The NOvA Collaboration

#  Measurement of the $\nu_{e}-$Nucleus Charged-Current
Double-Differential Cross Section at $\left<E_{\nu}\right>=$
$2.4\text{\,}\mathrm{GeV}$ using NOvA

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  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  A. C. Booth
Particle Physics Research Centre, Department of Physics and Astronomy, Queen
Mary University of London, London E1 4NS, United Kingdom  Department of
Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United
Kingdom  R. Bowles  Indiana University, Bloomington, Indiana 47405, USA  B.
Brahma  Department of Physics, IIT Hyderabad, Hyderabad, 502 205, India  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  S. Childress  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  A. Chatla  School of Physics,
University of Hyderabad, Hyderabad, 500 046, India  R. Chirco  Illinois
Institute of Technology, Chicago IL 60616, 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  Particle Physics
Research Centre, Department of Physics and Astronomy, Queen Mary University of
London, London E1 4NS, 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  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  V Hewes  Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, USA  A. Himmel  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, 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  D. M. Kaplan  Illinois Institute of Technology, Chicago
IL 60616, USA  A. Kalitkina  Joint Institute for Nuclear Research, Dubna,
Moscow region 141980, Russia  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  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  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  J. M. C. Lopez  Physics and Astronomy
Department, University College London, Gower Street, London WC1E 6BT, United
Kingdom  R. Mahji  School of Physics, University of Hyderabad, Hyderabad, 500
046, India  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. T. Manoharan  Department of Physics, Cochin
University of Science and Technology, Kochi 682 022, India  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
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  V. Mikola  Physics and Astronomy Department,
University College London, Gower Street, London WC1E 6BT, United Kingdom  W.
H. Miller  School of Physics and Astronomy, University of Minnesota Twin
Cities, Minneapolis, Minnesota 55455, USA  S. Mishra  Department of Physics,
Institute of Science, Banaras Hindu University, Varanasi, 221 005, India  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  A. Nath  Department of Physics, IIT
Guwahati, Guwahati, 781 039, India  N. Nayak  Department of Physics and
Astronomy, University of California at Irvine, Irvine, California 92697, USA
S. Nelleri  Department of Physics, Cochin University of Science and
Technology, Kochi 682 022, India  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  A. Pal  National Institute of Science Education
and Research, Khurda, 752050, Odisha, India  J. Paley  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  L. Panda  National
Institute of Science Education and Research, Khurda, 752050, Odisha, India  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  A. Pobedimov  Joint Institute for Nuclear Research, Dubna, Moscow region
141980, Russia  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  L. R. Prais
University of Mississippi, University, Mississippi 38677, 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  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  S. Shukla
Department of Physics, Institute of Science, Banaras Hindu University,
Varanasi, 221 005, India  A. Sheshukov  Joint Institute for Nuclear Research,
Dubna, Moscow region 141980, Russia  I. Singh  Department of Physics and
Astrophysics, University of Delhi, Delhi 110007, India  P. Singh  Particle
Physics Research Centre, Department of Physics and Astronomy, Queen Mary
University of London, London E1 4NS, United Kingdom  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  R. L. Talaga
Argonne National Laboratory, Argonne, Illinois 60439, USA  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  B. N. Temizel  Illinois
Institute of Technology, Chicago IL 60616, USA  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  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  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  T. Wieber  School of Physics and
Astronomy, University of Minnesota Twin Cities, Minneapolis, Minnesota 55455,
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  B. Yaeggy  Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, 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

###### Abstract

The inclusive electron neutrino charged-current cross section is measured in
the NOvA near detector using $8.02\times 10^{20}$ protons-on-target (POT) in
the NuMI beam. The sample of GeV electron neutrino interactions is the largest
analyzed to date and is limited by $\simeq$ 17% systematic rather than the
$\simeq$ 7.4% statistical uncertainties. The double-differential cross section
in final-state electron energy and angle is presented for the first time,
together with the single-differential dependence on $Q^{2}$ (squared four-
momentum transfer) and energy, in the range $1\text{\,}\mathrm{GeV}$ $\leq
E_{\nu}<$ $6\text{\,}\mathrm{GeV}$. Detailed comparisons are made to the
predictions of the GENIE, GiBUU, NEUT, and NuWro neutrino event generators.
The data do not strongly favor a model over the others consistently across all
three cross sections measured, though some models have especially good or poor
agreement in the single differential cross section vs. $Q^{2}$.

††preprint: FERMILAB-PUB-22-446-ND

Precision measurements of neutrino oscillation parameters Acero  _et al._
(2019); Abe  _et al._ (2021); Abi  _et al._ (2020); Yokoyama (2017), such as
$\delta_{\rm{CP}}$ and the neutrino mass ordering use $\nu_{e}$
($\mathrm{bar}\nu_{e}$) appearance in predominantly $\nu_{\mu}$
($\mathrm{bar}\nu_{\mu}$) beams and depend on detailed understanding of
neutrino interaction models. Despite the two-detector technique that largely
mitigates the impact of cross-section uncertainties, model predictions are
still required by oscillation measurements to estimate the energy spectrum and
selection efficiencies of $\nu_{e}$ interactions, requiring accurate models of
interaction rate and outgoing-particle kinematics of $\nu_{e}$ interactions.

Lepton universality suggests that $\nu_{\mu}$ and $\nu_{e}$ interactions are
largely similar, differences in final-state lepton mass, tree-level radiative
corrections, and imprecisely known form factors of the nucleon reaction lead
to differences in the predicted cross sections and their uncertainties Day and
McFarland (2012). There are few direct $\nu_{e}$ cross-section measurements at
the GeV energy scale as $\nu_{e}$ are a small component ($\leq 1\%$) of the
neutrino beams produced at accelerators. The dominant $\nu_{\mu}$ component
produces significant backgrounds. Previous measurements in this energy range
have been performed by Gargamelle Blietschau  _et al._ (1978) and T2K Abe  _et
al._ (2014, 2020), both of which reported the total integrated cross section
as a function of energy. MINERvA has reported single-differential cross
sections for the quasi-elastic component of the $\nu_{e}$ \- carbon cross
section Wolcott  _et al._ (2016). The measurement presented here uses the
largest $\text{\,}\mathrm{GeV}$ sample of electron neutrino-initiated
interactions to date, with which the simultaneous differential dependence on
electron energy, $E_{e}$, and angle, $\theta_{e}$, is extracted for the first
time. The cross sections as a function of squared four-momentum transfer,
$Q^{2}$, and as a function of neutrino energy, $E_{\nu}$, is also presented.

Generators, such as Andreopoulos  _et al._ (2010); Lalakulich  _et al._
(2013); Hayato (2009); Golan  _et al._ (2012), group the predictions of
neutrino-nucleus interaction cross sections into three processes relevant in
this energy range: quasielastic (QE) scattering, baryon resonance production
(RES), and deep inelastic scattering (DIS). The modeling of these processes
must be altered in the presence of a nuclear medium. A fourth process due to
meson exchange current (MEC), occurs only inside the nuclei. These nuclear
effects are not completely understood and the implementation of existing
models in generators is currently incomplete Martini  _et al._ (2009); Nieves
_et al._ (2011); Megias  _et al._ (2016); Cuyck  _et al._ (2017). Measured
differential dependences using electron neutrinos can be used to constrain the
underlying interaction channels and elucidate nuclear effects. These
measurements can also probe key aspects of neutrino-nucleus interaction models
used to calculate oscillation-experiment acceptances.

The NOvA experiment is designed to measure neutrino flavor oscillations Acero
_et al._ (2019) using two detectors separated by $809\text{\,}\mathrm{km}$,
placed $14.6\text{\,}\mathrm{mrad}$ off-axis from the central beam direction
of Fermilab’s NuMI beam Adamson  _et al._ (2016). Magnetic focusing horns in
the beamline charge-select neutrino parents giving 96% pure $\nu_{\mu}$ flux
between 1 and $6\text{\,}\mathrm{GeV}$. The energy spectrum peak at
$1.8\text{\,}\mathrm{GeV}$ at the near detector (ND) as shown in Figure 1. The
largest contamination is $\mathrm{bar}\nu_{\mu}$; the $<1\%$
$\nu_{e}+\mathrm{bar}\nu_{e}$ arises mostly from muon decays below
$3\text{\,}\mathrm{GeV}$ neutrino energy and kaon decays above.

Figure 1:  NUMI neutrino-mode beam component flux spectra at the NOvA near
detector below $6\text{\,}\mathrm{GeV}$. The bands represent the total flux
uncertainty on hadron production and beam optics for each component.

The ND is a tracking calorimeter composed of liquid scintillator contained in
PVC cells. The portion of the detector relevant to this measurement is
$3.9\text{\,}\mathrm{m}$ $\times$ $3.9\text{\,}\mathrm{m}$ $\times$
$12.8\text{\,}\mathrm{m}$ (long dimension along beam direction) in size and is
segmented into cells that are $6.6\text{\,}\mathrm{cm}$ $\times$
$3.9\text{\,}\mathrm{cm}$ (0.15 radiation length $\times$ 0.45 Molière radius)
in cross section and span the height and width of the detector in planes of
alternating vertical and horizontal orientation. Each cell is filled with a
blend of 95% mineral oil and 5% pseudocumene with trace concentrations of
wavelength shifting fluors Mufson  _et al._ (2015). The resulting composition
by mass is 67% carbon, 16% chlorine, 11% hydrogen, 3% titanium, 3% oxygen with
other trace elements. When a particle traverses the detector, wavelength
shifting fiber in the cell collect and deliver scintillation light to
avalanche photodiodes. The result is digitized by custom front-end
electronics. All signals above a noise-vetoing threshold are sent to a data
buffer. This Letter presents data corresponding to $8.02\times 10^{20}$
protons delivered to the NuMI production target (POT) between November 2014
and February 2017.

This analysis relies on simulation to calculate the integrated flux, optimize
event selection criteria, estimate selection efficiency, assess the effects of
detector resolution and acceptance, and assess systematic uncertainties. The
simulation proceeds in several steps: first neutrinos are generated from
simulated mesons in the NuMI beam, then those neutrinos interact with nuclei
in the detector, after which the final-state particles are transported through
the detector.

The NuMI flux is predicted using Geant4 v9.03 Agostinelli  _et al._ (2003)
with the FTFP BERT hadronic model. The hadron production model is adjusted and
constrained using external measurements by the PPFX package Aliaga  _et al._
(2016). Neutrino interactions are simulated using the GENIE v2.12.2
Andreopoulos  _et al._ (2010) event generator, hereinafter referred to as
GENIE v2. The neutrino-interaction model output is adjusted to incorporate
advances in theory and external data to better match a sample of $\nu_{\mu}$
charged-current(CC) interactions from NOvA ND data as detailed in Ref. (Acero
_et al._ , 2020). In this dedicated tune, hereinafter referred to as the NOvA-
tune, there are three substantial adjustments to the GENIE v2 prediction: (1)
the QE axial mass ($M_{A}$ CCQE) is changed from 0.99 to
$1.04\text{\,}\mathrm{G}\mathrm{eV}\mathrm{/}\mathrm{c}^{2}$ Meyer  _et al._
(2016); (2) soft non-resonant single pion production events from neutrinos are
reduced by 57%; (3) the empirical MEC model Katori (2015) is tuned to ND data
using a two-dimensional fit in hadronic energy and momentum transfer. This
procedure changes the shape of the MEC prediction and increases the cross
section for that process by an average of 50%. The NOvA-tune is applied to all
simulated CC interactions.

Geant4 v10.1.p3 is used to simulate energy deposited in the NOvA ND from the
particles generated by neutrino interactions. Photon generation and
propogation in the cells is modeled separately as follows: (1) the modeling
and transfer of scintillation light uses NOvA measurements of scintillator
response and fiber attenuation properties Aurisano  _et al._ (2015); (2) the
Birks suppression of the scintillation light is tuned using test-stand
measurements and validated using a custom simulated response of the readout
electronics Anfimov  _et al._ (2020).

The signature of $\nu_{e}$ CC interactions is an electron in the final state
that produces an electromagnetic cascade within the detector. The major
backgrounds are neutral current (NC) and $\nu_{\mu}$ CC interactions with
final-state $\pi^{0}$s that decay into two photons. These can mimic electron
energy depositions if the showers overlap or one photon is low energy.

Candidate neutrino interactions (events) are reconstructed by forming
collections of observed energy deposits (hits) from final-state charged
particles correlated in space and time Ester  _et al._ (1996). Events are
assigned a vertex by minimizing the angular spread of hits relative to
candidate vertices Baird  _et al._ (2015). Using a fuzzy K-means algorithm,
hits within the event are clustered in regions of space emanating from the
reconstructed interaction vertex into reconstructed particle trajectories
(prongs) Dunn (1973). Event primary vertices are required to be contained in a
$49\text{\,}\mathrm{t}$ fiducial volume spanning $2.8\text{\,}\mathrm{m}$ in
both width and height and $6.5\text{\,}\mathrm{m}$ in length. All prongs are
required to stop at least $30\text{\,}\mathrm{cm}$ from any edge of the
detector to ensure containment. Events containing between 20 and 200 hits and
at least one prong are selected for this analysis. Events outside this range
are generally low visible energy NC events or high multiplicity events in
which the electromagnetic cascade is difficult to isolate among the energy
depositions from the other particles in the final state.

Prongs are identified as candidate electrons using a Boosted Decision Tree
(BDT) algorithm Hoecker  _et al._ (2007). One input to the BDT is the output
of a convolutional neural network (CNN) trained using simulated samples of
single particles which labels prongs as muonic, electromagnetic, or hadronic
in origin. Other inputs include the energy-weighted transverse width of the
prong and the distance from the vertex to the prong start which aid in the
separation of photon- from electron-induced showers. The performance of this
BDT was validated against a sideband of data enriched with EM showers in
$\nu_{\mu}$CC$\pi^{0}$ events. Candidate events are assigned an ElectronID
score equal to the most electron-like BDT score of their constituent prongs.
Figure 2 shows the ElectronID distribution in data and simulation after all
selection criteria (and kinematic restrictions, defined below) are applied.

Figure 2: Simulated ElectronID distributions of
$\nu_{e}+\mathrm{bar}{\nu}_{e}$ CC (blue), $\nu_{\mu}+\mathrm{bar}{\nu}_{\mu}$
CC (orange), and NC (green), compared to data integrated over the reported
electron kinematic phase space. The dashed (solid) line shows the total
prediction from simulation (after extracted fit normalization corrections).
Statistical uncertainties on data are too small to be seen. The vertical
errors represents the $\pm 1\sigma$ systematic range. All spectra are
normalized to the data exposure.

The selected sample is binned according to the candidate electron shower’s
energy $E_{e}$, and its angle $\cos\theta_{e}$. Within each electron kinematic
bin, the simulated ElectronID distribution is used to generate signal and
background predictions, or templates. Each bin’s templates are broken further
into three components, whose normalizations are constrained in a simultaneous
fit to the data: $\nu_{e}$ +$\mathrm{bar}\nu_{e}$, $\nu_{\mu}$
+$\mathrm{bar}\nu_{\mu}$, and NC. Neutrinos and antineutrinos are not
separated due to similarities in the shapes of their ElectronID templates.

Electron-kinematic bins that satisfy two criteria are included in the fit: at
least 100 predicted signal events and an estimated signal-to-background ratio
$>0.4$ in the ElectronID $>0.2$ region. These requirements ensure adequate
shape discrimination between the signal and background templates and limit the
measured kinematic range to:

$\begin{gathered}0.85\leq\cos\theta_{e}<0.90\;\cap\;1.0\leq
E_{e}(\mathrm{GeV})<1.65\\\ 0.90\leq\cos\theta_{e}<0.94\;\cap\;1.0\leq
E_{e}(\mathrm{GeV})<2.0\;\\\ 0.94\leq\cos\theta_{e}<0.97\;\cap\;1.4\leq
E_{e}(\mathrm{GeV})<3.0\;\\\ 0.97\leq\cos\theta_{e}\leq 1.00\;\cap\;1.4\leq
E_{e}(\mathrm{GeV})<6.0\;\end{gathered}$ (1)

A $\chi^{2}$ minimization procedure James and Roos (1975) fits the
normalization of the signal and two background templates as described
previously. A covariance matrix encodes the correlations between templates
across all kinematic and template bins using systematic uncertainties which
affect the shape of the ElectronID templates and the reconstructed electron
kinematics within the fit. The fit produces a $\chi^{2}$ of 131 for 141
degrees of freedom. Using simulation with the best-fit parameters applied the
sample purity is $13\pm 1.3$%. Uncertainties on the purity prediction rise to
1.5% in kinematic regions where the $\nu_{\mu}$ +$\mathrm{bar}\nu_{\mu}$ and
NC normalization parameters become highly correlated due to templates no
longer being as discriminating. The contribution of $\mathrm{bar}\nu_{e}$ CC
background is removed from the renormalized signal template using the
predicted contribution rate of 5% from simulation. The resulting sample is
estimated to contain $9,200\pm 1,000$ $\nu_{e}$ CC interaction candidates,
selected with 35% efficiency. The signal sample is estimated to be 28% QE, 20%
MEC, 31% RES, 20% DIS, and 1% CC coherent pion production.

The flux-averaged double-differential $\nu_{e}$ CC cross section in final
state electron kinematic variables is constructed using

$\left(\frac{d^{2}\sigma}{d\cos\theta
dE}\right)_{i}=\frac{\sum_{j}U_{ij}^{-1}N_{\mathrm{\nu_{e}}}(\cos\theta,E)_{j}}{N_{\mathrm{t}}\;\phi\;\epsilon(\cos\theta,E)_{i}\;\Delta\cos\theta_{i}\;\Delta
E_{i}},$ (2)

where $N_{\mathrm{\nu_{e}}}$ the estimated $\nu_{e}$ CC events from the
template fit. $U_{ij}$, an unfolding matrix used to relate the reconstructed
value in bin $j$ to the true value in bin $i$. The data are iteratively
unfolded according to d’Agostini’s method D’Agostini (1995) using two
iterations as implemented in RooUnfold Adye (2011). The optimal number was
found by minimizing the average mean square error Eadie  _et al._ (1971)
calculated across simulated samples with random variations of systematic
uncertainties. $N_{t}$ the number of nuclear targets in the fiducial volume,
$\phi$ the integrated neutrino flux, $\epsilon$ the selection efficiency
correction factor, and $\Delta E$ and $\Delta\cos\theta$ are the bin widths
used for the electron kinematic variables. Bin widths are chosen small enough
to match the detector resolution and large enough to include a statistically
significant event sample. The average $E_{e}$ resolution is 350 MeV, and the
angular resolution ranges from $2\text{\,}\mathrm{\SIUnitSymbolDegree}$ for
forward-going electrons to $11\text{\,}\mathrm{\SIUnitSymbolDegree}$ for less
forward-going electrons.

LABEL:table:uncertainties summarizes the effects of various sources of
uncertainty on the measurements. Systematic uncertainties are evaluated by
varying the parameters used to model the neutrino flux, neutrino-nucleus
interactions ($\nu$-A), detector response, and re-extracting the differential
cross section. The difference between the cross section extracted using the
nominal simulation and that extracted using the simulation with a varied
parameter is taken as the uncertainty due to each parameter. The procedure
accounts for changes in the compositions of backgrounds, selection efficiency,
and event reconstruction due to the variations considered.

The dominant sources of systematic uncertainty are from the neutrino flux and
$\nu$-A predictions. Uncertainties on the flux arise from hadron production
uncertainties (9%) Aliaga  _et al._ (2016) and beam optics modeling (4%). A
custom set of NOvA-specific uncertainties Acero  _et al._ (2020) and
uncertainties available from the GENIE generator are used to assess
uncertainties on $\nu$-A interaction modeling. Model parameters affecting RES,
DIS, and MEC predictions have the largest impact and become dominant at
$E_{e}>$ $3.0\text{\,}\mathrm{GeV}$.

Non-leading sources of uncertainty come from detector calibration and
modeling. These sources become dominant for $\cos\theta_{e}<0.94$ and
$E_{e}<\;$$1.5\text{\,}\mathrm{GeV}$. Minor sources of uncertainty, which
include detector mass, integrated beam exposure, beam intensity modeling, and
the modeling of diffractive (DFR) $\pi^{0}$ production, are combined in the
“Other” category of LABEL:table:uncertainties. DFR modeling uncertainties are
evaluated by reweighting the default $\nu$-H NC interactions producing a
$\pi^{0}$ prediction from GENIE to an estimate based upon the calculation by
Kopeliovich et al. Kopeliovich  _et al._ (2012); Kopeliovich  as a function of
$E_{\nu}$ and the Björken scaling variables. The average uncertainty on DFR
modeling is 2.6% across all bins.

LABEL:table:uncertainties also shows the weighted average bin-to-bin
correlations for each source of systematic uncertainty. The large bin-to-bin
correlations from the flux uncertainty indicate that it mainly impacts
normalization. Interaction modeling also exhibits strong correlations across
all bins, due to a combination of the template fitting procedure and the model
parameters, such as the axial mass from the RES model and DIS model parameters
that impact the selection efficiency prediction.

Table 1: Fractional uncertainties and correlations, broken down by source. Averages are taken across all bins reported in this measurement, weighted by the measured cross section. Source | Avg. Uncertainty (%) | Avg. Correlation
---|---|---
Flux | 10.3 | 0.90
$\nu$-A Model | 9.8 | 0.64
Calibration | 5.9 | 0.05
Detector Model | 5.6 | 0.21
Other | 2.8 | 0.03
Statistical | 7.4 | 0.02
Total | 18.2 | 0.59

Three results are presented: the flux-integrated double-differential cross
section vs. electron energy and angle shown in Figure 3, the cross section vs.
$E_{\nu}$ shown in Figure 4, and the differential cross section vs. $Q^{2}$
shown in Figure 5. All results are calculated from data using unfolding and
efficiency corrections derived from the NOvA-tune. For the one-dimensional
results we first apply the phase space restrictions in $E_{e}$ and
$\cos\theta_{e}$ given in Eqn. 1. The result is then extracted with unfolding
and efficiency corrections in $E_{\nu}$ or $Q^{2}$.

The extracted cross sections are compared to predictions from several
generators: GENIE models (NOvA-tune, v2.12.2, and v3.00.06 with a global
configuration N18_10j_02_11a), NuWro 2019 Golan  _et al._ (2012), GiBUU 2019
Lalakulich  _et al._ (2013), and NEUT 5.4.0 Hayato (2009). Table 2 summarizes
$\chi^{2}$ values for each model compared to data with a full treatment of
bin-to-bin correlations. The values show no model is consistently favored
across the ensemble of measurements.

Though the GiBUU and NuWro predictions have normalizations systematically
lower than the data in Figures 3, 4 and 5 (15% and 10%, respectively) their
$\chi^{2}$s in Table 2 are comparable to other models’ due to the phenomenon
of Peelle’s Pertinent Puzzle (PPP) Smith (1991); Frühwirth, R.  _et al._
(2012). PPP arises when the dominant uncertainty of a result is from highly
correlated normalization uncertainties, like the flux uncertainty in
LABEL:table:uncertainties. Under these circumstances, the best-fitting model
as reckoned by $\chi^{2}$ can be well outside the data points. On the other
hand, disagreements with the GENIE predictions are dominated by the
discrepancy in the $E_{e}>$ $4.75\text{\,}\mathrm{GeV}$ bin and its
correlations with the other bins. When this bin is excluded from the
calculation, the $\chi^{2}$’s for the GENIE predictions approach those of the
NuWro and GiBUU predictions. NEUT shows slight disagreement throughout that
angular slice, predicting a softer energy spectrum than is observed in data.

In $E_{\nu}$ comparisons NuWro and GiBUU plateau at a lower total cross
section than the data as described above. The differential-cross section vs.
$Q^{2}$ shows preference towards the NOvA-tune and GENIE v3 predictions. NEUT
predicts a softer $Q^{2}$ distribution than seen in data, while GiBUU and
NuWro accurately predict the low $Q^{2}$ behavior but tension is seen at high
$Q^{2}$. The grey band in Figure 5 shows the size of the flux-related
normalization uncertainties, which illustrates that differences seen between
data and the generators are not consistent with an overall change in the
cross-section normalization.

Table 2: $\chi^{2}$ calculated for each model presented in Figures 3, 4 and 5 compared to data. Degrees of freedom are 17, 12, and 9 for the double-differential, $E_{\nu}$, and $Q^{2}$ results, respectively. Generator | $\frac{d^{2}\sigma}{d\cos\theta_{e}dE_{e}}$ | $\sigma(E_{\nu})$ | $\frac{d\sigma}{dQ^{2}}$
---|---|---|---
GENIE v2 - NOvA-tune | 24.1 | 13.4 | 1.3
GENIE v2.12.2 | 24.3 | 14.3 | 19.6
GENIE v3.00.06 | 27.4 | 21.6 | 3.4
GiBUU 2019 | 17.5 | 16.0 | 14.7
NEUT 5.4.0 | 25.1 | 16.9 | 45.0
NuWro 2019 | 18.7 | 15.3 | 10.0
Figure 3: Extracted double differential cross section, subdivided in slices of
electron angle. The outer error bars of the data represent total
uncertainties, while the inner error bars are statistical only. The data are
compared to several models. Figure 4: Top: The differential cross section vs.
$E_{\nu}$. Bottom: Comparisons as a ratio to the NOvA-tune prediction. The
grey band represents the normalization uncertainty from the neutrino flux
prediction. The data are presented showing statistical and total
uncertainties. Figure 5: Top: The differential cross section vs. $Q^{2}$.
Bottom: As in Fig. 4, but for $Q^{2}$.

This paper presents the first measurement of the inclusive charged-current
double-differential electron-neutrino cross section vs. electron energy and
angle, using the NOvA near detector. This provides new information concerning
directly observable final-state electron kinematic variables necessary for
neutrino energy estimation and efficiency correction in $\nu_{e}$ appearance
measurements. Measured cross sections are shown compared to multiple versions
of GENIE, and to GiBUU, NEUT, and NuWro event generators, in the neutrino
energy range from 1 to $6\text{\,}\mathrm{GeV}$. The models show general
agreement with the data in the two-dimensional phase space as well as various
levels of success in the neutrino energy and $Q^{2}$ measurements.
Disagreements are seen in both overall normalization across all reported
measurements and with certain shape differences, primarily seen in $Q^{2}$.
The data related to this measurement and systematic covariance matrices can be
found at Acero  _et al._ (2022).

##  I Acknowledgements

###### 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, MSHE,
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. For the purpose of open access, the author
has applied a Creative Commons Attribution (CC BY) license to any Author
Accepted Manuscript version arising.

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