We present an efficient quantum circuit for block encoding pairing Hamiltonians often studied in nuclear physics. Our block encoding scheme does not require mapping the creation and annihilation operators to the Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode the Hamiltonian directly using controlled swap operations. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with the quantum singular value transformation to construct an efficient quantum circuit for approximating the density of states of a pairing Hamiltonian.

The techniques presented can be extended to encode more general second-quantized Hamiltonians.

§ ACKNOWLEDGMENTS

This work was supported by US DOE Grant DE-SC0023707 under the Office of Nuclear Physics Quantum Horizons program for the "NuNuclei and Hadrons with Quantum computers (NuHaQ)" project (W.D., J.V.). This work was supported by Computing Sciences 2023 Summer Program at Berkeley Lab (D.L.). This work was also supported by the Quantum Systems Accelerator program  (L.L., C.Y.),  by  the  Applied Mathematics Program of the US Department of Energy (DOE) Office of Advanced Scientific Computing Research under contract number DE-AC02-05CH1123 (L.L.), the DOE Office of Advanced Scientific Computing Research Accelerated Research for Quantum Computing (ARQC) Program (C.Y.) under U.S. Department of Energy Contract No. DE-AC02-05CH11231. L.L. is a Simons Investigator in Mathematics.

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The NOvA Collaboration

# Measurement of the Double-Differential Muon-neutrino Charged-Current
Inclusive Cross Section in the NOvA Near Detector

M. A. Acero  Universidad del Atlantico, Carrera 30 No. 8-49, Puerto Colombia,
Atlantico, Colombia  P. Adamson  Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA  L. Aliaga  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  N. Anfimov  Joint Institute for
Nuclear Research, Dubna, Moscow region 141980, Russia  A. Antoshkin  Joint
Institute for Nuclear Research, Dubna, Moscow region 141980, Russia  E.
Arrieta-Diaz  Universidad del Magdalena, Carrera 32 No 22-08 Santa Marta,
Colombia  L. Asquith  Department of Physics and Astronomy, University of
Sussex, Falmer, Brighton BN1 9QH, United Kingdom  A. Aurisano  Department of
Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA  A. Back
Indiana University, Bloomington, Indiana 47405, USA  Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  C. Backhouse  Physics
and Astronomy Department, University College London, Gower Street, London WC1E
6BT, United Kingdom  M. Baird  Indiana University, Bloomington, Indiana 47405,
USA  Department of Physics and Astronomy, University of Sussex, Falmer,
Brighton BN1 9QH, United Kingdom  Department of Physics, University of
Virginia, Charlottesville, Virginia 22904, USA  N. Balashov  Joint Institute
for Nuclear Research, Dubna, Moscow region 141980, Russia  P. Baldi
Department of Physics and Astronomy, University of California at Irvine,
Irvine, California 92697, USA  B. A. Bambah  School of Physics, University of
Hyderabad, Hyderabad, 500 046, India  S. Bashar  Department of Physics and
Astronomy, Tufts University, Medford, Massachusetts 02155, USA  K. Bays
California Institute of Technology, Pasadena, California 91125, USA  Illinois
Institute of Technology, Chicago IL 60616, USA  B. Behera  Department of
Physics, IIT Hyderabad, Hyderabad, 502 205, India  R. Bernstein  Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA  V. Bhatnagar
Department of Physics, Panjab University, Chandigarh, 160 014, India  D.
Bhattarai  University of Mississippi, University, Mississippi 38677, USA  B.
Bhuyan  Department of Physics, IIT Guwahati, Guwahati, 781 039, India  J. Bian
Department of Physics and Astronomy, University of California at Irvine,
Irvine, California 92697, USA  School of Physics and Astronomy, University of
Minnesota Twin Cities, Minneapolis, Minnesota 55455, USA  J. Blair  Department
of Physics, University of Houston, Houston, Texas 77204, USA  A. C. Booth
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  R. Bowles  Indiana University, Bloomington, Indiana
47405, USA  C. Bromberg  Department of Physics and Astronomy, Michigan State
University, East Lansing, Michigan 48824, USA  N. Buchanan  Department of
Physics, Colorado State University, Fort Collins, CO 80523-1875, USA  A.
Butkevich  Institute for Nuclear Research of Russia, Academy of Sciences 7a,
60th October Anniversary prospect, Moscow 117312, Russia  S. Calvez
Department of Physics, Colorado State University, Fort Collins, CO 80523-1875,
USA  T. J. Carroll  Department of Physics, University of Texas at Austin,
Austin, Texas 78712, USA  Department of Physics, University of Wisconsin-
Madison, Madison, Wisconsin 53706, USA  E. Catano-Mur  Department of Physics,
William & Mary, Williamsburg, Virginia 23187, USA  B. C. Choudhary  Department
of Physics and Astrophysics, University of Delhi, Delhi 110007, India  A.
Christensen  Department of Physics, Colorado State University, Fort Collins,
CO 80523-1875, USA  T. E. Coan  Department of Physics, Southern Methodist
University, Dallas, Texas 75275, USA  M. Colo  Department of Physics, William
& Mary, Williamsburg, Virginia 23187, USA  L. Cremonesi  School of Physics and
Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom
Physics and Astronomy Department, University College London, Gower Street,
London WC1E 6BT, United Kingdom  G. S. Davies  University of Mississippi,
University, Mississippi 38677, USA  Indiana University, Bloomington, Indiana
47405, USA  P. F. Derwent  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  P. Ding  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  Z. Djurcic  Argonne National Laboratory, Argonne,
Illinois 60439, USA  M. Dolce  Department of Physics and Astronomy, Tufts
University, Medford, Massachusetts 02155, USA  D. Doyle  Department of
Physics, Colorado State University, Fort Collins, CO 80523-1875, USA  D.
Dueñas Tonguino  Department of Physics, University of Cincinnati, Cincinnati,
Ohio 45221, USA  E. C. Dukes  Department of Physics, University of Virginia,
Charlottesville, Virginia 22904, USA  H. Duyang  Department of Physics and
Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA
S. Edayath  Department of Physics, Cochin University of Science and
Technology, Kochi 682 022, India  R. Ehrlich  Department of Physics,
University of Virginia, Charlottesville, Virginia 22904, USA  M. Elkins
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  E. Ewart  Indiana University, Bloomington, Indiana 47405, USA  G. J.
Feldman  Department of Physics, Harvard University, Cambridge, Massachusetts
02138, USA  P. Filip  Institute of Physics, The Czech Academy of Sciences, 182
21 Prague, Czech Republic  J. Franc  Czech Technical University in Prague,
Brehova 7, 115 19 Prague 1, Czech Republic  M. J. Frank  Department of
Physics, University of South Alabama, Mobile, Alabama 36688, USA  H. R.
Gallagher  Department of Physics and Astronomy, Tufts University, Medford,
Massachusetts 02155, USA  R. Gandrajula  Department of Physics and Astronomy,
Michigan State University, East Lansing, Michigan 48824, USA  Department of
Physics, University of Virginia, Charlottesville, Virginia 22904, USA  F. Gao
Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania
15260, USA  A. Giri  Department of Physics, IIT Hyderabad, Hyderabad, 502 205,
India  R. A. Gomes  Instituto de Física, Universidade Federal de Goiás,
Goiânia, Goiás, 74690-900, Brazil  M. C. Goodman  Argonne National Laboratory,
Argonne, Illinois 60439, USA  V. Grichine  Nuclear Physics and Astrophysics
Division, Lebedev Physical Institute, Leninsky Prospect 53, 119991 Moscow,
Russia  M. Groh  Department of Physics, Colorado State University, Fort
Collins, CO 80523-1875, USA  Indiana University, Bloomington, Indiana 47405,
USA  R. Group  Department of Physics, University of Virginia, Charlottesville,
Virginia 22904, USA  B. Guo  Department of Physics and Astronomy, University
of South Carolina, Columbia, South Carolina 29208, USA  A. Habig  Department
of Physics and Astronomy, University of Minnesota Duluth, Duluth, Minnesota
55812, USA  F. Hakl  Institute of Computer Science, The Czech Academy of
Sciences, 182 07 Prague, Czech Republic  A. Hall  Department of Physics,
University of Virginia, Charlottesville, Virginia 22904, USA  J. Hartnell
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  R. Hatcher  Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA  H. Hausner  Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA  M. He  Department of
Physics, University of Houston, Houston, Texas 77204, USA  K. Heller  School
of Physics and Astronomy, University of Minnesota Twin Cities, Minneapolis,
Minnesota 55455, USA  J. Hewes  Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, USA  A. Himmel  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  A. Holin  Physics and Astronomy
Department, University College London, Gower Street, London WC1E 6BT, United
Kingdom  J. Huang  Department of Physics, University of Texas at Austin,
Austin, Texas 78712, USA  B. Jargowsky  Department of Physics and Astronomy,
University of California at Irvine, Irvine, California 92697, USA  J. Jarosz
Department of Physics, Colorado State University, Fort Collins, CO 80523-1875,
USA  F. Jediny  Czech Technical University in Prague, Brehova 7, 115 19 Prague
1, Czech Republic  C. Johnson  Department of Physics, Colorado State
University, Fort Collins, CO 80523-1875, USA  M. Judah  Department of Physics,
Colorado State University, Fort Collins, CO 80523-1875, USA  Department of
Physics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  I.
Kakorin  Joint Institute for Nuclear Research, Dubna, Moscow region 141980,
Russia  A. Kalitkina  Joint Institute for Nuclear Research, Dubna, Moscow
region 141980, Russia  D. Kalra  Department of Physics, Panjab University,
Chandigarh, 160 014, India  D. M. Kaplan  Illinois Institute of Technology,
Chicago IL 60616, USA  R. Keloth  Department of Physics, Cochin University of
Science and Technology, Kochi 682 022, India  O. Klimov  Joint Institute for
Nuclear Research, Dubna, Moscow region 141980, Russia  L. W. Koerner
Department of Physics, University of Houston, Houston, Texas 77204, USA  L.
Kolupaeva  Joint Institute for Nuclear Research, Dubna, Moscow region 141980,
Russia  S. Kotelnikov  Nuclear Physics and Astrophysics Division, Lebedev
Physical Institute, Leninsky Prospect 53, 119991 Moscow, Russia  R. Kralik
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  Ch. Kullenberg  Joint Institute for Nuclear Research,
Dubna, Moscow region 141980, Russia  M. Kubu  Czech Technical University in
Prague, Brehova 7, 115 19 Prague 1, Czech Republic  A. Kumar  Department of
Physics, Panjab University, Chandigarh, 160 014, India  C. D. Kuruppu
Department of Physics and Astronomy, University of South Carolina, Columbia,
South Carolina 29208, USA  V. Kus  Czech Technical University in Prague,
Brehova 7, 115 19 Prague 1, Czech Republic  T. Lackey  Indiana University,
Bloomington, Indiana 47405, USA  K. Lang  Department of Physics, University of
Texas at Austin, Austin, Texas 78712, USA  P. Lasorak  Department of Physics
and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
J. Lesmeister  Department of Physics, University of Houston, Houston, Texas
77204, USA  S. Lin  Department of Physics, Colorado State University, Fort
Collins, CO 80523-1875, USA  A. Lister  Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA  J. Liu  Department of
Physics and Astronomy, University of California at Irvine, Irvine, California
92697, USA  M. Lokajicek  Institute of Physics, The Czech Academy of Sciences,
182 21 Prague, Czech Republic  S. Magill  Argonne National Laboratory,
Argonne, Illinois 60439, USA  M. Manrique Plata  Indiana University,
Bloomington, Indiana 47405, USA  W. A. Mann  Department of Physics and
Astronomy, Tufts University, Medford, Massachusetts 02155, USA  M. L. Marshak
School of Physics and Astronomy, University of Minnesota Twin Cities,
Minneapolis, Minnesota 55455, USA  M. Martinez-Casales  Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA  V. Matveev
Institute for Nuclear Research of Russia, Academy of Sciences 7a, 60th October
Anniversary prospect, Moscow 117312, Russia  B. Mayes  Department of Physics
and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
D. P. Méndez  Department of Physics and Astronomy, University of Sussex,
Falmer, Brighton BN1 9QH, United Kingdom  M. D. Messier  Indiana University,
Bloomington, Indiana 47405, USA  H. Meyer  Department of Mathematics,
Statistics, and Physics, Wichita State University, Wichita, Kansas 67206, USA
T. Miao  Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
W. H. Miller  School of Physics and Astronomy, University of Minnesota Twin
Cities, Minneapolis, Minnesota 55455, USA  S. R. Mishra  Department of Physics
and Astronomy, University of South Carolina, Columbia, South Carolina 29208,
USA  A. Mislivec  School of Physics and Astronomy, University of Minnesota
Twin Cities, Minneapolis, Minnesota 55455, USA  R. Mohanta  School of Physics,
University of Hyderabad, Hyderabad, 500 046, India  A. Moren  Department of
Physics and Astronomy, University of Minnesota Duluth, Duluth, Minnesota
55812, USA  A. Morozova  Joint Institute for Nuclear Research, Dubna, Moscow
region 141980, Russia  W. Mu  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  L. Mualem  California Institute of Technology, Pasadena,
California 91125, USA  M. Muether  Department of Mathematics, Statistics, and
Physics, Wichita State University, Wichita, Kansas 67206, USA  K. Mulder
Physics and Astronomy Department, University College London, Gower Street,
London WC1E 6BT, United Kingdom  D. Naples  Department of Physics, University
of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  N. Nayak  Department of
Physics and Astronomy, University of California at Irvine, Irvine, California
92697, USA  J. K. Nelson  Department of Physics, William & Mary, Williamsburg,
Virginia 23187, USA  R. Nichol  Physics and Astronomy Department, University
College London, Gower Street, London WC1E 6BT, United Kingdom  E. Niner  Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA  A. Norman
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA  A.
Norrick  Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
T. Nosek  Charles University, Faculty of Mathematics and Physics, Institute of
Particle and Nuclear Physics, Prague, Czech Republic  H. Oh  Department of
Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA  A. Olshevskiy
Joint Institute for Nuclear Research, Dubna, Moscow region 141980, Russia  T.
Olson  Department of Physics and Astronomy, Tufts University, Medford,
Massachusetts 02155, USA  J. Ott  Department of Physics and Astronomy,
University of California at Irvine, Irvine, California 92697, USA  J. Paley
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA  R. B.
Patterson  California Institute of Technology, Pasadena, California 91125, USA
G. Pawloski  School of Physics and Astronomy, University of Minnesota Twin
Cities, Minneapolis, Minnesota 55455, USA  O. Petrova  Joint Institute for
Nuclear Research, Dubna, Moscow region 141980, Russia  R. Petti  Department of
Physics and Astronomy, University of South Carolina, Columbia, South Carolina
29208, USA  D. D. Phan  Department of Physics, University of Texas at Austin,
Austin, Texas 78712, USA  Physics and Astronomy Department, University College
London, Gower Street, London WC1E 6BT, United Kingdom  R. K. Plunkett  Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA  J. C. C. Porter
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  A. Rafique  Argonne National Laboratory, Argonne,
Illinois 60439, USA  V. Raj  California Institute of Technology, Pasadena,
California 91125, USA  M. Rajaoalisoa  Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, USA  B. Ramson  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  B. Rebel  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA  P. Rojas  Department of
Physics, Colorado State University, Fort Collins, CO 80523-1875, USA  P. Roy
Department of Mathematics, Statistics, and Physics, Wichita State University,
Wichita, Kansas 67206, USA  V. Ryabov  Nuclear Physics and Astrophysics
Division, Lebedev Physical Institute, Leninsky Prospect 53, 119991 Moscow,
Russia  K. Sachdev  Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA  O. Samoylov  Joint Institute for Nuclear Research, Dubna, Moscow
region 141980, Russia  M. C. Sanchez  Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA  S. Sánchez Falero  Department of
Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  P.
Shanahan  Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
A. Sheshukov  Joint Institute for Nuclear Research, Dubna, Moscow region
141980, Russia  P. Singh  Department of Physics and Astrophysics, University
of Delhi, Delhi 110007, India  V. Singh  Department of Physics, Institute of
Science, Banaras Hindu University, Varanasi, 221 005, India  E. Smith  Indiana
University, Bloomington, Indiana 47405, USA  J. Smolik  Czech Technical
University in Prague, Brehova 7, 115 19 Prague 1, Czech Republic  P. Snopok
Illinois Institute of Technology, Chicago IL 60616, USA  N. Solomey
Department of Mathematics, Statistics, and Physics, Wichita State University,
Wichita, Kansas 67206, USA  A. Sousa  Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, USA  K. Soustruznik  Charles University,
Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics,
Prague, Czech Republic  M. Strait  School of Physics and Astronomy, University
of Minnesota Twin Cities, Minneapolis, Minnesota 55455, USA  L. Suter  Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA  A. Sutton
Department of Physics, University of Virginia, Charlottesville, Virginia
22904, USA  S. Swain  National Institute of Science Education and Research,
Khurda, 752050, Odisha, India  C. Sweeney  Physics and Astronomy Department,
University College London, Gower Street, London WC1E 6BT, United Kingdom  A.
Sztuc  Physics and Astronomy Department, University College London, Gower
Street, London WC1E 6BT, United Kingdom  B. Tapia Oregui  Department of
Physics, University of Texas at Austin, Austin, Texas 78712, USA  P. Tas
Charles University, Faculty of Mathematics and Physics, Institute of Particle
and Nuclear Physics, Prague, Czech Republic  T. Thakore  Department of
Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA  R. B.
Thayyullathil  Department of Physics, Cochin University of Science and
Technology, Kochi 682 022, India  J. Thomas  Physics and Astronomy Department,
University College London, Gower Street, London WC1E 6BT, United Kingdom
Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin
53706, USA  E. Tiras  Department of Physics, Erciyes University, Kayseri
38030, Turkey  Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA  J. Tripathi  Department of Physics, Panjab University,
Chandigarh, 160 014, India  J. Trokan-Tenorio  Department of Physics, William
& Mary, Williamsburg, Virginia 23187, USA  A. Tsaris  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  Y. Torun  Illinois
Institute of Technology, Chicago IL 60616, USA  J. Urheim  Indiana University,
Bloomington, Indiana 47405, USA  P. Vahle  Department of Physics, William &
Mary, Williamsburg, Virginia 23187, USA  Z. Vallari  California Institute of
Technology, Pasadena, California 91125, USA  J. Vasel  Indiana University,
Bloomington, Indiana 47405, USA  P. Vokac  Czech Technical University in
Prague, Brehova 7, 115 19 Prague 1, Czech Republic  T. Vrba  Czech Technical
University in Prague, Brehova 7, 115 19 Prague 1, Czech Republic  M. Wallbank
Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA
T. K. Warburton  Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA  M. Wetstein  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  D. Whittington  Department of
Physics, Syracuse University, Syracuse NY 13210, USA  Indiana University,
Bloomington, Indiana 47405, USA  D. A. Wickremasinghe  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  S. G. Wojcicki
Department of Physics, Stanford University, Stanford, California 94305, USA
J. Wolcott  Department of Physics and Astronomy, Tufts University, Medford,
Massachusetts 02155, USA  W. Wu  Department of Physics and Astronomy,
University of California at Irvine, Irvine, California 92697, USA  Y. Xiao
Department of Physics and Astronomy, University of California at Irvine,
Irvine, California 92697, USA  A. Yallappa Dombara  Department of Physics,
Syracuse University, Syracuse NY 13210, USA  A. Yankelevich  Department of
Physics and Astronomy, University of California at Irvine, Irvine, California
92697, USA  K. Yonehara  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  S. Yu  Argonne National Laboratory, Argonne, Illinois
60439, USA  Illinois Institute of Technology, Chicago IL 60616, USA  Y. Yu
Illinois Institute of Technology, Chicago IL 60616, USA  S. Zadorozhnyy
Institute for Nuclear Research of Russia, Academy of Sciences 7a, 60th October
Anniversary prospect, Moscow 117312, Russia  J. Zalesak  Institute of Physics,
The Czech Academy of Sciences, 182 21 Prague, Czech Republic  Y. Zhang
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  R. Zwaska  Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA

(August 27, 2024)

###### Abstract

We report cross-section measurements of the final-state muon kinematics for
$\nu_{\mu}$ charged-current interactions in the NOvA near detector using an
accumulated 8.09$\times 10^{20}$ protons-on-target (POT) in the NuMI beam. We
present the results as a double-differential cross section in the observed
outgoing muon energy and angle, as well as single-differential cross sections
in the derived neutrino energy, $E_{\nu}$, and square of the four-momentum
transfer, $Q^{2}$. We compare the results to inclusive cross-section
predictions from various neutrino event generators via $\chi^{2}$ calculations
using a covariance matrix that accounts for bin-to-bin correlations of
systematic uncertainties. These comparisons show a clear discrepancy between
the data and each of the tested predictions at forward muon angle and low
$Q^{2}$, indicating a missing suppression of the cross section in current
neutrino-nucleus scattering models.

††preprint: PUB-21-455-ND-PPD-SCD

##  1 Introduction

Neutrino scattering on nuclei is a rich topic with many challenges, both
experimentally and theoretically. Experimentally it is challenging to produce
a well characterized source of neutrinos and to collect high statistics with
high resolution detectors. Accordingly, many previous measurements are limited
by large statistical and/or systematic uncertainties Filkins  _et al._ (2020);
Coplowe  _et al._ (2020); Abratenko  _et al._ (2020, 2019); Abe  _et al._
(2017, 2020a). Theoretical challenges arise from a lack of accurate models
that are valid across a large range of energies and account for the initial
state of the nuclear environment and final-state interactions Alvarez-Ruso
_et al._ (2018).

In current and future long-baseline neutrino flavor oscillation experiments,
beams of muon (anti)neutrinos are used to precisely measure the rate of muon
(anti)neutrino disappearance and electron (anti)neutrino appearance Acero  _et
al._ (2018); Abe  _et al._ (2020b); Abi  _et al._ (2020); Abe  _et al._
(2018). Weak charged-current (CC) interactions, in which a charged lepton is
produced in the final state, are used to identify neutrino flavor and measure
the neutrino energy. The accuracy of these measurements depends explicitly on
the kinematics of the lepton and hadrons visible in the final state.

To relate these final state observables to the energy of the neutrino,
accurate knowledge of neutrino-nucleon interaction cross sections and the
dynamics of the propagation of particles through nuclear matter is necessary.
Many of the uncertainties in neutrino oscillation parameters that arise from
limited understanding of neutrino cross sections are reduced by using a two
detector scheme with a near detector placed close to the beam source to
characterize interactions prior to oscillation Abe  _et al._ (2015); Wolcott
(2019), and a far detector placed much farther away to measure the oscillated
neutrino spectra. However, the near and far detectors are typically
substantially different in size and have differing acceptances of the final-
state particles produced by neutrino interactions in the detector. Knowledge
of kinematic distributions of the final-state leptons is crucial to correctly
account for differences in event selection efficiency and purity between the
two detectors. In practice, experiments rely on neutrino event generators for
this knowledge.

Neutrino interactions are typically characterized by the type of target (e.g.,
individual nucleons, pairs of nucleons, the nucleus as a whole or electrons)
and the particles produced in the interaction. At around
$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ in neutrino energy, quasielastic
(QE) scattering dominates, in which the neutrino scatters off a single
nucleon, producing a lepton and a single unbound nucleon. At these energies,
meson exchange currents (MEC) between pairs of correlated nucleons resulting
in 2-particle-2-hole (2p2h) interactions also significantly contribute to the
neutrino scattering rate Acero  _et al._ (2020); Rodrigues  _et al._ (2016);
Dolan  _et al._ (2017). Around 2 GeV, resonant (Res) interactions contribute
significantly to the total cross section. In these interactions, intermediate
hadronic excited states are created inside the nucleus (predominantly those
associated with $\Delta(1232)$ resonances) that decay to a baryon and a meson.
At energies above 3 GeV, shallow- and deep-inelastic scattering become more
prevalent. Other rarer interactions, such as neutrino-electron and coherent
scattering off the entire nucleus (COH) also contribute to the total cross
section.

Nuclear effects play a significant role in the initial and final states of the
interaction. In addition to introducing new primary processes such as MEC as
noted above, at GeV energies the initial state of the nucleus influences the
kinematics of the particles produced in the interaction Rodrigues  _et al._
(2016); Gran  _et al._ (2018); Dolan  _et al._ (2017). Furthermore, these
particles must traverse the nuclear medium, during which scattering and
interactions may occur. These final-state interactions (FSI) alter the
kinematics and possibly the composition of the final state Alvarez-Ruso  _et
al._ (2018).

The inclusive cross section, $\sigma_{\mathrm{incl}}$, is the sum of the cross
sections of all of the individual processes. As such, predictions of the
inclusive cross section must properly combine the individual processes,
including interference terms. Measurements of inclusive cross sections serve
to constrain the quantum-mechanical sum of these processes, as well as their
dependence on the neutrino energy ($E_{\nu}$) and square of the four-momentum
transfer from the lepton system ($Q^{2}$), and the impact of final-state
interactions.

We report the flux-integrated double-differential inclusive cross section of
neutrino-nucleus CC interactions in the NOvA near detector, $\nu_{\mu}$ \+ A
$\rightarrow\mu+\mathrm{X}$, where A is a target nucleus (see Table 1) and X
represents all other final state particles. The measurement is differential
with respect to the final-state muon’s kinetic energy and angle relative to
the neutrino beam direction. We also report the inclusive cross section as a
function of the derived $E_{\nu}$ and $Q^{2}$, integrated over the range of
muon kinematics reported in the double-differential measurement.

##  2 The NOvA Experiment

Figure 1: Neutrino beam component spectra integrated over the NOvA near
detector fiducial volume. From top to bottom: muon-neutrinos (solid line),
anti-muon-neutrinos (dashed line), electron- neutrinos (dotted line) and anti-
electron-neutrinos (dashed-dotted line).

NOvA is a long-baseline neutrino experiment Acero  _et al._ (2019) designed to
measure neutrino flavor oscillations. A 96% pure muon-neutrino beam is
produced at Fermilab. Two functionally identical detectors are directly
exposed to the beam: the near detector located
$1\text{\,}\mathrm{k}\mathrm{m}$ downstream of the beam target, and the far
detector located $810\text{\,}\mathrm{k}\mathrm{m}$ away from the target near
Ash River, Minnesota. The primary measurements of electron (anti)neutrino
appearance and muon (anti)neutrino disappearance provide constraints on the
neutrino mixing parameters, $\theta_{23}$ and $\Delta\text{m}^{2}_{32}$, the
neutrino mass ordering, and the CP-violating parameter, $\delta$. The high
statistics neutrino and antineutrino samples gathered at the near detector
constrain the flux and neutrino cross-section parameters for the oscillation
analyses, and are also ideal for measurements of various neutrino interaction
cross sections.

Neutrinos for NOvA are provided by the Fermilab NuMI beam Adamson  _et al._
(2016). The Fermilab Main Injector protons at
$120\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ strike a graphite target,
producing pions and kaons. These hadrons are focused by two magnetic horns and
directed towards a $650\text{\,}\mathrm{m}$ drift region where they decay to
produce primarily muons and muon neutrinos. The horn polarity can be changed
to focus positive (negative) mesons and produce a primarily (anti)neutrino
beam. The NOvA detectors are located
$14.6\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ off-axis from the
central beam direction, resulting in an incident neutrino energy spectrum
narrowly peaked at $1.8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. Figure 1
shows the flux at the NOvA near detector in the neutrino beam configuration.
The neutrino beam includes a 1.8$\%$ intrinsic $\bar{\nu}_{\mu}$ component
coming from opposite-sign meson decay in the energy range of interest for this
measurement, between 1 and 5 GeV. There is also an electron neutrino and
antineutrino contribution of 0.7% in this energy range.

The NOvA near detector is a tracking calorimeter with $193\text{\,}\mathrm{t}$
of active mass, located $100\text{\,}\mathrm{m}$ underground. The detector is
composed of planes of hollow cells made from a custom formulation of extruded
PVC. The planes are segmented into $3.9\text{\,}\mathrm{c}\mathrm{m}$ wide
cells that are $3.9\text{\,}\mathrm{m}$ long; the depth of each plane in the
beam direction is $6.6\text{\,}\mathrm{c}\mathrm{m}$. The planes are
alternated in horizontal and vertical orientations perpendicular to the beam,
allowing full 3D tracking for $12.7\text{\,}\mathrm{m}$ along the beam axis.
Each cell is filled with liquid scintillator, a blend of 95% mineral oil and
5% pseudocumene with trace concentrations of wavelength shifting fluors. The
resulting composition by mass is about 63% scintillator and 37% PVC with
nuclear targets for neutrino interactions in the detector as described in Tab.
1. When a particle traverses the detector, wavelength shifting fiber in the
PVC cells collect and deliver scintillation light to avalanche photodiodes.
The resulting signals are digitized by custom front-end electronics and all
signals above a noise-vetoing threshold are sent to a data buffer. A timestamp
sent from the Fermilab accelerator prior to the pulsed delivery of a
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$-long beam spill starts the
recording of $550\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ of data,
which is saved for analysis.

The downstream end of the detector is a “muon catcher” designed to improve
containment of muons produced in neutrino interactions up to
$\sim$$2.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. The muon catcher consists
of 10 layers of $10\text{\,}\mathrm{c}\mathrm{m}$ thick steel absorbers
interleaved with pairs of PVC/scintillator. The muon catcher planes span the
full detector width and the lower 2/3rds of the detector height.

Table 1: Mass contributions from various elements in the fiducial volume used in this analysis.  Element | Mass [kg] | Fraction of Total
---|---|---
C | 43,061 | 0.67
Cl | 10,408 | 0.16
H | 6,943 | 0.11
Ti | 2,085 | 0.03
O | 1,930 | 0.03
Others | 174 |  $<$ 0.01

##  3 Simulation

Simulation is used in this analysis to calculate the integrated flux,
selection efficiencies and purities, estimate energies, and effects of
detector resolutions. The analysis also relies on simulation to optimize event
selection criteria and assess various systematic uncertainties that can impact
event rates and selection efficiency and purity. The simulation is a chain of
steps that begins with the generation of the neutrino beam and transport of
all particles through the beamline to the detector. Interactions of the
neutrinos with the detector are then generated, after which the final-state
particles are transported through the detector. The generation, detection and
digitization of light in the detector are the final steps of the simulation
chain. Each step of the simulation chain, described below, is matched to the
real data-taking conditions in beam intensity and total protons-on-target,
wherever appropriate.

The NuMI flux predictions start with a detailed simulation of the beamline
components and the hadronic showers that follow the primary proton striking a
long graphite target until the mesons decay to neutrinos. The simulation is
based on GEANT4 v9.2.p03 Agostinelli  _et al._ (2003) with the FTFP BERT
hadronic model. The hadron production model is adjusted using the PPFX
package, which uses external measurements on thin targets with the procedure
outlined in Ref. Aliaga  _et al._ (2016). The NuMI flux prediction for the
neutrino beam mode at the NOvA near detector is shown in Fig. 2.

Figure 2: Muon-neutrino flux spectrum at the NOvA near detector below
$5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. The shaded band represents the
total 1-$\sigma$ flux uncertainty (hadron production and beam optics). The
ratio in the lower panel is the size of the flux correction using PPFX Aliaga
_et al._ (2016) with respect to the GEANT4 v9.2.p03 Agostinelli  _et al._
(2003) FTFP BERT hadronic model.

The simulated neutrino flux is passed through a detailed description of the
NOvA near detector geometry, including surrounding rock, where interactions
are simulated with the GENIE v2.12.2 Andreopoulos  _et al._ (2010, 2015) event
generator. The initial state is simulated via the default Smith and Moniz
global relativistic Fermi gas (RFG) model Smith and Moniz (1975). Short-range
nuclear correlations in the initial state Subedi  _et al._ (2008) are
accounted for by the addition of a high-momentum tail of the Fermi momentum
distribution for single nucleons Bodek and Ritchie (1981). The QE interactions
are simulated according to the formalism of Llewellyn Smith Llewellyn Smith
(1972). 2p2h interactions are simulated using the Empirical MEC model Katori
(2015). Charged-current Res interactions are simulated via the Rein and Sehgal
model Rein and Sehgal (1981). Inelastic scattering over a large range of
hadronic invariant masses, resulting in a range of final state hadrons, is
simulated using the Bodek-Yang scaling formalism Bodek and Yang (2003) coupled
to a custom hadronization model Yang  _et al._ (2009) and PYTHIA 6 T.
Sjostrand and Skands (2006). Charged-curret COH interactions are simulated
using Rein and Sehgal (1983, 2007).

The GENIE output has been adjusted to incorporate advances in theory and
experimental data Acero  _et al._ (2020). These modifications include
adjustments to the CCQE and non-resonant pion production interactions based on
reevaluated bubble chamber measurements; improved nuclear models of CCQE
kinematics; and suppression at low $Q^{2}$ of resonant pion production. After
applying these modifications, differences in the shapes of the energy and
three-momentum transfer ($q_{0}$,$|\vec{q}|$) distributions of near detector
data and simulation were used to tune the Empirical MEC model. These
adjustments significantly enhance the agreement between selected muon-neutrino
candidates in the NOvA near detector data and simulation across multiple
kinematic variables such as visible hadronic energy and reconstructed four-
momentum transfer squared.

GEANT4 v10.1.p3 is used to simulate energy deposited in the NOvA near detector
from the particles generated by neutrino interactions. A custom simulation
tuned to reproduce measured scintillator response and fiber attenuation
properties is then used to model and transfer scintillation and Cherenkov
light Aurisano  _et al._ (2015). Test-stand measurements are used to tune the
Birk’s suppression of the scintillation light and to validate the response of
the readout electronics in the simulation Anfimov  _et al._ (2020).

Figure 3: Simulated muon (hashed blue) and non-muon (dashed red) track
distributions in: $dE/dx$ log-likelihood differences between that of a muon
and a pion (top left), multiple scattering log-likelihood differences (top
right), average $dE/dx$ in last $10\text{\,}\mathrm{c}\mathrm{m}$ (bottom
left) and average $dE/dx$ in last $40\text{\,}\mathrm{c}\mathrm{m}$ (bottom
right) used in the MuonID selector. Muon distributions are normalized to data
exposure ($8.09\times 10^{20}$ POT), non-muon distributions are normalized by
area to the muon distributions.

##  4 Event Reconstruction and Calibration

Figure 4: Left: Stacked distributions of the simulated maximum MuonID in each
reconstructed event for signal (hashed blue), neutral current (NC, dashed red)
and electron-neutrino (solid green) backgrounds. Right: Fractional uncertainty
on the selection efficiency, selection purity and FOM versus the required
largest MuonID value per event. Candidate $\nu_{\mu}$ CC events are retained
with a requirement of largest MuonID value in an event $>$ 0.24.

Energy deposits (hits) in the detector are recorded with pulse height, time
and channel location information. Cell-to-cell variations in pulse height are
first corrected using through-going muons, followed by the calibration of
absolute energy deposition using minimum-ionizing portions of stopping cosmic
ray muon tracks. The reconstruction of neutrino interactions first clusters
hits that are correlated in space and time Baird (2015). These clusters of
hits are all assumed to be associated with a single neutrino interaction,
referred to as an event. Hits in an event are then grouped into possible
particle trajectories (tracks) via a Kalman filter-based algorithm in both the
horizontal and vertical two-dimensional detector views Baird  _et al._ (2015).
Three-dimensional tracks are formed by combining tracks from the two views
based on their overlap in the longitudinal direction. The track reconstruction
algorithm assumes a start point at the most upstream hit, and requires a
minimum of 4 hits in each detector view. A different algorithm Niner (2015) is
used to form particle trajectories (prongs) from hits associated with a
reconstructed vertex, and requires a minimum of 1 hit in each detector view.
As described later, tracks and prongs are used for different purposes in this
analysis.

##  5 Event Selection

Candidate events are required to have a reconstructed track which crosses more
than 4 contiguous planes, made up of hits in at least 20 unique cells. The
candidate muon track, described below, is required to start inside a
$2.7\text{\,}\mathrm{m}$ $\times$ $2.7\text{\,}\mathrm{m}$ $\times$
$9\text{\,}\mathrm{m}$ fiducial volume inside the detector upstream of the
muon catcher. We require all tracks and prongs identified in the event to have
stopped several cm before reaching any detector edge to ensure containment of
all the neutrino energy. We further require that no track or shower other than
the selected muon enter the muon catcher. These criteria select 23% of signal
events.

The signature of a $\nu_{\mu}$ CC interaction is the presence of a muon in the
final state. This analysis implements a multivariate muon identification
algorithm, MuonID, based on energy deposition and scattering observables (see
Fig. 3). For energy deposition, we use the difference between log-likelihood
functions based on the $dE/dx$ of a muon and a pion, the average $dE/dx$ in
the cells of the last $10\text{\,}\mathrm{c}\mathrm{m}$ of the track
trajectory, and average $dE/dx$ in the cells of the last
$40\text{\,}\mathrm{c}\mathrm{m}$ of the track trajectory. We also use
distributions of the difference between log-likelihood functions based on the
angular deflections along the trajectory of the reconstructed track for muons
and pions. These reconstructed variables are used as input to a boosted
decision tree (BDT) algorithm, the output of which is a MuonID score. The BDT
is trained on all true muon tracks and true non-muon tracks using
reconstructed simulated neutrino interactions that have passed the
preselection criteria described above. The samples used to train and test
these algorithms are drawn from non-overlaping subsamples that each comprise
10% of the overall simulated sample. The distributions of the highest MuonID
score in signal and background events passing the preselection are shown in
the left-side plot of Fig. 4.

As this measurement is systematically limited, we optimize the MuonID
selection criteria by minimizing a figure-of-merit (FOM) that is approximately
the fractional uncertainty on the total cross section:

$\mathrm{FOM}=\left(\frac{\delta_{\epsilon}}{\epsilon}\right)^{2}+\left(\frac{\delta_{P}}{P}\right)^{2}\,,$
(1)

where $\epsilon$ is the selection efficiency and $P$ is the selection purity.
The sources of uncertainty considered for the selection criteria optimization
are neutrino interaction modeling, energy scale uncertainties, and the
modeling of light generation and propagation in the detector. These sources of
systematic uncertainty, described in more detail in Sec. 8, have the greatest
impact on muon identification. The right plot in Fig. 4 shows the uncertainty
on the purity and efficiency, and the FOM as a function of the minimum MuonID
value in the event. Signal events with MuonID greater than 0.24 are retained
as candidate $\nu_{\mu}$ CC interactions, resulting in an overall 98%
selection efficiency and overall 97% selection purity after the previously
described selection criteria are applied. The muon is correctly identified in
98.8% of signal events. The neutrino interaction vertex is taken as the most
upstream position of the selected muon track.

##  6 Energy Reconstruction, Resolution and Binning

The muon and muon-neutrino energy estimators developed for this analysis rely
on the simulation to relate the muon energy to the length of the reconstructed
muon track. Muons that stop before reaching the muon catcher are reconstructed
with a typical energy resolution of 4%; those that stop in the muon catcher
have a resolution of 5-6%.

We reconstruct the visible hadronic energy as the sum of calibrated energy of
hits in the event that are not associated with the muon track, plus any
additional energy that may be deposited by hadrons on and near the start of
the muon track. The latter is reconstructed by subtracting the energy of a
minimum ionizing particle from the first few planes of the muon track. We then
use the simulation to convert the visible hadronic energy to an estimate of
$E_{\mathrm{avail}}$ Coplowe  _et al._ (2020), the total true energy of the
visible hadrons in the final state.

Figure 5: Relative cross-section contributions for different interaction modes
(QE - purple dotted filling, MEC - red right diagonal lines, Res - orange left
diagonal lines, DIS - green horizontal lines, Other - all non $\nu_{\mu}$ CC
contributions in blue vertical lines) in the NOvA-tuned version of GENIE
v2.12.2 as a function of $T_{\mu}$ for each bin of $\cos\theta_{\mu}$.

The cross section is reported as a function of the kinetic energy of the muon,
$T_{\mu}$, and the cosine of the angle of the muon with respect to the
neutrino beam direction, $\cos\theta_{\mu}$. All bins are at least as wide as
the resolution estimated from simulated signal events that pass the selection.
The average $T_{\mu}$ resolution is
$50\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$, and the resolution of the average
muon angle is typically less than 4∘. We use 20 equal-sized bins from
$0.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}2.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$
for reconstructed $T_{\mu}$, and 13 variable-sized bins for reconstructed
$\cos\theta_{\mu}$ between 0.5 and 1. The choice of the variable-sized binning
in $\cos\theta_{\mu}$ accounts for both resolution and statistics, with
smaller bins in the most forward angles. The binning can be seen in Fig. 5,
discussed below.

##  7 The Measurement and Results

Figure 6: Event selection purity (top) and efficiency (bottom) versus
$T_{\mu}$ for each bin of $\cos\theta_{\mu}$ in representative ranges of
$E_{\mathrm{avail}}$. Solid lines are for 0.00 GeV $<E_{\mathrm{avail}}<$ 0.25
GeV, dashed oranges line are for 1.00 GeV $<E_{\mathrm{avail}}<$ 1.25 GeV,
dotted green lines are for 1.50 GeV $<E_{\mathrm{avail}}<$ 1.75 GeV, and
dashed-dotted blue lines are for 2.50 GeV $<E_{\mathrm{avail}}<$ 120 GeV.

The results presented in this paper use $8.09\times 10^{20}$ protons-on-target
(POT) collected between August 2014 and February 2017 in the neutrino beam
configuration. The double-differential cross section is determined as:

$\left(\frac{d^{2}\sigma_{\mathrm{incl}}}{d\cos\theta_{\mu}\,dT_{\mu}}\right)_{i}=\frac{1}{N_{\mathrm{target}}\,\phi\;}\sum_{E_{\mathrm{avail}}}\left(\frac{\sum_{j}U^{-1}_{ij}[N_{\mathrm{sel}}(\cos\theta_{\mu},T_{\mu},E_{\mathrm{avail}})_{j}\,P(\cos\theta_{\mu},T_{\mu},E_{\mathrm{avail}})_{j}]}{\epsilon(\cos\theta_{\mu},T_{\mu},E_{\mathrm{avail}})_{i}\,\Delta\cos\theta_{\mu_{i}}\,\Delta
T_{\mu_{i}}}\right)\,.$ (2)

Here $N_{\mathrm{sel}}$ is the number of selected events, $P$ is the selection
purity (the fraction of signal events among selected events), $\epsilon$ is
the selection efficiency (the fraction of signal events selected), $\phi$ is
the integrated neutrino flux, $N_{\mathrm{target}}$ is the number of nucleon
targets in the fiducial volume, $\Delta\cos\theta_{\mu}$ is the width of the
angle bin, and $\Delta T_{\mu}$ is the width of the muon kinetic energy bin.
An unfolding matrix, $U^{-1}_{ij}$, is used to relate the reconstructed
observable in bin $j$ to the true observable in bin $i$. As seen in Fig. 5,
the analysis receives non-negligible contributions from different interaction
modes, each with varying amounts of hadronic energy in the final state.
Hadrons in the final state can influence the purity, unfolding and efficiency,
for example when charged pions are misidentified as muons, or if a hadronic
shower hides the presence of the muon, or if the hadronic system is too close
to the edge of the detector and the event fails the containment criteria.
Therefore, a three-dimensional space involving the muon kinematics and
$E_{\mathrm{avail}}$ is used in applying the purity, unfolding, and efficiency
corrections to reduce potential model dependences on the final-state hadronic
system. Ten 250 MeV-wide bins (and one overflow bin) are used for
$E_{\mathrm{avail}}$. The corrected three-dimensional result is then
integrated over $E_{\mathrm{avail}}$.

In order to compare this measurement to theoretically based predictions, we
use the D’Agostini iterative unfolding algorithm D’Agostini (1995) as
implemented by the RooUnfold package Adye (2011) to correct for bin-to-bin
migrations of selected signal events. The number of iterations performed is a
regularization parameter that serves to reduce extreme variations in the
unfolded distribution that are consistent with the data given the predicted
smearing but are implausible given the underlying physics of the system. We
choose the number of iterations which minimizes the weighted mean of the
relative bias and variance across all bins using independent simulation
samples with randomized systematic shifts. The generation of systematically
shifted simulations is described in Sec. 8. The nominal simulation prediction
for the response matrix was used to unfold the shifted simulation data sets.
We found the optimum number of iterations for this analysis to be between 2
and 4 for a variety of shifted simulation data sets where both signal and
background normalizations and shapes were systematically shifted, and chose 3
iterations for the unfolding applied to the data.

The purity, $P$, and the efficiency, $\epsilon$, are shown in Fig. 6 vs.
$T_{\mu}$ for each bin of $\cos\theta_{\mu}$. Curves are drawn separately for
various representative ranges of $E_{\mathrm{avail}}$. Each angular grouping
shows a clear dependence of the purity on the muon kinetic energy. At low
$T_{\mu}$ our selection suffers from contamination by NC interactions. This
effect is more evident at higher available energy, as a higher fraction of
hadronic activity increases the chances of misidentification. The efficiency
increases with increasing $\cos\theta_{\mu}$, as at larger angles the muon is
more likely to escape via the side of the detector or less likely to be
clearly separated from hadronic activity in the detector and therefore less
likely to be reconstructed as a track or identified as a muon. The efficiency
decreases as a function of muon kinetic energy as higher energy muons are less
likely to be contained in the detector. We also observe a clear dependence on
$E_{\mathrm{avail}}$, as a larger fraction of hadronic activity makes the
event reconstruction and identification of the muon more difficult.
Comparisons of the purity and efficiency of the event selection with and
without the NOvA tune of the simulation were found to be in agreement within
systematic uncertainties.

Figure 7 shows the extracted double-differential cross section in slices of
muon angle. Figure 8 shows the extracted single differential cross section vs.
$Q^{2}$ and vs. of $E_{\nu}$, restricted to the phase-space of the double-
differential measurement. Efficiency, purity, and unfolding corrections are
simple functions of $Q^{2}$ ($E_{\nu}$) for these derived quantities. The data
are presented with total and statistical error bars in the plot, and the
values are also available in the table in Appendix A, and in electronic format
at the NOvA Experiment Data Releases webpage
111https://novaexperiment.fnal.gov/data-releases/. The data are compared to
predictions from GENIE v2.12.2 with and without the tune described in Sec. 3.
We observe better than 5% agreement between the data and the NOvA-tuned GENIE
v2.12.2 prediction across all muon angle slices, although small discrepancies
are still present, especially at low ($\sim$1 GeV) muon kinetic energies and
very forward angles. The tuning procedure does not significantly impact the
predictions at larger muon angles and so the untuned predictions are very
similar to the tuned predictions. However a clear discrepancy between the data
and the untuned GENIE v2.12.2 prediction is evident in the three most forward-
going angle bins ($\cos\theta_{\mu}>0.96$). As shown in Fig. 5, these three
bins are heavily dominated by QE, MEC and resonant interactions and are
particularly sensitive to the low-$Q^{2}$ suppression discussed in Sec. 3. The
$Q^{2}$ discrepancy between the data and the GENIE v2.12.2 predictions is
shown in the left side of Fig. 8, where the data imply a need for an even
stronger suppression of the cross section at very low values of $Q^{2}$ than
is currently achieved via the simulation tuning procedure. The $Q^{2}$
discrepancy is washed out as a function of neutrino energy, so there is broad
agreement between the data and predictions in the right-side of Fig. 8.

##  8 Uncertainties

Several sources of systematic uncertainty impact this measurement: the
neutrino flux prediction, detector response, muon energy scale, muon angle,
normalization, modeling of neutrino-nucleus interactions, and modeling of
neutron interactions in the detector. In general, for each source of
uncertainty, we use the difference between our nominal simulation and
systematically modified simulations to estimate the uncertainty on the
selection efficiency and purity. If the source of systematic uncertainty could
impact the reconstruction or particle identification algorithms, then the
effect is applied to the same simulated neutrino interactions at the relevant
point in the simulation-reconstruction chain, and the effect is propagated
through the reconstruction and analysis chain. Otherwise, the impact of a
systematic source is estimated by applying weights to events in the
simulation.

Figure 7: Extracted double-differential cross section, in slices of muon
angle. The data are presented showing statistical and total uncertainties, and
compared to GENIE v2.12.2 - NOvA Tune Acero  _et al._ (2020) (solid red line)
and GENIE v2.12.2 - Untuned Andreopoulos  _et al._ (2010, 2015) (dashed red
line). The inner error bars are from statistics, and may not be visible on
this scale.

For each individual source of systematic uncertainty, a systematically shifted
“universe” is simulated with a $\pm 1\sigma$ shift to the systematic source
parameter. Calibration and muon energy scale are examples of systematic
uncertainties for which this approach is used. Other uncertainties, such as
neutrino cross-section modeling and flux, are impacted by many sources and are
calculated with a multi-universe method. In this method, a hundred or more
universes are generated where parameters influencing the uncertainty are drawn
from a normal distribution, with a width that corresponds to the $1\sigma$
uncertainty on each systematic source parameter.

Uncertainties in the neutrino flux prediction arise from the modeling of
hadron production in the target, horns and decay pipe, and from the modeling
of the beam optics. The hadron production uncertainty on the neutrino flux
after the adjustments explained in Sec. 3 is $\sim$7% at the spectrum peak.
This uncertainty is dominated by interactions for which there are no relevant
external data to be included in the adjustment procedure (mostly meson and
proton elastic and quasielastic scattering). Uncertainties in beam optics are
incorporated by propagating uncertainties in the alignment and focusing of
beamline elements; this uncertainty is $\sim$4% at the peak.

Detector response uncertainties include uncertainties in the calibration of
the visible hadronic energy scale and simulation of light production and
transport from the liquid scintillator and wavelength-shifting fibers to the
photodetectors. A $5\%$ difference in the recorded energy deposition as a
function of distance traveled of candidate proton prongs measured between
simulation and data is used as the uncertainty in the hadronic energy
response. We use systematically shifted simulation samples where the absolute
energy scale is shifted by $\pm 5\%$ to evaluate the impact on this analysis.
An observed non-uniformity in the calibrated energy response as a function of
distance from the readout is included as a calibration shape uncertainty. The
uncertainty on the light model arises from the uncertainty on overall light
yield of the scintillator and the efficiency with which Cherenkov photons are
absorbed by the scintillator and re-emitted at wavelengths that can be
detected. A simulation sample where Cherenkov light production is disabled is
used to assess an upper limit on the uncertainty on this aspect of the light
model.

Uncertainties in the muon energy scale arise from modeling the energy loss of
muons in the detector. A detailed analysis of muon energy loss in the NOvA
near detector material composition in Geant4 indicates a $\pm 0.8\%$
uncertainty for the portion of the track that traverses the fully active
region, and $\pm 1.2\%$ for the portion of the track that traverses the muon
catcher Strait  _et al._ (2019). We conservatively assume the worst-case
scenario and scale the reconstructed muon energy in these fractions by either
all positive or all negative directions in both regions of the detector in
assessing this uncertainty.

Figure 8: Single-differential cross section as a function of $Q^{2}$, (left)
and $E_{\nu}$ (right) calculated only in the muon kinematics space specified
by the double-differential measurement (see Sec. 6). The data are presented
showing statistical and total uncertainties, and compared to GENIE v2.12.2 -
NOvA Tune Acero  _et al._ (2020) (solid red line) and GENIE v2.12.2 - Untuned
Andreopoulos  _et al._ (2010, 2015) (dashed red line). The inner error bars
are from statistics, and may not be visible on this scale.

Uncertainties in the muon angle arise from misalignments of the PVC cells in
the near detector. To estimate the impact of these misalignments on the muon
direction, an alternative simulation sample was generated with randomly
shifted cell positions according to the construction tolerances of the
detector. A comparison of this systematically shifted simulation to the
nominal simulation shows a
$2.5\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ spread in the
reconstructed muon angle distribution, and negligible spreads in the muon and
hadron energy distributions. We implement a
$2.5\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ systematic shift to the
reconstructed angle of the muon to determine the impact on the measured cross
section.

Table 2: Fractional uncertainties and correlations across all bins, broken down by source. Averages are taken across all bins reported in this measurement, weighted by the measured cross section, as described in Eqns. 3 and 4. | Weighted Avg. | Weighted Avg.
---|---|---
Source | Fractional | Correlation
| Uncertainty (%) |
Flux | 9.1 | $1.0$
Detector Response | 3.7 | $0.16$
Muon Energy Scale | 3.6 | $0.028$
Muon Angle | 2.4 | $0.087$
Normalization | 2.1 | $1.0$
$\nu$-A Modeling | 1.9 | $0.15$
Neutron Modeling | 1.5 | $0.92$
Total Systematic | 12 | $0.71$
Statistical | 1.6 | $0.0031$

Normalization uncertainties in the measured cross section arise from
uncertainties in the detector mass, integrated POT exposure and modeling of
beam intensity. Data from the manufacturing and construction processes are
used to constrain the uncertainty on the mass of the detector to 0.28%, and
the uncertainty on the POT accrual in the NuMI beamline is 0.5% based on
measurements of beam current through a toroid magnet. The simulation accounts
for time-dependent variations of beam intensity. An observed $\sim$2%
difference between shapes and normalizations of data and simulation selection
efficiencies as a function interaction vertex position in this analysis is
used as the uncertainty due to beam intensity modeling effects on the
normalization. The combined uncertainty on the normalization of the reported
cross section is 2.1%.

We use a reweighting approach to estimate the impact of neutrino-nucleus
scattering uncertainties. The weights applied are a mix of NOvA-specific
uncertainties and uncertainties available from the GENIE event generator
Andreopoulos  _et al._ (2015). The NOvA-specific uncertainties include a 5%
uncertainty on the value of the CCQE $M_{A}$ parameter and a 100% one-sided
uncertainty on the $Q^{2}$ suppression of resonant pion production applied to
the simulation. For MEC interactions, uncertainties in the fraction of target
nucleon pairs ($np$ vs. $pp$) in the nucleus and the dependence of the MEC
cross section as a function of $q_{0}$ and $q_{3}$ are taken into account.
Additional NOvA-specific uncertainties are included for DIS interactions.
Further details of the NOvA-specific uncertainties are described in Acero  _et
al._ (2020).

An energy uncertainty is assigned to the detector’s response to neutrons. This
uncertainty is driven by comparison of data to simulation in a neutron-rich
subsample of the antineutrino dataset. An excess of neutrons with low visible
energy is observed. A sample where one third of the neutron candidates with
energy below $40\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ had their visible
energy scaled down by a factor of 3.6 produces better data-simulation
agreement. The sample is used to set the size of a conservative two-sided
neutron response uncertainty.

Bin-to-bin correlations from all sources of systematic uncertainty are derived
from using the difference between $10^{5}$ systematically-shifted simulations
and the nominal simulation to calculate a total systematic uncertainty
covariance matrix. The unfolding procedure also induces small bin-to-bin
correlations. We calculate the statistical covariance matrix using a multi-
universe procedure similar to that described above, with 4000 toy simulations
with Poisson-fluctuated event counts in each measurement bin in reconstructed
space. The total uncertainty covariance matrix is a linear sum of the total
systematic and the statistical covariance matrices. Table 2 shows the
breakdown of the weighted average fractional uncertainties and correlations in
the double-differential cross-section measurement. The weighted average
fractional uncertainty is defined as

$\bigg{\langle}\frac{\delta\sigma^{\prime\prime}}{\sigma^{\prime\prime}}\bigg{\rangle}=\frac{\sum_{i}\sqrt{V_{ii}}}{\sum_{i}\sigma^{\prime\prime}_{i}},$
(3)

where $i$ is a measurement bin, $V$ is the covariance matrix, and
$\sigma^{\prime\prime}$ is the measured double-differential cross section. The
weighted average correlation is defined as

$\langle\mathrm{corr}\rangle=\frac{\sum_{i<j}C_{ij}\times\sigma^{\prime\prime}_{i}\times\sigma^{\prime\prime}_{j}}{\sum_{i<j}\sigma^{\prime\prime}_{i}\times\sigma^{\prime\prime}_{j}}$
(4)

where $i$ is a measurement bin, $j$ is a different measurement bin (so that
diagonal elements are excluded), and $C$ is the correlation matrix.

The dominant source of uncertainty comes from the flux prediction. As the
average correlation indicates, this is mainly a normalization uncertainty and
is significantly reduced in the shape-only analysis, where normalization
differences have been removed. We note that the statistical uncertainties are
at the level of a few percent per bin, and that the interaction modeling
uncertainties are subdominant. The normalization uncertainties that are 100%
correlated across all bins are removed in a shape-only covariance matrix. The
typical total uncertainty of our measurement is around 12%, which is reduced
to 7% in the shape-only analysis.

Table 3: Summary of the neutrino event generators and choice of models used by each to generate the inclusive cross section predictions against which comparisons to this measurement are made. RFG = Relativistic Fermi Gas, LFG = Local Fermi Gas, L-S = Llewellyn Smith, RPA = Random Phase Approximation, R-S = Rein-Sehgal, B-S = Berger-Sehgal, B-Y = Bodek-Yang, PY = PYTHIA, BUU = Boltzmann-Uehling-Uhlenbeck. Generator | QE/MEC | QE | MEC | Res | DIS | FSI
---|---|---|---|---|---|---
Initial State
GENIE v2.12.2 | RFG | L-S | Empirical | R-S | B-Y + PY 6 | hA (data-driven empirical cascade)
GENIE v3.00.06 | LFG | València | València | B-S | B-Y + PY 6 | hN (Oset (pions) + GENIE (nucleons))
NEUT v5.4.0 | LFG | València | València | B-S | B-Y + PY 5 | Oset + external data
NuWro 2019 | LFG | L-S + RPA | València | B-S | B-Y + PY 6 | Oset (pions) + NuWro (nucleons)
GiBUU 2019 | LFG | ——- GiBUU Model ——- | B-Y + PY 6 | BUU Equations

##  9 Comparisons to Generators

Figure 9: Extracted double-differential cross section divided by the GENIE
v2.12.2 - NOvA Tune prediction. Ratios are shown in slices of muon angle and
compared to the ratio obtained from GENIE v3.00.06 (dashed red line), GiBUU
2019 (dot-dashed cyan line), NEUT v5.4.0 (solid green line), and NuWro 2019
(solid thin purple line). Top: data are shown with total uncertainties,
predictions are taken directly from the generators. Bottom: data are shown
with shape-only uncertainties, predictions are area-normalized to the data.

Figure 10: Single differential cross section as a function of $Q^{2}$ (left)
and cross section as a function of $E_{\nu}$ (right), divided by the GENIE
v2.12.2 - NOvA TUNE prediction. Quantities are calculated over the reported
muon kinematics space specified by the double-differential measurement. The
data are presented showing statistical and total uncertainties, and compared
to the ratio obtained from GENIE v3.00.06 (dashed red line), GiBUU 2019 (dot-
dashed cyan line), NEUT v5.4.0 (solid green line), and NuWro 2019 (solid thin
purple line). In the top plots, the predictions are unmodified and the error
bars on the data represent the total uncertainties, which includes
normalization uncertainties that are 100% correlated across bins. In the
bottom plots, the predictions are first area-normalized to the data, and the
error bars on the data represent shape-only uncertainties, where
100%-correlated normalization uncertainties have been removed.

The results of the double-differential and single differential cross-section
measurement are presented in this section and compared to GENIE versions
v2.12.2 and v3.00.06, NEUT v5.4.0 Hayato (2002), NuWro 2019 Golan  _et al._
(2012a, b) and GiBUU 2019 Buss  _et al._ (2012); Gallmeister  _et al._ (2016).
Table 3 lists the models used in each for the initial state, interaction
modes, and final state interactions in the generators. GENIE v2.12.2 is the
neutrino event generator used in the simulation for this analysis and is
described above in Sec. 3. GENIE v3.00.06 is a more recent version of GENIE,
and we use a configuration chosen by the NOvA experiment for its 2020
oscillation analysis, `18_10j_02_11a }, a combination of {\verb G18_10j_00_000
} and {\verb G18_10b_02_11a } which in practice results in predictions that
are nearly identical to the out-of-the-box predictions from the {\verb
G18_10b_02_11a } tune. We note however that the GE`IE v3.00.06 tune shown here
has no other NOvA-specific tuning applied. Whereas all other event generators
use a local Fermi-gas model (LFG) for the initial state, GENIE v2.12.2 uses a
relativistic Fermi-gas (RFG) model. QE and MEC interactions are implemented
via the València group model from Nieves, et al. Nieves  _et al._ (2011);
Schwehr  _et al._ (2016) in GENIE v3.00.06 and NEUT v5.4.0. NuWro implements
QE interactions based on the Llewellyn Smith Llewellyn Smith (1972) model with
an additional Random Phase Approximation (RPA) suppression, but implements MEC
interactions based on the València model. Resonant interactions are based on
the Berger-Sehgal Berger and Sehgal (2009) model, and DIS interactions use
PYTHIA 6 T. Sjostrand and Skands (2006) in GENIE v3.00.06 and NuWro 2019 and
PYTHIA 5 in NEUT v5.4.0. GiBUU implements its own unique model for neutrino
interactions across the QE and Resonant regions based on many of the same
principles as the models mentioned aboveLeitner  _et al._ (2006, 2009). Final-
state interactions are implemented via a variety of models, including that of
Oset et al. Salcedo  _et al._ (1988), cascade models in GENIE and the
Boltzmann-Uehling-Uhlenbeck (BUU) equations in GiBUU. It is worth noting that
most of the models listed above use form factors and cross sections extracted
from very similar data sets, and in principle should be highly correlated.
However, as we show below, and has been noted elsewhere (see e.g., Alvarez-
Ruso  _et al._ (2018)), the inclusive charged-current neutrino-nucleus cross
section predictions from these different generators differ considerably,
likely due to differences in implementation.

The top panel of Fig. 9 shows the ratio of the extracted double-differential
cross section to the GENIE v2.12.2 - NOvA Tune Acero  _et al._ (2020)
prediction, in slices of muon angle. The outer error bars of the data
represent total uncertainties, while the inner error bars of the data are
statistical only. The solid histograms are ratios of the predictions from
different neutrino event generators to the GENIE v2.12.2 - NOvA Tune
prediction. In the lower panel of Fig. 9 the predictions are first area-
normalized to the data across the reported double-differential measurement
space before taking the ratio with respect to the GENIE v2.12.2 - NOvA Tune
prediction, and the outer error bars of the data represent shape-only
uncertainties. These comparisons indicate 5-10% agreement between the
measurement and the various generators at high angle slices. Discrepancies
become more apparent at more forward-going angles and lower muon energies.

Figure 10 shows similar comparisons of the the differential cross section as a
function of $Q^{2}$ and the cross section as a function of $E_{\nu}$. These
model-dependent variables are calculated only in the muon kinematics space
specified by the double-differential measurement. The top plots show
unmodified predictions and the data with total error bars. The bottom plots
show predictions that are area-normalized to the data and the data with shape-
only error bars, where the normalization uncertainties that are 100%
correlated across bins have been subtracted. As was the case for GENIE
v2.12.2, the large suppression at $Q^{2}<$
$0.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{2}$ is not described by any of
the generators. As the generators use very similar models for interactions
that contribute most at low values of $Q^{2}$ (QE and MEC), this strongly
indicates that some additional suppression of the cross section at low $Q^{2}$
is lacking from the underlying theory. Furthermore, many of the predictions
prefer a stronger suppression of the cross section at $Q^{2}$ values ranging
from 100-800 MeV than is observed in the data, which suggests the need for
improved modeling of resonant interactions. The differences between the data
and predictions as a function of $Q^{2}$ are washed out when looking at the
cross section as a function of the neutrino energy, and we see good agreement
between our measurement and most neutrino generators.

Table 4: Summary of global $\chi^{2}$ calculations for different neutrino generators for the double-differential cross section measurement across 172 bins of muon energy and angle. The scale-factors and $\chi^{2}$s for shape-only comparisons are also shown. The $\chi^{2}$ calculation accounts for bin-to-bin correlations using the statistical and systematic covariance matrix described in Sec. 8. Generator | Tune | Total Uncertainty | Shape-only | Shape Uncertainty
---|---|---|---|---
|  | Global $\chi^{2}$ | Scale factor | Global $\chi^{2}$
GENIE v2.12.2 | NOvA | 347 | 1.01 | 353
GENIE v2.12.2 | Default | 1350 | 0.98 | 1301
GENIE v3.00.06 | N18-10j-02-11a | 1332 | 1.15 | 1834
GiBUU 2019 | Default | 1318 | 1.28 | 2170
NuWro 2019 | Default | 705 | 1.15 | 955
NEUT v5.4.0 | Default | 1941 | 1.02 | 2080

In order to make a more quantitative assessment of the agreement between our
measurement and the various event generators, we calculate the global
$\chi^{2}$ between our measurement and different generators across all
measurement bins. We use the systematic uncertainty covariance matrix
described in Sec. 8 to account for bin-to-bin correlations in the $\chi^{2}$
calculation. Table 4 shows a summary of the global $\chi^{2}$ calculations for
both total and shape-only uncertainties. The normalization factor used to
area-normalize the predictions to the data for the shape-only comparisons is
also shown. GENIE v2.12.2 with the NOvA tune results in the best $\chi^{2}$,
however we note that the $\chi^{2}$ per degree of freedom (dof) of $\sim$2
(there are 172 degrees of freedom) is yet another reflection of the remaining
discrepancies between the measured and tuned predicted cross section. As
expected from the data-generator comparisons in Fig. 9, the global $\chi^{2}$s
are very high, but vary across generator predictions. The global $\chi^{2}$s
for the shape-only comparison are even larger, which implies that shape
differences are significant and that a simple normalization correction to the
predictions is insufficient to reduce the discrepancies. The combination of
the differences seen in Fig. 9 with the information on the interaction types
in Fig. 5 emphasizes the regions of muon kinematic phase space and perhaps the
particular models in each generator that need the most attention by the
neutrino-nucleus scattering community.

##  10 Conclusion

We have presented a measurement of the double-differential $\nu_{\mu}$ CC
inclusive cross section in the NOvA near detector in 172 bins of muon momentum
and angle. The measurement applies purity, unfolding and efficiency
corrections based on muon energy, muon angle and amount of observable hadronic
energy in the detector, reducing the neutrino-nucleus interaction model
dependence on the measurement. The measured cross sections and the covariance
matrices are available in digital format on the NOvA Experiment Data Releases
webpage 222https://novaexperiment.fnal.gov/data-releases/. The weighted
average fractional total uncertainty of 12% is driven primarily by a 9.1% flux
normalization uncertainty. The flux normalization uncertainty is expected to
decrease by more than a factor of two over the next few years as new
constraints become available from external hadron production experiments such
as NA61/SHINE and EMPHATIC as well as neutrino-electron scattering
measurements in the NOvA near detector. The weighted average fractional shape-
only uncertainty of 8.1% is driven by muon and hadronic energy scale
uncertainties. Comparisons to generator predictions are made by calculating
$\chi^{2}$ using both the total and shape-only covariance matrices. There is
an apparent tension between the measurement and predictions at very forward
angles, consistent with a large observed discrepancy between the measurement
and predictions at very low $Q^{2}$. This discrepancy at low $Q^{2}$ is seen
in all generators regardless of normalization uncertainties, and is consistent
across all neutrino event generators to which the data are compared. The
region of phase space covered by very forward muon-scattering angles receives
contributions from QE-like and resonant interactions. Consequently, the data
strongly suggest that a fundamental component responsible for greater
low-$Q^{2}$ suppression of the cross section is missing from the interaction
models. We note too that since a presentation of preliminary results of this
analysis Cremonesi (2020), the Giessen group has modified the resonant and
shallow-inelastic scattering region in the GiBUU simulation improving
agreement with our data Mosel (2021), exemplifying the rapid pace of neutrino
generator development and the need for additional data. Future measurements of
neutrino interactions by the NOvA collaboration (e.g. see Soplin (2021);
Ramson (2021)) aim to isolate the exclusive final states that could be
contributing to the large discrepancies observed in the inclusive channel
presentated in this paper.

##  11 Acknowledgements

This document was prepared by the NOvA 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.
This work was supported by the U.S. Department of Energy; the U.S. National
Science Foundation; the Department of Science and Technology, India; the
European Research Council; the MSMT CR, GA UK, Czech Republic; the RAS, RMES,
and RFBR, Russia; CNPq and FAPEG, Brazil; UKRI, STFC and the Royal Society,
United Kingdom; and the state and University of Minnesota. We are grateful for
the contributions of the staffs of the University of Minnesota at the Ash
River Laboratory, and of Fermilab.

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##  Appendix A Results in Table Format

[ longtable=ccccc, table head=Double-differential cross-section
$\left(\dfrac{\mathrm{d}^{2}\sigma}{\mathrm{d}\cos\theta_{\mu}\mathrm{d}T_{\mu}}\right)$
results table including total and systematic errors
$\left(\dfrac{\mathrm{cm}^{2}}{\mathrm{GeV\leavevmode\nobreak\ nucleon}}\times
10^{-39}\right)$ .
$\cos\theta_{\mu}$ range $T_{\mu}$ range (GeV) Cross section Total Error Stat.
Error

, late after line=
, ] Results/Table_MuKin_total.csv thetaMin=, thetaMax=, TmuMin=, TmuMax=,
xsec=, totalErr=, statErr= [,) [,) $\xsec$ $\totalErr$ $\statErr$