Download File

A Deep-Learning-Based a` CCQE Selection for Searches Beyond the Standard Model with MicroBooNE

Davio Cianci

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

Davio Cianci

A Deep-Learning-Based a` CCQE Selection for Searches Beyond the Standard Model with MicroBooNE Davio Cianci

The anomalous Low Energy Excess (LEE) of electron neutrinos and antineutrinos in MiniBooNE has inspired both theories and entire experiments to probe the heart of its mystery. One such experiment is MicroBooNE. This dissertation presents an important facet of its LEE investigation: how a powerful systematic can be levied on this signal through parallel study of a highly correlated channel in muon neutrinos. This constraint serves to strengthen MicroBooNE’s ability to conﬁrm or validate the cause of the LEE and will lay the groundwork for future oscillation experiments in Liquid Argon Time Projection Chamber (LArTPC) detector experiments like SBN and DUNE. In addition, this muon channel can be used to test oscillations directly, demonstrated through the world’s ﬁrst a` disappearance search with LArTPC data. Table of Contents

List of Figures ...... vii

List of Tables ...... xxiv

Acknowledgments ...... xxvii

Prologue ...... 1

I Introductions 2

Chapter 1: Neutrinos ...... 3

1.1 A Strange Position within the Standard Model ...... 3

1.1.1 The Standard Model of Particle Physics ...... 3

1.1.2 How Neutrinos Fit In ...... 5

1.2 Neutrino Oscillation Formalism ...... 6

1.3 Leading Experimental Constraints ...... 9

1.4 eV Scale Neutrino Masses ...... 11

Chapter 2: The MicroBooNE Experiment ...... 14

2.1 The BNB ...... 15

2.1.1 The Proton Beam ...... 15

i 2.1.2 Proton Target and Focusing Horn ...... 16

2.1.3 Beam Composition and Flux Uncertainty ...... 18

2.2 The MicroBooNE Detector ...... 19

2.2.1 The Time Projection Chamber ...... 20

2.2.2 The Optical System ...... 23

2.2.3 Triggering ...... 23

2.2.4 The Readout ...... 24

2.2.5 2D Deconvolution ...... 26

2.3 Neutrino Interactions and Their Signatures in MicroBooNE ...... 28

Chapter 3: On Notation ...... 33

II Rising Action 34

Chapter 4: Sterile Neutrinos ...... 35

4.1 What We Talk About When We Talk About the LEE ...... 35

4.1.1 What has MiniBooNE Measured? ...... 35

4.1.2 Limits of Cherenkov technology ...... 36

4.2 MiniBooNE’s eLEE and the Sterile Neutrino Hypothesis ...... 38

4.2.1 Further Extending the Extended Standard Model ...... 38

4.2.2 Oscillation Probabilities ...... 40

4.2.3 Predicting an Oscillated Spectrum ...... 43

4.2.4 Predicting 3+1 a` Disappearance in MicroBooNE ...... 45

4.2.5 Drawing Limits and Conﬁdence Intervals ...... 46

4.2.6 Sensitivities ...... 50

ii 4.3 Sterile Neutrino Compatibility with Global Data ...... 50

4.3.1 a4/a¯4 Appearance Experiments ...... 53

4.3.2 a4/a¯4 Disappearance Experiments ...... 55

4.3.3 a`/a¯` Disappearance Experiments ...... 58

4.3.4 Global Short Baseline Data ...... 60

4.3.5 Tensions and Limitations ...... 62

4.4 The Future of Global Fit Analyses ...... 66

4.4.1 The 2016 Global Fit ...... 66

4.4.2 Coverage Plots ...... 70

Chapter 5: A Deep Learning-Based LEE Search with MicroBooNE ...... 74

5.1 The MicroBooNE Path ...... 74

5.2 DL LEE Philosophy and Reconstruction ...... 77

5.2.1 The DL Reconstruction Strategy ...... 77

5.2.2 Wire-Cell Cosmic Tagging and Semantic Segmentation ...... 79

5.2.3 3D Muon Neutrino Vertex Finding and Reconstruction ...... 81

5.3 Quantifying the eLEE Measurement ...... 82

5.3.1 GENIE Models ...... 82

5.3.2 A Simple LEE Model ...... 83

5.3.3 A Hypothesis Test for the LEE ...... 85

5.3.4 Motivation for the 1`1? Constraint ...... 87

III Conﬂict 91

Chapter 6: The 1`1? Selection ...... 92

iii 6.1 Pre-Selection ...... 94

6.1.1 Data and Simulation Samples ...... 94

6.1.2 Two-Body Scattering, Kinematics, and Boosting ...... 96

6.1.3 The Pre-Selection Cuts ...... 100

6.1.4 What’s in a Plot? ...... 104

6.1.5 Truth labeling ...... 105

6.1.6 Training the BDTs ...... 109

6.1.7 MPID ...... 112

6.2 Selection ...... 113

6.2.1 Applying the BDT Weights ...... 113

6.2.2 MPID Cut ...... 115

6.2.3 Final selection ...... 116

6.3 Systematics ...... 118

6.3.1 Reweightable Systematics ...... 121

6.3.2 Detector Systematics ...... 122

IV Resolution 124

Chapter 7: Quantifying Final Uncertainty ...... 125

7.1 Relative Systematic Contributions to Uncertainty ...... 125

7.1.1 Theoretical Limit on a` Disappearance Sensitivity ...... 127

7.2 Strength of the a` Constraint ...... 128

Chapter 8: Quantifying MicroBooNE’s Sensitivity to a MiniBooNE eLEE ...... 133

8.1 SBNﬁt ...... 133

iv 8.1.1 Drawing Pseudo-Experiments with SBNﬁt ...... 134

8.2 Method of Frequentist Hypothesis Testing with SBNﬁt ...... 135

8.3 MicroBooNE Sensitivity to a MiniBooNE eLEE ...... 137

8.4 Further Extrapolation ...... 138

Chapter 9: Measurement of a` Disappearance with MicroBooNE ...... 140

9.1 Calculating Sensitivity ...... 140

9.1.1 A Parallel Shape-Only Analysis ...... 142

9.2 Applying a Frequentist Correction ...... 143

9.3 Fitting to Data ...... 147

V Denouement 154

Chapter 10: Reﬂections On the Long Way Home ...... 155

Epilogue ...... 157

References ...... 158

Appendix A: Sterile Neutrino Fits to Global Data ...... 165

A.1 IceCube 2017 Oscillation Result Reproduction ...... 165

A.1.1 Constructing a Predicted Spectrum ...... 167

A.2 DANSS 2018 Oscillation Result Reproduction ...... 170

A.2.1 Calculate Ratio of Predicted Events in True Positron Energy ...... 171

A.2.2 Smear Ratios into Reconstructed Positron Energy ...... 172

A.2.3 Calculate j2 ...... 172

v A.3 NEOS 2017 Oscillation Result ...... 173

A.3.1 Calculate Predicted Event rates in NEOS and Daya Bay ...... 174

A.3.2 Calculate the Ratio of the Event Rates in NEOS and Daya Bay ...... 176

A.4 Comparisons of Fits with Reference Examples ...... 177

Appendix B: 1`1? Selection Asides ...... 180

B.1 Boosted Decision Tree Primer ...... 180

B.2 Selection Code Release ...... 181

B.3 U) Study ...... 182

B.4 Inter-Run Compatibility in BDT Selection ...... 186

B.4.1 Compatibility Tests ...... 187

B.4.2 Consequences of Incompatibility ...... 189

B.4.3 Conclusions ...... 189

B.5 BDT Input Distributions At Preselection ...... 191

B.6 1`1? Distributions After Selection ...... 199

Appendix C: a` Disappearance Validation ...... 215

C.1 Signal Injection Closure Tests ...... 215

vi List of Figures

1.1 Table of the bosons and three generations of fermions that currently make up the Standard Model with their quantum properties [3]...... 4 1.2 Feynman diagrams for charged-current neutrino interactions (left) and neutral current neutrino interactions (right). In the diagrams, = is a neutron, ? is a proton, # is either nucleon, and ; is any lepton. The shaded circles on each diagram account for nuclear interactions, which may complicate the final-state outputs. Time progresses from left to right in each diagram...... 6 1.3 A cartoon demonstrating how the three SM neutrino mass states (colored bars) are divided between three flavor components (individual color components). Neither the flavor compositions nor mass positions are to scale, but up-to-date measurements can be found in [4] ...... 7 1.4 Plot of the excess of electron antineutrinos observed by LSND (dots). The SM prediction is given as the sum of the red and green histograms. The blue hatched histograms illustrate hypothetical predictions beyond the ESM to attempt to explain the excess [18]...... 11 1.5 Plot of MiniBooNE’s observed (dots) versus their predicted (stacked histogram) counts of electron neutrinos and antineutrinos. A clear excess is visible in the leftmost four bins in both neutrino modes [19]...... 12 1.6 antineutrino observed excesses from both MiniBooNE and LSND overlaid on a common axis. The dotted and dashed lines illustrate hypothetical, oscillatory distributions to provoke curiosity [19, 18]...... 13

vii 2.1 An aerial illustration of Fermilab’s Neutrino Campus where one can see the trajectory of the Booster Neutrino Beam through several detectors, including Mini- BooNE and MicroBooNE [23]...... 14 2.2 A cartoon of the several stages of the BNB, from protons to mesons to mostly neutrinos...... 16 2.3 A schematic illustration of the focusing horn used by MiniBooNE and Micro- BooNE. The proton beam is incident on the left with respect to this diagram. The target is inserted rod-first along the central axis from the left. An electric current travels across the inner surface of the horn outward towards the front and loops over to return back along the upper surface (during neutrino-mode, and with opposite direction for antineutrinos), generating a toroidal magnetic field...... 17 2.4 A cartoon of what happens to the proton beam when it enters the target hall and hits the beryllium target...... 17 2.5 Prediction of the neutrino components of the BNB flux in neutrino-mode, modeled with Geant-4 simulation [26, 24]...... 18 2.6 Prediction of the BNB flux in neutrino-mode, modeled with Geant-4 simulation, of

a` (left) and a4 (right) [24]...... 19 2.7 The fully-constructed MicroBooNE cryostat on its way from the Fermilab Detector Assembly Building (DAB) to its home in the Liquid Argon Test Facility (LArTF). Photo by Cindy Arnold, Fermilab...... 20 2.8 Schematic model of TPC (left) alongside the fully-constructed TPC, ready for insertion into the cryostat (right) with overlaid axes. The MicroBooNE TPC measures 10 meters along the z axis, 2.6 meters along the x axis, and 2.3 meters along the y axis...... 21

viii 2.9 Cartoon of the detection mechanism of a LArTPC. Incoming neutrinos interact with an argon nucleus, releasing charged particles that ionize argon atoms along their path. Electrons drift along the electric ﬁeld towards the wire planes (on the right), where they leave a charge signal. The waveforms of these charges on the wire grid can be used to determine coordinates of a neutrino interaction within the detector...... 22 2.10 Schematic of the components of MicroBooNE’s optical system. Importantly, one can see how PMTs ("optical units") are spaced across the anode plane...... 23 2.11 Detailed diagram of the full TPC and PMT readout systems for MicroBooNE. Ana- log signals are measured in the TPC on the left and transported via long cables to the electronics and readout systems presented on the right for digitization (done in hardware) and event assembly...... 25 2.12 Three LArTPC event displays illustrating the stages of signal reconstruction across a single wire plane. On the left, we have several messy particle signatures; in the center, we have applied 1D deconvolution to clean up the tracks; and on the right, we see the well-deﬁned, high-contrast signature that analyzers will pick apart [33]. 28 2.13 Sample event display of a candidate neutrino event observed in MicroBooNE across the collection plane (top) and two induction planes (middle, bottom) [34]...... 30

2.14 A sample event display of BNB data where we see a a` candidate event, identiﬁed by its long muon track and two stubby proton tracks. One can see a higher-energy cosmic muon crossing through the upper-right corner...... 31 2.15 A sample event display data from Fermilab’s Neutrino Main Injector (NuMI) beam. It is a bit harder to identify the tracks in this event display because many may have arisen from ﬁnal-state-interactions within the argon nucleus, but the centerpiece is a very clear look at an electromagnetic shower signature...... 31

ix 2.16 An event display of BNB data that appears to feature a a` NC event. At the 1 o’clock and 7 o’clock positions relative to the central vertex, one sees two electromagnetic shower-like clusters. Neither are connected to the vertex, but both point back to a common origin where a c0 may have decayed...... 32

4.1 The MiniBooNE a4 distribution from their most recent, 2020 paper, comprising over three times as much data as the initial result while still exhibiting a strong excess in the four lowest energy bins [20]...... 36 4.2 Cartoon examples of potential particle signatures as would be detected in a Cherenkov- based detector. On the top, one sees a clean, ﬁlled circle, caused by a muon track, and on the bottom, one sees the jagged-edged circle that an EM event would leave [3]. 37 4.3 Diagram of a 3+1 neutrino model, expanding upon Figure 1.3 with the addition of

one new neutrino ﬂavor state aB, and one new neutrino mass state a4...... 39 4.4 A cartoon demonstrating an example of sterile neutrino appearance and disappear-

ance across both a` (left) and a4 (right) channels. In this example, the solid lines show the energy distribution in each channel, assuming there are no sterile neutrino oscillations, while the dotted and dashed lines illustrate exaggerated oscillation scenarios...... 43

4.5 A cartoon of the method by which one builds a a` disappearance spectrum from an initial prediction under the 3a hypothesis. The center plot contains identical events to those in the left spectrum, where each individual event was scaled by the

a` disappearance probability...... 46 4.6 A fake (though qualitatively representative) relation between Δ j2 and our fit parameters. This plot features a set of parameters, indicated on the horizontal axis by the best fit point, which most-closely model the underlying data. By defining a Δ j2 j2 threshold, , one can divide the parameter space into allowed (better agreement) and excluded (worse agreement) regions...... 48

x 4.7 Fake sterile neutrino oscillation contours to illustrate allowed regions of 90% and 99% conﬁdence (left) and an exclusion limit of 95% conﬁdence (right). Oscillation models with parameters lying within the allowed regions are favored by data, while those to the right of the exclusion limit are disfavored...... 49

4.8 A collage of individual 3+1 a4 appearance results from each experiment included within this dissertation’s global ﬁt, and the allowed region of a combined ﬁt over all of those experiments together...... 54

4.9 A collage of individual 3+1 a4 disappearance results from each experiment included within this dissertation’s global ﬁt, and the exclusion contour of a combined ﬁt over all of those experiments together...... 56

4.10 A collage of individual 3+1 a` disappearance results from each experiment included within this dissertation’s global fit, and the exclusion contour of a combined fit over all of those experiments together...... 59 4.11 Two dimensional projections of a globally allowed region of 3+1 sterile neutrino oscillation, determined via a combined fit of each aforementioned data set, across

three oscillation parametrizations: a4 appearance (left), a` disappearance (middle),

a4 disappearance (right). For convenient comparison and reference, the data set that dominates each channel is overlaid...... 61

4.12 MiniBooNE data [35] (black dots) compared with three predictions: that of a 3a hypothesis, that of the best fit across 3+1 oscillations to MiniBooNE data, and that of the best fit across 3+1 oscillations to data from all experiments in the study. While both best fits demonstrate better agreement than the null hypothesis, neither adequately resolves disagreement at low energy...... 63

4.13 The allowed region of a combined ﬁt of a4 appearance data sets to a 3+1 sterile neutrino hypothesis in contrast with the exclusion region of all disappearance data sets combined together...... 64

xi 4.14 The allowed region of a combined ﬁt of MiniBooNE and LSND data sets to a 3+1 sterile neutrino hypothesis in contrast with the exclusion region of all other included data sets combined together...... 65 4.15 Globally allowed regions for 3+1 (left) and 3+2 (right) sterile neutrino oscillation as derived in [39]. The 3+1 allowed region is overlaid with contemporary ﬁts to comparable SBL data sets. Note that the 3+2 oscillation plot is marginalized so as <2 <2 to only portray Δ 41 vs Δ 51...... 67 4.16 Globally allowed region for 3+3 sterile neutrino oscillation as derived in [39]. The 12 parameters of 3+3 SBL oscillation have been marginalized in two ways, show- <2 <2 <2 <2 ing Δ 41 vs Δ 51 (left) and Δ 41 vs Δ 61 (right)...... 67 4.17 A demonstration of our coverage concept in the 2-dimensional case, showing an allowed region on the left, a sensitivity contour in the middle, and the resulting overlap on the right. This can be extrapolated to any number of dimensions, where the allowed region is replaced by a corresponding allowed hypervolume, and the contour is replaced by a sensitivity hypersurface...... 70 4.18 Coverage of the 2016 globally allowed region of 3+1 parameter space (Figure 4.15, left) by SBN’s sensitivity under four different oscillation study methods [39]. . . . 71 4.19 Coverage of the 2016 globally allowed region of 3+2 (Figure 4.15, right) and 3+3 (Figure 4.16) parameter space by SBN’s sensitivity under four different oscillation study methods [39]...... 73

5.1 A flowchart of MicroBooNE’s blind analysis strategy, designed to both eliminate analytical bias and to provide a clear road map for achieving a final result...... 75 5.2 A flowchart of the DL LEE analysis reconstruction and selection strategy. Deep learning techniques are utilized in the Sparse SSNet and 3D reconstruction stages. 78

xii 5.3 A demonstration of SparseSSNet’s classification ability as used on MC simulation. The plot on the left illustrates the TPC image (what is observed by the detector), the center plot is the true neutrino event, as provided by the Geant4 simulation toolkit [26], and the right plot is the pixel-level classification output by SparseSS- Net [68]...... 80 5.4 A TPC image of a shower event as observed by MicroBooNE (left), the true neutrino event signature from our simulation (center), and the SparseSSNet classification output (right) [68]...... 81

5.5 Differential cross section of CC 0c in MiniBooNE data compared to GENIE predictions [70]...... 83 5.6 On the upper plot, we see the the unfolded MiniBooNE LEE observation overlaid

with the unfolded MiniBooNE MC under a a4 CCQE assumption, plotted across true neutrino energy. The lower plot illustrates the ratio of the LEE to the MC in the upper, giving the unfolded model used in this dissertation. An isolated version of this lower plot is shown in Figure 5.7 [71]...... 84 5.7 The unfolded MiniBooNE eLEE model used by MicroBooNE and in this dissertation, given as the ratio of the unfolded MiniBooNE LEE observation to the un-

folded MiniBooNE MC under a a4 CCQE assumption. A cross-check using Sin- gular Value Decomposition is overlaid alongside the D’Agostini unfolding [71]...... 85

6.1 Toy examples of the trade-off between efﬁciency and purity, as demonstrated with

an arbitrary cut. On the left plot, one sees 83% a` CCQE purity (in blue), but a meagre 185 events. On the right plot, one sees the efﬁciency increased massively

with 3349 a` CCQE events, but less than 50% purity...... 93

xiii 6.2 A ﬂow chart of the stages of selection. Pre-selection is broken down into three component parts based on their utility to the process: cleaning the distribution, pro-

viding orthogonality from the 141? selection, and ensuring only one reconstructed vertex per neutrino candidate event...... 94 6.3 Diagram of MicroBooNE coordinate system. The anode and three wire planes are positioned at x=0, with the cathode at the opposite end...... 97 6.4 Distribution of reconstructed neutrino energy after pre-selection. Note that systematic uncertainties were not calculated for events outside the ﬁnal selection, so all pre-selection plots feature statistical error only...... 104 6.5 Two plots containing equivalent distributions under different labeling schemes. In

the lower plot, our a` CCQE Signal is narrowed to only include events with good positional and energy reconstruction. Comparing the plots, we see that our main backgrounds are not non-CCQE events, but rather events that are not reconstructed adequately...... 106 6.6 An illustration of 2-body elastic scattering of particles with pre-collision four-

momenta ?1 and ?2...... 107 6.7 Relative importance of each feature used in training both the Run 1 BDT (left) and Run 3 BDT (right). A variable with a higher importance was chosen more often by the training algorithm for cuts and can be interpreted as a particularly strong discriminator between signal and background...... 111 6.8 Distribution of the ratio of reconstructed neutrino transverse momentum to total reconstructed neutrino momentum, measured in practice as the sum of the corre-

sponding muon and proton momenta under 1`1? assumption...... 112 6.9 Two plots demonstrating the separation of the bkgBDT variable between signal (in blue) and various backgrounds (in orange, red, and green). The left plot considers a BDT trained on Run 1 MC (to score Run 1 MC), and the right is the same for Run 3 MC...... 113

xiv 6.10 Plot of the relationship between well-reconstructed CCQE purity and BDT cut efficiency for different cut strengths -. Efficiency is defined specifically for the

BDT cut such that we have 100% efﬁciency at - = 1.0 where all preselection and non-BDT selection cuts have been applied. The horizontal red lines contextualize

the efﬁciency by indicating where along the orange line 1000, 2000, or 3000 a` CCQE events will pass ﬁnal selection (in 6.8 × 1020 POT)...... 114 6.11 Preliminary plot of sensitivity to the LEE using the 1`1? sample to constrain uncertainty. One should note that the values shown on the vertical sensitivity axis should be regarded as incomplete, this plot having been created before elements of

the 141? selection and detector systematics were ﬁnalized and comprising a fraction of available data. Relevant to study is that the BDT cut value between 0.3 and 0.4 proceeds a plateau of sensitivity increase. Plot provided by Lauren Yates (MIT). 115 6.12 The reconstructed neutrino energy distribution after pre-selection and the BDT score cut, shown with both labeling schemes to highlight the source of a strange bump below reconstructed energy of 400 MeV...... 116 6.13 Maximum proton MPID score for events of reconstructed neutrino energy less than 400 MeV...... 116

6.14 Reconstructed a` energy after ﬁnal selection, overlaid with all currently-available data...... 117

6.15 Reconstructed a` energy after ﬁnal selection, overlaid with all currently-available data, but our reconstruction-based labeling scheme has been pared back to a purely interaction-based scheme...... 118

6.16 Combined fractional covariance matrix of a` candidate event counts, binned ac-

cording to reconstructed a` energy, including all reweightable systematics. Note: the empty 200-250 MeV bin is omitted, giving us a 19 × 19 matrix...... 122

xv 6.17 Combined fractional covariance matrix for reconstructed a` energy, including all detector systematics. Note: the empty 200-250 MeV bin is omitted, giving us a

19 × 19 matrix...... 123

7.1 Plot of the fractional uncertainty in each bin of our 1`1? distribution, divided into contributions from each collection of systematics. Systematic groups are plotted as dashed lines, but note that each contribution is overlaid, not stacked. The solid lines combine individual systematics via a sum in quadrature to give the total fractional uncertainty in each bin...... 125

20 7.2 Plot of selected 141? events in 6.8 × 10 POT with no a` constraint. The ESM prediction is given as a magenta histogram and the unfolded LEE signal is outlined

in blue. The vertical axis illustrates the state of a4 statistics in MicroBooNE. . . . . 128 7.3 The complete fractional covariance matrix for the combined 141? (upper-left) and 1`1? (lower-right) selections. The two selections are visually separated by red bars, with the upper-right and lower-left quadrants containing the correlations be-

tween the 1`1? and 141? selections. Note that there are 10 bins in reconstructed

a4 energy (each 100 MeV wide), and 19 bins in reconstructed a` energy (each 50 MeV wide)...... 130

7.4 The complete, correlation matrix for the combined 141? (upper-left) and 1`1? (lower-right) selections. The two selections are visually separated by red bars, with the upper-right and lower-left quadrants containing the correlations between

the 1`1? and 141? selections. Note that there are 10 bins in reconstructed a4

energy (each 100 MeV wide), and 19 bins in reconstructed a` energy (each 50 MeV wide)...... 131

7.5 Plot of selected 141? events in 6.8 × 1020 POT. The ESM prediction is given as a magenta histogram and the unfolded LEE signal is outlined in blue. In this exam-

ple, the a` CCQE selection (as presented in Chapter 6) has been used to constrain

both the a4 prediction and the uncertainty on that prediction...... 132

xvi 8.1 Plot of Δ j2 PDFs of our two-hypotheses (null versus an LEE hypothesis) with the

1`1? constraint applied. The red histogram is a PDF for observing Δ j2 under a true null hypothesis, and the blue is a PDF for observing Δ j2 under a true LEE

hypothesis. Vertical lines are drawn to indicate observations of median, ±1f, and ±2f likelihoods under the LEE hypothesis...... 136 8.2 Plot of Δ j2 PDFs of our two-hypotheses (null versus an LEE hypothesis) for only

the 141? selection. The red histogram is a PDF for observing Δ j2 under a true null hypothesis, and the blue is a PDF for observing Δ j2 under a true LEE hypoth-

esis. Vertical lines are drawn to indicate observations of median, ±1f, and ±2f likelihoods under the LEE hypothesis...... 139 8.3 Plot of Δ j2 PDFs for our two hypothesis (null versus an LEE hypothesis) with

MC scaled to account for Runs 1-5 data, 1.3 × 1021 POT. On the left, only a 141? selection is used for the LEE ﬁt, and on the right, we add the 1`1? constraint [81]. 139

2 9.1 j surface across the parameter space of a` disappearance under 3+1 sterile neutrino oscillations. At each point on the grid, the coordinates deﬁne a sterile neutrino hypothesis, which is compared with the null spectrum. The color on the grid is given by the natural log of the j2. Two points are selected to demonstrate how the predicted spectrum (in blue) varies across parameter space versus a constant, no-osc hypothesis (orange points)...... 141 9.2 On the left, we see the full systematic covariance matrix used for the DL LEE

1`1? analysis including all reweightable and detector systematic components and scaled to the null hypothesis. On the right, we have removed the normalization component from each individual element according to Equation 9.3...... 143

xvii 9.3 MicroBooNE’s 90% sensitivity contours for a shape-only (SO) and shape-and-rate (S+R) global scan. The contours are drawn assuming a Δ j2 distribution with 2

degrees of freedom. MiniBooNE’s a` disappearance sensitivity is overlaid [56],

2 2 as well as a vertical line at sin 2\`` = 0.35, the predicted high-Δ< sensitivity calculated from our normalization uncertainty...... 144

9.4 Distribution of Δ j2 across N = 1000 pseudo-experiments for the null, no-osc,

2 hypothesis as %) , shown in red. A j distribution with 2 degrees of freedom is overlaid in blue. Vertical lines are drawn for the 90% CL critical j2 of each distribution, such that 90% of the distribution is to the left of each line. This distribution is the result of a shape-and-rate ﬁt...... 147

9.5 Distribution of Δ j2 across N = 1000 pseudo-experiments for the null, no-osc,

2 hypothesis as %) , shown in red. A j distribution with 1 degree of freedom is overlaid in blue. Vertical lines are drawn for the 90% CL critical j2 of each distribution, such that 90% of the distribution is to the left of each line. This distribution is the result of a shape-only ﬁt...... 148

9.6 The critical chi-squared for U = 0.1 across every point in our parameter space. The j2 left plot displays the raw value, while the right shows the fractional difference j2 in from a 2-DoF scenario...... 149 9.7 The critical chi-squared for U = 0.1 across every point in our parameter space j2 under a shape-only analysis. The left plot displays the raw value, while the j2 right shows the fractional difference in from a 2-DoF scenario...... 150 9.8 On the left, MicroBooNE’s 90% sensitivity contours for a shape-only (SO) and shape-and-rate (S+R) global scan with the frequentist correction applied. Mini-

BooNE’s a` disappearance sensitivity is overlaid, which was also drawn with a frequentist contour [56]. On the right, we see the MicroBooNE S+R sensitivity with and without the frequentist correction...... 151

xviii 9.9 MicroBooNE’s 90% limit for a shape-and-rate (S+R) ﬁt to data and a 90% allowed region resulting from a shape-only (SO) ﬁt to data. Both the limit and allowed region are constructed using the Wilks’ Theorem (2 degrees of freedom) approxi-

mation. MiniBooNE’s a` disappearance limit is overlaid, drawn with a frequentist contour [56]. Notably, the SO ﬁt rejects the null hypothesis at 90% conﬁdence. . . 152 9.10 MicroBooNE’s 90% limits for a shape-and-rate (S+R, in light blue) and shape-only

(SO, in dark blue) frequentist fits to data. MiniBooNE’s a` disappearance limit is overlaid, also drawn with a frequentist contour [56]. Where the non-frequentist- corrected iteration of this plot (as seen in Figure 9.9) illustrated an allowed region for the SO fit, the frequentist correction does not. Despite a Δ j2 of 6.38 favoring the SO best fit point, the critical j2 diverges greatly from a 2 DoF assumption. This effect is highlighted in Table 9.2...... 153

A.1 A 2D histogram of IceCube’s recorded data from [59]. Note that the axes are marked according to bin number, but the horizontal axis, cosine of the zenith angle, is spaced linearly from 0.24 to -1.0, and the vertical axis, reconstructed neutrino energy, is spaced linearly from 400 GeV to 20 TeV. The bin contents are printed in text at the center of each bin for convenience...... 167 A.2 The 90% global exclusion limit as published in [59], overlaid with that of my reproduction...... 169

A.3 Ratio of observed a¯4 events in the DANSS detector in the far position (down) to those observed in the near position (up) relative to the reactor source [52]...... 170 A.4 On the left, positron energy resolution in the DANSS detector, from [53]. The resolutions were used as Gaussian widths, illustrated in the right figure (before normalization) and used to form an energy smearing matrix...... 172 A.5 The predicted best-fit spectrum as published by the DANSS collaboration (in red) and my reproducted best-fit prediction (in yellow) against DANSS data[53]. . . . . 173

xix A.6 NEOS data, presented in two forms: b) the ratio of counted events in NEOS to that of a theoretical Herbert-Mueler-Vogel prediction, and c) the ratio of counted events in NEOS to the Daya-Bay unfolded spectrum [10]. Plot c) shows the data we will ﬁt to [47]...... 174

A.7 Published 90% global exclusion contours for a` disappearance under the 3+1 sterile neutrino oscillation model in MiniBooNE (left) and IceCube (right). Overlaid are the reproductions by the author of this dissertation as used in the global ﬁt analysis of Section 4.3...... 177

A.8 Published global exclusion contours and allowed regions for a4 appearance under the 3+1 sterile neutrino oscillation model in LSND (upper-left), MiniBooNE (upper-right), KARMEN (mid-left), MiniBooNE using the NuMI Beam (mid-right),

and NOMAD (lower-left). Published global exclusion contour for a4 disappearance under the 3+1 sterile neutrino oscillation model in Gallex and SAGE (lower- right). Overlaid are the reproductions by the author of this dissertation as used in the global ﬁt analysis of Section 4.3...... 178

A.9 Published global exclusion contours and allowed regions for a4 disappearance under the 3+1 sterile neutrino oscillation model in NEOS (upper-left), DANSS (upper-right), the LSND and KARMEN joint cross-section analysis (mid-left), and

Bugey (mid-right).Published global exclusion contour for a` disappearance under the 3+1 sterile neutrino oscillation model in CDHS (lower-left) and CCFR (lower- right). Overlaid are the reproductions by the author of this dissertation as used in the global ﬁt analysis of Section 4.3...... 179

xx B.1 A cartoon of two decision trees comprising an ensemble. The yellow boxes contain the conditions upon which the tree must decide, with possible options shown as arrows. One can see how a tree with more than two outputs can still be reduced to two binary choices. The end of each decision path displays the score awarded to any observable that reaches that point. The total score of an observable from a tree ensemble is the sum of scores received from each tree...... 180

19 B.2 An out-of-date plot of U) , using 5 × 10 POT of Run 1 data and with box cuts applied to shrink cosmic contributions for a clearer view of the prediction (the stacked histogram, comprising only Run 1 MC)...... 182

B.3 A diagnostic plot created by the MINERVA collaboration comparing U) (some-

times, as in this case, labeled XU) ) in data with a prediction using GENIE truth variables. The dark green prediction shows a distribution of events with no FSIs included, while other colors include FSIs and gain strong, asymmetric shapes [94]. 183

B.4 A cartoon rendition of a 1`1? event with a crossing, un-tagged cosmic track. The left image shows the three tracks in "truth" as they are seen by the detector, and the right image portrays a simulated output of our reconstruction algorithm. In this example, the tracker follows the wrong track after reaching an intersection and the resulting track length will be substantially truncated...... 185

B.5 Distribution of U) with reconstruction-based signal deﬁnition using Runs 1-3 Data and MC corresponding to 6.8 × 1020 POT...... 186 B.6 The reconstructed neutrino energy of the BNB Overlay MC samples for each of Runs 1-3 after selection, overlaid, with j2 comparisons between each pair and the ratios of Runs 1 and 2 over Run 3 on the lower frame...... 187 B.7 The reconstructed neutrino energy of the 1m1p ﬁltered data for each of Runs 1-3 after selection, overlaid, with j2 comparisons between each pair and the ratios of Runs 1 and 2 over Run 3 on the lower frame...... 188

xxi B.8 The error of the training (orange) and test (blue) samples for the Run 3 BDT with increasing iterations. The pale, dashed line marks the number of iterations used for the ﬁnal training...... 189 B.9 Difference between proton and lepton reconstructed azimuths...... 191 B.10 Reconstructed neutrino energy using range-based deﬁnition (See Section 6.1.2), and the transverse momentum contribution with respect to a perfectly forward-

going interaction (q) )...... 192

B.11 Transverse momentum asymmetry with respect to the lepton (U) ) and the ratio of reconstructed transverse momentum to total neutrino momentum...... 193 B.12 Bjorken’s X and Y scalings, both bossted into the nucleon rest frame...... 194 B.13 The square of the momentum-transfer four-vector and QE Consistency (also labeled as Δ& in Section 6.1.2...... 195 B.14 The ﬁrst and the beam-direction components of the reconstructed momentum- transfer four-vector...... 196 B.15 Lepton candidate azimuthal angle distribution and track length...... 197 B.16 Proton candidate angular distributions...... 198 B.17 Deposited charge (in ADC units) within ﬁve centimeters of the neutrino candidate vertex...... 199 B.18 The p-values of each of the following distributions plotted in descending order of Data-MC agreement...... 200

B.19 The transverse momentum asymmetry with respect to the lepton (U) ) and the transverse momentum contribution with respect to a perfectly forward-going interaction

(q) )...... 201 B.20 Bjorken’s X and Y scalings, boosted into the nucleon rest frame...... 202 B.21 Muon candidate angular distributions...... 203 B.22 Proton candidate angular distributions ...... 204

xxii B.23 More angular distributions, combining components of both particles to explore "forward-ness" of candidate neutrino events...... 205 B.24 QE Consistency (also labeled as Δ& in Section 6.1.2), boosted into the nucleon rest frame, and the square of the momentum-transfer four-vector...... 206 B.25 The ﬁrst and the beam-direction components of the reconstructed momentum- transfer vector...... 207 B.26 Reconstructed neutrino transverse momentum, and the ration of the transverse momentum to the total reconstructed momentum...... 208 B.27 Deposited charge (in ADC units) within ﬁve centimeters of the neutrino candidate vertex, and reconstructed neutrino energy...... 209 B.28 Maximum MPID score for the presence of an electron, and that of a photon across all three planes...... 210 B.29 Maximum MPID score for the presence of a pion, and that of a proton across all three planes...... 211 B.30 Maximum MPID score for the presence of a muon across all three planes, and the BDT scores. Note that a low BDT score corresponds to a very signal-like event. . . 212 B.31 Reconstructed vertex X and Y coordinates within the active TPC volume...... 213 B.32 Reconstructed vertex Z coordinate (note that the conspicuous dip in events at around 700 cm corresponds to a large region of dead wires in the detector)...... 214

C.1 MicroBooNE’s sensitivity to a` disappearance at 90% conﬁdence, with four points indicated where signal was injected for the crosschecks described in this chapter. Interpretations of the signal injection can be found in the text...... 216 C.2 90% globally allowed regions to MicroBooNE fake data. A central value was

2 constructed by injecting an oscillation signal with parameters B8= 2\`` = 0.8 and <2 2 Δ 41 = 2 eV and drawing 20 different pseudo-experiments with SBNfit. On each plot, the best fit point from a shape+rate fit is marked by a red star and the injected signal CV is marked by a black star...... 218

xxiii List of Tables

4.1 Experiments whose data sets comprise the presented global ﬁt to 3+1 sterile neutrino oscillation models, sub-divided by their channels of measurable oscillation...... 52

4.2 Table of a4 appearance experiments, the degrees of freedom each dataset contributes, the parameters of a best ﬁt to 3+1 sterile neutrino oscillation to each data set, and several j2 measurements explicitly detailed in the text...... 55

4.3 Table of a4 disappearance experiments, the degrees of freedom each dataset contributes, and several ﬁt measurements, each of which is explicitly detailed in the text...... 57

4.4 Table of a` disappearance experiments, the degrees of freedom each dataset contributes, and several ﬁt measurements, each of which is explicitly detailed in the text...... 59 4.5 Mass-squared splittings are presented in eV2, and CP-violating factors are given in

radians. The null hypothesis has a j2/3> 5 = 299.5/243...... 68

6.1 Table of the cut progression through each individual pre-selection substage in terms of candidate neutrino vertices and candidate neutrino events. To clarify, there may be multiple candidate vertices for a single event until the ﬁnal substage where du-

plicates are removed. The Δ% columns reﬂect the percent change in count from the previous row...... 103

xxiv 6.2 Monte Carlo event counts of our a` CCQE signal, our backgrounds, and the total of all events remaining after the pre-selection stage. Event counts are scaled to match

the POT of open Run 1-3 data, 6.8 × 1020 POT. Beside the signal and background counts, in parentheses, are each subsample’s fractional contribution to the total event count...... 104 6.3 Reconstructed neutrino event variables used for BDT training...... 110

6.4 Monte Carlo event counts of our well-reconstructed a` CCQE signal, our backgrounds, and the total of all events remaining after each selection stage. Event

counts are scaled to match the POT of open Run 1-3 data, 6.8 × 1020 POT. Beside the signal and background counts, in parentheses, are each subsample’s fractional

contribution to the total event count. Additionally, the Δ% column beside each event count records the fractional decrease in events of that subsample from the previous stage. One should note that these counts span all reconstructed energy and therefore will not match the counts in Figure 6.14, which are speciﬁc to the plotted energy region...... 119

6.5 Monte Carlo event counts of our GENIE true a` CCQE signal, our backgrounds, and the total of all events remaining after each selection stage. Event counts are

scaled to match the POT of open Run 1-3 data, 6.8 × 1020. Beside the signal and background counts, in parentheses, are each subsample’s fractional contribution to

the total event count. Additionally, the Δ% column beside each event count records the fractional decrease in events of that subsample from the previous stage...... 119

7.1 Contributions of each source of systematic uncertainty in MicroBooNE, as used to plot the reconstructed energy distribution in Figure 6.14, to normalization uncertainty. The shape component of each systematic is removed according to Equa- tion 7.1...... 126

xxv 9.1 This table lists the critical j2 as predicted by Wilks’ theorem (using a standard two- j2 j2 sided distribution with 2 degrees of freedom), and the same as derived from our Feldman-Cousins, shape-and-rate analysis at two different points in parameter

2 2 2 space: the Null, no-oscillation point (sin 2\`` = 0.01, Δ< = 0.014+ ) and the

2 2 2 best ﬁt point (sin 2\`` = 0.30, Δ< = 244+ )...... 149 9.2 This table lists the critical j2 as predicted by Wilks’ theorem (using a standard j2 j2 two-sided distribution with 2 degrees of freedom), and the same as derived from our Feldman-Cousins, shape-only analysis at two different points in param-

2 2 2 eter space: the Null, no-oscillation point (sin 2\`` = 0.01, Δ< = 0.014+ ) and

2 2 2 the best ﬁt point (sin 2\`` = 0.69, Δ< = 244+ )...... 150

xxvi Acknowledgements

By all rights, this dissertation should not have happened. It is not without the love, support, and motivation from so many that I was able to survive this program and put together this culmination of so many years of work. Thank you Georgia, for being the best advisor I could have asked for and for showing endless patience and support every step of the way. Had you not invited me to join you in Manchester, I may have abandoned Physics in 2015. Thank you to all my physicist friends from Manchester to New York to Batavia (and just about everywhere else) for accepting me in your ranks. Thanks to my family for trying their best, despite only understanding bits and pieces of what I’ve been doing for so many years. Thanks to my comrades in New York and Chicago. What I learned through selling communist newspapers, attending talks, and spending time with you helped set me on the path to organize for a better world, and gave me hope to keep proceeding through some very trying times. Thanks to those select friends and family members who have promised they would try to read this dissertation (bless their hearts) and who will reﬂect kindly upon this call-out (I hope), even if they make it no further than the Acknowledgements. Thanks to the small army of therapists who picked me up and kept me up, and to every friend, neurodivergent or otherwise, who supported this particular journey. Thanks to my two cats for making this dissertation so much harder to write by sitting on my keyboard and not letting me work: Jacob (AKA Prince Fart, AKA Thomas Mouth-us, AKA

xxvii Robert Ze-smack-is, AKA the Gray Menace), and Syd (AKA Orange Baby, AKA Sissy Space-Cadet, AKA George Without-a-Clue-ney). You are awful and the love I feel when you plop on my arm is immense beyond compare. Lastly, thanks to my fartner, Michelle. Much like a cosmic horror, words cannot describe what you mean to me and how much you’ve done. Unlike a cosmic horror, your mention is not a thinly veiled allegory for racial intolerance.

xxviii Prologue

This dissertation contains work conducted by the author and many collaborators from the years 2015 to 2021 AD on the topics of neutrino oscillation phenomenology and experimentation. Early chapters will provide context vital to the understanding and appreciation of the presented material, with final results saved for the end. The work is organized in a theatrical five-act structure to facilitate a gentle narrative arc through the material. Act I will introduce important background on relevant physics concepts and the workings of the MicroBooNE detector; Act II will raise the main queries this work aims to address and the strategies taken to that end; Act III will outline the premier analysis—conducting pure selection of a` CCQE neutrino candidate events in MicroBooNE data; Act IV will use the fruits of this analysis to respond to the questions posed in Act II and posit final results; and Act V will take a step back to discuss the place of this analysis within the field of neutrino physics. As part of large and active collaborations like MicroBooNE and SBND, some of the work presented is not wholly original, but rather a review to frame the author’s main contributions. These original contributions are (though not explicitly limited to) the following: the global fit of sterile oscillation models presented in Chapter 3, the entirety of the a` selection presented in Act

III, and the a` disappearance search presented in Chapter 8.

1 Part I

Introductions

2 Chapter 1: Neutrinos

Neutrinos have spent their relatively brief time in public consciousness dancing on the edge of human understanding. They are elementary particles—fundamental components of the reactions that orchestrate the universe—and yet their story is one of befuddlement. Neutrinos ﬁnd themselves at the center of anomaly-after-anomaly; for every new facet physicists probe, something else in our conception breaks. They test our hypotheses and ourselves. This work hopes to accomplish the Wile-Ethelbert-Coyotean* feat of getting even a little closer to our quarry and, furthermore, to bring our reader with us.

1.1 A Strange Position within the Standard Model

Particle physicists in the early twentieth century faced a conundrum: the common radioactive reaction of beta decay seemingly violated energy conservation. An unstable nucleus emitted an electron (4−) or positron (4+), but the energy of this radiated particle did not add up to what was lost by the nucleus—energy was missing. And so, in 1930, Wolfgang Pauli committed the cardinal sin of particle physics: he proposed the existence of a new, neutral particle, which carried this lost energy and thereby satisﬁed conservation laws. But crucially, this particle could never actually be detected [1]. Neutrinos—as they would later be called—were not experimentally observed for over 25 years [2].

1.1.1 The Standard Model of Particle Physics

By the 1970s, particle physics was in a booming era. The collected knowledge of elementary particles and their interactions were compiled into a single uniﬁed theory called the Standard

*Wile E. Coyote is a hapless cartoon character created by artist Chuck Jones and director Michael Maltese who forever hunts (and fails to capture) his prey, The Road Runner, in the Looney Tunes animated series.

3 Model (SM), the main pieces of which are displayed in Figure 1.1. To this day, this model provides a compelling framework for understanding these particles and their inter-relations: known matter is made up of fermions (particles of spin 1/2); nuclear and electromagnetic forces are carried by bosons (particles of spin 1); and fundamental forces can be understood through the exchange of bosons between fermions. Force interactions manifest mathematically as couplings between vector fermion ﬁelds and boson ﬁelds and can be written out in the form of a Lagrangian. Despite some limitations, the Standard Model has been reinforced by countless experiments and has even facilitated the prediction of particles, like the top quark and the Higgs boson, long before the experimental capacity had been developed to observe them in nature.

Figure 1.1: Table of the bosons and three generations of fermions that currently make up the Standard Model with their quantum properties [3].

4 The electromagnetic (EM) force is mediated via the photon (W, also called the gamma particle); the strong nuclear force is mediated via the gluon; and the weak nuclear force is mediated via either the Z or W bosons, whose primary difference is that Z is neutral and W has electric charge ±1. The Higgs does not explicitly transmute force, but engaging with the Higgs mechanism through coupling with this boson grants a particle mass. Elementary fermions can be further subdivided by their capacity for gluon coupling. Quarks are fermions that can participate in strong nuclear interactions, while leptons cannot.

1.1.2 How Neutrinos Fit In

Neutrinos (a) quickly press the limits of the Standard Model. For example, as a consequence of their 1/2 spin, fermions may exist in either of two chiralities: left-handed or right-handed. Thus far, only left-handed neutrinos and right handed antineutrinos (a¯) have ever been observed. This single-chirality means that neutrinos are not able to couple with the Higgs and subsequently do not receive their mass through the Higgs mechanism. There is currently no conclusive explanation for where neutrino mass comes from. An exception to the Standard Model that allows for massive neutrinos is called the Extended Standard Model (ESM). The mass states corresponding to the three types (ﬂavors) of neutrinos are lighter than those of any other known particle. These minuscule masses of unknown origin have never been directly measured and neutrinos were long-believed to be entirely massless [1]. The three ﬂavors of neutrinos are each neutral leptons, only capable of interacting via the weak nuclear force, which (as its name suggests) is a very faint force with a cross section many orders of magnitude less than either the EM or strong interactions. Neutrinos are therefore quite adept at propagating through matter, very seldom interacting with any given nucleus or electron they may pass by. They are thus incredibly hard to detect and study. A neutrino’s emission of a W boson to produce a charged lepton is called a Charged-Current (CC) interaction (whose Feynman diagram is sketched in the left of Figure 1.2), and the emission of a Z boson is a Neutral-Current (NC) interaction (Feynman diagram on the right). Neutrinos can

5 only engage in couplings with leptons of their same generation, which is how the different ﬂavors

get their names. A muon neutrino (a`) will only undergo a CC exchange producing a muon (`),

an electron neutrino (a4) producing an electron, and a tau neutrino (ag) producing a tauon (g) [4].

Figure 1.2: Feynman diagrams for charged-current neutrino interactions (left) and neutral current neutrino interactions (right). In the diagrams, = is a neutron, ? is a proton, # is either nucleon, and ; is any lepton. The shaded circles on each diagram account for nuclear interactions, which may complicate the ﬁnal-state outputs. Time progresses from left to right in each diagram.

Neutrinos occupy a strange place within the standard model: they are small, their mass having unknown origin; they only interact via the weakest force, making them incredibly elusive; there are no right-handed neutrinos or left-handed antineutrinos. What’s more, we must discuss oscillations.

1.2 Neutrino Oscillation Formalism

Those three neutrino ﬂavor states (aU for U ∈ (4, `, g)) don’t exist as unique mass states. Each ﬂavor state is actually a superposition of three neutrino mass states in linear combination.

We will refer to these neutrino mass states in the abstract with Latin numerals (i.e. a8 for 8 ∈ Z).

We will write the vector of neutrino ﬂavor state aU as

a *∗ a *∗ a *∗ a , | Ui = U1| 1i + U2| 2i + U3| 3i (1.1) or more generally, for # = 3 neutrino mass states,

# a Õ *∗ a . | Ui = U8 | 8i (1.2) 8

6 Here, we have introduced the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix *, a 3 × 3 uni- tary matrix that gives the parameters for lepton mixing.

Figure 1.3: A cartoon demonstrating how the three SM neutrino mass states (colored bars) are divided between three ﬂavor components (individual color components)‡. Neither the ﬂavor compositions nor mass positions are to scale, but up-to-date measurements can be found in [4].

The makeup of each mass state is represented visually with the diagram in Figure 1.3. Each tri-colored bar represents a single neutrino mass state, and each color within the bar shows the

2 proportional makeup of the contributing flavor states. PNMS element |*41| for example is the fraction of green (the electron neutrino flavor) in the lowest bar (first neutrino mass state), element

2 |*`3| is the fraction of red in the upper-most (highest mass) bar, and so on. Next, we will evolve our neutrino state through time as though it were a plane wave in the

‡Two neutrino mass-squared terms have been measured, but without the third, one cannot divine the ordering of the three neutrino mass states. This ﬁgure presents the Normal Ordering of masses which places <3 > <2 > <1, but an inverted ordering <2 > <1 > <3 is equally valid with current knowledge [4].

7 Schrödinger picture.

Õ ∗ − ( 2+ 2/ ) a C * 4 8C ? <8 2? a | U ( )i = U8 | 8i 8 (1.3) − 2 Õ ∗ − 2/ 4 8C ? * 4 8C<8 2? a = U8 | 8i 8

The mass and momentum of neutrino mass state vector |a8i are given as <8 and ? respectively. As a convenience, we have also adopted a natural unit system such that 2 = \ = 1, which we will correct for later.

The expectation of observing a neutrino in some ﬂavor state |aVi at time C is now:

a a C 2 Õ *∗ 8C<2 ? * * 8C<2 ? *∗ |h V | U ( )i| = U8 exp(− 8 /2 ) V8 U 9 exp( 9 /2 ) V 9 8 9 (1.4) X Õ *∗ * * *∗ 8 <2 ! = UV − U8 V8 U 9 V 9 (1 − exp( Δ 8 9 /2 )) 8 9

<2 <2 <2 8 > 9 where we have Δ 8 9 = 8 − 9 (with ). For light relativistic neutrinos, we may further substitute ? ≈ and C = G/2 ≈ G = ! (both utilizing 2 = 1). In the second line of Equation 1.4, <2 one may observe that so long as Δ 8 9 is not equal to zero, we will have a non-zero probability of observing a neutrino of flavor V that originated in flavor U. In other words, as a neutrino propagates some distance ! with energy , it mixes—or, oscillates—through each other flavor state, such that it may be observed as a different flavor from how it began with some non-zero probability. We will a a 2 % relabel the probability |h V | Ui| as aU→aV . Experimental observations of neutrino oscillation by the Sudbury Neutrino Observatory (SNO) and Super Kamiokande (SuperK) conclusively proved that these mass differences are not zero [5, 6]. And if the mass differences are not zero, then at least two neutrinos must not be massless, even if the origin of that mass is unknown.

Using the property that 48\ = cos(\) + 8 sin(\) and the trigonometric identity 1 − cos(\) =

8 2 sin2(\/2), we can parametrize the expression in Equation 1.4 to a more useful form:

% a a X Õ *∗ * * *∗ 2 <2 ! 8 <2 ! ( U → V) = UV − U8 V8 U 9 V 9 (2 sin (Δ 8 9 /4 ) + sin(Δ 8 9 /2 )) 8 9 X Õ *∗ * * *∗ 2 . <2 ! = UV − 4<[ U8 V8 U 9 V 9 ] sin (1 27Δ 8 9 / ) (1.5) 8 9 Õ *∗ * * *∗ . <2 ! + 2=[ U8 V8 U 9 V 9 ] sin(2 55Δ 8 9 / ) 8 9

We arrive at the coefﬁcients within the sin and sin2 terms by reintroducing 23/\ to remove dimensionality for L/E of units km/GeV.

1.3 Leading Experimental Constraints

The PNMS matrix can be refactored as three rotations between neutrino mass states as follows:

©* * * ª 41 42 43® ® ® ® * = * * * ® (1.6) `1 `2 `3® ® ® ® *g1 *g2 *g3® « ¬ © ª © 2 B 48Xª © 2 B ª 1 0 0 ® 13 0 13 ® 12 12 0® ® ® ® ® ® ® ® ® ® = 2 B ® ® B 2 ® , (1.7) 0 23 23® 0 1 0 ® − 12 12 0® ® ® ® ® ® ® ® ® ® 8X 0 −B23 223® −B134 0 213 ® 0 0 1® « ¬ « ¬ « ¬ where 28 9 and B8 9 are shorthand for cos \8 9 and sin \8 9 respectively, and X is a CP-violating phase (i.e. one which facilitates matter/antimatter asymmetry). Together with two independent mass-squared <2 <2 \ ,\ ,\ mixing terms Δ 12 and Δ 23, these three mixing angles 23 13 12, and the CP-violating phase X completely describe the parameters of neutrino oscillation in the extended standard model and allow one to predict neutrino oscillation probability.

9 Some of the earliest observed neutrinos originated in the Sun: byproducts of nuclear fusion reactions that produce solar energy [1]. The simplest such reaction is that of a pp chain, wherein two protons fuse to form a deuterium isotope, a positron, and an electron neutrino.

+ ? + ? → 3 + 4 + a4 (1.8)

This electron neutrino ﬂux was measured in the seminal Homestake Mine experiment by R. Davis Jr. in the 1960s, but turned up far fewer neutrinos than one would expect from the Sun’s energy output [7]. It wasn’t until 2001 when SNO presented direct evidence that the solar ﬂux contained additional a` and ag contributions that this Solar Neutrino Problem had a solution: oscillation [5].

An initial solar a4 could potentially be observed in any of the three neutrino ﬂavor states by the time it reached earth. Solar neutrino experiments like SAGE [8], Gallex [9], and SNO [5] have so <2 . −54+2 2 \ . far constrained Δ 21 ≈ 7 5 × 10 and sin 12 ≈ 0 3 [4]. Similarly, there are many experiments observing neutrinos from nuclear reactors on Earth like Daya Bay [10], Double Chooz [11], and KamLAND [12]. Reactor experiments have helped con-

2 −2 strain sin \13 ≈ 2.2 × 10 , while also (especially in the case of KamLAND) providing additional measurements of solar oscillation parameters [4]. Naturally-occurring neutrinos are also produced in Earth’s atmosphere. Cosmic rays interact with nucleons in the atmosphere to produce pions and kaons, whose decay chains lead to the creation of both electron and muon neutrinos. As an example, pion decay follows the chain:

+ + c → ` + a` (1.9)

+ + ` → 4 + a4 + a¯` (1.10)

<2 . −34+2 Atmospheric neutrino experiments like Super-Kamiokande have constrained Δ 23 ≈ 2 5×10 ◦ and \23 ≈ 45 . These measurements have been further corroborated by accelerator experiments like MINOS [13], NOVA [14], K2K [15], and T2K [16, 4].

10 1.4 eV Scale Neutrino Masses

The standard model, even under the extended paradigm is still far from settled. The Liquid Scintilator Neutrino Detector (LSND) was an experiment designed to measure electron antineutrinos having oscillated from a nearby beam of primarily a¯` ﬂux at Los Alamos National Labora- tory [17, 18].

LSND observed a 3.8 f excess of electron antineutrinos beyond what one would predict under the standard model [17]. Their measurement is shown in Figure 1.4. The vertical axis of this ﬁgure directly plots the "beam excess": the count of observed electron neutrino events above the initial beam content. According to the neutrino oscillation framework prescribed by the ESM, this excess should be in line with the top of the green histogram.

Figure 1.4: Plot of the excess of electron antineutrinos observed by LSND (dots). The SM prediction is given as the sum of the red and green histograms. The blue hatched histograms illustrate hypothetical predictions beyond the ESM to attempt to explain the excess [18].

11 The Mini Booster Neutrino Experiment (MiniBooNE) used a spherical Cherenkov detector to measure electron neutrinos and antineutrinos having oscillated from the Booster Neutrino Beam

(BNB), of a` and a¯` ﬂuxes respectively, at Fermi National Accelerator Laboratory (Fermilab) [19]. Taking direct inspiration from the LSND anomaly, MiniBooNE was situated at a comparable distance-to-neutrino-energy ratio (!/) to that of LSND and hoped to observe the same oscillations. MiniBooNE’s early results are shown in Figure 1.5.

Figure 1.5: Plot of MiniBooNE’s observed (dots) versus their predicted (stacked histogram) counts of electron neutrinos and antineutrinos. A clear excess is visible in the leftmost four bins in both neutrino modes [19].

MiniBooNE’s ﬁnd did not exactly match LSND’s anomaly: MiniBooNE still observed an excess, but one which was far more focused on lower energy electron neutrinos and antineutrinos. This new anomaly was dubbed the Low Energy Excess (LEE). In the decade since the ﬁrst Mini-

12 BooNE oscillation data release, additional data has only strengthened the disparity of the LEE [20].

Figure 1.6: antineutrino observed excesses from both MiniBooNE and LSND overlaid on a common axis. The dotted and dashed lines illustrate hypothetical, oscillatory distributions to provoke curiosity [19, 18].

In plotting the antineutrino observations of both experiments on the same !/ axis, we see the pattern in Figure 1.6. The vertical axis here records each data point in terms of an excess. We could convince ourselves that there may be some apparent trend: that there is perhaps some new kind of oscillation present, but at an !/ scale unlike what the ESM neutrinos would allow within their framework. If these two anomalies are considered as oscillatory in nature, then those hypothetical oscillations could only be caused by a much heavier neutrino—one with mass of the order 1eV§. To gain further insight, we would need a detector with more powerful resolution and the capability of understanding backgrounds that were irreducible in MiniBooNE or LSND.

§It is worth noting that cosmological and astrophysical measurements are capable of placing constraints on neutrino mass and oscillation. Pertinent to the claims made in this section, cosmological studies have excluded eV2-scale sterile neutrinos and must be further reconciled, but that is outside the scope of this dissertation [21].

13 Chapter 2: The MicroBooNE Experiment

The Micro Booster Neutrino Experiment (MicroBooNE) was born with two speciﬁc physics goals: to make high-statistics measurements of the neutrino cross-section in liquid argon, and, centrally, to resolve the MiniBooNE low energy excess [22]. This yen to understand the LEE has been factored into every aspect of the experiment: from its placement on the same neutrino beam as MiniBooNE, to its state-of-the-art technology that is uniquely capable of seeing what its forebear could not. MicroBooNE is part of the Short Baseline Neutrino (SBN) program at Fermilab that, with the impending operations of two additional detectors, SBND and ICARUS, intends to make high- precision measurements of eV-scale neutrino oscillation in the BNB [23]. Figure 2.1 illustrates the landscape of experiments receiving neutrinos from the BNB at Fermilab. Towards the center, we see the MicroBooNE detector hall and its neighbor, MiniBooNE, just to its left.

Figure 2.1: An aerial illustration of Fermilab’s Neutrino Campus where one can see the trajectory of the Booster Neutrino Beam through several detectors, including MiniBooNE and Micro- BooNE [23].

14 2.1 The BNB

The neutrino source for MicroBooNE (and MiniBooNE before it) is the Booster Neutrino Beam [24]. In practice, the BNB is actually a series of beams, beginning with protons, proceeding into mesons through inelastic collision with a heavy target, and then ﬁnally decaying into neutrinos. This section will provide an overview of the BNB and the mechanisms by which neutrinos are produced for our experiment. The beam delivers protons of 8 GeV energy to a Beryllium target and uses an electromagnetic "horn" to focus secondary mesons from the collision down a decay tunnel. Over the course of travel, many of these secondary particles—mostly positive pions (c+) and kaons ( +)—will decay into forward-going neutrinos. The rest may be absorbed by a concrete and steel beam dump at 50 m, through which neutrinos can easily pass as they proceed to MicroBooNE.

2.1.1 The Proton Beam

Protons (of 400 MeV energy) produced by Fermilab’s Linear Accelerator (Linac) are loaded into the Booster synchrotron in "bunches" of approximately 4 × 1012 protons (over 1.6 microsec- onds each). These bunches are accelerated in a loop to 8 GeV before each bunch is released— spilled—on a track, steered along the way by electromagnets. The BNB is capable of spilling proton bunches at a rate close to 15 Hz, but the design of the focusing horn limits this rate to 5 Hz [25]. Proton spills are monitored for intensity and release time by a pair of toroids: one in the Booster loop and one along the track. Additional Beam Position Monitors (BPMs) and a multi-wire chamber are present to measure properties of beam quality and position before the target is reached. The total count of protons spilled from the Booster is measured as Protons On Target (POT) and is the primary metric for how many neutrinos are delivered and available to an experiment. Micro-

BooNE was intended, for example, to receive neutrinos from 6.6 × 1020 POT over the course of a three year period, but through fortuitous accelerator conditions has surpassed that. However, only

15 6.8 × 1020 POT are both processed and available for the analysis presented in this dissertation.

Figure 2.2: A cartoon of the several stages of the BNB, from protons to mesons to mostly neutrinos.

2.1.2 Proton Target and Focusing Horn

The proton target comprises seven ribbed, tri-finned Beryllium cylinders placed end-to-end, totalling 71.1 centimeters in length (corresponding to 1.7 proton interaction lengths) and 1.1 centimeters in diameter. Inelastic collisions from proton bombardment of the Beryllium nuclei produce secondary mesons, including charged pions and charged and neutral kaons, as well as protons and neutrons. This target is housed within a focusing horn—an aluminum alloy structure capable of producing a pulsed magnetic field to coincide with the beam from a current peaking at approximately 170 kA. The current produces a toroidal magnetic field perpendicular to the direction of the incoming proton beam as strong as 1.5 Tesla and weakening with increased distance from the central (beam) axis. Charged particles traveling through the horn are subjected to a magnetic force either focusing them (pulling inward, towards the beam) or unfocusing them (pushing outward, away from the beam), allowing for charge-sign discrimination. By operating the magnetic field in one direction, the horn focuses positively charged mesons into a narrow beam that may then decay into a beam of mostly neutrinos, while negatively charged particles are skewed away. If the horn’s polarity were to be reversed,then one would produce a beam of mostly antineutrinos.

16 Figure 2.3: A schematic illustration of the focusing horn used by MiniBooNE and MicroBooNE. The proton beam is incident on the left with respect to this diagram. The target is inserted rod- ﬁrst along the central axis from the left. An electric current travels across the inner surface of the horn outward towards the front and loops over to return back along the upper surface (during neutrino-mode, and with opposite direction for antineutrinos), generating a toroidal magnetic ﬁeld.

Behind the horn is a concrete collimator. Any particles sufficiently off-axis so as to miss its 30 centimeter diameter aperture are absorbed into the concrete, along with any other excess radiation from meson production. Energized particles then fly freely through a 45 meter long decay pipe filled with air and either decay into forward-going neutrinos (and muons) or are lost to the tunnel’s corrugated steel shell and packed dolomite surrounding. At the end of the pipe is the beam dump: a thick mass of concrete and steel that acts as a final means of halting passage of non-neutrinos. Only neutrinos (and some high-energy muons) will pass through this final absorber.

Figure 2.4: A cartoon of what happens to the proton beam when it enters the target hall and hits the beryllium target.

Lastly, there are approximately 420 meters of "dirt"—soil, concrete, rock, whatever is in the

17 path between the beam dump and detector. MicroBooNE’s baseline of 470 meters is measured from the front of the target. This leg of the process from 8 GeV protons to ﬁnal neutrinos is illustrated for reference in Figure 2.4.

2.1.3 Beam Composition and Flux Uncertainty

The primary neutrino production mechanism of the BNB is the decay of charged pions, neutral

+ + and charged kaons, and muons. The most common mode of pion decay is via c → ` a`, which has a branching fraction of ∼99.98% [4]. As discussed with atmospheric neutrinos (in Section 1.3), muons that are not absorbed by the beam dump may decay further into a a¯` and a a4. Pions can

+ 4 also directly decay into a a4, 4 pair, but this mode is suppressed by 10 times compared to a` production.

Figure 2.5: Prediction of the neutrino components of the BNB ﬂux in neutrino-mode, modeled with Geant-4 simulation [26, 24].

Predicted ﬂux contributions from a`, a¯`, a4, and a¯4 in the BNB with the focusing horn set to

"neutrino-mode" are shown in Figure 2.5. We can see that the a` (and a¯`) ﬂuxes strongly dominate our beam with less than 1% a4 (and a¯4) contamination. Neutrino ﬂux contributions are further subdivided by their decay sources in Figure 2.6. At energies above ∼2.5 GeV, kaon decay becomes dominant, but for the purpose of this dissertation

18 Figure 2.6: Prediction of the BNB flux in neutrino-mode, modeled with Geant-4 simulation, of a` (left) and a4 (right) [24]. and the MicroBooNE LEE search, we are only interested in sub-2 GeV neutrinos. Specific details on the BNB flux and calculation of its uncertainty have been published in [24], including the above plots. We will return to discuss flux contributions to systematic uncertainty in Section 6.3.

2.2 The MicroBooNE Detector

MicroBooNE’s dedication to resolving the LEE goes beyond its placement on the same beam line (not to mention the large overlap of scientists who have served on both projects) as Mini- BooNE. The MicroBooNE detector is speciﬁcally designed to grant particular advantages in this endeavor. Here, we present an overview of the detector as a neutrino interaction target, how it receives and processes data, and what that data looks like after processing. MicroBooNE is a 170 tonne Liquid Argon (LAr) Time Projection Chamber (TPC) installed on the Earth’s surface at a base line of 470 meters from the BNB proton target. LArTPCs are a novel detector technology able to produce 3D images of neutrino interactions with sub-millimeter spatial resolution and ﬁne calorimetric resolution via the collection of electrons freed by ionization (both of which can be combined to measure the rate by which energy deposition drops off along the path of a particle track, 3/3G). These precision measurements allow MicroBooNE an unprecedented opportunity to resolve backgrounds that were irreducible in MiniBooNE.

19 Figure 2.7: The fully-constructed MicroBooNE cryostat on its way from the Fermilab Detector Assembly Building (DAB) to its home in the Liquid Argon Test Facility (LArTF). Photo by Cindy Arnold, Fermilab.

This section will summarize important aspects of MicroBooNE’s detector design, but more speciﬁc details can be found in [27].

2.2.1 The Time Projection Chamber

MicroBooNE’s TPC is a rectangular prism that is 10 meters in length (measured along the neutrino beam axis), 2.6 meters wide, and 2.3 meters tall, encompassing an active volume of 85 tonnes of liquid Argon. The TPC structure comprises a steel scaffold housing several critical parts, most of which are visible in both schematic and photographic form in Figure 2.8, and all of which are housed within an outer cylindrical steel cryostat.

• Along the TPC face at X=2.6m is a cathode plane, made up of three steel sheets, and along the opposite (X=0m) face is an anode plane. A current is fed into the detector, creating and

maintaining a constant 273 V/cm electric field between the two planes*. A field cage of steel tubes looping lengthwise around the TPC shapes the electric field to be uniform, even at the

20 Figure 2.8: Schematic model of TPC (left) alongside the fully-constructed TPC, ready for insertion into the cryostat (right) with overlaid axes. The MicroBooNE TPC measures 10 meters along the z axis, 2.6 meters along the x axis, and 2.3 meters along the y axis.

edges.

• The anode plane itself comprises a set of three, parallel wire arrays labeled the U, V, and Y wire planes. The outside-most Y plane contains 3456 wires arranged in vertical orientation with 3 mm between them, while the U and V planes house 2400 wires each of orienta-

tion ±60◦ from vertical and 3 mm between.

• Behind the wire planes are 32 photo-multiplier tubes (PMTs) for light collection.

Neutrinos pass through the active detector volume with each spill of the BNB. When a neutrino interacts with an Argon nucleus, any charged final-state particles from that interaction propagate through the detector, ionizing the electrons from any Argon atoms they pass. This process is illustrated in Figure 2.9. Charged final-state particles leave a trail of ionized electrons in their wake, drawing a 3D image of the scattering event in Argon. The constant electric field then pulls electrons uniformly towards the anode face. These electrons pass by the U and V induction planes, inducing an electromagnetic waveform on those wires before they are collected by the Y collection plane.

*MicroBooNE was designed and tested to withstand a 500 V/cm electric ﬁeld across the TPC active volume, but has not yet been operated to those speciﬁcations during normal data-taking [22].

21 Figure 2.9: Cartoon of the detection mechanism of a LArTPC. Incoming neutrinos interact with an argon nucleus, releasing charged particles that ionize argon atoms along their path. Electrons drift along the electric ﬁeld towards the wire planes (on the right), where they leave a charge signal. The waveforms of these charges on the wire grid can be used to determine coordinates of a neutrino interaction within the detector.

The charge received by each wire corresponds to the energy deposited by the inciting ionizing particle, and by combining information about which wires observed particular waveforms, one can reconstruct the 2D coordinates of each ionized electron in the detector (on the YZ plane). Immediately after a neutrino interaction takes place in Argon, a flash of scintillation light is emitted and received by the PMT array—by factoring in this final piece of information with the constant speed at which electrons drift through the TPC, one can derive the final (X) dimension, allowing for 3D construction of these interactions within the detector.

22 2.2.2 The Optical System

MicroBooNE’s optical system comprises 32 Hamamatsu 5912-02MOD PMTs mounted behind the anode (wire) plane and arranged so as to have a full view of the active TPC volume without obstruction by the TPC scaffolding (see Figure 2.10).

Figure 2.10: Schematic of the components of MicroBooNE’s optical system. Importantly, one can see how PMTs ("optical units") are spaced across the anode plane.

Light-collection plays a critical role in isolating the time and position of a neutrino interaction. The bright ﬂash of scintillation light from a neutrino interaction in Argon will reach PMTs far faster (on the order of nanoseconds) than the electron drift time (on the order of milliseconds), giving a precise account of the exact time an interaction took place.

2.2.3 Triggering

The MicroBooNE detector sits very close to the Earth’s surface with very little material sep- arating it from the sky overhead. This positioning makes MicroBooNE incredibly vulnerable to cosmic rays and other atmospheric particle contamination. Clever reconstruction and event selection can only accomplish so much; the most signiﬁcant means of cosmic reduction come from carefully tuned triggers. The BNB spills neutrinos at a rate of approximately 5 Hz, and each spill is approximately

1.6`B wide. During the other 99.9992% of the time the beam is active, no neutrinos are being

23 sent. When MicroBooNE receives a trigger from the BNB, a small time window is opened during

which the data is read out and processed. This window includes one 1.6

2.2.4 The Readout

The TPC and PMT readout (RO) subsystems convert received signals in the detector wires to digital signals for later processing and analysis. Both the TPC and PMT ROs operate by similar means, but the TPC RO has some properties that speak more to the scalability of LArTPCs for future, much larger experiments. For this reason, and because Nevis Labs and the author of this dissertation played a large role in the design, construction, and maintenance of this subsystem for MicroBooNE and in the commissioning and installation for the forthcoming Short Baseline Near Detector (SBND) experiment, we will take a brief aside to elaborate further on the TPC readout. The schematic shown in Figure 2.11 will act as a helpful guide that outlines the entire RO process. MicroBooNE utilizes application-specific integrated circuit (ASIC) cards to collect analog signals from each wire and to apply pre-amplification to each waveform. If the signal has to travel too far before amplification, then noise will overwhelm it, so ASICs must live on the side and top of the TPC within the cryogenic liquid Argon. These pieces are hence called "cold" electronics.

24 Figure 2.11: Detailed diagram of the full TPC and PMT readout systems for MicroBooNE. Analog signals are measured in the TPC on the left and transported via long cables to the electronics and readout systems presented on the right for digitization (done in hardware) and event assembly.

Next, the signal must travel out of the cryostat via "warm" cables towards readout crates to be digitized. When the signal leaves the cryostat, it is ampliﬁed again to ensure safe passage across the 20 meters of cabling to its destination: the TPC readout crates. Each of nine TPC readout crates house analog-digital-converters (ADCs) and TPC Front End Modules (FEMs). The ADCs receive analog signals from every wire and save them as 12 bit sample values at 2 MHz. The FEMs take the digitized signal and load it into an SRAM circular buffer where it can be processed. Here, the data splits into two streams:

• a triggered neutrino stream, which assembles data into events when a trigger is received. Data from this stream is trimmed into 4.8 ms, losslessly compressed (via a Huffman algorithm) binary fragments and assembled into events by the Data Acquisition (DAQ) subsystem to be later used for analysis.

• a continuous, supernova (SN) stream, which does not separate data into events, but cease- lessly writes (and overwrites) zero-suppressed data. If a supernova were announced by the Supernova Early Warning System (SNEWS) collaboration [29], then we could look for this

25 very small signal in our data. The SN stream takes in approximately 33GB/s, so quick processing and zero suppression are very important to reduce this input by at least 80 times [30]. Larger experiments, like the Deep Underground Neutrino Experiment (DUNE) [31], will have orders of magnitude more input and a much higher chance of actually resolving a supernova, so work done in MicroBooNE will set an important precedent.

The PMT readout follows similarly. However, PMT data is always zero-suppressed, has fewer individual channels than TPC data, and is digitized at a 64 MHz rate.

SBND

Over the course of MicroBooNE’s period of data-taking, the Short-Baseline Near Detector (SBND) [23] has been under construction, and I have been responsible for both porting Micro- BooNE’s RO code to a new DAQ software framework for the new experiment and also for early steps of installing the SBND TPC RO subsystem. SBND features an important shift in RO design that will become relevant in all larger LArTPC experiments—with so many more wires, transport- ing the analog signal outside of the detector and to the readout crates is unfeasible. The cables are too long, and the opportunity for noise contamination is greater. As such, some of the formerly "warm" electronics, like the ADCs, will be adapted for the cold. The signal will be digitized within the cryostat and cleanly fed out through ﬁber optic cables. A vertical slice test of this setup was orchestrated in the Summer of 2018 as a proof of concept, using the former cryostat and TPC of a much smaller (in size) experiment, ArgoNeuT, ﬁtted with SBND electronics. I was one of the lead coordinators of the Nevis equipment (all the RO electronics and software outside the cryostat) [32].

2.2.5 2D Deconvolution

The path from electron waveform to analysis-ready data has one more important step. Before proper reconstruction can begin, one must be able to recover how many ionized electrons actually comprise a given signal in the detector, taking into account the multitude of factors that could

26 smear or skew this reading [33]. As electrons pass by the two induction planes and get absorbed by the collection plane, we are not only measuring the charges of those electrons after a simple ionization from their Argon atoms, but we are also seeing the consequences of various detector effects, the high-voltage electric ﬁeld, and background noise. To recover the true signal (or as close as one can hope), we must deconvolve all these added effects.

In practice, this means solving the equation below where "8 (C0) is the measured electric signal

on wire 8 at a time C0 and ((C) is the true, real electrical signal transferred to that wire.

¹ ∞ "8 (C0) = '8 (C,C0)((C)3C (2.1) −∞

Here, '8 (C,C0) is the detector response function, which in this case describes how a signal varies

in the detector over a time interval C − C0. This solution works well for the collection plane, but one must recall that the induction planes receive charge via passing electrons. A single electron will not simply be observed in a single wire. One must expand Equation 2.1 to account for this, by integrating over both time and wires. This is referred to as 2D deconvolution and the expanded equation for a measured signal is

¹ ∞ "8 (C0) = ('0(C,C0)(8 (C) + '1(C,C0)(8+1(C) + ... )3C (2.2) −∞ where one now integrates between any number of neighboring wires. This whole process is easier to parse and solve through the Fourier transformed versions of Equations 2.1 and 2.2. However, this parametrization of the detector response function reveals a limitation: here, we only depend upon time and wire. One could conceive of a response function that took position within the detector into account, for example, but the ensuing calculations and calibration would be an unremitting nightmare. One can see the striking improvements in resolution and clarity of tracks that result from this technique in Figure 2.12, which shows the effects on an example neutrino candidate event from

27 Figure 2.12: Three LArTPC event displays illustrating the stages of signal reconstruction across a single wire plane. On the left, we have several messy particle signatures; in the center, we have applied 1D deconvolution to clean up the tracks; and on the right, we see the well-deﬁned, high-contrast signature that analyzers will pick apart [33]. data. From this clear starting point, analysers may begin reconstructing neutrino candidate signatures in MicroBooNE.

2.3 Neutrino Interactions and Their Signatures in MicroBooNE

When all these pieces (and truly, so many more) come together, MicroBooNE can produce beautiful "photos" of interactions within the TPC. These events are reconstructed in 3D, but often viewed as 2D projections on each wire plane with wire number as the abcissa and drift time as the ordinate. Color intensity illustrates the total deposited energy at that coordinate. An example of one neutrino event observed across three planes is shown in Figure 2.13. From the information in these displays, we will eventually attempt to reconstruct the full four- momenta of each ﬁnal state particle to intuit the source neutrino. This process is done algorith- mically, but it helps to have a visual intuition of what the most common signatures look like in MicroBooNE. Particle signatures are divided between tracks and showers.

28 Tracks

Tracks are the long, mostly-straight signatures of a particle propagating directly through the detector, hemorrhaging energy along the way. Tracks are easy to identify because they are straight lines and are most often the result of a proton or a muon (either cosmic or CC-induced). Tracks can be further subdivided as being caused by Highly Ionizing Particles (or HIP, most often associated with protons) or Minimally Ionizing Particles (or MIP, most often associated with muons). These two types can be differentiated by their pixel intensities, as seen in Figure 2.14.

Showers

Showers are much more chaotic. As an energetic electron slows down (like it would when ionizing a bunch of Argon atoms), it undergoes Bremsstrahlung wherein it emits a photon. Because a photon has no charge, it will not create any trace from ionization. The photon will propagate invisibly and, if it has enough energy, will then pair-produce an electron and a positron. The electron and positron will also Bremsstrahlung into photons, which will pair produce again until all the energy is gone. This leaves a very distinct, conical, shower-like signature in the detector as seen in Figure 2.15.

The shower process can start with a photon rather than an electron. In the case of a a` NC interaction, one may have a neutral pion (c0) that will immediately decay into two photons, creating their own electromagnetic showers. If these two showers were to overlap, then it would be incredibly hard to tell them apart, and this a` NC signature would mimic that of a a4 CC. There are two important giveaways to discern whether or not a shower was instigated by a photon or an electron in a TPC:

• the instigating photon in a NC interaction will go some distance before pair-producing, leaving a gap between the shower and vertex, as seen in Figure 2.16;

• the beginning of a photon shower is marked by a charged electron-positron pair that will have double the ionization rate as that of an electron shower (beginning with a single electron).

29 Figure 2.13: Sample event display of a candidate neutrino event observed in MicroBooNE across the collection plane (top) and two induction planes (middle, bottom) [34].

30 Figure 2.14: A sample event display of BNB data where we see a a` candidate event, identiﬁed by its long muon track and two stubby proton tracks. One can see a higher-energy cosmic muon crossing through the upper-right corner.

NuMI DATA: RUN 10811, EVENT 2549. APRIL 9, 2017.

Figure 2.15: A sample event display data from Fermilab’s Neutrino Main Injector (NuMI) beam. It is a bit harder to identify the tracks in this event display because many may have arisen from ﬁnal-state-interactions within the argon nucleus, but the centerpiece is a very clear look at an electromagnetic shower signature.

31 BNB DATA : RUN 5187 EVENT 2727. FEBRUARY 28, 2016.

Figure 2.16: An event display of BNB data that appears to feature a a` NC event. At the 1 o’clock and 7 o’clock positions relative to the central vertex, one sees two electromagnetic shower-like clusters. Neither are connected to the vertex, but both point back to a common origin where a c0 may have decayed.

32 Chapter 3: On Notation

The statistical concepts described in this dissertation require immaculate bookkeeping to do correctly, let alone carry a passenger through. To facilitate clarity, the author has chosen to adapt an internally consistent notation, which shall be outlined in this brief interlude. This dissertation will adhere to the following principles:

• Many symbols will have a superscript or subscript label. A superscript will always present the identiﬁer (or type) of a given symbol. Subscripts will always present an index.

• We will often discuss vector elements, but a vector in its totality will always appear identically to one of its elements, but emboldened and without an index. For example, NG = {#G,#G,...} 1 2 .

• The symbol N will always stand for the distribution of a histogram, with each element thus the event count for a given bin.

• The symbol (f2)G will always stand for the total, systematic variation of distribution #G.

• The full covariance matrix for some distribution NG will always be given by a bold ", to be deﬁned in situ by " = 2>E(G). The fractional covariance matrix is deﬁned per-element by "8 9 = . (3.1) 8 9 #G #G 8 9

On every occasion when shorthand or symbols arise, they will be addressed in the text, but this list should provide a helpful reference to avoid ambiguity.

33 Part II

Rising Action

34 Chapter 4: Sterile Neutrinos

The MiniBooNE LEE measurement raised far more questions than it answered. One of the most important theories to receive attention from the LSND and MiniBooNE anomalies was that this chaos was stirred by a brand new particle: the sterile neutrino.

4.1 What We Talk About When We Talk About the LEE

We have already alluded to the impact of MiniBooNE’s LEE, but in this section we will peer more closely at the signal itself, review what MiniBooNE was actually aiming for, and discuss how we have come to interpret the LEE today.

4.1.1 What has MiniBooNE Measured?

The MiniBooNE detector comprises a 12.2 meter diameter sphere ﬁlled with 818 tonnes of mineral oil (2). Within the main structure is another spherical shell containing the active detector volume, with an inside surface covered with a radial lattice of 1280 PMTs. The space between the outer and inner shells is peppered with 240 additional PMTs and is perfectly optically isolated from the inner-volume. This outer region acts as a veto—scintillating light from a neutrino interaction must be wholly contained within the active volume; coincident light outside implies a through-going particle that cannot be fully reconstructed. In addition to scintillation light, the main mechanism of neutrino detection and reconstruction in MiniBooNE is via Cherenkov radiation. A particle going faster than the speed of light in a substance will release the light-equivalent of a sonic boom: a cone of photons bursting forth in the direction of the particle’s momentum. The width and intensity of a Cherenkov light cone, as measured by MiniBooNE’s PMTs, can determine the energy of its source, while the scintillation

35 light and geometry of the projected circle on the detector interior can determine the neutrino angle and vertex origin.

With this detector, the MiniBooNE collaboration counted a4 and a¯4 candidate events within the & & reconstructed energy range of 200 MeV < a < 1250 MeV, where a is the neutrino energy under the assumption of Quasi-Elastic (QE) scattering conditions, using reconstructed angles and energy of the ﬁnal-state lepton.

Figure 4.1: The MiniBooNE a4 distribution from their most recent, 2020 paper, comprising over three times as much data as the initial result while still exhibiting a strong excess in the four lowest energy bins [20].

The results of this measurement in neutrino-mode, as of 2020, are shown in Figure 4.1. From this plot, one can see the LEE disparity manifest in the lowest four energy bins, as well as qualitatively great agreement in the upper energy bins. This LEE measurement comes over a decade after the original LEE observation, with all the increased data that time affords. Over this period, the LEE has only grown in statistical signiﬁcance, now rejecting the standard model prediction at

4.8f [20].

4.1.2 Limits of Cherenkov technology

Neutrino candidate ﬂavor in a Cherenkov experiment is determined via the distinct light pat- terns produced by source particles. A track particle (like a muon or proton) will project a circle on the inner detector wall straight ahead of its path. As the particle continues to propagate, the radius of the projection will shrink and ﬁll in the circle. This is illustrated in the top diagram of

36 Figure 4.2. A particle that travels a short distance, like a proton, will leave the outline of a circle, while a farther-going muon may ﬁll in the circle completely. The signature of an EM shower is, just like in MicroBooNE’s LArTPC displays, messier. Each electron and positron in the shower will emit cones in a diffuse pattern, leaving a jagged ﬁgure on the wall like what one sees in the bottom of Figure 4.2.

Figure 4.2: Cartoon examples of potential particle signatures as would be detected in a Cherenkov- based detector. On the top, one sees a clean, ﬁlled circle, caused by a muon track, and on the bottom, one sees the jagged-edged circle that an EM event would leave [3].

The roughness of this EM signature is at the heart of the LEE question for one key reason: photons and electrons look identical, making certain backgrounds irreducible. There are three dominant backgrounds indicated in Figure 4.1—"dirt," which refers to any event where a neutrino interacts outside the detector, but a final-state particle (usually a photon) somehow sneaks past the veto region; NC Δ radiative decay, where a Δ particle decays into a single photon; and, most significantly, NC c0 misidentification, where the two photons from pion decay only create one observable signature.

37 One must decide if they wish to interpret the LEE as electron-like (this is called the eLEE), proton(gamma)-like (gLEE), or some combination, and then form their theories from there.

4.2 MiniBooNE’s eLEE and the Sterile Neutrino Hypothesis

This dissertation is interested in the LEE as prescribed by the eLEE interpretation: resulting from an excess of electron-like events. A leading hypothesis to explain the physical origin of this excess is the existence of sterile neutrinos: theorized neutrinos comprising new sterile flavor states and mostly-inactive mass states, through which their SM counterparts may freely oscillate. Their sterile naming arises from their inability to couple with W or Z bosons; they are invisible to even the weak interaction. This makes sterile neutrinos impossible to directly observe because they will never actually interact in a detector. Their contribution to mixing, however, introduces the opportunity for oscillation-based effects to influence measurement of other neutrino flavors. Sterile neutrinos can therefore be observed through mixing, specifically in the deviations their presence creates from ESM predictions. The sterile neutrino hypothesis poses a question echoing Wolfgang Pauli’s initial musing that birthed neutrino physics: What if there is a particle that is impossible to observe directly, but whose effects permeate through other physics we can see?

4.2.1 Further Extending the Extended Standard Model

The framework of neutrino oscillation laid out in Section 1.2 allows for a simple extrapolation of the 3a model to accommodate any number of sterile neutrinos. Standard nomenclature dictates that such a model be referred to as 3+N, which contains the 3 ESM neutrinos and N additional sterile ﬂavor states. Adding N new sterile ﬂavors to the SM involves contributing three expansions to the framework:

1. N new sterile ﬂavor states, the ﬁrst of which we will label as aB;

38 2. N new mostly inactive mass states which we will label as a3+8, ..., a3+# ; and

3. N new rows and columns of the PNMS lepton mixing matrix to account for the additional dimensions of mixing.

The sterile mass states are considered to be mostly inactive. Because the PNMS matrix is uni- tary, ESM ﬂavor contributions to any sterile state must lie within the bounds of present uncertainty.

Figure 4.3: Diagram of a 3+1 neutrino model, expanding upon Figure 1.3 with the addition of one new neutrino ﬂavor state aB, and one new neutrino mass state a4.

In Figure 4.3, we illustrate the addition of a single new state in what is called a 3+1 model.

Here, one sees the three standard model neutrino mass states, a1,2,3, as colored bars spaced verti-

cally according to their mass-squared. The mass-squared-differences, Δ<8 9 , that appear in oscillation probabilities are the vertical distances between any two of these bars. In the context of sterile

neutrino oscillation, we will call hypotheses which engage in only ESM mixing as 3a or no-osc (short for no-sterile-oscillation) models.

39 4.2.2 Oscillation Probabilities

The neutrino oscillation formalism outlined in Equation 1.5 can be easily generalized to accommodate sterile mixing. In practice, however, each additional added neutrino greatly compounds complexity. The 3+1 sterile neutrino oscillation model that best resolves tensions within the MiniBooNE and LSND data has a sterile mass on the order of 1 eV [35, 18], which is far more massive than the <2 <2 −54+2 −34+2 three ESM neutrino mass states (with Δ 21 and Δ 32 being constrained to 10 and 10 respectively). With this knowledge, we can approximate these three ESM neutrino masses to be degenerate at 0 eV relative to this proposed sterile neutrino mass, such that we have

<2 <2 <2 4+2. Δ 21 ≈ Δ 31 ≈ Δ 32 ≈ 0 (4.1)

Since the oscillation probability for a given neutrino oscillation model depends only upon energy (E) and distance from the neutrino source (L), this assumption can be reasonably applied to other experiments with designs similar to LSND and MiniBooNE, having !/ ∼ 1 :

2 2 2 %(aU → aV) = 4|*U4| |*V4| sin G41, (4.2)

40 and the survival probability becomes

2 2 2 %(aU → aU) = 1 − 4|*U4| (1 − |*U4| ) sin G41 , (4.3)

G . <2 ! where 8 9 ≡ 1 27Δ 8 9 / . One may notice that with the SBL approximation, 3+1 oscillation <2 ,* , * manifests as a 2-neutrino oscillation with only three parameters, Δ 41 U4 and V4, with no CP violating phases present, dictating that a and a¯ oscillations operate identically across this model. The 3+2 appearance and survival probabilities get more complicated and are given by

2 2 2 %(aU → aV) = 4|*U4| |*V4| sin G41

2 2 2 + 4|*U5| |*V5| sin G51

+ 8|*U4||*V4||*U5||*V5| · sin G41 sin G51 cos(G54 − q54) (4.4) and

2 2 2 2 2 2 %(aU → aU) = 1 − 4(1 − |*U4| − |*U5| ) · (|*U4| sin G41 + |*U5| sin G51)

2 2 2 − 4|*U4| |*U5| sin G54 (4.5) respectively. In addition to the new mass-squared and extended PNMS mixing terms, the biggest addition from a 3+2 SBL model is the reintegration of CP violating phases, totalling seven parameters in all.

41 Lastly, the 3+3 appearance and survival probabilities are

%(aU → aV) =

2 − 4|*U5||*V5||*U4||*V4| cos q54 sin G54

2 − 4|*U6||*V6||*U4||*V4| cos q64 sin G64

2 − 4|*U5||*V5||*U6||*V6| cos q65 sin G65

2 + 4(|*U4||*V4| + |*U5||*V5| cos q54 + |*U6||*V6| cos q64)|*U4||*V4| sin G41

2 + 4(|*U4||*V4| cos q54 + |*U5||*V5| + |*U6||*V6| cos q65)|*U5||*V5| sin G51

2 + 4(|*U4||*V4| cos q64 + |*U5||*V5| cos q65 + |*U6||*V6|)|*U6||*V6| sin G61

+ 2|*V5||*U5||*V4||*U4| sin q54 sin 2G54

+ 2|*V6||*U6||*V4||*U4| sin q64 sin 2G64

+ 2|*V6||*U6||*V5||*U5| sin q65 sin 2G65

+ 2(|*U5||*V5| sin q54 + |*U6||*V6| sin q64) · |*U4||*V4| sin 2G41

+ 2(−|*U4||*V4| sin q54 + |*U6||*V6| sin q65) · |*U5||*V5| sin 2G51

+ 2(−|*U4||*V4| sin q64 − |*U4||*V5| sin q65) · |*U6||*V6| sin 2G61, (4.6) and

2 2 2 %(aU → aU) = 1 − 4|*U4| |*U5| sin G54

2 2 2 − 4|*U4| |*U6| sin G64

2 2 2 − 4|*U5| |*U6| sin G65

2 2 2 − 4(1 − |*U4| − |*U5| − |*U6| )

2 2 2 2 2 2 (|*U4| sin G41 + |*U5| sin G51 + |*U6| sin G61) (4.7)

respectively. The 3+3 case comprises a whopping 12 parameters including 3 CP violating phases.

42 4.2.3 Predicting an Oscillated Spectrum

Provided with the tools and MC necessary to make an accurate 3a prediction, one can create a similar prediction for any given neutrino oscillation model by applying the appropriate oscillation probability. See, for example, the toy spectra in Figure 4.4. The solid black lines trace a standard model hypothesis—one with no sterile oscillation—while the dotted and dashed lines show potential appearance and disappearance of neutrinos. This example is speciﬁc to an experiment able to measure appearance of electron neutrinos from a beam of primarily muon neutrinos.

Figure 4.4: A cartoon demonstrating an example of sterile neutrino appearance and disappearance across both a` (left) and a4 (right) channels. In this example, the solid lines show the energy distribution in each channel, assuming there are no sterile neutrino oscillations, while the dotted and dashed lines illustrate exaggerated oscillation scenarios.

In such a case, where both a` and a4 channels are measurable, one could potentially observe three types of sterile neutrino oscillation:

1. muon neutrino disappearance, where an initial a` flux experiences oscillation through another flavor state such that fewer muon flavor neutrinos are observed in the final distribution than expected;

2. electron neutrino disappearance, which follows analagously; and

3. electron neutrino appearance, wherein neutrinos initially in the a` state oscillate through

sterile states and are ﬁnally observed as an excessive a4 ﬂux.

43 There is a fundamental difference between appearance and disappearance channels: for appearance, one must know both the starting and end ﬂavor states of a neutrino ﬂux (for example,

a a4 appearance measurement from a a` beam); but disappearance is less model dependent—one only cares that the neutrinos of one flavor are gone. By introducing sterile flavor states, we bring about the possibility that some portion of neutrinos from the initial flux may be oscillating through sterile states at time of observation and subsequently go unmeasured, changing the total number of neutrino events across all flavors one would expect*. To perform a disappearance search, one must take a predicted 3a spectrum and oscillate away neutrinos based on the oscillation probability. A disappearance study manifests in practice as a scaling down of predicted events according to their true !/ (with true referring to variables used to simulate an event, as opposed to reconstructed variables that are determined by reconstruction algorithms). This procedure is illustrated in Equation 4.9 for some set of sterile neutrino parameters

Θ. The number of events in bin 8 of the ﬁnal spectrum, #8, is the sum of the number of events in that bin for both the background and the predicted signal. The predicted signal is the product

of the number of events predicted by the 3a hypothesis scaled down according to the oscillation probability of each individual event, 4, in bin 8.

# # 1:6 # ?A43 8 = 8 + 8 (4.8) Õ # 1:6 + #3a × % ( ! , ) = 8 8 aU→aU Θ; 4 4 (4.9) 4∈8

A a4 appearance search is built upon a very specific interaction assumption—that an initial flux of muon (anti)neutrinos oscillates through some indeterminate number of flavor states (both sterile and active) and is finally measured in the electron (anti)neutrino state. To facilitate this oscillation model, many analyses begin with what is called a full-oscillation or "full-osc" sample. A full-osc sample is one that convolves the muon neutrino beam flux with the electron neutrino detector cross section. This creates a sample prediction wherein every single muon neutrino of the initial muon

*As an aside, this normalization difference from oscillation effects can be reliably observed by counting NC events and disregarding ﬂavor entirely. In a 3a paradigm, the total number of NC events should not vary with !/.

44 neutrino flux is observed in the electron neutrino flavor state, or in other words, it describes a % scenario where the oscillation probability a`→a4 = 1. One can then scale events in this full-osc channel according to their true energy and distance from the neutrino source and construct a final

prediction by adding it on top of the a4 background. An illustration of this method is presented in Equation 4.10.

Õ # # 1:6 + #3a + # 5 D;;−>B2 × % ( ! , ) 8 = 8 8 8 aU→aV Θ; 4 4 (4.10) 4∈8

In any case of a4 appearance, one mustn’t neglect the simultaneous a4 disappearance. If one

were to use a sterile neutrino oscillation probability to predict a4 appearance without also considering disappearance, then they would be viewing an incomplete picture that could prove radically different from what the model should prescribe. This concept will be revived and elaborated on later, but the solution to this premise is demonstrated by combining Equations 4.9 and 4.10 in the following way:

Õ Õ # # 1:6 + #3a × % ( ! , ) + # 5 D;;−>B2 × % ( ! , ) 8 = 8 8 aV→aV Θ; 4 4 8 aU→aV Θ; 4 4 (4.11) 4∈8 4∈8

4.2.4 Predicting 3+1 a` Disappearance in MicroBooNE

As an illustrative example, we will present the process for creating the predicted spectrum for a a` disappearance search in MicroBooNE. In MicroBooNE (as with many experiments) reconstructed neutrino energy is the variable most commonly used for oscillation ﬁts because it’s the most obviously sensitive to oscillation (via !/). One begins with an MC prediction of the reconstructed neutrino energy spectrum in Micro- BooNE, comprising simulated events that are observed by the detector and selected for the analy-

sis. Each simulated event has a true neutrino energy and distance from the neutrino source, CAD4 and !CAD4, as well as the corresponding reconstructed energy as seen by analyzers, A42>. An initial, 3a prediction is shown on the left of Figure 4.5.

45 Figure 4.5: A cartoon of the method by which one builds a a` disappearance spectrum from an initial prediction under the 3a hypothesis. The center plot contains identical events to those in the left spectrum, where each individual event was scaled by the a` disappearance probability.

For a set of oscillation parameters, Θ, one can calculate the disappearance probability for each MC event,

1 − %`→` (Θ; CAD4,!CAD4), (4.12)

where %`→` is the 3+1 a` survival probability from Equation 4.3. To predict the spectrum of a`

events that oscillated out of the a` ﬂavor—disappeared—at the time of observation (seen in the middle of Figure 4.5), one must weigh each individual event by its disappearance probability. The ﬁnal disappearance prediction is the result of simple subtraction of this disappeared histogram from the No-Osc histogram, giving something like what we see in the dashed line on the right of Figure 4.5.

4.2.5 Drawing Limits and Conﬁdence Intervals

To assess the viability of any model as an accurate descriptor of physical phenomena, one must deﬁne some metric by which we can compare it to real, observed data. <2 ,* A 3+1 sterile neutrino oscillation model (in the short baseline) has three parameters, Δ 41 44,

and *`4, that form a three-dimensional parameter space: a volume, across which any point—any set of coordinates—represents a set of values for each parameter and hence, a different oscillation model. This concept can be expanded to 3+2 and 3+3 oscillation frameworks by adding more dimensions (thus making the math much harder). Let Θˆ be the full set of all points across parameter

46 space for sterile neutrino oscillation. For each sterile neutrino oscillation model in Θˆ , one can create a predicted spectrum and compare it to what was actually observed using a strictly defined metric. In this dissertation, all spectrum comparisons will be performed with some j2, which will be defined specifically for each case. By performing this comparison across all of parameter space, one arrives at a scalar j2 field across all dimensions of the space. Models with a low j2, such that one cannot readily claim the data and prediction as coming from different underlying populations, best describe the real physics. A model with a high j2 is the opposite: a poor explanation of what an experiment saw. We can use our j2 at each point to carve out regions of parameter space where enclosed oscillation models are favored or disfavored according to some degree of confidence, U. The point across all of parameter space that gives the lowest j2 is called the Best Fit (BF) point, which demonstrates the best agreement between observation and prediction. The value of j2 j2 this minimum will be labeled 1 5 . For any point P within our parameter space, we can then j2 calculate Δ %, such that j2 j2 j2 , Δ % = % − 1 5 (4.13)

j2 j2 % ˆ where % is the of the oscillation prediction at point ∈ Θ. Now, to build a confidence interval, j2 U we will assert some critical chi-squared, , for our desired . We will say that point P is "allowed" U j2 j2 j2 > j2 at (1- ) confidence if Δ % ≤ , and we will say that P is "excluded" if Δ % . This concept is illustrated in Figure 4.6. In order to build full excluded and allowed regions, one must compare the Δ j2 at each point in parameter space with the critical j2. This method of constructing confidence regions is referred to as a global scan, because we rely on a Δ j2 calculated using the global best fit across all dimensions. If one were to assume that each Δ j2 followed a smooth, Gaussian distribution with degrees j2 of freedom (DoF) equal to the dimensions of parameter space, then the corresponding to use for any U can be found in a statistics textbook. For the analyses in this chapter, this assumption is upheld despite being a massive oversimplification. By the frequentist definition of confidence intervals, the 99% allowed region represents a portion of parameter space such that if some very

47 Figure 4.6: A fake (though qualitatively representative) relation between Δ j2 and our fit parameters. This plot features a set of parameters, indicated on the horizontal axis by the best fit point, j2 j2 which most-closely model the underlying data. By defining a Δ threshold, , one can divide the parameter space into allowed (better agreement) and excluded (worse agreement) regions. large number of pseudo-experiments were run with the underlying assumption of a true oscillation model, in 99% of experiments, then the chosen true oscillation model would be somewhere within. In order to truly build a confidence interval fitting this definition, one would need to perform a full j2 frequentist study to find the effective degrees of freedom and a la Feldman and Cousins (see Section 9.2 for this study in action) [36].

Conﬁdence Intervals of a Toy Example

A toy allowed region can be seen in Figure 4.7 on the left. For easier plotting 3+1 sterile neutrino oscillations are often marginalized into two dimensions: a frequency term (Δ<2) describing how wiggly an oscillation is, and an amplitude term (sin2 2\) describing how big those wiggles are. If an allowed region does not form a closed contour and instead expands to include the entire low sin2 2\ portion of the plot (because as normalization approaches zero, a model approaches a 3a hypothesis), then none of the sterile neutrino oscillation models may be favored and one may

48 Figure 4.7: Fake sterile neutrino oscillation contours to illustrate allowed regions of 90% and 99% confidence (left) and an exclusion limit of 95% confidence (right). Oscillation models with parameters lying within the allowed regions are favored by data, while those to the right of the exclusion limit are disfavored. opt to construct an exclusion limit, as seen in Figure 4.7 on the right. A model < is excluded at U j2 > j2 j2 j2 confidence if Δ < Δ , where is defined for a 1-sided distribution with degrees of freedom equal to the number of dimensions. On such plots, any model to the right of the limit is said to be excluded by an experimental observation and should be rejected with some confidence. Creating allowed and excluded regions for different parameter spaces follows easily from the definitions above, but trouble arises when one is restricted to the two-dimensions afforded by flat media. Even in the 3+1 case, there are three dimensions, and things only get more complicated as we add sterile neutrinos. Some degree of marginalization is necessary to communicate these allowed regions in a figure. One may follow the same procedure to calculate a scalar field of j2 across all dimensions of parameter space, but when constructing the allowed and excluded region, j2 j2 should correspond to a distribution with two degrees of freedom (for a two-dimensional projection). Points in these regions may then be plotted as projections onto the 2D space. One last way of constructing an exclusion limit is with a raster scan. This assumes, as above, that one’s allowed regions do not form closed contours and therefore stretch all the way to sin2 2\ → 0. With this approach, one slices out a single Δ<2 and defines that as the true Δ<2 for sterile neutrino oscillation. All models with this Δ<2 are lined up and scanned from lowest to

49 highest B8=22\ until a j2 minimum is found. One defines Δ j2 ≡ j2 − j2 , where j2 <,Δ<2 < 1 5 ,Δ<2 1 5 ,Δ<2 is the j2 minimum of a given Δ<2 parameter. One may then define an excluded region just as j2 j2 j2 . before, but with defined for a 1-sided with 1 degree of freedom ( = 1 64). In cases where the best fit is a 3a hypothesis, a raster scan will provide an identical contour to the global exclusion limit described above, but if there is any point in 3+N parameter space that is preferred by data, a raster scan will underestimate compatibility.

4.2.6 Sensitivities

A sensitivity study follows very similar principles, but for an experiment that has not yet been conducted (therefore using hypothetical observed scenarios). The global fit study presented in this chapter is part of a larger study to assess the sensitivity of SBN and DUNE experiments to sterile neutrino oscillations [37, 38]. The key difference between a sensitivity and a limit or allowed region is that one does not have data—only predictions. For a given sterile neutrino oscillation model, one constructs a prediction by applying the appropriate probabilities to MC as in Equations 4.9-4.11 and then compares this prediction to that of a 3a SM prediction to calculate a j2. The goal in this case is to find the regions of parameter space where, were sterile neutrino oscillations truly present with a model described by those parameters, one would be able to confidently tell them apart from a 3a hypothesis. An experiment is said to be sensitive to sterile neutrino oscillations with parameters in the enclosed region, whereas those models outside will be unreachable: indistinguishable from a null hypothesis.

4.3 Sterile Neutrino Compatibility with Global Data

A sterile neutrino hypothesis that adequately resolves the anomalies of one experiment is an exciting development, but if this theory vehemently disagrees with data observed elsewhere, it is of no practical use. To this end, if one wishes to analyze the state of a 3+N hypothesis in a broader scheme, one can perform a combined fit to all relevant oscillation experiments. This global fit study aims to find the region in parameter space that best satisfies all data sets, and expands upon methods

50 described in the 2016 paper, Prospects of Light Sterile Neutrino Oscillation and CP Violation Searches at the Fermilab Short Baseline Neutrino Facility, which I co-authored [39]. The work presented in this dissertation combines that of two individual projects: a fit of 3+1, 3+2, and 3+3 sterile neutrino oscillation models to global data and exploration of potential new physics in long baseline detectors from the aforementioned publication; and a fit of 3+1 sterile neutrino oscillation models to an updated set of global data tied to an oscillation sensitivity study for the upcoming DUNE and SBN experiments, from an in-progress paper [38]. A selection of (mostly) short baseline neutrino experiments was chosen as a representative subset of all data, comprising historically important results, anomalous results, and ones that set world-leading limits on the largest areas of parameter space. A list of included experiments is provided in Table 4.1 and will be described in greater detail as to their contributions below†. Performing this study entailed collecting reference work on all selected experiments—including papers, conference talks, email correspondence, and some personal conversations—and approximately reproducing each collaboration’s sterile neutrino oscillation analysis with enough detail to effectively manipulate each experiment’s prediction into our intended oscillation model. An outline of how each reproduction was performed is written below and comparisons of my reproductions with the published results of corresponding experiments are shown in Appendix A.4. The fit methods used for several of these approximations have been previously documented within [39] and its predecessor [41], so the ink spilled here will be in service of recent contributions. Where possible, procedures for generating predictions and calculating likelihoods are exactly in accordance with collaboration-endorsed methods. Even though each result used throughout this chapter has been shown publicly by its collaboration either at a conference or in a paper, not every experiment has released their data or methods publicly. In such cases where details are scarce, my best approximation based on physics principles, other experiments, and pieces of detector setup information were combined with a digitization of public figures to approximate the results [42].

†A reader in the ﬁeld may recognize the absence of recent results from the MINOS+ and Daya Bay [40]. When taking on this project, the author assumed that IceCube would stake a dominating claim in the a` disappearance channel, rendering the addition of MINOS+ and Daya-Bay moot. As recent results have demonstrated otherwise, these data sets will be considered in upcoming studies [38].

51 Dataset Oscillation Channel

Appearance

KARMEN a¯` → a¯4

LSND a¯` → a¯4 (−) (−) MiniBooNE - BNB a ` → a 4

MiniBooNE - NuMI a` → a4

NOMAD a` → a4 Disappearance

KARMEN, LSND (xsec) a4 → a4

Gallium (GALLEX and SAGE) a4 → a4

Bugey a¯4 → a¯4

DANSS a¯4 → a¯4

NEOS a¯4 → a¯4 (−) (−) MiniBooNE - BNB a ` → a `

CCFR84 a` → a`

CDHS a` → a`

IceCube a` → a`

Table 4.1: Experiments whose data sets comprise the presented global ﬁt to 3+1 sterile neutrino oscillation models, sub-divided by their channels of measurable oscillation.

52 4.3.1 a4/a¯4 Appearance Experiments

The following experiments all made measurements of a4/a¯4 appearance from an initial, mostly

a`/a¯`, ﬂux: LSND [18], the KArlsruhe Rutherford Medium Energy Neutrino experiment (KAR- MEN) [43], MiniBooNE (both 2018 data [35] and measurements of neutrinos from the Neutrino Main Injector (NuMI) beam at Fermilab [44])‡, and the Neutrino Oscillation MAgnetic Detector (NOMAD) experiment [45].

Different parametrizations of the 3+1 a4 appearance oscillation probability are listed in Equa- tions 4.14-4.16 below. Each equation is equivalent, but has different use cases. The parameters in Equation 4.16 are most often utilized for sensitivity plots because all mixing elements are marginal-

2 ized to a single parameter (sin 2\`4) for convenient 2D display.

% a a * 2 * 2 B8=2 . <2 ! ( ` → 4) = 4 | `4| | 44| (1 27Δ 41 / ) (4.14) B8=2 \ B8=2 \ B8=2 . <2 ! = (2 14) (2 24) (1 27Δ 41 / ) (4.15) B8=2 \ B8=2 . <2 ! = (2 `4) (1 27Δ 41 / ) (4.16)

Allowed regions or exclusion limits for each individual a4 appearance data set are superimposed in Figure 4.8 with the combined allowed region of all a4 appearance data together. A helpful summary of this ﬁt is presented in Table 4.2, with each column deﬁned accordingly:

• DoF: Degrees of freedom for a dataset, which is equivalent to the number of bins used in the ﬁt less any nuisance parameters for that individual experiment.

<2 4+2 B8=2 \ • Δ 41 [ ] and (2 `4): Oscillation parameters of the best ﬁt point of a given data set, using the parametrization shown in Equation 4.16.

‡Despite a more recent MiniBooNE data release being made public [20], this work uses the previous iteration from 2018 [35]. At the time of writing, the 2020 MiniBooNE data release contained broken and incorrect data sets making accurate reproduction of published results impossible. The MiniBooNE collaboration was informed and has taken steps to ﬁx the release.

53 Figure 4.8: A collage of individual 3+1 a4 appearance results from each experiment included within this dissertation’s global ﬁt, and the allowed region of a combined ﬁt over all of those experiments together.

j2 j2 • 1 5 ,B8=6;4: The minimum (i.e. that of the best ﬁt) resulting from a comparison of observed data with predicted oscillations across all of parameter space for this individual experiment.

j2 j2 a • =>>B2: The of a 3 hypothesis compared with data in this individual experiment.

j2 j2 • 1 5 ,6;>10;: The of a prediction made at the global best ﬁt point across all experiments in Table 4.1 compared with data in this individual experiment. This can be seen as each data set’s contribution to the global j2 minimum.

j2 j2 • 1 5 ,38B: The of a prediction made at the best ﬁt point across all disappearance experiments compared with data in this individual experiment, shown to highlight particular tensions between appearance and disappearance data sets.

All appearance experiments in this study are used exactly as presented in [39] with one exception: the MiniBooNE results not only use a data set updated as of 2018, but also their inclusion has been adjusted to a combined analysis of both a4 and a¯4 appearance simultaneously. As demonstrated by the allowed regions in Figure 4.8, LSND and MiniBooNE were the only two experiments with data that could effectively reject the no-osc hypothesis and demonstrate evidence of sterile neutrino oscillation—each of the others could only provide exclusion, ﬁnding

54 <2 4+2 B8=2 \ j2 j2 j2 j2 Dataset DoF Δ 41 [ ] (2 `4) 1 5 ,B8=6;4 =>>B2 1 5 ,6;>10; 1 5 ,38B LSND 5 0.059 0.73 3.4 52 17 6.0 KARMEN 9 6.8 0.0010 6.0 7.1 9.3 30 MiniBooNE 37 0.041 0.84 39 51 42 94 NuMI 10 6.8 0.020 2.0 6.7 6.0 4.5 NOMAD 30 51 0.0 34 35 40 132

Table 4.2: Table of a4 appearance experiments, the degrees of freedom each dataset contributes, the parameters of a best ﬁt to 3+1 sterile neutrino oscillation to each data set, and several j2 measurements explicitly detailed in the text. no such evidence. Even so, the allowed region of the combination of all data sets on the right plot shows that the LSND and MiniBooNE data sets hold the most sway over this oscillation channel.

4.3.2 a4/a¯4 Disappearance Experiments

The following experiments all made measurements of a4/a¯4 disappearance: The Soviet–American Gallium Experiment (SAGE)[8] and the Gallium Experiment (GALLEX) [9], the Neutrino Exper- iment for Oscillations at Short base line (NEOS) [46, 47, 48, 49, 50, 51], DANSS [46, 52, 53], Bugey [54], and a combined LSND and KARMEN cross-section study [55]. The two gallium experiments are frequently paired under the shared moniker "Gallium," as is the case in this analysis. The most common type of a¯4 disappearance experiment (and the case for NEOS, DANSS, and Bugey) places a detector somewhere outside an active nuclear reactor to measure low-energy electron neutrinos from beta decay. Unlike in the appearance scenario, disappearance does not ask which ﬂavors these neutrinos oscillated to, but merely how many are gone. As given with appearance, common parametrizations of a4 disappearance in a 3+1 sterile neutrino model are shown in Equations 4.17-4.19. One will notice that there is no explicit dependence upon \24 or |*`4|. Experiments measuring a4 disappearance will therefore only show sensitivity to two effective oscillation parameters.

55 % a a * 2 * 2 B8=2 . <2 ! ( 4 → 4) = 1 − 4 | 44| (1 − | 44| ) (1 27Δ 41 / ) (4.17) B8=2 \ B8=2 . <2 ! = 1 − (2 14) (1 27Δ 41 / ) (4.18) B8=2 \ B8=2 . <2 ! = 1 − (2 44) (1 27Δ 41 / ) (4.19)

Figure 4.9: A collage of individual 3+1 a4 disappearance results from each experiment included within this dissertation’s global ﬁt, and the exclusion contour of a combined ﬁt over all of those experiments together.

Exclusion limits for each individual a4 disappearance data set are superimposed in Figure 4.9 with the combined allowed region of all a4 appearance data together. A helpful summary of this ﬁt is presented in Table 4.3, with each column deﬁned exactly as in the a4 appearance case with two exceptions:

<2 4+2 B8=2 \ • Δ 41 [ ] and (2 44): We use the oscillation parameters according to the parametrization in Equation 4.19.

j2 j2 • 1 5 ,0??: We identify the of a prediction made at the best ﬁt point across all appearance experiments.

All of these a4 disappearance experiments are used exactly as presented in [39] excepting the

56 <2 4+2 B8=2 \ j2 j2 j2 j2 Dataset DoF Δ 41 [ ] (2 44) 1 5 ,B8=6;4 =>>B2 1 5 ,6;>10; 1 5 ,0?? Gallium 4 2.0 0.22 2.9 7.7 4.8 7.2 NEOS 60 0.091 0.048 40 47 46 174 DANSS 24 4.4 0.84 9.2 18 19 24 Bugey 60 0.90 0.052 47 51 52 135 LSND/KAR. XSec 11 7.4 0.30 5.5 10 10 10

Table 4.3: Table of a4 disappearance experiments, the degrees of freedom each dataset contributes, and several ﬁt measurements, each of which is explicitly detailed in the text.

addition of NEOS and DANSS, whose data were only made public in recent years. More detailed descriptions of these two experiments are given below for context.

Unlike with the appearance case, a4 disappearance shows only exclusion, with NEOS and Bugey acting as dominant contributors (unsurprising given their many bins, indicating an abun- dance of data with which to ﬂood a combined j2). None of these experiments witness any evidence of sterile neutrino oscillation at the short base line.

NEOS

The Neutrino Experiment for Oscillations at Short base line—or NEOS—uses a 1 ton liquid scintillator detector positioned 23 meters from a reactor at the Hanbit Nuclear Power Complex in Yeong-gwang, Korea. To account for the known bias in prediction stemming from the unresolved

reactor anomaly [46], NEOS instead compares their data to a 3a prediction based on observations made by the Daya Bay collaboration—a similar experiment at a nuclear reactor complex in China. NEOS concluded no strong evidence for 3+1 oscillations. A detailed overview of the NEOS oscillation analysis performed for this dissertation is in Ap- pendix A.3.

57 DANSS

DANSS uses a plastic scintillator detector underneath the Lalinin Nuclear Power Plant near Moscow, Russia. The detector volume is moved between three positions from the nuclear reactor, and the differences in efﬁciency between these distances allow measurement of oscillations. An exclusion constraint has been placed, but actual signiﬁcance of potential sterile neutrino oscillations are forthcoming. A detailed overview of the DANSS oscillation analysis performed for this dissertation is in Appendix A.2.

4.3.3 a`/a¯` Disappearance Experiments

The following experiments all made measurements of a`/a¯` disappearance: MiniBooNE [56], the CERN-Dortmund-Heidelberg-Saclay (CDHS) experiment [57], the Columbia-Chicago-Fermilab- Rochester (CCFR) experiment [58], and IceCube [59]. Of note here is the inclusion of IceCube, a very-long-base line experiment and hence an outlier in this project, but one that nonetheless posits a powerful constraint on this oscillation and thereby warrants inclusion.

Common parametrizations of a` disappearance in a 3+1 sterile neutrino model are given in equations 4.20-4.22.

2 2 2 %(a` → a`) = 1 − 4 |*`4| (1 − |*`4| ) B8= (G41) (4.20)

2 2 2 2 2 = 1 − 4 2>B (\14) B8= (\24)(1 − 2>B (\14) B8= (\24)) B8= (G41) (4.21)

2 2 = 1 − B8= (2\``) B8= (G41) (4.22)

G . <2 ! where 41 abbreviates 1 27Δ 41 / .

Exclusion limits for each individual a` disappearance data set are superimposed in Figure 4.10. A summary of this ﬁt is presented in Table 4.4, with each column deﬁned as in each previous oscillation scenario with the singular exception:

58 Figure 4.10: A collage of individual 3+1 a` disappearance results from each experiment included within this dissertation’s global ﬁt, and the exclusion contour of a combined ﬁt over all of those experiments together.

<2 4+2 B8=2 \ j2 j2 j2 j2 Dataset DoF Δ 41 [ ] (2 ``) 1 5 ,B8=6;4 =>>B2 1 5 ,6;>10; 1 5 ,0?? MiniBooNE a Dis 16 16 0.15 13 19 19 19 MiniBooNE a¯ Dis 16 23 0.30 7.6 11 11 10 CDHS 15 25 0.28 9.0 14 14 15 CCFR 18 36 0.076 15 18 17 18 IceCube 207 0.012 0.0 224 224 240 925

Table 4.4: Table of a` disappearance experiments, the degrees of freedom each dataset contributes, and several ﬁt measurements, each of which is explicitly detailed in the text.

<2 4+2 B8=2 \ • Δ 41 [ ] and (2 ``): We use the oscillation parameters according to the parametrization in Equation 4.22.

The only addition to this set for the present analysis is IceCube, which, as demonstrated in

Figure 4.10, strongly dominates the a` disappearance channel with its exclusion (see the light- orange contour on the left plot). None of the a` disappearance experiments ﬁnd any substantial evidence to support sterile neutrino oscillation at the short base line.

59 IceCube

IceCube is the only long-base line experiment included, but its recent results and strong \24 bound were too compelling to ignore on those grounds. IceCube is a novel water-Cherenkov detector whose active volume comprises 1 :<3 of Antarctic ice. Neutrinos are measured using an array of PMTs strategically buried throughout the ice sheet. The experiment places a powerful constraint on a` disappearance, ﬁnding no strong evidence of 3+1 sterile neutrino oscillation.

Reproducing the IceCube result involves propogating the a`/a¯` flux (containing both atmospheric pion and kaon decay contributions) through the Earth, to account for matter effects in oscillation. This intense calculation is carried out via nuSQuIDS, an integration and propagation tool built by members of the IceCube collaboration specifically for this challenge. Armed with the oscillated fluxes for a given 3+1 sterile neutrino oscillation model, one must then find the total number of events per bin by performing a fit across a series of nuisance parameters to account for systematic effects. This full process is outlined in great detail within Appendix A.1. Some important differences between our implementation of the IceCube result and what is presented by the IceCube collaboration are the following:

• To save processing time, only a subset of nuisance parameters were used and detector effects were neglected. This approximation shows a slight improvement in the limit relative to what IceCube has shown.

• The IceCube oscillation study in the reference does not vary each oscillation parameter for

its ﬁnal result, choosing instead to set \14 = \34 = 0 and therefore reducing a` disappearance

2 senstivity to primarily depend on only Δ< and \14. For the purposes of this study, it was important to oscillate over each parameter pertaining to sterile neutrino oscillation.

4.3.4 Global Short Baseline Data

The 3+1 sterile neutrino allowed regions for the data sets listed in Table 4.1 are presented in Figure 4.11, marginalized across appearance and both disappearance parametrizations and juxta-

60 posed with leading contributors to those respective oscillation channels.

Figure 4.11: Two dimensional projections of a globally allowed region of 3+1 sterile neutrino oscillation, determined via a combined ﬁt of each aforementioned data set, across three oscillation parametrizations: a4 appearance (left), a` disappearance (middle), a4 disappearance (right). For convenient comparison and reference, the data set that dominates each channel is overlaid.

For this, specifically a 3+1 study, only three variables were needed, so performing a sufficiently <2 \ \ dense grid scan across three dimensions was computationally reasonable. Δ 41, 14, and 24 are all varied logarithmically across their respective parameter ranges: [0.01, 100.0]4+2 for the mass- squared parameter and [0.01, c/4] radians for each of the mixing angles. A 100 × 100 × 100 grid was scanned and at each point in this volume, the j2 was calculated for the given oscillation model to each data set. The j2 minimum is 518.128 over 519 degrees <2 ,\ ,\ ,\ . 4+2, . , . , . of freedom, found at (Δ 41 14 24 34) = (51 84 0 150 0 0731 0 0122). In terms of the oscillation amplitudes for each individual oscillation channel, the best fit point corresponds to

2 −3 2 −2 2 −2 sin 2\`4 = 2.08 × 10 , sin 2\`` = 2.14 × 10 , and sin 2\44 = 9.46 × 10 . This best fit comes as an alternative to the null-hypothesis, a 3a prediction, which gives j2 =573.17 (with 522 degrees of freedom). From these plots, a striking revelation is that the best fit Δ<2 is quite high. That is believable in the context that disappearance experiments place very strong exclusion constraints on the mid-to- low Δ<2 regions, but when one considers the fits to each individual experiment (tabulated above), they will notice that no individual data set strongly favors a mass term of this energy (except for

61 NOMAD, which has a negligible j2 contribution). In fact, the global fit point is not a particularly strong fit for any of the individual experiments. One can also see how the methodology employed in summing the j2 for each experiment substantially favors experiments with more bins and therefore more data. That is correct, but only serves to reflect the dearth of strong appearance data against an overwhelming avalanche of exclusionary disappearance observations.

4.3.5 Tensions and Limitations

Even if one’s global fit results in a contour, the line of questioning regarding sterile neutrino oscillations may not be neatly wrapped up without reconsidering some of the initial drivers. Simply put, the most likely set of parameters for 3+1 sterile neutrino oscillation may have a use case that ends there: being the least-bad fit to an inapt hypothesis. The j2/> of our 3+1 global fit comes to 1.00, demonstrating good agreement—notably better than what is seen for the 3a SM hypothesis (with j2/>=1.09). However, as seen in Table 4.2, MiniBooNE and LSND are the strongest drivers of disagreement with the SM hypothesis with other experiments showing little-to-no preference for any 3+1 oscillation. Considering this list of experiments as a gestalt is thereby misleading. It would behoove us to investigate this fit at a more granular level. Here, we explore a few examples that reflect the limitations of our methods, assumptions, and the concept of performing a global fit altogether. This discussion does not intend to undermine my own work (or the work of many others), but to emphasize that while global fits are a very useful tool for understanding the relationships between individual experiments and the wider field, care must be taken in their interpretation.

Return to MiniBooNE

The inciting event for this sterile neutrino hypothesis and search was an anomalous excess observed by LSND, later compounded by MiniBooNE. One can conduct a helpful exercise by bringing their ﬁnal result back to the source. In Figure 4.12, we consider the MiniBooNE result

62 speciﬁcally, for its convenient data structure and versatility.

Figure 4.12: MiniBooNE data [35] (black dots) compared with three predictions: that of a 3a hypothesis, that of the best fit across 3+1 oscillations to MiniBooNE data, and that of the best fit across 3+1 oscillations to data from all experiments in the study. While both best fits demonstrate better agreement than the null hypothesis, neither adequately resolves disagreement at low energy.

The observed and predicted neutrino energy spectra are plotted for three cases: a no oscillation

(3a) hypothesis, the MiniBooNE individual best fit parameters, and the global best fit parameters. The MiniBooNE best fit point touts vastly improved agreement with the data by the j2 metric, but neither of the 3+1 sterile neutrino oscillation hypotheses is able to effectively address the excess in the first energy bin. To this end, the global best fit point hardly addresses the LEE at all, very likely being pulled by the intense pressure of high-sensitivity disappearance experiments like IceCube and NEOS. Whether or not 4+-scale sterile neutrinos exist and however they may smooth discrepancies across some experimental observations, they are unlikely to be the sole cause of the LEE, at least under the 3+1 hypothesis.

63 Tensions Between Subsets

One may also consider whether or not individual subsets of experiments directly contradict one another. Figure 4.13 shows the 90% and 99% allowed regions for all appearance data sets

on the same plot as the 95% exclusion contour for all of the disappearance experiments (both a4

2 and a` disappearance). Despite achieving 3+1 global best ﬁt with a j lower than that of the null hypothesis, the disappearance experiments soundly exclude all of the space best allowed by the appearance experiments.

Figure 4.13: The allowed region of a combined ﬁt of a4 appearance data sets to a 3+1 sterile neutrino hypothesis in contrast with the exclusion region of all disappearance data sets combined together.

As another example, Figure 4.14 carves two similar subsets; one can compare the allowed

region for MiniBooNE and LSND data sets (the originators of the 14+2 sterile neutrino hypothesis) with the excluded region of each other experiment included in the study. MiniBooNE and LSND cooperate very well with each other in these ﬁts: their combined ﬁt exists in the overlap between both experiments’ allowed regions. Apart from these two experiments, one must grapple with the incredible weight of exclusion from disappearance experiments, which

contain vastly more data. Even with MiniBooNE’s and LSND’s very decisive Δ j2 (−12.84 and

64 Figure 4.14: The allowed region of a combined ﬁt of MiniBooNE and LSND data sets to a 3+1 sterile neutrino hypothesis in contrast with the exclusion region of all other included data sets combined together.

−49.19 respectively, disfavoring a 3a prediction), their conviction in a sterile neutrino hypothesis is handily disregarded by global data.

More Statistical Rigor

There are some core truths that this global fit analysis has taken for granted that must be ex- amined more closely, at the forefront of which is the naive use of a summation of the j2 for each individual experiment into one grand total for the global fit. This assumption posits both that each experiment is entirely independent despite potential correlations in flux, cross-section model, or other systematic similarities, and that the distribution of the global Δ j2 follows that of a j2 with degrees of freedom equal to the sum of all bins less the number of fit parameters. These are big assumptions, yet both could be addressed through a more rigorous frequentist analysis using the Feldman and Cousins framework [36] introduced in Section 4.2.5 that will later be used for MicroBooNE a` disappearance in Section 9.2. In such an approach, for each given sterile oscillation model, one would consider not one single observation from a single experiment, but rather some incredibly large number of observations from some incredibly large number of

65 pseudo-experiments. Contours corresponding to an X% conﬁdence level (CL) could then be drawn more explicitly around regions where the truth is found X% of the time. This project would be very computationally taxing, especially over large parameter spaces, but it is under active discussion between collaborators who aim to translate this presented work into a GPU-compatible (and thus parallelizable) form for the future [60].

4.4 The Future of Global Fit Analyses

Global ﬁts are an imperfect way of exploring sterile neutrino oscillation, but if one is undeterred by those limits, then there is a lot more useful physics one can explore in this phenomenological space. The presented ﬁts are only the tip of the iceberg, and we can look at our previous work for directions in which to expand with future experiments like the Short Baseline Neutrino (SBN) experiment and DUNE.

4.4.1 The 2016 Global Fit

The global fit presented in Section 4.3 comes as the second iteration of a project begun in 2016 [39]. In that project, a similar global fit was performed (on a slightly different group of data sets) for the purpose of assessing the sensitivity of the forthcoming SBN experiment relative to important regions of parameter space. Global fits were made across 3+1, 3+2, and 3+3 sterile neutrino oscillation models, and the results are given in Figures 4.15 and 4.16. Notably, this is the first time that SBN’s sensitivities to 3+2 and 3+3 sterile neutrino oscillation scenarios have been assessed. Important differences between the studies are as follows:

• In the 2016 study, MiniBooNE results are shown with 2012 MiniBooNE data, which analyze

a4 and a¯4 appearance separately, rather than the combined study used in the 2018 release.

Also, results of the MINOS charged current a` disappearance study and the T2K data are neglected in the more recent paper, while those from DANSS, NEOS, and IceCube have been added.

66 Figure 4.15: Globally allowed regions for 3+1 (left) and 3+2 (right) sterile neutrino oscillation as derived in [39]. The 3+1 allowed region is overlaid with contemporary ﬁts to comparable SBL <2 <2 data sets. Note that the 3+2 oscillation plot is marginalized so as to only portray Δ 41 vs Δ 51.

Figure 4.16: Globally allowed region for 3+3 sterile neutrino oscillation as derived in [39]. The 12 <2 <2 parameters of 3+3 SBL oscillation have been marginalized in two ways, showing Δ 41 vs Δ 51 <2 <2 (left) and Δ 41 vs Δ 61 (right).

• Our contemporary study intends to use global ﬁts to both update the sensitivity of SBN and benchmark the sensitivity for the long base line (LBL) experiment, DUNE. Incorporating DUNE led us to limit consideration of oscillation models to 3+1, to include IceCube (another LBL experiment), and to change our oscillation formalism from PNMS matrix elements to

mixing angles for convenience. IceCube’s dominance over the a` disappearance channel allowed the dropping of the atmospheric constraints and MINOS.

Best fit points from the 2016 global fits are recorded in Table 4.5 for reference. As one would expect, when more parameters are added, one finds models that allow better

67 <2 * * j2 3> 5 (3+1) Δ 41 `4 44 / Best Fit 0.92 0.17 0.15 245.6/240

<2 * * <2 * * q j2 3> 5 (3+2) Δ 41 `4 44 Δ 51 `5 45 54 / Best Fit 0.46 0.15 0.13 0.77 0.13 0.14 5.56 238.2/236

<2 * * <2 * * <2 * * (3+3) Δ 41 `4 44 Δ 51 `5 45 Δ 61 `6 46 Best Fit 0.68 0.18 0.12 0.90 0.13 0.14 1.55 0.03 0.12 2 q54 q64 q65 j /3> 5 5.60 4.31 3.93 232.5/231

Table 4.5: Mass-squared splittings are presented in eV2, and CP-violating factors are given in radians. The null hypothesis has a j2/3> 5 = 299.5/243.

agreement, and the j2 minimum decreases with each added sterile mass state. The biggest gains, however, emerge out of the jump from 3+1 to 3+2, where the addition of a CP violating phase introduces ﬂexibility to relieve some of the tension between neutrino and antineutrino datasets.

A Note on Scanning Parameter Space

Performing a grid scan across 3 dimensions, as in a 3+1 scenario, is relatively fast and easy when one’s code is efficient. Difficulties arise in the scanning and plotting of 3+2 and 3+3 analyses. With 7 or 12 parameters, as one would encounter respectively in these cases, a grid scan of the same density would take an incredibly long time (on the order of centuries) on a single CPU, so a reliable solution was found to probe as much of the parameter space as possible while not taking hundreds of years. These scans were performed using Markov chains: a model of scanning through a space where each link of the chain represents a step through parameter space that seeks out a global minimum and each step depends only on the one preceding. <2 For these fits, Δ 81 were allowed to cover the same range as in the 3+1 case, but PNMS matrix elements were used in lieu of mixing angles, and CP-violating phases were introduced

68 with the higher dimensions, with ranges *U8 ∈ [0, 0.5] and q8 9 ∈ [0, c). Here, U ∈ (4, `) and 8, 9 ∈ [1, 2, 3]. The starting point of each chain is generated randomly, with each oscillation parameter taken from a ﬂat distribution across its corresponding bounds. The coordinates of the next step in a chain, \ are given by

\=4F = \>;3 + (' − 0.5)(\<0G − \<8=)B, (4.23)

where ' is a random number in (0,1), generated with the ROOT TRandom3 package, B is a conﬁg-

urable step size, and \<0G and \<8= are the upper and lower bounds of parameter \. For each step, a j2 is calculated for the set of oscillation parameters. This new ﬁt is then compared to that of the j2 previous step in the chain, >;3, to determine the probability of accepting this point as the next in

the Markov chain, %) , such that

% , 4G? j2 j2 ) , ) = min(1 (−( − >;3)/ )) (4.24)

where ) is also a conﬁgurable parameter in the Markov chain. By randomly varying the values of ', B, and ), one can combine multiple minimization chains to reach the global minimum j2 while evading local minima. Each individual chain will therefore roll down the j2 hypersurface gradient until resting at a local minimum. We next needed to convince ourselves that the Markov chain method was actually probing the fullness of the space and not missing any vast areas. We could tweak the length of an individual chain to achieve optimal depth—ensuring a local minimum is reached by every chain—and we could tweak the step size, temperature, and number of chains—ensuring that the full parameter space is explored and that our chain is more likely to ﬁnd the true minimum. Closure tests to verify the completeness of our Markov chain implementation included val- idations that each individual chain be long enough to settle upon a minimum point, that each individual parameter’s range be spanned across all chains, and that a plateau of minimization was

69 reached when adding additional chains to our scan.

4.4.2 Coverage Plots

Marginalization of multidimensional spaces—for example, projecting a 3D parameter space to a more digestible 2D format—is widely accepted in the field for illustrating 3+1 sterile neutrino oscillations and is trivially expanded to more parameters. However there is currently no consensus on how to best interpret the 7 or 12 parameters of 3+2 and 3+3 oscillation allowed regions with any degree of scrutability. How does one then meaningfully communicate the outcome of such a fit? This problem was circumvented by entangling our global fit results with a sensitivity study of the future SBN experiment and creating coverage plots. The construction of these plots is as follows. One begins with a globally allowed hypervolume of some confidence, comprising as many dimensions as are needed for the given sterile neutrino oscillation model. It helps to imagine this as a blob, where every oscillation model with a Δ j2 j2 below the relevant U comprises its volume. We then normalize the extent of this blob by superim- posing each point to an even grid across every dimension. One is left, in the case of our study, with two blobs of uniform density, each made up of several individual coordinate points, representing the 90% CL and 99% CL allowed regions for 3+N oscillations across all data sets.

Figure 4.17: A demonstration of our coverage concept in the 2-dimensional case, showing an allowed region on the left, a sensitivity contour in the middle, and the resulting overlap on the right. This can be extrapolated to any number of dimensions, where the allowed region is replaced by a corresponding allowed hypervolume, and the contour is replaced by a sensitivity hypersurface.

70 For each point in a blob, one may take the corresponding oscillation model and measure the sensitivity of SBN to resolve those sterile neutrino oscillations. Some fraction of this blob’s points will be covered at some sensitivity by SBN. One can think of this sensitivity curve as taking a huge bite out of the blob. The fraction of the blob covered at a given sensitivity is its coverage. This is illustrated in Figure 4.17, where the fraction of purple on the right plot in relation to the entire blob is the coverage.

Signiﬁcance(σ) 1 2 3 4 5 6 7

140 3+1 νe Appearance Only 120 νμ Disappearance Only

100 νe App/Dis &ν μ Dis

80 νe App &ν μ Dis

60 99 % C.L Coverage 40

20 3 5 σ σ

Global 0 5 10 50 100 Δχ2

Figure 4.18: Coverage of the 2016 globally allowed region of 3+1 parameter space (Figure 4.15, left) by SBN’s sensitivity under four different oscillation study methods [39].

These steps all come together in the final plot shown in Figure 4.18, where the vertical axis is percent coverage of the global 99% allowed region for 3+1 sterile neutrino oscillation, and the horizontal axis is the strength of that coverage in terms of the Δ j2 of SBN to covered oscillation models. If we follow the solid red line from the left across the plot, then we note, for example, that SBN covers 87% of the 99% globally allowed region from the 2016 study at 5f confidence. And SBN covers close to 100% of the allowed region at 3f confidence. This plot does not directly display the shape or size of a global fit, but it effectively reveals that SBN will be incredibly sensitive to regions of parameter space that are highly favored by global data. In other words,

71 if the universe reflected a 3+1 sterile neutrino hypothesis, then our global fit tells us where the parameters are most likely to be, given global data, and our coverage plot tells us that SBN will be able to very strongly discern most of those likely parameter coordinates. If one of those points describes true sterile oscillation, then SBN will be able to resolve it. Analogous coverage plots for the 3+2 and 3+3 hypotheses are shown in Figure 4.19, where one can clearly see that the dimensionality of an oscillation hypothesis does not affect how one interprets or draws conclusions from these plots. One last illustrative point is demonstrated by the differently styled lines in each coverage plot, which show the kinds of sensitivity studies one can perform with SBN. By performing a combined search of a4 appearance, a4 disappearance, and a` disappearance, one gains substantially in sensitivity compared to a4 appearance or a` disappearance alone. The full combined study does not appear to be the most sensitive, however—if one does not consider a4 disappearance, then they are failing to adequately predict their a4 background and will therefore artificially inflate sensitivity. Similar coverage studies are in progress, tying our new globally allowed region (as seen in Figure 4.11) to a new combined sensitivity analysis of both SBN and DUNE [38].

72 Signiﬁcance(σ) 1 2 3 4 5 6 7

140 3+2 νe Appearance Only 120 νμ Disappearance Only

100 νe App/Dis &ν μ Dis

80 νe App &ν μ Dis

60 99 % C.L Coverage 40

20 3 5 σ σ

Global 0 10 50 100 Δχ2 Signiﬁcance(σ) 1 2 3 4 5 6 7 120 3 5

σ σ 3+3 ν Appearance Only 100 e νμ Disappearance Only

80 νe App/Dis &ν μ Dis

νe App &ν μ Dis 60

40 99 % C.L Coverage

Global 0 10 50 100 Δχ2

Figure 4.19: Coverage of the 2016 globally allowed region of 3+2 (Figure 4.15, right) and 3+3 (Figure 4.16) parameter space by SBN’s sensitivity under four different oscillation study methods [39].

73 Chapter 5: A Deep Learning-Based LEE Search with MicroBooNE

MiniBooNE’s irreducible c0 background has a considerably more distinct signal in Micro- BooNE. Topologically, a photon-instigated EM shower can be distinguished from one that is electron-instigated by a resolvable gap between the neutrino vertex and the start of the shower (as demonstrated in Section 2.3). Calorimetrically, the start of a photon shower will arise from the charged electron-positron pair and therefore distinguish itself with double the ionization rate as an electron shower. If we couple this background differentiation with the fact that MicroBooNE, at 470 meters from the beam source, will see comparable oscillation effects to MiniBooNE, itself at 571 meters from the same beam, then we can expect MicroBooNE to be in a prime position to provide some answers for the LEE. In this section, we present the MicroBooNE collaboration’s grand strategy to tackle this anomaly and introduce the Deep Learning (DL) LEE analysis: a search for the MiniBooNE low energy excess under the eLEE hypothesis, utilizing deep-learning based reconstruction and selection techniques.

5.1 The MicroBooNE Path

MicroBooNE’s path from onset to result is mapped out in Figure 5.1. MicroBooNE operates according to a blind analysis strategy, wherein data corresponding to the speciﬁc signal deﬁnition— here a4 CCQE-like electron candidate events with reconstructed neutrino energy below 1.5 GeV— is not accessible until all algorithms have been frozen, locked, and thoroughly vetted. Samples of data adjacent to the signal (and meeting appropriate blindness criteria*) are called sidebands, and are used to prove an analysis’ robustness and accuracy before access is given to the signal data.

The largest sideband, that of a` CCQE events, is the focus of this thesis. Other notable sidebands

74 Figure 5.1: A ﬂowchart of MicroBooNE’s blind analysis strategy, designed to both eliminate analytical bias and to provide a clear road map for achieving a ﬁnal result.

0 are high-energy a4 CCQE candidate events and reconstructed NC c events. At the time of writing, MicroBooNE is preparing to open near sidebands—ones even closer to the LEE signal. That said, as the a` CCQE candidate events constitute a signal-blind data set with virtually no risk of signal contamination, all necessary data is available, and the analysis presented is unaffected and ﬁnal. With such new technology, the very dedicated MicroBooNE collaboration was not content to adopt traditional means for undergoing its LEE search. In fact, four independent analyses have been enacted, each taking a unique approach and probing the LEE signal in different, comple- mentary ways. These four separate analyses are referred to as PeLEE [61], DL LEE [62], Wire- Cell [63], and Single Photon [64]. This dissertation is preoccupied with the DL LEE analysis and thus will describe that analysis in greater detail elsewhere. For completeness, summaries of the other three analyses follow.

PeLEE

PeLEE, short for Pandora-based eLEE search [61], relies on the Pandora reconstruction algorithm suite [65] for its analysis. This reconstruction model groups ionized electrons drifted towards

*To maintain data blindness, analyzers must prove a sideband contains no more than 45 a4 CCQE candidate events below 1.5 GeV before any data may be opened.

75 the TPC wire array as "hits" in 3D space and reconstructs tracks and showers using topological and calorimetric information. This technique is entirely agnostic to the underlying kinematics that may have caused interactions, but those physics principles can be applied during event selection by PeLEE analyzers to isolate a pure signal from their reconstructed events.

Wire-Cell

The Wire-Cell analysis [63, 66] divides the detector into 3D voxels and performs a tomographic image scan of the charge density within the detector. For a given time slice, the wire plane is divided into "wire-cells," 2D regions of overlapping wires. Advanced signal processing allows for very accurate matching of an individual charge across wires. Speciﬁcally, Wire-Cell utilizes geometry, time information, and prior knowledge of the sparsity of neutrino events to help minimize ambiguities that would otherwise impede such reconstruction. This voxel view of the detector also plays a part in Wire-Cell’s strategy of cosmic ray rejection [67]. The scintillation light signature is measured from neutrino events simulated across each voxel in the detector and saved into a large library. When an actual neutrino interacts, Wire-Cell can check the observed light against its library to pinpoint the neutrino vertex. Other stray particles can be tagged for removal as cosmic contamination. Wire-Cell’s cosmic-tagging algorithm is so effective that it has been adopted by the DL LEE analysis.

Single Photon

As the name portends, the photon LEE (or gLEE) search is the single MicroBooNE analysis searching for the LEE under a photon hypothesis [64]. This analysis explores the possibility that the LEE is caused by mistaken measurements of single-photon events resulting from neutral current Δ radiative decay (NC Δ → #W where # is a ﬁnal-state nucleon). The gLEE analysis utilizes Pandora reconstruction tools to entirely different ends: to create a world-leading constraint on the cross section of NC Δ and NC c0 interactions in liquid Argon, and also to test the viability of a single-photon interpretation of the LEE hypothesis.

76 5.2 DL LEE Philosophy and Reconstruction

Each of the four LEE analyses begins with the same samples of data and Monte Carlo (MC) with the most generic reconstruction. These samples are run through the Common Optical Filter, also called "PMT Precuts," which requires that 20 photo-electrons are received in a short period during the beam spill and that no photo-electrons are received in a short period before. With this noise reduction measure applied, analyzers are provided with waveforms for each event across all channels (both wire and PMT). From this point forward, each analysis team diverges according to their independent strategies.

5.2.1 The DL Reconstruction Strategy

The DL LEE analysis differentiates itself from other MicroBooNE strategies in three primary ways:

• The use of deep learning techniques for categorizing pixels as either shower-like or track- like, reconstructing shower energy, and multi-particle identiﬁcation (MPID).

• An additive reconstruction tactic, which builds out from candidate vertices based on two- body QE kinematics and is explicitly model-dependent.

• A narrow focus on one-lepton-one-proton (1;1?) a4 and a` CCQE events, for a smaller, yet better-understood signal. The lepton (;) in question is either an electron or a muon.

This analysis follows the assumption that the LEE is a4 CC-like, so a specific focus on CCQE interactions both places our analysis in the signal region and also creates the opportunity for a constraining sideband in the a` CCQE—whose events share a flux parentage and are thus strongly correlated (See Section 5.3.4). If particle tracks are sufficiently long (with an explicit threshold set at 3cm), then events consisting of two bodies—one lepton and one proton—are relatively easy to reconstruct. As an added

77 bonus, the resolving capability of MicroBooNE allows for reconstruction of full four-momenta for the lepton and proton, which opens the door for kinematic study. By following our two-body QE assumption about our signal, we can neglect final-state interactions and impose specific kinematic expectations on reconstructed neutrino-candidate events. For example, neutrino energy can be reconstructed from a two-body scattering interaction as the sum of the reconstructed energies from both component particles (minus that of the neutron at rest); under a QE assertion, there are more, equally viable ways to calculate energy via the four-momenta of either final-state particle. When our premise is upheld, we see meaningfully reconstructed variables; elsewhere, the reconstruction may produce nonsense, unphysical variables. This disparity allows us to further isolate CCQE events from a broader selection of two-body events. Figure 5.2 illustrates the DL LEE analysis chain from this beginning until a final result.

Figure 5.2: A ﬂowchart of the DL LEE analysis reconstruction and selection strategy. Deep learning techniques are utilized in the Sparse SSNet and 3D reconstruction stages.

78 5.2.2 Wire-Cell Cosmic Tagging and Semantic Segmentation

After signal processing and deconvolution, we have three arrays of waveforms from electron induction (on planes U,V) and collection (on plane Y). The Wire-Cell cosmic tagger [63] tags background cosmic particle tracks for removal and identifies regions of interest (ROIs) on the planes where the candidate neutrino interaction is most likely to have taken place. These waveforms are then converted to images with each pixel having a width corresponding to a single wire (3 mm) and a height corresponding to the time over which an electron waveform is integrated (6 time tics = 3 `s = 3.3 mm drift distance). A pixel’s intensity in the image is assigned according to the total number of electrons deposited at its corresponding time-space position in the detector. Pixels outside the given ROI are given a value of zero, as are pixels whose intensities fall below a threshold. At this point, we have three images very much like those presented in our LArTPC event displays, but much more sparse. Convolutional Neural Networks (CNNs) are machine-learning algorithms that have proven across many applications in recent years to be adept at image processing and classification. Recog- nizing the potential that this technology could have with the photograph-like signatures of a TPC, the DL LEE analysis was built around this concept. Deep learning is used in a few places within the DL LEE analysis, but the most prominent case is through a Sparse Submanifold Segmentation Network called SparseSSNet. SparseSSNet is a deep-learning algorithm trained to classify pixels in an image as originating from one of five different particle categories: Tracks (either a highly ionizing (HIP) or a minimally ionizing particle (MIP)), or Showers (either an electromagnetic shower, delta ray, or Michel electron). While not part of this dissertation work, specific details pertaining to the training of this network can be found in [68]. When an event is processed by SparseSSNet, its output takes the form of three matrices (one for each wire plane) that assign a classification to each pixel with non-zero charge-deposition. Despite the availability of five categories, the one-muon-one-proton (1`1?) analysis presented in this dissertation only takes advantage of whether a particle is classified as track-like or shower-like. In particular, we will later define the variable shower_fraction

79 as the fraction of pixels in a given reconstructed prong (ie: either the proton or lepton candidate fork from the vertex), which is classiﬁed as being part of a shower. A hard cut on shower_- fraction is made to ensure orthogonality between 1`1? and one-electron-one-proton (141?) candidate event selections (as will be described in Section 6), thus also keeping the 1`1? selection blind to potential LEE signal.

Figure 5.3: A demonstration of SparseSSNet’s classiﬁcation ability as used on MC simulation. The plot on the left illustrates the TPC image (what is observed by the detector), the center plot is the true neutrino event, as provided by the Geant4 simulation toolkit [26], and the right plot is the pixel-level classiﬁcation output by SparseSSNet [68].

The effectiveness of SparseSSNet on Monte Carlo is demonstrated in Figures 5.3 and 5.4. Comparing the SparseSSNet Outputs (right plots) with the true particle categories used by the MC shows qualitatively strong accuracy. In Figure 5.3, the heavier, highly-ionizing proton is correctly classified as a HIP, while the longer, lighter muon track is classified as a MIP. Furthermore, there is a third track present: one that crosses through the region of interest and therefore is likely a cosmic muon background event, which is also correctly categorized as a MIP. One should note that in Figure 5.4, some pixels of the electron shower are misclassified as track-like from their ambiguity within the cluster. However, as long as the shower fraction of this particle signature is greater than 20% (and it is), this example will be successfully removed from the 1`1? selection.

80 Figure 5.4: A TPC image of a shower event as observed by MicroBooNE (left), the true neutrino event signature from our simulation (center), and the SparseSSNet classiﬁcation output (right) [68].

5.2.3 3D Muon Neutrino Vertex Finding and Reconstruction

By demanding two-body scattering, we conveniently restrict the shapes of neutrino interaction candidate signatures to a "vee" shape: a vertex with two prongs. For each event, our 2D vertex-ﬁnding algorithm is applied to each plane independently. All 2D candidate vertices are projected into 3D detector coordinates and compared between the planes: any vertex that has an overlap in at least two of the three planes is passed on to track reconstruction

(and, additionally, shower reconstruction in the 141? case, which is outside the scope of this thesis). Each plane image is scanned for 2D vertex seeds, which are selected geometrically through analysis of unbroken clusters of pixels. Clusters that form an unbroken line are called contours. To find our two-body V-shaped signature, the algorithm first places seeds at likely vertex locations via defect scans, which mark points where these contours exhibit "kinks," or via principal component analysis (PCA) crossings. In the latter method, each contour is fit to a straight line, and a seed is placed at each intersection of these lines. Next, every seed is passed through a series of consistency checks, including one ensuring that the vertex is not placed on a charge-less pixel. A seed must pass all checks to be considered as a 3D vertex candidate. The 3D position of each vertex candidate is calculated and compared across the three wire planes for overlap. To account for cosmic interference and unresponsive (dead) TPC wires, we

81 only require 3D agreement between vertex candidates on two wire planes. Candidate 3D vertices are then passed onto our 3D track reconstruction algorithm (called our "tracker") that iteratively walks out from the vertex, connecting 3D points with consistent deposited charge and favoring the forward direction. Tracks are smoothed through regularization, which ﬁnds the minimum number of points that describe the track at our resolution capability.

The full method of vertex ﬁnding and reconstruction is explained in [69]. The 1`1? selection, wherein candidate vertices are whittled down to candidate a` CCQE events, will be described in the following chapter.

5.3 Quantifying the eLEE Measurement

In this section, we will outline what is actually meant by an "LEE search" and where the 1`1? selection at the heart of this work ﬁts into MicroBooNE’s strategy. MicroBooNE’s speciﬁc aim is to count a4 CCQE events in terms of their reconstructed neutrino energy.

5.3.1 GENIE Models

The starting point of neutrino interaction simulation—the basis by which MicroBooNE must construct a predicted neutrino distribution—is a firm understanding of what is physically hap- pening when an interaction occurs. One has theory, but unfortunately, there is a distinct lack of experimental constraints to the neutrino cross section in Argon atoms. MicroBooNE utilizes the GENIE simulation suite [70], which boasts calibrations from many experimental measurements combined with cutting-edge theoretical interpretations of nuclear interactions. The purpose of an effective simulation is to produce believable background and signal distributions that match well with measured data. This dearth of experimental constraints befitting MicroBooNE’s particulars means that one may encounter edge cases where predictions do not match up with observation. One such disagreement arose in early data/MC comparisons of the CC 0c (charged current interaction that produces zero final-state pions) cross section, illustrated for MiniBooNE in Figure 5.5. In this plot, one may note

82 that the discrepancy manifests as a slight shift in energy that could perhaps be corrected.

Figure 5.5: Differential cross section of CC 0c in MiniBooNE data compared to GENIE predictions [70].

To apply a correction, one wants to minimize any additional bias. It is inadvisable, for example, to make an adjustment according to MicroBooNE data if one intends to later make a physical claim about that data. This energy-shift effect has also been seen in the CC 0c cross section observed by the T2K experiment [16], so GENIE uses T2K data, which has a cross section in a similar range to MicroBooNE, to make what is called a MicroBooNE Tune for GENIE. This tune corrects only the CCQE and meson exchange current (MEC) models for all MicroBooNE predictions used in this thesis. This data-driven approach was adopted because there is still theoretical disagreement as to the cause of this CC c0 shift [70].

5.3.2 A Simple LEE Model

MicroBooNE’s ultimate question is whether or not our data agrees with a particular ESM hypothesis; we wish to test against the LEE (speciﬁcally, The LEE, as was recorded in MiniBooNE) and not some other excess of a4-like events in the low energy. We must make a few initial signal assumptions that will undergird our entire search: that this eLEE is made up of neutrino events in

83 the a4 flux, interacting via the a4 charged-current cross section. To make an appropriate prediction, we must go straight to the MiniBooNE data and "unfold" it—deconvolve any MiniBooNE-specific detector effects to approximate what the raw signal may have looked like under our hypothesis. This is done via D’Agostini’s Iterative Bayesian Unfolding and is described in great length in [71]. The unfolded excess is shown over the unfolded Monte Carlo prediction in Figure 5.6 in terms of true neutrino energy. One could construct an eLEE signal by then convolving this model— via the ratio shown in Figure 5.7—with a a4 flux, a a4 CCQE cross section, and one’s detector reconstruction and selection efficiency.

Figure 5.6: On the upper plot, we see the the unfolded MiniBooNE LEE observation overlaid with the unfolded MiniBooNE MC under a a4 CCQE assumption, plotted across true neutrino energy. The lower plot illustrates the ratio of the LEE to the MC in the upper, giving the unfolded model used in this dissertation. An isolated version of this lower plot is shown in Figure 5.7 [71].

We refold this signal through the MicroBooNE simulation in practice by weighing a sample of selected a4 CCQE candidate MC events according to their true energy—thereby accounting for energy dependencies as a result of the unfolding. The weights we apply are the same ratios shown in Figure 5.7. The result is a prediction of MicroBooNE’s a4 spectrum under the LEE hypothesis.

84 Figure 5.7: The unfolded MiniBooNE eLEE model used by MicroBooNE and in this dissertation, given as the ratio of the unfolded MiniBooNE LEE observation to the unfolded MiniBooNE MC under a a4 CCQE assumption. A cross-check using Singular Value Decomposition is overlaid alongside the D’Agostini unfolding [71].

5.3.3 A Hypothesis Test for the LEE

This dissertation is concerned with a 1`1? selection, which by design should have no trace of the LEE, but works in tandem with a 141? selection for a higher sensitivity measurement. We compare our data to our prediction using the j2 test statistic, deﬁned below

#18=B j2 Õ #>1B # ?A43 "−1 #>1B # ?A43 , = ( 8 − 8 ) 8 9 ( 9 − 9 ) (5.1) 8, 9

#>1B 8 # ?A43 where 8 is the content of the th bin of our observation and 8 is that of our predicted spectrum. Importantly, 8 and 9 in Equation 5.1 span the bins of the 1`1? selection and also those of the parallel 141? selection, positioned side-by-side. The covariance matrix M can be subdivided into statistical and systematic components "BHB and "BC0C.

M = "BHB + "BC0C (5.2)

85 This covariance matrix therefore includes the statistical and systematic uncertainties of both the

a` and a4 selections and also their correlated components. It is in those correlations that our 1`1? selection gains meaning and relevance. As an instructional example, a simple two-bin covariance matrix " takes the form:

" = "BHB + "BC0C (5.3)

© f2 df f ª ©B2 ª 1 1 2® 1 0 ® ® + ® = ® ® (5.4) ® ® df f f2 ® B2® 1 2 2 0 2 « ¬ « ¬

©B2 + f2 df f ª 1 1 1 2 ® ® , = ® (5.5) ® df f B2 f2® 1 2 2 + 2 « ¬

where B1 is the statistical uncertainty for bin 1, f1 is the systematic uncertainty for that bin, and d is the correlation factor between the two bins. Each element of this standard example can be extrapolated, so long as we have two parts.

If there is no correlation between the two bins in question, then d = 0 and the resulting matrix contains only diagonal terms.

©B2 + f2 ª 1 1 0 ® " ® . = ® (5.6) ® B2 f2® 0 2 + 2 « ¬

Calculating the j2 with the above, diagonal-only matrix would carry the assumption that each bin is entirely independent—that variation of the count in bin 1 has absolutely no bearing on the

movement of bin 2. A case of d = +1 indicates perfectly positive correlation and one of d = −1 indicates perfectly negative correlation. We will explore what that means for our LEE search and why we care in the following section.

86 The Combined-Neyman-Pearson j2

The 141? event selection predicts that it will count on the order of 10 candidate neutrino interactions or fewer per bin in the ﬁnal set of open data. This low event count is well-modeled by thousands of MC interactions, positing far greater accuracy than the mere statistical uncertainty from the data would imply. Moreover, in a standard j2 test between data and MC, the high statistical uncertainty from the low data count would overwhelm the calculation and drastically overestimate the agreement, rendering the comparison virtually useless. To account for this, the DL analysis group has opted to adopt the Combined-Neyman-Pierson j2 j2 , given as #%, for data/MC comparisons. This formulation is tailored to distributions with low event counts and leads to the following changes:

j2 • In the #% calculation, the statistical component of the covariance matrix is replaced with

(3#>1B # ?A43)/(# ?A43 + 2#>1B). (5.7)

• In bins where there are zero data events, yet some MC events, the statistical error is replaced instead with

(# ?A43/2). (5.8)

This approximation is estimated to hold true when we have both good data/MC agreement and also at least ﬁve times as many MC events as data events in each bin [72]†. All event selection plots in Section 6 and the Appendices use j2 CNP for their data-to-MC comparisons. Similarly, all LEE sensitivities (as shown in Section 8) are calculated using the j2 CNP formalism.

5.3.4 Motivation for the 1`1? Constraint

The speciﬁcs of our reconstruction and selection for both the 1`1? and 141? sides of the analysis are designed to hone speciﬁcally on 1;1? CCQE neutrino events. As described in Section 2.1.3,

†These demands are met throughout this analysis.

87 the primary component of our BNB ﬂux is from c+ decay. The foremost mode of this decay results

+ in a a` and a ` , which then further decays into an e, a¯`, and a4. From Figure 2.6, we saw that the a4 channel contribution of the BNB is dominated by this source at our desired energy levels (below 1 GeV), with other sources being suppressed.

Any a4 flux measured by MicroBooNE has a most likely origin of oscillation from the initial a` flux, or of this decay chain. The fact that massive contributions from both neutrino channels originate via the same flux parentage (a c+ produced at the proton target) means that their fluxes are highly correlated. The BNB flux is a major source of uncertainty, but this correlation can be exploited to our advantage. The a4 CCQE count will be very small compared to what is seen of a`

ﬂux in MicroBooNE. Our hope is that by leveraging the higher statistics of the a` sample, we can place a constraint on the ﬂux uncertainty of the a4 count, granting better sensitivity to the LEE.

Statistical Justiﬁcation: A 2-Bin Example

To demonstrate the mechanics of this constraint in practice, please consider the barest case of only two bins: one for a4 events and one for a`. For this example, we will represent the statistical and systematic uncertainties as Ba and fa respectively, and the contents of the two bins are related by the correlation coefﬁcient d [73, 74]. The covariance matrix M is then given by:

©B2 + f2 df f ª a4 a4 a4 a` ® ® , M = ® (5.9) ® df f B2 + f2 ® a4 a` a` a` « ¬ which is trivially invertable to

©B2 + f2 df f ª © B2 + f2 −df f ª 1 1 1 2 ® 1 a4 a4 a4 a` ® ® −1 = ® . ® " ® (5.10) ® | | ® df f B2 + f2® −df f B2 + f2 ® 1 2 2 2 a4 a` a` a` « ¬ « ¬

88 Now, the LEE signal will lie in the a4 channel, so one could break down the predicted event # 1:6 count in that channel into two components: the predicted background event count, a4 , and the ( # ?A43 # 1:6 ( signal events, , such that a4 = a4 + . Let us deﬁne the difference between observation and background as they will be used in the j2 as Δ = #>1B − # 1:6, such that:

! ©Δ − (ª a4 ® j2 = M−1 ® (5.11) Δa − ( Δa ® 4 ` ® ® Δa` « ¬

! © B2 + f2 −df f ª ©Δ − (ª 1 a4 a4 a4 a` ® a4 ® = ® ® (5.12) Δa − ( Δa " ® ® 4 ` | | ® ® −df f B2 + f2 ® Δ ® a4 a` a` a` a` « ¬ « ¬ 1 = [(B2 + f2 )(Δa − ()2 − 2df f Δ (Δ − () + (B2 + f2 )Δ2 ] (5.13) |"| a` a` 4 a4 a` a` a4 a4 a4 a`

j2 m j2 By minimizing the , setting m( = 0, we arrive at the best ﬁt signal,

" # fa fa ( = Δ − 4 ` dΔ . (5.14) 1 5 a4 B2 f2 a` a` + a`

And we can get the inverted variance of ( using the Taylor expansion of j2.

1 1 m2 j2 = (5.15) 2 f( 2 m( B2 f2 B2 f2 d2f2 f2 |"| ( a + a )( a + a ) − a a f = = ` ` 4 4 ` ` (5.16) ( B2 f2 B2 f2 ( a` + a` ) ( a` + a` )    d2  = Ba + 1 −  fa (5.17) 4  B2  4  a` − 1  f2   a` 

f2 From Equation 5.17, we see how correlations can affect our signal’s variance ( . If there is no correlation between the a4 and a` observations (d = 0), then the uncertainty on our signal would

89 predictably reduce to that of our a4 bin. In the case where we see non-zero correlation, we can a B << f draw a powerful insight: if our ` sample is incredibly statistics-dominated, such that a` a` , B2 a` a d f then we see the 2 approach zero, and the 4 systematics term will approach (1 − ) a4 . In other fa` words, high a` statistics can reduce (constrain) the effects of systematic uncertainty on a correlated

a4 sample.

The a` flux incident on the MicroBooNE detector is approximately 100 times greater than that of the a4 flux from the BNB. To get the greatest benefit from this constraint, our selection must have high statistics (and therefore low statistical uncertainty), relatively low systematic uncertainty, and a high correlation factor. These conditions all depend on the purity and efficiency of our 1`1? selection.

90 Part III

Conﬂict

91 Chapter 6: The 1`1? Selection

The core of the author’s MicroBooNE analysis work is focused on presenting a pure, efficient selection of 1`1? events to constrain the systematic uncertainties of the a4 selection for a broader DL LEE search. For the purposes of this chapter, we will ignore the 141? side of the DL LEE analysis and, to this end, redefine our desired signal to be a` CCQE events with one proton and one muon in the final state. This selection has the specific goal of taking a sample of observed events and whittling away background until all that remains is a well-understood, highly-correlated CCQE signal. This process involves applying identical selection algorithms on data and MC, using the simulated events to approximate the physics of our observation. Some metrics of a successful selection are the following:

• Data-to-Monte Carlo (Data/MC) Agreement: To what degree do the distributions of the data and MC agree with one another? Bad data/MC agreement implies that either the underlying physics of the observation are not being properly modeled by the MC or that our analysis is somehow treating the samples differently. Either scenario would drastically undermine the validity of any hypothesis we could make about the data. Data/MC agreement is expressed on each plot as a Combined Neyman-Pearson j2 [72].

• Efﬁciency: What fraction of the available signal events is one able to select? To achieve an

effective constraint on the 141? selection, the 1`1? selection must have substantially higher statistics than its electron neutrino counterpart.

• Purity: What fraction of the selected events corresponds to the specific, desired signal? Purity is the common trade-off with efficiency. If many events are allowed to pass through a loose selection, then some deceptive backgrounds may sneak their way into the final sample.

92 Alternatively, if the selection is tightened to allow only the most perfectly signal-like signal events to pass through, then the resulting sample will be incredibly pure, but risk having low

statistics. Purity is important in this selection for two main reasons: the CCQE 1`1? is most highly correlated with the CCQE 141? and we need that constraint, and 1;1? CCQE events follow the most well-understood physics, lowering expected systematic uncertainties.

Figure 6.1: Toy examples of the trade-off between efﬁciency and purity, as demonstrated with an arbitrary cut. On the left plot, one sees 83% a` CCQE purity (in blue), but a meagre 185 events. On the right plot, one sees the efﬁciency increased massively with 3349 a` CCQE events, but less than 50% purity.

Figure 6.1 illustrates two extreme cases resulting from a very tight, conservative selection (left) versus a much looser, low-purity selection (right). On the left, our signal, shown in blue, makes up 83% of a very small sample; on the right, our signal makes up less than 50% of the sample, but there is almost 20 times as much available. These examples feature arbitrary cuts, but this balancing act between efﬁciency and purity will be revisited.

The DL LEE 1`1? selection comprises two main pieces: a stage of Pre-Cuts to clean up the data and isolate the 1`1? events from those that will appear in the 141? selection, and a stage of Final Cuts that directly assault backgrounds. This chapter will detail the full process, and one can follow the ﬂow of the selection with the chart in Figure 6.2.

93 Figure 6.2: A ﬂow chart of the stages of selection. Pre-selection is broken down into three component parts based on their utility to the process: cleaning the distribution, providing orthogonality from the 141? selection, and ensuring only one reconstructed vertex per neutrino candidate event.

6.1 Pre-Selection

After the vertex finding and reconstruction described in the previous chapter, analyzers are presented with large samples of data and MC from which the signal must be carved. Before the selection can take place, these samples must first be cleaned and subjected to a number of pre- selection cuts. In addition, the most substantial cuts in the final selection are on scores applied by a pair of Boosted Decision Trees (BDTs), which must be trained on MC that has passed pre- selection. BDT implementation uses the python XGBoost library [75]. For a brief primer on how BDTs work and for where to find the selection code framework for deeper perusal, see Appendix B.

6.1.1 Data and Simulation Samples

Data presented in the following plots throughout this dissertation constitute a subset of the total data taken over the course of MicroBooNE’s running period. There are two important considera- tions when discussing open data on this experiment:

• This subset of data is further narrowed by a a4 filter to remove any potential eLEE signal events, thus preserving blindness. A Monte Carlo test affirmed that the final 1`1? selection

passes virtually no a4 CCQE events (0.14 out of 3690.29 MC events, or .0040%), rendering

94 this ﬁlter moot.

• The MicroBooNE data-taking period is divided into a number of Runs, each of which is further subdivided into epochs, denoting unique periods of consistency across data-taking conditions. For example, Run 2 and beyond distinguish themselves from Run 1 by the addition of a Cosmic Ray Tagger (CRT) to the detector. Further epochs are separated by hardware replacements or software improvements. No change in runs is expected to manifest a substantial divergence in core physics models, but as a caution, these differences are noted and utilized in analysis.

Our Monte Carlo comprise millions of simulated neutrino interactions and their ensuing propagation through the detector. Neutrinos are simulated with the GENIE suite and paths of their final state particles through Argon are modeled via a software package called Geant4 [26]. To compensate for the Run divisions noted above, a bespoke set of MC are produced for each run under identical circumstances to the data. Rather than simulating cosmic backgrounds, MC are overlaid with cosmic data events taken from beam-off observations by MicroBooNE. These cosmic overlays are also matched by Run and are detailed in [76]. The term Run X Monte Carlo, where X is 1, 2, or 3, refers to MC simulated under the conditions of that corresponding Run of data. All data-to-MC comparisons are carried out between corresponding sets of data and MC (i.e. we would not necessarily expect Run 1 MC to perfectly match Run 3 data). One must take care not to confuse Runs—eras consisting of several months of data-taking— with runs (note lower-case)—individual periods of data-taking lasting up to 7 hours. Despite this troublesome naming convention, Runs discussed in this thesis will never exceed 5 and are often discussed in more general terms, while runs will never fall below 4952 and will only be invoked as part of the identifier for individual neutrino events (indexed by [run, subrun, event] or [RSE]). Specifically, the Runs are differentiated in the following terms:

• Run 1 comprises runs 4952-7770 and approximately 1.7 × 1020 POT. This Run is the ﬁrst

95 with an integrated software trigger and is the earliest data set that will be used in analysis.

• Run 2 comprises runs 8317-13696 and approximately 2.6 × 1020 POT. This Run took place after a service-board replacement and has a partially-integrated CRT. This second distinction is irrelevant for the DL LEE analysis, which does not make use of the CRT.

• Run 3 comprises runs 13697-18960 and approximately 2.4 × 1020 POT. This run is charac- terized by the full installation of the CRT.

"Good Runs"

An additional qualifier on the data is whether or not a given run is counted as a Good run.A "good run" is a run (lower-case) that is fit for scientific analysis. Over the course of data-taking, occasions arise that compromise the current run. Elements on the Good Run list are chosen by the following criteria:

1. Selection is made based on detector conditions as recorded during data-taking. As each run is taken, summary metrics are noted and must be approved by whomever is on-shift at that time.

2. A manual cut is made of known bad periods in data-taking. For example, a delivery of dirty argon during Run 4 fundamentally changed the shape and nature of MicroBooNE’s observations. All runs taken before this issue was resolved were cut.

3. Certain beam metrics are required to be within 3f of their per-Run mean.

The runs that pass are compiled into a list for analyzers to integrate with their studies [77].

6.1.2 Two-Body Scattering, Kinematics, and Boosting

Neutrino interactions are reconstructed as tracks and showers emanating from a vertex, presenting us with a heap of topological and calorimetric information about the ﬁnal state. This information is recorded as a long list of variables: physical position of the vertex within the ﬁducial

96 volume of the detector, angular trajectory of a track, transverse momentum of a given ﬁnal-state particle, etc.

With our 1;1? focus, we can utilize the properties of physical kinematics to gain even more information about each neutrino candidate event. By asserting that our selected events are CCQE, we can impart our knowledge of two-body QE physics on these reconstructed events.

Figure 6.3: Diagram of MicroBooNE coordinate system. The anode and three wire planes are positioned at x=0, with the cathode at the opposite end.

The most important variables to emerge from 3D reconstruction are tabulated as follows:

Variable Name Deﬁnition

?, ` Proton and muon kinetic energies Reconstructed based on track range [4, 78]

?, ` Proton and muon energies ? +

−→ −→ ??, ?` Proton and muon momenta Reconstructed via 3D tracker [69]

−→I −→ −→I −→ cos \ ?, cos \` Proton and muon forwardness ? 4/| ? 4 |, ? `/| ? ` |

q , q 0C0= ?H ,?G , 0C0= ?H ,?G ? ` Proton and muon azimuth 2( ? H) 2( ` `)

−→ −→ ??,?` Proton and muon four-momenta (?, ??), (`, ?`)

97 For this analysis, we also use <` = 105.6584 MeV,

simulation as a constant = 29.5 "4+ [70]. Positions and angles within the detector should be understood in the context of the coordinate scheme illustrated in Figure 6.3. A particle’s polar (\) and azimuthal (q) angles are measured in relation to the z-component (beam-direction) of the reconstructed momentum vector. Kinetic energies of the ﬁnal state track particles are both determined by the length of their tracks measured linearly from vertex to endpoint and compared with the known stopping power of both particles in liquid argon [4, 78]. At these energies, we expect ﬁnal-state protons and muons to travel across a relatively straight path. By combining conservation of energy and the assumption

of two-body scattering, one can construct a range-based reconstructed neutrino energy a using

these components: a = ? + ` − (<= − ). With our quasi-elastic assumption, we can deﬁne reconstructed neutrino energy by other means, &−? &−` in terms of both the proton (a ) and the lepton (a ) as shown in the following table [62].

Variable Deﬁnition

a ? + ` − (<= − )

1 2 2 2 &−? ? (<=−)+ 2 (<`−(<=−) −< ?) a −→ (<=−)+|? ? | cos \ ?− ?

1 2 2 2 &−` ` (<=−)+ 2 (< ?−(<=−) −<`) a −→ (<=−)+|?` | cos \`−`

q & &−? &−` 2 &−? 2 &−` 2 Δ (a − a ) + (a − a) + (a − a)

We introduce a QE consistency variable, Δ& , to exploit the fact that we have three theoret- ically equivalent energies, and deﬁne it as the 3D distance between each of these energies. In a perfectly quasi-elastic scenario of a neutrino scattering off a nucleon at rest and producing a

1;1? ﬁnal state signature, Δ& should be identically zero. If any of our assumptions up to this

98 point are not met by the underlying particles, then these three energies will demonstrate substantial disagreement, and the resulting Δ& will be enormous. This disagreement could result from ﬁnal- state interactions where lost energy could not be accounted for, kinked muon tracks that may look like a two-pronged vertex but have non-physical kinematics in a two-body scheme, interactions that produce a resonant charged pion, or many more edge cases. With our toolbox of energies and momenta, we can fully reconstruct the momentum four- vectors of each constituent particle in this interaction. From here, a smattering of kinematic variables are at our disposal—the most important of which are the following:

Variable Deﬁnition

&2 %I <2 2 a ( ` − `) − `

Hadronic Mass (<ℎ03) a − `

2 Bjorken’s X Scaling & /2<=<ℎ03

Bjorken’s Y Scaling <ℎ03/a

−1 Opening Angle cos (?ˆ` · ?ˆ?)

q G G 2 H H 2 ?) (?` + ??) + (?` + ??) )

? ?I ?I ! ` + `

−→ ?` ·−→? U cos−1 − ) ) ) −→ −→` |?) ||? |

−→ −→ ` · ? −1 ?) ?) q) cos − −→ ? −→` |?) ||? |

All equations presented so far have hinged upon the additional, unspoken assumption that our nucleon is at rest when struck by an incident neutrino. However, because we can fully reconstruct

99 −→ the ﬁnal-state particle four-momenta, we can also reconstruct the nucleon’s fermi-momentum ? 5 −→ directly. We need not rely on this at-rest approximation. With ? 5 , we can remove the effects of pre-interaction nucleon momentum by Lorentz boosting each of our kinematic formulae into the −→ −→ nucleon rest frame using V = ? 5 where =

= = <= − − )1 ≈ <= − (6.1) −→? −−−−→? −→? ?G ?G ,?H ?H ,?I ?I . 5 = 8=0; − a = ( ? + ` ? + ` ? + ` − a) (6.2)

In Equation 6.1, we estimate that the ﬁnal-state nuclear recoil kinetic energy )1 will be very small

compared to (<= − ) in the case of a QE collision. In Equation 6.2, we let the neutrino be ? ?I perfectly aligned with the z axis, such that a = a ≈ a. Variables that have been boosted into the nucleon rest frame have been marked with an asterisk in plot axis labels. Further discussion of boosting, or the derivation of QE formulae, can be found in [62].

6.1.3 The Pre-Selection Cuts

After vertex ﬁnding and reconstruction, events included in our samples contain any candidate neutrino interactions that were reconstructed as a single vertex with two prongs. This distinction

alone is not enough to achieve 1;1? CCQE purity, but it sets the baseline for our efﬁciency. No more events will be added—we will only be trimming from this point forward. These candidate events are each differentiated by a unique vertex and carry a full array of reconstructed variables, assembled under the assumption that the vertex signiﬁes a neutrino inter-

action with a 1;1? ﬁnal state. As vertices act as an effective index, we will often synecdochically refer to individual neutrino interaction candidates simply as vertices (and later, as events).

Being merely reconstructable does not guarantee that our vertices lie within our desired 1;1? CCQE topology. There are plenty of backgrounds that must be pruned before making any strong claims about the signal. Furthermore, at this stage the list of vertices will be quite messy. Some

100 vertices will contain artifacts of processing failure, some will be very easy to disregard immediately, and some will lie within a region of ambiguity that could compromise further selection efforts. We must engage in the process of Pre-Selection: a series of cuts that will not necessarily move the sample closer to CCQE purity, but will remove any events that the analysis cannot (or chooses not to) handle. These cuts are all small, but even if they only removed a single vertex, then they would allow us to make clear claims about the vertices that remain. The pre-selection cuts are as follows:

• PMT Precuts. Any neutrino interaction of reconstructable energy will produce a ﬂash of scintillation light co-synchronous with the vertex. This pair of precuts demands both that the PMTs receive at least (a very conservative) 20 photo-electrons over the course of the interaction and also less than 20 photo-electrons in the immediate lead-up.

• Two Prong Requirement. Our reconstruction suite is only designed to handle vertices with two prongs—two protrusions, either tracks or showers, jutting out from the interaction point. Some vertices with more or fewer than two prongs may have passed through reconstruction and the algorithm would have ﬂaccidly attempted to assign meaning to an interaction it could not understand. We therefore eliminate any vertices without exactly two prongs.

• Restrict vertex to be within detector ﬁducial volume. MicroBooNE is a large detector, but not all of that volume is instrumented to make measurements. This cut makes sure that the reconstructed vertex is within our trusted measurement area of the detector, with a 10 cm buffer from each edge. This buffer exists to increase the likelihood that the entire interaction is contained within the ﬁducial volume. For example, an NC c0 interaction close enough to the walls that one photon escapes the detector will unambiguously appear as a single photon event.

• Restrict track to minimum distance from detector walls. In addition to keeping the reconstructed vertex within bounds, another cut is imposed, requiring that the end points of

101 each track be greater than 15 cm from the edges of the detector. An experimenter can only see energy that was deposited within the detector; if a track exits before concluding, then the neutrino event cannot be fully reconstructed.

• Require total charge within 5cm of the vertex to be more than 0 ADC.

• Require all interactions be boostable into the nucleon rest frame. As described in Sec- tion 6.1.2, the DL analysis takes advantage of the kinematic properties of a CCQE interaction. To perform any Lorentz boost, one must calculate W; however, when taking into account the off-shell-ness of nucleons in these interactions, a possibility for W to be undeﬁned (i.e.

for V > 1) is introduced. Unboostability indicates that a vertex does not follow the physics of a two-particle interaction and is almost certainly some background.

• Restrict the opening angle between the two prongs to be greater than .5 radians. This restriction is purely a limitation of our resolution and reconstructive capabilities. If the two tracks are too close together, then there is substantial room for error, so these are removed for ease.

• Require that proton be forward-going. This requirement is explicitly stated as cos \ ? > 0. In the case of two-body QE scattering, the much-heavier proton must be going forward to conserve momentum.

• Require the fraction of shower-like pixels in each track, according to SparseSSNet, to

be less than 20%. To achieve our goal of a systematic constraint on the 141? selection, we must be absolutely certain that there are no events which appear in both the 141? and 1`1? selections. This is executed with a binary orthogonality cut, where the 141? analyzer performs the exact inverse requirement (that the fraction of shower-like pixels be greater than 20%).

Until this point, we have been indexing our individual counts as vertices, thereby allowing the potential for multiple vertices to have passed the prior tests for a single neutrino candidate event. A

102 neutrino event, particularly under our 1;1? CCQE signal deﬁnition, can only have a single vertex. In these cases of vertex multiplicity, we need a consistent way to remove all but one vertex for each event. To this end, we only keep the single candidate vertex for each event that has the best (most signal-like) BDT score (to be deﬁned later in this chapter).

Pre-Selection Substage Vertices Δ% Candidate Events Δ%

Before Presel 160,708 NA 103,461 NA Two-Prong and Coordinate Cuts 55,635 -65% 41,181 -60% Charge Near Trunk Cut 54,555 -2% 40,479 -2% Opening Angle Cut 54,233 -1% 40,304 -0% Boostability Cut 53,386 -2% 39,969 -1% Orthogonality Cut 39,120 -27% 31,863 -20% PMT Precuts 38,334 -2% 31,199 -2% Remove Duplicate Vertices 31,199 -19% 31,199

Table 6.1: Table of the cut progression through each individual pre-selection substage in terms of candidate neutrino vertices and candidate neutrino events. To clarify, there may be multiple candidate vertices for a single event until the ﬁnal substage where duplicates are removed. The Δ% columns reﬂect the percent change in count from the previous row.

The resulting distribution in reconstructed neutrino energy is shown in Figure 6.4 and the spe- ciﬁc event counts for our a` CCQE signal and background are listed in Table 6.2. One will notice that after pre-cuts, the distribution has twice as much background as signal. Despite this background-dominance, the distribution is representative of what our algorithms can reliably reconstruct— a much better canvas on which to work. In this and all following plots of this format, the signal will be represented as the blue contribution to the histogram. Following the pre-cuts, we have determined the maximum amount of signal we can reconstruct: the state where our selection efﬁciency is 100%.

103 Figure 6.4: Distribution of reconstructed neutrino energy after pre-selection. Note that systematic uncertainties were not calculated for events outside the ﬁnal selection, so all pre-selection plots feature statistical error only.

Selection Stage Total Events a` CCQE Events Background Events Pre-Selection 52518.54 17050.31 (32%) 35468.23 (68%)

Table 6.2: Monte Carlo event counts of our a` CCQE signal, our backgrounds, and the total of all events remaining after the pre-selection stage. Event counts are scaled to match the POT of open Run 1-3 data, 6.8 × 1020 POT. Beside the signal and background counts, in parentheses, are each subsample’s fractional contribution to the total event count.

6.1.4 What’s in a Plot?

The plot in Figure 6.4 is conveniently structured to be representative of all 1`1? distribution plots to follow. There is a lot of information contained in these ﬁgures, so we will walk through the purpose of each piece in this section. The predicted distribution is drawn as a stacked histogram that features both MC and cosmic backgrounds. The vertical-axis label indicates the total POT of contributing data, to which the prediction is scaled. The legend details each individual component of the stacked prediction histogram, with the total event count (scaled to the current POT) in parentheses. Any prediction components that individually make up less than one percent of the total distribution are consoli- dated into the "Other" category for improved readability. The legend also displays the total count

104 of observed (data) events in parentheses beside its marker. The lower plot shows the ratio of data to MC for each bin in the above histogram. Observed event counts are given Poisson error bars in each plot. Poisson errors are not used in j2 the corresponding data/MC #% calculation, but are used to aid visual intuition. Systematic and statistical errors are represented as hashed, colored bars protruding from the stacked histogram. The plotted systematic error is given by the square root of the covariance matrix diagonal. Errors are propagated to the ratio plot, and statistical error in the prediction is illustrated via thick bars across the data/MC=1 horizontal. At the top left of the ﬁgure is the ratio of the total count of observed events to the total count of predicted events in the shown histogram. The error given for this ratio is the total normalization uncertainty (deﬁned in Chapter 7). j2 Lastly, in a small box at the upper-right corner of the ratio plot, we print the ratio of the #% to the number of bins and also the corresponding p-value. The p-value is calculated for a j2 distribution of degrees of freedom equal to the number of bins, using the SciPy stats library in Python [79].

6.1.5 Truth labeling

In response to individual variable studies, particularly of the transverse momentum asymmetry with respect to the lepton (U) ) and the total ADC charge within 5cm of the reconstructed vertex, we discovered that the 1`1? analysis and selection is particularly sensitive to events where we either fail to reconstruct the full energy or mistake a non-1;1? interaction to be 1;1?. To amend this, we further narrow our signal deﬁnition, which we use for both interpreting purity and for training BDTs. Three new Monte Carlo qualiﬁers are introduced: 1L1P, On Vertex, and Good Reco. For detail on the variable studies and conclusions drawn that provoked this change, please see Appendix B.3. An MC event is called 1L1P at truth-level if it contains exactly one lepton with true kinetic energy greater than 35 MeV and exactly one proton with true kinetic energy greater than 60 MeV.

105 An event is called On Vertex if the reconstructed vertex position is within 5 cm of the true vertex position. An event is called Good Reco if it has total neutrino energy reconstructed within 20% of true neutrino energy.

Figure 6.5: Two plots containing equivalent distributions under different labeling schemes. In the lower plot, our a` CCQE Signal is narrowed to only include events with good positional and energy reconstruction. Comparing the plots, we see that our main backgrounds are not non-CCQE events, but rather events that are not reconstructed adequately.

The two plots in Figure 6.5 contain identical events, but the distribution on the bottom uses this new labeling scheme. Predictably, the signal—now under the narrower deﬁnition: CCQE a` interactions that are reliably reconstructed in both position and energy, or in fewer words, well-

reconstructed a` CCQE interactions—has shrunk substantially. All the blue that vanishes from the

106 upper plot was poorly reconstructed from the start and could only have introduced greater uncertainty. The dominant backgrounds are also no longer these esoteric other neutrino interactions, but rather ones with fundamentally different topology (not-1L1P) or ones that did not get reconstructed adequately (Off-Vertex or Bad Reco). We will see later that this labeling leads to a very favorable distinction between signal and background as measured by our BDTs. A ﬁnal point worth noting involves the variable whose distribution has been illustrated— Bjorken’s X, boosted into the nucleon rest frame—as described below.

Brief Aside on Bjorken’s X Scaling

Please consider a simple 2-body scattering scenario as presented in Figure 6.6. For the rest of this interlude we will again set natural constants to unity such that ?2 = −2 + |? ì|2 = −<2, where ? is the momentum four-vector ? = (, ?ì) and < is the corresponding invariant mass.

Figure 6.6: An illustration of 2-body elastic scattering of particles with pre-collision four-momenta ?1 and ?2.

2 2 Through conservation of momentum, one sees that (?1 − ?3) = (?2 − ?4) is an invariant quantity, and we can call this quantity the square of the momentum-transfer four-vector, or &2 as we have already seen. The ﬁrst component of & is then (1 − 3) = (2 − 4), or the energy transfer, which we will call &0.

107 ? ? 2 ?2 ?2 ? ? ( 1 − 3) = 1 + 3 − 2 1 3 (6.3) <2 <2 ? ? = − 1 − 3 − 2(− 1 3 + ì1 · ì3) (6.4) <2 <2 ? ? = − 1 + 1 3 − 3 + 1 3 − 2 ì1 · ì3 (6.5) <2 & <2 & ? ? = − 1 + 1( 1 − 0) − 3 + 3( 3 + 0) − 2 ì1 · ì3 (6.6) <2 2 & <2 2 & ? ? = − 1 + 1 − 1 0 − 3 + 3 + 3 0 − 2 ì1 · ì3 (6.7)

2 2 = |? ì1| + |? ì3| − 2?ì1 ·? ì3 + &0(3 − 1) (6.8) ? ? 2 &2 = ( ì1 − ì3) − 0 (6.9)

So now Equation 6.9 gives a simple form for &2 in 2-body, perfectly-elastic scattering, where

&0 = (1 − 3) has been substituted in for convenience. &2 is invariant, so we may consider this same interaction in a different frame. Take Equation 6.9

in the case where our target particle is in its rest frame (i.e. ?ì2 = 0). Furthermore, let us call this

target particle a free nucleon hit by a neutrino and give its invariant mass as <# . If all energy is

transferred from the neutrino to the outgoing nucleon, then &0 = 4 − <# gives the kinetic energy of the nucleon.

&2 ? 2 &2 = (0 − ì4) − 0 (6.10) ? 2 2 <2 < = | ì4| − ( 4 + # − 2 4 # ) (6.11) <2 2 2 <2 < = (− # + 4) − 4 − # + 2 4 # (6.12)

= 2<# (4 − <" ) = 2<#&0 (6.13)

2 As deﬁned in Section 6.1.2, Bjorken’s X Scaling is given by & /2"# ℎ03. Substituting in our result for Equation 6.13 and therefore assuming a perfectly-elastic, 2-body collision, one would ﬁnd Bjorken’s X to be exactly equal to one. In MicroBooNE, our nucleons are bound, so the re-

108 sulting Bjorken’s X will differ from this idealized case, yet we can see a peak of the a` CCQE signal on the lower plot of Figure 6.5 that lends some agreement to this theory. Notably, a peak

manifests where there was no such shape before. By redeﬁning our signal to reﬂect true a` CCQE events, which are well understood and should be fairly well-modeled by 2-body scattering kinematics, the resulting distribution corroborates theory.

6.1.6 Training the BDTs

Our selection’s most decisive cut is on a score assigned by our boosted decision trees. To

achieve a pure and plentiful a` CCQE sample, we train a single network to discriminate our desired

a` CCQE signal against cosmic and a` non-CCQE events. With this in mind, the signal and background used to train the BDTs are given as the following, with the background split into two subcategories for easier discussion, both of which are combined for training:

• Signal: a` CCQE - Our signal deﬁnition, for the purpose of training the BDT network,

is a` MC events that are 1L1P, On Vertex, Good Reco, and feature an interaction mode corresponding to CCQE.

• Background Component 1: a` non-CCQE - Our non-CCQE deﬁnition is a` MC events that either: a) are 1L1P, On Vertex, and Good Reco, but have an interaction mode corresponding to anything besides CCQE, b) are 1L1P and On Vertex, but not Good Reco, or c) are not 1L1P.

• Background Component 2: Cosmic-Ray-Only Events - Due to the excellent performance of the Wire-Cell tagger, the cosmic rays are a secondary background. However, it is still necessary to cut remaining events. These events are identical to the EXT data set taken with the BNB trigger applied. Therefore, this off-beam sample can be used to accurately describe these events.

Two separate BDTs are trained for use in our analysis—one on Run 1 MC to be used with Run 1 data and one on Run 3 MC to be used with Run 2 and Run 3 data. This decision stems from

109 Feature Name Deﬁnition

Phis |q? − q` | ChargeNearTrunk Total charge deposited within 5 cm of vertex

Enu_1m1p Reconstructed a` Energy

PhiT_1m1p q)

AlphaT_1m1p U)

PT_1m1p Reconstructed a` Transverse Momentum ?)

PTRat_1m1p ?) /|?| BjXB_1m1p Bjorken’s X Scaling, boosted BjYB_1m1p Bjorken’s Y Scaling, boosted SphB_1m1p QE Consistency, boosted Q0_1m1p Energy Transfer Q3_1m1p z-component of Momentum Transfer Four-Vector

Lepton_PhiReco q` Lepton_TrackLength Length of Reconstructed Muon Track

Proton_PhiReco q?

Proton_ThetaReco \ ?

Table 6.3: Reconstructed neutrino event variables used for BDT training. pertinent differences in post-Run 1 data attributed to a replaced service-board. Further clarification and diagnosis of this two-BDT split are outlined in Appendix B.4. Variables used as features of the neutrino event candidates were chosen after the pre-selection stage. Each potential feature was plotted and both data/MC agreement and signal/background separation were considered. Because our analysis depends upon 2-body scattering and kinematics, our a` CCQE sample is expected to follow certain trends (see Bjorken’s X Scaling in Section 6.1.5), while backgrounds give nonsense distributions when subject to these assumptions. The full suite of selected features is listed in Table 6.3, and their corresponding distributions after pre-selection are all collected in Appendix B.5. Any variables not explicitly defined in the table have been previously defined to greater precision in Section 6.1.2.

110 Training the BDT involves providing XGBoost with a list of values for each feature of each event in both the signal and background samples. We set training parameters for the ﬁt to avoid over- or under-training. XGBoost also has a useful built-in diagnostic tool, the function feature _importance, which lists how heavily each feature is relied upon by the training algorithm cuts, averaged across all estimators. This metric reports whether any variables are utterly useless, or if any could shoulder the entire selection on their own, rendering the use of a BDT overkill.

Figure 6.7: Relative importance of each feature used in training both the Run 1 BDT (left) and Run 3 BDT (right). A variable with a higher importance was chosen more often by the training algorithm for cuts and can be interpreted as a particularly strong discriminator between signal and background.

Feature importances for each of the two BDTs are presented in Figure 6.7. One can see that in the case of either run, the most important variable to the training is PTRat_1m1p, which is the ratio of reconstructed neutrino transverse momentum to total reconstructed neutrino momentum.

One should expect 1`1? CCQE events to be strongly forward-going in the beam direction to conserve momentum in a two-body scatter with a (mostly) at-rest target. Background events will be much more diffuse, with downward-going cosmic backgrounds piling up with a very high transverse momentum fraction. That each BDT found this variable of prime importance is a testament to our high-purity methodology within the DL analysis. The distribution of the ratio of transverse to total momentum is given in Figure 6.8, demonstrating our predicted separation—signal piles at the far left and backgrounds spread across the right.

111 Figure 6.8: Distribution of the ratio of reconstructed neutrino transverse momentum to total reconstructed neutrino momentum, measured in practice as the sum of the corresponding muon and proton momenta under 1`1? assumption.

Despite the very high importance positing this feature as over ﬁve times more useful than others used in training, no viable cut on this variable alone was found to be as effective as its combination with others in the BDT.

6.1.7 MPID

The DL LEE analysis further employs deep learning through a Multi Particle Identiﬁcation

(MPID) algorithm. MPID uses a CNN framework to take a LArTPC image of 512×512 pixels and to simultaneously predict the likelihood of five different types of particles being present. MPID is trained to classify the existence probabilities for electrons, photons, muons, pions, and protons. An analysis of each image through MPID returns a series of five floats pertaining to the different particle scores. Because the scores reflect presence within the image, scores are not exclusive (and thus needn’t sum to unity).

The MPID algorithm is utilized to a small, but powerful, extent in the 1`1? selection. More details on the MPID can be found in [80].

112 6.2 Selection

With the pre-selection applied and BDTs trained, we begin the ﬁnal selection.

6.2.1 Applying the BDT Weights

BDT Training with XGBoost outputs a set of weights, which will assign a BDT score to each reconstructed event based on its reconstructed feature variables. This score is a float value in [0,1] that we call bkgBDT. A neutrino candidate event with bkgBDT closer to zero is thought to be more signal-like (by the signal definition in Section 6.1.6), and a bkgBDT closer to one signifies the opposite classification. To comprehend the power of these BDTs, it is useful to consider the plots in Figure 6.9, in which we see the normalized distributions of bkgBDT for our signal and key background samples. The fraction of each sample whose events have bkgBDT< 0.4 are given in each plot. While shape separation is apparent, with so much more background than signal events, it is hard to intuit where to place our analysis’ final cut from these plots alone.

Figure 6.9: Two plots demonstrating the separation of the bkgBDT variable between signal (in blue) and various backgrounds (in orange, red, and green). The left plot considers a BDT trained on Run 1 MC (to score Run 1 MC), and the right is the same for Run 3 MC.

To this end, two independent studies were performed: one to weigh the trade-off between efﬁciency and purity across various cuts on the BDT Score (see Figure 6.10); and the other to estimate the dependence of the LEE sensitivity on this BDT Score (see Figure 6.11).

113 Conclusions of each study are as follows:

• Figure 6.10 explicitly illustrates the relationship between purity and efﬁciency across selection cuts. A cut at bkgBDT < 0.4 gives a purity of 67% with a 56% efﬁciency.

• From Figure 6.11, one can see that for a low bkgBDT, the low statistics of the resulting a` CCQE candidate events are not enough to compensate for any dramatic demonstration of purity. Loosening the cut to allow events with bkgBDT score > 0.4 shows only marginal improvements in sensitivity.

Figure 6.10: Plot of the relationship between well-reconstructed CCQE purity and BDT cut efficiency for different cut strengths -. Efficiency is defined specifically for the BDT cut such that we have 100% efficiency at - = 1.0 where all preselection and non-BDT selection cuts have been applied. The horizontal red lines contextualize the efficiency by indicating where along the orange 20 line 1000, 2000, or 3000 a` CCQE events will pass final selection (in 6.8 × 10 POT).

A selection cut of bkgBDT < 0.4 was chosen for its capability to grant high sensitivity and purity without sacriﬁcing statistics too greatly. Before delays in data processing due to the COVID-

19 pandemic, the total opened data was to span Runs 1-5 and encompass 1.3 × 1021 POT, which would result in a selection of over 4000 a` CCQE events under this selection cut. The presented selection is only a portion of that, but this selection will continue through analyses in the immediate future.

114 Figure 6.11: Preliminary plot of sensitivity to the LEE using the 1`1? sample to constrain uncertainty. One should note that the values shown on the vertical sensitivity axis should be regarded as incomplete, this plot having been created before elements of the 141? selection and detector systematics were ﬁnalized and comprising a fraction of available data. Relevant to study is that the BDT cut value between 0.3 and 0.4 proceeds a plateau of sensitivity increase. Plot provided by Lauren Yates (MIT).

6.2.2 MPID Cut

After the BDT cut, one ﬁnal idiosyncrasy remains, which may be seen in Figure 6.12: on the far left of the histogram—speciﬁcally, for reconstructed neutrino energy less than 400 MeV—one sees a large protuberance of BNB Not 1L1P background. The plot on the right returns to a pure

GENIE true interaction-based labeling system and reveals that this bump heavily comprises a` resonant c events. Further investigation into this low energy Not 1L1P background led to the plot seen in Fig- ure 6.13, showing the maximum proton MPID score for low energy neutrino events in our sample. Protons leave a very distinct signature in a LArTPC: a short, highly ionizing track. Nonetheless, there exists a broad array of proton MPID scores well below the expected value of 1 for a 1;1? event. If we hypothesize that at low energies, resonant pions are being convincingly reconstructed

115 Figure 6.12: The reconstructed neutrino energy distribution after pre-selection and the BDT score cut, shown with both labeling schemes to highlight the source of a strange bump below reconstructed energy of 400 MeV. as though they were protons, then a tight cut on this proton MPID score should effectively trim this bump of misreconstructed interactions. The proton MPID score very highly peaks close to a score of 1, so a selection of events with only such a score above 0.9 will result in minimal efﬁciency loss.

Figure 6.13: Maximum proton MPID score for events of reconstructed neutrino energy less than 400 MeV.

6.2.3 Final selection

The ﬁnal 1`1? selection comprises two cuts after pre-selection:

1. BDT Score < 0.4—only select events with sufﬁciently low (i.e. signal-like) BDT Score.

116 A42> 2. a > 400MeV or Proton MPID Score > 0.9—cut out all low energy events that do not have a sufﬁciently proton-like ﬁnal-state particle.

Figure 6.14: Reconstructed a` energy after ﬁnal selection, overlaid with all currently-available data.

The reconstructed energy distribution of the ﬁnal 1`1? selection is shown in Figure 6.14 and the speciﬁc event counts of signal and background after each stage are given in Table 6.4. In

this final 1`1? selection, we see a well-reconstructed a` CCQE signal with 67% purity and 57% efficiency with respect to the available signal after pre-selection. Our dominant backgrounds are events with mis-reconstructed energies, which pile more towards the center of our energy range. Most cases that lead to Bad Reco result from the truncated muon tracks of higher-energy neutrino events. The Not 1L1P background is spread more flatly across the distribution. One may also note that we have reduced all non-beam backgrounds to less than 2%. While our efficiency is relatively low, owing to our incredibly specific signal, this high purity should grant higher correlation between the 1`1? and 141? selections.

117 Figure 6.15: Reconstructed a` energy after ﬁnal selection, overlaid with all currently-available data, but our reconstruction-based labeling scheme has been pared back to a purely interaction- based scheme.

If one takes a step back from our unique labeling convention and considers each neutrino event in our prediction purely in terms of the physical interaction according to the GENIE simulation, thus disregarding reconstruction-based qualiﬁers, then one would see the plot in Figure 6.15 (whose event counts are recorded in Table 6.5). This plot features the same exact events, but now our blue signal includes a` CCQE events that our analysis chain did not adequately reconstruct.

Considering the quality of this analysis in purely these terms reveals a a` CCQE purity of 82%, but with only 19% efﬁciency.

6.3 Systematics

This section will describe how sources of systematic uncertainty (or simply, systematics) are handled in MicroBooNE. Further quantitative assessment will follow in Chapter 7. All systematics are combined into a single covariance matrix for the ﬁnal hypothesis test. Each

118 Selection Stage Total Δ% Well Reco a` CCQE Δ% Backgrounds Δ% Pre-Selection 52518.54 NA 4634.53 (9%) NA 47884.01 (91%) NA BDT Cut 4193.07 -92% 2694.63 (64%) -42% 1498.44 (36%) -97% MPID Cut 3963.38 -5% 2650.90 (67%) -2% 1312.48 (33%) -12%

Table 6.4: Monte Carlo event counts of our well-reconstructed a` CCQE signal, our backgrounds, and the total of all events remaining after each selection stage. Event counts are scaled to match the POT of open Run 1-3 data, 6.8 × 1020 POT. Beside the signal and background counts, in parentheses, are each subsample’s fractional contribution to the total event count. Additionally, the Δ% column beside each event count records the fractional decrease in events of that subsample from the previous stage. One should note that these counts span all reconstructed energy and therefore will not match the counts in Figure 6.14, which are speciﬁc to the plotted energy region.

Selection Stage Total Δ% a` CCQE Δ% Backgrounds Δ% Pre-Selection 52518.54 NA 17050.31 (32%) NA 35468.23 (68%) NA BDT Cut 4193.07 -92% 3331.69 (79%) -80% 861.38 (21%) -98% MPID Cut 3963.38 -5% 3230.84 (82%) -3% 732.54 (18%) -15%

Table 6.5: Monte Carlo event counts of our GENIE true a` CCQE signal, our backgrounds, and the total of all events remaining after each selection stage. Event counts are scaled to match the POT of open Run 1-3 data, 6.8 × 1020. Beside the signal and background counts, in parentheses, are each subsample’s fractional contribution to the total event count. Additionally, the Δ% column beside each event count records the fractional decrease in events of that subsample from the previous stage. individual systematic refers to a core assumption made in our simulation and reconstruction. We address these assumptions by tweaking individual parts of the simulation and measuring how far our ﬁnal neutrino event spectra may deviate from its nominal state (called our Central Value (CV)) across many variations. One variation of a given systematic is called a universe because we perform our full selection on the ensuing spectrum, treating the variation as though it were a wholly possible outcome if our experiment were run in a parallel universe. In our present universe, the one where you are reading this dissertation, the central value is all we have; we will only make a single observation. But we can explore the different outcomes that could arise in a statistical fashion. A covariance matrix summarizes how our spectrum may spread or deviate across different universes, and crucially, how correlations arise between different bins.

119 The elements of covariance matrix M for a given systematic are given by

N 1 Õ "BHB = ((#E0A ) − #+ )((#E0A ) − #+ ) (6.14) 8 9 N 8 D 8 9 D 9 D=1

#E0A 8 D where ( 8 )D is the predicted event count in the th bin of our distribution in the th universe of D #+ 8 variation, and spans a total of N thrown universes. 8 is the event count of the th bin in the CV—the nominal distribution assuming no variation. The total covariance matrix, taking into account all systematics is then the sum of the covariance matrices for each uncorrelated systematic component:

"BHB C>C0; = "BHB 1 + "BHB 2 + "BHB 3 + ... (6.15)

Flux, Cross-Section (XSec), and Nuclear Re-interaction systematics are all handled through a method of event weights. Because there are so many individual systematics, it would be very taxing to simulate a statistically viable number of universes for each one. Instead, a single central value distribution is simulated, and each variation is considered as a set of weights. For context, we simulate approximately one million neutrino events for our central value, there are dozens of individual systematic variations, and each variation requires on the order of 1000 universes for a statistically meaningful spread. Therefore it is much more sensible to simulate the neutrino events once, and then, for each variation, change how much each event contributes to the whole distribution. Calculating systematics via reweighting does raise a limitation. The relative contribution of an event within its bin may change, but that event will never be moved to another bin. This method will fail to model effects that substantially affect the kinematics of an event. For this reason, detector systematics—variations in our TPC’s ability to resolve neutrino events—are considered differently.

120 6.3.1 Reweightable Systematics

Flux systematics describe uncertainty related to the simulation of hadron production at the proton target, horn effects, and secondary particle interactions. Each individual systematic parameter is varied across 1000 universes. The full list and detailed discussion of these systematic contributions can be found in [25]. The most substantial contribution from the flux uncertainty is c+ production, which is the dominant neutrino production mechanism. XSec systematics describe uncertainties related to how neutrinos interact with argon nuclei. They are handled by the GENIE simulation package. Variations are each treated as functions of the Geant4 [26] truth information about each MC event (i.e. true neutrino energy, true interaction type, etc). Some of these systematics are handled like the flux, with 1000 universes each, but we also take advantage of a super-parameter called Genie_All, which simultaneously varies several correlated systematics across 100 universes. Lastly, there are some systematics that the GENIE collaboration suggests we only vary across two universes: "minmax" parameters. All GENIE systematics are expounded upon in [70]. Hadron re-interaction systematics describe uncertainties related to final-state particles’ (particularly protons and charged pions) tendencies to interact with another argon nucleus. A covariance matrix is formed for each systematic (or collection of systematics) and all are finally combined into one singular covariance matrix. To better intuit the fractional uncertainty in each bin, it is useful to consider the fractional covariance matrix, K, composed via

"8 9 8 9 = (6.16) # ?A43 # ?A43 8 9

where #8 is the count of events in the prediction’s 8th bin. The fractional covariance matrix for all reweightable systematics is given in Figure 6.16. One can glean the fractional error of each bin in our distribution from the square root of each f ?A43 √ 8 f?A43 ?A43 diagonal element: ?A43 = 88, where is the total systematic uncertainty of N . #8

121 Figure 6.16: Combined fractional covariance matrix of a` candidate event counts, binned according to reconstructed a` energy, including all reweightable systematics. Note: the empty 200- 250 MeV bin is omitted, giving us a 19 × 19 matrix.

6.3.2 Detector Systematics

Detector systematics describe physics and signal-observation effects within the TPC and may be divided into three main categories:

• Wire modiﬁcation (WireMod) systematics, which consider how the ionization electron charge waveform on each wire plane may be distorted as functions of different topological or calorimetric variables.

• Light yield, which uses different lighting models to alter the quantity and directionality of light that may be observed in a given event.

• Physics parameters, like those that may alter the electron drift. These include a variation in the likelihood that drifting electrons may get re-absorbed into argon atoms, as well as what we call the Space Charge Effect (SCE).

The SCE is an effect that must be accounted for when making any topological measurements within

122 the TPC. The MicroBooNE detector is under constant bombardment from cosmic rays due to its position on the Earth’s surface. While smart triggering removes these (mostly) muon tracks as backgrounds, unrecorded tracks still produce large numbers of ions within the detector. Therefore, cosmic rays will ﬁll the detector with more and more ions, which we do not measure, but that will exert some EM force upon other drifting electrons from actual neutrino signals. This manifests as a distortion in position, with electrons being pushed in different directions over the course of their drift. These three categories make up a total of nine different detector systematics that are considered, and each one requires an entirely new MC simulation and reconstruction. This poses a particular challenge, because, as in the case of reweightable-systematics, simulating and reconstructing all these neutrinos is a computationally and time-intensive process. We therefore simulate only a single universe for each detector systematic, and then compare it to the central value after applying Kernel Density Estimation (KDE) smoothing to account for the high statistical ﬂuctuation [81]. The fractional covariance matrix for the combined detector systematics is given in Figure 6.17.

Figure 6.17: Combined fractional covariance matrix for reconstructed a` energy, including all detector systematics. Note: the empty 200-250 MeV bin is omitted, giving us a 19 × 19 matrix.

123 Part IV

Resolution

124 Chapter 7: Quantifying Final Uncertainty

With all the pieces in place, we may ﬁnally begin to answer our inciting questions. In this chap-

ter, before we directly address the LEE and perform our a` disappearance search in MicroBooNE, we will reﬂect on our uncertainties and begin to assess whether or not our a` constraint will work as predicted.

7.1 Relative Systematic Contributions to Uncertainty

The individual systematic covariance matrices were deﬁned and discussed in Section 6.3, but to gain a stronger intuition for the underlying components, the errors are broken down further and shown in relation to reconstructed neutrino energy in Figure 7.1.

Figure 7.1: Plot of the fractional uncertainty in each bin of our 1`1? distribution, divided into contributions from each collection of systematics. Systematic groups are plotted as dashed lines, but note that each contribution is overlaid, not stacked. The solid lines combine individual systematics via a sum in quadrature to give the total fractional uncertainty in each bin.

125 This plot features individual contributors to systematic uncertainty (each taken from the square root of the diagonals of each component covariance matrix) as dashed lines, overlaid to compare relative effects. The bold, solid lines are calculated as the sum of their listed systematics in quadrature. From this ﬁgure, we see that the largest individual source of uncertainty is the cross-section model: the purple dashed line that plateaus above 10% in the median energy range. We have far fewer events on either edge of our plotted energy range than we do at the median, which serves to shape the previous plot and make it difﬁcult to judge uncertainty as a whole. It may be easier to remove that shape and consider the normalization on its own by looking at the normalization uncertainty 5 , which we can quantify with

Í8=B "BHB 8 9 8 9 5 2 = . (7.1) 2 Í8=B # 8 8

Here, 5 2 is equivalent to the singular element of a one-bin fractional systematic covariance matrix, and its square root will give us the fractional uncertainty of our 1`1? selection. The fractional normalization uncertainties for each of the individual systematic components are collected in Ta- ble 7.1.

Uncertainty Source Normalization Uncertainty

Flux 6.3% Cross-Section 10.2% Re-Interaction 1.5% Detector Systematics 6.4% Total Systematic 13.67%

Table 7.1: Contributions of each source of systematic uncertainty in MicroBooNE, as used to plot the reconstructed energy distribution in Figure 6.14, to normalization uncertainty. The shape component of each systematic is removed according to Equation 7.1.

126 7.1.1 Theoretical Limit on a` Disappearance Sensitivity

The normalization uncertainty reveals how well our analysis can resolve the total number of neutrinos we observe, a parameter that is very relevant to oscillation analyses that will be detailed

in Section 9. In fact, the normalization uncertainty can inform our a` disappearance sensitivity

before that particular study even begins. Recall the probability for a` disappearance in a 3+1 sterile neutrino scenario, given in its two-term parametrization below.

% 2 \ 2 . <2! a`→a` = sin 2 `` sin (1 27Δ / ) (7.2)

For sufﬁciently high Δ<2, oscillations across !/ will be so rapid that they will average out above a detector’s energy resolution threshold—peaks and valleys will be smeared within the same

bin indistinguishably*. 2 . <2! 1 In this case, the sin (1 27Δ / ) term will average to a ﬂat 2 , leaving only the oscillation 2 amplitude term (sin 2\``) to dictate oscillation probability. The point where our sensitivity is suppressed by normalization uncertainty is given by

% 2 \ 2 . <2! < j2 5 a`→a` = sin 2 `` sin (1 27Δ / ) (7.3)

j2 j2 <2 where is the critical for the conﬁdence of our sensitivity. As described above, at high Δ , this reduces to 2 \ < j2 5 . sin 2 `` 2 (7.4)

j2 . To demonstrate our sensitivity at 90% conﬁdence, we will use = 1 28. Here, we assume that we have a j2 distribution with 1 degree of freedom (our amplitude term), and that we are placing an exclusion contour, therefore looking at a 1-sided distribution.

2 Our ability to exclude a` disappearance to 90% conﬁdence therefore ends at sin 2\`` = 0.35.

*This effect is inevitable and will come into effect when energy resolution is no longer meaningfully ﬁner than c 1.27Δ<2 ! , given by the period of 3+1 sterile neutrino oscillation at a short baseline.

127 In Section 9, we will validate this claim with a full a` disappearance analysis [82].

7.2 Strength of the a` Constraint

We have frequently alluded to the diligent work of other analyzers on the DL LEE team who have been performing the 141? selection. This 141? selection requires a few different techniques from what has been described in this dissertation, particularly in order to parse the EM showers

endemic to the a4 CCQE signature and to account for the small a4 ﬂux. A full, detailed outline

of the 141? selection is presented in the DL LEE internal note [81]. The ﬁnal a4 selection of this parallel analysis, at the time of writing, is presented in Figure 7.2 †.

20 Figure 7.2: Plot of selected 141? events in 6.8 × 10 POT with no a` constraint. The ESM prediction is given as a magenta histogram and the unfolded LEE signal is outlined in blue. The vertical axis illustrates the state of a4 statistics in MicroBooNE.

This ﬁgure shows a rather simpliﬁed view, purely illustrating the nominal MC prediction (the magenta histogram) and the LEE signal prediction (the blue dashed line). The full uncertainty on

†The DL LEE analysis is currently under ﬁnal collaboration review before the near-sidebands and, eventually, the full Run 1-3 a4 data, can be unblinded.

128 the prediction is drawn as grayed hatched bars. The LEE signal is created by weighing a separate

sample of a4 CCQE events by the MiniBooNE LEE Unfolded distribution according to their true energies [71]. MicroBooNE’s sensitivity to an eLEE signal with the distribution in Figure 7.2 would follow as a comparison between the blue dashed line and the nominal ESM prediction in pink. We have not yet, however, factored in the relevant correlations between the a4 and a` distributions and incorporated our constraint. Instead of considering these two distributions—the 141? a4 CCQE selection and the 1`1? a` CCQE selection—as two separate spectra, it is useful to place them side-by-side, forming one single, long histogram. Every signal event in both selections originates from the same beam and, crucially, the same interaction chain within the ﬂux (speciﬁcally, c+

decay). Our a4 and a` observations are not independent from one another, and neither are their predictions. The combined fractional covariance matrix of both selections is shown in Figure 7.3. The upper-left and lower-right quadrants are the covariance matrices for the 141? distribution and 1`1? distributions respectively—the bottom-right is explicitly identical to the sum of the reweightable and detector systematic covariance matrices shown in Figures 6.16 and 6.17. The off-diagonal

quadrants (upper-right and lower-left) both contain correlated elements between the a4 and a` selections, just as the off-diagonal elements within the 1`1? covariance matrix contains correlated

elements between bins in the a`-only selection. The strength of our analysis lies in correlations. If we recall the covariance matrix formalism given in Equation 5.5, then we can create a matrix

containing only the correlation coefﬁcients d8 9 for each pair of bins 8 9.

"8 9 d8 9 = (7.5) f8f9

This correlation matrix for 141? and 1`1? selections is shown in Figure 7.4. The diagonal has a correlation of 1 by deﬁnition—every bin is perfectly correlated with itself—but things get interesting in the off-diagonals. From either of the correlated quadrants (they are symmetrical),

129 Figure 7.3: The complete fractional covariance matrix for the combined 141? (upper-left) and 1`1? (lower-right) selections. The two selections are visually separated by red bars, with the upper-right and lower-left quadrants containing the correlations between the 1`1? and 141? selections. Note that there are 10 bins in reconstructed a4 energy (each 100 MeV wide), and 19 bins in reconstructed a` energy (each 50 MeV wide). we see that the majority of bins are over 50% correlated between the two distributions. Many bins have correlations surpassing that, getting as high as 75% in the mid-energy range.

We can see how these correlations actually manifest in the 141? distribution in Figure 7.5. Not only have the error bars shrunk (as discussed in Section 5.3.4), but correlations have also been conditionally applied to the overall a4 prediction. The a4 and LEE signal predictions can both be constrained, as they are now being made under the condition of our correlated 1`1? observation and they are effectively connected to that a` observation. The mechanics of this constraint and how systematic errors are reduced is described in [83].

130 Figure 7.4: The complete, correlation matrix for the combined 141? (upper-left) and 1`1? (lower- right) selections. The two selections are visually separated by red bars, with the upper-right and lower-left quadrants containing the correlations between the 1`1? and 141? selections. Note that there are 10 bins in reconstructed a4 energy (each 100 MeV wide), and 19 bins in reconstructed a` energy (each 50 MeV wide).

131 Figure 7.5: Plot of selected 141? events in 6.8 × 1020 POT. The ESM prediction is given as a magenta histogram and the unfolded LEE signal is outlined in blue. In this example, the a` CCQE selection (as presented in Chapter 6) has been used to constrain both the a4 prediction and the uncertainty on that prediction.

132 Chapter 8: Quantifying MicroBooNE’s Sensitivity to a MiniBooNE eLEE

At time of writing, the MicroBooNE collaboration has not yet approved opening the signal box for the DL LEE analysis (nor any LEE analysis). When the data is freely unblinded, Micro- BooNE will be able to speciﬁcally address two worthwhile questions: ﬁrst, and most straightfor- ward, whether or not MicroBooNE data agrees with the standard model prediction in the signal region. By comparing data and MC, as with the j2 in each plot—albeit with a much more rigorous, frequentist approach—we can determine how well MicroBooNE’s data can exclude the null hypothesis for a given signal observation, and vice versa. Second, we can compare MicroBooNE data to the unfolded LEE prediction to see how well data cottons to that unique signal prediction.

With no such data, we can still use the ﬁnely honed a` and a4 selections to predict how sensitive MicroBooNE will be to the unfolded LEE signal, purely comparing an MC prediction under two hypotheses. The following analysis utilizes the SBNﬁt software suite for statistical analysis.

8.1 SBNﬁt

This entire frequentist procedure and the machinery for constructing covariance matrices are conveniently facilitated by the SBNfit package, which can be found at https://github.com/ NevisUB/whipping_star/releases/tag/v2.0.0. As an aside, the author made contributions to SBNfit in its earlier iterations, particularly in its functionality pertaining to oscillation studies. SBNfit is a multi-detector, multi-mode, multi-channel framework for performing statistical analyses on neutrino data, specially designed for a streamlined, iterative, and consistent means of performing the LEE signal statistical analysis and testing across all LEE analyses undergone within MicroBooNE.

133 8.1.1 Drawing Pseudo-Experiments with SBNﬁt

Ultimately, any experiment will make a single observation of a given truth in our universe and that observation will carry a burden of systematic and statistical uncertainty. One will never know if their observation is close to the expected value or if it is a wildly-unlikely edge case. It is helpful, then, to use simulated pseudo-experiments to perform a more rigorous statistical analysis of potential outcomes. Consider some MC distribution, #+ , which we’ll call our central value. #+ is a prediction intended to model some true, underlying physics, but we can never have access to the actual, true distribution. We can simulate, however, different possibilities for what the true distribution may be. We assume our central value to be within some uncertainty of the underlying truth. For any given simulation, which we will call universe D, we will take #+ and for each bin 8, draw a random #+ value from a Gaussian centered around 8 with variance given by the uncertainty in that bin, f+ 8 , thereby building a new spectrum systematically ﬂuctuated around our CV. We will call this new spectrum (#+ )D, and it will represent the true distribution that #+ set out to model in one example universe D. Now we will consider an observation made in this new universe by an identical experiment to ours, which we will call a pseudo-experiment. The pseudo-experiment in universe D will make a measurement of (#+ )D, but will be subject to standard statistical ﬂuctuations. We then Poisson smear this true distribution to simulate a fake data observation.

Fake data is what a pseudo-experiment observes, or in other words, it is a simulation of what our experiment could have observed, given the statistical and systematic ﬂuctuations of another universe.

In the case of our 1`1? and 141? selections, these distributions carry many inter-bin correlations that would be neglected with a purely Gaussian draw. To account for this, SBNﬁt utilizes the properties of Cholesky decomposition. The speciﬁc procedure is outlined in greater detail in [71], but Cholesky decomposition allows us to draw random distributions using the correlated

134 uncertainties given by a covariance matrix. For every example in this dissertation where pseudo-experiments are drawn, they are drawn from a covariance matrix.

8.2 Method of Frequentist Hypothesis Testing with SBNﬁt

We do not simply wish to know whether or not we see the LEE, or whether or not we reject our null hypothesis (the ESM with no LEE): rather, we wish to communicate a joint statement that one hypothesis is preferred over the other. This concept gains complexity when one considers that, like every physics experiment, our observations will account for a single measurement in one of infinite universes of probabilistic fluctuation. If the ESM is true, then how often will some statistical or systematic variation mimic an LEE-like effect? What if our universe is one that provides a very ambiguous outcome—how much is our analysis subject to the fragile whim of an uncaring god? It is easier to fully explain the statistical motivations for the following method after describing its process. In short, we use a frequentist technique [84, 36] that attempts to bridge the gap between two ideas, represented by two quantities: confidence (U), the probability that an observation falsely

disfavors the null (ESM) hypothesis; and power (1−V), the probability that an observation correctly favors the alternative (LEE) hypothesis. j2 j2 j2 First, we reassert our test statistic: Δ = =D;; − ! , where we have claimed our Null Hypothesis to be one with no LEE, described exactly as the ESM prescribes,

8=B j2 Õ #>1B # ℎH? "−1 #>1B # ℎH? . ℎH? = ( − 8 ) 8 9 ( 9 − 9 ) (8.1) 8, 9

j2 8 #>1B A ℎH? is deﬁned in terms of the event count in each bin of an observed spectrum ( 8 ), the 9 # ℎH? event count in each bin of a predicted spectrum under some hypothesis ( 9 ), and the covariance matrix for the prediction under that hypothesis (M = cov(ℎH?)). Note that " ℎH? is the full systematics covariance matrix (as presented in Section 7.2) with statistical uncertainty along the diagonals as prescribed under a Combined-Neyman-Pearson j2 (as presented in Section 5.3.3).

135 We then simulate 100,000 pseudo-experiments under the assumption that the null hypothesis models the truth. This process of generating pseudo-experiments is facilitated by SBNﬁt and operates speciﬁcally as outlined earlier, in Section 8.1.1. For each universe, we calculate our test statistic and form a probability density function for Δ j2. This distribution is plotted in red on Figure 8.1.

Figure 8.1: Plot of Δ j2 PDFs of our two-hypotheses (null versus an LEE hypothesis) with the 1`1? constraint applied. The red histogram is a PDF for observing Δ j2 under a true null hypothesis, and the blue is a PDF for observing Δ j2 under a true LEE hypothesis. Vertical lines are drawn to indicate observations of median, ±1f, and ±2f likelihoods under the LEE hypothesis.

This red distribution illustrates the span of outcomes for our Δ j2 measurement and, if one takes the integral, the probability of any particular observation. For example, in Figure 8.1, we see that under the assumption that our null hypothesis is true, we are most likely to see a Δ j2 ∼ −5, which is negative, and thus in favor of the null hypothesis. There is, however, still the possibility that our measured test statistic lies in the positive range, favoring the LEE hypothesis, which we

136 know to be incorrect by our deﬁnition. We calculate U for a given measurement by taking the integral of the red distribution from that point on the Δ j2 axis rightward: it is the probability that one could make an observation that was worse than or equal to what was measured. Next, we simulate 100,000 fake observations under the assumption that the LEE hypothesis models the truth. Just as before, we calculate a test statistic for each generated universe and plot them, forming the blue histogram on Figure 8.1. In this scenario, a positive Δ j2 would favor the true hypothesis. The power is then the probability that an observation could be better than or equal to what was measured under the LEE hypothesis: the integral of the blue histogram from a given measurement to the left. Thus, wherever our observation of Δ j2 may land, we can calculate these two quantities: we can assess how well our measurement lies with either hypothesis.

8.3 MicroBooNE Sensitivity to a MiniBooNE eLEE

2 Assuming the LEE hypothesis (! ) is true, there is a 68% chance we will observe a Δ j somewhere between those two green bars, but until a measurement is made, there is no singular statement we can make that reconciles all of the information in Figure 8.1. For better understanding, consider the following cases:

• If we see Δ j2 ≈ 20, then that measurement rejects our null hypothesis to 3.6f while being within the 1f allowed region for the LEE hypothesis. Such an observation would signiﬁ- cantly favor the LEE hypothesis, but is not quite strong enough to completely rule out the null.

• If we see Δ j2 ≈ 0, then that measurement rejects our null hypothesis to only 1.2f while being within the 1f allowed region for the LEE hypothesis. This observation shows no clear favorite and a measurement in this area would be so ambiguous as to be a worst-case scenario.

137 • If we see Δ j2 ≈ 40, then that measurement rejects our null hypothesis implicitly (the null hypothesis is so unlikely that we are limited by the 100,000 fake observations used to con-

struct the plot). However, it also lies outside 2f rejection of our LEE hypothesis, so we could not conﬁdently claim that we favor the LEE hypothesis over the null.

One must consider each of the previous scenarios to understand MicroBooNE’s sensitivity to the LEE, but a helpful, single (though, incomplete) summary is that MicroBooNE has a median signiﬁcance to reject the Null hypothesis (=D;;) of 2.5f.

If we were to omit the a` constraint and perform the LEE search with only a 141? distribution, our Δ j2 PDFs would appear as shown in Figure 8.2. In this case, the median signiﬁcance to reject

=D;; assuming the LEE hypothesis is 2.3f. Our constraint gains us a 0.2f improvement to the median, but is better illustrated in how the blue histogram is shaped. The left edge of the LEE PDF in the unconstrained case reaches so low that our zone of ambiguity, where there is overlap between highly-likely portions of each histogram, is much bigger when unconstrained. The 1`1? constraint lessens the probability that we are thrust into an irreconcilable scenario.

8.4 Further Extrapolation

A sensitivity improvement of 0.2f may be read as an anticlimax—an unarguable boon, but hardly, and hardly commensurate to the effort spent. The presented analysis has demonstrated MicroBooNE’s capability to resolve an eLEE in Runs 1-3 of data, a paltry fraction of what will soon be available. If we project forward to a scenario where all data in Runs 1-5 are included, amounting to a 1.3 × 1021 POT, approximately double what has been considered thus far, then we see the LEE ﬁts presented in Figure 8.3. On the left, there is no constraint in place, and on the right, the 1`1? selection is being utilized, granting an improvement of 0.7f to the median sensitivity, raising it to 3.4f.

As more data is unblinded, the a` constraint will only increase in strength.

138 Figure 8.2: Plot of Δ j2 PDFs of our two-hypotheses (null versus an LEE hypothesis) for only the 141? selection. The red histogram is a PDF for observing Δ j2 under a true null hypothesis, and the blue is a PDF for observing Δ j2 under a true LEE hypothesis. Vertical lines are drawn to indicate observations of median, ±1f, and ±2f likelihoods under the LEE hypothesis.

Figure 8.3: Plot of Δ j2 PDFs for our two hypothesis (null versus an LEE hypothesis) with MC scaled to account for Runs 1-5 data, 1.3 × 1021 POT. On the left, only a 141? selection is used for the LEE ﬁt, and on the right, we add the 1`1? constraint [81].

139 Chapter 9: Measurement of a` Disappearance with MicroBooNE

An additional benefit to having this high-purity a` selection is that we can perform a direct search for a` disappearance due to sterile neutrino oscillations at a short baseline. With this analysis, we will both demonstrate that MicroBooNE’s technology and analytical methods are sufficiently matured to perform an entirely separate analysis from its intended goals, and look for sterile neutrino oscillations under a 3+1 hypothesis independently from our LEE search. This process follows an identical framework to what was used in performing the global fits described in Chapter 4, but will go into much greater detail.

9.1 Calculating Sensitivity

Before considering data, we will explore MicroBooNE’s sensitivity to 3+1 sterile neutrino oscillations by measuring against our MC prediction, which we will call our null (or no-osc, or 3a) hypothesis. We will compare the predicted distribution of reconstructed neutrino energy in Micro-

BooNE under various sterile neutrino hypotheses with that of our 3a spectrum, thereby posing the question: for what sets of oscillation parameters will MicroBooNE be able to differentiate a sterile neutrino signal from our null hypothesis?

The test statistic for our comparison will be a Pearson j2, given by

#18=B j2 Õ #>1B # ?A43 2 \ , <2 "−1 #>1B # ?A43 2 \ , <2 = ( 8 − 8 (sin 2 `` Δ )) ( 9 − 9 (sin 2 `` Δ )) (9.1) 8, 9

# ?A43 2 \ , <2 8 where 8 (sin 2 `` Δ ) is the count of predicted events in the th bin of a distribution under <2 2 \ #>1B a sterile neutrino hypothesis defined by parameters Δ and sin 2 ``, and 8 is the count of events of our no-osc hypothesis in the same bin. Our covariance matrix, M, is specifically defined

140 as cov(?A43). The method of creating a prediction of MicroBooNE MC oscillated under different sterile neutrino hypotheses is described in Section 4.2.4. At each point in parameter space, we construct our oscillated spectrum and calculate the j2 between that sterile neutrino prediction and our null hypothesis. The fruits of this process are illustrated in Figure 9.1, where we see our full, 2D parameter space, and at each point, a color representing the log of the j2 calculated with the prediction at that set of coordinates. For added clarity, two arbitrary points on the grid have been highlighted and their predicted spectra are plotted on the right.

2 Figure 9.1: j surface across the parameter space of a` disappearance under 3+1 sterile neutrino oscillations. At each point on the grid, the coordinates deﬁne a sterile neutrino hypothesis, which is compared with the null spectrum. The color on the grid is given by the natural log of the j2. Two points are selected to demonstrate how the predicted spectrum (in blue) varies across parameter space versus a constant, no-osc hypothesis (orange points).

Conﬁdence intervals may be drawn exactly as described in Section 4.2.5. This type of analysis, where we are comparing the shape and normalization components of two spectra is called a Shape and Rate (shape+rate, or S+R) ﬁt.

141 9.1.1 A Parallel Shape-Only Analysis

Recall that our total normalization uncertainty from systematics, at 13.67% is quite large. It is valuable to perform a parallel analysis assuming that we have no usable normalization information and must rely entirely on the shape of a distribution. A Shape-Only (SO) ﬁt operates in much the same way as the shape+rate analysis, save for two key changes:

• At every point in parameter space, the total count of events in the prediction is scaled to exactly match the total count of observed events.

• The normalization component of the covariance matrix is removed for every j2 calculation.

Speciﬁcally, the shape-only j2 is given by

#18=B j2 ($ Õ #>1B ^# ?A43 2 \ , <2 "($ −1 #>1B ^# ?A43 2 \ , <2 , ( ) = ( 8 − 8 (sin 2 `` Δ ))( ) ( 9 − 9 (sin 2 `` Δ )) (9.2) 8, 9

^ Í # ?A43 Í #>1B "(+' where = 8 8 / 8 8 is a normalizing factor. The covariance matrix M = has had the normalization component of the systematic matrix removed on a per-element basis, via

"($ BHBC "(+' BHBC 5 2 # ?A43 # ?A43 . ( )8 9 = ( )8 9 − 8 9 (9.3)

The normalization uncertainty 5 2 is deﬁned by Equation 7.1. A side-by-side comparison of the shape-only and shape-plus-rate matrices is presented in Figure 9.2. The resulting sensitivity from both a shape+rate and a shape-only ﬁts are presented in Fig-

2 ure 9.3. A vertical red line is drawn at sin 2\`` = 0.35 to illustrate predicted sensitivity in the high-Δ<2 region. We see that our shape+rate 90% sensitivity contour corroborates our prediction.

On the same plot, MiniBooNE’s ﬁrst a` disappearance sensitivity result, performed with over 20 times as much data as MicroBooNE has available, is overlaid [56].

142 Figure 9.2: On the left, we see the full systematic covariance matrix used for the DL LEE 1`1? analysis including all reweightable and detector systematic components and scaled to the null hypothesis. On the right, we have removed the normalization component from each individual element according to Equation 9.3.

9.2 Applying a Frequentist Correction

Harkening back to our global fit discourse, we recall that building disappearance limits involves finding a Δ j2 with respect to the best-fit point and drawing an exclusion region via a comparison to j2 j2 U some . The critical depended upon two factors: a chosen to determine the confidence and the DoF of the search. One’s naive assumption is that a disappearance fit across two parameters should operate with two degrees of freedom. In practice, things are much messier. Consider two truths:

2 • Our amplitude term, sin 2\``, is naturally bound between 0 and 1. As this term approaches zero and the amplitude diminishes, any disappearance effects will vanish. Regardless of

the Δ<2 value, any disappearance model would be indistinguishable from a 3a hypothesis, rendering both parameters meaningless.

• Our frequency term, Δ<2, is only bound by detector energy resolution (not counting cosmic

constraints to mass). As Δ<2 increases, sin2(1.27Δ<2!/) will oscillate rapidly. Beyond a Δ<2 threshold, the oscillation wavelength will shrink below a detector’s resolution capability 2 . <2! 1 and resulting measurements of sin (1 27Δ / ) will smear to 2 (see Section 7.1.1).

To adhere to Wilks’ Theorem and use two DoF avers that our Δ j2 distributions are perfectly

143 Figure 9.3: MicroBooNE’s 90% sensitivity contours for a shape-only (SO) and shape-and-rate (S+R) global scan. The contours are drawn assuming a Δ j2 distribution with 2 degrees of freedom. 2 MiniBooNE’s a` disappearance sensitivity is overlaid [56], as well as a vertical line at sin 2\`` = 0.35, the predicted high-Δ<2 sensitivity calculated from our normalization uncertainty.

Gaussian. With either of the above cases, one could not claim to have two effective degrees of freedom across parameter space, and there may be even more underlying correlations we cannot as readily consider. In order to proceed, we require a way to find the effective degrees of freedom j2 and hence as a function of our oscillation parameters. A frequentist definition of a confidence interval with some U can be interpreted as a region of parameter space that, across some very large number of experiments, will contain the true value of the parameter in (1-U) percent of experiments. Our strategy will proceed as follows:

144 2 2 For each point % = (sin 2\``, Δ< ) in our parameter space Θˆ , we will consider % to be the true value of the parameter set, perfectly modeling the underlying physics. We will then construct

a a` distribution whose prediction is oscillated according to the prediction model at P. To consider different true scenarios that would give this prediction, we simulate N pseudo-experiments, each j2 with their own fake observations. By performing a fit to each fake observation, one can find by j2 U j2 j2 < j2 counting. We define such that (1- ) percent of Δ satisfies Δ .

A Step-by-Step Frequentist Study

With the big picture loosely described, we will now walk through the process of this analysis exactly as performed by the author. Our sterile neutrino oscillation parameter space Θˆ is divided into 625 discrete points: 25 points

logarithmically spaced across Δ<2 in (0.014+2, 1004+2) and 25 points logarithmically spaced across sin2 2\ in (0.01, 1.0). These ranges, particularly that of sin2 2\, were selected based on the initial non-frequentist disappearance search (i.e. one where we assumed two DoF, as in Figure 9.3), and the coarseness of the grid is limited by the processing time.

We begin by selecting one point %) ∈ Θˆ —our true point. For the purpose of this exercise, we

assert that 3+1 sterile neutrino oscillation exists and that %) are exactly the parameters that model

the true, underlying mechanics. From %) , one can construct a predicted spectrum of reconstructed

%) ?A43 a` energy N = # (%) ) following the means outlined in Section 4.2.4. As in our LEE search, we will take advantage of SBNﬁt’s faculty to simulate fake experiments

(using the methods articulated in Section 8.1.1). We will draw N pseudo-experiments around %)

from covariance matrix " %) for some large value of N. The analysis presented in this disserta-

tion uses N = 1000 fake universes, limited, again, by processing time. Each pseudo-experiment operates as though one had conducted an identical oscillation experiment and observed an independent spectrum of fake data. We can then perform a ﬁt to 3+1 muon neutrino disappearance

models across Θˆ . We do this by scanning across the same 25 × 25 grid outlined above. For each pseudo-experiment :, and for each on this grid, we build predicted spectrum N and calculate

145 the j2 using #18=B Õ j2 # # %) 0 "−1 # # %) 0 ( ): = ( 8 − ( )8: ) 8 9 ( 9 − ( )9 : ) (9.4) 8 9

# %) 0 8 : where ( )8: is the number of events in the th bin of the fake observation made in the th pseudo- experiment drawn from # %) , and M is the covariance matrix of the prediction point , given as cov(). For each pseudo-experiment :, we perform a ﬁt and settle upon a best ﬁt point 1 5 ∈ Θˆ . We combine this with the j2 at our actual truth point, % , into (Δ j2) = (j2 − j2 ) , a distribution of ) : %) 1 5 :

how similarly the oscillation models of the best fit and %) describe data. It is important to clarify that each j2 is calculated with a covariance matrix updated to match the predicted spectrum, such that j2 uses cov(% ) and j2 uses cov(1 5 ). %) ) 1 5 2 2 2 Spectra of (Δ j ) for %) = (sin 2\, Δ< ) = (0.01, 0.01 eV) (the null point in our search) in both shape-and-rate and shape-only analyses are plotted in Figures 9.4 and 9.5. These Δ j2 spectra, shown as red histograms, are overlaid with theoretical j2 distributions (2 DoF for shape- and-rate, 1 DoF for shape-only) to illustrate the folly of our uncorrected assumption. Furthermore, j2 the positions of each are marked with vertical lines, each one representing the point in its respective distribution where 90% of events are to its left. U j2 j2 < j2 Next, we choose the confidence, , with which we define our critical such that (Δ ): for (1 − U)% of pseudo-experiments : ∈ N. One must then repeat this entire process for every % ˆ j2 point ) ∈ Θ. The 90% gleaned from counting in this frequentist method is shown as bin color

across all 625 points (%) ) across parameter space in the left plot of Figure 9.6. We then proceed through the entire frequentist analysis again, using a shape-only procedure. j2 ˆ The resulting 90% CL spectrum across Θ in this scenario is given in Figure 9.7. j2 With defined at each point in parameter space, we may redraw our confidence intervals j2 j2 exactly as before—but instead Δ for every prediction with the at that prediction. To account for the coarseness of our frequentist analysis, the author linearly interpolates across the surfaces presented in Figures 9.6 and 9.7 to draw finer contours.

146 Figure 9.4: Distribution of Δ j2 across N = 1000 pseudo-experiments for the null, no-osc, hypoth- 2 esis as %) , shown in red. A j distribution with 2 degrees of freedom is overlaid in blue. Vertical lines are drawn for the 90% CL critical j2 of each distribution, such that 90% of the distribution is to the left of each line. This distribution is the result of a shape-and-rate ﬁt.

MicroBooNE’s corrected sensitivity is thus presented on the left plot of Figure 9.8. To the right, we directly compare our 90% shape+rate sensitivity before and after applying this frequentist cor- j2 rection to our methods. Using a statistically driven grid of across parameter space substantially curbs our sensitivity, reﬂecting how important this consideration is to assessing MicroBooNE’s sensitivity to 3+1 sterile neutrino oscillation. Further validation of our sensitivity through signal- injection tests may be read in Appendix C.

9.3 Fitting to Data

Lastly, we may perform a ﬁt of MicroBooNE unblinded data, chosen by our DL LEE 1`1? selection, to 3+1 sterile neutrino oscillation via the a` disappearance channel. Instead of using a no-osc hypothesis, we will now ask for which parameters of 3+1 sterile neutrino oscillation will an oscillated prediction compare favorably with our data. Alternatively, we will demonstrate that

147 Figure 9.5: Distribution of Δ j2 across N = 1000 pseudo-experiments for the null, no-osc, hypoth- 2 esis as %) , shown in red. A j distribution with 1 degree of freedom is overlaid in blue. Vertical lines are drawn for the 90% CL critical j2 of each distribution, such that 90% of the distribution is to the left of each line. This distribution is the result of a shape-only fit. our data rules out a region of this parameter space. As a reminder, the data being fit is presented in Figure 6.14. Assuming (incorrectly, naively) that Δ j2 follows a uniform 2 DoF distribution across parameter space, a fit to data gives the exclusion contours and allowed region at 90% confidence shown in Figure 9.9. The shape-only fit forms closed contours around an allowed region of parameter space with a best fit at (sin2 2\, Δ<2) = (0.72, 244+2). The j2 of the null hypothesis with data in a shape-only comparison is 13.26, compared with a j2 of 6.88 at the best-fit point. The shape-only analysis actually rules out the null no-osc hypothesis at 90% confidence. j2 We follow up by applying our frequentist correction to the fit, using the grids determined by counting pseudo-experiments. The resulting fig is presented in Figure 9.10. With the correction to our method, we no longer see a closed contour for either the shape+rate or the shape-only fits—the no-osc hypothesis is no longer ruled out at 90% confidence.

148 Figure 9.6: The critical chi-squared for U = 0.1 across every point in our parameter space. The left j2 j2 plot displays the raw value, while the right shows the fractional difference in from a 2-DoF scenario.

j2 The at both the null hypothesis and best fit oscillation model are listed in Table 9.1 (shape+rate), j2 j2 j2 and Table 9.2 (shape-only). Despite the relatively large Δ in the shape-only fit, (Δ = =>−>B2 − j2 . j2 1 5 = 6 8), at the best fit for 90% is twice as large as in the 2 DoF assumption.

j2 j2 j2 Conﬁdence Level (2DoF) at Null (FC) at BF (FC) 90% 4.61 1.94 2.06 99% 9.21 7.13 6.13

Table 9.1: This table lists the critical j2 as predicted by Wilks’ theorem (using a standard two-sided j2 j2 distribution with 2 degrees of freedom), and the same as derived from our Feldman-Cousins, shape-and-rate analysis at two different points in parameter space: the Null, no-oscillation point 2 2 2 2 2 2 (sin 2\`` = 0.01, Δ< = 0.014+ ) and the best ﬁt point (sin 2\`` = 0.30, Δ< = 244+ ).

The MicroBooNE limit on 3+1 oscillation through a` disappearance is not world-leading, but one must recall that this entire ﬁt is built on the back of an entirely separate analysis. MicroBooNE

places a meaningful limit on 3+1 oscillation using only a pure selection of a` CCQE events that have been optimized for an LEE search. We are excited for the prospect of a MicroBooNE os-

cillation limit performed with combined a4 appearance and disappearance searches and with a a` search that is reoptimized for maximal oscillation sensitivity (via looser cuts and a higher-stat se-

149 Figure 9.7: The critical chi-squared for U = 0.1 across every point in our parameter space under a j2 shape-only analysis. The left plot displays the raw value, while the right shows the fractional j2 difference in from a 2-DoF scenario.

j2 j2 j2 Conﬁdence Level (2DoF) at Null (FC) at BF (FC) 90% 4.61 9.32 9.86 99% 9.21 16.31 18.73

Table 9.2: This table lists the critical j2 as predicted by Wilks’ theorem (using a standard two- j2 j2 sided distribution with 2 degrees of freedom), and the same as derived from our Feldman- Cousins, shape-only analysis at two different points in parameter space: the Null, no-oscillation 2 2 2 2 2 2 point (sin 2\`` = 0.01, Δ< = 0.014+ ) and the best ﬁt point (sin 2\`` = 0.69, Δ< = 244+ ).

lection). Until that point, this has been the ﬁrst-ever a` disappearance analysis with LArTPC data and analysis techniques.

150 Figure 9.8: On the left, MicroBooNE’s 90% sensitivity contours for a shape-only (SO) and shape- and-rate (S+R) global scan with the frequentist correction applied. MiniBooNE’s a` disappearance sensitivity is overlaid, which was also drawn with a frequentist contour [56]. On the right, we see the MicroBooNE S+R sensitivity with and without the frequentist correction.

151 Figure 9.9: MicroBooNE’s 90% limit for a shape-and-rate (S+R) fit to data and a 90% allowed region resulting from a shape-only (SO) fit to data. Both the limit and allowed region are constructed using the Wilks’ Theorem (2 degrees of freedom) approximation. MiniBooNE’s a` disappearance limit is overlaid, drawn with a frequentist contour [56]. Notably, the SO fit rejects the null hypothesis at 90% confidence.

152 Figure 9.10: MicroBooNE’s 90% limits for a shape-and-rate (S+R, in light blue) and shape-only (SO, in dark blue) frequentist fits to data. MiniBooNE’s a` disappearance limit is overlaid, also drawn with a frequentist contour [56]. Where the non-frequentist-corrected iteration of this plot (as seen in Figure 9.9) illustrated an allowed region for the SO fit, the frequentist correction does not. Despite a Δ j2 of 6.38 favoring the SO best fit point, the critical j2 diverges greatly from a 2 DoF assumption. This effect is highlighted in Table 9.2.

153 Part V

Denouement

154 Chapter 10: Reﬂections On the Long Way Home

At the time of writing, April of 2021, MicroBooNE has not opened data and so the fate of the LEE is unknown. The DL LEE analysis has progressed in its 1e1p selection, but the most current sensitivities are as shown in this dissertation. It is yet to be seen if MicroBooNE will be very lucky or very unlucky in its hypothesis test measurement. The effort put forth towards a a` constraint boasts marginal improvements now and greater increases to LEE sensitivity as more data is unblinded. Regardless of outcome, MicroBooNE has set an important milestone in its field, being the longest-running LArTPC and the first to have a fully automated event selection. MicroBooNE has placed a constraint on the neutrino cross section in Liquid Argon, which continues to strengthen [85]. The DL LEE analysis specifically has pioneered the use of deep learning techniques for LArTPC reconstruction and classification. While the MicroBooNE LEE result will be very quickly outdone when SBND and Icarus begin taking data, thereby completing the SBN project, the work done by MicroBooNE sets an important precedent. As for sterile neutrinos: the global fit presented in this dissertation signals that such analyses need to reconsider their approach to the question of global compatibility. We find a best-fit point, but tensions in global data void that accomplishment of its classical meaning. It should be un- controversial to conclude that 3+1 sterile neutrino oscillations (or, for that matter, 3+2, 3+3, or 3+N) do not resolve the LEE and other anomalies on their own. An issue taking many forms will likely require many solutions. Still, there is much to learn from combining results of disparate experiments and that practice should continue, but with increased statistical rigor and more nuanced questions. Despite statistical disadvantages, MicroBooNE is able to perform an oscillation fit with a selection that was never intended for that purpose and extract meaningful limits in both shape-only

155 and shape+rate analyses—a testament to the technological and analytical capabilities of the Mi- croBooNE collaboration and detector. MicroBooNE does not have the high-statistics of oscillation experiments that came before [18, 19] or ones that will follow [23, 31], but this dissertation demonstrates that LArTPC technology, when paired with deep-learning analysis techniques, grants us an opportunity to exploit our inti- mate knowledge of CCQE physics and to watch those interactions manifest throughout our data and predictions. We can utilize our physics principles to lend both strength and credence to our understandings of the LEE, the ESM, and beyond.

156 Epilogue

Neutrinos remain elusive and this is largely because neutrino physics is incredibly hard. The author hopes that this work has carved away at least some of the mystery and presented a snapshot of what this process of inquiry entails.

157 References

[1] Introduction to Elementary Particles. Second, Revised Edition. Weinheim: Wiley-VCH Ver- lag GmbH & Co KGaA, 2008.

[2] C. Cowan Jr. et al., “Detection of the free neutrino: A conﬁrmation,” Science, vol. 124, p. 103, 1956.

[3] G. S. Karagiorgi, “Searches for New Physics at MiniBooNE: Sterile Neutrinos and Mixing Freedom,” PhD thesis, 2010.

[4] P. Zyla et al., “Review of Particle Physics,” PTEP, vol. 2020, no. 8, p. 083C01, 2020.

− [5] Q. R. Ahmad et al., “Measurement of the rate of a4 + 3 → ? + ? + 4 interactions produced by 8B solar neutrinos at the Sudbury Neutrino Observatory,” Phys. Rev. Lett., vol. 87, p. 071 301, 2001. arXiv: nucl-ex/0106015.

[6] Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett., vol. 81, pp. 1562–1567, 1998. arXiv: hep-ex/9807003.

[7] R. Davis and D. S. Harmer, “Attempt to observe the cl37(a-bar, e-)ar37 reaction induced by reactor antineutrinos,” Bulletin of the American Physiological Society, vol. 4, p. 217, 1956.

[8] J. N. Abdurashitov, V.N. Gavrin, V.V.Gorbachev, et al., “Measurement of the solar neutrino capture rate with gallium metal. iii. results for the 2002–2007 data-taking period,” Phys. Rev. C, vol. 80, p. 015 807, 1 2009.

[9] F. Kaether, W. Hampel, G. Heusser, et al., “Reanalysis of the GALLEX solar neutrino ﬂux and source experiments,” Phys. Lett. B, vol. 685, pp. 47–54, 2010. arXiv: 1001.2731 [hep-ex].

[10] D. Adey et al., “Improved Measurement of the Reactor Antineutrino Flux at Daya Bay,” Phys. Rev. D, vol. 100, no. 5, p. 052 004, 2019. arXiv: 1808.10836 [hep-ex].

[11] F. Ardellier et al., “Double Chooz: A Search for the neutrino mixing angle theta(13),” Jun. 2006. arXiv: hep-ex/0606025.

[12] S. Abe et al., “Precision Measurement of Neutrino Oscillation Parameters with KamLAND,” Phys. Rev. Lett., vol. 100, p. 221 803, 2008. arXiv: 0801.4589 [hep-ex].

[13] J. Evans, “The MINOS Experiment: Results and Prospects,” Adv. High Energy Phys., vol. 2013, p. 182 537, 2013. arXiv: 1307.0721 [hep-ex].

158 [14] P. Adamson et al., “Measurement of the neutrino mixing angle \23 in NOvA,” Phys. Rev. Lett., vol. 118, no. 15, p. 151 802, 2017. arXiv: 1701.05891 [hep-ex].

[15] M. H. Ahn et al., “Measurement of Neutrino Oscillation by the K2K Experiment,” Phys. Rev. D, vol. 74, p. 072 003, 2006. arXiv: hep-ex/0606032.

[16] K. Abe et al., “The T2K Experiment,” Nucl. Instrum. Meth. A, vol. 659, pp. 106–135, 2011. arXiv: 1106.1238 [physics.ins-det].

[17] C. Athanassopoulos, L. B. Auerbach, R. L. Burman, et al., “Evidence for a`→ae oscillations from the lsnd experiment at the los alamos meson physics facility,” Phys. Rev. Lett., vol. 77, pp. 3082–3085, 15 1996.

[18] A. Aguilar-Arevalo et al., “Evidence for neutrino oscillations from the observation of a¯4 appearance in a a¯` beam,” Phys. Rev. D, vol. 64, p. 112 007, 2001. arXiv: hep-ex/0104049.

[19] A. A. Aguilar-Arevalo et al., “A Search for Electron Neutrino Appearance at the Δ<2 ∼ 14+2 Scale,” Phys. Rev. Lett., vol. 98, p. 231 801, 2007. arXiv: 0704.1500 [hep-ex].

[20] ——, “Updated MiniBooNE neutrino oscillation results with increased data and new background studies,” Phys. Rev. D, vol. 103, no. 5, p. 052 002, 2021. arXiv: 2006 . 16883 [hep-ex].

[21] A. M. Knee, D. Contreras, and D. Scott, “Cosmological constraints on sterile neutrino oscillations from Planck,” JCAP, vol. 07, p. 039, 2019. arXiv: 1812 . 02102 [astro- ph.CO].

[22] MicroBooNE Collaboration, “The microboone technical design report,” MICROBOONE- Internal-DocDB-1821, 2012.

[23] M. Antonello et al., “A Proposal for a Three Detector Short-Baseline Neutrino Oscilla- tion Program in the Fermilab Booster Neutrino Beam,” Mar. 2015. arXiv: 1503.01520 [physics.ins-det].

[24] A. A. Aguilar-Arevalo et al., “The Neutrino Flux prediction at MiniBooNE,” Phys. Rev. D, vol. 79, p. 072 002, 2009. arXiv: 0806.1449 [hep-ex].

[25] J. Zennamo and Z. Pavlovic, “Microboone ﬂux and ﬂux uncertainties,” MICROBOONE- Internal-DocDB–8622 v2.0, 2017.

[26] S. Agostinelli et al., “Geant4—a simulation toolkit,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equip- ment, vol. 506, no. 3, pp. 250–303, 2003.

159 [27] R. Acciarri et al., “Design and Construction of the MicroBooNE Detector,” JINST, vol. 12, no. 02, P02017, 2017. arXiv: 1612.05824 [physics.ins-det].

[28] A. Furmanski, “Triggering in microboone,” MicroBOONE-Internal-DocDB-17143, 2018.

[29] S. Collaboration. (2016). What is snews? (Visited on 03/27/2021).

[30] P. Abratenko et al., “The continuous readout stream of the MicroBooNE liquid argon time projection chamber for detection of supernova burst neutrinos,” JINST, vol. 16, no. 02, P02008, 2021. arXiv: 2008.13761 [physics.ins-det].

[31] B. Abi et al., “Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume I Introduction to DUNE,” JINST, vol. 15, no. 08, T08008, 2020. arXiv: 2002.02967 [physics.ins-det].

[32] M. Betancourt, “Sbnd vertical slice test,” SBN-Internal-DocDB–8580, 2018.

[33] MicroBooNE Collaboration, “A method to extract the charge distribution arriving at the tpc wire planes in microboone,” MicroBooNE Public Note 1017, 2020.

[34] ——, “First neutrino interactions observed with themicroboone liquid-argon tpc detector,” MicroBooNE Public Note 1002, 2015.

[35] A. Aguilar-Arevalo, B. Brown, L. Bugel, et al., “Signiﬁcant excess of electronlike events in the miniboone short-baseline neutrino experiment,” Physical Review Letters, vol. 121, no. 22, 2018.

[36] G. J. Feldman and R. D. Cousins, “A Uniﬁed approach to the classical statistical analysis of small signals,” Phys. Rev. D, vol. 57, pp. 3873–3889, 1998. arXiv: physics/9711021.

[37] Y.-J. Kwa, “Neutron-antineutron oscillation search with microboone and dune,” Interna- tional Conference on High Energy Physics, 2020.

[38] D. Cianci, Y.-J. Kwa, M. Ross-Lonergan, et al., “Resolving oscillation parameter ambiguities in dune under the 3+1 sterile neutrino oscillation scenario using,” In Progress, to be submitted to Phys. Rev. D, 2021.

[39] D. Cianci, A. Furmanski, G. Karagiorgi, et al., “Prospects of Light Sterile Neutrino Oscil- lation and CP Violation Searches at the Fermilab Short Baseline Neutrino Facility,” Phys. Rev. D, vol. 96, no. 5, p. 055 001, 2017. arXiv: 1702.01758 [hep-ph].

[40] P. Adamson et al., “Improved Constraints on Sterile Neutrino Mixing from Disappearance Searches in the MINOS, MINOS+, Daya Bay, and Bugey-3 Experiments,” Phys. Rev. Lett., vol. 125, no. 7, p. 071 801, 2020. arXiv: 2002.00301 [hep-ex].

160 [41] J. M. Conrad, C. M. Ignarra, G. Karagiorgi, et al., “Sterile Neutrino Fits to Short Baseline Neutrino Oscillation Measurements,” Adv. High Energy Phys., vol. 2013, p. 163 897, 2013. arXiv: 1207.4765 [hep-ex].

[42] A. Rohatgi, Webplotdigitizer: Version 4.4, 2020.

[43] B. Armbruster, I. M. Blair, B. A. Bodmann, et al., “Upper limits for neutrino oscillations a`→a4 from muon decay at rest,” Phys. Rev. D, vol. 65, p. 112 001, 11 2002.

[44] P. Adamson, A. A. Aguilar-Arevalo, C. E. Anderson, et al., “Measurement of a` and a4 events in an off-axis horn-focused neutrino beam,” Phys. Rev. Lett., vol. 102, p. 211 801, 21 2009.

[45] P. Astier et al., “Search for nu(mu) —> nu(e) oscillations in the NOMAD experiment,” Phys. Lett. B, vol. 570, pp. 19–31, 2003. arXiv: hep-ex/0306037.

[46] M. Dentler, A. Hernández-Cabezudo, J. Kopp, et al., “Sterile neutrinos or ﬂux uncertainties? — Status of the reactor anti-neutrino anomaly,” JHEP, vol. 11, p. 099, 2017. arXiv: 1709. 04294 [hep-ph].

[47] Y. Ko et al., “Sterile Neutrino Search at the NEOS Experiment,” Phys. Rev. Lett., vol. 118, no. 12, p. 121 802, 2017. arXiv: 1610.05134 [hep-ex].

[48] P. Huber, “NEOS Data and the Origin of the 5 MeV Bump in the Reactor Antineutrino Spectrum,” Phys. Rev. Lett., vol. 118, no. 4, p. 042 502, 2017. arXiv: 1609.03910 [hep- ph].

[49] T. Mueller et al., “Improved Predictions of Reactor Antineutrino Spectra,” Phys. Rev. C, vol. 83, p. 054 615, 2011. arXiv: 1101.2663 [hep-ex].

[50] P. Huber, “On the determination of anti-neutrino spectra from nuclear reactors,” Phys. Rev. C, vol. 84, p. 024 617, 2011, [Erratum: Phys.Rev.C 85, 029901 (2012)]. arXiv: 1106.0687 [hep-ph].

[51] P. Vogeland J. F. Beacom, “Angular distribution of neutron inverse beta decay, anti-neutrino(e) + p —> e+ + n,” Phys. Rev. D, vol. 60, p. 053 003, 1999. arXiv: hep-ph/9903554.

[52] I Alekseev et al., “Search for sterile neutrinos at the DANSS experiment,” Phys. Lett. B, vol. 787, pp. 56–63, 2018. arXiv: 1804.04046 [hep-ex].

[53] M. Danilov, “Sensitivity of DANSS detector to short range neutrino oscillations,” Dec. 2014. arXiv: 1412.0817 [physics.ins-det].

[54] Y. Declais et al., “Search for neutrino oscillations at 15-meters, 40-meters, and 95-meters from a nuclear power reactor at Bugey,” Nucl. Phys. B, vol. 434, pp. 503–534, 1995.

161 [55] J. Conrad and M. Shaevitz, “Limits on Electron Neutrino Disappearance from the KARMEN and LSND a4 - Carbon Cross Section Data,” Phys. Rev. D, vol. 85, p. 013 017, 2012. arXiv: 1106.5552 [hep-ex].

[56] A. A. Aguilar-Arevalo et al., “A Search for muon neutrino and antineutrino disappearance in MiniBooNE,” Phys. Rev. Lett., vol. 103, p. 061 802, 2009. arXiv: 0903.2465 [hep-ex].

[57] F. Dydak et al., “A Search for Muon-neutrino Oscillations in the Delta m**2 Range 0.3- eV**2 to 90-eV**2,” Phys. Lett. B, vol. 134, p. 281, 1984.

2 4 [58] I. Stockdale et al., “Search for a` and a¯ Oscillations in the Mass Range 15 − eV /2 < X"2 < 1000 − eV2/24,” Z. Phys. C, vol. 27, p. 53, 1985.

[59] M. Aartsen et al., “Searches for Sterile Neutrinos with the IceCube Detector,” Phys. Rev. Lett., vol. 117, no. 7, p. 071 801, 2016. arXiv: 1605.01990 [hep-ex].

[60] H. Schulz, M. Wospakrik, M. Ross-Lonergan, et al., “Grid-based minimization at scale: Feldman-Cousins corrections for SBN,” Feb. 2020. arXiv: 2002.07858 [physics.comp- ph].

[61] MicroBooNE Collaboration, “Search for electron neutrinos in multiple topologies with themicroboone experiment,” MicroBooNE Public Note 1085, 2020.

[62] ——, “Event selection in the microboone deep learning based low energy excess analysis using two-body scattering criteria,” MicroBooNE Public Note 1086, 2020.

[63] P. Abratenko et al., “Neutrino Event Selection in the MicroBooNE Liquid Argon Time Pro- jection Chamber using Wire-Cell 3-D Imaging, Clustering, and Charge-Light Matching,” Nov. 2020. arXiv: 2011.01375 [physics.ins-det].

[64] ——, “The microboone single-photon low-energy excess search,” MicroBooNE Public Note 1087, 2020.

[65] MicroBooNE Collaboration, “The pandora multi-algorithm approach to automated pattern recognition in lar tpc detectors,” MicroBooNE Public Note 1015, 2016.

[66] X. Qian, C. Zhang, B. Viren, et al., “Three-dimensional Imaging for Large LArTPCs,” JINST, vol. 13, no. 05, P05032, 2018. arXiv: 1803.04850 [physics.ins-det].

[67] The MicroBooNE Collaboration, “Cosmic ray background rejection with wire-cell lartpc event reconstruction inmicroboone,” MicroBooNE Public Note 1084, 2020.

[68] P. Abratenko et al., “Semantic Segmentation with a Sparse Convolutional Neural Network for Event Reconstruction in MicroBooNE,” Dec. 2020. arXiv: 2012.08513 [physics.ins- det].

162 [69] ——, “Vertex-Finding and Reconstruction of Contained Two-track Neutrino Events in the MicroBooNE Detector,” JINST, vol. 16, no. 02, P02017, 2021. arXiv: 2002.09375 [physics.ins- det].

[70] MicroBooNE Collaboration, “Neutrino interaction model and uncertainties for microboone analyses,” MicroBooNE Public Note 1074, 2020.

[71] ——, “Microboone low-energy excess signal prediction from unfolding miniboone monte- carlo and data,” MicroBooNE Public Note 1043, 2018.

[72] X. Ji, W. Gu, X. Qian, et al., “Combined Neyman–Pearson chi-square: An improved approximation to the Poisson-likelihood chi-square,” Nucl. Instrum. Meth. A, vol. 961, p. 163 677, 2020. arXiv: 1903.07185 [physics.data-an].

[73] D. W. Schmitz, “A measurement of hadron production cross sections for the simulation of 2 accelerator neutrino beams and a search for a` to a4 oscillations in the X< about equals 1 − 4E2 region,” PhD thesis, Columbia University, New York, NY, 2008.

[74] M. H. Shaevitz, “Constraining the a4 background uncertainties with the observed a` events,” BOONE-Internal-DocDB-221, 2007.

[75] T. Chen and C. Guestrin, “XGBoost: A scalable tree boosting system,” in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ser. KDD ’16, ACM, 2016, pp. 785–794, ISBN: 978-1-4503-4232-2.

[76] MicroBooNE Collaboration, “Supporting note: Mcc9 overlay genie simulated bnb and cosmic data version 1.0,” MICROBOONE-Internal-DocDB–27003, 2019.

[77] P. Guzowski, “Good runs in mcc9 for neutrino 2020,” MICROBOONE-Internal-DocDB- 26979, 2019.

[78] N. I. of Standards and Technology, Pstar stopping power and range tables for protons, 2019.

[79] P. Virtanen, R. Gommers, T. E. Oliphant, et al., “SciPy 1.0: Fundamental Algorithms for Scientiﬁc Computing in Python,” Nature Methods, vol. 17, pp. 261–272, 2020.

[80] P. Abratenko et al., “A Convolutional Neural Network for Multiple Particle Identiﬁcation in the MicroBooNE Liquid Argon Time Projection Chamber,” Oct. 2020. arXiv: 2010. 08653 [hep-ex].

[81] ——, “Final dl lee internal note for near box opening,” MICROBOONE-Internal-DocDB–32720, 2021.

[82] A. Diaz, C. A. Argüelles, G. H. Collin, et al., “Where Are We With Light Sterile Neutrinos?” Phys. Rept., vol. 884, pp. 1–59, 2020. arXiv: 1906.00045 [hep-ex].

163 [83] X. Qian, M. Toups, and M. Shaevitz, “New method for applying the a` constraint on the a4 cov-matrix and prediction,” MICROBOONE-Internal-DocDB–32672, 2020.

[84] A. L. Read, “Presentation of search results: The CL(s) technique,” J. Phys. G, vol. 28, M. R. Whalley and L. Lyons, Eds., pp. 2693–2704, 2002.

[85] P. Abratenko et al., “First Measurement of Differential Charged Current Quasielastic-like a`-Argon Scattering Cross Sections with the MicroBooNE Detector,” Phys. Rev. Lett., vol. 125, no. 20, p. 201 803, 2020. arXiv: 2006.00108 [hep-ex].

[86] IceCube Collaboration. (2016). Search for sterile neutrinos with one year of icecube data, (visited on 04/06/2021).

[87] A. Esmaili and A. Y. Smirnov, “Restricting the LSND and MiniBooNE sterile neutrinos with the IceCube atmospheric neutrino data,” JHEP, vol. 12, p. 014, 2013. arXiv: 1307.6824 [hep-ph].

[88] M. Lindner, W. Rodejohann, and X.-J. Xu, “Sterile neutrinos in the light of IceCube,” JHEP, vol. 01, p. 124, 2016. arXiv: 1510.00666 [hep-ph].

[89] X. Developers. (2021). Introduction to boosted trees, (visited on 01/12/2021).

[90] The pandas development team, Pandas-dev/pandas: Pandas, version latest, Feb. 2020.

[91] Wes McKinney, “Data Structures for Statistical Computing in Python,” in Proceedings of the 9th Python in Science Conference, Stéfan van der Walt and Jarrod Millman, Eds., 2010, pp. 56 –61.

[92] C. R. Harris, K. J. Millman, S. J. van der Walt, et al., “Array programming with NumPy,” Nature, vol. 585, no. 7825, pp. 357–362, Sep. 2020.

[93] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Computing in Science & Engineer- ing, vol. 9, no. 3, pp. 90–95, 2007.

[94] X.-G. Lu, M. Betancourt, T. Walton, et al., “Measurement of ﬁnal-state correlations in neutrino muon-proton mesonless production on hydrocarbon at < a> = 3 GeV,” Phys. Rev. Lett., vol. 121, p. 022 504, 2 2018.

164 Appendix A: Sterile Neutrino Fits to Global Data

This appendix contains supplementary material relevant to the global ﬁt analysis presented in Chapter 4. The global ﬁt analysis presented in this dissertation is a spiritual successor to work done by Conrad et al in [41]. As such, the methods for reproducing the Bugey, CCFR, CDHS, Gallium, KARMEN, LSND, MINOS, MiniBooNE, NOMAD, NuMI, and the KARMEN+LSND XSec results were adapted directly from source code handed down by original authors of that paper. In undergoing this study, the author ported all code from the original FORTRAN into C++, during which, small amendments were made but the methodology and data remained the same. The most notable change within the original, listed data sets is in MiniBooNE, which now both includes data from a more recent run period (up 2018) and a new implementation that takes advantage of

MiniBooNE’s, combined a4/a¯4 appearance analysis [35]. The code used to perform these global ﬁts is released on https://github.com/georgiak/ SBN_3plusN/releases/tag/v1.1, and the code for the IceCube ﬁt is released on https:// github.com/dcianci/IcecubePack/releases/tag/v1.0.

A.1 IceCube 2017 Oscillation Result Reproduction

The IceCube oscillation experiment [59] is the only long baseline (LBL) dataset included in our global ﬁt. This exception to our SBL trend was made for two reasons:

• IceCube places a world-leading constraint on 3+1 sterile neutrino oscillations in the a` disappearance channel. Neglecting to include such a prominent result in an otherwise sparse channel would be reckless and compromise the integrity of our ﬁt.

• We intend to use this global ﬁt to benchmark the sensitivity of the Deep Underground Neu-

165 trino Experiment (DUNE, a forthcoming LBL experiment). IceCube’s result can place a

constraint on \34 and on CP violation—two effects to which SBL experiments lack any sensitivity, but that DUNE may be able to measure.

Oscillations are therefore signiﬁcantly more complex. We can no longer make the short baseline approximation and, additionally, we must account for the MSW (matter) effects incident on a neutrino propagating through a non-uniform Earth. Oscillation probabilities are calculated analytically with the stand-alone, open-source, C++ neutrino oscillation package nuSQuIDS (developed by IceCube collaborators and found at https://github.com/arguelles/nuSQuIDS/releases/tag/0.1). The IceCube collaboration has also produced a generous data release with some instruction for reproducing their oscillation result (paper: [59]; data release: [86]). These resources were helpful, but the author of this dissertation would have been lost without the assistance of Dr. Carlos Delgado who graciously outlined the complex and often arcane process of performing this oscillation analysis. The broad strokes of the analysis are as follows:

1. Oscillate a a` ﬂux through the Earth according to a given set of 3+1 sterile neutrino oscillation parameters;

2. Weight MC events by the oscillation probabilities of the oscillated ﬂux, with respect to their

true neutrino energy, a,CAD4, and true cosine of the neutrino’s zenith angle, cos /CAD4;

3. Compare our prediction to data in two dimensions (across reconstructed muon energy, `,A42>,

2 and reconstructed cosine of the neutrino’s zenith angle, cos /A42>) with a j that requires a minimization of various nuisance parameters.

IceCube’s data is presented in Figure A.1 exactly as it is used in our ﬁtting: with 10 bins in energy, spaced linearly from 400 GeV to 20 TeV; and 21 bins in cosine zenith angle, spaced linearly from 0.24 to -1.0.

In this section, neutrino energy a will always refer to true neutrino energy, and muon energy

` will always refer to reconstructed muon energy.

166 Figure A.1: A 2D histogram of IceCube’s recorded data from [59]. Note that the axes are marked according to bin number, but the horizontal axis, cosine of the zenith angle, is spaced linearly from 0.24 to -1.0, and the vertical axis, reconstructed neutrino energy, is spaced linearly from 400 GeV to 20 TeV. The bin contents are printed in text at the center of each bin for convenience.

A.1.1 Constructing a Predicted Spectrum

The IceCube data release provides initial neutrino ﬂuxes constructed via various neutrino models. We chose to exclusively use the nominal, Honda-Glassier (HG), model. The initial ﬂuxes

comprise a` and a¯` contributions from atmospheric c and decays, totalling four independent ﬂuxes. Since nuSQuIDS is designed to oscillate both neutrino and antineutrino contributions simul-

taneously, two instances of nuSQuIDS are initialized with a` starting fluxes set to either the pion or kaon contributions from the HG model. All other flavor states are initialized with zero flux contribution. ESM and sterile neutrino oscillation parameters are set according to our desired model, \˜, and nuSQuIDS commences its process of oscillation. a` flux probabilities may be extracted from nuSQuIDS at any (a,cos /) within the set bounds. We then loop over each of approximately 8 million MC events to fill our 2D histogram such

167 that the predicted count of neutrino and antineutrino events in reconstructed bin A is given by

# Õ , qa,>B2 \˜ Õ , qa,>B2¯ \˜ A = < < ( ) + < < ( ) (A.1)

where the two summations are over each MC neutrino event and antineutrino event < respectively (−) a ,>B2 and ,< is that event’s MC weight. q< (\˜) is the propagated ﬂux of (anti-)neutrinos, contribut-

ing at the true a and cos / of event <, given speciﬁcally by

qa,>B2 \˜ qa \˜ qa \˜ ' < ( ) = =>A< ( <, ( ) + <,c ( ) c/ ) (A.2) qa,>B2¯ \˜ ' qa¯ \˜ qa¯ \˜ ' . < ( ) = =>A< a¯/a ( <, ( ) + <,c ( ) c/ ) (A.3)

Here, the ﬂux has been further divided into its kaon (q ) and pion (qc) contributions. We have

also introduced three nuisance parameters: =>A<, a total normalization term; 'c/: , the ratio of

pion to kaon a` contribution; and '=D¯ /a, the ratio of the antineutrino to neutrino contribution. These nuisance parameters [ must be ﬁt to data by minimizing the negative log likelihood ln L of the prediction according to the following formula:

0;;18=B ! Õ 1 Õ ([ − [¯)2 ln L = <8= [$ ln # − # − ln $ !] + (A.4) [ 8 8 8 8 2 f2 8=1 [ [ where $8 is the observed event count in bin 8. This minimization must be done for each individual set of oscillation parameters \˜. This analysis chain is very computationally intensive, demanding the propagation of multiple ﬂuxes through the Earth and a complicated minimization for each model. The purpose of this reproduction is speciﬁcally to test the sensitivity of SBN and DUNE, the latter of which will be sensitive to ag contributions to oscillation. This necessitates the consideration of an additional oscillation parameter, \34, to which no other SBL experiments would have sensitivity. To accommodate these processing demands, the IceCube j2 was calculated across variations

2 of the four 3+1 sterile neutrino oscillation parameters, Δ< ,\14,\24, and \34, with the parameter

168 Figure A.2: The 90% global exclusion limit as published in [59], overlaid with that of my reproduction.

ranges split into 40, 15, 40, and 10 bins respectively. We chose to reduce the density of the scan