MSc Physics and Astronomy
GrAPPA
Master Thesis
!
Models of Particle Signatures
in
KM3NeT ORCA
!
by
Jordan Seneca Ralston
11410388
September 2018
60 ECTS
School year 2017 - 2018
Supervisor/Examiner:
Examiner:
UNIVERSITY OF AMSTERDAM
Abstract
Department of Physics and Astronomy Masters of Science
Models of Particle Signatures in KM3NeT ORCA by Jordan Seneca Ralston
The KM3NeT ORCA detector aims at uncovering the neutrino mass ordering (NMO), for which accurate predictive particle signal models are needed. Photo-electron yield models of product particles in KM3NeT ORCA are produced. The models are presented and shown to accurately replicate expected behaviour of electromagnetic particles, hadronic particles, and muons. The use of the models for reconstruction of individual showers and entire neutrino events in ORCA is then explored, and shown to be competitive in the best cases. Finally, the models are used to perform rapid Monte Carlo simulations, showing good agreement to the original sample in the number of photo-electrons and number of hits, with deviations in time arrival. Avenues towards future use and improvements of the models are suggested.
Abstract ii
List of Figures v
List of Tables vii
Introduction 2
1 Background 3
1.1 Theory of Neutrinos . . . 3
1.2 Neutrino Interactions in Water . . . 8
1.3 Particle Showers . . . 12
1.4 Atmospheric Neutrinos . . . 16
1.5 Cherenkov Neutrino Detection. . . 18
1.6 KM3NeT ORCA . . . 20
2 Photo-Electron Pattern Probability Density Functions 25 2.1 Motivation. . . 25
2.2 Approach . . . 26
2.3 PEPP Building . . . 27
2.4 Results. . . 31
3 Particle Shower and Neutrino Reconstruction 41 3.1 Statistical Methods . . . 41
3.2 Model Building . . . 43
3.3 Informed Reconstructions . . . 45
3.4 Event pre-selection . . . 46
3.5 Data and data cuts . . . 47
3.6 Reconstruction Methodology . . . 48
3.7 Reconstruction resolution of single shower . . . 50
3.8 Neutrino Reconstruction . . . 56
4 Rapid PEPP Monte Carlo Simulations 65 4.1 Inverse Transformation Method . . . 65
4.2 PEPP MC-method . . . 66
4.3 Trueness to full Monte Carlo sample . . . 66
Charged Particles in KM3NeT ORCA CONTENTS
5 Outlook 69
5.1 Re-building the PEPP . . . 69
5.2 Reconstructions . . . 70
5.3 Outer half volume . . . 72
5.4 Different Optimisation Algorithm . . . 72
6 Conclusion 75 A Appendix 77 A.1 Cherenkov Threshold . . . 77
A.2 Chapter 1 Figures . . . 77
A.3 Chapter 2 Figures . . . 77
A.4 Chapter 3 Figures . . . 77
A.5 Tables . . . 77
Bibliography 87
1.1 Fundamental neutrino interactions . . . 4
1.2 Neutrino-electron scattering . . . 9
1.3 Quasi-elastic scattering . . . 10
1.4 Resonant scattering. . . 11
1.5 Deep inelastic scattering . . . 11
1.6 Deep inelastic scattering cross sections . . . 11
1.7 Total cross sections for neutrino-nucleus interactions . . . 12
1.8 Heitler model . . . 14
1.9 Cosmic ray energy flux . . . 16
1.10 Atmospheric neutrino production . . . 17
1.11 Atmospheric neutrino flux before (A) and after baseline (B). . . 17
1.12 Cherenkov diagram . . . 18
1.13 Path lengths of charged particles . . . 19
1.14 Oscillation at ORCA . . . 21
1.15 Comparison of KM3NeT PMT and propagated Cherenkov light . . . 23
1.16 The KM3NeT DOM . . . 23
2.1 PEPP dimensions . . . 29
2.2 2 GeV electron PEPP . . . 32
2.3 3 GeV electromagnetic PEPP . . . 32
2.4 3 GeV hadronic PEPP . . . 33
2.5 3 GeV neutron PEPP . . . 33
2.6 3 GeV neutral pion PEPP . . . 33
2.7 electron and proton PEPP, energy scaling . . . 34
2.8 electron and proton PEPP, energy comparison . . . 35
2.9 muon PEPP . . . 35
2.10 muon, electron PEPP, energy comparison . . . 36
2.11 muon PEPP energy scaling . . . 36
2.12 PEPP directions . . . 37
2.13 PEPP direction 1 and 4 . . . 37
2.14 PEPP direction 2 and 3 . . . 38
2.15 PEPP time arrival difference direction 3 . . . 38
2.16 PEPP time arrival difference direction 0 and 2. . . 39
2.17 PEPP time arrival difference electromagnetic particles direction 0 and 2 . 39 3.1 Reconstruction model building . . . 44
3.2 Event distribution in ORCA . . . 47 v
List of Figures LIST OF FIGURES
3.3 Relative energy difference of 1-2 GeV showers . . . 51
3.4 Relative energy difference resolution of hadronic showers . . . 52
3.5 Angle difference of 5 GeV hadronic showers . . . 53
3.6 Angle difference resolution of hadronic showers . . . 53
3.7 Angle difference resolution of EM showers and muon . . . 54
3.8 Translational position difference distribution of showers . . . 54
3.9 68% quantile translational position difference of showers . . . 54
3.10 Longitudinal position difference distribution of showers . . . 55
3.11 Longitudinal position difference resolution distribution of showers . . . 55
3.12 Time difference resolution of showers . . . 55
3.13 Classification of showers for 25-35 hits . . . 56
3.14 Classification of showers . . . 56
3.15 Relative energy difference distribution of 3-4 GeV neutrino events . . . 58
3.16 Energy difference resolution of neutrino events. . . 59
3.17 Angle difference distribution of 3-4 GeV neutrino events . . . 59
3.18 Angle difference resolution of neutrino events . . . 60
3.19 68% quantile translational position difference of neutrino events . . . 60
3.20 Longitudinal position difference resolution of neutrino events. . . 61
3.21 Time difference distribution of 3-4 GeV neutrino events. . . 61
3.22 Time difference resolution of neutrino events . . . 62
3.23 Classification of neutrino events for few hits . . . 62
3.24 Classification of neutrino events . . . 63
4.1 Number of hit comparison to Monte Carlo . . . 67
4.2 Number of p.e. comparison to Monte Carlo . . . 67
4.3 Time arrival difference comparison to Monte Carlo . . . 68
5.1 Likelihood phase space . . . 73
A.1 Sub-dominant neutrino cross sections . . . 78
A.2 Quasi-elastic scattering cross section . . . 78
A.3 Cosmic ray composition . . . 79
A.4 ORCA neutrino flux ratio . . . 79
A.5 Neutron PEPP . . . 80
A.6 Positive pion PEPP . . . 80
A.7 Proton PEPP, directions 2 and 3 . . . 81
A.8 Electron PEPP, directions 5 and 6 . . . 81
A.9 Energy resolution of ORCA outer half . . . 82
A.10 Diretion resolution of ORCA outer half. . . 82
A.11 Log relative energy difference of 1-2 GeV showers . . . 82
A.12 Relative energy difference resolution of EM showers and muon . . . 83
A.13 Angle difference of 1-2 GeV showers. . . 83
A.14 Position difference distribution of showers . . . 83
A.15 68% quantile position difference of showers . . . 84
A.16 Time difference distribution of showers . . . 84
A.17 Log relative energy difference distribution of 3-4 GeV neutrino events . . . 84
3.1 Domain of the PEPP . . . 45
3.2 Reconstructed quantities for single showers . . . 51
3.3 Parameter spread . . . 58
5.1 Reconstructed quantities for neutrino events . . . 71
A.1 Neutrino parameters . . . 78
A.2 Domain of the PEPP . . . 79
A.3 Types of PEPP, PEPE and PEPC . . . 80
The neutrino is a curious particle which is under current scrutiny, suspected of being responsible for the matter-antimatter asymmetry of the universe, being its own
anti-particle, energising stellar explosions, and participating in other exotic physics scenarios[1].
Interacting very weakly, neutrinos can also be used as a unique tool to probe the interior of the Earth, the Sun, distant active black holes and other celestial objects.
The properties of the neutrinos are not yet all understood. The fact that neutrinos are
massive was recently discovered with the confirmation of the neutrino oscillations[2][3][4][5].
One of the properties which has not yet been determined is whether one of the masses of the neutrinos is either much larger or much smaller than the two other masses. This ques-tion is called the Neutrino Mass Ordering (NMO). Knowing the NMO will be tremen-dously helpful in discovering other properties of neutrinos and bears implications in many contexts involving neutrinos.
The Cubic Kilometre Neutrino Telescope (KM3NeT) is a project under construction in
the Mediterranean sea[6]. KM3NeT is composed of two water Cherenkov neutrino
detec-tors. Oscillation Research with Cosmics in the Abyss (ORCA) is located off the French coast 40 km from Toulon, at a depth of 2450 m. Astroparticle Research with Cosmics in
the Abyss (ARCA) is located off the southern Sicilian coast in Italy 100 km from Porto
Palo di Capo Passero, at a depth of 3500 m. The Earth is constantly bombarded with
high energy particles called cosmic rays[7][8], which create a large amount of energetic
neutrinos in the Earth’s atmosphere[9][10], called atmospheric neutrinos. ORCA is a
dense, lower energy neutrino detector, which takes advantage of the flux of atmospheric neutrinos with the goal of measuring the NMO. ORCA achieves its goals by observing
the light produced in neutrino interactions with water[11] through the Cherenkov effect.
KM3NeT is unique as it is one of the two largest neutrino detector of its size in water (along with Baikal-GVD), and currently the only to use digital optical modules (DOM) with directional photomultiplier tubes (PMT). ORCA is also much denser than other Cherenkov neutrino telescopes. These features permit the development of new tools to aid the physics goals of the experiment.
LIST OF TABLES 2 KM3NeT ORCA is currently in possession of predictive Cherenkov light global models for various types of neutrino events. Global models can be used to estimate the energy and direction of a neutrino event, and give limited information on the neutrino type. Measuring the energy, direction, and type of the neutrino interactions as accurately as possible is essential for achieving the discovery of the NMO. Certain desirable parameters of neutrino events, such as the portion of energy which goes into the leading lepton, the Bjorken inelasticity parameter (Bjorken y), or the neutrino interaction flavour, are currently inaccessible. The models are also dependent on neutrino nucleon interaction models, which describe highly complex processes. These model uncertainties and large fluctuations involved in the particle yield of neutrino interactions are all folded into the global models.
This thesis brings to light the signal that product particles in neutrino events display in ORCA. For KM3NeT, this would be the first time the signal of product particles will be available for lookup. This work aims at producing such predictive, detailed models of product particles, and exploring their use in the KM3NeT ORCA detector.
First, the models will be compared to expectations of hadronic and electromagnetic particle behaviour, and how the behaviour changes at different energies. The detailed models will then be used to explore whether current neutrino event parameters can be measured with better resolution, and whether the measurement of new parameters, such as the Bjorken y and the neutrino interaction flavour are possible. It is hypothesised that by resolving details of the neutrino event, improvements can be made. Success on any of these fronts will couple directly to the NMO sensitivity of KM3NeT ORCA. A final question is whether the detailed models can be used to perform reliable, fast Monte-Carlo simulations of ORCA signals from particles in the detector. Simulations of product particle propagation in the detector are extremely computationally expensive, finding a rapid alternative would economise time and computational resources.
The thesis first presents the theoretical background for neutrino oscillations, neutrino interactions, their sources, and their detection using water Cherenkov detectors. In the second chapter, the development of the detailed models is explained, and the models presented. The third chapter shows the detailed models used to perform particle shower and neutrino event reconstructions. Finally, the models are used to perform fast Monte-Carlo simulations. The work was done for the KM3NeT group of Nikhef in Amsterdam under the supervision of Dr. Ronald Bruijn.
1.1
Theory of Neutrinos
Neutrinos are elementary particles which are not fully understood and extremely difficult to observe. In the current electroweak Standard Model of particle physics (SM), the neutrinos are left-chiral neutral spin-½particles. There exist six neutrinos, one for every
lepton flavour, νe, νµ, ντ, and the anti-neutrino partners. Having no electromagnetic
charge, the neutrino interacts only through the weak interaction. Since the coupling constant of the weak interaction is small, and mass of the gauge bosons is large, the
interaction cross sections of neutrinos are also minuscule1.
To give an idea of the magnitude of the cross-sections at play in neutrino interactions, take neutrinos produced in the Sun, which typically have an energy O(1 MeV). At the Earth’s
equator, the column number density of electrons is approximately 1.5 × 1033 cm−2[12].
With a cross section of around 2 × 10−44 cm2 at these energies, over 3 × 1011 electron
neutrinos would need to traverse the entire width of the Earth to interact once through charged current electron scattering.
1.1.1 Neutrino Masses
Neutrinos have all been observed to have left-handed helicity2, and their anti-neutrino
partners to have right-handed helicity. In the case of massive particles, helicity is not Lorentz invariant. In the limit of E m however, the chiral and helicity states of the
1
The coupling strength of weak interactions is primarily dictated by the Fermi coupling constant in the Lorentz invariant matrix element,
GF ≡
g2 4p2M2
W
,
where g is the coupling constant of the weak interaction and MW is the mass of the weak boson. 2
Helicity is defined as the sign of the projection of the spin of a particle onto its direction of motion. Particles which have their spin pointing in the direction (opposite direction) of motion are said to be right-handed (left-handed).
1.1. Theory of Neutrinos 4 particle become the same. This condition is always satisfied for massless fermions; in the SM, neutrinos were therefore taken to be massless.
In the seventies, the Homestake experiment found that the νe flux from the sun was
smaller than expected at the time[13]. After decades of new generations of detectors,
the Sudbury Neutrino Observatory observed neutrino appearance in Solar neutrinos in
2002[2] and Super-Kamiokande observed νµ disappearance in atmospheric neutrinos in
1998[3][4]. The deficit of Solar νe was explained: the interaction probability of neutrinos
oscillates through flavour states. This means that a neutrino born through an e.g. µ-interaction, could later interact with an electron. This effect is currently best explained
by giving the neutrinos three non-zero mass3eigenstates, which calls for a new description
of the neutrinos and an extension of the SM.
1.1.2 Neutrino Interactions
Neutrinos interact through the exchange of a weak boson. Interactions involving the
charged W− and W+ bosons are called Charged Current (CC) interactions. Interactions
with the neutral Z0 boson are called Neutral Current (NC) interactions, see Fig. 1.1.
νl l−
W+
(a) CC interaction of neu-trino. ¯ νl ¯l W− (b) CC interaction of anti-neutrino. νl νl Z0 (c) NC interaction of either neutrino or anti neutrino.
Figure 1.1: Fundamental neutrino interactions.
1.1.3 Neutrino Mixing
A proper framework to describe the oscillations of neutrinos is needed, which can be built
using one-particle quantum mechanics[4]. Describing the neutrino flavour eigenstates as
a superposition of neutrino mass eigenstates (also known as quantum mixing),
h|νli = Al0h|ν1i + Al1h|ν2i + Al3h|ν3i , (1.1)
3
In order to be consistent with the apparent lack of right-handed neutrinos, these masses have to be very small. Current estimates put the total neutrino mass at ∼ 0.25 eV[14].
νe νµ ντ = Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ 1 Uτ 2 Uτ 3 ν1 ν2 ν3 . (1.2)
Since a neutrino flavour state must be composed of the existing neutrino mass states, the unitarity condition must be applied,
U†U = U U†= I, (1.3)
where I is the identity matrix. The unitary matrix can now be parametrised into three
mixing angles, θ12, θ13, θ23, and three mixing phases. However, two of the phases are
non-zero only in lepton-number violating processes, in which neutrinos are Majorana
particles5. The third phase is non-zero only if Charge-Parity (CP) is violated in neutrino
interactions[5]. Keeping only the CP phase, δCP, the neutrino mixing matrix becomes
U = c13c12 c13s12 s13e−iδCP −c23s12− s13s23c12eiδCP c23c12− s13s23s12eiδCP c13s23 s23s12− c12c23s13eiδCP c12s23− s13c23s12eiδCP c13c12 , (1.4)
where sij ≡ sin θij, and cij ≡ cos θij.
1.1.4 Two flavour vacuum oscillations
Using the established framework for the neutrino mixing, we can now study the evolution for a neutrino system. To demonstrate the probability oscillation between flavour states, a simplified model where we assume there to be only two neutrino flavours is studied. In this model, the relation between the flavour and mass eigenstates can be parametrised into a single angle θ:
" νe νµ # = " cos θ sin θ − sin θ cos θ # " ν1 ν2 # . (1.5)
In vacuum, the hamiltonian for the neutrino is then
Hvacuum= 1 2EU ˆm 2U†= ∆m2 4E " − cos 2θ sin 2θ sin 2θ cos 2θ # , (1.6) 4
Note that the number of flavours here is chosen, as a fourth generation of neutrinos has not been excluded by current constraints.
5
1.1. Theory of Neutrinos 6
where an identity matrix multiple has been subtracted and ∆m2≡ m2
2− m21[15]. An νe
and νµ in their initial flavour states are then
|νei = cos θ |ν1i + sin θ |ν2i , |νµi = − sin θ |ν1i + cos θ |ν2i . (1.7)
The Shrödinger equation ∀x,
i d
dx|νi = H
matter|νi , |νi =
"
ae
aµ
#
. (1.8)
where ae and aµ are the probability amplitudes of each flavour, has a solution for the
time evolved νe
|νe(x)i = cos θ |ν1i e−ix+ sin θ |ν2i e−ix, (1.9)
where time has been rearranged and approximated in the ultra-relativistic regime, which
is valid for neutrinos with very small mass, into p1 and p2, the four momenta of the mass
eigenstates ν1 and ν2, and L, the distance travelled of the neutrino, so that
|νe(L)i = cos θ |ν1i e−ip1L+ sin θ |ν2i e−ip2L. (1.10)
To demonstrate the oscillation of neutrinos, the probability for a neutrino born through an electron flavoured interaction to be observed through a µ flavoured interaction is used as an example, and found by
P2ν(νe → νµ) = |hνµ|νe(L)i|2= | − sin θ cos θe−ip1L+ cos θ sin θe−ip2L|2, (1.11)
using Eq. 1.7. In the ultra-relativistic approximation, p ' E−m2/2E, and thus p
2−p1 =
∆m2/2E, where ∆m2 ≡ m2
2 − m21. This approximation simplifies the evolved state
equation,
P2ν(νe→ νµ) =
1
4sin
22θ(1 − cos(L∆m2/2E) − i sin(L∆m2/2E))2, (1.12)
which, in the limit of 1 − cos 2
1/2 ' 0for 1 2 becomes
P2ν(νe→ νµ) ' sin22θ sin2(L∆m2/4E). (1.13)
Eq.
refeqn:oscillation shows that a non-zero mass squared difference is necessary to cause neutrino oscillations. This means that at most one of the mass eigenstates can be zero, motivating a new description of the neutrinos to account for their non-zero masses. Note here that oscillations in vacuum are only sensitive to the absolute value of the mass
1.1.5 Two flavour oscillations in matter
In 1985, building upon the work of Lincoln Wolfenstein, it was proposed by Stanislav
Mikheyev and Alexei Smirnov neutrino oscillations could be affected by ambient matter[16].
Common matter is composed of electrons, causing interactions with the neutrino electron
flavour state only6, interfering with oscillations. The two flavour neutrino model will be
used again to compare the way oscillations evolve in media.
Taking into account the interactions with the medium using the properties of ordinary matter and electroweak interactions, the effective Hamiltonian for a two-neutrino model in matter becomes Hmatter = 1 2EU ˆm 2U†+√2G Fdiag Ne− 1 2Nn, 1 2Nn, 1 2Nn , (1.14)
where Ne is the electron number density, appearing due to CC interactions, and Nn is
the neutrino number density, appearing due to NC interactions[17]. Following the same
steps as in Eq. 1.6, the neutron density term disappears, leaving
Hmatter= 1
4E "
A − ∆m2cos 2θ ∆m2sin 2θ
∆m2sin 2θ −A + ∆m2cos 2θ
#
, (1.15)
where A = 2√2GFENe. For comparison with oscillations in vacuum, it is desirable to
seek the form of the Hamiltonian from Eq. 1.6:
Hmatter = 1 4E " ∆m2mcos 2θm ∆m2msin 2θm ∆m2msin 2θm ∆m2mcos 2θm # , (1.16) where ∆m2m ≡ ∆m2 q
sin22θ + (cos 2θ − A/∆m2)2, and sin22θ
m≡
sin22θ
sin22θ + (cos 2θ − A/∆m2)2
6Neutrinos of all flavours can also interact with electrons, protons, or neutrons through NC. Since this
interaction affects all neutrino flavours equally, the neutron density term appears as an added diagonal matrix. This matrix can be safely subtracted away as it ultimately does not affect the relative energy eigenvalues of the Hamiltonian, leaving oscillations unaffected. This fact is also used to simplify the Hamiltonian with the results shown below.
1.2. Neutrino Interactions in Water 8
have been used. [15] In this case, the oscillation probability from an νe to a νµ is,
following the same steps in Eq. (1.11– 1.13),
P2ν(νe → νµ) = sin22θmsin2(∆m2mL/4E). (1.17)
Due to the new Hamiltonian in matter, the sign of ∆m2 is now measurable.
1.1.6 Neutrino Mass Ordering
In reality, three flavours of neutrinos are known to exist, electron, µ, and τ neutrinos, implying the existence of three mass states, and therefore two independent mass
differ-ences, ∆m2
21 ≡ m22 − m21, and ∆m2 ≡ m23 − (m22m21)/2. m1 and m2 were named such
that ∆m2
21> 0 [18], leaving the sign of ∆m2 to be measured.
As of the writing of this thesis, the sign of ∆m2 is unknown. The sign of ∆m2determines
the neutrino mass ordering (NMO)7. If the third mass state turns out to be heavier than
the two other mass states, ∆m2 > 0, the NMO is said to be normal (NO). If this is
not the case, ∆m2 < 0, the NMO is said to be inverted (IO). Apart from narrowing
down on the properties of an elementary particle, the determination of the NMO bears implication in early cosmology, supernovae models, and the behaviour of neutrinos in
other ultra-dense media[6].
1.2
Neutrino Interactions in Water
Water is a molecule used as a target in water Cherenkov neutrino detectors composed of
an oxygen atom,16O and two hydrogen atoms, 1H. In water, a neutrino or anti-neutrino
can conceivably interact with an electron, an atomic nucleus, or one of its component particles.
For MeV neutrino energies and upwards8, elastic scattering off nuclei, quasi-elastic and
elastic scattering off nucleons (QE), resonance scatterings, deep inelastic scattering (DIS), and neutrino-electron scattering occur. Each interaction is shortly described and assessed for relevance in ORCA observations.
7A hierarchy is an ordered structure where each element is subordinate to the element above. It is
the opinion of the author that this nomenclature is a bit odd if it turns out that that m3 < m1 < m2,
so the term ordering is preferred in this thesis.
8
e− νe
W+
e e
Z0
Figure 1.2: Neutrino-electron scattering through charged current (left), and neutral current (right).
1.2.1 Neutrino-electron scattering
Neutrino-electron scattering is a purely leptonic process in which a neutrino or anti-neutrino scatters off a positron or electron, the Feynman diagrams are shown in Fig. 1.2.
In the context of negligible neutrino masses (mνl me) and high neutrino energies
(meEν >> m2l), the cross section can be approximated as
σ ' 2meG
2 FEν
π ∝ Eν,
where GF ' 1.66 × 10−5 GeV−2 is the Fermi coupling constant. This proportionality
to the neutrino energy is a property of four body interactions between elementary
parti-cles, which will become relevant when discussing deep inelastic scattering in 1.2.3. The
neutrino-electron scattering cross section per neutrino energy is then upper-bounded,
σ(νle− → lνe)/Eν . 2 × 10−41 cm2/GeV, while the cross section for neutrino-nucleon
interactions lies in the O(10−38 cm2/GeV) region for E
ν & 300 MeV, see Fig. 1.7a.
A water molecule will on average contain 18 nucleons and 10 electrons. This implies
that roughly 1 in 900 neutrino interactions in the Eν & 300 MeV region will be from
neutrino-electron scattering. Neutrino-electron scattering is therefore not prioritised in this work.
1.2.2 Nucleus scattering
At very low energies (< 1 MeV), neutrinos can elastically scatter off an entire nucleus A with atomic number Z and neutron number N,
ν + AZN → ν + A∗ZN,
or be captured by the nucleus,
1.2. Neutrino Interactions in Water 10 u d u u d d ¯ νl ¯l W− p n u u u u d d ν ν Z0 p p
Figure 1.3: Quasi-elastic scattering (left) and elastic scattering (right) off nucleon.
While theoretically established, these reactions are extremely rare and remain to be
detected experimentally[11], and will be ignored for the rest of this thesis.
Neutrinos can also coherently emit a pion when scattering off a nucleus,
ν + A → ν + A + π0,
νe+ A → e−+ A + π+,
contributing significantly to the inclusive neutrino cross-section, especially at low
ener-gies, see Fig. A.1b.
1.2.3 Nucleon scattering
Neutrinos can scatter off nucleons in three broad categories that become dominant in their respective regions of highest amplitude.
At low energies (1 − 100 MeV), neutrinos start interacting with individual nucleons. Quasi-elastic (QE) scattering involves a neutrino scattering off a nucleon through charged current without exciting it,
¯
νl+ p → ¯l + n, (1.18)
while elastic scattering sees a neutrino scatter off a nucleon through neutral current without exciting it,
ν + p → ν + p. (1.19)
The Feynman diagrams for these interactions are shown in Fig. 1.3. The cross section
QE scattering increases quadratically with energy before plateauing at ∼ 1 GeV at a
value of ∼ 10−38 cm2/GeVas seen in Fig. A.2.
When going to higher energies (0.5 − 10 GeV), the neutrino can also inelastically scatter off the nucleon, exciting the nucleon and producing ∆ resonance states which quickly decay. These resonance states decay into single pions the vast majority of the time, but can also decay into multiple pions, or, albeit more rarely, into kaons, photons, or other
d d u d d ¯ d u u W+ n n π+ u u u u u ¯ u d d Z0 p p π0
Figure 1.4: Examples of resonant scattering through charged current (left) and neutral current (right). After interaction with the boson, the nucleon is in a ∆ resonance state.
q q0 νl l W h X q q νl νl Z0 h X
Figure 1.5: Deep inelastic scattering through charged current (left) and neutral cur-rent (right).
Figure 1.6: Cross section for deep inelastic scattering (DIS) of neutrinos off quarks divided by neutrino energy. Note that at higher energies, the cross section follows a
linear dependence to the neutrino energy.
Finally, when reaching high energies (> 20 GeV), the neutrino accesses quarks in the
nucleon, producing a shower of hadronic particles, see Fig. 1.5. This interaction can
happen through charged and neutral currents from a neutrino of any flavour. Like neutrino-electron scattering, DIS is a four-elementary particle scattering expected to approach a linear scaling with energy at higher energies. This behaviour is observed in
experimental results, see Fig. 1.6.
1.3. Particle Showers 12
(a) Cross section for neutrinos (b) Cross section for anti-neutrinos
Figure 1.7: Total cross section for neutrino-nucleon interactions.
point in the energy spectrum of the neutrino, which is therefore called the transition
region, see Fig. 1.7a. Anti-neutrinos experience a slightly smaller total cross section in
this region, see Fig. 1.7b. Understanding the interactions happening in this region is
of particular interest for experiments working with neutrinos at these energies such as
KM3NeT ORCA9, CHiPs, NOνA, and more.
1.3
Particle Showers
When interacting in a medium, high energy particles will produce new high energy particles, which will themselves re-interact with the medium. This increase in particle multiplicity in the medium is called a particle cascade or shower. Particle showers are typically classified into two categories, electromagnetic showers, which do not contain hadronic particles, and hadronic showers.
1.3.1 Electromagnetic Showers
In media, energetic electrons can lose energy through bremsstrahlung, where it emits a single photon,
e−+ γ → e−+ γ.
If such particles are present, electrons can also annihilate with a positron, producing a
pair of photons10,
e−+ e+→ γ + γ.
9In1.6.1, the 1 − 10 GeV region is shown to be the most sensitive to the desired ORCA signal.
10
An electron-positron pair will never produce a single photon. It will always be possible to find a reference frame in which the electron and positron have a combined null momentum, which would yield a photon of non-zero momentum, violating conservation of momentum.
γ + γ → e−+ e+.
Electrons can also lose energy through ionisation11. The energy at which bremsstrahlung
becomes the dominant energy loss channel for the electron is called the critical energy,
Ec. In air and for electrons, Ec= 86 MeV, in water, Ec= 78 MeV[19]. An idiosyncratic
feature of electromagnetic showers is that the dominant particle production processes, bremsstrahlung and pair production, are invariant of the shower energy.
With these properties in mind, the Heitler model was developed by John F. Carlson
and J. Robert Oppenheimer[20], in which a single particle of energy E produces two new
particles of energy E/2 at every interaction length in the medium, X012. The multiplicity
of particles can be calculated for a given depth X = nX0, where n ∈ Z+is the generation
number, as
N (X) = 2n= 2X/X0, (1.20)
see Fig. 1.8. Likewise, the energy of each particle at depth X is
E(X) = E0
2n =
E0
2X/X0, (1.21)
where E0 is the energy of the primary particle. Production of particles is stopped when
the particles have reached the critical energy, E = Ec, where the multiplicity of particles
is maximal, called the shower maximum. The depth of the shower maximum is then
XmaxEM(E0) = X0log2
E0
Ec
, (1.22)
and its maximum multiplicity,
NmaxEM(E0) =
E0
Ec
. (1.23)
The Heitler model has been validated by more detailed simulations of electromagnetic
showers[21]. The main take-away point of electromagnetic showers is their energy scaling
NmaxEM ∝ E0, and XmaxEM ∝ log2E0. (1.24)
11
When energetic electrons travel through a medium, they impart some of their energy to ambient atoms, which ionise as a result.
12The interaction length is the expected path length of a particle at which there is a probability of
1.3. Particle Showers 14
Figure 1.8: The Heitler model simplification of electromagnetic (left) and hadronic showers (right). In the case of electromagnetic shower, every particle splits in two equal
energy parts at every generation, until the critical energy is reached. [21]
1.3.2 Hadronic Showers
When an energetic hadronic particle collides with the medium, the most frequent prod-ucts are charged and neutral pions, and more rarely, kaons,
X + Amedium → X0+ π+/−+ π+/−+ · · · + π0+ π0+ · · · + κ0, (1.25)
where Amedium is an atom from the medium, X and X0 are the initial hadronic particle
and its remnant (if any). The neutral pions with an extremely short lifetime13 decay
almost immediately into a pair of photons,
π0 → γ + γ,
while the other particles can re-interact with the medium, producing more particles
following Eqn.1.25. Ultimately, charged pions also decay into a muon and muon neutrino,
π+(−)→ µ+(−)+ ν
µ(¯νµ),
and the muon finally decays into stable particles,
µ−(+)→ νµ(¯νµ) + e−(+)+ ¯νe(νe)
[21].
Unlike with electromagnetic showers, hadronic showers cannot be solved analytically due to the complex production of multiple types of particles, and the presence of de-cays. Typically, hadronic showers are numerically simulated in detail, and the result
13
Under the assumption that one third of the energy of the shower core gets transmitted into neutral pions at every stage of interaction, and all particles receive equal portions of
energy on average, see Fig. 1.8, the total energy of the primary hadron dissipated into
hadronic particles becomes
Ehadr=
2 3
n
E0,
and into electromagnetic particles becomes
EEM = 1 −2 3 n E0= E0− Ehadr.
At lower energies, and thus fewer numbers of production generations n, a larger portion
of the energy is dissipated through hadronic particle, with a maximum of 2
3 for n = 1
14.
As energy goes up with the number of generations, all of the energy of the hadronic
primary is dissipated through electromagnetic particles, limn→∞EEM = E0.
It is also interesting to get an idea of the scale of hadronic showers. Here, it will be assumed that since the multiplicity of electromagnetic particles outweigh that of hadronic particles (especially at high energies), electromagnetic particles will dictate the position of the hadronic shower maximum to a reasonable approximation. Electromagnetic particles are produced in the first interaction of the hadronic primary (assuming instantaneous decay of neutral pions),
Xmaxhadr∼ X0hadr+ XmaxEM. (1.26)
At each point of interaction, a certain number of particles, nprod, are produced. Similarly
as before, one third of the number of particles are assumed to be neutral pions. Since
each neutral pion decays into a pair of photon, each photon receives EEM = 2nE0
prod. We
can now use the shower maximum depth equation for electromagnetic showers again,
Eqn.1.22, in combination with Eqn.1.26,
Xmaxhadr ∼ Xhadr
0 + X0 log2 E0 nprodEc − 1 . (1.27)
Eqn.1.27 shows that in the case of Xhadr
0 ∼ X0, the shower maximum of hadronic and
electromagnetic showers should be similar.
14n = 0 meaning that the initial energy of the hadron is below the critical energy, meaning of course
1.4. Atmospheric Neutrinos 16
Figure 1.9: Cosmic ray flux of energy per nucleon[8]. The spectrum is multiplied by E2.6 to make it flatter. From a few hundred of GeVs until the first knee appearing at a few PeV, the spectrum follows a slope of ∼ −2.7, appearing nearly flat in this figure.
1.4
Atmospheric Neutrinos
In the Solar system, the Milky Way, and other galaxies, charged particles are accelerated in a variety of objects, yielding a considerable amount of particles permeating space with a wide energy spectrum. These cosmic rays (CR) interact with particles in the Earth’s atmosphere, causing particle showers which produce a large flux of energetic neutrinos.
1.4.1 Cosmic Rays
At lower energies, the CR flux is dominated by ionised H and He, while heavier elements
such as Fe, Si and C dominate the higher energy flux, see Fig. A.3.
The CR spectrum has been mapped out up to extremely high energies per nucleon, see
Fig. 1.9.
As described in1.3.2, high energy hadrons produce hadronic showers in the atmosphere.
These showers produce pions, which decay into other pions, muons, and neutrinos. A
diagram of neutrino production in hadronic showers is given in Fig. 1.10. Every neutral
pion decay results in three neutrinos, π0 → ν
e+ 2νµ, implying a flux ratio ννµe+¯+¯ννµe ' 2
before oscillations. Thanks to the distribution the incoming CR energies, product
neu-trinos represent a flux covering a wide energy range, see Fig. 1.11a. This information
π μ e νe π π νμ Cosmic ray νμ νμ e νe νμ μ
Figure 1.10: Production of neutrinos in hadronic showers in the Earth’s atmosphere.
(a) Calculated flux of atmospheric neutrinos at sea-level, before oscillations. Horizontal neu-trinos are going straight down, while vertical neutrinos are going tangentially to the Earth. This means that ORCA can potentially detect neutrinos coming from anywhere in the Earth
atmosphere[10].
(b) Expected flux of matter-oscillated at-mospheric neutrinos at ORCA.
1.5. Cherenkov Neutrino Detection 18
Figure 1.12: Electromagnetic wavefronts caused by a charged particle moving slower (left) and faster (right) than the speed of light in the medium. When exceeding the speed of light in the medium, the wavefronts constructively add, yielding an observable
front of light, called Cherenkov light.
1.5
Cherenkov Neutrino Detection
This section details the principles of Cherenkov neutrino detection. This technique is the principal working mechanism of KM3NeT ORCA and ARCA, as well as other neutrino telescopes such as Antares, IceCube, and GVD-Baikal, but also other neutrino detectors
such as Super Kamiokande, CHiPs, and the retired Sudbury Neutrino Observatory15.
1.5.1 The Cherenkov Effect
When charged particles move through a polarisable medium, ambient molecules get po-larised isotropically emitting electromagnetic perturbations. If the particle is moving slower than the phase velocity of light in the medium, the perturbations travel faster the particle, adding destructively. However if the charged particle has a speed greater than that of light in the medium, the perturbations constructively add, causing a
coher-ent wavefront of electromagnetic radiation[27], see Fig. 1.12. This effect is called the
Cherenkov effect, after the Russian physicist who first described it, Pavel Alekseyevich
Cherenkov. This light is observable and useful in detecting relativistic charged particles in media.
15There also exist other techniques for neutrino detection, such as using liquid scintillators as in
NOνA[22], or time projection chambers such as in DUNE[23], but also others which are in early stages of development. These include radio neutrino detection through the Askaryan effect[24], Mossbauer neutrino detection[25], or even acoustic neutrino detection[26].
Figure 1.13: Path length of muons, and electromagnetic and hadronic showers. The path length of muons is vastly superior to that of electromagnetic and hadronic showers
for energies & 3 GeV[28]
In electromagnetic and hadronic showers, many particles are responsible for producing Cherenkov light, among which are electrons, protons, charged pions, and muons. The signature from these particles comes primarily from the angle of emission of Cherenkov light, and the particle’s path length. The angle of emittance of Cherenkov light is called the Cherenkov angle,
θc= arccos(1/nβ), (1.28)
where n is the index of refraction of the medium, and β is the speed of the particle.
In water, and in the ultra-relativistic limit, the Cherenkov angle is limβ→1θc = 41.4
deg. The fixed Cherenkov angle causes a cone to propagate through the detector, which permits acute sensitivity to the direction of the mother particle. Another interesting signature is that of the muon, which has a path length vastly superior to that of other
particles, see Fig. 1.13. At high energies, the Cherenkov cone of the muon will be
strikingly elongated, permitting excellent detection of high energy muons in Cherenkov detectors.
1.5.2 Water Cherenkov Neutrino Detectors
The strategy of water Cherenkov neutrino detectors is to observe the Cherenkov light emanating from charged particles produced in neutrino-nucleus interactions. Water is a useful detection material for this purpose as it serves both the purpose of target for the neutrino interaction and propagator for the product particles. It easy to produce a very large mass (and hence a large amount of targets) of water. Water is transparent to visible light and polarisable, which means that Cherenkov light can be produced and
1.6. KM3NeT ORCA 20 propagated without being absorbed. Liquid water also has the advantage over ice that light does not scatter as much.
1.6
KM3NeT ORCA
KM3NeT ORCA is the dense, 0.004-cube-kilometre detector of the KM3NeT experiment. While ARCA’s focus is to identify and study galactic and extra-galactic neutrino sources,
ORCA’s main purpose is to measure the NMO[6]. This thesis will focus on the physics
relevant for the ORCA detector.
1.6.1 ORCA’s Strategy for the NMO
In the physical three-neutrino model, the oscillation equations become more complicated[6].
The transition probability of a muon neutrino into electron neutrino, and survival prob-ability of a muon neutrino are given, respectively, as
P3ν(νµ→ νe) ' sin2θ23sin22θ13msin2(∆mm2L/4E) (1.29)
P3ν(νµ→ νµ) '1 − sin22θ23cos2θ13msin2((∆m231+ ∆mm2L)/8E + AL/4)
− sin22θ23sin2θ13msin2((∆m231− ∆mm2L)/8E + AL/4)
− sin4θ23sin22θ13msin2(∆mm2L/4E), (1.30)
where sin22θ13m ≡ sin22θ13 ∆m2 31 ∆mm2 2 and ∆mm2≡ q (∆m2
31cos 2θ13− 2EA)2+ (∆m231sin 2θ13)2
are the effective parameters of the neutrinos in matter. From Eqn.1.29, one can see that
matter effects due to the parameter of interest, ∆mm2, are exacerbated in the case of a
long baseline L. Advantages of using muon neutrino oscillations are that they provide two sensitive channels, and are not as demanding of the energy resolution to detect the signal.
With this information in mind, ORCA’s strategy to determine the NMO is to observe the oscillation of atmospheric muon neutrinos propagating through the Earth. This fulfils the requirements of observing muon neutrino survival and disappearance through a long baseline. For this baseline, O(GeV) neutrinos represent the energy at which the
Figure 1.14: Expected oscillations for ORCA at various zenith angles[6].
differences between the IO and NO are the most noticeable. Conveniently, the flux of
atmospheric neutrinos contains muon neutrinos of those energies, see Fig. 1.11a.
Assuming a spherical Earth, the baseline can then be estimated as
L = 2 cos θνR⊕, for θν ∈ (0,
π
2), (1.31)
where θν is the zenith angle of the neutrino in ORCA16, and R⊕ is the radius of the
Earth.
From Eqns.1.29and 1.31, it is clear that the energy and direction resolutions, as well as
the flavour identification of ORCA all feed directly into the sensitivity of the experiment to the NMO. Another parameter of interest is the Bjorken inelasticity parameter for CC (NC)
y ≡ Eνl− El(νl)
Eνl
, (1.32)
where l = (e, µ, τ), which indicates the fraction of the primary neutrino energy transferred through the weak boson. The cross sections of neutrino and antineutrino interactions differ for a given y, meaning that the measurement of this parameter further improves the sensitivity of ORCA to the NMO signal.
ORCA is designed to detect atmospheric neutrinos interactions, and measure their en-ergies, directions and flavour. The directions of the neutrinos gives information on the amount of matter traversed. The flux of given flavours for energies and amount of matter
traversed will then tell us which of the IO or NO is correct, see Fig. 1.14
16
Here, θν = 0 is defined to be pointing upwards, i.e. a neutrino which was created at the exact
1.6. KM3NeT ORCA 22
1.6.2 Digital Optical Module
The sensitive elements of the KM3NeT detectors are Photomultiplier Tubes (PMT) with
a 72 mm photocathode area17[29][30]. The PMTs are photo-sensors sensitive to photons
in the [300, 650] nm range, chosen for their ability to detect Cherenkov light in water. The wavelength dependent sensitivity of the PMTs can be shown to match the spectrum of the expected signal. The number of Cherenkov photons per unit path length per unit wavelength of the photons is
d2N dxdλ = 2παz2 λ2 sin 2(θ c) nm−1cm−1, (1.33) where sin2(θ
c) = 1 − (η21β2) is the sin2 Cherenkov angle, where n is the refractive index
of water, and β is the ratio of the speed of the particle to the speed of light, v
c [18].
After integrating, the total number of photons produced by a super-relativistic particle (β ' 1) over a given path length ∆x, at a wavelength λ and for a wavelength interval
∆λ can be calculated as N = 2παz2 1 − 1 n2 ∆λ λ2+ λ∆λ∆x. (1.34)
Photons also experience attenuation in water, which can be measured as the λ dependent
loss of intensity of a light wave per unit distance, Aw(λ) cm−1. This quantity has been
measured over the [100, 1000] nm range[31][32][33]. At one unit distance from the source,
T0 = I0(1 − Aw(λ)) photons are transmitted for I0 incoming photons. The next unit
of distance receives the transmitted photons, I1 = T0. Then, the number of surviving
photons after k units of distance is Ik = Ik−1(1 − Aw(λ) = I0(1 − Aw(λ))
k unit 1 unit. The
relative number of Cherenkov photons transmitted in water at a distance d from the
source to the total number of emitted photons over the [λ0, λ1]range for any path length
is then calculated as N Ntotal (λ, d) = (1 − Aw(λ)) d 1 cm ∆λ λ2+ λ∆λ λ0λ1 λ0− λ1 (1.35)
over the [100, 1000] nm range for various distances from the particle, and compared to
the sensitivity of the KM3NeT PMTs, see Fig. 1.15.
31 PMTs are compactly and approximately uniformly arranged into a 17-inch glass sphere
called a Digital Optical Module (DOM), seen in Fig. 1.16. Experiments such as Antares
and IceCube have previously used a single-PMT DOM, with a PMT about three times larger than those of KM3NeT. The multi-PMT DOM design, however, offers a cathode
17
Figure 1.15: Proportion of total Cherenkov photons transmitted in water after dis-tance and quantum efficiency of the Hamamatsu R12199-02 PMT[29]. KM3NeT PMTs’
spectral range matches the expected Cherenkov light signal.
Figure 1.16: The KM3NeT DOM (bottom view)[35].
surface that is about three times larger, a quasi-isotropic field of view, and the advantage
of having small PMTs which are negligibly affected by the Earth’s magnetic field[34].
The PMTs have undergone tests to characterise their performance. The quantum
effi-ciency18 of the PMTs is also shown in Fig. 1.15. When a photon successfully kicks an
electron off the photo-cathode of the PMT, an error is introduced as the electron must travel stochastically through the PMT dynodes before triggering the output signal,
re-ferred as the Transit Time Spread (TTS). The TTS is measured as the FWHM19 of the
18
Quantum efficiency is the ratio of number of incident photons to the number of electrons converted by the PMT. The tests of this quantity for the Hamamatsu PMTs were performed by comparing the output current of the PMT and a reference photodiode with known Q.E. for an identical light source.
1.6. KM3NeT ORCA 24
arrival time of the converted electrons, and has been measured to be strictly < 5 ns[29].
1.6.3 Detection Unit
KM3NeT is composed of Detection Units (DU)[6], each of which comprise 18 DOMs.
ORCA’s DUs are 200 m tall with DOMs vertically spaced 9 m apart with the first DOM placed 40 m from the seabed. The DUs are anchored by a heavy base which lies on the sea floor, and are kept upright by the buoyancy of the DOMs. The buoyancy of the DUs is adjusted at the top in order to minimise lateral movement due to sea currents. 115 DUs placed 20 m apart will constitute KM3NeT ORCA phase I.
As seen in Fig. 1.14, the < 10 GeV neutrino energy region displays the largest differences
between NO and IO. There is a trade-off between maximising ORCA’s density to resolve low energy neutrino-interactions, and ORCA’s size, to increase the number of neutrino interactions happening inside the detector. The flexible positioning of the DUs permits the density and volume of ORCA to be optimised to account for these factors.
1.6.4 Hits
When a photon causes a converted electron, the electron causes a PMT response and is called a photo-electron (p.e.). This p.e. will cause a pulsed response from the PMT. A threshold of 30% of the mean single p.e. pulse height is set, which causes a triggered PMT hit if exceeded. The length of time the pulse is above this threshold is measured and is called the Time over Threshold (ToT). An FPGA stores the ToT and arrival time information of the hit. Along with information on the position of the DOM and the identifier of the PMT, the following information is available for each hit:
• thit, trigger time.
• ToT, time over threshold. • A, amplitude of the pulse. • θ and φ direction of the PMT. • x, y, z position of the PMT.
2.1
Motivation
At ORCA energies (≤ 100 GeV) and on a detector scale resolution of O(100 m), ν-interactions either look like spherical showers, or elongated,“track-like” showers, if they contain an energetic µ. Details start emerging for smaller scale resolution, such as the
electromagnetic component of a νe-CC interaction, or the decays of a τ from a ντ-CC
interaction. Current models used for reconstruction in KM3NeT, which will from now on be referred to as global models, are efficient in reconstructing large scale topologies, but are too coarse for interaction details. KM3NeT does not currently possess models for reconstruction of these details in ν-interactions.
In addition to this, global models used for event reconstruction of ν-interactions use an EM shower-like dependence of shower size and luminosity on energy. Although this dependence is largely applicable for high energy interactions, hadronic showers do not follow this dependence in reality, particularly for low energies. For interaction detail reconstruction models, energy must be set as a free parameter to track the real and non-trivial dependence of details to the energy.
The motivation for this thesis was to remedy to these two shortcomings by producing energy-aware models of interaction details, which will from now on be referred to as
detailed models.
2.1.1 Detailed Reconstruction
A basic concern for ORCA is to reconstruct as much information as possible from
inter-actions happening in the detector. As seen from eq. 1.29, an improvement in resolution
2.2. Approach 26 in the flavour tagging of the interaction, the energy resolution, and the direction of the
neutrino, all feed directly into the sensitivity of the experiment to measure the NMO1.
Improving the resolutions of the detector is possible by using the detailed models to perform reconstructions of ν-interaction events which have several advantages over the usual reconstruction methods.
2.1.1.1 Interaction level fluctuations
Products of ν-nucleus interactions experience large fluctuations in their products due to hadronic showering. These fluctuations are currently folded in the global models. With detailed reconstructions of neutrino events, the initial fluctuations can be measured instead of limiting the resolving power of the models. Among those measurements is the
Bjorken y parameter, and identification of νNC interactions, which have previously been
largely inaccessible.
2.1.1.2 Neutrino interaction model independence
Global models depend on Monte Carlo simulations of ν-interactions. These include simulations of ν-nucleus interactions, which are notoriously difficult to model due to
the transition region described in 1.2 and currently represent one of the largest source
of systematic uncertainties in neutrino experiments[36]. Detailed models depend only
on the behaviour of the products of the initial interaction, which means that the true behaviour of ν-nucleus interactions is measured by reconstructions using the detailed models.
2.2
Approach
The idea for this thesis is to create models of every common charged particle in ORCA. Different types of particles are expected to produce different light signatures in the detector, which will be referred as the light pattern. The only observable KM3NeT has access to is the information coming from PMT p.e. caused by this light pattern, which will be globally referred to as the photo-electron pattern. Concretely, the model is a probability density function (PDF) for the p.e. pattern expected for a particle type at a given energy, and will be from now on referred to as a Photo-Electron Pattern PDF, or PEPP.
1
The particle p.e. patterns from KM3NeT ν-interaction samples are translated into PEPPs. In this section, the building chain of the PEPP is described.
2.3.1 Simulated sample
As seen in1.3and1.5.1, the propagation of energetic particles in water is highly complex.
Mapping out the light pattern of particles analytically is therefore unfeasible. Stepwise Monte-Carlo simulations are well suited for this task. The Monte Carlo simulations also simulate the p.e. creation, providing the p.e. pattern.
In this thesis, samples made with the GSeaGen[37] and KM3SiM[38][39] packages are
used, which contain the propagation of particles and Cherenkov light from neutrino interactions in the detector. GSeaGen is an underwater neutrino interaction generator. It simulates rapid decays, re-interactions and hadronisations. GSeaGen outputs the particles products. KM3SiM is a Geant4 based particle propagation simulator which has been built using the geometry of the KM3NeT detectors and the relevant physics list. Geant4 has a long history of use in particle physics, particularly at LHC experiments, and repeated validation of the software package have established it as a very reliable
simulator[40][41][42]. KM3Sim inputs a particle type, energy, direction, and position
in the detector, and outputs a list of photon-electrons (p.e. ), as well as the identifier of the PMT. KM3SiM simulates the propagation of the particle in the medium, its re-interactions with other particles, and the Cherenkov light produced by all charged
particles. Using the quantum efficiency[29] and the angular response[43], KM3SiM also
simulates the creation of p.e. at the PMT photocathode. The results of the simulation at this point are used.
A full ORCA sample is used which contains a low-energy heavy distribution of [1, 100]
GeV νCC and νN C interactions of all three flavours. The interactions have a flat
distri-bution in positions and directions inside of the instrumented volume.
2.3.2 Histogram building
One 6-dimensional histogram, and one 5-dimensional histogram are created for each of
12 particles: e−, e+, γ, µ−, µ+, p, ¯p, n, ¯n, π
0, π+, and π−. Each dimension of the
histograms uses variable binning to increase precision in regions of interest2. For every
2
The highest bin density is found for the region where ∆t = 0, E and D are small, and cos η is close to 1. TableA.2contains more details on the histogram phase space.
2.3. PEPP Building 28 new product particles from GSeaGen, an expanding sphere region of D ≤ 80 m and
−15 < ∆t < 20ns3 is scanned for PMTs. Each PMT in this region represents a potential
hit (p.h. ). From each PMT in the expanding sphere, 5 quantities are extracted:
1. E, the total energy of the particle
2. D ≡ |rhit− r0|, the distance between the starting position of the particle and the
hit position4
3. cos η ≡ ˆp·(rhit− r0), the cosine of the angle separating the direction of the particle
from the direction to the hit
4. θP M T, the zenithal direction of the hit PMT on the DOM, where 0 is upwards.
5. φP M T, the azimuthal direction of the hit PMT on the DOM, where 0 is to the
right.
The p.h. histogram is filled with 1 for every p.h. in the phase space position defined by these 5 quantities. If a PMT has been hit in the expanding sphere, two additional quantities are extracted:
1. ∆t ≡ thit− t0, where t0 ≡ n · D/c, the difference between the expected light arrival
time and the actual hit time, where n is the index of refraction of water
2. Np.e., the number of p.e.
In this case, the Np.e. is filled in the p.e. histogram at the position defined by the six
other quantities. The quantities are displayed visually in Fig. 2.1.
2.3.3 Photo-Electron Pattern Expected Value Function
The p.e. histograms bins contain the total number of p.e. over the dataset in the phase space region established by the bin dimensions. The p.e. histogram is now divided by the p.h. histogram. The resulting bin fills represent the average number of p.e. per particle
per unit dimensions, Np.e. ns−1m−1GeV−1rad−2. This average is taken to be the true
3∆t is defined by the expected arrival time of a photon. Using a fixed speed of light in water, a ∆t
interval is then equivalent to an interval in space expanding spherically at the speed of light in water.
4The "starting position" of the particle in the simulation refers to the vertex. Which vertex is in
question here, the vertex of the interaction producing the product particle in question or the neutrino interaction vertex? GSeaGen relays pseudo-stable particles to KM3Sim which result from a long in-teraction chain. This chain evolves however on extremely short timescales, which are negligible in the context of KM3NeT. For this reason, distinction between the two in the context of this thesis can be safely neglected. The defintions t0 ≡ tvertexν ' tvertexparticle and r0 ≡ rvertexν ' rvertexparticle are
D
φ
PMT, E
Δt, N
p.e.e-cos η
0η
0Figure 2.1: Visual representation of the PEPP dimensions for an electron (yellow) and a PMT of interest (purple) as an example. Note that the PMT orientations are defined in the frame of the particle and not in that of the detector. η is labeled for
clarity even though cos(η) is the parameter used in the PEPPs.
average of the underlying distribution, or the expected value of the number of p.e. causing
a hit given x = (E, D, cos η, θP M T, φP M T, ∆t), the position in the phase space,5
E(Np.e.|x) = Np.e.(x) ≡
R
NeventsNp.e.(x)
R
NeventsNp.h.(x)
. (2.1)
To obtain smoother values between bins, the histograms are finally interpolated using a
1st degree polynomial6to produce what will be referred to as the Photo-Electron Pattern
Expected Value Functions(Photo-Electron Pattern EVF, or PEPE). The PEPE is a
mul-tivariate function which inputs the property of the particle (type, energy), the properties of the PMT of interest (distance from vertex, cosine angle from direction, directions of
5The Law of Large Numbers states that the average results of a repeated experiment approaches the
expected value of the random variable being measured,
E(X) = X = lim N →∞ N X i=0 Xi N.
This equivalence is thus valid if a large enough dataset is used. The histograms in this thesis are built from [HOW MANY?] events, which yields an error of [?????] on the mean. One must keep in mind that the dataset is not uniform, and that bins with lower statistics (high energy particles, and rarer particles) might experience spurious deviations from the true mean.
6
It is assumed that the light yield does not fluctuate wildly within the bin regions imposed by Table
2.3. PEPP Building 30 the PMT, expected arrival time difference), and outputs the expected number of p.e.
causing a hit at the given position in the phase space, E(Np.e.|x).
2.3.4 Integrated Photo-Electron Pattern EVFs
Variants of the PEPEs where some of the dimensions have been integrated are useful for reconstructions and plotting purposes. Three new EVFs are derived from the PEPE, the
PEPEDOM, and the All-time PEPE and All-time PEPEDOM. The PEPEDOM is the
5-dimensional PEPE where the θP M T and φP M T dimensions have been integrated and
normalised over the number of PMTs in a DOM, NP M T to give the expected Np.e. per
PMT on the DOM,
E(Np.e.DOM) =
R
DOME(Np.e.)dθP M TdφP M T
NP M T
.
The All-time PEPE and All-time PEPEDOM are the PEPE and PEPEDOM where the
∆tdimension is integrated to give the total E(Np.e.) and E(Np.e.DOM) over all time.
2.3.5 Photo-Electron Pattern PDFs and CDFs
The expected values of the PEPEs are now used to extract the probabilities under the
assumption that the Np.e. is Poisson distributed7:
P (Np.e.|x) =Poisson(Np.e.|E(Np.e.|x)) =
e−E(Np.e.|x)E(N
p.e.|x)Np.e.
Np.e.!
, (2.2)
which are the PEPPs. PEPPs were produced from all four PEPE variants, to yield
the PEPP, the All-time PEPP, the PEPPDOM, and the All-time PEPPDOM. Next, the
Photo-Electron Pattern CDFs (PEPCs) are produced from the PEPPs. The Cumulative
Distribution Function (CDF) of a PDF is calculated as F (x) =
Z x
a
f (x)dx, (2.3)
where f(x) is the PDF of random variable X normalised on (a, b), and F (x) is its CDF. The CDF’s range is (0, 1); for a position x ∈ (a, b), F (x) returns the proportion of
events, u, which are expected to occur for xevent ≤ x. PEPCs are made from the the
7
Hits are counted in positive integers, and are independent from one another, but do not have a constant rate per event, which changes with a change in the phase space position, making their measurement a non-homogeneous Poisson process. In this case, Poisson distributions can be constructed using the event average for the interval of interest, Xxi =
Rxi+δx
xi X(x)dx. The value returned by the
PEPEs is exactly this value, the average event occurrence per dimensions, and can be used to construct the Poisson distributions.
All PEPPs, PEPEs, and PEPCs are described in Table A.3. In this thesis, only P (0|x), the probability of obtaining 0 hits, and P ([1, ∞]|x) ≡ 1 − P (0|x), the probability of obtaining at least 1 hit, will be used. Prospects of using the full PEPPs is discussed in 5.3.1. The PEPPs give a poor description of the muon p.e. pattern in the far forward
direction, and requires a different coordinate system, see5.1.2. Details close to the origin
of the particle are however present in the current parametrisation, and can be analysed.
2.4
Results
Aspects of a variety of PEPPs are shown, with emphasis on the most striking and interesting features.
2.4.1 Dimension flattening
The PEPPs are 7-dimensional: P (Np.e.|E, D, cos η, θP M T, φP M T, ∆t). In order to
repre-sent them on a 2-d image, 5 dimensions must be flattened. First, the EVFs are used, so
that the expected number of p.e. E(Np.e.) are displayed rather than the probability for
a specific number of Np.e. to be measured, P (Np.e.). First, the All-time PEPEDOM are
used, which have been integrated over ∆t, θP M T and φP M T. Next, All-time PEPE are
used, shown for specific PMT directions. Finally, the time dimension of the PEPEs are shown for specific PMT directions, distance, and cos η. All plots are shown for specific energies.
2.4.2 Features in space
The electron is used as a reference particle to which other particles will be compared. The 2 GeV electron PEPP is displayed in cartesian coordinates, as well as in the native
coordinates of the PEPP in Fig. 2.2. The majority of the p.e. received from the electron
are found in the forward direction8. The bulk of the p.e. from the electron also appear
at an angle from the direction of the particle, identified as the Cherenkov angle.
Photons and anti-electrons are also electromagnetic particles. To compare two PEPPs, their 2-d plots of interest are divided by one another. The result is a 2-d plot which indicates that the PEPP in the numerator is brighter (dimmer) for regions of value > 1
8
2.4. Results 32 4 − 10 3 − 10 2 − 10 1 − 10 1 10 2 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 D (m) 5 10 15 20 25 30 35 40 45 50 pmt N > pe <N DOM Σ Weight: 4 − 10 3 − 10 2 − 10 1 − 10 1 10 2 10 R (m) 0 5 10 15 20 25 30 35 40 45 50 Z (m) 10 − 0 10 20 30 40 50 pmt N > pe <N DOM Σ Weight:
Figure 2.2: 2 GeV electron PEPP in the (cos η, D) plane (left) and (R, Z) plane (right). A large excess of p.e. is visible at η ∼ 40deg, indicating the Cherenkov angle.
(< 1). The anti-electron and photon PEPPs are compared to that of the electron in Fig.
2.3, where all three particles are shown to behave strikingly similarly. Since electrons
at ORCA energies can only produce photons through annihilation and bremstrahlung, and photons only pair-produce electrons, the electron and photon PEPPs are an effective description of an electromagnetic shower.
1 − 10 1 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 D (m) 5 10 15 20 25 30 35 40 45 50 -e > p.e. <N + e > p.e. <N DOM Σ Weight: 1 − 10 1 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 D (m) 5 10 15 20 25 30 35 40 45 50 -e > p.e. <N γ > p.e. <N DOM Σ Weight:
Figure 2.3: Left (Right): comparison of the 3 GeV anti-electron (photon) PEPP to the electron PEPP in the (cos η, D) plane. The intensity of the showers differs strictly
less than a factor of 2 from that of an electron.
Hadronic particles such as the proton and the positive pion display a p.e. pattern distinct
from that of the electron, which appears more isotropic, see Fig. 2.4. Moreover, unlike
the electromagnetic particles studied so far, hadronic particles behave distinctly from
4 − 10 3 − 10 2 − 10 1 − 10 1 10 2 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 pmt N > pe <N DOM Σ Weight: 4 − 10 3 − 10 2 − 10 1 − 10 1 10 2 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 pmt N > pe <N DOM Σ Weight:
Figure 2.4: Left (Right): 3 GeV proton (positive pion) PEPPs in the (cos η, D) plane. The p.e. of the particles are received somewhat isotropically, with a sizeable excess in
the Cherenkov angle region.
other hand, behave nearly electromagnetically9, see the comparison in Fig. 2.6. Since
hadronic particles at ORCA energies can interact and decay into one another, none of the PEPPs can represent an entire hadronic shower accurately. However, the proton will be used from now on as an archetypical hadronic particle because of its stability, ensuring the maximal presence of hadronic particles in its shower.
4 − 10 3 − 10 2 − 10 1 − 10 1 10 2 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 D (m) 5 10 15 20 25 30 35 40 45 50 pmt N > pe <N DOM Σ Weight:
Figure 2.5: Comparison of 3 GeV neutron (neutral pion) PEPPs to the electron PEPP in the (cos η, D) plane. The neutron exhibits a strik-ingly isotropic distribution of p.e. with a slight excess in the forward
di-rection. 1 − 10 1 10 ) η cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 D (m) 5 10 15 20 25 30 35 40 45 50 -e > p.e. <N 0 π > p.e. <N DOM Σ Weight:
Figure 2.6: Comparison of the 3 GeV neutral pion PEPP to the elec-tron PEPP in the (cos η, D) plane. The neutral pion behaves remarkably
similarly to the electron.
