Energy Dependence in the Neutrino Scattering Data of ...

Energy Dependence in the Neutrino Scattering Data ofMINERvA

A THESIS

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTA

BY

Ishmam Mahbub

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

Advisor: Dr. Richard Gran

July, 2021

FERMILAB-MASTERS-2021-05

Acknowledgements

First, I would like to acknowledge the support of my advisor Dr. Richard Gran. The

work would not have been possible without his constant guidance and mentorship, which

kept me pushing throughout the research.

A special thanks go to Dr. Eric West. He has been a great teacher and deepened my

understanding of physics during my stay at UMD.

Finally, I would like to thank my collaborator at MINERvA for their invaluable input

on my work, and to all my classmates for being supportive throughout.

i

Abstract

As we prepare for lower neutrino energy beams where physics is highly energy-

dependent, it is important to isolate factors that can contribute largely to these low

energy neutrino experiments. The world of neutrino oscillation experiments uses a wide

range of energies: from 0.7 GeV beam of T2K and MicroBooNE, 2 GeV beam of NOvA,

to the 0.6 to 6.0 GeV beam produced by DUNE in the future. MINERvA’s data are

currently the best place to test the upper end of the range for DUNE and can be extrap-

olated into both DUNE’s and NOvA’s oscillation maximum. The goal of the research is

to constrain energy dependence in the neutrino scattering experiment using data from

MINERvA and then determine what factors contribute to the observed energy depen-

dence. The research is divided into two parts. First, I will analyze different theoretical

models in different channels of interaction, such as the quasi-elastic and delta resonance

channels, in terms of neutrino energy dependence. The goal is to study the models at

the level of the structure functions, which emphasizes the W2 structure function (the C

function for QE) dominates the MINERvA data, but the W3 is increasingly important

for NOvA and DUNE. The second half of the research is data driven and focuses on the

different detectors and systematic effects. An experimental effect, the angle acceptance,

is larger than the structure function effects. However, it is a detector geometry effect

and is well measured and well modeled. Other effects such as the muon energy scale are

small and localized. Additionally, the research will determine if the MINERvA GENIE

model correctly predicts all of the energy dependence seen in the data, and identify

the remaining unmodeled energy dependence between the MINERvA data and its best

simulation.

ii

Contents

Acknowledgements i

Abstract ii

List of Tables vi

List of Figures viii

1 Introduction 1

1.1 Three-Momentum Transfer & Energy Transfer . . . . . . . . . . . . . . 2

1.2 Different Channels of Neutrino Interaction . . . . . . . . . . . . . . . . . 3

1.2.1 Quasi-Elastic Scattering (QE) . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Delta Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Two particle, Two hole (2p2h) . . . . . . . . . . . . . . . . . . . 5

1.2.4 Deep Inelastic Scattering (DIS) . . . . . . . . . . . . . . . . . . 6

1.2.5 Triangle Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Monte Carlo Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Random Phase Approximation (RPA): . . . . . . . . . . . . . . 11

1.4.2 2p2h enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.3 QE enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.4 Low-Q2 resonance suppression: . . . . . . . . . . . . . . . . . . . 12

1.4.5 SuSA Valencia Tuning . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.6 Removal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 MINERA detector and detection system . . . . . . . . . . . . . . . . . . 14

iii

1.5.1 Neutrino Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.2 MINERvA Active Tracker Region . . . . . . . . . . . . . . . . . 15

1.5.3 Electro-magnetic calorimeter (ECAL) . . . . . . . . . . . . . . . 16

1.5.4 Hadron Calorimeter (HCAL) . . . . . . . . . . . . . . . . . . . . 17

1.5.5 Outer Detector (OD) . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.6 Muon Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 MINERA event reconstruction and MC simulation . . . . . . . . . . . . 18

2 Energy Dependence in Delta Resonance Models 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Berger-Sehgal (BS) Model . . . . . . . . . . . . . . . . . . . . . . 22

2.1.2 Lalakulich-Paschos Model . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Comparison of Berger-Sehgal and Lalakulich-Paschos Model . . . . . . . 30

2.2.1 Eν dependence of BS and LP model . . . . . . . . . . . . . . . . 32

3 Energy Dependence in the Structure Functions 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Energy Dependence of ∆++ Structure Functions . . . . . . . . . . . . . 41

3.3 Quasi-Elastic Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Energy dependence in the MINERvA data 50

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Background and Review of Literature . . . . . . . . . . . . . . . . . . . 51

4.3 Method of Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Looking for Energy Dependence in the MINERvA Data . . . . . . . . . 56

4.4.1 Energy Dependence in MidEnu and HighEnu data . . . . . . . . 57

4.4.2 Sources of Energy Dependence for HighEnu and MidEnu . . . . 60

4.4.3 Energy Dependence in LowEnu and MidEnu data . . . . . . . . 63

4.5 Looking Further into Angle Acceptance with Imposed Angle Cuts . . . 66

4.6 Muon Kludge and Energy Dependence . . . . . . . . . . . . . . . . . . . 70

4.6.1 Expected Muon Kludge Pattern . . . . . . . . . . . . . . . . . . 71

4.6.2 Recoil energy Plots without the Muon Kludge . . . . . . . . . . . 72

4.6.3 Energy Dependence due to Muon Kludge . . . . . . . . . . . . . 74

iv

4.7 Flux Measurements and a Necessity for a Flux Shift? . . . . . . . . . . . 77

5 Uncertainties in the Energy Dependence Modeling 80

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Putting in artificial energy dependence . . . . . . . . . . . . . . . . . . 81

5.2.1 Quasi-Elastic Channel . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2 Delta Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.3 2p2h Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3 Conclusions and future considerations . . . . . . . . . . . . . . . . . . . 93

6 Conclusion and Discussion 94

References 96

Appendix A. The Hadronic Rich Component in Muon Fuzz 101

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 What Does Event by Event Tell Us . . . . . . . . . . . . . . . . . . . . . 103

A.3 Can We Completely Ignore Muon Fuzz . . . . . . . . . . . . . . . . . . . 104

A.4 Selected Muon Fuzz Subtraction . . . . . . . . . . . . . . . . . . . . . . 106

A.5 Looking into high muon fuzz events . . . . . . . . . . . . . . . . . . . . . 107

A.6 Further Event Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.6.1 Events with Negative Bias . . . . . . . . . . . . . . . . . . . . . . 110

A.6.2 Events with Positive Bias . . . . . . . . . . . . . . . . . . . . . . 112

A.6.3 Event with High Muon Fuzz in the ECAL . . . . . . . . . . . . . 113

A.7 Looking into MuonFuzz2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

A.8 Recoil Plots and Muon Fuzz . . . . . . . . . . . . . . . . . . . . . . . . . 116

v

List of Tables

2.1 Integrated incoherent neutrino scattering cross-section in units of 10−38cm2.

The cross section σQ2 refers to dσdQ2 for Q2 < 0.1GeV . In all of Q2, there

is a 14% suppression coming from adding low-Q2 effects of Berger-Sehgal

model (taken from table 1 in ref [1]). . . . . . . . . . . . . . . . . . . . 23

4.1 The table summarizes R. Fine’s work using low-ν method [2]. His work

suggests a 5% positive shift for MidEnu and HighEnu energies, thereby

increase MidEnu flux by 5% or decrease HighEnu flux by 5%. On the

hand, he determined a 5% positive shift for LowEnu and MidEnu. So,

decrease LowEnu Flux by 5% or increase MidEnu flux by 5% . . . . . . 78

4.2 The table outlines the double ratio for low q3, mid q3 and high q3 for

different energy cuts in my work. As can be seen from the table, the dou-

ble ratio for HighEnu/MidEnu is around 0.95 for all the q3, and suggests

a flux shift. By contrast, the double ratio for MidEnu/LowEnu do not

suggest any concrete evidence for a flux shift. . . . . . . . . . . . . . . 78

5.1 The table summarizes the lower and upper 2σ bounds for unmodeled QE

energy dependence for different energy and q3 slices . . . . . . . . . . . . 86

5.2 The table summarizes the best fit energy dependence model for the QE

channel and the associated 2σ width for different energy and q3 slices . 86

5.3 The table summarizes the nominal χ2 (current GENIE setting) and the

best fit χ2 for QE at six different energy and q3. . . . . . . . . . . . . . 86

5.4 The table summarizes the lower and upper 2σ bounds for delta channel

energy dependence for different energy and q3 slices . . . . . . . . . . . . 89

5.5 The table summarizes the best fit energy dependence model for the delta


vi

5.6 The table summarizes the lower and upper 2σ bounds for unmodeled

2p2h energy dependence for different energy and q3 slices . . . . . . . . 92

5.7 The table summarizes the best fit energy dependence model for the 2p2h


A.1 Events with muon fuzz > 40 MeV in the Ecal region for 4 < Eν < 7

GeV. We randomly selected five events in each of low q3,mid q3 and high

q3 region. From the table we can see that in the ECAL, bremstrahlung

activity increases by 20% and neutral hadron activity decreases by 20%

from our previous analysis of looking into the whole detector . . . . . . 113

A.2 A comparison between using muonfuzz1 and muonfuzz1 + muonfuzz2 for

various event selection and for 4 < Eν < 7 GeV, 1000 < q3 < 1200MeV .

The resolution is improved when the mufuzz2 is included in the subtraction.114

A.3 A comparison between using muonfuzz1 and muonfuzz1 + muonfuzz2 . 116

A.4 A comparison between using muonfuzz1 in Tracker and ECAL region for

4 < Eν < 7 GeV, 1000 < q3 < 1200MeV . . . . . . . . . . . . . . . . . 116

vii

List of Figures

1.1 Feynman diagram for quasi-elastic scattering (left), and a QE event can-

didate seen in the MINERvA detector (right). [3] . . . . . . . . . . . . 4

1.2 Feynmann diagram for Delta scattering (left), and a ∆++ candidate seen

in the MINERvA detector (right). [3] . . . . . . . . . . . . . . . . . . . 5

1.3 Feynman diagram for 2p2h scattering (left), and a candidate 2p2h event

seen in the MINERvA detector (right). [3] . . . . . . . . . . . . . . . . 6

1.4 Feynman diagram for DIS scattering (left), and a DIS event seen in the

MINERvA detector (right). [3] . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 The figure shows the triangle diagram which is a 2-D histogram of the

phase space where the color axis represents the double-differential cross

section and also the expected relative event rate. It provides a view of

the events in the three-momentum transfer and energy transfer space

simultaneously. The QE white line is the invariant mass line for proton

(938 MeV), and the delta white line is the delta invaraint mass line (1232

MeV). The quasi-elastic events are gather around the QE line, and delta

events are around the delta white line which can also be seen by the high

event counts around these lines. . . . . . . . . . . . . . . . . . . . . . . 8

1.6 The figure shows the theoretical energy dependence modeled in GENIE

with lines of constant W at 0.938GeV, 1.232GeV and 1.535GeV. The left

plot is the ratio of 15GeV prediction to 5GeV prediction (15GeV/5GeV),

and the right plot is the ratio of 5GeV and 3GeV prediction (5GeV/3GeV). 9

viii

1.7 The figure shows low energy and medium energy neutrino (left) and anti-

neutrino (right) fluxes per proton on target (POT) as a function of Energy

as predicted by a GEANT4 simulation reweighted using results from the

NA49 hadron production experiment [4]. . . . . . . . . . . . . . . . . . . 10

1.8 The figure depicts how a core distribution is altered by the SuSA and

2p2h tuning [5]. The structure of the plot is introduced in chapter 4. . 13

1.9 The top diagram depicts the schematics of the NUMI neutrino beam

creation process. The outgoing beam (shown by blue line) is directed to

the Soudan Mine and to the NOvA experiment in Northern Minnesota.

The bottom figure depicts the beam traveling through the earth, as if you

were viewing it from the side somewhere in orbit. There is no tunnel;

most neutrinos pass through the earth without interacting. . . . . . . . 15

1.10 The figure shows the front view of the MINERvA detector (left) and side

view of the detector (right)). From the front view, we notice the hexag-

onal shape of the inner detector and how it is wrapped by side ECAL

and outer detector. From the side view, the reader should note the order

of the inner detector. A little left of center is the active tracker region

mainly composed of scintillators for the best resolution of collision, then

ECAL to force electromagnetic interactions, then HCAL to stop hadrons,

and finally muon spectrometer to measure outgoing muon momentum. 16

1.11 The figure illustrates a reconstructed MINERvA event, and ach triangle

is a scintillator with measured activity. Different colors represent differ-

ent amounts of energy deposited in the scintillator, darker representing

more energy. The green line is the MINERvA reconstructed track of the

particles. The long one is muon track (since it has the most energy and

loses it energy slowly and therefore can travel a long distance), and the

event also has two other hadrons having a shorter trajectory. . . . . . . 17

1.12 The figure shows twelve reconstructed time slices for 10 microseconds

through an open gate. Each colored slice indicates the occurrence of

scintillator and particle interaction. The vertical axis also depicts the

total number of hits in each time slice. . . . . . . . . . . . . . . . . . . . 19

ix

2.1 Differential cross secrtion dσdQ2 for charged current pion production by

muon neutrinos using Berger-Sehgal model from the Berger-Sehgal paper

[1]. a) Eν = 0.7GeV, b) Eν = 1.3GeV [1] . . . . . . . . . . . . . . . . . 22

2.2 The figure shows a comparison between the Rein-Sehgal model used in

GENIE v2.12 and the Berger-Sehgal model used in GENIE v3.0 at neu-

trino energy 3GeV (left) and 5GeV (right). Ratio shown at the bottom

panel is relatively independent of neutrino energy, and a suppression ef-

fect averaging around 15% is observed at Q2 < 0.1GeV 2. It should be

noted that all of suppression is not due to lepton mass effects . . . . . . 24

2.3 The figure is from the work done by J.Nowak [6], comparing Berger-

Sehgal and Rein-Sehgal model with MiniBooNE data. This uses the

MiniBooNE neutrino flux whose energies are around 0.7 GeV, similar to

the left plot of Fig. 2.1. The 20% suppression observed in my analysis is

present in his work as well. . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 The figure shows extracted suppression factors for charged pion inter-

actions from low-Q2 tuning to each channel, taken from ref [7]. When

compared to Fig. 2.2, the extracted low-Q2 suppression is much wider

and deeper than suppression from BS and RS ratio, and also to a result

from the MINOS experiment [8] . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Calculated cross-section dσ/dQ2 for ANL neutrino energy disctribution

using Lalakulich-Paschos model from their original paper [9]. The dotted

line shows the theoretical prediction neglecting the muon mass (effectively

an electron neutrino interaction), and the solid line shows expected cross-

section using mµ = 0.105GeV . The plot on the left uses axial mass

MA = 1.10GeV , and the plot on the right uses MA = 0.84GeV . We can

observe from the figure that reducing axial mass in LP model allows us

to fit the ANL data [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 The figure shows a cross-section for axial mass MA = 0.84GeV (blue)

and MA = 1.10GeV (red) at neutrino energy Eν = 5GeV (left) and

Eν = 3GeV (right). From the ratio of the two cross-section, we discern

that it is relatively flat in Q2 dependence above Q2 = 0.5GeV 2, and has

a sharp dip below Q2 = 0.5GeV 2. . . . . . . . . . . . . . . . . . . . . . 30

x

2.7 The figure shows a comparison between Lalakulich-Paschos model and

Berger-Sehgal model at neutrino energies 15GeV (a), 5GeV (b), 3GeV

(c), 2GeV (d) and 1GeV (e). (Add more explanation) . . . . . . . . . . 32

2.8 The figure shows the differences in energy dependence the Lalakulich-

Paschos model and Berger-Sehgal model from energies 1GeV to 15GeV.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


with lines of constant Q2 at 0.45, 0.64, 0.78 and 1 GeV2 . The left plot

is the ratio of 15GeV prediction to 5GeV prediction (15GeV/5GeV), and

the right plot is the ratio of 5GeV and 3GeV prediction (5GeV/3GeV). 38


with lines of constant W at 0.938GeV, 1.232GeV and 1.535GeV. The left

plot is the ratio of 15GeV prediction to 5GeV prediction (15GeV/5GeV),

and the right plot is the ratio of 5GeV and 3GeV prediction (5GeV/3GeV).

We could see that in the QE and delta region, the energy dependence is

within 10% limit theoretically. . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 The figure depicts the Q2 dependence of W1, W2, W3, W4 and W5 at

neutrino energy 5GeV, 3GeV, 2GeV, and 1GeV as well as the total cross

section at these energies. Each term is actually the structure function

multiplied by its kinematic factors in the cross-section formula. For E >

3GeV , almost the entire cross-section contribution comes from W2. As

we go to lower neutrino energies Eν < 3GeV , W3 starts becoming more

relevant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 The figure shows the energy dependence of each structure function. To

find the energy dependence of the W1, W2, W3, W4 and W5, we take

the ratio each structure function at different energies. From the plot,

we could see that only W2 has a significant Q2 dependence in the in its

energy dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 The figure shows the energy dependence of the total cross-section for the

Delta channel. The cross-section is not dependent (or within 5%) on

neutrino energy at high Q2 and high energy, which can be inferred from

the ratio being close to one. . . . . . . . . . . . . . . . . . . . . . . . . . 44

xi

3.6 The figure depicts the Q2 dependence of A, B and C temrs at neutrino

energy 5GeV, 3GeV, 2GeV, and 1GeV. The plot includes all the kinematic

factors such that the total cross section is the sum of the three parts.

Similar to the Delta channel, for E > 3GeV , almost the entire cross-

section contribution comes from the C term. As we go to lower neutrino

energies Eν < 3GeV , B term starts becoming non negligible. . . . . . . 46

3.7 The figure uses ratios to show the energy dependence of the three QE

structure functions and the total cross-section. From the plot, we can

notice that the A and B terms have a flat energy dependence, and the C

term has a curved Q2 dependent energy dependence similar to W2 for the

Delta channel. The total energy dependence is also similar to the Delta

case, is around one for most of the neutrino energy range of interest. . 47

3.8 The figure shows the total cross-section energy dependence for 15GeV/5GeV

(left plot) and 5GeV/3GeV (right plot) using GENIE v3.0. The total

cross-section energy dependence is very similar to W2 energy dependence

in Fig. 3.4 and has a Q2 dependence. This brings in the question whether

such a dependence is coming from the energy dependent pre-factor similar

to that of W2 at high energies? Then, one can speculate a way to ex-

amine the energy factor of W2 by studying the total cross-section energy

dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 The figure shows MiniBooNE CCQE cross-section per nucleon with re-

spect to true neutrino energy, taken from ref [11] for long range of ener-

gies. Cross-section results from the LSND [12], the NOMAD experiment

[13], and cross-sections for a free nucleon are reported in the plot. . . . 53

4.2 The figure shows the energy and three momentum cuts. The energy

spectrum is divided into 0-4GeV (LowEnu), 4-7GeV (MidEnu) and 7-

20GeV (HighEnu). The recoil q3 is broken into three regions: 0-0.4GeV

(low q3), 0.4-0.8GeV (mid q3) and 0.8-1.2GeV (high q3). One more thing

to notice in the plot is that we will be using the MINERvA ME beam

which peaks at around 5GeV. . . . . . . . . . . . . . . . . . . . . . . . 54

xii

4.3 The figure shows the distribution of MidEnu (4-7GeV) and HighEnu (7-

20GeV) data and MC distribution for high q3 (0.8 < q3 < 1.2GeV). The

left panel also shows the ratio of HighEnu and MidEnu ratio for Data

and MC distribution. While the right panel shows the double ratio of

Data and MC as defined in the text. Data HighEnu and MidEnu ratio

(left plot) shows that there is indeed energy dependence in the data in

these phase space. But, the double ratio near one indicates the energy

dependence being well modelled. . . . . . . . . . . . . . . . . . . . . . . 55

4.4 The figure depicts the data and MC distribution for MidEnu and High-

Enu, and for high q3 (top panel), mid q3 (moiddle panel) and low q3

(bottom panel). The left plots show the HighEnu/MidEnu ratio and

the right plots show the double ratio. Gradient in the HighEnu/MidEnu

shows existing energy dependence in the data. We also observe that these

gradients go down at lower q3 showing the energy dependence is milder

at lower q3. The double ratio around one for all range of q3 outlines that

we are modelling these energy dependence in the data well. . . . . . . . 58

4.5 The figure shows the detector setup of the MINERvA experiment. Due

to the limited size of MINOS Near Detector, some events will not be

counted if if they have too large a muon angle large and miss the MINOS

detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 The figure shows how the angle of outgoing muons changes with neutrino

energy and three momentum transfer. When the muon misses the MINOS

detector, we do not count the event. . . . . . . . . . . . . . . . . . . . . 62

4.7 The figure depicts the data and MC distribution for LowEnu and MidEnu,

and for high q3 (top panel), mid q3 (middle panel) and low q3 (bottom

panel). The left plots show the MidEnu/LowEnu ratio and the right

plots show the double ratio. The angle acceptance pattern is negligible

in mid q3 and low q3 due to lower neutrino energy in these cases. Another

noticeable feature in all three ratio is the bumping up of the ratio in the

middle of the spectrum and the data ratio falling off near the tail. . . . 65

xiii

4.8 The figure depicts the data and MC distribution with angle cut of 10

degrees (left panels) and angle cuts of 5 degrees (right panel) for MidEnu

and HighEnu, and for high q3 (top panel), mid q3 (middle panel) and low

q3 (bottom panel). The angle accpetance pattern detailed in the previous

section becomes more prominent in the 10 degree cuts for high q3. The 5

degree cuts seem to have an added pattern not seen before. There is an

extremely large bump in the middle of the spectrum. . . . . . . . . . . 68

4.9 The figure depicts the data and MC distribution with angle cut of 10

degrees (left panels) and angle cuts of 5 degrees (right panel) for LowEnu

and MidEnu, and for high q3 (top panel), mid q3 (middle panel) and low

q3 (bottom panel). The angle acceptance pattern should present itself

more clearly at lower energies, and so we should expect to see it better

in these energy cuts. But, the number of events are cut down drastically

for LowEnu events for it to be of good statistical value, and it has limited

statistical power compared to the 10 degree cut. . . . . . . . . . . . . . 69

4.10 Movement due to a resolution sized shift in the muon energy for the low

q3 on the left and the mid q3 on the right [14]. The effect shown is the

opposite effect of what is discussed in the section, it turns on the muon

energy scale rather than turning it off. So, for our case we should expect

the arrow the face the opposite direction. . . . . . . . . . . . . . . . . . 72

4.11 The figure depicts the effect of muon scale for MidEnu and LowEnu. The

top panel is for high q3, mid panel is mid q3 and the bottom panel is

low q3 plots. The left column shows the data and MC distribution with

muon kludge turned on, and the right column of plots illustrates the

distributions with muon kludge turned off. . . . . . . . . . . . . . . . . 75

4.12 The figure depicts the effect of muon scale for LowEnu and MidEnu. The

top panel is for high q3, mid panel is mid q3 and the bottom panel is

low q3 plots. The left column shows the data and MC distribution with

muon kludge turned on, and the right column of plots illustrates the

distributions with muon kludge turned off. . . . . . . . . . . . . . . . . 76

xiv

5.1 The blue dots in the figure represent the data distribution at midEnu

(4-7GeV) and the red dots represent data distribution at highEnu (7-

20GeV). The solid lines that follow them are the corresponding MC pre-

dictions. The bottom panel depicts the corresponding double ratio and

the best fit for the double ratio. . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 The figure shows the comparison between a nominal distribution and

the corresponding distorted distribution where a total of 14% energy

dependence is added only to the QE channel. The vertical scale on the

two plots are different. On the right plot, midEnu MC has gone up,

highEnu MC has gone down, and so the double ratio went up. The chi-

square is taken with respect to the best fit of line of the double ratio.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 The figure shows a chi-square scan performed for midEnu and highEnu

data at lowq3. The nominal χ2 = 6.73, best fit χ2 = 6.13, and best

fit model has 0.9% increased QE cross-section for midEnu and 0.9% de-

creased QE cross-section for highEnu. . . . . . . . . . . . . . . . . . . . 84

5.4 The figure shows the distribution and the corresponding double ratio for

lowEnu and midEnu at high q3, which is the same as the top right plot

in Fig. 4.7. We can see that double ratio in the first bins are equally

distributed above and below the best fit, and this leads to a nominal

p-value of 0.0001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 The left plot shows data and MC distribution for lowEnu and midEnu at

midq3 after increasing delta cross-section for lowEnu by 5% and decreas-

ing midEnu delta cross-section by 5%. We can observe the deviation in

the middle of the double ratio where delta channel is most prominent.

The right plot shows the chi-square scan performed in this region. The

nominal χ2 = 37.11, best fit χ2 = 35.273, and best fit model is at +1%

artificial energy added to lowEnu with similar decrease in midEnu. . . 90

xv

5.6 The left plot shows data and MC distribution for lowEnu and midEnu

at midq3 after increasing 2p2h cross-section for lowEnu by 10% and de-

creasing midEnu 2p2h cross-section by 10%. The right plot shows the

chi-square scan performed in this region. The nominal χ2 = 37.11, best

fit χ2 = 36.12, and best fit model is at -4% artificial energy added to

lowEnu with similar increase in midEnu. The slow variation in chi-square

is due to high width and low amplitude of 2p2h distribution. . . . . . . 91

A.1 Normalized muon fuzz distribution for 800 < q3 < 1200 MeV, 4 < Eν <

7 GeV(left plot) and 800 < q3 < 1200 MeV, 7 < Eν < 20 GeV (right plot)103

A.2 An event with 413 MeV bremsstralung electron that generates 181 MeV

muon fuzz energy. These compose 20% of events with muon fuzz > 40

MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A.3 An event with proton activity in the muon fuzz region . . . . . . . . . . 105

A.4 An event with neutron activity in the muon fuzz region. 80% of events

with muon fuzz > 40 MeV are neutron,proton or pion events. . . . . . . 105

A.5 Resolution plot with muon fuzz subtraction (left plot) and without muon

fuzz subtraction (right plot) for 4 < Eν < 7 GeV and 600 < q3 < 800

MeV. RMS increase by almost 7% when we stop subtracting muon fuzz 106

A.6 Reolution plot with muon fuzz subtraction (left plot) and without muon

fuzz subtraction (right plot) for 7 < Eν < 20 GeV and 600 < q3 < 800

MeV. RMS increase by almost 8% when we stop subtracting muon fuzz. 107

A.7 Resolution plot with several muon fuzz event selection for 4 < Eν < 7

GeV and 600 < q3 < 800 MeV. RMS is the lowest for selected events

with muon fuzz < 60 MeV (top left) and greatest for events with muon

fuzz < 10 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.8 Resolution plot with several muon fuzz event selection for 7 < Eν < 20

GeV and 600 < q3 < 800 MeV. Similar to the previous figure, in this

energy range RMS is the lowest for selected events with muon fuzz < 60

MeV (top left) and greatest for events with muon fuzz < 10 MeV . . . . 109

xvi

A.9 RMS comparison of selected events with muon fuzz > 40MeV (right plot)

which has a small sample size and events without any selction (left plot).

The increased RMS indicates the high uncertainty in the high muon fuzz

events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.10 Resolution plot with muon fuzz subtraction (left plot) and without muon

fuzz subtraction (right plot) for events with muon fuzz > 40MeV. The

increased RMS here shows the necessity for muon fuzz subtraction even

for these high muon fuzz events. . . . . . . . . . . . . . . . . . . . . . . 110

A.11 An event pion and proton in muon fuzz region causing negative resid-

ual. Negative residual is partly caused by muon fuzz subtracting these

hadronic energy and partly by the proton going off the side . . . . . . . 111

A.12 An event pion in muon fuzz region causing negative bias. Negative residue

is partly caused by muon fuzz subtracting these hadronic energy . . . . 111

A.13 An event neutron in muon fuzz region causing positive resolution. Since

the truth value does not include neutron energy, the residue here is pos-

itive. Subtracting these energy as muon fuzz increases our resolution. . . 112

A.14 Recoil plot comparing muonfuzz1 for all events and muonfuzz1 for events

with fuzz energy < 40 MeV (left plot), fuzz energy < 5 MeV. Double

ratio for the left plot goes up indicating lower muon fuzz events have a

lower data/MC ratio. Also, the first bin has a slight interesting dip which

shows up in various other works . . . . . . . . . . . . . . . . . . . . . . 117

A.15 Recoil plot comparing muonfuzz1 for all events and muonfuzz1+muonfuzz2

for events with fuzz energy < 40 MeV . The first bin has a slight inter-

esting dip here as well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xvii

Chapter 1

Introduction

How neutrinos and anti-neutrinos interact with other particles has been vital to studies

of neutrino oscillations and the weak interaction. Over decades, researchers have devel-

oped several models to predict how neutrinos interact; however, these models do not

describe the data precisely, and these significant shortcomings are not totally under-

stood yet. We still need to improve our model to match our prediction with data. In

this thesis, three different channels of neutrino interactions are broadly analyzed, and

the central mis-modeling of interest is the neutrino energy dependence.

The sample we will study is the neutrino interaction with the neutrons and protons

bound in the carbon nucleus or neutrino interaction with hydrogen atom. Currently,

we use the Fermi gas assumption to build our model for the nuclear environment. In

this model, the nucleon is considered free inside the nucleus. The nucleons have several

hundred MeV/c of momentum with a distribution following a gas of fermions, and the

removal of a nucleon from the strong force potential requires some average energy.

The simplest reaction of a neutrino with a neutron is the quasi-elastic (QE) scattering,

which produces a muon and a proton. This reaction is listed in equation 1.1:

νµ + n −→ µ− + p (1.1)

1

2

1.1 Three-Momentum Transfer & Energy Transfer

Three-momentum transfer and energy transfer are observables that will play a crucial

role in the analysis throughout the entire thesis. First, let us start with the definition

of a four-momentum. Four-momentum is a relativistic four-vector with energy as the

zeroth component (time-like component), and momentum in the three independent di-

rections as the rest of the three components (space-like component). So, the definition

is as follows:

p → (E, px, py, pz)

A four-momentum transfer in the case of neutrino scattering is defined as the difference

of momentum of neutrino and the outgoing muon (see equation 1.1). So, a four mo-

mentum transfer describing the positive energy lost in an interaction would be:

pν − pµ → ( Eν − Eµ, pxν − pxµ, pyν − pyµ, pzν − pzµ )

The zeroth component in the above expression is called the energy transfer, which is

the difference of energy of neutrino and muon. Energy transfer is sometimes labelled as

ω or q0, or mixing notation the lab frame energy transfer is also labelled the relativistic

invariant ν. So, energy transfer is defined as:

q0 = Eν − Eµ

The last three components of four-momentum transfer are the momentum transfer along

x,y and z direction. The three space quantity is called the three-momentum transfer,

and its magnitude is labelled as q3. So, three-momentum transfer is defined as:

q3 =√

(pxν − pxµ)2 + (pyν − pyµ)2 + (pzν − pzµ)2

Finally, the invariant square of four-momentum is defined as:

3

p2 = E2 − p2x − p2y − p2z

Therefore the square of the four-momentum transfer q2 is defined as:

q2 = q20 − q23

q2 is an intrinsically negative quantity because energy transfer is always less than mo-

mentum transfer and so for convenience we define a positive quantity Q2 = −q2. Like all

squares of a four-vector, the four-momentum transfer Q2 is a Lorentz invariant quantity,

and so it is more theoretically motivating to use Q2 for scattering on a free particle like

a proton or neutron. In our investigation of theoretical models, we will use Q2 as often

as we use energy and momentum transfer separately to probe the energy dependence in

our model. It should be noted that, later when we look at data, we will use q3 and q0.

1.2 Different Channels of Neutrino Interaction

1.2.1 Quasi-Elastic Scattering (QE)

Quasi-elastic interactions were briefly presented in the introduction. The quasi-elastic

interaction is written down again for reference and Fig. 1.1 shows the quasi-elastic

interaction Feynman diagram.

νµ + n −→ µ + p (1.2)

When the neutrino energy transfer is not high, the most it can do is kick a single nucleon

out of the carbon nucleus, and this results in a quasi-elastic collision. The interaction is

mediated by intermediate W boson, and requires conservation of internal and external

quantities, similar to billiard ball collisions in introductory physics. Since the neutron

converts to a proton during the process, this is not an elastic collision. Sometimes

4

Figure 1.1: Feynman diagram for quasi-elastic scattering (left), and a QE event can-didate seen in the MINERvA detector (right). [3]

the interaction doesn’t look as simple as the one on the right because the proton can

proceed to interact with other particles in the same nucleus, or with other nuclei as it

travels through the detector to produce more or different hadrons than the proton in

the simplest example.

1.2.2 Delta Interaction

Another prominent channel of interaction is the delta resonance interaction:

νµ + p −→ µ− + ∆++ (1.3)

νµ + n −→ µ− + ∆+ (1.4)

The ∆ particles in these examples are the excited state of a neutron or a proton. This

excited state has 0.3GeV more energy than either the proton or the neutron and so

requires at least that much energy transfer from the neutrino. Both reactions happen

in carbon nuclei, but only top reaction can occur on the proton present in a hydrogen

atom. Furthermore, this is considered as an inelastic process because there are three

particles in the final state, not two. But the process looks like a two-particle process

until the Delta decays, and can be treated similar to a quasi elastic reaction.

The ∆ particles live for a short amount of time, they wouldn’t even make it to the other

side of the nucleus, and decay to lower energy particles. Delta particles usually decay

5

into a nucleon and a pion. In the case of the ∆++ particle, it decays to a proton and

π+. On the other hand, ∆+ can decay to either a proton or a neutron.

∆++ −→ π+ + p (1.5)

∆+ −→ π+ + n (1.6)

∆+ −→ π0 + p (1.7)

∆+ −→ γ + p (1.8)

The Feynman diagram of the entire reaction is shown in Fig. 1.2

Figure 1.2: Feynmann diagram for Delta scattering (left), and a ∆++ candidate seenin the MINERvA detector (right). [3]

1.2.3 Two particle, Two hole (2p2h)

Two particle, Two hole (2p2h) is a particular interaction that only occurs in a bound

nucleon because it interacts with two nucleons at once. During a 2p2h interaction, a

neutrino kicks two nucleons and produces two holes. Examples of 2p2h interactions are:

νµ + n + n −→ µ− + n + p

νµ + n + p −→ µ− + p + p

Fig. 1.3 depicts an interaction in which two neutrons convert into a proton and a neu-

tron. The figure can be deceiving in the sense that we can only observe a single hadron

6

track which is from the proton. The neutron track is not visible due to the neutral

charge of neutrons. This is because only charged particle leave a track in the detector

via electromagnetic interaction that cause simple ionization. Neutrons and other neu-

tral particles leave traces only when they collide with the atoms in the detector or if

they decay to charged particles. In Fig.1.3, there is extra activity to the right of the

proton that is possibly from a neutron in the event.

Figure 1.3: Feynman diagram for 2p2h scattering (left), and a candidate 2p2h eventseen in the MINERvA detector (right). [3]

1.2.4 Deep Inelastic Scattering (DIS)

Deep Inelastic Scattering (DIS) is a high energy-transfer process, and therefore usually

a high number of hadrons are created during this type of interaction. Fig. 1.4 depicts a

DIS event, and from the figure we notice the abundance of particle tracks in such inter-

actions. A typical DIS process can create one, two or three protons, neutrons, pions or

even kaons, and sigma particles. These reactions happen at higher momentum transfers

and are subdominant in the studies presented in this thesis.

7

Figure 1.4: Feynman diagram for DIS scattering (left), and a DIS event seen in theMINERvA detector (right). [3]

1.2.5 Triangle Diagram

Fig. 1.5 illustrates the the QE and ∆ channels of neutrino interaction and their intensity

in the energy and momentum-transfer phase space by using the triangle diagram. The

triangle diagram is a 2-D outline of the phase space, and provides an overview of all the

different channels of interaction at the same time. Each channel lives in the different

section of the triangle diagram which makes it an useful visualization for investigation

as we will see later. The triangle diagram in Fig. 1.5 covers the three momentum

transfer (q3) range from 0 < q3 < 1.2 GeV, and energy transfer (q0) range from 0 < q0

< 1.2GeV. Throughout the whole thesis, this is the range we will mostly investigate.

The white lines through the diagram are the invariant mass line for proton rest mass (938

MeV), and delta particle rest mass (1232 MeV). Since QE process is associated with the

production of a proton, the reaction should be exactly on the QE white line. However,

these reactions in carbon nuclei are interacting with nucleons in motion, creating a wide

distribution around the white line. Similar to the QE line, most of the delta events

are around delta invariant mass line. In addition to involving nucleons in motion in

the carbon nucleus, this width is also due to the intrinsically short lifetime of the delta

resonance. This leads to a spread in the energy of the resonance, even for interactions

on the proton at rest in a hydrogen atom.

The 2p2h events are not pictured in this diagram but are present throughout the entire

range. Of particular interest, they are the only process that populates the dark “dip”

regions between the QE and delta, and also the region in the lower right corner. The

8

Figure 1.5: The figure shows the triangle diagram which is a 2-D histogram of the phasespace where the color axis represents the double-differential cross section and also theexpected relative event rate. It provides a view of the events in the three-momentumtransfer and energy transfer space simultaneously. The QE white line is the invariantmass line for proton (938 MeV), and the delta white line is the delta invaraint mass line(1232 MeV). The quasi-elastic events are gather around the QE line, and delta eventsare around the delta white line which can also be seen by the high event counts aroundthese lines.

DIS events can be found in small amounts toward the middle and upper right of this

diagram, above the delta resonance, and become more prevalent as the triangle continues

off the plot to even higher momentum transfers.

1.3 Motivation

Now that we have introduced the basic concepts and interaction channels relevant to

MINERvA, the motivation of the study is easy to demonstrate. The goal is to analyze

the neutrino energy dependence present theoretically in our current model of neutrino

interaction as well as making an initial study of the actual energy dependence in the

MINERvA data.

9

Figure 1.6: The figure shows the theoretical energy dependence modeled in GENIE withlines of constant W at 0.938GeV, 1.232GeV and 1.535GeV. The left plot is the ratio of15GeV prediction to 5GeV prediction (15GeV/5GeV), and the right plot is the ratio of5GeV and 3GeV prediction (5GeV/3GeV).

Let us start with the topic of theoretical energy dependence in our models. The software

package GENIE encodes our theoretical predictions of neutrino-nucleus interactions and

our expectation for the energy dependence [15]. In Fig. 1.6, we show the GENIE

prediction of energy dependence for 3GeV, 5GeV and 15GeV. The left plot which is

marked as 15GeV/5GeV, we take the ratio of GENIE 15GeV prediction and 5GeV

prediction. On the right one we take the ratio of GENIE 5GeV prediction and 3GeV

prediction. If the whole plot had a ratio of 1.0 (an orange color), there would be no

energy dependence at all. Instead, our theoretical model predicts energy dependence is

±5% through most of the QE and Delta range, rising past +10% along the diagonal.

Comparing to Fig. 1.5, one can further look into the energy dependence in separate

channels of interaction, which we will do in the later chapters. What is important to

note now is that our aim is to identify what our theory predicts. It seems that our

theory tells us there should not be more than 5% energy dependence in most of our

data. Then if we see more than 10% energy dependence anywhere, we would conclude

that they origin either from experimental effects, systematic uncertainties, or from our

misunderstanding of neutrino-nucleus interaction. Chapter 2 and chapter 3 will be

devoted to identifying energy dependence in our theories.

After a description of the model energy dependence, we will revert our focus to energy

dependence present in different detector or experimental effects in the data. These

effects are angle acceptance, muon scale, and non-hadronic energy near the muon “muon

10

fuzz” from bremsstrahlung and knock-on electrons. As a preview, angle acceptance is a

detector limitation where muons with angles greater than 25 degrees are not completely

measured and can not be used for analysis of momentum and energy transfer. It will be

shown that muons from higher neutrino energies have lower angle for the same energy

and momentum transfer. So, such a detector effect naturally imposes energy dependence

in our data. Chapter 4 and chapter 5 as well as appendix A is devoted to finding the

energy dependence due the three systematics: angle acceptance, muon scale and muon

fuzz.

Figure 1.7: The figure shows low energy and medium energy neutrino (left) and anti-neutrino (right) fluxes per proton on target (POT) as a function of Energy as predictedby a GEANT4 simulation reweighted using results from the NA49 hadron productionexperiment [4].

Now, it is natural to ask a question such as what can one do with the knowledge of

energy dependence in the theory and data. One of the simplest examples where such

information can be useful is the comparison of the MINERvA Low-Energy (LE) and

Medium-Energy (ME) beams, shown in Fig. 1.7. The MINERvA LE beam peaks

around 3GeV and the ME beam peaks around 5.5GeV. So a comparison between the

two beams should reveal an energy dependence similar to the 5 GeV / 3 GeV compar-

ison presented earlier in Fig. 1.6. Such is done in the thesis work of MINERvA Ph.D.

11

student Marvin Ascencio.

As will be shown, the energy dependence is stronger at lower energies. So, the results

of this research will assist us to isolate factors that can contribute largely to neutrino

experiments at and below MINERvA energies. The world of neutrino oscillation ex-

periments use a wide-range of energies: from 0.7 GeV beam of T2K and MicroBooNE,

2 GeV beam of NOvA, to the 0.6 to 6.0 GeV very wide beam produced by DUNE in

the future. MINERvA’s data are currently the best place to test the upper end of the

range for DUNE and can be extrapolated into both DUNE’s and NOvA’s oscillation

maximum. My work can be used for any such analysis. My analysis will show energy

dependence between 2GeV and 1GeV, 3GeV and 2GeV (which would be a comparison

between the MINERvA LE beam and the NOvA beam), 5GeV and 3GeV, and 15GeV

and 5GeV. So, one can use any of these analysis to extrapolate MINERvA data to other

experiments.

1.4 Monte Carlo Tuning

MINERvA uses the GENIE version 2.12.6 Monte Carlo event generator to predict the

outcome of the scattering. But the Monte Carlo (MC) does not describe the MINERvA

data accurately. To describe the data closely, we introduce additional tunings on top

of the GENIE base model. A few of these tunings are added empirically to describe

the data better, and some are based on improved theoretical assumptions compared

to the original GENIE model. Most of them are implemented as scale factors without

their own energy dependence, so they retain the energy dependence of the model being

modified. In this section, we will describe a few of these tunings. The tunings described

here are the ones used in the data section of the thesis, or are the ones that may be

interesting for a future study.

1.4.1 Random Phase Approximation (RPA):

Random Phase Approximation is a mathematical method to implement a type of screen-

ing effect. The idea behind it is very similar to screening introduced for an electrically

12

polarizable medium, but RPA is used for screening for the the weak interactions. When

RPA is turned on, the screened neutron looks like less of a neutron, the quasi-elastic

cross section or event rate goes down, and the effect is most prominent at low energy

transfer.

1.4.2 2p2h enhancement

This tuning is necessary because the existing 2p2h model underpredicts the data in the

“dip” region between delta and quasi-elastic scattering. We tune both the n+n −→ n+p

and n+ p −→ p+ p interaction. The tune is empirically determined from a low energy

data fit. It is to be noted that needing to tune this process is not a surprise, but we

are cautious that some of this additional event rate may really be due to QE or delta

production.

1.4.3 QE enhancement

The quasi-elastic enhancement is an alternative for a portion of the 2p2h tunings de-

scribed before. The tuning is done by enhancing the rate of QE in the “dip” region

to describe the data. The implementation enhances the component of the QE model

that comes from the high momentum tail of the struck nucleon distribution. That high

momentum is added to the rest of the momentum transfer leading to extreme proton

momenta and the event lands far from the center of the quasi elastic distribution. This

model is not used in this analysis but is available for future study.

1.4.4 Low-Q2 resonance suppression:

The tune is obtained from empirical MINOS data [16], and the tune suppresses reso-

nance production at low-Q2. MINERvA has also made its own tune [7] and it is very

similar in magnitude.

13

1.4.5 SuSA Valencia Tuning

SuSA is a 1p1h and 2p2h interaction model [17], [18]. Improvements due to relativistic

and other effects are included in the SuSA 2p2h model, compared to the Valencia model

[19], [20], [21]. The SuSA 2p2h model increases event rates in the dip region between

QE and Delta peak, and decreases the rate at the tail. MINERvA has started using

the SuSA 2p2h prediction as an alternative to the Valencia prediction in use since 2015.

Fig. 1.8 depicts the total event rate (in a kinematic space described in chapter 4) when

the SuSA 2p2h model is substituted for the Valencia 2p2h model. This model is not

used in this analysis, but is available for future study. Though it fell out of the scope of

this thesis, it will be very interesting to make a comparison of the structure functions

for the two models, using a similar strategy as for the QE and delta models presented

in chapter 3.

Figure 1.8: The figure depicts how a core distribution is altered by the SuSA and 2p2htuning [5]. The structure of the plot is introduced in chapter 4.

14

1.4.6 Removal Energy

There is an energy cost to knocking a nucleon out of the nucleus. In GENIE, this energy

is incorporated as the removal energy. For practical purposes, 25MeV is assumed to be

the average energy needed to get a nucleon out of the attractive potential of a bound

nucleus. The effect on this analysis is to make the outgoing protons and neutrons less

energetic. The effect was already in place for QE, though it’s implementation is contro-

versial [22], [23]. Our modification implements the effect for resonance production. It

is an alternative to the low Q2 suppression of resonances. This number has also been

verified by MINERvA student Marvin Ascencio (phd thesis forthcoming in 2021) in his

study of removal energy. Again I want to mention that this model is not used in this

analysis, but is available for future study. Especially it impacts a region described later

as having significant unmodeled energy dependence.

1.5 MINERA detector and detection system

1.5.1 Neutrino Beam

To generate neutrinos for the MINERvA experiment, we use the Fermilab’s NuMI beam

[24]. Fig. 1.9 shows the complex setup of NuMI ν-beam, and how it produces a flux of

incoming neutrinos for the MINERvA detector. During a duration of 10 microseconds,

the Main Injector (MI) discharges a total of 25-50 trillion protons. The process is then

repeated every 1.33 seconds for how long we want to accumulate data. To create the

neutrino in NuMI, protons are delivered to the NuMI target from a chain of increasingly

powerful accelerators, the LINAC, the Booster, and the Main Injector. The total number

of protons hitting the NUMI target is called Proton On Target (POT). In the target,

these protons collide with carbon to produce pions, which further-on decays to create

muon neutrinos via π+ =⇒ µ+ + νµ. Magnets are used to channel these pions, and

therefore the resulted muon neutrinos, towards the MINOS and NOvA detectors located

in Northern Minnesota. These neutrinos first travel through MINERvA detector, which

is less than 1km from the NuMI.

15

Figure 1.9: The top diagram depicts the schematics of the NUMI neutrino beamcreation process. The outgoing beam (shown by blue line) is directed to the SoudanMine and to the NOvA experiment in Northern Minnesota. The bottom figure depictsthe beam traveling through the earth, as if you were viewing it from the side somewherein orbit. There is no tunnel; most neutrinos pass through the earth without interacting.

1.5.2 MINERvA Active Tracker Region

The active tracker is mainly composed of plastic scintillators, which light up whenever

a charged particle deposits energy in it. Higher the energy deposited during a parti-

cle scintillator interaction, higher the number of photons are produced. So, from the

lighted path, we can trace the direction of a particle in the active tracker region. The

scintillator is made mostly of polystyrene which has equal numbers of carbon and hy-

drogen nuclei, and the active detector elements also serve as the neutrino interaction

target. A diagram of the detector is in Fig. 1.10, and each interaction of a neutrino

with carbon in this region is considered as an event. The energy of particles inside this

region is calculated by finding energy loss, correcting for the passive material, and by

using calorimetric techniques to sum it all up and remove known or calibrated biases..

The energy can sometimes also be estimated from how far the particle travels. This

requires us to know if the particle is a proton or pion, the former can be identified by

its prominent Bragg peak. Active region is where we can get the best resolution of a

16

Figure 1.10: The figure shows the front view of the MINERvA detector (left) and sideview of the detector (right)). From the front view, we notice the hexagonal shape ofthe inner detector and how it is wrapped by side ECAL and outer detector. From theside view, the reader should note the order of the inner detector. A little left of centeris the active tracker region mainly composed of scintillators for the best resolution ofcollision, then ECAL to force electromagnetic interactions, then HCAL to stop hadrons,and finally muon spectrometer to measure outgoing muon momentum.

particles energy, and it’s where the most events occur, and usually it is desired to make

this region as large as possible if finances allow. Fig. 1.11 shows how an event looks

in our detector. The tracker part at the center-left is active region or the Inner Detector.

1.5.3 Electro-magnetic calorimeter (ECAL)

ECAL or electro-magnetic calorimeter is built out of alternating scintillator and lead. A

heavier metal leads to more energy lost in lead due to ionization compared to the Inner

Detector. This is due to the high Z onset of an electromagnetic shower for electrons,

positrons, and photons, which ensures a robust calorimetric measurement of their en-

ergy. The passive lead material also turns this part into a sampling calorimeter, with

lower resolution but requiring half as much electronics as the tracker region. When elec-

trons, muons, or photons travel through the ECAL, processes such as bremsstrahlung

and pair production create an electromagnetic shower. ECAL region comes right after

the active tracker region (Fig. 1.10). Bremsstrahlung will be a core part of our analysis

of Muon Fuzz in the appendix, and so the ECAL region will be important there since

bremsstrahlung is large in ECAL. Also in Chapter 4, some plots labeled “energy in the

tracker” are actually energy in the tracker and the ECAL.

17

Figure 1.11: The figure illustrates a reconstructed MINERvA event, and ach triangleis a scintillator with measured activity. Different colors represent different amounts ofenergy deposited in the scintillator, darker representing more energy. The green line isthe MINERvA reconstructed track of the particles. The long one is muon track (since ithas the most energy and loses it energy slowly and therefore can travel a long distance),and the event also has two other hadrons having a shorter trajectory.

1.5.4 Hadron Calorimeter (HCAL)

Hadronic Calorimeter (HCAL) is placed after ECAL (Fig. 1.10). HCAL serves to con-

tain the hadrons and calculate the hadronic energy. It is composed of scintillators and

steel in a ratio of 1:10. So the calorimetric energy is measured with even lower precision

than the ECAL, but in 1/10th the space and with 1/10th the electronics.

1.5.5 Outer Detector (OD)

The Outer detector (OD) is used to track particles leaving the detector and also as a

frame. Between the outer detector and inner detector, there is a layer of side ECAL

(Fig. 1.10 ) It is mainly built of steel and scintillators similar to HCAL. It is not used

directly in this analysis.

18

1.5.6 Muon Spectrometer

The muon spectrometer utilizes magnets to curve outgoing muons and find their mo-

mentum from the curvature. For this purpose, MINOS near detector is used in the

MINERvA experiment and it is placed after the MINERvA detector. It should be

noted that The MINOS Near Detector was already there for many years, and the MIN-

ERvA detector was placed in front of it in such a way as to best measure the muons.

1.6 MINERA event reconstruction and MC simulation

Now that we have a brief understanding of the detector setup, the next step is to re-

construct the events in the detector and find the energy and momentum of the muon

and separately the total energy of the hadrons. To find the particle energy, a photo-

multiplier tube (PMT) is used which measures energy deposited to each scintillator. It

utilizes the fact that the number of photons produced is proportional to ionization in a

scintillator, and so the number of photons detected is proportional to the electric signal

that is digitized and saved. Each time a scintillator lights up is considered a “hit”.

Information related to a hit are: the time of earliest interaction, location of the scintil-

lator, amount of activity in the scintillator which is proportinal to the energy deposited

in it.

In the last section, we learned about the NuMI beam and Proton On Target (POT).

As discussed, every 10 microseconds, the Main Injector (MI) discharges a total of 25-50

trillion protons, and the process is repeated every 1.33 seconds. During data acquisi-

tion of a single electronics “gate”, one NuMI beam pulse hits the MINERvA detector

for around 16 microseconds time frame (10 microseconds beam width, 5 microseconds

after, and a few before). Every scintillator-particle interaction results in a ’hit’ and the

time of every hit are recorded with a 3ns precision. An example of an open gate with

associated time slices is shown in Fig. 1.12. Particles traveling at the speed of light

should finish all their interaction inside the detector within a 35 nanoseconds time span.

So in a typical beam pulse, several interactions will be present and an algorithm (or

19

your eye) can easily separate them in time. This is the basis for what in jargon is called

a time slice.

Figure 1.12: The figure shows twelve reconstructed time slices for 10 microsecondsthrough an open gate. Each colored slice indicates the occurrence of scintillator andparticle interaction. The vertical axis also depicts the total number of hits in each timeslice.

Finally, the reconstruction algorithm of MINERvA takes all of the ’hit’ information into

account and uses a tracking algorithm to reconstruct the particle trajectories based on

the timing of activities on various scintillators. At first, the algorithm looks for the

muon track around because it goes out the side or back of the detector. The algorithm

can find muon track below 25 degrees from the z-axis (z-axis is shifted 3 degrees off the

beam axis). It can find higher angle tracks too, but the muon usually misses the MINOS

detector at the back which is used to find the muon momentum. This limitation is called

angle acceptance, and it is one of the systematic effects that will be explored in detail

in the thesis. If the muon track is identified, then the algorithm infers the vertex of

interaction, and after that other particle, trajectories are recognized. Charged hadrons

interact with almost every scintillator along its track and deposit measurable energy

through ionization, and this energy can be summed to get the total hadronic energy.

Once the hadron track is detected, we use the presence of a Bragg peak to resolve that

it is likely a proton rather than a pion.

MINERvA experiment uses GENIE v2.12.6 Monte Carlo event generator to model neu-

trino interactions and detector effects. Elsewhere in this thesis, I also used samples from

GENIE 3.0.6 using the G18 10b configuration most similar to the one MINERvA uses.

Another software framework called GEANT4 is used to model how particles produced by

20

GENIE interact inside the MINERvA detector. The simulations model neutrino-nucleus

interactions using existing theories for cross-section models, the complex geometry of

the MINERvA detector, along with data-driven scintillator and electronic responses,

and uncertainties on all of these. It is assumed in the community that GEANT4 and

detector models are are very precise, often more so than the statistical power of the data

set, and any mismatch of data and model comes from the GENIE model. It is a major

goal of the thesis to look into the energy dependence modeling of GENIE in detail and

find whether GENIE is modeling neutrino energy dependence satisfactorily or not.

Chapter 2

Energy Dependence in Delta

Resonance Models

2.1 Introduction

Charged current single charged pion (CC1π+) interactions are a major contributor to

the neutrino-nucleus cross-section. Very often, this process proceeds by intermediate

baryon-resonance production. The simplest resonance production on proton producing

∆++ is of the following form: νp p → µ− ∆++ → µ− p π+, where the resonant

delta particle decays into two charged particles in the final state. Historically, a com-

prehensive experimental study of one pion production was conducted in the 70’s using

deuterium bubble chambers at the Argonne National Laboratory (ANL) [10] and the

Brookhaven National Laboratory (BNL) [25]. A detailed Q2 dependence of the cross-

section was measured in these experiments. In more recent times, the MiniBooNE

experiment collected an even larger sample of CC1π+ events on liquid scintillator which

has a proportion of two hydrogens to each carbon. [26], [6]. GENIE v2.12 and GENIE

v3.0 both use fit parameters obtained from the MiniBooNE data.

Theoretically, there are two different approaches to calculating resonance production,

and we have access to one example of each: Berger-Sehgal model [1] and Lalakulich-

Paschos model [9]. The Berger-Sehgal model, which improves upon the Rein-Sehgal

model [27], is based on quark harmonic oscillator model developed by Feynman, Kislinger,

21

22

Randall [28]. The Berger-Sehgal model includes the effects of lepton mass and the pion-

pole contribution to the hadronic axial vector current. This results in a reduced cross-

section at low-Q2. Another approach to theoretical calculations is to express the interac-

tion vertex utilizing the phenomenological form factors [29]. This method of calculating

scattering amplitude is utilized in the model of Lalakulich-Paschos. Lalakulich-Paschos

also take into account non-zero muon mass, and Pauli blocking. In this chapter, we will

investigate these neutrino interaction models and their variations to estimate how they

would compare with the MINERvA data. Primarily, my research work will focus on

identifying how different the two models are in Q2 and Eν dependence.

Figure 2.1: Differential cross secrtion dσdQ2 for charged current pion production by muon

neutrinos using Berger-Sehgal model from the Berger-Sehgal paper [1]. a) Eν = 0.7GeV,b) Eν = 1.3GeV [1]

2.1.1 Berger-Sehgal (BS) Model

In their paper, Berger and Sehgal extends the original Rein-Sehgal (RS) model [27] by

including non-zero lepton mass and pion pole production effects. This model is based on

the work done by Feynman, Kislinger and Ravndal [28], which has been an attractive

model for resonance calculations in the last 35 years due to its economy: the model

23

uses quasi-elastic vector and axial vector form factors as an input, and provides unified

description of nearly 20 resonances up to W = 2GeV. Furthermore, Berger and Seh-

gal introduced lepton mass corrections by making use of formalism formed by Kuzmin,

Lyubushkin and Naumov [30], but changed it in a notable way that respects weak axial

vector PCAC properties.

For the scope of the work presented in the chapter, we mainly focus on the Q2 de-

pendence of the cross-section for ∆++. Because Q2 is a relativistic invariant, it is useful

for probing for probing an interaction on nearly free nucleons practically at rest, such as

∆++ in hydrogen and deuterium bubble chamber experiments. Fig. 2.1 shows the Q2

distribution for neutrino energy of Eν = 0.7GeV, and Eν = 1.3GeV from the original

Berger-Sehgal paper [1]. The figure additionally depicts the coherent pion production,

a topic that would not be a focus for us, we are concerned only with the dashed line in

these plots. Table 5.1 summarizes the low-Q2 suppression of BS model over RS model

(Table 1 in ref [1]). As can be seen in the table, in the interval Q2 < 0.1GeV, the

Berger-Sehgal model adds an additional 14% suppression to the Rein-Sehgal model but

the effect decreases with increasing neutrino energy. Note that this suppression comes

from the addition of non-zero muon mass in the resonance calculations.

Table 2.1: Integrated incoherent neutrino scattering cross-

section in units of 10−38cm2. The cross section σQ2 refers todσdQ2 for Q2 < 0.1GeV . In all of Q2, there is a 14% suppression

coming from adding low-Q2 effects of Berger-Sehgal model

(taken from table 1 in ref [1]).

Incoherent scattering

E = 0.7GeV E = 1.3GeV

σ σQ2 < 0.1 σ σQ2 < 0.1

RS 0.227 0.049 0.504 0.073

BS 0.194 0.042 0.483 0.067

24

Comparing Rein-Sehgal and Berger-Sehgal Model

A comparison between the Rein-Sehgal and Berger-Sehgal model is interesting because

GENIE v2.12, which is used by MINERvA, implements the Rein-Sehgal method for

calculating resonance productions. As mentioned before, RS model does not implement

low-Q2 suppression effects. So, we wish to analyze the improvements in our predictions if

we try to incorporate the BS model used in GENIE v3.0. Also, if we are using the older

model but nature was like the improved model, what anomalous energy dependence

would we see? All the comparison between the two models will be made by comparing

the default versions in GENIE v2.12 with GENIE v3.0. Additional comparisons of

non-default configurations are available in MINERvA docdb:28974 [31].

Figure 2.2: The figure shows a comparison between the Rein-Sehgal model used inGENIE v2.12 and the Berger-Sehgal model used in GENIE v3.0 at neutrino energy 3GeV(left) and 5GeV (right). Ratio shown at the bottom panel is relatively independent ofneutrino energy, and a suppression effect averaging around 15% is observed at Q2 <0.1GeV 2. It should be noted that all of suppression is not due to lepton mass effects

Fig. 2.2 shows the cross-section prediction for charged pion production for the two

models. Given that low energy neutrino beam for MINERvA centers around 3GeV,

and medium energy neutrino beam centers around 5.5GeV, we chose 3GeV and 5GeV

neutrino energies for our comparison. From the figure, we can discern that the two

cross-section profiles are similar at higher-Q2, which is to be expected. From the ratios

25

Figure 2.3: The figure is from the work done by J.Nowak [6], comparing Berger-Sehgaland Rein-Sehgal model with MiniBooNE data. This uses the MiniBooNE neutrinoflux whose energies are around 0.7 GeV, similar to the left plot of Fig. 2.1. The 20%suppression observed in my analysis is present in his work as well.

of the two models, we can see that there is a 15% suppression factor missing in the RS

model compared to BS model.

To verify our results, we looked into J.Nowak’s work [6] at MiniBooNE. Fig. 2.3 is

taken from his work where he analyzed the best fit model for MiniBooNE CC1π+ data.

The comparison of the default configurations of GENIE in Fig. 2.3 is not just the BS

changes from RS model. It also includes a tuning of the form factors to better describe

MiniBooNE data (the purple line in Nowak’s plot), which were also incorporated into

the GENIE 3.0 configuration. We notice from the figure that at 1GeV, the same 15%

low-Q2 suppression between the RS model (red line) and the BS model (green line) is

observed as in my work (Fig. 2.2). For references, the model in GENIE 3.0 (which

includes the BS model) is actually the magenta line where two additional parameters

are tuned to different values. At MINERvA energies, this change is the main effect, and

not the lepton mass effect when we go from GENIE v2.12 to GENIE v3.0.

Although the distribution of BS is shown after lepton mass effects, form factor tuning,

and other changes, Fig. 2.2 shows that the resonance model still retains the original

26

energy dependence. This is because in the figure at both 3GeV and 5GeV, the pattern

of the ratio R-S/B-S remains the same at these two energies. Therefore, using one model

when the data was the other would change the cross section and event rate, but not

introduce a mis-modeled energy dependence. Also, I have extracted this ratio shown

for MINERvA (Fig. 2.2) to use as a cross-section weight. This allows a comparison of

the prediction in GENIE 3.0.6 that many experiments use to the fully simulated Rein

Sehgal prediction in GENIE 2.12.6 that MINERvA uses.

In MINERvA using Rein Sehgal in GENIE 2.12, we need fewer low Q2 resonance events

Figure 2.4: The figure shows extracted suppression factors for charged pion interactionsfrom low-Q2 tuning to each channel, taken from ref [7]. When compared to Fig. 2.2,the extracted low-Q2 suppression is much wider and deeper than suppression from BSand RS ratio, and also to a result from the MINOS experiment [8]

compared to data [7]. MINERvA has two ways to incorporate an ad-hoc low-Q2 suppres-

sion. One is introducing an external suppression factors obtained from low-Q2 tuning

analysis in each channel. Fig. 2.4 shows the low-Q2 suppression factors taken from that

same MINERvA paper, and is typically implemented for all resonance pion interactions

together. In the figure, the suppression is compared to one obtained by the MINOS

experiment [8] for neutrino interactions in iron nuclei. It is also similar to what Mini-

BooNE observed and Nowak was fitting illustrated in Fig. 2.3. How does the change in

GENIE 3.0 compare to the empirical suppression functions MINERvA is using? From

the figure we observed that this suppression factor averages around 30%. Furthermore,

27

we also noticed that the curve has a much wider and deeper suppression than the ratio

of RS and BS model in Fig. 2.2. This analysis shows that the extracted suppression

factors obtained in Stowell paper [7] is much steeper and wider than the suppression

factors from BS and RS analysis. Though not the same, the proposed weight from RS

to BS would account for part of the effect needed to describe the MINERvA data.

This brings in the question: what is the real origin of the extra suppression in the MIN-

ERvA data? The BS to RS effect is largely from an empirical change to form factor

parameters, but we rarely assume they should change with energy or from hydrogen

to carbon. That fit is likely using form factor parameters to account for some other

effect. A possible answer could lie in coherent scattering, which is a large fraction of π+

production. A possible nuclear effect not used by default in GENIE is Pauli-blocking.

Pauli blocking is a nuclear medium effect, and it occurs when final fermion states are

already occupied in the nuclear medium. Therefore the scattered proton cannot enter

those states due to Pauli-exclusion principle. It gives a small additional suppression.

Another nuclear effect is the cost to unbind nucleons from the nucleus, an idea explored

by Marvin Ascencio in his upcoming PHD thesis. Finally the structure functions or

form factors themselves may be incorrect. This is the one we will explore in the next

chapter using actual code from another model.

2.1.2 Lalakulich-Paschos Model

Lalakulich and Paschos utilizes the phenomenological form factor strategy to formulate

the neutrino-nucleon interaction vertex [9]. In this case, the cross-section is written

using the helicity amplitudes [32], [29]. As mentioned already, the Lalakulich-Paschos

model includes the non-zero muon mass and Pauli blocking. In addition to being a

different formalism, we have code that produces the Lalakulisch-Paschos model at the

structure-function level thanks to Daniel Norman [33].

Fig. 2.5 shows the predicted cross-section from the original Lalakulch-Paschos paper,

and the data is taken from the Argonne National Lab (ANL) bubble chamber experi-

ment. The interactions in the ANL experiment were on deuterium, and so the nuclear

medium effects should be significantly less than in carbon, argon, or iron. Moreover,

28

Figure 2.5: Calculated cross-section dσ/dQ2 for ANL neutrino energy disctributionusing Lalakulich-Paschos model from their original paper [9]. The dotted line showsthe theoretical prediction neglecting the muon mass (effectively an electron neutrinointeraction), and the solid line shows expected cross-section using mµ = 0.105GeV .The plot on the left uses axial mass MA = 1.10GeV , and the plot on the right usesMA = 0.84GeV . We can observe from the figure that reducing axial mass in LP modelallows us to fit the ANL data [10].

Nowak’s MiniBooNE tuning, which was introduced in the last section, should not apply

to this comparison. The dotted line shows the model prediction without muon mass

corrections and Pauli blocking. The low-Q2 improvements due to the inclusion of these

two factors can be clearly distinguished in the figure.

In their formalism, Lalkulich and Paschos initially used a modified dipole form factor

formula to fit the ANL data (Fig. 2.5a), which has a second factor compared to a simple

dipole:

CA5 =CA5 (0)

(1 + Q2

M2A

)2

1

(1 + 3M2

A)

; MA = 1.05GeV

It is important to note here that the axial mass (MA) used here is 1.05 GeV. The inte-

grated cross-section approaches 0.7×10−38cm2 at higher energies, being consistent with

the experimental data. But, there is still a discrepancy between data and model in the

29

Q2 dependence as seen in Fig. 2.5a, which cannot be fixed by an overall normalization

factor. To obtain a better description of data, they changed the axial mass to 0.84

GeV, and achieved a good fit, shown in Fig. 2.5b. Again, the figure shows the model

predictions with (solid line) and without (dotted line) muon mass corrections. The im-

proved Q2 dependence of the model is clearly visible in the figure. For the rest of the

chapter, we will be using MA = 0.84 GeV and the complicated form factor whenever

we use the Lalakulich-Paschos model. The integrated cross-section in this case reduces

to 0.55× 10−38cm2, but is still consistent with the data. Finally, we want to direct the

readers focus to low-Q2 suppression due to muon mass correction and Pauli blocking in

the Lalakulich-Paschos model. By comparing the dotted and solid line in Fig. 2.5, we

observe a 20% suppression due to the inclusion of the these two factors. This is com-

parable to the 15% suppression factor observed between the RS and BS model, though

by somewhat different means.

The Effect of Changing Axial Mass in the LP Model

In this section, we explore whether changing the axial mass parameter introduces a mis-

modeling of the energy dependence. A precise determination of axial mass (MA) has

an experimental and theoretical importance, and is generally determined from fitting

model parameters to data. GENIE v2.12 and GENIE v3.0 both use MA = 1.12GeV

to fit the MiniBooNE CC1π+ data on CH2. On the other hand, the Lalakulich-Paschos

model used MA = 0.84GeV to fit their prediction with the ANL deuterium data. In

fact in the previous section, we described why the original authors changed the axial

mass to MA = 0.84GeV , which is to fit ANL data using LP model. The effect of this

one parameter in isolation will be important for later comparison of the LP model with

the BS model.

Fig. 2.6 depicts the cross-section prediction using MA = 1.10GeV (red) and MA =

0.84GeV (blue) at neutrino energies 3GeV and 5GeV. For Fig. 2.6, I used Dan Nor-

man’s code directly with different choices of parameter. In the figure, we discern that

increasing axial mass increases our cross-section, as the red line is above the blue line

through the Q2 limit. In the bottom panel, this is shown as the ratio of MA = 1.10GeV

and MA = 0.84GeV . The ratio has a relatively flat Q2 dependence but is far from

30

Figure 2.6: The figure shows a cross-section for axial mass MA = 0.84GeV (blue)and MA = 1.10GeV (red) at neutrino energy Eν = 5GeV (left) and Eν = 3GeV(right). From the ratio of the two cross-section, we discern that it is relatively flat inQ2 dependence above Q2 = 0.5GeV 2, and has a sharp dip below Q2 = 0.5GeV 2.

1.0 above Q2 = 0.5GeV 2, and the ratio approaches 1.0 below Q2 = 0.5GeV 2. We will

observe similar Q2 dependence again when we compare the LP model with BS model in

Fig. 2.7. We will see this pattern again because the LP model uses MA = 0.84GeV , and

BS model uses MA = 1.12GeV . Most important for the topic of my thesis, the ratios

the ratios at 3GeV and 5GeV have almost identical Q2 dependence, which implies that

the ratio is energy independent.

2.2 Comparison of Berger-Sehgal and Lalakulich-Paschos

Model

Q2 dependence of BS and LP models

In this section, we will compare two models that are about as different as they could

be. We will estimate how badly mis-modeled is the energy dependence if one was the

MINERvA model and the other was nature’s actual interaction. The BS model closely

resembles the MiniBooNE data on liquid scintillator CH2, and the LP model has its

axial form factor tuned to describe the ANL deuterium data. Now at MINERvA we use

31

GENIEv2.12 as our Monte Carlo event generator, which uses the Rein-Sehgal model

for resonance production. Both the RS and BS models in GENIE are parameterized to

duplicate the MiniBooNE one pion data. Therefore, If the data observed in MINERvA

follows the ANL data pattern, we should expect our data/MC ratio to have the same

pattern as LP/BS ratio.

What Q2 differences are to be expected between the two models? We should predict a

disagreement between BS and LP model because they take two different approaches to

calculating the form factors for cross-section calculations. The two models will use two

different axial masses to fit two different experimental data, but the GENIE 3 version

of BS also modifies two other form factor parameters.. In the previous section, we saw

the expected dissimilarities when using two separate axial mass values. Therefore, we

should expect similar pattern to show up again in BS and LP comparison.

Fig. 2.7 depicts the cross-section prediction for the two models at neutrino energies

15GeV, 5GeV, 3GeV, 2GeV and 1GeV. The first point to notice in these figures is that

the ratio has similar pattern from 2GeV neutrino energy and above, and that the two

cross sections are very different than each other. The reason the 1GeV plot ratio rapidly

goes to one above Q2 > 0.7GeV 2 is that there is a kinematic limit at some Q2 when

the incoming neutrino energy is Eν = 1GeV . So, the cross-section at both cases goes

to zero near Q2 = 1GeV 2. This is the reason why the 1GeV plot tail has an unique

feature not present in other curves.

Now let us compare the plots in Fig. 2.7 with the plots in Fig. 2.6. As mentioned

earlier in this section, we predicted that the ratio between BS and LP should follow the

same pattern in Fig. 2.6 because of the differences in axial mass used. We do indeed

see similar pattern in both of the ratio curves, but the ratio BS/LP has a bit more

steeper Q2 dependence. At higher Q2, the BS/LP ratio is higher than Fig. 2.6 by 50%.

The excess 50% difference that is not present in Fig. 2.6 comes from the differences in

formalism and other parameters between the theories.

32

Figure 2.7: The figure shows a comparison between Lalakulich-Paschos model andBerger-Sehgal model at neutrino energies 15GeV (a), 5GeV (b), 3GeV (c), 2GeV (d)and 1GeV (e). (Add more explanation)

The data want even more low-Q2 suppression than Rein-Sehgal, suggesting the Berger-

Sehgal prediction tuned to carbon is preferred to the free-nucleon Lalakulich-Paschos

prediction tuned to deuterium. A parameterization of the ratio in fig. 2.7 has been

coded into the MINERvA framework as a weight to be used in future studies. The idea

is similar to the weight described for RS and BS comparison in Fig. 2.2. It was first

used by Marvin Ascensio in his thesis as one of many modifications to the resonance

model and can be used in future MINERvA studies.

2.2.1 Eν dependence of BS and LP model

Finally, let us focus on the energy dependence between the BS and LP model. In the

last section, we saw the ratio at different energies above Eν > 2GeV follow a similar

pattern regardless of how the parameters had been tuned or what additional theoreti-

cal details had been added. We are therefore expecting similar energy dependence at

high neutrino energies in both of the models. In this section, we will introduce a new

33

visualization of the energy dependence of the two models by taking double ratio. Let

us start with the definition of double ratio in this analysis. I will refer to the top left

plot in Fig. 2.8, where we have defined the double ratio as:

DoubleRatio =BS(15GeV/5GeV )

LP (15GeV/5GeV )

So in the numerator, we have the ratio of BS prediction at neutrino energy 15GeV and

5GeV, and in the denominator we have the LP prediction at neutrino energy 15GeV and

5GeV. So, if hypothetically, the ratio comes out to be 1.10, this would mean that BS

model has an excess 10% energy dependence over the LP model when increasing neutrino

energy from 5GeV to 15GeV. The double ratio could be flat, the energy dependence is

different in a uniform way, or it could have shape effects or Q2 dependence that follow

features of the theoretical model.

With the idea of double ratio to probe energy dependence in the two models, let us go

back to Fig. 2.8. We can see at the top two plots that the double ratio is near one.

This means that both the models have similar energy dependence above E > 3GeV . It

would suggest the MINERvA RS model could have the energy dependence right even

though it has the cross section very wrong. Let us now focus specifically on the plot

on bottom left in Fig. 2.8. The double ratio is 1.10. Therefore, there is an excess 10%

energy dependence between 2GeV < E < 3GeV in the BS model. With similar analysis

technique, we could also conclude that between 1GeV < E < 2GeV , the BS has a lower

energy dependence by a factor of 10%.

Since almost all the MINERvA data is above 3 GeV, these models predict the same

energy dependence, even though they predict very different cross sections. With the

MINERvA data, we can compare 5 GeV (ME beam) to 3 GeV (LE beam), but in that

case we would expect to see a tiny 2% energy dependence. However, if MINERvA data

was used to project to the NOvA data at 2 GeV, and we observed a 10% effect similar

to Fig. 2.8, we could conclude that our data follow some feature of the LP model

34

more closely than the BS model. Similarly, if we have enough data at 2GeV and 1GeV

neutrino energy, we could perform a similar analysis at these energy ranges.

Furthermore, the DUNE experiment will have resonance interactions at all of these

energies, with the primary oscillation effects around 2.5 GeV. Having the correct model,

or at least having the correct energy dependence will be crucial for analyzing the DUNE

oscillation spectrum.

35

Figure 2.8: The figure shows the differences in energy dependence the Lalakulich-Paschos model and Berger-Sehgal model from energies 1GeV to 15GeV.

Chapter 3

Energy Dependence in the

Structure Functions

3.1 Introduction

In this section, we will focus on the structure functions for neutrino-nucleus scatter-

ing. We will try to identify the energy dependence pattern of each structure function

and figure out which structure functions are the most significant in the energy range

of MINERvA. This chapter utilizes only the theoretical models for QE and delta pro-

cesses, and does not yet include data from MINERvA. One of the reasons to divide the

total cross-section into individual structure function is that it can provide us with new

insights in our analysis. Since using structure functions in data analysis is not common

in the neutrino community, I hope this section would encourage a new way to view the

data in the future using structure functions in neutrino experimental analysis. Results

from this section can be used as reference or be a starting point for future works.

Traditionally, form factors are introduced to describe a nucleon and how the nucleon

scattering cross section deviates from a point particle. In particle physics, form factors

are a distribution function used to encapsulate particle interactions and its distribution

without incorporating all the underlying physics. It is always a function of Q2 and

represents a Fourier transform between a spherically symmetric spatial structure and a

momentum space structure. The simplest example of form factor is the electric form

36

37

factor GE which describes the spatial charge distribution, and magnetic form factor GM

which describes the magnetic moment distribution inside a proton. Another example of

form factor is the axial form factor, which has already come up in the previous chapter.

The “axial form factor” parameterized with the axial mass (MA) details how the proton

look like to an axial current from the weak nuclear interaction. With an appropriate

study of the neutron or proton, one could predict the form factors.

On the other hand, the structure functions are similar to form factors, but are usually

two dimensional and in principle encode additional information about the interaction

and not just how the proton or neutron looks. In fact, the axial and vector form factors

(electric and magnetic) are ingredients to calculate and predict the structure functions

for nucleons. When we investigate the quark-level “deep inelastic scattering” structure

functions, they are written as F1, F2, and F3 and are functions of Q2 and Bjorken x

scaling variable. When we study nucleon-level structure functions like QE and Delta,

they are usually functions of energy and momentum transfer, which is just a change of

variables away from Q2 and x, and label them W1, W2, W3, W4, and W5 for inclusive

calculations.

To reiterate the motivation of studying the energy dependence of the model, let us start

from a figure that summarizes the conclusion of the study in the next two chapters.

Figure 3.2 repeats figure 1.6, and depicts the energy dependence by using a two di-

mensional kinematic space of energy and momentum transfer. There is no resolution

smearing or angle cuts in the figure. This is the expected cross-section energy depen-

dence theoretically using the tuned version of the Berger Sehgal model in GENIE 3.0.6.

Now let us analyze the details of the figure!

First, we will explore how constant lines of Q2 look like in a 2-D plot from Fig. 3.1.

Recall that Q2 = q23 − q20. So, Q2 = 0 when q3 = q0, and Q2 = 0 line is along the

diagonal. Consider true energy transfer q0 = 0, then Q2 = q23. So, the line along the

horizontal axis is of increasing Q2, or the right of the figure is high-Q2. To reiterate

graphically, the Q2 = 1 line (dropping units temporarily) in Fig. 3.1 varies as follows:

it starts from the horizontal axis at q3 = 1 where q0 = 0 in Fig. 3.1. The line ends at

38

Figure 3.1: The figure shows the theoretical energy dependence modeled in GENIE withlines of constant Q2 at 0.45, 0.64, 0.78 and 1 GeV2 . The left plot is the ratio of 15GeVprediction to 5GeV prediction (15GeV/5GeV), and the right plot is the ratio of 5GeVand 3GeV prediction (5GeV/3GeV).

Figure 3.2: The figure shows the theoretical energy dependence modeled in GENIE withlines of constant W at 0.938GeV, 1.232GeV and 1.535GeV. The left plot is the ratio of15GeV prediction to 5GeV prediction (15GeV/5GeV), and the right plot is the ratioof 5GeV and 3GeV prediction (5GeV/3GeV). We could see that in the QE and deltaregion, the energy dependence is within 10% limit theoretically.

q3 = 1.2 and q0 = 0.66 keeping Q2 = 1. A constant Q2 line is parabolic, and satisfies

Q2 = q23 − q20. Any point to the right of this Q2 = 1 line will have Q2 > 1, and any

point to the left will have Q2 < 1 in Fig. 3.1. To summarize, for a 2-D plot such as this

is the low-Q2 is to the left and up of the figure and high-Q2 is to the right and bottom

of the figure.

Fig. 3.2 shows the 2-D plot with invarant mass lines at 0.938GeV, 1.232GeV and

1.535GeV. The QE events in Fig. 3.2 are populated around the invariant mass line of

0.938GeV (the first white line at the bottom). From the figure we notice that the QE

39

energy dependence is mostly around 0.95, and we should expect 5% energy dependence

from theoretical cross-section calculations in the QE region.

The delta events are around 1.232GeV line in Fig. 3.2 (above the QE line). For both

15GeV/5GeV comparison and 5GeV/3GeV comparison, the delta energy dependence is

mostly around 1. Contrary to low-Q2 QE events, low-Q2 delta events show high energy

dependence.

The highest energy dependence is observed at the top part of the 2-D spectrum. This

is where most low-Q2 delta and coherent resonance production events are. The energy

dependence is around 1.1-1.2 here. Another pattern to notice here is how the energy

dependence goes from 0.95 at the QE peak to 1.2 at the top of the 2-D plot. So, low-Q2

and high W events have higher positive energy dependence.

The most important conclusion of the diagram is that whatever the energy dependence

from the theory, it is mostly within 5-10% for almost the entirety of the spectrum.

If higher energy dependence is observed in the data, it would indicate an unmodeled

energy dependence in the theoretical cross section, or a systematic effect of the experi-

ment. MINERvA is a precision experiment designed to control the latter and measure

the former.

To explore the origin of these patterns, we will begin with the general double differential

cross-section formula for neutrino-nucleus scattering [34]:

d2σ

dωdq3=G2F cos

2θcq32πE2

LµνWµν

where

LµνWµν = W1(Q

2 +m2) + W2(2E(E − ω)− M2 +Q2

2) ± W3

M(EQ2 − ω

2(m2 +Q2))

+W4

M2(1

2Q2m2 +

1

2m4) − W5

Mm2E

(3.1)

40

Here, W1, W2, W3, W4 and W5 are the five structure functions which are a function en-

ergy transfer (ω) and momentum transfer (q3). E is the neutrino energy, M is the hadron

mass (which could be proton, neutron or their resonances depending on the type of in-

teraction), m is the lepton mass (muon mass for MINERvA), q3 is the three-momentum

transfer from the incoming neutrino to the lepton, and ω is the energy transfer from

neutrino to lepton, and Q2 = q23 − ω2 .

The energy dependence of this general formula holds for all the channels of interaction.

The only modification we would need to make for different interaction channels is the

functional form of the structure functions. So, the form of W1, W2, W3, W4 and W5

will be different for Quasi-elastic, delta and 2p2h channel. But the energy dependent

factors of each structure function will remain the same for all the channels. Also, the

plus sign in front of W3 is for neutrino scattering, and the minus sign for W3 is for

anti-neutrino scattering. This term is always the vector-axial interference term, and is

only accessible in neutrino experiments, and could make for interesting future analysis

with anti-neutrino data.

The formula can be rearranged according to the energy dependence:

LµνWµν

E2=

1

E2[(W1(Q

2 +m2) +W4

M2(1

2Q2m2 +

1

2m4)]

± 1

E

W3

M[Q2 − ω

2E(m2 +Q2)] − W5 m

2

ME+ W2 [(2− 2ω

E2)− M2 +Q2

2E2]

=1

E2[(W1(Q

2+m2) +W4

M2(1

2Q2m2+

1

2m4) ∓ ωW3

2M(m2+Q2) − W2

2(4ω + M2+Q2)]

+1

E[ ± W3

MQ2 − W5 m

2

M] + 2 W2

41

(3.2)

From this equation, the most important observation is that W2 is the only structure

function with an energy independent factor associated with it. So, at the highest neu-

trino energies, we should expect W2 to be the most dominant structure function for

neutrino scattering since the other terms are suppressed. The structure functions W3

and W5 both have 1/E energy dependence. Therefore, as we lower neutrino energies, we

should anticipate these two form factors to play crucial role in the energy dependence

of the total cross-section. Finally, W1 and W4 both have 1/E2 energy dependence, and

should not be relevant for the energy ranges we are interested in.

3.2 Energy Dependence of ∆++ Structure Functions

The reason to choose the delta channel initially is because I have access to the five

structure functions through Daniel Norman’s code [33] of the Lalakulich-Paschos model

[9]. The variation due to the energy and Q2 factors of W1, W2, W3, W4 and W5 in

the delta channel can be easily identified through the code. Moreover, the introductory

study done in the previous chapter can be interpreted in terms of the structure functions

in the Lalakulich-Paschos model.

Looking at Fig. 3.3, we see that for 5GeV neutrinos, the cross-section is almost all

from W2 for Q2 < 1GeV 2. Fig. 3.3 shows the contribution of each structure function

term in the cross-section formula for four different energies. In each case, the structure

function is multiplied by all the kinematic factors, energy dependence, and constants

in Eq 3.1, so the total differential cross section is the simple sum. All other structure

functions have 1/E or 1/E2 energy dependence including additional terms for W2, and

their contribution drops off as we go to higher energies. In contrast, at higher Q2,

some energy dependent terms are multiplied by Q2 and equal or exceed the energy

independent part of the W2 term.

42

Figure 3.3: The figure depicts the Q2 dependence of W1, W2, W3, W4 and W5 atneutrino energy 5GeV, 3GeV, 2GeV, and 1GeV as well as the total cross section atthese energies. Each term is actually the structure function multiplied by its kinematicfactors in the cross-section formula. For E > 3GeV , almost the entire cross-sectioncontribution comes from W2. As we go to lower neutrino energies Eν < 3GeV , W3

starts becoming more relevant

W2 stays dominant for neutrino energies larger than 3GeV. As we go lower into neutrino

energy E < 3GeV , we could see that W3 starts playing an important role and cannot

be neglected. This is because W3 has a 1/E dependence. Only when the energy is as

low as 1GeV, W3 has a larger contribution than W2 in high Q2 region (Q2 > 0.3GeV 2).

W5 has 1/E dependence too, but it is very localized at low Q2 and is negative. Finally,

W1 and W4 both have 1/E2 dependence, and the contribution of W1 comes into play

at E < 1GeV , but W4 is negligible everywhere.

Now let us look into Fig. 3.4, where we have depicted the energy dependence of each

term as a ratio of two neutrino energies. So, for example in Fig. 3.4, 15GeV/5GeV

for W2 (top left plot) means that we are taking the W2 contribution at 15GeV and

dividing it by it’s contribution at 5GeV. This will tell us how the magnitude of W2

43

Figure 3.4: The figure shows the energy dependence of each structure function. To findthe energy dependence of the W1, W2, W3, W4 and W5, we take the ratio each structurefunction at different energies. From the plot, we could see that only W2 has a significantQ2 dependence in the in its energy dependence.

varies as we change neutrino energy from 5GeV to 15GeV, similar to the ratios formed

in Chapter two for different models. From the figure, we observe that all structure

functions except for W2 have a relatively flat Q2 dependence. The flat line clearly

shows the 1/E dependence for W3 and W5, and 1/E2 dependence for W1. For instance,

in the plot with 5GeV and 15GeV comparison (top left plot), we notice that the W3

and W5 lines are around 1/3 (= 5/15) showing the 1/E dependence.

W2 has an energy independent term and two 1/E2 terms in Eq. 3.1. The curvy Q2

dependence of W2 in Fig 3.4 comes from the −W22E2 (4ω + M2 + Q2) term associated

with W2 in Eq 3.1. Lastly, the Q2 dependence gets stronger at lower energies. For

example compare 2GeV/1GeV plot with 15GeV/5GeV. This is again due to the 1/E2

factor in this part of W2 which makes the Q2 dependent terms even more important.

Finally, Fig. 3.5 illustrates the energy dependence of the total delta cross-section on its

44

Figure 3.5: The figure shows the energy dependence of the total cross-section for theDelta channel. The cross-section is not dependent (or within 5%) on neutrino energyat high Q2 and high energy, which can be inferred from the ratio being close to one.

own, the sum of all these terms. For high neutrino energy (>3GeV), the total cross-

section seems to have very small energy dependence in high Q2 region (Q2 > 0.1GeV 2),

where the ratio is flat around one. Only at very low-Q2 (Q2 < 0.1GeV 2) do we observe

a +10% energy dependence in the total cross section for neutrino energies greater than

2GeV. All of this agrees with our conclusion from Fig 3.2. For reference to the later

chapters, we will focus on the subset of MINERvA data with 0 < q3 < 1.2GeV and

0 < q0 < 1.2GeV . So, the range of Q2 we are interested in is 0 < Q2 < 1.44GeV 2.

In Fig. 3.3 and Fig. 3.4, we looked at the Q2 energy dependence of W2, and found that

W2 provides the dominant contribution to the total cross-section at higher energies.

So, it is logical to suppose that the total cross-section energy dependence should also

have similar Q2 pattern of W2 as in Fig. 3.4. But from Fig. 3.5, it seems that the

increasing energy dependence of W2 (even though it is a mild one) is counteracted by

decreasing energy dependence of other structure functions. If the magnitudes of the

structure functions were different (from the axial mass or other parameters inside the

structure function terms) the cancellation would be less perfect and we would predict a

larger energy dependence.

45

This explains the energy dependence seen in the GENIE model in Fig. 3.2. For the delta

process, it is around 1.0 in the middle of the range in the plot, which corresponds to Q2 =

1.0GeV 2 and below. At the lowest Q2, the ratio increases sharply to 1.1. Even though

GENIE 3.0.6 sample uses the Berger-Sehgal model, and not the Lalakulich-Paschos used

to create the 1-D plots in Fig 3.5, the study in chapter 2 and the universality of Eq. 3.1

suggests the energy dependence pattern is the same.

3.3 Quasi-Elastic Channel

For the Quasi-Elastic channel, I would not be packaging the cross-section in terms of the

five structure functions used in the last section. Instead, I will follow the calculations

done by C. H. Llewellyn-Smith [35], [36]. The cross-section results can be expressed as:

dσ

dQ2(νµn → µ−p) =

G2F cos

2θCM2

8πE2[ A(Q2) ∓B(Q2)

4ME + Q2 − m2

M2

+ C(Q2)16E2M2 + 8EM(Q2 −m2) + (Q2 −m2)2

M4]

(3.3)

Here, E is neutrino energy, M is the hadron mass, m is the lepton mass. The total cross-

section contribution is divided into three structure functions called the A, B and C term,

instead of the W1, W2, W3, W4, W5. Structure function A has a term with 1/E2 energy

dependence, structure function B has a term with 1/E dependence (as well as a term

with 1/E2 dependence) and structure function C has a term that is energy independent

(as well as a term with 1/E and 1/E2 dependence). Comparing with the total cross-

section formula written with the five structure functions in Eq. 3.1, we observe that the

B term is most like W3 and changes sign between neutrino and anti-neutrino. The C

46

term is the QE equivalent to W2. Further details of the calculation could be found in

[35], [36].

Figure 3.6: The figure depicts the Q2 dependence of A, B and C temrs at neutrinoenergy 5GeV, 3GeV, 2GeV, and 1GeV. The plot includes all the kinematic factors suchthat the total cross section is the sum of the three parts. Similar to the Delta channel,for E > 3GeV , almost the entire cross-section contribution comes from the C term. Aswe go to lower neutrino energies Eν < 3GeV , B term starts becoming non negligible.

Following the analysis done in the delta channel, let us start with absolute magnitude

of A, B and C and compare them to the total QE cross-section. Figure 3.6 shows the

Q2 dependence of terms with A, B and C at different neutrino energies. In the figure,

the “A term” refers to the structure function A(Q2) times the factor of neutrino energy

E, cabbibo angle θC , hadron mass M and other constants in Eq. 3.3. The analysis

here can be paralleled with similar analysis in Fig 3.3 for the delta channel with C

replaced with W2 and B replaced with W3. The delta production is actually a two body

interaction like QE, but with the nucleon in an upper level quark state, so the structure

functions themselves are similar to the ∆++ case but not identical. The shapes of the Q2

47

dependence of B and C terms are similar to those observed in Fig. 3.3 for W2 and W3,

and their kinematic factors containing E and Q2. For the QE case, the C term remains

the most dominant term for E > 3GeV , and B term starts becoming non-negligible

for E < 3GeV . This is expected since the general cross-section formula written in the

introductory section is applicable to different channels, and the energy factors of each

structure function (or A,B,C term for QE case) remains the same for all the channels.

Figure 3.7: The figure uses ratios to show the energy dependence of the three QEstructure functions and the total cross-section. From the plot, we can notice that the Aand B terms have a flat energy dependence, and the C term has a curved Q2 dependentenergy dependence similar to W2 for the Delta channel. The total energy dependenceis also similar to the Delta case, is around one for most of the neutrino energy range ofinterest.

Finally, we will use ratios to look into the energy dependence of the A, B and C terms

which is comparable to the analysis in Fig. 3.4 for the delta case. From the observed

similarities between the delta and QE channel structure functions, we should expect

48

energy dependence for the QE case to parallel that of the delta channel. Fig 3.7 illus-

trates the energy dependence of the QE form factors. We notice that A and B terms

have a flat Q2 energy dependence, and the C term has the same curved Q2 energy

dependence as W2 for the delta case. As with the delta channel, the total cross section

is almost a flat line around 0.95. This implies that the increasing contribution of the C

term with increasing energy is counteracted by the decreasing contribution the A and

B term. But the structure functions are intrinsically different, so the balance ends near

0.95 compared to 1.0 for the delta case. The QE channel does not have a rise in the

ratio at low Q2. Finally, referring back to the 2-D plot in Fig. 3.2, we see that in the

QE region, the ratio is also around 0.95 which is now explained by the plots in Fig 3.7.

I will finish the analysis of the structure functions of the QE and delta channel by

presenting the modelled energy dependence of the differential cross section prediction

dσ/dQ2 for MINERvA using GENIE 3.0.6. For clarification, the total cross-section here

includes contributions from all the channels of interaction, including the 2p2h, coherent,

and deep inelastic (quark) scattering as well as the QE and delta channel (using and

the Berger-Sehgal resonance model). Fig 3.8 illustrates the energy dependence of the

total cross-section. Comparing the plot to Fig 3.5 (Delta channel) and Fig 3.7 (QE

channel), the total cross-section ratio diverges from 1.0 with a Q2 dependence. What

is interesting to note here is that the Q2 dependence seen in Fig 3.8 is similar to Q2

dependence observed for W2 in Fig 3.3 and the C term in Fig 3.7. Now, we know that

the total cross-section formula written in the introductory section can be extended to all

the channels of interaction. So we conclude the shape in Fig. 3.8 is largely the result of

the W2 contribution from each channel. We speculate the MINERvA data can therefore

be used to constrain the W2 structure function. Projecting the cross section to lower

energies for NOvA and DUNE then picks up mostly the model uncertainty on W3 and

perhaps W5.

49

Figure 3.8: The figure shows the total cross-section energy dependence for 15GeV/5GeV(left plot) and 5GeV/3GeV (right plot) using GENIE v3.0. The total cross-sectionenergy dependence is very similar to W2 energy dependence in Fig. 3.4 and has a Q2

dependence. This brings in the question whether such a dependence is coming fromthe energy dependent pre-factor similar to that of W2 at high energies? Then, one canspeculate a way to examine the energy factor of W2 by studying the total cross-sectionenergy dependence.

Chapter 4

Energy dependence in the

MINERvA data

4.1 Introduction

The goal of this chapter is to find energy dependence in the neutrino scattering exper-

iment data of MINERvA and then determine what factors contribute to the observed

energy dependence. A further aim of this research is to isolate the intrinsic physical fac-

tors such as cross-section dependence from the extrinsic factors such as detector effects.

Additionally, I intend to ascertain if the MINERvA GENIE model correctly predicts all

of the energy dependence seen in the data.

To reach my objective, I will break the whole MINERvA neutrino energy spectrum

into three distinct regions of 0-4 GeV, 4-7 GeV, and 7-20 GeV. These correspond to

the 3 GeV, 5 GeV, and 15 GeV choices for the theoretical cross sections in chapters 2

and 3. Then, I will separately plot the data and Monte Carlo (MC) distributions as a

function of energy in tracker for all three of my energy sub-samples and then calculate

the ratios of these distributions over different energy spectra. Variations of these ratios

can only be caused by the dependence of energy. As illustrated in the triangle diagram

Fig. 3.2, subsets of three-momentum transfer and energy transfer can resolve which

physical scattering process contributes to observed energy dependence.

50

51

4.2 Background and Review of Literature

Within MINERvA, my analysis is building on some prior work on the energy depen-

dence in the QE and delta region.

• A study of cross-section of Quasi-Elastic like event as a function of neutrino energy

for MiniBooNE experiment can be found in ref [11]. Figure 4.1, which is taken from

the paper, shows the flux-unfolded CCQE cross-section per nucleon with respect to

true neutrino energy. The figure also depicts similar cross-section from the LSND

[12], the NOMAD experiment [13], and cross-sections from a free nucleon which

assumes the QE interaction occurs in a free nucleon. In agreement with our study

in chapter 2 and chapter 3, the energy dependence only becomes non-negligible

for neutrino energy less than 1-2GeV. MINERvA also produced a cross section

in this format [37]. By this time the apparent discrepancy between MiniBooNE

and NOMAD was understood. The former (and MINERvA’s) analysis strategy

included the previously unsimulated 2p2h process as signal (called QE-like or

CC0π) and NOMAD’s strategy had a stricter requirement that would reject 2p2h

events.

• Alec Lovlein, an undergraduate at the University of Minnesota Duluth, [38] first

introduced the proposed techniques of my research to explore the energy depen-

dence problem. The method of using hadronic energy in different q3 regions and

plotting the associated histograms can also be found in the works of Miltenberger

[39], Demgen [40], and Rodrigues et al.[41]. Lovlein’s work divides the neutrino

energy spectrum into two regions: 4-7 GeV and 7-20 GeV. The research did not

include the 0-4 GeV energy region, where the most compelling energy dependence

is likely to manifest itself. Furthermore, Lovlein put in different levels of artificial

energy dependence in the quasi-elastic scattering channel and predicted an upper

limit of unmodeled energy dependence in the MINERvA GENIE. His analysis

was based on MINERvA ME1A playlist, which is one-tenth of the full MINERvA

52

dataset. In these next two chapters I extend Lovlein’s study by probing the full

dataset, and by examining other scattering channels such as the delta resonance

channel or the two-particle two-hole (2p2h) channel. Also, MINERvA now has

reprocessed, updated and finalized a version of the Medium Energy data to use

for a complete analysis.

• The work of Rob Fine [2] on MINERvA shows that we are able to correct our

calculation of beam flux for different energy regions. My techniques are similar

in some ways to Fine’s low-ν method, and complementary in other ways. The

low-ν method has complicated sensitivity to the uncertainty on the magnitude

and energy dependence of the QE, 2p2h, and delta resonance cross sections. The

method assumes there is little unmodeled cross section energy dependence below

some energy transfer (the ν of the low-ν method), and it can be adequately cor-

rected with the model. Then any remaining energy dependent discrepancy with

the data is due to errors in the flux. My work directly seeks additional insight

and constraint on on the QE, 2p2h, and Delta contributions.

• Miltenberger [39] in his study of neutrino-nuclear interaction at low momentum

transfer showed energy-dependence in the distribution when he applied an overall

muon energy scale shift. Analysis done on my thesis provides a way to test the

pattern of energy dependence of an muon energy scale shift.

4.3 Method of Data Analysis

I will break the whole neutrino beam energy spectrum into three distinct regions: 0-

4 GeV(LowEnu), 4-7 GeV (MidEnu), and 7-20 GeV (HighEnu). Additionally, three

momentum transfer will be subdivided into three sections as well: 0-0.4 GeV (low q3),

0.4-0.8 GeV (mid q3) and 0.8-1.2 GeV (high q3). For each of these nine samples, the dis-

tribution of reconstructed hadronic energy is made following the strategy of Rodrigues et

al. [41], and also the upcoming thesis of Marvin Ascencio. The reconstructed hadronic

energy is the passive corrected energy in the tracker and ECAL regions of the detector,

with a region around the muon excluded, and an unbiasing calorimetric correction of a

factor of 1.17 applied. For the purposes of this analysis, I will ignore the interactions

53

Figure 4.1: The figure shows MiniBooNE CCQE cross-section per nucleon with respectto true neutrino energy, taken from ref [11] for long range of energies. Cross-sectionresults from the LSND [12], the NOMAD experiment [13], and cross-sections for a freenucleon are reported in the plot.

with higher three momentum transfer. The subsample boundaries are are illustrated in

Fig. 4.2.

Here, I will make a few comments about the figure and my data analysis as a whole. I

will be using the MINERvA Medium Energy beam for all the analysis in this chapter.

The MINERvA ME beam peaks around 6GeV, which can be seen in the histogram in

Fig 4.2. Reader should also note that the low q3 interactions (black dots) occur less

frequently than the mid q3 and high q3 interactions. So, we will have a lower amount

of low q3 events compared to mid q3 and high q3 events. Let us now focus on the data

and Monte Carlo ratio in the figure. We could observe that over the whole spectrum of

neutrino energy and momentum transfer, this ratio is above one. Therefore, our Monte

Carlo seems to be underpredicting the data over the total spectrum. The data/MC

has discrepancies up to 20% from a combination of flux and cross section errors, and

possibly other systematic uncertainties.

I will be using the whole MINERvA ME playlist and MINERvA GENIE v2.12.8 with

MINERvA Tune v1.0 to do my research. Furthermore, Duluth code will be used to per-

form my energy dependence search. Researchers at the University of Minnesota Duluth

who are associated with MINERvA use their own code to analyse the data, and it is

54

often called the “Duluth code.” It uses a reduced file optimized for these analyses. It

has a partial set of systematic uncertainties but can run very fast and without the full

computing infrastructure at Fermilab.

Figure 4.2: The figure shows the energy and three momentum cuts. The energyspectrum is divided into 0-4GeV (LowEnu), 4-7GeV (MidEnu) and 7-20GeV (HighEnu).The recoil q3 is broken into three regions: 0-0.4GeV (low q3), 0.4-0.8GeV (mid q3) and0.8-1.2GeV (high q3). One more thing to notice in the plot is that we will be using theMINERvA ME beam which peaks at around 5GeV.

Now I will illustrate my plotting methods and how I will analyze the results. I will be

making separate plots for each of the energy ranges given a three-momentum range.

For example, I will have a distinct plot for the MidEnu (4-7GeV) high q3 (0.8 < q3 <

1.2GeV ) and a plot for HighEnu (7-20GeV) high q3. Then, I will take ratios of these

distributions. The idea is depicted on the left plot of Fig. 4.3. The blue dots represent

the MidEnu data distribution at high q3 and the red dots show the HighEnu data at high

q3. The solid lines show the Monte Carlo predictions at these energy and momentum

transfer. Since our beam peaks between 4-7GeV, the higher event count at MidEnu

should not surprise us.

55

The lower panel outlines the ratio of these distribution and is vital for identifying sub-

tle differences in the upper part of each plot. In the figure, Data- HighEnu/MidEnu

means that we are dividing the HighEnu data by MidEnu data. MC- HighEnu/MidEnu

corresponds to similar ratio with Monte Carlo predictions. If these ratios are flat, that

would correspond to no energy dependence. Variation of these ratios can only come

from the energy dependence. Since in Fig. 4.3, there is a noticeable gradient in these

ratio, we can conclude that there is a energy dependence in the data at high q3 which

is a function of neutrino energy transfer (or recoil energy in tracker summed from the

tracker and ECAL).

Figure 4.3: The figure shows the distribution of MidEnu (4-7GeV) and HighEnu (7-20GeV) data and MC distribution for high q3 (0.8 < q3 < 1.2GeV). The left panel alsoshows the ratio of HighEnu and MidEnu ratio for Data and MC distribution. Whilethe right panel shows the double ratio of Data and MC as defined in the text. DataHighEnu and MidEnu ratio (left plot) shows that there is indeed energy dependencein the data in these phase space. But, the double ratio near one indicates the energydependence being well modelled.

Is our model able to predict the data ? No, the upper panels indicate it is about as

bad as the LP vs. BS delta comparison. Is our model able to predict the energy de-

pendence in the data? From Fig. 4.3, we could see that the MC ratio follows the data

56

ratio closely. So, we could conclude that our model is doing a good job in predicting

the existing energy dependence in our data. Although, we have not started to predict

what could be the cause of such large energy dependence in our data, we could strongly

reason that these energy dependence sources are well modelled by us already for for

high q3 4-20GeV neutrino energy range.

The fact that our MC has modelled these well can again be illustrated by using double

ratio. Here, we define double ratio as:

DoubleRatio =HighEnu(Data/MC)

MidEnu(Data/MC)

A double ratio of one would indicate a perfect energy dependence modeling and a per-

fectly modeled flux. Any deviation would indicate incorrect modeling if the deviation

is beyond statistically acceptable limit. In Fig. 4.3, we could observe that the double

ratio is best fitted by a line around 0.96. This indicates a 4% mismodeling of energy

dependence throughout the whole spectrum of energy transfers for high q3. A key idea

her is that mismodeled flux produces a constant offset in the double ratio. Most other

mismodeled energy dependence will cause a distortion away from a flat double ratio.

Much of that 4% shift in the double ratio could possibly be attributed to a flux mismod-

elling or a model error persistent in the whole spectrum. For a complex model such as

this, the flux might be a 3% effect and the remaining 1% effect and any shape distortion

could come from the model or some non-flux effects.

4.4 Looking for Energy Dependence in the MINERvA

Data

Now that I have introduced the method of looking for energy dependence, I will now

focus on the analysis. I will start with HighEnu (7-20GeV) and MidEnu(4-7GeV) data

57

first. Then I will move my focus to MidEnu (4-7GeV) and LowEnu (0-4GeV) data.

4.4.1 Energy Dependence in MidEnu and HighEnu data

Fig. 4.4 shows the data and MC distribution for HighEnu and MidEnu. The top plot

is for high q3, the middle plot is for mid q3 and the bottom plot is for low q3. The

figure shows both the data HighEnu and MidEnu ratio as well as the double ratio. The

biggest effect is that we naturally have fewer events in low q3. This fact could also be

observed in Fig. 4.2, where we could see that low q3 has a much lower event count than

higher q3. The difference in distribution for MidEnu and HighEnu is because our beam

peaks at MidEnu. Let us now return our focus to energy dependence analysis:

58

Figure 4.4: The figure depicts the data and MC distribution for MidEnu and HighEnu,and for high q3 (top panel), mid q3 (moiddle panel) and low q3 (bottom panel). Theleft plots show the HighEnu/MidEnu ratio and the right plots show the double ratio.Gradient in the HighEnu/MidEnu shows existing energy dependence in the data. Wealso observe that these gradients go down at lower q3 showing the energy dependence ismilder at lower q3. The double ratio around one for all range of q3 outlines that we aremodelling these energy dependence in the data well.

59

In the left plots of Fig. 4.4, our data has a higher energy dependence in higher q3,

there is no energy dependence at low q3. For example, the top left plot which is for high

q3 has the largest gradient in the data HighEnu/MidEnu ratio. Compare this to the

middle left plot which is for mid q3, we could see that the ratio is flatter than it is for

high q3. I have drawn a dotted line in the middle plot to show how the high q3 ratio has

a much larger slope. When we go to the lowest q3 (bottom left plot), we could see that

the ratio is relatively flat around 0.4. Here the dotted line is drawn to show the energy

dependence of mid q3. Therefore, we observe that there is energy dependence in our

data as a function of energy in tracker (or energy transfer ω). The energy dependence

also seem to be a function of q3, the larger the q3 the larger the energy dependence in

our data.

Now let us move our attention to the Monte Carlo simulation and how well it models

this energy dependence. From the left plots in Fig. 4.4, we could see that the MC ratio

(red dots) follow the data ratio (blue dots) very well. Therefore our model is doing a

good job in predicting these energy dependencies in the data. The right plot shows the

double ratio for all three q3 cuts. In all cases, we notice that the double ratio is near

1.0 and mostly flat, which is the characteristic of modeling the energy dependence well,

even if the cross section itself is poorly modeled.

The fit line is close but not perfectly at 1.0. For example, the top right plot which is for

high q3, we see that the double ratio is around 0.96, for mid q3 (middle right plot) the

double ratio is around 1.00 and for low q3 (bottom right plot) the double ratio is around

1.02 as well. A shift in double ratio from one indicates a error in energy dependence

modelling that exists throughout the whole spectrum. A single overall scale common

to all three double ratios would be the signature of a mismodeling of the flux. We will

delve into this problem of possible flux mismodelling in the later section of this chapter.

60

4.4.2 Sources of Energy Dependence for HighEnu and MidEnu

Angle Acceptance

We do not observe all the events in MINERvA because of the size of our detector. Fig.

4.5 repeats the detector setup of MINERvA from Fig 1.10. When the outgoing muon

misses the MINOS detector at the back (Fig. 4.5), we have no way to measure the muon

energy for these events. And so, an event is only counted when the muon goes through

the MINOS detector. This is the idea of angle acceptance.

Now what influences the the angle of an outgoing muon? This could be calculated by

using simple ideas in physics and geometry. The result is that the angle of muon θ is

dependent on:

sinθ ∝ pTptotal

pT is the transverse momentum of the muon, and ptotal is the total momentum of the

muon. This can be further approximated by assuming mµ = 0, and so neutrino energy

Eν = ptotal. Then we can write:

sinθ ∝ pTEν

Since neutrino momentum along the neutrino beam should be proportional to neutrino

energy (Eν =pνc), a higher neutrino energy would increase the muon momentum along

the direction of neutrino beam as well, and thus decrease the muon angle for an event

with the same transverse momentum. Equivalently a larger transverse momentum in-

creases the outgoing muon angle for the same neutrino energy. All of these now could

be related to our analysis in the previous section.

A higher total three momentum transfer q3 almost always means the transverse com-

ponent pT is larger too, and a higher transverse momentum would lead to higher muon

61

Figure 4.5: The figure shows the detector setup of the MINERvA experiment. Dueto the limited size of MINOS Near Detector, some events will not be counted if if theyhave too large a muon angle large and miss the MINOS detector.

angle. Therefore, an event with higher q3 is likely to increase missing events due to angle

acceptance. Similarly, a lower neutrino energy is also likely to cause a larger amount of

events missing due to angle acceptance. These ideas area illustrated in Fig. 4.6.

I have not introduced any additional angle cuts in the first pass of the analysis, and so

the angle acceptance pattern seen are due to MINOS angle acceptance. Reverting back

to Fig. 4.4, do we observe any angle acceptance pattern? We do! Let me start with the

high q3 plot in Fig. 4.4 (top left plot) and break down and refine the interpretation of

the ratio seen in Data-HighEnu/MidEnu. For low recoil energy in tracker ( 0.3GeV), we

see that the ratio is around 0.6 which is higher than expected 0.4 value due to flux dif-

ferences between HighEnu and MidEnu. The ratio slowly goes towards 0.4 and becomes

flat at high recoil energy in tracker. Increasing neutrino energy decreases the likelyhood

of an event missing due to angle acceptance. This is why the HighEnu/MidEnu ratio

is higher than expected because an MidEnu event is more likely to be missing than an

HighEnu event. Such an effect is more present at high q3 because muon angle is larger

due to high transverse momentum of these events. This is why we see the gradient is

reduced for mid q3 and low q3 in Fig. 4.4.

62

Figure 4.6: The figure shows how the angle of outgoing muons changes with neutrinoenergy and three momentum transfer. When the muon misses the MINOS detector, wedo not count the event.

Finally, we see that energy dependence is higher at lower recoil energy in tracker, the

left side of each plot. This is because for a lower energy transfer events at a given

total three momentum transfer, the muon energy is higher and thereby the possibility

of a higher transverse momentum. For all these reasons, muon angle acceptance plays a

critical role in these patterns observed in Fig. 4.4. Finally, I want to focus on the Monte

Carlo predictions. Since, we have noted that the MC models these energy dependence

within satisfactory limits (double ratio near one), we conclude that angle acceptance is

well modelled by Monte Carlo simulation. Geometry effects are almost always modeled

accurately by mature experiments like MINERvA, who have used a combination of pre-

cision laser surveys.

63

4.4.3 Energy Dependence in LowEnu and MidEnu data

Now let us look into the energy dependence in the LowEnu (0-4GeV) and MidEnu (4-

7GeV) data. From our previous discussion on angle acceptance, we expect as we lower

the neutrino energy, we will have higher muon angle. Fig. 4.7 outlines the data and

MC distribution with their corresponding ratio for LowEnu and MidEnu. But the ratio

in Fig. 4.9 do not exhibit the expected angle acceptance pattern which now should be

higher. We can see in the high q3 plot in top left, we can see that a new pattern emerges

in low recoil energy, and the gradient angle acceptance pattern is lost. The same goes

for mid q3 plot as well. The low q3 is not affected by muon angle due to lower angle of

these events.

Looking at the high q3 plot (top left) in detail, angle acceptance is causing the gradient

in low recoil energy in tracker (0.2GeV). But, rather than going down to a relatively

flat value at high recoil energy in tracker like in the MidEnu and HighEnu case, the

ratio actually bumps up a bit in the middle (0.2-0.5GeV) in Fig. 4.7. This bumping up

feature of the ratio around 0.2-0.5GeV recoil energy in tracker is present in mid q3 plot.

The reason for such a feature is not totally understood. In the next section when we

impose artificial angle cuts, we will see such patterns are even more dominant. So, such

a feature must be coming from angle cuts, but we are not sure if it is directly associated

with angle acceptance. The reason being that such a pattern cannot be understood by

the theory presented in section 4.4.2.

Finally, at the tail of high q3 plot, we see that the data ratio falls off but the MC ratio

does not. This falling off in the tail is also a present in all three q3 plots. This is the

region where most low-Q2 delta events live, and we seem to be underpredicting the

energy dependence in these region. Why does the MC flattens while the data falls off

in the region? The simplest explanation could be that the number of events count is

not statistically significant to make a conclusion about the region. But, we can see in

the figure that the MC is outside the statistical-only uncertainty of the data points.

Moreover, such a feature is consistent throughout all q3. It was extensively discussed

in chapters 2 and 3 that the low-Q2 resonance events are suppressed in the theory as

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well as from empirical study. It could be that the low-Q2 suppression has a energy

dependence that is not modeled which is why the MC line is flat and the data is not.

A future study could be done to characterize this energy dependence of the data at the

tail of the spectrum using more of the resonance model tools available to MINERvA

or creating new ones, and determine the cause of such energy dependence not modeled

by our theory. Since, this effect is at the low-Q2 part of the data where we introduce

empirical low-Q2 suppression which in neutrino energy independent, this suggests to

reconsider possible energy dependent low-Q2 suppression in the data. But, detector

effects are still not out of question.

I have not yet figured out why, for LowEnu and MidEnu, the ratio would go up rather

then flattening at high energy in tracker as in the MidEnu and HighEnu case. Similarly,

I am still uncertain as to why the data ratio would fall of so drastically at the tail for

LowEnu and HighEnu, and such a pattern to be present for MidEnu and HighEnu. I

will explore possible causes in the upcoming sections, and a detailed evaluation of this

will be a task for future work.

65

Figure 4.7: The figure depicts the data and MC distribution for LowEnu and MidEnu,and for high q3 (top panel), mid q3 (middle panel) and low q3 (bottom panel). The leftplots show the MidEnu/LowEnu ratio and the right plots show the double ratio. Theangle acceptance pattern is negligible in mid q3 and low q3 due to lower neutrino energyin these cases. Another noticeable feature in all three ratio is the bumping up of theratio in the middle of the spectrum and the data ratio falling off near the tail.

66

4.5 Looking Further into Angle Acceptance with Imposed

Angle Cuts

To explore how angle acceptance reveals the pattern shown in Fig. 4.4 and Fig. 4.7, we

will impose artificial angle cuts of 10 degrees and 5 degrees in this section. So, whenever

we say there is an angle cut of 10 degrees, it means that any event with muon angle

greater than 10 degrees is not included in the sample. The previous section did not

impose an explicit cut, but the geometry of MINERvA and the MINOS detector lead

to selecting angles within about 20 degrees.

First of all, we are going to look at angle cuts for MidEnu and HighEnu. This is shown

in Fig. 4.8. The left panels show ratios for angle cut of 10 degrees and the right panels

show the ratios for an angle cut of 5 degrees. The top panels are for high q3, the middle

panels are for mid q3, and the bottom panels are for low q3.

Going back to the ideas presented in the previous section, we stated that the gradient

seen with varying recoil energy in tracker is due to the limited size of the MINOS

detector being able to accept events within certain threshold of muon angle. This led

us to the idea of angle acceptance. Now that we are looking into even stricter cuts of

10 degrees and 5 degrees. The pattern of angle acceptance should be evident in these

plots. Do we see the expected trend in these plots? Well, if we look into our top left

plot in Fig. 4.8, which is for high q3, we do see that the gradient is much larger with

a stricter angle cut, which does imply that the gradient seen is a result of angle cuts.

And the midEnu is significantly decreased, but the highEnu is about the same as in Fig.

4.4, so the highest energy data is almost all within 10 degrees. Great! So, we must be

on the right track. Not totally. When we look at the plots on the right panel with 5

degree cuts, our pattern is somewhat lost!

Looking into the plots with 5 degree cuts in Fig. 4.8, we observe that the high q3

plot (top right) has a distinct bump in the middle. Although the mid q3 plot does

not have the bump present in the middle, you can see that there is an effect from our

known angle acceptance pattern which makes the ratio go down, and there is another

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effect of increasing ratio in the middle. The overall effect of which is to keep the ratio

constant around 0.9. Then after 0.2 GeV recoil energy in tracker, the ratio goes down

according to our known angle acceptance pattern. Then in the lowest q3 there is no

angle acceptance pattern. Let us analyze all of these.

First of all, angle acceptance pattern does not show up in the lowest q3 because the

muon angle for these events are almost all lower than 5 degrees. Reading the vertical

axis of the upper plots, the total number of events for 10 degree and 5 degree hardly

changes for either of these cases. So, low q3 is free from an angle acceptance pattern.

The mid q3 region should be affected more by angle cuts than the low q3 region but

less than the high q3 region. This can be seen by observing that the total number of

events for an angle cut of 10 degrees stays almost the same compared to no angle cuts.

So, angle acceptance pattern does not show up in the 10 degree plot or is very minimal.

When we introduce 5 degree cut, we see the number of events is significantly reduced.

We talked about the lower energy events having larger muon angle. Therefore, it can

be seen that the MidEnu events going down more than the HighEnu events with the

HighEnu remaining practically unchanged. In fact, at 5 degree cuts, the total number

of events in these two region is comparable even though our flux actually peaks in the

MidEnu region. In fact we do see such pattern rising. We observe a gradient with

varying recoil energy in tracker, which is our expected trend of angle acceptance. But,

looking closer into the plot the gradient pattern is counteracted by another rising pattern

around 0.2 GeV. Is this new pattern, which can also be seen in the high q3 plot, due to

angle acceptance or some other systematics that comes in when we impose strict angle

cuts? The answer to which we are still unsure about.

68

Figure 4.8: The figure depicts the data and MC distribution with angle cut of 10degrees (left panels) and angle cuts of 5 degrees (right panel) for MidEnu and HighEnu,and for high q3 (top panel), mid q3 (middle panel) and low q3 (bottom panel). Theangle accpetance pattern detailed in the previous section becomes more prominent inthe 10 degree cuts for high q3. The 5 degree cuts seem to have an added pattern notseen before. There is an extremely large bump in the middle of the spectrum.

69

Figure 4.9: The figure depicts the data and MC distribution with angle cut of 10degrees (left panels) and angle cuts of 5 degrees (right panel) for LowEnu and MidEnu,and for high q3 (top panel), mid q3 (middle panel) and low q3 (bottom panel). Theangle acceptance pattern should present itself more clearly at lower energies, and so weshould expect to see it better in these energy cuts. But, the number of events are cutdown drastically for LowEnu events for it to be of good statistical value, and it haslimited statistical power compared to the 10 degree cut.

70

Finally, let us look into high q3 plot on top right with 5 degree angle cut. Here, the

gradient pattern of the angle cut only shows up near the tail of the spectrum, and is

totally counteracted by the other pattern we talked about in the previous paragraph. If

we look into the number of events, we observe the significant cut of events for MidEnu

below 0.4 GeV compared to 10 degree cuts.

Going back to the question of what could give rise to such an unexpected pattern? In

both cases of high q3 and mid q3, the pattern has a distinct peak. For high q3, this

is at 0.3 GeV and for mid q3, this is at 0.2 GeV. This is also the region where quasi-

elastic spectrum peaks and also has a significant 2p2h component. But, we also have

to remember that these trends show up only when we impose strict angle cuts. What

is confusing about the trend is that it has a distinct peak which is not natural for an

angle pattern. We expected a mostly linear pattern with recoil energy in tracker. It

could be the combination of the angle effect with a quasi elastic specific effect or 2p2h

specific effect. Importantly, it appears relatively well modeled.

Finally, I will focus on the LowEnu and MidEnu plots in Fig. 4.9. We know that angle

cuts affect the spectrum more for lower neutrino energy. So, the number of events are

drastically cut down for LowEnu in all of these plots. So much so that the 5 degree cuts

have almost negligible event count for LowEnu. So, the statistical power is lower, but

the trend of MidEnu and HighEnu are still present throughout the plots in Fig. 4.9.

There is a particularly strong effect at high q3 at 0.3 where the data ratio is significantly

lower than the MC ratio, indicating an unmodeled energy dependence. This was not

present without the special angle selection, nor is it there after imposing the tighter 5

degree cut in Fig. 4.9.

4.6 Muon Kludge and Energy Dependence

The final systematic effect that we will focus on is the muon energy scale known in

MINERvA as the “muon kludge”. The muon kludge is a correction we make late in the

analysis to update a quantity with an improvement we learned about after processing

the data and/or MC. The motivation behind muon kludge is that we are uncertain

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about reconstructed muon momentum and energy using the MINOS detector. In addi-

tion to the regular uncertainty, MINERvA found evidence that our muon energy is off

by 3.6% and increase the muon energy by 3.6% to account for the supposedly miscal-

culated muon energy using the MINOS magnetic field [42]. After the muon energy of

each event is changed to account for this, we fix the muon momentum, neutrino energy,

q3 and Q2. The effect is applied only to data and not to Monte Carlo since detector

miscalculations only apply to the data distribution. The Monte Carlo, by construction,

has a self consistent muon simulation and reconstruction, even if it does not match the

real data muons in some way.

The cause of the MINOS detector miscalculation is somewhat unclear. The systemat-

ics shifted our data closer to the MC in a sophisticated version of the neutrino energy

distribution shown in Fig. 4.2 [42]. Other than that there was no reason to believe

we needed such a change to muon energy. It should also be noted the muon kludge is

applied only to the Medium Energy MINERvA beam, and was not known when low

energy beam data was being analyzed. Here one can argue that LE beam should be be

treated as the ME beam, the muon kludge should be applied to both of them or none

at all.

The goal of the section is not to question the necessity of the muon kludge but to inspect

if such an error would give rise to an unmodeled energy dependence. We will turn off

the muon kludge, that is we will look into the spectrum before muon kludge was applied

and analyze the two spectrum before and after muon kludge.

4.6.1 Expected Muon Kludge Pattern

First of all, we are going to discuss the expected muon scale trends. When we turn

off the muon scale, we are decreasing the muon energy calculated uniformly by 3.6%.

It is desired to decrease muon energy calculated from the MINOS detector only. But

for simpliciry, the 3.6% decrease is in effect for both the MINOS and MINERvA muon

energy contributions. So, we expect q3 to go down, and so there will be a migration of

events from higher q3 to lower q3. But, the migration will not be uniform with recoil

72

energy. Events with high recoil energy would be shifted less and events with low recoil

energy will be shifted more.

Figure 4.10: Movement due to a resolution sized shift in the muon energy for the lowq3 on the left and the mid q3 on the right [14]. The effect shown is the opposite effect ofwhat is discussed in the section, it turns on the muon energy scale rather than turningit off. So, for our case we should expect the arrow the face the opposite direction.

The migration of events can be clearly understood in Fig. 4.10 which is from Mil-

tenberger’s thesis work at UMD [14]. He plotted the migration of events due to a +2%

muon energy uncertainty in the space of q3 and recoil energy using an arrow to repre-

sent the magnitude and direction in this 2D space. Unlike other systematics he studied,

the migration is only horizontal and has no effect on the tracker energy quantity. Mil-

tenberger was looking at an effect with the opposite sign from the of the one we are

interested in. So, for our purposes, we can picture the arrow facing the opposite direc-

tion, other than that the migration of events will have the same relative magnitude. He

noticed that events with higher q3 had a larger migration and events with lower recoil

energy also had stronger shifts.

4.6.2 Recoil energy Plots without the Muon Kludge

The effect in the ME analysis matches the effect on the neutrino energy and also Mil-

tenberger’s demonstration of q3. Fig. 4.11 depicts the effect of muon kludge for MidEnu

and HighEnu events. In the figure, the left plots are with muon scale turned on, which

73

is our current setup in MINERvA, and the right plots are with muon kludge turned

off. The bottom panel in each plot shows the double ratio that highlights mismodeled

energy dependence. We can visually inspect which one is better modeled and where the

effect is significant. In a future version of this analysis, one can imagine turning this

comparison into a systematic error band.

The first pattern to observe in all of these plots is that the data and MC ratio gets

much further than one by turning off the muon scale. All of these should not come as

a surprise since we have already mentioned that the motivation behind muon scale was

the observed improvement of the data distribution compared to MC in a neutrino en-

ergy plot like Fig. 4.2. This is effectively by construction or by tuning. In this section,

we want to look for more subtle effects of muon kludge.

Now let us try to search for the trend reported by Miltenberger. According to his re-

ports, when we turn off muon kludge, events will be shifted to the low q3. In other

words, the number of events will increase in low q3. This is indeed the case in our plots

shown in Fig. 4.11.

But, there is another expected pattern hidden in these plots, parts of the data spectrum

go up and others go down. The fact that decreasing recoil energy increases the migration

of events from one q3 region to another. Let us try to look for that in the high q3 plot

(top panel). We can notice that below 0.4 GeV, the effect of turning off muon kludge

decreases events in that region. On the other hand, events with recoil energy greater

than 0.4 increases events in that region. The reason for that is low recoil energy events

are shifted more to the lower q3 bins. On the other hand, events with higher recoil

energy are shifted in less amount to lower q3 bins. In fact, the higher events coming in

from higher q3 bins overwhelms the migration to lower q3 bins. All of these causes the

number of events to rise for events greater than 0.4 GeV in high q3 (top panel). Similar

pattern is seen in the mid q3 panel as well.

Fig. 4.12 depicts the result of turning off muon kludge for LowEnu and MidEnu. All of

the trend seen in the MidEnu and HighEnu region are observed in these region of the

74

phase space, except the low enu samples gain overall from the muon kludge migration,

even as the q3 migration is complicated.

4.6.3 Energy Dependence due to Muon Kludge

The energy dependence of muon kludge is hard to analyze because the effect is so small.

The best tool to investigate energy dependence is to see how the double ratio changes,

effectively imagining a kind of triple ratio. So, let us try to analyze the double ratios in

Fig. 4.11 and Fig. 4.12

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Figure 4.11: The figure depicts the effect of muon scale for MidEnu and LowEnu. Thetop panel is for high q3, mid panel is mid q3 and the bottom panel is low q3 plots. Theleft column shows the data and MC distribution with muon kludge turned on, and theright column of plots illustrates the distributions with muon kludge turned off.

76

Figure 4.12: The figure depicts the effect of muon scale for LowEnu and MidEnu. Thetop panel is for high q3, mid panel is mid q3 and the bottom panel is low q3 plots. Theleft column shows the data and MC distribution with muon kludge turned on, and theright column of plots illustrates the distributions with muon kludge turned off.

77

The first result of turning off muon kludge is to reduce more events HighEnu com-

pared to MidEnu. In other words, events are likely to be migrated from the HighEnu

region to the MidEnu region. Since the ratio of HighEnu (data)/ MidEnu (data) de-

creases, the double ratio decreases. This is very easy to understand. Since we are

decreasing the muon energy when we turn of muon kludge, we are also decreasing the

calculated neutrino energy. Thereby, there is a migration of events from high neutrino

energy to low neutrino energy which causes the double ratio to go down.

Now let us try to find any change in pattern in recoil energy by turning off muon scale.

Except for the first few bins in all of the plots in Fig. 4.11 and Fig. 4.12, the pattern of

double ratio almost stays the same throughout the whole spectrum. It seems that muon

kludge has a uniform energy dependence as opposed to a local energy dependence in a

certain region of phase space. Some small gradients can be noticed in few of the plots.

For example, the high q3 plot in Fig. 4.12, the one with the kludge is appears to be

flatter by 2 to 5% compared to removing the kludge. But, the pattern is not universal

in all of the plots and energy dependence pattern that comes from muon energy scale

is very mild.

4.7 Flux Measurements and a Necessity for a Flux Shift?

I will end this chapter with a short discussion on flux measurements and how my work

could relate to the research being done on flux uncertainties. I have already mention

in the literature review section how the work of Rob Fine [2] on MINERvA shows that

we need to correct our calculation of beam flux for different energy regions. His low-ν

method uses anchor flux values to predict such a necessity for improvements. Table

4.1 summarizes the reported flux shifts suggested by R. Fine in his work but mapped

on to my definitions of HighEnu, MidEnu, and LowEnu. Fine’s low-ν flux uses largely

the same sample as mine, but the high-ν flux is cleverly designed to be practically

independent from our samples.

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Table 4.1: The table summarizes R. Fine’s work using low-ν

method [2]. His work suggests a 5% positive shift for MidEnu

and HighEnu energies, thereby increase MidEnu flux by 5% or

decrease HighEnu flux by 5%. On the hand, he determined

a 5% positive shift for LowEnu and MidEnu. So, decrease

LowEnu Flux by 5% or increase MidEnu flux by 5%

Rob Fine’s Low-ν Method MidEnu & HighEnu LowEnu & MidEnu

Low-ν Flux 0.95 1.06

High-ν Flux 0.89 1.04

Through his sensitivity test to determine the model error of the magnitude and energy

dependence of the cross sections, he reported a flux model error of around -5% for

HighEnu/MidEnu ratio. Similarly, he presented a model error of around +5% for the

ratio of MidEnu/LowEnu. I have tried to use the results from my work to complement

the findings of his research.

Table 4.2: The table outlines the double ratio for low q3, mid

q3 and high q3 for different energy cuts in my work. As can be

seen from the table, the double ratio for HighEnu/MidEnu is

around 0.95 for all the q3, and suggests a flux shift. By con-

trast, the double ratio for MidEnu/LowEnu do not suggest

any concrete evidence for a flux shift.

MidEnu & HighEnu LowEnu & MidEnu

Low q3 0.95 1.02

Mid q3 0.95 1.00

High q3 0.97 0.96

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The way I used my analysis to connect the flux model errors is to use the double ratio.

As I have mentioned, a double ratio around 0.95 would indicate an error in the model

that is persistent throughout the whole spectrum and not localized to any part of the

recoil energy in tracker or a process like QE or delta. Table 4.2 outlines the double

ratio for different energy and momentum cuts in my work. It is clearly evident in my

research that for all the q3 ranges that I worked with, the HighEnu/MidEnu double ratio

is around 0.95. This double ratio could be shifted to one by increasing the MidEnu MC

distribution by +5% or by decreasing HighEnu MC distribution by -5%. Such a constant

shift throughout the spectrum could be could be easily explained by a flux shift. My

samples suggest a need for a 5% flux shift which complements the reports provided by

R.Fine, and disfavors any unexpected cross section effect as a cause.

Finally, I will focus on the LowEnu and MidEnu region. Here, my results do not provide

any strong suggestions for a flux shift since the double ratio is not around any constant

value for the different q3 cuts. On the other hand, Fine’s work suggest a 5% shift.

In fact, my analysis adds evidence that other systematic effects are at work, probably

affecting both our samples. For example, I have pointed out how the data tail falls off

whereas the MC tail does not fall off in Fig. 4.7.

Chapter 5

Uncertainties in the Energy

Dependence Modeling

5.1 Motivation

Neutrino interaction cross-section models are one of the largest sources of uncertainty

in neutrino oscillation experiments. The high precision measurements of the MINERvA

data can help us to characterize these uncertainties. Future neutrino oscillation exper-

iments are going to be in the 1 to 20GeV range, and so MINERvA low energy and

medium energy beam lie in a good spot to study neutrino interaction model at energies

relevant to future experiments. Furthermore, interactions in the quasi-elastic channel,

delta channel, and 2p2h channel are likely to be signal and/or background for these

oscillation experiments.

In the previous chapters, we have shown that the energy dependence from the cross-

section model is mild for most of the event in QE, Delta, and 2p2h (Fig 1.6). But, if we

were mismodeling the energy dependence by 10%, would we notice that? To test this,

I will add artificial energy dependence into our existing model, and put an uncertainty

on possible unmodeled energy dependence. And so, an upper limit could be imposed

on the uncertainty of the theoretical parameters of the model.

Upper limits and standard fit uncertainty bounds will be obtained by performing a full

80

81

chi-square scan where our added artificial energy dependence will be the parameter of

variation. The chi-squares will be taken with respect to best zeroth order polyfit of

double ratios. From the chi-square scan, we will also be able to extract information

regarding the best-fit energy dependence for the MINERvA data, and understand if our

nominal model is within the statistical limits of the best fit model.

To reiterate the core part of the analysis method, we will use the double ratio. We will

define the double ratio for MidEnu and HighEnu sample as:

Double Ratio =highEnu Data/midEnu Data

highEnu MC/midEnu MC

And double ratio for lowEnu and midEnu will have the following form:

Double Ratio =midEnu Data/lowEnu Data

midEnu MC/lowEnu MC

In either case, a double ratio of one would mean the energy dependence and the flux

is well modeled by GENIE. The straight line shown in the bottom panel of Fig. 5.1 is

the zeroth-order polyfit of the double ratio. The best fits line will usually be shifted

upwards or downwards from one due to our flux miscalculations. A flux miscalculation

would be present in the whole spectrum, and will not be localized in any specific region

in the spectrum. So, it is more motivating to take chi-square with respect to this best

fit line rather than a straight line along one to make our data analysis insensitive to

flux errors.

5.2 Putting in artificial energy dependence

5.2.1 Quasi-Elastic Channel

Quasi-elastic scattering represents a large fraction of the total cross-section. So, we

should expect a small variation in the model by adding artificial energy dependence will

produce a large deviation in double ratio and the resulting chi-square. One example

82

Figure 5.1: The blue dots in the figure represent the data distribution at midEnu (4-7GeV) and the red dots represent data distribution at highEnu (7-20GeV). The solidlines that follow them are the corresponding MC predictions. The bottom panel depictsthe corresponding double ratio and the best fit for the double ratio.

where we increase midEnu QE cross-section by 7%, and decrease highEnu QE cross-

section by 7% is shown in Fig. 5.2. First, let us note that we take the chi-square with

respect to the zeroth order polyfit of the double ratio. As can be seen in the figure, the

original best fit is shifted away from 1.0 by a few percent (left plot), but the new best

fit is shifted upward significantly (right plot). The larger shift is following the slope

created by the artificial energy dependence.

Now, looking at Fig. 5.2, adding artificial energy of 14% decreased the p-value from

0.94 to 0.003. A p-value around 0.003 is where the three sigma deviation lies in, and it

is considered the ’discovery’ threshold for many experiments in particle physics. What

threshold of p-value should be set to signal that a deviation is not of random statistical

origin is different for different experiments. For the discussions in this chapter, we would

use 2σ deviation as the limit of statistical uncertainty. If we set our level of statistical

significance at p = 0.003, then we would conclude that there cannot be more than 14%

energy dependence unmodeled in GENIE for midEnu and highEnu data at low q3.

83

Figure 5.2: The figure shows the comparison between a nominal distribution and thecorresponding distorted distribution where a total of 14% energy dependence is addedonly to the QE channel. The vertical scale on the two plots are different. On the rightplot, midEnu MC has gone up, highEnu MC has gone down, and so the double ratiowent up. The chi-square is taken with respect to the best fit of line of the double ratio.

A second style of analysis is to to figure out if the energy dependence of our existing

model is the best fit for MINERvA data. And if the nominal model is not the best fit

model, is it within satisfactory statistical limits? To do that, we will be performing a

chi-square scan. Fig. 5.3 shows an example of such an analysis for midEnu, highEnu

data at lowq3. Here being at 10% on the horizontal axis means there is an increase

by 10% in midEnu cross section and a corresponding 10% decrease in the highEnu QE

cross-section. The chi-square again is taken with respect to the best zeroth order polyfit

of the double ratios.

It can be deduced from the chi-square scan that the nominal χ2 = 6.73, the minimum

χ2 = 6.13 for the data in this region. Lowq3 sample has 14 DOF with our binning.

Furthermore, the best fit model has 0.9% increased QE cross-section for midEnu and

0.9% decreased QE cross-section for highEnu. I will write this in short as 1.009 artifi-

cial energy added to QE midEnu, and a decrease in highEnu QE cross-section will be

implied.

84

The model’s nominal is within ∆χ2 = 1 of the best fit, even with no experimental sys-

tematics considered. The 2σ limits are at 0.96 and 1.059 which means the 2σ limit has

a width of ±2%. These are smaller than the predicted energy dependence described in

chapter 2. A fitting technique like this has some power to constrain even small effects,

if other systematic effects were well controlled. Also, traditionally it is desired to report

the one sigma bounds as an error on the result. However the exploration in this thesis

does not include systematic effects like the muon energy scale, so the regular error bars

would look like an unnaturally strong constraint. As a placeholder I am instead report-

ing the 2 sigma errors obtained with this method.

Figure 5.3: The figure shows a chi-square scan performed for midEnu and highEnudata at lowq3. The nominal χ2 = 6.73, best fit χ2 = 6.13, and best fit model has0.9% increased QE cross-section for midEnu and 0.9% decreased QE cross-section forhighEnu.

85

Conclusions from QE analysis

Now that we are familiar with the methods of adding extra energy dependence and the

associated data analysis, the results of the of all six comparisons for QE are summa-

rized in table 5.1 and table 5.2. In table 5.1, I have listed 2σ upper bound of possible

unmodeled energy dependence in our existing model. In table 5.2, I have summarized

the energy dependence of the best fit model and the associated 2σ width.

86

Table 5.1: The table summarizes the lower and upper 2σ

bounds for unmodeled QE energy dependence for different

energy and q3 slices

lowq3 midq3 highq3

Lower Upper Lower Upper Lower Upper

Bound Bound Bound Bound Bound Bound

lowEnu & midEnu 0.976 1.040 0.964 1.010 0.965 1.039

midEnu & highEnu 0.960 1.059 0.969 0.999 0.953 0.987

Table 5.2: The table summarizes the best fit energy depen-

dence model for the QE channel and the associated 2σ width

for different energy and q3 slices

lowq3 midq3 highq3

lowEnu & midEnu 0.8% ±2% -1.5% ±2% 0.1% ±2.5%

midEnu & highEnu 0.9% ±2% -1.5% ±1.5% -3% ±2%

Table 5.3: The table summarizes the nominal χ2 (current

GENIE setting) and the best fit χ2 for QE at six different

energy and q3.

Nominal χ2 Best Fit χ2

low q3 mid q3 high q3 low q3 mid q3 high q3

lowEnu & midEnu 16.1975 37.116 51.0562 15.679 34.502 51.032

midEnu & highEnu 6.73699 44.0903 79.8525 6.13163 38.8067 67.267

87

First, let us focus on table 5.2. We see that other than midEnu & highEnu data at

high q3, our current model is within 2σ of the best fit model. Especially, we should note

that in the lowEnu and midEnu region, our model seems to have a well-modeled energy

dependence.

The problems start to arise in midEnu and highEnu at higher q3. The question that

should be asked here is whether this is due to systematics such as muon energy scale

or is it due to our QE interaction model. If we look at Fig. 5.4, which is in the region

we are discussing, we observe that the nominal model already presents us with a high

chi-square. Table 5.3 summerizes the nominal and best fit χ2 for the six possible energy

and q3 configurations. The fact that we observe the minimum chi-square after adding

negative artificial energy to midEnu QE should not come as a surprise here since the first

few double ratio bins in the nominal model are above the best fit line. If the deviation

in the first few bins are due to a phenomenon other than interaction modeling, then the

limits set cannot be related to the parameters of the model. In particular, the deviation

in the tail of figure is not reduced with the fit and the χ2 stays high since the QE

distribution does not distort distribution at that range.

Let us expand on this topic a bit further. If we look at table 5.2, we can begin

to notice a pattern. In midEnu and highEnu, as we start increasing q3, the minimum

starts to shift to the left. This means that we are underpredicting the increasing energy

dependence factor at higher q3. This could mean the model energy dependence lacks

a q3 dependent factor. The pattern could also be a result of an unknown systematics

which has a q3 dependent energy dependence.

A natural question now is what pattern is present for lowEnu and midEnu. We see

that the same pattern is present for low q3 and mid q3, but high q3 does not follow

the trend. This forces us to look more closely at the nominal highq3 distribution for

lowEnu and midEnu. This is shown in Fig. 5.4. Looking at the figure, we can quickly

realize why the minimum is at 0.1% instead of the expected -3%. This is because the

first bins are equally distributed below and above the best fit line. If we can account

88

Figure 5.4: The figure shows the distribution and the corresponding double ratio forlowEnu and midEnu at high q3, which is the same as the top right plot in Fig. 4.7. Wecan see that double ratio in the first bins are equally distributed above and below thebest fit, and this leads to a nominal p-value of 0.0001.

for the distortions in the first few bins, then we will be able to conclude whether or not

the trend seen in midEnu and highEnu is also present in lowEnu and midEnu.

5.2.2 Delta Channel

We now shift our attention to the delta channel. Fig. 5.5 shows an example of increasing

lowEnu delta predictions by 5% and decreasing midEnu delta predictions by 5% at

midq3. The deviation due to this distortion is visible in the middle of the spectrum

where delta channel contribution is the highest. From the discussion at the end of the

last section, we should expect our chi-square scan for delta channel not to be affected

by the systematics that may be present in the previous section, because they are in the

first few bins where delta interaction is not prominent.

89


bounds for delta channel energy dependence for different en-

ergy and q3 slices

lowq3 midq3 highq3



lowEnu & midEnu 0.952 1.036 0.986 1.039 0.930 1.012

midEnu & highEnu 0.92 1.05 1.001 1.037 N/A N/A


dence model for the delta channel and the associated 2σ width


lowq3 midq3 highq3

lowEnu & midEnu -0.6% ±3% +1% ±2% -3% ±3.5%

midEnu & highEnu -1.5% ±3% 2% ±2% 2.5% ±3.0%

We now shift our attention to the delta channel. Fig. 5.5 shows an example of increas-

ing lowEnu delta predictions by 5% and decreasing midEnu delta predictions by 5% at

midq3. The deviation due to this distortion is visible in the middle of the spectrum

where delta channel contribution is the highest. From the discussion at the end of the

last section, we should expect our chi-square scan for delta channel not to be affected

by the systematics that may be present in the previous section, because they are in the

first few bins where delta interaction is not prominent.

Table 5.4 and table 5.5 summarizes the analysis for the delta channel. Looking at

table 5.5, we observe that the left shifting trend that was present in the QE analysis

90

Figure 5.5: The left plot shows data and MC distribution for lowEnu and midEnu atmidq3 after increasing delta cross-section for lowEnu by 5% and decreasing midEnudelta cross-section by 5%. We can observe the deviation in the middle of the doubleratio where delta channel is most prominent. The right plot shows the chi-square scanperformed in this region. The nominal χ2 = 37.11, best fit χ2 = 35.273, and best fitmodel is at +1% artificial energy added to lowEnu with similar decrease in midEnu.

is now gone. The lack of any discernible pattern in table 5.5 suggests that they have

a statistical origin. Therefore, we expect these numbers to remain the same or change

slightly after we include more systematic uncertainties.

Now let us inspect if our nominal model is within satisfactory statistical limits from

the minimum. From table 5.5, we see that for midEnu and highEnu data at midq3 and

highq3, our current model is on the edge of the 2σ limit. Similar can be said regarding

lowEnu and midEnu high q3 sample.

In Chapter 3 we showed that the W3 and W5 form factors had their largest contribu-

tions at very low Q2, which is the far right of these plots. For future work, rather than

scaling the whole delta process, its possible that only one of the structure function is

mismodeled. Also, MINERvA has been exploring other model effects in this region such

as an incorrect treatment of removal energy.

91

5.2.3 2p2h Channel

Finally, we focus on the 2p2h channel. In comparison to QE and delta interactions, the

2p2h channel has the smallest contribution to the total cross-section. Fig. 5.6 shows an

example of increasing lowEnu delta predictions by 10% and decreasing midEnu delta

predictions by 10% at midq3. Reader would be quick to notice that the chi-square values

vary slowly in the case of 2p2h. This is aniticipated because 2p2h has a wide distri-

bution and the maximum amplitude of the distribution compared to the total sample

is relatively minor. And so one should not be alarmed by higher shift from nominal

energy dependence in the case of 2p2h.

Figure 5.6: The left plot shows data and MC distribution for lowEnu and midEnu atmidq3 after increasing 2p2h cross-section for lowEnu by 10% and decreasing midEnu2p2h cross-section by 10%. The right plot shows the chi-square scan performed in thisregion. The nominal χ2 = 37.11, best fit χ2 = 36.12, and best fit model is at -4%artificial energy added to lowEnu with similar increase in midEnu. The slow variationin chi-square is due to high width and low amplitude of 2p2h distribution.

92


bounds for unmodeled 2p2h energy dependence for different

energy and q3 slices

lowq3 midq3 highq3



lowEnu & midEnu <0.900 1.066 <0.900 1.077 <0.900 1.095

midEnu & highEnu <0.900 1.095 <0.900 1.050 N/A N/A


dence model for the 2p2h channel and the associated 2σ width


lowq3 midq3 highq3

lowEnu & midEnu -4% ±7% -4.5% ±8% <-10%

midEnu & highEnu -1.0% ±7% -6% ±8% <-10%

Summary of the study for 2p2h can be found in table 5.6 and table 5.7. First I want to

point the readers attention to table 5.6, where many of the lower bounds are not listed.

This is because the chi-square scan is conducted by adding artificial energy dependence

of ±10%. Let us now turn our focus to table 5.7 where shifts from nominal energy

dependences can be found. Here we see that for highq3 samples, the best fit model is

highly left shifted from the nominal one. The origin of this can be found in the negligible

presence of 2p2h contribution in the highq3 region. Looking at the table 5.7, we see

that all the values are higher than previously encountered, but all of them lie within

our statistical limits. Regardless, these values are less affected by systematics than QE

region since 2p2h contributions are in the middle of the spectrum. In a search for any

93

noticeable pattern in table 5.7, I found that there is a left shifting pattern present in

this table similar to QE.

5.3 Conclusions and future considerations

The goal of the sub-project was to set initial upper bounds on unmodeled energy depen-

dence considering only statistical uncertainty. The unmodeled effects in the QE region

and delta region are clearly affecting the fit, but the simple scaling produces too broad

an effect to fully fit them out. They may be due to systematic effects, or potentially to

specific structure functions. For example, the first look at this problem showed us that

putting such a bound is difficult in the QE region where potential systematics at the

two ends of the spectrum complicates our data analysis. For future considerations, one

can add the muon energy scale (studied in chapter 4) and hadron (and nucleon removal)

energy systematic effects to this study. Amother future approach would be to try scale

the structure functions individually instead of the process as a whole. This may have

more narrow effects on the distributions or how the energy dependence evolves with q3

and Q2.

Chapter 6

Conclusion and Discussion

In conclusion the predicted and actual energy dependence from cross-section model for

most of the events in all of the channels is mild. This is summarized in the 2-D plot

in Fig 1.6. From the Quasi-elastic cross-section models, one should expect around 5%

energy dependence at the energy range from 3GeV and above for Q2 < 1GeV 2. Only for

a comparison between 2GeV and 1GeV events would one start seeing energy dependence

around 10% for the QE events (Fig. 3.7). For the structure functions of QE, almost

all of the contribution to the total cross-section at MINERvA energies comes from the

C term, and most Q2 variation of the energy dependence also originates from the C

term when Q2 < 1GeV 2. The A term and B term are negligible at the energy range of

MINERvA, but the B term cannot be neglected even at NOvA energy of 2GeV.

The energy dependence from the delta channel cross-section model is non-existent above

3GeV for Q2 < 1GeV 2. Similar to QE, there is around 10-15% energy dependence

below 2GeV (fig 3.5). The study of delta structure function reveals that the energy

independent factor of W2 makes it dominant in the MINERvA energies when Q2 is below

1GeV, and the energy dependence at these energies comes from the W2 contributions

as well. But for NOvA and low energy part of the DUNE spectrum, the 1/E dependent

factor of W3 and W5 must be accounted for.

So, for a comparison between the MINERvA LE beam and NOvA beam, we would

mainly be interested in the uncertainty on the W3 (delta) or the B (QE) contributions to

94

95

the energy dependence. On the other hand, DUNE will have primary oscillation effects

around 2.5 GeV with the low end of the spectrum (and the second oscillation maximum)

just below 1.0 GeV where the energy dependence clearly cannot be neglected. So,

having a correct understanding and modelling of energy dependence will be important

for studying the DUNE spectrum.

The study of angle acceptance shows that our data has a higher energy dependence in

higher q3, and there is almost no energy dependence at low q3 (Fig 4.3 and Fig 4.6).

Lower energy events also show higher energy dependence due to angle acceptance. But,

the Monte Carlo is doing a good job in predicting these energy dependencies in the

data (Fig 4.3 and Fig 4.6). A curious and poorly modeled pattern observed in the angle

acceptance study was the data falling off at the tail of the distribution where the low

Q2 resonance process is very difficult to model. Also, we observed an unexpected bump

in the middle of the spectrum in the high q3 plot in fig 4.6. The later pattern was seen

to be more dominant when we introduced imposed angle cuts of 5 degree and 10 degree.

What is confusing about the trend is that it has a distinct peak which is not natural

for an angle pattern. It could be the combination of the angle effect with a quasi elastic

specific effect. It appears relatively well modeled.

Finally, we studied the energy dependence and the migration of events in the recoil plots

due to muon kludge. The energy dependence of muon kludge is hard to analyze because

the effect is so small. It was found that muon kludge has a uniform energy dependence

as opposed to a local energy dependence because we are decreasing the muon energy

when we turn of muon kludge. But, some small gradients can be noticed in few of the

plots in fig 4.10 and 4.11, but they are 2-5% effect.

References

[1] Ch. Berger and L. M. Sehgal. Lepton mass effects in single pion production by

neutrinos. Phys. Rev. D, 76:113004, 2007, 0709.4378.

[2] R. Fine. Low-ν and High-ν Analysis Update @ Feb 2020 Collaboration Meeting.

MINERvA- doc-26057-v1, 2020.

[3] N. Fuad. Analysis of MINERνA tunings of GENIE based on Hadron Multiplicity

using Machine Learning. Master’s thesis, Minnesota U., Duluth, 2020.

[4] MINERvA Public Paper. MINERvA’s Medium Energy Physics Program.

[5] M. Ascencio, A. Bashyal, and R. Gran. Implementation of SuSA 2p2h and QE

weights for MINERvA. MINERvA-doc-29414-v2, 2021.

[6] J. A. Nowak. Four Momentum Transfer Discrepancy in the Charged Current π+

Production in the MiniBooNE: Data vs. Theory. AIP Conf. Proc., 1189(1):243–248,

2009, 0909.3659.

[7] P. Stowell et al. Tuning the GENIE Pion Production Model with MINERνA Data.

Phys. Rev. D, 100(7):072005, 2019, 1903.01558.

[8] P. Adamson et al. Study of quasielastic scattering using charged-current νµ -

iron interactions in the MINOS near detector. Phys. Rev. D, 91(1):012005, 2015,

1410.8613.

[9] O. Lalakulich and E. A. Paschos. Resonance production by neutrinos. I. J = 3/2

resonances. Phys. Rev. D, 71:074003, 2005, hep-ph/0501109.

96

97

[10] G. M. Radecky et al. Study of Single Pion Production by Weak Charged Currents in

Low-energy Neutrino d Interactions. Phys. Rev. D, 25:1161–1173, 1982. [Erratum:

Phys.Rev.D 26, 3297 (1982)].

[11] A. A. Aguilar-Arevalo et al. First Measurement of the Muon Neutrino Charged

Current Quasielastic Double Differential Cross Section. Phys. Rev. D, 81:092005,

2010, 1002.2680.

[12] L. B. Auerbach et al. Measurements of charged current reactions of muon neutrinos

on C-12. Phys. Rev. C, 66:015501, 2002, nucl-ex/0203011.

[13] V. Lyubushkin et al. A Study of quasi-elastic muon neutrino and antineutrino scat-

tering in the NOMAD experiment. Eur. Phys. J. C, 63:355–381, 2009, 0812.4543.

[14] E. R. Miltenberger. Nuclear Effects in Neutrino Interactions at Low Momentum

Transfer. Master’s thesis, Minnesota U., Duluth, 2015.

[15] C. Andreopoulos et al. The GENIE Neutrino Monte Carlo Generator. Nucl. In-

strum. Meth. A, 614:87–104, 2010, 0905.2517.

[16] P. Adamson et al. Study of quasielastic scattering using charged-current νµ -

iron interactions in the MINOS near detector. Phys. Rev. D, 91(1):012005, 2015,

1410.8613.

[17] M. F. Carneiro et al. High-Statistics Measurement of Neutrino Quasielasticlike

Scattering at 6 GeV on a Hydrocarbon Target. Phys. Rev. Lett., 124(12):121801,

2020, 1912.09890.

[18] S. Dolan, G. D. Megias, and S. Bolognesi. Implementation of the SuSAv2-meson

exchange current 1p1h and 2p2h models in GENIE and analysis of nuclear effects

in T2K measurements. Phys. Rev. D, 101(3):033003, 2020, 1905.08556.

[19] J. Nieves, I. Ruiz Simo, and M. J. Vicente Vacas. Inclusive Charged–Current

Neutrino–Nucleus Reactions. Phys. Rev. C, 83:045501, 2011, 1102.2777.

[20] R. Gran, J. Nieves, F. Sanchez, and M. J. Vicente Vacas. Neutrino-nucleus quasi-

elastic and 2p2h interactions up to 10 GeV. Phys. Rev. D, 88(11):113007, 2013,

1307.8105.

98

[21] J. Schwehr, D. Cherdack, and R. Gran. GENIE implementation of IFIC Valencia

model for QE-like 2p2h neutrino-nucleus cross section. 1 2016, 1601.02038.

[22] A. Bodek and T. Cai. Removal Energies and Final State Interaction in Lepton

Nucleus Scattering. Eur. Phys. J. C, 79(4):293, 2019, 1801.07975.

[23] Arie Bodek and Tejin Cai. Comparison of optical potential for nucleons and

resonances: In electron scattering from nuclear targets. Eur. Phys. J. C, 80(7):655,

2020, 2004.00087.

[24] P. Adamson et al. The NuMI Neutrino Beam. Nucl. Instrum. Meth. A, 806:279–

306, 2016, 1507.06690.

[25] T. Kitagaki et al. Study of neutrino d —> mu- p p(s) and neutrino d —> mu-

Delta++ (1232) n(s) using the BNL 7-foot deuterium filled bubble chamber. Phys.

Rev. D, 42:1331–1338, 1990.

[26] A. A. Aguilar-Arevalo et al. The MiniBooNE Detector. Nucl. Instrum. Meth. A,

599:28–46, 2009, 0806.4201.

[27] Dieter R. and Sehgal L. M. Neutrino-excitation of baryon resonances and single

pion production. Annals of Physics, 133(1):79–153, 1981.

[28] R. P. Feynman, M. Kislinger, and F. Ravndal. Current matrix elements from a

relativistic quark model. Phys. Rev. D, 3:2706–2732, 1971.

[29] P. A. Schreiner and F. Von Hippel. Neutrino production of the Delta (1236). Nucl.

Phys. B, 58:333–362, 1973.

[30] K. S. Kuzmin, V. V. Lyubushkin, and V. A. Naumov. Lepton polarization in

neutrino nucleon interactions. Mod. Phys. Lett. A, 19:2815–2829, 2004, hep-

ph/0312107.

[31] R. Gran, I. Mahbub, and A. R. Mislivec. Rein Sehgal and Berger Sehgal Reso-

nances. MINERvA- doc-26057-v1, 2021.

[32] P. A. Zucker. Weak Production of Nucleon Resonances. Phys. Rev. D, 4:3350,

1971.

99

[33] D. Norman. Phenomenological Study of Baryon-Resonance Production in Neutrino-

Nucleon Scattering. Tufts Undergraduate Senior Thesis, 2016.

[34] J. Sobczyk. Recent Work on Multi-Nucleon Correlations. MINERvA-doc-26551-v2,

2020.

[35] C. H. Llewellyn Smith. Neutrino Reactions at Accelerator Energies. Phys. Rept.,

3:261–379, 1972.

[36] A. Mann. Calculate like a Llewellyn. NOVA-doc-28289-v2, 2018.

[37] D. Ruterbories, K. Hurtado, J. Osta, et al. Measurement of Quasielastic-Like

Neutrino Scattering at 〈Eν〉 ∼ 3.5 GeV on a Hydrocarbon Target. Phys. Rev. D,

99(1):012004, 2019, 1811.02774.

[38] A. Lovlein. Exploring Enu Dependence with ME beam. MINERvA-doc-12404-v5,

2016.

[39] E. R. Miltenberger. Nuclear Effects in Neutrino Interactions at Low Momentum

Transfer. Master’s thesis, Minnesota U., Duluth, 2015.

[40] J. G. Demgen. Nuclear Effects in Quasi-Elastic and Delta Resonance Production

at Low Momentum Transfer. Master’s thesis, Minnesota U., Duluth, 2015.

[41] P. A. Rodrigues, J. Demgen, E. Miltenberger, et al. Identification of nuclear effects

in neutrino-carbon interactions at low three-momentum transfer. Phys. Rev. Lett.,

116:071802, 2016, 1511.05944. [Addendum: Phys.Rev.Lett. 121, 209902 (2018)].

[42] A. Bashyal, D. Rimal, B. Messerly, et al. Use of Neutrino Scattering Events with

Low Hadronic Recoil to Inform Neutrino Flux and Detector Energy Scale. Submit-

ted to JINST, 4 2021, 2104.05769.

[43] I. Mahbub. Looking into Hadronic Rich Component in Muon Fuzz. MINERvA-

doc-27965-v1, 2020.

[44] L. Ren. Measurement of Neutrino and Antineutrino Total Charged-Current Cross

Sections on Carbon with MINERvA. PhD thesis, U. Pittsburgh (main), 2017.

100

[45] Passage of Particles Through Matter (Particle Data Group).

[46] R. Gran. Distortion in the First Bin of the Low Recoil Analysis. MINERvA-doc-

25523-v2, 2020.

Appendix A

The Hadronic Rich Component

in Muon Fuzz

A.1 Introduction

The initial motivation was to look for the energy dependence expected because the

muon bremsstrahlung process has inherent energy dependence. The plan was to use the

same techniques present in chapter 4. However, most of the Bremsstrahlung activity

is is associated with the muon track and was not saved to the hadronic activity data

structures available for this analysis. However the attempt was useful to quantify the

mix of hadronic and muon energy in the muon fuzz region. This is informative for a

number of MINERvA analyses. The rest of this appendix is nearly identical to MIN-

ERvA Technical Note 96, docdb:27965 [43].

Our aim of the study is to put the hadronic energy in the muon fuzz region back to

the hadrons and improve the resolution of our data. For this, we are going to focus on

the low recoil sample with three momentum transfers less than 0.8 GeV, where many

events have very small amounts of hadron activity. The criteria that we use are muon

fuzz energy and examining different geometric definition of the muon fuzz region. The

overall goal of the study is to understand and categorize the proton or pion activity,

neutron activity, and muon activity in the muon fuzz region.

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Currently, the first 300mm after the interaction in the muon fuzz track is not included

in the muon fuzz section which excludes majority of hadron activity [44]. This is about

10 modules in Arachne or 20 planes at 2 planes per module. The radius is 80mm around

the muon track in the Tracker region and 100mm in the ECAL region (Fig A.2). Also,

the energy on the muon track itself is not included; clusters of energy must be separated

from the track node. The radius is larger in the ECAL because bremsstrahlung is more

likely and downstream hadronic activity is less likely in this region. Using the larger

exclusion radius for low recoil analysis is our main practical recommendation.

The initial motivation of the study stems from looking for energy dependence in the

muon fuzz region. Now, Bremsstrahlung is an inherently energy-dependent phenomenon,

its cross section is proportional to the lepton energy squared which in this case are the

muons. So, as the energy of the neutrino increases and thereby the energy of the muon

increase, so does the frequency and energy of bremsstrahlung activity [45]. Hence, we

initially decided to analyze this particular energy dependence in our data and plotted

the distribution of muon fuzz energy for different neutrino energy ranges and compared

the results (Fig A.1). Looking into the plots for neutrino energy of 4-7 GeV shows

that around 18% of events have no muon fuzz activity, 39% of them have 0-5 MeV

of muon fuzz activity, 10% of them have 5-10 MeV of muon fuzz activity and so on.

Surprisingly, we see similar statistics for events with neutrino energy of 7-20 GeV rather

than the expected increase of muon fuzz activity. So, we concluded that there must be

large hadronic activity along with bremsstrahlung activity in the muon fuzz region and

the energy dependent bremsstrahlung activity might be largely confined to the tracking

nodes and not part of the mufuzz energy.

If the hadronic activity is more than the bremsstrahlung activity in the muon fuzz under

some circumstances, we might improve the resolution with a different treatment of the

muon fuzz region. Therefore, the relative amount of hadronic energy in the muon fuzz

region is the main topic of interest in this technical note. As this note will show, the

treatment of neutrons will prevent us from achieving the desired improvement. Although

adding proton and pion activity in the mufuzz region back into the hadron energy does

make an improvement, but adding back overlay, or neutron activity makes a bias the

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other way. Furthermore, the resolution is driven in part by activity that leaves out the

side of the detector which can not be corrected with any change in muon fuzz treatment.

Figure A.1: Normalized muon fuzz distribution for 800 < q3 < 1200 MeV, 4 < Eν < 7GeV(left plot) and 800 < q3 < 1200 MeV, 7 < Eν < 20 GeV (right plot)

A.2 What Does Event by Event Tell Us

To get an insight into the kind of activity in the muon fuzz region, we first looked into

50 data and Monte Carlo events. The goal was to ascertain our initial assumption of

high hadronic activity in the muon fuzz section. Fig A.2 depicts an event with 413

MeV bremsstrahlung electron. We found that when imposing the criteria of muon fuzz

energy > 40 MeV, only 2 out of 10 were bremsstrahlung events. This is an important

first observation: that hadronic activities become more important in higher muon fuzz

events. We will use these criteria to isolate the hadronic activity in the muon fuzz

region.

Examples of events with hadronic components in the muon fuzz region are shown in Fig

A.3 and Fig A.4. As can be seen in the figures, there two categories of events here: one

where the muon fuzz energy comes from proton or pion (Fig A.3) and the second kind

where muon fuzz energy comes from neutrons (Fig A.4). Since, neutron energy is not

considered in the truth energy, we will see that these two classifications will fall into

two separate domains of our resolution plot and will shift differently under our event

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selection criteria.

To reiterate, When we look at events with muon fuzz > 40MeV, 8 out of 10 of them

have hadronic energy contributing to them, and 2 out of 10 of the have bremsstrahlung

component contributing to muon fuzz energy. So, in an effort to put hadronic energy

component in the muon fuzz region back to the hadrons, we set forward the three

following hypothesis:

• We can ignore muon fuzz completely and stop any muon fuzz subtraction.

• We hypothesize that we could set a limit to muon fuzz subtraction since these

energy contributions are from hadrons rather than from muon. For example, we

would exclude muon fuzz subtraction for events with muon fuzz > 40 MeV or

muon fuzz > 20 MeV whichever produces a better resolution plots.

• We can subtract muon fuzz for all events (our current way of analysis).

Figure A.2: An event with 413 MeV bremsstralung electron that generates 181 MeVmuon fuzz energy. These compose 20% of events with muon fuzz > 40 MeV

A.3 Can We Completely Ignore Muon Fuzz

The answer is no. Even though it seems at higher muon fuzz energies almost all energies

are coming from hadronic activities, muon fuzz subtraction still appears to be necessary.

This is evident in Fig A.5 where we plotted resolution plots with muon fuzz subtracted

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Figure A.3: An event with proton activity in the muon fuzz region

Figure A.4: An event with neutron activity in the muon fuzz region. 80% of events withmuon fuzz > 40 MeV are neutron,proton or pion events.

for all events (left plot) and with muon fuzz ignored for all events (right plot) for 4

< Eν < 7 GeV and 600 < q3 < 800 MeV. We see that the RMS increases from 0.454

to 0.487 when we disregard muon fuzz completely. We observe similar results for other

energy and q3 ranges as well. Fig ?? shows another evidence for a similar conclusion for

7 < Eν < 20 GeV and 600 < q3 < 800 MeV. Therefore, conclusion from this analysis is

that we cannot ignore muon fuzz subtraction completely and we will have to use event

selection method to isolate the hadronic components.

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Figure A.5: Resolution plot with muon fuzz subtraction (left plot) and without muonfuzz subtraction (right plot) for 4 < Eν < 7 GeV and 600 < q3 < 800 MeV. RMSincrease by almost 7% when we stop subtracting muon fuzz

A.4 Selected Muon Fuzz Subtraction

So far we have discerned that events with high muon fuzz energy have a large hadronic

component to them. Therefore, to put the hadronic energies back to hadrons, we de-

cided to stop muon fuzz subtraction only for these high muon fuzz energy events. The

reasoning here is that in the previous section, we recognized that completely neglecting

muon fuzz deteriorates our resolution, and so low muon fuzz events must be mostly

bremsstrahlung events.

Now, the basic problem in this way of selected muon fuzz subtraction is to decide what

to call a high muon fuzz event. So, we chose to perform a variety of different event

selections. Fig A.7 shows such selected muon fuzz subtraction for 4 < Eν < 7 GeV and

600 < q3 < 800 MeV. To interpret Fig A.7, in the top left plot, we subtract muon fuzz

only for event with muon fuzz < 60 MeV. Likewise, for the plot right next to it, we do

similar analysis but subtract muon fuzz only for events with muon fuzz < 50. After

that, we do similar muon fuzz subtraction with the ceiling being set to 40,30,20 or 10

MeV and then compare the results. Our goal here is to a limit to muon fuzz subtraction

by comparing the RMS values of these plots. We can observe that the RMS is the lowest

for the top left plot and it progressively gets larger as decrease the limit of muon fuzz

subtraction. For example, the bottom right plot shows us that when subtracting muon

fuzz for events with muon fuzz < 10 MeV, the RMS is 0.486 which is the highest of the

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Figure A.6: Reolution plot with muon fuzz subtraction (left plot) and without muonfuzz subtraction (right plot) for 7 < Eν < 20 GeV and 600 < q3 < 800 MeV. RMSincrease by almost 8% when we stop subtracting muon fuzz.

six of them. What is noticeable in all six variations is that all of them have RMS greater

than 0.454 which is when we subtract muon fuzz for all events. This is contradictory to

what we observed in our event by event analysis where it was obvious that higher muon

fuzz events are dominantly hadronic events in the muon fuzz region, but as have been

shown putting these high muon fuzz energies back to available energy does not improve

the statistics. We are led to comparable conclusion for 7 < Eν < 20 GeV and 600 < q3

< 800 MeV as shown in Fig A.8. The conclusion from this analysis:

• We cannot perform selected muon fuzz subtraction even though majority of high

muon fuzz events have dominant hadronic component to them. This necessitate a

further look into muon fuzz region as there must be more than hadronic activity

effecting our analysis .

A.5 Looking into high muon fuzz events

The analysis in the previous section prompted us to take a closer look into high muon

fuzz event separately. We plotted resolution plots only for events with muon fuzz >

40MeV for 4 < Eν < 7 GeV and 0 < q3 < 200 MeV (Fig A.9). Here, the plot on the

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Figure A.7: Resolution plot with several muon fuzz event selection for 4 < Eν < 7 GeVand 600 < q3 < 800 MeV. RMS is the lowest for selected events with muon fuzz < 60MeV (top left) and greatest for events with muon fuzz < 10 MeV

left-hand side depicts resolution plot for all events and the plot on the right-hand size

only takes the sample with muon fuzz > 40 MeV which is why the plot on the right-hand

side has a very small sample size.

As can be seen in Fig A.9, RMS is significantly higher for events with muon fuzz >

40 MeV which indicates the inherent uncertainty in such events. Naturally, we would

think that this higher RMS for these high muon fuzz events stems from the hadronic

activity in them, and not subtracting muon fuzz for these events is the correct way to

go about it. But as can be seen in Fig A.10, without subtracting muon fuzz for these

events leads to a significantly higher mean with the same RMS. In Fig A.10, the plot

on the left-hand side shows resolution plot with muon fuzz subtracted (what we are

currently doing now) and the plot on the right-hand side shows our proposed method

of not subtracting muon fuzz for these high muon fuzz events.

Looking into these plots, the mean jumps from 0.55 to 1.73. Although the percentage

increase in mean is quite high here, note that, the increase in mean is to be expected since

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Figure A.8: Resolution plot with several muon fuzz event selection for 7 < Eν < 20GeV and 600 < q3 < 800 MeV. Similar to the previous figure, in this energy range RMSis the lowest for selected events with muon fuzz < 60 MeV (top left) and greatest forevents with muon fuzz < 10 MeV

we are adding energy back to available energy. Now, events with a negative resolution

on the left-hand plot end up having a better resolution when we stop subtracting muon

fuzz. In contrast, events with a positive bias on the left-hand plot end up having an even

worse resolution when we stop subtracting muon fuzz. Therefore, it appears that these

events with initial positive bias caused our RMS to rise when we decided to perform

selected muon fuzz subtraction.

A.6 Further Event Analysis

With the new knowledge of our analysis, we went back to look into the events again.

The goal now is to look into events with high muon fuzz only. As was pointed out in

the previous section, we want to categorize these events into positive resolution and

negative resolution.

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Figure A.9: RMS comparison of selected events with muon fuzz > 40MeV (right plot)which has a small sample size and events without any selction (left plot). The increasedRMS indicates the high uncertainty in the high muon fuzz events.

Figure A.10: Resolution plot with muon fuzz subtraction (left plot) and without muonfuzz subtraction (right plot) for events with muon fuzz > 40MeV. The increased RMShere shows the necessity for muon fuzz subtraction even for these high muon fuzz events.

A.6.1 Events with Negative Bias

Consider the event in Figure A.11, where we can see a pion going alongside the muon.

The true energy for this event is 1.82 GeV, and Energy in Tracker is 0.696 GeV. Hence,

there is missing energy in the tracker, and the event has a resolution (Reco Energy-True

Energy/True Energy) of -0.62. Part of the missing energy is in the muon fuzz region

which accounts for 0.2 GeV of hadronic energy from the pion. Now, these kinds of

events with a negative bias are shifted closer to zero when we stop subtracting muon

fuzz. A similar example is shown in Fig A.12 where a pion goes in the direction of the

muon, and its activity is considered as muon energy.

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One more thing to notice in figure A.11 is that the proton from the vertex goes out

through the side of the detector. Rest of the missing energy in our tracker is due to the

fact that we do not have a full track of this proton. The negative bias from this can

not be corrected by changing the muon fuzz treatment. A proton or pion that interacts

and creates a neutron that leaves out through the side has the same effect.

Figure A.11: An event pion and proton in muon fuzz region causing negative residual.Negative residual is partly caused by muon fuzz subtracting these hadronic energy andpartly by the proton going off the side

Figure A.12: An event pion in muon fuzz region causing negative bias. Negative residueis partly caused by muon fuzz subtracting these hadronic energy

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Figure A.13: An event neutron in muon fuzz region causing positive resolution. Sincethe truth value does not include neutron energy, the residue here is positive. Subtractingthese energy as muon fuzz increases our resolution.

A.6.2 Events with Positive Bias

Now the question comes what lies in the positive resolution such that they are shifted

farther to the right when we stop subtracting muon fuzz. Fig A.13 shows an event with

positive bias and high muon fuzz. The true available energy for the event is 0.308 GeV

while the energy in tracker is 1.171 GeV. The excess energy in the tracker comes from

1.6 GeV neutron which also disposes some of its energy in the muon fuzz region. Now,

the true energy does not include neutron energy and so the resolution here is positive.

For events such as this, subtracting muon fuzz improves our resolution because we are

removing neutron energy in the tracker and it is not included in the truth value. So,

even though these events do not have muon energy in them, stopping subtraction makes

the resolution plot shift extravagantly to the right.

To reiterate, the truth value does not include neutron energy and events with positive

resolution with high muon fuzz energy have energy component from neutron. Subtract-

ing the muon fuzz energy from the tracker energy brings the total energy in tracker

closer to the truth energy for these events. It is these events that has been causing our

RMS to go up when we perform our proposed selected muon fuzz subtraction.

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A.6.3 Event with High Muon Fuzz in the ECAL

In this section, we focus on the ECAL region. ECAL being likely to be away from the

vertex, we expect events with bremsstrahlung to go up. To investigate muon fuzz in the

ECAL region, we looked into events with muon fuzz > 40 MeV in the ECAL region for

4 < Eν < 7 GeV. Specifically, we looked into five events in low q3, five events in mid q3

and five events in high q3. The results are listed in table 1. We could note that 40% of

the events are now bremsstrahlung ones. This is a 20% increase from our previous high

muon fuzz event analysis which included the whole detector range, an increase which

was expected. A similar decrease in neutron event is also observed which likely due to

the fact that ECAL is a relatively small portion of the whole detector in terms of area.

In the table, we have separated the events into low q3, mid q3 and high q3 region. The

sample size is too small for us to tell if going from low q3 to high q3 changes the amount

of bremsstrahlung. It would be an interesting topic to look further into if we want to

determine the optimum definition of muon fuzz in the ECAL.

Table A.1: Events with muon fuzz > 40 MeV in the Ecal re-

gion for 4 < Eν < 7 GeV. We randomly selected five events in

each of low q3,mid q3 and high q3 region. From the table we

can see that in the ECAL, bremstrahlung activity increases

by 20% and neutral hadron activity decreases by 20% from

our previous analysis of looking into the whole detector

Bremsstrahlung Charged Hadron π0 Neutron

0 < q3 < 0.4 GeV 2 2 1 0

0.4 < q3 < 0.8 GeV 2 2 0 1

0.8 < q3 < 1.2 GeV 2 2 1 1

0 < q3 < 1.2 GeV 6 6 2 2

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A.7 Looking into MuonFuzz2

Table A.2: A comparison between using muonfuzz1 and

muonfuzz1 + muonfuzz2 for various event selection and for

4 < Eν < 7 GeV, 1000 < q3 < 1200MeV . The resolution is

improved when the mufuzz2 is included in the subtraction.

Mufuzz1 Mufuzz1 + Mufuzz2

Mean RMS Mean RMS

All 0.219 0.670 0.140 0.642

Fuzz <40 MeV 0.21 0.666 0.134 0.636

Fuzz <30 MeV 0.209 0.663 0.131 0.634

Fuzz <20 MeV 0.203 0.649 0.124 0.628

Fuzz <10 MeV 0.189 0.649 0.112 0.619

Fuzz <5 MeV 0.171 0.640 0.968 0.610

In the code, muonfuzz2 enlarges the muon fuzz region. The extended region should have

less muon fuzz than the original, because it is farther away from the muon. However,

especially downstream, it may have even less charged hadron activity because it is much

farther from the event vertex. The goal of this section is to use a combination of the two

muon fuzz sections and event selection to compare what produces the best statistics.

In table 2 we can see comparison between different muon fuzz and event selections. In

this table, we compare results between using muonfuzz1 only and using muonfuzz1 and

muonfuzz2 together. Also, the leftmost column of the table indicates the type of event

selection being used. ’All’ here means that we have included all events in our sample.

Simlarly, fuzz < 40 MeV means we have inlcuded only events with muon fuzz < 40 MeV

in our sample.

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Now, when we compare our means and RMS between mufuzz1 and muonfuzz1 + mu-

fuzz2, we can see that both of them go down as we include both muonfuzzs in our

analysis. This was predicted in our previous analysis since including muonfuzz2 is likely

to include more events with neutron, it improves our statistics. Even though a larger

area of muon fuzz results in including more hadronic events, it eventually reduces the

RMS because we have more neutron events in muonfuzz area than hadronic events.

Secondly, from comparing the RMS values of different event selection, we can see that

as we select events with lower muon fuzz our resolution gets better. And so, we get

the best values of RMS when event selection criteria is Fuzz Energy < 5 MeV. But, the

problem with using such an extreme event selection is that we will lose a lot of events.

So, if one wants to improve both the statistics and keep most of the data one must

look for some event selection criteria in between. For example going from our current

settings to using muonfuzz1 + muonfuzz2 with only events with fuzz < 30 MeV will

improve our RMS by almost 6% without much loss of events. So, we do recommend

producing a cross section selecting only the low muon fuzz high resolution events and

compare the results with our full sample ones.

Now, we turn our interest individually to ECAL and tracker region. The mufuzz quan-

tities are saved separately for the tracker, ECAL, and HCAL regions, so we can enlarge

just the ECAL region. We used a special mufuzz definition, using mufuzz1 for all region

and using mufuzz2 only for either ECAL or tracker region. The mean and RMS are

shown in table A3 and table A4. As can be seen, the best improvements come when we

use mufuzz2 only in the ECAL region.

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Table A.3: A comparison between using muonfuzz1 and

muonfuzz1 + muonfuzz2

Mufuzz1 Mufuzz1 + Mufuzz2

Mean RMS Mean RMS

All 0.219 0.670 0.140 0.642

Fuzz <5 MeV 0.171 0.640 0.968 0.610

Table A.4: A comparison between using muonfuzz1 in

Tracker and ECAL region for 4 < Eν < 7 GeV, 1000 < q3

< 1200MeV

ECAL Tracker

Mean RMS Mean RMS

All 0.164 0.65 0.195 0.6623

Fuzz <5 MeV 0.112 0.615 0.1368 0.623

A.8 Recoil Plots and Muon Fuzz

We will end our technical note by connecting our muon fuzz analysis with low recoil

analysis. In figure A.14 and figure A.15, we made distribution plots for 0 < q3< 400MeV

and 7 < Eν < 20GeV. In these plots the red lines represent all events with our current

muon fuzz settings. In Figure A.14, the blue dots on the left plot takes into account

events with fuzz energy < 40MeV and the right plot takes into consideration events

with fuzz energy < 5MeV (which is the best resolution sub sample) using muonfuzz1

definition. Similarly, in figure A.15, the blue dots takes events with fuzz energy <

40MeV but uses muonfuzz1 and muonfuzz2 to define fuzz energy.

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Figure A.14: Recoil plot comparing muonfuzz1 for all events and muonfuzz1 for eventswith fuzz energy < 40 MeV (left plot), fuzz energy < 5 MeV. Double ratio for the leftplot goes up indicating lower muon fuzz events have a lower data/MC ratio. Also, thefirst bin has a slight interesting dip which shows up in various other works

The effect of the selection is easier to see with the double ratio. If the MC described the

data perfectly for both the whole and the sub sample, the double ratio would be flat

at 1.0, up to statistical fluctuations. But, events with greater than 40 MeV is such a

small part of the sample, no change in the double ratio is noticeable. In the right plot,

the double ratio is shifted by 5%, indicating the modeling of the sub sample is different

than the whole. In the previous section we have shown that events with fuzz < 5 MeV

has a better resolution than the rest of the sample and the 5% change most probably

comes from the mismodeling of overlay, charged hadron, neutron activity.

One further interesting point to note in here is that there is also a distortion in the first

bin relative to the rest of the distribution. This distortion has been reported in other

works as well. The reason for such distortion is has not yet been well reported and as

shown in the figures here, muon fuzz could be a potential reason for such deformation

[46].

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Figure A.15: Recoil plot comparing muonfuzz1 for all events and muon-fuzz1+muonfuzz2 for events with fuzz energy < 40 MeV . The first bin has a slightinteresting dip here as well

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