Improved constraints on neutrino mixing from the T2K experiment with \mathbf{3.13\times10^{21}} protons on target
T2K Collaboration, K. Abe, N. Akhlaq, R. Akutsu, A. Ali, C. Alt, C. Andreopoulos, M. Antonova, S. Aoki, T. Arihara, Y. Asada, Y. Ashida, E.T. Atkin, Y. Awataguchi, G.J. Barker, G. Barr, D. Barrow, M. Batkiewicz-Kwasniak, A. Beloshapkin, F. Bench, V. Berardi, L. Berns, S. Bhadra, A. Blanchet, A. Blondel, S. Bolognesi, T. Bonus, B. Bourguille, S.B. Boyd, A. Bravar, D. Bravo Bergu, C. Bronner, S. Bron, A. Bubak, M. Buizza Avanzini, S. Cao, S.L. Cartwright, M.G. Catanesi, A. Cervera, D. Cherdack, G. Christodoulou, M. Cicerchia, J. Coleman, G. Collazuol, L. Cook, D. Coplowe, A. Cudd, G. De Rosa, T. Dealtry, C.C. Delogu, S.R. Dennis, C. Densham, A. Dergacheva, F. Di Lodovico, S. Dolan, D. Douqa, T.A. Doyle, J. Dumarchez, P. Dunne, A. Eguchi, L. Eklund, S. Emery-Schrenk, A. Ereditato, A.J. Finch, G. Fiorillo, C. Francois, M. Friend, Y. Fujii, R. Fukuda, Y. Fukuda, K. Fusshoeller, C. Giganti, M. Gonin, A. Gorin, M. Grassi, M. Guigue, D.R. Hadley, P. Hamacher-Baumann, D.A. Harris, M. Hartz, T. Hasegawa, S. Hassani, N.C. Hastings, Y. Hayato, A. Hiramoto, M. Hogan, J. Holeczek, N.T. Hong Van, T. Honjo, F. Iacob, A.K. Ichikawa, M. Ikeda, T. Ishida, M. Ishitsuka, K. Iwamoto, A. Izmaylov, N. Izumi, M. Jakkapu, B. Jamieson, S.J. Jenkins, et al. (210 additional authors not shown)
IImproved constraints on neutrino mixing from the T2K experiment with 3 . × protons on target K. Abe, N. Akhlaq, R. Akutsu, A. Ali, C. Alt, C. Andreopoulos,
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M. Antonova, S. Aoki, T. Arihara, Y. Asada, Y. Ashida, E.T. Atkin, Y. Awataguchi, G.J. Barker, G. Barr, D. Barrow, M. Batkiewicz-Kwasniak, A. Beloshapkin, F. Bench, V. Berardi, L. Berns, S. Bhadra, A. Blanchet, A. Blondel,
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S. Bolognesi, T. Bonus, B. Bourguille, S.B. Boyd, A. Bravar, D. Bravo Bergu˜no, C. Bronner, S. Bron, A. Bubak, M. Buizza Avanzini, S. Cao, S.L. Cartwright, M.G. Catanesi, A. Cervera, D. Cherdack, G. Christodoulou, M. Cicerchia, ∗ J. Coleman, G. Collazuol, L. Cook,
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D. Coplowe, A. Cudd, G. De Rosa, T. Dealtry, C.C. Delogu, S.R. Dennis, C. Densham, A. Dergacheva, F. Di Lodovico, S. Dolan, D. Douqa, T.A. Doyle, J. Dumarchez, P. Dunne, A. Eguchi, L. Eklund, S. Emery-Schrenk, A. Ereditato, A.J. Finch, G. Fiorillo, C. Francois, M. Friend, † Y. Fujii, † R. Fukuda, Y. Fukuda, K. Fusshoeller, C. Giganti, M. Gonin, A. Gorin, M. Grassi, M. Guigue, D.R. Hadley, P. Hamacher-Baumann, D.A. Harris, M. Hartz,
59, 27
T. Hasegawa, † S. Hassani, N.C. Hastings, Y. Hayato,
53, 27
A. Hiramoto, M. Hogan, J. Holeczek, N.T. Hong Van,
19, 26
T. Honjo, F. Iacob, A.K. Ichikawa, M. Ikeda, T. Ishida, † M. Ishitsuka, K. Iwamoto, A. Izmaylov, N. Izumi, M. Jakkapu, B. Jamieson, S.J. Jenkins, C. Jes´us-Valls, P. Jonsson, C.K. Jung, ‡ P.B. Jurj, M. Kabirnezhad, H. Kakuno, J. Kameda, S.P. Kasetti, Y. Kataoka, Y. Katayama, T. Katori, E. Kearns,
3, 27, ‡ M. Khabibullin, A. Khotjantsev, T. Kikawa, H. Kikutani, S. King, J. Kisiel, T. Kobata, T. Kobayashi, † L. Koch, A. Konaka, L.L. Kormos, Y. Koshio, ‡ A. Kostin, K. Kowalik, Y. Kudenko, § S. Kuribayashi, R. Kurjata, T. Kutter, M. Kuze, L. Labarga, J. Lagoda, M. Lamoureux, D. Last, M. Laveder, M. Lawe, R.P. Litchfield, S.L. Liu, A. Longhin, L. Ludovici, X. Lu, T. Lux, L.N. Machado, L. Magaletti, K. Mahn, M. Malek, S. Manly, L. Maret, A.D. Marino, L. Marti-Magro,
53, 27
T. Maruyama, † T. Matsubara, K. Matsushita, C. Mauger, K. Mavrokoridis, E. Mazzucato, N. McCauley, J. McElwee, K.S. McFarland, C. McGrew, A. Mefodiev, M. Mezzetto, A. Minamino, O. Mineev, S. Mine, M. Miura, ‡ L. Molina Bueno, S. Moriyama, ‡ Th.A. Mueller, L. Munteanu, Y. Nagai, T. Nakadaira, † M. Nakahata,
53, 27
Y. Nakajima, A. Nakamura, K. Nakamura,
27, 15, † Y. Nakano, S. Nakayama,
53, 27
T. Nakaya,
31, 27
K. Nakayoshi, † C.E.R. Naseby, T.V. Ngoc, ¶ V.Q. Nguyen, K. Niewczas, Y. Nishimura, E. Noah, T.S. Nonnenmacher, F. Nova, J. Nowak, J.C. Nugent, H.M. O’Keeffe, L. O’Sullivan, T. Odagawa, T. Ogawa, R. Okada, K. Okumura,
54, 27
T. Okusawa, R.A. Owen, Y. Oyama, † V. Palladino, V. Paolone, M. Pari, W.C. Parker, S. Parsa, J. Pasternak, M. Pavin, D. Payne, G.C. Penn, L. Pickering, C. Pidcott, G. Pintaudi, C. Pistillo, B. Popov, ∗∗ K. Porwit, M. Posiadala-Zezula, A. Pritchard, B. Quilain, T. Radermacher, E. Radicioni, B. Radics, P.N. Ratoff, C. Riccio, E. Rondio, S. Roth, A. Rubbia, A.C. Ruggeri, C. Ruggles, A. Rychter, K. Sakashita, † F. S´anchez, G. Santucci, C.M. Schloesser, K. Scholberg, ‡ M. Scott, Y. Seiya, †† T. Sekiguchi, † H. Sekiya,
53, 27, ‡ D. Sgalaberna, A. Shaikhiev, A. Shaykina, M. Shiozawa,
53, 27
W. Shorrock, A. Shvartsman, K. Skwarczynski, M. Smy, J.T. Sobczyk, H. Sobel,
4, 27
F.J.P. Soler, Y. Sonoda, R. Spina, S. Suvorov,
25, 50
A. Suzuki, S.Y. Suzuki, † Y. Suzuki, A.A. Sztuc, M. Tada, † M. Tajima, A. Takeda, Y. Takeuchi,
30, 27
H.K. Tanaka, ‡ Y. Tanihara, M. Tani, N. Teshima, L.F. Thompson, W. Toki, C. Touramanis, T. Towstego, K.M. Tsui, T. Tsukamoto, † M. Tzanov, Y. Uchida, M. Vagins,
27, 4
S. Valder, D. Vargas, G. Vasseur, C. Vilela, W.G.S. Vinning, T. Vladisavljevic, T. Wachala, J. Walker, J.G. Walsh, Y. Wang, D. Wark,
51, 41
M.O. Wascko, A. Weber,
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R. Wendell, ‡ M.J. Wilking, C. Wilkinson, J.R. Wilson, K. Wood, C. Wret, J. Xia, K. Yamamoto, †† C. Yanagisawa, ‡‡ G. Yang, T. Yano, K. Yasutome, N. Yershov, M. Yokoyama, ‡ T. Yoshida, Y. Yoshimoto, M. Yu, A. Zalewska, J. Zalipska, K. Zaremba, G. Zarnecki, M. Ziembicki, M. Zito, and S. Zsoldos (The T2K Collaboration) University Autonoma Madrid, Department of Theoretical Physics, 28049 Madrid, Spain University of Bern, Albert Einstein Center for Fundamental Physics,Laboratory for High Energy Physics (LHEP), Bern, Switzerland Boston University, Department of Physics, Boston, Massachusetts, U.S.A. University of California, Irvine, Department of Physics and Astronomy, Irvine, California, U.S.A. IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A. Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. a r X i v : . [ h e p - e x ] F e b Duke University, Department of Physics, Durham, North Carolina, U.S.A. Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France ETH Zurich, Institute for Particle Physics and Astrophysics, Zurich, Switzerland CERN European Organization for Nuclear Research, CH-1211 Gen`eve 23, Switzerland University of Geneva, Section de Physique, DPNC, Geneva, Switzerland University of Glasgow, School of Physics and Astronomy, Glasgow, United Kingdom H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan University of Houston, Department of Physics, Houston, Texas, U.S.A. Institut de Fisica d’Altes Energies (IFAE) - The Barcelona Instituteof Science and Technology, Campus UAB, Bellaterra (Barcelona) Spain IFIC (CSIC & University of Valencia), Valencia, Spain Institute For Interdisciplinary Research in Science and Education (IFIRSE), ICISE, Quy Nhon, Vietnam Imperial College London, Department of Physics, London, United Kingdom INFN Sezione di Bari and Universit`a e Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy INFN Sezione di Napoli and Universit`a di Napoli, Dipartimento di Fisica, Napoli, Italy INFN Sezione di Padova and Universit`a di Padova, Dipartimento di Fisica, Padova, Italy INFN Sezione di Roma and Universit`a di Roma “La Sapienza”, Roma, Italy Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia International Centre of Physics, Institute of Physics (IOP), Vietnam Academyof Science and Technology (VAST), 10 Dao Tan, Ba Dinh, Hanoi, Vietnam Kavli Institute for the Physics and Mathematics of the Universe (WPI), The Universityof Tokyo Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan Keio University, Department of Physics, Kanagawa, Japan King’s College London, Department of Physics, Strand, London WC2R 2LS, United Kingdom Kobe University, Kobe, Japan Kyoto University, Department of Physics, Kyoto, Japan Lancaster University, Physics Department, Lancaster, United Kingdom University of Liverpool, Department of Physics, Liverpool, United Kingdom Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A. Michigan State University, Department of Physics and Astronomy, East Lansing, Michigan, U.S.A. Miyagi University of Education, Department of Physics, Sendai, Japan National Centre for Nuclear Research, Warsaw, Poland State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. Okayama University, Department of Physics, Okayama, Japan Osaka City University, Department of Physics, Osaka, Japan Oxford University, Department of Physics, Oxford, United Kingdom University of Pennsylvania, Department of Physics and Astronomy, Philadelphia, PA, 19104, USA. University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A. Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom University of Rochester, Department of Physics and Astronomy, Rochester, New York, U.S.A. Royal Holloway University of London, Department of Physics, Egham, Surrey, United Kingdom RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom University of Silesia, Institute of Physics, Katowice, Poland Sorbonne Universit´e, Universit´e Paris Diderot, CNRS/IN2P3, Laboratoirede Physique Nucl´eaire et de Hautes Energies (LPNHE), Paris, France STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom University of Tokyo, Department of Physics, Tokyo, Japan University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan Tokyo Institute of Technology, Department of Physics, Tokyo, Japan Tokyo Metropolitan University, Department of Physics, Tokyo, Japan Tokyo University of Science, Faculty of Science and Technology, Department of Physics, Noda, Chiba, Japan University of Toronto, Department of Physics, Toronto, Ontario, Canada TRIUMF, Vancouver, British Columbia, Canada University of Warsaw, Faculty of Physics, Warsaw, Poland Warsaw University of Technology, Institute of Radioelectronics and Multimedia Technology, Warsaw, Poland University of Warwick, Department of Physics, Coventry, United Kingdom University of Winnipeg, Department of Physics, Winnipeg, Manitoba, Canada Wroclaw University, Faculty of Physics and Astronomy, Wroclaw, Poland Yokohama National University, Department of Physics, Yokohama, Japan York University, Department of Physics and Astronomy, Toronto, Ontario, Canada (Dated: February 25, 2021)
The T2K experiment reports updated measurements of neutrino and antineutrino oscillationsusing both appearance and disappearance channels. This result comes from an exposure of14 . . × protons on target in neutrino (antineutrino) mode. Significant improvementshave been made to the neutrino interaction model and far detector reconstruction. An extensiveset of simulated data studies have also been performed to quantify the effect interaction modeluncertainties have on the T2K oscillation parameter sensitivity. T2K performs multiple oscillationanalyses that present both frequentist and Bayesian intervals for the PMNS parameters. For fitsincluding a constraint on sin θ from reactor data and assuming normal mass ordering T2K mea-sures sin θ = 0 . +0 . − . and ∆ m = (2 . ± . × − eV c − . The Bayesian analyses show aweak preference for normal mass ordering (89% posterior probability) and the upper sin θ octant(80% posterior probability), with a uniform prior probability assumed in both cases. The T2K dataexclude CP conservation in neutrino oscillations at the 2 σ level. I. INTRODUCTION
The fact that neutrino flavor mixing [1] and oscilla-tions [2] account for the apparent depletion of neutrinofluxes from natural sources is now well-established bydetailed observations of these sources [3–5], and veri-fied by experiments using monitored artificial sources [6–8]. Neutrino mixing requires that at least two of theneutrino masses ( m , m and m ) be non-zero which,in turn requires expanding upon the Standard Model.Masses require either new gauge singlets—right handedneutrinos—or a different mass generation mechanismfrom other Standard Model fermions, or a combinationof both. The observed pattern of neutrino masses andmixing is therefore of great interest as a window ontophysics beyond the Standard Model. Unanswered questions in neutrino oscillations
The generally accepted explanation of leptonic mixingand neutrino oscillation phenomena centers around the3 × ν , ν and ν ) in terms of the weak flavoreigenstates ( ν e , ν µ , and ν τ ). Under the assumption thatneutrinos are Dirac particles, the matrix is convention-ally parameterized as the product of three Tait–Bryanrotations (by θ ij ) and a phase transformation (by δ CP ), ∗ also at INFN-Laboratori Nazionali di Legnaro † also at J-PARC, Tokai, Japan ‡ affiliated member at Kavli IPMU (WPI), the University ofTokyo, Japan § also at National Research Nuclear University ”MEPhI” andMoscow Institute of Physics and Technology, Moscow, Russia ¶ also at the Graduate University of Science and Technology, Viet-nam Academy of Science and Technology ∗∗ also at JINR, Dubna, Russia †† also at Nambu Yoichiro Institute of Theoretical and Experimen-tal Physics (NITEP) ‡‡ also at BMCC/CUNY, Science Department, New York, NewYork, U.S.A. as in Eq. 6 of [9]: U PMNS = U e U e U e U µ U µ U µ U τ U τ U τ = R ( θ ) U ( θ , δ CP ) R ( θ ) . (1) It is well-established that all of these elements mustbe large, with the smallest | U e | ∼ /
45 and the ma-jority of elements having magnitude-squared of at least1/4. The top row elements are well-constrained bymeasurements of ν e disappearance [3, 8, 10–12]. The U µ element is likewise constrained by disappearance of ν µ [4, 7, 13, 14], but the dependence is of the approx-imate form | U µ | (1 − | U µ | ), leading to a degeneracy,commonly expressed in terms of the octant of the mixingangle θ .Since the matrix may be complex, a wide range ofvalues for the the magnitudes of the four elements U µ , U µ , U τ and U τ are possible, depending on thephase parameter δ CP . A purely real matrix correspondsto δ CP being an integer multiple of π ; any other value ismanifested as violation of Charge-Parity (CP) symmetryin any neutrino appearance channel, via the Kobayashi–Maskawa mechanism [15]. The discovery of CP violationin the lepton sector is of great interest and is a majorfocus of current experiments [13, 16]. The fact that CPviolation is controlled by a single parameter means thatthe rate of ν e appearance is not independent of ν e ap-pearance, so studying both channels provides a test ofthe standard PMNS picture.Another approximate degeneracy is in the ordering ofneutrino masses. It is known that ∆ m = m − m > (cid:12)(cid:12) ∆ m (cid:12)(cid:12) > ∆ m , but the sign of ∆ m is asyet unknown. Neutrino masses (and the correspondingeigenstates) are numbered in order of decreasing ν e con-tent. In the case where m > m , the predominant part-ner of the lightest charged lepton is the lightest neutrino.As the analogous pattern is seen in the quark sector, itis known as Normal Ordering (NO), with Inverted Or-dering (IO) corresponding to a light ν . This Mass Or-dering (MO) degeneracy is partially lifted at higher neu-trino energies by the interactions of neutrinos with mat-ter [17, 18]. This matter effect changes both the prop-agation eigenstates (‘effective masses’) of the neutrinosand their mixing with the flavor states.These three remaining unknowns (octant, δ CP , MO)are all accessible to current-generation long-baseline neu-trino experiments such as T2K, through the ν µ → ν e appearance channel and its CP conjugate ν µ → ν e . Al-though CP violation has the most obvious significance,the general structure of the matrix may give us a windowinto the deeper problem of neutrino mass and a broadrange of new physics. An inverted ordering would im-ply the lightest neutrino is relatively weakly coupled tothe lightest charged lepton. Similar in character wouldbe resolving the octant degeneracy in favor of the upperoctant, as this implies ν is not the predominant part-ner of the heaviest charged lepton. More generally, thehighly non-diagonal structure of the PMNS matrix is sug-gestive of an origin for neutrino masses that is separatefrom the electroweak physics that dominates the massesof the heavier fermions, and precision measurements ofthe ( ν ) µ → ( ν ) e channel can help to determine the remainingelements. II. THE T2K EXPERIMENT
T2K is a second-generation accelerator neutrino os-cillation experiment [19], utilizing a narrow-band beamand a 295 km baseline from J-PARC in T¯okai, Ibarakito Super-Kamiokande (commonly Super-K or just SK)in Hida, Gifu. The primary proton beam is acceleratedto 30 GeV by J-PARC’s Main Ring. In each cycle eightbunches are extracted in a single turn and directed duewest at a downward angle of − . ◦ . The intensity ofthe extracted beam is monitored by five current trans-formers that are also used to normalize the neutrino ex-posure between the various detectors of the experiment,while the beam profile is monitored with secondary emis-sion monitors—most of which are removed during physicsruns—and an optical transition radiation monitor. Thebeam power has increased over time, reaching 500 kW bythe end of May 2018.The protons impinge on a 91 . ν µ . This is referredto as ‘Forward Horn Current’ (FHC) or neutrino mode.Alternatively the horn current can be reversed, to give abeam of primarily ν µ , which is referred to as ‘ReversedHorn Current’ (RHC) or antineutrino mode. In eithercase, secondary hadrons produced in the very forwarddirection pass through the field-free necks of the hornsand contribute to a ‘wrong-sign’ flux that is of order 1% of the intended ‘right-sign’ component in FHC mode, andorder 10% in RHC mode.The horn configuration is most effective at focusingpions with momenta between 2 and 2 . /c , resultingin a broadband neutrino flux along the beamline axis thatpeaks at around 1 GeV. However, the beam is directedso that its center passes roughly 4 km south of and 12 kmbelow the Super-Kamiokande detector, equivalent to anangle 2 . ◦ from the beam axis, as measured from thetarget. This results in a narrow-band flux at the detectorthat peaks more strongly at 0 . ν e signal events through neutral currentinteractions.The experiment utilizes a suite of Near Detectors(NDs) at a site 280 m downstream of the target to charac-terize the initial flux of neutrinos and their interactions.A high-mass monitoring detector, INGRID, is centeredon the nominal beam axis and samples the beam outto about 1 ◦ around the beam axis, as measured from thetarget. This allows the intensity and direction of the neu-trino beam to be monitored on a daily basis, the latterto a precision of a few centimeters.The second near detector, ND280, is optimized formeasuring interaction rates and the properties of neu-trino interactions. The detector is positioned off-axis atthe same azimuthal angle as Super-Kamiokande to min-imize the effect of any directional shift of the beam, andat a distance from the beam axis to make the neutrinospectrum as similar as possible to what would be seen inSK in the absence of oscillations. Residual differences inspectra exist because the decay volume covers a signifi-cant fraction of the 280 m baseline; these are automati-cally accounted for in the way the ND280 data is used inthe analysis. The ND280 off-axis detector provides themost relevant information on the neutrino flux and in-teractions, but more importantly allows the analysis toform a multi-dimensional constraint, incorporating corre-lations between flux and interaction uncertainties. TheND280 additionally has a 0 . ∼ µ ± or e ± , so events arecategorized as muon- or electron-like based on the pat-tern of Cherenkov light in the detector. SK has no mag-netic field, so the sign of the horn current is used to cat-egorize events as a proxy for neutrino/antineutrino iden-tification, and the kinematics give some discriminationbetween signal events and backgrounds. Events withouta lepton-like Cherenkov ring and events where there isevidence of additional hadronic activity are not used, ex-cept for a fifth class of events which selects FHC ν e -likeevents with evidence of a low energy positron from anejected π + . These selections will be discussed in greaterdetail in Sec. IX B. III. THE OSCILLATION PROBABILITY
Following [21], and assuming three neutrinos masseslabeled with i, j ∈ { , , } , the vacuum oscillation prob-ability between neutrino flavor states ν α and ν β can bewritten P ( ν α → ν β ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j (cid:104) ν β | e − i H L | ν α (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2)where H L = 2 T ∆ + 2 T ∆ , (3)and ∆ ji = ∆ m ji L/ E . For states ν α and ν β the T i takevalues T αβi = U αi U ∗ βi and, considering the specific case of ν µ to ν e oscillations, the standard parameterization canbe used to derive approximate forms for the two ampli-tudes: T µe = sin 2 θ sin θ e i δ CP = |T µe | e i δ CP T µe (cid:39) sin 2 θ cos θ cos θ . (4)The appearance probability in vacuum can then be ap-proximated as a sum of three terms, P ( ν µ → ν e ) (cid:39) |T µe | sin ∆ + 4 |T µe | sin ∆ + 8 |T µe | |T µe | sin ∆ sin ∆ cos(∆ + δ CP ) , (5)which includes: • A dominant ‘atmospheric’ (sin ∆ ) term, whichis independent of the CP-phase. • A ‘solar’ (sin ∆ ) term that is small at T2K’s L/E . • An ‘interference’ term that depends onsin ∆ sin ∆ cos(∆ + δ CP ).Because the atmospheric term is proportional to | U µ | | U e | , the ν e appearance probability is free of theoctant degeneracy seen in the ν µ disappearance prob-ability. In practice the ν µ channel remains an impor-tant part of the fit, because there is a larger number ofevents, which gives higher precision on ∆ m i and sin θ .The disappearance channel is also important because itis relatively insensitive to sin θ , δ CP , and mass ordering,which helps to reduce the impact of degeneracies in theappearance channel.The amplitude of the interference term is about 20%of the atmospheric term, making it possible to measure the phase δ CP . This interference term is of particularphysical importance as the relative sign of δ CP and ∆ ji changes between neutrinos and antineutrinos, leading toCP violation if δ CP is not an integer multiple of π . SinceT2K’s event spectrum peaks very close to the oscillationmaximum at ∆ ∼ π/
2, the contribution from the in-terference term is a direct measure of the amount of CPviolation in the neutrino sector.The interference term also depends directly onsign(∆ m ), through the sin ∆ part. Up to a smalldifference between ∆ m and ∆ m , this is degeneratewith a substitution δ CP → π − δ CP . This means the samedata will prefer opposite signs of ( δ CP − π/
2) for normaland inverted orderings. This degeneracy is lifted if themass ordering is assumed, or if one or other ordering isstrongly preferred in the fit.
A. Matter effects
The (approximately constant) density of electrons inthe Earth’s crust along the T2K baseline changes [22]the propagation Hamiltonian (c.f. Eq. (3)): H L = 2 T ∆ + 2 T ∆ ± diag(1 , , √ G F n e , (6)where n e is the number density of electrons, G F is theFermi coupling constant, and the minus sign appliesfor antineutrinos. The impact can be understood usingT2K’s beam energy of ∼ . . − , in which case:∆ m (cid:29) √ G F n e E . > ∆ m . . ( × − eV ) (7)Because of the size ordering of terms in the Hamiltonian,the matter effect dominates the solar term, but is only asmall perturbation compared to the atmospheric and in-terference terms. In practice, because the resulting oscil-lation probability is dominated by the atmospheric term,T2K’s sensitivity to the mass ordering comes mostly fromthis perturbative effect on | U e | and ∆ m . In this regimethe matter effect can be described with a dimensionlessparameter ξ ( E ) = ± √ G F n e ∆ m E (cid:39) ± . E/ GeV, (8)which is positive for neutrinos in the normal ordering andfor antineutrinos in the inverted ordering. The resultingprobability for neutrinos or antineutrinos is P ( ( ν ) µ → ( ν ) e ) (cid:39) |T µe | sin ([1 − ξ ]∆ )[1 − ξ ] + 4 |T µe | sin ( ξ ∆ ) ξ +8 |T µe | |T µe | sin ([1 − ξ ]∆ )1 − ξ sin ( ξ ∆ ) ξ cos ∆ cos δ CP ∓ |T µe | |T µe | sin ([1 − ξ ]∆ )1 − ξ sin ( ξ ∆ ) ξ sin ∆ sin δ CP ,(9)where the CP-violating sin δ CP term takes a negative signfor neutrinos and positive sign for antineutrinos.For T2K, the largest observable effect from propaga-tion in matter is the [1 − ξ ] − scaling of the atmosphericterm. This modification to the leading atmospheric termis about 5%, leading to a roughly 10% difference in theappearance probability for neutrino and antineutrinos.Since this difference is about half the amplitude of theCP-violating term, it is in general difficult to disentanglethe two phenomena if they have opposite effects on thetotal number of events observed, or if the value of sin δ CP is close to zero. However if both phenomena enhance (orboth suppress) the total number of events, then the neteffect can be too large to attributable to either sourcealone, and there will be much less ambiguity. B. The survival probability
In the same notation, the ( ν ) µ survival probability atT2K is to a good approximation given by: P ( ( ν ) µ → ( ν ) µ ) (cid:39) − T µµ (1 − T µµ ) sin ∆ Atm , (10)where ∆
Atm = ∆ + ∆ × T µµ / (1 − T µµ ). So althoughthe observable survival probability is not sensitive to themass ordering, the best-fit value of ∆ is different fornormal and inverted orderings. As for the oscillation am-plitude, in terms of the standard parameterization |T µµ | = sin θ cos θ , (11)so the amplitude reaches a maximum value aroundsin θ = 1 / (2 cos θ ), and values on either side of thisare degenerate.Propagation in matter does not change the survivalprobability by much; matter dependent effects are sup-pressed by a factor of T ee (1 − T µµ ) [23]. Since the mat-ter density can be approximated as symmetric, the prob-ability also has no dependence on sin δ CP [24]. Howeverthe relationship between ∆ and the observable ∆ Atm does depend on cos δ CP through T µµ , which can for someparameter combinations give rise to a correlation betweenthe measured ∆ and δ CP . IV. UPDATES SINCE THE PREVIOUSRESULTS
This analysis uses a SK data set collected up to theend of May 2018. This corresponds to an exposure of14 . × Protons on Target (POT) in FHC mode and16 . × POT in RHC mode, the same as used toreport indications of CP violation in [16]. A detailedbreakdown is given in Tab. I. Compared to the previousupdate [25] this is a nominal increase of 1% in FHC mode,but 116% in RHC mode, which is particularly of interestfor indications of ν e appearance, described in Sec. XIV.In parallel with the statistical increase, our event se-lection has been refined since it was last described indetail [26]. Event reconstruction is now based on an al-gorithm that matches the pattern of light observed in SKdirectly [27]. This makes use of more information aboutthe event, providing better discrimination between eventcategories, and improving the resolution of the leptonmomentum and vertex location. As a result, the fidu-cial volume can also be expanded, roughly equivalent toa 20% increase in statistics for the ν e samples, as de-scribed in Sec. IX. The newer reconstruction algorithmhas previously been used for rejecting neutral current π events in the ν e samples [28, 29] and for all aspects ofevent selection and reconstruction in more recent publi-cations [16, 25].A large fraction of the analysis development focuseson the interaction model, which incorporates constraintsfrom a number of new external data sets and theoreti-cal improvements. Since reported in [26], the dominantcharged-current quasi-elastic (CCQE) models have beenupdated in various respects, including: the handling ofweak charge screening in nuclei; the handling of nucleonremoval energy and its effect on lepton kinematics; andadditional freedom allowed in the kinematic dependenceof interactions involving correlated nucleon pairs (2 p h ).Modeling of (and uncertainties assigned to) subdominantprocesses have also been improved, including coherentscattering and neutral current interactions.Constraints on neutrino oscillations and the associatedparameters come from a combined analysis of disappear-ance and appearance channels across FHC and RHC con-figurations, using the same approach as in [16]. Thispaper provides a fuller description of the method and abroader range of results. V. ANALYSIS OVERVIEW
The T2K near and far detectors have different targetnuclei and are based on different particle detection tech-niques. The T2K oscillation analysis therefore uses pa-rameterized models of the neutrino beam flux and theneutrino interaction cross section to propagate near de-tector information to predict the far detector event rate.The neutrino flux prediction has been described in de-tail in Ref. [30]. The collision of 30 GeV protons fromTABLE I: T2K Run periods and exposure used in thisanalysis, for ND280 and SK.
T2K run End date SK POT / ND280 POT / FHC RHC FHC RHCRun 1 Jun. 2010 0 .
33 — — —Run 2 Mar. 2011 1 .
12 — 0 .
78 —Run 3 Jun. 2012 1 .
60 — 1 .
56 —Run 4 May 2013 3 .
60 — 3 .
47 —Run 5 Jun. 2014 0 .
24 0 .
51 — 0 . .
19 3 .
55 — 3 . .
48 3 .
50 — —Run 8 Apr. 2017 7 .
17 — — —Run 9 May 2018 0 .
20 8 .
79 — —Total 14 .
94 16 .
35 5.80 3.86 the J-PARC main ring with the T2K neutrino productiontarget is simulated using FLUKA [31–33]. The resultantsecondary particles are passed to a GEANT3 [34] simu-lation of the magnetic focusing horns and decay volumedownstream of the target. GCALOR [35, 36] is used tomodel hadronic interactions of the secondary particles asthey traverse the focusing horns and decay volume. Par-ticles are then allowed to decay to produce neutrinos.Data from proton beam monitors is used to tunethe initial proton beam parameters in the simulation.NA61/SHINE, a fixed-target experiment at CERN’s Su-per Proton Synchrotron, measures particle production innucleus and hadron collisions with a large acceptancespectrometer. This includes measurements of the col-lisions of 30 GeV protons with graphite. Data from theNA61/SHINE [37–40] experiment are then used to tunethe secondary particles produced from the target. Fi-nally, the INGRID [41] on-axis near detector is used tomonitor the neutrino beam direction. The uncertaintyfrom each of these measurements are combined with un-certainties from the beam simulation to give the final fluxuncertainty. This is parameterized as a function of neu-trino energy, neutrino species and whether the beam isoperating in FHC or RHC mode.Figure 1 shows the predicted neutrino fluxes at SK forboth the FHC and RHC modes. Previous T2K flux esti-mates [30] used thin target hadron production data col-lected in the NA61/SHINE experiment in 2007 [37–39],where a 31 GeV proton beam impinged upon a graphitetarget with a thickness of 4% of a nuclear interactionlength (the so-called thin target). The work presentedhere uses an updated flux prediction based on higherstatistics thin target hadron production data collectedin the NA61/SHINE experiment in 2009 [40], includingthe yields of π + , π − , K + , K − , K s , Λ and p . Future anal-yses will include NA61/SHINE hadron production mea-surements on a replica of the T2K neutrino productiontarget.Figure 2 shows the fractional uncertainty on the ν µ flux at SK in FHC mode, on the wrong-sign ν µ flux atSK in FHC mode and on the right-sign ν µ flux at SK inRHC mode. The improvement obtained by including the (GeV) n E P O T ) / M e V / F l ux ( / c m Neutrino mode flux at the far detector m n m n e n e n Neutrino mode flux at the far detector (GeV) n E P O T ) / M e V / F l ux ( / c m Antineutrino mode flux at the far detector m n m n e n e n Antineutrino mode flux at the far detector
FIG. 1: The SK flux prediction for Runs 1–9a withhorns operating in FHC (250 kA) mode (upper) andRHC ( −
250 kA) mode (lower).2009 NA61/SHINE thin target data is also indicated.The T2K neutrino interaction model is described ex-tensively in Section VI. The model incorporates a num-ber of tunable parameters whose prior uncertainties andnominal values come from comparisons to electron andneutrino scattering data sets. There are a number ofmodels that agree with existing data equally well, andit is not always possible to parameterize the differencesbetween these models. In this case simulated data stud-ies have been performed using these alternate models, asdescribed in Section XII.The neutrino flux and interaction models are fit to datacollected by the T2K off-axis near detector, ND280 [19].The fit varies the parameters within both models simul- (GeV) n E -
10 1 10 F r ac ti on a l E rr o r m n SK: Neutrino Mode,
Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment , Arb. Norm. n E ·F Material ModelingNumber of ProtonsNA61 2009 DataNA61 2007 Data m n SK: Neutrino Mode, (GeV) n E -
10 1 10 F r ac ti on a l E rr o r m n SK: Antineutrino Mode,
Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment , Arb. Norm. n E ·F Material ModelingNumber of ProtonsNA61 2009 DataNA61 2007 Data m n SK: Antineutrino Mode, (GeV) n E -
10 1 10 F r ac ti on a l E rr o r m n SK: Antineutrino Mode,
Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment , Arb. Norm. n E ·F Material ModelingNumber of ProtonsNA61 2009 DataNA61 2007 Data m n SK: Antineutrino Mode,
FIG. 2: The fractional systematic uncertainty on the ν µ flux at SK in FHC mode (top), on the right-sign ν µ fluxat SK in RHC mode (middle), and on the wrong-sign ν µ flux at SK in RHC mode (bottom). The solid black lineshows the current total fractional uncertainty(NA61/SHINE 2009 Data), while the dashed black linein the top panel shows the fractional uncertainty froman earlier flux prediction (NA61/SHINE 2007 Data).taneously to best match the ND280 data, as described inSection X. This results in tuned flux and interaction mod-els with correlated uncertainties, providing a more accu-rate and precise prediction of the event rate at Super-Kamiokande.At the far detector a simultaneous fit of muon-like and electron-like samples from both neutrino and antineu-trino beams is used to constrain the PMNS oscillation pa-rameters. The conventional { θ ij , δ CP } parameterizationis used, enforcing unitarity, and the effect of propagationin matter is included. Data from ν e and ν e disappearanceexperiments are used to constrain the parameters ( θ and ∆ m ) [42] that T2K has little sensitivity to. Fitsare performed both with and without an external con-straint on θ [43]. Systematic uncertainties are treatedby a numerical marginalization technique: all parameter-ized uncertainties are randomly sampled many times ac-cording to prior constraints, including ND280 data, andthe likelihoods averaged over the ensemble. This pro-cess is described in detail in Section XI, with the resultsdescribed in Section XIII. VI. NEUTRINO INTERACTION MODELING
Oscillation parameter values are inferred from spec-tra of observable quantities, herein either reconstructedcharged-lepton kinematics, ( p (cid:96) , θ (cid:96) ), or reconstructed neu-trino energy. The reconstructed neutrino energy is esti-mated from final-state charged-lepton kinematics only as E Rec QE ( p (cid:96) , θ (cid:96) ) = 2 M N,i E (cid:96) − M (cid:96) + M N,f − M N,i M N,i − E (cid:96) + p (cid:96) cos θ (cid:96) ) , (12)where M N,i , M N,f , and M (cid:96) are the mass of the initial-state nucleon (an effective, off-shell mass that includesthe ‘nucleon removal energy’ is often used), final-statenucleon, and final-state charged lepton respectively; E (cid:96) , p (cid:96) , and θ (cid:96) are the energy, three-momentum, and angle ofthe final-state charged-lepton respectively. E Rec QE providesa smeared but minimally-biased estimate of the neutrinoenergy for quasi-elastic neutrino scattering off bound nu-cleons (CCQE). For other interaction channels, such asthose that produce extra hadrons, E Rec QE underestimatesthe energy of the interacting neutrino.The procedure of inferring oscillation parameter valuesfrom observable quantities implicitly relies on an accurateunderstanding of the rate of background processes and amapping between true energy and observable quantities, e.g. E Rec QE ( E ν ), both of which are derived from simula-tion. As a result, accurate neutrino interaction modelingis critical. Event selections are trained on the simulateddistribution of final-state particles and the predicted rateof various signal and background processes. The pre-dicted rate of a number of neutrino interaction processes,which exhibit different true-to-observable mappings, isconstrained by near detector data and then used to inter-pret the observed far detector data. This section brieflydescribes the neutrino interaction model, accounted-forfreedom within the model, and specific studies used totest resilience to known weaknesses of the model. A. The Base Interaction Model
The samples of simulated neutrino interactions used inthis analysis were made with version 5.3.3 of the neut interaction generator [44]. neut simulates known neu-trino interaction channels relevant for few GeV neutri-nos, these channels are broadly categorized as: 1p1h,2p2h, single pion production, and deep inelastic scatter-ing (DIS). In addition to the ‘primary’ interaction chan-nels, the effect of using nuclear targets, where the strucknucleons are bound within a nuclear potential, needs tobe modeled well. These effects can be separated intoinitial-state and final-state effects. Most updates to theinteraction model since the previous analysis [26], arein the treatment of systematic uncertainties; however, ashort description of the whole model is included here forcompleteness. As the ‘base’ model has not changed, theinterested reader is directed to Ref. [26] and Ref. [45] fora discussion of the motivations behind any specific modelchoices. a. Initial-state nuclear effects:
Nucleons boundwithin a nuclear potential undergo non-negligible ‘Fermimotion’. For carbon, this means bound nucleons have amomentum of p f < ∼
217 MeV /c , or equivalently, a Fermienergy of E f < ∼
25 MeV. A Global Relativistic FermiGas (GRFG) is used to model the initial-state nucleonmomentum distribution in this analysis. Neutrino inter-actions with bound nucleons are largely handled underthe impulse approximation, whereby a single ‘struck’ nu-cleon receives a four-momentum kick while the rest of thetarget nucleus acts as a group of non-interacting ‘spec-tator’ nucleons. This rudimentary nuclear model is asimple approximation for the correct modeling of the ini-tial nucleon momentum distribution and nucleon removalenergy, a study, presented below, accounts for the effectof this approximation. b. 1p1h:
One particle, one hole interactions arethose where the neutrino interacts quasi-elastically witha single bound nucleon—the interaction is only quasi -elastic because of the bound nature of the target nucleonand, for charged current events (CCQE), the initial-to-final-state charged-lepton and nucleon rest mass differ-ence. Such interactions are modeled in the Lewellyn–Smith formalism [46], using the BBBA05 [47] descrip-tion for the vector part of the nucleon form factors, anda simple dipole form for the axial part. The neut modelincludes two additional features of note: the nucleon re-moval energy, ‘NRE’, and in-medium modifications tothe W boson propagator via the Random Phase Approx-imation (‘RPA’). Variations in the average nucleon re-moval energy modify the predicted kinematics of final-state particles, most importantly charged leptons. Whencomparing predictions based on Fermi Gas nuclear mod-els to 1p1h-like cross-section data, a suppression at lowfour-momentum transfer is favored relative to the free-nucleon-target calculation [48]. This is often attributedto a weak-charge screening effect as a result of the nuclearmedium [49]. The effect is termed ‘RPA’ after the ‘Ran- dom Phase Approximation’ technique used to sum upthe series of contributing W-boson self-energy diagrams.Here, the distribution of four-momentum transfer is mod-ified by the RPA calculation from Nieves et. al. [49]. Ascan be seen in Fig. 3, 1p1h is the dominant interactionchannel at T2K energies. E ν ( GeV ) ν µ F l u x ( / c m / G e V / P O T ) σ ( E ν ) / E ν ( − c m / G e V / N u c l e o n s ) neut , ν µ − CCC-Inc CC-1p1hCC-2p2h CC-SPPCC-DIS ν -mode beamFGD SK Osc. ( × ) FIG. 3: The total charged-current cross section formuon neutrinos interacting with a carbon nucleus, aspredicted by neut , overlaid on the ND280 muonneutrino flux, and an example oscillated muon neutrinoflux at SK. The oscillation parameters used here are thebest fit from the previous analysis [26]. The total (Inc)cross section is separated into 1p1h, 2p2h, single pionproduction (SPP), and deep inelastic scattering (DIS)channels. c. 2p2h:
Two particle, two hole interactions are aninherently nuclear-target process, whereby the incomingneutrino interacts with a bound pair of nucleons, knock-ing both out of the nuclear potential. The Nieves et. al. model [50] is used to predict the cross-section as a func-tion of lepton kinematics. While this process is sub-dominant, it produces observable final states that areindistinguishable from 1p1h interactions in the T2K de-tectors, but with different observed lepton kinematics asa function of neutrino energy. In the Nieves et. al. q < ∼ . q > ∼ . E Rec QE biases can be seen in Fig. 4. d. Single pion production: Single pion productioncan be separated into three sub-processes: resonant, non-resonant, and coherent single pion production. The reso-nant and non-resonant processes describe the productionof a pion involving neutrino scattering of a single nu-0 d σ / d q d q − ( c m c / G e V / N u c l e o n s ) Momentum transfer, q ( GeV /c ) E n e r g y t r a n s f e r , q ( G e V ) -0.8 -0.6 -0.4 -0.2 0 0.2 E Rec QE /E ν − E v e n t s ( A . U . ) CC-1p1h ( × . CC-2p2hCC-2p2h QE-likeCC-2p2h ∆ -like FIG. 4: Top: The energy and momentum transferdistribution for the Nieves et. al. neut .The two peak structure is clear, with QE-likekinematics corresponding to the lower left peak, andDelta-like kinematics to the stronger central peak.Bottom: The reconstructed energy bias at SK is shownfor 1p1h and 2p2h events for an oscillated muonneutrino flux. The different reconstructed energysmearing for 2p2h events with QE-like and Delta-likeinteraction kinematics can be seen.cleon, either via an intermediate baryon resonance (res-onant), or not (non-resonant). These processes are mod-eled in the Rein–Sehgal formalism [51], with an improve-ment that includes the effect of the final-state charged-lepton mass [52], and updated nucleon axial form factorsfrom Graczyk & Sobczyk [53]. The contributions from17 baryon resonances are considered, with the ∆(1232) being dominant, and interference terms between the reso-nances are taken into account. The non-resonant channelaugments the production of half-unit isospin final states( e.g. ν + n → (cid:96) − + p + π and ν + n → (cid:96) − + nπ + );any interference between the resonant and non-resonantcontributions is ignored. These processes are used tomodel final states with an invariant hadronic mass of W ≤ . /c . The modeling of the so-called ’transi-tion region’ between single pion production off a nucleonand shallow- and deep-inelastic scattering is an unsolvedtheoretical problem [54]. In the neut model, the re-gion 1 . ≤ W ≤ . /c contains contributions fromboth the Rein–Sehgal single pion model, described above,and the deep-inelastic-scattering model, described below.For higher invariant masses the deep inelastic scatteringmodel is used.The axial form factors and the strength of the non-resonant contribution in the Rein–Sehgal model weretuned to published cross-section data using the NUI-SANCE framework [55]. As these parameters controlonly the nucleon-level interaction, the central values weredetermined from fits to deuterium-target bubble chamberdata, which is largely free from nuclear effects. Data fromANL [56] (with some reanalyzed distributions taken fromRef. [57]) and BNL [58] was used. The uncertainties de-termined from the fits to bubble chamber data were theninflated to approximately cover cross-section data fromMiniBooNE [59] and MINER ν A [60, 61].Coherent single pion production describes the interac-tion of a neutrino coherently with a whole nucleus. Thisis a sub-dominant pion production process, observed atlow energy for the first time by the MINER ν A experi-ment [62] and is characterized by very little four momen-tum transfer to the struck nucleus. We follow the prefer-ence of the MINER ν A data and use the Bergher–Sehgalmodel [63]. e. Deep inelastic scattering:
For interactions pro-ducing hadronic systems with two or more pions and in-variant hadronic masses of
W > . /c , the crosssection is constructed from nucleon structure functionsthat depend on the Bjorken scaling variables x and y . The structure functions are calculated from theGRV98 [64] parton distribution functions, with modifi-cations from Bodek et. al. [65] to account for the rela-tively low momentum transfers involved. For interactionswith 1 . < W ≤ . /c the hadronic state is gener-ated by a custom multi-pion production model, above W = 2 . /c PYTHIA 5.72 is used [66]. f. Final-state nuclear effects:
After the primary neutrino interaction has been simulated, a number ofadditional ‘nuclear effects’ are included. For interac-tions that produce a final-state proton or neutron, thePauli exclusion principle is applied, rejecting any eventsthat produce a nucleon below the Fermi energy. Thisresults in a suppression at low four-momentum transferfor 1p1h events. Final-state hadrons produced at theneutrino interaction vertex are stepped through the nu-clear medium in a classical cascade, in which they may:1interact and produce secondary particles, be absorbed,or undergo charge exchange ( e.g. π + + n → π + p ).Such re-interactions are often called ‘Final-State Inter-actions’, or FSIs. Finally, after the primary interactionand hadronic cascade, the remnant nucleus can be leftin an excited state that will subsequently decay. For in-teractions on oxygen, nuclear de-excitations that resultin secondary, low energy photons ( O (1 −
10) MeV) aremodeled following Ref. [67].With the exception of 2p2h interactions, these chan-nels and effects are also implemented for neutral currentinteractions, but the details are not repeated here forbrevity. The total charged-current cross-sections, brokendown by interaction channel, are shown in Fig. 3.
B. The Uncertainty Model
As the number of observed events included in the anal-ysis grows with exposure, a robust interaction uncer-tainty model is required to assess the significance of theresults. The uncertainty model for 1p1h and 2p2h inter-actions have seen recent improvements and will be dis-cussed in detail here. For details on other, unchangedsources of interaction uncertainty see Ref. [26] SectionIII. a. 1p1h
The neut M QEA ), the effect of RPAon the cross section as a function of four-momentumtransfer, and the Nucleon Removal Energy associatedwith scattering off a bound nucleons.In this analysis, M QEA does not have a prior uncertaintyand is constrained by near detector data alone. The pa-rameterization of the uncertainty on the RPA suppres-sion has been updated in this analysis; the previous im-plementation proved problematic because variations ofdifferent free parameters effected a similar response in Q . For this analysis, Bernstein polynomials were usedto model the shape below some Q cutoff, U , above whichan exponential decay form is used: f ( x ) = A (1 − x (cid:48) ) + 3 B (1 − x (cid:48) ) x (cid:48) + 3 p (1 − x (cid:48) ) x (cid:48) + Cx (cid:48) , x < U p exp( − D ( x − U )) , x > U , where x = Q and x (cid:48) = Q / U . To ensure continuity at Q = U , the conditions: p = C + U D ( C − p = C − A, B, C, D .The fifth, U , is kept fixed at 1 . . This parame-terization has been termed ‘BeRPA’ after the Bernsteinpolynomials on which it is based. The effect of varyingeach of the four free parameters relative to the theoretical uncertainty calculated by following Ref. [68] can be seenin Fig. 5. Together, the four free BeRPA parameters andthe M QEA parameter give effective freedom over a rangeof Q . The Q distribution is then largely constrainedby the fit to near detector samples. Q ( GeV ) B e R P A W e i g h t Central value Total uncertainty‘A’ variation ‘B’ variation‘C’ variation ‘D’ variation
FIG. 5: The central value BeRPA suppression factor(solid) and the total prior uncertainty (dashed)determined from 10 uncorrelated throws of the freeparameters, overlaid on the corresponding envelopesgenerated by varying each parameter. The continuityconditions result in a perhaps unintuitive totaluncertainty envelope given the individual parametervariations. b. 2p2h The details of the 2p2h process are highlyuncertain. The total cross-section, evolution with neu-trino energy, and energy and momentum transfer char-acteristics of the process are all predicted differentlyby the available models ( e.g.
Nieves et. al. [50], Mar-tini et. al. [69], SUSAv2-MEC [70], GiBUU [71]). Whiledata from the T2K near detector [72–74], MINER ν A [75,76], and NO ν A [77] favor a process with similar interac-tion kinematics to such a multi-nucleon process , experi-mental sensitivity to this exclusive channel is weak. As aresult, significant freedom is afforded to the 2p2h processin this analysis.The uncertainty on the 2p2h process is separated intonormalization and shape components. An overall 100%normalization uncertainty is separately assigned to inter-actions involving neutrinos and those involving antineu-trinos. An additional parameter that introduces freedomin the relative normalization of 2p2h interactions with i.e. at fixed momentum transfer, a process that occurs betweenthe quasi-elastic and pion production peaks in energy transfer. d σ / d q d q − ( c m c / G e V / N u c l e o n s ) Momentum transfer, q ( GeV /c ) E n e r g y t r a n s f e r , q ( G e V ) d σ / d q d q − ( c m c / G e V / N u c l e o n s ) Momentum transfer, q ( GeV /c ) E n e r g y t r a n s f e r , q ( G e V ) FIG. 6: The differential cross section for the twoextreme variations of the 2p2h ‘shape’parameter—QE-like (top) and Delta-like (bottom)— c.f.
Fig. 4 (top) for the central-value-predicted cross section.
C. Simulated data studies
It is strongly suspected that the described uncertaintymodel may not cover all differences between nature andthe interaction model described above. To begin to ad-dress this, we perform fits of the model to targeted ‘sim-ulated data sets’ that test the robustness of the model and associated uncertainties to known missing features.In some cases the results of these studies are used to mo-tivate additional uncertainties. This section introducesthe simulated data sets that were analyzed to addressspecific concerns, the results of the fits will be discussedin Section XII. a. Alternative 1p1h Nuclear Models
The GRFGused to model the nuclear initial state is a simple modelthat contains no correlations between initial momentumand Nucleon Removal Energy ( nre ). Such correlationsmay be important for correctly modeling the observedcharged lepton spectrum [78] and are seen in nuclearresponse measurements from electron scattering exper-iments. To test the robustness of the implemented un-certainty model to such details, two simulated data setsare used: the Nieves et. al. et. al. [78] (BSF). Both containsome correlation between the initial momentum and the nre . The Nieves et. al. model differs from the base modelby implementing a local, rather than a global, FermiGas (LFG), in which the concept of a radially-dependentnuclear density profile introduces such correlations. Inthe BSF model, initial nucleons are chosen from a full,two dimensional nuclear response distribution, which isconstructed from ( e , e (cid:48) p ) data [78]. It should be notedthat the BSF model contains no ‘RPA’-like, low four-momentum-transfer suppression effect. In constructingthe simulated data, only the 1p1h cross section is modi-fied. The predicted final-state muon kinematics for eachmodel are shown in Fig. 7. b. Nucleon Removal Energy The implementation ofthe nre parameter was revised for this analysis. Theprevious implementation relied on calculating the changein the predicted differential cross-section for a varia-tion nre → nre (cid:48) , σ CCQE ( E ν , p (cid:96) , θ (cid:96) , nre ); this provedproblematic as variations of nre modify the availablekinematic phase space for the production of final-statemuons. For some simulated interaction, ( E ν , p (cid:96) , θ (cid:96) ),and binding-energy variation, nre (cid:48) , the variation weight, w = σ CCQE ( E ν ,p (cid:96) ,θ (cid:96) , nre (cid:48) ) / σ CCQE ( E ν ,p (cid:96) ,θ (cid:96) , nre ) , will be ill-defined when the denominator is vanishingly small. In-stead, an effective implementation was used that shiftsthe final-state charged-lepton momentum in response tovariations of nre . The momentum shifts were calcu-lated in bins of true neutrino energy and true final-statecharged-lepton polar angle—in the neut implementa-tion, variations of the binding energy effect only smallchanges in the final-state lepton angular distribution. Anexample of such a variation can be seen in Fig. 8. Theprior uncertainty on the new nre parameterization wastaken as 18 MeV—this large uncertainty is motivated inpart because of implementation choices in neut [79] andin part because of uncertainties on the analyzed electron-scattering measurements.The uncertain nre parameter was not included as afree parameter in the near detector fit for this analysis.Instead, an extremal variation based on the results inRef. [79] was included as a simulated data set.3 d σ / d p ℓ c o s ( θ ℓ ) − c m c / G e V / N c o s ( θ ℓ ) p ℓ ( GeV /c ) dσ/d cos( θ ℓ ) 10 − cm / N d σ / d p ℓ − c m c / G e V / N neut , GRFG, CC-1p1h, SK Flux, oscillated | ∆ m | = 2 . × − eV , sin ( θ ) = 0 . BSFGRFGLFG
FIG. 7: The predicted flux-averaged cross section forthree different nuclear response models. The differentmodels result in different predictions of the observedfinal-state lepton kinematics for the same oscillatedneutrino flux. If such variations are not accounted for inthe uncertainty model, extracted oscillation parametersmay be biased. The contours contain the region ofphase space with a differential cross sectiond σ/ d p (cid:96) cos( θ (cid:96) ) > . × − cm c/ GeV / N. c. Martini et. al. As previously mentioned,the modeling of 2p2h interactions is highly uncertain.We include a simulated data set based on an alternate2p2h calculation by Martini et. al. [69]. This calculationpredicts a larger inclusive 2p2h cross section than theNieves et. al. calculation—importantly increasing the rel-ative neutrino/antineutrino 2p2h strength—and is thusan instructive alternate model. d. Kabirnezhad Single Pion Production
The Rein–Sehgal model accounts for interference between pion pro-duction channels that include a baryon resonance, how-ever, interference between resonance and non-resonancechannels is neglected. A new model, developed byKabirnezhad [80], overcomes this limitation and is usedto build a simulated data set. e. Data-driven CC π E ν − Q dependence The un-tuned ND280 CC 0 π sample prediction underestimatesthe data by approximately 5%. This sample is largelycomposed of 1p1h and 2p2h interactions but with a signif-icant contribution from interactions that produce a pionwhich is then absorbed before leaving the nucleus. The2p2h interaction can be further classified into events withand without a virtual ∆(1232) particle. Simulated datasets are created by assigning the observed CC 0 π data– FIG. 8: Top: The predicted muon momentum spectrumfor a number of different values of nre in an oscillatedSK flux. The shift towards lower momentum may beconfused for a shift in | ∆ m | . Bottom: The effect ofvarying nre on the reconstructed energy bias; highervalues result in more energy ‘feed down’.simulation discrepancy to either the 1p1h or 2p2h eventcategories. At the near detector the event category isweighted in bins of lepton momentum and angle so thatthe simulation matches the data. This weighting is thenprojected as a function of neutrino energy and Q andapplied to the far detector simulation to create the sim-ulated far detector data. f. Coulomb Correction As the final-state chargedlepton leaves the nuclear potential, it undergoes a smallmomentum shift because of interaction with the Coulombfield of the nucleus. In addition, the Coulomb po-4tential results in a small variation in the relative neu-trino/antineutrino cross section. The effect of theCoulomb potential was not included in the base model,and thus a simulated data set was included in which amomentum shift was applied to final-state (anti-) muonsand (anti-) electrons, following Ref. [81], and the relativecharged-current cross section for neutrinos and antineu-trinos was varied by 3%.
VII. NEAR DETECTOR DATA
The history of protons on target delivered to the T2Kbeamline until the end of May 2018 is shown in Fig. 9.Data for runs 1–9 in the muon monitors and the on-
Year A cc u m u l a t e d P O T B ea m P o w e r / k W × Total Accumulated POT for Physics Mode Accumulated POT for Physics ν Mode Accumulated POT for Physics ν Mode Beam Power ν Mode Beam Power ν FIG. 9: The T2K data-taking periods, showing theaccumulated beam protons on target delivered as afunction of time.axis INGRID detector are shown in Fig. 10. The rateis stable throughout the run periods, and the horizontaland vertical beam positions are stable to less than 1 mradthroughout all of the run periods.The off-axis near detector, ND280, is located 280 mupstream of the beam source. It consists of several sub-detectors inside a 0.2 T magnet. Charged current (CC) ν µ and ν µ neutrino interactions are selected in the trackerregion, which is composed of two fine-grained detectors(FGD1 and FGD2) [82] interleaving three time projec-tion chambers (TPC) [83]. The FGDs provide the targetmass for neutrino interactions. The first FGD consists of30 layers each composed of 192 plastic scintillator bars.The bars in each layer are oriented perpendicularly tothe neutrino beam direction and to the bars in the pre-ceding layer. Each pair of layers forms a single moduleproviding a three-dimensional position for charged par-ticles passing through it. The second FGD consists ofalternating plastic scintillator modules and water pan-els. There are seven scintillator modules interspersedwith six water panels, providing a water target for neu-trinos to interact within. This allows effects relating toneutrino interactions on water to be isolated from thoseon carbon, reducing the uncertainty in extrapolating the ( E v e n t s / P O T ) ( m r a d ) − − Day ( m r a d ) − − Run1 Run2 Run3 Run4 Run6 Run7 Run8 Run9
Event rate Horn250kAHorn205kAHorn-250kAHorizontal beam direction
Vertical beam direction
Run5
INGRIDMUMON
INGRIDMUMON FIG. 10: Data in the muon monitors (MUMON) andINGRID on-axis detector for runs 1–9. The top rowshows the event rate in both detectors, which is reducedas expected for RHC mode. The middle and bottomrows show the horizontal and vertical beam direction inboth detectors.event rate measurement from ND280 to SK. The TPCsmeasure both the curvature of charged particles in themagnetic field of ND280 and the energy lost by the par-ticles as they travel through the TPC gas. The curvatureof the particles provides a precise measurement of theirmomentum and charge, while the energy loss allows theparticle species to be identified.Only ND280 data from runs 2–6 are used in this anal-ysis, a smaller sample than for SK. Data quality is as-sessed weekly, and the total ND280 data taking efficiencyacross runs 2–6 was ∼ × POT in FHC and2.84 × POT in RHC, as shown in Tab. I.The event selection for FHC is unchanged since theanalysis described in Ref. [26]. The highest momentum,negatively charged track in each event is selected as thelepton candidate. The candidate track must start withinthe fiducial volume of FGD1 or FGD2 and be identi-fied as muon-like by the TPC. This produces a selectionof charged-current (CC) ν µ interactions. The selectedevents are divided into three samples for each FGD, basedon the reconstructed pion multiplicity. Positive pions areidentified in three ways: a positive charged FGD-TPCtrack with energy loss consistent with a pion; a posi-tively charged FGD-contained track with charge deposi-tion consistent with a pion; or a delayed energy depositin the FGD due to stopped π + → µ + → e + decays. Neg-atively charged, minimally ionising TPC tracks are iden-tified as negatively charged pions. Neutral pions decayinstantaneously to pairs of photons, which can then con-vert to electron-positron pairs. TPC tracks with chargedepositions consistent with an electron are used to iden-tify these decays.The three FHC CC sub-samples are CC 0 π , whichis dominated by CCQE interactions, CC 1 π + , which is5dominated by CC resonant single pion production, andCC Other, which is dominated by interactions producingmultiple pions. The reconstructed muon momentum andangle of the selected data and simulation events in theFHC CC 0 π and CC 1 π + samples are shown in Fig. 11and Fig. 12, for both FGD1 and FGD2. The numbers ofevents recorded in each sample and the expectation priorto the ND280 fit are shown in Tab. II.TABLE II: ND280 samples, with the observed andexpected numbers of events (before and after fitting atND280). Beam Topology Target Data Prediction Postfit ν µ CC 0 π FGD 1
17 136 16 724 17 122
FGD 2
17 443 16 959 17 495FHC ν µ CC 1 π + FGD 1
FGD 2 ν µ CC other
FGD 1
FGD 2 ν µ CC 1-track
FGD 1
FGD 2 ν µ CC N -track FGD 1
FGD 2 ν µ CC 1-track
FGD 1
FGD 2 ν µ CC N -track FGD 1
FGD 2
The event selection for ν µ and ν µ interactions in theRHC beam mode is unchanged since the previous analy-sis [26]. These selections differ from the FHC selectionsdescribed above in two important ways. As a larger num-ber of interactions are produced by “wrong-sign” neutri-nos, selections of both ν µ and ν µ interactions are used inthe RHC beam. Taking into account differences in theflux and cross-section, the wrong-sign contamination isapproximately 30% in the selected RHC samples com-pared to 4% in the FHC samples.The selected ν µ ( ν µ ) CC candidate events are dividedinto two samples for each FGD, based on the numberof reconstructed tracks crossing a TPC. These are CC1-track, which is dominated by CCQE-like interactions,and CC N -track, which is dominated by interactions pro-ducing pions. The events are not divided according to thenumber of observed pions, unlike the FHC selections, dueto the lower interaction rate for antineutrinos. The re-constructed muon momentum and angle of the selectedevents in these samples for FGD1 are shown in Fig. 13and Fig. 14 respectively.In total there are 14 ND280 event samples: six for FHC(CC 0 π , 1 π + and Other, for FGD1 and FGD2), four forright-sign RHC (CC 1-track and CC N -Track, for FGD1and FGD2) and four for wrong sign RHC (CC 1-Trackand CC N -Track, for FGD1 and FGD2). The number ofobserved and predicted events for each sample are shown in Tab. II. VIII. SUPER-KAMIOKANDE DATA ANDSIMULATION
The Super-Kamiokande detector [84] consists of acylindrical tank filled with 50 kilotonnes of pure water,located in the Mozumi mine in Hida, Gifu. An overbur-den of 2700 meter-water-equivalent provided by MountIkeno suppresses the cosmic ray muon flux by five or-ders of magnitude. Photo-multiplier tubes (PMTs) aresupported by a 55 cm wide steel structure, placed 2 maway from the tank walls, which divides the detectorinto two optically separated regions. The outer detec-tor (OD) region, used to identify events with enteringparticles, is lined with reflective material and viewed by1885 8 (cid:48)(cid:48)
PMTs. The inner detector (ID) region contains32 kilotons of water and is instrumented with 11146 20 (cid:48)(cid:48)
PMTs which make up 40% of the detector’s inner sur-face. The high density of PMTs in the ID allows for theimaging of the ring-like light patterns projected on thedetector walls by particles traveling above the Cherenkovthreshold in the water.
A. Super-Kamiokande data
Pulses on PMTs exceeding a charge threshold corre-sponding to roughly 0.1 photo-electrons are registered ashits, all of which are processed by a software trigger sys-tem [85]. For T2K analyses, all hits occurring in the 1 mswindows centered on each beam spill arrival are writtento disk. Beam spills are excluded from the analysis if theycoincide with problems in the data acquisition system orthe GPS system used to synchronize SK with the accel-erator at J-PARC. Additionally, spills that occur within100 µ s of a beam-unrelated event are rejected to reducethe contamination of T2K data with cosmic ray muondecay electrons. The beam spill selection introduces aninefficiency of 1%, with roughly half of this being due tothe preceding detector activity criterion.For the analysis presented here, events associated withaccepted spills are further required to have a recon-structed energy corresponding to an electron of at least30 MeV and no more than 15 hits in the largest OD hitcluster. Additional criteria are used to reject spuriousevents that originate from spontaneous corona dischargesin PMTs. Only events reconstructed in the [ − , µ swindow around the leading edge of the beam spill areused in the analysis.Distributions of the reconstructed times for events inboth 1 ms and [ − . , . µ s windows around the beamspill arrival are shown in Fig. 15. In the 1 ms window, apeak of events coincident with the beam arrival is clearlyseen; after applying the OD and minimum energy criteriavery few events remain outside this peak. The eight-bunch structure of the J-PARC beam is clearly seen in6 E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . FIG. 11: Final state muon momentum distributions of the FHC ν µ CC 0 π (top) and ν µ CC 1 π + (bottom) data andsimulation samples in FGD1 (left) and FGD2 (right).the narrow window event time distribution. B. Super-Kamiokande event simulation
Events at SK are simulated using J-PARC (anti-) neu-trino flux predictions and the neutrino interaction gen-erator
NEUT , which implements the neutrino interac-tion model described in detail in Section VI. Particlesresulting from the neutrino interactions are propagatedthrough an SK detector model using the same Geant3-based [34] SKDETSIM 13.90 package as in [26]. Thedetector model, including the optical properties of theultra-pure water and detector materials, is tuned to cal-ibration data [86].
IX. SUPER-KAMIOKANDE EVENTRECONSTRUCTION AND SELECTION
Events at SK are reconstructed with the FiTQun max-imum likelihood estimation algorithm [27]. While thisalgorithm was initially used exclusively for NC π back-ground suppression in the ν e appearance channel [26, 28],in recent T2K publications [16, 25] FiTQun was used for all aspects of event reconstruction. Updating the recon-struction tools prompted a re-optimization of the eventselection criteria, including an expansion of the fiducialvolume (FV).In this section, the reconstruction algorithm is brieflydescribed, as well as the updated event selection criteriaand the procedure for their optimization. A discussion ofthe systematic uncertainties related to the SK detectorconcludes the section. A. Event reconstruction algorithm
The FiTQun likelihood function consists of the prob-ability of each PMT registering a hit in a given event,and for hit PMTs, the probability density functions forthe charge and time of the hit. Particles in an event aredescribed by tracks (or track segments) parameterized byparticle type, momentum, direction and initial position.The FiTQun likelihood is a function of these track pa-rameters and multiple tracks can be combined to formcomplex event hypotheses.In an initial pre-fitting stage, the approximate loca-tion of the neutrino interaction is found with a simpli-fied likelihood using only the time of the PMT hits. A7 )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . FIG. 12: Distributions of the final state muon angle of the FHC ν µ CC 0 π (top) and ν µ CC 1 π + (bottom) data andsimulation samples in FGD1 (left) and FGD2 (right).residual time is calculated for each PMT hit by subtract-ing the Cherenkov photon time-of-flight, calculated usingthe straight-line distance from the vertex position to thePMT, from the hit time. Hits are associated to one ormore clusters in residual time, with the initial clustercontaining hits due to particles produced in the neutrinointeraction and subsequent clusters containing hits dueto products of weakly decaying prompt particles. Eachhit cluster is then reconstructed separately by maximiz-ing the likelihood function for the e , µ , π + and p single-particle hypotheses.For the earliest hit cluster only, multiple-track eventhypotheses are also reconstructed using the results of thesingle-particle fits as the starting point. A multi-particlesearch algorithm is used to determine the number of par-ticles observed in the event. This algorithm proceeds byiteratively adding a new electron-like or π + -like track tothe event until the best-fit likelihood after adding thenew track fails to improve beyond a set threshold. Inthe analysis described here, additional event hypothesestargeting neutral current backgrounds are used: a π hy-pothesis consisting of two electron-like tracks consistentwith a π → γγ decay, and a π + hypothesis with twotrack segments compatible with a π + undergoing a hardscatter. B. Event selection
Events are selected into samples using cuts on best-fitlikelihood ratios between signal-like and background-likehypotheses: Λ αβ def = log L α L β , where α and β are competinghypotheses. The cut points are typically parameterizedas a function of reconstructed kinematics, such asthe best-fit electron momentum or the reconstructedinvariant mass obtained from the π hypothesis best-fitkinematics.Five signal-enriched SK samples are used in the anal-ysis. Samples of events containing a single reconstructed µ -like ring (1R µ ) and a single reconstructed e -like ring(1R e ) target ν µ and ν e CCQE interactions in both FHCand RHC beam modes. An additional sample, used inFHC data only, targets CC 1 π + interactions where the π + is below Cherenkov threshold. The π + is identifiedby the detection of a delayed µ -decay electron followingthe single prompt electron which results from the CCinteraction (1R e + 1 d.e). The CCQE-like selectioncriteria are the same for FHC and RHC samples.Events in all samples are required to be fully contained(FC) in the ID using the cut on OD activity described8 E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || ||
FIG. 13: Final state muon momentum distributions for the RHC ν µ (top) and ν µ (bottom) CC 1-track (left) andCC N -track (right) FGD1 simulation samples. These distributions are before the ND280 fit.in Sec. VIII above and to have only one prompt recon-structed particle identified by the multi-particle iterativesearch algorithm. Events are separated into e -like and µ -like with a criterion based on the likelihood ratio ofthe best-fit e -like to µ -like hypothesis (Λ eµ ) and the re-constructed momentum for the e hypothesis ( p e ).The FV criteria are defined in terms of the distancefrom the event vertex to its closest point on the detectorwalls ( wall ) and the distance from the event vertex tothe detector wall along the track direction ( towall ). Thisparameterization of the FV allows for a larger volume ofthe detector to be used by reducing the wall thresholdcompared to previous T2K neutrino oscillation analyses,while ensuring that Cherenkov rings projected on thedetector walls illuminate a large number of PMTs withthe towall criterion, introduced for the first time in theanalysis described here. The wall and towall criteriaare chosen separately for each sample to maximize thesensitivity to θ and δ CP , as described in Sec. IX D. Forthe µ -like samples, a minimum wall of 50 cm is required,with a minimum towall of 250 cm. The requirements forthe e -like samples with no decay- e are wall >
80 cm and towall >
170 cm, while for the sample with one decay- ewall >
50 cm and towall >
270 cm are required.For both FHC and RHC FC events, the distributions of the number of reconstructed particle tracks are shownin Fig. 16. For events with a single reconstructed track,the distributions of the e / µ discriminator and number ofidentified µ -decay electrons are shown in Figs. 17 and 18respectively. In these figures, and throughout this sec-tion, the MC predictions are produced with the neutrinomixing parameters given in Tab. III, and the flux andcross-section parameters set to the best-fit value result-ing from the near detector analysis described in Sec. XI.The reconstructed momentum is required to be largerthan 100 MeV/ c for the e -like samples to reduce contam-ination from below-threshold- µ decays, and larger than200 MeV/ c for the µ -like samples.Events in the µ -like samples can have up to one recon-structed decay- e , while the e -like samples are requiredto have zero and one decay- e for the samples targetingCCQE and CC1 π + interactions, respectively.Neutral current π production events are a backgroundin both e -like and µ -like samples. In the former, elec-tromagnetic showers resulting from π → γγ decays canmimic an electron-like event; while in the latter, chargedpion detector signatures are only significantly differentfrom those of muons through their hadronic interactions,which are not always present. Additional criteria areused to remove these backgrounds in all analysis sam-ples.9 )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . FIG. 14: Distributions of the final state muon angle for the RHC ν µ (top) and ν µ (bottom) CC 1-track (left) andCC N -track (right) FGD1 simulation samples. These distributions are before the ND280 fit.TABLE III: Reference set of values of the oscillationparameters used to evaluate the expected sensitivity,number of events and effect of systematic uncertainties.The values of sin θ and ∆ m are taken from [42],the value of sin θ from [43], and all the other valuesare set to the most probable value found in a previousT2K measurements [29]. Parameter Valuesin θ θ θ δ CP − m (NO)∆ m (IO) 2.509 × − eV /c ∆ m × − eV /c Mass Ordering Normal
For the µ -like samples a cut is applied on the likelihoodratio of the best-fit single- π + to the single- µ hypotheses(Λ π + µ ) as a function of reconstructed µ momentum ( p µ ).This selection criterion is only available in the FiTQunreconstruction algorithm and is deployed for the first timein the current analysis. The NC π rejection criterion for the e -like samples,based on the likelihood ratio of the best-fit π hypothesisto the e -like hypothesis (Λ π e ) and the reconstructed π mass ( m γγ ), is unchanged from previous analyses, whereit was the only use of FiTQun reconstruction. Distribu-tions of the neutral current π rejection discriminator forthe 1R e and 1R e + 1 d.e. samples are shown in Figs. 19and 20 respectively. For 1R µ events, the distribution ofthe neutral current π + discriminator is shown in Fig. 21.Finally, e -like samples are required to have a recon-structed neutrino energy ( E rec ) lower than 1250 MeV,as events beyond this energy are insensitive to neutrinooscillations and are more susceptible to systematic un-certainties associated with the neutral current rejectioncut.The selection criteria described above are summarizedin Tab. IV and the breakdown of data and MC eventspassing each cut is given in Tabs. V, VI and VII, for thesamples targeting ν e CCQE, ν e CC 1 π + and ν µ CCQEinteractions, respectively.Distributions of reconstructed vertices of 1R e eventsin FHC and RHC data are shown in Figs. 22 and 23respectively, and for 1R e + 1 d.e. events in Fig. 24.Reconstructed neutrino energy distributions are shownin Figs. 25 and 26 for 1R e and 1R e + 1 d.e. events,0 - - s) m ( T D s ) m N u m b e r o f e v e n t s / ( Low energy eventsHigh energy events with OD activityHigh energy events without OD activity < 30 MeV vis and EHigh energy events without OD activity 30 MeV ‡ vis and E - (ns) T D N u m b e r o f e v e n t s / ( n s ) High energy events without OD activity
FIG. 15: Reconstructed event time distributions in the1 ms (top) and [ − . , . µ s (bottom) windows aroundthe beam spill leading edge.respectively, and in Fig. 27 for 1R µ events. C. Optimization of selection criteria
The likelihood ratio of best-fit e and µ hypotheses givesvery good separation between these classes of events,with the separation improving at higher momentum. Thecut line chosen to select e -like and µ -like events accountsfor the momentum dependence of the likelihood ratio andachieves mis-identification rates smaller than 1% for trueCCQE events across the T2K energy range.Pion production in neutral current events forms one of Number of reconstructed particles N u m b e r o f e v e n t s DataCCQECC non-QENeutral current
Number of reconstructed particles N u m b e r o f e v e n t s DataCCQECC non-QENeutral current
FIG. 16: Distribution of number of reconstructedparticles for events passing the wall >
80 cm and towall >
170 cm FV criteria in FHC (top) and RHC(bottom) data.the main backgrounds to both µ -like and e -like selections.Furthermore, since the cross sections for these processesare not known precisely, these contributions carry signif-icant systematic uncertainties into the signal samples. Asimplified neutrino oscillation analysis framework is usedto optimize the neutral current rejection criteria takinginto account systematic uncertainties and with statisticscorresponding to an exposure of 7 . × POT, evenlysplit between neutrino and antineutrino modes. In thissimplified analysis framework, the systematic uncertain-ties (taking into account the near detector constraints)are propagated to the SK prediction as a covariance ma-1TABLE IV: SK selection criteria for the five fully-contained, single prompt particle, analysis samples. µ e e + 1 d.e. wall >
50 cm >
80 cm >
50 cm towall >
250 cm >
170 cm >
270 cm e / µ identification Λ eµ < p e /c Λ eµ > p e /c Λ eµ > p e /c e / µ momentum >
200 MeV /c >
100 MeV /c >
100 MeV /c Number of decay- e ≤ π rejection Λ π + µ < p µ
40 MeV /c Λ π e < − m γγ
40 MeV /c Λ π e < − m γγ
40 MeV /c E Rec QE — < < TABLE V: Expected number of 1R e signal and background events passing each selection stage, compared to thedata. ν µ + ν µ ν e + ν e ν + ν ν µ → ν e ν µ → ν e FHC CC CC NC CC CC MC total DataFC and FV 692.28 43.10 241.98 87.18 0.80 1065.34 1077Single particle 307.47 22.18 44.31 73.09 0.61 447.65 451Electron-like 8.72 22.16 26.38 72.99 0.61 130.85 151p e >
100 MeV/c 3.22 22.00 18.45 71.56 0.61 115.84 131No decay-e 0.88 18.73 15.57 64.60 0.59 100.37 108Erec < π e >
100 MeV/c 1.41 10.83 9.92 4.06 8.75 34.97 32No decay-e 0.41 9.48 8.60 3.47 8.58 30.53 28Erec < π trix in reconstructed neutrino energy. As a result, theSK samples do not constrain the nuisance parameters,and no correlations between these and the neutrino mix-ing parameters are taken into account. The four samplestargeting e -like and µ -like CCQE events in both FHCand RHC neutrino beam mode are fit simultaneously toan Asimov data set [87] to determine the experiment’ssensitivity under different neutral current rejection cutpoints.The criterion to reject NC π + events in the µ -like sam-ples, a line cut on the Λ π + µ vs p µ plane, is chosen tominimize the width of the sin θ σ confidence interval.This cut, which was not available with the reconstruc-tion techniques used in previous T2K neutrino oscillationanalyses, reduces the NC contribution to the µ -like sam-ples by a factor of two, while selecting CCQE events with99% efficiency.The NC π rejection line cut in the Λ π e vs m γγ plane,applied to the e -like samples, is optimized based on thesignificance to exclude δ CP = 0. As the optimal cut lineis very close to the one used in previous T2K neutrinooscillation analyses, this criterion is not updated for the analysis described here. It should be noted that the rel-ative impact of this cut on the selected sample is sig-nificantly smaller in the analysis described here, whereit reduces the NC contribution in the e -like samples bya factor of three, compared to previous analyses, wherethe reduction is of a factor of 6. This is due to the excel-lent performance of the multi-particle search algorithm,which reduces the NC background in e -like samples bya factor of five by correctly identifying the two γ s from π decays with higher efficiency than the previously usedalgorithms.Optimization metrics for both neutral current rejectioncriteria are shown as a function of the cut parameters inFigs. 28 and 29, along with the chosen cut points and dis-tributions of the signal and background events in the cutvariable planes. While this optimization is performed as-suming the neutrino mixing parameters preferred by pre-vious T2K results as given in Tab. III, it should be notedthat in both cases the optimization metrics show largeregions around the optimal points where they are con-sistently good. Therefore, the sensitivity of this analysisdoes not depend strongly on the exact value of the cutpoints and the experimental sensitivity is not expected2TABLE VI: Expected number of 1R e + 1 d.e. signal and background events passing each selection stage, comparedto the data. ν µ + ν µ ν e + ν e ν + ν ν µ → ν e ν µ → ν e FHC CC CC NC CC CC MC total DataFC and FV 697.81 43.87 247.50 87.30 0.81 1077.27 1085Single particle 303.60 22.24 44.39 72.83 0.61 443.67 443Electron like 8.46 22.22 26.98 72.74 0.61 131.01 148p e >
100 MeV/c 2.81 22.05 18.73 71.24 0.61 115.44 129One decay-e 1.35 3.04 2.18 6.81 0.02 13.40 23E rec < π TABLE VII: Expected number of 1R µ signal and background events passing each selection stage, compared to thedata. ν e + ν e ν + ν ν µ + ν µ ν µ ν µ FHC CC NC CC non-QE CCQE CCQE MC total DataFC and FV 125.04 234.01 373.95 251.21 14.20 998.41 1002Single particle 92.94 43.30 63.00 220.14 12.53 431.91 429Muon like 0.10 17.78 58.93 215.84 12.44 305.08 285p µ >
200 MeV/c 0.10 17.66 58.89 215.63 12.44 304.71 2840 or 1 decay-e 0.10 17.07 37.99 213.41 12.31 280.88 255Not π + µ >
200 MeV/c 0.02 8.63 33.07 36.18 66.40 144.30 1590 or 1 decay-e 0.02 8.37 25.22 35.76 65.71 135.07 144Not π + to depend strongly on the choice of neutrino mixing pa-rameters used in the optimization procedure. D. Fiducial volume expansion
In previous T2K neutrino oscillation analyses, eventsat SK were required to have wall ≥
200 cm in orderto remove entering backgrounds and ensure the qualityof reconstructed quantities. The new event selection pre-sented here, benefiting from improvements in reconstruc-tion, provides an opportunity to re-optimize the FV cri-terion for the T2K analysis samples, with the objectiveof increasing the event yield while maintaining a highpurity of signal events in the selected samples and a lowimpact of detector systematic uncertainty.A two-dimensional parameterization of the FV crite-ria is chosen to allow for balancing two classes of effects.On one hand, the reduction of entering backgrounds andmitigation of the impact of known shortcomings of thesimulated detector geometry are achieved with a wall threshold, as in previous analyses of T2K data. On theother hand, important aspects of event reconstruction,such as particle identification, improve with the numberof PMTs illuminated by the Cherenkov ring patterns. Asthis number grows with the distance to the detector walls along the particle direction of travel, towall , a thresholdon this distance is used to select events with a high re-construction performance.The FV criteria are optimized in a fit to SK atmo-spheric neutrino data, from which the systematic uncer-tainty associated to the particle counting and identifica-tion is also extracted.The SK atmospheric neutrino FC data is divided into18 samples consisting of combinations of six detector re-gions, defined with wall and towall as shown in Fig. 30,and three classes of events discriminated by the num-ber (0, 1 or 2+) of detected Michel electrons. The sixdetector regions were chosen to isolate areas where thesystematic uncertainty associated to the detector modelis expected to differ, while maintaining an adequate levelof statistics in the SK atmospheric neutrino data sample.These samples are projected into three particle identifi-cation variables (Λ eµ , Λ π e and Λ π + µ ) and a continuous vari-able that discriminates single-particle from multi-particleevents (Λ − particles − particle ).In each of the samples the MC is split into six trueevent topologies consisting of: a single visible e , a singlevisible µ , a visible e with other visible particles, a visi-ble µ with other visible particles, a single π , and finallyevents with a single visible p or π + . For each topology,3 - - - - PID discriminator m e/ N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like e n -like m n - - - - PID discriminator m e/ N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like e n -like m n FIG. 17: Distribution of the e / µ PID discriminator forsingle-particle events passing the wall >
80 cm and towall >
170 cm FV criteria in FHC (top) and RHC(bottom) data.the particle identification and counting variables are lin-early transformed with two nuisance parameters each:Λ (cid:48) αβ = a Λ αβ + b (13)where Λ (cid:48) αβ is the transformed variable and a and b are“scale” and “shift” nuisance parameters, respectively.These “scale” and “shift” parameters are estimated witha Markov Chain Monte Carlo (MCMC) method [88] thatsamples the Poisson likelihood for the observed datagiven the model that includes, in addition, parametersto capture the uncertainty on the atmospheric neutrinoflux and cross sections. The SK atmospheric neutrino Number of decay-electrons N u m b e r o f e v e n t s Data CCQE e n and e n CCQE m n and m n CC non-QENeutral current
Number of decay-electrons N u m b e r o f e v e n t s Data CCQE e n and e n CCQE m n and m n CC non-QENeutral current
FIG. 18: Distribution of the number of identifieddecay-electrons for single-particle events passing the wall >
80 cm and towall >
170 cm FV criteria in FHC(top) and RHC (bottom) data.data used in this analysis were collected between Octo-ber 2010 and May 2015. To ensure the validity of theatmospheric neutrino fit for the T2K oscillation analysis,high-statistics control sample data collected over the en-tire beam data period are used to monitor the detectorstability.The flux and cross-section parameterizations used inthis procedure are simpler than those used in oscillationanalyses and are based on Refs. [89] and [90]. The at-mospheric neutrino flux is scaled by two parameters, oneaffecting events with neutrino energy lower than 1 GeV,with a prior uncertainty of 25% and the other for events4 - - - PID discriminator p e/ N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like e n -like p - - - PID discriminator p e/ N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like e n -like p FIG. 19: Distribution of the e / π PID discriminator for1R e events in FHC (top) and RHC (bottom) data.with energies higher than 1 GeV, with a prior uncertaintyof 15%. The ratio of ν e + ν e to ν µ + ν µ events is con-trolled by a parameter with 5% prior uncertainty. A prioruncertainty of 20% is assigned to both CC non-QE andNC cross sections. The CCQE cross section has a moredetailed parameterization, with events below 190 MeVhaving a prior uncertainty of 100% and events in the 190MeV to 1 GeV range being characterized by 11 param-eters which scale the cross section in unevenly spacedenergy bins with gradually decreasing prior uncertaintiesfrom 41% to 2.2%, with 0.6 GeV events being assigned5.4% prior uncertainty. CCQE events with energy in the1 to 2 GeV and 2 to 3 GeV ranges are assigned 1.7% and0.9% prior uncertainty, respectively. Finally, each sample - - - PID discriminator p e/ N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like e n -like p FIG. 20: Distribution of the e / π PID discriminator for1R e + 1 d.e. in FHC data.used in the fit is assigned an unconstrained overall scaleparameter.The choice of fitted variables and conservative flux andcross-section prior uncertainties reduces the sensitivity ofthis fit to the neutrino oscillation parameters, which arefixed at the reference values given in Tab. III.Examples of the posterior predictive distributions re-sulting from the MCMC sampling are shown in Fig. 31.To quantify the impact of the disagreement betweendata and MC on the T2K samples, the “shift” and “scale”parameters are sampled from the MCMC posterior andapplied to the T2K beam MC. The T2K selection criteriaother than FV are applied, and a fractional uncertainty iscalculated for each analysis sample in each of the detec-tor regions, with the MC separated in five true categories:CCQE, CC non-QE, CC events with mis-identified lep-ton flavor, neutral current, and entering backgrounds. Inthis procedure, the atmospheric neutrino flux and cross-section parameters used in the MCMC fit are marginal-ized over. The resulting fractional uncertainty on theexpected number of events is taken to be the systematicuncertainty associated to the SK detector model.With the detector systematic uncertainty estimated foreach detector region, the optimal FV criteria are found bymaximizing the following figure of merit, which quantifiesthe sensitivity of the sample with respect to changes in θ and δ CP for µ -like and e -like samples, respectively: F.O.M = (cid:16) ∂ ˆ N∂θ (cid:17) ˆ N + σ syst (14)where ˆ N is the expected number of events in a givensample, θ is the parameter of interest ( θ for the µ -like samples and δ CP for the e -like samples) and σ syst is5 - - - - - - PID discriminator + p / m N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like m n -like + p - - - - - - PID discriminator + p / m N u m b e r o f e v e n t s Data CC e n and e n CC m n and m n Neutral current -like m n -like + p FIG. 21: Distribution of the µ / π + PID discriminatorfor 1R µ events in FHC (top) and RHC (bottom) data.the systematic uncertainty, including uncertainties asso-ciated to the detector and cross-section models as well asthe uncertainty on the number of entering backgrounds.A minimum requirement of at least 50 cm in wall and150 cm in towall is chosen based on MC studies whichshow deterioration in momentum reconstruction beyondthose regions. Different FV criteria are chosen for the1R µ , 1R e and 1R e + 1 d.e. samples, with the same cutsapplied in equivalent selections in neutrino and antineu-trino modes. The figure of merit is shown as a function of wall and towall for the three samples in Fig. 32. The op-timal criteria resulting from this procedure are describedin Tab. IV.The combined effect of the new reconstruction algo- ) cm (10 r z ( c m ) -2000-1500-1000-5000500100015002000 Accepted Rejected 80 cm from wall x (cm) -2000 -1500 -1000 -500 0 500 1000 1500 2000 y ( c m ) -2000-1500-1000-5000500100015002000 AcceptedRejected
80 cm from wall
FIG. 22: Reconstructed vertices of selected FHC data1R e events projected in z vs r (top) and y vs x (bottom). The neutrino beam direction is shown as ared arrow and events which do not pass the fiducialvolume criteria are shown as hollow crosses.rithm, optimized neutral current rejection cuts and ex-panded FV volume has a significant impact across allanalysis samples. The signal acceptance in the 1R e sam-ple increases by 20%, with the same purity as in pre-vious analyses, while the signal events in the 1R e + 1d.e. sample increase by 30%, with a reduction in themis-identified muon background of 70%. The 1R µ sam-ples have an increase in signal efficiency of 15% and areduction in backgrounds of 40%.6 ) cm (10 r z ( c m ) -2000-1500-1000-5000500100015002000 Accepted Rejected 80 cm from wall x (cm) -2000 -1500 -1000 -500 0 500 1000 1500 2000 y ( c m ) -2000-1500-1000-5000500100015002000 AcceptedRejected
80 cm from wall
FIG. 23: Reconstructed vertices of selected RHC data1R e events projected in z vs r (top) and y vs x (bottom). The neutrino beam direction is shown as ared arrow and events which do not pass the fiducialvolume criteria are shown as hollow crosses. E. Systematic uncertainty
The systematic uncertainty associated with the SKevent selection is propagated to the oscillation analysisfitting frameworks as a covariance matrix in bins of ei-ther reconstructed neutrino energy or reconstructed lep-ton momentum, and broken down in true event topolo-gies.Systematic uncertainties on the particle count andidentification are extracted from the atmospheric neu-trino data with the MCMC method described above. ) cm (10 r z ( c m ) -2000-1500-1000-5000500100015002000 Accepted Rejected 50 cm from wall x (cm) -2000 -1500 -1000 -500 0 500 1000 1500 2000 y ( c m ) -2000-1500-1000-5000500100015002000 AcceptedRejected
50 cm from wall
FIG. 24: Reconstructed vertices of selected FHC data1R e + 1 d.e. events projected in z vs r (top) and y vs x (bottom). The neutrino beam direction is shown as ared arrow and events which do not pass the fiducialvolume criteria are shown as hollow crosses.The uncertainties on decay- e tagging and mis-identification of muons as electrons are extracted fromdifferences between the data and the MC in a controlsample consisting of cosmic ray muons that stop withinthe ID. The cosmic ray muon events are weighted in mo-mentum and towall to match the expected distributionof beam-induced muons. The uncertainty on the decay- e identification efficiency is 1% and the uncertainty on therate of spurious decay- e tags is 0.2%. The relative uncer-tainty on the mis-identification of muons as electrons is30%, though the contamination of ν µ CC events is smaller7 (MeV)
QErec E N u m b e r o f e v e n t s / ( M e V ) Data CC e n fi m n CC e n fi m n CC e n fi e n and e n fi e n CC m n and m n Neutral Current (MeV)
QErec E N u m b e r o f e v e n t s / ( M e V ) Data CC e n fi m n CC e n fi m n CC e n fi e n and e n fi e n CC m n and m n Neutral Current
FIG. 25: Reconstructed energy distribution for 1R e events in FHC (top) and RHC (bottom) data.than 1% in the ν e samples without decay-electrons andaround 2% in the 1R e + 1 d.e. sample.Uncertainties introduced by the FV criteria are also es-timated with MC to data comparisons in the cosmic raymuon sample, with both vertex and direction uncertain-ties taken into account. The reconstructed vertices in thecosmic ray muon sample cluster at the top or side walls ofthe detector, allowing for shifts in the MC relative to thedata to be identified. The uncertainty on the directionis estimated by comparing the reconstructed muon direc-tion to the equivalent quantity estimated using the muonand subsequent decay- e vertices. The uncertainties are2.5 cm for the vertex position and 0.24 degrees for thedirection, corresponding to a 0.3% to 0.4% systematic un- (MeV) D rec E N u m b e r o f e v e n t s / ( M e V ) Data CC e n fi m n CC e n fi m n CC e n fi e n and e n fi e n CC m n and m n Neutral Current
FIG. 26: Reconstructed energy distribution for 1R e + 1d.e. events in FHC data. The ∆(1232) mass is used forthe final-state nucleon in the reconstructed neutrinoenergy calculation.certainty on the FV, depending on the analysis sample.This uncertainty is dominated by the uncertainty on thevertex position, with the direction playing a negligiblerole.The uncertainty on the π rejection efficiency in 1R e samples is estimated using hybrid π sample constructedby superimposing an e -like event from the atmosphericneutrino or decay- e from cosmic ray muon data with asimulated γ with kinematics taken from NC π events inthe MC. The procedure is performed using both the MCand the real data as the source of the event and the dif-ference in π rejection efficiency between the data–MCand MC–MC samples is taken as the systematic uncer-tainty, binned in reconstructed lepton momentum andangle with respect to the beam. The overall uncertaintyon the π rejection efficiency is 26%.A summary of the uncertainties associated with theSK detector model is given in Tab. VIII.To propagate the systematic uncertainty on the SKevent selection to the neutrino oscillation analysis frame-works, a covariance matrix is computed using the beamMC and the uncertainties quoted above. The MC isweighted with the flux and cross-section parameters attheir central value from the fit to the near detector dataand neutrino oscillation weights using the parameters inTab. III are applied. Variations of the MC are then pro-duced with random throws of the systematic effects de-scribed above and projected into bins of true event topol-ogy and reconstructed neutrino energy or lepton momen-tum to produce the covariance matrix. The diagonal ele-ments of the covariance matrix in reconstructed neutrinoenergy are shown in Fig. 33.8TABLE VIII: Super-Kamiokande detector systematic uncertainties Source 1 σ uncertainty SampleDecay- e tagging efficiency 1.0% Cosmic ray muonSpurious decay- e tagging rate 0.2 / event Cosmic ray muon µ → e mis-identification 30% Cosmic ray muonFiducial volume acceptance 0.3 - 0.4% Cosmic ray muonNC π rejection efficiency 26 % Hybrid π e / µ , π / e , π + / µ , single/multi particle identification — Atmospheric neutrino X. NEAR/FAR EXTRAPOLATION FIT
Parameterized flux and cross-section models are usedto calculate the predicted event rates at ND280 and SK.These models are fit to the high statistics, unoscillatednear detector data to constrain the parameter uncertain-ties and tune their central values. An additional uncer-tainty is included in the flux covariance to account forthe fact that the near detector fit results for Runs 2–6are extrapolated to the far detector, which uses a largerdata set from Runs 2–9. As in Ref. [26], additional un-certainties which affect ( − ) ν e events have been introduced.These account for effects which may potentially affect ( − ) ν e but not ( − ) ν µ cross sections.The ND280 likelihood and fitting methods are un-changed since the analysis described in Ref. [26]. The14 event samples are binned in p µ and cos θ µ , giving 1624bins in total. The full likelihood includes a contributionfrom the binned χ data-model comparison, and a priorpenalty contribution for each parameter.There are two near detector fitting frameworks used inthis analysis. One fitter uses MINUIT to find the param-eters which maximize the likelihood, while the other usesMarkov Chain Monte Carlo (MCMC) methods to samplethe parameter space. Both frameworks treat the system-atics identically, and apply parameter variations on anevent-by-event basis in the fit. The resulting parametervalues from the MINUIT-based fit are used by two of theoscillation analyses (detailed in Sec. XI), with a covari-ance matrix describing their uncertainties. The MCMCanalysis performs joint near and far detector data fits,but can also run near detector only fits for cross groupvalidation.The only change to the fitting frameworks since thelast analysis was the treatment of the Fermi surface mo-mentum systematics near their physical boundary. Pre-viously, the covariance could not be calculated when theparameter was at its limit, causing the fit to not con-verge. For this analysis, the penalty contribution to thelikelihood was ‘mirrored’ around the physical boundariesfor the p F parameters. This meant the parameters wereallowed to pass beyond their physical boundaries. Themirroring was performed by setting the likelihood for val-ues beyond the boundary to the value of the likelihoodthe same distance from the boundary on the other side.For example, for a physical boundary at +1 .
0, the like- lihood at +1 . .
8. This allowed the uncertainty to be calculated atthe limit, and the fit to converge.The prefit and postfit SK flux and cross-section pa-rameter values and uncertainties are shown in Figs. 34and 35, as a fraction of the nominal values. The cen-tral values and uncertainties for all parameters are tab-ulated in the Appendix. There is a significant reductionin the postfit uncertainty for the majority of parame-ters. Those that are not constrained by the near detec-tor fit are uncertainties that only apply to interactionswith low statistics in the near detector. In the last anal-ysis, Ref. [26], the neutrino flux increased for all samplesand species, but this effect is no longer present. The dif-ference between the nominal simulation and data is nowbeing absorbed by the movement of other parameters, inparticular the BeRPA model.The goodness-of-fit for the near detector analysis wasestimated by calculating the p -value in the MINUIT-based framework. Toy data sets were produced by throw-ing all systematics according to their prior covariance,and applying them to the nominal simulation prediction.The likelihood for each toy data set is shown in Fig. 36,along with the likelihood from the data fit. The overall p -value for the fit is 47.3%.The postfit muon momentum and angle distributionsof events are produced by applying the best-fit parame-ter values to the nominal simulation. These are shownbroken down by interaction mode for each sample in Fig-ures 37, 38, 39 and 40, along with the observed distri-butions. There is much better agreement with the datathan for the prefit distributions shown in Figures 11, 12,13 and 14. The numbers of postfit predicted events forall the 14 samples are shown in Tab. II.The best-fit values are also used to produce SK eventrate predictions, shown for the FHC 1R µ and 1R e SKsamples in Figs. 41 and 42. The total predicted andobserved event rates are shown in Tab. IX.Near detector only fits with the MCMC analysis frame-work were used to cross-check the two analyses. Thepostfit cross-section parameters for the two fits are com-pared in Fig. 43, showing good agreement.9 (MeV)
QERec E N u m b e r o f e v e n t s / ( M e V ) Data CC non-QE m n CCQE m n CC non-QE m n CCQE m n CC e n and e n Neutral Current (MeV)
QERec E N u m b e r o f e v e n t s / ( M e V ) Data CC non-QE m n CCQE m n CC non-QE m n CCQE m n CC e n and e n Neutral Current
FIG. 27: Reconstructed energy distribution for 1R µ events in FHC (top) and RHC (bottom) data. XI. OSCILLATION ANALYSIS FITTERS
To produce constraints on the three-flavor PMNS os-cillation parameters, the rate and kinematic distributionsof all five SK event samples are analyzed simultaneously.Systematic uncertainties in the flux, interaction and de-tector models are accounted for using systematic param-eters applied as weights to the nominal prediction as de-scribed in [26]. Confidence regions and intervals are pro-duced from marginal likelihood distributions as a func-tion of parameters of interest.The predicted kinematic distributions of each SK sam-ple are generated from the nominal SK simulation towhich weights are applied for each set of oscillation and - Cut line slope (c/MeV) - - - C u t li n e i n t e r ce p t ) · i n t e r v a l ( s q W i d t h o f s i n Chosen cut (MeV/c) m Reconstructed p - - m + pL N u m b e r o f N C e v e n t s Number of CC events
FIG. 28: The width of sin θ σ confidence interval asa function of the slope and intercept of the NC π + rejection cut line is shown on the top and thedistribution of signal and background events is shownon the bottom, along with the line below which µ -likeevents are selected.TABLE IX: Observed and predicted event rates for SKsamples, pre- and post-ND280 fit. Sample Data Prefit ND280 PostfitFHC 1R µ
243 250 .
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73 16.77 ± - - - - /MeV) Cut line slope (c C u t li n e i n t e r ce p t = C P d t o e x c l ud e cD Chosen cut ) (MeV/c gg Reconstructed m e pL N u m b e r o f N C e v e n t s Number of CC events
FIG. 29: The sensitivity to exclude δ CP = 0 as afunction of the slope and intercept of the NC π rejection cut line is shown on the top and thedistribution of signal and background events is shownon the bottom, along with the line below which e -likeevents are selected.systematic parameter values. The oscillation weights cor-respond to the three-flavor oscillation probability, calcu-lated with matter effects and dependent on the true neu-trino energy and flavor [22], while the systematic weightsare multiplicative factors.A likelihood, L , is calculated according to Eq. 15 as theproduct of the Poisson likelihood ratios for the number ofevents in each bin of the kinematic variables consideredfor each SK sample: towall (cm) w a ll ( c m ) FIG. 30: Detector regions used in detector systematicuncertainty estimation. − L ( o , f ) = 2 (cid:88) s =0 N s − (cid:88) i =0 (cid:16) n obss,i · ln (cid:0) n obss,i /n exps,i (cid:1) + (cid:0) n exps,i − n obss,i (cid:1) (cid:17) . (15)In Eq. 15, o is a vector of the parameters of inter-est, f is a vector of nuisance parameters, n obss,i is the ob-served number of events in kinematic bin i of SK sample s which has a total of N s bins, n exps,i = n exps,i ( o , f ) is thecorresponding expected number of events. The param-eter(s) of interest correspond to one or more oscillationparameters among sin θ , ∆ m / , sin θ , and δ CP .The nuisance parameters correspond to the systematicparameters, and the oscillation parameters not chosen asparameters of interest in a given fit. To obtain a likeli-hood which only depends on the parameter(s) of interest o and a data set x (the set of n obss,i in Eq. 15), a marginallikelihood L marg is computed: the full likelihood, madeof the product of the likelihood L defined in Eq. 15 withthe prior constraint on some of the parameters π ( o , f ),is numerically integrated over the nuisance parameters: L marg = L marg ( o ; x ) = (cid:90) L ( o , f ; x ) π ( o , f ) d f . (16)External constraints are used for some of the oscil-lation parameters. The solar parameters, which havelimited impact on the observed event distributions atT2K, are either kept fixed to their nominal values orhave a Gaussian constraint applied, depending on theanalysis considered. The nominal values and uncertain-ties used are sin θ = 0 . ± .
021 and ∆ m =1 - - - - · PID discriminator m e/ N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MCSingle electron m Single (a) - - - - · PID discriminator m e/ N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MCSingle electron m Single (b) - Single/multiple particle discriminator N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MCMultiple particlesSingle particle (c) - Single/multiple particle discriminator N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MCMultiple particlesSingle particle (d) - PID discriminator p e/ N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MC p Single Single e (e) - - - - PID discriminator + p / m N u m b e r o f e v e n t s n Atmospheric Postfit MCTotal prefit MCSingle hadron m Single (f)
FIG. 31: Posterior predictive distributions of SK atmospheric MCMC fit to determine detector systematicuncertainties. The total nominal MC is shown in gray, with components targeted by each of the distributions shownin red and blue. The 68.27% intervals of the posterior predictive distribution is shown in green. Observed data areshown as black circles. All distributions shown are for the largest detector region, 5. Distributions of the e / µ PIDdiscriminator for 0 and 1 decay- e events are shown in (a) and (b). In (c) and (d), the particle-counting parameter isshown for events with 1 and 2 decay-electrons, respectively. The e / π discriminator distribution is shown in (e) for e -like events and and the µ / π + distribution is shown in (f) for µ -like events.(7 . ± . × − eV / c [42]. The three parame-ters sin θ , ∆ m and δ CP are unconstrained. T2K issensitive to sin θ , but to date, the world’s most accu-rate measurements of this parameter come from reactorneutrino experiments [10–12]. To obtain increased sen-sitivity to the other oscillation parameters, the reactoraverage of sin θ = 0 . ± . θ were also performed.Three different analysis frameworks are used to per-form the fit of the far detector data. They will be labeledas A, B and C, and follow the general procedure describedabove, but differ on a number of points summarized inTab. X: a. Kinematic information used to fit the data fromthe electron-like samples Two-dimensional distributionsare used for those samples, either the combination of themomentum and angle with respect to the beam directionof the particle reconstructed as the lepton ( p lep , θ lep ),or the reconstructed energy assuming CCQE kinematics(E rec ) combined to this angle θ lep . b. Oscillation probability calculation The events caneither be binned in true neutrino energy with an oscilla-tion probability corresponding to the mean true neutrinoenergy of the bin, or have individual oscillation probabil-ities computed for each event’s true neutrino energy. c. Use of the near detector data
The near detectordata are either used in a simultaneous fit with the fardetector data, or they can be fit separately to constrainthe neutrino flux and interaction systematic uncertain-ties. In this second case, the constraint is propagatedto the far detector analysis through a covariance matrix.2 F i gu r e o f m e r it towall (cm) w a ll ( c m ) ExcludedChosen cut F i gu r e o f m e r it towall (cm) w a ll ( c m ) ExcludedChosen cut F i gu r e o f m e r it towall (cm) w a ll ( c m ) ExcludedChosen cut
FIG. 32: Scans of the figures of merit used to optimize the FV criteria in the wall vs towall space that defines theFV for the analysis described here. Figures of merit corresponding to the 1R e (top left), 1R e + 1 d.e. (top right)and 1R µ (bottom) samples are shown here. The chosen cut point is indicated in the figures as is the region which isruled out by reconstruction performance considerations.The two methods are expected to lead to different resultsfor the far detector fit if the constraint on the systematicparameters obtained with the near detector data cannotbe properly described by a multi-variate normal distri-bution. d. Fitting method Two of the analyses use a gridsearch method, where the likelihood in terms of the pa-rameters of interest is computed for different fixed valuesof those parameters while marginalizing over the otherparameters. This marginalization over the nuisance pa-rameters is done through numerical integration. The lastanalysis uses an MCMC method to sample the parameter space, where the density of the obtained samples followsthe joint posterior probability density of the parameters.Those samples can then be binned to produce 1D or 2Dposterior distributions for the parameter(s) of interest,effectively marginalizing over the nuisance parameters.A full description of the MCMC analysis can be foundin [91].The T2K experiment’s expected median constraints onthe oscillation parameters can be evaluated by fitting toan Asimov data set, generated for the true values of theoscillation parameters given by Tab. III, and nominal val-ues of the systematic parameters. Using Analysis A, the3 -
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FHC Single- e FHC Single- RHC Single- e RHC Single- FHC Single- e (1 decay- e ) Analysis sample
FIG. 33: Square root of the diagonal elements of the covariance matrix describing the systematic uncertaintyassociated to the modeling of the SK detector.TABLE X: Differences between the three far detectoranalyses.
A B C e -like sampleanalysis bins ( E rec , θ lep ) ( E rec , θ lep ) ( p lep , θ lep ) P ( ν → ν )calculation Binned Event-by-event BinnedND280constraint Covariancematrix Simultaneousfit CovariancematrixFittingmethod Grid search MCMC Grid search expected sensitivity to δ CP is shown in Fig. 45, and for∆ m vs. sin θ in Fig. 46. The predicted and observedevent rates in each of the five SK samples are shownin Tab. XI at various δ CP values, with other oscillationparameters fixed to the values in Tab. III and with allsystematic parameters set to their nominal values. Simi-larly, the predicted and observed kinematic distributionsare shown in Fig. 44, with all oscillation parameters fixedto the values in Tab. III.Using the differences between the three analyses, it canbe checked that the results are not too sensitive to thedetails of how the analysis was performed. The expectedsensitivities obtained by each analysis for δ CP and ∆ m vs. sin θ are shown in Fig. 47. They show good agree-ment between each analysis, thus the differences in the TABLE XI: Predicted and observed total number ofevents in each Super-K sample. The oscillationparameters other than δ CP were fixed to the valuesgiven in Table III and all systematic parameters wereset to their nominal values. δ CP FHC RHC FHC RHC FHC ν e µ -like µ -like e -like e -like CC 1 π − π/ . . . . .
00 272 . . . . . π/ . . . . . π . . . . . analysis methods do not significantly impact the results.Additional comparisons between the three analyses wereperformed for each combination of parameter(s) of inter-est, mass ordering, use of reactor constraint. In each ofthese, the results produced by the three analyses werefound to be consistent. XII. SIMULATED DATA STUDIES
Equation 2 shows that the neutrino oscillation prob-ability depends upon the energy of the neutrino andits path length from creation to interaction. In long-baseline accelerator-based neutrino experiments the neu-trino path length is fixed. Accurately measuring the neu-4 (GeV) n E -
10 1 10 P a r a m e t e r V a l u e Prior to ND280 ConstraintAfter ND280 Constraint -mode n , m n SK (GeV) n E -
10 1 10 P a r a m e t e r V a l u e Prior to ND280 ConstraintAfter ND280 Constraint -mode n , e n SK (GeV) n E -
10 1 10 P a r a m e t e r V a l u e Prior to ND280 ConstraintAfter ND280 Constraint -mode n , m n SK (GeV) n E -
10 1 10 P a r a m e t e r V a l u e Prior to ND280 ConstraintAfter ND280 Constraint -mode n , e n SK FIG. 34: The SK flux parameters for the ν µ (top left) and ν e (top right) neutrino species in FHC, and for the ν µ (bottom left) and ν e (bottom right) neutrino species in RHC, as a fraction of the nominal value. The bands indicatethe 1 σ uncertainty on the parameters before (solid, red) and after (hatched, blue) the near detector fit. Q E M A C p F O p F n N o r m n N o r m N o r m C t o O S h a p e C S h a p e O B e R P A A B e R P A B B e R P A D B e R P A E B e R P A U C A R E S M A I s o B kg mn / e n mn / e n CC D I S CC C oh C CC C oh O N C C oh g N C N C O t h e r N ea r N C O t h e r F a r FS I I n e l . L o w E FS I I n e l . H i gh E P r od . p FS I A b s . p FS I FS I C h a r g e E x . L o w E FS I C h a r g e E x . H i gh E P a r a m e t e r V a l u e Prior to ND280 ConstraintAfter ND280 Constraint
FIG. 35: The cross-section parameters as a fraction ofthe nominal value. The bands indicate the 1 σ uncertainty on the parameters before (solid, red) andafter (hatched, blue) the near detector fit.trino oscillation parameters therefore requires a preciseunderstanding of the neutrino energy spectrum. All neu-trino experiments use models to link the observed finalstates back to the initial neutrino energy. There is no Minimum negative log-likelihood N u m b e r o f t oy e xp e r i m e n t s Expected distributionData valuep-value = 47.3%
FIG. 36: Distribution of the minimum negativelog-likelihood values from fits to the mock data sets(black), with the value from the fit to the datasuperimposed in red. The p -value is 47 . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . E v e n t s / ( M e V / c ) Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n Reconstructed muon momentum (MeV/c) D a t a / S i m . FIG. 37: Post ND280 fit muon momentum distributions of the FHC ν µ CC 0 π (top), and FHC ν µ CC 1 π (bottom)samples in FGD1 (left) and FGD2 (right).constructed neutrino energy in different ways. This isshown in Fig. 4, where the bottom plot compares thereconstructed energy bias between quasi-elastic-like and∆-like 2p2h interactions. The choice of interaction modeltherefore affects the neutrino energy distribution that ex-periments infer from their observed neutrino events. Thisin turn can affect their measurement of the neutrino os-cillation parameters, as has been shown in Ref. [92].In this analysis a comprehensive set of neutrino interac-tion models have been tested using simulated data studiesto quantify their effect on the T2K oscillation result. Thesimulated data procedure is described in Ref. [26], whilethe model changes that are tested here are described inSection VI. Simulated data is created for both the nearand far detectors, where a fit is performed in the sameway as for the real data. The resultant oscillation pa-rameter contours are then compared to those extractedfrom a fit to the Asimov data set. If the T2K oscillationanalysis is insensitive to the model change, or has thefreedom to account for it correctly, then the simulateddata contours and the Asimov contours should be verysimilar. The oscillation parameter values used for thisstudy are shown in Tab. III. Other parameter sets with δ CP = 0 and sin θ = 0 .
45 were also studied, but showedno significant difference to the results presented here.The likelihood distributions for each oscillation param- eter are created for both the simulated data fit and theAsimov fit. Any change in the center of the 2 σ confidenceinterval for each parameter is taken as a bias due to thechange in the interaction model introduced to the simu-lated data. This is compared to the uncertainty on theparameter coming from the systematic uncertainties in-cluded in the analysis. If a simulated data bias is greaterthan 50% of the systematic uncertainty on a parameterthen an additional uncertainty is added to the analysisto account for this. A. Simulated data study of the nucleon removalenergy
As described in Section VI the T2K neutrino event gen-erator, NEUT 5.3.3, implements a Relativistic Fermi Gas(RFG) nuclear model. To remove a nucleon from the nu-cleus requires energy to overcome the nuclear potential.The nucleon removal energy (NRE), can be measured byelectron scattering experiments, but is not known per-fectly. Even the definition of this quantity is not sim-ple [79]. In addition, the RFG is a very simple modelof the nuclear structure. More advanced models, such asspectral function models, provide a much more detaileddescription of the nucleon energies within the nucleus,6 )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n CC 2p-2h n p
CC Res 1 n p
CC Coh 1 n CC Other n NC modes n modes n -mode n ) m q cos( D a t a / S i m . FIG. 38: Post ND280 fit distributions of the final state muon angle of the FHC ν µ CC 0 π (top), and FHC ν µ CC 1 π (bottom) samples in FGD1 (left) and FGD2 (right).each of which has its own NRE.The T2K oscillation analysis does not have a parame-ter to account for the NRE uncertainty directly in the fit.Therefore the effect of the NRE uncertainty on the oscil-lation parameter measurements is evaluated using a sim-ulated data study. Neutrino events were generated withthe RFG nucleon removal energy increased by 18 MeVfrom the nominal value (25 MeV for interactions on Car-bon and 27 MeV for interactions on Oxygen). Increasingthe NRE results in a decrease in the energy availableto the lepton produced in the neutrino interaction. Asa consequence, the events simulated with a larger NREproduced leptons with lower momenta than the nominalMC. This momentum shift was calculated as a functionof neutrino energy and lepton angle, shown in Fig. 48.The momentum shift in Fig. 48 was applied to the recon-structed momenta of the charged lepton for the full nearand far detector MC. The MC was then scaled to matchthe POT exposure of the data, creating a simulated dataset. The ratio of the simulated data to the nominal MCis shown in Fig. 49 for the ND280 CC 0 π sample, high-lighting the shift in events from high momentum bins tolower momentum bins. A fit is then performed on thissimulated data using the analysis framework describedearlier in this paper. There are no parameters in theT2K cross-section model that change the lepton momen- tum directly, but combinations of parameters are ableto mimic this effect. The result of fitting the simulatednear detector data is presented in Fig. 50, which showsthe best fit values for the flux and cross-section modelparameters. Figure 50 shows that the low energy flux isincreased, to increase the rate of low momentum events.The 2p2h shape parameters control whether the 2p2hevents in the MC are produced more by ∆(1232) reso-nances (values above 1.0) or other modes (values below1.0). The ∆(1232) resonance produces leptons with alower momentum than the other modes. This means thatincreasing the shape parameter increases the rate of 2p2hevents in the low momentum region.The near detector fit result is used to predict the oscil-lated event distribution at the far detector. This is shownin Fig. 51, which compares the nominal MC to the sim-ulated data and the near detector prediction. The neardetector prediction is closer to the simulated SK datathan the nominal MC simulation, but does not matchwell below the oscillation dip. The nominal prediction isfit to the simulated SK data to extract the best fit os-cillation parameter values and their 2 σ confidence inter-vals. These are compared to oscillation parameter valuesand confidence intervals produced by fitting to the Asi-mov data set, as described in Sec. XI. This comparison7 E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || E v e n t s / ( M e V / c ) Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n D a t a / S i m . Reconstructed muon momentum (MeV/c) || || ||
FIG. 39: Post ND280 fit muon momentum distributions for the RHC ν µ (top) and ν µ (bottom) CC 1-track (left)and CC N -track (right) FGD1 samples.is shown in Fig. 52.Changing the NRE changes the shape of the simulateddata, not the normalization. Since the constraint onsin θ and δ CP is largely driven by the normalizationof the electron-like sample, it is unsurprising that theseparameters are relatively unaffected. In addition sin θ is strongly constrained by the reactor experiment mea-surements. However, both sin θ and ∆ m show asignificant difference between the nominal and the simu-lated data results, with sin θ shifting towards maximaldisappearance and ∆ m decreasing.The changes in the oscillation parameter contours indi-cate that the T2K cross-section model parameterizationcannot account for changes to the NRE. The near de-tector fit mis-attributes the NRE change to the flux and2p2h model parameters. The near detector post-fit pre-diction has a different neutrino energy distribution to thenominal MC, and so produces different far detector eventdistributions when the neutrino oscillation probability isapplied. As a result the oscillation parameters extractedfrom the simulated data fit no longer match those fromthe nominal MC analysis.To account for this in the T2K analysis an additionaluncertainty is introduced. A spline is created for each binof the five far detector sample histograms. The value ofthe spline is the ratio between the simulated far detector data and the far detector prediction calculated using thenear detector simulated data fit result. Spline knots arecreated using NREs of 18 MeV, 27 MeV (nominal) and45 MeV at both near and far detector. The splines are100% correlated across all sample bins, and produce amultiplicative weight to scale the far detector predictionin each bin. The spline takes into account the changein the far detector prediction due to both the changingNRE and the mis-fitting at the near detector. This meansthat it does not provide a measurement of the true NRE,but an ‘effective NRE’ taking into account both of theseeffects. A prior constraint is placed on this effective NREparameter, setting the central value to 27 MeV, with anuncertainty of ±
18 MeV.The result of including this parameter in the simulateddata study is shown in Fig. 53. The new parameter in-creases the size of the oscillation contours whilst shiftingthe simulated data result to be in much better agreementwith the expectation. There is some residual differencebetween the contours, particularly in the ∆ m best fitpoint. This difference is included as an additional uncer-tainty in the analysis by smearing the ∆ m likelihoodsurface. The far detector oscillation fit likelihood as afunction of ∆ m has a Gaussian distribution. This dis-tribution is convolved with a Gaussian of unit area, cen-tered at 0, with a width given by the shift in the ∆ m )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . )) q E v e n t s / ( . c o s ( Data CCQE n non-CCQE n CCQE n non-CCQE n -mode n ) m q cos( D a t a / S i m . FIG. 40: Post ND280 fit distributions of the final state muon angle for the RHC ν µ (top) and ν µ (bottom) CC1-track (left) and CC N -track (right) FGD1 samples.best fit point between the simulated data fit and the ex-pectation. This is equivalent to adding this shift as anuncertainty on ∆ m in quadrature with the existing un-certainties in the analysis. The result of this smearing isshown in Fig. 53. B. Summary of simulated data studies
Table XII shows the final bias table for simulated datasets studied in the analysis, after the addition of the NREuncertainty parameter. For the data-driven E ν − Q cat-egory the largest effect from the three simulated datastudies is shown. In all cases the observed bias on sin θ and δ CP was insignificant compared to existing system-atic uncertainty on the parameter and so no additionaluncertainty was introduced. Non-negligible bias was ob-served for ∆ m . The quadrature sum of the observedbiases, 4 . × − eV c − , was added as an additionaluncertainty on ∆ m using the method described above.The effect of the systematic parameters on the predictedevent rates on each SK event sample, including the addi-tional NRE uncertainty, is shown in Tab. XIII. The effectof the prior uncertainties on the (typically marginalized)oscillation parameters sin θ , ∆ m and sin θ is also TABLE XII: Oscillation parameter biases (aspercentages of the total and systematic uncertainties)observed in the simulated data studies including theadditional uncertainties on the NRE. Simulated data set Relative to sin θ ∆ m δ CP Martini 2p2h Total 9.0 % 16 % 0.1 %Syst. 20 % 22 % 0.3 %Data-driven CC 0 π Total 15 % 14 % 4.0 % E ν − Q dependence Syst. 34 % 20 % 17 %BSF 1p1h Total 1.5 % 22 % 0.1 %Syst. 3.4 % 31 % 0.3 %Nieves LFG 1p1h Total 4.0 % 25 % 7.0 %Syst. 8.3 % 35 % 20 %Nucleon Total 5.0 % 33 % 0.1 %removal energy Syst. 10 % 46 % 0.6 %Coulomb correction Total 1.0 % 0.1 % 0.1 %Syst. 2.3 % 0.1 % 0.3 %Kabirnezhad Total 8.0 % 34 % 0.0 %single pion Syst. 20 % 50 % 1.0 % shown.9TABLE XIII: Fractional uncertainty (%) on event rate by error source and sample, calculated with expected eventsrates generated according to the nominal oscillation parameter values from Table III. Final column is the fractionaluncertainty (%) on the ratio of FHC/RHC events in the one-ring e sample. The final row, ‘All Systematics’, doesnot include the effects of any oscillation parameters. µ e Error source FHC RHC FHC RHC
FHC1 d.e. FHC / RHC
SK Detector 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . σ ( ν e ) /σ ( ν e ) 0 . . . . . . γ . . . . . . . . . . . . θ + ∆ m . . . . . . θ PDG2018 0 . . . . . . . . . . . . XIII. OSCILLATION ANALYSIS RESULTS
Following the recommendations in [93], we produce re-sults using different statistical approaches, both frequen-tist and Bayesian (with analysis of sensitivity to prior)and test the frequentist properties of our Bayesian meth-ods when possible. Using the three far detector analysesdescribed previously, point and interval estimations aremade for the parameters sin θ , δ CP and ∆ m (nor-mal ordering) or ∆ m (inverted ordering). Two typesof intervals are produced: confidence intervals (with ap-proximate coverage based on the constant ∆ χ method inmost cases, and with exact coverage using the Feldman–Cousins unified approach for δ CP ) and credible intervals.The mass ordering was studied using mainly Bayesianhypothesis testing, with additional frequentist checks. A. Measurements of the parameters of the 3 flavoroscillation model ∆ χ and frequentist results Intervals based on the constant ∆ χ method are pro-duced for the different parameters using analyses A andC (analysis B can produce similar intervals for compar-ison purpose, although its main results are the credibleintervals described in XIII A 2). As their results are ingood agreement, only the results obtained with analysisA are shown in this section, unless otherwise indicated.The best fit values and 1 σ confidence intervals obtainedfor the different parameters in both mass ordering sce-narios are summarized in Tab. XIV and in Tab. XV withand without using the results of reactor experiments toconstrain sin θ , respectively. The global best fit wasfound to be for the NO, and the data show a preferencefor the upper octant. These preferences will be quanti-fied in part XIII C. The ∆ χ = − L/L max ) functions obtained for δ CP with and without using the results of re-actor experiments to constrain sin θ are displayed inFig. 54. The favored and disfavored values of δ CP are sim-ilar between the two cases, but the constraint on δ CP be-comes stronger when the reactor experiments results areused. The obtained 90% confidence regions for (sin θ , δ CP ) are displayed in Fig. 55. The largest parts of the con-fidence regions are located in the upper octant, especiallywhen the constraint from reactor experiments is used, butthe results are still compatible with maximal mixing. Forthe atmospheric parameters, the obtained normal order-ing 90% confidence region for (sin θ , ∆ m ) is shown inFig. 56 together with the measurements from other neu-trino oscillation experiments. Good agreement is seenbetween all of the experiments.TABLE XIV: The measured oscillation parameterbest-fit and the ± σ intervals, shown for the T2K-only(without reactor constraint) fit and for normal andinverted hierarchies with respect to the hierarchybest-fit. The ± σ interval corresponds to the values forwhich ∆ χ ≤ Parameter Best-fit and 1 σ intervalNO IO δ CP − . +0 . − . − . +0 . − . sin θ / − . +5 . − . . +5 . − . sin θ . +0 . − . . +0 . − . ∆ m / − eV c − . +0 . − . | ∆ m | / − eV c − . +0 . − . Section XI demonstrated that analyses A, B and Chave similar sensitivities (Fig. 47), their data fit resultsare now compared for δ CP vs sin θ in Fig. 57 and ∆ m vs sin θ in Fig. 58. The largest differences are observedfor analysis C, in particular for the atmospheric parame-0 [GeV] rec E E v e n t s p e r B i n [GeV] rec E E v e n t s p e r B i n [GeV] rec E E v e n t s p e r B i n FIG. 41: Post-ND280-fit predicted event spectrum forthe FHC beam SK samples as a function of E Rec QE . Thesamples are: single-ring muon-like (top), single-ringelectron-like without decay electrons (middle) andsingle-ring electron-like with a single decay electron(bottom).ters. Those differences reduce to a negligible level if anal-ysis C is repeated with the appearance samples binnedin the same E rec variable as analyses A and B, indicat-ing this is primarily an effect coming from the choice ofkinematic variables.The intervals listed in Tabs. XIV and XV wereconstructed using constant ∆ χ critical values. Thistreatment gives proper coverage when Wilks’ theorem [GeV] rec E E v e n t s p e r B i n [GeV] rec E E v e n t s p e r B i n FIG. 42: Post-ND280-fit predicted event spectrum forthe RHC SK samples as a function of E Rec QE . Thesamples are: single-ring muon-like (top) and single-ringelectron-like without decay electrons (bottom).TABLE XV: The measured oscillation parameterbest-fit and the ± σ intervals, shown for the T2K +reactor fit and for normal and inverted hierarchies withrespect to the hierarchy best-fit. The ± σ intervalcorresponds to the values for which ∆ χ ≤ Parameter Best-fit and 1 σ intervalNO IO δ CP − . +0 . − . − . +0 . − . sin θ . +0 . − . . +0 . − . ∆ m / − eV c − . +0 . − . | ∆ m | / − eV c − . +0 . − . holds [94], but can result in poor coverage when this isnot the case. In particular, it is not expected to giveproper coverage for δ CP , due to the cyclic nature of theparameter and the presence of physical boundaries at ± π/
2. In these cases intervals with exact coverage canbe formed directly from the likelihood ratio, by comput-ing the appropriate ∆ χ critical value for each value ofthe parameters considered, as proposed by Feldman and1 (GeV) n E -
10 1 10 P a r a m e t e r V a l u e MINUIT Based FitMCMC Based Fit -mode n , m n SK FIG. 43: Comparisons of the fitted SK FHC fluxparameters (top) and cross-section parameters (bottom)between the ND280 fit (red, solid) and the MCMCanalysis (blue, hashed).Cousins [95]. This method is very CPU-intensive, so it isonly used for the one-dimensional δ CP interval.For analyses A and C, critical values of ∆ χ (values ofthe parameter of interest for which the ∆ χ is lower thanthe critical value for a given confidence level are includedin the confidence interval for this level) were calculatedfor multiple values of δ CP as follows. Pseudo-experimentswere generated at 9 evenly-spaced grid points of δ CP ineach mass ordering, and at two additional points near theintersection of the 3 σ critical value and ∆ χ curves. Thesystematic and other oscillation parameters are randomlyvaried to generate the pseudo-experiments, with differentprocedures for the different parameters. The systematicparameters, sin θ , ∆ m and sin θ are drawn fromthe prior probabilities described in Sec. XI, with sin θ constrained by reactor data. The two oscillation parame-ters which do not have a prior constraint in the analysis,sin θ and ∆ m , are drawn from the 2D likelihood re-sulting from the fit of an Asimov data set generated atthe best fit point obtained in the T2K+reactor data fit.Each set of parameter values is used to generate predictedkinematic distributions for each sample, which are thensampled assuming a Poisson probability in each recon- structed variable(s) bin to obtain a pseudo-experiment.The pseudo-experiments are fit in the same manner asthe real data, and the ∆ χ between the true and best fit δ CP and mass ordering is recorded. The N th percentileof this distribution then forms the N% critical value forthis combination of δ CP and mass ordering.The obtained critical values for δ CP are displayed inFig. 59. For all confidence levels, significant deviationsfrom the values expected for a parabolic log likelihoodfunction (as assumed by the constant ∆ χ method) areobserved, demonstrating the necessity of the method.The critical values obtained by Analysis A were com-pared to those of Analysis C and were generally found tobe in good agreement, with minor differences mostly ex-plained by the different kinematic variables used by thetwo analyses for appearance samples.In order to better understand the structure of thecritical values, the effects of the δ CP -mass ordering andcos( δ CP ) degeneracies, the physical boundaries around δ CP = ± π/ ν µ → ν e and ν µ → ν e oscillation probabilities can be changed by vary-ing δ CP is limited. This creates an effect similarto physical boundaries at the values of δ CP corre-sponding to the maximum and minimum number ofevents in the appearance samples. The critical val-ues in those regions are therefore lower than thoseexpected under the assumption of a parabolic log-likelihood.2. Mass ordering is a discrete parameter and worksas an additional degree of freedom, raising criticalvalues, though without sufficient freedom to makethe critical values behave as if the problem had 2degrees of freedom.3. The effect of physical boundaries is more visiblefor the 3 σ critical values, since the critical valuesat this confidence level are usually determined bythe pseudo-experiments corresponding to more ex-treme statistical fluctuations.4. The critical values increase with statistics, and thevalues obtained have increased relative to their val-ues in previous T2K analyses [25, 26, 29].To understand this last point, critical values for the1 , , σ and 90% confidence levels were computed assum-ing different exposures, and were found to non-linearlyincrease with exposure in all cases. The leading cause wasfound to be the approximate degeneracy between δ CP and π − δ CP (T2K observables are mainly sensitive to sin δ CP ,with cos δ CP having a much smaller effect), which acts asan additional pseudo-degree-of-freedom that is negligibleat low exposures but becomes more important as moredata is taken. The above physical boundary effects alsocontribute at all exposures; however, for true δ CP values2 Reconstructed Energy (GeV) n N u m b e r o f E v e n t s CC e n osc CC e n osc NC CC e n CC e n CC m n CC m n (a) FHC 1-Ring µ -like Reconstructed Energy (GeV) n N u m b e r o f E v e n t s CC e n osc CC e n osc NC CC e n CC e n CC m n CC m n (b) RHC 1-Ring µ -like Reconstructed Energy (GeV) n ( d e g r ee s ) q N u m b e r o f E v e n t s (c) FHC 1-Ring e-like Reconstructed Energy (GeV) n ( d e g r ee s ) q N u m b e r o f E v e n t s (d) RHC 1-Ring e-like Reconstructed Energy (GeV) n ( d e g r ee s ) q N u m b e r o f E v e n t s (e) FHC 1-Ring e-like + 1 d.e FIG. 44: Observed kinematic distributions compared to the expectations generated with oscillation parameters setto the values in Tab. III for the different samples. The uncertainty shown around the data points in (a) and (b)accounts for statistical uncertainty only. The uncertainty range is chosen to include all points for which the measurednumber of data events is inside the 68% confidence interval of a Poisson distribution centered at that point.away from the boundaries, the boundary effect decreaseswith increased exposure.The finite number of pseudo-experiments used to com-pute the critical values introduces a Monte Carlo statis- tical uncertainty, and the number of pseudo-experimentsis chosen to make the size of this uncertainty negligiblefor the 2 σ critical values; however, the number requiredto reduce the 3 σ critical value statistical uncertainties to3 - - - (Radians) CP d c D undefinedclNormal - T2K + reactorInverted - T2K + reactorNormal - T2K-onlyInverted - T2K-only FIG. 45: The predicted ∆ χ = − L/L max ] functionas a function of δ CP with and without reactorconstraint, for both mass orderings. ) q ( sin - · ) - c (I O ) ( e V m D ( NO ) , m D Best fit - T2K + reactorBest fit - T2K-onlyNormal - T2K + reactorInverted - T2K + reactorNormal - T2K-onlyInverted - T2K-only
FIG. 46: Predicted 90% CL regions for ∆ m vs. sin θ with and without the reactor constraint, for both massorderings. Normal and inverted mass ordering contoursare independent.this level would be computationally intractable, so ad-ditional care is taken to ensure the validity of the 3 σ confidence intervals. The confidence intervals are calcu-lated for the different combinations of the nominal and ± σ binomial error values of the critical ∆ χ , and thelargest interval is kept. A second source of error comesfrom the interpolation of the critical ∆ χ between thepoints where it was computed. To minimize this effect,critical values have been calculated at two additional δ CP values, each close to the boundaries of the 3 σ confidenceinterval.The resulting confidence intervals obtained for δ CP arelisted in Tab. XVI and the obtained 3 σ intervals aredisplayed in Fig. 60, superimposed on to the observed∆ χ function.In both mass orderings, the two CP conservingvalues of 0 and π are outside of the 2 σ confidenceintervals. Our data therefore exclude the conservation - - - CP d cD Analysis AAnalysis BAnalysis C q sin - · ) ( e V m D
90% Confidence Level Analysis A68% Confidence Level Analysis BBest Fit Point Analysis C
FIG. 47: Comparison between analyses A, B and C.The normal mass ordering is assumed and the reactorconstraint is applied. Top: the predicted ∆ χ functionsfor δ CP . Bottom: the 1 σ (dashed) and 90% (solid) CLregions for ∆ m vs. sin θ .TABLE XVI: Feldman–Cousins confidence intervals for δ CP with the reactor constraint applied in both normaland inverted mass orderings. Confidence Interval (Radians)Level Normal Ordering Inverted Ordering1 σ [ − . , − .
26] —90% [ − . , − .
84] —2 σ [ − . , − .
63] [ − . , − . σ [ − π, − . ∪ [2 . , π ] [ − . , − . of CP symmetry in neutrino oscillation at the 2 σ level.For the inverted mass ordering, both CP-conservingvalues are outside of the 3 σ confidence intervals. Forthe normal mass ordering, δ CP = 0 is just outside ofthe 3 σ confidence interval, while δ CP = ± π is inside.The robustness of the exclusion of δ CP = 0 giventhe model uncertainties is studied in more details inSec. XIII B 3, where it was found that the boundary ofthe 3 σ interval is so close to δ CP = 0 that this point canmove in and out of the 3 σ interval due to changes inthe model. On the contrary, the exclusion of the CP-conserving values at the 2 σ level was found to be robust.4 -35-30-25-20-15-10-5 ∆ (cid:104) p µ (cid:105) ∆ N R E : → M e V ( M e V / c ) E ν ( GeV ) -1-0.500.51 c o s ( θ µ ) NEUT 5.3.3, GRFG, C ∆ NRE: (25 →
43) MeV
FIG. 48: The momentum shift caused by increasing thenucleon removal energy in NEUT 5.3.3 by 18 MeV.
Reconstructed lepton momentum (MeV/c) ) l e p t on q R ec on s t r u c t e d c o s ( R a ti o t o no m i n a l p r e d i c ti on FIG. 49: The ratio of the NRE simulated data to thenominal prediction for the ND280 CC 0 π FGD1 sample.The histogram axes have been truncated for clarity.
2. Bayesian results
Bayesian results for the oscillation parameters wereproduced using Analysis B. The posterior probability of δ CP marginalized over both mass orderings obtained in afit of T2K data with sin θ constrained by the reactorexperiments is shown in Fig. 61 with two different priorprobabilities for the parameter of interest. The CP con-serving values of δ CP are found to be outside of the 2 σ credible interval in both cases, with the 1 σ range stillcovering the maximal CP violation value of δ CP = − π/ δ CP , the alternative prior probability wastested to see the effect of the choice of prior. The alter-native was chosen uniform in sin( δ CP ) as this is both thevariable involving δ CP to which our observables are mostsensitive to, and the one relevant for CP violation. The R a ti o t o g e n e r a t e d v a l u e Asimov data setNRE sim. data set
Neutrino Energy (GeV) Q E M A C p F O p F n N o r m n N o r m N o r m C t o O S h a p e C S h a p e O B e R P A A B e R P A B B e R P A D B e R P A E B e R P A U C A R E S M A I s o B kg mn / e n mn / e n CC D I S CC C oh C CC C oh O N C C oh g N C N C O t h e r N ea r N C O t h e r F a r R a ti o t o g e n e r a t e d v a l u e FIG. 50: Results of the fit to the near detectorsimulated data with an increased nucleon removalenergy. The best fit values for the ν µ flux parameters atSK (top) and cross-section parameters (bottom) areshown for the NRE fit (blue) compared to the Asimovdata set fit (red).detailed comparison between the intervals obtained withthe two prior probabilities can be found in Tab. XVII,where it can be seen that this choice affects the size ofthe intervals obtained, but does not change the main con-clusions of the analysis.TABLE XVII: Credible intervals for δ CP with uniformprior probabilities on either δ CP or on sin( δ CP ). Prior uniform in δ CP sin δ CP σ ) [ − . , − .
13] [ − . , − . σ ) [ − . , − .
72] [ − . , − . σ ) [ − π, . ∪ [2 . , π ] [ − π, . ∪ [2 . , π ] The proposal function for choosing the random stepsused in Analysis B’s MCMC can jump between mass or-derings, with a 50% chance at each step to propose apoint with the opposite sign of ∆ m . It therefore pro-duces a single posterior probability for ∆ m , coveringboth positive and negative values. The posterior proba-bility obtained in a fit of T2K data with constraint fromthe reactor experiments, is shown in Fig. 62. A clear5 (GeV) QErec E P r e d i c t e d e v e n t s NominalNRE sim. dataND280 pred. (GeV)
QErec E P r e d i c t e d e v e n t s NominalNRE sim. dataND280 pred.
FIG. 51: Comparison of the nominal MC prediction(dashed black), simulated data (solid blue) andprediction from the near detector fit (shaded red), forthe far detector muon-like (top) and electron-like(bottom) neutrino beam event samples.preference towards the normal ordering, which contains89% of the total posterior density, can be seen, with inparticular the lack of an observed 1 σ credible region inthe ∆ m < θ posterior distribution has alargely non-Gaussian shape, the marginal likelihood (forthe other parameters) used in the analyses described inthis paper is expected to have some differences with theprofile likelihood commonly used in the neutrino com-munity. Correlations are mostly seen between the at-mospheric parameters, sin θ and ∆ m , and between∆ m and δ CP . The correlation between the atmosphericparameters is clear from Eq. 10. For ∆ m and cos( δ CP ),both parameters shift the energy of the peak (dip) in theelectron (muon) neutrino oscillation probability. Com-bined with the energy profile of the T2K beam this energy shift produces a change in event rate for the electron-likesamples, and a smaller change in the muon-like samples.This change in both rate and shape introduces a correla-tion between the two parameters. B. Additional checks on the validity of δ CP results The main result of the analysis described in this paperis the measurement of δ CP , with in particular the exclu-sion of CP conservation in neutrino oscillations at the2 σ level, and the fact that some of the possible valuesof δ CP are outside of the 3 σ confidence intervals. Theobserved constraint on δ CP is considerably stronger thanthat expected from sensitivity studies, as can be seen bycomparing Figs. 45 and 54. A number of additional stud-ies were done to check the likelihood and robustness ofthis result.
1. Probability of the δ CP results The probability of getting a constraint this strongor stronger considering systematic and statistical uncer-tainties was evaluated assuming normal ordering and atrue value of δ CP of − π/
2. This case was chosen as itgave the best agreement between data and predictions inTab. XI. 10 000 pseudo-experiments were generated fol-lowing the same method as used in the determinationof the Feldman–Cousins critical values. The pseudo-experiments were then fitted for δ CP , and the result-ing ∆ χ distributions were compared to the ∆ χ func-tions obtained for the data. Figure 65 shows the ob-served ∆ χ = −
2∆ ln L overlaid on the one-sided 1 σ and2 σ bands. Those bands are built from the ensemble of∆ χ values from the different pseudo-experiments at eachvalue of δ CP , and cover the interval between zero and the68.27% (1 σ band) or 95.45% (2 σ band) quantile. Theobserved ∆ χ function lies within the one-sided 2 σ bandfor the fits done in the inverted mass ordering scenario,whilst falling just outside the 2 σ band in the normal massordering scenario in the region around δ CP = 0. An al-ternative way to consider the results is to check whatfraction of pseudo-experiments exclude one or both ofthe CP conserving values at a specified confidence level.As shown in Tab. XVIII, for the normal mass ordering, δ CP = 0 and δ CP = π are separately excluded at 90%confidence level in around 46–48% of pseudo-experimentsand excluded at 2 σ confidence level in around 32–34%.Both CP-conserving values are excluded at the 2 σ con-fidence level in about 25% of the pseudo-experiments.Overall, a 2 σ exclusion of CP conservation is not unlikelyaccording to the model used in this analysis.6 ) q ( sin cD NominalNRE sim. data | m D | cD NominalNRE sim. data ) q (2 sin cD NominalNRE sim. data CP d - - - cD NominalNRE sim. data
FIG. 52: Comparison of the nominal MC oscillation parameter ∆ χ function (black) to those from the simulateddata fits (red).TABLE XVIII: Fraction of the pseudo data sets forwhich the ∆ χ at δ CP = 0 , π and both CP conservingvalues are above the critical values for 90% and 2 σ CL.Pseudo data sets were generated assuming true normalordering and a value of δ CP = − π/ δ CP Mass orderingscenario 90% 2 σ π Normal 0.46 0.320 and π π Inverted 0.82 0.730 and π
2. Contributions from individual samples
The discrepancies between predictions and observa-tions seen in some of the samples used in the analysis werestudied in more detail. In Tab. XI, the observed num-bers of events in the far detector samples are generallyin good agreement with the predicted number of eventsfor δ CP = − π/
2. Two samples show a difference with thenominal prediction: for the FHC 1R µ -like sample a small deficit of events can be seen in the data, corresponding to1.33 times the statistical and systematic uncertainty onthe predicted number of events. For the FHC ν e CC 1 π + -like sample, the discrepancy is larger with the observednumber of events being twice the predicted number. Tounderstand the impact on the δ CP result of the differencesbetween predictions and observations from each sample,hybrid MC/data fits were performed: the data were re-placed by the MC predictions for each sample in turn, asshown in Fig. 66.It can be seen that in all cases including data, theconstraint on δ CP is stronger than that obtained fromthe MC predictions alone. For the CP conserving value δ CP = − π , the most significant changes happen whendata are replaced by MC predictions for the two sampleswhich showed differences between observations and pre-dictions. In particular, in the case of the FHC ν e CC1 π + -like sample, the ∆ χ at δ CP = − π becomes lowerthan the 2 σ critical value obtained with the Feldman–Cousins method, meaning that the conservation of CPsymmetry is no longer excluded with 2 σ significance inthis case.Additional studies were therefore performed to esti-mate the likelihood of the observations for the FHC ν e CC 1 π + -like sample. First, the probability to obtain15 events or more in this sample when taking into ac-7 ) q ( sin ] / c [ e V m D - · NominalNRE sim. data ) q ( sin ] / c [ e V m D - · Nominal w/ NRE parameterNRE sim. data w/ NRE parameter ) q ( sin ] / c [ e V m D - · Nominal w/ NRE parameter and smearingNRE sim. data w/ NRE parameter and smearing
FIG. 53: Comparison of the sin θ -∆ m parametercontours using the nominal cross-section model (top),after the addition of the ‘effective NRE’ parameter(middle) and after the additional smearing is applied to∆ m (bottom). In all cases the expected result, asdescribed in the text, is shown in black and thesimulated data result is in red. The solid lines representthe 90% confidence region and the dashed linesrepresent the 1 σ confidence region.count statistical and systematic uncertainties was eval-uated. This probability was found to be 2.49% for truevalues of the oscillation parameters corresponding to theT2K-only best fit, and 1.34% for the T2K with reactorconstraint best fit point. As there are 5 samples in the - - - (Radians) CP d c D undefinedclNormal - T2K + reactorInverted - T2K + reactorNormal - T2K-onlyInverted - T2K-only FIG. 54: The observed ∆ χ function of δ CP , with andwithout the reactor constraint. The ∆ χ is computedwith respect to the best fit over the two mass orderings,and separate best fit points are used for the T2K-onlyand the T2K+reactor cases. ) q ( sin - - - ( R a d i a n s ) C P d Best fit - T2K + reactorBest fit - T2K-onlyNormal - T2K + reactorInverted - T2K + reactorNormal - T2K-onlyInverted - T2K-only
FIG. 55: The observed constant ∆ χ
90% confidenceregions of sin θ and δ CP with normal and invertedmass orderings and with and without the reactorconstraint. Normal and inverted mass ordering contoursare independent. ∆ χ values are calculatedindependently for the functions with and without thereactor constraint.analysis, a trial factor should be taken into account. Theprobability to have such an excess in at least one of the 5samples (meaning an excess of events in the sample cor-responding to a p -value smaller or equal to the p -value forthe FHC ν e CC 1 π + -like sample) was found to be 11.3%for the T2K only best fit, and 5.8% for the T2K+reactorbest fit point. p -values were also calculated when taking into accountnot only the number of events, but also the distribution ofthe kinematic variables of the observed events, and werefound to be equal to the rate-only p -value. As can beseen in Fig. 67, the kinematic distributions of the dataevents are in good agreement with the prediction, and8 ) q ( sin - · ] / c [ e V m D T2K run 1-9 Super-K 2018A 2019 n NO IceCube 2017
Best fits
FIG. 56: The observed T2K constant ∆ χ θ and∆ m with the reactor constraint and compared to theresults from Super-K [4], NO ν A [13] and IceCube [14].
19 20 21 22 23 24 25 26 - · q sin - - - C P d
19 20 21 22 23 24 25 26 - · q sin - - - C P d FIG. 57: Comparison between the observed 1 σ (dashed)and 90% (solid) CL regions for δ CP vs sin θ producedusing analyses A, B and C, assuming normal massordering (top) or inverted mass ordering (bottom). Thereactor constraint is applied. q sin - · ) ( e V m D
90% Confidence Level Analysis A68% Confidence Level Analysis BBest Fit Point Analysis C q sin - - - - - - - · ) ( e V m D
90% Confidence Level Analysis A68% Confidence Level Analysis BBest Fit Point Analysis C
FIG. 58: Comparison between the observed 1 σ (dashed)and 90% (solid) CL regions for ∆ m vs sin θ produced using analyses A, B and C, assuming normalmass ordering (top) or inverted mass ordering (bottom).The reactor constraint is applied.so taking into account the shape information does notincrease the disagreement between data and prediction.
3. Additional interaction model checks
Section XII described checks for possible biases comingfrom the choice of interaction model. Those checks werebased on comparisons of MC sensitivities obtained withdifferent interaction models. However, some model un-certainties affect primarily the FHC ν e CC 1 π + sample,which brings only a small contribution to the δ CP sen-sitivity compared to the single ring e -like samples, andtherefore do not have a significant effect on the simulateddata studies. They can nevertheless have a sizable effecton the predictions for the FHC ν e CC 1 π + sample, andgiven the large excess of data events in this sample, andits effect on the δ CP measurement, the possible impactof additional model uncertainties were considered. Theadditional studies were done in the context of the datafit and not the MC based sensitivity studies.The first source of uncertainty studied is the data/MCdiscrepancy in the pion spectrum for the near detector9 - - - (Radians) CP d c r it c D s s s FIG. 59: Feldman–Cousins critical values for δ CP . Theshaded bands represent the ± σ Monte Carlo statisticaluncertainty on the calculated critical values. - - - (Radians) CP d c D Normal - T2K + reactorInverted - T2K + reactor conf. region s FC 3
FIG. 60: The observed 3 σ Feldman–Cousins (FC)confidence intervals for δ CP . The ∆ χ is computed withrespect to the best fit over the 2 mass orderings. (rad.) CP d - - - P o s t e r i o r p r ob a b ilit y d e n s it y - - - -
10 110 ) CP d C.I. (flat s CP d C.I. (flat sin( s CP d C.I. (flat s CP d C.I. (flat sin( s CP d C.I. (flat s CP d C.I. (flat sin( s CP d C.I. (flat s CP d C.I. (flat sin( s CP d C.I. (flat s CP d C.I. (flat sin( s CP d C.I. (flat s CP d C.I. (flat sin( s FIG. 61: Posterior probability density for δ CP andcredible intervals obtained using a prior uniform in δ CP compared against credible intervals obtained with aprior uniform in sin( δ CP ). The CP conserving values areoutside of the 2 σ credible intervals in both cases. ) (eV m D - - - P o s t e r i o r p r ob a b ilit y d e n s it y - - - -
10 110 Credible Interval s
3 Credible Interval s
2 Credible Interval s FIG. 62: Posterior probability distribution for ∆ m covering both mass orderings with 1, 2 and 3 σ BayesianCredible Intervals marked. The normal orderingcontains 89% of the posterior probability. q sin ) ( e V m D - - - - · credible interval s
1 credible interval s
2 credible interval s FIG. 63: Bayesian credible regions for ∆ m againstsin θ covering both mass orderings.CC 1 π sample. Even after tuning using the results ofthe near detector data fit, the predicted pion momentumspectrum did not reproduce the data for the near de-tector FHC CC 1 π + sample: the data events had lowerpion momentum than predicted by the model. This dis-crepancy could have an impact on the prediction for thefar detector FHC CC 1 π + sample: charged pions canappear at SK either as rings if they have high enoughmomentum ( >
156 MeV /c ), or as Michel electrons fromthe decay of the pion. If the pions produced in CC 1 π + interactions have lower momentum than our model pre-dicts, a larger fraction of the CC 1 π + events at SK willenter the FHC CC 1 π + sample than our MC predicts.Studies showed that a discrepancy of the size observedat the near detector could lead to a 10% increase in thenumber of events in the far detector CC 1 π + sample. To0 p- p p- p q sin q sin m D CP d C P d m D q s i npo s t . p r ob a . FIG. 64: Posteriors probabilities together with 1, 2 and 3 σ credible intervals for all the oscillation parameters ofinterest and their combinations in the normal mass ordering. A logarithmic scale is used for the axis correspondingto the posterior probability density, and darker colors correspond to larger probabilities.evaluate the possible impact of this uncertainty on our δ CP results, the fit of the run 1–9 data was redone withtwo different modifications: • adding a 10% normalization uncertainty on thenumber of events in the far detector ν e CC 1 π + sample. • increasing the number of events in the SK ν e CC1 π + sample by 10% in the MC predictions used to fit the data.The second model uncertainty concerns the number ofhadrons produced in deep inelastic interactions in thelow invariant mass region W <
NEUT generator used to produce the MC, and have atleast two pions produced at the interaction level. Theycan nevertheless enter the ν e CC 1 π + sample if some ofthose pions re-interact in the nucleus (through the final1 (Radians) CP d - - - c D (Radians) CP d - - - c D FIG. 65: The observed ∆ χ = −
2∆ ln L functionscompared with one-sided distributions of ∆ χ valuescorresponding to 68.27% and 95.45% of 10 000pseudo-experiments generated for δ CP = − π/ - - - CP d c D Run 1-9 data fitMC expectationReplace FHC 1-Ring eReplace RHC 1-Ring eReplace FHC 1-Ring e 1 d.e. m Replace FHC 1-Ring m Replace RHC 1-Ring CL) s (2 crit.2 c D FC CL) s (3 crit.2 c D FC FIG. 66: Change of the results of the fit of the T2K run1–9 data for δ CP when observations get replaced bypredictions for the different samples. The gray dashedlines correspond to the critical values for 2 σ and 3 σ obtained using the Feldman–Cousins method. Lepton momentum (MeV/c) ( d e g r ee s ) q N u m b e r o f e v e n t s FIG. 67: Lepton momentum and angle with respect tothe beam direction for the 15 events observed in theFHC ν e CC 1 π + -like sample, overlaid with the MCpredictions.state interactions) or are not detected. The uncertaintieson the number of hadrons produced in those interactionsproduce an uncertainty on the fraction that will enterthe ν e CC 1 π + sample. To assess the potential impacton the δ CP result, the fit of the run 1–9 data was redoneusing two alternative models for the hadron multiplic-ity model in the MC: the first (M1) is based on fits ofdata from deuterium bubble chamber experiments [96],while the second (M2) is based on the multiplicity partof the AGKY model [97]. Only the alternative modelM1 was found to significantly change the expected num-ber of events, with the biggest effect seen for the ν e CC1 π + sample (+13 . χ obtained in the datafit for δ CP changed when an alternative model was usedto fit the data. For each of the two model uncertain-ties, only the case which gave the largest effect is shown:10% increase in the SK ν e CC 1 π + sample MC predic-tions for the pion spectrum case, and M1 model for thehadron multiplicity model case. Both those changes pre-dict an increase in the number of events in the ν e CC1 π + sample, reducing the discrepancy with the data, andtherefore weakening the constraint on δ CP . The magni-tude of the change in ∆ χ is small enough that it doesnot change the main conclusions obtained in the fit of thedata for δ CP : the CP-conserving values are excluded atthe 2 σ level, and some values of δ CP are outside of the 3 σ confidence level intervals. It is large enough however tochange whether δ CP = 0 is excluded at the 3 σ confidencelevel or not in the results of analyses A and C.2 (Radians) CP d - - - ) c D ( D Pion momentum alt. model (NO) Pion momentum alt. model (IO) DIS hadron multiplicity alt. model (NO) DIS hadron multiplicity alt. model (IO)
FIG. 68: Change in the ∆ χ function for δ CP whenfitting data using an alternative model for DIS hadronmultiplicity (red) and pion momentum spectra (black).The ∆ χ was reduced by the value shown on the plot ineach case. The difference is shown for the model changethat gave the largest difference in each case. C. Neutrino mass ordering
The question of the mass ordering was studied in aBayesian framework, by computing posterior probabili-ties and Bayes factors for each ordering hypothesis. Re-sults shown in this section use the constraint from reactorexperiments on sin θ unless otherwise stated.
1. Posterior probabilities and Bayes factor
All three analyses evaluated posterior probabilities toestimate the preference for the neutrino mass orderings.Analyses B and C additionally looked at the posteriorprobabilities for the octant of sin θ . The practical cal-culation of the posterior probabilities differs between theanalyses, due to differences in the fitting techniques used.Analyses A and C first compute the marginal likelihoodfor each hypothesis, and compute the posterior probabil-ity for hypothesis H i from: P ( H i | N obs , x ) = L marg ( H i ; N obs , x ) P ( H i ) (cid:80) j L marg ( H j ; N obs , x ) P ( H j ) . (17)Where the denominator sums over all the possible hy-pothesis combinations (either the two mass orderings,or the 4 combinations of mass ordering and octant),( N obs , x ) is the observed measurement and P ( H ) is theprior probability, taken to be equal for all the hypotheses.The MCMC-based Analysis B calculates the hypotheses’posterior probabilities by counting the number of MCMCsteps in the selected hypothesis against the total numberof MCMC steps. That ratio gives a fully marginalizedposterior probability for a given hypothesis. Table XIX shows the posterior probabilities for themass orderings and the octants of sin θ obtained withAnalysis B. Most of the posterior probability lies in theupper octant and normal mass ordering. The obtainedvalues of the Bayes factors, corresponding to the ratio ofthe marginal likelihoods of the two hypothesis, are of 8.0for the normal over the inverted mass orderings, and 3.9for the upper over the lower octant. The commonly-usedJeffreys’ scale [98] classifies both results as “substantial”.TABLE XIX: Model comparison posterior probabilitiesfor normal and inverted mass orderings, and for theupper and the lower octants of sin θ , from AnalysisB. There is a preference for the normal mass orderingand upper octant of sin θ . sin θ < . θ > . m >
0) 0.184 0.705 0.889IO (∆ m <
0) 0.021 0.090 0.111Sum 0.205 0.795 1
The impact of the details of the analysis methods onthe mass ordering results were checked by comparing theresults of the different analyses. Analysis C prefers thenormal mass ordering and the upper octant of sin θ with 91.1% and 80.4% posterior probability, respectively,whereas Analysis A finds a posterior probability of 87.7%for the normal mass ordering. As before, the largest dif-ference is seen between Analysis C on one side and Aand B on the other, with the main contribution being thechoice of variables used for the kinematic information ofthe candidate events from the appearance samples.The posterior probabilities above assumed equal priorprobabilities for the different hypothesis. The effect ofthe choice of prior probabilities was checked by lookingat how the posterior probabilities obtained in the data fitfrom Analysis C changed as a function of the prior prob-abilities assumed (Fig. 69). As expected when testingdiscrete hypotheses using Bayesian methods, the choiceof the prior probability has a significant effect on theobtained posterior probabilities. However, the obtainedcurves are different from y = ± x , demonstrating that thedata contain information about the mass ordering.
2. Frequentist properties of the Bayesian results for themass ordering
As advocated in [99], the frequentist properties of theBayesian mass ordering results were studied. More pre-cisely, it was checked whether excluding a given orderinghypothesis (based on the other ordering having a poste-rior probability superior or equal to α %) was selectingthe true ordering approximately α % of the time. Forthis purpose, 20,000 pseudo-experiments were generated,both for each mass ordering hypothesis and for different3 Prior NO probability P o s t e r i o r p r ob a b ilit y Normal orderingInverted ordering
FIG. 69: Mass ordering posterior probabilities as afunction of the prior probability assumed for the normalordering, obtained in the data fit using Analysis C. Thedashed black line corresponds to the equal priorprobability case used for the main result.true values of δ CP . Then, the fraction of the pseudo-experiments which had a posterior probability higherthan 95% for one of the two orderings were studied.It was found that this fraction depended strongly onthe true value of δ CP assumed. Only for true values of δ CP close to − π/ δ CP = − π/
2, the true ordering was excluded 5.67%of the time the wrong one was. This shows that usingthe posterior probability to select the mass ordering, al-though a Bayesian method, nevertheless has reasonablefrequentist properties in this case.The pattern seen highlights the degeneracy betweenthe measurement of δ CP and the determination of themass ordering (also visible on Fig. 72 of the next section),which have similar effects on the predictions: they bothchange the two observables (the number of ν µ → ν e and ν µ → ν e events) in opposite directions. The sensitivityto the mass ordering as a function of δ CP (Fig. 70) showsthat for δ CP < δ CP >
3. Frequentist results for the mass ordering
For completeness, frequentist results for the massordering were derived by computing p -values for the (Radians) CP d - - - I O c - NO c = cD - - - - - FIG. 70: Ability to distinguish between the two massorderings as a function of the true value of δ CP . The barindicates the range of values of ∆ χ = χ − χ whichcontains 95% of the values obtained for thepseudo-experiments generated in each case. The blackline corresponds to the value obtained in the data fit byAnalysis A.Bayes factor. 100,000 pseudo-experiments were gener-ated, randomizing over the nuisance parameters (includ-ing sin θ , δ CP and ∆ m , following a similar methodas for the frequentist 1D δ CP results), and the value ofthe Bayes factor P (NO) /P (IO) obtained in the data fitwas compared to the values obtained for those pseudo-experiments (Fig. 71) Bayes(NO/IO) - - - - -
10 1 F r ac ti on o f t oy d a t a s e t s True NOTrue IORun 1-9 data
FIG. 71: Expected distributions of the Bayes factorbetween the mass ordering hypotheses, compared to thevalue obtained in the data fit.The fraction of the pseudo-experiments for which theBayes factor was larger than what was observed in thedata (corresponding to a result more NO-like) was foundto be 4 . × − assuming true inverted ordering (in-verted ordering p -value), and 6 . × − assuming true4normal ordering (1 minus normal ordering p -value usingstandard definitions). Those two values are both low, andit would be misleading to claim exclusion of the invertedordering based solely on the low p -value obtained for thishypothesis. An alternative tool sometimes used in thecollider community in such cases is CL s [100], in whichthe p -value obtained for one hypothesis is penalized byone minus the p -value for the other hypothesis:CL s (IO) = p (IO)1 − p (NO) (18)where p (IO) and p (NO) are the p -values for respec-tively the inverted and normal orderings. In this case,CL s (IO) = 0 .
075 is obtained. It should be noted thatthe more commonly used test statistic ∆ χ = χ − χ is equal to − / IO)). The p -values obtainedwith this ∆ χ test statistics are therefore the same as theones presented here for the Bayes factor. D. Summary of results
To summarize the oscillation parameter constraints,fits to δ CP , sin θ , sin θ and ∆ m have been pro-duced using constant ∆ χ critical values, and their1 σ confidence intervals are listed in Tab. XIV and inTab. XV with and without using the results of reactorexperiments to constrain sin θ , respectively. Addition-ally, δ CP critical values have been calculated using theFeldman–Cousins method, and several confidence inter-vals are listed in Tab. XVI.It is valuable to see how different values of δ CP , sin θ and the mass ordering affect the predicted event rates.Figure 72 shows the predicted ν e event rate vs ν e eventrate for true values of oscillation parameters where δ CP is varied between CP conserving and maximally CP vio-lating values, and sin θ is varied around maximal mix-ing, for both mass orderings. Predicted event rates for agiven value of sin θ and mass ordering are linearly in-terpolated between those computed for 9 evenly-spacedvalues of δ CP from − π to + π to indicate the behaviorproduced by varying δ CP . The observed number of FHCand RHC 1-ring electron-like candidate events falls on theedge of the 1 σ uncertainty region generated around true δ CP = − π/
2, sin θ = 0 .
5, ∆ m = 2 . × − eV /c and normal mass ordering, and is therefore consistentwith this point. XIV. ν e APPEARANCE ANALYSIS
This section, which differs from the main oscillationanalysis reported above, evaluates the significance of the ν µ → ν e oscillation under the assumption of two differ-ent hypotheses, corresponding to no ν e appearance, andto ν e appearance consistent with our current knowledgeof the PMNS mixing parameters. To date, the world’s Neutrino mode 1Re candidates
30 40 50 60 70 80 90 100 A n ti n e u t r i no m od e R e ca nd i d a t e s q sin /c eV -3 · = 2.45 m D /c eV -3 · = 2.43 m D p = CP d /2 p = + CP d = 0 CP d /2 p = - CP d syst errstat + syst errData FIG. 72: Candidate RHC one-ring e -like event rate vsthe candidate FHC one-ring e -like event rate (includingCC 1 π events) for a variety of different oscillationparameter values. Predictions are generated for thegiven values of δ CP , sin θ and mass ordering withremaining oscillation parameters fixed at the centralvalues of the prior probability density functions definedin Sec. XIII A 1. The uncertainty regions are createdassuming that δ CP = − π/
2, sin θ = 0 . m = 2 . × − eV /c and normal mass ordering.best measurement of ν e appearance has been made bythe NO ν A collaboration [13], which reports an excess of4 . σ over the expected background. The T2K measure-ment of ν e appearance has been made using the sameanalysis framework described in Section XI and has beenperformed by analyses A and C.The ν e appearance analysis is performed by multiply-ing the ν µ → ν e PMNS oscillation probability by a fac-tor, β , i.e. P ( ν µ → ν e ) = β × P PMNS ( ν µ → ν e ). Theparameter β is set to either 0 or 1 to select a null hy-pothesis for two independent tests: when β = 0, thenull hypothesis under consideration is that there is no ν e appearance, while for β = 1 the null hypothesis isthat ν e appearance occurs according to the current bestknowledge of the PMNS parameters. For each hypothe-sis, p -values are produced from two analyses: a rate-onlywhose test statistic is the number of candidate eventsin the RHC one-ring e -like sample, and one rate+shapeanalysis whose test statistic is the difference in marginalnegative log-likelihood values between the β = 0 and β = 1 cases, denoted ∆ χ , as in Eq. 19.∆ χ = χ ( β = 0) − χ ( β = 1) (19)Here the use of χ is taken to be synonymous with − L marg . Unlike in the main oscillation analysis, thelikelihoods are not only marginalized over the flux, cross-section and detector parameters, but also over all oscil-lation parameters except β , including δ CP and the massordering, using 2 × samples of the nuisance parameterspace. The number of pseudo-experiments and the num-ber of samples used during marginalization were selected5to ensure the stability of the p -values.To calculate p -values, the data are compared to distri-butions of the test statistics from ensembles of pseudo-experiments produced under the assumption of either β = 0 or β = 1. Each pseudo-experiment is produced byrandomizing nuisance parameters according to the priorprobability density functions (PDF) in Section XI. Addi-tionally, a uniform PDF in the range [ − π, + π ] is used for δ CP , and a two-point PDF is used for the mass orderingwith equal probability for normal and inverted ordering.T2K data from four “control samples” (FHC single-ring e -like and ν e CC1 π -like and both neutrino and RHCsingle-ring µ -like) are used to constrain the oscillationand systematic model parameters. The impact of thesefour data control samples is estimated differently in thetwo analyses, A and C. In the latter, each of the 2 × pseudo-experiments used to build the distributions of therate-only and rate+shape test statistics is weighted by itslikelihood over the four control samples, given the T2Kdata. Analysis A uses an alternative method, applyingthis constraint by using rejection sampling to select thepseudo-experiments that are most probable according tothe data in the control samples.Due to the presence of the four control samples,the marginal likelihood used to calculate ∆ χ , therate+shape test statistic, is also constructed differentlyfrom the main oscillation analysis. For each pseudo-experiment, the marginal likelihood is constructed basedon the prediction in the RHC single-ring e -like sample,while its background is constrained using the four controlsamples. This is reflected in Eq. 20, where L ν e denotesthe likelihood of the RHC single-ring e -like sample com-pared to the pseudo-experiment, E , L c is the product ofthe likelihoods of the four control samples compared tothe T2K data, D , and f j the set of nuisance parameters. χ ( β ) = − (cid:20) n n (cid:88) j =1 L ν e (cid:0) β, f j ; E (cid:1) · L c (cid:0) β, f j ; D (cid:1) (cid:21) (20)The expected and observed test statistic distributionsproduced using analysis A are shown in Fig. 73 and thecorresponding p -values are shown in Tab. XX. The hy-pothesis of no ν e appearance ( β = 0) is excluded at the1 . σ and 2 . σ levels, respectively using the rate-only andrate+shape analyses. The observed p -values provide aweaker exclusion of the β = 0 case than expected withAsimov data set, in both rate-only and rate+shape anal-yses. This is primarily due to observing fewer events(15) in the RHC single-ring e -like sample than expected(16.8), and strengthened by their relatively background-like spectrum in the rate+shape analysis, as shown inFig. 44 (d). Our data are consistent with the PMNS ν e appearance hypothesis ( β = 1), with p -values of 0.32 and0.30 for the rate-only and rate+shape analyses, respec-tively (corresponding to a 1 σ exclusion). Analyses A andC both produce test statistic distributions and p -values that are in good agreement with each other, where mi-nor differences between them were determined to resultfrom the difference in kinematic variables used to bin theanalysis templates. events N F r ac ti on o f p s e udo - e xp e r i m e n t s =1) b ( c =0) - b ( c = c D - = 0 b = 1 b Expected events N F r ac ti on o f p s e udo - e xp e r i m e n t s =1) b ( c =0) - b ( c = c D - - = 0 b = 1 b T2K Data
FIG. 73: Distributions of the expected (top) andobserved (bottom) rate-only (left) and rate+shape(right) test statistics compared to theexpected/observed test statistics. Here N events denotesthe number of observed events in the RHC single-ring e -like sample.TABLE XX: Expected and observed p -values andsignificance of the β = 0 and β = 1 hypotheses usingboth the rate-only and rate+shape analyses β Analysis p -value Significance ( σ )Expected Observed Expected Observed0 Rate-only 0 .
019 0 .
059 2 .
36 1 . .
006 0 .
016 2 .
76 2 .
401 Rate-only 0 .
379 0 .
321 0 .
88 0 . .
409 0 .
300 0 .
83 1 . It is also desirable to test the robustness of these re-sults to various alternative choices of parts of the in-teraction model which have a non-negligible effect onthe kinematic distributions of the RHC single-ring e -like6sample. This is done using Analysis C by weighting thenominal near and far detector MC (see Sec. XII). Threesimulated data sets are studied: the Kabirnezhad singlepion production model, the Nieves LFG model and thedata-driven CC 0 π E ν − Q simulated data, describedin Sec. VI. To test their effects on the observed resultsof the ν e appearance analysis, the test statistic distribu-tions are weighted by the ratio of the distribution pro-duced using the alternative model to the expected distri-bution. Similarly, the observed test statistics are shiftedby the difference between the median expected pseudo-experiment with and without the use of the alternativemodel. The change in the observed p -values from thesealternative models are shown in Tabs. XXI and XXIIfor the rate and rate+shape analyses respectively. Theobserved changes are small compared to the nominal p -value and the changes do not affect the conclusions, so theanalyses are considered robust against alternative modelchoices.TABLE XXI: Nominal and alternative model p -valuesof the β = 0 and β = 1 hypotheses using the rate-onlyanalysis. Model β = 0 β = 1 p -value ( σ ) p -value ( σ )Nominal 0.0686 (1.82) 0.246 (1.16)Kabirnezhad single pion 0.0824 (1.74) 0.176 (1.35)Nieves LFG 1p1h 0.0804 (1.75) 0.222 (1.22)Data-driven 2p2h-∆ 0.0859 (1.72) 0.211 (1.25) TABLE XXII: Nominal and alternative model p -valuesof the β = 0 and β = 1 hypotheses using therate+shape analysis. Model β = 0 β = 1 p -value ( σ ) p -value ( σ )Nominal 0.0244 (2.25) 0.261 (1.12)Kabirnezhad single pion 0.0227 (2.28) 0.225 (1.21)Nieves LFG 1p1h 0.0201 (2.32) 0.277 (1.09)Data-driven 2p2h-∆ 0.0178 (2.37) 0.301 (1.03) XV. CONCLUSIONS
The T2K collaboration has analyzed the full data setcollected by the experiment between 2010 and 2018 toproduce measurements of sin θ , ∆ m , sin θ , δ CP and the mass ordering. The parameter values and un-certainties are taken from a simultaneous fit to bothmuon-like and electron-like event samples at Super-Kamiokande, collected from both neutrino-dominatedand antineutrino-dominated beam operation. This anal-ysis uses a new event selection at SK to increase the ef-fective fiducial volume of the detector, resulting in a 20%increase in the electron-like sample efficiency and a 40% reduction in the background contamination in the muon-like samples.The neutrino interaction model used for this work isimproved relative to Ref. [26], incorporating in-mediumeffects (RPA) and 2p2h shape uncertainties in thecharged-current zero-pion signal channel. A detailed setof simulated data studies were also performed to assessthe robustness of the analysis to alternative choices ofneutrino interaction model. These studies demonstratedthat the nucleon removal energy uncertainty can have asignificant effect on the allowed regions for sin θ and∆ m and, to a lesser extent, ∆ m was also sensitiveto all of the alternative models studied. This led to theaddition of new systematic uncertainties to the SK sam-ples and oscillation contours to account for these effects.The results of these simulated data studies are an impor-tant outcome of this analysis: long-baseline experimentsare now entering the precision era, and changes of a fewpercent in the reconstructed neutrino energy can havesignificant effects on the oscillation parameters observedby the experiment.The T2K oscillation parameter measurements are,however, limited by statistics. T2K will collect moredata in both neutrino and antineutrino beam operationmode. More complex event topologies will be included inthe analyses at both ND280 and Super-Kamiokande, andthe collaboration is working towards joint analyses withboth the Super-Kamiokande and the NO ν A experiments.The combination of new topologies and differing neutrinoenergies will lift the degeneracies between oscillation pa-rameters present in a single experiment. This will enablethe most sensitive measurements of neutrino oscillationsto date, and will provide a template for future analysesat the next generation of long-baseline experiments. Thedata related to the measurement and results presented inthis paper can be found in [101].
ACKNOWLEDGMENTS
We thank the J-PARC staff for superb acceleratorperformance. We thank the CERN NA61/SHINECollaboration for providing valuable particle productiondata. We acknowledge the support of MEXT, JSPSKAKENHI (JP16H06288, JP18K03682, JP18H03701,JP18H05537, JP19J01119, JP19J22440, JP19J22258,JP20H00162, JP20H00149, JP20J20304) and bilat-eral programs(JPJSBP120204806, JPJSBP120209601),Japan; NSERC, the NRC, and CFI, Canada; the CEAand CNRS/IN2P3, France; the DFG (RO 3625/2), Ger-many; the INFN, Italy; the Ministry of Education andScience(DIR/WK/2017/05) and the National ScienceCentre (UMO-2018/30/E/ST2/00441 ), Poland; theRSF (19-12-00325), RFBR(JSPS-RFBR 20-52-5001020) and the Ministry of Science and HigherEducation(075-15-2020-778), Russia; MICINN (SEV-2016-0588, PID2019-107564GB-I00, PGC2018-099388-BI00) and ERDF funds and CERCA program, Spain;7the SNSF and SERI (200021 185012, 200020 188533,20FL21 186178I), Switzerland; the STFC, UK; and theDOE, USA. 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The nominal and post-ND280-fit values are shown inTables XXIII and XXIV for the flux parameters, andTable XXV for the cross section parameters.TABLE XXIII: Prefit and postfit weights for the SKFHC flux parameters.
FHC SK flux Prefit ND280 Postfitparameter /GeVSK ν µ [0.0, 0.4] 1.00 ± ± ν µ [0.4, 0.5] 1.00 ± ± ν µ [0.5, 0.6] 1.00 ± ± ν µ [0.6, 0.7] 1.00 ± ± ν µ [0.7, 1.0] 1.00 ± ± ν µ [1.0, 1.5] 1.00 ± ± ν µ [1.5, 2.5] 1.00 ± ± ν µ [2.5, 3.5] 1.00 ± ± ν µ [3.5, 5.0] 1.00 ± ± ν µ [5.0, 7.0] 1.00 ± ± ν µ [7.0, ∞ ] 1.00 ± ± ν µ [0.0, 0.7] 1.00 ± ± ν µ [0.7, 1.0] 1.00 ± ± ν µ [1.0, 1.5] 1.00 ± ± ν µ [1.5, 2.5] 1.00 ± ± ν µ [2.5, ∞ ] 1.00 ± ± ν e [0.0, 0.5] 1.00 ± ± ν e [0.5, 0.7] 1.00 ± ± ν e [0.7, 0.8] 1.00 ± ± ν e [0.8, 1.5] 1.00 ± ± ν e [1.5, 2.5] 1.00 ± ± ν e [2.5, 4.0] 1.00 ± ± ν e [4.0, ∞ ] 1.00 ± ± ν e [0.0, 2.5] 1.00 ± ± ν e [2.5, ∞ ] 1.00 ± ± RHC SK flux Prefit ND280 Postfitparameter /GeVSK ν µ [0.0, 0.7] 1.00 ± ± ν µ [0.7, 1.0] 1.00 ± ± ν µ [1.0, 1.5] 1.00 ± ± ν µ [1.5, 2.5] 1.00 ± ± ν µ [2.5, ∞ ] 1.00 ± ± ν µ [0.0, 0.4] 1.00 ± ± ν µ [0.4, 0.5] 1.00 ± ± ν µ [0.5, 0.6] 1.00 ± ± ν µ [0.6, 0.7] 1.00 ± ± ν µ [0.7, 1.0] 1.00 ± ± ν µ [1.0, 1.5] 1.00 ± ± ν µ [1.5, 2.5] 1.00 ± ± ν µ [2.5, 3.5] 1.00 ± ± ν µ [3.5, 5.0] 1.00 ± ± ν µ [5.0, 7.0] 1.00 ± ± ν µ [7.0, ∞ ] 1.00 ± ± ν e [0.0, 2.5] 1.00 ± ± ν e [2.5, ∞ ] 1.00 ± ± ν e [0.0, 0.5] 1.00 ± ± ν e [0.5, 0.7] 1.00 ± ± ν e [0.7, 0.8] 1.00 ± ± ν e [0.8, 1.5] 1.00 ± ± ν e [1.5, 2.5] 1.00 ± ± ν e [2.5, 4.0] 1.00 ± ± ν e [4.0, ∞ ] 1.00 ± ± Cross-section Prefit ND280 Postfitparameter M QEA (GeV/c ) 1.20 ± ± C(MeV/c) 217 ±
13 224 ± O(MeV/c) 225 ±
13 205 ± ν ± ± ν ± ± C / O ratio 1.00 ± ± C 1.00 ± ± O 1.00 ± ± ± ± ± ± ± ± ± ± ± ± C A ± ± M RESA (GeV/c ) 1.07 ± ± I = background 0.96 ± ± ν e / ν µ ± ± ν e / ν µ ± ± ± ± C 1.00 ± ± O 1.00 ± ± ± ± γ ± ± ± ± ± ± ± − . ± ± − . ± ± ± ± − . ± ± − . ± ± ±±