The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# Search for an anomalous excess of inclusive charged-current $\nu_{e}$
interactions
in the MicroBooNE experiment using Wire-Cell reconstruction

P. Abratenko  Tufts University, Medford, MA, 02155, USA  R. An  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  J. Anthony  University
of Cambridge, Cambridge CB3 0HE, United Kingdom  L. Arellano  The University
of Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi  University of
Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv University, Tel Aviv,
Israel, 69978  S. Balasubramanian  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  B. Baller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  C. Barnes  University of Michigan,
Ann Arbor, MI, 48109, USA  G. Barr  University of Oxford, Oxford OX1 3RH,
United Kingdom  V. Basque  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Bathe-Peters  Harvard University, Cambridge, MA 02138, USA
O. Benevides Rodrigues  Syracuse University, Syracuse, NY, 13244, USA  S.
Berkman  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
A. Bhanderi  The University of Manchester, Manchester M13 9PL, United Kingdom
A. Bhat  Syracuse University, Syracuse, NY, 13244, USA  M. Bishai  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster
University, Lancaster LA1 4YW, United Kingdom  T. Bolton  Kansas State
University (KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University,
Cambridge, MA 02138, USA  L. Camilleri  Columbia University, New York, NY,
10027, USA  D. Caratelli  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  I. Caro Terrazas  Colorado State University, Fort
Collins, CO, 80523, USA  F. Cavanna  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  G. Cerati  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  Y. Chen  Universität Bern, Bern
CH-3012, Switzerland  D. Cianci  Columbia University, New York, NY, 10027, USA
J. M. Conrad  Massachusetts Institute of Technology (MIT), Cambridge, MA,
02139, USA  M. Convery  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  L. Cooper-Troendle  Wright Laboratory, Department of Physics, Yale
University, New Haven, CT, 06520, USA  J. I. Crespo-Anadón  Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid
E-28040, Spain  M. Del Tutto  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  S. R. Dennis  University of Cambridge, Cambridge CB3
0HE, United Kingdom  P. Detje  University of Cambridge, Cambridge CB3 0HE,
United Kingdom  A. Devitt  Lancaster University, Lancaster LA1 4YW, United
Kingdom  R. Diurba  University of Minnesota, Minneapolis, MN, 55455, USA  R.
Dorrill  Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  K.
Duffy  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
S. Dytman  University of Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly
University of Southern Maine, Portland, ME, 04104, USA  A. Ereditato
Universität Bern, Bern CH-3012, Switzerland  J. J. Evans  The University of
Manchester, Manchester M13 9PL, United Kingdom  R. Fine  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  G. A. Fiorentini Aguirre  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  R.
S. Fitzpatrick  University of Michigan, Ann Arbor, MI, 48109, USA  B. T.
Fleming  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  N. Foppiani  Harvard University, Cambridge, MA 02138, USA  D.
Franco  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  A. P. Furmanski  University of Minnesota, Minneapolis, MN,
55455, USA  D. Garcia-Gamez  Universidad de Granada, Granada E-18071, Spain
S. Gardiner  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  G. Ge  Columbia University, New York, NY, 10027, USA  S. Gollapinni
University of Tennessee, Knoxville, TN, 37996, USA  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  O. Goodwin  The University of
Manchester, Manchester M13 9PL, United Kingdom  E. Gramellini  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  P. Green  The
University of Manchester, Manchester M13 9PL, United Kingdom  H. Greenlee
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W. Gu
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  R. Guenette
Harvard University, Cambridge, MA 02138, USA  P. Guzowski  The University of
Manchester, Manchester M13 9PL, United Kingdom  L. Hagaman  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  O. Hen
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  C.
Hilgenberg  University of Minnesota, Minneapolis, MN, 55455, USA  G. A.
Horton-Smith  Kansas State University (KSU), Manhattan, KS, 66506, USA  A.
Hourlier  Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139,
USA  R. Itay  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA
C. James  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
X. Ji  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  L. Jiang
Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA  J. H.
Jo  Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  R. A. Johnson  University of Cincinnati, Cincinnati, OH, 45221,
USA  Y.-J. Jwa  Columbia University, New York, NY, 10027, USA  D. Kalra
Columbia University, New York, NY, 10027, USA  N. Kamp  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  N. Kaneshige
University of California, Santa Barbara, CA, 93106, USA  G. Karagiorgi
Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Kobilarcik
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  I.
Kreslo  Universität Bern, Bern CH-3012, Switzerland  I. Lepetic  Rutgers
University, Piscataway, NJ, 08854, USA  K. Li  Wright Laboratory, Department
of Physics, Yale University, New Haven, CT, 06520, USA  Y. Li  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  K. Lin  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  B. R. Littlejohn  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  W. C. Louis  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  X. Luo  University of
California, Santa Barbara, CA, 93106, USA  K. Manivannan  Syracuse University,
Syracuse, NY, 13244, USA  C. Mariani  Center for Neutrino Physics, Virginia
Tech, Blacksburg, VA, 24061, USA  D. Marsden  The University of Manchester,
Manchester M13 9PL, United Kingdom  J. Marshall  University of Warwick,
Coventry CV4 7AL, United Kingdom  D. A. Martinez Caicedo  South Dakota School
of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  K. Mason  Tufts
University, Medford, MA, 02155, USA  A. Mastbaum  Rutgers University,
Piscataway, NJ, 08854, USA  N. McConkey  The University of Manchester,
Manchester M13 9PL, United Kingdom  V. Meddage  Kansas State University (KSU),
Manhattan, KS, 66506, USA  T. Mettler  Universität Bern, Bern CH-3012,
Switzerland  K. Miller  University of Chicago, Chicago, IL, 60637, USA  J.
Mills  Tufts University, Medford, MA, 02155, USA  K. Mistry  The University of
Manchester, Manchester M13 9PL, United Kingdom  A. Mogan  University of
Tennessee, Knoxville, TN, 37996, USA  T. Mohayai  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. Moon  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  M. Mooney  Colorado State
University, Fort Collins, CO, 80523, USA  A. F. Moor  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Mousseau  University of
Michigan, Ann Arbor, MI, 48109, USA  M. Murphy  Center for Neutrino Physics,
Virginia Tech, Blacksburg, VA, 24061, USA  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  R. K. Neely  Kansas State
University (KSU), Manhattan, KS, 66506, USA  D. A. Newmark  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  J. Nowak  Lancaster
University, Lancaster LA1 4YW, United Kingdom  M. Nunes  Syracuse University,
Syracuse, NY, 13244, USA  O. Palamara  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  V. Paolone  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  A. Papadopoulou  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  V. Papavassiliou  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  S. F. Pate  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  N. Patel  Lancaster
University, Lancaster LA1 4YW, United Kingdom  A. Paudel  Kansas State
University (KSU), Manhattan, KS, 66506, USA  Z. Pavlovic  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv
University, Tel Aviv, Israel, 69978  I. D. Ponce-Pinto  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  S. Prince
Harvard University, Cambridge, MA 02138, USA  X. Qian  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V. Radeka  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Rafique  Kansas State
University (KSU), Manhattan, KS, 66506, USA  M. Reggiani-Guzzo  The University
of Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New Mexico State
University (NMSU), Las Cruces, NM, 88003, USA  L. C. J. Rice  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  L. Rochester  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Rodriguez Rondon  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  M.
Rosenberg  University of Pittsburgh, Pittsburgh, PA, 15260, USA  M. Ross-
Lonergan  Columbia University, New York, NY, 10027, USA  B. Russell  Wright
Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA
G. Scanavini  Wright Laboratory, Department of Physics, Yale University, New
Haven, CT, 06520, USA  D. W. Schmitz  University of Chicago, Chicago, IL,
60637, USA  A. Schukraft  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  W. Seligman  Columbia University, New York, NY, 10027,
USA  M. H. Shaevitz  Columbia University, New York, NY, 10027, USA  R.
Sharankova  Tufts University, Medford, MA, 02155, USA  J. Shi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  J. Sinclair  Universität Bern,
Bern CH-3012, Switzerland  A. Smith  University of Cambridge, Cambridge CB3
0HE, United Kingdom  E. L. Snider  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  M. Soderberg  Syracuse University, Syracuse,
NY, 13244, USA  S. Söldner-Rembold  The University of Manchester, Manchester
M13 9PL, United Kingdom  P. Spentzouris  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  J. Spitz  University of Michigan, Ann Arbor,
MI, 48109, USA  M. Stancari  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  J. St. John  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  T. Strauss  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  K. Sutton  Columbia University, New
York, NY, 10027, USA  S. Sword-Fehlberg  New Mexico State University (NMSU),
Las Cruces, NM, 88003, USA  A. M. Szelc  University of Edinburgh, Edinburgh
EH9 3FD, United Kingdom  W. Tang  University of Tennessee, Knoxville, TN,
37996, USA  K. Terao  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  C. Thorpe  Lancaster University, Lancaster LA1 4YW, United Kingdom
D. Totani  University of California, Santa Barbara, CA, 93106, USA  M. Toups
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y.-T.
Tsai  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  M. A.
Uchida  University of Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  W. Van De
Pontseele  University of Oxford, Oxford OX1 3RH, United Kingdom  Harvard
University, Cambridge, MA 02138, USA  B. Viren  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern, Bern CH-3012,
Switzerland  H. Wei  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  Z. Williams  University of Texas, Arlington, TX, 76019, USA  S. Wolbers
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T.
Wongjirad  Tufts University, Medford, MA, 02155, USA  M. Wospakrik  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  K. Wresilo
University of Cambridge, Cambridge CB3 0HE, United Kingdom  N. Wright
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  W. Wu
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E.
Yandel  University of California, Santa Barbara, CA, 93106, USA  T. Yang
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G.
Yarbrough  University of Tennessee, Knoxville, TN, 37996, USA  L. E. Yates
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  H. W.
Yu  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  G. P. Zeller
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J.
Zennamo  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
C. Zhang  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We report a search for an anomalous excess of inclusive charged-current (CC)
$\nu_{e}$ interactions using the Wire-Cell event reconstruction package in the
MicroBooNE experiment, which is motivated by the previous observation of a
low-energy excess (LEE) of electromagnetic events from the MiniBooNE
experiment. With a single liquid argon time projection chamber detector, the
measurements of $\nu_{\mu}$ CC interactions as well as $\pi^{0}$ interactions
are used to constrain signal and background predictions of $\nu_{e}$ CC
interactions. A data set collected from February 2016 to July 2018
corresponding to an exposure of 6.369 $\times$ 1020 protons on target from the
Booster Neutrino Beam at FNAL is analyzed. With $x$ representing an overall
normalization factor and referred to as the LEE strength parameter, we select
56 fully contained $\nu_{e}$ CC candidates while expecting 69.6 $\pm$ 8.0
(stat.) $\pm$ 5.0 (sys.) and 103.8 $\pm$ 9.0 (stat.) $\pm$ 7.4 (sys.)
candidates after constraints for the absence (eLEEx=0) of the median signal
strength derived from the MiniBooNE observation and the presence (eLEEx=1) of
that signal strength, respectively. Under a nested hypothesis test using both
rate and shape information in all available channels, the best-fit $x$ is
determined to be 0 (eLEEx=0) with a 95.5% confidence level upper limit of $x$
at 0.502. Under a simple-vs-simple hypotheses test, the eLEEx=1 hypothesis is
rejected at 3.75$\sigma$, while the eLEEx=0 hypothesis is shown to be
consistent with the observation at 0.45$\sigma$. In the context of the eLEE
model, the estimated 68.3% confidence interval of the $\nu_{e}$ CC hypothesis
to explain the LEE observed in the MiniBooNE experiment is disfavored at a
significance level of more than 2.6$\sigma$ (3.0$\sigma$) considering
MiniBooNE’s full (statistical) uncertainties.

##  I Introduction

Neutrino flavor oscillation is one of the few observations in particle physics
for evidence of physics beyond the standard model. While the majority of
neutrino oscillation data can be successfully explained by a three-neutrino
framework [1], the exact mechanism for neutrinos to acquire their masses
remains a puzzle. In addition, the fact that the mass of the electron neutrino
is at least five orders of magnitude smaller than that of the electron [2] is
an interesting mystery. The possible existence of additional neutrino flavors
may provide a natural explanation of the small neutrino mass [3]. Constrained
by precision electroweak measurements [4], these additional neutrinos are
expected to be sterile [5] not participating in any fundamental interaction of
the standard model. Despite the fact that there is no known mechanism to
detect them directly, they may be indirectly observed in neutrino oscillation
experiments where sterile neutrinos could mix with the three active neutrinos
and affect the way they oscillate.

Because of strong theoretical motivations, there are many dedicated programs
searching for sterile neutrinos. While most of the results are consistent with
the three-neutrino framework without sterile neutrinos (see Ref. [6, 7] among
others), there are several experimental anomalies suggesting the existence of
an eV-mass-scale sterile neutrino: i) the observation that calibrated
$\nu_{e}$ sources (51Cr for GALLEX [8] and BEST [9], 51Cr and 37Ar for SAGE
[10]) produced lower rates of measured $\nu_{e}$ than expected in the three-
neutrino framework, which could be explained by $\nu_{e}$ disappearance
considering light sterile neutrinos; ii) the reactor anti-neutrino anomaly
[11], which suggests that the observed deficit in the measured $\bar{\nu}_{e}$
events relative to the expectation based on the recent reactor anti-neutrino
flux calculations [12, 13] could be explained by $\bar{\nu}_{e}$ disappearance
considering light sterile neutrinos, although there are recent experimental
measurements [14, 15] and improved flux calculations [16, 17] that disfavor
this explanation; iii) the Neutrino-4 [18] anomaly, which suggests reactor
$\bar{\nu}_{e}$ oscillation at a few meters; and iv) the LSND [19] and
MiniBooNE [20, 21] anomalies, which suggest $\nu_{e}$ appearance from
$\nu_{\mu}$ to $\nu_{e}$ oscillations considering light sterile neutrinos.
However, there are significant challenges in explaining all available
experimental results with a sterile neutrino oscillation model in a global fit
[22].

The MiniBooNE experiment observes an anomalous excess of electromagnetic
(electron- or photon-like) events in the data above the prediction for
reconstructed neutrino energies below 800 MeV, which is commonly referred to
as the MiniBooNE low-energy excess (LEE). The interpretation of the MiniBooNE
LEE observation is limited by the capability of its detector technology, which
primarily uses the pattern of Cherenkov rings to differentiate between muons
and electrons. However, the Cherenkov technology does not perform well in
separating an electron from a photon leading to the possibility that the LEE
may come from an excess of photon background rather than from the $\nu_{e}$
charged-current (CC) interactions. The Liquid Argon Time Projection Chamber
(LArTPC) [23, 24, 25, 26] has superb imaging capabilities through the
combination of both tracking and calorimetry in a fully active volume.
Throughout the years since its inception, many LArTPCs ranging in size from
hundreds of liters to hundreds of cubic meters have been constructed and
operated for neutrino experiments [27, 28, 29, 30, 31, 32, 33]. Although
electrons and photons with energy greater than tens of MeV both induce
electromagnetic (EM) showers in LArTPCs, electrons can be differentiated from
photons in neutrino interactions through identifying a gap between neutrino
and EM shower vertices, topological pattern recognition, and reconstructed
ionization energy loss per unit length ($dE/dx$) measurement [34]. This
improves the identification of $\nu_{e}$ CC interactions and will enable
sensitive measurements such as searches for
$\overset{\scalebox{0.3}{(}\raisebox{-1.7pt}[0.0pt][0.0pt]{$-$}\scalebox{0.3}{)}}{\nu}_{\mu}\rightarrow\overset{\scalebox{0.3}{(}\raisebox{-1.7pt}[0.0pt][0.0pt]{$-$}\scalebox{0.3}{)}}{\nu}_{e}$
oscillations. LArTPCs can also reconstruct and identify hadrons at lower
energies than Cherenkov detectors, which can only detect charged particles
with energies above their respective Cherenkov thresholds. The extended
capability of LArTPCs to detect these lower energy hadrons enables the study
of various exclusive final states in the detector.

The MicroBooNE experiment was designed primarily to explore the nature of the
MiniBooNE LEE observation using an 85-ton active volume LArTPC [32] through
its excellent electron versus photon separation. The MicroBooNE detector is
located at the Booster Neutrino Beam (BNB) [35] at the Fermi National
Accelerator Laboratory in Batavia, IL, USA. It is in the same beam line with a
similar distance to the neutrino source as the MiniBooNE detector. While
MicroBooNE is capable of searching for exotic signatures, such as electron-
positron pairs [36], the MicroBooNE experiment’s main goal is to search for an
LEE signal in $\nu_{e}$ CC interactions with an electron in the final state
(referred to as the eLEE search) as well as in neutral-current (NC)
interactions with a single photon in the final state. In these LEE searches,
the MicroBooNE experiment has developed four analyses: i) a photon-based-
search focusing on the $\Delta$-decay hypothesis [37]; ii) an exclusive search
using the $\nu_{e}$ charged-current quasi-elastic (CCQE) channel [38]; iii) a
semi-inclusive search for events with one electron and no pions in the final
state [39]; and iv) a fully inclusive search for events with one electron and
any final state, which is what this article describes. The results of the
three eLEE searches are individually quantified with an empirical model
constructed based on MiniBooNE data (see Sec. II) instead of performing a
sterile neutrino oscillation model fit and summarized together in Ref. [40].
Looking ahead, a dedicated search of sterile neutrino(s) oscillations through
comparison of multiple detectors at different baselines is the goal of the
upcoming Short-Baseline Neutrino (SBN) program [41].

This paper is organized as follows. An overview of the search strategy is
presented in Sec. II. The MicroBooNE experiment and the Wire-Cell event
reconstruction package is introduced in Sec. III. The event selections
including $\nu_{\mu}$ CC and $\nu_{e}$ CC selections111In this paper,
$\nu_{x}$ refers to both $\nu_{x}$ and its counterpart $\bar{\nu}_{x}$. are
described in Sec. IV. Section V describes the systematic uncertainties
considered in this work. To search for new physics, validations of the overall
model including the predictions of both signal and background are crucial.
Various examinations of the model including i) signal prediction, ii)
background prediction, and iii) modeling of neutrino energy reconstruction are
summarized in Sec. VI. The physics sensitivities and the final result of the
eLEE search are presented in Sec. VII followed by the summary in Sec. VIII.

##  II Overview of the analysis

In this paper, we report the results of the eLEE search using the Wire-Cell
event reconstruction package [42, 43] after analyzing the MicroBooNE data set
collected from February 2016 to July 2018. The search for an eLEE signal is
equivalent to testing the null hypothesis, which corresponds to the nominal
prediction without an eLEE signal. This prediction is formed based on the
state-of-the-art understanding of the BNB flux [35, 44], the neutrino-argon
interaction cross section [45, 46], the detector simulation [47, 48, 49], and
data-driven constraints on $\nu_{e}$ CC signal and background predictions. An
alternative hypothesis is formed based on an eLEE model which represents an
anomalous enhancement in the rate of intrinsic $\nu_{e}$ CC events at true
neutrino energies between 200 MeV and 800 MeV with a fixed spectral shape
extracted from the MiniBooNE experiment. This alternative hypothesis has
several advantages over a dedicated sterile neutrino oscillation model [50,
51]. Firstly, this model is constructed from the MiniBooNE data with a minimum
set of assumptions and is agnostic of sterile neutrino oscillation model
parameters. Secondly, this model contains many fewer free parameters, which is
beneficial for the single detector configuration of MicroBooNE. Thirdly, this
model avoids complications introduced by simultaneous modifications in
multiple channels including $\nu_{\mu}$ disappearance, $\nu_{\mu}$ to
$\nu_{e}$ appearance, and $\nu_{e}$ disappearance in data. Finally, it allows
for a quantitative comparison of results between MicroBooNE and MiniBooNE,
particularly given the difficulty in explaining the shape of the MiniBooNE LEE
in global fits with a sterile neutrino model [22].

This eLEE model used as the alternative hypothesis is obtained by unfolding
the observed excess of electron-like events in MiniBooNE to true neutrino
energy under a CCQE hypothesis and applying the excess-to-intrinsic $\nu_{e}$
ratios directly to the flux of intrinsic $\nu_{e}$ expected in the MicroBooNE
detector. The MiniBooNE data and simulation results reported in 2018 [52] were
used as an input to the D’Agostini’s multi-dimensional unfolding procedure
[53]. We note that this eLEE model is constructed using the true neutrino
energy and is limited below 800 MeV, beyond which the unfolded LEE signal is
negligible. Furthermore, the additional information regarding the lepton
kinematics reported by MiniBooNE in 2020 [21] is not taken into account. In
addition, a 68.3% confidence interval of the eLEE strength parameter of $1\pm
0.21$ ($1\pm 0.08$) considering both statistical and systematic (only
statistical) uncertainties is estimated from the reported 4.8$\sigma$
(12.2$\sigma$) LEE significance from the latest MiniBooNE 2020 result [21]
considering both neutrino and anti-neutrino data taking. If only neutrino data
taking is considered, the LEE significance would slightly reduce to
4.7$\sigma$ (11.7$\sigma$) considering full (statistical) uncertainty,
respectively.

In the hypothesis test, we allow the normalization of this eLEE model to float
and define a non-negative strength parameter $x$, such that $x=0$ corresponds
to the nominal prediction without an eLEE signal (eLEEx=0 hypothesis) and
$x=1$ corresponds to the prediction with an eLEE signal with magnitude equal
to that of the unfolded median of the MiniBooNE LEE result (eLEEx=1
hypothesis). Figure 1 illustrates the prediction of the eLEEx=1 hypothesis
with respect to that of the eLEEx=0 hypothesis for $\nu_{e}$ CC interactions
in the TPC active volume in MicroBooNE assuming 100% detection efficiency.

MicroBooNE Simulation

Figure 1: Expected $\nu_{e}$ CC events in the MicroBooNE detector for eLEEx=1
hypothesis and eLEEx=0 hypothesis as a function of true neutrino energy at
6.4$\times$1020 proton-on-target (POT) exposure, assuming 100% detection
efficiency.

We fit this eLEE model to our data by minimizing a combined Neyman-Pearson
(CNP) $\chi^{2}$ test statistic [54] that incorporates the MicroBooNE’s
experimental uncertainties into a covariance matrix thereby obtaining a best
fit value of $x=x_{\text{min}}$. We compute the primary nested likelihood
ratio test statistic, $\Delta\chi^{2}_{\rm nested}=\chi^{2}_{{\rm
eLEEx}=x_{0}}-\chi^{2}_{{\rm eLEEx}=x_{\rm min}}$, for our data given a
varying eLEE strength $x=x_{0}$ and obtain frequentist confidence intervals
for the eLEE strength $x$ following the Feldman-Cousins procedure [55]. In
addition to the $\Delta\chi^{2}_{\text{nested}}$, several other statistical
tests are performed to provide additional information. They are i) goodness-
of-fit (GoF) tests based on a Pearson $\chi^{2}$ to examine the compatibility
between data measurement and Monte-Carlo (MC) prediction, especially with
available constraints using the conditional covariance method [56], and ii) a
simple-vs-simple likelihood ratio test,
$\Delta\chi^{2}_{\text{simple}}=\chi^{2}_{\text{eLEEx=1}}-\chi^{2}_{\text{eLEEx=0}}$
with the CNP $\chi^{2}$ to demonstrate the compatibility between data and the
two signature hypotheses (eLEEx=1 and eLEEx=0).

Without a near detector to measure the neutrino flux and neutrino-argon
interaction cross section, the search for an eLEE in the $\nu_{e}$ CC channel
would suffer from large systematic uncertainties. In order to compensate for
the lack of a near detector and to maximize the physics sensitivity of this
search, $\nu_{\mu}$ CC channel is used to constrain the $\nu_{e}$
CCprediction. As to be elaborated in Sec. III.1, the prediction of $\nu_{e}$
and $\nu_{\mu}$ flux are strongly correlated, given the parent hadron species
are mostly common. In addition, the $\nu_{e}$ CC and $\nu_{\mu}$ CC
interaction cross section is also strongly correlated, given the lepton
universality. Therefore, the measurement of $\nu_{\mu}$ CC rates at different
energies, which is proportional to the product of $\nu_{\mu}$ flux and
$\nu_{\mu}$-Ar cross section, is efficient in constraining the prediction of
$\nu_{e}$ CC rates at different energies, which is proportional to the
production of $\nu_{e}$ flux and $\nu_{e}$-Ar cross section. In practice, a
seven-channel fit strategy is adopted. The seven channels are:

  1. 1.

fully contained (FC) $\nu_{e}$ CC,

  2. 2.

partially contained (PC) $\nu_{e}$ CC,

  3. 3.

FC $\nu_{\mu}$ CC,

  4. 4.

PC $\nu_{\mu}$ CC,

  5. 5.

FC $\nu_{\mu}$ CC with $\pi^{0}$ in the final state (FC CC$\pi^{0}$),

  6. 6.

PC CC$\pi^{0}$, and

  7. 7.

NC interactions with $\pi^{0}$ in the final state (NC$\pi^{0}$).

Here, the FC events are defined to be events with the reconstructed TPC
activity fully contained within the fiducial volume (3 cm inside the effective
TPC boundary [57], which is the corrected boundary that takes a space charge
effect [58, 48] into account). All non-FC events are defined as PC events. FC
events have many advantages such as higher signal-to-background ratio and
better energy resolution when compared to the PC events. The primary channel
that is sensitive to the eLEE search is FC $\nu_{e}$ CC. FC $\nu_{\mu}$ CC and
PC $\nu_{\mu}$ CC channels are used to provide constraints on the prediction
of $\nu_{e}$ CC interaction rate and its systematic uncertainties. PC
$\nu_{e}$ CC channel, which is less sensitive to the eLEE search, also
provides constraints to some extent. The other three channels, FC CC$\pi^{0}$,
PC CC$\pi^{0}$, and NC$\pi^{0}$, are used to constrain/improve the background
prediction since $\pi^{0}$ events are one of the major backgrounds in the
$\nu_{e}$ CC selection. These seven channels are designed to be orthogonal to
each other. The $\nu_{\mu}$ CC channel excludes $\nu_{e}$ CC and CC$\pi^{0}$
candidates, and the CC$\pi^{0}$ channel excludes the $\nu_{e}$ CC candidates.

To reduce bias, we adopted a blind analysis for this eLEE search. Two
sidebands (near and far) are defined in addition to the signal region. The
signal region is defined to contain the events with reconstructed neutrino
energy $E^{rec}_{\nu}$ lower than 600 MeV and passing the $\nu_{e}$ CC
selection. The far sideband is defined to contain the events that fail a
looser $\nu_{e}$ CC selection or the events with $E^{rec}_{\nu}$ higher than
800 MeV. The far sideband essentially includes $\nu_{\mu}$ CC events as well
as high-energy $\nu_{e}$ CC events, which are not sensitive to the eLEE
signal. The near sideband covers the remainder of the events between the far
sideband and the signal region. A completely open data set with an exposure of
5.33$\times 10^{19}$ proton-on-target (POT) is available for development of
the event reconstruction, selection, and statistical analysis. This open data
set is less than 10% of the eventual data set of an exposure of $6.369\times
10^{20}$ POT which is used in this analysis. The selection and analysis
procedure were frozen after analyzing the open data set, and no changes were
made after checking the consistency between the data measurement and MC
prediction using the far sideband data set. The overall model including the
central value predictions and their associated uncertainties is quantitatively
validated with extensive goodness-of-fit tests. The power of these validations
are further enhanced through a conditional covariance matrix formalism, which
allows for a comparison between data and a constrained model prediction.
Constrained by the sideband data, the systematic uncertainties of the model
prediction can be significantly reduced, allowing for a in-depth examination
of the systematic uncertainties and estimated correlations between data
samples. The near sideband was openned subsequently for further examination
and, only then, was the signal region unblinded. During the finalization of
the analysis procedure, several fake data sets were generated based on
different cross section models and injected with or without eLEE signals to
test the robustness of the analysis procedure. All details of the fake data
sets were initially blind to the analysers, and the analysis results of these
data sets were as expected, either extracting or excluding an eLEE signal as
appropriate given the truth-level information released afterwards.

##  III The MicroBooNE Experiment

###  III.1 BNB Neutrino Flux

The booster neutrino beam line uses 8 GeV kinetic energy protons from the
Booster accelerator to bombard a beryllium target. The primary hadrons
produced in these interactions and the secondary hadrons further produced
through additional interactions of primary hadrons and the surrounding
materials are focused by a magnetic horn into a 50-m-long decay pipe. Decays
of these hadrons produce a neutrino beam. The neutrino horn current polarity
has been set to focus positive charged hadrons, resulting in a beam with a
small antineutrino component for all data taking to date.

\begin{overpic}[width=201.85172pt]{canv_h1_flux_total.png}
\put(35.0,64.0){\text{(a)}\ \ \ \ \ \ \ \ MicroBooNE Simulation} \end{overpic}

\begin{overpic}[width=201.85172pt]{canv_h1_flux_numu.png}
\put(35.0,64.0){\text{(b)}\ \ \ \ \ \ \ \ MicroBooNE Simulation} \end{overpic}

\begin{overpic}[width=201.85172pt]{canv_h1_flux_nue_Yaxis.png}
\put(35.0,64.0){\text{(c)}\ \ \ \ \ \ \ \ MicroBooNE Simulation} \end{overpic}

Figure 2: Predicted neutrino flux from different decay modes at the MicroBooNE
detector: (a) flux of different neutrino flavors, (b) $\nu_{\mu}$ flux of
different decay modes (the peak at 236 MeV corresponds to Kaon decay at rest),
(c) $\nu_{e}$ flux of different decay modes.

The BNB flux is simulated based on a Geant4 framework [59] following the
earlier work by the MiniBooNE collaboration [35]. The top panel of Fig. 2
shows the composition of the BNB flux in terms of different neutrino flavors
seen by the MicroBooNE detector, which is located at 468.5 m on axis from the
target. The dominant neutrino species is the $\nu_{\mu}$, which is mostly
produced by a $\pi^{+}$ two-body decay mode as shown in the middle panel in
the same figure. For energies higher than $\sim$2.3 GeV, another two-body
decay mode, $K^{+}\rightarrow\mu^{+}+\nu_{\mu}$, becomes the main mechanism to
generate $\nu_{\mu}$. The $\nu_{e}$ flux is about 0.5% of the overall neutrino
flux. For neutrino energies below about 1.2 GeV, $\nu_{e}$’s are mostly
produced by a three-body decay mode, $\mu^{+}\rightarrow
e^{+}+\bar{\nu}_{\mu}+\nu_{e}$, with $\mu^{+}$ originating from the $\pi^{+}$
two-body decay mode as shown in the bottom panel of Fig. 2. For neutrino
energies higher than 1.2 GeV, $\nu_{e}$s are mostly produced by another three-
body decay mode, $K^{+}\rightarrow\pi^{0}+e^{+}+\nu_{e}$. Since the parent
hadron species are mostly common, the predictions of $\nu_{\mu}$ and $\nu_{e}$
flux are strongly correlated, which supports the overall strategy of using
$\nu_{\mu}$ CC events to provide constraints on the prediction of $\nu_{e}$ CC
events.

(a)(b)

Figure 3:  Taken from Ref. [57]. (a) The MicroBooNE detector cryostat with the
field cage shown inside. (b) Inside the cryostat of the MicroBooNE detector,
visualized with the VENu software [60]. The maximum drift distance is 2.56 m
with a drift electric field of 273 V/cm. The light-collection system, which
consists of 32 PMTs, is located behind the three anode wire planes, which
detect ionization charge.

###  III.2 MicroBooNE Detector

The MicroBooNE detector [32] is a 10.4 m long, 2.6 m wide, and 2.3 m high
LArTPC, located on-axis along the BNB at a distance of 468.5 m from the
beryllium target, 72.5 m upstream of the MiniBooNE detector. It consists of
approximately 85 metric tons of liquid argon in the active volume for
ionization charge detection along with an array of 32 photomultiplier tubes
(PMTs) [61] for the scintillation light detection222One PMT stopped working
after the run 1 data taking.. Figure 3(a) shows the MicroBooNE TPC, which is
housed in a foam-insulated evacuable cryostat vessel. The cathode-plane high
voltage is set at -70 kV during normal operation, creating a drift field of
273 V/cm. The ionization electrons drift at a speed of 1.1 mm/$\mu$s [62] in
the drift field. This corresponds to 2.3 ms drift time for the maximum 2.56 m
drift distance.

As shown in Figure 3(b), there are three parallel wire readout planes at the
anode side of the TPC. These planes are labeled as the “U”, “V”, and “W”
planes333The “W” plane is sometimes also referred to as the “Y” plane., and
each plane contains 2368, 2368, and 3456 wires, respectively. The wire spacing
within a plane is 3 mm, and the planes are spaced 3 mm apart. The wires in the
W plane are aligned vertically and the wires in the U and V planes are
oriented at $\pm$60∘ with respect to the vertical direction. The different
orientations of the wires allow for determination of the positions of the
ionization electrons within the plane that is transverse to the drift
direction. Bias voltages for the U, V, and W planes are -110 V, 0 V, and 230
V, respectively, which satisfies the transparency condition that all drifting
electrons pass through the U and V (induction) wire planes and are fully
collected on the W (collection) plane. The induced current on each wire is
amplified, shaped, and digitized through a custom designed front-end
application-specific integrated circuit [63] operating at 89 K in the liquid
argon. The direct implementation of readout electronics in the cold liquid
significantly reduces electronics noise. The equivalent noise charge on each
wire is generally below 400 electrons. A minimum ionizing particle usually
produces in total 13,000 electrons at a single wire if the particle trajectory
is perpendicular to the wire orientation [64].

Figure 3(b) also shows the light-collection system behind the anode wire
planes. This light-collection system is used to detect scintillation light
from the liquid argon providing the precise timing of particle activity, which
is crucial to rejection of the high-rate cosmic-ray background necessary in a
surface-operating LArTPC detector like MicroBooNE. Thirty-two 8-inch Hamamatsu
R5912-02MOD PMTs [61] provide uniform coverage of the anode plane. An acrylic
plate coated with tetraphenyl butadiene is installed in front of each PMT to
shift the ultraviolet argon scintillation light to the visible part of the
spectrum to which the PMT is sensitive. The magnitude of the detected light on
each PMT provides position information for time-isolated particle activities,
which is compared with the light pattern predicted from the ionization charge
signals in the TPC. A successful match [43] of charge and light signals
determines the association between individual TPC activity and light detection
and, therefore, the time of the corresponding TPC activity.

###  III.3 Event Trigger and Readout

The BNB delivers proton pulses at a rate of 5 Hz and approximately
4$\times$1012 POT per pulse. Each pulse is called a beam spill and lasts
$\sim$1.6 $\mu$s with 82 2-ns wide bunches of protons. The MicroBooNE detector
is expected to have one neutrino interaction inside the TPC active volume per
about 600 spills. Each beam spill initiates a hardware trigger in the
MicroBooNE data acquisition (DAQ), which records 4.8 ms (digitized at 2 MHz)
of TPC data of ionization charge signals and 6.4 ms (digitized at 64 MHz) PMT
data of scintillation light signals. The PMT data contain two separate trigger
streams, a forced trigger covering the beam spill which is referred to as the
beam discriminator and a self-trigger which is referred to as the cosmic
discriminator. In each event, there are on average 26 cosmic-ray muon induced
activities observed in the 4.8 ms TPC readout window.

One such record of TPC and PMT data corresponds to one event which covers not
only the beam spill but also the cosmic-ray muon activity in the proximity of
the beam spill. The time window of an event accounts for the few-millisecond
delay of the TPC readout signal coming from the relatively slow drift of
ionization electrons. Following the hardware trigger, a software trigger is
applied to reduce the data size. It requires distinct PMT signals within the
beam spill window. This results in a reduction of the event rate by a factor
of 22 but a negligible efficiency loss for neutrino interactions.

MicroBooNE data run 6195 event 1020(a)(b)

Figure 4:  A $\nu_{e}$ CC interaction candidate from MicroBooNE data. The X
axis is the drift electric field direction from the TPC anode to the cathode.
The Y axis is vertical up, and the Z axis is along the neutrino beam
direction. Panel (a) shows three 2D projections of reconstructed 3D clusters
in the full TPC readout window before charge-light matching. Each cluster is
shown in a different color. The gray box represents the TPC active volume
while the two ends along the X axis correspond to the trigger time and the
maximum drift time relative to the trigger. Panel (b) shows the 2D projections
of the $\nu_{e}$ CC candidate cluster after applying the charge-light
matching. The red (green) circles represent the observed (predicted) number of
photoelectrons (PEs) at each PMT, where the area of the circle is proportional
to the number of PEs. The effective detector boundary as a result of the space
charge effect is indicated by the red dashed lines in the corner of the TPC
active volume as shown in the “Y-X view” and “X-Z view”.

###  III.4 Monte Carlo Simulation

The simulated neutrino flux (introduced in Sec. III.1) is provided to the
event generator Genie v3 [45, 46] to generate neutrino-argon interactions
inside444Here “inside” also includes the 11.1 mm thick stainless steel wall of
the cryostat but excludes the foam insulation covering the cryostat outer
surfaces. and outside the cryostat. The latter case is also referred to as the
dirt background. Genie v3.0.6, G18_10a_02_11a, was used, which includes
improvements on the usage of the Valencia model [65, 66, 67] for the local
Fermi gas nucleon momentum distributions, improvements in the CCQE and CC two-
particles-two-holes (CC2p2h) interactions, and improvements in the treatments
of final state interaction (FSI) and pion production, when compared to Genie
v2_12_2 (used by previous MicroBooNE analyses [68, 69, 70, 71]). In addition
to the default configuration, the parameters governing the CCQE and CC2p2h
models are adjusted according to the T2K CC0$\pi$ cross-section results [72].
Given that T2K has a neutrino flux in a similar energy range to that of
MicroBooNE, this additional adjustment represents an improved model prediction
and uncertainty treatment [73], despite the fact that T2K data is on carbon-
hydrogen and MicroBooNE’s measurement is on argon. Corrections from carbon to
argon targets are applied based on a smooth nuclear dependence (or
$A$-dependence) that are determined in fits to inclusive and semi-inclusive
electron scattering data.

The resulting final state particles of each simulated MC555Throughout this
paper, we use the acronym “MC” to identify the output from our event generator
simulation. event are processed using the LArSoft [74] software framework,
which is a toolkit to perform simulation, reconstruction, and analysis of
LArTPC events. The final state particles are propagated through the detector
using the Geant4 toolkit [59] v4_10_3_03c. The resulting energy depositions
are further ported to dedicated detector simulation programs taking into
account all detector effects to simulate the ionization charge and
scintillation light signals after taking into account the space charge effect.
The space charge effect [58, 48] is caused by the accumulation of positively
charged ions inside the active volume. For on-surface LArTPC detectors such as
MicroBooNE, cosmic-ray muons provide a constant source of positively charged
ions, which distort the local drift electric field. Consequently, the
ionization electrons bend toward the detector center when drifting to the
anode plane and the reconstructed positions of ionization electrons appear to
be closer to the detector center compared to their true position making the
effective detector boundary smaller than the actual active TPC boundary. In
addition to the deformation of reconstructed positions, the distortion in the
electric field also changes the amount of ionization electrons and
scintillation photons through the charge recombination process [75, 76]. After
comparing simulation (with these effects implemented) and calibration run
data, the amount of ionization electrons is further scaled down in the
simulation in order to include the effects from conversion of analog-to-
digital converter values to the number of ionization electrons and position-
dependent energy calibrations [49], which results in an average scaling factor
of 0.86.

The position and number of ionization electrons modified by recombination and
the space charge effect is ported to the TPC detector simulation [47], which
takes into account the charge transportation and diffusion [62]. The induced
currents on the wires are simulated by convolving the ionization charge
distribution at the wire plane with the position-dependent (at 1/10th of the
wire pitch resolution) field response function, which are calculated by the
dedicated Garfield simulation [77] following Ramo’s theorem [78]. The induced
current is further convolved with the electronics response before adding the
inherent electronics noise from data and is then digitized to generate the
final waveform on each wire channel.

The optical detector simulation models the light emitted by charged particles
interacting with the detector and produces signals in photomultiplier tubes. A
full optical simulation implements the Geant4 simulation of individual optical
photons produced along the path of charged particles through both the
scintillation and Cherenkov processes. These photons are stepped through
several processes with the detector medium including Rayleigh scattering,
reflection, and partial absorption in order to produce a realistic detector
response to the light source. However, because of the vast number of photons
typically produced in a neutrino physics event, the full optical simulation
can take hours or days per event. Therefore, a fast optical simulation was
developed to overcome this problem for routine simulation tasks. This mode
utilizes a library of stored data that represent the PMT acceptance of
scintillation light signals666The contribution from Cherenkov light is
insignificant at this stage. to sample an expected detector response given an
isotropic emission of light at a certain point in the volume. The PMT response
is further convolved with the time distribution of these photons to generate
the digitized waveform.

Compared to early MicroBooNE analyses [68, 69, 70, 71], the simulation used in
this eLEE search adopts a scheme of overlaying the simulated neutrino
interactions with dedicated beam-off data. These data are taken without the
neutrino beam and are triggered by a random signal to mimic the neutrino beam
gate. The simulated TPC and PMT waveforms from neutrino interactions are
overlaid with the data waveform of a beam-off event. Such simulation is also
referred to as overlay MC simulation. This scheme eliminates the systematic
uncertainties in the simulation of excessive electronics noise and cosmic-ray
muon activity. This scheme limits the statistics of the overlay MC sample
because of the finite size of available beam-off data sample to be overlaid.
Nevertheless, the statistics of the MC background sample is more than a factor
of three larger than that of the data. As elaborated in Sec. V.5, a Bayesian
approach was developed to obtain an optimal estimation of MC statistical
uncertainties, even when the predicted background is zero.

###  III.5 Event Reconstruction

An end-to-end automated event reconstruction chain containing various
fundamental reconstruction techniques was developed and implemented in this
analysis. TPC signal processing, which mitigates the excess noise [64] and
deconvolves the detector response from the drift electric field and the
electronics readout [47, 79], provides the reconstructed ionization charge
distributions for each wire to the subsequent calorimetry and topology
reconstruction algorithms. A tomographic three-dimensional (3D) image
reconstruction algorithm, Wire-Cell [42], is used as the core algorithm of the
reconstruction chain. Wire-Cell uses reconstructed ionization charge at
different times and readout wire 1D positions to reconstruct the 3D images of
ionization electrons without topology heuristic assumptions (e.g. tracks from
muons/pions/hadrons or EM showers from electrons/photons) prior to the pattern
recognition stage. Other algorithms such as clustering and de-ghosting [43]
are implemented to further improve the quality of the 3D images particularly
addressing the challenge that gaps occur in the 2D view of the charge signals
because of the inefficiency of TPC signal processing or non-functional wires.
The space points (representing the 3D voxels with non-zero reconstructed
charge) of the reconstructed 3D image are grouped into clusters, each of which
represents individual physical activity in the TPC from a cosmic-ray muon or a
neutrino interaction. Figure 4(a) shows each 2D projection view of an event’s
3D image after imaging and clustering.

MicroBooNE data run 6195 event 1020

Figure 5: Displays of the Wire-Cell pattern recognition results at different
stages. (a) In-beam candidate neutrino cluster selected by the generic
neutrino selection. The color scale represents the reconstructed charge
associated with each space point, where blue and cyan are lower in charge and
yellow and orange higher in charge. (b) Identified tracks and EM showers,
which are displayed in blue and red, respectively. (c) Identified particles
(or track segments), which are displayed in different colors. (d) Fitted
$dQ/dx$ associated with each piece ($\sim$6 mm) along the trajectories. The
blue, cyan, green, yellow, and red colors roughly correspond to 1/3, 1, 2, 3,
and 4 times the $dQ/dx$ of a minimum ionizing particle (MIP). (e)
Reconstructed particle flow starting from the primary neutrino interaction
vertex, which is displayed in a rainbow-colored wheel.

As previously discussed, the TPC ionization charge signal data, which provide
the topology and calorimetry information, are collected separately from the
PMT scintillation light signals, which provide the timing information, because
of the longer drift time of ionization electrons relative to the light
propagation. This results in a challenge, especially for surface-operating
LArTPCs such as MicroBooNE, in identifying neutrino interactions from numerous
cosmic-ray muon interactions [80, 81, 82, 83, 68]. A many-to-many charge-light
matching algorithm was developed to overcome this challenge by finding the
corresponding light signals and providing the interaction time for each charge
cluster [43]. About 70% of the cosmic-ray muon events that pass the software
trigger are rejected by requiring a charge cluster to have its start time
coincide with the beam spill. These clusters are referred to as in-beam
clusters. Figure 4(b) shows an example from data, where the in-beam $\nu_{e}$
CC candidate cluster is selected out of about 20 cosmic-ray muon clusters
after charge-light matching.

The majority of in-beam clusters still originate from cosmic-ray muons after
charge-light matching for MicroBooNE events. Additional algorithms were
developed to reject cosmic-ray backgrounds with 5-10% efficiency loss for
neutrino interactions [84]. The effective boundary of the TPC active volume
considering the space charge effect [58, 48] is used to define the fiducial
volume in the cosmic-ray rejection as well as the subsequent neutrino
selection. The fiducial boundary is defined as 3 cm inside the effective
boundary, which leads to a fiducial volume of 94.2% of the active TPC. The
through-going muons, which traverse the TPC active volume, are rejected if the
two ends of the track exit the fiducial boundary. The rejection of stopped
muons, which enter the active volume and stop inside, is based on the
identification of an increase of ionization charge loss per unit length
($dQ/dx$) near the end of the track (i.e., Bragg peak). This is obtained by a
newly developed 3D track trajectory and $dQ/dx$ fitting procedure [57], which
is also an important ingredient in pattern recognition and particle
identification. Note that the external cosmic-ray-tagger [85] system may
provide additional rejection of cosmic-ray muon events, but is not included in
this work as the system was not installed until late 2017. Using the above
mentioned cosmic-ray rejection techniques including charge-light matching, the
cosmic-ray background is reduced significantly resulting in less than 15%
cosmic-ray contamination in the selected neutrino candidate events while the
original neutrino to cosmic-ray muon ratio was about 1:200 for the events
passing the software trigger. The efficiency loss for CC neutrino interactions
is 10-20% for different flavors of neutrinos up to this stage [84, 57].

Pattern recognition is vital for the identification of different flavors of
neutrinos, e.g. $\nu_{e}$ CC and $\nu_{\mu}$ CC, for a variety of physics
analyses. The details of Wire-Cell pattern recognition can be found in Ref.
[86] and highlights are provided in the following. Wire-Cell pattern
recognition starts by finding initial end points of track segments by
searching for kinks and splits in the selected 3D in-beam cluster. Track
segments and their end points are then determined by iterative multi-track
trajectory and $dQ/dx$ fitting where linear algebra algorithms and graph
theory operations are utilized to achieve a robust performance. Particle
identification (PID) is performed based on the $dQ/dx$, topology information
(direction, track or shower, etc.), and allowable particle flow relationships
for each track segment. Candidate primary neutrino interaction vertices are
concurrently identified as parts of the particle flow tree, which is a series
of particles that starts from the neutrino interaction vertex and loops over
all identified particles following the particle flow relationship. A deep
neural network of SparseConvNet [87] is used to boost the performance of the
neutrino vertex identification by predicting the distance from each 3D voxel
to the neutrino vertex. It chooses from the neutrino vertex candidates, which
are identified based on the above traditional algorithms, and determines the
final reconstructed neutrino interaction vertex. The particle flow is then
refined if needed. $\pi^{0}$ particles are reconstructed relying on the
opening angle and topological information of the two decay $\gamma$’s and
other $\pi^{0}$ decay modes like Dalitz decay plays a very minor role in this
analysis. A proton ($p$) can be reconstructed if its length is $\gtrsim$1 cm
which corresponds to a kinetic energy threshold of 35 MeV. Neutrons ($n$) are
invisible in LArTPCs as they are neutral, but they could be suggested by off-
vertex protons that result from the neutron scattering. Figure 5 illustrates
the results of pattern recognition at different stages. As reported in Ref.
[86], this pattern recognition achieves 60-75% efficiencies of good neutrino
vertices (distance between reconstructed and true vertices below 1 cm) and
subsequently a 80-90% reconstruction efficiency for primary leptons for
charged-current neutrino interactions.

###  III.6 Neutrino Energy Reconstruction

MicroBooNE Simulation

(a) $\nu_{\mu}$ CC candidates, FC

MicroBooNE Simulation

(b) $\nu_{\mu}$ CC candidates, PC

MicroBooNE Simulation

(c) $\nu_{e}$ CC candidates, FC

MicroBooNE Simulation

(d) $\nu_{e}$ CC candidates, PC

MicroBooNE Simulation

(e) $\nu_{\mu}$ CC candidates, FC

MicroBooNE Simulation

(f) $\nu_{\mu}$ CC candidates, PC

MicroBooNE Simulation

(g) $\nu_{e}$ CC candidates, FC

MicroBooNE Simulation

(h) $\nu_{e}$ CC candidates, PC

Figure 6:  Reconstructed neutrino energy vs. true neutrino energy [(a)-(d)]
and reconstructed neutrino energy resolution and its bias [(e)-(h)] for FC and
PC $\nu_{\mu}$ CC and FC and PC $\nu_{e}$ CC candidates. The black points in
the energy resolution plots represent the peak positions for each bin
indicating the typical bias, and the error bars represent 68.3% quantiles from
each bin’s peak position.

Neutrino energy reconstruction for Wire-Cell inclusive neutrino selection
adopts a calorimetric approach, essentially adding up the visible energy
deposited in the TPC active volume. The reconstructed neutrino energy is given
by the following formula:

$E^{\text{rec}}_{\nu}=\sum_{i}\left(K^{\text{rec}}_{i}+m_{i}+B_{i}\right),$
(1)

where $i$ represents each identified particle in the reconstructed particle
flow, $K^{\text{rec}}_{i}$ represents the reconstructed kinetic energy,
$m_{i}$ represents the rest mass value from the PDG [1], and $B_{i}$
represents the average binding energy (8.6 MeV) per nucleon after which an
argon-40 nucleus is completely disassembled into its constituent nucleons.
$m_{i}$ is only added for particles reconstructed as $\mu^{\pm}$, $\pi^{\pm}$,
and $e^{\pm}$ particles, and $B_{i}$ is added for each reconstructed proton in
the particle flow, either a primary proton connected to the neutrino vertex or
a secondary proton scattered off from an argon nucleus by a neutron. There are
three methods to calculate the kinetic energy, $K^{\text{rec}}_{i}$, for each
particle:

  * •

Range: the range is used to calculate the kinetic energy of track-like
particles $\mu^{\pm}$, $\pi^{\pm}$, and $p$ if they stop inside the active
volume. This is based on the NIST PSTAR database [88] with a correction for
different particle masses.

  * •

Summation of $dE/dx$: for short ($<$4 cm) track-like particles that stop
inside the active volume or any exiting particles, the (visible) kinetic
energy is estimated by summing up $dE/dx$ for each piece ($\sim$6 mm) of the
track. The energy loss per unit length $dE/dx$ is converted from the
ionization charge per unit length $dQ/dx$ considering the recombination effect
when the ionization electrons are produced. This method is also used to
estimate the energy of long muons with significant delta-rays emitted along
the trajectory. This avoids bias in calculating the range. Note that an
effective recombination model that also takes into account the overall
normalization difference between MicroBooNE data and simulation charge signals
[49] is built by tuning the parameters of the modified box model from ArgoNeuT
[76]. This effective recombination model ($\alpha$=1.0 and $\beta$=0.255), as
is used in current Wire-Cell energy reconstruction, has an improved agreement
with MicroBooNE data; however, it still underestimates the energy by about
10%. Given that the ionization charge in simulation has been scaled to match
that in the data, this bias from the effective recombination model appears in
both data and simulation in the same way. The residual difference between data
and simulation is covered by the systematic uncertainties as discussed in Sec.
V.

  * •

Overall charge-energy scaling: for EM showers (non-track topologies), the
energy is estimated by scaling the total reconstructed charge of the shower
cluster by a factor of 2.50 after multiplying by the 23.6 eV per ionization
pair [89, 90]. This factor is derived from the nominal simulation and takes
into account the bias in the reconstructed charge with respect to the true
charge, and the average recombination effect [43] which converts the deposited
energy to the true charge. For data events, an additional scaling factor of
0.95, which is calibrated from the reconstructed $\pi^{0}$ invariant mass
(Sec. IV.3), is applied.

Performance of this neutrino energy reconstruction is evaluated using MC
samples by comparing the reconstructed values with the true values. The
reconstructed neutrino energy versus the true neutrino energy as well as the
energy resolution and bias relative to the true values are shown in Fig. 6.
These figures present the results for selected $\nu_{\mu}$ CC and $\nu_{e}$ CC
candidates (see Sec. IV), respectively. The neutrino candidates are further
divided into FC and PC samples (See Sec. II) based on the containment of the
selected in-beam cluster. We should note that since an energetic muon track is
much more extended than an electron shower with the same energy, the FC cut
actually selects $\nu_{\mu}$ CC events with higher energy transfer (thus
larger missing hadronic energy on average) than the corresponding $\nu_{e}$ CC
events at the same neutrino energy. For the FC samples, the reconstructed
neutrino energy resolution is 15-20% for $\nu_{\mu}$ CC candidates with
$\sim$10% bias (towards lower energies) and 10-15% for $\nu_{e}$ CC candidates
with $\sim$7% bias (towards lower energies). The slightly worse energy
resolution of $\nu_{\mu}$ CC than that of $\nu_{e}$ CC reflects the fact that
$\nu_{\mu}$ CC event selection can tolerate more imperfect event
reconstruction given its higher signal to background ratio than that of
$\nu_{e}$ CC. Energy deposited outside the active volume by charged particles
or carried away by neutral particles such as neutrons is missing in this
calorimetric energy reconstruction. In the energy range of interest, below 800
MeV in true neutrino energy, we have achieved uniform performance in the
resolution of reconstructed neutrino energy and its bias, especially in
$\nu_{e}$ CC FC signal event selection (Fig. 6(g)). More validation of this
neutrino energy reconstruction in terms of the modeling of missing energy
using $\nu_{\mu}$ CC events and data-MC comparisons can be found in Sec. VI.2.

##  IV Event Selection

The starting point of the neutrino event selection is the generic neutrino
selection [84, 57], in which the cosmic-ray backgrounds are reduced resulting
in an overall contamination below 15%. After the generic neutrino selection,
the efficiencies777In this article, the selection efficiency is defined as the
number of selected events relative to the true neutrino interactions with
vertices inside the fiducial volume. for $\nu_{\mu}$ CC and $\nu_{e}$ CC
events are approximately 80% and 90% with signal-to-background ratios of about
2:1 (purity of 66%) and 1:250 (purity of 0.4%), respectively.

For the search of the low-energy excess in the $\nu_{e}$ CC channel, the event
selections are designed to be as general as possible (i.e., inclusive
$\nu_{e}$ CC), so that more freedom in examining exclusive channels would be
available at later stages of the analysis if an excess was to be observed.
Since the $\nu_{\mu}$ CC events are used to constrain the systematic
uncertainties in neutrino flux, neutrino-argon interaction cross section, and
detector effects, an inclusive $\nu_{\mu}$ CC selection is also adopted.

###  IV.1 Charged-Current $\nu_{e}$ Selection

The development of $\nu_{e}$ CC selection involves two stages. The first stage
is the categorization of non-$\nu_{e}$ CC backgrounds, which is informed by
hand scans of a small amount of background events. Then, variables with
signal-background discrimination capability which represent the characteristic
features of each type of background in the first stage are used as input into
boosted decision trees (BDTs) trained on large MC simulation samples. Events
used in BDT training are removed in making predictions.

The basic selection of inclusive $\nu_{e}$ CC events requires an EM shower
with a reconstructed energy higher than 60 MeV connected to the (primary)
neutrino vertex [86]. The energy threshold is applied in order to exclude
Michel electrons. When there are multiple reconstructed EM showers connected
to the same neutrino vertex, the EM shower with the highest energy is taken as
the primary electron candidate for further examinations.

The backgrounds are categorized into five major types. The first type focuses
on primary electron identification, including the examination of the $dQ/dx$
profile at the first few centimeters of the shower (i.e. shower stem) and the
identification of a gap between the shower and the neutrino vertex. This gap
occurs in photon showers due to the photon conversion length of approximately
18 cm in liquid argon. The second type of background focuses on interactions
with multiple EM showers in the final state, most likely from $\pi^{0}$
production. The third type focuses on muon-related misidentification as
electrons. The fourth type focuses on more general background rejection using
kinematic information, e.g. comparison of lepton kinematics between an
electron candidate and its competing muon candidate in the same event888While
this situation does not exist for true physics event, it can happen due to an
imperfect event reconstruction.. The last type focuses on interactions with
poor pattern recognition, which includes several different failure modes
leading to incorrect pattern recognition. Beside these five major
categorizations of background featuring in the $\nu_{e}$ CC selection, there
is a another set of dedicated taggers removing residual cosmic-ray muon
induced backgrounds, which will be described in detail in the next section.

The primary electron identification includes:

  * •

Gap cut: the beginning of the EM shower in each 2D projection view is examined
to search for a gap to the identified neutrino interaction vertex.

  * •

Stem quality cut to remove backgrounds: the beginning of the shower is
examined to ensure the quality of the shower stem. The checks include
examinations of i) other minimum ionizing particle (MIP) tracks overlapping
with the shower stem, and ii) possible track splitting, e.g. the splitting of
pair-produced electron and positron instead of traveling in the same
direction.

  * •

MIP $dQ/dx$ cut: we examine the $dQ/dx$ profile of the shower stem to ensure a
MIP (electron-like) event. We calculate the length of the MIP-like track below
a MIP threshold cut (i.e. 1.3 times the expected MIP $dQ/dx$). The calculation
of the length also considers the possibility of a delta ray (i.e. a single
sample with high dQ/dx). In addition, a high $dQ/dx$ value at the vertex
because of additional vertex activities must be taken into account.

With the hand scanned features selected, we apply BDT techniques to high-
statistics MC simulation samples to finalize the $\nu_{e}$ CC selection. The
usage of machine learning techniques mitigates the limitation of human
learning when processing a large number of events. From among the many
different machine learning tools, the BDT technique is chosen because it is
more robust and approachable for general users compared to other multivariate
analysis tools. The BDT package XGBoost [91], which provides fast and robust
training through parallel tree boosting, is used. XGBoost also improves the
model generalization and overcomes the issues of over-fitting in gradient
boosting enabling the use of a large pool of variables (over 300 variables
from all categorizations of backgrounds in this $\nu_{e}$ CC selection) in a
single model. To train the BDT, the true $\nu_{e}$ CC events in the fiducial
volume that pass the generic neutrino selection and have at least one
reconstructed electron EM shower are used to define the signal. In order to
enhance the performance, the events with bad reconstruction, when the
reconstructed neutrino and EM shower vertices are incorrectly reconstructed,
are removed from the signal events. The overlay MC simulation after excluding
true $\nu_{e}$ CC events and the dedicated beam-off data are used as
background in training the BDT. Figure 7(a) shows the signal efficiency and
purity as a function of the $\nu_{e}$ CC BDT score, and the distribution of
$\nu_{e}$ BDT score is shown in Fig. 7(b). Here, the efficiency is defined
with respect to all true $\nu_{e}$ CC events with their neutrino interaction
vertices inside the fiducial volume. The cut value of 7.0 was chosen in order
to maximize the $\nu_{e}$ selection efficiency$\times$purity value. A final
$\nu_{e}$ CC selection with 46% efficiency and 82% purity is achieved.

MicroBooNE Simulation

(a)

MicroBooNE

(b)

Figure 7: (a) $\nu_{e}$ CC selection efficiency and purity at different BDT
cut values, with the finalized cut value of 7.0 indicated. (b) $\nu_{e}$ BDT
score distribution for events with BDT score$>$0.

MicroBooNEFC+PC

(a) Neutrino energy, broken down with event types

MicroBooNEFC+PC

(b) Shower cos$\theta$, broken down with event types

MicroBooNEFC+PC

(c) Neutrino energy, broken down with interaction types

MicroBooNEFC+PC

(d) Shower cos$\theta$, broken down with interaction types

MicroBooNE Simulation

(e) Efficiency as a function of true neutrino energy

MicroBooNE Simulation

(f) Efficiency as a function of true shower cos$\theta$

Figure 8: The final $\nu_{e}$ CC selections as a function of reconstructed
neutrino energy [(a) and (c)] and reconstructed shower cos$\theta$ [(b) and
(d)]. (a) and (b) are categorized by event types, and (c) and (d) by
interaction types. The number of events correspond to the range shown in the
plot. The bottom sub-panels show both the statistical and systematic
uncertainties. The pink band includes the statistical, cross section, and flux
uncertainties. The purple band corresponds to the full uncertainty with the
addition of detector systematic uncertainty. The selection efficiencies are
shown as a function of (e) true neutrino energy and (f) true shower
cos$\theta$ with only statistical uncertainty considered. The other dimensions
are integrated in calculating these efficiencies.

The selected events are categorized into different types which are determined
by the truth information. Each category, which will be used throughout this
paper, as shown in Fig. 8(a) and Fig. 8(b), is defined as follows: (1) “EXT”:
cosmic-ray background from the beam-off data set that is external to the BNB
data stream and triggered by cosmic-ray activity in coincidence with a fake
beam spill, in which case the events have no BNB neutrino interactions; (2)
“Cosmic”: mistakenly selected cosmic-ray background from BNB overlay MC
simulation, in which case each event has a simulated BNB neutrino interaction
overlaid with dedicated beam-off data; (3) “Dirt”: neutrino interactions with
their true neutrino interaction vertices outside the cryostat, as defined in
Sec. III.4; (4) “out FV”: neutrino-argon interactions with vertices outside
the fiducial volume but within cryostat; (5) “$\nu_{\mu}$ CC $\pi^{0}$ in FV”:
$\nu_{\mu}$ CC interactions with vertices inside the fiducial volume and with
at least one true $\pi^{0}$ in the final state; (6) “$\nu_{\mu}$ CC in FV”:
$\nu_{\mu}$ CC interactions with vertices inside the fiducial volume and with
no $\pi^{0}$ in the final state; (7) “NC $\pi^{0}$ in FV”: NC interactions
with vertices inside the fiducial volume and with at least one true $\pi^{0}$
in the final state; (8) “NC in FV”: NC interactions with vertices inside the
fiducial volume and with no $\pi^{0}$ in the final state; (9) “$\nu_{e}$ CC in
FV”: beam intrinsic $\nu_{e}$ CC interactions with vertices in the fiducial
volume; and (10) “LEE”: all categories of excessive events originating from
eLEE neutrino interactions, which accounts for the difference between the
eLEEx=1 hypothesis and eLEEx=0 hypothesis. Except for “EXT”, the other
categories correspond to MC events which overlay simulated neutrino
interactions with randomly triggered beam-off (cosmic) data. As shown in Fig.
8(c) and Fig. 8(d), the other categories for the selected neutrino
interactions correspond to different interaction types obtained from the event
generator. “CC” or “NC” represent charged-current or neutral-current. “QE”,
“RES”, “MEC”, and “DIS” represent quasi-elastic, resonance, meson exchange
current, and deep inelastic scattering, respectively. Figure 8(e) and Figure
8(f) show the $\nu_{e}$ CC selection efficiency as a function of neutrino
energy ($E_{\nu}$) and the cosine of the polar angle (cos$\theta$, relative to
the BNB direction) of the electron EM shower, respectively.

Before applying the $\nu_{e}$ CC selection to the BNB data stream, we
validated its performance using the off-axis Neutrinos at the Main Injector
(NuMI) [92] neutrino beam at FNAL. The NuMI beam is created from collisions of
protons accelerated to an energy of 120 GeV with a graphite target. Similar to
that of BNB, the charged hadrons are focused by a magnetic field into a 675 m
long decay pipe. The distance between the NuMI target and the MicroBooNE
detector is about 679 m. At an off-axis location of $\sim 8^{\circ}$, the
$\nu_{e}$’s with a sizable amount of $\bar{\nu}_{e}$’s above 200 MeV are
mostly coming from the 3-body $K^{+}$ and unfocused $K^{0}_{L}$ decays.
Compared to that of the BNB, the percentage of $\nu_{e}$ in the flux is an
order of magnitude higher at $\sim$5 %, which makes it ideal to validate the
performance of the $\nu_{e}$ CC selection. With 1.917$\times 10^{20}$ POT
exposure, we select 269 FC $\nu_{e}$ CC and 162 PC $\nu_{e}$ CC candidates
with reconstructed neutrino energy below 2.5 GeV and with an overall
efficiency of 42% and purity of 91%. The overall ratio between data and
nominal NuMI MC prediction for FC and PC $\nu_{e}$ CC without considering any
anomalous enhancement is 0.99$\pm$0.06 (stat.) and $1.08\pm 0.08$ (stat.)
indicating an overall good agreement.

###  IV.2 Charged-Current $\nu_{\mu}$ Selection

The generic neutrino selection results in a 88.4% selection efficiency and
65.0% purity for $\nu_{\mu}$ CC events [84, 57]. Here, the selection
efficiency is defined with respect to all true $\nu_{\mu}$ CC events with
their neutrino interaction vertices inside the fiducial volume. The achieved
purity is limited by the residual cosmic-ray muon background, neutrino-induced
background originating outside the fiducial volume, and NC interactions inside
the fiducial volume. The precise reconstruction of the $\nu_{\mu}$ CC neutrino
vertex (resolution is less than 1 cm) and particle identification (an
integrated efficiency of 90% for primary muons) are leveraged to suppress
these backgrounds. The reconstructed neutrino vertex is required to be inside
the fiducial volume and the length of primary muons is required to be greater
than 5 cm before applying the BDT selection.

In analogy to the $\nu_{e}$ CC selection, human scans of the remaining
backgrounds were performed to extract the main features of each type of
background. The residual cosmic-ray background is typically the result of
incorrect charge-light matching where the TPC cluster is placed at an
incorrect location along the electric field direction. A through-going muon
could have only one track end reconstructed at the detector boundary instead
of two track ends reconstructed at the detector boundary, mimicking a single
muon starting inside the TPC and exiting the detector. A stopped muon might
also appear to be fully contained and, alternatively, be reconstructed with
the candidate neutrino vertex at the muon decay vertex connecting the muon and
the Michele electron. These topological features are leveraged to do this
background rejection. For neutrino-induced background originating outside the
fiducial volume, a charged hadron usually enters the detector and undergoes a
hadronic interaction. For this kind of events, the neutrino vertex is
typically reconstructed at the hadronic interaction point, and the event could
then appear to originate inside the fiducial volume with a misidentified muon
candidate. Note, with an exiting high-energy charged particle track, one may
not achieve a reliable PID for MIP particles such as muons. The kinematics,
especially the direction of the muon candidate, can be used to reject such
background since most of the hadrons entering the detectors from outside of
the detector are not as forward-going as expected.

MicroBooNE Simulation

(a)

MicroBooNE

(b)

Figure 9: (a) $\nu_{\mu}$ CC selection efficiency and purity at different BDT
cut values with the finalized cut value of 0.9 indicated. (b) $\nu_{\mu}$ BDT
score distribution for events with BDT score$>$0.

MicroBooNEFC+PC

(a) Neutrino energy, broken down with event types

MicroBooNEFC+PC

(b) Muon cos$\theta$, broken down with event types

MicroBooNEFC+PC

(c) Neutrino energy, broken down with interaction types

MicroBooNEFC+PC

(d) Muon cos$\theta$, broken down with interaction types

MicroBooNE Simulation

(e) Efficiency as a function of true neutrino energy

MicroBooNE Simulation

(f) Efficiency as a function of true muon cos$\theta$

Figure 10: The final $\nu_{\mu}$ CC selections as a function of reconstructed
neutrino energy [(a) and (c)] and reconstructed muon cos$\theta$ [(b) and
(d)]. (a) and (b) are categorized by event types, and (c) and (d) by
interaction types. The number of events correspond to the range shown in the
plot. The bottom sub-panels present both the statistical and systematic
uncertainties. The pink band includes the statistical, cross section, and flux
uncertainties. The purple band corresponds to the full uncertainty with the
addition of the detector systematic uncertainty. The selection efficiencies
are shown as a function of (e) true neutrino energy and (f) true muon
cos$\theta$ with only statistical uncertainty considered. The other dimensions
are integrated in calculating these efficiencies.

For NC neutrino interaction background inside the fiducial volume, the main
difference from $\nu_{\mu}$ CC events is the absence of a primary muon at the
neutrino vertex. However, the separation of $\nu_{\mu}$ CC and this NC
background, which mainly relies on the discrimination of charged pions and
muons, is very difficult if only the $dQ/dx$ information is used. To further
reject such NC background, the activities associated with charged pions, e.g.
proton scattering, and the relatively large-angle deflection ($\sim$10
degrees) of the trajectory of charged pions can be used to provide additional
separation power.

With the identification of the major features of the residual cosmic-ray
background, neutrino-induced background originating outside the fiducial
volume, and NC events inside the fiducial volume, the BDT was trained and
applied to improve the $\nu_{\mu}$ CC selection. A similar training strategy
as discussed in Sec. IV.1 is used with a signal definition switched to the
$\nu_{\mu}$ CC events. Figure 9 shows the $\nu_{\mu}$ CC selection efficiency
and purity as a function of the $\nu_{\mu}$ CC BDT score as well as the
distribution of the $\nu_{\mu}$ CC BDT score. The final cut value of 0.9 was
chosen for the $\nu_{\mu}$ CC selection with a 68% efficiency and 92% purity.
Figure 10 shows the selected $\nu_{\mu}$ CC events and selection efficiency as
a function of neutrino energy and muon cos$\theta$. The efficiency is
generally higher for more forward-going angles as events with forward-going
angles are more likely to have a typical topology of a $\nu_{\mu}$ CC event to
which the BDT input variables are tuned. The “slope” of data-prediction ratios
present in the bottom panel of Fig. 10(a) will be discussed in Sec. VI.3. For
some specific backgrounds, the BDT could select a small number of poorly
reconstructed events. This is because the $\nu_{\mu}$ CC BDT was trained on a
data set without an explicit request of a muon being present. Therefore, some
$\nu_{\mu}$ CC candidates may not have a primary muon identified. This results
in a non-zero number of entries in the first bin (less than 100 MeV) in Figs.
10(a) and 10(c) in which case the muon rest mass is not considered in the
neutrino energy reconstruction and, also, in an absence of some events that
have no reconstructed muon cos$\theta$ in Figs. 10(b) and 10(d). We can do
this because of the high initial signal-to-background ratio of the events.

###  IV.3 $\pi^{0}$ Selection

MicroBooNEFC+PC

(a) CC$\pi^{0}$reconstructed $\pi^{0}$ mass

MicroBooNEFC+PC

(b) NC$\pi^{0}$reconstructed $\pi^{0}$ mass

MicroBooNEFC+PC

(c) CC$\pi^{0}$reconstructed $\pi^{0}$ kinetic energy

MicroBooNEFC+PC

(d) NC$\pi^{0}$reconstructed $\pi^{0}$ kinetic energy

MicroBooNE Simulation

(e) CC$\pi^{0}$ selection efficiency as a function of reconstructed $\pi^{0}$
kinetic energy

MicroBooNE Simulation

(f) NC$\pi^{0}$ selection efficiency as a function of reconstructed $\pi^{0}$
kinetic energy

Figure 11:  Distributions of the reconstructed $\pi^{0}$ invariant mass [(a)
and (b)], reconstructed $\pi^{0}$ kinematic energy [(c) and (d)] and selection
efficiency [(e) and (f)] as a function of reconstructed $\pi^{0}$ kinetic
energy for CC$\pi^{0}$ [(a), (c), and (e)] and NC$\pi^{0}$ [(b), (d), and
(f)]. Only the statistical uncertainty was considered in the efficiency plots
(e) and (f). The bottom sub-panels of plots [(a)-(d)] present both the
statistical and systematic uncertainties. The pink band includes the
statistical, cross section, and flux uncertainties. The purple band
corresponds to the full uncertainty with the addition of the detector
systematic uncertainty. A consistency is observed between the data and
simulation validating the energy scale reconstruction for EM showers.

The $\nu_{\mu}$ CC selection described in the previous section is employed to
select CC$\pi^{0}$ events from $\nu_{\mu}$ CC interactions. Additionally, a
NC$\pi^{0}$ selection is constructed by considering the non-cosmic events that
fail the $\nu_{\mu}$ CC selection ($\nu_{\mu}$ BDT score smaller than zero).
In the reconstruction of $\pi^{0}$ events, the pair of EM showers with highest
energies, supposedly from $\pi^{0}$ two-$\gamma$ decay, is chosen to be the
ones pointing to the same vertex. $\pi^{0}$ particles are identified as
primary particles by placing a maximum distance cut between the neutrino
vertex and the $\pi^{0}$ vertex. More details on the vertexing, clustering,
and pattern recognition of $\pi^{0}$ events can be found in Ref. [86]. Further
selection cuts use the $\gamma$ energies, the distances between the neutrino
vertex and the $\gamma$ vertices, the opening angle between the two
$\gamma$’s, and the reconstructed $\pi^{0}$ invariant mass. The distribution
of the reconstructed $\pi^{0}$ mass of the selected CC and NC $\pi^{0}$ events
can be found in Fig. 11 where a Gaussian fit over the peak region of data
returns a mean$\pm$width of 131.2$\pm$22.1 MeV/c2 for the CC$\pi^{0}$
selection and 130.4$\pm$19.3 MeV/c2 for the NC$\pi^{0}$ selection. The best-
fit mass values have $<$1 MeV differences compared to the MC reconstructed
mass values and are consistent with the expected $\pi^{0}$ invariant mass of
135 MeV/c2. The best-fit peak width does not include the contribution from the
long tails, which is the result of imperfect event reconstruction. Table 1
lists the efficiency and purity of the $\pi^{0}$ selections. Here, the
efficiency is defined with respect to all CC or NC $\pi^{0}$ events with their
neutrino interaction vertex inside the fiducial volume. The dominant
background component for the CC$\pi^{0}$ selection is $\nu_{\mu}$ CC events
without a $\pi^{0}$ in the final state, and the dominant background for the
NC$\pi^{0}$ selection is external events originating from cosmic-ray muons or
neutrino interactions outside the fiducial volume.

Event selection | Containment | Efficiency | Purity
---|---|---|---
CC$\pi^{0}$ | FC+PC | 32% | 72%
| FC | 12% | 75%
| PC | 20% | 71%
NC$\pi^{0}$ | FC+PC | 25% | 44%
Table 1: Selection efficiency and purity for CC$\pi^{0}$ FC+PC, CC$\pi^{0}$
FC, CC$\pi^{0}$ PC, and NC$\pi^{0}$ FC+PC samples.

##  V Systematic Uncertainties

In this analysis, we consider sources of systematic uncertainties from i) the
neutrino flux of the BNB, ii) neutrino-argon cross sections of the Genie event
generator, iii) hadron-argon interactions of the Geant4 simulation, iv)
detector response resulting from imperfect calibration, and v) the finite
statistics of MC samples used for prediction, as well as vi) additional
uncertainty for dirt events, which originate from neutrino interactions
outside the cryostat. The various sources of systematic uncertainty each have
different impacts on the reconstruction or selection efficiency (for both
signal and background) as well as the reconstruction of kinematic variables of
the predicted events. The uncertainty due to limited statistics of MC samples
and beam-off data is particularly important for estimating backgrounds of rare
event searches, e.g. the selection of BNB $\nu_{e}$ CC, which is $\sim$0.5% of
the total flux. Figure 12 summarizes the relative uncertainties from all
systematic uncertainty sources for the seven channels. In the following, we
describe each uncertainty in detail.

MicroBooNE Simulation

Figure 12: Summary of relative uncertainties ($\frac{\mathrm{absolute\
error}}{\mathrm{prediction}}$) of all systematic uncertainty sources for the
seven channels. Spikes at very low or high energy regions for detector
systematic uncertainty are largely attributed to statistical errors stemming
from the finite size of related MC samples.

###  V.1 Uncertainties from the Model of Neutrino Beam Flux

The calculation of the BNB neutrino flux and its uncertainties follows earlier
work from the MiniBooNE collaboration, which includes a well constrained
beamline simulation based on the Geant4 framework [59] as well as techniques
to handle systematic uncertainties [35]. Our flux prediction uses the updated
flux calculation that takes into account the SciBooNE measurement of
$p+Be\rightarrow K^{+}$ production in the BNB [93, 94], which provided a
better constraint on kaons produced in the BNB. In addition, our flux
prediction evaluates $\pi^{+}$ and $\pi^{-}$ production uncertainties directly
from HARP pion production data [95] rather than using a fit parameterization.
This technique allows the HARP measurement uncertainties to be more properly
propagated to the calculated neutrino flux.

Tuning parameter name | Parameter type
---|---
$\pi^{+}$ hadron production | FLUX
$\pi^{-}$ hadron production | FLUX
$K^{+}$ hadron production | FLUX
$K^{-}$ hadron production | FLUX
$K^{0}_{L}$ hadron production | FLUX
horn current distribution | FLUX
horn current calibration | FLUX
nucleon total scattering Xs | FLUX
nucleon inelastic scattering Xs | FLUX
nucleon quasi-elastic scattering Xs | FLUX
pion total scattering Xs | FLUX
pion inelastic scattering Xs | FLUX
pion quasi-elastic scattering Xs | FLUX
MicroBooNE Genie All |  Genie Xs ($\mu$B tune)
Strength of the CCQE RPA correction |  Genie Xs ($\mu$B tune)
Parameterization of the CCQE nucleon axial form factor |  Genie Xs
Parameterization of the CCQE nucleon vector form factors |  Genie Xs
Changes angular distribution of nucleon cluster in MEC |  Genie Xs ($\mu$B tune)
CCMEC Cross-section Shape |  Genie Xs ($\mu$B tune)
Angular distribution for RES $\Delta\rightarrow N+\pi$ |  Genie Xs
Angular distribution for RES $\Delta\rightarrow N+\gamma$ |  Genie Xs ($\mu$B tune)
Scaling factor for CC coherent $\pi$ production |  Genie Xs ($\mu$B tune)
Scaling factor for NC coherent $\pi$ production |  Genie Xs ($\mu$B tune)
Second-class vector current | Xs
Second-class axial current | Xs
$\pi^{-}$ interactions | Geant4
$\pi^{+}$ interactions | Geant4
proton interactions | Geant4
Table 2: Summary of tuning parameters used in generating the covariance matrix
for systematic uncertainties. For each tuning parameter, a covariance matrix
is generated according to its number of universes. The final covariance matrix
is the summation of all individual covariance matrices. For Genie Xs, the
label “$\mu$B tune” indicates that the tuning parameter and/or its uncertainty
is from MicroBooNE Genie tune other than the default treatment from Genie
v3.0.6.

As summarized in Table 2, the systematic uncertainties in the predicted flux
include effects from i) hadron production of $\pi^{+}$, $\pi^{-}$, $K^{+}$,
$K^{-}$, and $K^{0}_{L}$ and ii) non-hadron production: modeling of the horn
current distribution, horn current calibration, and pion and nucleon total,
inelastic, and quasielastic scattering cross-sections on beryllium and
aluminum. There is also an overall 2% normalization uncertainty associated
with the POT counting. Figure 13 shows the fractional contributions to the
overall flux systematic uncertainty from each source, for the seven channels
of the analysis. In the low-energy region, e.g. $E^{\text{rec}}_{\nu}<500$
MeV, which is relevant to the search for a $\nu_{e}$ low-energy excess, the
flux systematic is limited by the hadron production of $\pi^{+}$, which, with
its decay to muons, produces most of the $\nu_{e}$’s and $\nu_{\mu}$’s in this
energy range. Figure 14 shows the correlations of flux systematics for the
seven channels. There are obviously strong correlations between the low energy
ranges of $\nu_{e}$ events and $\nu_{\mu}$ events given that they all
originate from $\pi^{+}$ decays. There are strong correlations in the high
energy range between $\nu_{e}$ and $\nu_{\mu}$ given that these largely
originate from $\pi^{+}$ and $K^{+}$ decays. There are also strong
correlations between the high energy $\nu_{\mu}$ and the entire energy range
of the $\pi^{0}$ events since the neutrino interactions that enter into the
$\pi^{0}$ channels are almost all from $\nu_{\mu}$ interactions (negligible
$\nu_{e}$) which pass the energy threshold of $\pi^{0}$ resonance
interactions.

MicroBooNE Simulation

Figure 13: Fraction ($\frac{\sigma_{i}^{2}}{\sigma^{2}_{\mathrm{flux}}}\times
100$) of uncertainties of total flux systematics for the seven channels.

MicroBooNE Simulation

Figure 14: Correlations of flux systematics for the seven channels.

###  V.2 Uncertainties from the Neutrino-Argon Interaction Cross Sections

As introduced in Sec. III.4, Genie v3.0.6 [46], with parameters governing the
CCQE and CC2p2h models adjusted [73] by MicroBooNE, is used to generate
exclusive neutrino-argon interactions.

MicroBooNE Simulation

Figure 15: Correlations of cross section systematics for the seven channels.

Table 2 summarizes the uncertainties considered in the reweighting procedure
for neutrino-argon interaction cross sections (Xs). In particular, the
contribution labeled “MicroBooNE Genie All” contains 46 tuning parameters that
are simultaneously varied in generating hundreds of universes. The variations
in these universes are used to construct the covariance matrix. These tuning
parameters cover a wide range of models including CCQE, CC resonance (CCRES),
CC non-resonance, CC transition, CC deep-inelastic scattering (CCDIS), NC
interactions, and final-state interactions. In addition to “MicroBooNE Genie
All”, as listed in Table 2, there are additional 9 “Genie Xs” parameters
taking into account other cross-section uncertainties in Genie, each of which
have only two universes corresponding to the nominal value and the 1$\sigma$
bound of the tuning parameter, respectively. The 1$\sigma$ uncertainty is
taken to be the difference. There are two other tuning parameters labeled “Xs”
focusing on second-class currents [96] that may contribute to
$\nu_{e}/\nu_{\mu}$ CCQE cross section differences as suggested by Ref. [97].

Figure 15 shows the correlations of cross section systematics for the seven
channels. There are generally strong correlations between $\nu_{e}$CC and
$\nu_{\mu}$CC across the entire energy range, which is a natural result of the
lepton universality assumption. In the low-energy region, e.g.
$E^{\text{rec}}_{\nu}<500$ MeV which is relevant to the search for a $\nu_{e}$
low-energy excess, the neutrino cross section systematic is limited by the
level of suppression of the CCQE cross section behavior at low $Q^{2}$ (four-
momentum transfer from the lepton) because of long-range nucleon-nucleon
correlations, which is poorly constrained by existing data. We should further
note that some inconsistencies were identified in the Genie v3.0.6 reweighing
code used to evaluate FSI-related systematics as also discussed in Ref. [73],
but the effect of these were found to have a negligible impact on the overall
analysis sensitivity and have been ignored.

###  V.3 Uncertainties from the Hadron-Argon Interaction

Charged hadrons can scatter, both elastically and inelastically, with external
argon nuclei via hadronic interactions. These interactions can lead to the
production of additional particles or can cause large angle changes in
particle trajectories that may affect the reconstructed neutrino energy.
Therefore, it is important to include the uncertainties of these secondary
hadron-argon interactions. Studies were performed separately for elastic and
inelastic interactions for protons and charged pions.

Geant4 is used to propagate all hadrons through the detector medium based on a
semi-classical cascade model [98]. Using the Geant4reweight [99] package,
events with inelastic hadronic interactions are reweighted independently for
interactions containing protons, positive pions, and negative pions. For each
particle type and independent of the particle’s energy, the inelastic cross
section is varied around its mean by $\mathcal{O}$(20%), based on the
uncertainties of world data. Figure 16 shows the correlations of hadron-argon
interaction cross section systematics for the seven channels. The contribution
(square of the uncertainty) from this source of systematic uncertainty to the
total systematic uncertainty is up to 1% as indicated in Fig. 12.

MicroBooNE Simulation

Figure 16: Correlations of hadron-argon interaction systematics for the seven
channels.

###  V.4 Uncertainties from the Model of Detector Effects

There are four major categories of detector systematic uncertainties each
targeting one area of the detector calibration effort: i) variations related
to the light yield (LY) and propagation simulation, ii) variation in the space
charge effect, iii) variation in the recombination model, and iv) variations
on the amplitude and width of the deconvolved ionization charge waveforms for
other discrepancies between the detector response used in simulation and that
in data. We vary the magnitude of the space charge effect based upon the
measurements of the spatial distortions at the edge of the TPC, extrapolated
to the bulk field [58, 48]. A different recombination model, which provides
slightly better agreement to the data, is used to estimate the data/MC
difference in the $dE/dx$ to $dQ/dx$ conversion. The variations in the wire
waveform include effects as a function of $x$ (drift direction),
$\left(y,z\right)$ (vertical/beam directions), $\theta_{xz}$ and $\theta_{yz}$
(the angular orientation of the particle’s trajectory with respect to the
global coordinate system); these variations are constructed by comparing the
waveforms in data and simulation. More details on the variations of the wire
waveform can be found in Ref. [100].

For each source of detector systematic uncertainty, the same set of MC
simulation events are re-simulated with a change to the detector modeling
parameter of interest. The systematic uncertainty is estimated by comparing
efficiency and reconstructed kinematic variables in the new and old
simulation. The change of each detector modeling parameter is treated as
1$\sigma$, meaning that there is, in principle, only one degree of freedom in
constructing each detector covariance matrix after factoring in the
statistical uncertainties. The usage of the same set of events in the old and
new detector simulation aims to reduce statistical fluctuation, which is
estimated using the bootstrapping method [101].

The central idea of bootstrapping is to re-sample existing MC events in order
to estimate the uncertainties on the central value of the prediction. The
basic procedure is illustrated in the following:

  * •

Choose a common set of events in the central value (CV) and 1$\sigma$
variation simulation samples.

  * •

Re-sample MC events to form distributions with statistical uncertainty
corresponding to that of the expected POT exposure. During this process, one
can naturally take into account the event weight in the re-sampling process.

  * •

Apply the event selection requirements to the re-sampled events in both the CV
and the 1$\sigma$ samples and calculate the difference in the spectra of
different channels. Each such vector is considered one “universe.” By
repeating this procedure many times, many universes are simulated to form the
covariance matrix, $M_{R}$. This covariance matrix essentially represents the
uncertainty on the difference between the CV and 1$\sigma$ sample (nominal
difference vector, $\vec{V}^{\text{nominal}}_{D}$). In other words, these are
uncertainties of the uncertainties.

  * •

In order to generate the detector covariance matrix ($M_{D}$), a 2-step
procedure is adopted. In the first step, based on $M_{R}$, generate a random
set of vectors (based on decomposition of the symmetric matrix)
$\delta\vec{V}_{D}$ to be applied to the nominal difference vector
($\vec{V}^{\text{nominal}}_{D}$) in order to obtain a new vector
$\vec{V}_{D}=\vec{V}^{\text{nominal}}_{D}+\delta\vec{V}_{D}$.

  * •

In the second step, generate a single random number $r$ using the normal
distribution. Then multiply $\vec{V}_{D}$ by this random number (bin-to-bin
fully correlated) to obtain $r\cdot\vec{V}_{D}$, which is treated as one
universe.

  * •

By multiple repetitions of the above two steps, many universes are simulated,
which can the be used to construct the detector covariance matrix.

In the aforementioned approach, the statistical uncertainties in the detector
variation samples are naturally taken into account in the $M_{R}$ matrix.
Correlation of the detector variation samples is ensured by using the single
random number $r$. The relative strengths of the correlated and uncorrelated
uncertainties are properly dealt with via this procedure.

MicroBooNE Simulation

Figure 17: Correlations of total detector systematics for the seven channels.
The correlations between the FC and PC $\nu_{\mu}$CC channels are dominated by
the components of light yield and recombination.

MicroBooNE Simulation

Figure 18: Fractional contribution
($\frac{\sigma_{i}^{2}}{\sigma^{2}_{\mathrm{det}}}\times 100$) to the total
detector systematic uncertainty for the seven channels.

Figure 17 shows correlations of all detector systematics for the seven
channels, including statistical uncertainties. The correlations seen between
the FC and PC $\nu_{\mu}$ CC channels and between the FC $\nu_{\mu}$ CC and
$\nu_{e}$ CC channels mostly come from the variations in light yield and
recombination. Figure 18 shows the fractional contributions to the overall
detector systematic uncertainty from each type of variation for the seven
channels of the analysis.

###  V.5 Uncertainties of Finite Statistics in Making Predictions

For a rare event search with low background predictions such as that in the
MicroBooNE eLEE analysis, statistical uncertainties of the MC simulation and
dedicated background data can be important. For example, when the predicted
background is zero, quoting a zero uncertainty on this prediction following
the general error propagation would lead to an underestimation of statistical
uncertainties. On the other hand, if we use an upper limit to quote the
statistical uncertainty, the size of statistical uncertainties would increase
rapidly through the general error propagation procedure when there are
multiple components in estimating the background. In order to solve this
problem and obtain an optimal estimation of MC statistical uncertainties, we
adopt a new approach by combining the Bayesian approach with the covariance
matrix. For the Poisson distribution assuming a unit step function as the
prior (1 for $\mu\geq 0$ and 0 for $\mu<0$), an accurate approximation of the
posterior distribution of the expectation $\mu$ given the observation of $N$
is given by

$P\left(\mu|N\right)\approx e^{N-\mu+N\cdot log\left(\frac{\mu}{N}\right)},$
(2)

where the high-order asymptotic expansions (based on Stirling’s approximation)
of the Gamma function, $\Gamma(\mu+1)$, are ignored. For example, in the case
of $N=0$, we have $P\left(\mu|0\right)=e^{-\mu}$. If we use the covariance
matrix formalism to estimate the RMS of $\mu$ given a central value of zero,
we would have an uncertainty of 1.41 (standard deviation relative to the
central value). Similarly, we can estimate the uncertainties for other values
of $N$.

In MC simulation, different events may be associated with a different weight
value, which can lead to deviations from the simple Poisson distribution. For
example, if a MC event is associated with a weight value of two, it would
present two events in making a prediction. In the literature, there are many
studies of the treatment of a likelihood function given this situation. For a
recent review, see Ref. [102]. In this section, we give a prescription for how
to deal with this situation. The actual implementation follows Ref. [103].
Given events (labeled by $i$) with different weights ($w_{i}$), we define the
mean and variance of the prediction as:

$m:=\sum_{i}w_{i}~{}~{}~{}\text{and}~{}~{}~{}\sigma^{2}:=\sum_{i}w_{i}^{2},$
(3)

which can be written using effective uniform weights as

$m=w_{\textrm{Eff}}\cdot
m_{\textrm{Eff}}~{}~{}~{}\text{and}~{}~{}~{}\sigma^{2}=w_{\textrm{Eff}}^{2}\cdot
m_{\textrm{Eff}}.$ (4)

Define

$\overline{m_{\textrm{Eff}}}:=\mu\cdot m/\sigma^{2},$ (5)

so that $m_{\textrm{Eff}}=m^{2}/\sigma^{2}$ approximately follows a Poisson
distribution with an expectation value of $\overline{m_{\textrm{Eff}}}$, and
then the likelihood function (or distribution) can be written as

$P(\mu|m,\sigma^{2})=\frac{e^{-\mu\cdot m/\sigma^{2}}\left(\mu\cdot
m/\sigma^{2}\right)^{m^{2}/\sigma^{2}}}{\Gamma\left(m^{2}/\sigma^{2}+1\right)},$
(6)

where $\mu$ is the expected number of events in data with non-uniform weights.
Accordingly, Eq. 2 can be modified by replacing “$\mu$” and “$N$” with
$\overline{m_{\textrm{Eff}}}$ and $m_{\textrm{Eff}}$.

The above estimation can easily be extended to the case where several
estimations are added together (e.g., a situation with multiple backgrounds).
In practice, we first divide all distributions into two categories: one with
non-zero event count predictions and one with predictions of exactly zero
event count. These two distributions are then convolved. Finally, we correct
for the Bayesian prior. For implementation, we give prior distributions when
we convolve multiple ($n$) distributions together. If the prior information of
each distribution is flat, the overall prior of the summation would be
max($\mu$,0)n-1, or $\mu^{n-1}$ for $\mu\geq 0$ and 0 for $\mu<0$. In this
case, it is corrected (divided) after the convolution of multiple
distributions.

###  V.6 Additional Systematics for Dirt Events

The dirt events are neutrino interactions originating outside the cryostat.
The biggest uncertainty associated with the dirt events is the modeling of the
geometry and materials of the foam insulation covering the cryostat outer
surfaces, the nozzle penetrations for cryogenic and electronic services, the
supporting structures, and the dirt around the detector facility. In addition
to the systematic uncertainties associated with neutrino flux, cross section,
and detector, we conservatively assign an additional 50% bin-to-bin
uncorrelated uncertainty to dirt events.

MicroBooNE Simulation

Figure 19: Correlations of all systematic uncertainties for the seven
channels.

MicroBooNE Simulation

Figure 20: Summary of fractional contributions to the overall systematic
uncertainty ($\frac{\sigma_{i}^{2}}{\sigma^{2}_{\mathrm{total}}}\times 100$)
for the seven channels.

###  V.7 Summary of Systematic Uncertainties

Figure 19 shows correlations of all sources of systematic uncertainty across
the seven channels. Figure 20 summarizes the fractional contribution of each
source of uncertainty to the total systematic uncertainty, for each of the
seven channels. In general, without any constraint, the cross section
systematic uncertainty dominates in the region where neutrino interaction rate
is relatively high, followed by the flux systematic uncertainty. With
constraint, as discussed in later sections, cross section and flux
uncertainties will be significantly reduced while the data statistical
uncertainty becomes predominant in this analysis. For $\nu_{e}$ CC channels,
the relative contribution of MC statistical uncertainty, which includes
statistical uncertainty from both beam-off data and MC simulation in making
predictions, is significantly higher than that of the $\nu_{\mu}$ CC and
$\pi^{0}$ channels mainly because of the limited statistics of the samples in
estimating the backgrounds. The equivalent exposure in POT of the beam-off
data sample to estimate the beam-off (EXT) background is only 2.5 times of the
exposure of the BNB data sample, and the POT of the MC samples to estimate the
beam-on background is only 3.7 times of the exposure of the BNB data sample.
The situation will be improved with more MC production or data taking. For the
$\nu_{\mu}$ CC and $\pi^{0}$ channels, the detector systematic uncertainties
become the dominating uncertainty at high energies where the number of events
in the distribution become smaller (See. Fig. 21). This effect is expected
with the bootstrapping method and is the result of the limited MC statistics
of the detector systematic CV and 1$\sigma$ detector variation samples in
these regions.

##  VI Model Validations

This section summarizes the validations of the overall model, which includes
i) the prediction of $\nu_{e}$CC signal through the examination of data and
model consistency in $\nu_{\mu}$ CC events assuming lepton universality with
the difference in lepton masses included in the neutrino-argon interaction
cross section model, ii) constraints on the $\nu_{e}$ CC backgrounds through
the examination of data and model consistency in the various $\pi^{0}$
channels, and iii) modeling in the neutrino energy reconstruction.

###  VI.1 Methodology

The covariance matrix formalism is adopted to construct the $\chi^{2}$ test
statistic:

$\chi^{2}=\left(M-P\right)^{T}\times
Cov_{\text{full}}^{-1}\left(M,P\right)\times\left(M-P\right),$ (7)

where $M$ and $P$ are vectors of measurement and prediction, respectively. The
$Cov\left(M,P\right)$ is the full covariance matrix:

$Cov_{\text{full}}=Cov^{\text{stat}}_{\text{CNP}}+Cov^{\text{sys}}_{\text{MC
stat}}+Cov^{\text{sys}}_{\text{xs}}+Cov^{\text{sys}}_{\text{flux}}+Cov^{\text{sys}}_{\text{det}}+Cov^{\text{sys}}_{\text{add}}.$
(8)

The $Cov^{\text{stat}}_{\text{CNP}}$ is the diagonal covariance matrix
constructed based on the combined-Neyman-Pearson (CNP) method [54] with the
statistical uncertainty square being $3/\left(1/M_{i}+2/P_{i}\right)$ for the
$i$th bin. The $Cov^{\text{sys}}_{\text{MC stat}}$ is the diagonal covariance
matrix containing the statistical uncertainties corresponding to finite
statistics in making predictions as described in Sec. V.5. The other four
covariance matrices $Cov^{\text{sys}}_{\text{xs}}$,
$Cov^{\text{sys}}_{\text{flux}}$, $Cov^{\text{sys}}_{\text{det}}$,
$Cov^{\text{sys}}_{\text{add}}$, are the covariance matrices corresponding to
uncertainties from cross section (Sec. V.2), neutrino flux (Sec. V.1),
detector performance (Sec. V.4), and dirt (Sec. V.6), respectively. The
dependence on the LEE strength $x$ is considered in calculating the
covariance. It is based on dedicated systematic and statistical covariance
matrices for the eLEEx=1 component and correlations between the eLEEx=1
component and the other channels.

MicroBooNEFC

(a) FC $\nu_{\mu}$ CC

MicroBooNEPC

(b) PC $\nu_{\mu}$ CC

MicroBooNEFC

(c) FC CC$\pi^{0}$

MicroBooNEPC

(d) PC CC$\pi^{0}$

MicroBooNEFC+PC

(e) NC$\pi^{0}$

Figure 21: Event distributions of FC $\nu_{\mu}$ CC, PC $\nu_{\mu}$ CC, FC
CC$\pi^{0}$, PC CC$\pi^{0}$, and NC$\pi^{0}$ in (a)-(e), respectively. These
five selections serve as constraints to reduce the systematic uncertainties in
searching for eLEE, thus maximizing the physics sensitivity. The breakdown of
each component for different final states for both signal and background
events is shown in the legend. The bottom sub-panels present the data-to-
prediction ratios as well as the statistical and systematic uncertainties. The
pink band includes the statistical, cross section, and flux uncertainties. The
purple band corresponds to the full uncertainty with an addition of detector
systematic uncertainty.

The goodness-of-fit (GoF) test can be performed to test the compatibility
between the data and the overall model. Following the recommendation of Ref.
[104], we adopt the Pearson $\chi^{2}$ construction (instead of the CNP
construction) for the data statistical uncertainty:

$Cov_{\text{GoF}}=Cov^{\text{stat}}_{\text{Pearson}}+Cov^{\text{sys}}_{\text{MC
stat}}+Cov^{\text{sys}}_{\text{xs}}+Cov^{\text{sys}}_{\text{flux}}+Cov^{\text{sys}}_{\text{det}}+Cov^{\text{sys}}_{\text{add}},$
(9)

with the statistical uncertainty square being $P_{i}$ for the $i$th bin. Given
the null hypothesis (i.e. eLEEx=0), the $\chi^{2}$ value can be used to
perform the GoF test by comparing to the $\chi^{2}$ distribution with the
associated number of degrees of freedom ($ndf$), which is the total number of
bins of the measurement.

Channel |  $\chi^{2}/ndf$ w/o constr. |  $\chi^{2}/ndf$ w/ constr. | Notes
---|---|---|---
FC $\nu_{\mu}$CC | 6.64/25 | N/A | No constraint, see other checks
PC $\nu_{\mu}$CC | 5.84/25 | 6.94/25 | Constrained by FC $\nu_{\mu}$CC
FC CC$\pi^{0}$ | 6.17/10 | 7.39/10 | Constrained by both FC and PC $\nu_{\mu}$CC
PC CC$\pi^{0}$ | 5.51/10 | 6.80/10
NC$\pi^{0}$ | 2.81/10 | 5.33/10
PC $\nu_{e}$CC | 24.93/25 | 24.19/25 | See Sec. VII.1; constrained by the above five channels; eLEEx=0 hypothesis
FC $\nu_{e}$CC | 12.55/25 | 17.86/25 | See Sec. VII.1; constrained by the other six channels; eLEEx=0 hypothesis
Table 3: $\chi^{2}/ndf$ without and with constraints, for various selection
channels. The central values and uncertainties (covariance) of the MC
prediction are both changed with the constraint. Pearson construction of
$\chi^{2}$, instead of CNP construction of $\chi^{2}$, is used for these
goodness-of-fit tests.

MicroBooNE 6.369$\times$1020 POTFCPC

(a) $\nu_{\mu}$ CC primary muon energy.

MicroBooNE 6.369$\times$1020 POTFCPC

(b) $\nu_{\mu}$ CC primary muon polar angle.

MicroBooNE 6.369$\times$1020 POTFCPC

(c) $\nu_{\mu}$ CC hadronic energy ($E_{\rm had}=E_{\nu}$-$E_{\mu}$).

Figure 22: Event distributions of reconstructed muon energy (a), muon angle
(b), hadronic energy (c), for selected $\nu_{\mu}$CC events with fully
contained (FC) events on the left half and partially contained (PC) events on
the right half in each sub-figure. Black points are from data measurements.
Red (blue) histograms and error bands are for the MC prediction before (after)
constraint. The last bin is the overflow bin for FC or PC muon energy or
hadronic energy distributions.

The above GoF test using a Pearson $\chi^{2}$ construction provides an overall
evaluation of the model and the null hypothesis compared to the data. This
evaluation can be used to study different parts of the model using the
conditional covariance matrix formalism [56]. For example, given the full
covariance (stat + sys) containing two channels (X, Y):

$\Sigma=\begin{pmatrix}\Sigma^{XX}&\Sigma^{XY}\\\
\Sigma^{YX}&\Sigma^{YY}\end{pmatrix},~{}~{}~{}n:{\rm
measurement},~{}~{}~{}\mu:~{}{\rm prediction},$ (10)

we can derive the conditional mean and conditional covariance of the
prediction of $X$ given the constraints from the measurement of $Y$:

$\displaystyle\mu^{X,\text{const.}}$ $\displaystyle=$
$\displaystyle\mu^{X}+\Sigma^{XY}\cdot\left(\Sigma^{YY}\right)^{-1}\cdot\left(n^{Y}-\mu^{Y}\right)$
(11) $\displaystyle\Sigma^{XX,\text{const.}}$ $\displaystyle=$
$\displaystyle\Sigma^{XX}-\Sigma^{XY}\cdot\left(\Sigma^{YY}\right)^{-1}\cdot\Sigma^{YX}.$
(12)

Thus, a goodness-of-fit test can be performed on $Y$ first, and then performed
on $X$ after the constraints from $Y$. This allows the examination of the
model on $X$ and $Y$ individually, which provides more information about the
compatibility between the model and data.

The current cross-section model we use has conservative uncertainties in
general, and along with correlated systematics, the reduced $\chi^{2}$ values,
which is the ratio between $\chi^{2}$ and number of degrees of freedom, are
generally low suggesting that the model describes the data well within the
given uncertainty.

###  VI.2 Validation of Monte Carlo Models with Goodness-of-fit Tests

As previously introduced, we use a seven-channel fit strategy in searching for
a low-energy excess. The seven channels are FC $\nu_{e}$ CC, PC $\nu_{e}$ CC,
FC $\nu_{\mu}$ CC, PC $\nu_{\mu}$ CC, FC CC$\pi^{0}$, PC CC$\pi^{0}$, and
NC$\pi^{0}$. The selections for these seven channels are exclusive from each
other. In particular, the candidates in the CC$\pi^{0}$ excludes the selected
$\nu_{e}$ CC candidates, and the candidates in the $\nu_{\mu}$ CC channel
excludes $\nu_{e}$ CC and CC$\pi^{0}$ candidates. Figure 21 shows the five
non-$\nu_{e}$ CC channels in the eLEE search that serve as constraints to
reduce the systematic effects in the measurement of $\nu_{e}$ CC. The
$\chi^{2}/ndf$ value in each plot is calculated based on the data-MC
difference and the full covariance matrix taking into account the bin-to-bin
correlations from the off-diagonal terms. The error bands in the plot
correspond to only diagonal terms in the covariance matrix.

Figure 21 shows that the data and MC simulation agree within uncertainties,
and the $\chi^{2}/ndf$ values without any constraint suggest consistency
albeit with conservative uncertainties. To further validate the MC models,
i.e. data-MC consistency, GoF tests using the conditional mean prediction and
conditional covariance of the prediction, as described in Sec. VI.1, are
performed on these channels before applying them in the final eLEE search.
After constraints, the $\chi^{2}/ndf$ values based on both of the reduced
systematic uncertainty and adjusted central values of the MC prediction are
summarized in Table 3. The $\chi^{2}/ndf$ values after constraints are
generally increased to a certain degree mostly because of the reduced
systematic uncertainty. For the PC $\nu_{\mu}$ CC, the uncertainty is reduced
by about 75% with constraints from the FC $\nu_{\mu}$CC. For the three
$\pi^{0}$ channels, the uncertainties are reduced by about 30% with
constraints from the FC and PC $\nu_{\mu}$ CC channels. All the $\chi^{2}/ndf$
values after constraints are still less than 1 indicating consistency between
data and MC within the estimated systematic uncertainties of the MC
prediction.

###  VI.3 Validation of energy reconstruction

One may note that the data-to-prediction ratios demonstrate a slope for
$\nu_{\mu}$CC channels, which motivates further energy reconstruction
validation, particularly on inclusive hadronic final states, in addition to
the validations on $dE/dx$ and EM shower energy scale using $dE/dx$ profiles
from various reconstructed particles and $\pi^{0}$ invariant mass.

As discussed in Appendix, this “slope” can be decomposed using different final
state topologies, such as via 0$p$X$\pi$/N$p$X$\pi$ (N$\geq$1 and X$\geq$0)
separation, where $p$ represents reconstructed protons with kinetic energy
greater than 35 MeV, and there is no requirement regarding whether charged
pions are reconstructed. The enhancement in the data measurement is found to
be concentrated in the 0$p$X$\pi$ channels which essentially correspond to the
events with low hadronic energy while N$p$X$\pi$ channels have an overall good
agreement between data and MC. To quantitatively assess this issue, a
conditional constraint study was performed with the high-statistics
$\nu_{\mu}$ CC events in this analysis following the methodology described in
Sec. VI.1. The more precise measurements of leptonic kinematics, such as muon
energy ($E_{\mu}$) and muon polar angle relative to the BNB direction
(cos$\theta_{\mu}$), are utilized to constrain/adjust the MC prediction on the
hadronic kinematics, such as the reconstructed hadronic energy (reconstructed
$E_{\nu}$-$E_{\mu}$), which is expected to contain significant missing energy
from invisible neutral particles.

The measurements of muon kinematics and hadronic energy constraint results are
shown in Fig. 22. The measurements of $E_{\mu}$ and cos$\theta_{\mu}$ show
good data-MC consistency for the MC predictions with or without certain
constraints. The measurement of hadronic energy shows an excess for low
hadronic energy events. With constraints from the measurements of $E_{\mu}$
and cos$\theta_{\mu}$, the measurement of hadronic energy, as in Fig. 22(c),
shows a good agreement ($\chi^{2}/ndf$=8.66/32) with the constrained MC
prediction even though the common systematic effect between leptonic and
hadronic systems (e.g. neutrino flux uncertainties) is significantly reduced
resulting in a much smaller residual systematic uncertainty (blue error band).
This is an important evidence that the origin of this issue is essentially
from the common systematics, most likely cross-section modeling, shared by
both leptonic and hadronic final states and can be adequately
described/allowed by the current models and their variations in MicroBooNE MC
simulation. More discussion can be found in Ref. [105]. This constraint study
is sensitive to the potential missing energy in the hadronic system. For
example, 5-10% (relative to the true energy transfer) additional invisible
energy would result in a noticeable increase of the $\chi^{2}/ndf$ values in
this test. See the supplementary material of Ref. [105] for more details. This
stringent test validates the MC model as well as our neutrino energy
reconstruction treatment (leptonic and hadronic energy) and gives us
confidence using the overall MC model in the seven-channel fit.

MicroBooNEFC, unconstrained

(a) $\nu_{e}$ CC shower energy, fully contained events.

MicroBooNEPC, unconstrained

(b) $\nu_{e}$ CC shower energy, partially contained events.

MicroBooNEFC, unconstrained

(c) $\nu_{e}$ CC shower cos$\theta$, fully contained events.

MicroBooNEPC, unconstrained

(d) $\nu_{e}$ CC shower cos$\theta$, partially contained events.

MicroBooNEFC, unconstrained

(e) $\nu_{e}$ CC hadronic energy, fully contained events.

MicroBooNEPC, unconstrained

(f) $\nu_{e}$ CC hadronic energy, partially contained events.

Figure 23: Event distributions of FC $\nu_{e}$ CC [(a), (c), and (e)] and PC
$\nu_{e}$ CC [(b), (d), and (f)] candidates as a function reconstructed shower
energy [(a) and (b)], shower cos $\theta$ [(c) and (d)], and hadronic energy
[(e) and (f)]. No constraint is used. Data-to-prediction ratios, $\chi^{2}$,
and error bands are calculated based on the eLEEx=0 hypothesis. The MC
expectation of the eLEEx=1 component is added on top of the energy spectrum as
represented by the dashed red curve. The breakdown of each component for
different final states for both signal and background events is shown in the
legend. The bottom sub-panels present the data-to-prediction ratios as well as
the statistical and systematic uncertainties. The pink band includes the
statistical, cross section, and flux uncertainties. The purple band
corresponds to the full uncertainty with an addition of detector systematic
uncertainty.

##  VII Results

In this section, eLEE sensitivity, nue CC selected data and MC comparisons
(Sec. VII.1), and various statistical analyses (Sec. VII.2 and Sec. VII.3)
including the best-fit of the eLEE strength and Feldman-Cousins confidence
intervals are reported. The eLEE model used here is generated by unfolding the
MiniBooNE’s observed excess to true neutrino energy under a CCQE hypothesis
and scaling the flux of intrinsic $\nu_{e}$ with the excess-to-intrinsic
$\nu_{e}$ ratios.

As explained in Sec. II, a blind analysis strategy was implemented for the
MicroBooNE LEE analysis which sequesters $\nu_{e}$CC candidates in BNB data
below a reconstructed neutrino energy of 600 MeV until the analysis had been
finalized. This procedure ensured that there was no bias in the reconstruction
or event selection while allowing for cross-checks of data selection and
simulation performance using sideband events with similar kinematic and/or
topological characteristics to those of $\nu_{e}$ CC signal.

###  VII.1 $\nu_{e}$ CC Selection Results

$\nu_{e}$CC events are identified as described in Sec. IV.1 starting from a
generic neutrino selection that is followed by various dedicated selection
criteria based on Wire-Cell event reconstruction and BDTs. The $\nu_{e}$ CC
events are selected with a high purity as shown in Fig. 8. To further maximize
the eLEE sensitivity, we divide the $\nu_{e}$CC candidates into two sub-
samples: a fully contained (FC) and a partially contained (PC) sample, both of
which follow the aforementioned seven-channel fit strategy. The FC and PC
$\nu_{e}$CC selection results are shown in Fig. 23 and Fig. 24. The green
histogram represents the MC predicted $\nu_{e}$ CC signal expectation obtained
from the beam simulation and the Genie interaction models tuned to MicroBooNE
(eLEEx=0 hypothesis). The figures also contain the data-to-prediction ratios,
$\chi^{2}$ per number of degrees-of-freedom, and error bands calculated based
on the simulation of the expected $\nu_{e}$ CC signal and background. The MC
model of the $\nu_{e}$ CC signal with the eLEEx=1 component added to the beam
expectation is indicated by the red dashed histogram. The $\chi^{2}/ndf$
values (assuming eLEEx=0 hypothesis) for FC and PC $\nu_{e}$ CC channels
without and with the constraints from the $\nu_{\mu}$ CC and $\pi^{0}$
channels are summarized in Table 3.

MicroBooNEFC, unconstrained

(a) $\nu_{e}$ CC fully contained events.

MicroBooNEPC, unconstrained

(b) $\nu_{e}$ CC partially contained events.

Figure 24: Event distributions of (a) FC $\nu_{e}$ CC and (b) PC $\nu_{e}$ CC
candidates as a function reconstructed $E_{\nu}$. No constraint is used. Data-
to-prediction ratios, $\chi^{2}$, and error bands are calculated based on the
eLEEx=0 hypothesis. The MC expectation of the eLEEx=1 component is added on
top of the energy spectrum as represented by the dashed red curve. The
breakdown of each component for different final states for both signal and
background events is shown in the legend. The bottom sub-panels present the
data-to-prediction ratios as well as the statistical and systematic
uncertainties. The pink band includes the statistical, cross section, and flux
uncertainties. The purple band corresponds to the full uncertainty with an
addition of detector systematic uncertainty.

The neutrino energy spectrum of the data compared to the constrained MC
prediction for the FC $\nu_{e}$ CC channel, of which the low energy region is
most sensitive to the eLEE search is shown in Fig. 25. The applied constraints
include the PC $\nu_{e}$ CC channel999The LEE component is included in
predicting the PC $\nu_{e}$ CC distribution for the corresponding hypothesis.
and various $\nu_{\mu}$ and $\pi^{0}$ channels. The constrained MC prediction
with the eLEE component added is indicated by the red dashed histogram in the
figure. The number of predicted events for each category of background, beam
intrinsic $\nu_{e}$ CC, and eLEE $\nu_{e}$ CC in the energy region of
reconstructed $E_{\nu}<$ 600 MeV are summarized in Table 4. Definitions of
each category can be found in Sec. IV.2. The main background comes from NC
$\pi^{0}$ events in which one of the $\pi^{0}$ decay $\gamma$’s (the other one
may exit the active volume) is mis-identified as a primary electron EM shower.
The $\nu_{e}$ CC selection is also contaminated by CC or NC interactions with
no $\pi^{0}$ in the final state in which case the typical failure mode is that
the decay electrons from muons or charged pions are mis-identified as primary
electron EM showers alongside bad reconstruction of the neutrino vertex. Note
that the “Cosmic” category corresponds to the cosmic-ray background estimated
from the overlay MC. This background is essentially a mismatch between charge
and light signals which accounts for the mis-selected cosmic-ray background
from neutrino activity in addition to the cosmic-ray background estimated from
beam-off data with no neutrino activity (“EXT” category). Since this “Cosmic”
background is associated with neutrino interactions in the overlay MC, the
predicted number will change accordingly in the constraint.

In Table. 4, expected number of events for each type of background, beam
intrinsic $\nu_{e}$ CC, and eLEEx=1 $\nu_{e}$ CC in the energy region of
reconstructed $E_{\nu}<$600 MeV for FC $\nu_{e}$ CC are shown. The
uncertainties for each category of the predicted background or $\nu_{e}$ CC
signal are systematic uncertainties originating from the flux, cross section
(including both $\nu$-Argon and hadron-Argon interactions), detector, and the
limited statistics of the samples used in making predictions (e.g. MC
statistical uncertainty). “Dirt” background has an additional 50% systematic
uncertainty. The statistical uncertainties are presented only for beam-on and
beam-off data results. With the constraints, the systematic uncertainty is
reduced significantly for the $\nu_{e}$ CC signal. However, the uncertainties
of the predictions of various backgrounds change only slightly because the
uncertainty of each background is dominated by the MC statistical uncertainty.
With the constraints, the contributions from flux and cross section systematic
uncertainties are largely reduced and the central values of the predictions
are adjusted as well. The constraining power is mostly from $\nu_{\mu}$ CC
channels, while $\pi^{0}$ channels help constrain the residual $\pi^{0}$
background in $\nu_{e}$ CC candidate events. The PC $\nu_{e}$ CC channel has
an insignificant effect in the constraint or in the eLEE search, and it serves
as a good validation check on the data and simulation. After such constraints,
the uncertainty of the result is dominated by statistical uncertainty followed
by the systematic uncertainties from the MC statistics, detector response,
cross section, and neutrino flux.

MicroBooNEFC, constrained