The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# Measurement of double-differential cross sections for mesonless charged-
current muon neutrino interactions on argon with final-state protons using the
MicroBooNE detector

P. Abratenko  Tufts University, Medford, MA, 02155, USA  O. Alterkait  Tufts
University, Medford, MA, 02155, USA  D. Andrade Aldana  Illinois Institute of
Technology (IIT), Chicago, IL 60616, USA  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  G. Barr  University of Oxford,
Oxford OX1 3RH, United Kingdom  D. Barrow  University of Oxford, Oxford OX1
3RH, United Kingdom  J. Barrow  University of Minnesota, Minneapolis, MN,
55455, USA  V. Basque  Fermi National Accelerator Laboratory (FNAL), Batavia,
IL 60510, USA  O. Benevides Rodrigues  Illinois Institute of Technology (IIT),
Chicago, IL 60616, USA  S. Berkman  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  Michigan State University, East Lansing, MI
48824, USA  A. Bhanderi  The University of Manchester, Manchester M13 9PL,
United Kingdom  A. Bhat  University of Chicago, Chicago, IL, 60637, USA  M.
Bhattacharya  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  M. Bishai  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA
A. Blake  Lancaster University, Lancaster LA1 4YW, United Kingdom  B. Bogart
University of Michigan, Ann Arbor, MI, 48109, USA  T. Bolton  Kansas State
University (KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University,
Cambridge, MA 02138, USA  M. B. Brunetti  University of Warwick, Coventry CV4
7AL, United Kingdom  L. Camilleri  Columbia University, New York, NY, 10027,
USA  Y. Cao  The University of Manchester, Manchester M13 9PL, United Kingdom
D. Caratelli  University of California, Santa Barbara, CA, 93106, USA  F.
Cavanna  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
G. Cerati  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  A. Chappell  University of Warwick, Coventry CV4 7AL, United Kingdom  Y.
Chen  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, 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  University of Pittsburgh, Pittsburgh, PA, 15260, USA  J.
I. Crespo-Anadón  Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Madrid E-28040, Spain  R. Cross  University of Warwick,
Coventry CV4 7AL, United Kingdom  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  Universität Bern, Bern CH-3012,
Switzerland  Z. Djurcic  Argonne National Laboratory (ANL), Lemont, IL, 60439,
USA  R. Dorrill  Illinois Institute of Technology (IIT), Chicago, IL 60616,
USA  K. Duffy  University of Oxford, Oxford OX1 3RH, United Kingdom  S. Dytman
University of Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly  University of
Southern Maine, Portland, ME, 04104, USA  P. Englezos  Rutgers University,
Piscataway, NJ, 08854, USA  A. Ereditato  University of Chicago, Chicago, IL,
60637, USA  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  J. J. Evans  The University of Manchester, Manchester M13 9PL, United
Kingdom  R. Fine  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  W. Foreman  Illinois Institute of Technology (IIT), Chicago, IL
60616, USA  B. T. Fleming  University of Chicago, Chicago, IL, 60637, USA  D.
Franco  University of Chicago, Chicago, IL, 60637, USA  A. P. Furmanski
University of Minnesota, Minneapolis, MN, 55455, USA  F. Gao  University of
California, Santa Barbara, CA, 93106, 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  Los Alamos National Laboratory (LANL),
Los Alamos, NM, 87545, USA  E. Gramellini  The University of Manchester,
Manchester M13 9PL, United Kingdom  P. Green  University of Oxford, Oxford OX1
3RH, United Kingdom  H. Greenlee  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  L. Gu  Lancaster University, Lancaster LA1
4YW, United Kingdom  W. Gu  Brookhaven National Laboratory (BNL), Upton, NY,
11973, USA  R. Guenette  The University of Manchester, Manchester M13 9PL,
United Kingdom  P. Guzowski  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Hagaman  University of Chicago, Chicago, IL, 60637, 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  Z.
Imani  Tufts University, Medford, MA, 02155, USA  B. Irwin  University of
Minnesota, Minneapolis, MN, 55455, USA  M. S. Ismail  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  C. James  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  X. Ji  Nankai University, Nankai
District, Tianjin 300071, China  J. H. Jo  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, 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  G.
Karagiorgi  Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  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  N. Lane  The University of
Manchester, Manchester M13 9PL, United Kingdom  I. Lepetic  Rutgers
University, Piscataway, NJ, 08854, USA  J.-Y. Li  University of Edinburgh,
Edinburgh EH9 3FD, United Kingdom  Y. Li  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  K. Lin  Rutgers University, Piscataway, NJ,
08854, USA  B. R. Littlejohn  Illinois Institute of Technology (IIT), Chicago,
IL 60616, USA  H. Liu  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  W. C. Louis  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  X. Luo  University of California, Santa Barbara, CA, 93106, 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
N. Martinez  Kansas State University (KSU), Manhattan, KS, 66506, USA  D. A.
Martinez Caicedo  South Dakota School of Mines and Technology (SDSMT), Rapid
City, SD, 57701, USA  S. Martynenko  Brookhaven National Laboratory (BNL),
Upton, NY, 11973, USA  A. Mastbaum  Rutgers University, Piscataway, NJ, 08854,
USA  I. Mawby  Lancaster University, Lancaster LA1 4YW, United Kingdom  N.
McConkey  University College London, London WC1E 6BT, United Kingdom  V.
Meddage  Kansas State University (KSU), Manhattan, KS, 66506, USA  J. Mendez
Louisiana State University, Baton Rouge, LA, 70803, USA  J. Micallef
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  Tufts
University, Medford, MA, 02155, USA  K. Miller  University of Chicago,
Chicago, IL, 60637, USA  A. Mogan  Colorado State University, Fort Collins,
CO, 80523, USA  T. Mohayai  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  Indiana University, Bloomington, IN 47405, 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  M. M.
Moudgalya  The University of Manchester, Manchester M13 9PL, United Kingdom
S. Mulleriababu  Universität Bern, Bern CH-3012, Switzerland  D. Naples
University of Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The
University of Manchester, Manchester M13 9PL, United Kingdom  N. Nayak
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  M. Nebot-Guinot
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  J. Nowak
Lancaster University, Lancaster LA1 4YW, United Kingdom  N. Oza  Columbia
University, New York, NY, 10027, USA  O. Palamara  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  N. Pallat  University of Minnesota,
Minneapolis, MN, 55455, USA  V. Paolone  University of Pittsburgh, Pittsburgh,
PA, 15260, USA  A. Papadopoulou  Argonne National Laboratory (ANL), Lemont,
IL, 60439, USA  V. Papavassiliou  New Mexico State University (NMSU), Las
Cruces, NM, 88003, USA  H. B. Parkinson  University of Edinburgh, Edinburgh
EH9 3FD, United Kingdom  S. F. Pate  New Mexico State University (NMSU), Las
Cruces, NM, 88003, USA  N. Patel  Lancaster University, Lancaster LA1 4YW,
United Kingdom  Z. Pavlovic  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv University, Tel Aviv, Israel,
69978  K. Pletcher  Michigan State University, East Lansing, MI 48824, USA  I.
Pophale  Lancaster University, Lancaster LA1 4YW, United Kingdom  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
Argonne National Laboratory (ANL), Lemont, IL, 60439, USA  M. Reggiani-Guzzo
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  The University of
Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New Mexico State
University (NMSU), Las Cruces, NM, 88003, 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  Tufts University, Medford, MA, 02155, USA  M. Ross-Lonergan  Los
Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  I. Safa
Columbia University, New York, NY, 10027, 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  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. Shi  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  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  A. M. Szelc
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  W. Tang
University of Tennessee, Knoxville, TN, 37996, USA  N. Taniuchi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  K. Terao  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  C. Thorpe  The University
of Manchester, Manchester M13 9PL, United Kingdom  D. Torbunov  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  D. Totani  University of
California, Santa Barbara, CA, 93106, USA  M. Toups  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  A. Trettin  The
University of Manchester, Manchester M13 9PL, United Kingdom  Y.-T. Tsai  SLAC
National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Tyler  Kansas
State University (KSU), Manhattan, KS, 66506, USA  M. A. Uchida  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  B. Viren  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern,
Bern CH-3012, Switzerland  H. Wei  Louisiana State University, Baton Rouge,
LA, 70803, USA  A. J. White  University of Chicago, Chicago, IL, 60637, 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  W. Wu
University of Pittsburgh, Pittsburgh, PA, 15260, USA  E. Yandel  University of
California, Santa Barbara, CA, 93106, USA  T. Yang  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. E. Yates  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, 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

Charged-current neutrino interactions with final states containing zero mesons
and at least one proton are of high interest for current and future
accelerator-based neutrino oscillation experiments. Using the Booster Neutrino
Beam and the MicroBooNE detector at Fermi National Accelerator Laboratory, we
have obtained the first double-differential cross section measurements of this
channel for muon neutrino scattering on an argon target with a proton momentum
threshold of $0.25\text{\ }\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$. We also
report a flux-averaged total cross section of
$\sigma=($11.8278$\pm$1.1627$)\times${10}^{-38}\text{\
}{\mathrm{cm}}^{2}\text{/}$\,/\,\mathrm{Ar}$ and several single-differential
measurements which extend and improve upon previous results. Statistical and
systematic uncertainties are quantified with a full treatment of correlations
across 359 kinematic bins, including correlations between distributions
describing different observables. The resulting data set provides the most
detailed information obtained to date for testing models of mesonless
neutrino-argon scattering.

##  I Introduction

Accelerator-based measurements of neutrino oscillations will be critical for
definitively answering key open questions in high-energy physics, including
whether charge-parity ($CP$) violation [1] occurs in the lepton sector,
whether sterile neutrino species [2] (which do not participate in the weak
interaction) exist, and whether $\nu_{3}$ is the heaviest or lightest of the
known neutrino mass eigenstates [3]. In the coming years, the Fermilab-based
Short Baseline Neutrino (SBN) program [4] and Deep Underground Neutrino
Experiment (DUNE) [5] will both pursue high-precision neutrino oscillation
analyses using liquid argon time projection chambers (LArTPCs) as the primary
detector technology. Maximizing the discovery potential of these large
experimental efforts will require substantial improvements to the present
understanding of $\mathrm{GeV}\text{/}$-scale neutrino-nucleus scattering
physics [6]. An emerging literature of neutrino-argon cross-section
measurements [7], pioneered by the ArgoNeuT [8] and MicroBooNE [9] LArTPC
experiments, will provide the most direct constraints for refining interaction
models to the precision needed for SBN and DUNE.

This article contributes to that effort by presenting measurements of
differential cross sections for charged-current (CC) $\nu_{\mu}$ scattering on
argon leading to final states containing no mesons and one or more protons.
This event topology (hereafter abbreviated as CC$0\pi Np$) is the dominant
neutrino interaction channel for the SBN program, and its contribution is
expected to remain important for DUNE oscillation analyses. The results
presented herein build upon a previous MicroBooNE investigation [10] which
measured single-differential neutrino-argon cross sections in the CC$0\pi Np$
channel for the first time. The present work leverages subsequent improvements
to the MicroBooNE simulation [11], reconstruction [12], and systematic
uncertainty quantification [13] to achieve more detailed measurements,
including multiple double-differential distributions. The analysis also
benefits from a significantly larger data set, which was recorded by the
MicroBooNE detector [9] with a total exposure of
$6.7949\text{\times}{10}^{20}$ protons on target ($\mathrm{POT}\text{/}$) from
the Fermilab Booster Neutrino Beam (BNB) [14].

Various kinematic distributions involving the three-momenta of the final-state
muon and leading proton are measured and compared to simulations in this
article. An innovative feature of the data set is the strategy used for its
presentation; covariances between all 359 kinematic bins studied in the
analysis are reported and used to quantify an overall goodness-of-fit of
theoretical predictions. The full group of cross-section measurements may thus
be regarded as a single result with greater combined model discrimination
power (due to the need to describe inter-distribution correlations) than its
constituent parts. Together with the inclusive cross-section measurements
recently reported by MicroBooNE in Refs. [15, 16], this analysis thus
represents a first application of the “blockwise unfolding” technique proposed
in Ref. [17].

The experimental setup and simulations used in the analysis are described in
Sec. II. Section III presents the signal definition and the event selection
criteria. Section IV discusses the strategy for reporting the results,
including the observables measured and the general approach to corrections for
imperfect event reconstruction. The procedures used to estimate uncertainties
and convert measured event distributions to cross sections are documented in
Secs. V and VI, respectively. Section VII reports the cross-section results
and compares them to several model predictions. Finally, Sec. VIII contains a
summary and conclusions.

##  II MicroBooNE experiment and simulation

The MicroBooNE detector [9] is a LArTPC that operated in the BNB from
2015–2021. The detector’s 85-tonne active mass of liquid argon was exposed to
a $273\text{\ }\mathrm{V}\text{/}\mathrm{cm}$ electric field and instrumented
with three wire planes and thirty-two photomultiplier tubes (PMTs). Acrylic
disks coated with tetraphenyl butadiene (TPB) were placed in front of the PMTs
to convert the 128-$\mathrm{nm}\text{/}$ argon scintillation light into
visible wavelengths for efficient detection.

Neutrinos from the BNB are generated by bombarding a beryllium target with
8-$\mathrm{GeV}\text{/}$ protons. Secondary particles produced by the proton
collisions are focused by a magnetic horn and decay in flight. The decay
products are directed toward a beam stop made of steel and concrete that
absorbs particles other than neutrinos. During operation, the MicroBooNE
detector was located $463\text{\ }\mathrm{m}\text{/}$ downstream from the
target along the beam axis. The BNB neutrino flux at this position is
dominated by muon neutrinos with a mean energy of $0.8\text{\
}\mathrm{GeV}\text{/}$. Minor contributions from $\bar{\nu}_{\mu}$ (5.8%) and
a mixture of $\nu_{e}$ and $\bar{\nu}_{e}$ (0.5% combined) are also present.

Interpretation of the data obtained from the MicroBooNE detector is enabled by
a suite of Monte Carlo simulations that provide comprehensive modeling of the
entire experiment. Neutrino beam production is simulated using the Geant4
framework [18, 19] and a detailed representation of the BNB apparatus
developed by the MiniBooNE collaboration [14].

###  II.1 Neutrino interaction model

Neutrino interactions are modeled in the MicroBooNE simulation chain using
version 3.0.6 of the GENIE neutrino event generator [20, 21, 22] with the
G18_10a_02_11a configuration. This configuration includes a local Fermi gas
(LFG) representation of the nuclear ground state [23] and the Valencia model
for quasielastic (QE) and two-particle two-hole (2p2h) interactions in the CC
channel [24, 25, 26, 27, 28]. Resonance production (RES) is simulated with the
model of Kuzmin-Lyubushkin-Naumov and Berger-Sehgal (KLN-BS) [29, 30, 31, 32],
deep inelastic scattering [33] (DIS) is described using the structure
functions of Bodek and Yang [34, 35], and coherent pion production (COH)
follows the Berger-Sehgal approach [36]. Intranuclear hadronic final-state
interactions (FSI) are simulated using the hA2018 model [37]. Several model
parameters relevant for calculations of neutrino-nucleon cross-sections were
tuned by the GENIE collaboration to data sets from bubble chamber experiments
[38].

The GENIE v3.0.6 G18_10a_02_11a configuration described above was modified
with MicroBooNE-specific tuning [11] of two CCQE and two CC2p2h model
parameters to data from the T2K experiment reported in Ref. [39]. The
resulting interaction model is referred to as the MicroBooNE Tune in the
remainder of this article.

###  II.2 Detector simulation

Final-state particles produced in GENIE events are transported through the
MicroBooNE detector geometry using version 10.3.3 of Geant4 [18, 19]. The
response of the detector electronics to these particles is modeled using a
custom simulation implemented within the LArSoft framework [40]. This detector
response model includes data-driven treatments of position dependence in the
wire responses, distortion of the time projection chamber (TPC) electric field
due to buildup of slow-moving positive ions (space-charge effects), and
dynamically-induced charge [41, 42, 43, 44, 45].

Backgrounds generated by cosmic rays and other sources unrelated to the
neutrino beam are directly measured in the analysis using beam-off data
samples collected at times when the BNB was not active. To ensure that these
backgrounds are properly treated when reconstructing simulated neutrino
interactions, an overlay technique is used; simulated TPC wire and PMT
waveforms induced by GENIE neutrino scattering events are superimposed on
measured waveforms from beam-off data before analysis [46].

##  III Signal definition and event selection

A neutrino scattering event is considered part of the signal for this analysis
if it fulfills the following criteria:

  1. (1)

A muon neutrino undergoes a CC interaction with an argon nucleus.

  2. (2)

The final state contains at least one proton.

  3. (3)

The momentum of the outgoing muon lies within the interval [0.1, 1.2]
$\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$.

  4. (4)

The momentum of the leading final-state proton (i.e., the proton with the
highest momentum) lies within the interval [0.25, 1.0]
$\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$.

  5. (5)

The final state contains zero mesons and zero antimesons.

The momentum limits on the muon and leading proton in signal requirements (3)
and (4) are motivated by considerations of efficiency, resolution, and
systematic uncertainties. Apart from adjustments to these momentum limits, the
signal definition given above is the same as the one adopted in the previous
MicroBooNE CC$0\pi Np$ analysis [10].

###  III.1 Reconstruction workflow

Identification of candidate CC$0\pi Np$ signal events in this article relies
upon an automated event reconstruction workflow implemented within the Pandora
multi-algorithm pattern-recognition toolkit [47]. The individual hits
reconstructed from the pulses recorded by each of the TPC wire planes are
clustered in 2D and then correlated to produce 3D particle candidates.
Collections of nearby particle candidates are grouped into slices, and
information from both the TPC and PMTs is used to identify a maximum of one
slice per event that contains a candidate neutrino interaction. Algorithms to
calculate additional reconstructed quantities, such as the particle
identification score described in Ref. [12] and the muon momentum estimator
described in Ref. [48], are combined with this generic neutrino selection to
perform the analysis.

###  III.2 Event categories

Plots of reconstructed event distributions in Fig. 1 and Figs. 7–14 show the
prediction of the MicroBooNE simulation as a stacked histogram that
distinguishes between the following categories of events:

Signal

All events which satisfy signal requirements (1) to (5) and have a true
neutrino vertex position that falls within the fiducial volume defined in
selection requirement (I) from Sec. III.3. Signal events are divided into
subcategories based on whether the primary interaction mode is QE, 2p2h, or
any other process. The last of these subcategories is dominated by resonant
pion production followed by intranuclear pion absorption.

Out FV

Events in which the true neutrino vertex falls outside of the fiducial volume
(FV). This category includes all final-state topologies since it is defined
solely in terms of the true vertex location.

CC$N\pi$

Events containing a charged-current $\nu_{\mu}$ interaction which produces one
or more final-state pions of any charge. These fail to satisfy signal
requirement (5). Note that $\nu_{\mu}$ CC events containing final-state mesons
other than pions fall into another event category.

CC0$\pi$0$p$

Charged-current $\nu_{\mu}$ events containing zero final-state pions of any
momentum and zero protons within the true momentum limits imposed by signal
requirement (4). Events with any number of mesons other than pions are allowed
in this category.

Other CC

Charged-current $\nu_{\mu}$ events which do not fall into any of the previous
categories. Among these are events which would otherwise be signal but do not
satisfy the muon momentum limits imposed by signal requirement (3). This
category also includes events containing at least one proton within the
momentum limits from signal requirement (4) together with one or more mesons
other than pions.

$\nu_{e}$ CC

Events involving a charged-current $\nu_{e}$ interaction inside the fiducial
volume. These fail to satisfy signal requirement (1).

NC

Events in which a neutrino or antineutrino of any flavor undergoes a neutral-
current interaction within the fiducial volume. These fail to satisfy signal
requirement (1).

Beam-off

Cosmic-ray-induced backgrounds.

Other

All events which do not fall into one of the other categories defined above.
This category is dominated by CC interactions of antineutrinos.

###  III.3 Inclusive $\pmb{\nu_{\mu}}$ CC preselection

Following automated reconstruction using the Pandora framework, twelve event-
selection criteria are applied to distinguish the CC$0\pi Np$ signal from
other event topologies. These criteria are divided into a preselection that
seeks to identify inclusive $\nu_{\mu}$ CC events (described in the following
paragraphs), quality checks to eliminate events with a poorly-reconstructed
final-state muon (Sec. III.4), and additional requirements designed to isolate
mesonless final states containing one or more protons (Sec. III.5). Overall
performance of the selection is then discussed in Sec. III.6, including
evolution of the efficiency and purity as each requirement is applied.

In order of application, the selection criteria used to identify $\nu_{\mu}$
CC candidate events are as follows:

  1. (I)

The position of the reconstructed neutrino vertex must lie within a fiducial
volume representing a region in which the efficiency of the remainder of the
selection is appreciable and the detector response is well-understood. The
fiducial volume chosen for this analysis is a rectangular prism with most
boundaries parallel to and $21.5\text{\ }\mathrm{cm}\text{/}$ inward from the
edges of the active volume of the MicroBooNE detector. The only exception is
the boundary that is perpendicular to the beam on the downstream side. This
boundary is chosen to be $70\text{\ }\mathrm{cm}\text{/}$ inward from the edge
of the active volume to enable better acceptance of forward-going muons.

  2. (II)

Starting points for all reconstructed primary particles (i.e., those labeled
by the reconstruction as direct progeny of the neutrino) are required to lie
within a looser containment volume offset inward from the active volume edges
by $10\text{\ }\mathrm{cm}\text{/}$ on all sides.

  3. (III)

The topological score [49] assigned to the event must be greater than 0.1.
This score represents the output of a support-vector machine designed to
classify events as either neutrino-like (scores near 1) or cosmic-ray-like
(scores near 0). Figure 1 shows the measured distribution of this variable,
obtained at this stage of the selection, compared to the prediction from the
MicroBooNE simulation chain. The topological score threshold of 0.1 is
sufficient to remove a large fraction of beam-off and out-of-fiducial-volume
backgrounds.

Figure 1: Distribution of topological score values obtained when applying the
$\nu_{\mu}$ CC inclusive preselection. The portion near zero is dominated by
beam-off events as expected. The green arrow points into the region in which
events are accepted by selection requirement (III). The dashed lines indicate
the total uncertainty on the simulation prediction evaluated according to the
prescription given in Sec. V. The lower panel shows the ratio of the data to
the simulation prediction. The same total uncertainty band is represented by
the shaded gray region.

  4. (IV)

The event must contain a muon candidate, which is defined as a reconstructed
primary particle that satisfies the following criteria:

    1. a.

The track score [49, 50] assigned to the particle by the reconstruction must
be greater than 0.8. This score classifies reconstructed particles as shower-
like (values near 0) and track-like (values near 1). This requirement slightly
increases the muon purity (see Fig. 2). The “cosmic” (“beam-off”) category in
the legend represents reconstructed particle candidates attributed to cosmic-
ray activity in events that include (do not include) a simulated neutrino
interaction. All other categories represent reconstructed particle candidates
that correspond to a specific kind of true particle produced by a simulated
neutrino interaction.

Figure 2: Track score criterion applied to all reconstructed primary particles
when searching for a muon candidate. The green arrow points into the region in
which particle candidates are accepted. The percentages in the legend indicate
the fraction of reconstructed particles of each type before the illustrated
selection is applied. The lower panel shows the ratio of the data to the
simulation prediction. The uncertainty on the simulation prediction is
represented by the dashed lines in the top panel and the gray region in the
bottom panel.

    2. b.

The Euclidean distance between the particle’s reconstructed starting position
and the reconstructed neutrino vertex must be less than $4\text{\
}\mathrm{cm}\text{/}$ (see Fig 3). This is a minor quality check to ensure
that the muon candidate is correctly associated with the reconstructed
neutrino vertex.

Figure 3: Distribution of distances between the reconstructed neutrino vertex
and reconstructed primary particles. The green arrow points into the region in
which particles are accepted as possible muon candidates. The percentages in
the legend indicate the fraction of reconstructed particles of each type
before the illustrated selection is applied. The lower panel shows the ratio
of the data to the simulation prediction. The uncertainty on the simulation
prediction is represented by the dashed lines in the top panel and the gray
region in the bottom panel.

    3. c.

The length of the particle track must be greater than $10\text{\
}\mathrm{cm}\text{/}$ (see Fig 4). Tracks below this length are overwhelmingly
generated by protons and cosmic activity. The lower limit on the muon momentum
applied in signal requirement (3) helps to mitigate the impact of this cut on
the efficiency for very low-energy muons.

Figure 4: Track length criterion applied to reconstructed primary particles
when searching for a muon candidate. The green arrow points into the region in
which particle candidates are accepted. The percentages in the legend indicate
the fraction of reconstructed particles of each type before the illustrated
selection is applied. The lower panel shows the ratio of the data to the
simulation prediction. The uncertainty on the simulation prediction is
represented by the dashed lines in the top panel and the gray region in the
bottom panel.

    4. d.

The log-likelihood ratio particle-identification (LLR PID) score assigned to
the particle must exceed 0.2 (see Fig. 5). This score is calculated by
comparing track hit information from all three TPC wire planes to theoretical
templates for muons and protons. The logarithm of a likelihood ratio for these
two particle identification hypotheses is then converted to a score where -1
is most proton-like and 1 is most muon-like. Reference [12] describes this
particle identification technique in greater detail.

Figure 5: Log-likelihood ratio particle ID score for reconstructed primary
particles under consideration as possible muon candidates. The green arrow
points into the region in which muon candidates are accepted by the selection.
The percentages in the legend indicate the fraction of reconstructed particles
of each type before the illustrated selection is applied. The lower panel
shows the ratio of the data to the simulation prediction. The uncertainty on
the simulation prediction is represented by the dashed lines in the top panel
and the gray region in the bottom panel.

In cases where two or more reconstructed primary particles satisfy these
criteria, the one with the highest LLR PID score (most muon-like) is
considered the muon candidate.

###  III.4 Muon momentum reconstruction quality checks

Three additional selection criteria are applied immediately after the
$\nu_{\mu}$ CC preselection to ensure adequate reconstruction of the outgoing
muon momentum. In order of application, they are as follows:

  1. (V)

The muon candidate track must have a reconstructed end point within the
containment volume defined in selection requirement (II). Requiring
containment allows for a reliable track-length-based estimate of the muon
momentum. This criterion substantially improves the resolution when
reconstructing the muon momentum and derived quantities (e.g., $\delta p_{T}$)
at the cost of a notable drop in efficiency.

  2. (VI)

Muon momentum estimators based on multiple Coulomb scattering [48]
($p_{\mu}^{\mathrm{MCS}}$) and based on track length [51]
($p_{\mu}^{\mathrm{range}}$) for the muon candidate track must agree with each
other within 25%, i.e., they must satisfy the relation

$\frac{\big{|}p_{\mu}^{\mathrm{range}}-p_{\mu}^{\mathrm{MCS}}\big{|}}{p_{\mu}^{\mathrm{range}}}<0.25\,.$
(1)

This requirement removes events in which only a portion of the full muon track
is successfully reconstructed.

  3. (VII)

The track-length-based estimator for the reconstructed muon momentum
$p_{\mu}^{\mathrm{range}}$ must lie within the limits imposed by requirement
(3) from the signal definition: $$0.1\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$\leq
p_{\mu}^{\mathrm{range}}\leq$1.2\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$$.

###  III.5 $\pmb{0\pi Np}$ selection

The remaining five criteria from the full CC$0\pi Np$ selection are intended
to isolate final states containing zero mesons and one or more protons. In
order of application, they are

  1. (VIII)

All reconstructed primary particles must have a track score higher than 0.5
(track-like). This requirement eliminates events containing electromagnetic
showers, which are not expected for mesonless $\nu_{\mu}$ interactions.

  2. (IX)

At least one reconstructed primary particle that is not the muon candidate
must be present in the event. All such particles are considered proton
candidates.

  3. (X)

All proton candidates must have reconstructed end points that lie within the
containment volume defined in selection requirement (II). This is a quality
requirement intended to ensure that a track-length-based estimator of the
proton momentum will be valid. Containment of the end point for the muon
candidate is already enforced by selection requirement (V).

  4. (XI)

All proton candidates must have an LLR PID score less than 0.2 (see Fig. 6).
The chosen cutoff value comes close to optimizing the product of efficiency
and purity while maintaining good acceptance of low-momentum protons.

Figure 6: Log-likelihood ratio particle ID requirement applied to proton
candidates in selection requirement (XI). The green arrow points into the
region in which proton candidates are accepted. The percentages in the legend
indicate the fraction of reconstructed particles of each type before the
illustrated selection is applied. The lower panel shows the ratio of the data
to the simulation prediction. The uncertainty on the simulation prediction is
represented by the dashed lines in the top panel and the gray region in the
bottom panel.

  5. (XII)

The longest proton candidate track must have a length-based momentum estimator
$p_{p}^{\mathrm{range}}$ that falls within the limits given in requirement (4)
from the signal definition: $$0.25\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$\leq
p_{p}^{\mathrm{range}}\leq$1.0\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$$.

###  III.6 Overall selection performance

Events that satisfy all twelve selection requirements described in the
previous subsections are considered the CC$0\pi Np$ candidates of interest for
this analysis. The full CC$0\pi Np$ selection achieves an estimated overall
efficiency of 12.3% and a purity of 78.5%. The evolution of these quantities
as each requirement is applied is shown in Table 1, as well as the relative
efficiency (“Rel. eff.”), which is defined as the ratio of the selection
efficiency after the current requirement is applied to the efficiency before
it is applied. Equivalently, it is the efficiency calculated considering only
those signal events accepted after applying all previous criteria.

Table 1: Evolution of the selection efficiency, purity, and relative
efficiency as the various requirements described in the text are applied.

##  IV Data presentation strategy

The selection defined in the previous section is used in the remainder of this
article to obtain single- and double-differential cross sections for a variety
of kinematic variables. To maximize the usefulness of these measurements for
the neutrino interaction modeling community, an innovative strategy for
reporting the results has been developed.

In previous MicroBooNE cross-section publications, and typically in the
experimental neutrino scattering literature as a whole, individual kinematic
distributions are treated as entirely separate entities. For example, in the
article describing the previous MicroBooNE CC$0\pi Np$ analysis [10], single-
differential cross sections are shown for both the muon momentum and
scattering angle (among several other observables), but the covariance
matrices used to report the measurement uncertainties consider only
correlations between bins of the same variable. Readers are thus obliged to
treat each reported differential cross section as if it was obtained in
isolation from all others. This reduces the power of the full multivariable
data set since external users must either ignore correlations between the
observables that are known to be important (e.g., those introduced by limited
data statistics and systematic uncertainties on the neutrino flux prediction),
invent ad hoc correlations that may be inaccurate, or consider only a single
variable at a time when performing model comparisons. This problem has been
encountered and discussed previously in the neutrino interaction literature
[52], but standard practices for experimental data releases have not yet been
adjusted to address it.

The present analysis overcomes this limitation by providing comprehensive
uncertainties for all final results; cross sections are reported in a binning
scheme that involves multiple variables, and an overall covariance matrix is
computed that accounts for the measurement uncertainties and correlations
between all pairs of bins. To avoid double-counting in the cross-section
extraction procedure, bins describing the same kinematic distribution are
grouped into distinct blocks. As described in the sections that follow, some
mathematical operations are applied to each block independently, but the
blockwise results are later merged together to form the final group of
measurements.

###  IV.1 Observables measured

Various observables related to the 3-momenta of the outgoing muon
($\mathbf{p}_{\mu}$) and leading proton ($\mathbf{p}_{p}$) are measured in
this analysis. Studies of sub-leading protons in the CC$0\pi Np$ channel are
reserved for later work. In addition to the magnitude of the momentum for the
muon ($p_{\mu}$) and leading proton ($p_{p}$) and their scattering angles
($\theta_{\mu}$ and $\theta_{p}$, defined with respect to the neutrino
direction $+\hat{\mathbf{z}}$), differential cross sections in several other
variables are reported. These include the opening angle $\theta_{\mu p}$
between the muon and leading proton

$\theta_{\mu
p}\equiv\arccos\left(\frac{\mathbf{p}_{\mu}\cdot\mathbf{p}_{p}}{p_{\mu}\,p_{p}}\right)\,,$
(2)

as well as the magnitude $\delta p_{T}$ of the transverse missing momentum

$\delta\mathbf{p}_{T}\equiv\mathbf{p}^{T}_{\mu}+\mathbf{p}^{T}_{p}\,.$ (3)

Here, $\mathbf{p}^{T}_{\mu}$ is the vector projection of the muon momentum
transverse to the neutrino beam direction and $\mathbf{p}^{T}_{p}$ is the same
for the leading proton. These vectors have magnitudes $p^{T}_{\mu}$ and
$p^{T}_{p}$ respectively. One may also describe the angular orientation of
$\delta\mathbf{p}_{T}$ with respect to $\mathbf{p}^{T}_{\mu}$ using the angle

$\delta\alpha_{T}\equiv\arccos\left(\frac{-\mathbf{p}^{T}_{\mu}\cdot\delta\mathbf{p}_{T}}{p^{T}_{\mu}\,\delta
p_{T}}\right)$ (4)

as well as the vector components

$\delta
p_{T_{x}}\equiv\frac{(\hat{\mathbf{z}}\times\mathbf{p}^{T}_{\mu})\cdot\delta\mathbf{p}_{T}}{p^{T}_{\mu}}\,,$
(5)

and

$\delta
p_{T_{y}}\equiv\frac{-\mathbf{p}^{T}_{\mu}\cdot\delta\mathbf{p}_{T}}{p^{T}_{\mu}}=\delta
p_{T}\cdot\cos\delta\alpha_{T}\,,$ (6)

where the hat on $\hat{\mathbf{z}}$ indicates that it is a unit vector.

Measurements of the transverse kinematic imbalance (TKI) variables $\delta
p_{T}$, $\delta\alpha_{T}$ were originally proposed [53] as a means of
exploring nuclear effects in neutrino scattering with a reduced dependence on
the neutrino energy. Cross sections as a function of these variables for a
hydrocarbon target have been reported by the T2K [54] and MINERvA [55]
collaborations, the latter also extending [56] the investigation to $\delta
p_{T_{x}}$ and $\delta p_{T_{y}}$. In response to community interest [57] in
measuring these observables using a LArTPC, a recent MicroBooNE analysis
reported first neutrino-argon differential cross sections in TKI variables for
an exclusive one-proton final state [58], including the first double-
differential TKI measurement for neutrino scattering on any nuclear target
[59]. The present work reports similar measurements for the more inclusive
CC$0\pi Np$ signal event topology.

Finally, this analysis measures an estimator $p_{n}$ for the momentum of the
initial struck neutron in the CC interaction. This estimator was proposed in
Ref. [60] with similar phenomenological motivations as the TKI variables, but
it is calculated using momentum components both transverse and longitudinal to
the neutrino beam direction. Specifically, $p_{n}$ is computed via the
expression

$p_{n}=\sqrt{\delta p_{L}^{2}+\delta p_{T}^{2}}$ (7)

where the reconstructed longitudinal missing momentum $\delta p_{L}$ is given
by

$\delta p_{L}\equiv\frac{R}{2}-\frac{m_{f}^{2}+\delta p_{T}^{2}}{2R}\,.$ (8)

Here

$m_{f}\equiv m_{\mathrm{Ar40}}-m_{n}+B$ (9)

and

$R\equiv m_{\mathrm{Ar40}}+p_{\mu}^{z}+p_{p}^{z}-E_{\mu}-E_{p}\,,$ (10)

where $m_{\mathrm{Ar40}}=37.215526$
$\mathrm{GeV}\text{/}{\mathrm{\text{$c$}}}^{2}$ is the mass of the [40]Ar
nuclear target (not the atomic mass), $m_{n}=0.93956541$
$\mathrm{GeV}\text{/}{\mathrm{\text{$c$}}}^{2}$ is the mass of a neutron, and
$B=0.02478$ $\mathrm{GeV}\text{/}$ is the assumed average binding energy for a
neutron in [40]Ar. The longitudinal momentum components and total energies of
the muon ($p_{\mu}^{z}$, $E_{\mu}$) and leading proton ($p_{p}^{z}$, $E_{p}$)
are also used above.

The value of $B$ chosen in the present work is a weighted average of the
neutron separation energies $E_{\alpha}$ given in Table I of Ref. [60] with
the shell occupancies $n_{\alpha}$ used as the weights. An equivalent
prescription for carbon (using Table II from the same reference) was adopted
to estimate $B$ in a prior MINERvA analysis that studied $p_{n}$ [55]. The
first measured neutrino cross sections in this observable for an argon target
were recently reported by MicroBooNE for a one-proton final state [61]
assuming a different value $B=$0.0309\text{\ }\mathrm{GeV}\text{/}$$ based on
a study of electron scattering data [62].

###  IV.2 Estimation of inefficiency and bin migrations

Corrections for imperfect detector and reconstruction performance in the
analysis are applied using an unfolding procedure documented in Sec. VI. The
implementation of this procedure requires a description of the relationship
between reconstructed and true values of the observables of interest. This
relationship is estimated quantitatively by assigning simulated CC$0\pi Np$
events to two duplicate sets of the bins defined in Table LABEL:tab:bin_defs
of Appendix A. The true bins contain all simulated CC$0\pi Np$ events that
have true values of the observables that fall within the relevant kinematic
limits. The reconstructed bins contain those events that pass the full
selection and have appropriate reconstructed values of the observables.

With these definitions, the connection between the reconstructed and true
observables is described using a response matrix $\Delta$, which transforms a
prediction of signal event counts in true bins $\mu$ into a corresponding
prediction in reconstructed bins $j$. Each element of the response matrix is
computed from the MicroBooNE simulation results according to the relation

$\Delta_{j\mu}\equiv\frac{\phi_{j\mu}}{\phi_{\mu}}\,,$ (11)

where $\phi_{\mu}$ is the total number of signal events in true bin $\mu$ and
$\phi_{j\mu}$ is the number of signal events which fall simultaneously into
true bin $\mu$ and reconstructed bin $j$. Thus, the response matrix quantifies
the effects of both inefficiency and smearing of the observables due to
imperfect reconstruction. This matrix is employed in systematic uncertainty
estimation and unfolding as described in Secs. V and VI respectively. It is
plotted in Sec. IV of the supplemental materials.

One may also quantify the effect of smearing separately from inefficiency by
calculating the migration matrix $M$. Each element of this matrix is estimated
from simulation via the expression

$M_{j\mu}=\frac{\phi_{j\mu}}{\sum_{\ell}\phi_{\ell\mu}}\,,$ (12)

where, to avoid double-counting, the sum over $\ell$ includes only those
reconstructed bins in the block of interest. The migration matrix element
$M_{j\mu}$ is thus an estimate of the probability that a selected signal event
belonging to true bin $\mu$ will be assigned to reconstructed bin $j$. Plots
of the migration matrices for each of the 14 blocks of bins defined in Table
LABEL:tab:bin_defs are given in Sec. II of the supplemental materials.

##  V Uncertainties

Systematic uncertainties are estimated by modifying the nominal MicroBooNE
simulation and calculating resultant variations in the expected number of
selected events $n_{j}$ in each reconstructed bin $j$. This expected event
count includes contributions from both simulated neutrino interactions and
measured constant-in-time backgrounds from beam-off data:

$n_{j}=\phi_{j}+O_{j}+B_{j}\,.$ (13)

Here the expected contents of the $j$-th reconstructed bin are divided into
$\phi_{j}$ simulated signal CC$0\pi Np$ events, $O_{j}$ measured beam-off
background events, and $B_{j}$ simulated beam-correlated background events.

Uncertainties are quantified using covariance matrices calculated according to
a multiple-universe procedure. Under this approach, the covariance between the
expected event counts $n_{a}$ and $n_{b}$ in the $a$-th and $b$-th
reconstructed bins is represented by the matrix element

$V_{ab}=\frac{1}{N_{\mathrm{univ}}}\sum_{u=1}^{N_{\mathrm{univ}}}\big{(}n_{a}^{\text{CV}}-n_{a}^{u}\big{)}\big{(}n_{b}^{\text{CV}}-n_{b}^{u}\big{)}\,.$
(14)

Here $n_{a}^{\text{CV}}$ is the total event count in reconstructed bin $a$
predicted by the central-value MicroBooNE simulation. The variable $n_{a}^{u}$
is a prediction of the same quantity computed based on an alternate simulation
(i.e., in an alternate universe) in which some aspect of the models used to
describe the neutrino beam, particle interactions, and the detector response
has been changed from the adopted central value. The total number of alternate
universes $N_{\mathrm{univ}}$ included when computing the average in Eq. (14)
depends on the systematic effect of interest. In some cases, only a single
variation ($N_{\mathrm{univ}}=1$) is used. Covariance matrices $V_{ab}$ are
calculated individually for each source of uncertainty. A total covariance
matrix is then obtained by summing the individual contributions.

###  V.1 Sources of systematic uncertainty

Several classes of systematic uncertainties are considered in this analysis.
Those related to the BNB flux prediction, which include variations related to
the horn current and hadron production modeling, follow the treatment
developed by the MiniBooNE collaboration [14]. Flux shape uncertainties are
handled in a way which allows the final cross-section results to be reported
based upon the predicted BNB $\nu_{\mu}$ flux (tabulated in the supplemental
materials) rather than the unknown true flux [63].

The GENIE-based neutrino interaction model and associated uncertainties used
by MicroBooNE are documented in a dedicated publication [11]. Following Ref.
[58], an additional single-universe variation to the neutrino interaction
model is adopted in which an alternative event generator (version 19.02.2 of
NuWro [64]) replaces GENIE in the MicroBooNE simulation chain and is used to
predict the reconstructed event counts.

Uncertainties related to propagation of the final-state particles emerging
from the neutrino interaction are calculated using the Geant4Reweight [65]
software package. Systematic variations of the Geant4 [18, 19] total cross
section model for positive pions, negative pions, and protons are considered.
For protons, variations are applied to the elastic and reaction (i.e., total
inelastic) channels separately. For pions, the elastic, quasielastic,
absorption, single charge-exchange, double charge-exchange, and pion
production channels are individually varied.

Systematic uncertainties on the detector response model are calculated for
both the MicroBooNE photon detection system and the time projection chamber.
For the former, three alternate universes are constructed in which the
scintillation light yield, attenuation, and Rayleigh scattering length are
individually varied. For the latter, space-charge effects, electron-ion
recombination, and data-driven modifications to the simulated wire response
[13] are considered.

Fully-correlated fractional uncertainties are also applied to the beam-related
portion of the reconstructed events ($\phi_{j}+B_{j}$) to account for limited
precision in the counting of protons delivered to the beam target (2%) and the
number of argon atoms in the fiducial volume (1%).

###  V.2 Signal model uncertainty

For most of the systematic uncertainties mentioned above, the simulated
reconstructed event counts $\phi_{j}$ and $B_{j}$ are varied directly in each
alternate universe. However, since uncertainties related to predicting the
CC$0\pi Np$ signal will only affect the final results via their impact on
estimating the detection efficiency and bin migrations, a different treatment
is used for neutrino interaction model variations. In this case, the expected
number of selected signal events $\phi_{j}$ is rewritten in the form

$\phi_{j}=\sum_{\mu}\Delta_{j\mu}\,\phi_{\mu}^{\mathrm{CV}}\,,$ (15)

where $\Delta_{j\mu}$ is the response matrix element connecting true bin $\mu$
and reconstructed bin $j$. This quantity is varied in each alternate universe,
while the expected signal event counts in the $\mu$-th true bin
$\phi_{\mu}^{\mathrm{CV}}$ are held constant at the central-value prediction
(as indicated by the superscript). To avoid double-counting events, the sum
over $\mu$ includes only those true bins that belong to the same block as the
$j$-th reconstructed bin. Details about the binning scheme and response matrix
definition are provided above in Sec. IV.

###  V.3 Statistical correlations

The data release strategy outlined in Sec. IV requires a slightly more
complicated treatment of statistical uncertainties than has been typical for
previous neutrino cross-section analyses. When every measured event belongs to
exactly one bin, statistical fluctuations of the bin contents are described by
independent Poisson distributions. In that case, the statistical covariance
matrix is diagonal, and the variance of each bin is estimated by the number of
observed counts.

However, reporting combined measurements over the multiple blocks of bins
defined in Table LABEL:tab:bin_defs introduces correlations in the statistical
uncertainties because the same event belongs to a unique bin in each block.
Fortunately, the derivation of an estimator for the statistical covariance
between two arbitrary bins is straightforward. As shown in Sec. III C 1 of
Ref. [17], the statistical covariance between any pair of bins is estimated as
simply the number of events that fall simultaneously into both of them. In the
case of weighted Monte Carlo events, the sum of the squares of the event
weights is used rather than the raw event count.

###  V.4 Impact on integrated cross section

The flux-averaged total CC$0\pi Np$ cross section obtained by this analysis is
$\sigma=($11.8278$\pm$1.1627$)\times${10}^{-38}\text{\
}{\mathrm{cm}}^{2}\text{/}$\,/\,\mathrm{Ar}$. The full fractional uncertainty
of 9.8% includes contributions from the neutrino flux prediction (6.6%),
neutrino interaction modeling (5.1%), detector response modeling (4.1%), beam
exposure measurements (2.2%), data statistics (1.4%), estimation of the number
of argon atoms in the fiducial volume (1.1%), modeling of final-state particle
propagation (0.9%), and Monte Carlo statistics (0.8%). The data statistical
uncertainty includes a contribution from the beam-off measurements used to
estimate cosmic-ray backgrounds. The Monte Carlo statistical uncertainty
includes a contribution from simulated neutrino-induced backgrounds from the
BNB.

For the total cross-section measurement, the additional neutrino interaction
modeling uncertainty estimated by using NuWro as an alternative event
generator (1.7%) is substantially smaller than the uncertainty arising from
GENIE systematic variations (4.8%). However, for many of the differential
measurements reported below, the NuWro-based systematic uncertainty is larger.
Full covariance matrices giving the contributions of both of these classes of
interaction model variations to the total measurement uncertainty are reported
in the supplemental materials. For the detector response model, the
uncertainty related to the simulation of electron recombination [66] has a
dominant impact on the total cross section result (3.3%).

###  V.5 Reconstructed event distributions

As an intermediate step in the analysis, the event selection presented in Sec.
III and the uncertainty treatment described above were applied to measure
candidate CC$0\pi Np$ event rates in each of the 359 kinematic bins defined in
Table LABEL:tab:bin_defs. The measured event rates and associated
uncertainties were then used as input to the unfolding procedure defined in
Sec. VI to obtain the final cross-section results.

The plots in Figs. 7 to 14 report the measured event rates (black data points)
as a function of reconstructed observables for a total beam exposure of
$6.7949\text{\times}{10}^{20}$ $\mathrm{POT}\text{/}$. The error bars on the
data points show only the statistical uncertainty. The stacked histograms in
the main panel of each plot show the central-value prediction from the
MicroBooNE simulation chain. Individual contributions from the event
categories defined in Sec. III are shown using the same color scheme as in
Fig. 1. The total uncertainty (including statistical and systematic
contributions) on the central-value prediction is indicated by the dashed
lines. Below the main panel of each plot, the ratio of the data points to the
central-value prediction is also shown, and the total uncertainty on the
prediction is indicated by the gray band.

Reasonable agreement between the measured event rates and the MicroBooNE
simulation prediction is seen across most of the phase space studied, yielding
an overall $\chi^{2}$ value of $355.251006$ for the 359 reconstructed bins.
The areas of greatest tension are the muon angular distribution at moderate
$p_{\mu}$ (see Fig. 7), as well as the individual and double-differential
distributions of $p_{n}$ and $\theta_{\mu p}$ (see Figs. 9 and 13).

[-0.23]

Figure 7: Reconstructed event distributions for block #0
($p_{\mu},\cos\theta_{\mu}$). The bottom panel of each plot shows the ratio of
the data to the MicroBooNE simulation prediction. The uncertainty on the
prediction is represented by the dashed lines in each top panel and the gray
region in each bottom panel.

Figure 8: Reconstructed event distributions for block #1
($p_{p},\cos\theta_{p}$). The bottom panel of each plot shows the ratio of the
data to the MicroBooNE simulation prediction. The uncertainty on the
prediction is represented by the dashed lines in each top panel and the gray
region in each bottom panel.

(a)  (b)  (c)  (d)

Figure 9: Reconstructed event distributions for (LABEL:sub@subfig:block2)
block #2 ($\delta p_{T}$), (LABEL:sub@subfig:block5) block #5 ($\delta
p_{T_{x}}$), (LABEL:sub@subfig:block7) block #7 ($\theta_{\mu p}$), and
(LABEL:sub@subfig:block8) block #8 ($p_{n}$). The bottom panel of each plot
shows the ratio of the data to the MicroBooNE simulation prediction. The
uncertainty on the prediction is represented by the dashed lines in each top
panel and the gray region in each bottom panel.

Figure 10: Reconstructed event distributions for block #3
$(\delta\alpha_{T},\delta p_{T})$. The overall $\chi^{2}$ value includes
contributions from four $\delta p_{T}$ overflow bins that are not plotted. The
bottom panel of each plot shows the ratio of the data to the MicroBooNE
simulation prediction. The uncertainty on the prediction is represented by the
dashed lines in each top panel and the gray region in each bottom panel.

Figure 11: Reconstructed event distributions for block #4 $(\delta
p_{T},\delta\alpha_{T})$. The bottom panel of each plot shows the ratio of the
data to the MicroBooNE simulation prediction. The uncertainty on the
prediction is represented by the dashed lines in each top panel and the gray
region in each bottom panel.

[-0.23]

Figure 12: Reconstructed event distributions for block #6 $(\delta
p_{T_{y}},\delta p_{T_{x}})$. The overall $\chi^{2}$ value includes
contributions from three underflow and three overflow $\delta p_{T_{x}}$ bins
that are not plotted. The bottom panel of each plot shows the ratio of the
data to the MicroBooNE simulation prediction. The uncertainty on the
prediction is represented by the dashed lines in each top panel and the gray
region in each bottom panel.

[-0.23]

Figure 13: Reconstructed event distributions for block #9 $(p_{n},\theta_{\mu
p})$. The bottom panel of each plot shows the ratio of the data to the
MicroBooNE simulation prediction. The uncertainty on the prediction is
represented by the dashed lines in each top panel and the gray region in each
bottom panel.

(a)  (b)  (c)  (d)

Figure 14: Reconstructed event distributions for (LABEL:sub@subfig:block10)
block #10 ($\cos\theta_{\mu}$), (LABEL:sub@subfig:block11) block #11
($\cos\theta_{p}$), (LABEL:sub@subfig:block12) block #12 ($p_{p}$), and
(LABEL:sub@subfig:block13) block #13 ($p_{\mu}$). The bottom panel of each
plot shows the ratio of the data to the MicroBooNE simulation prediction. The
uncertainty on the prediction is represented by the dashed lines in each top
panel and the gray region in each bottom panel.

###  V.6 Sideband-based validation of background model

Removal of the beam-correlated backgrounds in this analysis relies upon a
simulation-based estimate $B_{j}$ of their contribution to the measured event
counts in each reconstructed bin $j$. A sideband study, described below, was
performed to demonstrate that the background predictions from the MicroBooNE
simulation are sufficiently precise to obtain the final measurements.

Three sets of alternative selection criteria were developed to enhance
sensitivity to the three dominant backgrounds: events outside of the fiducial
volume (Out FV, see Sec. III), neutral-current events (NC), and $\nu_{\mu}$
charged-current events with one or more final-state pions (CC$N\pi$). In each
case, the full CC$0\pi Np$ selection was used as a starting point, and minor
adjustments were made to improve acceptance of the background category of
interest. By construction, all events which pass at least one set of
alternative selection criteria are rejected by the original CC$0\pi Np$
selection. The alternative selection criteria for each class of background are
given below.

Out FV

The full CC$0\pi Np$ selection is unaltered except that the event must fail
requirement (I) (i.e., the reconstructed neutrino vertex position must lie
outside the fiducial volume).

NC

The event must fail requirement (IV) (there must not be a muon candidate). The
additional requirements (V)–(VII) applied to the muon candidate are also
removed. All other CC$0\pi Np$ selection criteria remain the same. The event
must also contain at least two reconstructed primary particles. To allow all
of the observables of interest to be defined for such events, the
reconstructed primary particle with the longest track length is treated as the
muon candidate, while the second-longest is treated as the leading proton
candidate. Checks of the simulated NC events that passed the original CC$0\pi
Np$ selection revealed that this ordering by track length occurred in a large
majority (88%) of cases.

CC$\pmb{N\pi}$

The full CC$0\pi Np$ selection is unaltered except that requirement (XI) is
reversed; at least one of the proton candidates must have an LLR PID score
greater than 0.2 (muon-like).

To assess the adequacy of the background model for this analysis, the logical
OR of these three alternative selections was applied to measure event rates.
That is, an event was accepted by this combined sideband selection if it
satisfied all of the criteria for at least one of the Out-FV-, NC-, or
CC$N\pi$-enhanced selections described above. Good agreement between the
MicroBooNE simulation prediction and the sideband data, within the
uncertainties defined above, was found across the full phase space defined in
Table LABEL:tab:bin_defs. An overall $\chi^{2}$ value of $177.935275$ was
obtained for the full set of 359 bins. Based on this successful comparison
with the sideband data set, the background predictions of MicroBooNE’s
simulation are used unaltered to obtain the final cross-section results. Plots
of the measured and predicted sideband event distributions are provided in
Sec. III of the supplemental materials.

##  VI Cross-section extraction

A neutrino cross-section measurement is ultimately the result of a counting
experiment: estimated backgrounds are subtracted from a measured number of
events in each bin, and scaling factors are then applied to obtain a quantity
with the appropriate units. Corrections to the measured event counts for
detector inefficiency and bin migrations (due to imperfect reconstruction)
must also be applied via a procedure called unfolding. Multiple standard
methods for unfolding are described in terms of the matrix transformation

$\hat{\phi}_{\mu}=\sum_{j}U_{\mu j}\,d_{j}\,,$ (16)

where a vector of measured, background-subtracted event counts $\mathbf{d}$ is
multiplied by an unfolding matrix $U$ to obtain an estimator
$\pmb{\hat{\phi}}$ for the vector of true signal event counts. The element of
$\mathbf{d}$ corresponding to the $j$-th reconstructed bin is given by

$d_{j}=D_{j}-O_{j}-B_{j}\,,$ (17)

where $D_{j}$ is the total number of measured events in the bin, and both
$O_{j}$ and $B_{j}$ are defined as in Eq. (13). In this context, $B_{j}$ is
evaluated in the central-value universe. The covariance between the measured
event counts in the $a$-th and $b$-th reconstructed bins is used to calculate
the covariance between the unfolded event counts in the $\mu$-th and $\nu$-th
true bins according to the relation

$V_{\mu\nu}=\mathrm{Cov}(\hat{\phi}_{\mu},\hat{\phi}_{\nu})=\sum_{a,b}\mathfrak{E}_{\mu
a}\,\mathrm{Cov}(d_{a},d_{b})\,\mathfrak{E}^{T}_{b\nu}\,,$ (18)

where the elements of the error propagation matrix $\mathfrak{E}$ are the
partial derivatives

$\mathfrak{E}_{\mu a}\equiv\frac{\partial\hat{\phi}_{\mu}}{\partial d_{a}}\,.$
(19)

An explicit expression for these, appropriate for this analysis, is given
below in Eq. (30). The contribution of systematic uncertainties to the
covariance on the background-subtracted measured event counts is estimated
from the expected event counts:

$\mathrm{Cov}(d_{a},d_{b})\approx\mathrm{Cov}(n_{a},n_{b})=V_{ab}\,,$ (20)

where $V_{ab}$ is calculated according to the prescription in Eq. (14).

###  VI.1 The additional smearing matrix $A_{C}$

A simple method of unfolding is to directly invert the response matrix
$\Delta$ via

$U^{\mathrm{direct}}=(\Delta^{T}\Delta)^{-1}\,\Delta^{T}\,.$ (21)

However, in most cases of practical interest, this approach leads to strong
anticorrelations between bins and large uncertainties. Overcoming these
difficulties requires introducing new information (specifically, a prior
prediction of the unfolded result) into the otherwise ill-posed unfolding
problem. The process of adding this information is known as regularization.
The various unfolding techniques defined in the literature differ in their
prescriptions for how the regularization should be carried out.

Multiple recent MicroBooNE analyses [67, 68, 59, 58, 61, 15, 16] have used the
Wiener-SVD [69] unfolding technique, which is based on an analogy with signal
processing: regularization is applied using a “Wiener filter” matrix designed
to mitigate “noise” arising from uncertainties on the measurement. The Wiener-
SVD method involves explicit construction of an additional smearing matrix
$A_{C}$ which is related to the unfolding matrix $U$ via

$U=A_{C}\cdot
U^{\mathrm{direct}}=A_{C}\cdot(\Delta^{T}\Delta)^{-1}\,\Delta^{T}\,.$ (22)

The additional smearing matrix thus encapsulates the effect of regularization
on the unfolding procedure. While an expression for $A_{C}$ specific to the
Wiener-SVD method is given in Eq. (3.23) of the original publication [69], it
follows from the definition in Eq. (22) that $A_{C}$ may be calculated for an
arbitrary unfolding method via

$A_{C}=U\cdot\Delta\,.$ (23)

Reference [69] also describes a method to avoid introducing new measurement
uncertainties related to the unfolding procedure itself. By multiplying
theoretical predictions by $A_{C}$ before comparisons are made to the unfolded
results, the effects of regularization-related bias on the measurement are
properly taken into account. This prescription is followed herein: all model
predictions are multiplied by $A_{C}$ when comparisons are made to the
unfolded data. The $A_{C}$ matrix elements needed to compare the final
measurements to new theoretical predictions are provided in the supplemental
materials.

###  VI.2 D’Agostini iterative unfolding

The present work adopts an iterative method of unfolding popularized in high-
energy physics by D’Agostini [70]. An initial estimate for the unfolded event
counts (iteration $i=0$) is obtained from the central-value prediction of the
MicroBooNE simulation in the true bins:

$\hat{\phi}_{\mu}^{0}=\phi_{\mu}^{\mathrm{CV}}\,.$ (24)

Superscripts on $\hat{\phi}$ and related quantities are used to denote the
number of iterations. Subsequent iterations of the method are used to refine
the initial estimate using the measured event rates in each reconstructed bin.
This is done via the formula

$\hat{\phi}_{\mu}^{i+1}=\sum_{j}U_{\mu j}^{i}\,d_{j}\,.$ (25)

where the unfolding matrix element

$U_{\mu j}^{i}=\frac{P^{i}_{j\mu}}{\epsilon_{\mu}}$ (26)

is defined in terms of the selection efficiency in the $\mu$-th true bin

$\epsilon_{\mu}=\sum_{\ell}\Delta_{\ell\mu}$ (27)

and the conditional probability that a signal event which belongs to the
$\mu$-th true bin will be assigned to the $j$-th reconstructed bin

$P^{i}_{j\mu}=\frac{\Delta_{j\mu}\,\hat{\phi}_{\mu}^{i}}{\sum_{\nu}\Delta_{j\nu}\,\hat{\phi}_{\nu}^{i}}\,.$
(28)

Because the unfolding matrix depends on the measured event counts for
iterations $i>0$, the error propagation matrix elements needed to evaluate
uncertainties on the unfolded result

$V_{\mu\nu}^{i}=\mathrm{Cov}(\hat{\phi}_{\mu}^{i},\hat{\phi}_{\nu}^{i})=\sum_{a,b}\mathfrak{E}_{\mu
a}^{i}\,\mathrm{Cov}(d_{a},d_{b})\,\mathfrak{E}_{\nu b}^{i}$ (29)

are given by [71, 72]

$\mathfrak{E}_{\mu a}^{i+1}=\frac{\partial\hat{\phi}_{\mu}^{i+1}}{\partial
d_{a}}=U_{\mu
a}^{i}+\frac{\hat{\phi}_{\mu}^{i+1}}{\hat{\phi}^{i}_{\mu}}\mathfrak{E}_{\mu
a}^{i}-\sum_{\nu,b}\epsilon_{\nu}\,\frac{d_{b}}{\hat{\phi}^{i}_{\nu}}\,U_{\mu
b}^{i}\,U_{\nu b}^{i}\,\mathfrak{E}_{\nu a}^{i}\,.$ (30)

For the first iteration, this reduces to

$\mathfrak{E}_{\mu a}^{1}=U_{\mu a}^{0}$ (31)

since the central-value prediction of the MicroBooNE simulation does not
depend on the data:

$\mathfrak{E}_{\mu a}^{0}=\frac{\partial\hat{\phi}_{\mu}^{0}}{\partial
d_{a}}=\frac{\partial\phi_{\mu}^{\mathrm{CV}}}{\partial d_{a}}=0\,.$ (32)

###  VI.3 Convergence criterion for unfolding iterations

In the limit of many iterations, the D’Agostini approach converges to the
direct inversion result discussed in Sec. VI.1. Regularization is thus applied
by stopping the method at a finite number of iterations.

Traditionally, the choice of the number of iterations to use in the D’Agostini
unfolding method is a critical step of an analysis: one must iterate enough
times to avoid a strong bias towards the initial estimate
$\hat{\phi}_{\mu}^{0}$ for the true event distribution without approaching the
direct inversion result too closely. These concerns are addressed in this
article by multiplying theoretical predictions by the additional smearing
matrix $A_{C}$ before comparing them to the unfolded data. The implicit
regularization in the D’Agostini method is thus applied to the predictions in
a consistent way, avoiding the need to introduce new unfolding-related
systematic uncertainties.

Nevertheless, even while using $A_{C}$, one must choose a definite number of
iterations in order to unfold via the D’Agostini approach. In this analysis,
the choice is made by continuing to iterate until the average fractional
deviation $\mathfrak{F}$ in the $T$ true bins

$\mathfrak{F}\equiv\frac{1}{T}\sum_{\mu}\frac{\big{|}\hat{\phi}_{\mu}^{i+1}-\hat{\phi}_{\mu}^{i}\big{|}}{\hat{\phi}_{\mu}^{i+1}}$
(33)

between neighboring iterations falls below 2.5%. This cutoff was chosen
empirically based on the unfolding performance in fake data studies and the
observation that fluctuations of the bin counts at this level will be well-
covered by typical systematic uncertainties. The fake data studies used to
validate the unfolding procedure treated the prediction in the NuWro alternate
universe (see Sec. V.1) as if it were real data. Recovery of the known true
bin counts $\phi_{\mu}$ predicted by NuWro was verified within the relevant
subset of the uncertainties, i.e., only the Monte Carlo statistical and
neutrino interaction model systematic uncertainties were included.

###  VI.4 Blockwise unfolding

The convergence criterion discussed in Sec. VI.3 leads to between two and five
iterations being used to unfold each individual block of bins.

To avoid the double-counting issues mentioned in Sec. IV (e.g., an incorrect
calculation of the selection efficiency from the elements of the response
matrix $\Delta$), the sums in Eqs. (27)–(30) and Eq. (33) should be understood
to include only true and reconstructed bins within the same block. Also, since
Eq. (33) is intended to be an average, $T$ should be interpreted as the number
of true bins in the block of interest.

An overall unfolding matrix $U$ used according to Eq. (16) to obtain final
results involving all bins is constructed as the direct sum of the blockwise
unfolding matrices:

$U=\bigoplus_{b}U_{b}=U_{0}\oplus
U_{1}\oplus\dots=\begin{pmatrix}U_{0}&0&0&\dots\\\ 0&U_{1}&0&\dots\\\
0&0&\ddots&\dots\\\ \vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix}\,.$ (34)

Here $U_{b}$ is the unfolding matrix for the $b$-th block of true and
reconstructed bins. A similar direct sum is calculated to build the combined
error propagation matrix needed to evaluate the unfolded covariance matrix in
Eq. (18):

$\mathfrak{E}=\bigoplus_{b}\mathfrak{E}_{b}=\mathfrak{E}_{0}\oplus\mathfrak{E}_{1}\oplus\dots$
(35)

###  VI.5 Decomposition of the blockwise covariance matrices

The covariance matrix $V_{\mu\nu}$ describing the unfolded event counts
$\hat{\phi}_{\mu}$ and $\hat{\phi}_{\nu}$ (see Eq. (18)) for an individual
block of bins can be decomposed into components representing normalization-
only, shape-only, and mixed uncertainties according to the relations [73]

$V_{\mu\nu}=V_{\mu\nu}^{\mathrm{norm}}+V_{\mu\nu}^{\mathrm{shape}}+V_{\mu\nu}^{\mathrm{mixed}}$
(36)
$V_{\mu\nu}^{\mathrm{norm}}=\frac{\hat{\phi}_{\mu}\,\hat{\phi}_{\nu}}{\hat{\phi}_{\mathrm{tot}}^{2}}\sum_{\alpha\beta}V_{\alpha\beta}$
(37)
$V_{\mu\nu}^{\mathrm{mixed}}=\frac{\hat{\phi}_{\mu}}{\hat{\phi}_{\mathrm{tot}}}\sum_{\alpha}V_{\alpha\nu}+\frac{\hat{\phi}_{\nu}}{\hat{\phi}_{\mathrm{tot}}}\sum_{\alpha}V_{\mu\alpha}-2V_{\mu\nu}^{\mathrm{norm}}$
(38)

where

$\hat{\phi}_{\mathrm{tot}}\equiv\sum_{\alpha}\hat{\phi}_{\alpha}\,,$ (39)

and the sums are understood to include all true bins belonging to the block of
interest and no others. Note that the same formulas cannot be meaningfully
applied to the multi-block covariance matrix due to double-counting when
computing the total number of events in Eq. (39). This decomposition of the
covariance matrix is used when presenting cross-section results for individual
blocks of bins in Sec. VII.2.

###  VI.6 Calculation of differential cross sections

The unfolded event counts $\hat{\phi}_{\mu}$ from the final iteration provide
an estimator for the true number of signal events in the $\mu$-th bin. They
may be converted to a flux-averaged differential cross section according to
the formula

$\Big{<}\frac{d^{n}\sigma}{d\mathbf{x}}\Big{>}_{\mu}=\frac{\hat{\phi}_{\mu}}{\Phi\,\mathcal{N}_{\mathrm{Ar}}\,\Delta\mathbf{x}_{\mu}}$
(40)

where $\Phi=$5.00846\text{\times}{10}^{11}$\text{
}\nu_{\mu}\,/${\mathrm{cm}}^{2}\text{/}$$ is the integrated flux of muon
neutrinos calculated for the beam exposure ($6.7949\text{\times}{10}^{20}$
$\mathrm{POT}\text{/}$) used in the analysis,
$\mathcal{N}_{\mathrm{Ar}}=$7.99249\text{\times}{10}^{29}$$ is the number of
argon nuclei in the fiducial volume, and $\Delta\mathbf{x}_{\mu}$ is the
product of the $n$ widths of the $\mu$-th bin of the $n$-dimensional
measurement.

The covariance between the $n$\- and $m$-dimensional differential cross
sections in the $\mu$-th and $\nu$-th true bins is likewise given by

$\mathrm{Cov}\left(\Big{<}\frac{d^{n}\sigma}{d\mathbf{x}}\Big{>}_{\mu},\Big{<}\frac{d^{m}\sigma}{d\mathbf{y}}\Big{>}_{\nu}\right)=\frac{\mathrm{Cov}(\hat{\phi}_{\mu},\hat{\phi}_{\nu})}{\Phi^{2}\,\mathcal{N}_{\mathrm{Ar}}^{2}\,\Delta\mathbf{x}_{\mu}\,\Delta\mathbf{y}_{\nu}}\,.$
(41)

##  VII Results

The measurements shown in the remainder of this article are flux-averaged
differential CC$0\pi Np$ cross sections obtained via the unfolding procedure
presented above. Several event generator predictions are compared to the data
using NUISANCE [74], and goodness-of-fit is quantified for the data set as a
whole and for the individual blocks of kinematic bins listed in Table
LABEL:tab:bin_defs.

In Sec. I of the supplemental materials, the full 359-bin data set is
presented in terms of flux-averaged total cross sections integrated over each
bin (trivially obtainable by multiplying by the bin widths
$\Delta\mathbf{x}_{\mu}$). This allows the measured cross sections and the
covariance matrix elements for all observables to be expressed using
consistent units.

###  VII.1 Interaction models

In addition to the MicroBooNE Tune model described in Sec. II.1 and used to
execute the analysis, the predictions of several other simulation-based
neutrino interaction models are compared to the measured cross-section
results. These models include multiple configurations of GENIE as well as
three alternative neutrino event generators.

The GENIE-based prediction labeled GENIE 3.0.6 is produced using the same code
version and model set (G18_10a_02_11a) as the MicroBooNE Tune, but the custom
MicroBooNE-specific modifications described in Ref. [11] are omitted. The
GENIE 2.12.10 prediction uses a near-default configuration of this older
version of the code, which includes the Bodek-Ritchie Fermi gas description of
the nuclear ground state [75], the Llewellyn Smith CCQE calculation [76], an
empirical model for 2p2h interactions [77], the Rein-Sehgal treatment of RES
and COH scattering [78], and the hA FSI model [79, 80]. The modeling
ingredients in GENIE 3.2.0 G18_02a, which uses the default configuration
(G18_02a_00_000) of this recent GENIE release, are largely similar to GENIE
2.12.10. In addition to code updates, however, the KLN-BS RES model [29, 30,
31, 32], Berger-Sehgal COH model [36], and hA2018 FSI model [37] replace their
prior counterparts. A final configuration, GENIE 3.2.0 G21_11b, uses the more
recently-added G21_11b_00_000 model set, which adopts the SuSAv2 calculation
[81, 82, 83, 84] of CCQE and 2p2h cross sections as well as the hN2018 FSI
model [37]. The remaining components of the model set are similar to GENIE
3.0.6.

The first prediction of an alternate neutrino event generator, NuWro 19.02.2,
uses the NuWro [64] implementations of the LFG nuclear ground state [23], the
Llewellyn Smith CCQE model [76], the Valencia CC2p2h model [27, 28], the
Adler-Rarita-Schwinger treatment of $\Delta$ resonance production [32], the
Berger-Sehgal COH model [36] and an intranuclear cascade approach to FSI. A
comparison to NEUT 5.6.0 is also provided, in which the NEUT event generator
[85, 86] is configured to use an LFG nuclear model [23], the Valencia model
for CCQE and 2p2h [24, 25, 26, 27, 28], the KLN-BS RES calculation [29, 30,
31, 32], the Berger-Sehgal COH model [36], and an FSI cascade treatment with
nuclear medium corrections for pions [87].

Finally, a prediction from the GiBUU event generator [88] (labeled GiBUU
2021.1) is studied using the 2021 “patch 1” release of the code announced to
users on 5 November 2021. Ingredients of the GiBUU physics model include an
LFG representation of the nuclear ground state [23], a standard expression for
the neutrino-nucleon CCQE cross section [89], an empirical 2p2h model, a
treatment of RES based upon the MAID analysis [90], and a DIS model from
PYTHIA [91]. A unique feature of the code is its dynamical model of
intranuclear hadron transport based upon numerical solution of the Boltzmann-
Uehling-Uhlenbeck equation. A consistent nuclear potential is used in both the
description of the target nucleus and FSI.

The $\chi^{2}$ metric used to assess goodness-of-fit when comparing these
models to data accounts for the total covariance matrix associated with the
measurements but neglects any theoretical uncertainties on the predictions
themselves. Documentation needed to enable new comparisons beyond those shown
here is provided in Sec. I of the supplemental materials. The data set from
this article has also recently been incorporated into the NUISANCE [74]
software framework for convenient use by the neutrino interaction modeling
community.

###  VII.2 Differential cross-section results

Figures 15 to 28 present the final results of the analysis as flux-averaged
differential cross sections. The measurements are compared to the event
generator models described above, each rendered in the plots as a distinct
colored line. The legends accompanying each figure list the models together
with their overall $\chi^{2}$ scores describing the level of agreement with
data. Each $\chi^{2}$ score is separated from the number of bins for which it
was calculated by a / character. A separate figure is included for each block
of bins defined in Table LABEL:tab:bin_defs.

Following recent MicroBooNE publications that report cross sections for an
exclusive single-proton final state [59, 58, 61], the covariance matrix
decomposition described in Sec. VI.5 is applied when displaying the
measurement uncertainties. For the data points shown in each plot, the inner
error bars represent the statistical uncertainty only, while the outer error
bars also include shape-only systematic uncertainties. The remaining portion
of the total uncertainty, composed of both normalization and mixed terms, is
shown by the gray band along the $x$-axis.

Table 2 presents the $\chi^{2}$ scores quantifying goodness-of-fit with the
full group of measurements for each of the event generator models studied.
None of the simulation predictions provides a satisfactory treatment of the
entire data set ($\chi^{2}\approx 1/$bin). The best $\chi^{2}$ value is
obtained by the GENIE 3.0.6 calculation, although poor agreement is seen
between this model and the data in some of the individual blocks of bins, and
its prediction is seen to be systematically low across the entire measured
phase space. The modest preference for GENIE 3.0.6 may thus be attributed to a
manifestation of Peelle’s Pertinent Puzzle [92] (in which implausibly low
predictions are favored using a least-squares metric like $\chi^{2}$) rather
than a particularly high-quality description of the data. The substantially
larger $\chi^{2}$ values seen for GENIE 2.12.10 and GENIE 3.2.0 G18_02a,
however, do suggest that some of their common model components (e.g., a global
rather than local Fermi gas description of the nuclear ground state) are
disfavored by the present measurements. As shown in the plots below, GiBUU
2021.1 has the best $\chi^{2}$ score for many of the individual blocks of bins
despite performing more poorly when confronted with the combined results.

Table 2: Overall $\chi^{2}$ scores for each of the neutrino interaction models
studied.

Figure 15 presents the double-differential cross-section measurement in terms
of muon momentum ($p_{\mu}$) and scattering angle ($\theta_{\mu}$) obtained
using the kinematic bins from block #0. The GiBUU 2021.1 model achieves the
best agreement with this distribution, partially due to its larger cross
section relative to most of the other plotted models in the region of moderate
$p_{\mu}$ (especially between 0.3 and 0.38 GeV$/c$) and intermediate
scattering angles. Interestingly, the data are compatible with an even larger
cross section in this region, to the extent that all studied predictions lie
noticeably below the measured central value in several relevant bins.

[-0.23]

Figure 15: Measured differential cross sections for block #0
($p_{\mu},\cos\theta_{\mu}$). Statistical (shape-only systematic)
uncertainties are included in the inner (outer) error bars. The remainder of
the total uncertainty is shown by the gray band along the $x$-axis.

Figure 16 presents the double-differential cross-section measurement in terms
of the leading proton momentum ($p_{p}$) and scattering angle ($\theta_{p}$)
obtained using the bins from block #1. At low $p_{p}$, the NuWro 19.02.2
calculation markedly undershoots the data, particularly at forward proton
angles. However, this model prediction becomes more similar to the others with
increasing $p_{p}$, as can also be seen in the single-differential $p_{p}$
results shown in Fig. 27. Significant model differences are seen in the shape
of the forward-angle region at relatively high $p_{p}$, but the systematic
uncertainty of the present measurement allows for only limited sensitivity to
these details.

Figure 16: Measured differential cross sections for block #1
($p_{p},\cos\theta_{p}$). Statistical (shape-only systematic) uncertainties
are included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

Figure 17 shows the single-differential cross section in the transverse
momentum imbalance $\delta p_{T}$ as obtained from the analysis bins in block
#2. A clear preference in the $\chi^{2}$ values is seen for models which can
simultaneously provide a relatively good treatment of the low-$\delta p_{T}$
region (dominated by CCQE events without proton FSI) and the high-$\delta
p_{T}$ tail (dominated by more inelastic events and CCQE with FSI). Notably,
two of the event generator models that show the poorest agreement with this
distribution, NuWro 19.02.2 and GENIE 3.2.0 G18_02a, have their greatest
tension with the data in opposite kinematic regions corresponding to high- and
low-$\delta p_{T}$, respectively.

[-0.23]

Figure 17: Measured differential cross sections for block #2 ($\delta p_{T}$).
Statistical (shape-only systematic) uncertainties are included in the inner
(outer) error bars. The remainder of the total uncertainty is shown by the
gray band along the $x$-axis.

A double-differential result in which the $\delta p_{T}$ measurement is sub-
divided into four $\delta\alpha_{T}$ bins (as defined in block #3) is plotted
in Fig. 18. Here the NuWro 19.02.2 model shows good agreement with data in the
lowest $\delta\alpha_{T}$ region, which gradually worsens with increasing
$\delta\alpha_{T}$. A similar trend is seen in Fig. 12 of Ref. [58], which
reports a similar MicroBooNE measurement studying a more exclusive single-
proton final state. Because the effect of FSI is to make the
$\delta\alpha_{T}$ distribution more asymmetric and weighted towards high
values [58, 93], these results may indicate an underestimation of proton FSI
in the NuWro 19.02.2 configuration considered herein. The GENIE 3.2.0 G18_02a
model, on the other hand, shows a roughly consistent deficit in the lowest
$\delta p_{T}$ bins across all $\delta\alpha_{T}$ values, which is suggestive
of missing CCQE strength. Similar conclusions follow from Fig. 19, in which
the bins from block #4 are used to report a measurement of double-differential
$\delta\alpha_{T}$ distributions in four coarse bins of $\delta p_{T}$.

Figure 18: Measured differential cross sections for block #3
$(\delta\alpha_{T},\delta p_{T})$. The overall $\chi^{2}$ value includes
contributions from four $\delta p_{T}$ overflow bins that are not plotted.
Statistical (shape-only systematic) uncertainties are included in the inner
(outer) error bars. The remainder of the total uncertainty is shown by the
gray band along the $x$-axis.

Figure 19: Measured differential cross sections for block #4 $(\delta
p_{T},\delta\alpha_{T})$. Statistical (shape-only systematic) uncertainties
are included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

Figure 20 presents a single-differential cross-section measurement in $\delta
p_{T_{x}}$, the component of $\delta\mathbf{p}_{T}$ that is orthogonal to the
transverse projection of the outgoing muon momentum. The GiBUU 2021.1 and NEUT
5.6.0 models provide particularly good predictions for this observable, which
follows a distribution that is nearly symmetric around zero. Figure 21
subdivides this measurement into three coarse bins of $\delta p_{T_{y}}$, the
other component of $\delta\mathbf{p}_{T}$. Here the effects of FSI enhance the
cross section in the region where $\delta p_{T_{y}}$ is negative. The strong
deficit seen in the NuWro 19.02.2 prediction in the lowest $\delta p_{T_{y}}$
bin is thus symptomatic of the same modeling deficiencies seen previously at
high values of $\delta p_{T}$ and $\delta\alpha_{T}$.

[-0.23]

Figure 20: Measured differential cross sections for block #5 ($\delta
p_{T_{x}}$). Statistical (shape-only systematic) uncertainties are included in
the inner (outer) error bars. The remainder of the total uncertainty is shown
by the gray band along the $x$-axis.

[-0.23]

Figure 21: Measured differential cross sections for block #6 $(\delta
p_{T_{y}},\delta p_{T_{x}})$. The overall $\chi^{2}$ value includes
contributions from three underflow and three overflow $\delta p_{T_{x}}$ bins
that are not plotted. Statistical (shape-only systematic) uncertainties are
included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

Figure 22 displays the measured single-differential CC$0\pi Np$ cross section
in bins of $\theta_{\mu p}$, the opening angle between the outgoing muon and
leading proton. Although all of the interaction models studied in this article
agree on the rough shape of this angular distribution, the position of the
peak is sensitive to the relative contributions of QE and 2p2h events, as seen
in the lower-left panel of Fig. 9. The data points provide a peak location
that is shifted slightly to the right of the bulk of the event generator
predictions, with NEUT 5.6.0 and especially GiBUU 2021.1 achieving the best
agreement.

[-0.23]

Figure 22: Measured differential cross sections for block #7 ($\theta_{\mu
p}$). Statistical (shape-only systematic) uncertainties are included in the
inner (outer) error bars. The remainder of the total uncertainty is shown by
the gray band along the $x$-axis.

Figure 23 reports the measured distribution of $p_{n}$, a three-dimensional
analog of $\delta p_{T}$ which includes the component of missing momentum
$\delta p_{L}$ longitudinal to the neutrino beam. The addition of this new
direction noticeably worsens the $\chi^{2}/$bin for all models studied,
particularly for GENIE 2.12.10 and NEUT 5.6.0, although the latter still
achieves the second-best $\chi^{2}$ score. When one considers the single-
differential cross section for $p_{n}$ in isolation, the GiBUU 2021.1, GENIE
3.2.0 G21_11b, and MicroBooNE Tune models also describe the data fairly well
($\chi^{2}/$bin $<1$) despite using distinct theoretical treatments of both
the QE and 2p2h interaction modes.

[-0.23]

Figure 23: Measured differential cross sections for block #8 ($p_{n}$).
Statistical (shape-only systematic) uncertainties are included in the inner
(outer) error bars. The remainder of the total uncertainty is shown by the
gray band along the $x$-axis.

Figure 24 presents double-differential cross sections as a function of
$\theta_{\mu p}$ in three bins of $p_{n}$. Here the preference for the peak
location predicted by GiBUU 2021.1 in Fig. 22 is partially explained by this
model’s remarkably good description of the data in the bin of moderate $p_{n}$
($0.21\text{\ }\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$ to $0.45\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$). As shown in Fig. 13, this is also
the kinematic region in which differences in the relative contributions of QE
and 2p2h events have the greatest impact on the shape of the $\theta_{\mu p}$
distribution. Although it obtains one of the larger $\chi^{2}$ values for this
group of bins (block #9), the NuWro 19.02.2 model provides nearly the best
agreement ($\chi^{2}=3.85/11$ bins) in the QE-dominated region of $p_{n}<$
$0.21\text{\ }\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$. Only the NEUT 5.6.0
model achieves slightly better performance ($\chi^{2}=3.32/11$ bins) for this
region of phase space.

[-0.23]

Figure 24: Measured differential cross sections for block #9
$(p_{n},\theta_{\mu p})$. Statistical (shape-only systematic) uncertainties
are included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

Figure 25 shows the measured single-differential cross section as a function
of the muon scattering angle $\theta_{\mu}$. The lowest $\chi^{2}$ values
obtained by GiBUU 2021.1 and NEUT 5.6.0 can be attributed to two features of
these model predictions. First, the data favor significantly more cross-
section strength in the $\cos\theta_{\mu}\lesssim 0.4$ region than is
predicted by the majority of the studied event generators. Here the GiBUU
2021.1 and NEUT 5.6.0 predictions are noticeably higher than the others. A
similar tendency for models to underpredict data in this angular region has
also been seen in recent MicroBooNE cross-section results for quasielastic-
like interactions [58] and inclusive charged-current $\nu_{\mu}$ scattering
with at least one proton in the final state [16]. Second, both GiBUU 2021.1
and NEUT 5.6.0 provide a good description within uncertainties of the forward-
angle region ($\cos\theta_{\mu}\gtrsim 0.8$), although GiBUU 2021.1 lies
substantially closer to the measured data point in the most-forward bin.

[-0.23]

Figure 25: Measured differential cross sections for block #10
($\cos\theta_{\mu}$). Statistical (shape-only systematic) uncertainties are
included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

Finally, Figs. 26, 27, and 28 present measured single-differential cross
sections in terms of the leading proton scattering angle ($\theta_{p}$),
leading proton momentum ($p_{p}$), and muon momentum ($p_{\mu}$),
respectively. For all three distributions, GiBUU 2021.1 provides the best
quantitative agreement with the data, driven in part by its prediction of a
higher CC$0\pi Np$ cross section at moderate proton scattering angles
($\cos\theta_{p}\in[0,0.6]$), low proton momenta ($p_{p}<$0.5\text{\
}\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$$) and moderate muon momenta
($p_{\mu}\in[0.25,0.5]~{}$\mathrm{GeV}\text{/}\mathrm{\text{$c$}}$$).

For the single-differential $p_{\mu}$ and $p_{p}$ results, similar trends can
be seen in MicroBooNE data for inclusive CC $\nu_{\mu}$ events with one or
more final-state protons [16]. For both the CC$0\pi Np$ cross sections shown
here and the inclusive results, data indicate a higher cross section near the
peak of the $p_{\mu}$ distribution than is predicted by a variety of neutrino
event generators, and models with a larger prediction at low $p_{p}$ are also
preferred.

[-0.23]

Figure 26: Measured differential cross sections for block #11
($\cos\theta_{p}$). Statistical (shape-only systematic) uncertainties are
included in the inner (outer) error bars. The remainder of the total
uncertainty is shown by the gray band along the $x$-axis.

[-0.23]

Figure 27: Measured differential cross sections for block #12 ($p_{p}$).
Statistical (shape-only systematic) uncertainties are included in the inner
(outer) error bars. The remainder of the total uncertainty is shown by the
gray band along the $x$-axis.

[-0.23]

Figure 28: Measured differential cross sections for block #13 ($p_{\mu}$).
Statistical (shape-only systematic) uncertainties are included in the inner
(outer) error bars. The remainder of the total uncertainty is shown by the
gray band along the $x$-axis.

##  VIII Summary and Conclusions

This article presents a detailed study of charged-current muon-neutrino
interactions with argon leading to mesonless final states containing one or
more protons. Flux-averaged cross sections for this CC$0\pi Np$ process were
measured using the Fermilab Booster Neutrino Beam and MicroBooNE detector. The
results are reported as a function of ten observables related to the three-
momenta of the outgoing muon and leading proton. A larger data set and
significant improvements to MicroBooNE’s software tools since the experiment’s
first CC$0\pi Np$ cross-section analysis was reported [10] allow the present
work to achieve higher precision and explore new kinematic distributions,
including the first double-differential measurements in the CC$0\pi Np$
channel for an argon target.

A covariance matrix describing correlated uncertainties between all 359
kinematic bins is provided in the supplemental materials, allowing goodness-
of-fit to be quantitatively assessed for theoretical predictions describing
the entire data set. This data release strategy represents an improvement over
the typical procedure employed to date in the neutrino scattering literature,
in which correlations between distinct kinematic distributions are not
disclosed.

The predictions of several standard neutrino event generators are presented
and compared to the MicroBooNE data. The GiBUU 2021.1 model provides a
relatively good description of all of the individual differential cross
sections studied, but its poor agreement with the combined set of measurements
(quantified with the $\chi^{2}$ score given in Table 2) suggests that the
correlations between kinematic distributions are less well-modeled. Overall
agreement is likewise unsatisfactory for the other neutrino interaction models
considered herein, although those with the least sophisticated treatments of
nuclear effects (GENIE 2.12.10 and GENIE 3.2.0 G18_02a) obtain substantially
higher $\chi^{2}$ values than the others. The GENIE 3.0.6 prediction achieves
the best overall goodness-of-fit of the models studied, but it is seen to
systematically underpredict the data across a large fraction of the measured
phase space.

Some of the kinematic regions for which the present measurements give the
greatest model discrimination power include moderate $p_{\mu}$, low to
moderate $\cos\theta_{\mu}$, and low $p_{p}$. The data show a clear preference
for the greater cross-section strength assigned to these regions by GiBUU
2021.1 and NEUT 5.6.0 compared with the other event generators studied. These
two models also describe the distribution of the muon-proton opening angle
$\theta_{\mu p}$ noticeably better than their counterparts. Especially when
combined with recent MicroBooNE measurements examining more exclusive [59, 58,
94] and inclusive [68, 95] final-state topologies in charged-current
$\nu_{\mu}$-argon scattering, the present data set provides a highly detailed
benchmark for the ongoing effort to improve event generators to the precision
needed for the future accelerator-based neutrino oscillation program.

##  IX Acknowledgments

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); the UK Research
and Innovation (UKRI) Future Leaders Fellowship; and the NSF AI Institute for
Artificial Intelligence and Fundamental Interactions. Additional support for
the laser calibration system and cosmic ray tagger was provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses,
without whom this work would not have been possible. For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC BY) public
copyright license to any Author Accepted Manuscript version arising from this
submission.

##  Appendix A Bin definitions

Table LABEL:tab:bin_defs presents the full set of 359 kinematic bins used to
report the CC$0\pi Np$ cross section measurements. For an observable $x$, the
bin with lower limit $x^{\mathrm{low}}$ and upper limit $x^{\mathrm{high}}$
will include events with $x^{\mathrm{low}}\leq x<x^{\mathrm{high}}$. The
blocks of related bins (indicated in the table) are defined so that a selected
event will belong to a unique bin within each block.

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