††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 neutrino and antineutrino oscillations by the T2K experiment
including a new additional sample of $\nu_{e}$ interactions at the far
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  M. Antonova  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow, Russia  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  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  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  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  R.G. Calland
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. 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  Oxford University,
Department of Physics, Oxford, United Kingdom  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  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  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  A. Hillairet  University of Victoria, Department of
Physics and Astronomy, Victoria, British Columbia, Canada  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  K. Huang  Kyoto University,
Department of Physics, Kyoto, 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  J.
Insler  Louisiana State University, Department of Physics and Astronomy, Baton
Rouge, Louisiana, U.S.A.  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  E. Iwai  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.  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  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. 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.  N.D.
Patel  Kyoto University, Department of Physics, Kyoto, Japan  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  Imperial College London, Department of Physics, London,
United Kingdom  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  J.-M. Poutissou  TRIUMF, Vancouver, British Columbia, Canada
A. Pritchard  University of Liverpool, Department of Physics, Liverpool,
United Kingdom  P. Przewlocki  National Centre for Nuclear Research, Warsaw,
Poland  B. Quilain  Kyoto University, Department of Physics, Kyoto, 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  P.A. Rodrigues  University of
Rochester, Department of Physics and Astronomy, Rochester, New York, U.S.A.
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
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  T. Shirahige  Okayama University,
Department of Physics, Okayama, Japan  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  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. Tomura  University of Tokyo,
Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan  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.  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  C.W. Walter  Duke
University, Department of Physics, Durham, North Carolina, 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  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

###### Abstract

The T2K experiment reports an updated analysis of neutrino and antineutrino
oscillations in appearance and disappearance channels. A sample of electron
neutrino candidates at Super-Kamiokande in which a pion decay has been tagged
is added to the four single-ring samples used in previous T2K oscillation
analyses. Through combined analyses of these five samples, simultaneous
measurements of four oscillation parameters, $|\Delta m^{2}_{32}|$,
$\textrm{sin}^{2}\theta_{23}$, $\textrm{sin}^{2}\theta_{13}$, and
$\delta_{\mathrm{CP}}$ and of the mass ordering are made. A set of studies of
simulated data indicates that the sensitivity to the oscillation parameters is
not limited by neutrino interaction model uncertainty. Multiple oscillation
analyses are performed, and frequentist and Bayesian intervals are presented
for combinations of the oscillation parameters with and without the inclusion
of reactor constraints on $\textrm{sin}^{2}\theta_{13}$. When combined with
reactor measurements, the hypothesis of CP conservation
($\delta_{\mathrm{CP}}$ $=0$ or $\pi$) is excluded at 90% confidence level.
The 90% confidence region for $\delta_{\mathrm{CP}}$ is [-2.95,-0.44] ([-1.47,
-1.27]) for normal (inverted) ordering. The central values and 68% confidence
intervals for the other oscillation parameters for normal (inverted) ordering
are $\Delta m^{2}_{32}=2.54\pm 0.08$ ($2.51\pm 0.08$) $\times 10^{-3}$
eV${}^{2}/c^{4}$ and $\textrm{sin}^{2}\theta_{23}=0.55^{+0.05}_{-0.09}$
($0.55^{+0.05}_{-0.08}$), compatible with maximal mixing. In the Bayesian
analysis, the data weakly prefer normal ordering (Bayes factor 3.7) and the
upper octant for $\textrm{sin}^{2}\theta_{23}$ (Bayes factor 2.4).

###### pacs:

Valid PACS appear here

††preprint: XXX

##  I Introduction

Neutrino oscillations have been firmly established by multiple experiments.
Super-Kamiokande (SK) observed an energy and pathlength dependent deficit in
the atmospheric muon neutrino flux Fukuda:1998mi ; and Sudbury Neutrino
Observatory (SNO) resolved the long-standing solar neutrino problem by
demonstrating that the previously observed deficit of electron neutrinos from
the Sun was due to flavor transitions Ahmad:2002jz . These two experiments,
together with accelerator-based (K2K Ahn:2006zza , MINOS Adamson:2014vgd ) and
reactor-based (KamLAND Eguchi:2002dm ) long-baseline experiments measured the
two mass-squared differences between mass eigenstates and two of the three
mixing angles in the PMNS matrix.

The mixing angle, $\textrm{\,}\theta_{13}$, has been measured as nonzero by
T2K Abe:2011sj ; Abe:2013hdq , by reactor experiments An:2012eh ; Ahn:2012nd ;
Abe:2011fz , and more recently by NO$\nu$A Adamson:2016tbq . Establishing that
all three mixing angles are nonzero opens a way to study CP violation in the
leptonic sector through neutrino oscillations. CP violation in neutrino
oscillations arises from $\delta_{\mathrm{CP}}$, an irreducible CP-odd phase
in the PMNS matrix. This phase introduces a difference in the appearance
probability between neutrinos and antineutrinos. To investigate this
phenomenon, after taking data with a beam predominantly composed of muon
neutrinos in order to observe the appearance of electron neutrinos at the far
detector, T2K has switched to taking data with a beam predominantly composed
of muon antineutrinos. A direct measurement of CP violation can then be
obtained by comparing $\nu_{\mu}\rightarrow\nu_{e}$ and
$\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}$ channels.

To produce neutrinos, protons extracted from the J-PARC main ring strike a
target producing hadrons which are then focused and selected by charge with a
system of magnetic horns. The hadrons decay in flight, producing an intense
neutrino beam. A beam predominantly composed of neutrinos or antineutrinos can
be produced by choosing the direction of the current in the magnetic horn. T2K
uses the so-called off-axis technique with the beam axis directed 2.5∘ away
from SK in order to produce a narrow–band neutrino beam, peaked at an energy
of 600 $\mathrm{\,Me\kern-1.00006ptV}$, where the effect of neutrino
oscillations is maximum for a baseline of 295 km. Neutrinos are also observed
at a near detector complex, installed 280 m from the target, comprising an on-
axis detector (INGRID) which provides day-to-day monitoring of the beam
profile and direction, and a magnetized off-axis detector (ND280), at the same
off-axis angle as SK, which measures neutrino interaction rates before
oscillation.

The analyses described in this paper are based on an exposure of 7.482$\times
10^{20}$ protons on target (POT) in neutrino mode ($\nu$-mode) and
7.471$\times 10^{20}$ POT in antineutrino mode ($\overline{\nu}$-mode)
collected at SK during seven physics runs as detailed in Tab. 1. The neutrino
oscillation parameters are measured by combining $\nu_{\mu}$ and
$\overline{\nu}_{\mu}$ disappearance channels with $\nu_{e}$ and
$\overline{\nu}_{e}$ appearance channels, using the same analysis techniques
described in Ref. Abe:2017uxa . The analyzed data set is the same as in Ref.
Abe:2017uxa , but an additional SK sample is included in the oscillation
analysis. Previously, for the appearance channel, only the SK single-ring
e-like interactions without additional activity in the detector were used for
the oscillation analysis. The analysis presented in this paper includes an
additional sample enriched in $\nu_{e}$ interactions in which the e-like ring
is accompanied by delayed Michel electron due to the decay chain
$\pi^{+}\rightarrow\mu^{+}\rightarrow e^{+}$ of $\pi^{+}$’s produced in the
neutrino interactions. This sample is currently only used in $\nu$-mode, and
increases the statistics of the $\nu_{e}$ sample in SK by roughly 10%.

Table 1:  T2K data-taking periods and collected POT used in the analyses presented in this paper.  Run | Dates |  $\nu$-mode POT |  $\bar{\nu}$-mode POT
---|---|---|---
Period |  | ($\times 10^{20}$) | ($\times 10^{20}$)
Run 1 | Jan. 2010-Jun. 2010 | 0.323 | –
Run 2 | Nov. 2010-Mar. 2011 | 1.108 | –
Run 3 | Mar. 2012-Jun. 2012 | 1.579 | –
Run 4 | Oct. 2012-May 2013 | 3.560 | –
Run 5 | May 2014-Jun. 2014 | 0.242 | 0.506
Run 6 | Nov. 2014-Jun. 2015 | 0.190 | 3.505
Run 7 | Feb. 2016-May 2016 | 0.480 | 3.460
Total | Jan. 2010-May 2016 | 7.482 | 7.471

The paper is organized as follows: the neutrino beam and the modeling of the
neutrino fluxes are described in Sec. II. The neutrino interaction model
developed for this analysis will then be described in Sec. III, followed by
the selection of neutrinos in the near detector complex in Sec. IV. The
neutrino flux and neutrino interaction inputs, and near detector selections
are combined to reduce flux and cross-section uncertainties at the far
detector as will be shown in Sec. V. The far detector selections are described
in Sec. VI. The neutrino oscillations and the T2K oscillation analyses
frameworks are then described in Sec. VII and in Sec. VIII respectively. Sec.
IX is dedicated to a description of the impact of the uncertainties of the
neutrino interaction model on the T2K oscillation analyses. Finally, the
results of the oscillation analyses are presented in Sec. X and in Sec. XI and
some concluding remarks are given in Sec. XII.

##  II The T2K beam

The neutrino beam is produced by the interaction of 30 GeV protons from the
J-PARC main ring accelerator on a 1.9 interaction-length graphite target.
Secondary hadrons, mainly pions and kaons, leaving the target pass through
three electromagnetic horns SEKIGUCHI201557 , which are operated at a current
of either +250 kA or $-$250 kA to focus positively or negatively charged
particles respectively. The outgoing hadrons decay in a 96-m long decay
volume, where a relatively pure beam of muon neutrinos is produced by the
decay of positively charged hadrons in positive focusing mode ($\nu$-mode),
and a beam mostly composed of muon antineutrinos is produced in negative
focusing mode ($\bar{\nu}$-mode). Protons and undecayed hadrons are stopped in
a beam dump, while muons above 5 GeV pass through and are detected in a muon
monitor (MUMON doi:10.1093/ptep/ptv054 ), and are used to monitor the
secondary beam stability. The T2K beamline hardware has been described in
detail elsewhere Abe:2011ks .

The T2K neutrino flux at the near and far detectors in case of no neutrino
oscillation is predicted by a simulation which has been described in detail in
Ref. Abe:2013fp . Interactions of the primary proton beam, whose profile is
measured for each run period by a suite of proton beam monitors, as well as
subsequently-produced pions and kaons, are simulated within the graphite
target by the FLUKA 2011 package Ferrari:2005zk ; Fluka:2014 . The predicted
hadron production rates inside and outside the target are then adjusted based
on the results from the latest analysis of the full 2009 thin-target dataset
by the NA61/SHINE experiment Abgrall:2011ae ; Abgrall:2011ts ; Abgrall:2016fs
, as well as other hadron production experiments eichten ; allaby ; e910 .
Particles which exit the target and subsequently decay are tracked through the
horns and decay volume by a GEANT3 GEANT3  simulation using the GCALOR GCALOR
package. The predicted (anti)neutrino fluxes at the far detector for T2K Run
1-7 is shown for both $\nu$\- and $\bar{\nu}$-modes in Fig. 1.

Figure 1: The T2K unoscillated neutrino flux prediction at SK for $\nu$\-
(left) and $\bar{\nu}$\- (right) modes. The binning used for the flux
systematic parameters is also shown.

Most of the “right-sign” $\nu_{\mu}$ flux (i.e. $\nu$’s in $\nu$-mode and
$\bar{\nu}$’s in $\bar{\nu}$-mode) comes from mesons produced inside the
target, dominantly “right-sign” (focused) pion and kaon production, which
subsequently decay to produce muons.Interactions producing “right-sign”
$\nu_{e}$’s also predominantly come from interactions in the target, with a
larger fraction of $\nu_{e}$ produced by kaon (rather than pion) decays.
Interactions producing the “wrong-sign” $\nu_{\mu}$ flux have a higher
fractional rate of out-of-target interactions, which are dominated by protons,
neutrons, and pions scattering in the horns and decay volume walls.
Interactions producing the “wrong-sign” $\nu_{e}$ flux have a significant
fraction of $K^{0}$ production from proton or neutron interactions, as well as
charged kaon production.

In general, the $\nu$\- and $\bar{\nu}$-mode fluxes are similar at low energy,
although the “right-sign” $\nu_{\mu}$ (and $\nu_{e}$) flux in $\nu$-mode is
$\sim$15% higher around the flux peak than the “right-sign” $\bar{\nu}_{\mu}$
(and $\bar{\nu}_{e}$) flux in $\bar{\nu}$-mode. The “wrong-sign” background
$\bar{\nu}$ flux is also lower in $\nu$-mode compared to the $\nu$ flux in
$\bar{\nu}$-mode, especially at high energy. These differences are due to the
higher production multiplicities of positive, rather than negative, parent
particles.

Uncertainties in the neutrino flux prediction arise from the hadron production
model, proton beam profile, off-axis angle, horn current, horn alignment, and
other factors. For each source of error, the underlying parameters in the
model are varied to evaluate the effect on the flux prediction in bins of
neutrino energy for each neutrino flavor as described in detail elsewhere
Abe:2013fp . The uncertainties on the unoscillated $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ beam fluxes at the far detector are shown in Fig. 2 and are
currently dominated by uncertainties on hadron production. The uncertainties
on the background $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes from the beam are 7–10%
in the relevant region.

Figure 2: The T2K fractional systematic uncertainties on the SK flux arising
from the beamline configuration and hadron production prior to constraints
from near detector data. Uncertainties are given for $\nu$’s in a $\nu$-mode
beam (top left), $\bar{\nu}$’s in a $\nu$-mode beam (top right),
$\overline{\nu}$’s in an $\bar{\nu}$-mode beam (bottom left), and $\nu$’s in
an $\bar{\nu}$-mode beam (bottom right). For the $\nu$-mode plots, the total
current uncertainties (NA61 2009 data) are compared to the total uncertainties
estimated for the previous T2K results (NA61 2007 data) Abe:2015awa .

##  III Neutrino interaction model

The neutrino interaction model used in this analysis is based on NEUT neut
version 5.3.2, which includes many significant improvements over the old
version 5.1.4.2, used in previous T2K oscillation analyses (described in
detail in Ref. Abe:2015awa ). This model is constrained where possible by
external experiments that are used to provide initial cross section parameter
uncertainties. Such uncertainties are reduced using ND280 data, as explained
in Sec. V. Alternative models are used to build simulated datasets to test the
robustness of the T2K analysis against model-dependent assumptions, as
explained in Sec. IX. This section describes the updated NEUT interaction
model and alternative models used for the oscillation analyses.

###  III.1 Neutrino interaction model used in the oscillation analyses

The interaction rate at T2K energies is dominated by charged current quasi-
elastic (CCQE) events, $\nu_{l}n\rightarrow l^{-}p$
($\overline{\nu}_{l}p\rightarrow l^{+}n$). Because CCQE is a two-body process
and the neutrino direction is known, the neutrino energy can be reconstructed
from the outgoing lepton kinematics alone. However, nuclear effects and other
processes, which have the same experimental signature of a single muon and no
final state pions (CC$0\pi$, or CCQE-like), are indistinguishable from CCQE
and can affect the reconstructed neutrino energy and thus the oscillation
result if not accounted for Ankowski:2016jdd ; Martini:2012fa ; Nieves:2012yz
; coloma_erec ; ankowski_erec ; lalakulich_erec_2012 . The T2K cross-section
modeling has been updated to include recent theoretical models of these
processes (full details can be found in Ref. Wilkinson:2016wmz ). In previous
analyses, the CCQE model was based on the Llewellyn-Smith neutrino-nucleon
scattering model llewellyn-smith  with a dipole axial form factor and BBBA05
vector form factors bbba05 , and used the Smith-Moniz Relativistic Fermi Gas
(RFG) model smith-moniz  to account for the fact that the nucleons are bound
in a nucleus. The main improvements available in NEUT v5.3.2 are the inclusion
of the Spectral Function (SF) model from Ref. Benhar:1999bg , which provides a
more sophisticated description of the initial state of the nucleus than the
RFG: the inclusion of the multi-nucleon interaction (2p2h) model from Refs.
Nieves:2011pp ; nievesExtension ; and the implementation of the Random Phase
Approximation (RPA) correction from Ref. Nieves:2011pp . The 2p2h model
includes interactions with more than one nucleon bound within the nucleus,
which contribute considerable strength to the CCQE-like cross section, and add
significant smearing to the reconstructed neutrino energy distribution (as it
is not a two-body process). RPA is a nuclear screening effect due to long-
range nucleon-nucleon correlations which modifies the interaction strength as
a function of four-momentum transfer, $Q^{2}$. The models make different
physical assumptions, so they cannot be combined arbitrarily. Two candidate
models were considered for the default model: SF and RFG+RPA+2p2h.
RFG+RPA+2p2h was selected as the default because it was most consistently able
to describe the available MiniBooNE mb-ccqe-2010 ; mb-ccqe-antinu-2013  and
MINERvA minerva-nu-ccqe ; minerva-antinu-ccqe  CCQE-like data (see Ref.
Wilkinson:2016wmz  for details).

Various parameters have been introduced to describe the theoretical
uncertainties and approximations in the RFG+RPA+2p2h model, which are
constrained using the near detector data. The variable parameters are the
axial-mass, $M_{\textrm{A}}$, the Fermi momentum, $p_{\mathrm{F}}$, the
binding energy, $E_{\mathrm{b}}$, and the 2p2h cross-section normalization.
Given the poor agreement between MINERvA and MiniBooNE datasets
Wilkinson:2016wmz , and slight inconsistencies between the signal definitions
from the two experiments, no external constraints are applied on the variable
parameters prior to the ND280 fit. Given the absence of firm predictions on
the scaling of various nuclear effects with the nucleus mass number, the Fermi
momentum, binding energy and 2p2h normalization are treated as uncorrelated
between 12C and 16O. The 2p2h normalization is also considered to be
uncorrelated between neutrino and antineutrino interactions. All of these
parameters can be separately constrained by the T2K near detector data with
the inclusion of samples with interactions on both 12C and on 16O, with
$\nu$\- and $\bar{\nu}$-mode data.

The NEUT model for resonant pion production is based on the Rein-Sehgal model
rein-sehgal  with updated nucleon form factors sobczyk_pion_FFs , and with the
invariant hadronic mass restricted to be $W\leq 2$ GeV to avoid double-
counting pions produced through Deep Inelastic Scattering (DIS). Three
variable model parameters are considered: the resonant axial mass,
$M_{\mathrm{A}}^{\mathrm{RES}}$; the value of the axial form factor at zero
transferred 4-momentum, $C^{\mathrm{A}}_{5}$; and the normalization of the
isospin non-resonant component predicted in the Rein-Sehgal model,
I$\frac{1}{2}$. Initial central values and uncertainties for these parameters
are obtained in a fit to low energy neutrino–deuterium single pion production
data from ANL ANL_Radecky_1982  and BNL BNL_Kitagaki_1986  for the resonant
pion production channels $\nu_{\mu}p\rightarrow\mu^{-}p\pi^{+}$,
$\nu_{\mu}n\rightarrow\mu^{-}p\pi^{0}$ and
$\nu_{\mu}n\rightarrow\mu^{-}n\pi^{+}$. For the dominant production channel,
$\nu_{\mu}p\rightarrow\mu^{-}p\pi^{+}$, the reanalyzed dataset from Ref.
anl_bnl_reanalysis  was used. Resonant kaon, photon and eta production is also
modeled using the Rein-Sehgal resonance production amplitudes, with modified
branching ratios to account for the decay of the resonances to kaons, photons,
or eta, rather than to pions. External neutrino-nucleus and antineutrino-
nucleus pion production data from MiniBooNE AguilarArevalo:2009ww ;
AguilarArevalo:2010xt ; AguilarArevalo:2010bm , MINERvA Eberly:2014mra ,
SciBooNE Kurimoto:2009wq  and K2K Nakayama:2004dp  were used as a cross check
to ensure that the broad features of all datasets were consistent with the
uncertainties on the interaction level parameters
($M_{\mathrm{A}}^{\mathrm{RES}}$, $C^{\mathrm{A}}_{5}$ and I$\frac{1}{2}$
component) or the uncertainties on Final State Interactions (FSI, which will
be described shortly). A full fit to the external data is difficult due to
strong correlations between FSI parameters and the neutrino-nucleus
interaction model parameters, and a lack of information on the correlations
between external data points.

The coherent pion production model used is the Rein-Sehgal model described in
Refs. rein-coh ; rein_2007 . However, recent results from MINERvA
minerva_coh_2014  are better described by the Berger-Sehgal model
Berger:2008xs , so a rough reweighting of the coherent events as a function of
the outgoing pion energy, $E_{\pi}$, is applied to approximate the Berger-
Sehgal model using the weights and binning given in Tab. 2. Normalization
uncertainties of 30% are introduced separately for CC- and NC-coherent events,
based on comparisons to the MINERvA data (after the weights in Table 2 are
applied), which are fully correlated between 12C and 16O.

Table 2: Weights applied to coherent pion interactions as a function of the pion energy, $E_{\pi}$. $E_{\pi}$ (GeV) |  Weight
---|---
0.00 - 0.25 |  0.135
0.25 - 0.50 |  0.400
0.50 - 0.75 |  0.294
0.75 - 1.00 |  1.206

The Deep Inelastic Scattering (DIS) model is unchanged from previous analyses
(described in Ref. Abe:2015awa ). The DIS cross section is calculated for
$W\geq 1.3$ GeV, using GRV98 structure functions Gluck1998  with Bodek-Yang
corrections bodek-yang . Single pion production through DIS is suppressed for
$W\leq 2$ GeV to avoid double counting with the resonant pion production
contributions, and uses a custom hadronization model described in Ref. neut .
For $W>2$ GeV, PYTHIA/Jetset SJOSTRAND199474  is used for hadronization. A CC-
other shape parameter, $x^{\mathrm{CC-Other}}$ was introduced to give
flexibility to the CC-DIS contribution. This parameter applies to CC resonant
kaon, photon and eta production, as well as CC-DIS events, and it scales the
cross section by $(1+x^{\mathrm{CC-Other}}/E_{\nu})$. It was designed to give
greater flexibility at low $E_{\nu}$, and the initial uncertainty was set by
NEUT comparisons with MINOS CC-inclusive data Adamson:2009ju .

In addition to the previously described NC-coherent parameter, two other NC-
specific parameters have been introduced in this analysis. A study in Ref.
Wang:2015ivq  showed that the NEUT neutral current single photon production
(NC1$\gamma$) cross section prediction was approximately a factor of two
smaller than a recent theoretical model Wang:2013wva . Because of this, the
NC1$\gamma$ cross section has been set to be 200% of the NEUT nominal
prediction, with an uncertainty of 100%. An NC-Other normalization parameter
is applied to neutral current elastic, NC resonant kaon and eta production, as
well as NC-DIS events, with an initial uncertainty set at 30%.

Hadrons produced inside the nucleus may undergo FSI before leaving the nuclear
environment, which changes the outgoing particle content and kinematics in the
final state. NEUT models FSI for pions, kaons, etas and nucleons using a
cascade model described in Ref. neut . Interactions are generated inside the
nucleus according to a Woods-Saxon density distribution woods-saxon , and all
outgoing hadrons are stepped through the nucleus with interaction
probabilities calculated at each step until they leave the nucleus. Particles
produced in DIS interactions are propagated some distance without interacting
to allow for a formation zone, where the initial step size is based on results
from the SKAT experiment skat_1979 . The allowed pion interactions in the
nucleus are: charge exchange, where the charge of the pion changes;
absorption, where the pion is absorbed through two or three body processes;
elastic scattering, where the pion only exchanges momentum and energy; and
inelastic scattering, where additional pions are produced. If an interaction
occurs, new and modified particles are also added to the cascade. For pion
momenta $p_{\pi}\geq 500$ MeV, nucleons are treated as free particles, and
separate high ($p_{\pi}\geq 500$ MeV) and low ($p_{\pi}<500$ MeV) energy
scattering parameters are introduced for charge exchange and elastic
scattering. Initial interaction uncertainties are obtained from fits to a
large body of pion–nucleon and pion–nucleus scattering data for nuclei ranging
from carbon to lead, as described in Ref. Abe:2015awa . The variable
parameters included to vary the pion FSI cross section are summarized in Tab.
9. Pion FSI parameters are assumed to be fully correlated between 12C and 16O.
Uncertainties on nucleon, kaon and eta FSI interaction probabilities are not
considered in the current analysis.

To account for effects which may potentially affect
$\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ but not $\nu^{\hbox{\kern-8.0pt\raise
1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ cross
sections, such as radiative corrections or second class currents (see, for
example, Ref. day_nue_numu_2012 ), which are not included in the NEUT cross
section model, additional uncertainties which affect
$\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ have been introduced. These include an
uncorrelated 2% uncertainty on the $\nu_{e}/\nu_{\mu}$ and
$\bar{\nu}_{e}/\bar{\nu}_{\mu}$ cross section ratios to account for radiative
corrections, and an additional 2% uncertainty which is fully anticorrelated
between $\nu_{e}$ and $\bar{\nu}_{e}$ to allow for second class currents.

The full list of cross-section uncertainties and their values before and after
the ND280 data constraints is provided in Tab. 9.

###  III.2 Alternative neutrino interaction models for studies of simulated
data

Neutrino interactions with 12C and 16O nuclear targets at the near and far
detectors may be affected by important nuclear effects which are not well
understood. Various theoretical models are available to describe such effects,
which are based on different approximations and with different ranges of
validity. None of the available models are capable of describing all the
available measurements of neutrino–nucleus cross sections from T2K and from
other experiments. It is therefore crucial to test that the T2K oscillation
analysis is insensitive to reasonable modifications of the neutrino
interaction model described in Sec.III.1, which will now be refered to as the
“reference model”. With this aim, various simulated datasets have been built
based on alternative models. The following effects have been considered:
variations of the distribution of the momentum of the initial nucleons in the
nucleus and of the energy needed to extract the nucleons from the nucleus (the
binding energy); uncertainties on the long-range nuclear correlations
modifying the cross section as a function of $Q^{2}$; and modifications of the
modeling of multi-nucleon interaction model (2p2h), including short-range
nuclear correlations and meson exchange currents.

To test the nuclear effects in the initial state two alternative models have
been considered beyond the RFG simulation used as reference: the SF developed
in Ref. Benhar:1999bg  and the Local Fermi Gas (LFG) model from Ref.
Nieves:2011pp . The LFG model also differs from the reference model in the
implementation of the binding energy. In the latter, an effective value is
considered, based on the average momentum of nucleons within the nucleus,
while the LFG model considers the different state of the initial and final
nucleus after the nucleon ejection, naturally including a different binding
energy for neutrino and antineutrino interactions. The simulated datasets
built with this alternative model will be referred to as the “alternative 1p1h
model”.

The correction to the CCQE cross section due to long-range nuclear
correlations, described by RPA in the reference model, has been parametrized
as a function of $Q^{2}$ in terms of five free parameters. A joint fit to the
MiniBooNE mb-ccqe-2010 ; mb-ccqe-antinu-2013  and MINERvA minerva-nu-ccqe ;
minerva-antinu-ccqe  $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ datasets has been
performed to extract an alternative, data-driven RPA correction, labeled
“effective-RPA” in the following. The effective-RPA correction deviates from
the reference RPA at high $Q^{2}$, as can be seen in Fig. 3.

Figure 3: Effective-RPA with error band from the fit to external data compared
with RPA corrections computed in Ref. Nieves:2011pp .

The model in Ref. Martini:2011wp  has been considered as an alternative 2p2h
model, which differs from the reference model in many respects. The
alternative 2p2h cross section is twice as large for neutrino interactions but
has a similar strength for antineutrino interactions, except at high neutrino
energies ($E_{\nu}\gtrsim 1$ GeV) where it is about 30% larger, as can be seen
in Fig. 4. The difference between the 2p2h normalization for neutrino and
antineutrino interactions is constrained with the ND280 data in order to avoid
biases in the CP asymmetry measurement in the oscillation analysis. The
alternative model has also been used for one of the studies of simulated data.
Another important difference between the two models consists in the relative
proportion of nucleon-nucleon correlations, meson exchange currents and their
interference, the first being strongly enhanced in the alternative model. This
difference affects the estimation of the neutrino energy from the outgoing
lepton kinematics. This estimation assumes the CCQE hypothesis, and it is well
known that the 2p2h contribution biases the neutrino energy reconstruction
Martini:2012fa ; Nieves:2012yz  if not properly taken into account in the
simulation. The reference model includes 2p2h events, and so this effect is
included in the T2K neutrino oscillation analysis. Nevertheless, the different
2p2h components produce different biases in the neutrino energy estimation, as
shown in Fig. 5, therefore incorrectly estimating the relative proportions of
nucleon–nucleon correlations and meson exchange current can cause a residual
bias in the neutrino energy estimation. To address this, three simulated
datasets have been built. In the first, the multi-nucleon interactions have
been reweighted as a function of neutrino energy, separately for neutrino and
antineutrino, to reproduce the alternative model (referred to as the
“alternative 2p2h model” in the following). In the other two simulated
datasets, the full 2p2h cross section has been assigned either to meson
exchange currents (“Delta-enhanced 2p2h”) or nucleon-nucleon correlations only
(“not-Delta 2p2h”) by reweighting the muon kinematics as a function of muon
angle, muon momentum and neutrino energy.

Figure 4: Multi-nucleon interactions (2p2h) cross section on 12C as a function
of energy from the models of Nieves (reference model in the text)
Nieves:2011pp  and Martini (alternative model in the text) Martini:2011wp .
Figure 5:  Neutrino energy calculated with the CCQE two-body assumption for
CCQE and 2p2h interactions of 600 MeV muon neutrinos on 12C simulated with the
reference model. The different components of 2p2h show differing amounts of
bias.

The results obtained by considering all the alternative models, SF,
alternative 1p1h, effective RPA, alternative 2p2h, Delta-enhanced 2p2h, not-
Delta 2p2h, are shown in Sec. IX.

##  IV The ND280 complex

The precise measurement of neutrino oscillations in T2K requires a good
understanding of the neutrino beam properties and of neutrino interactions.
The two previous sections have described the neutrino flux model and neutrino-
nucleus interaction model, and constraints on those models based on external
measurements. As we will show in Sec. VIII.4, with only that information, the
precision on the measurements of neutrino oscillations parameters would be
limited. In order to reduce the model uncertainties a near detector complex
has been built 280 m downstream of the production target. The goal of the near
detectors is to directly measure the neutrino beam properties and the neutrino
interaction rate. The near detector complex comprises an on-axis detector
(INGRID) and an off-axis detector (ND280). INGRID is composed of a set of
modules with sufficient target mass and transverse extent to monitor the beam
direction and profile on a day-to-day basis. The ND280 is composed of a set of
subdetectors, installed inside a magnet, and is able to measure the products
of neutrino interactions in detail.

In this section, the methods used to select high purity samples of neutrino
and antineutrino interactions in INGRID and ND280 will be described, and the
results are compared with the predictions obtained from the beam line
simulation and the interaction models. The use of the ND280 data to reduce the
systematic uncertainties in the T2K oscillation analysis will be described in
Sec. V.

###  IV.1 On-axis near detector

The INGRID detector is used to monitor the neutrino beam rate, profile and
center. Those parameters are used to determine the off-axis angle at SK.
INGRID is centered on the neutrino beam axis and samples the neutrino beam
with a transverse cross section of $10~{}\text{m}\times 10~{}\text{m}$ using
14 modules positioned in the shape of a cross. Each INGRID module holds 11
tracking segments built from pairs of orthogonally oriented scintillator
planes interleaved with nine iron planes. There are also three veto planes,
located on the top, bottom and one side of each module. The most upstream
tracking plane is used as a front veto plane. The scintillator planes are
built from 24 plastic scintillator bars instrumented with fibers connected to
multi-pixel photon counters (MPPCs) to detect scintillation light. More
details can be found in Ref. Abe:2011xv .

####  IV.1.1 Event selection and corrections

Neutrino and antineutrino interactions within INGRID modules are first
reconstructed independently in the horizontal and vertical layers of
scintillators. Pairs of tracks in the two different orientations are then
matched by comparing the most upstream point to form 3D tracks. The upstream
edges of the different 3D tracks are then compared in the longitudinal and
transverse direction with respect to the beam direction in order to construct
a common vertex. The subsequent reconstructed event is rejected if the vertex
is reconstructed out of the fiducial volume, if the external veto planes have
hits within 8 cm from the upstream extrapolated position of a reconstructed
track, or if the event timing deviates from more than $100~{}$ns to the
expected event timing.

In order to reduce the systematic uncertainty on the track reconstruction in
$\overline{\nu}$-mode, the selection has been improved from the one used in
Ref. Abe:2015awa . To reduce the impact of MPPC dark noise, the reconstruction
is only applied to events where two consecutive tracking planes have a hit
coincidence on their horizontal and vertical planes. This condition was not
used in Ref. Abe:2015awa  and has been applied only to the
$\overline{\nu}$-mode in the analyses presented here. A total of $12.8\times
10^{6}$ and $4.1\times 10^{6}$ neutrino events are reconstructed respectively
in $\nu$\- and $\overline{\nu}$-mode, with estimated purities of $99.6\%$ and
$98.0\%$ respectively.

The selected number of events in each module is corrected to take into account
the impact of the detector dead channels, the event loss due to non-
reconstructed neutrino interactions caused by pile-up, the variation of the
iron mass between the modules, the time variation of the MPPC noise during the
data taking and the contamination from external background as the previous
INGRID analysis Abe:2011xv ; Abe:2015awa .

####  IV.1.2 Systematic uncertainties

The systematic uncertainties on the event selection are estimated using the
simulation and control samples. The sources of error are the same as those
identified in Ref. Abe:2015awa  and include the neutrino target mass, the
accidental coincidence with MPPC dark noise, the hit efficiency, the event
pile-up, the cosmic and beam-induced backgrounds along with errors associated
to event selection cuts. The method for estimating the uncertainty has not
been changed since Ref. Abe:2015awa  for $\nu$-mode, and is also applied here
for $\overline{\nu}$-mode data. The uncertainties are evaluated to be $0.9\%$
and $1.7\%$ for neutrino and antineutrino data respectively. The larger
uncertainty in $\overline{\nu}$-mode mainly arises from a discrepancy between
data and simulation for interactions producing a track that cross less than
four tracking planes.

####  IV.1.3 Results of neutrino beam measurement

The stability of the neutrino flux is monitored by measuring the event rate,
that is the total number of selected events per protons on target. Fig. 6
shows the intensity stability as a function of time for both $\nu$\- and
$\overline{\nu}$-modes. Most of the data have been taken with the horn
currents set to an absolute value of 250 kA, except for a small fraction of
$\nu$-mode data taken during T2K run 3 in which horns were operated at 205 kA.

Figure 6: The event rate at INGRID as a function of time is shown in the top
panel. The horizontal dashed redline corresponds to the best fit for the
different horn currents with a constant function. The central and bottom
panels show the neutrino beam direction measured at INGRID and at MUMON along
the horizontal and vertical transverse directions with respect to the beam as
a function of time. The dashed vertical lines separate the seven T2K physics
runs.

The average event rates are compared with the simulation, and the ratios are:

$\displaystyle\begin{aligned} \frac{N^{\text{data,
$\nu$}}_{250kA}}{N^{\text{MC, $\nu$}}_{250kA}}&=1.010\pm 0.001(stat.)\pm
0.009(syst.),\\\ \frac{N^{\text{data, $\nu$}}_{205kA}}{N^{\text{MC,
$\nu$}}_{205kA}}&=1.026\pm 0.002(stat.)\pm 0.009(syst.),\\\
\frac{N^{\text{data, $\overline{\nu}$}}_{-250kA}}{N^{\text{MC,
$\overline{\nu}$}}_{-250kA}}&=0.984\pm 0.001(stat.)\pm 0.017(syst.).\\\
\end{aligned}$ (1)

The quoted systematic uncertainties do not include the uncertainties on flux
and cross section model, but they only include INGRID detector systematic
uncertainties. The numbers of expected events in the Monte Carlo are obtained
with the cross section models described in Sect. III.1. The spatial spread of
the neutrino beam is measured using the number of reconstructed events in each
INGRID module. This produces a measurement of the number of events as function
of the distance from the center in both vertical and horizontal directions.
The two distributions are fit with a Gaussian, and the neutrino beam center
and width are given by the mean and the sigma of the fit.

The measurement of the position of the beam center is crucial to determine the
off-axis angle, and therefore, the neutrino beam energy at SK. A deviation of
$1~{}$mrad of the beam direction would shift the peak neutrino energy by
$2\%$. Fig. 6 shows the beam direction stability for all data taking periods.
The variations are well within the design goal of 1 mrad. The average angles
are

$\displaystyle\begin{aligned} \overline{\theta}^{\text{beam,
$\nu$}}_{X}&=0.027\pm 0.010(stat.)\pm 0.095(syst.)~{}{\rm mrad}\\\
\overline{\theta}^{\text{beam, $\nu$}}_{Y}&=0.036\pm 0.011(stat.)\pm
0.105(syst.)~{}{\rm mrad}\\\ \end{aligned}$ (2)

for $\nu$-mode, and

$\displaystyle\begin{aligned} \overline{\theta}^{\text{beam,
$\overline{\nu}$}}_{X}&=-0.032\pm 0.012(stat.)\pm 0.121(syst.)~{}{\rm mrad}\\\
\overline{\theta}^{\text{beam, $\overline{\nu}$}}_{Y}&=0.137\pm
0.020(stat.)\pm 0.140(syst.)~{}{\rm mrad}\\\ \end{aligned}$ (3)

for $\overline{\nu}$-mode. All values are compatible with the expected beam
direction.

###  IV.2 The off-axis ND280 detector

The off-axis near detector ND280 measures the neutrino energy spectrum, flavor
content, and interaction rates of the unoscillated beam. These measurements
are crucial to reduce the uncertainties on neutrino flux and interaction
models which affect the prediction on the number of expected events at the far
detector.

The ND280 detector consists of a set of subdetectors installed inside the
refurbished UA1/NOMAD magnet, which provides a 0.2 T field, used to measure
the charge and the momentum of particles passing through ND280. For the
analyses described in this paper, $\nu_{\mu}$ and $\overline{\nu}_{\mu}$
charged current interactions are selected in the tracker region of ND280,
which consists of three Time Projection Chambers (TPC1, 2, 3) Abgrall:2010hi ,
interleaved with two Fine-Grained Detectors (FGD1, 2) Amaudruz:2012agx .

The upstream FGD1 detector consists of fifteen polystyrene scintillator
modules, while the downstream FGD2 contains seven polystyrene scintillator
modules interleaved with six water modules. The FGDs provide target mass for
neutrino interactions and track the charged particles coming from the
interaction vertex, while the TPCs provide 3D tracking and determine the
momentum and energy loss of each charged particle traversing them. The
observed energy loss in the TPCs, combined with the measurement of the
momentum, is used for particle identification of the charged tracks produced
in neutrino interactions in order to measure exclusive CC event rates. The
major updates in the near detector analysis with respect to Ref. Abe:2015awa
are the use of interactions in FGD2 and the inclusion of data taken with the
$\overline{\nu}$-mode beam.

The charge and particle identification ability of the tracker is important
because it provides separation between $\mu^{+}$ (produced by
$\bar{\nu}_{\mu}$ CC interactions) and $\mu^{-}$ (produced by $\nu_{\mu}$ CC
interactions) when T2K runs in $\bar{\nu}$-mode. Moreover, by including both
FGD1 and FGD2 samples, the properties of neutrino interactions on water can be
effectively isolated from those on carbon, reducing the uncertainties related
to extrapolating across differing nuclear targets in the near and far
detectors. The near detector analysis described here uses a reduced dataset
comprising $5.81\times 10^{20}$ POT in $\nu$-mode and $2.84\times 10^{20}$ POT
in $\bar{\nu}$-mode, as shown in Tab. 3.

Table 3:  Collected POT for each dataset used in the ND280 analysis.  Run | Dates |  $\nu$-Mode POT |  $\bar{\nu}$-Mode POT
---|---|---|---
Period |  | ($\times 10^{20}$) | ($\times 10^{20}$)
Run 2 | Nov. 2010-Mar. 2011 | 0.78 | –
Run 3 | Mar. 2012-Jun. 2012 | 1.56 | –
Run 4 | Oct. 2012-May 2013 | 3.47 | –
Run 5 | Jun. 2014 | – | 0.43
Run 6 | Nov. 2014-Apr. 2015 | – | 2.41
Total | Nov. 2010-Apr. 2015 | 5.81 | 2.84

####  IV.2.1 ND280 $\nu_{\mu}$ CC selection in $\nu$-mode

The event selection in $\nu$-mode beam is unchanged since the previous
analysis described in Ref. Abe:2015awa . Muon-neutrino-induced CC interactions
are selected by identifying the $\mu^{-}$ produced in the final state as the
highest-momentum, negative-curvature track in each event with a vertex in FGD1
(FGD2) Fiducial Volume (FV) and crossing the middle (last) TPC. The energy
lost by the selected track in the TPC must be consistent with a muon.

All the events generated upstream of FGD1 are rejected by excluding events
with a track in the first TPC. The selected $\nu_{\mu}$ CC candidates are then
divided into three subsamples, according to the number of identified pions in
the event: CC-$0\pi$, CC-$1\pi^{+}$ and CC-Other, which are dominated by
“quasi-elastic” (CCQE), CC resonant pion production, and DIS interactions,
respectively. Pions are selected in different ways according to their charge.
A $\pi^{+}$ can be identified in three ways: an FGD+TPC track with positive-
curvature and an energy loss in the TPC consistent with a pion; an FGD-
contained track with a charge deposition consistent with a pion; or a delayed
energy deposit in the FGD due to a decay electron from stopped
$\pi^{+}\rightarrow\mu^{+}$. In this analysis, $\pi^{-}$’s are only identified
by selecting negative-curvature FGD+TPC tracks, while $\pi^{0}$’s are
identified by looking for tracks in the TPC with charge depositions consistent
with an electron from a $\gamma$ conversion. The output of the $\nu$-mode
tracker selection are six samples, three per FGD. The selected CC-0$\pi$ and
CC-$1\pi^{+}$ samples in both FGDs before the ND280 fit are shown in Fig. 7.
For each of the selected samples, the number of observed and predicted events
are shown in Tab. 4.

Table 4: Observed and predicted event rates for different ND280 samples collected in $\nu$-mode beam. Before the ND280 fit that will be described in Sec. V, uncertainties of $\sim$20% on the event rates are expected. FGD1 sample | Data | Prediction
---|---|---
$\nu_{\mu}$ CC-0$\pi$ | 17354 | 16951
$\nu_{\mu}$ CC-1$\pi^{+}$ | 3984 | 4460
$\nu_{\mu}$ CC-Other | 4220 | 4010
FGD2 sample | Data | Prediction
$\nu_{\mu}$ CC-0$\pi$ | 17650 | 17212
$\nu_{\mu}$ CC-1$\pi^{+}$ | 3383 | 3617
$\nu_{\mu}$ CC-Other | 4118 | 3627

Figure 7: Top: muon momentum distributions of the $\nu$-mode $\nu_{\mu}$
CC-0$\pi$ samples in FGD1 (left) and FGD2 (right). Bottom: muon momentum
distributions of the $\nu$-mode $\nu_{\mu}$ CC-1$\pi^{+}$ samples in FGD1
(left) and FGD2 (right). All distributions are shown prior to the ND280 fit.

####  IV.2.2 ND280 $\overline{\nu}_{\mu}$ and $\nu_{\mu}$ CC selections in
$\overline{\nu}$-mode

The main difference between $\nu$\- and $\overline{\nu}$-modes is the increase
in the number of interactions produced by “wrong-sign” neutrinos. Once
differences in the flux and the cross section are taken into account, the
wrong-sign contamination in $\overline{\nu}$-mode is expected to be
approximately 30%, while the wrong-sign contamination in $\nu$-mode is
approximately 4%.

The lepton selection criteria of $\bar{\nu}_{\mu}$ ( $\nu_{\mu}$) CC
interactions is similar to the one used in the neutrino beam mode, except for
the condition that the highest-momentum, positively (negatively) charged
particles must also be the highest momentum track in the event. This
additional cut is essential to reduce the background due to $\pi^{+}$
($\pi^{-}$) generated in neutrino (antineutrino) interactions that can be
misidentified as the muon candidate. The selected $\bar{\nu}_{\mu}$ CC
($\nu_{\mu}$ CC) candidate events are divided in two subsamples: CC-1-Track,
dominated by CCQE-like interactions; and CC-N-Tracks (N$>$1), a mixture of
resonant production and DIS. These two subsamples are defined by the number of
reconstructed tracks crossing the TPC. For these selections the CC candidates
are not divided into three subsamples as in Sec. IV.2.1, according to the
number of identified pions in the event in order to avoid samples with low
statistics.

The output of the $\overline{\nu}$-mode tracker selection are eight samples,
four per FGD. For each of the selected sample, the number of predicted events
and the ones observed in data are shown in Tab. 5. The four selected samples
in FGD1, before the ND280 fit, are shown in Fig. 8.

Table 5: Observed and predicted event rates for different ND280 samples collected in $\bar{\nu}$-mode beam. Before the ND280 fit that will be described in Sec. V uncertainties of 20% on the event rates are expected. FGD1 sample | Data | Prediction
---|---|---
$\bar{\nu}_{\mu}$ CC-1-Track | 2663 | 2709
$\bar{\nu}_{\mu}$ CC-N-Tracks | 775 | 798
$\nu_{\mu}$ CC-1-Track | 989 | 938
$\nu_{\mu}$ CC-N-Tracks | 1001 | 995
FGD2 sample | Data | Prediction
$\bar{\nu}_{\mu}$ CC-1-Track | 2762 | 2730
$\bar{\nu}_{\mu}$ CC-N-Tracks | 737 | 804
$\nu_{\mu}$ CC-1-Track | 980 | 944
$\nu_{\mu}$ CC-N-Tracks | 936 | 917

Figure 8: Top: muon momentum distributions for the $\bar{\nu}$-mode
$\bar{\nu}_{\mu}$ CC-1-Track (left) and CC-N-Tracks (right) samples. Bottom:
muon momentum distributions for the $\bar{\nu}$-mode $\nu_{\mu}$ CC-1-Track
(left) and CC-N-Tracks (right) samples. All distributions are shown prior to
the ND280 fit.

###  IV.3 ND280 detector systematic uncertainties

In order to assess systematic uncertainties related to the ND280 detector
modeling, various different control samples are used, as described in Ref.
Abe:2015awa . The control samples include muons produced in neutrino
interactions outside ND280, cosmic muons, interactions upstream of TPC1, and
stopping muons. All control samples are independent of the samples used for
the ND280 analyses described earlier. The method to propagate the systematic
uncertainties in the near detector analysis is also unchanged with respect to
Ref. Abe:2015awa : a vector of systematic parameters $\vec{d}$ scales the
expected number of events in bins of $p_{\mu}$ and $\cos\theta_{\mu}$. The
covariance of $\vec{d}$, $V_{d}$, is evaluated by varying each systematic
parameter.

The difference with respect to the previous analysis are the inclusion of a
time of flight (ToF) systematic and new methods used to evaluate charge
misidentification and FGD tracking efficiency uncertainties and uncertainties
due to interactions outside the fiducial volume. The ToF between FGD1 and FGD2
is used to select events with a backward muon candidate in the FGD2 samples.
The ToF systematic uncertainty is $\sim$0.1% for the $\nu$-mode FGD2 samples,
and $\sim$0.01% for the $\bar{\nu}$-mode FGD2 samples. The ToF uncertainty is
smaller in $\bar{\nu}$-mode because fewer backward-going $\mu^{+}$ are
produced in $\overline{\nu}_{\mu}$ interactions than backward-going $\mu^{-}$
in $\nu_{\mu}$ interactions.

The charge mis-identification uncertainty is parametrized as a function of the
momentum resolution in the TPCs.

The FGD tracking efficiency for CC-events, where either a short pion or proton
track is also produced, is estimated using a hybrid data-MC sample. This
sample uses events with a long FGD-TPC matched muon candidate track with the
addition of an FGD-isolated track generated via particle gun with a common
vertex.

The method used to estimate the number of out-of-fiducial volume (OOFV) events
has been refined by estimating the number of events, and the error, separately
for each detector in which the OOFV events occur, rather than averaging over
the number of OOFV events produced in all of the detectors outside the tracker
as previously.

Most sources of systematic error are common between $\nu$\- and
$\overline{\nu}$-modes because the selection criteria are similar, as
described in Sec. IV.2.2. However, as $\overline{\nu}$-mode data are divided
into CC 1-Track and CC N-Track samples, only based on the number of
reconstructed FGD-TPC matched tracks, most uncertainties relating to the FGD
reconstruction are not relevant. The exceptions are the FGD-TPC matching and
ToF uncertainties, which apply to both modes. Other differences between modes
arise because some errors change with the beam conditions (sand muons, pile-up
and OOFV) and are evaluated independently for each run period.

The total systematic uncertainties are shown in Tab. 6.

Table 6: Systematic uncertainty on the total event rate affecting the near detector samples. ND280 sample | Total systematic
---|---
| uncertainty (%)
$\nu$-mode
FGD1 $\nu_{\mu}$ CC-0$\pi$ | 1.7
FGD1 $\nu_{\mu}$ CC-1$\pi^{+}$ | 3.3
FGD1 $\nu_{\mu}$ CC-Other | 6.5
FGD2 $\nu_{\mu}$ CC-0$\pi$ | 1.7
FGD2 $\nu_{\mu}$ CC-1$\pi^{+}$ | 3.9
FGD2 $\nu_{\mu}$ CC-Other | 5.9
$\bar{\nu}$-mode
FGD1 $\bar{\nu}_{\mu}$ CC-1-Track | 5.4
FGD1 $\bar{\nu}_{\mu}$ CC-N-Tracks | 10.4
FGD1 $\nu_{\mu}$ CC-1-Track | 2.5
FGD1 $\nu_{\mu}$ CC-N-Tracks | 4.8
FGD2 $\bar{\nu}_{\mu}$ CC-1-Track | 3.5
FGD2 $\bar{\nu}_{\mu}$ CC-N-Tracks | 7.3
FGD2 $\nu_{\mu}$ CC-1-Track | 2.0
FGD2 $\nu_{\mu}$ CC-N-Tracks | 4.0

The dominant source of uncertainty for all ND280 samples comes from the pion
re-interaction model, used to estimate the rate of pion interactions in the
FGDs. This is due to differences between the GEANT4 model, used to simulate
pion re-interactions outside the nucleus, and the available experimental data.
For example, the systematic uncertainty related to pion interactions affecting
the FGD1 $\nu_{\mu}$ CC-0$\pi$ ($\bar{\nu}_{\mu}$ CC-1-Track) sample is 1.4%
(4.9%), with a total error of 1.7% (5.4%). The pion re-interaction uncertainty
is larger for $\overline{\nu}$-mode samples than for $\nu$-mode samples
because $\pi^{-}$ interactions on carbon and water are less well understood
than $\pi^{+}$ interactions at the relevant energies, and because the fraction
of $\pi^{+}$ from wrong-sign contamination in $\overline{\nu}$-mode
misidentified as $\mu^{+}$ candidate is larger than the fraction of $\pi^{-}$
misidentified as $\mu^{-}$ in the $\nu$-mode.

##  V Near Detector data analysis

The predicted event rates at both the ND280 and SK are based on parametrized
neutrino flux and interaction models, described in Sec. II and Sec. III.1.
These models are fit to the precisely measured, high statistics data at the
ND280, producing both a better central prediction of the SK event rate and
reducing the systematic uncertainties associated with the flux and interaction
models. The near detector analysis uses event samples from both FGD1 and FGD2,
and from the $\nu$-mode and $\overline{\nu}$-mode data, giving 14 samples in
total. These, along with their associated systematic uncertainty, were
described in Sec. IV.2.

###  V.1 Near detector likelihood and fitting methods

The form of the ND280 likelihood and the fitting method are the same as
described in Ref. Abe:2015awa . The 14 event samples are binned in $p_{\mu}$
and $\cos\theta_{\mu}$, giving 1062 fit bins in total, though only the
$p_{\mu}$ projection is shown for clarity. The likelihood assumes that the
observed number of events in each bin follows a Poisson distribution, with an
expectation calculated according to the flux, cross-section and detector
systematic parameters discussed above. A multivariate Gaussian likelihood
function is used to constrain these parameters in the fit, with the initial
constraints that are described in Secs. II, III.1 and IV.2. The near detector
systematic and near detector flux parameters are treated as nuisance
parameters, as are the cross-section systematic parameters governing neutral
current and electron neutrino interactions. The fitted neutrino cross-section
and unoscillated SK flux parameters are passed to the oscillation analysis,
using a covariance matrix to describe their uncertainties.

One significant difference with respect to Ref. Abe:2015awa  is that, as
discussed in Sec. III.1, the CCQE cross-section parameters (except the nucleon
binding energy, $E_{\mathrm{b}}$) have no external constraint. These
parameters are constrained solely by the ND280 data. In addition, in order to
alleviate possible biases on the estimation of the oscillation parameters (see
Sec. IX for more details), the differences between the reference model and the
alternative model for the 1p1h component of the neutrino-nucleus interaction
cross section described in Sec. III are taken into account in the likelihood.
This is done by adding the difference in the expected number of events between
the two models in each $p_{\mu}$ and $\cos\theta_{\mu}$ bin to the diagonal of
the ND280 detector covariance matrix $V_{d}$. Finally, another significant
difference is the inclusion of event samples from FGD2, which contains a water
target, and $\overline{\nu}$-mode data samples.

###  V.2 Fit results

The fit produces central values for the flux, cross-section and detector
systematic parameters along with a covariance. Fig. 9 shows the values of the
unoscillated SK flux parameters and Fig. 10 the cross-section parameters
before and after the fit as a fraction of the nominal value, along with their
prior constraints. These parameter values are listed in Tables 7, 8 and 9,
showing the best-fit point for each along with its uncertainty, calculated as
the square root of the diagonal of the covariance.

Figure 9: The SK flux parameters for the $\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ (left) and $\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ (right) neutrino species in $\nu$\- (top) and $\overline{\nu}$-modes (bottom), shown as a fraction of the nominal value. The bands indicate the 1$\sigma$ uncertainty on the parameters before (solid, red) and after (hatched, blue) the near detector fit. Figure 10: Cross-section parameters before (solid, red) and after (hatched, blue) the near detector fit, shown as a fraction of the nominal value (given in Tab. 9). The extent of the colored band shows the 1$\sigma$ uncertainty.  Table 7: Prefit and postfit values for the SK $\nu$-mode flux parameters $\nu$-mode | Prefit | ND280 postfit
---|---|---
flux parameter (GeV) |  |
SK $\nu_{\mu}$ [0.0–0.4] | 1.000 $\pm$ 0.099 | 1.128 $\pm$ 0.064
SK $\nu_{\mu}$ [0.4–0.5] | 1.000 $\pm$ 0.103 | 1.156 $\pm$ 0.061
SK $\nu_{\mu}$ [0.5–0.6] | 1.000 $\pm$ 0.096 | 1.148 $\pm$ 0.051
SK $\nu_{\mu}$ [0.6–0.7] | 1.000 $\pm$ 0.087 | 1.128 $\pm$ 0.043
SK $\nu_{\mu}$ [0.7–1.0] | 1.000 $\pm$ 0.113 | 1.104 $\pm$ 0.047
SK $\nu_{\mu}$ [1.0–1.5] | 1.000 $\pm$ 0.092 | 1.100 $\pm$ 0.045
SK $\nu_{\mu}$ [1.5–2.5] | 1.000 $\pm$ 0.070 | 1.127 $\pm$ 0.044
SK $\nu_{\mu}$ [2.5–3.5] | 1.000 $\pm$ 0.074 | 1.124 $\pm$ 0.048
SK $\nu_{\mu}$ [3.5–5.0] | 1.000 $\pm$ 0.087 | 1.121 $\pm$ 0.049
SK $\nu_{\mu}$ [5.0–7.0] | 1.000 $\pm$ 0.098 | 1.075 $\pm$ 0.053
SK $\nu_{\mu}$ $>$7.0 | 1.000 $\pm$ 0.114 | 1.064 $\pm$ 0.065
SK $\bar{\nu}_{\mu}$ [0.0–0.7] | 1.000 $\pm$ 0.103 | 1.100 $\pm$ 0.081
SK $\bar{\nu}_{\mu}$ [0.7–1.0] | 1.000 $\pm$ 0.079 | 1.112 $\pm$ 0.048
SK $\bar{\nu}_{\mu}$ [1.0–1.5] | 1.000 $\pm$ 0.084 | 1.111 $\pm$ 0.060
SK $\bar{\nu}_{\mu}$ [1.5–2.5] | 1.000 $\pm$ 0.086 | 1.116 $\pm$ 0.070
SK $\bar{\nu}_{\mu}$ $>$2.5 | 1.000 $\pm$ 0.086 | 1.162 $\pm$ 0.069
SK $\nu_{e}$ [0.0–0.5] | 1.000 $\pm$ 0.090 | 1.134 $\pm$ 0.052
SK $\nu_{e}$ [0.5–0.7] | 1.000 $\pm$ 0.090 | 1.135 $\pm$ 0.049
SK $\nu_{e}$ [0.7–0.8] | 1.000 $\pm$ 0.086 | 1.135 $\pm$ 0.047
SK $\nu_{e}$ [0.8–1.5] | 1.000 $\pm$ 0.081 | 1.119 $\pm$ 0.043
SK $\nu_{e}$ [1.5–2.5] | 1.000 $\pm$ 0.079 | 1.115 $\pm$ 0.046
SK $\nu_{e}$ [2.5–4.0] | 1.000 $\pm$ 0.084 | 1.111 $\pm$ 0.050
SK $\nu_{e}$ $>$4.0 | 1.000 $\pm$ 0.094 | 1.118 $\pm$ 0.067
SK $\bar{\nu}_{e}$ [0.0–2.5] | 1.000 $\pm$ 0.074 | 1.121 $\pm$ 0.057
SK $\bar{\nu}_{e}$ $>$2.5 | 1.000 $\pm$ 0.128 | 1.153 $\pm$ 0.117
Table 8: Prefit and postfit values for the SK $\overline{\nu}$-mode flux parameters $\overline{\nu}$-mode | Prefit | ND280 postfit
---|---|---
flux parameter (GeV) |  |
SK $\nu_{\mu}$ [0.0–0.7] | 1.000 $\pm$ 0.094 | 1.098 $\pm$ 0.072
SK $\nu_{\mu}$ [0.7–1.0] | 1.000 $\pm$ 0.079 | 1.121 $\pm$ 0.052
SK $\nu_{\mu}$ [1.0–1.5] | 1.000 $\pm$ 0.077 | 1.130 $\pm$ 0.048
SK $\nu_{\mu}$ [1.5–2.5] | 1.000 $\pm$ 0.081 | 1.155 $\pm$ 0.054
SK $\nu_{\mu}$ $>$2.5 | 1.000 $\pm$ 0.080 | 1.111 $\pm$ 0.055
SK $\bar{\nu}_{\mu}$ [0.0–0.4] | 1.000 $\pm$ 0.104 | 1.118 $\pm$ 0.071
SK $\bar{\nu}_{\mu}$ [0.4–0.5] | 1.000 $\pm$ 0.102 | 1.127 $\pm$ 0.060
SK $\bar{\nu}_{\mu}$ [0.5–0.6] | 1.000 $\pm$ 0.096 | 1.117 $\pm$ 0.052
SK $\bar{\nu}_{\mu}$ [0.6–0.7] | 1.000 $\pm$ 0.085 | 1.121 $\pm$ 0.044
SK $\bar{\nu}_{\mu}$ [0.7–1.0] | 1.000 $\pm$ 0.125 | 1.155 $\pm$ 0.066
SK $\bar{\nu}_{\mu}$ [1.0–1.5] | 1.000 $\pm$ 0.105 | 1.132 $\pm$ 0.057
SK $\bar{\nu}_{\mu}$ [1.5–2.5] | 1.000 $\pm$ 0.078 | 1.139 $\pm$ 0.053
SK $\bar{\nu}_{\mu}$ [2.5–3.5] | 1.000 $\pm$ 0.074 | 1.141 $\pm$ 0.054
SK $\bar{\nu}_{\mu}$ [3.5–5.0] | 1.000 $\pm$ 0.094 | 1.151 $\pm$ 0.071
SK $\bar{\nu}_{\mu}$ [5.0–7.0] | 1.000 $\pm$ 0.093 | 1.133 $\pm$ 0.070
SK $\bar{\nu}_{\mu}$ $>$7.0 | 1.000 $\pm$ 0.130 | 1.082 $\pm$ 0.110
SK $\nu_{e}$ [0.0–2.5] | 1.000 $\pm$ 0.069 | 1.118 $\pm$ 0.051
SK $\nu_{e}$ $>$2.5 | 1.000 $\pm$ 0.085 | 1.112 $\pm$ 0.071
SK $\bar{\nu}_{e}$ [0.0–0.5] | 1.000 $\pm$ 0.095 | 1.126 $\pm$ 0.058
SK $\bar{\nu}_{e}$ [0.5–0.7] | 1.000 $\pm$ 0.091 | 1.127 $\pm$ 0.051
SK $\bar{\nu}_{e}$ [0.7–0.8] | 1.000 $\pm$ 0.091 | 1.133 $\pm$ 0.052
SK $\bar{\nu}_{e}$ [0.8–1.5] | 1.000 $\pm$ 0.084 | 1.132 $\pm$ 0.046
SK $\bar{\nu}_{e}$ [1.5–2.5] | 1.000 $\pm$ 0.080 | 1.125 $\pm$ 0.056
SK $\bar{\nu}_{e}$ [2.5–4.0] | 1.000 $\pm$ 0.089 | 1.119 $\pm$ 0.071
SK $\bar{\nu}_{e}$ $>$4.0 | 1.000 $\pm$ 0.156 | 1.166 $\pm$ 0.141
Table 9: Prefit and postfit values for the cross-section parameters used in the oscillation fits. If no prefit uncertainty is shown then the parameter had a flat prior assigned. If a parameter was not constrained by the ND280 fit this is noted in the postfit column. Cross section | Prefit | ND280 postfit
---|---|---
parameter |  |
$M_{\textrm{A}}^{\mbox{\scriptsize{QE}}}$ (GeV/$c^{2}$) | 1.20 | 1.12 $\pm$ 0.03
$p_{\mathrm{F}}$ 12C (MeV/c) | 217.0 | 243.9 $\pm$ 16.6
2p2h 12C | 100.0 | 154.5 $\pm$ 22.7
$E_{\mathrm{b}}$ 12C (MeV) | 25.0 $\pm$ 9.00 | 16.5 $\pm$ 7.53
$p_{\mathrm{F}}$ 16O (MeV/c) | 225.0 | 234.2 $\pm$ 23.7
2p2h 16O | 100.0 | 154.6 $\pm$ 34.3
$E_{\mathrm{b}}$ 16O (MeV) | 27.0 $\pm$ 9.00 | 23.8 $\pm$ 7.61
$C_{\mathrm{A}}^{5}$ | 1.01 $\pm$ 0.12 | 0.80 $\pm$ 0.06
$M_{\textrm{A}}^{\mathrm{RES}}$ (GeV/$c^{2}$) | 0.95 $\pm$ 0.15 | 0.84 $\pm$ 0.04
I${}_{\frac{1}{2}}$ background | 1.30 $\pm$ 0.20 | 1.36 $\pm$ 0.17
CC other shape | 0.00 $\pm$ 0.40 | -0.02 $\pm$ 0.21
CC coherent | 1.00 $\pm$ 0.30 | 0.86 $\pm$ 0.23
NC coherent | 1.00 $\pm$ 0.30 | 0.93 $\pm$ 0.30
2p2h $\bar{\nu}$ | 1.00 | 0.58 $\pm$ 0.18
NC other | 1.00 $\pm$ 0.30 | Not constrained
NC 1-$\gamma$ | 1.00 $\pm$ 1.00 | Not constrained
$\nu_{e}$/$\nu_{\mu}$ ratio | 1.00 $\pm$ 0.02 | Not constrained
$\bar{\nu}_{e}$/$\bar{\nu}_{\mu}$ ratio | 1.00 $\pm$ 0.02 | Not constrained
FSI elastic low-E | 1.00 $\pm$ 0.41 | Not constrained
FSI elastic high-E | 1.00 $\pm$ 0.34 | Not constrained
FSI pion production | 1.00 $\pm$ 0.50 | Not constrained
FSI pion absorption | 1.00 $\pm$ 0.41 | Not constrained
FSI charge exchange low-E | 1.00 $\pm$ 0.57 | Not constrained
FSI charge exchange high-E | 1.00 $\pm$ 0.28 | Not constrained

Most noticeable in these results is the 10–15 % increase in the neutrino flux,
seen across all species and energies in both $\nu$\- and
$\overline{\nu}$-modes.

Small changes are seen in the central values of the CCQE cross-section
parameters, with the fit increasing the Fermi momentum parameter while
reducing the nucleon binding energy and axial mass parameters. More
interestingly, the 2p2h normalization is increased to approximately 1.5 times
its nominal value, indicating that the fit is sensitive to differences in
lepton kinematics between CCQE and 2p2h interactions. The antineutrino 2p2h
normalization is reduced compared to the neutrino parameter, highlighting a
difference in the neutrino and antineutrino CC–0$\pi$ event rates that cannot
be explained by flux or detector systematics. The fit also reduces the value
of the charged current single pion parameters, as seen in the previous
analysis Abe:2015awa . This accounts for the relative deficit observed in the
CC–1$\pi$ sample compared to the CC–0$\pi$ sample.

####  V.2.1 Goodness of fit and fit validation

The goodness of fit for the near detector analysis was estimated using mock
datasets including statistical uncertainties. Mock datasets are generated by
simultaneously varying the systematic parameters in the fit according to their
prior covariance then applying these to the nominal MC. These were then fit
and the minimum negative log-likelihood value found. The distribution of the
minimum negative log-likelihood values is shown by the histogram in Fig. 11,
with the value from the data fit indicated with a red line. The overall
p-value for the fit is 8.6%. Figure 12 shows the same distribution for the
flux and cross-section parameter priors, demonstrating that the fitted
parameter values propagated to the oscillation analysis are reasonable.

Figure 11: Distribution of the minimum negative log-likelihood values from
fits to the mock datasets (black), with the value from the fit to the data
superimposed in red.

Figure 12: Distribution of the minimum negative log-likelihood values from
fits to the mock datasets (black), with the value from the fit to the data
superimposed in red. The distributions shown make up the contribution from the
flux (top) and cross-section (bottom) prior terms.

In addition, the Bayesian analysis which simultaneously fits both near and far
detector samples, that will be described in Sec. VIII.2, was used to cross-
check the primary result by only fitting near detector data. The results of
this fit are compared to the best-fit parameters from the near detector
analysis in Fig. 13, showing excellent agreement between the two.

Figure 13: Comparison of the fitted SK $\nu$-mode flux parameters (top) and
cross-section parameters (bottom) between the ND280 fit (red, solid) and the
Bayesian analysis (blue, hashed).

###  V.3 ND280 postfit distributions

The expected muon momentum spectrum after the ND280 fit for the CC-0$\pi$ and
CC-1$\pi^{+}$ samples in $\nu$-mode and the FGD1 samples in
$\overline{\nu}$-mode are shown in Fig. 14 and Fig. 15 respectively. After the
ND280 fit, the expected distributions show in general a better agreement with
the data. The numbers of postfit predicted events for all the 14 samples are
shown in Tab. 10. The effects of the ND280 fit on the different neutrino
interactions contributing to each ND280 sample are detailed in Tab. 11 and
Tab. 12.

Figure 14: Data-MC comparison of $\nu$-mode $\nu_{\mu}$ CC-0$\pi$ (top) and
CC-1$\pi^{+}$ (bottom) samples in FGD1 (left) and FGD2 (right) after the ND280
fit. The simulation is broken-down by neutrino reaction type.

Figure 15: Top: data-MC comparison of $\bar{\nu}_{\mu}$ (right-sign) CC-1-Track (left) and CC-N-Tracks (right) samples after the ND280 fit. The simulation is broken-down by neutrino reaction type. Bottom: same muon momentum distributions for $\nu_{\mu}$ background samples. Table 10: Observed and predicted events rates for different ND280 samples after the ND280 fit. ND280 Sample | FGD1 | FGD1 | FGD2 | FGD2
---|---|---|---|---
| Data | Postfit | Data | Postfit
$\nu_{\mu}$ CC-0$\pi$ | 17354 | 17345 | 17650 | 17638
$\nu_{\mu}$ CC-1$\pi^{+}$ | 3984 | 4113 | 3383 | 3449
$\nu_{\mu}$ CC-Other | 4220 | 4150 | 4118 | 3965
$\bar{\nu}_{\mu}$ CC-1-Track | 2663 | 2639 | 2762 | 2728
$\bar{\nu}_{\mu}$ CC-N-Tracks | 775 | 785 | 737 | 814
$\nu_{\mu}$ CC-1-Track | 989 | 966 | 980 | 987
$\nu_{\mu}$ CC-N-Tracks | 1001 | 989 | 936 | 937
Table 11: Prefit and postfit expected fraction of events for each neutrino interactions for samples taken in $\nu$-mode. | FGD1 $\nu_{\mu}$ CC-0$\pi$ | FGD2 $\nu_{\mu}$ CC-0$\pi$
---|---|---
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.562 | 0.540 | 0.547 | 0.533
$\nu$ CC 2p2h | 0.100 | 0.165 | 0.094 | 0.155
$\nu$ CC Res $1\pi$ | 0.202 | 0.152 | 0.219 | 0.164
$\nu$ CC Coh $1\pi$ | 0.003 | 0.003 | 0.003 | 0.003
$\nu$ CC Other | 0.096 | 0.098 | 0.101 | 0.103
$\nu$ NC modes | 0.032 | 0.037 | 0.032 | 0.037
$\bar{\nu}$ modes | 0.003 | 0.003 | 0.004 | 0.004
| FGD1 $\nu_{\mu}$ CC-1$\pi^{+}$ | FGD2 $\nu_{\mu}$ CC-1$\pi^{+}$
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.054 | 0.053 | 0.055 | 0.054
$\nu$ CC 2p2h | 0.008 | 0.014 | 0.007 | 0.013
$\nu$ CC Res $1\pi$ | 0.485 | 0.409 | 0.476 | 0.401
$\nu$ CC Coh $1\pi$ | 0.025 | 0.025 | 0.026 | 0.025
$\nu$ CC Other | 0.348 | 0.389 | 0.353 | 0.395
$\nu$ NC modes | 0.059 | 0.087 | 0.60 | 0.089
$\bar{\nu}$ modes | 0.022 | 0.024 | 0.022 | 0.024
| FGD1 $\nu_{\mu}$ CC-Other | FGD2 $\nu_{\mu}$ CC-Other
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.047 | 0.043 | 0.049 | 0.045
$\nu$ CC 2p2h | 0.010 | 0.015 | 0.010 | 0.016
$\nu$ CC Res $1\pi$ | 0.148 | 0.112 | 0.154 | 0.117
$\nu$ CC Coh $1\pi$ | 0.004 | 0.003 | 0.003 | 0.003
$\nu$ CC Other | 0.701 | 0.712 | 0.696 | 0.707
$\nu$ NC modes | 0.078 | 0.102 | 0.075 | 0.099
$\bar{\nu}$ modes | 0.013 | 0.014 | 0.11 | 0.013
Table 12: Prefit and postfit expected fraction of events for each neutrino interactions for samples taken in $\overline{\nu}$-mode. | FGD1 $\bar{\nu}_{\mu}$ CC-1-Track | FGD2 $\bar{\nu}_{\mu}$ CC-1-Track
---|---|---
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.011 | 0.012 | 0.015 | 0.014
$\nu$ non-CCQE | 0.069 | 0.077 | 0.074 | 0.080
$\bar{\nu}$ CCQE | 0.733 | 0.751 | 0.731 | 0.751
$\bar{\nu}$ non-CCQE | 0.186 | 0.161 | 0.181 | 0.156
| FGD1 $\bar{\nu}_{\mu}$ CC-N-Tracks | FGD2 $\bar{\nu}_{\mu}$ CC-N-Tracks
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.017 | 0.017 | 0.021 | 0.021
$\nu$ non-CCQE | 0.422 | 0.447 | 0.412 | 0.437
$\bar{\nu}$ CCQE | 0.085 | 0.083 | 0.088 | 0.087
$\bar{\nu}$ non-CCQE | 0.474 | 0.452 | 0.478 | 0.453
| FGD1 $\nu_{\mu}$ CC-1-Track | FGD2 $\nu_{\mu}$ CC-1-Track
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.480 | 0.516 | 0.466 | 0.499
$\nu$ non-CCQE | 0.412 | 0.372 | 0.411 | 0.363
$\bar{\nu}$ CCQE | 0.037 | 0.033 | 0.042 | 0.038
$\bar{\nu}$ non-CCQE | 0.071 | 0.079 | 0.081 | 0.100
| FGD1 $\nu_{\mu}$ CC-N-Tracks | FGD2 $\nu_{\mu}$ CC-N-Tracks
Reaction | Prefit | Postfit | Prefit | Postfit
$\nu$ CCQE | 0.150 | 0.163 | 0.143 | 0.154
$\nu$ non-CCQE | 0.771 | 0.756 | 0.777 | 0.764
$\bar{\nu}$ CCQE | 0.003 | 0.003 | 0.004 | 0.003
$\bar{\nu}$ non-CCQE | 0.076 | 0.078 | 0.077 | 0.078

##  VI Far detector event selection and systematics

The T2K far detector is Super-Kamiokande (SK) FUKUDA2003418 , a 50 kton water
Cherenkov detector located in the Kamioka Observatory, Gifu, Japan. SK is
divided into two concentric cylinders, defining an inner detector (ID)
instrumented with 11,129 20–inch photomultiplier tubes (PMT) and an outer
detector (OD) instrumented with 1885 PMTs. The outer detector is mainly used
as a veto for entering backgrounds while neutrino interactions are selected in
a fiducial volume inside the ID.

In order to precisely measure neutrino oscillations parameters, together with
the large target volume, high acceptance and efficient discrimination is
necessary to distinguish the leptons produced in $\nu_{\mu}$ and $\nu_{e}$
interactions. Vertex, momentum reconstruction, and Particle identification
(PID) in SK are done by observing the Cherenkov radiation produced by charged
particles traversing the detector. These particles produce ring patterns that
are recorded by the PMTs and are the primary tool used for the PID. Muons
produced by $\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ CC interactions are usually unscattered
thanks to their large mass and produce a clear ring pattern. In contrast,
electrons from $\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ CC interactions produce electromagnetic
showers resulting in diffuse ring edges. In addition to the shape of the
Cherenkov ring, the opening angle also helps to distinguish between electrons
and muons. At the typical energies of leptons produced by neutrinos from the
T2K beam, the probability to misidentify a single electron (muon) as a muon
(electron) is 0.7% (0.8%). In SK it is not possible to distinguish neutrinos
from antineutrinos event-by-event since the charge of the outgoing leptons
cannot be reconstructed. For this reason the selection that will be described
in this section is identical for data taken in $\nu$-mode and in
$\overline{\nu}$-mode.

T2K data are extracted from the incoming stream in $\pm 500\,\,\mu{\rm s}$
windows centered on the beam trigger. A scan of each window recovers
individual events which are then classified.

Events in which Cherenkov light is deposited exclusively in the ID comprise
the fully contained (FC) sample. PMTs in the OD that register light are
grouped into clusters. If the largest such cluster contains more than 15 PMTs
the event is moved to an OD sample. Low energy (LE) events are separated into
a dedicated sample by requiring that the total charge from the ID PMT hits in
a 300 ns window be greater than 200 photoelectrons (p.e.), which corresponds
to the charge observed from a 20 MeV electromagnetic shower. Events are also
designated as LE if a single ID PMT hit constitutes more than half of the
total p.e. observed to reject events due to noise. LE events are not included
in the analysis presented here.

Events at the far detector are timed with respect to the leading edge of the
beam spill, taking into account the neutrino time of flight, the Cherenkov
photon propagation time and delays in the DAQ electronics. Fig. 16 shows the
event timing ($\Delta T_{0}$) distribution for all OD, LE, and FC events
within $\pm 500\,\mu{\rm s}$ of the beam arrival time. For FC events a visible
energy ($E_{\mathrm{vis}}$) greater than 30 MeV is also requested. A clear
peak is observed around $\Delta T_{0}=0$ in the FC sample. For an event to be
incorporated into the analysis, $\Delta T_{0}$ must lie in the interval
[-2,10] $\,\mu{\rm s}$. A total of 4 events have been observed outside this
range. The expected number of such events is 4.17, estimated by using data
taken with no beam. FC events within the spill window can be seen in Fig. 17
where the beam structure with eight bunches is clearly visible. The dotted
lines represent the fitted bunch center times with a fixed bunch interval of
581${\rm\,ns}$.

Figure 16: $\Delta T_{0}$ distribution of all FC, OD and LE events within
$\pm$500$\,\mu{\rm s}$ of the expected beam arrival time observed during T2K
Run 1-7. The histograms are stacked in that order. Figure 17: $\Delta T_{0}$
distribution all FC events observed during T2K Run 1-7 zoomed in on the
expected beam arrival time.

A fiducial volume is defined within the ID, 2 m away from the detector wall,
with a fiducial mass of 22.5 kt. Events whose vertex is reconstructed within
this volume are selected into the fully contained fiducial volume sample
(FCFV). Visible energy is defined as the energy of an electromagnetic shower
that produces the observed amount of Cherenkov light. In total, 608 events are
classified as FCFV. The expected number of background events from non-beam
related sources in accidental coincidence is estimated to be 0.0145.

###  VI.1 SK charged-current quasi-elastic selection

Charged-current interactions $(\;\nu^{\hbox{\kern-8.0pt\raise
1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}+N\rightarrow
l^{\pm}+X)$ in the narrow energy range of the T2K beam most commonly produce
single-ring events at SK because most of the resulting particles, except for
the primary lepton, do not escape the nucleus, or are below detection
threshold. The energy of the incoming neutrino can be calculated assuming the
kinematics of a CCQE interaction and neglecting Fermi motion:

$E_{\nu}^{rec}=\frac{m^{2}_{f}-(m_{i}^{\prime})^{2}-m^{2}_{l}+2m_{i}^{\prime}E_{l}}{2(m_{i}^{\prime}-E_{l}+p_{l}\cos\theta_{l})}$
(4)

where $E_{\nu}^{rec}$ is the reconstructed neutrino energy, $m_{i}$ and
$m_{f}$ are the initial and final nucleon masses respectively, and
$m_{i}^{\prime}=m_{i}-E_{\mathrm{b}}$, where
$E_{\mathrm{b}}=27$$\mathrm{\,Me\kern-1.00006ptV}$ is the binding energy of a
nucleon inside ${}^{16}\mathrm{O}$ nuclei. $E_{l}$, $p_{l}$ and $\theta_{l}$
are the reconstructed lepton energy, momentum, and angle with respect to the
beam, respectively. The selection criteria for both
$\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ CC and $\nu^{\hbox{\kern-8.0pt\raise
1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ CC events were
fixed using MC studies before being applied to data. Events are determined to
be $e$-like or $\mu$-like based on the PID of the brightest Cherenkov ring.
The PID of each ring is determined by a likelihood incorporating information
on the charge distribution and the opening angle of the Cherenkov cone. The
PID likelihood distribution for $\nu$-mode FCFV single-ring events is shown in
Fig. 18. The same criteria are applied to events observed for both $\nu$\- and
$\overline{\nu}$-mode data taking.

Figure 18: The PID likelihood distribution of the observed $\nu$-mode CC event samples after FCFV and single-ring cuts have been applied. The data are shown as points with statistical error bars and the shaded, stacked histograms are the MC predictions. The expectation is based on the parameters of Tab. 13. Table 13: Values of the oscillation parameters used for the Monte Carlo simulation at SK. The values of $\sin^{2}\theta_{12}$, $\Delta m^{2}_{21}$ and $\sin^{2}\theta_{13}$ are taken from Ref. PDG2015 , while all the other oscillation parameters correspond to the most probable values obtained by the Bayesian analysis in Ref. Abe:2015awa . Parameter | Value
---|---
$\sin^{2}2\theta_{12}$ | 0.846
$\Delta m^{2}_{21}$ | $7.53\times 10^{-5}~{}\text{eV}^{2}/\text{c}^{4}$
$\sin^{2}\theta_{23}$ | 0.528
$\Delta m^{2}_{32}$ | $2.509\times 10^{-3}\,\rm{eV}^{2}/\rm{c}^{4}$
$\sin^{2}2\theta_{13}$ | 0.085
$\delta_{\mathrm{CP}}$ | -1.601
Mass ordering | normal

$\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ CC candidate events are selected using the
criteria listed in Tab. 14. The $E_{\mathrm{vis}}$ requirement removes low
energy NC interactions and electrons from the decay of unseen parents that are
below Cherenkov threshold or fall outside the beam time window. The
$\pi^{0}$-like event rejection uses an independent reconstruction algorithm
which was introduced in previous analyses Abe:2015awa . The cut
$E_{\nu}^{rec}<1.25\mathrm{\,Ge\kern-1.00006ptV}$ is required, as above this
energy the intrinsic beam $\nu^{\hbox{\kern-8.0pt\raise
1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ background is
dominant. The numbers of events remaining in the neutrino and antineutrino
beam data after successive selection criteria for a simulation sample produced
with the oscillation parameters of Tab. 13 are shown in Tab. 14.

After all cuts, 32 events remain in the $\nu_{e}$ CC candidate sample and 4 in
the $\overline{\nu}_{e}$ CC candidate sample, as shown in Fig. 19. Kolmogorov-
Smirnov (KS) tests of the accumulated events as a function of POT are
compatible with a constant rate, with p-values of 0.99 and 0.25 respectively.

The vertex distributions of the candidate event samples are checked for signs
of bias that might suggest background contamination. Fig. 20 shows the vertex
distribution of the $\nu_{e}$ CC candidate events in the SK tank coordinate
system. Combined KS tests for uniformity in $r^{2}$ and $z$ yield p-values of
0.1 and 0.6 for the $\nu_{e}$ and $\overline{\nu}_{e}$ samples respectively.

Figure 19: The reconstructed energy spectra of the observed $\nu_{e}$ in
$\nu$-mode (left) and $\overline{\nu}_{e}$ in $\overline{\nu}$-mode (right) CC
candidate event samples assuming CCQE interaction kinematics. The data are
shown as points with statistical error bars and the shaded, stacked histograms
are the MC predictions. The expectation is based on the parameters of Tab. 13.

Figure 20: Two-dimensional vertex distributions of the observed $\nu_{e}$ CC candidate events in $(X,Y)$ and $(R^{2},Z)$ for $\nu$-mode (top) and $\overline{\nu}$-mode (bottom). The arrow indicates the neutrino beam direction and the dashed line indicates the fiducial volume boundary. Events indicated by open square markers passed all of the $\nu_{e}$ selection cuts except for the fiducial volume cut. Table 14:  Event reduction for the $\nu_{e}$ CC selection at the far detector. The numbers of expected MC events divided into five categories are shown after each selection criterion is applied. The MC expectation is based upon three-neutrino oscillations with the parameters as shown in Tab. 13.  |  | $\nu_{\mu}+\overline{\nu}_{\mu}$ | $\nu_{e}+\overline{\nu}_{e}$ | $\nu+\bar{\nu}$ | $\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}$ | $\nu_{\mu}\rightarrow\nu_{e}$ |
---|---|---|---|---|---|---|---
$\nu$-beam mode | MC total | CC | CC | NC | CC | CC | Data
interactions in FV | 744.89 | 364.32 | 18.55 | 326.16 | 0.39 | 35.47 | -
FCFV | 431.85 | 279.88 | 18.09 | 98.72 | 0.38 | 34.78 | 438
single ring111There is only one reconstructed Cherenkov ring | 223.49 | 153.40 | 11.15 | 28.68 | 0.32 | 29.95 | 220
electron-like222The ring is $e$-like | 66.94 | 6.46 | 11.06 | 19.53 | 0.31 | 29.57 | 70
$E_{\rm vis}>100{\rm\mathrm{\,Me\kern-1.00006ptV}}$333The visible energy, $E_{\mathrm{vis}}$, is greater than 100 $\mathrm{\,Me\kern-1.00006ptV}$ | 61.78 | 4.59 | 11.01 | 16.81 | 0.31 | 29.06 | 66
$N_{\rm Michel-e}=0$444There is no reconstructed Michel electron | 50.60 | 0.97 | 8.97 | 14.24 | 0.31 | 26.11 | 51
$E_{\nu}^{\rm rec}<1250{\rm\mathrm{\,Me\kern-1.00006ptV}}$555The reconstructed energy, $E_{\nu}^{rec}$, is less than 1.25 $\mathrm{\,Ge\kern-1.00006ptV}$ | 40.71 | 0.25 | 4.26 | 10.85 | 0.22 | 25.14 | 46
not $\pi^{0}$-like666The event is not consistent with a $\pi^{0}$ hypothesis | 28.55 | 0.09 | 3.68 | 1.35 | 0.18 | 23.25 | 32
$\bar{\nu}$-beam mode |  |  |  |  |  |  |
interactions in FV | 312.38 | 164.04 | 9.00 | 132.75 | 4.30 | 2.29 | -
FCFV | 180.48 | 123.24 | 8.75 | 42.05 | 4.20 | 2.24 | 170
single ring | 96.06 | 73.21 | 5.51 | 11.87 | 3.74 | 1.73 | 94
electron-like | 21.55 | 2.31 | 5.48 | 8.36 | 3.70 | 1.71 | 16
$E_{\rm vis}>100{\rm\mathrm{\,Me\kern-1.00006ptV}}$ | 20.05 | 1.83 | 5.46 | 7.39 | 3.68 | 1.69 | 14
$N_{\rm Michel-e}=0$ | 16.40 | 0.33 | 4.71 | 6.24 | 3.66 | 1.46 | 12
$E_{\nu}^{rec}<1250{\rm\mathrm{\,Me\kern-1.00006ptV}}$ | 11.40 | 0.08 | 1.89 | 4.83 | 3.42 | 1.19 | 9
not $\pi^{0}$-like | 6.28 | 0.02 | 1.58 | 0.60 | 3.04 | 1.05 | 4
Table 15:  Event reduction for the $\nu_{\mu}$ CC selection at the far detector. The numbers of expected MC events divided into four categories are shown after each selection criterion is applied. The MC expectation is based upon three-neutrino oscillations with the parameters as shown in Tab. 13.  |  | $\nu_{\mu}$ | $\overline{\nu}_{\mu}$ | $\nu_{\mu}+\overline{\nu}_{\mu}$ | $\nu_{e}+\overline{\nu}_{e}$ | $\nu+\bar{\nu}$ |
---|---|---|---|---|---|---|---
$\nu$-beam mode | MC total | CCQE | CCQE | CC nonQE | CC | NC | Data
interactions in FV | 744.89 | 100.17 | 6.45 | 257.70 | 54.41 | 326.16 | -
FCFV | 431.85 | 78.75 | 4.85 | 196.28 | 53.25 | 98.72 | 438
single ring777There is only one reconstructed Cherenkov ring | 223.49 | 73.49 | 4.70 | 75.21 | 41.41 | 28.68 | 220
muon-like888The ring is $\mu$-like | 156.56 | 72.22 | 4.65 | 70.06 | 0.47 | 9.16 | 150
$p_{\mu}>200{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$999The reconstructed momentum, $p_{\mu}$, is greater than 200 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ | 156.24 | 72.03 | 4.65 | 70.00 | 0.47 | 9.08 | 150
$N_{\rm Michel-e}\leq 1$101010There are less than two reconstructed Michel electrons | 137.76 | 71.28 | 4.63 | 52.61 | 0.46 | 8.78 | 135
$\bar{\nu}$-beam mode |  |  |  |  |  |  |
interactions in FV | 312.38 | 20.04 | 30.77 | 113.23 | 15.59 | 132.75 | -
FCFV | 180.48 | 15.04 | 24.95 | 83.26 | 15.19 | 42.05 | 170
single ring | 96.06 | 13.52 | 24.28 | 35.41 | 10.98 | 11.87 | 94
muon-like | 74.52 | 13.40 | 23.96 | 33.56 | 0.09 | 3.52 | 78
$p_{\mu}>200{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ | 74.42 | 13.39 | 23.92 | 33.54 | 0.09 | 3.48 | 78
$N_{\rm Michel-e}\leq 1$ | 68.26 | 13.18 | 23.85 | 27.79 | 0.09 | 3.35 | 66

$\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ CC candidate events are selected using
the criteria shown in Tab. 15. The momentum cut rejects charged pions and
misidentified electrons from the decay of unobserved muons and pions. Fewer
than two Michel electrons are required to reject events with additional unseen
muons or pions. After all cuts are applied, 135 events remain in the
$\nu_{\mu}$ CC candidate sample and 66 in the $\overline{\nu}_{\mu}$ CC
candidate sample as shown in Fig. 21.

Figure 21: The reconstructed energy spectra of the observed $\nu_{\mu}$ in
$\nu$-mode (left) and $\overline{\nu}_{\mu}$ in $\overline{\nu}$-mode (right)
CC candidate event samples assuming CCQE interaction kinematics. The data are
shown as points with statistical error bars and the shaded, stacked histograms
are the MC predictions, and the rightmost bin includes overflow. The
expectation is based on the parameters of Tab. 13.

###  VI.2 SK charged-current single pion selection

A new far detector event sample has been included in the oscillation analysis
described here. As mentioned previously, single-ring events produced by quasi-
elastic interactions are the most common at T2K neutrino energies. By
modifying the event selection criteria to include an additional Michel
electron candidate, it is possible to select events with a pion produced below
Cherenkov threshold. Michel electrons are tagged at SK by searching for
secondary hit clusters within the same trigger window and in subsequent
trigger windows that pass time, charge and vertex goodness criteria. Note that
this selection tags only positive $\pi^{+}$’s as $\rm\pi^{-}$’s are absorbed
by nuclei before they decay when stopped in water. Fig. 22 shows the true
Michel electron momentum distribution for true CC1$\pi^{+}$ simulated events
reconstructed inside the fiducial volume. Fig. 23 shows the true pion momentum
distribution for selected signal events and the selection efficiency. The
Cherenkov threshold for charged pions is also shown. The efficiency falls
above this threshold as the pion produces more light and the event-fitting
algorithm is increasingly likely to find a second ring, thus disqualifying
events from the sample. The event reduction for the full selection is shown in
Tab. 16. There is a larger background from misidentified muons in this sample
compared with the single-ring selection as such events are more likely to
contain a Michel electron candidate. It can be seen in Tab. 16 that there is
an apparent difference between the data and the MC expectation after cut
number five. Applying an equivalent cut sequence to atmospheric neutrino data
and MC (in this case the neutrino direction is not known so a cut on $E_{vis}$
is used instead of $E_{\nu}^{rec}$) yielded no such discrepancy. This single-
ring CC1$\pi^{+}$ selection has been implemented as a new sample for $\nu_{e}$
appearance with neutrino beam data. The reconstructed energy equation is
modified from the CCQE case by recognizing that the outgoing baryon is a
$\Delta^{++}$ instead of a proton and neglecting nuclear effects:

$E_{\nu}^{rec}=\frac{m^{2}_{\Delta^{++}}-m^{2}_{p}-m^{2}_{l}+2m_{p}E_{l}}{2(m_{p}-E_{l}+p_{l}\cos\theta_{l})}$
(5)

where $m_{\Delta^{++}}$ is the mass of the $\Delta^{++}$ (1232.0 MeV/c2). Fig.
24 shows the difference in the true and reconstructed neutrino energy for the
final CC1$\pi^{+}$ candidate selection along with that for the single-ring
selection for comparison. Fig. 25 shows the reconstructed energy distribution
for the final sample. Five $\nu_{e}$ CC1$\pi^{+}$ candidates are reconstructed
in the data while 3.1 events are expected for the oscillation parameters of
Tab. 13.

Figure 22: The true momentum distribution for tagged Michel electrons from
true CC1$\pi^{+}$ simulated events reconstructed in the fiducial volume (left)
and the tagging efficiency for these events (right). The expectation is based
on the parameters of Tab. 13.

Figure 23: The true momentum distribution for selected simulated signal events in the CC1$\pi^{+}$ candidate sample (left) and the selection efficiency for these events (right). The expectation is based on the parameters of Tab. 13. The red dashed line indicates the Cherenkov threshold for charged pions. Table 16:  Event reduction for the $\nu_{e}$ CC1$\pi^{+}$ selection at the far detector. The numbers of expected MC events divided into five categories are shown after each selection criterion is applied. The MC expectation is based on the parameters of Tab. 13.  |  | $\nu_{\mu}+\overline{\nu}_{\mu}$ | $\nu_{e}+\overline{\nu}_{e}$ | $\nu+\bar{\nu}$ | $\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}$ | $\nu_{\mu}\rightarrow\nu_{e}$ |
---|---|---|---|---|---|---|---
$\nu$-beam mode | MC total | CC | CC | NC | CC | CC | Data
interactions in FV | 744.89 | 364.32 | 18.55 | 326.16 | 0.39 | 35.47 | -
FCFV | 431.85 | 279.88 | 18.09 | 98.72 | 0.38 | 34.78 | 438
(1) single ring111There is only one reconstructed Cherenkov ring | 223.49 | 153.40 | 11.15 | 28.68 | 0.32 | 29.95 | 220
(2) electron-like222The ring is $e$-like | 66.94 | 6.46 | 11.06 | 19.53 | 0.31 | 29.57 | 70
(3) $E_{\rm vis}>100{\rm\mathrm{\,Me\kern-1.00006ptV}}$333The visible energy, $E_{\mathrm{vis}}$, is greater than 100 $\mathrm{\,Me\kern-1.00006ptV}$ | 61.78 | 4.59 | 11.01 | 16.81 | 0.31 | 29.06 | 66
(4) $N_{\rm Michel-e}=1$444There is one reconstructed Michel electron | 9.36 | 2.42 | 1.87 | 2.14 | 0.01 | 2.92 | 14
(5) $E_{\nu}^{rec}<1250{\rm\mathrm{\,Me\kern-1.00006ptV}}$555The reconstructed energy, $E_{\nu}^{rec}$, is less than 1.25 $\mathrm{\,Ge\kern-1.00006ptV}$ | 4.66 | 0.70 | 0.50 | 0.78 | $<0.01$ | 2.66 | 11
(6) not $\pi^{0}$-like666The event is not consistent with a $\pi^{0}$ hypothesis | 3.14 | 0.29 | 0.39 | 0.15 | $<0.01$ | 2.31 | 5

Figure 24: Difference between true and reconstructed energy of the $\nu_{e}$
CCQE-like (left) and CC1$\pi^{+}$ (right) simulated samples. The energy for
the CCQE-like sample is reconstructed using Eq. 4. For the CC1$\pi^{+}$ sample
the modified interaction kinematics in Eq. 5 are used. The expectation is
based on the parameters of Tab. 13. Figure 25: The reconstructed energy
spectrum of the observed $\nu_{e}$ CC1$\pi^{+}$ candidate events assuming the
modified interaction kinematics in equation 5. The data are shown as points
with statistical error bars and the shaded, stacked histograms are the MC
predictions. The expectation is based on the parameters of Tab. 13.

Figure 26: Two-dimensional vertex distributions of the observed $\nu_{e}$
CC1$\pi^{+}$ candidate events in $(X,Y)$ and $(R^{2},Z)$. The arrow indicates
the neutrino beam direction and the dashed line indicates the fiducial volume
boundary. Events indicated by open square markers passed all of the $\nu_{e}$
CC1$\pi^{+}$ selection cuts except for the fiducial volume cut.

Fig. 26 shows the vertex distribution of the $\nu_{e}$ CC1$\pi^{+}$ candidate
events in the SK tank coordinate system.

###  VI.3 SK detector systematic uncertainties

This section discusses the estimation of the uncertainty in the selection
efficiency and background for the oscillation samples that result from
modeling of the SK detector. This topic has been covered in detail in previous
publications Abe:2015awa  but there have been a number of updates,
particularly related to the addition of the $\nu_{e}$ CC1$\pi^{+}$ sample.

Control samples unrelated to the T2K beam are used to assess the
uncertainties. Cosmic-ray muon samples are used to estimate uncertainties
related to the FC, fiducial-volume and decay-electron requirements, for the
selections of both $\nu^{\hbox{\kern-8.0pt\raise 1.0pt\hbox{{\tiny(}{\lower
1.4pt\hbox{\bf--}}{\tiny)}}}}_{e}$ and $\nu^{\hbox{\kern-8.0pt\raise
1.0pt\hbox{{\tiny(}{\lower 1.4pt\hbox{\bf--}}{\tiny)}}}}_{\mu}$ CC candidates.
The error from the initial FC event selection is negligible. The uncertainty
in the fiducial volume is estimated to be 1% using the vertex distribution of
cosmic ray muons which have been independently determined to have stopped
inside the ID. The uncertainty due to the Michel electron tagging efficiency
is estimated by comparing cosmic-ray stopped muon data with MC. The rate of
falsely identified Michel electrons is estimated from MC and a 100%
uncertainty on that rate is assumed.

Table 17: Criteria for categorization of simulated events by final state topology for systematic studies. $N_{x}$ is the number of particles of type $x$ and the number of charged pions ($N_{\pi^{\pm}}$) and protons ($N_{p}$) only includes those particles produced with momentum above Cherenkov threshold set at 156.0 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and 1051.0 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ respectively. Event type |  MC truth selection criteria
---|---
CC 1$e$ |  $\nu_{e}$ CC and $N_{\pi^{0}}=0$ and $N_{\pi^{\pm}}=0$ and $N_{P}=0$
CC $e$ other |  $\nu_{e}$ CC and not $\nu_{e}$ CC1$e$
CC 1$\mu$ |  $\nu_{\mu}$ CC and $N_{\pi^{0}}=0$ and $N_{\pi^{\pm}}=0$ and $N_{P}=0$
CC $\mu$ other |  $\nu_{\mu}$ CC and $N_{\pi^{0}}=0$
CC $\mu$ $\pi^{0}$ other |  $\nu_{\mu}$ CC and $N_{\pi^{0}}>0$
NC 1$\pi^{0}$ |  NC and not NC 1$\gamma$ and $N_{\pi^{0}}=1$ and $N_{\pi^{\pm}}=0$ and $N_{P}=0$
NC $\pi^{0}$ other |  NC and not NC 1$\gamma$ and $N_{\pi^{0}}\geq 1$ and not NC 1$\pi^{0}$
NC 1$\gamma$ |  NC 1$\gamma$ in the MC generator
NC 1$\pi^{\pm}$ |  NC and not NC 1$\gamma$ and $N_{\pi^{0}}=0$ and $N_{\pi^{\pm}}=1$ and $N_{P}=0$
NC other |  NC and not NC 1$\gamma$ and not NC 1$\pi^{0}$ and not NC 1$\pi^{\pm}$ and not NC $\pi^{0}$ other

Other studies of systematic uncertainty in SK modeling divide simulated events
into categories according to their final state topologies, with the criteria
shown in Tab. 17. These topologies do not correspond exactly with true
interaction modes due to subsequent interactions within the nucleus or with
neighboring nuclei or because one or more particles are produced below
Cherenkov threshold. The dominant uncertainties are described in the following
paragraphs. Atmospheric neutrino data are used to assess possible mismodeling
of the ring counting (RC), particle identification (PID), and $\pi^{0}$
rejection for the first four topologies shown in Tab. 17. In addition to flux
and cross-section uncertainties from the atmospheric neutrino analysis,
additional parameters are included in the fit to alter the cut values applied
to the MC for the three classifiers above, thereby allowing for possible
mismodeling of the events. Separate parameters are used for each of the four
final state topologies and for different ranges of visible energy. A
likelihood is defined comparing the data and MC which is then marginalized
over flux and cross section parameters using a Markov chain Monte Carlo to
estimate corrected efficiencies for the four final state topologies in bins of
$E_{vis}$ and their covariance. The measured shifts between the nominal and
fitted efficiencies are included in the final uncertainty assuming full
correlation between samples.

To estimate the uncertainty in modeling $\pi^{0}$’s we construct a set of
hybrid data-MC control samples. Events in these samples are created by
overlaying a single electron-like ring from the SK data with a simulated
photon ring. The kinematics of the photon ring are chosen such that when
combined, the two rings follow the decay kinematics of $\pi^{0}$ events from
the T2K MC. Hybrid samples with both rings from the SK MC are also produced
for comparison with the hybrid data. The difference in the selection
efficiency when using the oscillation analysis sample candidate criteria is
used to determine the systematic error. This procedure is performed twice,
with the data electron as the higher energy and lower energy ring, and the
uncertainties are combined.

In previous analyses, the CC1$\mu$ events in the $\nu_{e}$ appearance sample
that did not involve muon decay-in-flight (81% of CC1$\mu$) were assigned a
conservative 150% error due to the ring-counting, PID and $\pi^{0}$ rejection
cuts. However while these events form only $\sim$1% of the $\nu_{e}$ CCQE-like
candidate sample, they are $\sim$10% of the $\nu_{e}$ CC1$\pi^{+}$ candidate
sample and thus the uncertainty was re-evaluated with a dedicated analysis
using stopped cosmic-ray muons and atmospheric neutrinos. Based upon this
work, this uncertainty has been reduced to 63.2%, where the error on the
particle identification is the dominant source of uncertainty. For CC1$\mu$
events involving a muon decay-in-flight, a 16% uncertainty is assigned,
unchanged with respect to the previous analysis.

All aspects of the SK detector simulation that can affect the modeling of the
candidate event selection described above are propagated using a vector of
systematic uncertainty parameters, $\vec{s}$, which scale the nominal expected
number of events in bins of the observable kinematic variables $E_{\nu}^{rec}$
or $p_{l}$ for the true neutrino interaction mode categories.

##  VII Neutrino oscillation framework

The previous sections have described how the near detector samples are used to