OBSERVATION OF MUON NEUTRINO CHARGED CURRENT EVENTS IN AN
OFF-AXIS HORN-FOCUSED NEUTRINO BEAM USING THE NOνA PROTOTYPE
DETECTOR
By
Enrique Arrieta D´ıaz

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2014

ABSTRACT
OBSERVATION OF MUON NEUTRINO CHARGED CURRENT EVENTS
IN AN OFF-AXIS HORN-FOCUSED NEUTRINO BEAM USING THE
NOνA PROTOTYPE DETECTOR
By
Enrique Arrieta D´ıaz
The NOνA is a long base-line neutrino oscillation experiment. It will study the oscillations between muon and electron neutrinos through the Earth. NOνA consists of two
detectors separated by 810 km. Each detector will measure the electron neutrino content of
the neutrino (NuMI) beam. Differences between the measurements will reveal details about
the oscillation channel. The NOνA collaboration built a prototype detector on the surface
at Fermilab in order to develop calibration, simulation, and reconstruction tools, using real
data. This 220 ton detector is 110 mrad off the NuMI beam axis. This off-axis location
allows the observation of neutrino interactions with energies around 2 GeV, where neutrinos
come predominantly from charged kaon decays. During the period between October 2011
and April 2012, the prototype detector collected neutrino data from 1.67 × 1020 protons on
target delivered by the NuMI beam. This analysis selected a number of candidate charged
current muon neutrino events from the prototype data, which is 30% lower than predicted by
the NOνA Monte Carlo simulation. The analysis suggests that the discrepancy comes from
an over estimation of the neutrino flux in the Monte Carlo simulation, and in particular,
from neutrinos generated in charged kaon decays. The ratio of measured divided by the sim+0.108 . The NOνA
ulated flux of muon neutrinos coming from charged kaon decays is: 0.70−0.094

collaboration may use the findings of this analysis to introduce a more accurate prediction
of the neutrino flux produced by the NuMI beam in future Monte Carlo simulations.

To my abuelo, Andr´es, Quique and Rochi.
The precious components of my high intensity beam of unconditional love.

iii

ACKNOWLEDGMENTS

I can not pinpoint a moment in my life as the time when I decided to be a scientist. However,
I can assure that my decision consolidated during my third year of elementary school in the
classroom of C. Cabezas. He impacted my life forever, and my incommensurable gratitude
accompanies him now in the stars, right where he belongs. Soon after I learned from M. Rios
that a successful man can only live under the strict regime of impeccable moral standards,
and I have tried to follow his teachings to the best of my capacities. During my junior and
senior years of high school I sat in the classroom of C. Mart´ınez where he elegantly provided
most of the knowledge in physics that I bear today. It was at this time when I told F. Casas
that if by 2007 I was not a rey de reyes in Valledupar, he could be certain that I would then
become a Ph.D. in physics. Thanks to him I was introduced to the books of Carl Sagan that
structured my scientific thinking. Thanks to all my teachers at Abraham Lincoln school,
and to my professors at FIT, Los Andes, and MSU for their wonderful lessons.
The long road that brought me to my present had many gratifying moments, and countless steep slopes and deep troughs. This road, that resembled that to Santa Ana during
heavy winter times, felt very smooth thanks to the love and support of those members of my
families Aar´on, Arrieta, D´ıaz, and Hamburger, that always believed in me. I never tried to
make them feel proud, however I have been working extremely hard not to make them feel
ashamed. My moments of happiness were doubled, and my periods of sadness were halved,
when I had the chance to share them with my siblings from the 16th street and my fellow
Wachakeros. When the wipers did not work due to the mud covering the windshield, only
the GPS guidance of E. M. Aar´on helped me to reach safe shelter. My personal oracle, A.
Op den Bosch, kept me safe and well advised. An especial gratefulness remains in my heart
iv

for the members of the Solano family that cared for me like one of their own. Many friends
have their two cents and their grains of sand deposited in the safe of my heart. I can not
name them all, but I do treasure all their support throughout the years.
I have felt like Achilles racing the tortoise many times. Nonetheless, there is always
someone out there to remind me that only a fool can not defeat a tortoise at a speed race,
and that I am not a fool. M. C. J´acome and M. Nowakowski made this very clear to me.
However, it is only with the help of extraordinary persons that extraordinary goals are
accomplished. I am lucky enough to be surrounded by extraordinary individuals like S. D.
Mahanti, J. Paley, and S. Pratt who supported me, beyond imagination, when I most needed
them. There is no word in human languages to express the infinite gratitude that I have for
them. My friendships with J. Clifford, E. Kessler, T. Olson, and L. C. Rodr´ıguez made of
my years at MSU a wonderful time.
The quest in NOνA started with the Fellowship of the Fiber, and my companions M.
Nila, R. Richards, and D. Shooltz. Days of darkness were illuminated by their amazing
personalities. Then I arrived to the land of Fermilab where I found the invaluable support
of M. Betancourt. After meeting her the locomotive became unstoppable. M. S´anchez and
P. Vahle provided endless fuel for my endeavor. My analysis was only possible with the
outstanding patronage of C. Backhouse, M. Baird, M. Frank, R. Hatcher, N. Mayer, M.
Messier, M. Muether, E. Niner, R. Patterson, G. Pawloski, N. Raddatz, P. Shanahan, and Z.
Wang. A. Sousa greatly contributed to my success with his knowledge about the divine and
human. Indubitably the entire NOνA collaboration made it happen. The last portion of my
journey, at Argonne Lab, was enlightened by M. Goodman, and S. Magill, R. Talaga, and the
members of the neutrino group. They made me feel like home. In particular, J. A. Sep´
ulveda
showed me the true meaning of humbleness and kindness. I will be forever thankful to D.
v

Barratt, E. Johnson, B. Wenzlick, and R. Young. Their magic always simplified my life.
Mission accomplished! Mabel can now host the party, and Manolo can prepare the hug
and the entertainment. Leandro made El Card´on Guajiro, and Diomedes sang it for me.
The great days are straight ahead because I am, indeed, made for great things. Thank you!
“Hay soledades que duelen mucho, hay un silencio para pensar, hoy quiero luces para que
alumbren lo que me falta por caminar. Soy andariego de los caminos, con palabritas quiero
decir, as´ı es mi vida siempre cambiando, un d´ıa muy triste y un d´ıa feliz. Solo le pido a la
vida, que me d´e felicidad, y que mi conciencia duerma siempre tranquila. Quiero sentir el
aprecio de los amigos, vivir feliz en mi Valle, no pido m´as. No pido m´as, no quiero ser
golondrina que errante va. No pido m´as, hoy quiero luces que alumbren mi oscuridad.”
Gustavo Guti´errez Cabello

vi

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

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x

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xi

Chapter 1 Introduction . . . . . . . . . . .
1.1 Brief History Of Neutrinos . . . . . . . .
1.2 Motivations For The NOνA Experiment
1.3 Outline . . . . . . . . . . . . . . . . . . .

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Chapter 2 Theoretical Framework for Neutrinos . . . .
2.1 Neutrinos In Weak Interactions . . . . . . . . . . . . .
2.1.1 Charged Current Interactions . . . . . . . . . .
2.1.1.1 Quasi-elastic Scattering . . . . . . . .
2.1.1.2 Resonant Pion Production . . . . . . .
2.1.1.3 Deep Inelastic Scattering . . . . . . . .
2.1.1.4 Inclusive Cross Section Measurements
2.1.2 Neutral Current Interactions . . . . . . . . . . .
2.2 Neutrino Masses . . . . . . . . . . . . . . . . . . . . .
2.2.1 Seesaw Mechanism . . . . . . . . . . . . . . . .
2.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . .

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3.1 Primary Beam System . . . . . . . . . . . . . . . .
3.2 Secondary Beam System . . . . . . . . . . . . . . .
3.3 NuMI Beam Composition And Energy Spectrum At
Chapter 4 The NOνA Prototype Detector . .
4.1 Experiment Overview . . . . . . . . . . . . .
4.2 The Near Detector On The Surface . . . . .
4.2.1 Liquid Scintillator . . . . . . . . . . .
4.2.2 Wavelength Shifting Fiber . . . . . .
4.2.3 Avalanche Photodiodes . . . . . . . .
4.2.4 Front-End Boards . . . . . . . . . . .
4.2.5 Data Acquisition System . . . . . . .
4.2.6 Performance And Calibration Of The
4.3 NOνA Software . . . . . . . . . . . . . . . .
4.3.1 Monte Carlo Simulation . . . . . . .
4.3.2 Reconstruction Tools . . . . . . . . .
4.3.2.1 The Slicer . . . . . . . . . .

vii

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4.4
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4.3.2.2 Track Reconstruction . . . . . . . . . . . . . . . . . . . . . .
Data Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5 Event Selection . . . . . . . . . . . . . . . .
5.1 Charged And Neutral Current Neutrino Interactions
5.2 Event Containment Criteria . . . . . . . . . . . . .
5.3 Charged Current Event Selection . . . . . . . . . .
5.3.1 Longest Track Length Cut . . . . . . . . . .
5.3.2 Cosmic Ray Cuts . . . . . . . . . . . . . . .
5.3.3 Minimum Ionizing Particle Cut . . . . . . .

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Chapter 6 Event Energy Reconstruction . . . . . .
6.1 Muon Energy Estimation . . . . . . . . . . . . . .
6.2 Hadronic Energy Estimation . . . . . . . . . . . .
6.3 Quasi-elastic And Non-quasi-elastic Classification
6.4 Neutrino Energy Estimation . . . . . . . . . . . .

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7.1 Energy Estimation Uncertainty . . . . . . . . . . . . . . . .
7.2 Prototype Detector’s Channels Configuration Uncertainty . .
7.3 GENIE Cross Sections And Final State Physics Uncertainty
7.4 Unfolding Systematic Uncertainty . . . . . . . . . . . . . . .
7.5 Total Systematic Uncertainty . . . . . . . . . . . . . . . . .

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Chapter 8 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.1 Data And Monte Carlo Simulation Comparison . . . . . . . . . . . . . . . . 152
8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Chapter 9 Discussion And Final Remarks . . . . . . . . . .
9.1 Flux Calculation . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Number Of Atoms In The Target Region . . . . . .
9.1.2 Monte Carlo Cross Sections . . . . . . . . . . . . .
9.1.3 Total Reconstruction Efficiency . . . . . . . . . . .
9.1.4 νµ + ν¯µ Flux Coming From Charged Kaon Decays
9.2 Inclusive νµ Charged Current Cross Section Calculation . .
9.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

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APPENDICES . . . . . . . . . . . . . .
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Appendix A Mass Terms In The Weak Interaction . . . . . . . . . . . . . .
Appendix B NOνA Kalman Tracker . . . . . . . . . . . . . . . . . . . . . . .
Appendix C Unfolding Algorithm, TSVDUnfold . . . . . . . . . . . . . . .
Appendix D List Of Muon Neutrino Charged Current Candidate Events

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BIBLIOGRAPHY

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ix

LIST OF TABLES

Table 2.1 Neutrino Oscillation Parameters. . . . . . . . . . . . . . . . . . .

32

Table 4.1 Kalman Tracker Performance. . . . . . . . . . . . . . . . . . . . .

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Table 5.1 Event Selection Performance. . . . . . . . . . . . . . . . . . . . . 114
Table 7.1 GENIE Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Table 8.1 Event Selection Cuts.

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Table 8.2 Quasi-elastic And Non-quasi-elastic Classification Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Table 8.3 Candidate Events With Background. . . . . . . . . . . . . . . . . 173
Table 8.4 Number Of Candidate Events. . . . . . . . . . . . . . . . . . . . . 174
Table 8.5 Neutral Current Background. . . . . . . . . . . . . . . . . . . . . 174
Table 8.6 Electron Neutrino Background. . . . . . . . . . . . . . . . . . . . 175
Table 9.1 Background To Neutrinos From Kaon Decays. . . . . . . . . . . 178
Table 9.2 Candidate Events From Kaon Decays. . . . . . . . . . . . . . . . 179
Table 9.3 Chemical Composition Of The Prototype Detector. . . . . . . 181
Table D.1 Quasi-elastic Candidate Events. . . . . . . . . . . . . . . . . . . . 220
Table D.2 Non-quasi-elastic Candidate Events, I. . . . . . . . . . . . . . . . 221
Table D.3 Non-quasi-elastic Candidate Events, II. . . . . . . . . . . . . . . 222

x

LIST OF FIGURES

Figure 2.1

Weak Interaction Charged Current Representation Of Neutrino-nucleon Scattering. . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.2

Experimental Results On Muon Neutrino Quasi-elastic Cross
Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Theoretical And Experimental Results On Muon Neutrino
Resonance Cross Sections. . . . . . . . . . . . . . . . . . . . . .

15

Experimental Results On The Total Inclusive Muon Neutrino Charged Current Cross Sections. . . . . . . . . . . . . .

17

Brookhaven National Laboratory Muon Neutrino Inclusive
Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.6

SciBooNE Muon Neutrino Inclusive Cross Sections. . . . . .

19

Figure 2.7

Weak Interaction Neutral Current Representation Of Neutrino-nucleon Scattering. . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 2.8

MiniBooNE Neutral Current Differential Cross Section. . .

22

Figure 2.9

Neutrino Mass Hierarchy. . . . . . . . . . . . . . . . . . . . . . .

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Figure 3.1

Fermilab Accelerator Campus. . . . . . . . . . . . . . . . . . . .

34

Figure 3.2

NuMI Beam Trajectory. . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 3.3

NuMI Beam Energy Spectrum. . . . . . . . . . . . . . . . . . .

35

Figure 3.4

Location Of The NOνA And MINOS Far Detectors. . . . . .

36

Figure 3.5

Schematics Of The NuMI Complex. . . . . . . . . . . . . . . .

37

Figure 3.6

NuMI Beam Schematics. . . . . . . . . . . . . . . . . . . . . . .

37

Figure 3.7

NuMI Target Drawing. . . . . . . . . . . . . . . . . . . . . . . .

39

Figure 3.8

Beam And Electric Current Directions Through The NuMI
Horns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Figure 2.3

Figure 2.4

Figure 2.5

xi

Figure 3.9

Meson Transverse And Longitudinal Momenta. . . . . . . . .

43

Figure 3.10

Neutrino Energy Distribution Discriminated By The Mother
Particle Of The Neutrino. . . . . . . . . . . . . . . . . . . . . .

44

Neutrino Energy Distribution Discriminated By The Neutrino Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos From Charged Pion Decays. . . . . . . . . . . . . . . . .

46

Figure 4.1

Electron Neutrino Bi-probability Plot. . . . . . . . . . . . . .

48

Figure 4.2

Drawing Of The NOνA Detectors. . . . . . . . . . . . . . . . .

50

Figure 4.3

Angle Of Neutrinos With Respect To The NuMI Beam. . .

51

Figure 4.4

Meson Momentum And Neutrino Energy As A Function Of
The NuMI Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 4.5

Drawing Of The NOνA Prototype Detector. . . . . . . . . . .

53

Figure 4.6

NOνA Extrusions And Cells Sample. . . . . . . . . . . . . . .

54

Figure 4.7

Side View Of The NOνA Prototype Detector. . . . . . . . . .

54

Figure 4.8

Emission Spectra Of The Wavelength Shifting Fiber. . . . .

57

Figure 4.9

Wavelength Shifting Fiber Transverse Area. . . . . . . . . . .

58

Figure 4.10

Avalanche Photodiode. . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 4.11

Schematic Overview Of The Data Acquisition System. . . .

62

Figure 4.12

Event Display Of The NOνA Prototype Detector. . . . . . .

64

Figure 4.13

Cosmic Data Of The NOνA Prototype Detector: Light Output vs. Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Figure 4.14

Energy Calibration Of The NOνA Prototype Detector. . . .

66

Figure 4.15

Energy Conversion Factors Of The NOνA Prototype Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Light Level Over Time On The NOνA Prototype Detector.

69

Figure 3.11

Figure 3.12

Figure 4.16

xii

Figure 4.17

Sample Cosmic Ray Event For Slicer Performance. . . . . . .

72

Figure 4.18

Event Vertex Resolution. . . . . . . . . . . . . . . . . . . . . . .

75

Figure 4.19

Longest Track Endpoint Resolution. . . . . . . . . . . . . . . .

76

Figure 4.20

Sample Event Of Tracking Algorithm Corrections. Case 1. .

77

Figure 4.21

Sample Events Of Tracking Algorithm Corrections. Cases 2
To 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.22

Sample Event Of Tracking Algorithm Corrections. . . . . . .

80

Figure 4.23

Number Of Active Channels In The NOνA Prototype Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Number Of Neutrino Candidates As A Function Of The Protons On Target In The NOνA Prototype Detector. . . . . . .

83

Time Of Event Slice In The Data Of The NOνA Prototype
Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Sample Trigger Time Window With A Selected Neutrino
Candidate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Time Of Slices In The Data Of The Prototype Detector After
Event Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Figure 4.28

Sample Trigger Time Window With Cosmic Ray Events. . .

87

Figure 4.29

Sample Cosmic Ray Event. . . . . . . . . . . . . . . . . . . . . .

88

Figure 4.30

Beam To NOνA Prototype Detector Coordinate Transformation Illustration. . . . . . . . . . . . . . . . . . . . . . . . . . .

89

The Direction Of Neutrinos From The NuMI Beam In The
Prototype Detector. . . . . . . . . . . . . . . . . . . . . . . . . .

91

Longest Track Angular Distributions For Data And Monte
Carlo Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Figure 5.1

Single Particle Events. . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 5.2

Sample Events For Neutral Current Background. . . . . . . .

95

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.31

Figure 4.32

xiii

Figure 5.3

Neutrino Energy Distributions From The NOνA Prototype
Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Monte Carlo Prediction Of The Ratio Of Charged Current
To Neutral Current Neutrino Interactions. . . . . . . . . . . .

97

Figure 5.5

Fraction of Reconstructed Charged Current Events. . . . . .

97

Figure 5.6

Ratio Of Reconstructed Over Simulated Events. . . . . . . .

98

Figure 5.7

Vertex Region Of The NOνA Prototype Detector. . . . . . . 100

Figure 5.8

Containment Region Of The NOνA Prototype Detector. . . 101

Figure 5.9

Monte Carlo Simulated Longest Track Length Distributions
For Muons and Non-muons. . . . . . . . . . . . . . . . . . . . . 103

Figure 5.10

Monte Carlo Simulation Of The Longest Track Length Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 5.11

Cosmic Rays Rejection Zone By The Cut In cos θY . . . . . . 106

Figure 5.12

Cosmic Rays Rejection Zone By The Cut In cos θNuMI . . . . 108

Figure 5.13

cos θY vs. cos θNuMI . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 5.14

Cosmic Track Mean dE/dX. . . . . . . . . . . . . . . . . . . . . 109

Figure 5.15

Minimum Ionizing Particle Fraction. . . . . . . . . . . . . . . . 110

Figure 5.16

Longest Track Particle Identity. . . . . . . . . . . . . . . . . . . 111

Figure 5.17

Consequences Of The Minimum Ionizing Particle Cut. . . . 112

Figure 6.1

Muon: True Energy vs. Reconstructed Track Length. . . . . 116

Figure 6.2

Profile Plots Of Figure 6.1a. . . . . . . . . . . . . . . . . . . . . 118

Figure 6.3

Muon Energy Resolution. . . . . . . . . . . . . . . . . . . . . . . 119

Figure 6.4

Muon Energy Distributions. . . . . . . . . . . . . . . . . . . . . 120

Figure 6.5

Hadronic Energy Classification Zones. . . . . . . . . . . . . . . 122

Figure 6.6

True Hadronic Energy vs. Deposited Energy. . . . . . . . . . 123

Figure 5.4

xiv

Figure 6.7

Hadronic Energy Resolution. . . . . . . . . . . . . . . . . . . . 124

Figure 6.8

Hadronic Energy Distributions. . . . . . . . . . . . . . . . . . . 125

Figure 6.9

Quasi-elastic And Non-quasi-elastic Hadronic Energy Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 6.10

Muon And Hadronic Energy Distributions For The Quasielastic And Non-quasi-elastic Events. . . . . . . . . . . . . . . 127

Figure 6.11

Muon Energy Resolutions For Quasi-elastic And Non-quasielastic Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 6.12

Overall Hadronic Energy Resolutions. . . . . . . . . . . . . . . 129

Figure 6.13

Neutrino Energy Resolution. . . . . . . . . . . . . . . . . . . . . 130

Figure 6.14

Neutrino Energy Distributions. . . . . . . . . . . . . . . . . . . 131

Figure 6.15

Quasi-elastic And Non-quasi-elastic True Minus Reconstructed Neutrino Energies. . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 6.16

Quasi-elastic And Non-quasi-elastic Neutrino Energy Resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 6.17

Comparison Of Neutrino Energy Resolutions. . . . . . . . . . 134

Figure 6.18

Quasi-elastic And Non-quasi-elastic Neutrino Energy Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 6.19

Neutrino Types: Fractions And Energy Distributions. . . . . 135

Figure 7.1

Mean Energy Deposition Per Unit Length. . . . . . . . . . . . 139

Figure 7.2

Comparison Of Data And Monte Carlo Simulated Longest
Track Length Distributions. . . . . . . . . . . . . . . . . . . . . 140

Figure 7.3

Energy Estimation Systematic Uncertainty As A Function
Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 140

Figure 7.4

Channels Configurations Systematic Uncertainty As A Function Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . 142

Figure 7.5

Example Of The ReWeight Package Output. . . . . . . . . . 144

xv

Figure 7.6

GENIE Parameters Systematic Uncertainty As A Function
Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 146

Figure 7.7

GENIE Parameters Total Systematic Uncertainty As A Function Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . 146

Figure 7.8

Unfolding Algorithm Systematic Uncertainty As A Function
Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 147

Figure 7.9

Summary Of Systematic Uncertainties. . . . . . . . . . . . . . 149

Figure 7.10

Systematic Uncertainty On The Number Of Events As A
Function Of The Neutrino Energy. . . . . . . . . . . . . . . . . 150

Figure 7.11

δN /N As A Function Of The Neutrino Energy. . . . . . . . . 151

Figure 8.1

Angles Of The Longest Track. . . . . . . . . . . . . . . . . . . . 152

Figure 8.2

Longest Track Angular Distributions Without Cosmic Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 8.3

Reconstructed Muon Energy Distributions, No Cosmic Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 8.4

Cumulative Distribution Functions For Muons. . . . . . . . . 157

Figure 8.5

Hadronic Energy Distributions, No Cosmic Background. . . 157

Figure 8.6

Selected Candidate Events Energy Distributions. . . . . . . . 158

Figure 8.7

Efficiency And Purity Matrix For The Quasi-elastic And
Non-quasi-elastic Samples. . . . . . . . . . . . . . . . . . . . . . 160

Figure 8.8

Sample Of Simulated Quasi-elastic Neutrino Events. . . . . . 162

Figure 8.9

Sample Of Simulated Non-quasi-elastic Neutrino Events. . . 164

Figure 8.10

Sample Of Quasi-elastic Neutrino Candidates. . . . . . . . . . 166

Figure 8.11

Sample Of Non-quasi-elastic Neutrino Candidates. . . . . . . 167

Figure 8.12

Number Of Neutrino Candidates vs. Neutrino Energy, No
Cosmic Background. . . . . . . . . . . . . . . . . . . . . . . . . . 169

xvi

Figure 8.13

Number Of Neutrino Candidates vs. Neutrino Energy, No
Cosmic Background, Shape Comparison. . . . . . . . . . . . . 169

Figure 8.14

Quasi-elastic And Non-quasi-elastic Unfolded Number Of
Neutrino Candidates vs. Neutrino Energy, No Cosmic Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 8.15

Quasi-elastic And Non-quasi-elastic Unfolded Number Of
Neutrino Candidates vs. Neutrino Energy, No Cosmic Background, Shape Comparison. . . . . . . . . . . . . . . . . . . . . 171

Figure 8.16

Number Of Neutrino Candidates vs. Neutrino Energy, Comparison Of Monte Carlo Simulation And Data. . . . . . . . . 172

Figure 8.17

Unfolded Number Of Neutrino Candidates vs. Neutrino Energy, Comparison Of Monte Carlo Simulation And Data. . . 172

Figure 9.1

Charged Kaon Longitudinal And Transverse Momenta. . . . 177

Figure 9.2

Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos From Charged Pion Decays After Event Selection Criteria Applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 9.3

GENIE Cross Sections As A Function Of Neutrino Energy.

183

Figure 9.4

GENIE Uncertainty In The Muon Neutrino Cross Section.

183

Figure 9.5

Reconstruction Efficiency As A Function Of The Neutrino
Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Figure 9.6

Total Flux Of Muon Neutrinos Coming From Charged Kaon
Decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Figure 9.7

Flux Systematic Uncertainties. . . . . . . . . . . . . . . . . . . 186

Figure 9.8

Data Over Monte Carlo Simulated Flux Ratio. . . . . . . . . 187

Figure 9.9

Total Flux Of Muon Neutrinos Coming From Charged Kaon
Decays Discriminated By Interaction Type. . . . . . . . . . . 187

Figure 9.10

Quasi-elastic Over Non-quasi-elastic Flux Ratio. . . . . . . . 188

Figure 9.11

Muon Neutrino Monte Carlo Simulated Flux Prediction. . . 191

xvii

Figure 9.12

Reconstruction Efficiency As A Function Of The Neutrino
Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Figure 9.13

Inclusive Muon Neutrino Charged Current Cross Section
Per Nucleon On A Carbon Target. . . . . . . . . . . . . . . . . 192

Figure 9.14

Cross Section Systematic Uncertainties. . . . . . . . . . . . . . 193

Figure 9.15

Data Over Monte Carlo Simulated Muon Neutrino Charged
Current Cross Section Ratio. . . . . . . . . . . . . . . . . . . . 193

Figure 9.16

Measurements Of Muon Neutrino Charged Current Inclusive Scattering Cross Sections. . . . . . . . . . . . . . . . . . . . 194

Figure A.1

Higgs Scalar Potential. . . . . . . . . . . . . . . . . . . . . . . . 200

Figure B.1

Sample Event Illustrating The Tracking Algorithm. . . . . . 205

Figure B.2

Flow Chart Of The Tracking Algorithm. . . . . . . . . . . . . 206

Figure B.3

Sample Event Of The Tracking Algorithm Output. . . . . . . 208

Figure C.1

Neutrino Energy Distributions Before And After Unfolding. 217

Figure C.2

Results From Unfolding Fake Data. . . . . . . . . . . . . . . . 218

xviii

Chapter 1
Introduction
Neutrinos are among the most mysterious of all the known particles. Neutrinos are weakly
interacting particles, which makes them extremely hard to detect, and yet neutrinos participate in a wide range of phenomena from the subatomic to the cosmological scales. Neutrinos
are key to type-II supernovae core collapse since these carry out of the exploding star about
99% of the gravitational energy release [1]. Neutrinos actively participate in the cooling of
stars that passed the He-burning stage and therefore strongly determine the lifetime of the
star [2]. Their finite mass presents the first evidence of physics beyond the Standard Model,
since the latter assumes that neutrinos are massless. This leads to ideas that would extend
the number of fundamental particles providing a mechanism for neutrinos to acquire their
mass; however through a mechanism not described in the Standard Model. As massive and
extremely abundant particles, neutrinos contribute significantly to the evolution of the Universe [3]. Massive neutrinos exhibit oscillations that change their identities, and the study
of this phenomenon could provide direction in the solutions of current mysteries, e.g. the
origin of the matter-antimatter asymmetry of the observable Universe.

1

1.1

Brief History Of Neutrinos

In the 1930’s experimental results showed that electrons in beta decays:

n → p + e− + energy,

(1.1)

were not monochromatic, but rather had a discrete energy spectrum. At that time, the
origins of the beta decay were unknown. However, it was certain that the final products of the
decay were a proton and an electron. Therefore, their energies ought to be fixed. To explain
the undetectable energy, N. Bohr suggested that the concept of conservation of energy could
be discarded. W. Pauli addressed his colleagues as “liebe radioaktive Damen un Herren” [4]
during the 1930 T¨
ubingen meeting to share with them his deep concern about the continuum
energy spectrum of electrons. Pauli suggested that an undetectable particle carried away
the missing energy, a hypothesis that would never be verified, should the particle be truly
undetectable. E. Fermi named Pauli’s hypothetical particle the neutrino 1 . Over twenty
years later, F. Reins and C. L. Cowan [5] finally discovered the neutrino. The neutrinos2
emitted by a nuclear reactor interacted with protons in two tanks of water, creating neutrons
and positrons. The gamma rays created in the annihilation of the positrons were detected
in the interspersed tanks of liquid scintillator. These neutrinos were later identified as the
electron neutrinos.
Neutrinos are produced in both pion and muon decays. However, it was not clear if the
neutrinos in these two decays were all electron neutrinos, or some belonged to a different
type of neutrinos. In 1962, L. Lederman and J. Steinberger discovered that there existed two
1 Italian for little neutral one, to distinguish it from the much heavier neutron.
2 The reaction is: ν
¯e + p → n + e+ , with antineutrinos.

2

types of neutrinos [6], the electron neutrino (νe ) and the muon neutrino (νµ ). The 1988 Nobel
Prize in physics was awarded to them for their discovery [7]. At this time, a model of hadrons
and leptons included: the electron, the muon, their corresponding neutrinos, and the up,
down, and strange quarks. The absence of kaon decays to a down quark via a strangeness
changing neutral current, along with symmetry arguments, suggested the existence of a
fourth quark [8], charm. This hypothesis was verified by the discovery of the bound state of
charm-anticharm quarks, the ψ/J meson [9, 10]. By 1977 the existence of the third family
was confirmed with the discovery of the bottom quark at Fermilab, by L. Lederman’s team
[11], and the discovery of the tau lepton, by the SLAC-LBL team [12]. Due to the rarity of
tau neutrinos (ντ ) in neutrino beams, it took over twenty years to experimentally confirm
its existence, by the DONUT collaboration, in 2000 [13].

1.2

Motivations For The NOνA Experiment

Decades of speculation about massive neutrinos3 ended when Super-Kamiokande [16] and
the Sudbury Neutrino Observatory [17] first reported strong evidence of atmospheric and
solar neutrino oscillations, respectively. Neutrino oscillations are only possible if neutrinos
are massive. The Super-Kamiokande result presented evidence that νµ oscillate primarily
into ντ , however they did not achieve a conclusive result on the νµ to νe oscillation. The lack
of evidence in the latter oscillation channel motivated, at that time, the neutrino community
to address the role of the νe in oscillations phenomena. Three experiments: Double Chooz
[18], Daya Bay [19], and RENO [20], presented positive results in 2012 on the νµ → νe
3 B. Pontecorvo first suggested, in 1957, the concept of neutrino oscillations [14].

R.
Davis found the first experimental indication of neutrino oscillations with his Homestake
experiment, where a third of the expected flux of neutrinos from the Sun was measured [15].
This is known as the solar neutrino problem.
3

oscillation channel showing that the oscillation indeed occurs. This last result presents an
unrivaled opportunity to study the occurrence of CP violation in the neutrino sector [21].
The NuMI Off-Axis νe Appearance (NOνA) experiment, will study the neutrino oscillations phenomenon. NOνA will investigate the oscillations of νµ to νe , and muon antineutrinos (¯
νµ ) to electron antineutrinos (¯
νe ), using a neutrino beam produced at Fermilab and
detected at the NOνA far detector located near Ash River, Minnesota. NOνA will determine
any differences that occur in the oscillations of neutrinos and antineutrinos when the beam
travels through matter. The neutrino beam at Fermilab is designed to run either in primarily
neutrino or primarily antineutrino modes. Studying the differences between the interactions
of neutrinos and antineutrinos with matter will help to address the following topics:
• The mass ordering of the three neutrino mass eigenstates.
• The value of a CP violating phase in the neutrino sector.
• Smaller uncertainties in the current values of the neutrino oscillation parameters.
The quarks and charged leptons sectors exhibit normal mass orderings, i.e. the member
of the third family is much more massive than the members of the other two families. In
the neutrino sector, however, the possibility of one light and two (quasi-degenerate) heavier
neutrinos could occur4 . Even though the absolute value of the mass gap between the lightest
and heaviest neutrinos is known, it is yet to be determined which eigenstate is the lightest
and which is the heaviest. This results in the uncertainty on the sign of the mass difference.
This sign has a profound influence in the νµ → νe oscillation enhancing (+) or suppressing
(-) it. NOνA is going to make a precise measurement of this oscillation channel, which will
determine the sign of the mass difference, and thus the neutrino mass ordering.
4 This is know as inverted mass ordering or hierarchy. See figure 2.9.

4

By taking neutrino and antineutrino measurements, NOνA will be able to establish
whether there is any significant difference in the two oscillation patterns beyond that introduced by propagation of neutrinos through matter, indicating that CP symmetry is violated
in neutrino oscillations. Therefore, neutrinos become prime candidates to explain the origin
of the observed matter-antimatter asymmetry at the time of the Big Bang.
NOνA is a two-detector experiment with one located underground at Fermilab (near
detector), and the other, with a mass of 14 kton, on the surface in northern Minnesota
(far detector). The two detectors are constructed from the same materials differing only in
mass (300 ton for the near detector), and use the same readout electronics differing only in
sampling rate (higher for the near detector). The similarity between the detectors allows
the initial event rate of νe and νµ , measured by the near detector, to yield a nearly biasfree normalization of the event rate at the far detector. The two detectors are 14 mrad
off the neutrino beam axis, which provides a narrow neutrino energy spectrum5 peaked
around 2 GeV. At the far detector, a 2 GeV neutrino beam combined with its baseline
length of 810 km produces a maximum in the νe appearance probability, and a minimum in
the νµ survival probability, with the flux dominated by oscillations to ντ . The exclusively
neutral current interactions of ντ at this energy, where the scattered ντ carries away a large
fraction of the energy, leaves little hadronic energy to produce backgrounds for the 2 GeV
νe interactions.
The two NOνA detectors are matrices of PVC tubes filled with liquid scintillator, resulting in detectors with approximately 70% of their weight being active detecting material.
The ionization left by charged particles created in the neutrino interactions generates light
in the scintillator, which is collected by a special kind of fiber optics that transport the
5 See figure 3.3.

5

wavelength-shifted light to avalanche photodiodes. The photodiode electrical pulses are amplified, digitized, and processed by a data acquisition system. The processed data are stored
on disc for later analysis.
The NOνA collaboration built a prototype detector at Fermilab exposed to the neutrino
beam to develop calibration, simulation, and reconstruction tools, using real data. A number
of construction issues were identified and solved. These influenced the final design of the
near and far detectors. Regardless of the location of the prototype on the surface, the
neutrino interaction signals can be separated from the enormous amounts of cosmic rays
that continuously illuminate the prototype detector. Cosmic rays are a background to the
νµ charged current neutrino signal6 . The neutrino beam is pulsed at regular intervals, and the
analysis of the data finds a peak at the time of the interaction candidates from the neutrino
beam. With the ability to distinguish neutrino events from cosmic rays, the prototype is
used to make a proof of principle: the collaboration searched for νµ interactions and these
were indeed found [22].
The νµ charged current quasi-elastic cross section7 was measured [22] in the NOνA
prototype detector using a selection of events with one and only one reconstructed track.
The neutrino energy was determined solely from the range and angle of the reconstructed
muon track. Using the calorimetric capabilities of the prototype detector’s design, and
the neutrino flux embedded in the NOνA Monte Carlo simulation, this thesis presents an
estimate of the inclusive νµ charged current cross section.
Charged kaons produced at the NuMI target8 are the source of the νe that contribute
6 See section 2.1.1 for a definition of the charged current interaction.
7 See section 2.1.1.1 for a definition of the charged current quasi-elastic interaction.
8 See section 3.1 for a detailed description of the NuMI beam, including the charged kaon

production.

6

to the total background of the νµ beam that reaches the far detector. Knowing the initial
content of νe in the neutrino beam will minimize the systematic uncertainty in the number
of νe expected from νµ oscillations. The charged kaon decay peak in the neutrino energy
is observed in the inclusive charge current interaction data to be presented. The off-axis
angle, and the two-body decay of the mesons, lead to a narrow peak in the energy of the
neutrinos at 2 GeV. However, the production rate of charged kaons in the NuMI target has
to be determined from Monte Carlo simulations as there are no direct measurements. At the
location of the prototype detector, the flux of 2 GeV neutrinos is nearly all from the decay
of charged kaons. The last two statements lead to a rather large normalization uncertainty
in the measured neutrino cross section. From the few other measurements of neutrino cross
sections in this energy regime, the level of uncertainty in the flux can be estimated. Therefore,
both interpretations of the data, as an inclusive cross section estimate, or as a flux estimate,
are presented.

1.3

Outline

In order to set the theoretical framework behind the neutrino interactions, chapter 2 presents
the basics of the weak interactions, emphasizing those involving neutrinos (section 2.1). In
addition, the possible origins of neutrino masses (section 2.2), and a review of the formalism
of neutrino oscillations (section 2.3) are presented.
Chapter 3 provides an overview of the Fermilab neutrino beam that supplies the neutrinos
to the NOνA experiment. Sections 3.1 and 3.2 summarize the components of the neutrino
beam, and the various steps involved in the neutrino production. The beam composition
and energy spectrum are presented in section 3.3.

7

The concepts behind the NOνA experiment are presented in chapter 4. Section 4.1 is an
overview of the experiment. The prototype detector is introduced in section 4.2. The various
software tools used for data processing and analysis are presented in section 4.3. The data
quality checks are reviewed in section 4.4. A brief study of cosmic rays, and its implications
in the experiment, is provided in section 4.5.
The νµ event selection procedure is described in chapter 5. The differences between
events from charged and neutral current interactions are discussed in section 5.1, the event
containment criteria are presented in section 5.2, and the charged current event selection is
explained in section 5.3.
The neutrino energy reconstruction process is discussed in chapter 6. There are two
steps in this process: muon energy reconstruction (section 6.1), and the hadronic energy
reconstruction (section 6.2). A classification between quasi-elastic and non-quasi-elastic
events (section 6.3) is necessary to better reconstruct the neutrino energy (section 6.4).
The various systematic uncertainties associated with the results are detailed in chapter
7. The analysis of the data is presented in chapter 8. A comparison between simulation
and data is provided in section 8.1, and the results are presented in section 8.2. Chapter 9
discusses the implications of the results on: the neutrino flux (section 9.1), and the inclusive
νµ charged current cross section (section 9.2). Final remarks are presented in section 9.3.

8

Chapter 2
Theoretical Framework for Neutrinos
The Standard Model provides the theoretical foundations for quantum chromodynamics
(QCD), and it also unifies electromagnetism and the weak interaction, based on the works
done by S. L. Glashow [23], S. Weinberg [24], and A. Salam [25]. Neutrinos have an important
role in the weak interaction, which is briefly reviewed in this chapter. Massive neutrinos are
not part of the Standard Model of particle physics. There are a few ideas to explain the
origins of neutrino masses, the most popular within the neutrino community is the See Saw
Mechanism, also reviewed in this chapter. Neutrino oscillations are one of the most relevant
consequences of massive neutrinos, and therefore are reviewed at the end of the chapter.

2.1

Neutrinos In Weak Interactions

The electroweak interaction was first developed in a phenomenological fashion, and later
complemented when additional measurements became available. In 1934 Fermi [26] introduced an effective Hamiltonian1 :

G
†
Heff = − √F Jµ (x)Jµ (x),
2
1 Fermi coupling constant: G /(¯
3
−5 GeV−2 .
F hc) = 1.166 × 10

9

(2.1)

to model the low energy charged current weak interaction of hadrons and leptons known at
the time as:
¯ ν γµ (1 − γ5 )Ψe + Ψ
¯ p γµ (1 − γ5 )Ψn .
Jµ (x) = Ψ
e

(2.2)

The model resembled that of electromagnetism given its vector part, however it also included
an axial part introduced after the discovery of parity violation [27].
At low neutrino energies, charged current interactions are characterized by quasi-elastic
scattering with a nucleon and lepton in the final state. Nucleon resonance production becomes important as their thresholds are crossed. At high energies, the cross section grows
linearly with energy and becomes dominated by deep inelastic scattering. The formalism
describing these interactions is described in the next section.

2.1.1

Charged Current Interactions

The interaction of the W boson with fermions has the following Lagrangian:

g
µ
µ†
LW = − √ JW Wµ− + JW Wµ+ ,
2 2

(2.3)

where the charged current (CC) is:

µ†

JW

3

=
k=1

ν¯k γ µ 1 − γ 5 lk + u¯k γ µ 1 − γ 5 dk ,

 
 
d
e
 
 
 
 
5
µ
5
µ
¯
u c¯ t)γ 1 − γ Vq s .
= (¯
νe ν¯µ ν¯τ )γ 1 − γ Ul µ + (¯
 
 
 
 
b
τ

10

(2.4)

Vq is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [28, 29], and Ul is the PontecorvoMaki-Nakagawa-Sakata (PMNS) leptonic mixing matrix [30, 31]. The vertices of interest,
from equations (2.3, 2.4), are: d¯j ui W − , u¯i dj W + , ¯lj νi W − , and ν¯i lj W + , which contribute to
the neutrino scattering with nucleons. The first two vertices correspond to the lower vertex
in figure 2.1, and the last two correspond to the upper vertex in figure 2.1. The interaction:
νµ + d → µ− + u, which is the underlying process of the νµ + n → µ− + p scattering, is an
example of the interactions that are generated with the four vertices mentioned above.

Figure 2.1 Weak Interaction Charged Current Representation Of Neutrinonucleon Scattering. In the drawing the index l represents all charged leptons, the letter
d represents all the Q = −e/3 quarks, and the letter u represents all the Q = 2e/3 quarks.

2.1.1.1

Quasi-elastic Scattering

For energies smaller than the W mass, the amplitude (M):

G
M = √F ¯lγµ (1 − γ5 )ν p|Jµ+ |n ,
2
11

(2.5)

of the neutrino interactions2 :

νl (k) + n(p) → l− (k′ ) + p(p′ ),
ν¯l (k) + p(p) → l+ (k′ ) + n(p′ ),

(2.6)

is the product of the hadronic and leptonic currents [27]:
The general hadronic current gets contributions from the motion of the quarks within
the hadrons, hence the introduction of form factors is useful to describe the physics of the
processes3 :
qµ
qµ γ5
pµ
qν γ5
+ hV
+ gA γµ γ5 + ıfA σµν
+ hA
p|Jµ+ |n = n
¯ gV γ µ + f V
2MN
2MN
2MN
2MN

p. (2.7)

Here the gA , fA , and hA are axial form factors, and the gV , fV , and hV are vector form
factors. The product hA qµ γ5 is proportional to the lepton’s mass, therefore is a negligible
term4 , and hV and fA need to be zero due to charge conjugation and time-reversal arguments
[27]. The simplified form of equation (2.7) is:

p|Jµ+ |n = n
¯ gV γµ + fV

pµ
+ gA γµ γ5 p.
2MN

(2.8)

The kinematics for the processes (see equation (2.6)) in the laboratory frame, for which the
2 k and p are momenta.
3M

N is the mass of the target nucleon.
4 True for light leptons.

12

nucleon is at rest, are given by:

p · k = p′ · k′ = MN Eν ,
p · k′ = p′ · k = MN Eν +

q2
,
2

m2l
q2
=
− ,
2
2
2
2
M
W −q ,
p · p′ =
2
2
k · k′

Q2 = 4Eν E ′ sin2

(2.9)

θ
2

,

where Eν is the energy of the incoming νl , E’ is the energy of the outgoing l, Q2 is the
momentum transfer, and θ is the angle between the momenta of the incoming neutrino and
the outgoing charged lepton. Equations (2.8, 2.9) lead to the differential cross section:
G2F MN
(gV − gA )2 + (gV + gA )2
4π

dσQE
=
dE ′
+

G2F MN 2
fV η + 2fV gV
8π

1+

E′
Eν

E′
Eν
2

−

2

2 ) MN
+ (gV2 − gA
Eν2

Q2
η ,
Eν2

(2.10)

with:
η =1+

Q2
.
2
4MN

(2.11)

Monte Carlo simulated quasi-elastic cross sections based on this function provide a reasonable representation of the data from experiments (MiniBooNE [32], ANL [33], BEBC [34],
BNL [35, 36], FNAL [37], Gargamelle [38, 39], LSND [40], NOMAD [41], Serpukhov [42],
and SKAT [43]), as seen in figure 2.2. The two solid curves are the Monte Carlo simulation
predictions of the cross sections based on equation (2.10). The quasi-elastic cross sections
become constant for energies above 10 GeV.
13

Figure 2.2 Muon Neutrino Quasi-elastic Cross Sections. Measurements of the νµ
(black) and ν¯µ (red) quasi-elastic scattering cross sections (per nucleon) as a function of
neutrino energy [44].

2.1.1.2

Resonant Pion Production

As neutrino energy gets larger, their scattering with nucleons can excite the latter to higher
energy levels; these interactions produce baryon resonances (N ∗ ):

νµ N → µ − + N ∗ ,

(2.12)

which rapidly decays (often) into a nucleon and a charged pion:

N ∗ → π + N ′,

(2.13)

where N and N ′ could be either a neutron or a proton in a nucleus. Higher order multiplicity
decay modes are also possible. These resonances were studied by D. Rein and L. M. Sehgal

14

(a)

(b)

Figure 2.3 Muon Neutrino Resonance Cross Sections. (a) Resonance production cross
section for differing nH [47]. (b) Existing measurements of the cross section for the process
in equation (2.15) as a function of energy [48].
[45], based on the model of baryon resonances of R. Feynman, M. Kislinger, and F. Ravndal
[46]. Their work describes the resonances in terms of excited states of three bounded quarks
(a, b, and c) with a relativistic harmonic oscillator potential:

Hres = 3(p2a + p2b + p2c ) +

1 2
Ω [(ua − ub )2 + (ub − uc )2 + (uc − ua )2 ],
36

(2.14)

with Ω2 = m2 ω02 , the u’s are conjugate positions of the quarks, and the p’s are fourmomentum vectors of the quarks5 . The eigenvalues of the harmonic oscillator presented
in [46] are a succession of integers, nH , times Ω. The first state, nH = 0, is the resonance ∆(1232), and its cross section saturates at energies around 2 GeV. In the cases where
1 ≤ nH ≤ 2, the theoretical cross sections for: nH = 1 saturates at energies higher than
20 GeV, and nH = 2 rises linearly with energy, as seen in figure 2.3a. Measurements of the
cross section for the process:
νµ + p → µ − + p + π + ,
5 u’s and p’s defined in [46].

15

(2.15)

have been made in bubble chamber experiments, and the results are shown in figure 2.3b.

2.1.1.3

Deep Inelastic Scattering

When hadron structures are examined on a very short distance scale, it is found that these
are a set of non-interacting quarks. Short distance, in this case, translates to high energy
electroweak interactions in deep inelastic (lepton-hadron) scattering of the form:

l + N → l′ + X,

(2.16)

where X includes single nucleon or nucleon resonance production. The kinematics of these
processes are described in terms of the following variables (recall equations (2.9)):
Q2
,
x˜ =
2MN Ey

y =

Ehad
,
Eν

(2.17)

where Ehad is the energy of the hadronic system. Using these variables, the deep inelastic
scattering (DIS) double differential cross section is:
d2 σ
=
d˜
xdy

G2F MN Eν
π

2
1 + Q2
MW

y2
2˜
x
(g + fV ) − fV
2
2 V

1−y−

MN x˜y
2Eν

.

(2.18)

Equation (2.18) is the core of the physics involved in DIS. However, additional effects must be
included in any realistic description. The inclusion of non-perturbative higher twist effects
[49], heavy quark production [50], target mass effects [51], nuclear effects and radiative
corrections [52], higher order QCD processes [53], and lepton masses [54], further modify the
scattering cross section in equation (2.18). In general, these contributions are known with

16

reasonable uncertainties [48].

(a)

(b)

Figure 2.4 Total Inclusive Muon Neutrino Charged Current Cross Sections. Total
(inclusive) (a) νµ and (b) ν¯µ charged current cross sections per nucleon over neutrino energy
as a function of neutrino energy. The total cross sections (solid lines) are the sum of: QE
(dashed), resonance (dot-dashed), and deep inelastic (dotted) [48].

A collection of data acquired over the last 50 years, using different experimental techniques [48], is presented in figure 2.4. These figures show the measured and predicted QE
and inclusive CC cross sections. For energies higher than 3 GeV the inclusive cross section is
dominated by the DIS processes, as can be seen in the region where the ratio σDIS /σQE > 1.
To isolate the DIS events, neutrino experiments usually apply kinematic cuts to remove QE
scattering and resonance contributions from the data.

2.1.1.4

Inclusive Cross Section Measurements

There are only two experiments that have measured the νµ CC inclusive cross section at
neutrino energies around 2 GeV: Brookhaven’s 7 ft bubble chamber [55] (1982) and SciBoone
(2011). The experiment at Brookhaven used a 7 ft deuterium bubble chamber exposed to the
Alternating Gradient Synchrotron wide-band neutrino beam. The value of the inclusive cross
section divided by the neutrino energy falls significantly from 0.4 GeV to 1.5 GeV, and then
17

becomes constant, as seen in figure 2.5. Above 1.6 GeV, the measured inclusive CC cross
section per nucleon divided by the neutrino energy is σ/E = (0.80±0.03)×10−38 × cm2 /GeV
[55], the average of the values shown in figure 2.5.

Figure 2.5 Brookhaven National Laboratory Muon Neutrino Inclusive Cross Sections. νµ inclusive cross section over neutrino energy as a function of the neutrino energy
for: ν p and ν n interactions. The solid lines are the best fits for neutrino energies higher
than 1.6 GeV [55].

At neutrino energies around 2 GeV, nuclear effects are important. When the W boson’s
four-momentum is low enough such that its de Broglie wavelength is comparable to the
size of the target nucleus, the scattering involves the entire nucleus. The cross sections on
deuterium targets are not directly applicable to the heavier nuclear target materials used in
current accelerator-based neutrino experiments. In neutrino scattering on a nucleon (in a
nuclear target), if there is a small momentum transfer to the final state nucleon, this can
only exist in excited states allowed by Pauli’s Exclusion Principle. In the Fermi gas model,
the excited nucleons undergo transitions across the Fermi surface, with momentum around

18

Figure 2.6 SciBooNE Muon Neutrino Inclusive Cross Sections. νµ inclusive cross
section per nucleon on a polystyrene target (C8 H8 ) [56]. The NEUT (dashed red line) [57]
and NUANCE (black line) [58] predictions are shown for reference.

250 MeV/c, from low energy states, which are all occupied, to high energy states. Thus, the
neutrino-nucleon interaction can only occur if there is an available higher energy state for
the nucleon to occupy. This is known as Pauli-Blocking [59]. When the nucleon involved
in the scattering interacts with other nucleons, the kinematics and multiplicity of the final
state are diverse. SciBoone measured the inclusive νµ CC interactions, using the Fermilab
Booster neutrino beam, on a polystyrene target. Their results are shown in figure 2.6. The
rise of the cross section with energy agrees somewhat better with the predictions of the
NEUT simulation. The smaller error bars show the uncertainties of the rate normalization
factors and the larger error bars represent the total error including the flux uncertainties.
The two results presented in this section are consistent with each other, and are useful
to tune various neutrino interaction models in the 2 GeV region.

19

2.1.2

Neutral Current Interactions

Though hints for the existence of neutrino scattering without final state leptons had been
around for a number of years, compelling experimental evidence for weak neutral current
interactions [60] became available at about the same time as the electroweak unification
theory was proposed, and it depended crucially on the existence of weak neutral currents.
The fermionic Lagrangian6 describes neutral and charged currents.

There is a La-

grangian, equivalent to that in equation (2.3), that describes the interaction of the Z boson
with fermions, given by [27]:
3

LZ =

GF
√
ν¯k γ µ 1 − γ 5 ν k u¯k γ µ ukL 1 − γ 5 + ukR (1 + γ 5 ) uk
2 k=1
3

G
+ √F
ν¯k γ µ 1 − γ 5 ν k d¯k γ µ dkL 1 − γ 5 + dkR (1 + γ 5 ) dk . (2.19)
2 k=1
The vertices of interest that come out from equation (2.19) are: u¯j ui Z, d¯j di Z, and ν¯j νi Z,
which contribute to the neutrino scattering with nucleons. The first two vertices correspond
to the lower vertex in figure 2.7, and the last vertex corresponds to the upper vertex in figure
2.7. The interaction νµ + d → νµ + d is an example of the scattering that is generated
with the three vertices mentioned above.
Equations (2.9, 2.19) lead to the elastic neutral current differential cross section [48]:
2
G2F MN
4MN Eν − Q2
dσ
4MN Eν − Q2
A
±
=
B
+
2
4
dQ2
8πEν2
MN
MN
6 See equation (A.12).

20

2

C ,

(2.20)

Figure 2.7 Weak Interaction Neutral Current Representation Of Neutrino-nucleon
Scattering. In the drawing the index l represents all charged leptons, the letter q represents
all quarks.

with:

A =

2
2
Q2
2 η − (g + f )2 η + f 2 Q η + (g − f )f Q
g
,
−
V
V
V
V V
V
A +
2
2 −
2
MN
4MN
MN

Q2
g f ,
2 A V
MN
1 2
g + (gV + fV )2 + fV2 (η− − 1) ,
C =
4 A
Q2
η± = 1 ±
.
2
4MN
B =

(2.21)

The MiniBooNE collaboration has published a measurement of the elastic neutral current
differential cross section for the process:

ν + N → ν + N,

21

(2.22)

on a CH2 target [61], as seen in figure 2.8.

Figure 2.8 MiniBooNE Flux-averaged Neutral Current Differential Cross Section.
The blue line is the predicted spectrum of elastic neutral current background which has been
subtracted out from the total differential cross section [61].

C. H. Llewellyn-Smith [62] showed that the semileptonic neutral and charged currents7
cross sections are related8 through sin θW by:

Rν =

σNC
=
σCC

5
1
− sin2 θW + sin4 θW (1 + r),
2
9

(2.23)

where r is the ratio of the antineutrino charge current cross section to the neutrino one. The
CHARM collaboration [63] measured this ratio for energies larger than 2 GeV to be:

Rν = 0.317 ± 0.006.
7 Charged current QE interactions.
8 See equation (A.22).

22

(2.24)

2.2

Neutrino Masses

The formalism9 by which particles acquire their mass in the Standard Model requires both
left-handed10 and right-handed11 states for a given particle to have a mass term in the
Lagrangian. Neutrinos are a special case since there is no evidence that supports the existence
of right-handed neutrinos. Yet, there is no argument or symmetry within the Standard Model
that explicitly forbids the existence of such neutrinos. In the case of the photon12 , the gauge
symmetry requires it to be massless.
In view of this, there are two possible mass terms: the Dirac term13 :

mD (ψ¯R ψL + h.c.),

(2.25)

c ψ + h.c.),
mM (ψ¯L
L

(2.26)

and the Majorana term:

c ≡ (1+γ )ψ c /2 is right-handed.
where ψ c = Cγ 0 ψ ∗ is the charge conjugated field of ψ, and ψL
5

The Majorana term in equation (2.26) violates lepton number conservation by two units14
and makes neutrinos indistinguishable from antineutrinos [64]. All the massive fundamental
particles in the Standard Model have some type of charge (even though gluons are massless,
these are linear combinations of color states), which changes sign under charge conjugation
9 See appendix A for a brief summary of the formalism.
10 See equation (A.2).
11 See equation (A.3).
12 See equation (A.19).
13 See equation (A.20).
14 In the Dirac mass term, the matter field has lepton number: +1, and the antimatter

field has lepton number: -1. In the Majorana mass term, both fields have lepton number:
+1.

23

providing a clear distinction between particles and antiparticles. A Majorana mass term is
therefore only possible for neutral particles, exhibiting no charges of any kind; a feature that
only neutrinos exhibit.

2.2.1

Seesaw Mechanism

The seesaw mechanism [65] is the simplest renormalizable model that introduces a mass term
for neutrinos in the Lagrangian:

m
c ψ + h.c..
LM = mD ψ¯R ψL + M ψ¯R
R
2


1 ¯c
 0 mD  ψL
(ψL ψR ) 
=

c + h.c..
2
ψR
mD mM

(2.27)

The Lagrangian in equation (2.27) is simplified by diagonalizing its mass matrix. The diagonalization procedure defines a new two component field νM , written as:

c ≡
νM ≡ ψL + ψL

νsm
,
νnsm

(2.28)

such that:


0  νsm
1
msm
LM = (¯
.
νsm ν¯nsm ) 

2
νnsm
0
mnsm

(2.29)

The mD scale is in the order of a typical Standard Model fermion, and mM ≫ mD . The
i are [66]:
masses of the νM

m2D
∼
,
msm =
mM

mnsm ∼
= mM .

24

(2.30)

The νsm states are associated with the neutrinos of the Standard Model15 , and the νnsm
states represent very massive neutrinos yet to be observed. Equation (2.30) suggests that
the msm are very small, and the model suggests that the mnsm are about 1015 GeV/c2
[66]. The direct laboratory limits on the msm establish an upper limit in the range: (0.30.9) eV /c2 , depending on the nuclear model considered [67]. The most relevant feature of the
seesaw mechanism is that it explains the lightness of the known neutrinos, the νsm states,
by setting a very high mass to the yet to-be-observed heavy neutrinos, the νnsm states.
The out-of-thermal-equilibrium decays of these heavy neutrinos are key to the attempts for
explaining the baryon-antibaryon asymmetry of the Universe in terms of leptogenesis [66].
The leptonic part of the Lagrangian can be written in the mass basis of the charged
leptons and of the singlet fermions as follows16 [68]:

c
¯ ∗ νnsm − 1 mnsm ν¯nsm νnsm
¯ c∗ lR − λ∗ Lφ
Lss = h∗ Lφ
+ h.c..
2

(2.31)

with:


0 −1 φ−
¯ c∗ = (¯
Lφ
νL ¯lL ) 
 ∗0 ,
φ
1 0

(2.32)

and where h and λ are matrices of Yukawa couplings. From the second Yukawa term in
1 ) can decay into:
equation (2.31), the lightest right-handed heavy neutrino (νnsm

1
νnsm
→ φlL .

(2.33)

1
In the early Universe, when the temperature (TU ) was TU ∼ mM , the νnsm
population was
15 There are three neutrino eigenstates associated with each of the two states in equation

(2.28).
16 See equations (A.1, A.2, and A.3).
25

1
stable. Once TU < mM the population of νnsm
is no longer stable, since they can not be
1
produced any more, and the νnsm
all decayed. If there is an asymmetry17 (A) in the decay of
1
the νnsm
that favors the production of charged leptons over charged antileptons, there would

be an excess of charged leptons in the early Universe. After charged leptons annihilation,
such excess would be the origin of the matter-dominated Universe seen nowadays. A is
defined in terms of the decay rates as follows [68]:

Akk ≡

1
k ) − Γ(ν 1
¯¯k
Γ(νnsm
→ φlL
nsm → φlL )
,
1
1
Γ(νnsm
→ φlL ) + Γ(νnsm
→ φ¯¯lL )

where k denotes the various lepton flavors.

By definition: |Akk | ≤ 1.

(2.34)

Usually, it is

much smaller than 1. To account for the observed baryon asymmetry, it is required that:
|Akk | > 10−7 [68].

2.3

Neutrino Oscillations

The charged current, given by equation (2.4), introduced the concept of quark and lepton
mixing. The leptonic part of the Lagrangian in equation (2.3) is:

g
LlW = Wµ+ ν¯e γ µ e + ν¯µ γ µ µ + ν¯τ γ µ τ + h.c..
2

(2.35)

Taking a basis in which the charged leptons’ mass term is diagonal, the neutrino flavor
eigenstates are a linear combination of the neutrino mass eigenstates:

α = T ν ′i ,
νR
αi R

α = U ν ′i ,
νL
αi L
17 If A > 0.

26

(2.36)

which leads to the mass term:

†

′i T m U ν ′i + h.c.,
LM = ν¯L
iα αβ βj L

(2.37)

T † mU = mdiag ,

(2.38)

with:

where U is the PMNS matrix [69] from equation (2.4).
Since neutrinos are only observed by their role in the weak interaction, all observed neutrinos are represented in the flavor basis; however, only mass eigenstates propagate through
space and time. If the resolution of real experiments allowed the direct measurement of
neutrino masses in individual processes, the oscillations would not be of scientific interest at
all, i.e. once a neutrino is produced in a mass eigenstate it remains in that mass eigenstate
forever. The current understanding of how neutrinos interact is through the weak interaction
and its flavor eigenstates, thus neutrino oscillations are unavoidable, under these conditions.
The weak interaction produces neutrinos in a given flavor eigenstate, and these are detected
by means of a weak interaction process which involves a given flavor eigenstate. The question
is: would those two states be the same?
In vacuum, a neutrino state of the generation α, after a time interval t, is given by [64]:

|ν α t =

i

Uαi e−ıEi t |ν ′i ,

(2.39)

and the transition amplitude to the state ν β is:

ν β |ν α t =

†

Uαi Uiβ e−ıEi t .
27

(2.40)

The lightness of neutrinos yields |p| ≫ mi , which means: Ei =

p2 + m2i ≃ p + m2i /2E.

With this approximation, and using equation (2.36), equation (2.39) becomes:

−ım2
1 t/2E
e



α
−ıpt
−ım22 t/2E
|ν t ≃ e
U
e




2

e−ım3 t/2E


m2
1t
1
−
ı

2E + · · ·


m22 t
= e−ıpt U 
1
−
ı
2E + · · ·





 † β
 U |ν ,



2

m3 t
1 − ı 2E
+ ···





 † β
 U |ν . (2.41)



Using equation (2.38) as:


2
0 
m1




†
†
2
2
U m mU = mdiag = 
,
m
2




0
m23

(2.42)

equation (2.41) becomes:
†
−ı m2Em t

|ν α t ≃ e−ıpt [e

]αβ |ν β .

(2.43)

Since NOνA will study the oscillation of νµ to νe , it is interesting to see the mixing between
two generations18 , which is a valid approximation in this case. The relevant mixing matrix
(U ) is [64]:




 cos θ sin θ 
U =
.
− sin θ cos θ

18 For the three generation mixing see e.g. [66].

28

(2.44)

Using equation (2.44), m† m can be written as:


m21 + m22 ∆m2 − cos 2θ sin 2θ 
m† m = U m2diag U † =
+

,
2
2
sin θ
cos 2θ

(2.45)

where ∆m2 = m22 − m21 . Equation (2.45) allows to rewrite equation (2.43) as:


2
2
2
∆m
∆m
∆m
−ı sin 4E t sin 2θ
 β
cos 4E t − ı sin 4E t cos 2θ
|ν α (t) = 
 |ν .
2
2
2
∆m
∆m
∆m
cos 4E t + ı sin 4E t cos 2θ
−ı sin 4E t sin 2θ

(2.46)

The transition from |ν µ to |ν e (t) is then given by:
ν µ |ν e (t)
=

=



∆m2 t − ı sin ∆m2 t cos 2θ
cos
4E
4E

(0 1) 
2
−ı sin ∆m
4E t sin 2θ
∆m2

−ı sin

4E

t sin 2θ.

The physical meaning of the transition:

 

2
−ı sin ∆m
4E t sin 2θ
 0
 ,
2
2
∆m
∆m
cos 4E t + ı sin 4E t cos 2θ
1

(2.47)

ν µ |ν e (t), is understood through the transition

probability (Pνµ →νe ) expressed as:
∆m2
Pνµ →νe = | ν µ |ν e (t)|2 = sin2
t sin2 2θ.
4E

(2.48)

Since neutrinos travel at nearly the speed of light, to a good approximation: t ≈ L/c, where
L is the distance covered by the neutrino between the creation and detection points. With
this small change, oscillation experiments might place their detectors at distances from the

29

source that fulfill19 :
π≤

∆m2 L
.
2E

(2.49)

∆m2 L
.
2E

(2.50)

The transition probability is maximal when:

π=

The ratio L/E is of paramount importance in neutrino oscillation experiments since it determines much of the logistics of their experimental setup. All neutrino oscillation experiments
that use a neutrino beam, send it through matter. Interactions with matter introduce changes
in the formalism described in this section since νe interacts with electrons via charged and
neutral current interactions, while νµ and ντ only interact with electrons via the neutral
current interaction. As a consequence, there is a coherent effect in the transitions. See [64]
for a formalism that describes neutrino oscillations in matter.
The formalism presented above is a simplification of the three neutrino oscillations. Nevertheless, this simplification is sufficiently good to allow the full problem to be represented
as a collection of two neutrino transitions for which U is:
 ϕ




ı 21
−ıδ
0 s13 e   c12 s12 0 e
0
0   c13
0
0
1





ϕ





ı 22
U = 0 c23 s23   0




,
−s
c
0
1
0
0
e
0
12
12










ıδ
0
0 1
−s13 e
0
c13
0 −s23 c23
0
0
1


ϕ
ϕ
ı 22
ı 21
s
c
e
s13 e−ıδ 
c
c
e
12 13
12 13



ϕ2
ϕ1


ı
ı
ıδ
ıδ
= (−s12 c23 − c12 s23 s13 e )e 2
 . (2.51)
2
(c
c
−
s
s
s
e
)e
s
c
12
23
12
23
13
23
13




ϕ2
ϕ1
(−c12 s23 − s12 c23 s13 eıδ )eı 2
c23 c13
(s12 s23 − c12 c23 s13 eıδ )eı 2

19 In natural units: c = 1 = h
¯.

30

Here: cij ≡ cos θij , and sij ≡ sin θij . θ12 , θ23 , and θ13 , are the mixing angles. The phase
δ is the leptonic analogue of the single phase that appears in the quark mixing matrix, and
ϕ1 , and ϕ2 are Majorana phases that are relevant if neutrinos are their own antiparticles.
The importance of the phases is evident when the Lagrangian in equation (2.35) is CP
transformed:
g
g
(CP )LlW (CP )−1 = Wµ− ¯ll γ µ U ∗ ν + Wµ+ ν¯l γ µ U T l.
2
2

(2.52)

If U ∗ = U , i.e. if U is complex due to δ = 0, the weak interaction is not CP invariant in
the neutrino sector. The existence of phases is of great relevance to leptogenesis since these
determine20 whether there is an asymmetry in the production of leptons and antileptons in
the early Universe [66].
In the three neutrino oscillations scenario, the parameter θ13 has an important role in
the transition probability (Pνµ →νe ) presented in equation (2.48), which is the probability in
vacuum for the two neutrino case. Once the matter effects are taken into account, equation
(2.48) becomes21 :

Pνmatter
µ →νe ≈
≈

E
ER
E
1±2
ER
1±2

Pνvacuum
µ →νe ,
2

∗ U C + 2U ∗ U C
2Uµ3
e3 1
µ2 e2 2 ,

(2.53)

where ER is the matter resonance energy associated with the atmospheric mass difference
(∆m232 ) and the electron number density in Earth (Ne ):
∆m2
ER = √ 32 .
2 2GF Ne

(2.54)

20 The decay of right-handed neutrinos in the seesaw mechanism provides the asymmetry.
21 The ± in equation (2.53) is: + for neutrinos and - for antineutrinos.

31

Parameter

Value

sin2 θ13
sin2 θ23
sin2 θ12
∆m231
∆m232

+0.0010
0.0219−0.0011
+0.027
0.451 ± 0.001 ⊕ 0.557−0.035
0.304 ± 0.012
(2.458 ± 0.002) × 10−3 eV2
(−2.488 ± 0.047) × 10−3 eV2

∆m221

+0.19 × 10−5 eV2
7.50−0.17

Table 2.1 Neutrino Oscillation Parameters. Best measurements of the neutrino oscillation parameters [70].
C1 and C2 are oscillation parameters not relevant at the moment. From the transition matrix
in equation (2.51), Pνvacuum
µ →νe depends on:
∗ U = e−ıδ sin 2θ sin θ .
2Uµ3
e3
13
23

(2.55)

The experimental results obtained by Double Chooz, Daya Bay, and RENO measuring a
non-zero θ13 allow the possibility for CP violation in the neutrino sector since the transition
probability presented in equation (2.53) depends on the phase δ. The transition probability
P¯ν¯vacuum
νe changes the sign of δ, which directly affects the measurement of the asymmetry
µ →¯
˜
(A):
P − P¯
.
A˜ =
P + P¯

(2.56)

If it is non-zero, δ is non-zero, resulting in the mentioned CP violation. Both A˜ and Akk
(recall equation (2.34)) are influenced by the phases δ and ϕ. A CP violation in the light
neutrino sector could have CP violation implications in the heavy neutrino sector, which is
the key to leptogenesis, as mentioned in section 2.2.1.
All the free parameters in the neutrino oscillation formalism have been measured [70],
except for the phases. Table 2.1 summarizes the most accurate measurements of these

32

parameters. Two open questions concerning these parameters remain unanswered, and are
the subject of study of the next generation of neutrino oscillation experiments, including
NOνA. The first is related to the octant of θ23 , since this is measured through the expression
sin2 2θ23 , its measured value only shows that it is close to π/4. However, whether θ23 is
smaller or greater than π/4 remains uncertain. The possibility for θ23 = π/4 implies that ν3
would have exactly the same amount of νµ and ντ . The second question concerns the mass
ordering, or mass hierarchy, of the mass eigenstates. Oscillation measurements are related
to sin2 ∆m2 /4E (see equation (2.48)), where the sign of the ∆m2 is undetermined. The
current knowledge is that there are two quasi-degenerate mass eigenstates, m1 and m2 , and
a third one, m3 , that is either much heavier (normal hierarchy), or much lighter (inverted
hierarchy) than the other two, as shown in figure 2.9. In the normal hierarchy scenario,
neutrino oscillations are enhanced and the antineutrino oscillations are attenuated, for the
inverted hierarchy the opposite is true.

Figure 2.9 Neutrino Mass Hierarchy.

33

Chapter 3
NuMI Beam Overview

Figure 3.1 Fermilab Accelerator Campus. Picture from Google Maps.

Figure 3.2 NuMI Beam Trajectory. Taken from [71].

The accelerator complex at Fermilab is currently dedicated to providing protons from
the Main Injector (MI) to seed the neutrino beam that is used by the NOνA experiment for
34

their neutrino oscillation investigations, and used by other experiments that study neutrino
interactions. In the next few years, accelerator-based neutrino experiments may resolve the
neutrino mass hierarchy, the octant ambiguity in θ23 , and begin to see hints that there is a
large violation of the CP symmetry in neutrino oscillations.

Figure 3.3 NuMI Beam Energy Spectrum. Neutrino event rates as a function of neutrino
energy and off-axis angle [72].

The MI [73] is a very high intensity1 proton accelerator located at Fermilab (see figure
3.1). The MI provides 120 GeV protons that serve as input to the Neutrinos at the Main
Injector (NuMI) beam, which has a power2 of 320 kW. This beam travels from the Fermilab
Accelerator Campus, through the Earth, to the MI Neutrino Oscillation Search (MINOS)
underground laboratory at Soudan, Minnesota (see figure 3.2). The energy spectrum of the
beam, shown in figure 3.3, allows experiments like NOνA to locate their detectors off the
1 The number of protons per pulse is on the order of 1013 [72].
2 The power cited is the one delivered to the NOνA prototype detector.

currently in the middle of an upgrade to deliver 700 kW.

35

The MI is

beam axis (see figure 3.4) to obtain narrower neutrino energy spectra.

Figure 3.4 Location Of The NOνA And MINOS Far Detectors. The MINOS far
detector in Minnesota is on-axis. Picture from Google Earth.

Protons from the MI are directed every 1.9 s, by single-turn extraction, into the NuMI
beamline. At 320 kW , there are 2.5 × 1013 protons delivered in spills of 8 µs, and focused
onto a target, producing secondary mesons, i.e. charged pions and kaons. For the neutrino
beam to be able to reach the Soudan MINOS far detector site, the proton beam is directed
downward at 58 mrad before it strikes the 0.95 m long NuMI graphite target, as shown in
figure 3.5. The forward-going mesons are focused and allowed to decay, producing the desired
neutrino beam. The focusing is performed by a set of two parabolic magnetic horns. The
charged pions and kaons selected by the horns propagate down a 675 m long (1 m radius)
decay tunnel. A hadron absorber (beam stop) is placed at the end of the decay tunnel
36

to remove the residual flux of protons and mesons, followed by a set of muon monitoring
detectors, as shown in figure 3.6 [71].

Figure 3.5 Schematics Of The NuMI Complex. Figure taken from [71].

Figure 3.6 NuMI Beam Schematics. Taken from [74].

In neutrino oscillation searches, the flavor composition of the neutrino beam should be
well known. The ideal case is to have a pure flavor beam, i.e. only one neutrino flavor. The NuMI beam design goals include: achieving the highest possible νµ intensity,
low backgrounds from other neutrino flavors, well understood spectra to control systematic uncertainties, and the selection of neutrino energy spectrum matched to the oscillation
physics [75].
37

3.1

Primary Beam System

The process of producing the neutrino beam starts with a set of primary protons that hits
a fixed target, which produces interactions that yield the mesons that decay to produce the
neutrinos.
The primary protons are extracted from the MI ring and transferred through the extraction enclosure through a steeply inclined carrier pipe to the target region located about
50 m underground. The extraction mechanism is a horizontal kick, leading to Lambertson
magnets3 deflecting the beam primarily in the vertical direction. The extracted protons are
focused and bent strongly downward by a string of quadrupoles and bending magnets so that
they enter the target hall located 122 m downstream of the extraction enclosure. Another
set of bending magnets brings the protons to the correct pitch, 58 mrad, for a zero targeting
angle beam directed toward the Soudan site [75].
To create the neutrino beam from the primary proton beam, these protons strike the
NuMI target to produce hadrons. This target is sufficiently long to enable most of the
primary protons to interact, as well as thin enough such that secondary interactions of the
charged pions and kaons are minimized by allowing them to escape through the sides, as
illustrated in figure 3.7. The depth of the field of the horn focusing system sets a limit on the
length of the target. The desired flux of charged pions and kaons out of the target decreases
with increasing radius due to particle re-absorption. The target stress due to the heat load
of the proton beam also decreases with increasing radius. As a consequence of the last two
statements, the target was designed to obtain the maximum yield and to ensure integrity
against mechanical failure [75].
3 Lambertson magnets are used to separate two adjacent beams by providing a bending

field for one beam and a field-free region for the other beam [76].
38

Figure 3.7 NuMI Target Drawing. Schematic drawing of a long and slim segmented
target designed for optimum production and decay of charged pions and kaons [75].

3.2

Secondary Beam System

The second stage in the production of the neutrino beam is to focus the mesons coming
out of the target in order to direct them toward the location of the detector. Therefore
the focusing horns produce toroidal magnetic fields and act as lenses to bend the secondary
particles back to the primary proton direction. The horns can either focus all the mesons at
a particular momenta or some of the mesons at all momenta. The parabolic shaped horns
produce magnetic fields that act as lenses, where the focal length is proportional to the
momenta of the mesons. The selection of a particular target position causes particles of a
certain momentum to be focused by the first horn. Mesons that were well focused by this
horn pass unaffected through a central aperture in the second horn. Other mesons move
to larger radii and are focused by the second horn, extending the momentum bite of the
system, as illustrated in figure 3.6. The horns are designed for 205 kA maximum current

39

pulses of 5.2 ms half-sine wave width, with a repetition rate of 1.87 s. The direction of the
current flowing in the horns determines which mesons are focused depending on the sign of
their electric charge. The forward horn current (FHC) focuses positive mesons which decay
into neutrinos, and the reverse horn current (RHC) focuses negative mesons which decay
into antineutrinos [75]. Figure 3.8 illustrates the direction of the two currents in the horns,
as well as the beam direction through the horns.

Figure 3.8 Beam And Electric Current Directions Through The NuMI Horns. The
drawing illustrates the direction of the beam through the horn. The FHC (RHC) flows in
the direction of the blue (red) arrows inside the conductors that make the horns.

The decay tunnel design allows for a sufficient flux of νµ within the energy band required
by MINOS and NOνA. Alignment is paramount part of the tunnel’s design. Misalignment
of the decay tunnel along its length can not occlude the aperture of the tunnel by more than
2%. The volume of the decay tunnel is carefully chosen to reach a compromise between the
neutrino flux and the cost of construction4 . The choice of tunnel’s radius balanced the loss
of secondary particles, that interact with the walls, with the cost of construction. A high
vacuum inside the tunnel is necessary to prevent unwanted interactions with air molecules.
The vacuum level in the tunnel ought to be 1 Torr or lower [75].
The main purpose of the hadron absorber is to eliminate hadrons that would overload the
4 Pions of energies about 50 GeV have a mean decay length of several km.

40

data acquisition systems of neutrino experiments on-site. All primary protons that did not
interact with the target are absorbed at this stage. This absorber is not thick enough to stop
the muons present in the beam, which are undesired background to the measurements of the
neutrino experiments. These muons can be eliminated by providing sufficient material to
absorb their energy. The NuMI beamline is located in dolomite, which is a dense rock. The
340 m of dolomite between the end of the hadron absorber and the MINOS hall is sufficient
to stop all muons coming from the decay tunnel [75].
The neutrino beam monitoring systems enables the beam users to measure the quality
of the neutrino beam. The neutrino flux is monitored through measuring: the spatial distribution of the hadrons directly upstream of the absorber, and the muons at various locations
within the dolomite shield, as seen in figure 3.6. In order to detect variations, the muon
intensity measurement is normalized to the number of incoming protons, and to each other,
while the measured profiles are compared to nominal profiles [75].

3.3

NuMI Beam Composition And Energy Spectrum
At 110 mrad

Neutrinos are produced in many weak processes, including particle decays. For NOνA,
the two most important sources of neutrinos are charged pion and kaon decays.

The

charged kaons are mesons made of two quarks: K + : u¯
s, and K − : u¯s, with mean life
time: τK = (1.2380 ± 0.0021) × 10−8 s, and mass: mK = (493.677 ± 0.016) MeV/c2
[77]. These have decay modes with only leptons (leptonic modes), with leptons and hadrons
(semileptonic modes), and with only hadrons (hadronic modes). The modes with the largest

41

branching ratios are5 [77]:

K ± → µ± + νµ (¯
νµ ),
→ π± + π0,

leptonic mode, branching ratio: 63.55 ± 0.11%,

(3.1)

hadronic mode, branching ratio: 20.66 ± 0.08%,

→ π 0 + e± + νe (¯
νe ),
→ π 0 + µ± + νµ (¯
νµ ),

semileptonic mode, branching ratio: 5.07 ± 0.04%,
semileptonic mode, branching ratio: 3.353 ± 0.034%.

About 96% of all the νµ , from charged kaon decays, reaching the NOνA prototype detector
at 110 mrad off the NuMI beam axis come from the leptonic mode. The rest of the νµ , at
the prototype’s location come from the semileptonic decay. A small νe component of the
FHC of the NuMI beam comes from the semileptonic charged kaon decays. At the NOνA far
detector, the knowledge of this νe initial component of the beam is extremely important since
it is the principal background to the νe appearance signal. There is no direct measurement
of the charged kaons flux from the NuMI beam, therefore a measurement that can constrain
the charged kaon production yield prediction could help to reduce the uncertainties in the
νe flux at 14 mrad.
¯ and π − : u¯d, with mean life time:
The charged pions contain two valence quarks: π + : ud,
τπ = (2.6033 ± 0.0005) × 10−8 s, and mass: mπ = (139.570 ± 0.00035) MeV/c2 [78]. The
decay mode with the largest branching ratio is6 [78]:

π ± → µ± + νµ (¯
νµ ),

leptonic mode, branching ratio: 99.98770 ± 0.00004%.

(3.2)

The leptonic decay modes of charged kaons and pions are two body decays that produce
5ν
¯µ (¯
νe ) are associated with µ− (e− ) and K − .
6ν
¯µ are associated with µ− and π − .

42

K & π True Transverse Momentum, p . NuMI MC.

K & π True Longitudinal Momentum, p . NuMI MC.

T

L

40

K

K

2

π

30

π

Mesons

Mesons

1.5

20

10

0
0

1

0.5

1

2

p (GeV/c)

3

0
0

4

5

10

15

p (GeV/c)

20

25

30

L

T

(a)

(b)

Figure 3.9 Meson Transverse And Longitudinal Momenta. Predicted (a) transverse
and (b) longitudinal momentum distributions of charged kaons (blue) and pions (red) that
decay into neutrinos detected in the NOνA prototype detector. Momenta evaluated at the
time of production of the meson in the target. MC simulation.
muons and neutrinos isotropically in the center of mass reference frame. In this reference
frame the neutrino energies are fixed. The decaying meson is boosted to translate the decay
into the laboratory reference frame. As a result, the neutrino energies will be relatively broad
since these are now a function of the relativistic parameter γ = E/m. The neutrino energies
in this case are also a function of the angle θ between the momentum of the decaying meson
and the momentum of the neutrino. For small θ, i.e. highly relativistic mesons, the flux and
energy of the neutrinos are given by:
2
A
2γ
,
2
2
1+γ θ
4πd2
Cm Em
Eν =
,
1 + γ 2 θ2

Φν =

(3.3)

where A is the transverse area of the detector that measures the neutrinos, d is the distance
between the decay point and the detector’s location, Cm is a constant that takes two values:
one for charged kaons, CK = 0.96, and one for charged pions, Cπ = 0.43; and Em is the

43

energy of the meson. Most of the mesons that yield the neutrinos seen at the location of the
prototype detector have transverse momenta (pT ) of the order of 300 MeV, as seen in figure
3.9a. The longitudinal momenta (pL ) of the mesons is mostly below 10 GeV/c, as seen in
figure 3.9b.
ν True Energy. Mother Particle. NDOS MC.
All
K±
π±
µ
KL

103 Events / 0.1 GeV

20

15

10

5

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 3.10 Neutrino Energy Distribution Discriminated By The Mother Particle
Of The Neutrino. All mother particles (black), π ± (red), K ± (blue), KL (light blue), and
µ± (green) energy distributions. MC simulation.

The neutrino energy spectrum observed with the prototype detector shows two peaks: one
comes from charged pion decays, around 0.2 GeV, and the second resulting from charged
kaon decays, around 2 GeV. The spectrum is produced mostly by νµ . However this also
receives contributions from ν¯µ , νe , and ν¯e which arise from the semileptonic charged kaon
decays, the leptonic decay of K − that were not defocused by the horns, and muons and KL
decays. 60.2% of all neutrinos come from charged kaon decays, 36.6% come from charged
pion decays, 2.9% resulting from KL decays, and 0.2% arising from decayed muons, as shown
in figure 3.10. The 4 neutrino components of the predicted energy spectrum, shown in figure
44

ν True Energy. All ν Types. NDOS MC.
All
νµ
νµ
νe + νe

103 Events / 0.1 GeV

20

15

10

5

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 3.11 Neutrino Energy Distribution Discriminated By The Neutrino Type.
All neutrino types (black), νµ (blue), ν¯µ (red), and νe + ν¯e (green) energy distributions.
MC simulation.

3.11, are distributed as follows: νµ 81%, ν¯µ 14.3%, νe 3.9%, and ν¯e 0.8%. The ratio K
π as
a function of energy is shown in figure 3.12. For energies higher than 1.8 GeV the ratio
remains above 10. This motivates a measurement of the neutrino flux coming from charged
kaon decays due to the low background from neutrinos produced in other decays.

45

ν True Energy. K/π Ratio. NDOS MC.
102

K/π Ratio

10

1

10-1

10-2
0

1

2

3

ν Energy (GeV)

4

5

Figure 3.12 Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos From
Charged Pion Decays. MC simulation.

46

Chapter 4
The NOνA Prototype Detector

4.1

Experiment Overview

The NOνA experiment will use the NuMI beam to measure νµ → νe oscillations. The
neutrino energies at the NuMI beam are a function of the energy of the meson, as well as of
the angle between the momentum of the meson that produces the neutrino and momentum
of the neutrino, as seen in equation (3.3). The MINOS experiment, located on the NuMI
beam axis, measured neutrinos on a wide energy range [79] (see figure 3.3). The NOνA
experiment, 14 mrad off the NuMI beam axis, will measure neutrinos at a much narrowed
energy band centered at 2 GeV. The far detector’s location, 810 km away from the neutrino
source (see figure 3.4), and the narrow neutrino energies observed there, produce the ideal
conditions to observe the νµ → νe oscillation very close to the maximum of probability.
Using equation (2.49), this maximum occurs at:

2c¯
hπ
km
L
≈ 534
,
=
2
E
GeV
∆m32

(4.1)

while NOνA has L/E = 505 km/GeV. Under such circumstances, NOνA will be capable of
determining the mass hierarchy and the CP violation with a significance up to 2σ, depending
on the real value of the oscillation parameters1 .
1 A sample bi-probability point fulfilling the previous statement is shown in figure 4.1.

47

Figure 4.1 Bi-probability Plot. Four values of the phase δ are presented of each of the mass
hierarchies: normal (blue) and inverted (red). Official NOνA figure [81]. 1σ and 2σ contours
are drawn around the estimated bi-probability points. The lower right point represented
by a star is at δ = 3π/2. The upper left point represented by a star is at δ = π/2. MC
simulation.

48

NOνA plans to collect three years of data in each mode2 . From each mode, the transition
probabilities (recall equation (2.53)): P (νe ) for νµ → νe , and P (¯
νe ) for ν¯µ → ν¯e , will
be estimated. Once these two probabilities are calculated, a point can be drawn in a biprobability plot [80] like the one shown in figure 4.1, and the mass hierarchy and the phase δ
could be resolved. 1σ and 2σ contours are shown around the calculated points. The ellipses
are representations of the various possible values of δ. The maximum CP violation occurs
at δ = π/2, 3π/2, while no CP violation occurs for δ = 0, π. Each mass hierarchy has
its own ellipse due to matter effects. The bi-probability plots summarize all the oscillation
parameters in an instructive fashion. Figure 4.1 assumes θ23 = π/4. Should θ23 > π/4, the
ellipses will move towards higher P (νe ) values along the P (νe ) = P (¯
νe ) line. On the other
hand, should θ23 < π/4, the ellipses will move downwards to lower P (νe ) values along the
P (νe ) = P (¯
νe ) line. The best case scenario for NOνA would be a normal hierarchy and
δ = 3π/2, or an inverted hierarchy and δ = π/2. These two points will exclude the other
mass hierarchy with a 2σ significance.
The measured value of θ13 (see table 2.1) gives NOνA the best sensitivity to: resolve
the mass hierarchy, measure δ, and resolve the octant of θ23 . In order to make these measurements, NOνA built two detectors. The near detector at Fermilab has a 3.9 × 3.9 m2
transverse area, and is 14.3 m long. The far detector has a 15.6 × 15.6 m2 transverse area,
and is 63.0 m long. Figure 4.2 shows to scale the three detectors. An Airbus A380-800 is
included in the figure for scale. The far detector is the largest free-standing PVC structure
in the world.
2 Neutrino and anti-neutrino modes.

49

Figure 4.2 Drawing Of The NOνA Detectors. NOνA detectors drawn to scale. An
average height person is shown to scale. An Airbus A380-800, drawn to scale, is included
for illustration purposes [82].

50

4.2

The Near Detector On The Surface
ν Angle From NuMI Beam. NuMI MC. ν Parent: K± & π±.

ν Angle From NuMI Beam. NuMI MC.
2

10

K±
π±

10

10 3 Events

10 3 Events

10

1

10-1

1

K± Normalized to π±

10-1

10-2

10-2

10-3

10-3
0

200

400

θNuMI (mrad)

600

800

0

200

(a)

400

θNuMI (mrad)

600

800

(b)

Figure 4.3 Angle Of Neutrinos With Respect To The NuMI Beam. (a) θNuMI for
neutrinos from all parents. (b) θNuMI for neutrinos from charged kaons (blue) and charged
pions (red) parents. The charged kaons area is normalized to that of the charged pions. MC
simulation.

The NOνA prototype detector is located on the surface, north from the MI at Fermilab
(see figure 3.1), and 110 mrad off the NuMI beam axis (see figure 3.5). This angle is measured
from the NuMI target to the origin of coordinates of the prototype detector. Not all neutrinos
are created at the same location along the decay pipe, their angular distribution is shown in
figure 4.3a for all neutrino parents. About 77% of all these neutrinos have:

110 mrad < θNuMI < 120 mrad,

(4.2)

The angular distributions for neutrinos from charged kaon and pion decays are shown in
figure 4.3b. Neutrinos from charged pion decays have a slightly higher rate between:

120 mrad < θNuMI < 200 mrad,

51

(4.3)

when compared with neutrinos from charged kaon decays. The peak of neutrinos around
650 mrad, in figure 4.3, comes predominantly from low longitudinal momentum mesons
(about 75% of which are charged kaons decaying into a νµ and a muon), as seen in figure
4.4a. The transverse momentum of these mesons is shown in figure 4.4b. Low energy
neutrinos make this peak, as seen in figure 4.4c.
Meson’s PL vs. ν Angle From NuMI Beam. NuMI MC.

Meson’s PT vs. ν Angle From NuMI Beam. NuMI MC.

30

4

103

104

25

102

PT (GeV/c)

PL (GeV/c)

20

15

10

3

103

2

102

1

10

10

5

0
0

200

400

600

θNuMI (mrad)

800

0
0

1

200

400

θNuMI (mrad)

(a)

600

800

1

(b)
ν: Energy vs. Angle From NuMI Beam. NuMI MC.
10

103

ν Energy (GeV)

8

6
102
4
10
2

0
0

200

400

θNuMI (mrad)

600

800

1

(c)
Figure 4.4 Meson Momentum And Neutrino Energy As A Function Of The NuMI
Angle. (a) Charged pions and kaons longitudinal momentum as a function of θNuMI , (b)
charged pions and kaons transverse momentum as a function of θNuMI , and (c) neutrino
energy as a function of θNuMI . MC simulation.
The initial purpose of this detector was to serve as a prototype, and then be moved
underground to become the near detector. For this reason it is known as the Near Detector
On the Surface (NDOS). However, the collaboration decided to build a totally new and
52

improved near detector. As a result the NDOS now serves as a testing facility for various
hardware and software purposes.

Figure 4.5 Drawing Of The Prototype Detector. The Prototype detector is a collection
of six blocks, each made of 31 planes. The neutrino beam goes, in the direction indicated
by the black arrow, upstream to downstream. At the end of the six blocks there is a
muon catcher that helps to reconstruct the energy of the most energetic muons. Artistic
representation.

The NDOS is 3.9 m high (Y ), 2.6 m wide (X), and 14.3 m long (Z), as shown in figure 4.5.
It is a modular structure made from highly reflective PVC extrusions [83]. Each extrusion
has 16 cells, and each cell has a 3.9 × 6.0 cm2 transverse area (see figure 4.6 for an illustration
of a single cell). Inside each cell there is a looped wavelength shifting (WLS) fiber, both
ends of the fiber are placed against the face of an avalanche photodiode (APD), and the
fiber covers the full cell length laid in a U-shape. Extrusions are arranged in modules, there
are two extrusions per module. Modules are arranged in planes, there are two (X) or three
(Y ) modules per plane. A block is an array of interspersed vertical (16) and horizontal (15)
planes. This arrangement of interspersed planes gives the X − Y coordinate system of the
53

Figure 4.6 NOνA Extrusions And Cells. Sample NOνA PVC extrusions arranged in a
modular collection of vertical and horizontal planes. Artistic representation of a single cell
[82].

Figure 4.7 Side View Of The Prototype Detector. Horizontal planes readouts.
Avalanche photodiode, front-end board and data concentrator modules [84].

54

detector. The NDOS bulk is 6 blocks long. Downstream of the detector there is a muon
catcher, which is a collection of interspersed extrusions and ten 10 cm thick steel plates. At
the end of the muon catcher, four more planes are positioned in the usual vertical-horizontal
arrangement used for containment purposes, as seen in figure 4.5.
Each APD contains 32 channels or pixels, where each pixel is exposed to both ends of a
single fiber in a cell. Each APD is mounted on a front-end board (FEB) which reads and
digitizes the signal3 . The digital output of each FEB is sent to a data concentrator module
(DCM) to consolidate the information. Each DCM reads 64 FEB (figure 4.7 illustrates the
arrangement of the various hardware components). All the electronics described above are
mounted on the west side of the detector, as seen in figure 4.7, for the horizontal planes, and
on top of the detector for the vertical planes.
The segmented nature of the NOνA detectors allows for multiple sampling of the energy
depositions of muons and electromagnetic showers4 . The radiation length of the detectors
(X0 ) is X0 ≈ 40 cm, and the Moli`ere radius is RM ≈ 10 cm. The X0 of the detectors allows
to get a very detailed sample of 2 GeV electron showers which are, on average, about 9X0
in length [85]. Photon showers are, on average, about 9X0 long as well, however the gap
these leave between the vertex and the start of the shower, which is X0 long, provides an
element of discrimination. With about 7 cells per X0 , the energy depositions are extremely
useful in particle identification. This same powerful feature is used in muon identification.
The signature of a νµ event in the detectors is a muon, and the experimental setup of NOνA
allows to identify and study them with great precision and detail. Identifying electrons is
extremely important for the νe appearance measurement. A background to electrons are the
3 See section 4.2.4.
4 The segmented geometry of the NOνA detectors is optimized for the identification and

measurement of νe CC interactions, as discussed at the end of section 4.2.
55

photons that result from π 0 decays.

4.2.1

Liquid Scintillator

The primary target in the detectors is the carbon nucleus. About 80% of the carbon nuclei
in the detectors come from the liquid scintillator (LS) blend that fills the PVC extrusions.
This blend is 95.79% mineral oil, used as a solvent, 4.11% scintillating pseudocumene, and
0.10% waveshifters [86]. When ionizing particles traverse the LS, ultraviolet light is emitted
[87]; the emitted wavelengths are then shifted by the waveshifters to the violet-blue range.
This light bounces, on average, 10 times inside a cell before it is captured by the WLS fiber.
In order to avoid light losses between emission and absorption, the PVC walls have 12%
TiO2 which gives a reflectance of about 90% for green wavelengths [21].

4.2.2

Wavelength Shifting Fiber

Wavelength shifting fibers are a critical component in the NOνA detectors. These collect
light emitted from the LS and transmit it into the APDs. Inside the far detector’s cells
the light travels as much as 16 m, which requires shifting of the wavelengths from the short
attenuation lengths of the blue (about 4 m) to the longer attenuation lengths of the green
(about 15 m). In addition, the APDs are more efficient in the green than in the blue light.
The spectra of the light emitted by the fiber as a function of the illumination point is
presented in figure 4.8. The highest light power is the spectrum for the light shined into the
fiber at 1 m from the readout. The lowest light power is the spectrum for the light shined
into the fiber at 26 m from the readout. Intermediate spectra are for illumination points at
1 m intervals. The various spectra show how the peak of the distribution rapidly transforms

56

Figure 4.8 Emission Spectra Of The Wavelength Shifting Fiber. Emission spectra
for the WLS fiber at various illumination points [88].

57

Figure 4.9 Wavelength Shifting Fiber Transverse Area.

from the 470 nm of the blue wavelength to the 580 nm of the green as the illumination point
moves father apart from the readout.
The WLS fiber used by NOνA is 0.7 mm in diameter (D). This features a polystyrene
(PS) core with a refractive index of 1.59, surrounded by a polymethylmethacrylate (PMMA)
inner cladding with a refractive index of 1.49, and a fluorinared polymer (FP) outer cladding
with refractive index of 1.42, as shown in figure 4.9. This multicladding structure produces
50% higher yield than a single cladding fiber due to a large trapping efficiency [89].

4.2.3

Avalanche Photodiodes

The NOνA collaboration selected APDs as photodetectors due to their high quantum efficiency and low cost. The 85% quantum efficiency enables the use of the long cells in the
far detector5 . The average signal for muons traversing a cell perpendicular to its walls is
5 Photomultiplier Tubes have a 10% quantum efficiency under similar circumstances.

58

30 photons at the face of the APD, which gives about 25 photoelectrons. This signal must
be distinguishable from high frequency noise. APDs are made from silicon, which features
a thermal generation of electron-hole pairs that mimic the signal from the fibers. These
electrons are amplified at the diode junction and appear as the input to the pre-amplifier,
thus contributing directly to the noise. In order to achieve a signal to noise ratio of 10:1,
the APDs are cooled to -15 o C.

Figure 4.10 Avalanche Photodiode.

The light output of the fiber is absorbed in the APD collection region, shown in figure
4.10, where electron-hole pairs are generated and, under the influence of an applied electric
field, the electrons propagate to the p-n junction. At the junction, the electric field, which
determines the multiplication of the current, is high enough to produce avalanche multiplication of electrons. The current generated from these electrons is the output signal of the
APD [21].

59

4.2.4

Front-End Boards

The readout of the APD, at the FEB, requires a pre-amplifier that can sample the signal
throughout a 10 µs time window. The signal from the APD is amplified and shaped by a
high-gain integrating amplifier with a shaping time of about 350 ns, and the output is stored
in a switched capacitor array (SCA) every 500 ns. The SCA contains 64 samples taken
500 ns apart for the 32 channels in the APD. The waveform is designed to have a 380 ns rise
time and 7000 ns fall time [90]:

F (t) ∝ 1 − e−t/380ns e−t/7000ns .

(4.4)

These choices of rise and fall times are intended to minimize the overall noise induced by the
electronics. Zero suppression is performed at 15 - 20 photoelectrons6 via a dual correlated
sampling (DCS) algorithm which subtracts the signal from the baseline. The DCS establishes
a rising edge triggered threshold under which the sampling points are zero suppressed. The
DCS algorithm then subtracts the sample taken right before the threshold is crossed from the
sample, with maximum amplitude, taken right after the threshold is crossed. This threshold
is set independently for each channel of the detector. Fits to the shape of the waveform
recover the timing and pulse height information [91].
√
The best timing resolution with this system is 500/ 12 ns ≈ 145 ns. The digitized
differences are stored in a field programmable gate array (FPGA) for transmission to the
data acquisition (DAQ) system. Digital signal processing algorithms could be encoded in the
FPGA firmware to improve both the signal-to-noise ratio and timing resolution. A FPGA
6 ADC and photoelectrons are proportional. See section 4.2.6 for a discussion on energy

calibration.

60

produces timing markers at periodic intervals of 50 µs interspersed with digitized hits [92].

4.2.5

Data Acquisition System

A DCM is a custom electronic component used to consolidate and concentrate the data
from 64 FEBs. A DCM programs, configures, and monitors each FEB, and passes the
timing system clock (TSC) and synchronization command to each FEB. The DCM-FEB
communication is through point-to-point serial data links with dedicated differential pair
lines for clock, synchronization, command and data. The digitized hits are consolidated by
the DCM FPGA to 50 µs time slices containing data from all 64 FEBs. An application
further consolidates this data to a longer 5 ms time slice and routes this time slice to a
downstream buffer node for processing. All DCMs throughout the detector send hit data
corresponding to one 5 ms time slice to a buffer node [92].
A buffer node is a commodity server residing in a 140 buffer nodes buffer farm. An
external global trigger system provides triggers to a buffer node in the form of a start
time and a time window, so that an application can select data corresponding to this time
window from buffered data. Only trigger-selected data are written by the buffer nodes to a
downstream data logger.
A data logger is a program that receives trigger-selected data from the buffer farm to
format into an event. A trigger issued to the buffer farm is also issued to the data logger
application for validation purposes. The event created by the data logger is written to disk,
and then archived to Fermilab’s mass storage. Events are also written by the data logger
to a shared memory segment, from which a dispatcher application can serve the data to a
quasi-online external client application.
All FEBs and DCMs throughout the DAQ system are synchronized through the use of
61

Figure 4.11 Schematic Overview Of The Data Acquisition System [92].

62

a sophisticated timing system. This system provides a stable master clock line that permit
the time stamp counters that are present on the FEBs, DCMs, and the timing system to
be loaded and synchronized with a universal wall time based off of a link to the GPS. The
timing system can achieve unit to unit synchronization that is accurate to within one clock
cycle. This system is also used to time stamp the beam spill information coming from the
Fermilab accelerator complex [91].
Each timing distribution unit (TDU) has four differential pair lines for communication:
synchronization, command, clock, and synchronization echo. The synchronization echo line
is used to compensate for cable length propagation delays. Refer to figure 4.11 for a DAQ
system schematic that illustrates the various steps and components involved in the data
processing.
Individual data output files are saved by SubRuns, which are either an hour long, or
512 MB in size, which ever happens first. SubRuns are grouped in Runs which can have up
to 24 SubRuns. Any time that the DAQ systems are stopped, that is the end of the SubRun
and Run in progress. The SubRuns that recorded the NuMI spills have one event per spill,
and each event is 500 µs long [92].

4.2.6

Performance And Calibration Of The Prototype Detector

The NDOS started its physics run in October 2010 during which cosmic ray data, and νµ and
ν¯µ data from the NuMI beam were collected. The collaboration intended to fully instrument
the detector with APDs. However, various problems occurred during commissioning, and the
majority of the APDs were damaged due to moisture on their face coming from condensed
water deposited when the detector was cooled down. About a third of the APD set survived
the problems. The APDs that survived were strategically located throughout the detector in
63

Figure 4.12 Event Display Of The Prototype Detector. Vertical planes make the top
view, and provide the horizontal (X) coordinate. Horizontal planes make the side view, and
provide the vertical (Y ) coordinate.

order to optimize the observation and measurement of neutrino interactions in the detector.
Most of the available APDs were gathered together in a region in order to identify the
neutrino interaction point, and the rest of the available APDs were located in strategic
individual planes in order to record the full length of the muon tracks. The plan with the
muon catcher was to have it fully instrumented. However there are some channels maps with
some regions that were also partially instrumented.
An event display of the detector is presented in figure 4.12. Cells with active APDs are
drawn with black lines, while inactive cells are drawn in light grey. The top view is made
from vertical planes which provide the X coordinate; the side view is made from horizontal
planes which provide the Y coordinate. Neutrinos from the NuMI beam enter the detector
from the bottom (Y = −200 cm). The bulk region, where most of the APDs are located, is
centered at Z = 400 cm. If APDs became available, these were installed to improve muon
identification, and if APDs failed, these were removed. This configuration is known as NDOS
run II, and the data recorded under this configuration goes from October 2011 until April
64

2012, during which there were 1.67 × 1020 protons on target (POT). The APDs operated
warm to avoid further losses. The active channels map for each SubRun is known, and in
general, such maps change frequently.

Figure 4.13 Cosmic Data Of The Prototype Detector. Distance from the center of
the detector vs. mean ADC value per cm. Attenuation corrections come from the fit. The
readouts are in the right. Official NOνA NDOS figure [81].

The performance and calibration of the detector were initially studied using cosmic ray
data. The energy calibration procedure used energy deposition of cosmic rays at different
locations in the detector. The light level is measured, initially, in analog to digital converter
(ADC) units of collected charge. Apart from a drop in the average light at short distances
from the APD (near the beginning of the PVC cell), the mean observed light level decreases
with increasing distance of the hit from the APD, as seen in figure 4.13. Here, the position
(W ) of a hit is measured from the center of the detector. The electronics are beyond the
W = 200 cm point. The goal of the energy calibration is to make hits depositing the same
energy, but located at different distances from the readouts, result in the same corrected
65

energy measurement. The mean ADC/cm is a representation of the light attenuation per
cell. The rapid fall at Y < −140 cm is due to the sum of two exponential functions that
represent the light attenuation in the fiber [93]:

A(r) ∼ AS e−r/RS + AL e−r/RL .

(4.5)

Here r is the distance traveled by the light inside the fiber, RS = 289.5 cm is a short
attenuation length, and RL = 852.3 cm is a long attenuation length. Light entering the fiber
is split up according to the constants AS = 0.3137 and AL = 0.1669, and propagated to the
readouts according to the two attenuation lengths [93]. The rapid fall at Y > 140 cm is due
to light absorption at the manifold that guides the fibers to the APDs.

(a)

(b)

Figure 4.14 Energy Calibration Of The Prototype Detector. ADC distributions for
various W slices (a) before and (b) after attenuation corrections. Official NOνA NDOS
figure [81].

Muons passing through the detector exhibit an uncorrected light level distribution with
a mean that shifts from about 400 ADC, at the closest distance to the readouts, to about
300 ADC at the farthest distance to the readouts, as shown in figure 4.14a. The small peak
at low light levels is the tail of the APDs noise distribution above threshold, and is due to
66

cells in which the muon passes entirely through the detector PVC. After the attenuation
corrections are applied the light level distributions overlap, as seen in figure 4.14b, so that
the energy measurement is then independent of W .
To approximate to photoelectron (PE) units, a simple rescaling of ADC is carried out:
ADC/P E = 2.275 [93]. The linear relation between ADC and PE is seen in figures 4.15a
and 4.15b, where the same conversion factor appears for the MC simulation and the data,
respectively. Application of the position dependent corrections discussed above yielded a
corrected number of photoelectrons (PECorr). The corrections are a function of W , which
means that ADC and PECorr are not proportional, as seen in figures 4.15c and 4.15d. The
initial assumption is that at W = 0: PE = 1 = PECorr [93]. The poor χ2 tests indicate
that the two quantities do not follow a straight line correlation.
To convert from PECorr to an absolute energy deposition, the range-energy relationship
provided by the Bethe Bloch formula [87] is used. A sample of cosmic ray tracks that clearly
stopped within the detector is selected and the reconstructed stopping point determined
for each track. With the distance (dc ) from the center of each cell in the track to its
endpoint, a relation between PECorr/cm and dc is established. The Bethe Bloch curves are
integrated to provide the expected dE/dX for every dc . The integration takes into account
the density of the material that receives the deposited energy, i.e. to distinguish between
PVC, scintillator, etc.. The best fit to the profile of the PECorr/cm plot yields a calibration
of: GeV/PECorr = 4.7807 × 10−5 [93], about 50 keV per corrected photoelectron.
The performance of NDOS over time is represented by the light level in a set of test cells.
The set of cells is located at the center of the detector in X -Y and at the south edge of the
bulk of APDs, Z = 280 cm. During the running of the NDOS there were times with the
APDs system cold or warm, and FEBs gains were varied; however, the performance of the
67

(a)

(b)

(c)

(d)

Figure 4.15 Energy Conversion Factors Of The Prototype Detector. ADC vs. PE
for (a) MC simulation and (b) data. ADC vs. PECorr for (c) MC simulation and (d) data
[94].

68

Figure 4.16 Light Level Over Time On The Prototype Detector. Light level on 8 test
channels of an APD as a function of time [95].

detector remained stable, as seen in figure 4.16. The time elapsed starts six months prior to
the NDOS run II, and ends a month before the end of this run. During the period shown,
there were times of cold and warm APD, and the gains varied in the FEB; however, the
performance of the detector remained stable.

4.3

NOνA Software

To evaluate the biases affecting various physics process involved in the analyses, the NOνA
collaboration developed a Monte Carlo (MC) detector simulation, and track reconstruction
programs which could be applied to both MC and data objects.

69

4.3.1

Monte Carlo Simulation

The NOνA simulation process begins at the NuMI beam level. The MC simulates the hadron
production in the NuMI target, which is followed by the propagation of the produced particles
through the target material, the magnetic horns and the decay pipe. The FLUKA software
[96] simulates interactions of 120 GeV protons in the NuMI target, and the interaction
of secondary particles created in the target as well. The simulation creates output files,
which contain the kinematic variables of the generated particles. These files are input to
the GEANT4 [97] software, which propagates the particles, and their decay products, from
the NuMI target, through the magnetic horns, and down the decay pipe, where weak decays
yield neutrinos. If the combination of FLUKA and GEANT4 (known as FLUGG [98])
generates a neutrino that reaches the NDOS, the GENIE [99] software simulates interactions
in proportion to predetermined cross sections. The interaction is simulated within the target
nucleus, and an intranuclear hadron transport yields identified final state particles with full
kinematics.
The NOνA collaboration has developed a full simulation of the NDOS using GEANT4 to
propagate GENIE generated particles through the detector. The software simulates energy
deposition, scattering, and decay processes that affect the particles produced during neutrino
interactions. The NOνA collaboration developed further simulation tools that use the energy
depositions in the scintillator to generate blue light, reflect it in the PVC cells until absorbed
by wavelength shifting fibers. Equation (4.5) parametrizes the attenuation of the green
shifted light as it propagates through the fibers to the APD and the readouts. All the values
of variables created by the MC simulations that are available for analysis are named: true
values.

70

To provide samples with the same biases as the real data, the MC SubRun are created
with the same active channels maps. Also, the number of POT used to produce the MC
sample is stored in the files, as it is done in the data.

4.3.2

Reconstruction Tools

The reconstruction tools developed by the NOνA collaboration are intended to work with
simulated and real values of the variables7 . The basic unit in the reconstruction process is a
hit. A hit is an energy deposition left in the detector by a particle. The simulation provides
hits for every interaction of a particle within the detector; there could be many simulated
hits inside an individual cell. Data also provides hits, which are energy depositions in a
detector cell that generate a detected amount of light, above the electronic noise threshold,
at a digitized time, and with a position provisionally given as the center of the cell. Any
group of real hits within a cell that share the same time tag will be collected into one hit:
a cell hit. Cell hits appear both in simulated and real data of the variables. Cell hits have
stored values of: energy, time, and position; which have been properly calibrated (see section
4.2.6).

4.3.2.1

The Slicer

In order to reconstruct a neutrino interaction, whether it is simulated or data, the first stage
in the reconstruction is to make slices of cell hits that share common features in space and
time. The Slicer sorts the cell hits in an event by time, after which subsets of cell hits (slices)
that are grouped by similar times are produced. Different slices are separated by at least
7 Real values of the variables are those provided by the DAQ, and come from real mea-

surements, or data.

71

Figure 4.17 Sample Cosmic Ray Event For Slicer Performance. Colors represent the
various physics slices.

530 ns. A physics slice is one where the number of cell hits is eight or greater.
Cell hits are considered noise, and removed from a physics slice, if they are isolated from
other cell hits in space and/or time. Cell hits with P E < 15 are also considered noise, and
removed from the physics slice: noise subtraction. Cell hits subtracted from a physics slice
are not recorded as noise, but are incorporated to the noise slice. After the classification
done by the Slicer there would be: physics slices, and a noise slice that collects all the noise
hits in the event. In events defined by a 500 µs time window, the performance of the Slicer
is determined by its efficiency (ǫ) and purity (p):

ǫ =

st
= 96%,
se

p =

st
= 99%,
sp

(4.6)

where se is the total number of true cell hits in the event that are not labeled as noise, st is
the total number of cell hits in the physics slices and not labeled as noise, and sp is the total
number of cell hits in the physics slices [100]. Colored dots are drawn on top of the cell hits

72

to represent each of the various physics slices, as shown in figure 4.17, for a sample cosmic
ray event. The noise slice is not drawn.

4.3.2.2

Track Reconstruction

In νµ CC interactions, the nominal path of the muon through the detector is a straight line,
though multiple scattering can cause noticeable deviations, especially near a stopping point.
Nevertheless, tracking muons in the detector is straightforward, and their initial energies
can be calculated from the paths lengths, if the muons stop in the detector. The NOνA
collaboration developed a series of tracking algorithms to reconstruct cosmic tracks, tracks
from neutrino interactions, and showers. In tests on MC simulated muon tracks in the NDOS,
with its large number of inactive channels, the Kalman Tracker algorithm performs better
than the alternatives at reconstructing muon tracks produced by neutrino interactions.
To test the performance of the tracker on various kinds of particles, the difference between
the true and the reconstructed track lengths (∆L) divided by the true track length (L) is
examined. The test shows that the tracker is most efficient at reconstructing muon track
lengths, and not so efficient at reconstructing the lengths of other particles. This test is most
relevant to this analysis of νµ CC interactions, where tracking is used to identify a candidate
muon track. The track length allows the energy of the muon to be calculated. The remainder
of the neutrino interaction energy is determined using the detector as a calorimeter. The
overall efficiency (ǫ) and purity (p) of the tracker are:

ǫ =

kt
,
ke

p =

kt
,
kp

(4.7)

where ke is the total number of true cell hits that belong to a particle, kt is the total number

73

of cell hits in the reconstructed track that belong to that particle, and kp is the total number
of cell hits in the reconstructed track. The efficiency and purity for muons are higher than
for protons and charged pions. A summary of the performance of the tracker is presented in
table 4.1. For a complete study on the performance of the various NOνA trackers see [101].
Kalman Tracker
Efficiency
Purity
100% × ∆L/L

All

µ

p

75% 92% 69%
61% 87% 37%
23% 131%

π±
68%
41%
78%

Table 4.1 Kalman Tracker Performance. Kalman Tracker’s efficiency, purity and ∆L/L
for various particles [101].

The Kalman Tracker uses a Kalman filter [102] for track pattern recognition and track
fitting. In this application, the filter has the ability to obtain pattern recognition and track
fitting in one module, and the capability to find multiple tracks within a group of correlated
cell hits. The input of the filter are slices from the Slicer, and the output are 2D and 3D
tracks. The 2D track reconstruction is performed on each physics slice, and separately in
each view. Whenever possible, the 2D tracks found in separate views are matched together
to form 3D tracks if these belong to the same physics slice. The Kalman Tracker algorithm
is described in appendix B.
The Kalman Tracker does not reconstruct a vertex. Therefore the interaction vertex is
placed at the starting (lowest Z coordinate) position of the longest track in the neutrino
interaction. The vertex resolution is very good in the three coordinates, about a cell, as seen
in figure 4.18. The tails in the distributions (beyond 1σ) account for 15% of all the tracks.
These tails are due to the large number of inactive channels in the detector, but they are
sufficiently small to make a reasonable fiducial volume cut based on the coordinates of the
vertex.
74

3

Vertex Resolution in x: ∆x. NDOS MC.

60

0.2717

Mean

-0.01085

RMS

11.76

RMS

11.87

60

Constant 7.882e+04

Sigma

Constant 7.652e+04
Mean

0.05367
3.344

40

20

Sigma

0.01203
3.422

40

20

-50

0

50

∆x (cm)

0
-100

100

-50

0

∆y (cm)

(a)

50

(b)
60 ×10

3

Vertex Resolution in z: ∆z. NDOS MC.

50

Mean

1.432

RMS

13.63

Constant 5.803e+04
40

Tracks

Tracks

3

Mean

Mean

0
-100

Vertex Resolution in y: ∆y. NDOS MC.

80 ×10

Tracks

80 ×10

Mean

-0.178

Sigma

4.264

30

20

10

0
-100

-50

0

∆z (cm)

50

100

(c)
Figure 4.18 Vertex Resolution. (a) X, (b) Y , and (c) Z. MC simulation.

75

100

3

End Point Resolution in x: ∆x. NDOS MC.
Entries
Mean
RMS
Underflow
Overflow
Integral
Constant
Mean
Sigma

10

3

End Point Resolution in y: ∆y. NDOS MC.
Entries
Mean
RMS
Underflow
Overflow
Integral
Constant
Mean
Sigma

15

10

159835
-15.86
56.29
1.595e+04
4.674e+04
9.714e+04
1.197e+04
-0.4022
7.899

5

5

0
-200

-100

0

100

∆x (cm)

0
-200

200

-100

0

∆y (cm)

(a)

100

(b)
×10

3

14

End Point Resolution in z: ∆z. NDOS MC.
Entries
Mean
RMS
Underflow
Overflow
Integral
Constant
Mean
Sigma

12
10

Tracks

Tracks

15

×10

159835
-0.4485
48.22
3.347e+04
3.276e+04
9.36e+04
1.308e+04
-0.008798
8.097

Tracks

×10

8

159835
11.4
68.94
8656
6.61e+04
8.508e+04
9669
2.207
7.133

6
4
2
0
-200

-100

0

∆z (cm)

100

200

(c)
Figure 4.19 Endpoint Resolution. (a) X, (b) Y , and (c) Z. MC simulation.

76

200

Finding the endpoint of the longest track has some significant problems. The large gaps
in Z, with no active channels in the detector downstream of the interaction region, presents
a serious obstacle to achieving acceptable performance with the tracker. Therefore, the
resolution for the endpoint of the longest track is not as good as the vertex resolution. When
muons reach the regions where only a few planes have active channels (Z > 600 cm), the
endpoint of long tracks has a large uncertainty, as seen in figure 4.19. The long tails in figures
4.19a, 4.19b, and 4.19c account for: 63.2%, 66.4%, and 75.7% of all tracks, respectively. The
overall endpoint resolution in X and Y is about 50 cm, and in Z is about 60 cm. For
reference, a 2 GeV muon deposits 120 MeV of energy in 60 cm of scintillator.

Figure 4.20 Sample Event. Case (1). MC simulation.

Reconstructed tracks have their starting and ending points in the center of an active cell.
However, a real muon is not guaranteed to end its trajectory in an active cell due to the
distribution of active cells. The trajectory of a long muon is more likely to end in an inactive
cell. The tracker places the end of the track in the last active cell with energy deposition
77

along the track’s path. In the following cases, however, the real endpoint of the muon would
not be accurately reconstructed:
1. The muon could have ended its path in the region between planes with active cells.
2. The muon could have passed a plane with active cells through the PVC.
3. The tracker could have missed a cell hit in an active plane.
4. The muon could have left the detector before reaching the next active plane.
About 30% of the muons exhibit one of these behaviors [103]. In case (1), shown in figure
4.20, the muon stops at the end of the dotted path in red. On average, however, a muon
with this topology will have an endpoint halfway between the end of the dotted path and the
blue “x”, as illustrated by the magenta line segment added to the end of the reconstructed
track (red). The length of this segment extends the track to a point half the distance (H)
between the adjacent active planes, so that H/2 is added to the Z coordinate of the track’s
endpoint. With this correction, the difference between the true and reconstructed endpoint
is, on average, centered at zero, and exhibits a resolution of 25 cm, or the thickness of 5 cells
[103].
The other three cases are more difficult to accurately reconstruct. Sample events of the
last three cases are presented in figure 4.21, where top views for each event are labeled: Reco
for the reconstructed track, and MC for the true trajectory. In case (2), illustrated in figure
4.21a, the true muon trajectory passes through the plastic (MC top view) of the active plane,
around Z = 825 cm, deposits no energy in that plane, and then stops around Z = 970 cm.
The reconstructed track (Reco top view) ends in the previous active plane, which represents
a loss of about a third of the muon’s reconstructed energy. In case (3), illustrated in figure
78

(a)

(b)

(c)
Figure 4.21 Sample Events. Cases: (a) 2, (b) 3, and (c) (4). Only the top view of the
detector in each event is shown. The red tracks are reconstructed objects, and the blue
tracks are the corresponding true objects. MC simulation.
79

4.21b, the tracker does not include in the muon track the cell hit left in the active plane
around Z = 825 cm, making the reconstructed red track around 3 m shorter than its true
length, and its energy about 600 MeV too low. In case (4), illustrated in figure 4.21c, the
true muon trajectory leaves the detector (MC top view), at Z = 1050 cm, before reaching
the next active plane at Z = 1075 cm. With this topology, the tracker incorrectly assumes
the muon stops in the detector, which results in an energy estimation that is too low. These
three cases are not properly represented by the correction applied to the track length in case
(1), nonetheless, these will be corrected by the method described above since, in data, it is
not possible to make a distinction from case (1).

Figure 4.22 Sample Event. From figure 4.20 with extra track length correction applied for
containment purposes. MC simulation.

For tracks that actually leave the detector (not contained), the length correction presented
above is insufficient to obtain the correct energy. Therefore, tracks with the potential to leave
the detector will be removed from the sample. The correction presented above is applied to

80

the reconstructed track length of muons to better estimate, on average, the energy of the
muon sample. The lack of information on the real endpoints of the muon tracks requires
further manipulation of the track lengths in order to determine their containment. Only for
containment purposes, an extra line segment is added to the muon tracks, with endpoints
beyond Z > 600 cm, so that the new reconstructed track length reaches the next active
plane, as seen in figure 4.22. The magenta line segment goes from the last cell hit in the
track to the next active plane in the projected muon trajectory. If the new endpoint of the
extension (magenta line segment) is within the detector volume in both views, the track will
be labeled as contained, while the uncontained tracks will be removed from the sample.

4.4

Data Quality

Figure 4.23 Number Of Active Channels Per Good Run. The good Run number
increases with time: 13067 = 10/29/2011, and 13782 = 04/30/2012.

A number of quality checks are performed to make a list of good data files to be used in
81

the physics analyses. A good data file, or SubRun, meets the following criteria:
• The first SubRun of a Run is longer than 30 min.
• The file is not a collection of empty events.
• The file has less than 10% of noise-only events.
• The number of cell hits per active channel per event (hace ) in the file is:
0 < hace < 0.06.
• The average number of slices (ns ) in the file is: 1 < ns < 6.
• The average slice duration (ts ) in the file is: 0 < ts < 1500 ns.
• The file shows a record of synchronized DCMs.
Also, good data files must have at least 4400 active channels, as seen in figure 4.23.
In the midst of changing conditions of the NuMI beam, the reconstruction process yields
a total number of neutrino candidate events that correlates well with the integrated number of POT [104], as shown in figure 4.24. Although the proton beam intensity varied
throughout the data taking period, the beamline components did not change. There are
a few times during the data taking period where the number of candidate events found
jumped or dropped slightly in a small region of integrated POT. The candidate event rate:
2 × 10−17 events/POT, correlates well with that expected on the basis of the neutrino beam
energy spectrum and the cross sections contained within the simulation program.
The DAQ reads out cell hits per event in 500 µs wide trigger time windows centered at
about the time of an expected neutrino arrival. The average time of the cell hits per physics

82

Figure 4.24 Number Of Neutrino Candidates As A Function Of The POT. Colors
represent various beam configurations [104].

Figure 4.25 Time Of Event Slice In The Data Of The NOνA Prototype Detector.
Full DAQ window. Official NOνA NDOS figure [105]. The bin size is 1 µs.

83

slice is shown in figure 4.25 [105], where a physics slice is considered an event8 . In the event
display, the trigger time window is represented by a histogram that starts at −50 µs and
ends at +550 µs. The data are read in blocks of 50 µs, and the boundaries of these blocks
generally do not match the boundary of the 500 µs time window, i.e. a portion of the first
and last blocks belongs to the 500 µs time window, and the other portion does not. Two full
blocks, one at the beginning and one at the end of the 500 µs time window, are presented in
each event display. The amount of hits recorded in these two blocks falls rapidly for times
away from the boundaries of the 500 µs time window. The neutrinos from the NuMI beam
are expected within the beam spill, 10 µs long, centered at 222 µs into the trigger time
window. A trigger time window containing a selected neutrino candidate event is shown
in figure 4.26. The red arrow points to the peak of cell hits that comes from the selected
neutrino candidate event at about 220 µs.

Figure 4.26 Sample Trigger Time Window. A selected neutrino candidate event with
no cosmic ray background.

The out-of-time NDOS data, all from cosmic rays, which is defined by (refer to figure
8 The word event used before to refer to collections of cell hits occurring within 500 µs is

also used to refer to a physics slice which is suppose to represent the cell hits of a neutrino
interaction.

84

4.25) the time intervals9 :

t < +216.5 µs, and t > +227.5 µs,

(4.8)

has a constant rate in the time intervals:

−3.5 µs < t < +216.5 µs, and + 227.5 µs < t < +436.5 µs.

(4.9)

The recorded cosmic ray rate rapidly goes to zero for:

t < −3.5 µs, and t > +436.5 µs.

(4.10)

The neutrino signal from the NuMI beam appears at about 222 µs in figure 4.25, as expected. As discussed above, the beam spill interval is 10 µs wide. However the NDOS
timing resolution is 0.5 µs, and consequently the in-time window is defined by:

+216.5 µs < t < +227.5 µs,

a window that is 11 µs wide for this NDOS analysis.

85

(4.11)

Slice Time. NDOS DATA.

Slices / 2 × 103 nsec

200

150

100

50

0

0

100000

200000

300000

400000

Time (nsec)

Figure 4.27 Time Of Physics Slices In The Data Of The Prototype Detector. Note
the in-time data at about 220 µs.

4.5

Cosmic Rays

As a consequence of its location in a surface building, NDOS records millions of cosmic rays.
A constant rate of cosmic rays is recorded in each trigger window for:

−3.5 µs < t < +436.5 µs.

(4.12)

Therefore, to establish the level of cosmic background in the neutrino data, cosmic rays
within this time interval, 429 µs wide10 , are considered. The time interval used to establish
the constant cosmic rate is shown in figure 4.27. The bin size is 2 µs, four times the NDOS
9 These time intervals exclude the beam window.
10 Note that the width of 429 µs results from the exclusion of the in-time data, which is

11 µs wide.

86

timing resolution11 . A straight line fit to the out-of-time data allows to determine that the
cosmic ray rate is flat within the shown time interval. The parameters of the fit are:

y-intercept:
slope:

(114.9 ± 1.5) Slices/2 × 103 ns,
(5.5 × 10−7 ) ± (6.1 × 10−6 )

Slices/2 × 103 ns2 .

(4.13)

The slope is consistent with zero, and its variation over the range of the time window
is of 0.2%. The timing histogram representing the trigger window of an event with four
reconstructed cosmic rays is shown in figure 4.28. The peaks coming from their cells hits
appear at about: 55 µs, 90 µs, 290 µs, and 380 µs. Note that there is no peak of hits at the
NuMI trigger time, around 222 µs. The event display of this particular event with cosmic
rays is presented in figure 4.29.

Figure 4.28 Sample Trigger Time Window. Only cosmic rays in the event.

In order to reduce the cosmic ray background that appears inside the NuMI trigger
window, the angular distribution of the out-of-time cosmic tracks is compared to that from
the MC simulated beam tracks. Three angles are studied: cos θX , cos θY , and cos θNuMI .
The first two angles are those between the unit vectors of the detector’s axes X and Y (ˆi
11 All the slices that passed the containment cuts discussed in section 5.2, and had a

reconstructed track with track length longer than 2 m are presented in figure 4.27. The time
assigned to a slice is an average of all its cell hits times.

87

0

200

400

600

800

1000

1200

1400

0

200

400

600

800

1000

1200

1400

x (cm)

100
50
0
-50
-100
200
150

y (cm)

100
50
0
-50
-100
-150
-200

UTC Sat Oct 29, 2011
17:34:16.348061696

102
10
1

102
10
1

hits

Run: 13068 / 3
Event: 138120 / NuMI

hits

NOvA - FNAL E929

z (cm)

0

100

200

300

400

500

t (µsec)

0

0.5

1

1.5

2

2.5

3

3.5

4

×10

q (ADC)

Figure 4.29 Sample Cosmic Ray Event. This figure shows a sample cosmic ray event
with four reconstructed cosmic rays.

88

Figure 4.30 Beam To Prototype Detector Coordinate Transformation Illustration.
Graphical representation of the beam reference frame (blue) and the NDOS reference frame
(black). The red arrow represents the direction of the neutrinos, from the NuMI beam, that
reach the NDOS. Not to scale.

and ˆj), and the unit vector of the longest track in the event, vˆt :
Xf − Xi
,
L
Y − Yi
ˆj · vˆt = cos θY = f
,
L
Zf − Zi
kˆ · vˆt = cos θZ =
,
L
ˆi · vˆt = cos θX =

(4.14)

where Xi , Yi , and Zi are the starting point coordinates of the longest track in the event; Xf ,
Yf and Zf are the ending point coordinates of the track, and L is the track’s length. The
cos θNuMI is the cosine of the angle between vˆt and the unit vector of the neutrinos coming

89

from the NuMI beam, vˆN :

y

y

x vx + v v + vz vz ,
vˆN · vˆt = cos θNuMI = vN
t
N t
N t
y

x cos θ + v cos θ + v z cos θ ,
= vN
Z
Y
X
N
N
dx
dy
dz
=
cos θX +
cos θY +
cos θZ ,
ds
ds
ds
dz ds dx
ds dy
=
cos θX +
cos θY + cos θZ
ds dz ds
dz ds
1 dx
dy
= ds
cos θX +
cos θY + cos θZ .
dz
dz

,
(4.15)

dz

The origin of the NDOS reference frame is: dx′ = -0.29 m, dy ′ = 92.21 m, and dz ′ = 841.76 m,
with ds′ = 846.80 m, in the beam coordinates. The coordinates transformation derived in
[106] (see figure 4.30) allows to get that point in the NDOS reference frame: dx = -29.0 cm,
dy = 4300.8 cm, and dz = 78070.2 cm, with ds = 78188.6 cm. In the NDOS reference frame:
dy
dx
ds
dz = 1.00151, dz = 0.00037, and dz = 0.055. From equation (4.15), the cos θNuMI is:

cos θNuMI =

0.055 cos θY + cos θZ
.
1.00151

(4.16)

The ratio dx
dz is assumed to be zero in the MC simulation as it is small compared to the other
fractions. The direction of vˆN is illustrated in figure 4.31.
Cosmic rays are a background to the neutrino signal. However the angle of a cosmic ray
muon is rarely parallel to the direction of the neutrino beam. The angular distribution of
the longest tracks in the simulated CC neutrino interactions are peaked in the direction of
the neutrino beam, as shown in figure 4.32. The cosmic rays and the simulated tracks from
neutrino events have peaks at: cos θX = 0, as shown in figure 4.32a, so that this variable does
not provide much discrimination power. Cosmic ray tracks have a strong preference for the
90

Figure 4.31 The Direction Of Neutrinos From The NuMI Beam In The NDOS.
The direction of the neutrinos from the NuMI beam at the NDOS location is represented by
red arrows drawn on top of the NDOS event display.

vertical direction, | cos θY | ≈ 1, while long tracks generated in neutrino interactions prefer
the horizontal direction, cos θY ∼ 0, as shown in figure 4.32b. The greatest discrimination
between cosmic rays and the longest tracks from neutrino interactions is in the angle with
respect to the beam direction. The neutrino interaction tracks have a peak at: cos θNuMI ≈ 1,
in contrast with the maximum exhibited by the cosmic rays at: cos θNuMI ≈ 0.1, as shown
in figure 4.32c.
Simple cuts in the angular distributions of the reconstructed tracks can make a clean
separation between neutrino data and the cosmic ray background, as will be discussed in
Chapter 5. All the reconstruction tools presented in this chapter aid to analyze the data
that passes the quality control checks summarized here.

91

Longest Track Cos θY . NDOS MC & Cosmic DATA.
Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

Reco MC

40

Cosmic
30

20

10

0

-1

-0.5

0

0.5

Cos θX

120

Reco MC

100

Cosmic

20

Tracks / (0.02 × 11 µ sec × 1.67 × 1020 POT)

Longest Track Cos θX. NDOS MC & Cosmic Data.

80
60
40
20
0

1

-1

-0.5

(a)

0

Cos θY

0.5

1

(b)

120

Reco MC

100

Cosmic

20

Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

Longest Track Cos θNuMI. NDOS MC & Cosmic Data.

80
60
40
20
0

0

0.2

0.4

0.6

Cos θNuMI

0.8

1

(c)
Figure 4.32 Longest Track Angular Distributions. NDOS MC (red) and cosmic ray
data (black). Longest track MC and out-of-time cosmic data (a) cos θX , (b) cos θY , and (c)
cos θNuMI distributions.

92

Chapter 5
Event Selection
As discussed in section 4.4, an event is a set of data collected within a 500 µs time window;
however, many different physical processes can occur within this time. The purpose of event
selection is to identify probable neutrino interactions within this time window, and study
them in order to determine their nature, i.e. their interaction type (CC or NC), and the
neutrino type (νµ or νe ). In the NDOS, the word event refers to a neutrino interaction
without confusion since the chances of two or more neutrino interactions in one 500 µs
window are extremely low.
Although muons are easy to identify in the NOνA detectors, energetic charged pions
and protons that behave like minimum ionizing particles (MIP) represent a background
to the muon sample1 . When protons and charged pions undergo hard scattering, these
leave in the detector characteristic signatures, figures 5.1a and 5.1c are examples of these
signatures. These two simulated single particle events have momenta around 2 GeV/c, and
do not represent a background to the muon signal because they are either too short or too
heavily ionizing. In contrast, when protons and charged pions behave like a MIP, as seen
in figures 5.1b and 5.1d, these do represent a background to the muon signal. These two
simulated single particle events also have momenta around 2 GeV/c, however, these leave
different signatures in the detector, similar to that of muons. Hadrons with momenta around
1 Electromagnetic showers also contribute, in smaller proportions, to the background.

93

(a)

(b)

(c)

(d)

(e)
Figure 5.1 Single Particle Events. (a) Proton: 1.9 GeV/c. (b) Proton: 1.9 GeV/c. (c)
π + : 2.1 GeV/c. (d) π + : 1.9 GeV/c. (e) µ: 0.5 GeV/c. Colored dots represent the various
energy depositions left in the detector by the particles. MC simulation.

94

2 GeV/c that behave like a MIP are not as much of a background to 2 GeV/c muons as
these are to 0.5 GeV/c muons, as seen in figure 5.1e.
Described in the sections to follow are the cuts applied to reject events without a final
state muon. The various efficiencies and purities of the selected sample are also presented
in the following sections. The cuts that remove cosmic rays from the selected event sample
are included as well.

5.1

Charged And Neutral Current Neutrino Interactions

(a)

(b)

Figure 5.2 Sample Events. (a) Sample NC event: νµ + p → νµ + K + + K 0 + n + π 0 . (b)
Sample CC event: νµ +12 C → µ + p + π − + π 0 . MC simulation.
CC neutrino interactions have a background that comes from NC neutrino interactions
which contain a hadronic track that mimics a muon. Two sample MC events are shown in
95

figure 5.2: one is a NC event (figure 5.2a), and the other one is a CC event (figure 5.2b). The
two events in figure 5.2 show a similar energy deposition, and a long MIP track. A longest
track with MIP behavior is shown in figure 5.2a; this particle is a K + (red track). The event
also features a K 0 , a neutron and a π 0 . The neutrino energy is 5.0 GeV. In contrast, figure
5.2b has a muon as its longest track (green track). The event also has a π 0 that decays into
two photons, a proton, and a π + . This is a DIS event. The neutrino energy is 1.9 GeV.
Figure 5.2a is an example of a NC MC simulated event that needs to be rejected in order to
achieve a clean CC sample. Section 5.3 is going to address the rejection criteria.
True ν Energy. Interaction Type. NDOS MC.
All

20

CC
NC

Events

15

10

5

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 5.3 NDOS Neutrino Energy Distributions. Predicted neutrino energy distributions discriminated by interaction type. All interactions (black), CC (blue), and NC (red)
energy distributions. MC simulation.

From the MC simulation it is known that 69.4% of the neutrino interactions in the NDOS
CC ratio rises rapidly with neutrino
are CC and 30.6% are NC, as seen in figure 5.3. The NC

energy, as shown in figure 5.4. For energies above 1.8 GeV the ratio is about 2.6, consistent
with measured cross sections ratios [44].
96

CC/NC Ratio vs. ν True Energy. NDOS MC.

(CC / NC) Ratio

3

2

1

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 5.4 MC Prediction Of The Ratio Of CC To NC Neutrino Interactions.
Ratio as a function of neutrino energy for all simulated events.

Reconstructed CC Fraction vs. ν True Energy. NDOS MC.
1

CC / (CC + NC)

0.8

0.6

0.4

0.2

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 5.5 Fraction of Reconstructed CC Events. Fraction of reconstructed CC events
out of the total sample: CC + NC, with at least one reconstructed track. MC simulation.

97

ν True Energy. Reconstructed/True Ratio. NDOS MC.

Reconstructed/True Ratio

1.2

1

0.8

0.6

0.4

All
0.2

CC
NC

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 5.6 Ratio Of Reconstructed Over Simulated Events. Reconstruction ratio as
a function of the predicted neutrino energy discriminated by interaction type. The ratio reconstructed/true for: all interactions (black), NC (red), and CC (blue) energy distributions.
All reconstructed events must have at least one reconstructed track. MC simulation.

As presented in section 4.3.2.2, the requirement of a minimum of 8 hits to reconstruct a
3D track is a strong cut resulting in an efficiency that strongly favors the desired CC events,
as shown in figure 5.5. At neutrino energies less than 0.1 GeV, no events are reconstructed.
For energies above 0.1 GeV the CC sample with at least one reconstructed track is about
four times larger than the reconstructed NC sample. From the total number of reconstructed
events: 81.4% are CC and 18.6% are NC. Within this sample there are three ratios of interest:
all reconstructed events to all simulated events, all reconstructed CC events to all simulated
CC events, and all reconstructed NC events to all simulated NC events, as seen figure
5.6. Events with energies less than 0.1 GeV are not reconstructed by the reconstruction
algorithms due to the lack of information. The NC event reconstruction rate drops below
50% for event energy less than 1.8 GeV, and the CC reconstruction rate dramatically drops

98

for event energy below 0.4 GeV. Summarizing, 26.4% of all simulated interactions are not
reconstructed. The effects of the remaining NC background is discussed in section 5.3.

5.2

Event Containment Criteria

Reconstructed neutrino interactions are required to originate2 in the vertex region (VR)
defined as (refer to figure 5.7):

|X| < 106 cm,
|Y | < 172 cm,
288 cm < Z < 452 cm.

(5.1)

These VR limits are chosen such that the number of neutrino interactions occurring in it
is maximum while the purity3 and efficiency4 of the limits are maximum as well. The VR
limits also ensure that full cell transverse areas are included.
The VR represents 8% of the total NDOS volume. The instrumented regions in front
(lower Z) and to the sides of the limits of the VR are used as veto regions in order to ensure
that the longest track of each event starts within the VR. Hits within these veto regions
imply that the track came from outside the VR. The reconstructed neutrino interaction 3D
coordinates, or vertex, are defined at the point with lowest Z coordinate of the longest track.
The simulation indicates that the containment of the reconstructed interaction vertex within
2 See section 4.2.
3 See equation (5.2).
4 See equation (5.3).

99

Figure 5.7 Vertex Region Of The Prototype Detector.

the VR has a purity (p) of:

p =

NR
= (96.7 ± 0.2)%,
NT

(5.2)

where NR is the number of reconstructed tracks starting within the VR that have a corresponding true track starting within the VR, and NT is the total number of reconstructed
tracks starting within the VR. The efficiency (ǫ) is:

ǫ =

NR
= (95.6 ± 0.2)%,
Ne

(5.3)

where Ne is the number of true tracks starting within the VR. The VR purity and efficiency
as a function of neutrino energy are summarized in table 5.1, at the end of this chapter.
To determine the energy of a muon it must range out and stop in the detector, therefore
it has to be contained within the detector. Containment of the longest track is also one of the
cuts designed to reduce background from cosmic ray muons. The longest track is required

100

Figure 5.8 Containment Region Of The Prototype Detector.

to be contained within a containment region (CR) defined as (refer to figure 5.8):

|X| < 119 cm,
|Y | < 184 cm,
288 cm < Z < 1406 cm.

(5.4)

These CR limits are chosen such that the number of neutrino events contained in it is
maximum while the purity5 and efficiency6 of the limits are maximum as well. The CR
limits also ensure that full cell transverse areas are included. In order for a longest track
to be fully contained, this has to start within the VR and end within the CR. Outside the
CR boundaries there are three veto cells between the edges of the CR and the edges of the
detector7 . Hits in these veto cells imply that the track is not fully contained, and therefore
5 See equation (5.5).
6 See equation (5.6).
7 From figure 5.8 it is clear that between Z = 0 and Z = 288 cm there are more than

three cells. All the instrumented volume in this region is used as veto.

101

the particle left the detector; or it came from outside the detector. The simulation indicates
that the containment of the longest track within the CR has a purity (p) of:

p =

NR
= (87.7 ± 0.4)%,
NT

(5.5)

where NR is the number of reconstructed tracks contained within the CR that have a corresponding true track contained within the CR, and NT is the total number of reconstructed
tracks contained within the CR. The efficiency (ǫ) is:

ǫ =

NR
= (76.1 ± 0.5)%,
Ne

(5.6)

where Ne is the number of true tracks contained within the CR. The CR purity and efficiency
as a function of neutrino energy are summarized in table 5.1, at the end of this chapter. These
purities and efficiencies are affected by the small number of active planes beyond the VR
that are used to identify if, and where, the muon stopped within the detector.
To verify the containment of a reconstructed longest track within the VR and the CR
the track must be 3D. The simulation indicates that the efficiency (ǫ) of reconstructed 3D
tracks is:
ǫ =

NR
= (79.6 ± 0.2)%,
Ne

(5.7)

where NR is the number of reconstructed 3D tracks with an associated true track, and Ne
is the total number of true tracks. The efficiencies as a function of neutrino energy are
summarized in table 5.1, at the end of this chapter. The low number of active channels
outside the VR is responsible for the 20% of longest tracks that are only reconstructed as
2D tracks. In these cases, one of the views does not have enough information to reconstruct
102

a 2D track, therefore, with only one 2D reconstructed track, a 3D reconstructed track is
impossible to make.

5.3

Charged Current Event Selection

In order to identify the CC neutrino events from the NC and cosmic ray backgrounds,
the NDOS MC simulation is used find the physical variables that are most sensitive to
the differences between them. To make such distinctions, various reconstructed physical
variables are investigated.

5.3.1

Longest Track Length Cut
Longest Track Length. NDOS MC.
Muon

600

Tracks

Non-muon
400

200

0
0

200

400

600

800

1000

1200

Track Length (cm)
Figure 5.9 Monte Carlo Simulated Longest Track Length Distributions For Muons
and Non-muons. Muon (blue), and non-muon (red) track length distributions. MC simulation.

103

The main characteristic of a νµ CC event of interest is that it has a muon. In these CC
events the longest track is typically that of the muon, and the longer the track, the lower is
the background from hadronic or electromagnetic tracks (non-muon), as seen in figure 5.9.
The next approach to identify CC events is to study the reconstructed longest track length
(LTL) for each event, and separately examine CC and NC samples. These tracks must start
within the VR and be 3D.

Longest Track Length. NDOS MC. Contained.
All
300

CC

Tracks

NC
200

100

0
0

200

400

600

800

Track Length (cm)

1000

1200

Figure 5.10 Monte Carlo Simulation Of The Longest Track Length Distributions.
All interactions (black), CC (blue), and NC (red) longest track length distributions. All
longest tracks are contained. MC simulation.

The LTL distributions are shown in figure 5.10 for each interaction type. Based on these
distributions, a cut is chosen at:
LT L > 200 cm.

(5.8)

Relation (5.8) divides the sample into two: the first one is that with LT L > 200 cm, which has
a 3.8% background coming from NC events, and the second one is that with LT L < 200 cm,

104

which has a 24.7% background coming from NC events. The contained reconstructed sample
shows that 63.9% of all the CC reconstructed tracks, and 17.1% of all the NC reconstructed
tracks are longer than 200 cm. The analysis sample has all the LT L > 200 cm and it is
assumed to be a CC sample with a 3.8% of NC background. The MC simulation indicates
that the purity (p) of the track length cut (TLC) is:

p =

NR
= (94.5 ± 0.1)%,
NT

(5.9)

where NR is the number of reconstructed tracks longer than 200 cm that have a corresponding
true track longer than 200 cm, and NT is the total number of reconstructed tracks longer
than 200 cm. The efficiency (ǫ) is:

ǫ =

NR
= (59.6 ± 0.3)%,
Ne

(5.10)

where Ne is the number of true tracks longer than 200 cm. The TLC purity and efficiency as
a function of neutrino energy are summarized in table 5.1, at the end of this chapter. These
relatively low efficiencies come from the combined effect of the reconstructed 3D tracks, the
reconstructed end point of track, and contained tracks efficiencies.

5.3.2

Cosmic Ray Cuts

The containment cut on the longest track in an event removes a large fraction of the cosmic
rays that are within the NuMI time window. However, those that appear to stop within
the vertex region and slip through the containment cut can be mistaken for a CC neutrino
event.

105

(a)

(b)

Figure 5.11 Cosmic Rays Rejection Zone By The Cut In cos θY . Reconstructed cosmic
tracks in red with (a) cos θY = 0.6, and with (b) cos θY = 1.0.

106

Additional cuts on the angles of the longest track, discussed in section 4.5, are necessary
to further reduce this background. In section 4.5 three cosines were defined. The cos θY
distributions for MC simulation and out-of-time data (see figure 4.32b) present clear differences between signal and background. The longest track cos θY ratio: background/signal,
becomes less than 1 for | cos θY | < 0.6. In order to reject cosmic rays, only in-time events
with longest tracks with | cos θY | < 0.6 are accepted. This cut rejects 90.7% of the cosmic
background, and keeps 94.4% of the neutrino data. The cos θNuMI distributions for MC
simulation and out-of-time data (see figure 4.32c) present a clear difference between signal
and background. The longest track cos θNuMI ratio: background/signal, becomes less than
1 for cos θNuMI > 0.6. To further reject cosmic rays, only in-time events with longest tracks
with cos θNuMI > 0.6 are accepted. This cut rejects 96.9% of the cosmic background, and
keeps 96.3% of the neutrino data. The two cuts combined reject: 98.5% of all cosmic rays,
and keep 93.4% of the neutrino data.
Two sample cosmic events that span the rejection region defined by the cut in cos θY are
shown in figure 5.11. A cosmic ray with cos θY = 0.6 is shown in figure 5.11a, and a cosmic
ray with cos θY = 1.0 is shown in figure 5.11b. Another two cosmic events that span the
rejection region, defined by the cut in cos θNuMI are presented in figure 5.12. A cosmic ray
with cos θNuMI = 0 is shown in figure 5.12a, and a cosmic ray with cos θNuMI = 0.6 is shown
in figure 5.12b.
To summarize, the cosmic cuts accept tracks that fulfill:

| cos θY | < 0.6, cos θNuMI > 0.6.

(5.11)

The correlations between cos θY and cos θNuMI are shown in figure 5.13. Most of the cosmic

107

(a)

(b)

Figure 5.12 Cosmic Rays Rejection Zone By The Cut In cos θNuMI . Reconstructed
cosmic tracks in red with (a) cos θNuMI = 0, and with (b) cos θNuMI = 0.6.

Longest Track: Cos θY vs. Cos θNuMI. NDOS Cosmic DATA.

Longest Track: Cos θY vs. Cos θNuMI. NDOS MC.

1

1

200

0.5

150

0

100

500

400

Cos θY

Cos θY

0.5

300

0

200

-0.5

-0.5

50

100

-1
0

0.2

0.4

0.6

Cos θNuMI

0.8

1

-1
0

0

(a)

0.2

0.4

0.6

Cos θNuMI

0.8

1

0

(b)

Figure 5.13 cos θY vs. cos θNuMI . (a) Cosmic rays. (b) MC simulation. The orange
rectangles define the acceptance region.

108

tracks occur in the rejection region defined above, as shown in figure 5.13a, while from figure
5.13b it is clear that the MC signal is mostly inside the acceptance region.
The out-of-time cosmic rays that pass the cuts in relations (5.11) are scaled to the size
of the trigger window8 , and then subtracted from the in-time data that pass these cuts to
obtain the final νµ + ν¯µ CC candidates sample. This subtraction is done bin by bin in the
number of selected neutrino candidates vs. neutrino energy figure9 .

5.3.3

Minimum Ionizing Particle Cut
Track
Track dE/dX.
dE/dX. NDOS Cosmic
CosmicDATA.
DATA.
0.12
0.1

Tracks

0.08
0.06
0.04
0.02
0
0

2

4

6

8

10

Track dE/dX (MeV/cm)

Figure 5.14 Cosmic Track Mean dE/dX. NDOS cosmic data.

The longest track in each event is assumed to be a muon. However, a background of
other particles, mostly hadrons, is present. The reconstruction algorithms can mistake an
electromagnetic shower as a long single track. CC interactions of the νe component in the
8 Recall relation (4.11).
9 See figure 8.6.

109

NuMI beam can produce an energetic electromagnetic shower. Most of the electromagnetic
tracks that fake a muon can be rejected on the basis on their energy deposition profile.
Longest Track MIP Fraction. NDOS MC.

-3

× 10

µ

80

Non-µ
Areas Normalized

60

40

20

0

0

0.2

0.4

0.6

0.8

1

MIP Fraction

Figure 5.15 MIP Fraction. Muon (blue) and non-muon (red). Areas normalized. MC
simulation.

The best way to know the energy deposition profile for muons is to investigate cosmic
muons. The distribution of the mean dE/dX of contained cosmic muons is presented in
figure 5.14. From this figure it is possible to define a MIP fraction (MIPfrac ) as the fraction
of cells within a track with:

1.1 MeV/cm ≤ dE/dX ≤ 2.7 MeV/cm,

(5.12)

range that encompasses 90% of all the cosmic muons, which have a mean of 1.95 MeV/cm.
A MIP must have a MIP fraction close to 1. The MIP fraction for simulated muon and
non-muon tracks is shown in figure 5.15. The mean MIP fraction for muons is 0.75, while
that of non-muons is 0.65. 99.8% of all muons have MIP fraction higher than 0.4, while
110

69.8% of all non-muons have MIP fraction higher than 0.4. These percentages motivate a
new cut, the MIP cut, defined as:
MIPfrac > 0.4.

(5.13)

The fractions of the distinct particles producing the longest reconstructed track (LRT) are
presented in figure 5.16. The fraction of events in which a muon produces the LRT is 0.85, the
other 0.15 is equally shared between protons, charged pions, electrons, and photons10 . Some
of these protons and charged pions come from NC interactions, however most of them are
secondary particles from high y (defined in equation (2.17)) interactions in the CC sample.
Electrons come from the νe CC background to the νµ CC signal, and can be rejected based
on their energy deposition profile.
Longest Track Particle ID. NDOS MC.
Before MIP Cut
0.8
After MIP Cut

Fraction

0.6

0.4

0.2

0

Muon

Proton

Pion

Electron

Photon

Other

Particle

Figure 5.16 Longest Track Particle Identity. Fraction of particles prior (red) and posterior (blue) to the MIP cut. MC simulation.

Rejecting tracks based on relation (5.13) considerably reduces the fraction of electromag10 Photons here refer to the primary particle that originates the electromagnetic shower.

111

Neutrino Type. NDOS MC.

Interaction Type. NDOS MC.

0.8
Before MIP Cut

After MIP Cut

After MIP Cut

Fraction

0.6

Fraction

Before MIP Cut

1

0.4

10-1

0.2

0

νµ

νe + νe

νµ

CC

Neutrino Type

(a)

Interaction Type

NC

(b)

Figure 5.17 Consequences Of The MIP Cut. Before (red) and after (blue) the MIP cut
for (a) neutrino types, and (b) interaction types. MC simulation.
netic tracks (from νe + ν¯e CC) in the νµ CC sample from about 4% to about 1%, as shown in
figure 5.16. The impact of this cut on other aspects of the analysis is minimal, as seen from
figure 5.17. The composition of the neutrino sample before and after the MIP cut is seen in
figure 5.17a, with 1.4% of νe + ν¯e CC background. The cut reduces the NC background to
a 2.9% level from the initial 3.5%, as seen in figure 5.17b. The MC simulation indicates that
the purity (p) of the MIP cut is:

p =

NR
= (91.3 ± 0.6)%,
NT

(5.14)

where NR is the number of reconstructed tracks with MIP fraction larger than 0.4 and
identified in true as a muon, and NT is the total number of reconstructed tracks with MIP
fraction larger than 0.4. The efficiency (ǫ) is:

ǫ =

NR
= (99.1 ± 0.2)%,
Ne

(5.15)

where Ne is the number of true muons. The MIP cut purity and efficiency per energy bins
are summarized in table 5.1, at the end of this chapter.
112

After the event selection procedure is complete, the sought set of νµ CC events is obtained.
Each of these events goes through a process of energy reconstruction that produces the
neutrino energy distribution of the selected sample.
Table 5.1 shows that the efficiency on the track length cut (TLC) increases with neutrino energy as the LTL reconstruction efficiency increases with length. The purity on the
containment cut (CR) decreases with increasing neutrino energy as the muons become more
energetic and leave the detector more frequently. Table 5.1 also shows how the MIP cut purity decreases with increasing neutrino energy as the contained reconstructed longest track
is not a muon. This is because the energetic muon escapes the detector more frequently for
high y CC interactions. All the efficiencies and purities shown in table 5.1 are calculated
to know the performance of each particular selection criterion. The calculations done for a
particular selection criterion have no dependency on previous or posterior selection criteria.

113

Cut\E (GeV)
3D
Efficiency
Error
TLC
Purity
Error
Efficiency
Error
VR
Purity
Error
Efficiency
Error
CR
Purity
Error
Efficiency
Error
MIP
Purity
Error
Efficiency
Error

0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-10.0
74.8%
0.3%

79.9%
0.3%

80.2%
0.2%

80.6%
0.3%

81.3%
0.5%

81.4%
0.7%

80.3%
0.9%

80.7%
0.5%

95.7%
0.2%
38.8%
0.4%

94.6%
0.2%
53.8%
0.4%

95.1%
0.1%
59.3%
0.3%

94.8%
0.2%
63.2%
0.4%

93.4%
0.4%
69.2%
0.7%

92.4%
0.5%
71.5%
0.9%

90.7%
0.7%
76.0%
1.0%

92.2%
0.4%
81.6%
0.6%

97.4%
0.3%
97.0%
0.4%

97.0%
0.2%
96.4%
0.3%

96.6%
0.2%
95.6%
0.2%

96.6%
0.2%
95.5%
0.2%

96.4%
0.4%
94.9%
0.5%

95.4%
0.5%
94.8%
0.6%

94.2%
0.7%
93.0%
0.8%

95.3%
0.4%
93.9%
0.4%

97.7%
0.4%
71.4%
1.1%

94.4%
0.4%
68.8%
0.7%

88.2%
0.4%
72.1%
0.5%

83.6%
0.5%
73.8%
0.5%

86.4%
0.8%
84.4%
0.9%

80.0%
1.2%
86.2%
1.0%

71.6%
1.6%
80.6%
1.4%

55.6%
1.1%
73.2%
1.0%

99.1%
0.3%
99.9%
0.1%

94.3%
0.5%
99.8%
0.1%

90.0%
0.6%
99.9%
0.1%

87.7%
0.7%
99.9%
0.1%

87.2%
1.4%
99.6%
0.2%

83.3%
1.8%
100%
0

77.7%
2.6%
99.4%
0.5%

58.6%
2.4%
98.4%
0.6%

Table 5.1 Event Selection Performance. Purity and efficiency for the various cuts made
during the event selection. The columns in this table represent the energy bins used in
this analysis. The last column includes energies from 4.0 GeV to 10 GeV. No statistically
significant data exist below 0.5 GeV.

114

Chapter 6
Event Energy Reconstruction
The energy of a neutrino interaction must be determined from the energy deposited in
the detector by the final state particles. In νµ CC events, different physical processes are
involved in the determination of the energy of the muon or the hadronic system. In some
cases, a muon will stop in the detector before it decays or is captured in an atomic orbit.
The ionization energy loss mechanism of a muon is well understood, so that the energy
of a muon can be directly related to its range, which is estimated by its track length in
the detector1 . The energy deposited in the detector by hadrons is reconstructed using the
detector as a calorimeter, i.e. the deposited energy is scaled, using the hadronic energy in
the MC simulation, in order to estimate the full energy encompassed within the hadronic
system.

6.1

Muon Energy Estimation

The MC simulation of the particles interacting with the detector incorporates the ionization
properties of the muon as well as the distortions introduced by the detector structure and
operation. For muon energies between 0.4 GeV to above 3 GeV, and the mixture of materials
of the NOνA detectors (average detector density of 1.02 g/cm3 [107]), the corresponding
ionization energy loss rates vary from 2.1 MeV cm2 /g to 2.3 MeV cm2 /g. As the muon
1 See [87] for a detailed discussion on passage of particles through matter.

115

µ: True Energy vs. Reconstructed Track Length. NDOS MC.

µ: True Energy vs. Reconstructed Track Length. NDOS MC. Profile.

Mean µ Energy (GeV)

µ Energy (GeV)

6
30

4

20

2

0
0

10

0

500

1000

Track Length (cm)

1500

6

4

2

0
0

2000

(a)

500

1000

Track Length (cm)

1500

2000

(b)

Figure 6.1 Muon: True Energy vs. Reconstructed Track Length. (a) True energy of
muons as a function of the reconstructed track length. (b) Profile plot of (a). MC simulation.
energy drops below 0.4 GeV, the energy loss rate rises and reaches higher ionization levels
shortly before stopping. Therefore, the experimentally determined muon energy and the
range are nearly linearly related. The true muon energy and its reconstructed track length
are strongly correlated, as shown in figure 6.1. The reconstructed track length is used to
determine the energy of each observed muon. The true energy of the muons (ET ) as a
function of their reconstructed track length (LR ) is (refer to figure 6.1a):

ET = p0 + p1 LR ,

(6.1)

where the pi are parameters to be determined. Deviations from equation (6.1), particularly
at short track lengths, come from uncontained muons, as described in section 4.3.2. Equation
(6.1) is a good approximation to describe the relationship between the true energy and the
reconstructed track length (longer than 200 cm) of the muons; however, the data shown in
figure 6.1a do not follow a straight line, these are scattered inside a narrow band with some
width. In order to reconstruct the energy of the muon, the most probable value of true energy
per reconstructed track length bin is calculated and drawn as a function of reconstructed

116

track length, as shown in figure 6.1b. The error bars are the spread of the energy per track
length bin.
The pi in equation (6.1) are functions of LR , and are set equal to parameters obtained
from straight line fits. More than one straight line fit are required to better represent different
regions in figure 6.1b. The region: LR < 580 cm (refer to figure 6.2a), has χ2 = 90.91 for
ndf = 74. The region: 580 cm < LR < 920 cm (refer to figure 6.2b), has χ2 = 72.34 for
ndf = 66. The last region: LR > 920 cm (refer to figure 6.2c), has χ2 = 2634.5 for ndf = 144,
and is more scattered than the first one. The three regions combined have χ2 = 9340.46 for
ndf = 284. In order to minimize the χ2 , a handful of straight line fits are calculated for the
first and last regions. For the last region, e.g., there are five fits with the following results:
• 920 cm < LR < 1115 cm: χ2 = 56.6 for ndf = 37.
• 1115 cm < LR < 1285 cm: χ2 = 53.4 for ndf = 32.
• 1285 cm < LR < 1375 cm: χ2 = 30.2 for ndf = 16.
• 1375 cm < LR < 1490 cm: χ2 = 30.6 for ndf = 20.
• 1490 cm < LR : χ2 = 132.3 for ndf = 31.
All the straight line fits are constrained to be continuous. The parameters of the fits are
used to obtain a first order approximation of the reconstructed energy.
The final version of the reconstructed muon energy is obtained after taking the difference
of the true energy and the approximated reconstructed energy. Small corrections, Cb , of the
order of 1% of the muon energy, are added to these approximated energies, per track length
bin, in order to achieve that the mean of the difference between the true energy (ET ) and
the reconstructed energy (ER ) of the muon is equal to zero: ∆E = ET − ER = 0, as seen in
117

figure 6.3a. Finally, the reconstructed energy of the muon is:

ER = p0 (LR ) + p1 (LR )LR + Cb (LR ).

(6.2)

The parameters of the fits and corrections are used to obtain the reconstructed muon energy
in data.

(a)

(b)

(c)
Figure 6.2 Profile Plots Of Figure 6.1a. (a) LR < 580 cm. (b) 580 cm < LR < 920 cm.
(c) LR > 920 cm.

The uniformity of the density of the NDOS is broken at the muon catcher, thus, muons
that reach the muon catcher require an additional term in equation (6.2) that would take
into account the portion of the muon’s trajectory that goes through the steel. For these

118

muons, equation (6.2) changes to:

ER = p0 (LR ) + p1 (Ls )Ls + p2 (Lµc )Lµc + Cb (LR ),

(6.3)

where Ls is the portion of the muon’s trajectory that goes through the plastic + scintillator
bulk, the p2 are fit parameters, and Lµc is the portion of the trajectory that goes through
the muon catcher. The reconstructed track length is: LR = Ls + Lµc .
µ Energy Resolution: ∆E. NDOS MC.

µ: Energy Resolution vs. Reconstructed Energy. NDOS MC.

4

800

120
100

2

600
80

0

Tracks

∆E (GeV)

Mean
0.009899
RMS
0.2966
Constant
726.7
Mean
-0.04572
Sigma
0.1303

60

400

40

-2

200
20

-40

1

2

µ Energy (GeV)

3

4

0

0

-4

(a)

-2

0

∆E (GeV)

2

4

(b)

Figure 6.3 Muon Energy Resolution. (a) True minus reconstructed muon energy (∆E)
as a function of the reconstructed muon energy. The average ∆E is effectively zero. (b)
Overall muon energy resolution. MC simulation.

The overall muon reconstructed energy resolution is obtained from the Gaussian fit shown
in figure 6.3b. The spread in the ∆E from the fit is the reconstructed energy resolution.
The tail of the distribution accounts for about 5% of the muons. These are considered
contained, but their true trajectory leaves the detector, therefore the difference between true
and reconstructed energies is positive and not close to zero. From the spread in the fit,
the resolution in reconstructed muon energy is 130 MeV. The muon energy resolution at
energies higher than 2 GeV is better than lower energies since the containment issues are
less frequent when compared to shorter, and less energetic, muon tracks. High energy muons
119

True & Reconstructed µ Energy Spectra. NDOS MC.
2.5

103 Muons / 0.2 GeV

True MC
Reco MC

2

1.5

1

0.5

0
0

1

2

3

µ Energy (GeV)

4

5

Figure 6.4 Muon Energy Distributions. Muon true (black), and reconstructed (red)
energy distributions. MC simulation.

often have their momentum vector almost parallel to the beam’s direction, and these reach
the muon catcher, which helps to contain them. Around 1 GeV the resolution is about 14%,
i.e. 140 MeV.
Even with the best attempts to properly reconstruct the muon energy, a comparison of
the simulated energy for muons in CC events with their reconstructed values shows that there
is a tendency to move muons with energies lower than 0.6 GeV to an energy above 0.6 GeV,
as shown in figure 6.4. The reconstructed muons with energies about 1 GeV sharply peak
while the true muon energy is more spread out. This is the result of the LTL corrections
described in section 4.3.2.

120

6.2

Hadronic Energy Estimation

The NOνA detectors were designed to obtain excellent electromagnetic shower identification
and energy resolution. This is accomplished with a detector comprised of primarily carbon
and hydrogen (low Z nuclei), resulting in an electromagnetic sampling fraction better than
65% [107]. These characteristics of the detector, however, result in a poor performance as a
hadronic calorimeter.
CC interactions produce hadrons in addition to the characteristic lepton. Hadronic energy
estimation is different from electromagnetic energy estimation (see [87] for details). Recalling
section 4.3.2, protons and charged pions often do not leave hit patterns to be reconstructed
into tracks in the NOνA detector, resulting in degraded energy resolution [108]. The presence
of hadronic showers combined with the neutrino event containment details studied in section
4.3.2, have as a consequence that the hadronic energy estimation is done by converting
deposited energy in the detector to real deposited energy.
The observed hadronic energy is defined as the sum of all the deposited energy in the
event that does not belong to the longest track, which is assumed to be the muon. Different
zones in the NDOS VR measure deposited energy differently depending on their location
and number of active channels. Since the neutrino event containment is only determined
by the longest track properties, it is necessary to introduce a method to take into account
hadronic energy that leaves the detector, is deposited in dead material or in cells without
APDs. Therefore, the energy corrections are determined by dividing the NDOS VR into 6
zones, as shown in figure 6.5. The limits of these zones are defined by:
• Zone 1: 53 cm > |X| > 106 cm, 288 cm < z < 345 cm.
• Zone 2: 53 cm > |X| > 106 cm, 345 cm < z < 402 cm.
121

Figure 6.5 Hadronic Energy Classification Zones. The coordinates of the edges of the
distinct zones are shown with arrows. All zones fulfill the condition: 172 cm > |Y|. Zone 1
is the union of the two zones labeled with the number: 1. These two zones are equivalent,
and therefore are united into one. The same is true for the zones 2 and 3. Zones 4 through
6 are in the middle of the detector.

• Zone 3: 53 cm > |X| > 106 cm, 402 cm < z < 452 cm.
• Zone 4: 53 cm < |X|, 288 cm < z < 345 cm.
• Zone 5: 53 cm < |X|, 345 cm < z < 402 cm.
• Zone 6: 53 cm < |X|, 402 cm < z < 452 cm.
All 6 zones are rectangular boxes fulfilling the condition: 172 cm > |Y|. Events are
classified by the zone in which their vertex is located. Events with an interaction occurring
in zones 1 through 3 will have a higher probability for energy to leave through the sides
of the detector, when compared to the other zones. In addition, zone 3 will have a higher
probability to leak energy into the region without active cells, when compared to the other
zones. Events with its neutrino interaction occurring in zone 6 will deposit a good fraction
of their energy into the region without active cells. Events with its neutrino interaction
122

occurring in zones 4 and 5 will deposit most of the hadronic energy inside the detector and
the NDOS VR. All these zones have inactive or missing channels. Due to the differences in
energy deposition mentioned above, each zone needs its own energy estimation parameters.
Energy estimation parameters for each zone are determined through a comparison between
the deposited hadronic energy (the energy read by detector) and the true energy of the
hadronic system.
Hadronic Energy: True vs. Deposited. NDOS MC. Zone 3. Profile.

Hadronic Energy: True vs. Deposited. NDOS MC. Zone 4. Profile.

8

Mean Hadronic Energy (GeV)

Mean Hadronic Energy (GeV)

8

6

4

2

0
0

0.5

1

1.5

Deposited Energy (GeV)

6

4

2

0
0

2

(a)

0.5

1

1.5

Deposited Energy (GeV)

2

2.5

(b)

Figure 6.6 True Hadronic Energy vs. Deposited Energy. Mean true hadronic energy
per deposited energy bin as a function of the deposited hadronic energy of (a) zone 3, and
(b) zone 4. MC simulation.

The mean true hadronic energy, per deposited hadronic energy bin, as a function of the
deposited hadronic energy of zone 3 is shown in figure 6.6a. Zone 3 is the sample of events
that deposits less energy into the detector, when compared to the other zones. Figure 6.6b
is the equivalent of figure 6.6a for zone 4, which is the zone that gets the largest hadronic
energy deposition of all 6 zones. The correlations between true hadronic energy (HT ) and
deposited hadronic energy (HR ), which are distinctly non-linear in all the 6 zones, help
to define the reconstructed hadronic energy (Ehad ). These correlations are presented by
polynomials. The degree (A) of the polynomial is a function of the zone. The coefficients of

123

the polynomials (qa ) are used to obtain the Ehad :
A
a,
qa H R

Ehad =

(6.4)

a=0

There is one equation (6.4) per zone.
Hadronic: ∆E vs. Reconstructed Energy. NDOS MC.
103

2

×103

Hadronic Energy Resolution: ∆E. NDOS MC.
Mean
RMS
Constant
Mean
Sigma

4

1.5

102

Events

∆ E (GeV)

2

0.06786
0.4268
1174
-0.02424
0.1909

0

1

10

-2

0.5

-4
0

1

2

3

Reconstruted Energy (GeV)

4

5

1

0
-4

-2

(a)

0

∆E (GeV)

2

4

(b)

Figure 6.7 Hadronic Energy Resolution. (a) True minus reconstructed hadronic energy
(∆E) as a function of the reconstructed hadronic energy. The average ∆E is effectively zero.
(b) Overall hadronic energy resolution. MC simulation.

Similar to the muon case, each equation (6.4) requires small corrections, Cg , of the order
of 5% of the hadronic energy, per deposited hadronic energy bin, in order to achieve that the
mean of the difference of energies (∆E = HT - Ehad = 0) in the hadronic system is equal
to zero, as seen in figure 6.7a. Finally, the hadronic reconstructed energy, Ehad , is:
A
a + C (H ).
qa H R
g
R

Ehad =

(6.5)

a=0

The hadronic energy resolution, in figure 6.7a, at 1 GeV is 330 MeV (33%), at 180 MeV is
0.5 GeV (36.5%), and it is 70 MeV at 0.2 GeV (35%). From the Gaussian fit shown in figure
6.7b, the overall hadronic energy resolution is 190 MeV. The tails account for about 5% of
124

True & Reconstructed Hadronic Energy Spectra. NDOS MC.

103 Muons / 0.2 GeV

2.5

True MC
Reco MC

2

1.5

1

0.5

0
0

1

2

3

4

5

Hadronic Energy (GeV)

Figure 6.8 Hadronic Energy Distributions. True hadronic (black) and reconstructed
(red) energy distributions. MC simulation.

all events.
The distributions in reconstructed and true hadronic energies shown in figure 6.8 have
bins of 200 MeV. From figure 6.6 it is clear that the spread of the data at energies above
1 GeV introduces deviations from the fits, and therefore underestimation of the hadronic
energies larger than 1.6 GeV is seen in figure 6.8.

6.3

Quasi-elastic And Non-quasi-elastic Classification

Section 2.1 presented differences between QE and non-QE interactions. Besides those differences, the two samples also present different efficiencies in the reconstruction process. These
two different neutrino interaction samples are used separately to calculate e.g. neutrino
fluxes and cross sections. Among the differences presented in section 2.1 is the hadronic
energy deposition, which is in general higher for non-QE interactions. This difference in
125

True Hadronic Energy Spectra. NDOS MC.

QE

1.5

Events

Non-QE
1

0.5

0
0

1

2

3

4

Hadronic Energy (GeV)

Figure 6.9 QE And Non-QE Hadronic Energy Distributions. True hadronic QE (blue)
and non-QE (red) energy distributions. MC simulation.

hadronic energy deposition is presented in figure 6.9. The mean true hadronic energy for
the QE sample is 150 MeV, while the mean for the non-QE sample is 910 MeV. The QE
true hadronic energy falls rapidly after 200 MeV, and it is at this same energy that the nonQE true hadronic energy starts to become important. Therefore, a separation is possible
following the criteria:

Ehad < 200 MeV → QE, Ehad > 200 MeV → non-QE.

(6.6)

With this separation, 84.1% of all true QE events, and 0.4% of all true non-QE events have
true hadronic energy lower than 200 MeV. Relations (6.6) are applied to the reconstructed
hadronic energy to make the separation between the QE and the non-QE samples.
The comparisons between true and reconstructed muon and hadronic energy are presented in figure 6.10. In these figures the true MC distributions represent the QE and
126

True & Reconstructed µ Energy Spectra. NDOS MC. QE.

True & Reconstructed µ Energy Spectra. NDOS MC. Non-QE.

Fraction of muons / 0.2 GeV

Fraction of muons / 0.2 GeV

True MC
0.2

Reco MC
Areas = 1

0.15

0.1

0.05

0
0

1

2

3

4

µ Energy (GeV)

Reco MC
Areas = 1
0.2

0.1

0
0

5

True MC

0.3

1

2

(a)

5

True & Reconstructed Hadronic Energy Spectra. NDOS MC. Non-QE.

True & Reconstructed Hadronic Energy Spectra. NDOS MC. QE.
0.3

True MC

True MC
Fraction of events / 0.2 GeV

Fraction of events / 0.2 GeV

4

(b)

1

Reco MC

0.8

Areas = 1
0.6

0.4

0.2

0
0

3

µ Energy (GeV)

0.2

0.4

0.6

0.8

Reco MC
0.2

0.1

0
0

1

Hadronic Energy (GeV)

Areas = 1

1

2

3

4

Hadronic Energy (GeV)

(c)

(d)

Figure 6.10 Muon And Hadronic Energy Distributions For The QE And Non-QE
Events. The energy distributions for the true (black) and reconstructed (red) samples, with
muon energy in (a) QE and (b) non-QE events, and hadronic energy in the (c) QE and (d)
non-QE events. Areas normalized to 1 in order to compare the shapes of the distributions.
MC simulation.

127

non-QE interactions as defined in the MC simulation. The reconstructed MC distributions
represent the interactions as defined by relations (6.6). The poor tracking coverage for muons
immediately downstream of the VR causes muon energies below 0.6 GeV to be systematically reconstructed to energies higher than the true values by 0.2 GeV (in the QE case), and
0.4 GeV (in the non-QE case), as shown in figures 6.10a and 6.10b, respectively. Apart from
this bias at low muon energy, the distributions are similar for muon energies above 1.0 GeV.
The reconstructed hadronic energy in the QE sample is forced by definition to be less than
0.2 GeV, as shown in figure 6.10c. The reconstructed hadronic energy in the non-QE sample
is shifted slightly toward energies lower than the truth, as shown in figure 6.10d, due to a
bias for large hadronic energies to be reconstructed slightly lower in energy.
µ Energy Resolution: ∆E. NDOS MC, QE.
600

Mean
RMS
Constant
Mean
Sigma

×10

-0.01601
0.2207
602.1
-0.04917
0.1337

3

µ Energy Resolution: ∆E. NDOS MC, Non-QE.
Mean
RMS
Constant
Mean
Sigma

1

0.8

0.02449
0.3841
1043
-0.04826
0.1636

Events

Events

400
0.6

0.4

200

0.2

0
-4

-2

0

∆ E (GeV)

2

0
-4

4

-2

(a)

0

∆ E (GeV)

2

4

(b)

µ: Energy Resolution vs. Reconstructed Energy. NDOS MC, QE.

µ : Energy Resolution vs. Reconstructed Energy. NDOS MC, Non-QE.

4

4

80

∆ E (GeV)

∆ E (GeV)

2
60
0
40
-2

2

150

0

100

-2

50

20

-4

0

1

2

µ Energy (GeV)

3

4

0

-4

(c)

0

1

2

µ Energy (GeV)

3

4

0

(d)

Figure 6.11 Muon Energy Resolutions For QE And Non-QE Events. Overall (a) QE
and (b) non-QE energy resolutions. Muon (c) QE and (d) non-QE true minus reconstructed
energy (∆E) as a function of muon reconstructed energy. The average ∆E are effectively
zero. MC simulation.
128

Following the procedure presented in section 6.2 gives the muon and hadronic energy
resolutions. The muon energy resolution for the QE sample at 1 GeV is 150 MeV (15%)
and at 2 GeV is 100 MeV (5%). The overall muon energy resolution for the QE sample
is 130 MeV, as shown in figure 6.11a. The muon energy resolution for the non-QE sample
at 1 GeV is 150 MeV (15%), and at 2 GeV is 130 MeV (6.5%). The overall muon energy
resolution for the non-QE sample is 160 MeV, as shown in figure 6.11b. The true minus
reconstructed muon energies as a function of reconstructed muon energy are shown in figure
6.11c, for QE, and in figure 6.11d, non-QE events.
A comparison between figures 6.3a and 6.11d shows that the muon energy resolution
around 0.8 GeV comes from non-QE muons that escape the detector, but are reconstructed
as contained. This effect was discussed in section 4.3.2. These muons deviate from the
original neutrino direction due to the high y of the event.
The hadronic energy resolutions in the QE and non-QE samples are different, as shown
in figure 6.12. The overall hadronic energy resolution is: 60 MeV for the QE sample, as
shown in figure 6.12a, and 240 MeV for the non-QE sample, as shown in figure 6.12b. Both
hadronic energy resolutions, at the average energy deposition in each case, are 30%.
×10

3

Hadronic Energy Resolution: ∆E. NDOS MC, QE.
Mean
RMS
Constant
Mean
Sigma

1

Mean
RMS
Constant
Mean
Sigma

800

600

Events

Events

0.8

Hadronic Energy Resolution: ∆E. NDOS MC, Non-QE.

0.002858
0.1607
1136
-0.03886
0.05601

0.6

0.09171
0.4872
710.8
-0.0129
0.2393

400

0.4
200
0.2

0
-3

-2

-1

0

∆E (GeV)

1

2

0
-4

3

(a)

-2

0

∆E (GeV)

2

4

(b)

Figure 6.12 Overall Hadronic Energy Resolutions. (a) QE and (b) non-QE events. MC
simulation.

129

6.4

Neutrino Energy Estimation
ν: ∆E vs. Reconstruted Energy. NDOS MC.

ν: ∆E vs. True Energy. NDOS MC.

3

3

2

2

300

300
1

∆E (GeV)

∆E (GeV)

1
200
0

-1

200

0

-1
100

100

-2

-3
0

-2

2

4

0

6

ν Energy (GeV)

-3
0

2

4

6

ν Energy (GeV)

(a)

8

10

0

(b)
1.2

× 10

3

ν Energy Resolution: ∆E. NDOS MC.
Mean
0.08721
RMS
0.4914
Constant
1008
Mean
-0.01775
Sigma
0.255

1

Events

0.8

0.6

0.4

0.2

0
-4

-2

0

∆E (GeV)

2

4

(c)
Figure 6.13 Neutrino Energy Resolution. Neutrino true minus reconstructed energy
(∆E) as a function of the (a) reconstructed (the average ∆E is effectively zero) and (b) true
neutrino energy. (c) Overall neutrino energy resolution. MC simulation.

The reconstructed neutrino energy (Eν ) in each event is the sum of the two previously
estimated energies:
Eν = ER + Ehad + Cu (ER + Ehad ),

(6.7)

where Cu are small corrections, of the order of 3% of the neutrino energy, per neutrino
energy bin, introduced in order to achieve the difference between true neutrino energy and
reconstructed neutrino energy centered at zero, as seen in figure 6.13a. At the peak of the
neutrino energy distribution, near 2.0 GeV, the neutrino energy resolution is 200 MeV (10%).
130

The neutrino energy resolution as a function of the true neutrino energy is presented in figure
6.13b, where the effect of uncontained muons reconstructed as contained is more evident for
energies higher than 3 GeV. The overall neutrino energy resolution is 255 MeV, as shown in
figure 6.13c. The tails account for about 5% of all events.
True & Reconstructed ν Energy Spectra. NDOS MC.

True MC

103 Events / 0.5 GeV

2.5

Reco MC
2

1.5

1

0.5

0
0

2

4

ν Energy (GeV)

6

Figure 6.14 Neutrino Energy Distributions. The neutrino true (black) and reconstructed
(red) energy distributions. Both distributions have the same number of events. MC simulation.

The neutrino energy distributions, with bins of 500 MeV, are in reasonable agreement, as
shown in figure 6.14. The effect observed earlier of muons below 0.6 GeV being reconstructed
at higher energies is reflected in the movement of events with true neutrino energies between
0.5 GeV and 1.0 GeV to higher values in the reconstructed neutrino energy. Also, inefficiencies in the containment of muons account for the underestimation of the neutrino energies
for events with energies higher than 3 GeV. The tendency observed earlier to reconstruct
hadronic energies above 2.0 GeV to slightly lower values contributes as well. Those events
are reconstructed between the 1 GeV and the 2 GeV energy bins.
131

ν: ∆E vs. True Energy. NDOS MC. QE.

ν: ∆E vs. Reconstruted Energy. NDOS MC, QE.
3

3

150

120

2

2
100

1
80

0

∆ E (GeV)

∆ E (GeV)

1

60

0

-1

40

-1

-2

20

-2

0

-3
0

-3
0

1

2

ν Energy (GeV)

3

4

100

50

1

2

ν Energy (GeV)

(a)

3

4

0

(b)
ν: ∆E vs. True Energy. NDOS MC. Non-QE.

ν: ∆E vs. Reconstruted Energy. NDOS MC, Non-QE.
3

4

2

200

200

2
150

∆ E (GeV)

∆ E (GeV)

1

0
100

150
0
100
-1

-2

-4

0

50

2

4

ν Energy (GeV)

6

50

-2

-3
0

0

(c)

2

4

ν Energy (GeV)

6

0

(d)

Figure 6.15 QE And Non-QE True Minus Reconstructed Neutrino Energies. QE
true minus reconstructed neutrino energy (∆E) as a function of (a) reconstructed and (b)
true neutrino energy. Non-QE ∆E as a function of (c) reconstructed and (d) true neutrino
energy. MC simulation.

132

From the separation of the neutrino sample between QE and non-QE events done in section 6.3, the neutrino energy resolution, as a function of the reconstructed neutrino energy,
are as shown in figures 6.15a and 6.15c for the QE and non-QE samples respectively. The
neutrino energy resolution at 2 GeV is: 130 GeV (6.5%) for the QE sample, and 220 GeV
(10.1%) for the non-QE sample. The neutrino energy resolution as a function of the true
neutrino energy are shown in figures 6.15b and 6.15d for the QE and non-QE samples respectively. From these figures, the effect of uncontained muons reconstructed as contained
is evident in the non-QE. These muons come from high y events, and their angles deviate
from the original neutrino angle, therefore the muons escape the detector before stopping.
ν Energy Resolution: ∆E. NDOS MC, QE.
500

Mean
RMS
Constant
Mean
Sigma

Mean
RMS
Constant
Mean
Sigma

600

Events

Events

400

ν Energy Resolution: ∆E. NDOS MC, Non-QE.

0.01951
0.2682
485
-0.02252
0.166

300

0.1167
0.5589
596.8
-0.00359
0.2972

400

200
200
100

0
-4

-2

0

∆ E (GeV)

2

0
-4

4

(a)

-2

0

∆ E (GeV)

2

4

(b)

Figure 6.16 QE And Non-QE Neutrino Energy Resolutions. Overall (a) QE and (b)
non-QE neutrino energy resolution. MC simulation.

The overall neutrino energy resolution for the QE sample is 170 MeV, as shown in figure
6.16a; and for the non-QE sample it is 300 MeV, as shown in figure 6.16b. The tails in
figure 6.16 account for about 5% of the events. The three neutrino energy resolutions, as a
function of the reconstructed neutrino energy, are compared in figure 6.17.
The neutrino energy distributions for QE and non-QE are presented in figure 6.18. In
these figures the true MC distributions represent the QE and non-QE interactions as defined
in the MC simulation. The reconstructed MC distributions represent the interactions as

133

ν : Energy Resolution vs. Reconstructed Energy. NDOS MC.

Energy Resolution (GeV)

0.4

0.3

0.2

0.1

All
QE
Non-QE

0

1

2

ν Energy (GeV)

3

4

Figure 6.17 Comparison Of Neutrino Energy Resolutions. The neutrino energy resolutions for all (black), QE (blue), and non-QE events. MC simulation.

True & Reconstructed ν Energy Spectra. NDOS MC. QE.

True & Reconstructed ν Energy Spectra. NDOS MC. Non-QE.

0.3

True MC
Fraction of events / 0.5 GeV

Fraction of events / 0.5 GeV

True MC
Reco MC
0.2

Areas = 1

0.1

0
0

2

4

ν Energy (GeV)

Areas = 1
0.2

0.1

00

6

(a)

Reco MC

0.3

2

4

ν Energy (GeV)

6

(b)

Figure 6.18 QE And Non-QE Neutrino Energy Distributions. (a) QE and (b) non-QE
neutrino true (black) and reconstructed (red) energy distributions. Areas normalized to 1 in
order to compare the shapes of the distributions. MC simulation.

134

defined by relations (6.6). In the QE and non-QE interactions, the area normalized true and
reconstructed neutrino energy distributions are in reasonable agreement. The distributions
for the QE and non-QE are distinct. The QE, with energy mean 28.4% smaller than that
of the non-QE, slowly varies between 0.5 GeV and 2.5 GeV, as shown in figure 6.18a. The
non-QE distribution is strongly peaked near 2.0 GeV, as shown in figure 6.18b. These effects
are the consequence of the cross sections dependence on the neutrino energy, as was shown
in figure 2.4. The two cross sections are equal (for νµ ) at about 1 GeV; at this point QE is
reaching a plateau and non-QE is increasing. The event rate in figure 6.18 is dominated by
non-QE for energies higher than 1 GeV.
ν µ & ν µ True Energy Spectra. NDOS MC. QE & Non-QE.

Neutrino Type. NDOS MC. QE & Non-QE.
1.5
0.8

ν µ QE
ν µ QE
ν µ Non-QE
ν µ Non-QE

QE
Non-QE

0.6

Events

Fraction

1

0.4

0.5
0.2

0

νµ

νµ

Neutrino Type

0
0

νe + νe

(a)

2

4

ν Energy (GeV)

6

(b)

Figure 6.19 Neutrino Types: Fractions And Energy Distributions. (a) Fraction of
neutrino types discriminated by the interaction type. (b) Neutrino energy distributions
discriminated by interaction type and neutrino type. No scale shown in order to compare
the shapes. MC simulation.
Each of the two interaction types is the sum of νµ and ν¯µ components. Figure 6.19
presents this composition. The overall fraction of the different neutrino types discriminated
by interaction type is presented in figure 6.19a. The non-QE fraction is higher for interactions
generated by a νµ , and the QE fraction is higher2 for interactions generated by a ν¯µ . The
neutrino energy distributions per neutrino type and interaction type are shown in figure
2 This is true because ν
¯µ interacts with Hydrogen nuclei in QE interactions and the νµ

does not.
135

6.19b. The ν¯µ contribution at the 2 GeV peak is about 10%. This contribution almost
triples at adjacent lower and higher energy bins from the peak, as a result of the various
cross sections that originate the interactions.

136

Chapter 7
Systematic Uncertainties
Systematic uncertainties are often related to measuring devices or measuring methods used
in experiments. A measuring device, or a detector, could consistently read the value of a
measured quantity with an offset from its real value. This offset could be due to an scale
factor, a calibration issue, or the materials and geometry of the detector. A measuring
method could under or over estimate a physical quantity due to the way it is designed or
implemented, which would result in a measurement that is consistently different from the
real value of the measured quantity.
There are six relevant systematic uncertainties on the number of events identified during
this analysis. The energy estimation of the neutrino candidate event relies on the absolute
energy calibration of the detector. Differences in the calibrations between data and MC
simulated events result in systematic uncertainties in the number of selected events per
energy bin. The various channels configurations of the detector during the period in which
the data was taken introduce systematic uncertainties in the number of selected events since
different configurations could disagree on whether a particular event is contained or not in
the detector. To minimize this uncertainty, each channels configuration in the data sample
was simulated in the MC sample. The uncertainty in the cross sections and the modeled final
state physics influence the number of selected events. In the case of the cross sections, the
measured number of events is directly proportional to the cross sections; and the model used

137

to describe the physics of the final state could increase or decrease the rate at which particular
types of events occur under the circumstances of the experiment. The mass of the target
region of the detector is also directly proportional to the measured number of events, and
therefore its uncertainty creates a systematic uncertainty on the number of selected events.
A mathematical procedure known as unfolding (see appendix C) shifts selected events from
the energy bins in which these are measured to the most probable energy bin for a particular
event. The shift is based on relations between true and reconstructed events that are built
into the algorithm. Since the rearrangement of events introduces a change in the number of
selected events per energy bin, the unfolding method also introduces a systematic uncertainty
on the number of selected events that is related to the parameters of the algorithm that are
subject of variation due to optimization criteria. Finally, the counting method of the POT
directly affects the relative number of selected events. The same absolute number of selected
events changes its interpretation depending on the reported POT. Thus, uncertainties in the
POT introduce systematic uncertainties in the number of selected events. Each one of these
major systematic uncertainties is presented in the following sections.

7.1

Energy Estimation Uncertainty

There are two key features in the energy estimation process: the length of the longest track
per event, and the deposited hadronic energy. To determine the systematic uncertainty of
the energy estimation of neutrino events in the NDOS, the mean energy deposited per unit
length in cosmic tracks are compared to MC simulated muon tracks, as seen in figure 7.1.
Differences in the two distributions imply a systematic difference in the estimated energy
of measured and simulated neutrinos. The two means of these distributions differ by 9.1%,

138

Logest Track dE/dX. NDOS MC & Cosmic DATA.

Reco MC

Tracks / 0.1 MeV cm-1

120

Cosmic

100

MC Area Normalized
80
60
40
20
0
0

2

4

6

Track dE/dX (MeV/cm)

8

10

Figure 7.1 Mean Energy Deposition Per Unit Length. dE/dX of cosmic (black) and
reconstructed MC simulated tracks (red). The MC simulation area is normalized to data.

which is interpreted as a 10% systematic uncertainty on the deposited energy.
This 10% systematic uncertainty is added to and subtracted from the deposited energy in
equation (6.5) to find how the hadronic energy estimation changes. These changes affect the
MIP fraction of the tracks as well, which is taken into account when evaluating the change
in the number of events. Finally, these systematic uncertainty affects the QE and non-QE
separation since it is carried out using the estimated energy of the hadronic system.
The muon energy estimation comes from the measured track length using equation (6.2).
The comparison between the longest track lengths of reconstructed MC simulated and data
tracks is presented in figure 7.2. The mean of the MC simulated track length distribution is
9.6% larger than that of data. This difference is assumed as a 10% systematic uncertainty on
the track length measurement. This 10% systematic uncertainty is added to and subtracted
from the track length in equation (6.2) to find how the muon energy estimation changes.

139

Tracks / (11 µsec × 1.67 × 1020 POT)

Track Length. NDOS MC & DATA.

Reco MC

40

Data
30

20

10

0
200

400

600

800

1000

1200

Track Length (cm)

Figure 7.2 Comparison Of Data And MC Simulated Longest Track Length Distributions. Reconstructed MC simulated (red) and data (black) tracks.

Energy Estimation Systematic Uncertainty. NDOS MC.

Uncertainty (%)

10

5

0

-5

-10
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.3 Energy Estimation Systematic Uncertainty As A Function Of The Neutrino Energy.

140

These changes combined with those presented above, for the deposited energy, give the
systematic uncertainties1 to the number of events shown in figure 7.3. The impact of these
variations (δE) is of the order of 8%.

7.2

Prototype Detector’s Channels Configuration Uncertainty

The NDOS active channels configuration varied over time, since APDs were added, removed,
and shifted during the period when the data were taken. To estimate the systematic uncertainties introduced to the number of events due to these changes in the channels configuration, three different configurations are examined (refer to figure 4.23): run numbers smaller
than 13250, run numbers bigger than 13250 and smaller than 13600, and run numbers bigger
than 13600. From the first set, the channels configuration of run 13220 is chosen because it
has the least number of active channels. From the second set, the channels configuration of
run 13258 is chosen because it is the one with the most active channels. Finally, from the
third set, the channels configuration of run 13703 is chosen because it has the most active
channels.
A set of MC simulated events is reconstructed with each of the three channels configurations, and the number of reconstructed events is examined as a function of the run number.
The systematic uncertainties2 on the number of events due to the impact of the various channels configurations are shown in figure 7.4. The same event reconstructed under different
channels configurations could be selected as contained in some of them, and not selected in
1 Figure 7.3 presents the uncertainties as percentages of the total number of events.
2 Figure 7.4 presents the uncertainties as percentages of the total number of events.

141

Channel Configuration Systematic Uncertainty. NDOS MC.

Uncertainty (%)

0

-5

-10

-15

-20
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.4 Channels Configurations Systematic Uncertainty As A Function Of The
Neutrino Energy.

others because it was not reconstructed as contained. The lack of instrumented cells in key
points of the detector is responsible for this issue. The variation in the number of selected
events introduced by each channels configuration is compared to the nominal one, and the
maximum variation is used as the systematic uncertainty. The impact of these variations
(δC) is of the order of 15%.

7.3

GENIE Cross Sections And Final State Physics
Uncertainty

In order to give the users the ability to verify the validity of new models explaining nuclear
interactions, or study how possible variations to the current models could affect the physics
output of the simulations, GENIE has features that allow the users to introduce variations to

142

key parameters that will result in changes to the simulated outputs produced by the software.
The developers provided a mechanism to assign weights to simulated neutrino interactions,
which is implemented by the ReWeight package. The main concept behind this process is
that for each input physics quantity P , with uncertainty3 δP , which is taken into account,
there is a parameter xP such that:

P → P′ = P

1 + xP

δP
P

.

(7.1)

P could be a single physical parameter, a simple function, or a prediction done by a MC
simulation. Some of these quantities may not be easy to write analytically or tabulated,
therefore it is preferable to formulate the problems in terms of xP . The parameter xP is an
integer, positive and negative values are allowed. When xP = 0, there would be no variation
to P . xP = ±1 indicates that P would change by one standard deviation, δP [109].
The reweighting package is named according to the final effect that it produces: it assigns
weights to individual MC simulated events based on the changes introduced by the xP . The
tool is very useful since it simplifies the analysis. Instead of running the MC simulation
as many times as changes are introduced to the models, the weights reflect the importance
of a given event on the sought final result. A weight equal to 1 means that the variation
in a given xP had no effect in the final result. A weight greater than 1 indicates that the
variation in xP produces an increase in the frequency of such weighted events proportional
to the size of the weight. A weight less than 1 means that the variation in xP produces
a decrease in the frequency of such weighted events proportional to the size of the weight.
ccqe

The Ma

parameter (defined in table 7.1) is used as an example of the ReWeight package

3 δP is the estimated standard deviation of P .

143

ccqe

Ma

GENIE ReWeight. NDOS MC.
+σ
Original
-σ

Events

600

400

200

0
0

1

2

3

ν Energy (GeV)

4

5

Figure 7.5 Example Of The ReWeight Package Output. The black distribution is the
ccqe
original, unaltered, MC sample. The +1σ variation to Ma
is shown in red, and the -1σ
variation in blue.

output, which is presented in figure 7.5. The original neutrino energy spectrum features a MC
simulated sample of events that passed all the event selection cuts. The ReWeight package
assigned weights to each of the events in the original distribution, and the consequences
of such weights are seen in red for the +1σ variation, and in blue for the -1σ variation.
Although the number of entries in the three distributions is the same, the integrals in the
reweighted distributions change as a result of the applied weights by 7.2% for +σ, and by
6.0% for −σ. The percentage of variation in the number of events is used as the systematic
uncertainty for this particular parameter. This procedure is the same for all the studied
parameters.
The dominant parameters that affect this analysis, include the axial mass for CC QE
scattering, the nuclear Pauli suppression effects in CC QE reactions4 , the non-resonance
4 Modifying the Fermi momentum is the way to affect the Pauli suppression.

144

xP

Description of P

δP /P

νµ interaction cross section systematic parameters
ccqe

Ma
Mccres
a
Rνncc1π
bkg

Axial mass for CC QE
Axial mass for CC resonance neutrino production
Non-resonance bkg in νn CC 1π reactions
Hadronization systematic parameters

+25% -15%
±20%
±50%

fz

Hadron formation zone
Intranuclear hadron transport systematic parameters

±50%

MFPπ
MFPn
Frn
inel
Cexn
Absn
Frπ

π mean free path (total rescattering probability)
Nucleon mean free path (total rescattering probability)
Nucleon inelastic reaction probability
Nucleon charge exchange probability
Nucleon absorption probability
Nucleon π production probability

±20%
±20%
±40%
±50%
±20%
±20%

Table 7.1 GENIE Parameters. Taken from [109].
background for all CC 1π and 2π production channels, CC DIS, intranuclear hadron transport
model, and total rescattering probability for hadrons within the target nucleus5 . Some of
these categories have more than one parameter available for the reweighting process. All the
parameters tabulated in tables 8.1, 8.2, and 8.3 of [109] are studied in this analysis, however
not all of them introduce finite changes to the number of events. All the parameters that
introduced finite variations to the number of events are presented in table 7.1, which also
shows their standard deviations.
There are three categories in table 7.1: neutrino interaction cross section systematic
parameters, intranuclear hadron transport systematic parameters, and hadronization systematic parameters. Hadrons produced in the nuclear environments are not obtained from
a single interaction, initially quarks propagate through the nucleus with low probability of
interaction as these are not yet part of a hadron. GENIE models this effect by defining
5 This systematic is related to the mean free path of the hadron within the nucleus.

145

GENIE Parameters Systematic Uncertainty. NDOS MC.
20

ccqe

Ma
Mccres
a
ncc1π
Rνbkg
Frninel
MFPπ
MFPn
Cexn
Absn
Frπ
fz

Uncertainty (%)

10

0

-10

-20
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.6 GENIE Parameters Systematic Uncertainty As A Function Of The
Neutrino Energy. The parameters shown are defined in table 7.1.

Total GENIE Systematic Uncertainty. NDOS MC.

Uncertainty (%)

20

10

0

-10

-20
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.7 GENIE Parameters Total Systematic Uncertainty As A Function Of
The Neutrino Energy. The sum of the parameters defined in table 7.1 is done in quadrature.

146

a free step, fz = pcτ0 /m, where p is the hadron’s momentum, τ0 = 0.342 fm/c is the
formation time, m is the mass of the hadron, and c is the speed of light. The formation zone
is defined by the distance between the intranuclear event vertex and the recorder position of
the hadron at the beginning of the intranuclear cascade step [109].
The systematic uncertainties6 in the number of events obtained for the parameters presented in table 7.1 are summarized in figure 7.6. The impact of the combined variations of
all the studied parameters (δG) is of the order of 15%, in the energy region where most of the
data occurs, i.e. 1.5 GeV to 2.5 GeV. The sum of these uncertainties is done in quadrature,
and is shown in figure 7.7.

7.4

Unfolding Systematic Uncertainty
Unfolding Systematic Uncertainty. NDOS MC.

Uncertainty (%)

5

0

-5

1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.8 Unfolding Algorithm Systematic Uncertainty As A Function Of The
Neutrino Energy.
6 Figure 7.6 presents the uncertainties as percentages of the total number of events.

147

The unfolding procedure presented in appendix C uses two parameters that can modify
the output of the algorithm, hence the number of events would vary accordingly. These
parameters are: k, which sets the number of statistically significant equations, and nTSVD ,
which is the number of toy MC simulated events generated by the TSVDUnfold tool to
calculate the uncertainties on the unfolded distribution. To find the systematic uncertainty
in the unfolded distribution of the number of events, the k parameter is modified to be: 4
and 6, the original value was 2; and the nTSVD is modified to be: 50 and 200, the original
value was 100. The systematic uncertainties7 in the number of events obtained after varying
these two parameters are shown in figure 7.8. The impact of these variations (δU ) is of the
order of 5%.

7.5

Total Systematic Uncertainty

Before discussing the total systematic uncertainties, there three more systematic uncertainties to be considered in this analysis: that related to the POT calculation, that related to the
fiducial mass, and that related to the fraction of neutrinos resulting from each charged meson
decay, i.e., the charged kaons to pions ratio, in the number of selected neutrino candidates.
NOνA uses the same POT counting scheme used by MINOS, thus the 2% systematic uncertainty (δP ) used in their study of νµ disappearance [110] is assumed in this analysis. The
fiducial mass has a 5% systematic uncertainty which comes from the comparison between
the physical measurements performed on the materials of the detector and the parameters
extracted from the MC simulation.
A. Lebedev [111] measured the ratio of charged pions and kaons production by 120 GeV/c
7 Figure 7.8 presents the uncertainties as percentages of the total number of events.

148

protons incident on a carbon target. The production ratios were measured in longitudinal
momenta from 20 GeV/c to 90 GeV/c. These are not the longitudinal momenta of the mesons
involved in the NDOS measurement, however, the lack of measurements in the longitudinal
momentum region concerning this analysis (around 7 GeV/c for kaons) leads to use the
measurements in [111] as a constraint on the K/π ratio at the momentum region of interest.
The ratios in [111] are in agreement with the ratios embedded in FLUKA within 10%. This
value is used as the systematic uncertainty on the number of neutrinos coming from charged
pion decays (δR) in this analysis.
Summary of Systematic Uncertainties. NDOS MC.

Uncertainty (%)

40

20

GENIE Parameters
Ch. Config.
Energy Est.
π/K
POT
Fiducial Mass

0

-20
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.9 Summary Of Systematic Uncertainties. The systematic uncertainties: δG
(magenta), δC (dark blue), δE (red), δR (green), δP (light blue), and δm (grey), are presented as a function of the neutrino energy.

The total systematic uncertainty associated to the number of events combines all the
systematic uncertainties discussed in the previous sections. The systematic uncertainties
presented in figure 7.3, 7.4, and 7.6, for energy estimation (δE), channels configuration
(δC), and GENIE parameters (δG), respectively, are combined along with the systematic
149

Number of Events Systematic Uncertainty. NDOS DATA.

Uncertainty Fraction

1

0.5

νµ
νµ
νµ
νµ

QE
Non-QE
QE
Non-QE

0

-0.5

-1
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.10 Systematic Uncertainty On The Number Of Events As A Function Of
The Neutrino Energy. Each neutrino kind and interaction type is represented by a color.

uncertainty associated to the POT (δP ), the fiducial mass (δm), and the charged kaons to
pions ratio (δR). All these uncertainties are dimensionless, and represent the uncertainty
in the number of events due to each of the variables mentioned above. The correlations
between all these uncertainties are of the order of the effect themselves, hence these second
order effects are ignored, and the uncertainties are considered independent of each other,
therefore these can be combined in quadrature:

δN =

δE 2 + δC 2 + δG2 + δP 2 + δm2 + δR2 .

(7.2)

The summary of all these uncertainties, which contribute to obtain the uncertainty in the
number of selected events (δN ) before the unfolding procedure is applied, is shown in figure
7.9. There are four equations (7.2) associated with each energy bin, one for each interaction
type and neutrino type: νµ QE, νµ non-QE, ν¯µ QE, and ν¯µ non-QE.
150

δN/N Ratio. NDOS DATA.
1

Uncertainty Fraction

Systematics

0.5

Statistics

0

-0.5

-1
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 7.11 δN /N As A Function Of The Neutrino Energy. Ratios between the systematic uncertainty and the measured number of events (red), and the statistical uncertainty
and the measured number of events (blue).

The combination of all the systematic uncertainties given by equation 7.2 are presented
in figure 7.10. Once equation 7.2 is computed, the unfolding algorithm is applied, and a new
unfolded uncertainty in the number of selected events (δN ′ ) is generated. The total systematic uncertainty on the unfolded number of events (δN ) is the combination in quadrature of
δN ′ and δU :
δN =

δN ′2 + δU 2 .

(7.3)

The ratios between the total systematic uncertainties, δN , and the measured number of
events, and the statistical uncertainties and the measured number of events, are compared
in figure 7.11. Statistical and systematic uncertainties are of the same order, about 12%, in
the region of the charged kaons peak in the neutrino energy, i.e. from 1.5 GeV to 2.5 GeV.

151

Chapter 8
Analysis
The event selection criteria presented in chapter 5, and the energy reconstruction process
presented in chapter 6, are applied to the NDOS data in order to find the observed number
of νµ + ν¯µ CC events.

8.1

Data And Monte Carlo Simulation Comparison
Longest Track Cos θNuMI. NDOS DATA.
Tracks / (0.02 × 11 µ sec × 1.67 × 1020 POT)

Tracks / (0.02 × 11 µ sec × 1.67 × 1020 POT)

Longest Track Cos θY. NDOS DATA.
In-time

120

Out-of-time

100
80
60
40
20
0

-1

-0.5

0

Cos θY

0.5

80

In-time
Out-of-time
60

40

20

0

1

(a)

0

0.2

0.4

0.6

Cos θNuMI

0.8

1

(b)

Figure 8.1 Angles Of The Longest Track. (a) Longest track’s cos θY for out-of-time
cosmic data (blue), and in-time data (black). (b) Longest track’s cos θNuMI .
NDOS data are separated (see section 4.4) into those recorded in-time (within the 11 µs
NuMI time window), where mixtures of neutrino interactions and cosmic rays are expected,
and out-of-time, where only cosmic rays are expected, as shown in figure 8.1. The cosmic
ray background expected in the in-time window is the out-of-time data normalized by the
152

ratio of the time windows widths, given by:

Cosmic → Cosmic ×

11
.
429

(8.1)

The regions where cosmic rays are the dominant contribution are: | cos θY | > 0.6, and
cos θNuMI < 0.6. The in-time distributions without a cosmic ray background, obtained by
subtracting the out-of-time data, can be compared with the distributions predicted by the
MC simulation, as shown in figures 8.2a and 8.2b. In these figures, all the cuts presented in
chapter 5 (see equations (5.1, 5.4, 5.8, and 5.13)) are applied, except the angular cuts. The
MC simulation distributions are normalized to the data POT such that:

MC → MC ×

POTData
1.67386 × 1020
.
= MC ×
POTMC
7.32109 × 1021

(8.2)

The background subtracted data has the general characteristics predicted by the MC
simulation with the cos θY distribution peaked near zero, and the cos θNuMI distribution
strongly peaked near 1. The only significant deviation is that the number of events predicted
by the MC simulation is higher than is seen in data, which is evident in the reduced height
of the peaks at: 1 in the cos θNuMI , and 0 in the cos θY distributions. When the shapes
of the distributions are compared, these are in good agreement, as shown in figures 8.2c
and 8.2d. Figure 8.2 illustrates the effect of eliminating the out-of-time data background.
However, in the analysis, the cosmic background subtraction of the data is done at the
number of events level (see figure 8.6). In order to verify that the subtraction of out-of-time
data from in-time data was adequate, five random 11 µs time windows from the out-of-time
data sample are chosen and treated as in-time data. The remaining out-of-time data are

153

Longest Track Cos θNuMI. NDOS MC & DATA.
Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

20

20

20

Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

Longest Track Cos θY. NDOS MC & DATA.

10

0

MC

-10

Data
-1

-0.5

0

Cos θY

0.5

MC

100

Data

50

0

1

0

0.2

0.4

(a)
Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

80

MC

20

20

Tracks / (0.02 × 11 µ sec × 1.67 × 10 POT)

10

0

MC
Data
MC Area Normalized

-0.5

0

Cos θY

0.5

MC Area Normalized

40

20

0

-20

1

Data

60

0

0.2

0.4

(c)

Tracks / (0.02 × 11 µsec)

Tracks / (0.02 × 11 µsec)

10

0

-10

0

Cos θY

0.8

1

Longest Track Cos θNuMI. NDOS Cosmic DATA.

20

-0.5

0.6

Cos θNuMI

(d)

Longest Track Cos θY . NDOS Cosmic DATA.

-1

1

Longest Track Cos θNuMI. NDOS MC & DATA.

20

-1

0.8

(b)

Longest Track Cos θY . NDOS MC & DATA.

-10

0.6

Cos θNuMI

0.5

10

0

-10

-20
0

1

(e)

0.2

0.4

0.6

Cos θNuMI

0.8

1

(f)

Figure 8.2 Longest Track Angular Distributions Without Cosmic Background. The
angular distributions for the longest track with a subtraction of the cosmic ray background
determined from out-of-time data. Longest track’s cos θY MC simulation (red) and data
(black) (a) POT normalized and (c) MC simulation area normalized to data. Longest track’s
cos θNuMI (b) POT normalized and (d) MC simulation area normalized to data. Data means:
in-time data minus out-of-time data. Longest track’s (e) cos θY and (f) cos θNuMI , for fake
in-time data minus normalized out-of-time data.
154

normalized as indicated in equation (8.1), however this time the factor of 429 is replaced by
a factor of 418 to take into account that now two, and not just one, 11 µs time windows are
subtracted from the full 440 µs time window. The out-of-time data are subtracted from the
fake in-time data, and the results are consistent with zero, as seen in figures 8.2e and 8.2f.
Each of the five fake in-time data samples chosen from the out-of-time data present similar
behaviors to the ones seen in these two figures. The deviations from zero seen in figure 8.2a,
for | cos θY | > 0.6, and in figure 8.2b, for cos θNuMI < 0.5, are consistent with the deviations
(of statistical origin) seen in figure 8.2e and 8.2f, respectively, for the same intervals. These
fluctuations occur in the regions where most of the cosmic data exists, however, the data in
these regions is rejected by the cosmic cuts presented in equation (5.11), hence, these do not
affect the analysis.
µ Reconstructed Energy. NDOS MC & DATA. Non-QE.

µ Reconstructed Energy. NDOS MC & DATA. QE.
60

Reco MC

20

Reco MC
Data

Muons / 0.2 GeV

Muons / 0.2 GeV

Data
15

MC Area Normalized

10

5

40

MC Area Normalized

20

0
0

1

2

3

µ Energy (GeV)

4

0
0

5

(a)

1

2

3

µ Energy (GeV)

4

5

(b)

Figure 8.3 Reconstructed Muon Energy Distributions, No Cosmic Background.
Reconstructed muon energy in the MC simulation (red) and in-time data (black) for (a) QE
and (b) non-QE interactions with a subtraction of the cosmic ray background determined
from out-of-time data. MC simulation area normalized to data.

The neutrino energy is reconstructed by adding the calorimetric energy and the muon
energy from range. These individual components in the data are compared to the MC
simulation, however, given the discrepancy in normalization between the MC simulation and
155

the recorded data described above, only shape comparisons of the individual components are
made. In addition, a cosmic background subtraction has been applied. The MC simulation
provides a good representation for the shape of the muon energy distribution in the non-QE
sample, as shown in figure 8.3b. The QE distributions require a deeper analysis because
there are several data points below the MC near 2 GeV. A Kolmogorov-Smirnov test [112]
is done in order to verify if the prediction of the MC simulation agrees with the data. The
test results give a maximum deviation (D) of: D = 0.0233, a Kolmogorov-Smirnov test
statistic (KS) of: KS = 0.628, and a p-value of: p = 0.825. Based on these results, the
two QE distributions are in good agreement. A p-value greater than 0.1 implies that there
is a low probability for the null hypothesis to be false. In this case, the null hypothesis is
that the two cumulative distribution functions (CDF) in figure 8.4 are drawn from the same
distribution, i.e. these represent the same physics. A CDF describes the probability that a
random variable (sample value) R, with a given probability distribution F (R), will be found
to have a value less than or equal to r [112].
The QE and non-QE distributions in figure 8.3 share the peak around 1 GeV. The muon
energy distribution for QE interactions exhibits another peak in energy just below 2 GeV.
This reflects the maximum in the neutrino flux at 2 GeV due to charged kaon decays, and
the general lack of energy in the hadronic system of a QE neutrino interaction. These 2 GeV
muons are likely to be contained only if these have cos θNuMI ≈ 1, otherwise these are likely
to leave the detector. There are more contained 2 GeV muons in the QE sample because
these are likely to keep the direction of the neutrino, therefore traveling towards the muon
catcher. Non-QE events have higher y, hence their muons are, in general, less energetic.
The simulated hadronic energy distribution in the non-QE sample is clearly consistent
with the recorded data, as shown in figure 8.5. This will enable a reliable neutrino energy
156

Figure 8.4 Cumulative Distribution Functions For Muons. Data (red) and MC simulation (green).

Hadronic Reconstructed Energy. NDOS MC & DATA. Non-QE.
50

Reco MC
Events / 0.2 GeV

40

Data
MC Area Normalized

30

20

10

0
0

1

2

3

4

Hadronic Energy (GeV)

Figure 8.5 Hadronic Energy Distributions, No Cosmic Background. Reconstructed
MC simulation (red) and in-time data (black) non-QE energy distributions. MC simulation
area normalized to data. Cosmic background subtracted.

157

calculation using the sum of the muon and hadronic energies. The mean hadronic energy is
around 0.7 GeV, which is about half of the mean energy of the muons.

Events / (0.5 GeV × 11 µ sec × 1.67 × 1020 POT)

Event Energy. NDOS DATA.
80

In-time
Out-of-time

60

40

20

0
0

1

2

3

4

5

Event Energy (GeV)

Figure 8.6 Selected Candidate Events Energy Distributions. In-time (black) and
out-of-time (blue) data energy distributions with cosmic cuts applied.

The level of the cosmic ray background remaining in the in-time data is obtained by applying all cuts to the out-of-time data. Near 1 GeV this cosmic ray background contribution
is approximately 10% of the number of events seen in the in-time data, as shown in figure
8.6. The consequences of the cuts, presented in table 5.1 and in equation (5.11) on the data
sample are summarized in table 8.1. 239 νµ + ν¯µ CC candidate events are selected. The
cosmic background subtraction is done to this set of selected candidate events.
The number of selected QE candidate events (Mq ) is the sum of a number of true QE
events (Tq ) and a number of true non-QE events, or fake QE events (Fq ), as seen in the first
row of the matrix in figure 8.7. Similarly, the number of selected non-QE candidate events
(Mn ) is the sum of the rest of the true non-QE events (Tn ) and the rest of the true QE

158

Cut Name
Number of Events
All events (no cut)
146085
3D
93722
LTL
54986
VR
3715
Containment
993
MIP
939
Cosmic
239
Table 8.1 Event Selection Cuts. Summary of the effect of all the event selection cuts on
the data sample. The number of events represents the number of selected neutrino candidate
events that passed each cut. Cosmic background subtraction is applied to the events set that
passed all these cuts.

(p, ǫ)\E (GeV)
QE
Purity
Error
Efficiency
Error
Non-QE
Purity
Error
Efficiency
Error

0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-10.0
97.4%
0.5%
78.4%
1.2%

89.1%
0.7%
71.7%
0.9%

86.2%
0.6%
71.6%
0.7%

84.0%
0.7%
68.1%
0.8%

91.2%
1.0%
64.4%
1.6%

87.4%
1.3%
48.1%
2.0%

50.0%
2.7%
22.0%
2.3%

85.7%
1.6%
35.3%
2.2%

34.6%
1.4%
84.5%
1.0%

79.1%
0.8%
92.4%
0.6%

88.1%
0.5%
94.8%
0.4%

87.5%
0.6%
94.5%
0.4%

85.8%
1.2%
97.2%
0.6%

82.0%
1.5%
97.2%
0.7%

89.8%
1.7%
96.9%
1.0%

97.7%
0.7%
99.8%
0.2%

Table 8.2 Quasi-elastic And Non-quasi-elastic Classification Performance. Purities
and efficiencies for QE and non-QE sample. The columns represent neutrino energy bins.

159

Figure 8.7 Efficiency And Purity Matrix For The QE And Non-QE Samples. This
matrix aids to construct the efficiencies and purities related to the selection of reconstructed
QE and non-QE candidate events.

events, or fake non-QE events (Fn ), as seen in the second row of the matrix in figure 8.7:

Mq = Tq + Fq ,

Mn = Tn + Fn .

(8.3)

The efficiencies (ǫq and ǫn ) and purities (pq and pn ) for the selection of QE and non-QE
candidate events are defined as:

ǫq =

Tq
,
Nq

ǫn =

Tn
,
Nn

pq =

Tq
,
Mq

pn =

Tn
,
Mn

160

(8.4)

where the total number of real QE events (Nq ) is:

Nq = Tq + Fn = Mq pq + Mn (1 − pn ),

(8.5)

and the total number of real non-QE events (Nn ) is:

Nn = Tn + Fq = Mn pn + Mq (1 − pq ).

(8.6)

The purity (see equation (8.4)) of the QE sample is (89.1 ± 0.8)%, and (84.4 ± 0.4)% for
the non-QE sample. The efficiency (see equation (8.4)) of the QE sample is (71.2 ± 1.0)%,
and (94.5 ± 0.3)% for the non-QE sample. The QE and non-QE samples purity and efficiency
per energy bins are summarized in table 8.2. Once the number of selected neutrino candidate
events (from the NDOS data), for QE and non-QE, is measured, the efficiencies and purities
in table 8.2 are used to obtain the real number of selected candidate events per interaction,
and per energy bin.
Three sample MC simulated events to illustrate the kind of topology that the selection,
presented in relation (6.6), allows in the QE sample are shown in figure 8.8. The event
display in figure 8.8a represents the following interaction:

νµ (0.8 GeV/c) +12 C → µ(0.7 GeV/c) + p(0.2 GeV/c).

(8.7)

The proton does not leave the interaction cell, and the muon’s track is about 3 m long. There
is no track reconstruction shown in these figures, and the colors of the cell hits represent the
amount of energy deposited, which follows the color code on the lower right histogram. The
reconstructed neutrino energy for this event is: 1.15 GeV, and the reconstructed hadronic
161

(a)

(b)

(c)
Figure 8.8 Sample Of Simulated QE Neutrino Events.
162

energy is: 0.1 GeV. The interaction in the event display shown in figure 8.8b is:

νµ (1.9 GeV/c) +12 C → µ(1.8 GeV/c) + p(0.2 GeV/c) + 2n(0.3 GeV/c).

(8.8)

The proton does not leave the interaction cell, and the muon’s track is about 7 m long. The
two neutrons do not deposit any reconstructable energy. This is the typical νµ CC event
expected to be found in the NDOS data, at the 2 GeV peak. The reconstructed neutrino
energy for this event is: 1.90 GeV, and the reconstructed hadronic energy is: 0.1 GeV. The
event display in figure 8.8c represents the following interaction:

νµ (3.0 GeV/c) +12 C → µ(2.8 GeV/c) + 4p + n(0.3 GeV/c).

(8.9)

There are four protons, each with about 0.2 GeV/c, that deposit all their energy in the
interaction cell, and a neutron. The muon has 2.8 GeV/c, which takes it half way into
the muon catcher. Muons of these energies are only likely to be contained if these reach
the muon catcher. The reconstructed neutrino energy for this event is: 2.85 GeV, and the
reconstructed hadronic energy is: 0.1 GeV. The three sample MC simulated events are good
representations of the QE sample, these deposit very little energy in cells that do not belong
to the reconstructed track.
In contrast, three sample events selected by relation (6.6) illustrate the kind of topologies
allowed in the non-QE sample. The event display in figure 8.9a represents the following
interaction:

νµ (2.0 GeV/c) + p → µ(1.3 GeV/c) + p(0.4 GeV/c) + π + (0.6 GeV/c).

163

(8.10)

(a)

(b)

(c)
Figure 8.9 Sample Of Simulated Non-QE Neutrino Events.
164

The proton does not leave the interaction cell, the muon’s track is about 6 m long, and the
π + leaves a MIP-like track with a hard scattering at the end. This is a resonance event. The
reconstructed neutrino energy for this event is: 2.15 GeV, and the reconstructed hadronic
energy is: 0.9 GeV. The event display in figure 8.9b represents the following interaction:

νµ (1.8 GeV/c) +12 C → µ(0.8 GeV/c) + p(0.9 GeV/c) + π + (0.6 GeV/c).

(8.11)

Although this interaction is similar to the previous one, in this case the proton leaves a track
with high energy depositions (top view, top track), and the π + leaves a shower with some
electromagnetic energy deposition associated with it (top view, bottom track). The muon’s
track is about 3 m long. This is a resonance event. The reconstructed neutrino energy
for this event is: 1.95 GeV, and the reconstructed hadronic energy is: 0.8 GeV. The event
display in figure 8.9c represents the following interaction:

νµ (2.1 GeV/c) +35 Cl → µ(0.6 GeV/c) + p(0.3 GeV/c) + n + π + (0.7 GeV/c) + π + (0.4 GeV/c).
(8.12)
The proton deposits most of its energy in the cell around: Y = 20 cm and Z = 375 cm,
the 0.7 GeV/c π + leaves a MIP-like track with a hard scattering at the end, the 0.4 GeV/c
π + leaves a scattered shower going downwards in the side view, the neutron deposits no
reconstructable energy, and the muon’s track is about 2 m long. This is a DIS event. The
reconstructed neutrino energy for this event is: 2.3 GeV, and the reconstructed hadronic
energy is: 1.55 GeV. The muons in the non-QE sample have energies around 1 GeV, and are
likely to be contained.
A sample of selected QE neutrino candidates, similar to the one presented in figure 8.8, is

165

(a)

(b)

(c)
Figure 8.10 Sample Of QE Neutrino Candidates. (a) Selected neutrino candidate with
1.0 GeV of estimated energy. (b) Selected neutrino candidate with 2.5 GeV of estimated
energy. (c) Selected neutrino candidate with 3.3 GeV of estimated energy.
166

(a)

(b)

(c)
Figure 8.11 Sample Of Non-QE Neutrino Candidates. (a) Selected neutrino candidate
with 1.9 GeV of estimated energy. (b) Selected neutrino candidate with 1.8 GeV of estimated
energy. (c) Selected neutrino candidate with 2.0 GeV of estimated energy.
167

shown in figure 8.10. The color of the cell hits represent times, and these follow the color code
of the timing histogram in the bottom left of figure 8.10. Figure 8.10a presents a selected QE
neutrino candidate with 1.0 GeV of reconstructed energy, similar to that presented in figure
8.8a. Figure 8.10b presents a selected QE neutrino candidate with 2.5 GeV of reconstructed
energy, similar to that presented in figure 8.8b. Figure 8.10c presents a selected QE neutrino
candidate with 3.3 GeV of reconstructed energy, similar to that presented in figure 8.8c.
Another sample of selected non-QE neutrino candidates, similar to the one presented in
figure 8.9, is shown in figure 8.11. Figure 8.11a presents a selected non-QE neutrino candidate
with 1.9 GeV of reconstructed energy, similar to that presented in figure 8.9a. Figure 8.11b
presents a selected non-QE neutrino candidate with 1.8 GeV of reconstructed energy, similar
to that presented in figure 8.9b. Figure 8.11c presents a selected non-QE neutrino candidate
with 2.0 GeV of reconstructed energy, similar to that presented in figure 8.9c. These six
selected neutrino candidates share similar topologies to the MC simulated ones, hence, MC
simulated and data events are likely to share the same type of neutrino interaction, as well
as the same final state particles. The full list of selected neutrino candidates is presented in
appendix D.
After the subtraction of the cosmic ray background, and the separation of the QE and
non-QE samples, the number of neutrino events predicted by the MC simulation is higher,
in each case, by about 30% than is observed in the data, as shown in figure 8.12. This is
agreement with the observation that the MC simulation predicted more muon tracks than
appeared in the data. When the MC simulated distribution is normalized to the number of
data events, the shapes are quite similar, as shown in figure 8.13.
The distributions presented in figure 8.12 are reconstructed MC simulation and data.
There are differences between reconstructed and true MC simulated distributions, as pre168

ν Energy. NDOS MC & DATA. Non-QE.
Events / (0.5 GeV × 1.67 × 1020 POT)

Events / (0.5 GeV × 1.67 × 1020 POT)

ν Energy. NDOS MC & DATA. QE.

Reco MC

30

Data
20

10

0
0

1

2

3

4

ν Energy (GeV)

5

Reco MC
80

Data
60

40

20

0
0

1

2

3

ν Energy (GeV)

(a)

4

5

(b)

Figure 8.12 Number Of Neutrino Candidates vs. Neutrino Energy, No Cosmic
Background. Number of selected neutrino candidate events vs. neutrino energy for (a) QE
and (b) non-QE interactions. Reconstructed MC simulation (red) and in-time data (black)
energy distributions with a cosmic background subtraction and normalized by POT.

ν Energy. NDOS MC & DATA. Non-QE.

Events / (0.5 GeV × 1.67 × 1020 POT)

Events / (0.5 GeV × 1.67 × 1020 POT)

ν Energy. NDOS MC & DATA. QE.

Reco MC

30

Data
20

MC Area Normalized

10

0
0

1

2

3

ν Energy (GeV)

4

5

Data
MC Area Normalized

40

20

0
0

(a)

Reco MC

60

1

2

3

ν Energy (GeV)

4

5

(b)

Figure 8.13 Number Of Neutrino Candidates vs. Neutrino Energy, No Cosmic
Background, Shape Comparison. (a) QE and (b) non-QE number of neutrino events
vs. neutrino energy. Reconstructed MC simulation (red) and in-time data (black) energy
distributions. MC simulation areas normalized to data.

169

sented in section 6.4, that can only be resolved by the unfolding procedure that accounts
for the inefficiencies and resolutions of the detector, and provides the transformation of the
reconstructed event energy into corrected energy bins.
ν Energy. NDOS MC & Data. Non-QE.

True MC

20

Events / (0.5 GeV × 1.67 × 1020 POT)

Events / (0.5 GeV × 1.67 × 1020 POT)

ν Energy. NDOS MC & Data. QE.

Unfolded Data

15

10

5

0
0

1

2

3

ν Energy (GeV)

4

True MC

40

20

0
0

5

(a)

Unfolded Data

60

1

2

3

ν Energy (GeV)

4

5

(b)

Figure 8.14 QE And Non-QE Unfolded Number Of Neutrino Candidates vs. Neutrino Energy, No Cosmic Background. (a) QE and (b) non-QE number of selected
neutrino candidate events vs. neutrino energy after unfolding. True MC simulation (green)
and unfolded data (black) energy distributions normalized by POT.
The unfolding algorithm is applied to the data distributions1 in figures 8.12a and 8.12b,
and the results are shown in figure 8.14. The unfolded distributions should be compared to
the true MC simulated distributions. There are 32.4% more QE predicted events by the MC
simulation than unfolded candidate events. There are 35.9% more non-QE predicted events
by the MC simulation than unfolded candidate events. These differences in the predicted and
candidate events come, thus, from an extra factor of 1/3 in the normalizations included in the
MC simulation. The shapes of the data and MC simulated distributions are in agreement,
as see in figure 8.15, where the MC simulation areas are normalized to data.
1 Figures 8.15 should be compared with figures 8.13 to appreciated the effect of the un-

folding procedure.

170

ν Energy. NDOS MC & DATA. QE.

ν Energy. NDOS MC & DATA. Non-QE.
Events / (0.5 GeV × 1.67 × 1020 POT)

True MC
Unfolded Data

20

Events / (0.5 GeV × 1.67 × 10 POT)

20

15

MC Area Normalized
10

5

True MC

50

Unfolded Data

40

MC Area Normalized
30

20

10

0
00

1

2

3

ν Energy (GeV)

4

5

0

(a)

1

2

3

ν Energy (GeV)

4

5

(b)

Figure 8.15 QE And Non-QE Unfolded Number Of Neutrino Candidates vs. Neutrino Energy, No Cosmic Background, Shape Comparison. (a) QE and (b) non-QE
number of neutrino events vs. neutrino energy after unfolding. True MC simulation (green)
and unfolded data (black) energy distributions. MC simulation areas normalized to data.

8.2

Results

The inclusive CC event sample is obtained by combining the QE and non-QE neutrino energy
distributions presented in figure 8.14. The total number of selected νµ + ν¯µ CC candidate
events is: 229.7 per 1.67 × 1020 POT. This number is not an integer because it comes from
the subtraction of out-of-time data from in-time data. The number of events in the in-time
data set is an integer. The out-of-time data set is scaled to the size of the in-time data time
window (11 µs), and therefore is not necessarily an integer. The sums of the two interactions,
which are the reconstructed MC simulation and the data (without unfolding), are shown in
figure 8.16a. With the MC simulation normalized to the number of data events, the shapes of
the inclusive sample are in good agreement, as seen in figure 8.16b. The true MC simulation,
with 328.1 predicted events per 1.67 × 1020 POT, exhibits 30% more predicted events than
the unfolded candidate events, as shown in figure 8.17a. Comparing the shapes between the
true MC simulated and unfolded data distributions, between 0.5 GeV and 4 GeV in figure
8.17b, indicates that there is a good agreement among them. The agreement is corroborated
171

by the χ2 test, which in this case concludes that χ2 = 5.549 for ndf = 6.
ν Energy Spectra. NDOS MC & DATA.

Events / (0.5 GeV × 1.67 × 1020 POT)

Events / (0.5 GeV × 1.67 × 1020 POT)

ν Energy Spectra. NDOS MC & DATA.

Reco MC

100

Data
80

60

40

20

0
0

1

2

3

ν Energy (GeV)

4

Data
60

MC Area Normalized

40

20

0
0

5

Reco MC

80

1

2

3

ν Energy (GeV)

(a)

4

5

(b)

Figure 8.16 Number Of Neutrino Candidates vs. Neutrino Energy, Comparison Of
MC Simulation And Data. Energy dependence of the reconstructed MC simulation (red)
and data without unfolding (black) normalized by (a) POT, and (b) with MC simulation
areas normalized to data.

ν Energy Spectra. NDOS MC & DATA.

Events / (0.5 GeV × 1.67 × 1020 POT)

Events / (0.5 GeV × 1.67 × 1020 POT)

ν Energy Spectra. NDOS MC & DATA.
100
True MC
Unfolded Data

80

60

40

20

0
0

1

2

3

ν Energy (GeV)

4

80
True MC

MC Area Normalized
40

20

0
0

5

(a)

Unfolded Data

60

1

2

3

ν Energy (GeV)

4

5

(b)

Figure 8.17 Unfolded Number Of Neutrino Candidates vs. Neutrino Energy, Comparison Of MC Simulation And Data. True MC simulation (green) and unfolded data
(black) normalized by (a) POT, and (b) with MC simulation areas normalized to data.
bin ) in figure 8.16 has
The number of selected CC candidate events per energy bin (NCC

no cosmic background2 , 2.9% of overall NC background, and 1.4% of overall νe + ν¯e CC
background (see section 6.1). Table 8.5, at the end of the chapter, summarizes the percent2 Due to cosmic background subtraction.

172

ages of NC background per energy bin, and table 8.6, at the end of the chapter, summarizes
the percentages of νe + ν¯e CC background per energy bin. These two tables feature the
bin .
backgrounds present in the NCC

ν kind\E (GeV)
νµ

0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

νµ QE: Mq
νµ
νµ Non-QE: Mn

16.8
1.8

13.2
19.8

14.9
39.0

11.2
34.1

2.3
8.4

1.3
5.2

0.2
1.6

ν¯µ QE: Mq
ν¯µ
ν¯µ Non-QE: Mn

8.9

6.5

3.4

2.5

1.6

1.0

0.1

1.2

5.5

4.7

3.7

3.2

1.9

0.6

bin
NCC

28.8

45.0

62.0

51.5

15.4

9.4

2.5

ν¯µ

Table 8.3 Candidate Events With Background. Number of selected candidate events
per neutrino type and interaction type. The cosmic background was subtracted, though
bin ,
these numbers still reflect the NC and νe + ν¯e CC background presence. The last row, NCC
is the total number of selected candidate events per energy bin. See equation (8.3) for the
definition of the Mqν and Mnν .
The total number of selected candidate events, NT , is the sum of the CC QE (Nq ) and
CC non-QE (Nn ) neutrino candidates (see equations (8.5) and (8.6)), and each of these two
numbers is the sum of the νµ and ν¯µ contributions:

νµ

νµ

ν¯µ

ν¯µ

NT = Nq + Nn + Nq + Nn .

(8.13)

After the background subtraction is applied, the Mqν and Mnν in table 8.3 are substituted
into equations (8.5, 8.6) yielding Nq and Nn respectively.
The final results, the number of selected νµ + ν¯µ CC candidate events per energy bin
(NTbin ), are summarized in table 8.4. Statistical and systematic uncertainties are of the
same order, with the systematic error typically larger. The total uncertainty is the sum in
quadrature of the two types of uncertainties, as presented in the full discussion on systematic
bin are about 2/3 of the prediction done by the MC
uncertainties in chapter 7. These NCC

173

ν Energy (GeV) Candidates/1.67×1020 POT stat. error sys. error total error
0.5
1.0
1.5
2.0
2.5
3.0
3.5

to
to
to
to
to
to
to

1.0
1.5
2.0
2.5
3.0
3.5
4.0

28.28
42.30
58.54
48.56
14.06
8.63
2.31

+7.60
−1.38
+5.83
−2.43
+8.91
−5.82
+6.62
−6.75
+4.13
−1.81
+4.66
−1.75
+1.08
−0.35

4.47
6.50
6.79
6.42
2.29
1.98
1.06

+8.82
−4.68
+8.73
−6.94
+11.2
−8.94
+9.22
−9.32
+4.74
−2.96
+5.07
−2.64
+1.51
−1.11

Table 8.4 Number Of Candidate Events. Selected νµ + ν¯µ CC candidates per neutrino
energy bin. Statistical and systematic errors are shown in independent columns. The total
error combining the statistical and systematic errors are shown in the last column.
simulation. Chapter 9 addresses the possible causes of the disagreement between NDOS
data and the MC simulation.
ν kind\E (GeV)
νµ QE
Uncertainty
νµ Non-QE
Uncertainty
ν¯µ Non-QE
Uncertainty
νe Non-QE
Uncertainty

0.5-1.0

1.0-1.5

1.5-2.0

2.0-2.5

2.5-3.0

3.0-3.5

3.5-4.0

0.1%
0
±0.1%
0
0
0.6%
0
±0.6%
0
0.1%
0
±0.1%
0
0
0
0

0.9%
±0.9%
1.7%
±1.3%
0.1%
±0.1%
0.1%
±0.1%

0.5%
±0.5%
3.9%
±2.0%
0.2%
±0.2%
0.1%
±0.1%

1.1%
±1.0%
4.0%
±2.1%
1.0%
±1.0%
0.3%
±0.3%

1.1%
0%
±1.0%
0%
6.4%
7.1%
±2.5% ±2.7%
0.5%
1.1%
±0.5% ±1.0%
0.2%
0.4%
±0.2% ±0.4%

Table 8.5 Neutral Current Background. The NC background per energy bin as a function
of the interaction type. The rows in this table represent the interaction type.

174

ν kind\E (GeV)

0.5-1.0

1.0-1.5

1.5-2.0

νe + ν¯e QE
1.6%
3.0%
2.4%
Uncertainty
±1.3% ±1.7% ±1.5%
νe + ν¯e Non-QE 0.8%
1.2%
0.6%
Uncertainty
±0.8% ±1.1% ±0.6%

2.0-2.5

2.5-3.0

3.0-3.5

3.5-4.0

0.9%
0.5%
1.1%
0±.9% ±0.5% ±1.0%
0.5%
1.7%
1.4%
±0.5% ±1.3% ±1.2%

0
0
0.2%
±0.2%

Table 8.6 Electron Neutrino Background. The νe + ν¯e CC background per energy bin
as a function of the interaction type. The rows in this table represent the interaction type.

175

Chapter 9
Discussion And Final Remarks
The number of selected events (Nνµ ) found in section 8.2 is a function of the neutrino flux,
and the cross section between the incident neutrinos and the number of target nuclei (T )
[113]:
Nνµ (Eν ) = T σ(Eν )Φ(Eν ).

(9.1)

To investigate which of the two considered quantities (σ(Eν ) or Φ(Eν )) is responsible for
the excess of neutrino events predicted by the MC simulation, as discussed in section 8.2,
the following sections present the calculations of both quantities using the results already
obtained.

9.1

Flux Calculation

The flux calculation presented in this section assumes that the cross sections used by GENIE
to simulated neutrino interactions are in agreement with the current measurements1 . The
flux of charged kaons produced at the NuMI target can be constrained using neutrinos
detected with NDOS since about 93% of the neutrinos in the 2 GeV peak come from charged
kaon decays. Figure 9.1 shows the longitudinal momentum (figure 9.1a) and the transverse
momentum (figure 9.1b) of the charged kaons produced at the NuMI target as a function of
1 See section 9.1.2 for a discussion on the cross sections used in the MC simulation.

176

K± Longitudinal Momentum vs. ν True Energy. NDOS MC.

K± Transverse Momentum vs. ν True Energy. NDOS MC.

30

3

103

100
25

2.5

K± pT (GeV/c)

20
60
15

L

K± p (GeV/c)

80

40
10
20

5

0
0

2

4

6

ν Energy (GeV)

8

10

2

102

1.5

1

10

0.5

0

0
0

2

(a)

4

6

ν Energy (GeV)

8

10

1

(b)

Figure 9.1 Charged Kaon Longitudinal And Transverse Momenta. (a) Longitudinal
and (b) transverse momenta of charged kaons produced at the NuMI target as a function
of the daughter neutrino energies. Event selection criteria applied to these MC simulated
sample. MC simulation.
the daughter neutrino energy2 . The average longitudinal momentum (pL ) of charged kaons
when the daughter neutrino energy is restricted to:

1.5 GeV < Eν < 3.5 GeV,

(9.2)

is 6.61 GeV/c. The average transverse momentum (pT ) of charged kaons for the neutrino
energies shown in equation (9.2) is 0.34 GeV. The average momentum of these charged
kaons is therefore 6.62 GeV/c. The neutrinos detected with NDOS at 110 mrad off NuMI
axis, thus, constrain the flux of charged kaons produced at the NuMI target with average
momentum of (6.62 ± 1.72) GeV/c.
The sum of all the neutrinos coming from: charged kaon, charged pion, muon, and KL
decays was presented in figure 3.10. In addition, the neutrinos coming from charged kaon
decays are dominant for energies higher than 1.5 GeV, as shown in figure 9.2. This figure is
an updated version of figure 3.12 after all the event selection is applied to the MC simulated
2 Here K means K + and K − added together.

177

ν True Energy. K/π Ratio. NDOS MC.

K/π Ratio

20

15

10

5

0

1

2

ν Energy (GeV)

3

4

Figure 9.2 Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos From
Charged Pion Decays After Event Selection Criteria Applied. MC simulation.

sample. For these energies, the overall (integrated) background of neutrinos coming from
charged pion decays is 7.7%, and from muons + KL decays is 1.1%. Table 9.1 summarizes the
percentages of charged pions and muons + KL background per energy bin. The statistically
significant data in figure 8.16 have energies less than 3.5 GeV, so the charged kaons flux
measurement is calculated using data between 1.5 GeV and 3.5 GeV.
Decay\E (GeV)
π + → νµ , QE
π + → νµ , Non-QE
π − → ν¯µ , QE
−
π → ν¯µ , Non-QE
µ− or KL → νµ , QE
µ− or KL → νµ , Non-QE
µ+ or KL → ν¯µ , QE
+
µ or KL → ν¯µ , Non-QE

0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0
62.9%
37.5%
40.5%
39.6%
2.2%
1.7%
1.9%
0.8%

20.6%
13.6%
17.9%
8.0%
1.1%
0.8%
1.2%
0.2%

5.7%
4.1%
8.9%
3.0%
0.6%
0.4%
1.1%
0.2%

3.1%
2.6%
7.8%
2.6%
0.5%
0.3%
0.5%
0.5%

2.1%
3.4%
19.1%
4.0%
1.1%
0.6%
0.5%
0.3%

6.3%
3.0%
7.4%
5.5%
1.1%
0%
1.0%
0.5%

33.3%
1.1%
33.3%
3.2%
0
0
0
0

Table 9.1 Background To Neutrinos From Kaon Decays. Percentage of neutrinos
not coming from charged kaon decays per energy bin. The rows in this table represent the
decaying particle that produces the neutrino.
178

int.\E (GeV)

0.5-1.0

1.0-1.5

1.5-2.0

2.0-2.5

2.5-3.0

3.0-3.5

3.5-4.0

νµ QE
νµ Non-QE
ν¯µ QE
ν¯µ Non-QE

9.8
1.3
6.5
0.9

9.9
14.9
5.2
4.8

13.8
35.6
3.1
4.5

10.8
31.5
2.4
3.6

2.3
7.4
1.3
3.0

1.3
4.7
1.0
1.8

0.2
1.5
0.1
0.6

bin
NK

+7.5
18.5−3.1

+7.3
34.7−5.6

+10.8
57.1−8.6

+5.1
8.7−2.7

+1.6
2.4−1.2

+4.8
48.3 ± 9.2 14.0−2.9

Table 9.2 Candidate Events From Kaon Decays. Number of candidate events per
neutrino type and interaction type, when the parent meson is a charged kaon. Each row
in this table represents a neutrino type associated with an interaction type. The last row,
bin , is the total number of selected candidate events per energy bin. These numbers come
NK
after NC, νe + ν¯e , and charged pions + muons + KL background subtraction. Tables 8.5
and 8.6 summarize these backgrounds.

The charged kaon decays of interest are:

K + → νµ + µ + ,
→ νµ + µ + + π 0 ,
K − → ν¯µ + µ− ,
→ ν¯µ + µ− + π 0 .

(9.3)

To find the total number of candidates coming from charged kaon decays (NK ), the charged
pions + muons + KL background is subtracted from NT (see equation (8.13)). Table 9.2
bin . The flux is a function of: the number of ν + ν
summarized the various NK
¯µ , NK ,
µ

detected with NDOS, their CC cross sections (σ) with respect to the nuclei in NDOS, the
number of target atoms (T ) in the VR, and the reconstruction efficiency (ǫ) [113]:

Φ=
i

NiK
.
ǫi σi T

(9.4)

All these quantities are a function of the energy bin, i. The total flux, Φ, is the sum of the
179

individual fluxes per energy bin.

9.1.1

Number Of Atoms In The Target Region

The number of atoms available to interact with neutrinos in the VR is calculated using
NOνA design information [114, 83], interacting with NOνA PVC extrusions experts [115],
and measuring dimensions and weights of spare NDOS PVC extrusions kept at Argonne
National Laboratory. This number of atoms is confined to the NDOS target region (TR)
with coordinates restricted by:

|X|

< 106 cm,

|Y |

< 172 cm,

288 cm <

Z

< 452 cm.

(9.5)

The PVC mass inside the TR (mpvc ) is 4.90% of the total PVC mass, Mpvc :
• Mpvc = (38.3 ± 1.9) × 103 Kg.
• mpvc = (1.9 ± 0.1) × 103 Kg.
The scintillator mass inside the TR (ml ) is 4.94% of the total scintillator mass, Ml :
• Ml = (91.3 ± 4.6) × 103 Kg.
• ml = (4.5 ± 0.2) × 103 Kg.
Even though the amount of WLS fiber, glue, and air within the TR is very small, it is
calculated as well:
• WLS fiber mass: mwls = (5.1 ± 0.3) Kg.
180

• Glue mass: mg = (3.1 ± 0.2) Kg.
• Air mass: ma = (6.0 ± 0.3) × 10−3 Kg.
The NOνA software has a function, GeometryBase::TotalMass, which provides the numbers
discussed above:
MC = 39.9 × 103 Kg.
• MC simulated PVC mass: Mpvc

• MC simulated scintillator mass: MlMC = 87.7 × 103 Kg.
Component
Scintillator
CH2
C9 H12
C15 H11 NO
C24 H22
PVC
C2 H3 Cl
TiO2
C36 H70 O4 Ca
C20 H24
C12 H20 O5
C21 H42 O4
C5 H8 O2
Air
N
O
Ar
WLS Fiber
C6 H5 CH
Glue
C

Molar Mass (g) fraction (f ) Mass (Kg)

Moles

Molecules

14
120
221
310

0.9464
0.0523
0.0014
0.0002

4270.6
236
6.3
0.7

3.02×105
1.97×103
2.84×10
2.33

1.83×1029
1.18×1027
1.71×1025
1.4×1024

62
80
607
264
244
359
100

0.8
0.15
0.006
0.009
0.002
0.002
0.031

1502
281.6
11.3
16.9
3.8
3.8
58.2

2.42×104
3.52×103
1.86×10
6.4×10
1.54×10
1.05×10
5.82×102

1.46×1028
2.12×1027
1.12×1025
3.85×1025
9.27×1024
6.3×1024
3.5×1026

14
16
40

0.78
0.21
0.01

5.0×10−3
1.0×10−3
1.0×10−4

0.33
0.08
1.35×10−3

2.01×1023
4.72×1022
8.13×1020

90

1

5.1

5.68×10

3.42×1025

12

1

3.1

2.59×102

1.56×1026

Table 9.3 Chemical Composition Of The Prototype Detector. NDOS chemical composition, and number of atoms or molecules in each of the elements or compounds that are
inside the target region defined in equation (9.5).
The chemical composition of the NDOS components [116] is summarized in table 9.3.
With the identity of all the molecules present in the NDOS compounds, the procedure to
181

obtain the target mass is straight forward. The molar mass (mm ) of molecules or atoms
is the mass of one mole of such substance. Table 9.3 contains the proportions (f ) of each
element or molecule in each of the compounds of NDOS. The number of target atoms or
molecules (Tx ), inside the TR, of a given element or compound with mass mx is:

Tx =

mx f na
,
mm

(9.6)

where na is Avogadro’s number. The known cross sections are for individual atoms, hence
the number of atoms in each molecule is known from its chemical composition. Table 9.3
summarizes the number of atoms or molecules that make up the mass of the NDOS TR.

9.1.2

Monte Carlo Cross Sections

GENIE has tables with the cross sections of many processes labeled under the categories:
QE and non-QE, where GENIE labels the individual non-QE processes as: resonance, deep
inelastic or coherent. Such tables contain the various cross sections as a function of the
neutrino energy as well as the target nucleus. Since the neutrino energies recorded in the
tables are discrete, GENIE interpolates the recorded cross sections in order to obtain the
cross section of a particular event for the given energy. The GENIE cross sections used in
this analysis are displayed in figure 9.3. All the non-QE cross sections per nucleus are added
together.
The cross sections shown in figure 9.3 are calculated assuming that experimental results
on cross sections per nucleon can be extrapolated to give nuclei cross sections. The uncertainties given by GENIE to these cross sections are around 10%, from the green band shown
in figure 9.4. This figure also summarizes the measurements used to calculate the GENIE
182

(a)

(b)

Figure 9.3 GENIE Cross Sections As A Function Of Neutrino Energy. Cross section
for each of the elements present in the NDOS target region’s mass. The color code is in the
right side of the figures, solid lines are for neutrinos and dashed lines are for antineutrinos.
Summary of (a) QE and (b) non-QE cross sections.

Figure 9.4 GENIE Uncertainty In The Muon Neutrino Cross Section. νµ CC
scattering from an isoscalar target. The shaded band indicates the estimated uncertainty on
the free nucleon cross section [109].

183

cross sections: [117], [118], [119], [120, 121], [122, 123], [124], [55], and [125].

9.1.3

Total Reconstruction Efficiency

The last piece to find the sought flux is the reconstruction efficiency. This efficiency was
calculated using the MC simulated sample, and reflects the fraction of events reconstructed
given the total number of events available:

ǫ=

NR
,
NMC

(9.7)

where NR is the number of reconstructed MC simulated events that passed all the event
selection cuts, and NMC is the number of MC simulated events which had their interaction
point inside the VR. The reconstruction efficiency as a function of the neutrino energy, and
discriminated per neutrino type and interaction type is presented in figure 9.5.
Reconstruction Efficiencies. NDOS MC.
0.6

Efficiencies

0.5

νµ
νµ
νµ
νµ

QE
Non-QE
QE
Non-QE

0.4

0.3

0.2

0.1

0
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 9.5 Reconstruction Efficiency As A Function Of The Neutrino Energy. Error
bars represent the binomial uncertainties. MC simulation.

184

9.1.4

νµ + ν¯µ Flux Coming From Charged Kaon Decays
ν: Flux vs. Energy. NDOS MC & DATA.

Flux (10-11 / POT cm2)

10

MC
Data

1

10-1
1.5

2

2.5

Energy (GeV)

3

3.5

Figure 9.6 Total Flux Of νµ + ν¯µ Coming From Charged Kaon Decays. NDOS data
(black) and MC simulation (red).

All components of equation (9.4) have been presented so that the flux of νµ + ν¯µ produced
in charged kaon decays can be determined. The cross sections are given discriminated by
interaction type and neutrino type, therefore there are four fluxes:
αβ
Ni
αβ
Φi = αβ αβ ; α = QE, non-QE; β = νµ , ν¯µ ,
ǫi σi T

(9.8)

and the total flux is the sum of the four:

αβ

Φ=

Φi ,
α,β

(9.9)

i

where i labels the energy bin. Here, N is the number of neutrino candidate events coming
from charged kaon decays (NK ) presented in table 9.2. T is the numbers of target nuclei
185

presented in table 9.3. σ are the cross sections presented in figure 9.3, and ǫ are the reconstruction efficiencies presented in figure 9.5. The total flux of νµ + ν¯µ coming from charged
kaon decays is shown in figure 9.6 in four energy bins of 0.5 GeV size. The uncertainties are
the sum in quadrature of statistical and systematic uncertainties. The systematic uncertainties are summarized in figure 9.7. The flux calculated from the NDOS data is systematically
lower than that predicted by the NDOS MC simulation (see figure 8.16), within the energy
region of interest defined in equation (9.2). The χ2 test for these distributions concludes
that χ2 = 9.336 for ndf = 4. The ratios between data and MC simulation for the total
flux, and also for QE and non-QE interactions, are shown in figure 9.8. The average ratio is
about 0.7.
Flux Systematic Uncertainties. NDOS MC.
20
Num. of events
Cross Sec.
Reco. Eff.
POT
Num. Of Targets

Uncertainty (%)

15

10

5

0

-5

-10
1.5

2

2.5

ν Energy (GeV)

3

3.5

Figure 9.7 Flux Systematic Uncertainties. MC simulation.

Figure 9.9 presents the fluxes discriminated by interaction type. The agreement within
error bars of the non-QE fluxes in figure 9.9b is better than the one in figure 9.9a with the

186

ν Flux Ratio: DATA / MC. NDOS MC & DATA.
1

0.8

Ratio

0.6

0.4
All

0.2

QE
Non-QE

0
1.5

2

2.5

3

ν Energy (GeV)

3.5

Figure 9.8 Data Over MC Simulated Flux Ratio. Total flux ratio (black), QE flux ratio
(blue), and non-QE flux ratio (red) as a function of neutrino energy.

ν: Flux vs. Energy. NDOS MC & DATA. QE.

ν: Flux vs. Energy. NDOS MC & DATA. Non - QE.
10

MC

MC

Data

Data

Flux (10-11 / POT cm2)

Flux (10-11 / POT cm2)

10

1

10-1

10-2
0.5

1

1.5

2

Energy (GeV)

2.5

3

1

10-1
0.5

3.5

(a)

1

1.5

2

Energy (GeV)

2.5

3

3.5

(b)

Figure 9.9 Total Flux Of νµ + ν¯µ Coming From Charged Kaon Decays Discriminated By Interaction Type. (a) QE and (b) non-QE fluxes. NDOS data (black) and
MC simulation (red).

187

QE fluxes, though the systematically lower measured flux is present in both interactions.
This is corroborated by comparing the two χ2 tests:

χ2QE = 6.506, χ2NQE = 3.995, ndf = 4,
χ2NQE
= 0.614.
χ2QE

(9.10)

Equation (9.10) shows a closer agreement between the non-QE data and the prediction by
the MC simulation than the QE data does. This result was expected since the non-QE data
sample is 60% larger than the QE one, which is a big difference in statistics, a determinant
factor in the analysis. The discrimination between QE and non-QE flux is useful to check if
the two results are consistent with each other, i.e. the flux must be the same, regardless of
the interaction, since the difference between them comes from the cross sections. The ratio
between the QE over the non-QE fluxes in figure 9.10, is consistent with 1, as expected.
Flux Ratio: QE / Non - QE. NDOS DATA.
2

QE / Non-QE

1.5

1

0.5

0
0.5

1

1.5

2

ν Energy (GeV)

2.5

3

Figure 9.10 QE Over Non-QE Flux Ratio.

188

3.5

The flux of charged kaons produced at the NuMI target can be constrained comparing
the data presented above. From the neutrinos detected with energies stated in equation
(9.2), the flux of charged kaons with total average momentum of 6.62 GeV/c compares to
that predicted by the MC simulation as follows:

+0.157 syst) × 10−10
ΦData = (1.462 ± 0.154 stat−0.113
+0.141 ) × 10−10
ΦMC = (2.086−0.139
+0.108 · Φ
ΦData = 0.701−0.094
MC

ν
cm2 POT

,

ν
,
cm2 POT

(9.11)

The observed data and the MC simulated distributions shapes agree, as seen in figures 9.6
and 9.9, though their normalization is not the same, these differ by the amount presented
in equation (9.11). This analysis suggests that the neutrino flux of νµ from charged kaon
decays obtained from the NDOS data is 30% lower that is predicted by the MC simulation.

9.2

Inclusive νµ Charged Current Cross Section Calculation

With the results presented in section 9.1, it becomes interesting to calculate the inclusive
νµ CC cross section using the selected candidates3 (Nνµ ) presented in table 8.3, and the
flux (Φ) predicted by the MC simulation. With the direct measurement of the number of
events done in this analysis it is possible to calculate either the flux, or the cross section,
using previous measurements incorporated in the MC simulation of the cross section or
the flux, respectively. The excess of predicted events observed in the MC simulation (see
3 The same procedure that led to equation (8.13) applies here for the number of ν , N .
µ
νµ

189

section 8.2) could be the result of overestimating either the flux or the cross section, or a
more complicated combination of over and underestimation of the two. A manipulation of
equation (9.4) results in the following relation between the cross section and the flux [113]:
Nνiµ

σ=
i

ǫi Φi T

.

(9.12)

The two previous direct measurements (see section 2.1.1.4) of the inclusive νµ CC cross
section around 2 GeV (refer to figure 2.4) reported their results per nucleon, therefore the
number of targets (T ) used in section 9.1.1 needs to be slightly modified from the number
of target nuclei to the number of target nucleons. The inclusive cross section per nucleon
(σN ) is the sum of the QE cross section per nucleon (σQE ) plus the non-QE cross section
per nucleon (σnQE ):
σN = σQE + σnQE .

(9.13)

The number of targets in the TR is:

T = 1.723 × 1030 neutrons, or
T = 3.854 × 1030 nucleons.

(9.14)

The FLUKA νµ flux prediction (Φi ), that is embedded in the MC simulated sample, is
shown as a function of neutrino energy in figure 9.11. The same FLUKA simulation was
used by the MiniBoone collaboration on their measurement of νµ and νe events [126]. The
MiniBoone detector is also located at 110 mrad off the NuMI beam axis. Their measurements have a 9% systematic uncertainty, on the NuMI flux, in the energy range between
0.9 GeV and 3.0 GeV. In this analysis the systematic uncertainty related to the FLUKA flux
190

νµ Flux. NDOS MC. QE & Non-QE.
6
QE
Non-QE

Flux (10-11 / POT cm2)

5

4

3

2

1

0

1.6

1.8

2

ν Energy (GeV)

2.2

2.4

Figure 9.11 Muon Neutrino MC Simulated Flux Prediction.

Reconstruction Efficiencies. NDOS MC.
0.4

QE
Non-QE

Efficiencies

0.3

0.2

0.1

0

1.6

1.8

2

ν Energy (GeV)

2.2

2.4

Figure 9.12 Reconstruction Efficiency As A Function Of The Neutrino Energy.
Error bars represent the binomial uncertainties. MC simulation.

191

prediction is assumed to be 10% based on the MiniBoone estimation.
Inclusive ν µ Charged Current Cross Section vs. ν Energy.

MC
Data

2

1.5

Cross Section (10

-38

cm2 / nucleon)

2.5

1

0.5

0

1.6

1.8

2

ν Energy (GeV)

2.2

2.4

Figure 9.13 Inclusive Muon Neutrino Charged Current Cross Section Per Nucleon
On A Carbon Target. NDOS data (black) and MC simulation (red).

The last pieces to calculate the inclusive νµ CC cross section are the reconstruction
efficiencies (defined in equation (9.7)) presented in figure 9.12. Following a procedure similar
to that shown in section 9.1.4 leads to the inclusive νµ CC cross section per nucleon on
a carbon target4 . The cross section is presented in two energy bins of 0.5 GeV, as seen
in figure 9.13. The uncertainties are the sum in quadrature of statistical and systematic
uncertainties. The systematic uncertainties are summarized in figure 9.14. The cross section
calculated using the NDOS data is systematically lower than that embedded in the NDOS
MC simulation. The χ2 test for these distributions concludes that χ2 = 6.803 for ndf = 2.
The ratio between the data and MC simulated cross sections is shown in figure 9.15, only
the error bars in the second bin are consistent with one, which is shown as a dashed line.
4 Refer to table 9.3.

192

Cross Section Systematic Uncertainties. NDOS MC.
20
Num. of Events
Reco. Eff.
Flux
Num. Of Targets
POT

Uncertainty (%)

10

0

-10

-20

1.6

1.8

2

ν Energy (GeV)

2.2

2.4

Figure 9.14 Cross Section Systematic Uncertainties. MC simulation.

Inclusive ν µ CC Cross Section Ratio: Data / MC. NDOS MC & DATA.
1.2

Ratio

1

0.8

0.6

0.4

1.6

1.8

2

ν Energy (GeV)

2.2

2.4

Figure 9.15 Data Over MC Simulated Muon Neutrino Charged Current Cross
Section Ratio.

193

Figure 9.16 Measurements Of Muon Neutrino Charged Current Inclusive Scattering Cross Sections Divided By Neutrino Energy. All the results are cited in the
figure. Vertical blue (red) error bars represent statistical (combined statistical and systematic) errors. The NDOS data point is inserted into a figure taken from [44].

The inclusive νµ CC cross section per nucleon at an average neutrino energy of 1.97 GeV
compares to that embedded in the MC simulation as follows:

+0.330 syst) × 10−38 cm2 ,
σData = (1.085 ± 0.219 stat−0.383
+0.216 ) × 10−38 cm2 ,
σMC = (1.588−0.210
+0.266 · σ
σData = 0.683−0.257
MC .

(9.15)

The comparison of the measured cross section with previous measurements is presented in
figure 9.16, which summarizes the measurements of νµ CC inclusive scattering cross sections
divided by the neutrino energy as a function of neutrino energy. Note the transition between
logarithmic and linear scales occurring at 100 GeV. The red error bars are the sum in
quadrature of the statistical and systematic uncertainties. The blue error bars are the
194

statistical uncertainties alone. The measured cross section agrees, within error bars, with the
two previous results [55, 56] mentioned above, and lies below the trend set by the collection
of all the results shown. The average cross section over neutrino energy of the wide range of
energies shown is represented by the horizontal dashed line.

9.3

Final Remarks

The two results presented in equations (9.11, 9.15) are the possible consequences of the
difference of about 1/3 seen in section 8.2 between the selected NDOS νµ + ν¯µ CC candidates
and the prediction done by MC simulation. The conclusion is that the FLUKA flux needs to
be adjusted, based on the two NDOS measurements available5 , by 30% since the inclusive
νµ CC cross section (over neutrino energy) embedded in the MC simulation is consistent
with the dashed line in figure 9.16 at 2 GeV.
NDOS was designed to serve as a testing instrument that would provide the NOνA
collaboration with a proof of principle for the concepts and technologies developed for the
experiment. Measuring the flux of νµ + ν¯µ coming from charged kaon decays was a worth
pursuing result since those neutrinos show a clean peak around 2 GeV, and the flux of
charged kaons had not been measured at the angle where NDOS sits. A 30% excess of
events predicted by the MC simulation is now reported by two different analyses of the
NDOS data, which should be taken into account by the neutrino oscillations community.
NOνA can verify the results presented in this analysis with its near detector underground
which is taking data now.

5 The result of this analysis and that from [22].

195

APPENDICES

196

Appendix A
Mass Terms In The Weak Interaction
All leptons participate in the weak interaction. However it is the only interaction in the
Standard Model where neutrinos are involved1 . Leptons and quarks enter in the weak
interaction as left-handed weak-isospin doublets2 :

L≡

ν
,
l

q≡

u
,
d

(A.1)

where the left-handed states are:

1
(1 − γ5 )ν,
2
1
uL =
(1 − γ5 )u,
2
νL =

1
(1 − γ5 )l,
2
1
dL = (1 − γ5 )d.
2

lL =

(A.2)

Here: u = u, c, t; d = d, s, b; l = e, µ, τ ; and ν = νe , νµ , ντ . There is no experimental
evidence of the existence of right-handed neutrinos3 , and since these only interact with other
1 Neutrinos do not participate in the electromagnetic interaction at the tree level.

If
neutrinos are Dirac particles these can have a magnetic moment, and therefore participate
in the electromagnetic interaction through one-loop diagrams. The transition moment, which
is relevant to νi → νj + γ (i = j), may exist for both Dirac and Majorana neutrinos [64].
For a detailed calculation of the one-loop diagram see [127, 128, 129].
2 The doublet in equation (A.1) is a general case which works for all three families.
3 Right-handed neutrinos have chirality eigenvalue: +1. Chirality is an intrinsic property,
independent of the reference frame of the observer. Since neutrinos are massive, they can
have both eigenvalues of helicity: ±1, which is a property that depends on the reference
frame of the observer. Chirality and helicity are only equivalent for massless particles [130].

197

particles through the weak interaction4 , the Standard Model excludes them. Right-handed
charged leptons and right-handed quarks enter the Standard Model weak interaction as
weak-isospin singlets given by:

1
(1 + γ5 )l,
2
1
1
uR = (1 + γ5 )u, dR = (1 + γ5 )d.
2
2
lR =

(A.3)

Weak-isospin and weak hypercharge are related by the Gell-Mann–Nishijima [131, 132] relation:
Q = I3 +

Y
,
2

(A.4)

where Q is the electric charge, I3 is the third component of the weak-isospin, and Y is the
weak hypercharge. Weak-isospin can only take two values I3 = ± 1/2, thus:
YL = −1,

YR = −2.

(A.5)

With the commutator:
[I3 , Y ] = 0,

(A.6)

the two quantities are commuting observables, and the product of the group transformations
generated by I and Y is the gauge group SU (2)L ⊗ U (1)Y of a gauge theory. Four massless
bosons and six massless leptons are the result of these constructions [130]. The Lagrangian
of this group is:
LG = Lg + Lφ + Lf + Lh ,
4 The weak interaction only operates on left-handed states [27].

198

(A.7)

where Lg is the gauge Lagrangian, Lφ is the scalar Lagrangian, Lf is the fermionic Lagrangian, and Lh is the Yukaga Lagrangian. The gauge part of the Lagrangian is:
1 i
1
µν
Lg = − Wµν
Wi − Bµν B µν ,
4
4

(A.8)

i is the SU(2) gauge field that has three and four-point self-interactions, and B
where Wµν
µν

is the U (1) gauge field associated with Y , which has no self-interaction [130].
The scalar part of the Lagrangian is:

Lφ = (Dµ φ)† Dµ φ − V (φ),

(A.9)

where Dµ is a covariant derivative [133], and V (φ) is a scalar potential. In order to generate
the masses of the weak bosons and all the charged leptons, a doublet of scalar fields (φ)
defined as:
φ≡

φ+
,
φ0

(A.10)

is introduced. The known masses are generated through a spontaneous symmetry breaking
in the potential V (φ) of the field. This transforms as a SU (2)L doublet which implies:
Yφ = 1. The invariance and renormalizability [134] of the group restrict V (φ) to the form:

V (φ) = µ2 φ† φ + |λ|(φ† φ)2 ,

(A.11)

where µ and λ are parameters. For µ > 0 the potential is quadratic with its only minimum at
zero, as shown in figure A.1a, and the vacuum is empty. An exact symmetry is characterized
by two conditions: the Lagrangian density is invariant under the symmetry, and the physical

199

vacuum is invariant under the symmetry transformations [130]. A model with an exact
symmetry contains a degenerate set of massless particles. The Standard Model has symmetry
breaking to account for the observed masses in the model. The case for µ < 0, in equation
(A.11), is shown in figure A.1b. There is a spontaneous symmetry breaking since the scalar
field takes one of all the possible minima in the potential.

(a)

(b)

Figure A.1 Scalar Potential. (a) µ > 0, and (b) µ < 0 [135].

The fermionic part of the Lagrangian is:
3

Lf =

k ıDlk ,
k +¯
k
k
k ıDq k + ¯
lR
¯kR ıDukL + d¯kR ıDqR
q¯L
R
L lL ıDlL + u
k=1

where k denotes the family, and all fields are weak eigenstates [130].

200

(A.12)

The Yukawa part of the Lagrangian is:
3

Lh = −

2 † k
l ¯n
2 † k
d ¯n
2 † k
Γukn u¯n
R ıτ φ qL + Γkn dR ıτ φ qL + Γkn lR ıτ φ lL

+ h.c.,

(A.13)

k,n=1

where the matrices Γkn describe the Yukawa couplings between the scalar doublet and the
various flavors k and n of quarks and leptons [130].
The vacuum expectation values of the various complex scalar fields are the components
φi of a complex vector υ:
υ = 0|φ|0 .

(A.14)

Without any loss of generality the four axes of this space can be chosen so that:

0|φi |0 = 0, for i = 1,2, and 4, and 0|φ3 |0 = v.

(A.15)

Here υ is determined by writing the scalar potential: V (φ) → V (υ), and choosing υ such
that V is minimized [136]. The quantum theory is obtained by considering the fluctuations
around this minimum: φ = υ + φ′ . Using equation (A.15), the potential in equation (A.11)
becomes:
V (φ) → V (υ) =

1 2 2 1
µ v + |λ|v 4 .
2
4

(A.16)

For µ > 0 in figure A.1a, the minimum of the potential occurs at v = 0, and no symmetry
breaking occurs. For µ < 0 in figure A.1b, the minimum is obtained at a non-zero value of
v:
dV (v)
= v(µ2 + |λ|v 2 ) = 0,
dv

201

(A.17)

which has solution5 :
v=

−

µ2
|λ|

1/2

.

(A.18)

With this result, the kinetic part of the scalar Lagrangian in equation (A.9) becomes:
2
g i i g′
1
†
µ
(Dµ φ) D φ = (0 v) τ Wµ + Bµ

2

2

2

0
+ ··· ,
v

(A.19)

where g and g ′ are coupling constants of the SU(2) and U(1) gauge groups. After some
algebraic manipulation [138], important mass terms are identified6 :

1:
2:

g2v2
(W 1 − ıW 2 )µ (W 1 + ıW 2 )µ ,
8
µ
(g 2 + g ′2 )v 2
gW 3
g′B
−
8
g 2 + g ′2
g 2 + g ′2

(A.20)
gW 3
g 2 + g ′2

W3 −

g′B
g 2 + g ′2 µ

.

The masses of the two weak interaction bosons are:

MW =

gv
,
2

MZ =

v

g 2 + g ′2
,
2

(A.21)

and the weak mixing angle (θW ) fulfills:

cos θW =

g
g 2 + g ′2

,

sin θW =

g′
g 2 + g ′2

.

(A.22)

1 and 2 in equation (A.20) can be written as:

2 W +µ W − , 2 :
1 : MW
µ

MZ2 µ
Z Zµ ,
2

(A.23)

5 The case for µ = 0 requires loop corrections, but the symmetry is again spontaneously

broken [137].
6 These are mass terms since the fields couple to themselves.
202

where:
1
W ± = √ (W 1 ∓ ıW 2 ), and Z = cos θW W 3 − sin θW B,
2

(A.24)

are the weak interaction bosons. From the form of the covariant derivative, the electron’s
charge is [133]:
e=

gg ′
g 2 + g ′2

,

(A.25)

which allows to estimate the weak interaction bosons’ masses. Notice that the term:

Aµ Aµ = (sin θW W 3 + cos θW B)µ (sin θW W 3 + cos θW B)µ ,

(A.26)

is multiplied by the zeroes in equation (A.19), thus the photon field (A), given by:

A = sin θW W 3 + cos θW B,

(A.27)

is massless. The coefficients MW and MZ (in equation (A.23)) depend on v, which is7 the
vacuum expectation value of the scalar field. This scalar field is known as the Higgs field
[139]. The spontaneous symmetry breaking that results in v = 0 produces three massive
weak interaction messengers and a massless electromagnetic interaction one [130]; unifying
the weak and electromagnetic interaction into one elegant electroweak interaction.

7 See equations (A.15, A.18).

203

Appendix B
NOνA Kalman Tracker
For track propagation in the Kalman Tracker, each track is represented as a system with
two parameters: position and slope. A minimum of two cell hits are required to estimate the
parameters. Track seeds are formed from pairs of cell hits separated by as much as 3 planes
in their view. The track starting point is assumed to be at the center of the cell hit with the
highest Z coordinate, and the track propagation is carried on towards lower Z coordinates.
Tracks are propagated plane by plane, using the current estimates of the track position and
slope, to estimate the location of the expected track’s cell hits in the next plane. For cell
hits on the projected plane, a ∆χ2 test is calculated from the inclusion of the new cell hit
in the track. If the ∆χ2 is less than a default value, the cell hit is added to the track. After
a new cell hit is added to the track, a new track fit is done to replace the older position and
slope parameters. When the estimated track position for the next cell hit falls in an inactive
cell, and no other cell hits in that plane are added to the track, the plane will not count for
the propagation algorithm. The track propagation continues until three consecutive planes
with no cells hits are found.
When multiple cell hits in a given plane are considered to be part of a track, the addition
of each cell hit is determined individually and independently from the inclusion of the other
cell hits on the plane. When multiple cell hits are added to a track on one plane, the
propagation to the next plane is calculated based on the average of all estimated states

204

found from adding each cell hit individually. After each propagation step the resulting
tracks are filtered. At the initial propagation, the filter provides the best estimated of the
track state for the propagation in the opposite direction. At the final propagation, the filter
provides the best estimate of the fit to the track.

Figure B.1 Sample Event. Illustrates the track finding process. MC simulation.

Before a 2D track is written in the output the following quality cuts are applied:
• Minimum number of cell hits per track: 4.
• Minimum number of valid planes crossed by the track: 3.
• Maximum χ2 per added cell hit: 10.
The cuts reject fake tracks or tracks with poor fits. From the tracks that pass the quality
control, the best track found is the one with the lowest χ2 /Nch , where Nch is the number
of cell hits. As mentioned in chapter 4, long muon tracks are expected to undergo multiple
scattering at the end of their path. This feature will raise the χ2 /Nch , which is taken into
205

account. The best 2D track found is written to the output, and its cell hits are removed
from the pool of cell hits in that particular slice. The entire 2D track finding is repeated
with the remaining cell hits in the slice to find new tracks.
The algorithm stops when no new tracks pass the cuts, or there are no more available
cell hits in the slice. A sample output of the algorithm is presented in figure B.1. From left
to right: the first iteration finds one track per view (red and green), the second iteration
finds one track per view (green and yellow), and the third iteration finds only one track in
the side view (light blue). A flow chart of the algorithm is shown in figure B.2.

Figure B.2 Flow Chart. Shows how the algorithm finds 2D tracks [140].

206

Within a single slice, 2D tracks from separate views are matched together to form 3D
tracks based on a scoring routine. From each pair of 2D tracks from separate views that
overlap in their Z coordinates, a match score is calculated based on the starting and ending
Z coordinates of each track:

score =

|Zlt − Zls | + |Zht − Zhs |
.
OZ

(B.1)

Here Zlt is the lowest Z coordinated of the 2D track in the top view, Zls is the lowest Z
coordinated of the 2D track in the side view, Zht is the highest Z coordinated of the 2D
track in the top view, Zhs is the highest Z coordinated of the 2D track in the side view, and
OZ is the length of the overlap of the two tracks in the Z coordinate. The two 2D tracks
with the lowest score are matched in each iteration to form a 3D track. The algorithm is
repeated until no more 2D tracks overlap in the Z coordinate. Each 2D track is only allowed
to be matched with one 2D track on the other view. All 3D tracks and unmatched 2D tracks
are written to the output. The final output of tracks from the sample event in figure B.1 is
presented in figure B.3. Two 3D tracks are reconstructed (red and green), and one 2D track
is left unmatched (blue). For a full description of the Kalman Tracker see [140].

207

Figure B.3 Sample Event. Final output of tracks from the sample event in figure B.1.

208

Appendix C
Unfolding Algorithm, TSVDUnfold
One of the many uses of the MC simulated sample is that the reconstructed neutrino energies can be compared to the corresponding true energies. Figure 6.14, for example, is a
comparison of true and reconstructed neutrino energies. From this figure it is clear that
some events do not have their reconstructed energy in the appropriate energy bin. There
are various reason for this to happen, most of them already mentioned. For example figures
6.10a and 6.10b show that muons with true energy less than 0.4 GeV attain a reconstructed
energy 25% higher than the true energy. A solution to this issue comes from the use of an
unfolding algorithm [141]. This is based on the knowledge of both true and reconstructed
energies, which allows to take into account the inefficiencies and resolutions of the detector
to try to place the reconstructed events in the appropriate energy bin.

Singular Value Decomposition Unfolding Algorithm
The unfolding algorithm incorporates the Singular Value Decomposition (SVD) method to
find the detector’s response matrix. The inputs to the algorithm are [141]:
• x ini : Vector of simulated events.
• b ini : Vector of reconstructed simulated events.

209

The components of the x ini vector are the MC simulated events that realistically represent
the underlying physics of the studied situation. The components of the b ini vector are
reconstructed MC simulated events which contain information about the performance of the
detector. The goal of the algorithm is to find a probability matrix Aˆ that will relate the two
vectors [141]:
ˆ ini = b ini .
Ax

(C.1)

Once the matrix is know, the algorithm performs the unfolding procedure that takes the real
life measurements gathered in the vector b to an unfolded vector x that will represent the
most probable distribution of events given the performance of the detector [141]:

ˆ = b.
Ax

(C.2)

ˆ which might not be
To obtain the vector x requires the knowledge of the determinant of A,
a straightforward endeavor since the components of the matrix could have values ranging
from big numbers to very small numbers. The inverses of these numbers could introduce a
wide variety of rapidly oscillating solutions to the mathematical problem. The SVD method
is a powerful tool that provides useful information about the matrix, e.g. if Aˆ is orthogonal,
all its singular values are 1. In this method the matrix Aˆ is represented as the product of
three matrices1 [141]:
Aˆ = U SV T ,

(C.3)

where U and V are orthogonal matrices, and S is a diagonal matrix with non-negative
entries called singular values. These represent the reconstruction efficiencies per energy bin.
1V V T = 1 = V T V , U U T = 1 = U T U .

210

As mentioned above, if some of these singular values are small, the system of equations could
become very difficult to solve.
With b and Aˆ known, solving for x requires the use of the SVD method [141]:

ˆ = b = U SV T x = U Sz,
Ax

U T U Sz = U T b = d = Sz,
S −1 d = z = V T x,
x = V z = V S −1 d = V S −1 U T b = Aˆ−1 b.

(C.4)

The solution means that b is a set of orthonormalized functions of a parameter i = 1, ..., nb .
The basis of these functions are the columns of the matrix U , and the elements of d are the
coefficients of the decomposition. The vector x is decomposed in a series of orthonormalized
functions of a parameter j = 1, ..., nx . These functions gather in the columns of the matrix
V . The coefficients of the decomposition are the components of the vector z. At the end,
the initial problem is reduced to solving a diagonal system:

z = S −1 d,

(C.5)

which is just a matter of inverting the diagonal matrix S, i.e. inverting the singular values.
If some of the singular values are small, and the measurements in b have associated
errors, the exact solution does not give any useful information. The following 2 by 2 system

211

is presented as an example:




 b1 
b=
,
b2





1  1 1 
U =V = √ 
,
2
1 −1








 1 0 
S=
,
0 ǫ




1  1 1  1 0  1 1  1
x = V S −1 U T b = √ 


 √ b,
2
2
1
1 −1
0 ǫ
1 −1
x = V S −1 U T b =

x =





2





1
1 1+ ǫ

1 − 1ǫ 





1  1 1  1 1 

b = 
 b,

2
2
1
−1
1
1
1− ǫ 1+ ǫ
1 −1
ǫ
ǫ


1
1
1  (1 + ǫ )b1 + (1 − ǫ )b2 







(1 − 1ǫ )b1 + (1 + 1ǫ )b2

 =





b1 + b2  1  b1 − b2  1 
,
 +

2
2ǫ
1
−1

x = z1 V1 + z2 V2 =

d
d1
V1 + 2 V2 .
s1
s2

(C.6)

To determine if the components of the vectors V1 and V2 are statistically significant to the
problem, the following relation between the coefficients d1 and d2 needs to be true [141]:

d22 ≤ d1 ,

(C.7)

If relation C.7 is true, the components of V2 are not statistically significant and d2 is simply
a random number. However, if:
1
s2 < √ ,
d2

(C.8)

V2 dominates over V1 , and the result is meaningless. Since the si are the efficiencies per
energy bin, what this means is that with very low efficiencies, the correct bin for a particular

212

event can not be resolved between bin 1 or 2.
In the general case, for a smooth distribution of measurements, only the first few di are
statistically significant. In a plot of log|di | vs. i, the values of di should exponentially decrease
with increasing i until these reach a point of rapid oscillation, therefore the di are a series of
random numbers. Only the di occurring before the region of rapid oscillation are statistically
significant, and useful in the solution of the problem. Once the number of statistically
significant di is established, the system is regularized to avoid possible meaningless results,
as mentioned above [141].
To measure the deviations of x from xini , a vector ω, with components: ωj = xj /xini
j , is
introduced. The product of the ωj with the columns of the probability matrix Aˆij results
in the number of events that were generated in bin j but belong to bin i. The result is the
matrix A, a number-of-events matrix. Rescaling the equations does not change the exact
solution. This unfolding problem is an over determined problem and its solution is obtained
by means of the least squares approach [141]:
nb
i=1



nx



j=1

2

Aˆij xj − bi  = min,

(C.9)

whenever the errors in the measurements are all equal. Since that is not the case here, i.e.
the errors vary from bin to bin, each equation should be weighted by its measured error
[141]:
2
nx ˆ
Aij xj − bi

 = min.
∆bi
i=1 j=1
nb

The general equation is:



ˆ −b
Ax

T

ˆ − b = min,
B −1 Ax

213

(C.10)

(C.11)

where B is the covariance matrix of b, which gathers the errors in the measurements. B is
a symmetric and positive-definite matrix, therefore its SVD is:

B = QRQT ,

(C.12)

Rii ≡ ri2 > 0 and Rij = 0, for i = j.

(C.13)

with:

If the statistics of the MC simulation are one or two orders of magnitude larger than the data
statistics, the unfolding error is dominated by the measurement errors, which is the case in
the this analysis. Dividing each equation by its measured error gives them equal footing in
the problem, and equation (C.11) transforms to [141]:

Aω − b

T

QR−1 QT Aω − b = min,

(C.14)

with:
1
Q A ,
A˜ij =
ri m im mj

˜bi = 1
Q bm ,
ri m im

(C.15)

such that:

Aω − b

T

QR−1 QT Aω − b

= min =

˜ − ˜b
Aω

T

˜ − ˜b .
Aω

(C.16)

Thus,
A˜ij ωj = ˜bi .
i

214

(C.17)

The regularization is achieved by introducing a stabilization term (see [142], [143], and
[144]) into equation (C.11):

˜ − ˜b
Aω

T

˜ − ˜b + τ · (Cω)T (Cω) = min,
Aω

(C.18)

where τ is a stabilization parameter, and C is the a priori condition matrix. These are
both problem dependent, and control the equations that present the irregular behavior.
The components Cij are chosen such that the components of ω with irregular behavior are
suppressed.
The solution to equation (C.18), for any τ , requires the damped least squares approach
[144]:










˜

 b 
Cω
=

 ,
√
ωI
0

˜ −1
AC

(C.19)

from which a similar result as in equation (C.5) is achieved2 :

d˜ = S˜−1 S d,

z˜ = S˜−1 d,

ω = C −1 V z˜,

x˜i = xini
i ωi ,

(C.20)

where:
s2 − τ
,
s˜i = i
si

(C.21)

are the new singular values. The choice of τ comes from the log|di | vs. i plot. The singular
value s˜k of the last statistically significant equation, k, determines it:

τ = s˜2k .
2 Detailed calculation in [144], Chapter 25, Section 4.

215

(C.22)

With all the pieces of the algorithm together, the unfolding procedure is the following:
1. Define the number of bins nb , and their boundaries.
2. Define the number of bins nx , and their boundaries.
3. Build the condition matrix C introduced in equation (C.18), and calculate its inverse
C −1 .
4. Generate the simulated events to find: x ini , reconstruct the simulated events to find:
b ini , and construct A, which is the number-of-events matrix introduced in equation
(C.14).
5. Calculate the covariance matrix B introduced in equation (C.11).
6. Calculate A˜ij , and ˜bi from equation (C.15).
˜ −1 to fulfill equation (C.19).
7. Multiply AC
8. Plot log|di | vs. i to determine the last statistically significant equation k, and set τ
(see equation (C.22)).
9. Obtain the unfolded distribution of events x˜ from equation (C.20).
ROOT (see [145, 146]) has a tool, TSVDUnfold, that implements the unfolding algorithm
in a very user friendly way. This tool requires the user to provide:
• x ini .
• b ini .
• A.
216

With these inputs, TSVDUnfold goes through steps 1 to 8 in the algorithm. Then it asks
the user to use the log|di | vs. i plot to provide the parameter k. Based on simulated events
and reconstructed simulated events, TSVDUnfold calculates the matrices B and C. At this
point it trained itself to be able to handle real data. Now the user provides a new vector:
b data , which contains all the real data events. From b data and a set3 of toy MC simulated
events, generated by TSVDUnfold itself, it calculates a new covariance matrix B data , which
is the error matrix for the unfolded distribution. Finally, after training, TSVDUnfold goes
through step 9 in the algorithm and gives the unfolded distributions x˜ and x˜data . x˜ is the
unfolded distribution for the reconstructed simulated events, and x˜data is that for the real
data events.

Unfolded Distributions
ν True Energy Spectra. NDOS MC. No Unfolding.

ν True Energy Spectra. NDOS MC. With Unfolding.

400

400

Reco MC
True MC

Fake Data

300

Events / 0.5 GeV

Events / 0.5 GeV

300

200

200

100

100

0
0

Unfolded Fake Data

2

4

6

ν Energy (GeV)

8

0
0

10

(a)

2

4

6

ν Energy (GeV)

8

10

(b)

Figure C.1 Neutrino Energy Distributions Before And After Unfolding. (a) Reconstructed MC simulation (red) normalized to fake data (black) area. (b) True MC simulation
(green) normalized to unfolded data (black) area. MC simulation.

Following the steps presented at the end of the previous section, and in order to train
3 The size of this toy MC simulated set is entered by the user. The recommended size is

nTSVD = 100 events.
217

the algorithm, the MC simulated sample is divided into two sets: one used to build x ini ,
training sample, and the other one4 used to build b ini , testing sample. The TSVDUnfold
is fed with the due inputs, and the results of the training process are presented in figure
C.1. As presented in the previous section, the parameter k comes from figure C.2b, which
shows that the last statistically significant equation is that with i = 2, since for higher i the
distribution oscillates randomly around |di | = 1. The parameter k is set to:
k = 2,

τ = s˜22 .

(C.23)

The migration of events from their reconstructed energy bins to their unfolded energy bins
is presented in figure C.2a, which is the covariance matrix B data mentioned above. The
diagonal entries between 1 GeV and 2.5 GeV are 20.1% of the total entries. In this case the
correct bin is reconstructed properly. The off diagonal bins in that same energy region, and
colored with light orange, account for 27.2% of all entries.
ν Energy. NDOS MC Covariance Matrix.

log|di| vs. i. TSVDUnfold.

10

10
100

1

6
i

|d |

ν Energy

True

(GeV)

8

4

10-1

50

10-2

2

0
0

2

4

6

ν Energy

Reco

8

10

-3

0

10

0

5

10

15

20

i

(GeV)

(a)

(b)

Figure C.2 Results From Unfolding Fake Data. (a) The covariance matrix B data . (b)
log|di | vs. i. MC simulation.
Figure C.1 shows 4 distributions: The true MC simulated events, the reconstructed MC
4 About 10% the size of the first sample.

218

simulated events, the fake data5 events, and the unfolded events. True MC and Reco MC
simulations resemble figure 6.14. The fake data, as part of the MC simulated sample, should
have the same shape as the Reco MC simulation does, which is the case in figure C.1a.
The unfolded distribution sits on top of the true MC simulation one, which means that the
unfolded procedure worked as expected, as seen in figure C.1b. The errors in the unfolded
distribution come from the input to the covariance matrix, which in this case are statistical
errors in the fake data.
The goal of the unfolding was to place all the fake data events into their most probable
energy bins given the performance of the detector embedded in the calculated covariance
matrices. This corrects for the discrepancies between reconstructed and simulated events
presented in chapter 6.
The training and testing of the unfolding algorithm are satisfactory, and the algorithm
is applied to the NDOS data. In order to find the unfolded data distributions, bdata is
filled with real NDOS events instead of fake data events, and the covariance matrix B data is
calculated based on the real data measured errors. Everything else is the same as presented
above, since it is the result of training the algorithm.

5 The small MC simulated testing sample.

219

Appendix D
List Of Muon Neutrino Charged
Current Candidate Events
Run
13067
13097
13131
13178
13228
13247
13289
13359
13405
13405
13417
13504
13521
13526
13531
13550
13648
13659
13679
13730
13732
13737
13743
13756
13767

Event
01
09
08
09
00
10
00
03
00
09
14
00
07
03
02
01
05
12
12
20
04
02
03
04
01

52339
401527
369633
423895
7822
468719
3856
149922
21468
423816
647470
5587
320076
173654
100521
67262
224998
576285
565281
948244
224730
112479
176473
219066
68409

Eν
(GeV)
1.1
2.0
2.3
1.0
0.9
1.0
1.8
1.3
2.1
1.0
2.0
1.6
1.7
1.4
1.1
1.0
1.9
1.2
1.8
1.0
1.4
1.2
1.0
3.3
1.2

Run

Event

Eν
(GeV)
13067 05 242768
1.3
13119 19 878982
1.4
13168 09 439180
2.0
13209 07 351868
1.5
13247 01 88704
1.0
13286 00 31278
0.9
13352 09 455006
1.5
13359 18 838462
2.1
13405 06 283899
2.5
13406 05 269806
1.1
13417 40 1815674
1.5
13508 11 502607
0.9
13523 09 429603
2.0
13528 16 704288
0.9
13540 09 432385
1.0
13557 20 853445
3.5
13659 03 154457
2.1
13675 05 251706
1.7
13679 13 624373
1.1
13731 18 858628
1.4
13732 05 233639
1.0
13737 04 188755
0.9
13754 01 62591
1.0
13759 05 248715
1.2
13767 02 81374
3.4

Run
13084
13120
13173
13210
13247
13286
13352
13390
13405
13406
13485
13508
13525
13531
13543
13567
13659
13679
13709
13732
13732
13737
13754
13763
-

Event
14
06
00
12
08
04
15
19
08
19
18
17
00
02
04
00
07
08
18
01
06
08
02
04

Table D.1 Selected QE Candidate Events.

220

636949
275972
21364
558497
381991
185526
686658
898936
373623
895717
898130
791080
30494
100521
197375
32484
333945
402161
838184
83757
281760
413309
123870
162894
-

Eν
(GeV)
1.4
1.5
1.0
0.8
1.3
1.0
2.1
1.2
2.1
2.0
0.9
1.4
3.1
1.1
0.9
1.1
1.2
1.9
0.9
1.5
3.1
1.0
2.5
0.6
-

All the selected QE neutrino candidates before cosmic background subtraction are presented in table D.1. All the selected non-QE neutrino candidates before cosmic background
subtraction are presented in tables D.2 and D.3.
Run
13068
13071
13084
13097
13117
13131
13160
13173
13178
13189
13209
13210
13220
13247
13262
13277
13312
13333
13333
13348
13352
13360
13389
13393
13399
13405
13406
13407
13441

Event
01
05
01
11
03
01
11
12
04
02
01
13
01
01
01
01
06
06
15
06
15
05
02
19
10
06
08
00
06

70583
273089
87787
516437
151666
73464
512920
576650
206510
95469
71489
608565
84251
55167
70110
79372
315859
285527
694421
271122
706818
260305
98800
880031
470075
281676
396159
40739
320584

Eν
(GeV)
1.9
1.5
1.6
2.4
0.9
2.4
1.5
1.6
1.6
1.5
1.5
3.1
2.2
1.6
1.6
1.9
1.7
1.9
3.0
1.5
1.2
2.0
2.9
2.3
1.7
2.3
2.6
1.9
1.4

Run
13068
13075
13087
13097
13120
13160
13160
13176
13178
13190
13209
13213
13236
13247
13265
13286
13320
13333
13339
13352
13353
13365
13389
13393
13399
13405
13406
13407
13443

Event
02
15
01
11
00
01
15
04
05
00
14
02
05
09
04
01
15
10
00
05
13
02
11
20
12
18
13
09
09

133375
704141
57985
517102
38220
61236
687140
244242
266544
7437
655627
94255
246984
430374
229559
48964
685032
488169
20224
269563
611234
118904
533620
933931
578886
857049
641569
435601
398666

Eν
(GeV)
1.1
2.0
3.2
1.8
1.3
1.3
2.3
1.7
2.1
2.4
2.0
1.8
2.4
1.8
2.6
2.3
1.6
1.3
2.2
1.7
1.1
1.7
2.8
1.6
1.5
3.0
1.1
2.1
1.9

Run
13068
13075
13094
13097
13123
13160
13166
13176
13178
13208
13209
13214
13241
13262
13265
13286
13330
13333
13339
13352
13359
13365
13393
13399
13399
13405
13406
13409
13443

Event
03
17
05
13
04
02
09
11
07
03
19
03
11
01
06
02
02
12
03
10
12
03
07
06
12
20
20
14
14

183770
781439
265850
604309
158468
112807
410288
539526
323898
140024
868527
177158
541964
55638
292238
120280
117981
546198
135737
494823
558228
141066
326670
300556
589675
936462
945997
668614
664933

Table D.2 Selected Non-QE Candidate Events, I.

221

Eν
(GeV)
1.8
2.5
2.0
2.0
1.3
2.0
1.3
1.4
4.5
1.9
0.8
1.4
1.1
4.0
1.3
1.5
1.9
2.7
1.9
2.0
1.7
1.3
1.6
2.0
1.4
2.1
1.5
1.8
1.3

Run
13444
13490
13504
13508
13512
13521
13523
13531
13534
13557
13611
13619
13660
13710
13717
13731
13732
13740
13743
13744
13744
13754
13754
13760
13764
13770

Event
09
00
03
00
06
04
12
08
00
07
04
18
07
18
12
14
05
00
00
04
10
01
16
17
00
01

445873
27697
163755
31211
292369
219941
573584
373797
18678
261561
184605
843304
338739
864719
589572
688158
236320
11613
7658
188689
495266
85788
773307
750963
4051
43260

Eν
(GeV)
2.0
2.2
2.9
3.4
2.2
1.3
1.7
1.3
1.3
2.4
1.2
1.2
1.6
2.2
2.3
2.0
1.3
3.0
1.0
1.2
1.4
1.2
1.8
2.2
1.1
2.0

Run

Event

Eν
(GeV)
13444 13 632045
1.3
13490 00 43841
1.9
13504 03 170439
2.8
13508 12 560534
1.7
13512 14 667424
2.4
13521 07 334779
1.7
13523 22 1005777
1.8
13533 07 337448
3.3
13539 03 169453
2.3
13557 14 619026
2.1
13619 04 205187
1.1
13659 11 534603
1.9
13660 10 493358
2.3
13711 03 150155
1.5
13717 15 717932
1.3
13731 19 917338
1.4
13737 04 197897
1.4
13740 21 972374
1.8
13743 08 406626
1.2
13744 05 272594
2.6
13746 23 1096829
2.6
13754 12 588024
2.8
13754 16 821744
1.5
13763 01 70828
1.9
13765 03 146213
1.5
13770 09 344492
2.3

Run

Event

Eν
(GeV)
13485 14 680500
1.5
13495 15 714932
2.2
13505 06 317921
2.4
13508 13 610384
2.7
13516 01 55536
1.5
13523 03 153709
3.7
13525 06 275923
1.1
13533 17 806866
1.6
13540 20 917878
1.6
13557 15 669256
1.8
13619 11 525351
1.5
13660 05 239565
1.5
13709 16 746592
1.8
13711 06 300659
4.1
13727 13 608348
3.3
13732 03 151136
1.7
13737 05 263039
1.5
13742 05 270585
2.0
13743 23 1069794
1.8
13744 07 362545
2.0
13751 15 700296
1.5
13754 15 691009
1.4
13759 19 888335
0.9
13763 09 358811
1.0
13767 02 91379
1.6
13778 22 831025
1.4

Table D.3 Selected Non-QE Candidate Events, II.

222

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232