††thanks: now at CERN††thanks: also at J-PARC, Tokai, Japan††thanks: also at
J-PARC, Tokai, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: also at
J-PARC, Tokai, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: affiliated
member at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: affiliated
member at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: affiliated
member at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: also at
J-PARC, Tokai, Japan††thanks: affiliated member at Kavli IPMU (WPI), the
University of Tokyo, Japan††thanks: also at National Research Nuclear
University ”MEPhI” and Moscow Institute of Physics and Technology, Moscow,
Russia††thanks: also at J-PARC, Tokai, Japan††thanks: affiliated member at
Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: affiliated member at
Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: also at J-PARC,
Tokai, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: affiliated member
at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: also at J-PARC,
Tokai, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: also at J-PARC,
Tokai, Japan††thanks: also at JINR, Dubna, Russia††thanks: also at J-PARC,
Tokai, Japan††thanks: affiliated member at Kavli IPMU (WPI), the University of
Tokyo, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: affiliated member
at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: also at J-PARC,
Tokai, Japan††thanks: also at J-PARC, Tokai, Japan††thanks: affiliated member
at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks: also at Institute
of Particle Physics, Canada††thanks: also at J-PARC, Tokai, Japan††thanks:
affiliated member at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks:
affiliated member at Kavli IPMU (WPI), the University of Tokyo, Japan††thanks:
also at J-PARC, Tokai, Japan††thanks: also at BMCC/CUNY, Science Department,
New York, New York, U.S.A.††thanks: affiliated member at Kavli IPMU (WPI), the
University of Tokyo, Japan††thanks: also at J-PARC, Tokai, Japan

The T2K Collaboration

# Measurement of inclusive double-differential $\nu_{\mu}$ charged-current
cross section with improved acceptance in the T2K off-axis near detector

K. Abe  University of Tokyo, Institute for Cosmic Ray Research, Kamioka
Observatory, Kamioka, Japan  J. Amey  Imperial College London, Department of
Physics, London, United Kingdom  C. Andreopoulos  STFC, Rutherford Appleton
Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United
Kingdom  University of Liverpool, Department of Physics, Liverpool, United
Kingdom  L. Anthony  University of Liverpool, Department of Physics,
Liverpool, United Kingdom  M. Antonova  IFIC (CSIC & University of Valencia),
Valencia, Spain  S. Aoki  Kobe University, Kobe, Japan  A. Ariga  University
of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High
Energy Physics (LHEP), Bern, Switzerland  Y. Ashida  Kyoto University,
Department of Physics, Kyoto, Japan  Y. Azuma  Osaka City University,
Department of Physics, Osaka, Japan  S. Ban  Kyoto University, Department of
Physics, Kyoto, Japan  M. Barbi  University of Regina, Department of Physics,
Regina, Saskatchewan, Canada  G.J. Barker  University of Warwick, Department
of Physics, Coventry, United Kingdom  G. Barr  Oxford University, Department
of Physics, Oxford, United Kingdom  C. Barry  University of Liverpool,
Department of Physics, Liverpool, United Kingdom  M. Batkiewicz  H.
Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland  V. Berardi
INFN Sezione di Bari and Università e Politecnico di Bari, Dipartimento
Interuniversitario di Fisica, Bari, Italy  S. Berkman  University of British
Columbia, Department of Physics and Astronomy, Vancouver, British Columbia,
Canada  TRIUMF, Vancouver, British Columbia, Canada  R.M. Berner  University
of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High
Energy Physics (LHEP), Bern, Switzerland  S. Bhadra  York University,
Department of Physics and Astronomy, Toronto, Ontario, Canada  S. Bienstock
UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire
et de Hautes Energies (LPNHE), Paris, France  A. Blondel  University of
Geneva, Section de Physique, DPNC, Geneva, Switzerland  S. Bolognesi  IRFU,
CEA Saclay, Gif-sur-Yvette, France  S. Bordoni  Institut de Fisica d’Altes
Energies (IFAE), The Barcelona Institute of Science and Technology, Campus
UAB, Bellaterra (Barcelona) Spain  B. Bourguille  Institut de Fisica d’Altes
Energies (IFAE), The Barcelona Institute of Science and Technology, Campus
UAB, Bellaterra (Barcelona) Spain  S.B. Boyd  University of Warwick,
Department of Physics, Coventry, United Kingdom  D. Brailsford  Lancaster
University, Physics Department, Lancaster, United Kingdom  A. Bravar
University of Geneva, Section de Physique, DPNC, Geneva, Switzerland  C.
Bronner  University of Tokyo, Institute for Cosmic Ray Research, Kamioka
Observatory, Kamioka, Japan  M. Buizza Avanzini  Ecole Polytechnique,
IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France  J. Calcutt
Michigan State University, Department of Physics and Astronomy, East Lansing,
Michigan, U.S.A.  T. Campbell  Colorado State University, Department of
Physics, Fort Collins, Colorado, U.S.A.  S. Cao  High Energy Accelerator
Research Organization (KEK), Tsukuba, Ibaraki, Japan  S.L. Cartwright
University of Sheffield, Department of Physics and Astronomy, Sheffield,
United Kingdom  M.G. Catanesi  INFN Sezione di Bari and Università e
Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy
A. Cervera  IFIC (CSIC & University of Valencia), Valencia, Spain  A. Chappell
University of Warwick, Department of Physics, Coventry, United Kingdom  C.
Checchia  INFN Sezione di Padova and Università di Padova, Dipartimento di
Fisica, Padova, Italy  D. Cherdack  Colorado State University, Department of
Physics, Fort Collins, Colorado, U.S.A.  N. Chikuma  University of Tokyo,
Department of Physics, Tokyo, Japan  G. Christodoulou  University of
Liverpool, Department of Physics, Liverpool, United Kingdom  J. Coleman
University of Liverpool, Department of Physics, Liverpool, United Kingdom  G.
Collazuol  INFN Sezione di Padova and Università di Padova, Dipartimento di
Fisica, Padova, Italy  D. Coplowe  Oxford University, Department of Physics,
Oxford, United Kingdom  A. Cudd  Michigan State University, Department of
Physics and Astronomy, East Lansing, Michigan, U.S.A.  A. Dabrowska  H.
Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland  G. De Rosa
INFN Sezione di Napoli and Università di Napoli, Dipartimento di Fisica,
Napoli, Italy  T. Dealtry  Lancaster University, Physics Department,
Lancaster, United Kingdom  P.F. Denner  University of Warwick, Department of
Physics, Coventry, United Kingdom  S.R. Dennis  University of Liverpool,
Department of Physics, Liverpool, United Kingdom  C. Densham  STFC, Rutherford
Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington,
United Kingdom  F. Di Lodovico  Queen Mary University of London, School of
Physics and Astronomy, London, United Kingdom  S. Dolan  Ecole Polytechnique,
IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France  IRFU, CEA Saclay,
Gif-sur-Yvette, France  O. Drapier  Ecole Polytechnique, IN2P3-CNRS,
Laboratoire Leprince-Ringuet, Palaiseau, France  K.E. Duffy  Oxford
University, Department of Physics, Oxford, United Kingdom  J. Dumarchez  UPMC,
Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), Paris, France  P. Dunne  Imperial College London,
Department of Physics, London, United Kingdom  S. Emery-Schrenk  IRFU, CEA
Saclay, Gif-sur-Yvette, France  A. Ereditato  University of Bern, Albert
Einstein Center for Fundamental Physics, Laboratory for High Energy Physics
(LHEP), Bern, Switzerland  T. Feusels  University of British Columbia,
Department of Physics and Astronomy, Vancouver, British Columbia, Canada
TRIUMF, Vancouver, British Columbia, Canada  A.J. Finch  Lancaster University,
Physics Department, Lancaster, United Kingdom  G.A. Fiorentini  York
University, Department of Physics and Astronomy, Toronto, Ontario, Canada  G.
Fiorillo  INFN Sezione di Napoli and Università di Napoli, Dipartimento di
Fisica, Napoli, Italy  C. Francois  University of Bern, Albert Einstein Center
for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern,
Switzerland  M. Friend  High Energy Accelerator Research Organization (KEK),
Tsukuba, Ibaraki, Japan  Y. Fujii  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki, Japan  D. Fukuda  Okayama University,
Department of Physics, Okayama, Japan  Y. Fukuda  Miyagi University of
Education, Department of Physics, Sendai, Japan  A. Garcia  Institut de Fisica
d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,
Campus UAB, Bellaterra (Barcelona) Spain  C. Giganti  UPMC, Université Paris
Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies
(LPNHE), Paris, France  F. Gizzarelli  IRFU, CEA Saclay, Gif-sur-Yvette,
France  T. Golan  Wroclaw University, Faculty of Physics and Astronomy,
Wroclaw, Poland  M. Gonin  Ecole Polytechnique, IN2P3-CNRS, Laboratoire
Leprince-Ringuet, Palaiseau, France  D.R. Hadley  University of Warwick,
Department of Physics, Coventry, United Kingdom  L. Haegel  University of
Geneva, Section de Physique, DPNC, Geneva, Switzerland  J.T. Haigh  University
of Warwick, Department of Physics, Coventry, United Kingdom  P. Hamacher-
Baumann  RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany
D. Hansen  University of Pittsburgh, Department of Physics and Astronomy,
Pittsburgh, Pennsylvania, U.S.A.  J. Harada  Osaka City University, Department
of Physics, Osaka, Japan  M. Hartz  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), The University of Tokyo Institutes for
Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan  TRIUMF, Vancouver,
British Columbia, Canada  T. Hasegawa  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki, Japan  N.C. Hastings  University of
Regina, Department of Physics, Regina, Saskatchewan, Canada  T. Hayashino
Kyoto University, Department of Physics, Kyoto, Japan  Y. Hayato  University
of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka,
Japan  Kavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba, Japan  T. Hiraki  Kyoto University, Department of Physics,
Kyoto, Japan  A. Hiramoto  Kyoto University, Department of Physics, Kyoto,
Japan  S. Hirota  Kyoto University, Department of Physics, Kyoto, Japan  M.
Hogan  Colorado State University, Department of Physics, Fort Collins,
Colorado, U.S.A.  J. Holeczek  University of Silesia, Institute of Physics,
Katowice, Poland  F. Hosomi  University of Tokyo, Department of Physics,
Tokyo, Japan  A.K. Ichikawa  Kyoto University, Department of Physics, Kyoto,
Japan  M. Ikeda  University of Tokyo, Institute for Cosmic Ray Research,
Kamioka Observatory, Kamioka, Japan  J. Imber  Ecole Polytechnique,
IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France  T. Inoue  Osaka
City University, Department of Physics, Osaka, Japan  R.A. Intonti  INFN
Sezione di Bari and Università e Politecnico di Bari, Dipartimento
Interuniversitario di Fisica, Bari, Italy  T. Ishida  High Energy Accelerator
Research Organization (KEK), Tsukuba, Ibaraki, Japan  T. Ishii  High Energy
Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan  K. Iwamoto
University of Tokyo, Department of Physics, Tokyo, Japan  A. Izmaylov  IFIC
(CSIC & University of Valencia), Valencia, Spain  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow, Russia  B. Jamieson
University of Winnipeg, Department of Physics, Winnipeg, Manitoba, Canada  M.
Jiang  Kyoto University, Department of Physics, Kyoto, Japan  S. Johnson
University of Colorado at Boulder, Department of Physics, Boulder, Colorado,
U.S.A.  P. Jonsson  Imperial College London, Department of Physics, London,
United Kingdom  C.K. Jung  State University of New York at Stony Brook,
Department of Physics and Astronomy, Stony Brook, New York, U.S.A.  M.
Kabirnezhad  National Centre for Nuclear Research, Warsaw, Poland  A.C. Kaboth
Royal Holloway University of London, Department of Physics, Egham, Surrey,
United Kingdom  STFC, Rutherford Appleton Laboratory, Harwell Oxford, and
Daresbury Laboratory, Warrington, United Kingdom  T. Kajita  University of
Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic
Neutrinos, Kashiwa, Japan  H. Kakuno  Tokyo Metropolitan University,
Department of Physics, Tokyo, Japan  J. Kameda  University of Tokyo, Institute
for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan  D. Karlen
University of Victoria, Department of Physics and Astronomy, Victoria, British
Columbia, Canada  TRIUMF, Vancouver, British Columbia, Canada  T. Katori
Queen Mary University of London, School of Physics and Astronomy, London,
United Kingdom  E. Kearns  Boston University, Department of Physics, Boston,
Massachusetts, U.S.A.  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo Institutes for Advanced Study,
University of Tokyo, Kashiwa, Chiba, Japan  M. Khabibullin  Institute for
Nuclear Research of the Russian Academy of Sciences, Moscow, Russia  A.
Khotjantsev  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow, Russia  H. Kim  Osaka City University, Department of
Physics, Osaka, Japan  J. Kim  University of British Columbia, Department of
Physics and Astronomy, Vancouver, British Columbia, Canada  TRIUMF, Vancouver,
British Columbia, Canada  S. King  Queen Mary University of London, School of
Physics and Astronomy, London, United Kingdom  J. Kisiel  University of
Silesia, Institute of Physics, Katowice, Poland  A. Knight  University of
Warwick, Department of Physics, Coventry, United Kingdom  A. Knox  Lancaster
University, Physics Department, Lancaster, United Kingdom  T. Kobayashi  High
Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan  L.
Koch  RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany
T. Koga  University of Tokyo, Department of Physics, Tokyo, Japan  P.P. Koller
University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory
for High Energy Physics (LHEP), Bern, Switzerland  A. Konaka  TRIUMF,
Vancouver, British Columbia, Canada  L.L. Kormos  Lancaster University,
Physics Department, Lancaster, United Kingdom  Y. Koshio  Okayama University,
Department of Physics, Okayama, Japan  K. Kowalik  National Centre for Nuclear
Research, Warsaw, Poland  Y. Kudenko  Institute for Nuclear Research of the
Russian Academy of Sciences, Moscow, Russia  R. Kurjata  Warsaw University of
Technology, Institute of Radioelectronics, Warsaw, Poland  T. Kutter
Louisiana State University, Department of Physics and Astronomy, Baton Rouge,
Louisiana, U.S.A.  L. Labarga  University Autonoma Madrid, Department of
Theoretical Physics, 28049 Madrid, Spain  J. Lagoda  National Centre for
Nuclear Research, Warsaw, Poland  I. Lamont  Lancaster University, Physics
Department, Lancaster, United Kingdom  M. Lamoureux  IRFU, CEA Saclay, Gif-
sur-Yvette, France  P. Lasorak  Queen Mary University of London, School of
Physics and Astronomy, London, United Kingdom  M. Laveder  INFN Sezione di
Padova and Università di Padova, Dipartimento di Fisica, Padova, Italy  M.
Lawe  Lancaster University, Physics Department, Lancaster, United Kingdom  M.
Licciardi  Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet,
Palaiseau, France  T. Lindner  TRIUMF, Vancouver, British Columbia, Canada
Z.J. Liptak  University of Colorado at Boulder, Department of Physics,
Boulder, Colorado, U.S.A.  R.P. Litchfield  Imperial College London,
Department of Physics, London, United Kingdom  X. Li  State University of New
York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New
York, U.S.A.  A. Longhin  INFN Sezione di Padova and Università di Padova,
Dipartimento di Fisica, Padova, Italy  J.P. Lopez  University of Colorado at
Boulder, Department of Physics, Boulder, Colorado, U.S.A.  T. Lou  University
of Tokyo, Department of Physics, Tokyo, Japan  L. Ludovici  INFN Sezione di
Roma and Università di Roma “La Sapienza”, Roma, Italy  X. Lu  Oxford
University, Department of Physics, Oxford, United Kingdom  L. Magaletti  INFN
Sezione di Bari and Università e Politecnico di Bari, Dipartimento
Interuniversitario di Fisica, Bari, Italy  K. Mahn  Michigan State University,
Department of Physics and Astronomy, East Lansing, Michigan, U.S.A.  M. Malek
University of Sheffield, Department of Physics and Astronomy, Sheffield,
United Kingdom  S. Manly  University of Rochester, Department of Physics and
Astronomy, Rochester, New York, U.S.A.  L. Maret  University of Geneva,
Section de Physique, DPNC, Geneva, Switzerland  A.D. Marino  University of
Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A.  J.F.
Martin  University of Toronto, Department of Physics, Toronto, Ontario, Canada
P. Martins  Queen Mary University of London, School of Physics and Astronomy,
London, United Kingdom  S. Martynenko  State University of New York at Stony
Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A.  T.
Maruyama  High Energy Accelerator Research Organization (KEK), Tsukuba,
Ibaraki, Japan  V. Matveev  Institute for Nuclear Research of the Russian
Academy of Sciences, Moscow, Russia  K. Mavrokoridis  University of Liverpool,
Department of Physics, Liverpool, United Kingdom  W.Y. Ma  Imperial College
London, Department of Physics, London, United Kingdom  E. Mazzucato  IRFU, CEA
Saclay, Gif-sur-Yvette, France  M. McCarthy  York University, Department of
Physics and Astronomy, Toronto, Ontario, Canada  N. McCauley  University of
Liverpool, Department of Physics, Liverpool, United Kingdom  K.S. McFarland
University of Rochester, Department of Physics and Astronomy, Rochester, New
York, U.S.A.  C. McGrew  State University of New York at Stony Brook,
Department of Physics and Astronomy, Stony Brook, New York, U.S.A.  A.
Mefodiev  Institute for Nuclear Research of the Russian Academy of Sciences,
Moscow, Russia  C. Metelko  University of Liverpool, Department of Physics,
Liverpool, United Kingdom  M. Mezzetto  INFN Sezione di Padova and Università
di Padova, Dipartimento di Fisica, Padova, Italy  A. Minamino  Yokohama
National University, Faculty of Engineering, Yokohama, Japan  O. Mineev
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow,
Russia  S. Mine  University of California, Irvine, Department of Physics and
Astronomy, Irvine, California, U.S.A.  A. Missert  University of Colorado at
Boulder, Department of Physics, Boulder, Colorado, U.S.A.  M. Miura
University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory,
Kamioka, Japan  S. Moriyama  University of Tokyo, Institute for Cosmic Ray
Research, Kamioka Observatory, Kamioka, Japan  J. Morrison  Michigan State
University, Department of Physics and Astronomy, East Lansing, Michigan,
U.S.A.  Th.A. Mueller  Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-
Ringuet, Palaiseau, France  Y. Nagai  University of Colorado at Boulder,
Department of Physics, Boulder, Colorado, U.S.A.  T. Nakadaira  High Energy
Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan  M. Nakahata
University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory,
Kamioka, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo Institutes for Advanced Study,
University of Tokyo, Kashiwa, Chiba, Japan  K.G. Nakamura  Kyoto University,
Department of Physics, Kyoto, Japan  K. Nakamura  Kavli Institute for the
Physics and Mathematics of the Universe (WPI), The University of Tokyo
Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan
K.D. Nakamura  Kyoto University, Department of Physics, Kyoto, Japan  Y.
Nakanishi  Kyoto University, Department of Physics, Kyoto, Japan  S. Nakayama
University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory,
Kamioka, Japan  T. Nakaya  Kyoto University, Department of Physics, Kyoto,
Japan  Kavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba, Japan  K. Nakayoshi  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki, Japan  C. Nantais  University of
Toronto, Department of Physics, Toronto, Ontario, Canada  C. Nielsen
University of British Columbia, Department of Physics and Astronomy,
Vancouver, British Columbia, Canada  TRIUMF, Vancouver, British Columbia,
Canada  K. Niewczas  Wroclaw University, Faculty of Physics and Astronomy,
Wroclaw, Poland  K. Nishikawa  High Energy Accelerator Research Organization
(KEK), Tsukuba, Ibaraki, Japan  Y. Nishimura  University of Tokyo, Institute
for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan
P. Novella  IFIC (CSIC & University of Valencia), Valencia, Spain  J. Nowak
Lancaster University, Physics Department, Lancaster, United Kingdom  H.M.
O’Keeffe  Lancaster University, Physics Department, Lancaster, United Kingdom
K. Okumura  University of Tokyo, Institute for Cosmic Ray Research, Research
Center for Cosmic Neutrinos, Kashiwa, Japan  Kavli Institute for the Physics
and Mathematics of the Universe (WPI), The University of Tokyo Institutes for
Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan  T. Okusawa  Osaka
City University, Department of Physics, Osaka, Japan  W. Oryszczak  University
of Warsaw, Faculty of Physics, Warsaw, Poland  S.M. Oser  University of
British Columbia, Department of Physics and Astronomy, Vancouver, British
Columbia, Canada  TRIUMF, Vancouver, British Columbia, Canada  T. Ovsyannikova
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow,
Russia  R.A. Owen  Queen Mary University of London, School of Physics and
Astronomy, London, United Kingdom  Y. Oyama  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki, Japan  V. Palladino  INFN Sezione di
Napoli and Università di Napoli, Dipartimento di Fisica, Napoli, Italy  J.L.
Palomino  State University of New York at Stony Brook, Department of Physics
and Astronomy, Stony Brook, New York, U.S.A.  V. Paolone  University of
Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania,
U.S.A.  P. Paudyal  University of Liverpool, Department of Physics, Liverpool,
United Kingdom  M. Pavin  UPMC, Université Paris Diderot, CNRS/IN2P3,
Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France
D. Payne  University of Liverpool, Department of Physics, Liverpool, United
Kingdom  Y. Petrov  University of British Columbia, Department of Physics and
Astronomy, Vancouver, British Columbia, Canada  TRIUMF, Vancouver, British
Columbia, Canada  L. Pickering  Michigan State University, Department of
Physics and Astronomy, East Lansing, Michigan, U.S.A.  E.S. Pinzon Guerra
York University, Department of Physics and Astronomy, Toronto, Ontario, Canada
C. Pistillo  University of Bern, Albert Einstein Center for Fundamental
Physics, Laboratory for High Energy Physics (LHEP), Bern, Switzerland  B.
Popov  UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique
Nucléaire et de Hautes Energies (LPNHE), Paris, France  M. Posiadala-Zezula
University of Warsaw, Faculty of Physics, Warsaw, Poland  A. Pritchard
University of Liverpool, Department of Physics, Liverpool, United Kingdom  P.
Przewlocki  National Centre for Nuclear Research, Warsaw, Poland  B. Quilain
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba, Japan  T. Radermacher  RWTH Aachen University, III.
Physikalisches Institut, Aachen, Germany  E. Radicioni  INFN Sezione di Bari
and Università e Politecnico di Bari, Dipartimento Interuniversitario di
Fisica, Bari, Italy  P.N. Ratoff  Lancaster University, Physics Department,
Lancaster, United Kingdom  M.A. Rayner  University of Geneva, Section de
Physique, DPNC, Geneva, Switzerland  E. Reinherz-Aronis  Colorado State
University, Department of Physics, Fort Collins, Colorado, U.S.A.  C. Riccio
INFN Sezione di Napoli and Università di Napoli, Dipartimento di Fisica,
Napoli, Italy  E. Rondio  National Centre for Nuclear Research, Warsaw, Poland
B. Rossi  INFN Sezione di Napoli and Università di Napoli, Dipartimento di
Fisica, Napoli, Italy  S. Roth  RWTH Aachen University, III. Physikalisches
Institut, Aachen, Germany  A.C. Ruggeri  INFN Sezione di Napoli and Università
di Napoli, Dipartimento di Fisica, Napoli, Italy  A. Rychter  Warsaw
University of Technology, Institute of Radioelectronics, Warsaw, Poland  K.
Sakashita  High Energy Accelerator Research Organization (KEK), Tsukuba,
Ibaraki, Japan  F. Sánchez  Institut de Fisica d’Altes Energies (IFAE), The
Barcelona Institute of Science and Technology, Campus UAB, Bellaterra
(Barcelona) Spain  S. Sasaki  Tokyo Metropolitan University, Department of
Physics, Tokyo, Japan  E. Scantamburlo  University of Geneva, Section de
Physique, DPNC, Geneva, Switzerland  K. Scholberg  Duke University, Department
of Physics, Durham, North Carolina, U.S.A.  J. Schwehr  Colorado State
University, Department of Physics, Fort Collins, Colorado, U.S.A.  M. Scott
TRIUMF, Vancouver, British Columbia, Canada  Y. Seiya  Osaka City University,
Department of Physics, Osaka, Japan  T. Sekiguchi  High Energy Accelerator
Research Organization (KEK), Tsukuba, Ibaraki, Japan  H. Sekiya  University of
Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba, Japan  D. Sgalaberna  University of Geneva, Section de
Physique, DPNC, Geneva, Switzerland  R. Shah  STFC, Rutherford Appleton
Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United
Kingdom  Oxford University, Department of Physics, Oxford, United Kingdom  A.
Shaikhiev  Institute for Nuclear Research of the Russian Academy of Sciences,
Moscow, Russia  F. Shaker  University of Winnipeg, Department of Physics,
Winnipeg, Manitoba, Canada  D. Shaw  Lancaster University, Physics Department,
Lancaster, United Kingdom  M. Shiozawa  University of Tokyo, Institute for
Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan  Kavli Institute for
the Physics and Mathematics of the Universe (WPI), The University of Tokyo
Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan  A.
Smirnov  Institute for Nuclear Research of the Russian Academy of Sciences,
Moscow, Russia  M. Smy  University of California, Irvine, Department of
Physics and Astronomy, Irvine, California, U.S.A.  J.T. Sobczyk  Wroclaw
University, Faculty of Physics and Astronomy, Wroclaw, Poland  H. Sobel
University of California, Irvine, Department of Physics and Astronomy, Irvine,
California, U.S.A.  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo Institutes for Advanced Study,
University of Tokyo, Kashiwa, Chiba, Japan  J. Steinmann  RWTH Aachen
University, III. Physikalisches Institut, Aachen, Germany  T. Stewart  STFC,
Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory,
Warrington, United Kingdom  P. Stowell  University of Sheffield, Department of
Physics and Astronomy, Sheffield, United Kingdom  Y. Suda  University of
Tokyo, Department of Physics, Tokyo, Japan  S. Suvorov  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow, Russia  IRFU, CEA Saclay,
Gif-sur-Yvette, France  A. Suzuki  Kobe University, Kobe, Japan  S.Y. Suzuki
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan
Y. Suzuki  Kavli Institute for the Physics and Mathematics of the Universe
(WPI), The University of Tokyo Institutes for Advanced Study, University of
Tokyo, Kashiwa, Chiba, Japan  R. Tacik  University of Regina, Department of
Physics, Regina, Saskatchewan, Canada  TRIUMF, Vancouver, British Columbia,
Canada  M. Tada  High Energy Accelerator Research Organization (KEK), Tsukuba,
Ibaraki, Japan  A. Takeda  University of Tokyo, Institute for Cosmic Ray
Research, Kamioka Observatory, Kamioka, Japan  Y. Takeuchi  Kobe University,
Kobe, Japan  Kavli Institute for the Physics and Mathematics of the Universe
(WPI), The University of Tokyo Institutes for Advanced Study, University of
Tokyo, Kashiwa, Chiba, Japan  R. Tamura  University of Tokyo, Department of
Physics, Tokyo, Japan  H.K. Tanaka  University of Tokyo, Institute for Cosmic
Ray Research, Kamioka Observatory, Kamioka, Japan  H.A. Tanaka  University of
Toronto, Department of Physics, Toronto, Ontario, Canada  TRIUMF, Vancouver,
British Columbia, Canada  T. Thakore  Louisiana State University, Department
of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A.  L.F. Thompson
University of Sheffield, Department of Physics and Astronomy, Sheffield,
United Kingdom  S. Tobayama  University of British Columbia, Department of
Physics and Astronomy, Vancouver, British Columbia, Canada  TRIUMF, Vancouver,
British Columbia, Canada  W. Toki  Colorado State University, Department of
Physics, Fort Collins, Colorado, U.S.A.  T. Tsukamoto  High Energy Accelerator
Research Organization (KEK), Tsukuba, Ibaraki, Japan  M. Tzanov  Louisiana
State University, Department of Physics and Astronomy, Baton Rouge, Louisiana,
U.S.A.  W. Uno  Kyoto University, Department of Physics, Kyoto, Japan  M.
Vagins  Kavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba, Japan  University of California, Irvine, Department of Physics
and Astronomy, Irvine, California, U.S.A.  Z. Vallari  State University of New
York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New
York, U.S.A.  G. Vasseur  IRFU, CEA Saclay, Gif-sur-Yvette, France  C. Vilela
State University of New York at Stony Brook, Department of Physics and
Astronomy, Stony Brook, New York, U.S.A.  T. Vladisavljevic  Oxford
University, Department of Physics, Oxford, United Kingdom  Kavli Institute for
the Physics and Mathematics of the Universe (WPI), The University of Tokyo
Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan  T.
Wachala  H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland
J. Walker  University of Winnipeg, Department of Physics, Winnipeg, Manitoba,
Canada  C.W. Walter  Duke University, Department of Physics, Durham, North
Carolina, U.S.A.  Y. Wang  State University of New York at Stony Brook,
Department of Physics and Astronomy, Stony Brook, New York, U.S.A.  D. Wark
STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury
Laboratory, Warrington, United Kingdom  Oxford University, Department of
Physics, Oxford, United Kingdom  M.O. Wascko  Imperial College London,
Department of Physics, London, United Kingdom  A. Weber  STFC, Rutherford
Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington,
United Kingdom  Oxford University, Department of Physics, Oxford, United
Kingdom  R. Wendell  Kyoto University, Department of Physics, Kyoto, Japan
M.J. Wilking  State University of New York at Stony Brook, Department of
Physics and Astronomy, Stony Brook, New York, U.S.A.  C. Wilkinson  University
of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High
Energy Physics (LHEP), Bern, Switzerland  J.R. Wilson  Queen Mary University
of London, School of Physics and Astronomy, London, United Kingdom  R.J.
Wilson  Colorado State University, Department of Physics, Fort Collins,
Colorado, U.S.A.  C. Wret  Imperial College London, Department of Physics,
London, United Kingdom  Y. Yamada  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki, Japan  K. Yamamoto  Osaka City
University, Department of Physics, Osaka, Japan  S. Yamasu  Okayama
University, Department of Physics, Okayama, Japan  C. Yanagisawa  State
University of New York at Stony Brook, Department of Physics and Astronomy,
Stony Brook, New York, U.S.A.  T. Yano  Kobe University, Kobe, Japan  S. Yen
TRIUMF, Vancouver, British Columbia, Canada  N. Yershov  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow, Russia  M. Yokoyama
University of Tokyo, Department of Physics, Tokyo, Japan  M. Yu  York
University, Department of Physics and Astronomy, Toronto, Ontario, Canada  A.
Zalewska  H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland
J. Zalipska  National Centre for Nuclear Research, Warsaw, Poland  L. Zambelli
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan
K. Zaremba  Warsaw University of Technology, Institute of Radioelectronics,
Warsaw, Poland  M. Ziembicki  Warsaw University of Technology, Institute of
Radioelectronics, Warsaw, Poland  E.D. Zimmerman  University of Colorado at
Boulder, Department of Physics, Boulder, Colorado, U.S.A.  M. Zito  IRFU, CEA
Saclay, Gif-sur-Yvette, France  A. Zykova  Institute for Nuclear Research of
the Russian Academy of Sciences, Moscow, Russia

###### Abstract

We report a measurement of the flux-integrated cross section for inclusive
muon neutrino charged-current interactions on carbon. The double differential
measurements are given as function of the muon momentum and angle. Relative to
our previous publication on this topic, these results have an increased
angular acceptance and higher statistics. The data sample presented here
corresponds to $5.7\times 10^{20}$ protons-on-target. The total flux-
integrated cross section is measured to be $(6.950\pm 0.662)\times
10^{-39}\text{cm}^{2}\text{nucleon}^{-1}$ and is consistent with our
simulation.

##  I INTRODUCTION

T2K is an experiment located in Japan with the primary aim of studying
neutrino oscillations Abe  _et al._ (2011). It was designed to measure with
high precision the $\nu_{\mu}\rightarrow\nu_{\mu}$ disappearance channel and
to discover the $\nu_{\mu}\rightarrow\nu_{e}$ appearance channel.

In addition to the oscillation measurements, T2K has an ongoing program to
study neutrino interactions using the near detector complex in order to
improve the understanding and modeling of these interactions. Results from
this program, as exemplified by those presented in this paper, are interesting
in their own right and can be used to constrain and reduce the systematic
errors arising from cross section uncertainties in the extraction of neutrino
oscillation parameters. Inclusive measurements provide a clear signals which
are very valuable to test different models.

Previously, T2K reported the measurement of the flux-integrated double
differential cross section for muon neutrino charged- current interactions on
carbon Abe  _et al._ (2013a). Since that time, many improvements have been
made in the analysis. The results presented in this paper were obtained with
more data, reduced neutrino flux uncertainties (thanks to new NA61/SHINE
measurements Abgrall  _et al._ (2016)), increased angular acceptance, reduced
background contamination and a different unfolding method. All the
improvements are described in more detail below.

The paper is organized as follows: we first summarize the experimental setup
in Sec. II, which contains the description of the off-axis beam, the near
detector and the neutrino event generators used in the present analysis. The
selection of the muon neutrino interaction samples is presented in Sec. III
together with the summary of the detector systematic uncertainties. The
analysis method is explained in Sec. IV and the results are given in Sec. V.

##  II EXPERIMENTAL APPARATUS

###  II.1 T2K beamline and flux prediction

The neutrino beam used by T2K is produced at the J-PARC Laboratory in Tokai,
Japan. In this process, 30 GeV/c protons are extracted from the main ring
accelerator at J-PARC onto a graphite target, producing secondary particles
consisting primarily of pions and kaons. The hadrons exiting the target are
focused by three magnetic horns and allowed to decay in a decay volume. The
decaying hadrons produce neutrinos (primarily of muon flavor) that continue to
the near and far detectors while the other particles range out. Depending on
the polarity of the electric current in the horns, a beam composed of mostly
neutrinos ($\nu$-mode) or antineutrinos ($\bar{\nu}$-mode) and with energy
peaked at 0.6 GeV is produced. The T2K beamline hardware has been described in
detail elsewhere Abe  _et al._ (2011).

The simulation that is used to predict the neutrino flux and its associated
uncertainty is described in detail in Abe  _et al._ (2013b). The uncertainties
are dominated by the hadron production model and, to second order, by the
beamline configuration. Currently, the uncertainty on the $\nu_{\mu}$ beam
flux at the near detector varies from 10$\%$ to 15$\%$ depending on the
neutrino energy. The error associated with the flux in the results presented
here has been reduced with respect to that used in the previous analysis Abe
_et al._ (2013a), in part, because the model of hadron production from the
target is tuned using the full 2009 thin-target dataset by the NA61/SHINE
experiment Abgrall  _et al._ (2016). The previous analysis used the 2007
dataset Abgrall  _et al._ (2012).

###  II.2 The off-axis near detector

The off-axis near detector (ND280) is made-up of two main components, the
$\pi^{0}$ detector (P0D Assylbekov  _et al._ (2012)) and the Tracker region.
Both parts are contained in a metal basket box surrounded by electromagnetic
calorimeters (ECal Allan  _et al._ (2013)) and a warm dipole magnet. The
magnet provides a 0.2 T field allowing for momentum measurement and charge
separation. Outside the ECal and magnet coil is the magnet flux return yoke
and the side muon range detector (SMRD Aoki  _et al._ (2013)).

The Tracker region contains two fine-grained detectors (FGDs Amaudruz  _et
al._ (2012)) sandwiched between three gas time projection chambers (TPCs
Abgrall  _et al._ (2011)). The TPCs contain a drift gas mixture which is
ionized when a charged particle crosses it. The TPCs provide excellent track
and momentum reconstruction. The observed energy loss in the TPCs, combined
with the measurement of the momentum, is used for particle identification.

The most upstream FGD (FGD1) consists of polystyrene scintillators bars, which
are oriented vertically and horizontally and perpendicular to the beam
direction. FGD1 is comprised of carbon ($86.1\%$), hydrogen ($7.4\%$) and
oxygen ($3.7\%$), where the percentages represent the mass fraction of each
element. The most downstream FGD (FGD2) is similar to FGD1 except that the
scintillators layers are interleaved with water layers. FGD1 is the active
target in this analysis. The fiducial volume (FV) begins 58 mm inward from the
lateral edges as shown in Fig. 1.

Figure 1: Schematic view of FGD1. The orange region indicates the fiducial
volume.

The P0D region of ND280, located upstream the Tracker region, is made of
layers of plastic scintillator, water, brass and lead. In this analysis, it is
used to veto interactions happening upstream of the active target.

The SMRD consists of 440 scintillator modules inserted in the air gaps between
sections of the magnet flux return yoke. Horizontal (vertical) modules are
composed of four (five) plastic scintillation counters. In this analysis, the
SMRD is used to identify and measure the range of muons at high angles with
respect to the beam direction. The range provides information about the muon
momentum.

The ECal consist of 13 modules surrounding the inner detectors. The tracker
module is covered by six modules in the sides (BarrelECal) and one module
downstream (DsECal). The modules are made up of plastic scintillator bars
interleaved with lead sheets. In this analysis, the ECal is used to complement
the reconstruction of the inner detectors. As with the SMRD, it is used to
measure the range/momentum of muons escaping, from inner detectors, at high
angles with respect to the beam direction. In addition, electromagnetic
showers and minimally ionizing tracks passing through the ECal can be
identified using a multivariate analysis quantity $R_{\text{MIP/EM}}$
determined by the features of the reconstructed clusters in the ECal Abe  _et
al._ (2014).

In this analysis, the timing information for particles crossing the different
detectors of ND280 is used for the first time. When a particle crosses a
detector composed by scintillators, the time information from each individual
hit is corrected for the light propagation time inside the fibers and for the
time offset of each slave clock module Abe  _et al._ (2011). Then, the
corrected time and position of the hits are used to define an average time
($T$). Finally, the time of flight (ToF) variable
(ToF$=T_{\text{X}}-T_{\text{Y}}$) between two detectors X and Y is
constructed. This information is used to determine the direction of tracks
crossing the following pairs of detectors: FGD1-FGD2, FGD1-P0D, and
FGD1-BarrelECal (see Fig. 2).

Figure 2: ToF between FGD1-BarrelECal for tracks crossing BarrelECal-
TPC1-FGD1. Stacked histograms indicate the prediction from NEUT of the true
direction and whether the true start position is inside FGD1. Data
distributions show their statistical error bars. The region indicated by the
red arrow shows tracks that are reconstructed as backward-going. They are
chosen that give the lowest wrong-sense fraction for each pair of detectors.

###  II.3 Event generators

Two event generators, NEUT 5.3.2 Hayato (2009) and GENIE 2.8.0 Andreopoulos
_et al._ (2010), are used to simulate the interaction of neutrinos in the near
detector and the effect of the nuclear medium on the produced particles. The
modeling of the main interaction channels and their associated uncertainties
is described below.

####  II.3.1 Charged-current interactions without pion production

Charged-current (CC) interactions without pion production are referred to here
as charged-current quasi-elastic-like, or CCQE-like, interactions. The sample
of such interactions is composed mainly of CCQE reactions. However, nuclear
effects can cause other processes to be included in this category.

For the CCQE channel, the primary neutrino-nucleon interaction is modeled in a
similar fashion by both generators. Each uses an implementation of the
Llewellyn-Smith formalism Smith (1972) through Lorentz-invariant form factors
(FFs). Both generators relate the vector FF to the electromagnetic FFs, for
which the parametrization BBA2005 is used Bradford  _et al._ (2006). For the
axial FF, a dipole shape with $g_{A}$=1.267 is used in both generators.
However, the default axial mass parameter, $M_{A}$, used in each generator
differs. In NEUT, $M_{A}=1.21$ GeV/c2, while in GENIE, $M_{A}=0.99$ GeV/c2.
Finally, they use the same pseudo-scalar FF suggested by the partially
conserved axial current (PCAC) hypothesis.

The majority of the CCQE interactions take place on bound nucleons. The
nuclear model differs between the two generators. In the case of GENIE, the
Bodek-Richie version of the Relativistic Fermi Gas (RFG) model is used, which
incorporates short range nucleon-nucleon correlations Bodek and Ritchie
(1981). For NEUT, a different nuclear model is used based on the spectral
functions from Benhar and Fabrocini (2000). Moreover, NEUT includes the multi-
nucleon interaction (2p2h) model from Nieves et al. Nieves  _et al._ (2011),
as it is thought that interactions on more than one bound nucleon contribute
significant strength to the signal relative to the single particle CCQE
interaction. Pauli blocking is implemented equally in both generators (reject
events with the momentum of the outgoing nucleon below the Fermi momentum of
the nucleus).

The CCQE and 2p2h interactions are parametrized in NEUT with several target-
dependent parameters (superscripts “C” and “O” represent parameters for carbon
and oxygen targets, respectively): the quasielastic axial mass ($M_{A}=1.21\pm
0.3$ GeV/c2), the binding energy ($E_{b}^{\text{C}}=25\pm 25$ MeV and
$E_{b}^{\text{O}}=27\pm 27$ MeV), the Fermi momentum ($p_{F}^{\text{C}}=217\pm
30$ MeV/c and $p_{F}^{\text{O}}=225\pm 30$ MeV/c) and the 2p2h cross-section
normalization ($\text{MEC}^{\text{C}}=1\pm 1$ and $\text{MEC}^{\text{O}}=1\pm
1$). The nominal values for these parameters and the associated uncertainties
were chosen based on a study of the MINERvA and MiniBooNE datasets Wilkinson
_et al._ (2016). Large uncertainties without correlations were assigned in
order to cover the tensions between the two datasets and different nuclear
models.

####  II.3.2 CC interactions with pion production

Pion production is treated differently in the two event generators. NEUT
generates interactions with single pion production using a resonant model when
$W<2$ GeV/c2. Single pion production above that value and the rest of pion
production channels are generated with a DIS model. In contrast, GENIE does
not restrict the resonant model to the single pion decay channel. This model
is switched off when $W>1.7$ GeV/c2 (to avoid double counting with its DIS
model). Below that value, the normalization of the single pion and two pions
production channels from its DIS model are tuned.

Resonant pion production is based on the Rein-Sehgal model for both generators
Rein and Sehgal (1981). In NEUT, the model uses 18 resonances taking into
account their interferences. The default parameters for the FFs are taken from
Graczyk and Sobczyk (2008). In contrast, GENIE incorporates 16 resonances
without including interference terms and the default FFs are taken from Kuzmin
_et al._ (2006).

The resonant model has three parameters in NEUT: the resonant axial mass
($M_{A}^{RES}=0.95\pm 0.15$ GeV/c2), the normalization of the axial form
factor for resonant pion production ($C_{5}^{A}=1.01\pm 0.12$) and the
normalization of the isospin non-resonant component predicted in the Rein-
Sehgal model ($I_{1/2}=1.3\pm 0.2$). Their nominal values and associated
uncertainties, with no correlation assumed, were obtained by comparison with
available low energy neutrino-deuterium single pion production data Wilkinson
_et al._ (2014).

Both NEUT and GENIE model deep inelastic scattering using the same GRV98 PDF
parametrization Glück  _et al._ (1998) including a Bodek-Yang correction to
describe scattering at low $Q^{2}$. The Bodek-Yang correction differs slightly
between the two generators, as NEUT uses Bodek and Yang (2003) and GENIE uses
Bodek and Yang (2005). An energy dependent normalisation uncertainty ($10\%$
at 4 GeV) is used based on MINOS CC-inclusive data Adamson  _et al._ (2010).

For coherent reactions, both generators use the Rein-Sehgal model Rein and
Sehgal (1983) including a correction that takes into account the lepton mass
Rein and Sehgal (2007). However, the implementation of the model differs
slightly. NEUT follows the prescriptions and data fit of pion scattering from
Rein and Sehgal (1983), leading to different cross sections for low momentum
pions. The MINERvA experiment has reported results which are consistent with
coherent pion production at $\nu$ energies around 1 GeV Higuera  _et al._
(2014). Considering that result, a $30\%$ normalization uncertainty in CC
coherent interactions is included.

####  II.3.3 Neutral-current interactions

Neutral-current (NC) interactions affect the background prediction in this
analysis. Therefore, an NC normalization parameter was included that scales
elastic, resonant kaon and eta production, and DIS events. A 30$\%$
uncertainty is assigned for those channels, motivated by poor constraints from
external data.

####  II.3.4 Hadronization and final state interactions

Hadron production and transport inside the nuclear medium are also simulated
by the event generators. In this analysis, the prediction of this processes is
particularly important for pions, as they contribute the main background.

The hadronization model (or fragmentation model) determines the kinematics of
the primary outgoing hadrons, prior to final state interactions (FSI), given a
particular interaction. In the high invariant mass region ($W_{\text{NEUT}}>2$
GeV/c2 and $W_{\text{GENIE}}>3$ GeV/c2), the hadronization is simulated using
the PYTHIA5 and PYTHIA6 predictions Sjöstrand (1994) in NEUT and GENIE,
respectively. These predictions are unsatisfactory near the pion production
threshold. So, both generators include a different phenomenological
description based on Koba-Nielsen-Olesen (KNO) scaling Koba  _et al._ (1972)
in the low invariant mass region. Moreover, the transition between the two
regions is handled differently between the two generators. Specifically, GENIE
includes the AGKY model Yang  _et al._ (2009) for $W<3$ GeV/c2 and the
transition region (2.3 GeV/c2 $<W<$ 3 GeV/c2) in which the PYTHIA model is
turned on gradually.

In GENIE, several parameters affect pion kinematics. In particular, for single
pion states four parameters are notable: the nucleon $x_{F}$ ($p_{T}^{2}$),
PDFs for $N\pi$ hadronic states, the nuclear formation zone, and the pion
angular distribution in $\Delta$ resonant pion production. Their nominal
values and associated uncertainties are estimated based on recommendations
from the GENIE Collaboration Andreopoulos  _et al._ (2010). These parameters
are treated as uncorrelated.

Near an energy of 1 GeV, pions immersed in a highly dense nuclear medium are
very likely to interact. Both generators simulate pion FSI using the intra-
nuclear cascade approach, though they use different predictions for the
interaction probabilities. In the case of NEUT, pion interaction probabilities
are dependent on the momentum of the pion: if $p_{\pi}<500$ MeV/c, NEUT uses a
density dependent model Salcedo  _et al._ (1988) and if $p_{\pi}>500$ MeV/c
the probabilities are extracted from pion-nuclear scattering experiments
Ashery  _et al._ (1981). GENIE uses a model called INTRANUKE hA which extracts
the interaction probabilities from several experiments up to 300 MeV/c, while
for higher energies it is based on the CEM03 predictions Mashnik  _et al._
(2006). The uncertainties associated with the pion interaction probabilities
and their correlations are estimated using the same methodology as in Abe  _et
al._ (2015).

##  III $\nu_{\mu}$ CC SAMPLES

This analysis uses data collected in $\nu$-mode between November 2010 and May
2013. The total sample comes from $5.7\times 10^{20}$ protons on target (POT),
which is a factor of five larger than that used in the similar previously
published analysis from T2K Abe  _et al._ (2013a).

Simulated Monte Carlo (MC) interactions within the ND280 subdetectors and
magnet were generated using both NEUT and GENIE. The background interactions
in the materials surrounding ND280, so-called sand interactions, were
generated using NEUT. Both interactions in ND280 and in the surrounding
material were generated using the same neutrino beam simulation, detector
simulation and reconstruction.

In this analysis, events containing muons emanating from interactions that
occur in the fiducial volume (FV) of FGD1 are selected. These events are
candidate $\nu_{\mu}$ CC interactions. The events within this sample that are
true $\nu_{\mu}$ CC events belong to the category referred to here as
$\nu_{\mu}$CC-$\mu$.

Background events in the initial selection include: interactions not happening
in the FV (either inside or outside the magnet volume, referred to as ‘out FV’
and ‘sand $\mu$’, respectively); interactions happening in the FV but not
actually a $\nu_{\mu}$ CC event, referred to as no$\nu_{\mu}$CC; or being
$\nu_{\mu}$ CC but where the muon candidate track is not the outgoing muon,
herein called $\nu_{\mu}$CC-no$\mu$.

The cross-section results presented here are based on the kinematics of the
outgoing muon. Specifically, the results are given as a function of the muon
momentum, $p_{\mu}$, and the cosine of the muon emission angle with respect to
the neutrino direction, $\cos\theta_{\mu}$. The event selection criteria and
performance, as well as the systematic uncertainties associated with the
detector response are described below.

###  III.1 Event selection

In previous T2K work on this topic, the analysis was optimized to select
forward-going muons originating from FGD1 and making a long track (at least 19
clusters as described in section III.1.1) through TPC2, which is downstream of
FGD1 Abe  _et al._ (2013a). The current work aims to include the so-called
high-angle tracks which miss or barely cross the TPCs, as well as long
backward-going tracks in TPC1 (upstream of FGD1). The addition of backward-
going muon candidates in the event selection is possible only with the
introduction of timing information correlated between subdetectors.

In this analysis, events are broken into samples according to the muon
direction. If the muon candidate in the event goes forward (in the direction
downstream of FGD1 into TPC2), the event is part of the forward (FWD) sample.
If the muon goes backward (in a direction upstream of FGD1 into TPC1), the
event is part of the backward (BWD) sample. Similarly, if the muon candidate
in the event is at a high angle in the forward or backward direction, the
event is categorized as high-angle forward (HAFWD) or high-angle backward
(HABWD), respectively. In the FWD/BWD selections, the muon candidate must have
long TPCs segments, while tracks with short or no TPC segment are used in the
HAFWD/HABWD (see Fig. 3).

Figure 3: Schematic representation of the regions of interest for each
selection.

For events to be considered in this analysis, they must occur within the time
window of one of the 8 beam bunches per 5 $\mu s$ spill RF structure of the
beam. The full spill is required to be of good quality. Events are resolved in
time by bunch and then processed. Given the beam intensity for these runs, the
frequency of multiple neutrino interactions happening in the same beam spill
(so-called pile-up events) is very low. This is ignored in the sample
selection and included in the systematic error treatment.

In order to avoid having multiple muon candidates, the analysis looks for
candidates sequentially in the different event selections. The ordering for
this process is FWD, BWD, and then the high angle categories. FWD and BWD have
a higher priority than the high angle categories because the muon PID from the
TPCs is more accurate than in the ECals. The FWD(HAFWD) selection has a higher
priority than the BWD(HABWD) because forward-going muon happen much more often
than backward-going ones.

Additionally, two control regions are selected to constrain neutral current
event rates and pion final state interactions. The control regions are non-
signal regions of phase space close enough to the signal region that the
backgrounds are similar to that in the signal region. The backgrounds used in
the model are tuned using the data observed in the control regions. The
control region selection is described in section III.1.4.

####  III.1.1 Forward selection

The selection criteria for the FWD sample are very similar to those used
previously, though some further optimization has been performed. The cuts used
to extract the FWD sample are described below.

  * •

Quality and FV: This selection considers negatively charged tracks originating
in the FGD1 FV which have TPC track segments containing more than 18 clustered
hits in the TPC. If multiple tracks satisfy these criteria, the muon candidate
is the one with highest momentum and going forward (by timing). In order to
reduce the contamination from events occurring outside the FV, tracks starting
in the most upstream layer of FGD1 are rejected.

  * •

Muon PID: This cut is applied to the muon candidate using discriminator
functions calculated for muon, pion and proton hypotheses based on the energy
loss and momentum measurement of the TPC. These functions are the same as used
in the previous analysis Abe  _et al._ (2013a). This cut rejects protons,
pions and low momentum electrons (below 500 MeV/c). Moreover, two new PID cuts
below have been developed in order to reduce the pion contamination of this
sample (which is the main background in this analysis).

    * –

Muon FGD2 PID: High energy pions are more likely to stop in FGD2 than muons.
Therefore, it is required that the muon candidate leave the FGD2 active volume
with a momentum above 280 MeV/c. This is expected to reduce the pion
contamination by $15\%$ while leading to a loss of $0.3\%$ of the muons.

    * –

Muon ECal PID: For tracks entering the BarrelECal or DsECal modules, the
multivariate analysis quantity $R_{\text{MIP/EM}}$ (based on the features of
the reconstructed clusters in the ECal Abe  _et al._ (2014)) is used. These
tracks must have $R_{\text{MIP/EM}}<15$, which is estimated to reduce the pion
contamination by $7\%$ while removing $0.3\%$ of the muons.

  * •

Veto: One of the main backgrounds in this analysis are interactions happening
outside the FV. This contamination can be reduced further by using the two
cuts described below:

    * –

Upstream background veto: Due to reconstruction failures and multiple
scattering, a reconstructed track can be broken into two unmatched segments.
One of those can have its beginning in the FV, mimicking an interaction that
originates in the FV. In the previous analysis, such events were rejected if
the second highest momentum track started more than 150 mm upstream of the
muon candidate. This cut was found to be too restrictive because it removed
events with a forward going muon and a second particle going backward. In the
current analysis, the ratio between the momentum of the muon candidate and the
other track is used. Ideally, if the muon candidate is a broken track, this
ratio should be bigger than one since the first segment of the track has a
higher momentum than the second segment. Therefore, the distance between both
tracks, or segments, as well as their momentum ratio are used. Cut values are
chosen that give the highest purity times efficiency.

    * –

Broken track cut: This cut rejects events where the reconstruction procedure
mistakenly breaks a single track into two tracks where the first is a FGD1
segment and the second is reconstructed to begin in the last layers of FGD1
and goes through the downstream TPC module. In this mis-reconstruction
pathology, the second track is considered a muon candidate. For such events,
the start position of muon candidate track is within the two most downstream
layers of FGD1. The broken track cut rejects these events by requiring that
there be no reconstructed track with only a FGD1 segment when the start
position of the muon candidate is in one of the last two layers of FGD1.

Fig. 4 shows the reconstructed kinematics for muon candidates in the FWD
sample in the data together with the prediction from NEUT and GENIE.

Figure 4: Momentum (top) and cosine of emission angle (bottom) for the muon
candidate when all selection criteria are fulfilled in the FWD selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

####  III.1.2 Backward selection

The selection criteria for the BWD sample are described below:

  * •

Quality and FV: This selection considers negatively charged tracks originating
in the FGD1 FV which have TPC track segments containing more than 18 clusters.
If the event contains multiple tracks of this type, the muon candidate is the
one with highest momentum and backward sense (by timing). In order to reduce
the contamination from events occurring outside the FV, tracks starting in the
most upstream layer of FGD1 are rejected.

  * •

Muon PID: For muon candidates in the BWD sample, the PID is based entirely on
the energy loss in the TPC. The value of the cut applied is the same as that
in the FWD selection. However, in this angular region the electron
contamination is very low and the discriminator function used to reduce the
low momentum electrons is not applied.

Fig. 5 shows the reconstructed kinematics for muon candidates in the BWD
sample in the data together with the prediction from NEUT and GENIE.

Figure 5: Momentum (top) and cosine of emission angle (bottom) for the muon
candidate when all selection criteria are fulfilled in the BWD selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

####  III.1.3 High Angle selection

In the selection for the high angle samples (HAFWD and HABWD), the muon
candidates are mostly (or all) contained in the FGD1, ECal and SMRD
subdetectors. A detailed explanation of the selection criteria is shown below.

  * •

Quality and FV: High angle tracks starting in FGD1 FV and stopping either in
SMRD or BarrelECal are considered. The stopping requirement is needed in order
to compute the momentum of the track by range. The contamination from events
occurring outside the FV is reduced by rejecting tracks starting in the most
upstream or downstream layers of FGD1.

  * •

Muon PID: The TPC PID information is not reliable for high angle tracks since
they have no (or short) TPC segments. The SMRD and BarrelECal information
forms the basis of the high angle track PID. Tracks that reach the SMRD in the
HAFWD sample are good muon candidates ($\sim$1200 tracks). In the HABWD
sample, most tracks reaching the SMRD come from out of the FV. Consequently,
tracks reaching the SMRD in the HABWD sample are rejected ($\sim$70 tracks).
Tracks not reaching the SMRD and stopping in the BarrelECal region of the
detector ($\sim$4250 and $\sim$1250 tracks for HAFWD and HABWD respectively)
are considered as muon candidates if the multivariate analysis quantity
$R_{\text{MIP/EM}}<0$. Besides, we reduce the contamination of protons
rejecting events that release high amount of energy in short distances within
the BarrelECal.

  * •

Veto: The upstream background veto, introduced in the FWD selection, is used
for the high angle samples. For this veto, the distance and momentum ratio
relation was optimized for forward going and backward going candidates
independently.

Fig. 6 and Fig. 7 show the reconstructed kinematics for the muon candidates in
the HAFWD and HABWD samples in the data together with the prediction from NEUT
and GENIE.

Figure 6: Momentum (top) and cosine of emission angle (bottom) for the muon
candidate when all selection criteria are fulfilled in the HAFWD selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

Figure 7: Momentum (top) and cosine of emission angle (bottom) for the muon
candidate when all selection criteria are fulfilled in the HABWD selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

####  III.1.4 Control regions selection

As mentioned earlier, uncertainties associated with the modeling of
backgrounds and pion kinematics, neutral current normalization and pion final
state interactions can be minimized using control regions. The backgrounds
used in the model are tuned using the data observed in the control regions.

Events that do not fulfill the muon ECal PID and muon FGD2 PID in the FWD
selection constitute the control region samples, CSECAL and CSFGD2,
respectively. Fig. 8 and Fig. 9 show the reconstructed kinematics for muon
candidates in the control region samples in data as well as the expectation
from NEUT and GENIE. A relative good agreement is observed within systematic
uncertainties, which are particularly large in these samples (mainly affected
by detector response). The main contribution (70$\%$) in both control samples
are negative pions formed in NC or CC deep inelastic interactions. The
fraction of signal events in each control sample is below 20$\%$.

Figure 8: Momentum (top) and cosine of emission angle (bottom) for the pion
candidate when all selection criteria are fulfilled in the CSFGD2 selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

Figure 9: Momentum (top) and cosine of emission angle (bottom) for the pion
candidate when all selection criteria are fulfilled in the CSECAL selection.
Stacked histograms indicate different reaction types predictions from NEUT.
Empty rectangles indicate the prediction from GENIE. Data distributions show
their statistical error bars.

###  III.2 Signal and background compositions of the selection

Table 1 summarizes how each step in the selection affects the number of events
and purity in each sample in both data and MC. Both the PID and veto cuts play
a significant role in increasing the purity in each sample.

Cut | FWD | BWD | HAFWD | HABWD
---|---|---|---|---
| DATA | NEUT | DATA | NEUT | DATA | NEUT | DATA | NEUT
Quality | 82155 | 81222 | 1861 | 1050 | 7225 | 7121 | 1582 | 1566
|  | 32.3 |  | 58.5 |  | 41.8 |  | 48.9
FV | 50519 | 51648 | 1165 | 1025 | 5669 | 5764 | 1356 | 1360
|  | 48.7 |  | 58.8 |  | 49.2 |  | 54.1
$\mu$ PID | 29140 | 29750 | 940 | 799 | 3712 | 3487 | 779 | 684
|  | 81.6 |  | 73.6 |  | 71.7 |  | 72.7
Veto | 25669 | 26656 | 940 | 799 | 3270 | 3107 | 730 | 645
|  | 89.4 |  | 73.6 |  | 79.2 |  | 75.9
Ordering | 25669 | 26656 | 940 | 799 | 3082 | 2857 | 682 | 591
$\nu_{\mu}$CC$-\mu[\%]$ |  | 89.4 |  | 73.6 |  | 81.9 |  | 78.9
Table 1: The selected number of events and signal purities percentage (in
bold) in each sample as successive requirements are added for data and MC. The
cut in last row refers to the priority order in cases where a muon candidate
has been found in two samples.

Table 2 breaks down each sample in the different reaction channels. In the low
angle selections, the dominant background is associated with negative pions
which are mis-identified as muons in the TPC. In the high angle selections, in
which there are no TPC segments, positive pions are the dominant background
because the charge of the track is not reconstructed. Those pions are coming
mainly from NC interactions or CC-DIS interactions. For the out of FV events,
a primary contribution arises from interactions taking place in the borders of
FGD1, where the hits closest to the interactions are not reconstructed. In the
case of interactions in BarrelECal, backgrounds arise when FGD and BarrelECal
reconstructed segments are not matched. Finally, the contribution from
interactions in the P0D is composed primarily by neutral particles that
scatter inside FGD1.

| FWD | BWD | HAFWD | HABWD
---|---|---|---|---
$\nu_{\mu}$CC-$\mu$ | 89.4 | 73.6 | 81.9 | 78.9
QE | 44.7 | 82.0 | 67.3 | 83.2
2p2h | 7.5 | 5.5 | 7.2 | 5.5
RES | 25.4 | 8.6 | 17.6 | 8.0
DIS | 19.9 | 3.8 | 7.2 | 3.4
COH | 2.5 | 0.0 | 0.7 | 0.0
$\nu_{\mu}$CC-no$\mu$ | 2.2 | 1.1 | 2.6 | 1.6
QE | 1.8 | 4.5 | 6.3 | 3.0
2p2h | 0.3 | 0.0 | 1.6 | 0.6
RES | 6.3 | 24.6 | 59.1 | 60.8
DIS | 91.4 | 70.3 | 31.7 | 35.6
COH | 0.3 | 0.6 | 1.3 | 0.0
no$\nu_{\mu}$CC | 3.3 | 1.4 | 3.7 | 1.9
NC | 75.5 | 67.2 | 51.4 | 69.1
$\bar{\nu}_{\mu}$ | 15.8 | 15.8 | 39.3 | 15.4
$\nu_{e},\bar{\nu}_{e}$ | 8.7 | 17.0 | 9.3 | 15.5
Out of FV | 4.4 | 21.5 | 11.3 | 16.9
$\nu_{\mu}$CC (in FGD1) | 12.4 | 16.4 | 33.3 | 34.6
$\nu_{\mu}$CC (out FGD1) | 65.2 | 69.2 | 51.7 | 55.9
NC | 17.0 | 11.0 | 11.3 | 7.5
Other | 5.4 | 3.4 | 3.8 | 2.0
Sand $\mu$ | 0.8 | 2.3 | 0.4 | 0.7
Table 2: Muon candidate composition in NEUT combining the true inclusive
reaction type and the true particle type of the muon candidate in bold. The
true reaction composition for each topology is shown as plain text.

###  III.3 Reconstruction efficiencies

The reconstruction efficiency for $\nu_{\mu}$ CC events as a function of the
kinematics of the outgoing muon is shown in Fig. 10. For low momentum muons
(below 500 MeV/c) the efficiency drops drastically because such low momentum
particles are unlikely to exit the FGD and pass the selection criteria. The
stopping requirement, necessary to determine muon momentum by range and the
timing, poses a significant limitation for high angle muons. This is
particularly true for backward going muons, which occur typically at very low
momentum and stop in the passive edges of material between subdetectors.

Figure 10: The reconstructed signal efficiency as a function of the momentum
and cosine of the emission angle of the true muon using NEUT (full dots) and
GENIE (empty dots). The colors indicate contributions from different samples.

Fig. 11 shows the signal reconstruction efficiency using the same binning in
$p_{\mu}$ and $\cos\theta_{\mu}$ as in the cross-section result (see Table 4).
The efficiency for high multiplicity events is reduced by the fact that
$\nu_{\mu}$ CC events in which the muon candidate is not the true muon (the so
called $\nu_{\mu}$CC-no$\mu$ sample) are not included as signal.

The efficiency as calculated in NEUT and GENIE is generally in agreement.
However, the predicted efficiency is different for low momentum muons going
very forward with respect to the neutrino direction. While generators are in
principle only used to correct for detector effects, this difference
highlights how the simulation of final state particles is important even for
an inclusive selection. In that region of phase space the two generators
differ in their predictions for CC deep inelastic and CC resonance channels,
particularly in the kinematics of the muon and hadrons.

Figure 11: The reconstructed signal efficiency as function of the momentum and
cosine of emission angle of the true muon using the same binning as that for
the cross-section result (see Table 4). Lines represent the efficiencies where
the signal is defined as $\nu_{\mu}$ CC events in which the muon candidate is
the true muon, so called $\nu_{\mu}$CC-$\mu$ events. Markers are efficiencies
when the muon candidate requirement is not imposed in the sample labeled as
$\nu_{\mu}$CC.

###  III.4 Detector systematic uncertainties

The uncertainties associated with the prediction of each subdetector response
(TPCs, FGDs, ECal modules, P0D and SMRD) are evaluated using dedicated control
samples in the data. This works since the events in the control samples share
many of the properties of the events in the $\nu_{\mu}$ CC selection.

The tracker systematic uncertainties are divided into four classes: selection
efficiency (TPC cluster finding, TPC track finding and charge assignment), TPC
momentum resolution, TPC PID, and TPC-FGD matching efficiency. They are all
assessed as in previous analyses from T2K using different control samples of
through-going muons Abe  _et al._ (2015).

Uncertainties associated with the ECal modules are computed for the ECal PID,
the energy resolution and scale, and the efficiency with which ECal objects
are reconstructed and matched to TPC tracks. The method to evaluate those
errors is unchanged with respect to Abe  _et al._ (2014), using high purity
control samples of muons crossing the TPCs and ECals.

Relative to the previous analysis, this work includes six additional
systematic errors. The new errors incorporated in this analysis are associated
with the ToF; the matching efficiency between TPC-P0D and FGD-ECal(SMRD); the
resolution of the momentum determined by range; vertex migration; and the
neutrino parent direction.

The ToF between FGD1 and FGD2 or BarrelECal or P0D is used to determine if the
track starts or ends in the FGD1, and infer the charge of the track. The
uncertainty is evaluated by comparing the ToF distribution in control samples
of tracks crossing the relevant subdetectors and starting/stopping in FGD1 for
data and MC. The ToF distributions are fit with Gaussian distributions for
data and simulation. To account for the differences in the means and widths of
the distributions between data and simulation, corrections are applied to the
simulation and the error is set to be equal to the maximum bias or resolution
correction. The error is not higher than $10\%$ for the Gaussian parameter in
any of the distributions.

The TPC-P0D matching efficiency is estimated using a control sample of cosmic
muons passing through part of the P0D and having a reconstructed segment in
TPC1. The efficiency is defined as the ratio between the number of events with
a matched TPC1-P0D segment and the total number of events in the control
sample. This efficiency is evaluated as function of the momentum of the track.
The data and MC are less than $5\%$ different when the momentum of the cosmic
is higher than 200 MeV/c.

To compute the FGD-ECal(SMRD) matching efficiency, a control sample is used
that contains through-going muons with a BarrelECal (SMRD) segment that points
to FGD. In order to mimic the kinematics of the muon candidate, it is required
that the muon stops within the FGD. The matching efficiency is computed from
the ratio between the number of events with a matched FGD-BarrelECal (or FGD-
BarrelECal-SMRD) segment and the total number of events in the relevant
control sample. The FGD-BarrelECal (FGD-BarrelECal-SMRD) efficiency is found
to be $52\%$ ($55\%$) for simulation and $47\%$ ($45\%$) for data. A
correction is applied to the simulation to account for this and the correction
uncertainty is included in the overall detector uncertainty.

The momentum by range resolution is studied using particles in a control
sample that are fully contained in ND280, stopping inside the FGD and
BarrelECal (or SMRD), and crossing at least one TPC. The distribution of the
difference between the momentum determined by curvature using the TPC segment
and the momentum by range are compared in data and MC. No bias is observed in
such distributions but some difference is seen in the width of the
distributions; this is used to set the uncertainty. In the case of the
BarrelECal (SMRD), the systematic uncertainty is around $10\%$ ($30\%$).

The vertex of the interaction is defined as the reconstructed position of the
start of the muon candidate inside the FGD. When the multiplicity of particles
increases, the reconstruction of the vertex becomes more difficult and the
vertex position can migrate. These migrations have a non negligible impact on
the BWD sample event vertices because back-to-back topologies are common in
that sample. The main effect is on the reconstructed momentum of the muon
candidate inside the FGD because it is proportional to track length. The
difference between the data and simulation for these migrations is difficult
to interpret since it is sensitive to the modeling of hadrons. An uncertainty
of 7 MeV/c (or $\sim$3 FGD layers), which was computed comparing the length of
the tracks inside the FGD1 for data and MC, is applied to the reconstructed
momentum of the muon candidate.

In this analysis, the angle of the outgoing muon is defined with respect to
the neutrino direction. The neutrino direction is determined from the position
of the vertex in FGD1 and the parent hadron decay point of the neutrino in the
decay tunnel. The mean position of hadron decays in the decay tunnel has an
associated uncertainty. This is taken into account by varying the mean parent
decay point according to the decay distribution in the beam simulation.

The detector systematic uncertainties are propagated in order to check their
impact in the rate of reconstructed events in $p_{\mu}$ and
$\cos\theta_{\mu}$. This analysis follows the methodology described in Abe
_et al._ (2015). The expected number of events are scaled using a vector of
systematic parameters. Then, the uncertainties in each reconstructed bin and
their correlations are computed using toy experiments in which the systematics
are varied simultaneously. Table 3 shows the full list of detector systematic
effects considered and the associated uncertainty in each.

| FWD | BWD | HAFWD | HABWD
---|---|---|---|---
Efficiency-like
TPC charge ID eff. | 0.1 | 0.2 | 0 | 0
TPC cluster eff. |  $<$0.002 |  $<$0.002 |  $<$0.002 |  $<$0.002
TPC tracking eff. | 0.8 | 0.4 | 0.05 | 0.02
ECal tracking eff. | 0.2 | 0.2 | 4.1 | 4.9
ECal PID eff. | 1.3 | 0 | 0.5 | 0.3
TPC-FGD match. eff. | 0.1 | 0.1 | 0.004 | 0.005
TPC-ECal match. eff. | 0.5 | 0.8 | 0.2 | 0.1
TPC-P0D match. eff. | - | 3.9 | - | -
FGD-ECal match. eff. | - | - | 4.7 | 6.5
FGD-SMRD match. eff. | - | - | 11.6 | -
Normalization-like
Pileup | 0.2 | - | 0.2 | 0.2
Out of fiducial volume | 0.5 | 1.9 | 1.0 | 2.0
Sand mu | 0.1 | 0.1 | 0.02 | 0.03
Pion secondary int | 0.3 | 0.3 | 0.4 | 0.3
Proton secondary int | 0.01 | 0.001 | 0.2 | 0.01
Observable variation
TPC Field Distortions | 0.007 | 0.008 | 0.001 | 0.004
TPC momentum scale | 0.007 | 0 | 0.004 | 0.01
TPC momentum res. | 0.02 | 0.015 | 0.01 | 0.01
Vertex migration | 0.003 | 0 | 0.01 | 0.01
TPC PID | 0.2 | 0.4 | 0.02 | 0.02
Momentum range res. | - | - | 0.1 | 0.1
ECal energy resolution | - | - | 0.1 | 0.2
ECal energy scale | - | - | 0.8 | 1.5
Time of flight | 0.1 | 2.6 | 2.4 | 7.3
$\nu$ direction | 0 | 0 | 0 | 0
Total | 1.8 | 5.9 | 14.3 | 14.3
Table 3: A summary of the fractional systematic uncertainty (in percentage)
associated with the detector response. The first column lists all the sources
taken into account and the other columns show the error size on the predicted
events in each sample.

The uncertainty associated to the matching among FGD, ECal and SMRD
subdetectors is dominant in both the HAFWD and HABWD selections. The reason is
that the misalignment between both subdetectors has not been properly
corrected in data, leading to discrepancies in the matching efficiency for
segments contained in those subdetectors. In the case of the BWD sample, the
matching between the TPC and P0D subdetectors and the ToF resolution
dominates. Meanwhile, in the FWD selection the uncertainty associated with the
particle identification in the BarrelECal and DsECal dominates.

##  IV CROSS-SECTION ANALYSIS

The following section describes the procedure to unfold the measured muon
kinematic distributions and to propagate uncertainties in the cross-section
measurement. After this, the flux-integrated, double-differential cross
section results for $\nu_{\mu}$ CC interactions are presented.

###  IV.1 Methodology

The flux-integrated, double-differential cross section is expressed as

$\frac{\text{d}\sigma_{f.i.}}{\text{d}p_{\mu}\text{d}\cos\theta_{\mu}}=\frac{S_{ij}^{\nu_{\mu}\text{CC}-\mu}}{\epsilon_{ij}^{\nu_{\mu}\text{CC}-\mu}\Phi\text{N}_{\text{FV}}\Delta
p_{\mu,i}\Delta\cos\theta_{\mu,j}}$ (1)

where $S_{ij}^{\nu_{\mu}\text{CC}-\mu}$ is the number of signal events with
momentum and angle bins $i$ and $j$, respectively.
$\epsilon_{ij}^{\nu_{\mu}\text{CC}-\mu}$ is the signal reconstruction
efficiency with momentum bin $i$ and angle bin $j$. $\Delta p_{\mu}$ and
$\Delta\cos\theta_{\mu}$ represent the bin widths. Finally, the normalization
factors are the total integrated flux and the number of target nucleons in the
FV.

The number of nucleons is computed using the areal density of the different
elements composing the FV ($\text{N}_{\text{FV}}=(5.93\pm 0.04)\times
10^{29}$) Amaudruz  _et al._ (2012). The integrated muon neutrino flux is
$\Phi=(1.107\pm 0.097)\times 10^{13}$ cm-2.

The reconstructed momentum and cosine of emission angle of the muon candidate
are not an exact representation of the true initial muon kinematics.
Therefore, an unfolding method is used to remove the detector effects in the
measurement. In this analysis, we unfold the muon kinematic quantities using a
binned likelihood fit as in Abe  _et al._ (2016). We vary the true spectrum of
the simulation (so called prior) and propagate its effect to the rate of
events in each reconstructed bin. Then, the predicted rate is compared with
the values from data. The variation of the true spectrum is performed scaling
up or down the rate of signal events simultaneously in the four signal and two
control regions for each true bin. Two of the parameters associated to the
background modeling (the normalizations of the neutral current cross section
and pion final state interactions) are included in the fit as nuisance
parameters.

This unfolding method is unregularised, which leads to strong anticorrelations
between neighboring bins if the binning is not properly defined. The different
samples described in Sec. III are well separated in the angular phase space.
In fact, the detector response is different for the selected events in each
sample. Thus, the angular binning is chosen (i.e. $\cos\theta_{\mu}$) to
separate the contribution from each sample as much as possible. The momentum
binning reflects the resolution of the detector and is chosen to maintain
sufficient statistics in each bin. Table 4 shows the binning used in this
analysis for the chosen muon kinematic variables.

$\cos\theta_{\mu}$ |  $p_{\mu}$ [GeV/c]
---|---
-1, -0.25 | 0, 30
-0.25, 0.25 | 0, 0.3, 0.4, 30
0.25, 0.45 | 0, 0.3, 0.4, 0.5, 0.6, 30
0.45, 0.6 | 0, 0.3, 0.4, 0.5, 0.6, 0.8, 30
0.6, 0.71 | 0, 0.3, 0.4, 0.5, 0.6, 0.75, 1, 30
0.71, 0.8 | 0, 0.3, 0.4, 0.5, 0.6, 0.7, 0.85, 1.1, 30
0.8, 0.87 | 0, 0.3, 0.4, 0.5, 0.6, 0.7, 0.85, 1.1, 1.5, 30
0.87, 0.92 | 0, 0.4, 0.5, 0.6, 0.7, 0.85, 1.1, 1.5, 2.1, 30
0.92, 0.96 | 0, 0.4, 0.5, 0.6, 0.7, 0.85, 1.1, 1.5, 2.1, 3, 30
0.96, 0.985 | 0, 0.5, 0.7, 1, 1.5, 2.3, 3.5, 30
0.985, 1 | 0, 0.7, 1.5, 2.5, 3.5, 4.8, 30
Table 4: Binning used for $\cos\theta_{\mu}$ and $p_{\mu}$ distributions in
both reconstructed and true phase space.

###  IV.2 Error propagation

Analytical computation for most of the uncertainties in this analysis is not
possible. So toy experiments are used to study their impact and determine
errors. In the toy experiments, some aspect of the simulated or real data is
changed depending on the source of uncertainty as described below.

To evaluate the uncertainty due to data statistics, toy experiments are
produced applying a Poisson fluctuation to the number of reconstructed events
in the data for each bin and sample. For each toy, the fluctuated data are
unfolded using as prior the nominal MC and the cross section is computed using
Eq. 1. The statistical error in each bin is taken as width of the cross
section distribution for many toys.

The methodology used to estimate systematic uncertainties involves reweighting
the MC prediction for each toy experiment. Parameters associated to each
systematic error are thrown according to a Gaussian distribution around the
nominal value, following the prior errors and taking into account
correlations. Then, for each toy, the data is unfolded using as prior the
reweighted MC. In addition, $\Phi$, $\text{N}_{\text{FV}}$ and
$\epsilon_{ij}^{\nu_{\mu}\text{CC}-\mu}$ are also weighted using the thrown
value of the parameters. Finally, the cross section is computed using Eq. 1
for each toy. The uncertainty in each bin is taken as width of the cross
section distribution for many toys.

Fig. 12 shows a comparison of the fractional error associated to each source
of uncertainty using 1500 toy models. Throughout most of the phase space, the
dominant systematic uncertainty is the flux. In the backward region, the
neutrino interaction modeling dominates, with the largest contribution coming
from the uncertainty assigned to the $M_{A}$ parameter. The detector
systematic becomes relevant in the high angle region
($-0.25<\cos\theta_{\mu}<0.25$) due to the large uncertainties in FGD-
ECal(SMRD) matching efficiencies, and at very low momentum where the out of FV
contribution is more pronounced. The statistical uncertainty is dominant in
the high momentum region where the number of reconstructed events is lower
(except at low angles in the forward direction).

It is interesting to note that the systematic uncertainties associated with
the signal and background modeling give a relatively unimportant contribution
to the overall inclusive cross section uncertainty because of the high purity
and efficiency for the signal sample. The systematic uncertainties associated
with the modeling of neutral-current interactions and pion final state
interactions are reduced by a factor of 2 thanks to the use of the control
samples.

Figure 12: The fractional error from each source of uncertainty on the flux-
integrated, double-differential cross section. The total error is computed
varying simultaneously both statistical and systematic uncertainties.

##  V RESULTS AND CONCLUSIONS

The flux-integrated total cross section is computed by integrating both the
number of signal events and the signal efficiency over the muon phase space.

$\displaystyle\sigma_{\text{DATA FIT W/ NEUT}}=\left(6.950\pm
0.049[\text{stat}]\pm 0.123[\text{syst}]\right.$ $\displaystyle\left.\pm
0.608[\text{flux}]\right)\times 10^{-39}\text{cm}^{2}\text{nucleon}^{-1}$
$\displaystyle\sigma_{\text{DATA FIT W/ GENIE}}=\left(6.850\pm
0.048[\text{stat}]\pm 0.121[\text{syst}]\right.$ $\displaystyle\left.\pm
0.599[\text{flux}]\right)\times 10^{-39}\text{cm}^{2}\text{nucleon}^{-1}$

This is compatible with predictions from the two event generators:
$\sigma_{\text{NEUT}}=7.108\times 10^{-39}\text{cm}^{2}\text{nucleon}^{-1}$
and $\sigma_{\text{GENIE}}=6.564\times
10^{-39}\text{cm}^{2}\text{nucleon}^{-1}$. It is known that the detector
performance varies substantially as a function of the momentum and angle of
the outgoing muon. Therefore, the extracted value using the total cross
section must be interpreted cautiously. This result shows good agreement with
the one obtained in Abe  _et al._ (2013a).

The flux-integrated, double-differential cross section is computed as function
of the outgoing muon kinematics using the methodology described in Sec. IV.1
and Sec. IV.2 using two independent MC generators detailed in Sec. II.3. Fig.
13 shows the results for the unfolded data as well as the NEUT and GENIE
predictions. A small disagreement is observed in the low momentum and very
forward regions when using different event generators as prior. This bias is
not due to unfolding but due to the different efficiency corrections in that
region of the phase space for NEUT and GENIE as shown in Fig. 11. The muon
neutrino flux used in this analysis and the measured cross section values,
errors and correlation matrix can be found in Abe  _et al._ .

This result is compared to the NEUT and GENIE predictions, showing in both
cases high $\chi^{2}$ values with respect to the total number of bins, 71. In
the new regions of phase space (high angle and backward-going muons) there is
good agreement but uncertainties are still large. For forward-going muons the
binning is finer and interesting structures are observed.

Figure 13: The flux-integrated, double-differential cross section per nucleon
for NEUT (continuous red line), for GENIE (dashed red line), and the unfolded-
data result using as prior either NEUT or GENIE. The bin of highest momentum
is scaled by the factor shown in each plot to make it visible. $\chi^{2}$
values are computed with unfolded-data result using as prior NEUT.

## ACKNOWLEDGEMENTS

We thank the J-PARC staff for superb accelerator performance. We thank the
CERN NA61/SHINE Collaboration for providing valuable particle production data.
We acknowledge the support of MEXT, Japan; NSERC (Grant No. SAPPJ-2014-00031),
NRC and CFI, Canada; CEA and CNRS/IN2P3, France; DFG, Germany; INFN, Italy;
National Science Centre (NCN) and Ministry of Science and Higher Education,
Poland; RSF, RFBR, and MES, Russia; MINECO and ERDF funds, Spain; SNSF and
SERI, Switzerland; STFC, UK; and DOE, USA. We also thank CERN for the
UA1/NOMAD magnet, DESY for the HERA-B magnet mover system, NII for SINET4, the
WestGrid and SciNet consortia in Compute Canada, and GridPP in the United
Kingdom. In addition, participation of individual researchers and institutions
has been further supported by funds from ERC (FP7), H2020 Grant No. RISE-
GA644294-JENNIFER, EU; JSPS, Japan; Royal Society, UK; the Alfred P. Sloan
Foundation and the DOE Early Career program, USA.

## References

  * Abe  _et al._ (2011) K. Abe _et al._ (T2K Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 659, 106 (2011).
  * Abe  _et al._ (2013a) K. Abe _et al._ (T2K Collaboration), Phys. Rev. D 87, 092003 (2013a).
  * Abgrall  _et al._ (2016) N. Abgrall _et al._ (NA61/SHINE Collaboration), Eur. Phys. J. C 76, 84 (2016).
  * Abe  _et al._ (2013b) K. Abe _et al._ (T2K Collaboration), Phys. Rev. D 87, 012001 (2013b).
  * Abgrall  _et al._ (2012) N. Abgrall _et al._ (NA61/SHINE Collaboration), Phys. Rev. C 85, 035210 (2012).
  * Assylbekov  _et al._ (2012) S. Assylbekov _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 686, 48 (2012).
  * Allan  _et al._ (2013) D. Allan _et al._ , J. Instrum. 8, P10019 (2013).
  * Aoki  _et al._ (2013) S. Aoki _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 698, 135 (2013).
  * Amaudruz  _et al._ (2012) P. Amaudruz _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 696, 1 (2012).
  * Abgrall  _et al._ (2011) N. Abgrall _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 637, 25 (2011).
  * Abe  _et al._ (2014) K. Abe _et al._ (T2K Collaboration), Phys. Rev. D 89, 092003 (2014).
  * Hayato (2009) Y. Hayato, Acta Phys. Pol. B 40, 2477 (2009).
  * Andreopoulos  _et al._ (2010) C. Andreopoulos _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 614, 87 (2010).
  * Smith (1972) C. L. Smith, Phys. Rep. 3, 261 (1972).
  * Bradford  _et al._ (2006) R. Bradford _et al._ , Nucl. Phys. B Proc. Suppl. 159, 127 (2006).
  * Bodek and Ritchie (1981) A. Bodek and J. L. Ritchie, Phys. Rev. D 23, 1070 (1981).
  * Benhar and Fabrocini (2000) O. Benhar and A. Fabrocini, Phys. Rev. C 62, 034304 (2000).
  * Nieves  _et al._ (2011) J. Nieves _et al._ , Phys. Rev. C 83, 045501 (2011).
  * Wilkinson  _et al._ (2016) C. Wilkinson _et al._ , Phys. Rev. D 93, 072010 (2016).
  * Rein and Sehgal (1981) D. Rein and L. M. Sehgal, Ann. Phys. (N.Y.) 133, 79 (1981).
  * Graczyk and Sobczyk (2008) K. M. Graczyk and J. T. Sobczyk, Phys. Rev. D 77, 053001 (2008).
  * Kuzmin  _et al._ (2006) K. S. Kuzmin _et al._ , Acta Phys. Pol. B 37, 2337 (2006).
  * Wilkinson  _et al._ (2014) C. Wilkinson _et al._ , Phys. Rev. D 90, 112017 (2014).
  * Glück  _et al._ (1998) M. Glück _et al._ , Eur. Phys. J. C 5, 461 (1998).
  * Bodek and Yang (2003) A. Bodek and U. K. Yang, AIP Conf. Proc. 670, 110 (2003).
  * Bodek and Yang (2005) A. Bodek and U. Yang, AIP Conf. Proc. 792, 257 (2005).
  * Adamson  _et al._ (2010) P. Adamson _et al._ (MINOS Collaboration), Phys. Rev. D 81, 072002 (2010).
  * Rein and Sehgal (1983) D. Rein and L. M. Sehgal, Nucl. Phys. B223, 29 (1983).
  * Rein and Sehgal (2007) D. Rein and L. Sehgal, Phys. Lett. B 657, 207 (2007).
  * Higuera  _et al._ (2014) A. Higuera _et al._ (MINERvA Collaboration), Phys. Rev. Lett. 113, 261802 (2014).
  * Sjöstrand (1994) T. Sjöstrand, Comput. Phys. Commun. 82, 74 (1994).
  * Koba  _et al._ (1972) Z. Koba _et al._ , Nucl. Phys. B40, 317 (1972).
  * Yang  _et al._ (2009) T. Yang _et al._ , Eur. Phys. J. C 63, 1 (2009).
  * Salcedo  _et al._ (1988) L. Salcedo _et al._ , Nucl. Phys. A484, 557 (1988).
  * Ashery  _et al._ (1981) D. Ashery _et al._ , Phys. Rev. C 23, 2173 (1981).
  * Mashnik  _et al._ (2006) S. G. Mashnik _et al._ , J. Phys. Conf. Ser. 41, 340 (2006).
  * Abe  _et al._ (2015) K. Abe _et al._ (T2K Collaboration), Phys. Rev. D 91, 072010 (2015).
  * Abe  _et al._ (2016) K. Abe _et al._ (T2K Collaboration), Phys. Rev. D 93, 112012 (2016).
  * (39) K. Abe _et al._ , “T2k public data,” http://t2k-experiment.org/results.