First T2K measurement of transverse kinematic imbalance in the muon-neutrino charged-current single-π^+ production channel containing at least one proton
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 Berguño, 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, C. Jesús-Valls, et al. (211 additional authors not shown)
FFirst T2K measurement of transverse kinematic imbalance in the muon-neutrinocharged-current single- π + production channel containing at least one proton K. Abe, N. Akhlaq, R. Akutsu, A. Ali, C. Alt, C. Andreopoulos,
52, 34
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,
51, 11
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,
42, 26
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,
60, 26
T. Hasegawa, † S. Hassani, N.C. Hastings, Y. Hayato,
54, 26
A. Hiramoto, M. Hogan, J. Holeczek, N.T. Hong Van,
18, 25
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, 26, ‡ 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,
54, 26
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,
54, 26
Y. Nakajima, A. Nakamura, K. Nakamura,
26, 14, † Y. Nakano, S. Nakayama,
54, 26
T. Nakaya,
30, 26
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,
55, 26
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, D. Pershey, L. Pickering, C. Pidcott, G. Pintaudi, C. Pistillo, B. Popov, ∗∗ K. Porwit, M. Posiadala-Zezula, 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,
54, 26, ‡ D. Sgalaberna, A. Shaikhiev, A. Shaykina, M. Shiozawa,
54, 26
W. Shorrock, A. Shvartsman, K. Skwarczynski, M. Smy, J.T. Sobczyk, H. Sobel,
4, 26
F.J.P. Soler, Y. Sonoda, R. Spina, S. Suvorov,
24, 51
A. Suzuki, S.Y. Suzuki, † Y. Suzuki, A.A. Sztuc, M. Tada, † M. Tajima, A. Takeda, Y. Takeuchi,
29, 26
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,
26, 4
S. Valder, D. Vargas, G. Vasseur, C. Vilela, W.G.S. Vinning, T. Vladisavljevic, T. Wachala, J. Walker, J.G. Walsh, Y. Wang, L. Wan, D. Wark,
52, 42
M.O. Wascko, A. Weber,
52, 42
R. Wendell, ‡ M.J. Wilking, C. Wilkinson, J.R. Wilson, K. Wood, C. Wret, J. Xia, Y.-h. Xu, 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. 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 Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France 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 8, 2021)This paper reports the first T2K measurement of the transverse kinematic imbalance in the single- π + production channel of neutrino interactions. We measure the differential cross sections in themuon-neutrino charged-current interaction on hydrocarbon with a single π + and at least one protonin the final state, at the ND280 off-axis near detector of the T2K experiment. The extracted crosssections are compared to the predictions from different neutrino-nucleus interaction event generators.Overall, the results show a preference for models which have a more realistic treatment of nuclearmedium effects including the initial nuclear state and final-state interactions. I. INTRODUCTION
In recent years, neutrino oscillation measurementshave reached unprecedented precision [1–7]. The nextgeneration of long-baseline (LBL) neutrino oscillation ex-periments, such as DUNE [8] and Hyper-Kamiokande [9],aim to measure important neutrino properties such asthe CP-violating phase and mass ordering [10, 11]. Thisrequires unprecedented constraints on the neutrino flux,neutrino cross sections and interaction model, and detec-tor response. Amongst all the systematic uncertainties,the limited knowledge of neutrino-nucleus interactions,especially those related to nuclear medium effects, is par-ticularly concerning because it can cause biases in eventclassification and neutrino energy reconstruction. In thelatest T2K oscillation analysis [12], the uncertainty innucleon removal energy in charged current quasielastic(CCQE) interactions is the dominant systematic compo-nent. In order to reduce its value, a more refined analysisis necessary to avoid potential biases in the next measure-ment of ∆ m .In the range of energies of current LBL experiments,neutrinos interact predominantly with nucleons. Theinitial state nucleon can be described by Fermi motiontogether with nucleon-nucleon correlations in a meanfield potential. After a neutrino interacts with a nu-cleon, the residual nucleus may be left in a simple one-particle-one-hole (1p1h) excited state, or collective 1p1hexcitations described by random phase approximations(RPA) [13–17]. It is also possible to have two-particle-two-hole (2p2h) excitations due to meson-exchange cur-rents (MEC) or short-range correlations [17–23]. How-ever, in most generators, these correlations are only im-plemented in the quasielastic (QE) channel, not for theresonant production (RES) or deep inelastic scattering ∗ 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. (DIS) channels.Moreover, after the primary neutrino-nucleon interac-tion, the outgoing hadrons have to propagate throughthe nuclear remnant before they can be detected. Final-state interactions (FSI) may cause energy dissipation andhadron absorption, or conversely induce the emission ofadditional hadrons. As a result, FSI can change the final-state topology of a neutrino-nucleon interaction, makingthe identification of primary neutrino-nucleon interac-tion and the measurement of primary hadronic kinemat-ics difficult. Neutrino cross sections are often measuredin terms of experimentally accessible final-state topolo-gies, e.g. in charged-current (CC) interactions, the CC0 π topology has only one charged lepton, any number of nu-cleons and no other particles; the CC1 π + topology hasonly one charged lepton, one π + , any number of nucleonsand no other particles.To achieve the designed sensitivity of future LBL ex-periments, nuclear effects have to be modelled accu-rately and consistently amongst all interaction channels.Experimental studies probing nuclear effects in carbon,through the measurement of transverse kinematic imbal-ance (TKI) in CC interactions [24, 25], have been per-formed in T2K [26] and MINER ν A [27–29]. TKI ex-plores the lepton-hadron correlations on the plane thatis transverse to the initial neutrino direction and helpsprecisely identify intranuclear dynamics [25–36], or theabsence thereof [24, 37–40], in neutrino-nucleus interac-tions. These measurements, in particular, either focusedon final-state topologies without any pions, or final-statetopologies with at least one neutral pion. These studiessuggest that modelling nuclear effects with Fermi gas ini-tial state models is insufficient, but more data is neededto draw solid conclusions.In this paper, we describe the first measurement of the ν µ cross section on hydrocarbon as a function of TKIvariables in CC production of exactly one π + and noother mesons, and at least one proton. We introduceTKI in Section II and the T2K experiment in Section III.The event simulation and event selection of the analysisare described in Section IV and Section V respectively.Then, the analysis procedure is explained in Section VI,followed by the interpretation of results in Section VII.We conclude in Section VIII. II. OBSERVABLES
In a ν µ CC RES π + interaction on a free proton p, ν µ + p → µ − + π + + p , (1)a ν µ interacts with an initial-state p to produce a final-state µ − , π + and p. This is the most important channelthat produces π + with the T2K neutrino beam which isnarrowly peaked at 0.6 GeV. However, in most neutrinoexperiments, the target involves some nucleus, A, heavierthan hydrogen. In general, a ν µ CC1 π + interaction withat least one proton in the final-state can be written as ν µ + A → µ − + π + + p + A (cid:48) , (2)where A (cid:48) is the final-state hadronic system consisting ofthe nuclear remnant and other possible knocked-out nu-cleons. Apart from the RES interaction in Eq. (1), thistopology also includes DIS interactions where multiplepions are produced and some are subsequently absorbedthrough FSI, leaving only one π + visible in the detector.Alternatively, CCQE interactions can be included in thistopology when an additional π + is produced through FSI.The kinematics of the µ − , π + and p tracks are used toconstruct the TKI. If there is more than one proton ob-served in the final state, only the highest momentum oneis considered.The set of three TKI variables, δp T T , p N and δα T ,were first introduced in Refs. [24, 25, 30, 33]. These ob-servables are designed to characterize the nuclear effectsthat are most relevant to oscillation experiments: theinitial nuclear state, such as the Fermi motion of initialstate nucleon and the nucleon removal energy, and theFSI of outgoing hadrons. The term “transverse” refersto the fact that all these observables are closely relatedto the transverse momentum component (cid:126)p iT (with respectto the incoming neutrino direction) of the final-state par-ticle i . In this analysis, the relevant transverse momentaare the transverse momenta of the muon, (cid:126)p µT , pion, (cid:126)p πT ,and proton, (cid:126)p p T .The first observable δp T T is the double-transverse mo-mentum imbalance [24], illustrated in Fig. 1a. A double-transverse axis is defined asˆ z T T ≡ (cid:126)p ν × (cid:126)p µ | (cid:126)p ν × (cid:126)p µ | , (3)and the pion and proton momenta are projected onto thisaxis: p πT T = ˆ z T T · (cid:126)p π ,p p T T = ˆ z T T · (cid:126)p p . (4)The imbalance is defined as δp T T = p πT T + p p T T . (5)In the absence of nuclear effects, δp T T = 0 is ex-pected due to momentum conservation. Inside a nuclear ~p p ~p π ˆ z TT ~p ν ~p µ p p TT p πTT (a) δp TT = p πTT + p p TT . ~p ν ~p µ ~p µ T − ~p µ T ~p h = ~p π + ~p p ~p hT δ~p T δα T (b) δ(cid:126)p T and δα T . FIG. 1. Schematic illustration of the TKI variables. The to-tal momentum of particle i is given by (cid:126)p i , while its transversecomponent with respect to the neutrino direction is repre-sented by (cid:126)p iT . In (b), the black circle represents the initialnucleon; the gray plane shows the transverse plane; the orangecircles and dashed lines indicate possible FSI experienced bythe outgoing hadrons. Figures adapted from Refs. [25, 29]. medium, an imbalance is caused by the initial state of thebound nucleon and the FSI experienced by the outgoingpion and proton.The second observable p N is the initial nucleon mo-mentum. Assuming the target nucleus is at rest and thereare no FSI, p N can be computed following the steps inRef. [33]. The transverse component of p N is equal to δ(cid:126)p T which is the sum of the transverse momenta [25](Fig. 1b): δ(cid:126)p T = (cid:126)p µT + (cid:126)p πT + (cid:126)p p T . (6)The longitudinal component of p N is given by [30] p L = 12 ( M A + p µL + p πL + p p L − E µ − E π − E p ) − δp T + M (cid:48) M A + p µL + p πL + p p L − E µ − E π − E p , (7)where p iL and E i are the longitudinal momentum andthe energy of the final-state particles. The target nucleusmass M A and the residual nucleus mass M A (cid:48) are relatedby M A (cid:48) = M A − M p + (cid:104) (cid:15) (cid:105) p , (8)where M p is the proton mass, and (cid:104) (cid:15) (cid:105) p = 26 . p N is given by [30] p N = (cid:113) δp T + p L , (9)which probes the Fermi motion inside the nucleus.Smearing by FSI can shift the peak position of p N , andcause a long tail in the region of large imbalance (simi-larly for δp T T ).The third observable δα T is the transverse boostingangle [25]: δα T = arccos (cid:18) − (cid:126)p µT · δ(cid:126)p T p µT δp T (cid:19) , (10)as illustrated in Fig. 1b. This observable quantifieswhether the hadronic system is accelerated or deceleratedby nuclear effects. Without FSI, the isotropic Fermi mo-tion of the initial-state nucleon would produce a ratherflat δα T distribution. However, FSI usually slows downthe outgoing hadrons, making δα T > o . Therefore, thestrength of FSI can be inferred from the shape of δα T .In the case where there are multiple nucleons emitted,these nucleons are not included in the above calculationand very likely result in a large imbalance in all the TKIvariables. III. THE T2K EXPERIMENT
The Tokai-to-Kamioka (T2K) experiment [42] is a LBLaccelerator-based neutrino experiment measuring oscilla-tions with a ν µ (¯ ν µ ) beam. The neutrino beam is pro-duced at the Japan Proton Accelerator Research Com-plex (J-PARC) which is located on the East coast ofJapan in T¯okai, Ibaraki. The neutrino beam is discussedin more detail in Section III A. J-PARC is also home toa suite of near detectors used to measure the propertiesof the unoscillated beam.The near detector complex is located at 280 m from theneutrino beam production target and consists of severaldetectors. INGRID [43] is an on-axis detector consistingof an array of 16 iron/scintillator modules, which pre-cisely measures the beam direction and intensity. The detector of primary interest for this analysis is the NearDetector at 280 m (ND280) which is placed 2 . ◦ awayfrom the beam axis and measures neutrino interactionsfor the off-axis flux. It is discussed in more detail inSection III B. The WAGASCI [44] and BabyMIND [45]detectors are located in the same near detector complexbut are situated 1 . ◦ off-axis.The far detector Super-Kamiokande [46] is a 50 ktwater Cherenkov detector located at a distance of 295km away from the J-PARC facility on the West coast ofJapan in Hida, Gifu. Super-Kamiokande is on the sameoff-axis angle as ND280. Neutrino CC interaction eventscan be classified into ν µ and ν e like, according to theshape of Cherenkov rings of the outgoing leptons. A. Neutrino Beam
The J-PARC facility utilizes a 30 GeV proton beamas the primary beamline. A proton spill consists of eightbunches spaced 580 ns apart and is produced every 2 .
48 s.The beam power has increased over time and reached520 kW during the latest data-taking period in 2019. Toproduce a neutrino beam, the proton beam is collidedwith a 91 . ν µ (¯ ν µ ). The magnetic horns are op-erated with a current of 250 kA ( −
250 kA) to produce a ν µ (¯ ν µ ) beam. The data taken while producing a ν µ (¯ ν µ )beam is qualified as neutrino-mode (antineutrino-mode).The focused beam of hadrons then enters a helium-filled,96 m long decay volume where they decay to produceneutrinos. At the end of the decay volume there is abeam dump and, behind this, a muon monitor [47, 48]which is used to monitor the stability of the secondarybeam.The neutrino beams are made up of ν µ , ¯ ν µ , ν e and ¯ ν e components. The neutrino flux predictions and the differ-ent flavour components at ND280 are shown in Fig. 2 [49].The off-axis configuration allows a narrow energy spec-trum with a peak energy of around 0.6 GeV. B. The off-axis Near Detector
In this analysis, we measure the ν µ differential crosssections as a function of TKI variables at the off-axis de-tector ND280. As shown in Fig. 3, ND280 is composed ofan upstream π detector (PØD) [50], followed by a cen-tral tracker region, all surrounded by an electromagneticcalorimeter (ECal) [51]. The outermost component is theformer UA1/NOMAD magnet, which provides a 0.2 Tdipole field, and contains scintillator modules in the airgaps acting as the side muon range detector (SMRD) [52]. (GeV) n E ] - P O T ) ( - ( M e V ) - F l ux [ c m m n m n e n e n FIG. 2. The flux prediction for ND280 in neutrino-mode isshown as well as the contributions from different neutrinoflavours.FIG. 3. Schematic showing an exploded view of the ND280off-axis detector. Each subdetector is labeled using theacronyms given in the text. FGD1 is placed upstream ofFGD2 and is shown in light green. The neutrino beam entersfrom the left of the figure.
The central tracker region contains two fine grained de-tectors (FGD1 and FGD2) [53] and three time projectionchambers (TPCs) [54]. The FGDs are instrumented withfinely segmented scintillator bars which act as both thetarget mass and particle tracker. The scintillator barsare made of 86 .
1% carbon, 7 .
4% hydrogen and 3 .
7% oxy- gen by mass. The bars are oriented alternately along thetwo detector coordinate axes (XY axes) transverse to theincoming neutrino beam (Z axis), and allow 3D trackingof charged particles. The most upstream FGD (FGD1) iscomposed of 15 XY planes of scintillator with each planehaving 2 ×
192 bars. The downstream FGD (FGD2) hasseven XY planes of scintillator with six 2 .
54 cm thick lay-ers of water in between, which allows cross section mea-surements to be made on water. This study focuses oncarbon interactions and only events occurring in FGD1are analyzed. For charged particles entering the TPCs,the curvature of the particle’s track and thus its momen-tum can be determined in the presence of the magneticfield with a resolution of 10% at 1 GeV. In combinationwith the measurement of energy loss per unit distance,TPCs provide high quality particle identification (PID)for charged particles.The ECal is a sampling calorimeter consisting of threekey parts: the PØD ECal which surrounds the PØD; theBarrel ECal which surrounds the FGDs and TPCs; andthe Downstream ECal which is located downstream ofthe FGDs and TPCs. The Barrel ECal and DownstreamECal together are referred to as the tracker-ECal. AllECals use layers of plastic scintillator bonded to leadsheets, and each alternating scintillator layer is rotatedby 90 ◦ to give 3D reconstruction. The tracker-ECal isdesigned to complement the FGDs and TPCs by givingdetailed reconstruction of electromagnetic showers and asecondary PID, with an energy resolution of 10% at 1GeV. IV. EVENT SIMULATION
For all T2K analyses, we need a reference Monte Carlo(MC) simulation to get a prediction based on the nominalneutrino flux, neutrino interaction model and detectoreffects. Data are then compared to MC to extract thephysics quantities of interest and estimate the systematicuncertainties.The modelling of the T2K neutrino flux [49] startswith the modelling of interactions of protons with thegraphite target, which is done using the FLUKA 2011package [55, 56]. Outside the target, the simulationof hadronic interactions and decays is done using theGEANT3 [57] and GCALOR [58] software packages.Hadronic interactions are further tuned with the recentmeasurements of π ± yields performed by NA61/SHINEexperiment using a T2K replica target [59]. The con-ditions of the proton beam, magnetic horn current andneutrino beam axis direction are continuously monitoredand incorporated into the simulation. This data-drivenstrategy helps to reduce the neutrino flux uncertaintynear the flux peak (0.5 - 0.6 GeV) to 5%. This results ina significant improvement with respect to previous T2Kcross-section analyses [60, 61] where the uncertainty wasaround 8.5% [62]. A comparison of the flux uncertaintyused in this analysis and the flux uncertainty used inprevious T2K analyses is shown in Fig. 4. -
10 1 10
Neutrino Energy (GeV) F r a c t i ona l E rr o r Updated (Replica target)Previous
FIG. 4. The fractional error on the muon neutrino flux atND280 as a function of energy used in this analysis (solid)and previous T2K analyses (dashed).
Neutrino-nucleus interactions and FSI of the outgoingparticles are simulated using the neutrino event genera-tor NEUT version 5.4.0 [63, 64]. NEUT uses the spectralfunction (SF) in Ref. [65] to describe the CCQE crosssection. The modelling of 2p2h interactions is based onthe model from Nieves et al. [66]. The RES pion produc-tion process is described by the Rein-Sehgal model [67]with updated nucleon form-factors [68] and an axial mass( M RESA ) of 1.07 GeV/c . The model contains contri-butions from non-resonant, I / pion-production chan-nels. The nuclear model used for RES is a relativis-tic global Fermi gas (RFG) [69], without a removal en-ergy and with a Fermi momentum of 217 MeV/c. DISinteractions are modelled using the GRV98 [70] partondistribution functions with corrections from Bodek andYang [71]. In the low invariant hadronic mass, W, re-gion (1 . < W ≤ . ) a custom hadronisationmodel [72] is used with suppressed single pion produc-tion to avoid double counting RES interactions. ForW > , PYTHIA/JETSET [73] is used as thehadronization model. The FSI, describing the transportof hadrons produced in elementary neutrino interactionthrough the nucleus, are simulated using a semi-classicalintranuclear cascade model. The NEUT cascade modelhas been tuned to external pion-scattering data, which isdescribed in Ref. [74].Outside the nucleus, final-state particles are then prop-agated through the detector material using GEANT4 ver-sion 4.9.4 [75]. The physics list [76] QGSP BERT is used forthe hadronic physics, emstandard opt3 for the electro-magnetic physics and
G4DecayPhysics for the particledecays. The pion secondary interactions are handled bythe cascade model in NEUT. The detector readout issimulated with a custom electronics simulation [42].
V. DATA AND EVENT SELECTION
In this analysis, the neutrino-mode data collected be-tween 2010 and 2017 is used, which corresponds to11 . × protons on target (POT) and an integratedmuon neutrino flux of 2 . × / cm . Events are re-quired to have an interaction vertex in the FGD1 fiducialvolume (FV), which includes all the XY planes of scin-tillator except for the most upstream one, and excludesthe outermost five bars on either end of each layer. Thisleaves the FV with 2 × ×
14 bars, and a total mass ofapproximately 973 kg. The MC contains simulated dataequivalent to 195 . × POT.
A. Signal definition
The goal of this analysis is to characterize nuclear ef-fects in ν µ CC1 π + interactions on carbon using neutrinointeractions inside FGD1, which is a hydrocarbon (CH)target. Since the CC1 π + production on carbon and onhydrogen cannot be clearly separated, the combined crosssection on CH is measured, with the TKI variables onhydrogen calculated in the same way as on carbon: forhydrogen signal events, in which there are no nuclear ef-fects, it is expected that δp T T = 0 and p N ≈
26 MeV/c. δα T is undefined for interactions on hydrogen because δp T = 0. A flat distribution across 0–180 o is assignedbecause it resembles the real δα T distribution due to thesmall but non-vanishing isotropic Fermi motion of a freeproton.To ensure the cross section results are not dependenton the signal model used in the reference T2K simulation,extensive precautions are taken in the analysis. A cru-cial one is to have the signal definition only be reliant onobservables experimentally accessible to ND280. There-fore, the signal is defined as any event with one µ − , one π + and no other mesons, and at least one proton in thefinal state, so that there is need to account for the pionand proton FSI. Hereafter, the signal topology is denotedas CC1 π + Xp, where X ≥
1. In order to mitigate modeldependence in the efficiency correction, phase-space re-strictions are applied in the signal definition to restrictthe measurement to the regions of kinematic phase spaceND280 is sensitive to. These restrictions are defined inTable I. However, the consideration of three-particle fi-nal states in this analysis necessitates the inclusion of ahigh dimensional kinematic phase space over which theefficiency cannot be kept entirely flat with simple phase-space constraints. This leads to a potential source of biasfrom the input neutrino interaction model predictions.To alleviate this concern, additional model uncertaintiesare added (discussed in Section VI B) to allow a varia-tion of the input simulation in regions of the underlyingparticle kinematics where the efficiency is not flat. Thesize of this uncertainty roughly double the largest varia-tion in the efficiency seen from a wide variety of differentgenerator predictions (broadly spanning those shown inSection VII A).
TABLE I. CC1 π + Xp signal phase-space restrictions for thepost-FSI final-state particles. The angle θ is relative to theneutrino direction. For events with multiple protons, only thehighest momentum proton is considered, and other protonsare ignored.Particle Momentum p Angle θµ − < o π + < o p 450-1200 MeV/c < o We select one signal sample for the events of interest,and four control samples to constrain the number of back-ground events in the signal sample. The five samples areshown schematically in Fig. 5.
B. Signal sample selection
The signal sample contains neutrino events with ex-actly one µ − track, one π + track, and at least one protontrack, maximizing the number of signal events selectedwith minimal background.The selection starts by searching for a good quality µ − track. Events within a 120 ns time window aroundone of the eight bunch centers per 5 µ s spill structureof the beam are considered. The highest momentum,negatively charged track originating from the FGD1 FVand making a long track through the downstream TPC ischosen to be a µ − candidate. Other detector activities inor around FGD1 are used as a veto to ensure the µ − trackis not a broken segment of another track from outsidethe FV. Then a muon PID cut is applied based on theenergy loss and momentum measurement in the TPC asin Ref. [77]. After this step a ν µ CC sample of 90.3%purity is obtained.Next, all other tracks originating from the FGD1 FVwith a long segment in the TPC are classified by theTPC PID. For positively charged tracks, three particlehypotheses are considered: π + , e + and proton; for neg-atively charged tracks, only two particle hypotheses areconsidered: π − and e − . Events with exactly one π + track, and at least one proton track are selected. Thosewith π − or e ± are rejected because they are likely to bethe products of DIS or other background interactions.Additional pions are identified in FGD1 and thetracker-ECal. Tracks fully contained inside FGD1 areclassified into pions or protons if the energy depositionand range are consistent with the corresponding parti-cle hypotheses. Michel electrons [78] are also identifiedby looking for a time-delayed FGD1 hit cluster, and areregarded as products of the pion-muon-electron decaychain. The tracker-ECal is employed to identify isolatedobjects that are consistent with a photon shower, andtags these as products of π → γ decay. Events withadditional charged pions in FGD1 or π in the ECal are rejected.In the final step, events with additional tracks in FGD1(either the fully contained tracks that are not classified,or the non-fully contained tracks without TPC PID) arerejected to reduce the low energy pion backgrounds thatare missed by the pion selection processes. Then we re-quire the µ − , π + and p tracks to have their starting po-sitions to be within a box of 50 mm ×
50 mm ×
30 mm inthe XY and Z planes. This ensures the tracks are comingfrom the same interaction vertex. Events that are not re-constructed to have matched the kinematic requirementsin Table II are put into an out-of-phase-space (OOPS)bin. Compared to the signal definition in Table I, thekinematic cuts have slightly larger ranges in momenta tocompensate for the finite momentum resolution. The ex-tremely good angular resolution (about 1 ◦ ) allows us touse the same angular restriction. TABLE II. Kinematic cuts for the reconstructed particles inthe analysis samples. The particle type and kinematics arethe reconstructed quantities. The angle θ is relative to theneutrino direction. For events with multiple reconstructedprotons, only the highest momentum proton is considered,and other protons are ignored.Particle Momentum p Angle θµ − < o π + < o p 405-1320 MeV/c < o Following the signal sample identification, the selectedevents (except the OOPS bin) are binned in one of thereconstructed TKI variables and the reconstructed high-est proton momentum, p p . The binning in TKI variablesis the same as that used in the cross section extractionin Section VI. The binning in p p helps to correct for thebias in estimating selection efficiencies. The binning in p p is chosen over other kinematic variables because nu-cleon emission from neutrino interactions is less under-stood than pion or muon emission. In addition, the TPCproton detection threshold is around 400 MeV, whichmight significantly affect the efficiency. Table III sum-marizes the signal sample binning. The CC1 π + Xp crosssections are measured as a function of a single TKI vari-able only, thus the number of reconstructed bins is muchmore than the number of cross-section bins. For exam-ple, in the δp T T measurement, there are six p p bins foreach of the five δp T T bins in the signal sample. In totalthere are 6 × δp T T .Fig. 6 shows the distribution of the reconstructedTKI variables and p p in the signal sample (without theOOPS bin). A total of 366 events are observed indata. The overall signal selection efficiency is around14%. When broken down by final-state topology, thetotal CC1 π +
1p (one proton) and CC1 π + Np (multipleprotons) signal purity is 61.1%. The four categories ofCC-other events with multiple pions in the final-state,CC1 π + π − , CC1 π + X π , CC-other-X π and CC-other- ECal µ − , π + , p in TPCFGD TPCECalSignal sample366 events µ − π + p ECalExtra π − in FGD/TPCFGD TPCECalCC1 π + π − enriched174 events µ − π + p π − ECalExtra π in ECalFGD TPCECalCC1 π + X π enriched404 events µ − π + p π ECalExtra π ± in FGD/TPC, π in ECalFGD TPCECalCC-other-X π enriched311 events µ − π + p π ± π ECalExtra π ± in FGD/TPC,except single π − FGD TPCECalCC-other-0 π enriched114 events µ − π + p π ± π ± FIG. 5. Schematic representation of the signal sample (left) and control samples (right) selection, together with the number ofevents observed in data. Details of the selection criteria are described in Sections V B and V C.TABLE III. Analysis bin edges for the CC1 π + Xp cross sec-tions as a function of the TKI variables. The signal sampleis binned in one of the reconstructed TKI variables vs. re-constructed p p . The control samples are binned in the recon-structed TKI variable only.Variable Number of bins Bin edges δp TT (MeV/c) 5 -700,-300,-100,100,300,700 p N (MeV/c) 4 0,120,240,600,1500 δα T (deg) 3 0,60,120,180 p p (MeV/c) 6 405,575,700,825,950,1075,1320 π , are mostly produced by DIS interactions and arethe dominant backgrounds. Details on how to constrainthese backgrounds are described in Sec.V C. There arealso small amounts of neutral current (NC) and ¯ ν µ /ν e / ¯ ν e events where a π − /e − is misidentified as a µ − . In mostcases the misidentification comes from NC interactions.The contribution from out of fiducial volume (OOFV)events is almost negligible. The OOPS background inFig. 6 refers to CC1 π + Xp events which do not satisfythe phase-space restrictions in Table I, and the separatedOOPS bin is used to constrain this background.
C. Control sample selection
To better constrain the CC-other background in thesignal sample, dedicated control samples (on the rightof Fig. 5) are selected based on the number of chargedand neutral pions identified in the events. Following theFGD1-TPC µ − , π + and p tracks selection described inSection V B, the control samples require the identificationof additional π ± tracks in the FGD/TPC or the identifi-cation of a π in the tracker-ECal. These events are thenclassified into four samples according to the additionalidentified pions:(i) CC1 π + π − enriched sample - events with one π − candidate from FGD1 or the TPC; (ii) CC1 π + X π enriched sample - events with π can-didates from the ECal;(iii) CC-other-X π enriched sample - events withcharged pion candidates from FGD1 or the TPC,and π candidates from the ECal;(iv) CC-other-0 π enriched sample - events with chargedpion candidates from FGD1 or the TPC, excludingthe case of single π − candidate.The four separate samples allow for better characteriza-tion of the pion emission model and detector responsesto different particles compared to a single CC-other sam-ple. The same kinematic cuts in Table II are appliedto the µ − , highest momentum π + and p tracks, and theTKI variables are calculated using only these tracks. Theselected events are binned in the reconstructed TKI vari-able only, using the same binning in Table III. Fig. 7shows the reconstructed TKI variable distributions forthe four control samples. The nominal MC shows adeficit of events and also some shape discrepancies withrespect to data, indicating the need for background cor-rection. VI. ANALYSIS METHODA. Binned likelihood fitting
The analysis is performed using an unregularizedbinned likelihood fit as in Refs. [26, 60, 61, 77, 79], withcontrol samples to constrain the background, to unfoldthe detector smearing and extract the number of selectedsignal events from the signal sample. Compared to previ-ous cross-section analyses, significant improvements havebeen achieved in the analysis framework, including theuse of principle component analysis to reduce the dimen-sionality of the fit, and the proper treatment of MC sta-tistical uncertainties. An unregularized fit means thatthere is no prior constraint on the shape of TKI from the0 (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n (MeV/c) N p Reconstructed E v e n t s p e r b i n Data ( 0.6%) p CC0 0p ( 0.7%) + p CC1 1p (53.6%) + p CC1 Np ( 7.5%) + p CC1 ( 7.4%) - p + p CC1 ( 8.0%) p X + p CC1 ( 2.2%) p CC-other-X ( 3.4%) p CC-other-0 ( 6.2%) e n , e n , m n NC, OOFV ( 1.1%)OOPS ( 9.4%) (deg) T ad Reconstructed E v e n t s p e r b i n (MeV/c) p p Reconstructed
500 600 700 800 900 1000 1100 1200 1300 E v e n t s p e r b i n FIG. 6. Distribution of events in the signal sample as a function of the reconstructed TKI variables and highest protonmomentum, broken down into true final-state topology predicted by the nominal MC. The legend shows the fraction of eventsin all plots. Histograms are stacked. The MC has been normalized to 11.6 × POT, the equivalent number of POT collectedfor the data. The error bars show the statistical uncertainty in data. input signal model, thus reducing model bias on the fittedcross sections. The numbers of signal events (and thuscross sections) as a function of the three TKI variablesare fitted independently in this study.The input MC is varied by a set of fit parameters, andthe set of parameters which best describes the observeddata is extracted together with its associated errors. Thefit parameters of primary interest are the “signal tem-plate parameters”, c i , which scale the number of signalevents in the truth TKI variable bin i without prior con-straints. The remaining parameters are the nuisance pa-rameters which describe plausible systematic variationsof the flux, detector response and neutrino interactionmodel. The effect of these parameters is propagated tothe number of selected events in the reconstructed bins.The best-fit parameters are found by minimizing thefollowing negative log-likelihood ( χ ): χ = − L ) = − L stat ) − L syst ) , (11) where χ = − L stat )= reco. bins (cid:88) j (cid:32) β j N MC j − N obs j + N obs j log N obs j β j N MC j + ( β j − σ j (cid:33) , (12)and χ = − L syst )= ( (cid:126)a syst − (cid:126)a systprior ) T ( V systcov ) − ( (cid:126)a syst − (cid:126)a systprior ) . (13)Eq. (12) is the modified Poisson likelihood ratio whichincludes the statistical uncertainty of finite MC statis-tics using the Barlow-Beeston method [80, 81]. N MCj and N obsj are the number of events in each reconstructed1 (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n enriched - p + p CC1 (MeV/c) N p Reconstructed E v e n t s p e r b i n enriched - p + p CC1 (deg) T ad Reconstructed E v e n t s p e r b i n enriched - p + p CC1
Data ( 0.4%) p CC0 0p ( 1.1%) + p CC1 1p ( 2.3%) + p CC1 Np ( 0.8%) + p CC1 (31.9%) - p + p CC1 ( 3.8%) p X + p CC1 (22.9%) p CC-other-X (25.7%) p CC-other-0 ( 6.1%) e n , e n , m n NC, OOFV ( 2.1%)OOPS ( 3.0%) (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n enriched p X + p CC1 (MeV/c) N p Reconstructed E v e n t s p e r b i n enriched p X + p CC1 (deg) T ad Reconstructed E v e n t s p e r b i n enriched p X + p CC1
Data ( 0.4%) p CC0 0p ( 1.2%) + p CC1 1p (12.2%) + p CC1 Np ( 2.1%) + p CC1 ( 7.5%) - p + p CC1 (29.5%) p X + p CC1 (19.1%) p CC-other-X ( 9.7%) p CC-other-0 (11.1%) e n , e n , m n NC, OOFV ( 2.2%)OOPS ( 5.0%) (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n enriched p CC-other-X (MeV/c) N p Reconstructed E v e n t s p e r b i n enriched p CC-other-X (deg) T ad Reconstructed E v e n t s p e r b i n enriched p CC-other-X
Data ( 0.2%) p CC0 0p ( 0.6%) + p CC1 1p ( 0.3%) + p CC1 Np ( 0.1%) + p CC1 (11.7%) - p + p CC1 ( 7.7%) p X + p CC1 (46.1%) p CC-other-X (21.1%) p CC-other-0 ( 9.2%) e n , e n , m n NC, OOFV ( 1.9%)OOPS ( 1.2%) (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n enriched p CC-other-0 (MeV/c) N p Reconstructed E v e n t s p e r b i n enriched p CC-other-0 (deg) T ad Reconstructed E v e n t s p e r b i n enriched p CC-other-0
Data ( 0.3%) p CC0 0p ( 1.0%) + p CC1 1p ( 1.2%) + p CC1 Np ( 0.6%) + p CC1 ( 7.7%) - p + p CC1 ( 4.5%) p X + p CC1 (27.8%) p CC-other-X (45.5%) p CC-other-0 ( 6.8%) e n , e n , m n NC, OOFV ( 1.5%)OOPS ( 3.0%)
FIG. 7. The distribution of events in the four control samples (top to bottom) as a function of reconstructed TKI variables (leftto right), broken down into true final-state topology predicted by the nominal MC. The legends show the fraction of events ineach control sample. Histograms are stacked. The MC has been normalized to 11.6 × POT, the equivalent number of POTcollected for the data. The error bars show the statistical uncertainty in data. j , for MC and data respectively. β j is the Barlow-Beeston scaling parameter given by β j = 12 (cid:16) − ( N MC j σ j −
1) + (cid:113) ( N MC j σ j − + 4 N obs j σ j (cid:17) , (14)and σ j is the relative variance of N MCj . In the limit ofinfinite MC statistics, σ j → β j → (cid:126)a syst agree with theirprior values (cid:126)a systprior , where V systcov is the covariance matrixdescribing the confidence in the prior values as well ascorrelations between parameters.The MC prediction N MC j in the signal and controlsamples is composed of both the signal and backgroundevents, which can be written as N MC j = true bins (cid:88) i ( c i w sig i,j N sig i,j + w bkg i,j N bkg i,j ) , (15)where N sig i,j and N bkg i,j are the number of signal and back-ground events in the truth bin i , contributing to the re-constructed bin j , predicted by the T2K MC; w sig i,j and w bkg i,j are the event weights coming from the same set ofsystematic variations and thus are correlated. B. Sources of systematic uncertainties
Three sources of systematic uncertainties are consid-ered in this analysis.
Neutrino flux uncertainty:
This is parametrized asscale factors in bins of true neutrino energy (samebinning as in Fig. 4). Such scale factors are con-strained by their prior uncertainty, encoded in acovariance matrix. At the same energy, identicalevent weights are applied on the signal and back-ground events.
Detector uncertainty:
The detector response (effi-ciency and resolution) is not perfectly modelled inthe simulation. Dedicated and independent controlsamples are used to evaluate each possible uncer-tainty based on the data-MC agreement. The over-all detector uncertainty is parametrized as a covari-ance matrix that describes the rate uncertainty andcorrelation between each reconstructed bin. Theuncertainty related to the modelling of the pionsecondary interactions, one of the largest detectorsystematics in previous T2K analyses, has been re-duced by around 40% using external data and thecascade model implemented in NEUT [74]. In thesignal sample and control samples without recon-structed π , the biggest uncertainty comes from themodelling of proton secondary interactions whichcauses a 5% uncertainty on the event rate. Onthe other hand, π -tagging uncertainty is dominant(around 10%) in the control samples with recon-structed π . Neutrino interaction model uncertainty:
Thistakes care of both the modelling of signal andbackground interactions, including FSI. In thisanalysis, the estimation of signal efficiency andbackground contamination are most significantlyaffected by the RES and DIS processes. In the RESchannel, there are three model parameters: the res-onant axial mass M RESA (1.07 ± ), thevalue of the axial form factor at zero transferred4-momentum C A5 (0.96 ± I / (0.96 ± ++ decay width with 50% uncer-tainty, and ad hoc scale parameters binned insignal particle momenta and angles with a 20%uncertainty, are included to give extra freedom tothe efficiency correction. The ad hoc variations arechosen to cover the efficiency’s dependency on theinitial state nuclear medium effects, which is nototherwise parametrized.In the DIS channel, a CC-other shape parameter x CC-Other with a 40% uncertainty is used, whichscales the cross section by (1 + x CC-Other /E ν )and gives greater flexibility at low E ν . Fournormalization parameters with a 50% uncertainty,with the same categorization as the four CC-othertopologies, are introduced to better parametrizemultiple pion production. The neutral currentand electron (anti)neutrino interactions, whichare not constrained by the control samples, aregiven a normalization uncertainty of 30% and 3%respectively.Finally, there are parameters varying the pionand proton FSI. The tunable pion interactionsin the nucleus are charge exchange, where thecharge of the pion changes; absorption, wherethe pion is absorbed through two- or three-bodyprocesses; elastic scattering, where the pion onlyexchanges momentum and energy; and inelasticscattering, where additional pions are produced.Their prior is given by Ref. [74]. For proton FSI,there is a single parameter scaling the overallinteraction probability inside the cascade with a50% uncertainty, without tuning specific processes.It is verified that with such comprehensive list ofparameters, the fit can cover the bias in signal effi-ciency and background subtraction under extrememodel variations as discussed in Section VI C.3 C. Cross section extraction, error propagation andvalidation
After the number of signal events is extracted from thefit, the differential cross section as a function of the trueTKI variable is calculated by the following formula: dσdx i = N signal i (cid:15) i Φ N FVnucleons ∆ x i , (16)where N signal i is the measured number of signal eventsin the i -th bin, for all CC1 π + Xp events on hydrocarbonsatisfying the kinematic phase restrictions in Table I. In-teractions on other elements are estimated by MC andsubtracted. Since the fraction of non-hydrocarbon eventsis small, the potential bias due to cross-section or detec-tor mismodelling is insignificant. (cid:15) i is the selection ef-ficiency in the i -th bin, contributed by both the signaland control samples. Φ is the overall flux integral, evalu-ated at the best-fit flux parameter values, and N FVnucleons is the number of target nucleons (only hydrocarbon) inthe fiducial volume. x i is one of the TKI variables and∆ x i is the bin width.We use a similar method as in Refs. [26, 89] to numeri-cally propagate the uncertainty of the fit to the cross sec-tion result, assuming the uncertainties of the fit parame-ters and cross sections are part of a Gaussian distribution.The covariance matrix of the fit parameters is Choleskydecomposed and multiplied by a vector of Gaussian ran-dom numbers to generate a set of random parameters.These random parameters are added to the best-fit pa-rameters to create 2000 sets of variations (“toys”) of pa-rameters. This effectively samples the likelihood spaceencoded in the covariance matrix, and represents thespread of the plausible parameters according to the sta-tistical and systematic uncertainties from the fit. Foreach toy, all variables in Eq. (16) (except ∆ x i ), and thus dσdx i , are re-evaluated with the toy parameters. The fluxintegral and selection efficiency are changed by the toyparameters. The resultant uncertainty of the flux inte-gral is around 5%, and Fig. 8 shows the mean values anduncertainties of the efficiency extracted from toys. Thenumber of target nucleons N FVnucleons is sampled indepen-dently with a mean value of 5 . × and an uncertaintyof 0.67% [53]. Finally, a covariance matrix V of dσdx i isbuilt from such toys. This method is different from theone used in previous analyses [60, 61], where the uncer-tainty was estimated by repeating the fit many times withtoys of input MC.To ensure our results are not biased, a plethora of mockdata studies with alternative neutrino event generators,nuclear ground state models, background models and al-tered flux models have been performed. It has been veri-fied that even in the case of extreme deviations from theinput MC model, such as doubling the signal/backgroundinteractions or completely turning off the FSI, the crosssection extraction method employed can always recoverthe truth values to within a 1 σ -uncertainty. The fit per-formance for every mock data study has been quantified (MeV/c) TT p d - - - E ff i c i e n c y (MeV/c) N p E ff i c i e n c y (deg) T ad E ff i c i e n c y FIG. 8. Mean values and uncertainties of the selection efficien-cies as a function of the TKI variables. The error bars includeboth the statistical and systematic uncertainties propagatedfrom the fit. by computing the post-fit p-value. First, 1000 sets ofMC data samples are produced as a result of statisticaland systematic variations of the nominal MC, which arethen fitted to build the distribution of the post-fit χ (Eq. (11)). The p-value for each mock data study hasbeen computed from this distribution and an acceptance4threshold of 5% has been chosen to quantify good fit-ter performances. All the mock data studies performed(without applying statistical fluctuations) have a p-valuearound 90%, showing that the model differences are wellcovered by the conservative systematic uncertainties. Onthe other hand, the agreement on the measured cross sec-tions is quantified by the χ statistic: χ = (cid:88) i (cid:88) j (cid:18) dσ truth dx i − dσ meas dx i (cid:19) · ( V − ) ij (cid:18) dσ truth dx j − dσ meas dx j (cid:19) , (17)where σ meas is the measured cross section, and σ truth isthe truth cross section in the mock data. All mock datafits return a χ /ndof less than 0.4, where ndof is thenumber of degrees of freedom, and a p-value greater 80%,showing the robustness of the cross section extractionmethod employed for this analysis. VII. RESULTS
Figs. 9 to 11 show the distributions of the recon-structed events in the signal and control samples, to-gether with the prediction from the pre-fit and post-fitMC. Overall, the fit is able to reproduce the observeddistributions, with a p-value greater than 10% for all theTKI variable fits, and is qualified to have a good data-MCagreement in the presence of statistical fluctuations. Allnuisance parameters are fitted within their prior uncer-tainties. The normalization difference in control samplesbefore the fit is well covered by the nuisance parameters,mostly through the CC-other normalization parameters.In the signal sample, there are few bins of reconstructed p p where the post-fit χ is worse than the pre-fit one.This indicates there might not be enough freedom in theshape of the signal particle kinematics. However, fromthe mock data studies, it is concluded that the potentialbias is much smaller than the statistical uncertainty andhas little impact on this analysis.Fig. 12 estimates the uncertainties of the cross sectionsas a function of the TKI variables, together with the cor-relation between bins. Contributions from each kind ofsystematic uncertainties are estimated by running the fitwith only the relevant nuisance parameters. As expected,the statistical error is much larger than the individualor combined systematic uncertainties. The largest sys-tematic uncertainties are those related to the neutrinointeraction model, which affect both the signal selectionefficiency and background estimation. The bin-by-bincorrelation in δα T is larger than that in δp T T and p N because the cross section on hydrogen is uniform acrossall bins of δα T . A. Comparisons to models
In the following, the measured cross sections are com-pared to different neutrino interaction models. Theagreement is quantified by the χ statistic in Eq. (17),with σ truth replaced by the model prediction σ model .On the other hand, the overall normalization uncer-tainty, which is fully correlated between bins, may consti-tute a relatively large fraction of the uncertainty. There-fore, the χ statistics may suffer from “Peelle’s perti-nent puzzle” [90, 91], in which the assumption in Eq. (17)that the variance is distributed as a multivariate Gaus-sian may not be valid for highly correlated results. Tomitigate this problem, the shape only χ is also pro-vided: χ = (cid:88) i (cid:88) j (cid:18) dσ model dx i σ modelint − dσ meas dx i σ measint (cid:19) · ( W − ) ij (cid:18) dσ model dx j σ modelint − dσ meas dx j σ modelmeas (cid:19) , (18)where σ modelint and σ measint are the total integrated crosssections per nucleon estimated from the model and datarespectively. The shape only covariance matrix W is builtby the same method as described in Section VI C but onthe shape variable dσ meas dx i σ measint instead. It is importantto notice that the ndof is one less for χ comparedto χ since the sum of the shape variables is equal toone by construction, reducing the number of independentdimensions.To compare the measured cross sections with modelpredictions, a sufficiently large number of events are gen-erated on hydrocarbon from each model using the T2Kflux. Events satisfying the CC1 π + Xp signal definition inTable I are selected to calculate the cross sections pertarget nucleon. The number of target nucleons for eachCH is equal to 13 which includes all seven protons andsix neutrons. The following models are considered.(i) NEUT version 5.4.0: models implemented in thisevent generator are described in Section IV. RFGis used as the nuclear ground state for pion produc-tion.(ii) GENIE [92, 93] version 3.0.6: two model configura-tions are compared: the “BRRFG+hA” model usesthe G18 01a physics configuration, with the Rein-Sehgal (RS) model for pion production, Bodek-Ritchie empirical corrections of RFG (BRRFG [94,95]) as the nuclear ground state model and the hA(“empirical”) FSI model; the “LFG+hN” modeluses the G18 10b physics configuration, with theBerger-Sehgal (BS) model [96] for pion production,local Fermi gas (LFG) as nuclear ground state andthe hN (“cascade”) FSI model. For both mod-els, the 2018a free nucleon cross section model re-tune [97] is used. Specific to pion production,5 (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n <575MeV/c p Signal sample, 405MeV/c
DataPre-fitPost-fit (MeV/c) TT p d Reconstructed - - - E v e n t s p e r b i n <700MeV/c p Signal sample, 575MeV/c
FIG. 9. Distribution of events in the signal and control samples in the δp TT fit. χ corresponds to the statistical contributionof the fit χ (Eq. (12)) in that sample. The MC prediction before (dashed) and after (solid) the fit are also shown. The errorbars show the statistical uncertainty in data. (MeV/c) N p Reconstructed E v e n t s p e r b i n <575MeV/c p Signal sample, 405MeV/c
DataPre-fitPost-fit (MeV/c) N p Reconstructed E v e n t s p e r b i n <700MeV/c p Signal sample, 575MeV/c
FIG. 10. Distribution of events in the signal and control samples in the p N fit. χ corresponds to the statistical contributionof the fit χ (Eq. (12)) in that sample. The MC prediction before (dashed) and after (solid) the fit are also shown. The errorbars show the statistical uncertainty in data. (deg) T ad Reconstructed E v e n t s p e r b i n <575MeV/c p Signal sample, 405MeV/c
DataPre-fitPost-fit (deg) T ad Reconstructed E v e n t s p e r b i n <700MeV/c p Signal sample, 575MeV/c
FIG. 11. Distribution of events in the signal and control samples in the δα T fit. χ corresponds to the statistical contributionof the fit χ (Eq. (12)) in that sample. The MC prediction before (dashed) and after (solid) the fit are also shown. The errorbars show the statistical uncertainty in data. (MeV/c) TT p d - - - F r ac ti on a l E rr o r - - - - - (MeV/c) TT p d - - - ( M e V / c ) TT p d - - - (MeV/c) N p F r ac ti on a l E rr o r - - - - - (MeV/c) N p ( M e V / c ) N p (deg) T ad F r ac ti on a l E rr o r Statistical + all systematic uncertaintiesStatistical uncertainties onlyStatistical + cross-section uncertaintiesStatistical + flux uncertaintiesStatistical + detector uncertainties - - - - - (deg) T ad ( d e g ) T ad FIG. 12. Error of the measured differential cross sections in each bin (left) and the correlation between bins (right), for the δp TT (top), p N (middle) and δα T (bottom) fit respectively. The statistical error is shown in red, and the total statistical andsystematic error in black. Contributions from each source of systematic uncertainties are shown one by one: neutrino flux inblue, detector in yellow, and neutrino interaction model in green. π and CC2 π cross section data on deuteriumtargets from ANL [82, 87, 98, 99], BNL [84, 87],BEBC [100–102] and FNAL [103] bubble chamberexperiments were used in the re-tune. This mostlyaffects the cross-section normalization ( ∼
15% re-duction in total cross section) and gives much better χ agreement.(iii) GiBUU [104] version 2019: it uses an LFG-basednuclear ground state to describe all neutrino inter-action modes and FSI consistently. In the RESchannel, 13 resonances are included and the non-resonant contribution is described by a phenomeno-logical model. Rather than a simple cascade model,GiBUU models FSI by solving the dynamical evo-lution of the particle phase space density in the nu-clear mean field potential.(iv) NuWro [105] version 19.02: four different nuclearground state models are implemented. These in-clude three Fermi gas models: LFG, RFG, and BR-RFG; and an effective approximation of a spectralfunction (ESF) [106]. The Adler-Rarita-Schwingersingle ∆ model [107, 108] is used for RES, and theFSI cascade model is based on the Metropolis algo-rithm [109].Fig. 13 shows the comparisons between the measuredcross sections and model predictions. The full χ andshape only χ are summarized in Table IV. It is ob-served that χ is usually much smaller than χ , im-plying a large part of the model separation power in thisanalysis comes from normalization differences. TABLE IV. χ and χ for the three TKI variable mea-surements. The ndof of χ is equal to 5, 4, and 3 for δp TT , p N and δα T respectively. The ndof of χ is one less thanthat of χ . χ ( χ )Generator δp TT p N δα T NEUT RFG 11.3 (5.1) 10.8 (2.7) 1.4 (0.4)GENIE BRRFG+hA 5.2 (4.8) 2.9 (2.2) 1.1 (0.5)GENIE LFG+hN 8.6 (4.2) 13.2 (2.7) 1.6 (0.8)GiBUU 3.6 (3.3) 1.7 (1.3) 1.2 (0.6)NuWro RFG 7.5 (5.9) 9.0 (5.4) 0.6 (0.4)NuWro BRRFG 4.9 (4.2) 2.7 (1.6) 2.7 (1.8)NuWro LFG 8.5 (6.7) 11.0 (4.9) 2.5 (1.7)NuWro ESF 5.2 (5.6) 3.5 (3.0) 3.3 (1.7)
B. Discussion
Amongst all the models compared, GiBUU showsmarginally better agreement with data. Both χ and χ are smaller than the corresponding ndof for eachTKI variable. It is explained in Ref. [110] that GiBUUuses a density- and momentum-dependent mean field to model the nucleon-nucleus potential and prepare the nu-clear ground state. Since the same potential is used inall interaction channels, it may provide a more accurateprediction than other generators which often treat QEand pion production processes differently. Also GiBUU’smodelling of FSI in the transport theory is a more com-plete approach than the commonly used cascade models,which might be a contributing factor to the overall agree-ment. The nice shape agreement at low p N suggests thatthe nuclear ground state modelling in GiBUU is betterthan other generators. In the tail all models have similarpredictions, meaning that we are not sensitive to the FSIdifferences there.Within the NuWro models, ESF and BRRFG have bet-ter agreement than LFG and RFG. In the pion channel,these nuclear models affect properties like the removalenergy and nucleon momentum distribution. This sug-gests that ESF and BRRFG may provide a more realisticnuclear ground state description. From the p N result, thecharacteristic nucleon momentum peak at the Fermi sur-face ( ∼
220 MeV/c) in RFG is strongly disfavored. Onthe contrary, LFG predicts a large number of events with p N <
120 MeV/c which is also incompatible with data.NEUT RFG, GENIE BRRFG and GENIE LFG usethe same types of Fermi gas nuclear ground state modelsas in NuWro. The choices of pion production and FSImodels make a difference in their predictions, but in gen-eral the same nuclear ground state model shows similarfeatures across generators in the small imbalance regionsof δp T T and p N . This indicates these observables are agood probe of the nuclear ground state models.In general the model separation in δp T T and p N is bet-ter than that in δα T , with most of the sensitivity comingfrom the central bin of δp T T and the first two bins in p N where the imbalance is small. While δα T is ratherinsensitive to the initial nuclear state, the hardening of δα T towards 180 ◦ is strongly affected by FSI which usu-ally slow down the final-state hadrons but not the lep-ton. However, with the present signal phase space restric-tions, in particular the high proton momentum thresh-old of 450 MeV/c, many of the CC1 π + Xp events thatundergo FSI are lost, making the measurement less effec-tive. With improved detector acceptance in the comingND280 upgrade [111], δα T will be an extremely usefuland independent probe of FSI.If only χ is considered, most models have a χ less than or roughly equal to the ndof. The worst caseis the p N prediction from NuWro RFG which has a p-value of 15%. Nevertheless, the large normalization dis-crepancy exhibited by the RFG and LFG nuclear groundstate models cannot be simply explained by flux or othernormalization uncertainties. Thus one should be carefulin interpreting the model agreement when the differencebetween χ and χ is large.It is not straight-forward to compare this study tothe T2K CC0 π [26] and MINER ν A [27, 29] TKI results,because of the different signal definition and, more im-portantly, a significant contribution from free nucleon0 (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K Result =11.3 tot2 c NEUT RFG, = 5.2 tot2 c GENIE BRRFG+hA, = 8.6 tot2 c GENIE LFG+hN, = 3.6 tot2 c GiBUU, (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K Result = 7.5 tot2 c NuWro RFG, = 4.9 tot2 c NuWro BRRFG, = 8.5 tot2 c NuWro LFG, = 5.2 tot2 c NuWro ESF, (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K Result =10.8 tot2 c NEUT RFG, = 2.9 tot2 c GENIE BRRFG+hA, =13.2 tot2 c GENIE LFG+hN, = 1.7 tot2 c GiBUU, (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K Result = 9.0 tot2 c NuWro RFG, = 2.7 tot2 c NuWro BRRFG, =11.0 tot2 c NuWro LFG, = 3.5 tot2 c NuWro ESF, (deg) T ad ] - ( d e g ) c m - [ N u c l e on T ad d s d - · T2K Result = 1.4 tot2 c NEUT RFG, = 1.1 tot2 c GENIE BRRFG+hA, = 1.6 tot2 c GENIE LFG+hN, = 1.2 tot2 c GiBUU, (deg) T ad ] - ( d e g ) c m - [ N u c l e on T ad d s d - · T2K Result = 0.6 tot2 c NuWro RFG, = 2.7 tot2 c NuWro BRRFG, = 2.5 tot2 c NuWro LFG, = 3.3 tot2 c NuWro ESF,
FIG. 13. Measured differential cross sections per nucleon as a function of δp TT (top), p N (middle) and δα T (bottom), togetherwith predictions from NEUT, GENIE, GiBUU (left) and NuWro (right). In the tails of δp TT and p N (beyond the magentalines), the cross sections are scaled by a factor of 5 for better visualization. The legend also shows the χ from Eq. (17). δp T T = 0 and p N ≈
26 MeV/c, and isflat in δα T . The cross section on hydrogen is related tothe carbon component via the common neutrino-nucleoncross section modelling; both components will scale simi-larly when the neutrino-nucleon cross section is changed.However the ratio between the hydrogen and carbon com-ponents is highly dependent on the modelling of the nu-clear medium effects.Qualitatively, almost all models are compatible withthe p N tail in both T2K and MINER ν A data, but havean over-prediction in the peak region. However, there arenot sufficient statistics to measure the peak of p N moreprecisely. MINER ν A also reported a mild asymmetryin δp T T , and attributed it to the interference between∆ and non-resonant amplitudes [35], but such an asym-metry is not observed within errors in this study. Thetight phase space restrictions used in this study reducesour sensitivity to FSI modelling. The rather flat dis-tribution of δα T compared to MINER ν A results can beattributed to the difference in phase space restrictions,where MINER ν A applied no phase space restriction onthe final-state π . The more energetic ( ∼ ν A also produces more energeticfinal-state particles and a more curved δα T .While GiBUU has a good agreement with thisCC1 π + Xp and MINER ν A CC π measurements, it showsan incompatibility with our CC0 π TKI results [26, 31].This incompatibility is not in the δp T (Eq. (6)) tail ornormalisation, suggesting this might be related to the nu-clear ground state. In our previous CC0 π cross sectionmeasurements as a function of outgoing muon kinemat-ics [60, 61], the GiBUU prediction also shows a large dis-crepancy, mainly in the most forward bin where the nu-clear physics governing low energy and momentum trans-fer interactions is the most important. VIII. CONCLUSION
In this paper, the CC1 π + Xp muon neutrino differen-tial cross sections on hydrocarbon as a function of thethree TKI variables, δp T T , p N and δα T , have been mea-sured independently in the ND280 tracker. δp T T and p N are most sensitive to the initial nuclear ground state, and δα T is an independent probe of FSI. The analysis is per-formed with a joint fit between the signal and controlsamples to minimize the uncertainties on background es-timation, and a maximum likelihood fit is used to unfoldthe detector smearing effect and extract cross sections inthe truth space. The reduced flux uncertainty and betterdetector modelling allow to have a reduced systematicuncertainty with respect to previous T2K cross sectionanalyses. Due to the complex and multifaceted natureof this analysis, exceptional care has been taken in mit- (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K ResultGiBUUCarbon CCQECarbon DISCarbon RESHydrogen (deg) T ad ] - ( d e g ) c m - [ N u c l e on T ad d s d - · FIG. 14. Measured cross sections as a function of the TKIvariables compared to GiBUU predictions. The GiBUU pre-dictions are decomposed into the contributions from carbonand hydrogen. In the tails of δp TT and p N (beyond the ma-genta lines), the cross sections are scaled by a factor of 5 forbetter visualization. p N with χ /ndof >
2, which indicates a mis-modelling ofthe nucleon Fermi motion. The similar data-MC compar-ison to the MINER ν A CC π results [29] seems to con-firm that the mis-modelling is general in pion productionchannels. While the tight phase space restrictions limitour sensitivity to FSI modelling, the relatively flat δα T in T2K results is in strong contrast to MINER ν A results,indicating a possible energy dependence of hadronic FSI.Future analyses will aim to unfold cross sections inmultiple TKI variables simultaneously and obtain theircorrelations which can then be used to separate effectsdue to the initial nuclear state and FSI. The upcom-ing ND280 upgrade is going to expand the measurablephase space, especially in the low energy and high an-gle regions. Thus the ND280 upgrade is expected to in-crease our statistics and model sensitivity significantly.Another possible extension is to isolate hydrogen inter-actions from carbon ones by selecting events with small δp T T and p N . With better detector resolution, this tech-nique could better identify and separate interactions onhydrogen on an event-by-event basis, and provide thefirst “free nucleon data” since the hydrogen bubble cham-ber experiments [24, 37–40].The data release for the results presented in this anal-ysis is posted in Ref. [112]. It contains the analysis bin-ning, the differential cross section best-fit values, and as-sociated covariance matrices. IX. ACKNOWLEDGEMENT
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 bilateralprograms (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; the RSF(19-12-00325), RFBR (JSPS-RFBR 20-52-50010 \ Appendix A: More model comparisons
This section shows a few more model comparisons todata with different physics configurations in the neutrinogenerators.
1. GENIE
GENIE provides a variety of model configurations forevent generation. Choices in the nuclear ground statemodel have a much larger effect on the TKI predictionsthan either the FSI models or pion production models.On the other hand, the GENIE 2018a free nucleon crosssection model re-tune reduces the CC1 π cross sectionsand increases the CC2 π cross sections relative to thebaseline tune. Fig. 15 shows the comparison amongstthese model configurations and physics tunes.
2. NuWro
Within NuWro, the BRRFG and ESF nuclear groundstate models show the best agreement with data. TheFSI configurations are varied to study their effects on thepredictions. These include a global scaling of the nucleonmean free path in the cascade, or the switch of the pion-nucleon interaction model from Ref. [113] to Ref. [114].As shown in Fig. 16, the change in χ is small, indicat-ing that there is limited sensitivity to FSI under currentstatistics and signal phase space restrictions.3 (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K ResultGENIE (tuned) = 5.2 tot2 c BRRFG+RS+hA, = 5.5 tot2 c BRRFG+RS+hN, = 6.1 tot2 c BRRFG+BS+hA, = 8.6 tot2 c LFG+BS+hN, (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K ResultGENIE (baseline) = 8.7 tot2 c BRRFG+RS+hA, =11.7 tot2 c BRRFG+RS+hN, =12.0 tot2 c BRRFG+BS+hA, =17.2 tot2 c LFG+BS+hN, (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K ResultGENIE (tuned) = 2.9 tot2 c BRRFG+RS+hA, = 3.3 tot2 c BRRFG+RS+hN, = 4.0 tot2 c BRRFG+BS+hA, =13.2 tot2 c LFG+BS+hN, (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K ResultGENIE (baseline) = 7.0 tot2 c BRRFG+RS+hA, = 9.9 tot2 c BRRFG+RS+hN, =11.8 tot2 c BRRFG+BS+hA, =27.9 tot2 c LFG+BS+hN, (deg) T ad )] - d e g c m - [( N u c l e on T ad d s d - · T2K ResultGENIE (tuned) = 1.1 tot2 c BRRFG+RS+hA, = 1.0 tot2 c BRRFG+RS+hN, = 1.2 tot2 c BRRFG+BS+hA, = 1.6 tot2 c LFG+BS+hN, (deg) T ad )] - d e g c m - [( N u c l e on T ad d s d - · T2K ResultGENIE (baseline) = 0.9 tot2 c BRRFG+RS+hA, = 2.3 tot2 c BRRFG+RS+hN, = 2.6 tot2 c BRRFG+BS+hA, = 4.6 tot2 c LFG+BS+hN,
FIG. 15. Measured differential cross sections per nucleon as a function of δp TT (top), p N (middle) and δα T (bottom), togetherwith predictions from the different model configurations of GENIE. The left plots use the free nucleon cross section re-tune,and the right plots use the baseline tune. In the tails of δp TT and p N (beyond the magenta lines), the cross sections are scaledby a factor of 5 for better visualization. The legend also shows the χ from Eq. (17). (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K ResultNuWro ESF = 5.2 tot2 c Nominal, = 4.7 tot2 c · NN MFP = 5.7 tot2 c · NN MFP = 5.2 tot2 c N cross section, p Alternative (MeV/c) TT p d - - - ] - ( M e V / c ) c m - [ N u c l e on TT p d d s d - · · · T2K ResultNuWro BRRFG= 4.9 tot2 c Nominal, = 4.1 tot2 c · NN MFP = 5.5 tot2 c · NN MFP = 4.5 tot2 c N cross section, p Alternative (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K ResultNuWro ESF = 3.5 tot2 c Nominal, = 2.7 tot2 c · NN MFP = 4.2 tot2 c · NN MFP = 3.2 tot2 c N cross section, p Alternative (MeV/c) N p ] - ( M e V / c ) c m - [ N u c l e on N dp s d - · · T2K ResultNuWro BRRFG= 2.7 tot2 c Nominal, = 2.1 tot2 c · NN MFP = 3.5 tot2 c · NN MFP = 2.2 tot2 c N cross section, p Alternative (deg) T ad )] - d e g c m - [( N u c l e on T ad d s d - · T2K ResultNuWro ESF = 3.3 tot2 c Nominal, = 3.8 tot2 c · NN MFP = 3.2 tot2 c · NN MFP = 3.5 tot2 c N cross section, p Alternative (deg) T ad )] - d e g c m - [( N u c l e on T ad d s d - · T2K ResultNuWro BRRFG= 2.7 tot2 c Nominal, = 2.8 tot2 c · NN MFP = 2.3 tot2 c · NN MFP = 2.3 tot2 c N cross section, p Alternative
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