inside of a water tank. However, there is one more mechanism through which neutrons are produced, and that is through muon-induced showers within this water tank. When a muon starts a hadronic shower inside of the water tank, the isotopes in table 2.1 can be produced. Although most of these only emit electromagnetic radiation, there are four possible isotopes that will decay through beta-delayed neutron emission. These could pose a problem for the XENON1T experiment and as of now would go undetected. Being able to track muons could help localize these radioisotopes in the water tank.

2.2 Neutrons in XENON1T 9

Table 2.1: Table of possible radioisotopes produced through muon-induced showers in water[11].

Radioactive isotope τ (s) Decay mode Ekin(MeV) Primary process

11_{Be 19.9 β}− _{(55%) 11.51} 16_{O(n, α + 2p)}11_Be β−γ (31%) 9.41 + 2.1(γ) 16_{N 10.3 β}− _{(28%) 10.44} 16_{O(n, p)}16_N β−γ (66%) 4.27 + 6.13(γ) 15_{C 3.53 β}− _{(37%) 9.77} 16_{O(n, 2p)}15_C β−γ (63%) 4.51 + 5.30(γ) 8_{Li 1.21 β}− _∼13.0 16_O(π−_{, α+}2_{H+p + n)}8_Li 8_{B 1.11 β}+ _∼13.9 16_O(π+_{, α + 2p + 2n)}8_B 16_{C 1.08 β}−_{+ n ∼4} 18_O(π−_{, n + p)}16_C 9_{Li 0.26 β}− _{(49.2%) 13.6} 16_O(π−_{, α + 2p + n)}9_Li β−+ n (50.8%) ∼10 9_{C 0.18 β}+_{+ p 3∼15} 16_{O(n, α + 4n)}9_C 8_{He 0.17 β}−_{γ (83.1%) 9.67 + 0.98(γ)} 16_O(π−_,3_{H+4p + n)}8_He β−+ n (16.0%) 12_{Be 0.034 β}− _11.71 18_O(π−_{, α + p + n)}12_Be 12_{B 0.029 β}− _13.37 16_{O(n, α + p)}12_B 13_{B 0.025 β}− _13.44 16_O(π−_{, 2p, n)}13_B 14_{B 0.02 β}−_{γ 14.55+6.09(γ)} 16_{O(n, 3p)}14_B 12_{N 0.016 β}+ _16.38 16_O(π+_{, 2p + 2n)}12_N 13_{O 0.013 β}+_{+ p 8∼14} 16_O(µ−_{, µ}−_{+ p + 2n + π}−₎13_O 11_{Li 0.012 β}− _{(8.07%) 20.62} 16_O(π+_{, 5p + π}0_{+ π}+₎11_Li β−+ n (84.9%) ∼16

Chapter 3

The XENON1T Experiment

In the search for dark matter, the XENON1T experiment has at the date of pub-lishing put the most stringent limits on the WIMP cross-section in the GeV mass-range[12]. Furthermore, the XENON1T collaboration could directly observe

two-neutrino double electron capture in 124_{Xe for the first time[13]. For these rare}

processes, it is vital to achieve a very low background. Therefore the experiment is located in the Laboratori Nazionali del Gran Sasso (LNGS) in Italy, an underground lab where the experiment is protected by 3400 m of water equivalent shielding. For further protection, the detector is placed inside a water tank, the so-called Muon Veto, which provides both passive active shielding, through the use of photomulti-plier tubes that detect the Cherenkov light emitted by charged particles as they cross the water faster than the speed of light. In this chapter, the XENON1T detector and its Muon Veto will be introduced in detail.

3.1

Photomultiplier Tubes

For both XENON1T and its Muon Veto, the sensitive part of the detector consists of photomultiplier tubes. Photomultiplier tubes (PMTs) are devices that turn photons into measurable electronic signals. When a photon hits the photocathode, as shown in fig. 3.1, an electron is emitted from the surface through the photoelectric effect. This photoelectron (PE) is then accelerated to the first dynode due to an electric field inside the PMT, where it kicks out more electrons. After multiple of these dynode stages, the multiplied electrons are collected at the anode and converted into an electronic signal. This analogue signal is then converted into a digital signal by an analogue to digital converter (ADC).

An important property of a PMT is the quantum efficiency (QE). As photons hit the photocathode, electrons in the valence band absorb the energy and get excited. These electrons then diffuse towards the surface of the photocathode, and if their

energy is higher than the vacuum level energy, EV, they are emitted into the vacuum

as photoelectrons. Thus for an electron to be converted into a photoelectron, it needs

to absorb at least the energy EF needed to reach the Fermi level plus the energy

difference between EV and EF, which is typically expressed as the work function φ.

This is a probabilistic process that is often expressed through the quantum efficiency, the ratio of photoelectrons to incident photons, which for a given PMT is a function of the photon wavelength. The quantum efficiency in the most sensitive range for a typical PMT is around 20 to 30%.

3.2 XENON1T 11

Figure 3.1: Schematic overview of a photomultiplier tube. A photon enters the PMT and deposits its energy into the photocathode, where the energy is converted into a photoelectron. This photoelectron travels to a series of dynodes, where it is multiplied into many more electrons. At the end the electrons are collected by an anode and converted into an electronic signal. From [3].

3.2

XENON1T

The XENON1T experiment is based on a dual-phase time projection chamber (TPC). It features 1.4 t of liquid xenon (LXe) as a target, with a small volume of gaseous xenon above the liquid phase. When a particle crosses the TPC, as shown in fig. 3.2, it can interact with the liquid xenon by transferring some of its energy to either a xenon nucleus or its electron shell. The recoiling xenon atom then moves through the liquid xenon, exciting and ionizing other xenon atoms along its path. As the excited particles decay to their ground states and freed electrons recombine with the ionized xenon, scintillation light is produced, which is referred to as the S1 signal. This is a simplification of a process that in reality is quite complicated. A full description of the process can be found in [14].

Due to an electric drift field of around 0.125 kV/cm inside of the TPC, the electrons that do not recombine drift upwards instead, towards the gaseous xenon volume. At the interface, a stronger electric field of 8.1 kV/cm is applied, extracting the electrons which then scintillate in the xenon gas. This second light signal is called the S2. As xenon is transparent to the 178 nm photons produced in these scintillation signals, they travel until they are captured by arrays of PMTs at the bottom and top of the TPC. As the S2 is very localized, it is used to reconstruct the x-y position of the event, while the time difference between the S1 and S2 allows for z-position reconstruction, as the drift velocity of the electrons is known. This thus allows for full 3D-position reconstruction.

12 The XENON1T Experiment

Figure 3.2: Schematic illustration of the XENON1T TPC and a particle crossing and interacting with the liquid xenon.

(Left) As the particle interacts, it recoils off the target and scintillation light is

produced, called the S1. Electrons freed through ionization drift upwards due to an electric field present inside of the TPC and produce a second scintillation signal as they are extracted into the volume of gaseous xenon, called the S2.

(Right) Differences between possible signals. Electronmagnetic radiation will provide

a much larger S2 for similar S1s than WIMPs or other neutral particles will, allowing for signal discrimination, and neutrons will likely scatter multiple times, producing multiple S2 signals. From [2].

There are two types of background events, shown in the right figure in fig. 3.2, which can be discriminated through the type of recoil. In electronic recoil (ER) events the incoming particle is charged and recoils off the electron shell through the electromagnetic interaction. When compared to a nuclear recoil (NR) event, in which a nucleus is struck, many more electrons are freed in ER events due to ionization resulting in a much larger S2 signal for a similarly sized S1. Calculating the charge to light ratio thus makes it possible to discriminate ER events from NR events.

WIMPs are expected to interact with ordinary matter through the weak inter-action. As xenon nuclei have a much larger weak interaction cross section than electrons, WIMPs are much more likely to strike a nucleus. Thus a WIMP signal is expected to be a NR event, which can be discriminated from an ER event through the aforementioned method.

NR background events, as produced by neutral particles like neutrons and neu-trinos are problematic as they produce a WIMP-like signature. The best way to discriminate between neutrons and WIMPs is by looking at multiple scatters. For neutrons, the mean free path in liquid xenon is less than 20 cm for fast neutrons in liquid xenon[3], which means neutrons are likely to scatter off multiple xenon nuclei

3.3 The XENON1T Muon Veto 13

while crossing the TPC. In the signal, this shows up as multiple S2s, allowing for discrimination between WIMP signals and background signals.

3.2.1

Neutron shielding

To reduce the neutron background mentioned in chapter 2, multiple techniques are used in XENON1T. Firstly, all material used in the experiment is screened for its radioactive content, ensuring all materials are extremely radiopure. Secondly, and more important for this work, there is the Muon Veto that passively and actively shields the experiment. Its PMTs are able to detect crossing muons thanks to the Cherenkov radiation these muons emit as they pass through the water. Based on this Cherenkov signal a veto is provided to the main detector that can negate most neutrons that were produced by muons. Furthermore the hydrogen atoms in the water slow down the neutrons due to their similar masses. In the TPC itself, more blocking is done through fiducialization. This means that the liquid xenon volume consists of two regions: an outer region and an inner, fiducial region. Due to its large density, the outer region shields the inner region, blocking a lot of background radiation. Combined with the rejection of multiple scatter events this negates most neutrons that enter the TPC, whether they are radiogenic or cosmogenic, and most other forms of background radiation.

3.3

The XENON1T Muon Veto

To obtain the low background needed for the extremely high sensitivity of the XENON1T experiment, the TPC was placed inside of a water tank, the so-called Muon Veto. In this section, the design of the Muon Veto will be discussed, followed by the data-taking procedure. For a detailed overview of the Muon Veto system, see [15].

3.3.1

Design of the Muon Veto

The XENON1T Muon Veto is a stainless steel tank with diameter D = 9.6 m and total height H = 10.55 m, shown in fig. 3.3. It consists of a 9 m high cylinder topped by a 1.55 m high cone frustrum that all cables are fed through. Inside it is clad with a highly reflective foil and it features 84 PMTs of the 8" R5912 ASSY model made by Hamamatsu. This PMT has a quantum efficiency of ≥10% in the wavelength range of 310 nm≤ λ ≤ 540 nm and a stable quantum efficiency of ∼30% in the wavelength range of 340 nm≤ λ ≤ 430 nm. The quantum efficiency is shown in fig. 3.4. The PMTs were placed in 5 rings at different heights. The bottom and top ring feature 24 PMTs, while the middle rings all contain 12 PMTs. The layout is shown in fig. 3.5.

The Muon Veto was designed to veto muons crossing the water tank and muon-induced showers. It does this by detecting the Cherenkov radiation produced by the muon, and vetoing any event in the TPC that occurs simultaneously with an event in the water tank.

14 The XENON1T Experiment

Figure 3.3: 3D model of the XENON1T water tank[15].

3.3.2

Readout and triggers

The signals measured by the PMTs are amplified and sent to digitizers. These have a sampling rate of 100 MHz, which means one sample, referred to as Sa, is 10 ns long. When the signal amplitude of one PMT is above a certain threshold thr in PE and for a certain duration tot in Sa, a single channel trigger is sent. A logic box receives these triggers, and when N triggers coincide within a time window δt, a global trigger is sent to all digitizers. The data of all 84 channels is then collected and saved in a 5.12 µs waveform. This waveform is then called an event. The values used in this work are: thr = 1 PE, tot = 1 Sa, N = 8 and δt = 300 ns. These triggering conditions make it possible to tag 99.78% of muon events and 70.6 % of muon induced shower events.[17]

3.3 The XENON1T Muon Veto 15

Figure 3.4: Quantum efficiency of the PMTs used in the Muon Veto, the R5912ASSY model made by Hamamatsu[16].

3.3.3

Calibration

To ensure that the PMT function remains constant, two calibration systems are used in the Muon Veto; an individual calibration and a global calibration. The individual calibration system consists of a single PMMA fiber per PMT. These are combined into 12 bundles of 7 fibers, each connected to a blue LED. The global calibration consists of so-called diffuser balls, 50 mm diameter PTFE spheres that are filled with silicone and glass. Inside there is a PMMA fiber connected to the same type of LED. Four of these diffuser balls are placed inside of the water tank to ensure full coverage.

Periodically a calibration run is done using both calibration systems, and the gain of each PMT is adjusted to make sure performance is kept constant.[15]

16 The XENON1T Experiment

Chapter 4

Muon track simulation and

reconstruction

To investigate the muons in the Muon Veto, the goal was to reconstruct their tracks using the information gathered by the PMTs. Before analysing the real data, a simulation algorithm was made to aid the reconstruction algorithm. In this simu-lation two properties of the tracks were simulated, namely the timing information of the signals for each PMT and the signal amplitude for each PMT. In the follow-ing chapter, the principles behind the simulation and reconstruction of muon tracks will be discussed, followed by some examples of simulated and reconstructed tracks. Then, some of the properties of muon tracks that were found with the aid of the simulation will be discussed. Lastly, the performance of both fitting algorithms will be compared to determine which is expected to perform best.

4.1

Muon track principles

As mentioned in chapter 2, the average energy of muons in Hall B of the LNGS cave is 270 GeV, so they can be assumed to be travelling at ultra-relativistic velocities, meaning β ≈ 1. For muons travelling through water, with n = 1.333, this means

that Cherenkov photons are emitted at a constant angle, θc = 41.25◦, along the

track. For a muon crossing the tank, there are two vectors that define the track:

the entry point, x0 = (x0, y0, z0), and the unit vector along its direction in spherical

coordinates, ˆ t =    sin(θ) · cos(φ) sin(θ) · sin(φ) cos(θ)   .

In other words, there are five parameters that define the muon track: x0, y0, z0, θ and φ.

From these five parameters, the time it takes for a Cherenkov photon to reach a PMT

at position xpmt can be calculated using geometry, following the diagram shown in

fig. 4.1. First the angle γ between the direction vector ˆt and the unit vector pointing

towards the PMT ˆδ, can be calculated from cosγ = ˆt · ˆδ, where

ˆ δ = L3 ||L3|| = xP M T − x0 ||xP M T − x0|| . 17

18 Muon track simulation and reconstruction

Figure 4.1: Geometry of a muon crossing the XENON1T water tank and emitting

Cherenkov radiation at an angle θc with respect to the muon’s direction. The muon

travels a distance of L1 until the Cherenkov cone passes the PMT. These photons

then travel a distance of L2, and from L1 and L2 the time it takes for a PMT to

first measure a signal after the muon enters the water tank can be calculated.

From this follows sin(γ) = q1 − cos(γ)2_{. Now the lengths L}

1 and L2 can be calculated through L1 = L3 cos(γ) − L3 sin(γ) tan(θc) L2 = L3 sin(γ) sin(θc) .

Here L1 is the distance travelled by the muon at v = c, and L2 is the distance

travelled by the Cherenkov photons with v = cm, the phase velocity of light in water.

From L1 and L2, the total time it takes after a muon enters the water tank for a

PMT to first measure a signal can be calculated through:

t = L1

c +

nL2

c . (4.1)

Calculating the amplitude measured by each PMT is a little more complicated. First, the number of photons emitted by the crossing muon over the distance tra-versed needs to be calculated. From eq. (2.2), the number of photons emitted per unit length and per unit wavelength is

4.2 The simulation algorithm 19 d2_N dxdλ = 2παz2 λ2 1 − 1 β2_n2 ! . (4.2)

Integrating this equation over the wavelength range of the PMTs where the quantum efficiency is larger than 10% and multiplying with the quantum efficiency in this wavelength range, gives a production rate of R = 52.23 photons per cm as it travels through the water. These photons are emitted isotropically in a cone at an

angle θc with respect to the muon’s direction. To calculate the number of photons

expected to strike a PMT, we can again use geometric considerations, as shown in fig. 4.2. As the Cherenkov cone moves past the PMT, a cone frustrum can be drawn, of which the lateral surface area is:

A = π(r1+ r2)

q

((r1− r2)2+ h2). (4.3)

The ratio between the surface area of the PMT and the total surface area of this cone frustrum is then equal to the ratio between the number of photons measured and the total number of Cherenkov photons emitted by the muon as the cone passes

the PMT. This is nγ,tot = ∆xR, where the photon rate is R = 52.23/cm. Now the

number of measured photons can be calculated, using:

nγ,P M T nγ,tot = πr 2 P M T π(r1+ r2) q ((r1− r2)2+ h2) (4.4) nγ,P M T = r2 P M T∆xR (r1+ r2) q ((r1− r2)2 + h2) . (4.5)

It needs to be noted that this is a simplified simulation. In the 300-600 nm wavelength range, the maximum photon energy is around 4 eV, which means that photons lose energy through the photoelectric effect and Rayleigh scattering[3]. This results in a smaller number of photons reaching the PMTs. Both of these effects have not been modelled. Further, the quantum efficiency of the PMTs is wavelength-dependent, but only regions with quantum efficiencies above 10% have been taken into account, which means again the actual number of effective photons is slightly different in the simulation. Keeping this in mind, we can now simulate muon tracks, using equations 4.1 to calculate the timing and 4.5 for the amplitude.

4.2

The simulation algorithm

To simulate a muon, the five track parameters, x0, y0, z0, cos(θ) and φ, are randomly

chosen, making sure the entry point lies on the detector surface and the track points

inwards. To ensure the entry point is on the surface, either r0 is kept fixed at 4.8

m for tracks through the side or z0 is kept fixed at 4 m for tracks through the top.

Then, for each PMT that contributes to the event, its spatial position, the time after muon entry and the signal amplitude are stored in a Hit object. To further emulate the conditions in the Muon Veto, the 10 ns sampling of the ADCs is simulated by rounding the time information up to the nearest factor of 10 and poissonian noise is added to simulate noise in the amplitude signals.

20 Muon track simulation and reconstruction

Figure 4.2: Geometry of the method used to calculate the number of measured photons per PMT. As the muon crosses the water tank, the PMT can see the part of the Cherenkov cone emitted in distance ∆x. From this, a cone frustrum can be constructed. The fraction of the surface area that the PMT covers is equal to the fraction of photons hitting the PMT with respect to the total amount of photons emitted along ∆x.

As an example, a simulated track and its projections in the three coordinate planes are shown in fig. 4.3(a) and (b) respectively. Here the black line is the sim-ulated muon track and the dots signify PMTs that received a signal. The color of the dot gives the time after the muon enters the water tank and the size of the dot the amplitude in that PMT. This clearly shows the effect the 10 ns sampling has on the data. Where in reality a gradient would be visible between the PMTs, now it is more like signals arriving in time steps. Although this is more than enough time resolution for the Muon Veto to function as intended, it will make track recon-struction using only time information difficult. For this reason the amplitude was also used for reconstruction. The timing algorithm and the amplitude algorithm are complementary methods and will be tested against each other.

4.2 The simulation algorithm 21

Figure 4.3: Event display of a simulated muon event. The black line is the simulated muon track and the dots signify PMTs that received a signal. The size of the dots is relative to the amplitude in that PMT and the color is the time after the muon enters the water tank.

22 Muon track simulation and reconstruction

As mentioned in chapter 3, the trigger condition for the Muon Veto is a coin-cidence of 8 PMTs receiving a signal of at least 1 PE in a 300 ns time window. As a muon will take at most 70 ns to cross the water tank, this means a lot of information in an event consists of noise from various sources, e.g. PMT noise and photons which reflected off the water tank walls. As these make reconstructing a track difficult, the simulation algorithm was used to find typical properties of muon tracks, to design cuts for the later analysis of real data. For this aim, 100,000 muon events were simulated, with a random entry point on the top of the detector and a random direction vector, and several properties of these events were investigated.

The reason that only events through the top are simulated is that most real muons travel almost vertically, which means most will enter through the top. The elevation angle distribution was chosen to resemble that of the muon flux as mea-sured by Borexino, as previously shown in fig. 2.2. The azimuthal angle was chosen as isotropic. The only constraint for the simulated events is that the amount of hits needs to be more than one, to ensure the track does not immediately leave the water tank.

Muon track properties

In the real data, the amplitude peak is fixed to the same position in each waveform. However, using this as a fixed point to find the muon data is not possible, as demon-strated by fig. 4.4. Here the time at which the total amplitude is at a maximum is shown. What makes using the time of maximum amplitude unfit to determine the offset, is that this distribution features two peaks. It appears that the amplitude is maximized when the muon passes closely by a PMT, which is during entry or exit as the PMTs are on the water tank walls. This shows that using the time step at which the the total amplitude is at a maximum will not be a good way to find the muon data in a given event.

0

10

20

30

40

50

60

70

80

Time of maximum amplitude [ns]

0

5000

10000

15000

20000

25000

Counts/bin

Figure 4.4: Time after entry at which the total amplitude is highest for 100,000 simulated muon events.

4.2 The simulation algorithm 23

0

10

20

30

40

50

60

70

80

Time after entry [ns]

0

10000

20000

30000

40000

50000

60000

Counts/bin

Figure 4.5: Time after muon entry at which most PMTs are hit concurrently for 100,000 simulated muon events.

A better statistic to use is the time after entry at which most PMTs are hit concurrently, shown in fig. 4.5. This distribution shows a clear peak at 50 ns, which means that at this timestep, PMT concurrency is at a maximum. With over 60% of events having a maximum at this time, using this statistic could help in identifying muons in the Muon Veto data.

0

5

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20

25

30

35

Max concurrent hits [n]

0

2000

4000

6000

8000

10000

Counts/bin

Figure 4.6: Time after muon entry at which most PMTs are hit concurrently for 100,000 simulated muon events.

24 Muon track simulation and reconstruction

least 5 concurrent hits, as can be seen in fig. 4.6. This shows the maximum amount of concurrent hits for each event. From this follows that discarding all events with less than 5 concurrent hits would only remove roughly 10% of all muon events.

0

10

20

30

40

50

60

70

80

Time of last signal [ns]

0

10000

20000

30000

40000

Counts/bin

Figure 4.7: The time at which the last signal is measured for 100,000 simulated muon events.

To further identify muons, the typical duration of a muon event also needs to be known. For this, we can take a look at the time at which the last signal is seen in an event, shown in fig. 4.7. It can be seen here that from first light to last light, a muon event will have a duration of 40 to 50 ns in 80% of all events where a muon enters through the top of the detector. Combining that knowledge with fig. 4.5, means that the time step of maximum concurrency is either the last time step, or the second to last time step.

Lastly, to get an idea of the expected number of hits, the number of hits per event is shown in fig. 4.8. This shows that events with less than 10 hits are quite unlikely, meaning they can be discarded without losing many events. What’s more is that a muon can only generate a signal in 55 PMTs at most. This is due the fact that any PMT that is not in the Cherenkov cone can never measure a signal, meaning that the PMTs on the side of the water tank where the muon enters will not be able to measure a signal.

4.3

Track reconstruction algorithm

To reconstruct a track, the simulation procedure discussed in the last section is reversed. Starting from the PMT hits, the entry point and track direction are found through the minimization of the residual sum over all PMTs that see a signal

R =

N

X

i=1

4.3 Track reconstruction algorithm 25

0

10

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30

40

50

60

70

80

Hits [n]

0

500

1000

1500

2000

2500

3000

3500

4000

Counts/bin

Figure 4.8: Amount of hits for 100,000 simulated muon events.

where ˆx is the expected value of the timing or amplitude, depending on which is

being used to reconstruct the track, using eq. (4.1) or eq. (4.5) respectively.

The procedure works as follows: First all signals are stored in Hit objects, which contain all important information in one object. Then, a grid scan is done on the array of Hits, to narrow down the parameter space and ensure the minimizer that is used afterwards does not get stuck in a local minimum. In this grid scan, the residual sum is calculated for combinations of parameter values over the full range of each parameter, with the following step sizes

(0 ≤ r0 ≤ 4.8), δr0 = 0.6 m

(0 ≤ θ0 ≤ 2π), δθ0 = 1 rad

(−1 ≤ cos(t) ≤ 0), δcos(t) = 0.1 (0 ≤ φ ≤ 2π), δφ = 0.5 rad.

The step sizes were determined through trial and error, and were the values for which most tracks could be reconstructed while giving acceptable performance. Note that the entry point is parametrized in polar coordinates. The parameters that provide the best results for the residual sum are then passed on to Minuit, a well-known numerical minimizer, as initial values.

An example of a reconstructed track is shown in section 4.3 and section 4.3, showing the previously used simulated track, the track reconstructed by the timeline fit and the track reconstructed by the amplitude fit. While both tracks follow the general direction of the real track, neither is able to perfectly reconstruct the track. This is no surprise, as the noise and sampling introduced to emulate the real data make it impossible to fully reconstruct it. It also appears that the amplitude fit was able to do a better reconstruction, and to quantify this, fit performance will be

26 Muon track simulation and reconstruction

tested in the following.

4.3.1

Fit performance

To determine which of the reconstruction methods performs best at reconstructing muon tracks, 10,000 events were simulated and reconstructed using both the timing and amplitude fitting algorithm. Again, the angular distribution of muons was chosen to be similar to the expected real distribution, and the entry points were distributed homogeneously over the top surface of the water tank. Note that events through the sides of the detector were not taken into account. After reconstructing 2,000 events that were simulated to enter through the top two times, once with

z0 fixed and once with r0 fixed, the method with r0 fixed gave a better result in

25% of all events. Knowing most muons will enter through the top, no further

reconstruction was done with r0 fixed.

First the results of reconstruction will be discussed separately for the two algo-rithms, followed by a comparison between the two.

Timeline fit

After reconstruction, the distribution of entry points for the timeline fit, shown in fig. 4.10, features a large excess at the domain edge, and a smaller excess at positive

x and y = 0. Upon closer inspection, the outer ring of the surface also shows a

deficiency of entry points. This means that when an event is in the outer ring, the algorithm reconstructs it as entering through the edge. The line excess can be explained as a consequence of the grid scan, meaning that there is a bias for the

grid scan towards θ0 = 0, resulting in the minimizer finding the best result near this

value. A track that shows this behaviour is shown in the Appendix.

Next, the distribution of elevation angles before and after reconstruction is shown in fig. 4.11. Also shown is the same data, with constraints on the number of hits in an event. This is done to showcase the effect of the number of hits on the form of

the distribution. In the full dataset as well as for events with nhits ≥ 12, there is

a strong excess at a value of cos(t)≈ -0.9. As the rest of the distribution more or less follows the distribution of the simulation, it appears that this is again a result of the grid scan. While the grid scan is done to help MINUIT avoid local minima, MINUIT still finds a minimum near the best result of the grid scan.

Further, when nhits ≥ 30 the shape of the distribution changes completely. After

inspection, it appears that paradoxically the algorithm performs worse when the number of data points increases. Due to the sampling in the time information, having more data points also greatly increases the residual sum, which can make it so the best result for the minimizer is to actually avoid some of the PMTs, meaning the residual in those PMTs is just the observed value squared. This is a consequence of not taking into account PMTs that do not receive a signal. Also, it was found

that many of the events in the entry point excess at r0 = 4.8 m discussed earlier are

actually events with nhits ≥ 30. This confirms that for large amounts of hits, the

algorithm will not reconstruct well.

Lastly, the distribution of azimuthal angles is shown in fig. 4.12, again together with the simulated distribution. Apart from a large excess at φ = 0, the distribution follows the simulation quite well. From this it can be concluded that the timeline

4.3 Track reconstruction algorithm 27

Figure 4.9: Event display of a reconstructed muon event. The black line is the simulated track, the red line the track reconstructed by the timeline fit and the blue line the track reconstructed by the amplitude fit.

28 Muon track simulation and reconstruction

4

2

0

2

4

x [m]

4

2

0

2

4

y [m]

0

2

4

6

8

> 10

Events/bin

Figure 4.10: Entry point of 10,000 simulated muon events reconstructed by the timeline fit.

fit will have a bias towards the value φ = 0, but for the rest will not have a bias towards the steps of the grid scan.

Amplitude fit

For the amplitude fit, the same analysis was done. Firstly, the entry point distribu-tion for the amplitude fit, shown in fig. 4.13, does not feature an excess as strong as that of the timeline fit along the edge. However, the line excess for positive x and y = 0 appears much more noticeably. Further, a bias for the steps of the grid scan

can be seen for θ0, through a slightly higher concentration of points near steps of

0.5.

For the angular distributions of data reconstructed using the amplitude fit, there are also some noticeable differences with the results of the time fit. Firstly, the distribution of elevation angles, shown in fig. 4.14, follows the simulation much more closely. Only for n > 30 does an excess appear, however this could simply be because high signal events travel more vertically.

On the other hand, the azimuthal angle distribution, shown in fig. 4.15, does feature more excesses. The same excess at φ = 0 appears, but also some smaller ones at φ = 1.5, 4.5, 5 and 2π. Again this is most likely a consequence of the grid scan, with MINUIT only looking for a minimum near the output of the grid scan. Apart from the excesses, the distribution looks similar to the simulated distribution.

Comparison

To quantify fit performance, the difference between the real value and the recon-structed value was calculated for each parameter in all events. The results of this

4.3 Track reconstruction algorithm 29

1.0

0.8

0.6

0.4

0.2

0.0

Elevation angle [cos ( )]

0

1

2

3

4

5

6

7

8

Normalized counts/bin

Timeline fit

Timeline fit, n

12

Timeline fit, n

30

Simulation

Figure 4.11: Elevation angle of 10,000 simulated muons and their tracks recon-structed by the time fitting algorithm. Also shown are the results with constraints on the number of hits.

procedure are shown in section 4.3.1.

It is clear that for all parameters, the amplitude fit has performed much better than the time fit. Although the azimuthal angle is reconstructed quite well, there is an average difference in the elevation angle of almost δcos(θ) = 0.2, and the

reconstructed entry point is often more than 2 m off. A noticeable amount of

tracks are even reconstructed as far as 9 m away, which is at the other side of

the top surface. As a final confirmation the reduced chi-squared, or χ2

ν, for both

distributions is shown in fig. 4.17. As a reminder, χ2_ν is a goodness of fit test, and is

given by

χ2_ν = χ

2

ν , (4.7)

where ν is the degree of freedom, ν = n − m, with n the number of observations and

m = 4 the number of fit parameters, and χ2 _{is the weighted residual sum}

χ2 = N X i=1 (xi− ˆxi)2 σ2 i . (4.8)

where xi is the observed value, ˆxi) the expected value and σi2 is the variance.

For this calculation, σt, the timing error, was taken to be 5 ns, as this is the average

error due to the 10 ns sampling, and the amplitude uncertainty was chosen to be 3 PE a posteriori, which is the error introduced by the poissonian noise.

30 Muon track simulation and reconstruction

0

1

2

3

4

5

6

Azimuthal angle [ ]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Normalized counts/bin

Timeline fit

Simulation

Figure 4.12: Elevation angle of 10,000 simulated muons and their tracks recon-structed by the time fitting algorithm.

For the timeline fit, the distribution is not continuous, which could indicate poor

reconstruction. Generally a value of χ2

ν < 1 indicates over-fitting, either through

fitting noise or because the error has been overestimated. However, there is no noise apart from the sampling, and using a bigger error would not make sense. Knowing how poorly the timeline fit performs from section 4.3.1, the only conclusion that can

be drawn is that for the real data, the χ2

ν will most likely not be a good test.

On the other hand, the distribution of reduced chi-squareds looks very promising for the amplitude fit algorithm. Although the distribution has a small tail towards larger values, the peak is nicely centered around 1, meaning that the model is fitting the data correctly. This was to be expected, as the model used to simulate the amplitude is the same model used to reconstruct the track, so any deviation from a value of 1 is only due to the poissonian noise. It can be concluded that the timeline fit will not perform well at reconstructing real muon tracks, while that remains to be seen for the amplitude fitting algorithm.

4.3 Track reconstruction algorithm 31

4

2

0

2

4

x [m]

4

2

0

2

4

y [m]

0

2

4

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8

> 10

Events/bin

Figure 4.13: Entry points of 10,000 muon tracks reconstructed by the amplitude fitting algorithm.

32 Muon track simulation and reconstruction

1.0

0.8

0.6

0.4

0.2

0.0

Elevation angle [cos ( )]

0

2

4

6

8

Normalized counts/bin

Amplitude fit

Amplitude fit, n

12

Amplitude fit, n

30

Simulation

Figure 4.14: Elevation angle of 10,000 simulated muons and their tracks restructed by the amplitude fitting algorithm. Also shown are the results with con-straints on the number of hits.

4.3 Track reconstruction algorithm 33

0

1

2

3

4

5

6

Azimuthal angle [ ]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized counts/bin

Amplitude fit

Simulation

Figure 4.15: Azimuthal angle of 10,000 simulated muons and their tracks recon-structed by the amplitude fitting algorithm.

34 Muon track simulation and reconstruction 0.0 0.2 0.4 0.6 0.8 cos( ) 0 200 400 600 800 1000 1200 1400 Counts/bin Timeline fit Amplitude fit 0 1 2 3 4 5 6 0 500 1000 1500 2000 Counts/bin Timeline fit Amplitude fit 0 2 4 6 8 10 x0 [m] 0 100 200 300 400 500 Counts/bin Timeline fit Amplitude fit

Figure 4.16: Difference between parameters of simulated tracks and tracks recon-structed using the two reconstruction algorithms. (a) Elevation angle. (b) Azimuthal angle. (c) Entry point.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2red 0 50 100 150 200 250 300 Counts/bin 0 2 4 6 8 10 2red 0 100 200 300 400 500 600 700 800 Counts/bin

Figure 4.17: Reduced chi-squareds of tracks reconstructed by (a) the time fitting algorithm and (b) the amplitude fitting algorithm.

Chapter 5

Data selection and cuts

For this work, data from the time range of June 18th to June 25th 2018 was used.

The total live-time adds up to ˜126 live-hours. For the XENON1T experiment, a

muon rate of Rµ = 144 h−1 is expected, according to [2]. In the total live-time of

the data used in this work, 691,863 events were registered. This is equal to a rate of 5,241 muons per hour, which is much higher than the expected rate. This means that only around 2.8% of events registered by the Muon Veto are expected to be muons crossing the water tank, with the rest being other types of events.

This comes as no surprise, as the triggering conditions used to collect the data are quite lax. As a reminder, a trigger is sent when 8 PMTs measure a signal of 1 PE during at least one 10 ns sample in a 300 ns time window. As seen in fig. 4.6 in the last chapter, most muon events have at least 5 concurrent hits at its peak, and as shown in fig. 4.8 at least 10 hits in total in a time window of 50 to 60 ns. Thus, for the reconstruction algorithms to work, some cuts need to be made on the data.

0

500

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2500

3000

3500

4000

Time [ns]

0

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1000

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3000

3500

Counts/bin

Figure 5.1: Time at which the signal amplitude is at its maximum.

When a trigger is sent, all data is captured. This means that signals under the threshold of 1 PE get saved too, and need to be removed. This is the first cut. Further, the time offset needs to be found. All waveforms are made to have the

36 Data selection and cuts

amplitude peak at a fixed position. As shown in fig. 5.1, this is around the 2000 ns mark. However, there are also events that peak at other times, which are most likely events that do not feature a muon. To show the difference between a muon event and a non-muon event, a muon event is shown in fig. 5.3 and a non-muon event is shown in fig. 5.2. In the first, multiple PMTs measure signals far above the 1 PE threshold, with one PMT measuring a signal of almost 140 PE. After the initial event, photons reflect around in the water tank until they either strike a PMT or get absorbed by the water or the water tank walls. In the non-muon event, there is no clear signal peak, and the PMTs that measure a signal only barely exceed the 1 PE threshold. While this could still be a muon event, it is most likely some other form of radiation, or coinciding noise producing a trigger. To get rid of these noise events, the cut algorithm looks for concurrence of at least 5 PMT signals within the 2150 - 2250 ns timeframe. If an event does not have a peak in this time frame, it is removed. 0 20 40 60 80 100 120 140

Hit Area (pe)

0 1000 2000 3000 4000 5000 Time (ns) 0 10 20 30 40 50 60 70 80 PMT Channel run ['180619_0925'], event 20 Figure 5.2

The time at which PMT signal concurrence peaks is then chosen as an offset, and all signals that occur more than 50 ns before it or 10 ns after it are removed as well. As shown in fig. 4.5 and fig. 4.7, concurrence peaks after 50 ns and this is either the last signal, or the second to last. Thus, assuming most tracks will look similar and assuming all tracks to enter through the top of the detector, all signals that do not satisfy these conditions are removed. Lastly, to aid the reconstruction algorithms, good events need to be selected. While at least 5 hits are needed in total, as there are 4 fit parameters, a minimum of 12 hits per event was used, in

5.0 37 0 20 40 60 80 100 120 140

Hit Area (pe)

0 1000 2000 3000 4000 5000 Time (ns) 0 20 40 60 80 PMT Channel run ['180619_0925'], event 19 Figure 5.3

hopes of improving the odds of the algorithms being able to reconstruct the tracks. From simulation we have seen that this has no adverse effect the results.

After applying these cuts, the 691,863 events that were started with have been

reduced to 15,829 events. This translates to a rate of Rµ = 125.6 h−1, which is

slightly lower than the expected number of 144 h−1. This is no surprise, as the cuts

used will also remove events with a low number of hits that do feature a muon. To see the effectiveness of these cuts at removing noise, we can take a look at the amount of hits per event, shown in fig. 5.4. Here we see a distribution with a shape that is similar to that of the simulated events, previously shown in fig. 4.8. However, one big difference is that in the simulation, the mean number of hits was 24.2, while the mean number of hits in the data is 42.3. Thus, even with these cuts not all noise has been removed. Knowing the timeline fit performs worse for events with more hits, this could impact performance.

38 Data selection and cuts

0

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100

Hits [n]

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Counts/bin

Chapter 6

Reconstructed muon tracks

Using the developed reconstruction algorithms, the muon events gathered were

re-constructed. In the following sections, the results of the reconstruction will be

shown, starting with the results for the time fitting algorithm.

6.1

Timeline fit

After reconstructing the dataset consisting of 15,829 events, 12,137 events were left. For the rest of the events, the fitting algorithm could not find a solution, which means the fitting algorithm could find a track in 76.7% of all events. From the remaining events, a distribution of entry points was made, shown in fig. 6.1.

4

2

0

2

4

x [m]

4

2

0

2

4

y [m]

0

2

4

6

8

> 10

Events/bin

Figure 6.1: Entry point of muon tracks reconstructed by the timeline fit.

As was seen previously in fig. 4.10, the timeline fit again has a bias towards the edge of the surface, as this gives a low residual sum in the event that the real track

40 Reconstructed muon tracks

can not be found. Further, there again appears a line along y = 0, which could be a result of the gridscan done. From this and from simulation, it is known that the

time fitting algorithm has a bias towards events on r0 = 4.8 m, and on a line at

x > 0 and y = 0. Further, all events with cos(θ) = 0.0 and φ = 0.0 and 2π were

removed, as these events are most likely poorly fitted. With all outliers removed, the angular distributions of the muon flux were made, shown in fig. 6.2 and fig. 6.3.

1.0

0.8

0.6

0.4

0.2

0.0

Elevation angle [cos ( )]

0

50

100

150

200

250

300

350

Counts/bin

Figure 6.2: Elevation angle of muon tracks reconstructed by the timeline fit.

Surprisingly, the distribution of elevation angles resembles the distribution made by Borexino (fig. 2.2) quite closely. In Borexino, the distribution is centered around an inclination angle of 0.9 (which translates to an elevation angle of -0.9), while for the timeline fit it is centered around -0.8. However, this result should be taken with a grain of salt.

From simulation, it is known that the timeline fit performs poorly due to the low time resolution. Therefore, something else must be going on. To illuminate this, the event display of a track with cos(θ) = -0.80 is shown in fig. 6.4. Although the algorithm manages to find a track, the data looks like it merely consists of noise. The event features 73 hits, even after cuts, which is known from simulation to be impossible. This is what is happening in most tracks, the cut does not remove all hits that are not part of the track, meaning the algorithm is mostly fitting noise. Thus, the only conclusion that can be made is that when events are very noisy, the timeline fit has a tendency towards the distribution shown in fig. 6.2. Likewise, the distribution of azimuthal angles, fig. 6.3, is not isotropic which is expected from the borexino results, but no real conclusions can be drawn from this as most events are simply noise.

6.2 Amplitude fit 41

0

1

2

3

4

5

6

Azimuthal angle [ ]

0

20

40

60

80

100

120

Counts/bin

Timeline fit

Figure 6.3: Azimuthal angle of muon tracks reconstructed by the timeline fit.

To proof this point, the reduced chi-squared was again calculated for all events, and is shown in fig. 6.5. The data is split up into two regions, the removed region

of r0 ≥ 4.75 m which most likely only has poor fits, and the region of r0 < 4.75

m. Unlike the distribution shown in the simulation (fig. 4.17), it is for both regions

a more continuous distribution, but now centered around χ2

ν ≈ 4, while for the

simulation all values were below 1. Although it was already determined to not be a good statistic to test the goodness of fit, this shows that the model is a poor method to capture the data. As the tracks in the outer region are most likely noise, and have a distribution with a sharper peak, centered closer to a value of 1, this means that in many cases a "wrong" solution is preferred over the correct solution.

6.2

Amplitude fit

For the amplitude fit, the same analysis was done as in the previous section. Firstly, the distribution of entry points is shown in fig. 6.6. As in the results of the simulation, a bias is shown towards a line at x > 0 and y = 0. Further, the same bias towards the edge of the surface that appeared for the timeline fit is visible. Assuming these events are poorly reconstructed, all events in these regions were removed. Moreover, all events at the domain edges of the elevation angle and azimuthal angle were removed, as a bias was also seen towards these points in the simulation. After these cuts, the resulting angular distributions are shown in fig. 6.7 and fig. 6.8.

For the elevation angle, one strong peak appears at -0.9, where a peak is expected to be, and one large excess appears for elevation angles closer to 0. This large

42 Reconstructed muon tracks

Figure 6.4: A poorly reconstructed muon event. Due to the presence of noise, both reconstruction algorithms were unable to find a good solution.

6.2 Amplitude fit 43

0

5

10

15

20

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35

40

0

50

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Counts/bin

r

4.75 m

r

< 4.75 m

ν of muon tracks reconstructed by the timeline fit.

4

2

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x [m]

4

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Events/bin

Figure 6.6: Entry point of muon tracks reconstructed by the amplitude fit.

excess can not be physical, as it is not expected for this many muons to travel almost horizontally. Taking a look again at fig. 6.4, where cos(θ) = -0.42, it is clear that the algorithm was not able to find a track due to the noise. When this occurs, the amplitude fit appears to have a tendency towards elevation angles closer to 0. Another thing to note is that the two fitting algorithms found a different solution, which actually is the case for almost all events. For the azimuthal angle,