Measurement of the muon neutrino charged-current single π + production on hydrocarbon using the T2K off-axis near detector ND280

We report the measurements of the single and double differential cross section of muon neutrino charged-current interactions on carbon with a single positively charged pion in the final state at the T2K off-axis near detector using 5 . 56 × 10 20 protons on target. The analysis uses data control samples for the background subtraction and the cross section signal, defined as a single negatively charged muon and a single positively charged pion exiting from the target nucleus, is extracted using an unfolding method. The model-dependent cross section, integrated over the T2K off-axis neutrino beam spectrum peaking at 0.6 GeV, is measured to be σ ¼ ð 11 . 76 (cid:2) 0 . 44 ð stat Þ (cid:2) 2 . 39 ð syst ÞÞ × 10 − 40 cm 2 nucleon − 1 . Various differential cross sections are measured, including the first measurement of the Adler angles for single charged pion production in neutrino interactions with heavy nuclei target.


I. INTRODUCTION
Precise knowledge of single charged pion production (CC1π þ ) induced by charged-current (CC) interactions of muon neutrinos with energy lower than a few GeV on nuclei is very relevant for current and upcoming neutrino oscillation experiments. In this energy range CC1π þ has the largest neutrino interaction cross section after the CC quasielastic (CCQE) process. In T2K it constitutes a background for the ν μ disappearance measurement when the charged pion is not observed and its precise knowledge is relevant for all current and planned neutrino oscillation experiments. Single pion production is sensitive mainly to resonant processes but also to nonresonant contributions as well as coherent pion production. Moreover, in a nuclear target, there are multinucleon contributions and final-state interactions to which the total and differential cross sections in pion kinematic variables are sensitive. The correct modeling of these effects, which is interesting in its own right, is also a key challenge to the reduction of the systematic uncertainties in neutrino oscillation experiments. A wide range of models exists and their validation requires well-understood cross section measurements, both absolute and differential, and possibly on different nuclear targets. To allow a meaningful comparison with different phenomenological models, the measured cross section data should be as independent as possible from the models themselves.
The first CC1π þ cross section measurements are from decades-old bubble chamber experiments [1][2][3][4]. Despite the unsurpassed detector spatial resolution of bubble chambers, these results disagree by as much as 30% due to large statistical uncertainties and poor modeling of the neutrino fluxes [5]. Moreover, the uncertainties in nuclear effects make it difficult to extrapolate the cross sections to the heavier nuclei used as targets in modern neutrino experiments. More recent measurements on different targets and energy ranges [6][7][8] are presented in the form of CC1π þ to CCQE cross section ratios rather than absolute cross section measurements.
In recent years MiniBooNE [9], MINERνA [10][11][12] and T2K [13] reported absolute CC1π þ cross sections, respectively in mineral oil, plastic scintillator and water, as a function of the relevant kinematic variables. These results show a significant disagreement, both in shape and in normalization [14,15]. The difficulty of getting simultaneous agreement between all available low-energy cross section data limits their effectiveness to constrain the uncertainty on cross section models and the corresponding systematic errors in neutrino oscillation experiments.
Since modern neutrino experiments use targets heavier than hydrogen and deuterium, it is not clear if the source of the discrepancy lies in the fundamental neutrino-nucleon cross section estimation or in the nuclear effects. In neutrino-nucleus interactions, the production of nucleons below the Fermi momentum is inhibited by the Pauli exclusion principle and collective nuclear effects have to be considered. Moreover, before leaving the target nucleus, interactions of the final-state particles with the nuclear medium change their observed spectrum and composition. In particular pion production, absorption and charge exchange processes, shift the event classification between CC1π þ and other final states and in experimental measurements these effects cannot be unfolded from the fundamental single nucleon cross section without relying on a specific model. A detailed understanding of the CC1π þ interaction, such as the left-right asymmetry of the final-state hadron with respect to the lepton scattering plane, may help to constrain the absorbed pion background contribution to the CCQE-like neutrino interactions [16].
Various models and implementations have been proposed [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] but since the size of the nuclear effects is large and there are discrepancies among models, it is important to provide experimental measurements that are as model-independent as possible. If the experimental signature is defined topologically by the particles leaving the target nucleus rather than the particles produced at the neutrino interaction vertex, the results can be compared with any specific model that combines the nucleon-level cross section, nuclear effects and final-state interactions. This allows a thorough comparison with different predictions, reducing the modeling systematic uncertainties and easing the task of comparing different experimental results on the same target. Robust experimental cross section data, and in particular CC1π þ data, are needed to pin down which model, if any, gives the more accurate predictions and to assign a systematic uncertainty to it. This paper describes the measurement of the CC1π þ neutrino interaction cross section using the ND280 off-axis near detector in the T2K beam. The target material is plastic scintillator (C 8 H 8 ) and the analysis selects charged-current events with a negatively charged muon and a single positively charged pion, with no additional mesons but any number of additional nucleons.
The paper is organized as follows. Section II describes the key aspects of the neutrino beam and the ND280 detector used for this measurement. Section III describes the analysis strategy, the event selection and the candidates and control samples. The results are presented in Sec. IV followed by conclusions in Sec. V.

II. EXPERIMENTAL SETUP
T2K is a long-baseline neutrino oscillation experiment located in Japan, whose goal is to make precise measurements of oscillation parameters via the observation of muon (anti)neutrino disappearance and electron (anti)neutrino appearance [33]. A muon (anti)neutrino beam, produced in the J-PARC accelerator in Tokai, Japan, is directed at Super-Kamiokande, a large water Cherenkov detector located 295 km away near Kamioka. The beam is monitored by a set of near detectors that are additionally used for cross section measurements.

A. Neutrino beam
The neutrino beam is initiated by collisions of 30 GeV=c protons on a graphite target [34]. The resulting mesons (mainly pions) are collimated by three magnetic horns and enter a 96 m decay tunnel, where they decay into (anti) neutrinos. Depending on the horns' polarity, mesons of a desired sign are selected to produce a neutrino or antineutrino beam of high purity. For the data presented in this paper, the horns were operating in neutrino mode, focusing π þ 's for a primarily ν μ beam.
The experiment uses an off-axis configuration, with detectors located away from the beam axis at an angle of 2.5°, to get a narrow spectrum shape, which is optimal for oscillation studies. Beam stability and direction are monitored by a muon detector located at the end of the decay tunnel and by the INGRID near detector, which samples the neutrino beam on its central axis at approximately 280 m from the target. The predicted neutrino fluxes at the ND280 near detector, also located 280 m from the target, peak at around 0.6 GeV and are shown in Fig. 1. Muon neutrinos represent the largest fraction of the beam, making up 92.6% of the total. The remaining species areν μ (6.2%), ν e (1.1%) andν e (0.1%).

B. Off-axis near detector ND280
The off-axis near detector ND280 is a magnetized particle-tracking apparatus (see Fig. 2). Placed inside a magnet with a uniform dipole magnetic field of 0.2 T, it consists of a tracker and a π 0 detector (P0D) [35], and is surrounded by electromagnetic calorimeters (ECals) [36] and side muon range detectors (SMRDs) [37]. The tracker, located downstream of the P0D, is made up of three gas time projection chambers (TPCs) [38] interleaved with two fine-grained detectors (FGDs) [39].
The FGDs are composed of finely segmented scintillator (C 8 H 8 ) bars organized in layers. The orientation of the layers alternates between the x and y directions almost perpendicular to the neutrino beam direction, allowing for the precise reconstruction of the neutrino interaction vertex and track directions. FGDs serve as the target for neutrino interactions in this analysis. Their tracking capabilities provide track reconstruction down to a length of a few centimeters and evidence for additional activity around interaction vertices when tracks are too short to be reconstructed. The upstream FGD (FGD1) contains only active scintillator layers, while the downstream FGD (FGD2) also incorporates inactive water layers. To measure cross sections on C 8 H 8 , only neutrino interactions occurring in FGD1 are selected for this analysis. There are 30 scintillator layers in FGD1, with each layer containing 192 bars. To reduce background from outside the FGD1 detector, a fiducial volume is defined by removing from the event selection events occurring inside any of the five bars at the edge of the detector in the transverse direction or in one of the two layers (one x and one y projection) upstream of the neutrino beam direction. The FGD1 fiducial volume has an elemental composition of 86.1% carbon and 7.35% hydrogen with the remaining contributions coming from oxygen (3.70%) and negligible quantities of other elements (Ti, Si, N).
Three TPCs provide trajectory and energy loss information for tracks entering and exiting the FGDs, predominantly from muons and pions. Their capabilities allow for precise three-dimensional track reconstruction, particle identification (PID) via the measurement of the ionization per unit length and determination of momentum and charge by looking at the curvature of tracks in the 0.2 T magnetic field.
The ECals are sampling calorimeters consisting of layers of plastic scintillator separated by layers of lead. Alternating layers are aligned orthogonally to one another to provide three-dimensional reconstruction of tracks and showers, both electromagnetic and hadronic. The topological characteristics of the energy deposited in the ECal provide additional particle identification capability.

A. Data sample definitions and observables
In order to reduce the dependence on the modeling of final-state particle reinteraction in the nuclear medium, the signal is defined in terms of the experimentally observable particles exiting the nucleus struck by the neutrino. The CC1π þ final state is defined as one negatively charged muon, one and only one positively charged pion and any number of additional nucleons. Several additional control samples are selected to directly constrain the background subtraction with data. Restrictions are applied to the muon and pion kinematics in order to exclude phase-space regions where the detection efficiency is low and the corresponding correction would introduce large model dependencies. Seven differential cross section measurements are performed: (1) d 2 σ=dp μ d cos θ μ , where p μ is the momentum of the muon and θ μ is the angle between the muon and the neutrino directions in the laboratory frame. (2) dσ=dQ 2 , where Q 2 is the reconstructed square of the 4-momentum transfer, defined from experimental observables in Eq. (2). (3) dσ=dp π , where p π is the momentum of the pion in the laboratory frame. (4) dσ=dθ πμ , where θ μπ is the angle between the muon and the pion directions in the laboratory frame. (5) dσ=dθ π , where θ π is the angle between the pion and the neutrino directions in the laboratory frame. (6) dσ=d cos θ Adler , where cos θ Adler is defined as the polar angle in the Adler coordinate system [40]. (7) dσ=dϕ Adler where ϕ Adler is defined as the azimuthal angle in the Adler coordinate system [40]. The pion is identified either by a reconstructed TPC track or by the presence of a Michel electron detected in the FGD. In the latter case, the direction of the pion and its momentum are unknown. For this reason the subsample of pions identified by the Michel electron is only used for d 2 σ=dp μ d cos θ μ . The flux-integrated differential cross sections are extracted using the D'Agostini unfolding method [41] to correct for detector effects.

B. Simulation
Detector response, acceptance and efficiency are corrected using simulated Monte Carlo (MC) events to model the specific detector and beam configuration of each run with a sample that is 10 times larger than the data statistics. The neutrino flux is predicted using simulations tuned to external measurements. Details of the beam simulation can be found in Ref. [34]. Interactions of protons in the graphite target and the resulting hadron production are simulated using the FLUKA 2011 package [42,43], weighted to match hadron production measurements [44][45][46][47][48][49]. The propagation and decay of those hadrons is performed in a GEANT3 [50] simulation, which uses the GCALOR package [51] to model hadron reinteractions and decays outside the target. Uncertainties on the proton beam properties, horn current, hadron production model and overall neutrino beam alignment are taken into account to assess an energy-dependent systematic uncertainty on the neutrino flux. Flux tuning using NA61/SHINE data [44][45][46] reduces the uncertainty on the flux-integrated overall normalization down to 8.5%.
Neutrinos are propagated through the ND280 detector and their interactions with matter are simulated with the NEUT event generator. NEUT [52,53] (version 5.1.4.2) uses the Llewellyn-Smith CCQE neutrino-nucleon cross section formalism [54] with the nuclear effects described by the Smith and Moniz [55] relativistic Fermi gas model. Dipole forms were used for both the axial and vector form factors. Tuning to Super-Kamiokande atmospheric data and K2K data leads to a nominal axial mass M QE A ¼ 1.21 GeV=c 2 . This version of the NEUT generator did not include a specific model of the two-particle twohole CCQE. NEUT emulates this contribution through a large value of the axial mass and the contribution from the pionless delta decay.
The resonant pion production in NEUT is based on the Rein-Sehgal model [18], taking into account 18 resonances with masses below 2 GeV=c 2 and their interference terms, with the axial mass M RES Neutral-current (NC) and charged-current (CC) coherent pion production was simulated using the Rein-Sehgal model in Ref. [56]. The CC coherent pion production includes the partially conserved axial-vector current and lepton mass corrections [57].
Deep inelastic scattering processes are simulated using the GRV98 [58] parton distribution with low-Q 2 corrections by the Bodek and Yang model [59].
Secondary interactions of pions inside the nucleus, socalled final-state interactions (FSIs) are simulated using an intranuclear cascade model based on the method described in Ref. [19], tuned to external π-12 C data.
The GENIE [60] (version 2.6.4) neutrino generator is used as an alternative simulation to test the dependence of the analyses on the assumed signal and background models. Among other differences, GENIE uses the different values . We did not observe any significant variation of the results using this alternative event generator.
The simulated final-state particles are then propagated through the detector material using GEANT4 [63].

C. Event selection
The analysis presented here uses data from the three T2K run periods between November 2010 and May 2013, when T2K was operating in neutrino mode. In total 5.56 × 10 20 protons on target (POT) are used, corresponding to all good quality data, with each subdetector working optimally.
Events with the highest momentum track consistent with a negatively charged particle passing the TPC track quality selection criteria and matched with a track originating in the upstream FGD are selected as muon neutrino interaction candidates. The energy deposition measured in the TPC is required to be compatible with the energy loss of a muonlike, minimum-ionizing particle. Further selection criteria are applied to remove events where the interactions occur outside the FGD fiducial volume. Further details on the ν μ CC inclusive selection can be found in Ref. [64].
To further select CC1π þ events, the presence of one and only one pion of positive charge is required. The pion is identified by a positively charged TPC track with an energy deposition compatible with a pion or by the presence of a Michel electron, tagged as a time-delayed energy deposition in the upstream FGD fiducial volume. Using cosmic-ray data samples, we estimate the efficiency of Michel electron tagging to be 58.6 AE 0.4%. Monte Carlo detector simulation predicts 61.9 AE 1.1%. We take the difference between these values as the systematic error of the efficiency. The event is rejected if additional pions, either charged or neutral, or photons are identified in the event by looking at either TPC tracks or electromagnetic showers in ECal. Table I shows the data and Monte Carlo reduction and the fraction of events that survive each selection criterion with respect to the previous one. Monte Carlo and data survival fractions are similar after the quality, fiducial and backward track removal selection criteria are applied. These selection cuts eliminate events outside of the detector fiducial volume that are not properly simulated in our event generation. The composition of the selected sample according to the π þ selection criteria is shown in Table II. The data sample has slightly more π þ events selected with the Michel electron criteria but it is still compatible within 1σ statistical error.
The MC events shown in Tables I and II are bare predictions: they are not corrected by several effects such as the detection efficiency and the reweighting of the event generator probabilities. The correction is applied later in the analysis leading to a modification of the reported final cross section. Table III shows the composition of the selected CC1π þ sample with respect to the true topology according to the NEUT Monte Carlo. The topologies are event classifications based on the number of pions leaving the nucleus: one μ − and zero pions (CC0π), one μ − and a single π þ (CC1π þ ), one μ − plus a π − ,aπ 0 or more than one pion (CC-Other), zero μ − (Background) or events produced outside the fiducial volume (OFV). Table III also shows the compositions of the full selected sample (second column) and the two subsamples where the pion is reconstructed in the TPC (third column) or identified by the Michel electron (fourth column). The largest contamination in the final sample comes from multipion interactions where the additional pions are absorbed in subsequent interactions with the detector material or simply not reconstructed. The main neutrino interaction process at the nucleon level contributing to the CC1π þ sample as predicted by the event generation is the pion resonant production (61.5%).

E. Kinematic observables
This section discusses the distributions of the reconstructed pion and muon basic kinematic variables for the selected sample. Data are compared with the expectations of the NEUT and GENIE Monte Carlo generators in terms of the topologies introduced in Sec. III D. Figure 3 shows the distributions of muon momentum (left plots) and angle (right plots) for the selected CC1π þ events compared with NEUT (upper plots) and GENIE (lower plots) Monte Carlo expectations.
The pion momentum distributions (see Fig. 4) are shown for the subsample of events where the pion is reconstructed in the TPC (left plots). Data are compared with NEUT 5.1.4.2 (upper plots) and GENIE 2.6.4 (lower plots). Similarly, Fig. 4 (right plots) shows the distribution of the pion angle with respect to the neutrino beam direction for the subsample with the pion direction reconstructed in the TPC. NEUT predictions before background subtraction show better agreement with the data than GENIE which predicts slightly more events.

Event kinematic observables
Kinematic variables like the neutrino energy and the momentum transfer are reconstructed from the muon and pion kinematics under the assumption that the nucleon struck by the neutrino is at rest, bound to the target nucleus by an energy E bind ð25 MeV=cÞ, and the final state contains, besides the pion and the muon, a single undetected proton. The neutrino energy is reconstructed using the equation where m p stands for the proton mass, ⃗ d ν is the predicted neutrino direction and ð ⃗p μ;π ;E μ;π Þ are the reconstructed muon and pion 4-momenta. Figure 5 (left) shows the reconstructed neutrino energy distribution for the CC1π þ events where the pion is reconstructed in the TPC. The neutrino direction ( ⃗ d ν ) is fixed along the neutrino flux thrust, although the Monte Carlo simulation includes an accurate description of its angular dispersion.
The 4-momentum transfer is defined as: where p μ and p ν are the 4-momentum vectors of the muon and neutrino respectively. Figure 5 shows the Q 2 distribution for the candidate CC1π þ events.

Adler angles
The angles, θ Adler and ϕ Adler , define the direction of the pion in the Adler system. The Adler reference system is the pπ þ rest frame as shown in Fig. 6 (left) where p Ã μ , p Ã π and p Ã p correspond to the muon, pion and final-state nucleon momentum. The angles θ Adler and ϕ Adler are sensitive respectively to the longitudinal and transverse polarization of the pπ þ final state for interactions mediated by the Δ þ , Δ þþ and nonresonant contributions. The experimental definition of the Adler system needs to be changed in terms of lepton and pion observables since the final-state nucleon is not usually detected [40]. The Adler rest frame and the angles θ Adler and ϕ Adler are redefined as shown in Fig. 6 (right), where the neutrino direction is assumed known and the neutrino energy is reconstructed from Eq. (1). It has been shown [40] that with this experimental definition the information of the original Adler angles is reasonably maintained when the neutrino interacts with light nuclei despite the need to determine the incoming neutrino energy from the lepton and pion observables and the effects of FSIs in the target nucleus. Existing models [20] predict an interference between the resonant and nonresonant pion production that leads to the transverse polarization as measured by the ANL data [1]. Figure 7 shows the distribution of cos θ Adler (left) and ϕ Adler (right) for the subsample of CC1π þ events with the pion reconstructed in the TPC.

F. Control samples for background subtraction
Control samples are selected in the data to constrain the normalization of several Monte Carlo background components listed in Table III. Each control sample is selected to be representative of a specific background and it is required to minimize the content of CC1π þ in order to be considered a sideband sample independent of the signal sample. They are also required to be independent from each other. The three control samples, described in the following subsections, correspond to the CC0π background and two subsamples of the CC-Other background: one with missing charged-pion detection and the other with misidentified electrons or positrons.
For the contamination from interactions taking place outside of the FGD fiducial volume, no control sample was found to reproduce the characteristics of this background. In this case the subtraction relies on the Monte Carlo prediction and the lack of a data constraint is taken into account in the systematic error estimation. Control samples are used to extract the normalization constants α k ¼ S data;k =S MC;k , where S data;k and S MC;k are the number of events in sideband k, respectively for data and Monte Carlo. These normalization constants α k are used to rescale the corresponding Monte Carlo background components before subtraction. The normalization constants are applied to each of the three background classes selected according to true Monte Carlo information.

Control sample A
One source of background is the CC0π misidentification; see Table III. Events where a proton is misidentified as a pion are a background in the CC1π þ selection. The misidentification arises from the similar ionization power of protons and pions around 1.5 GeV=c in the TPC. The first control sample aims to select CC0π events requiring a muon and no pions in the final state, with a proton identified in the final state and any number of additional nucleons [64]. The selection requires one and only one additional TPC track, other than the muon track, with an energy deposition not compatible with a pion or an electron. The angle with respect to the muon is required to be between 0.5 and 1.5 rad and the momentum must be between 0.6 and 1.8 GeV=c, which corresponds to the range where the misidentification between pions and protons is larger. Table IV shows the topological composition of the control sample A. Figure 8 shows the muon candidate momentum and angle (top row) along with the proton momentum distribution for the selected sample in data and MC. From this control sample the extracted normalization value for the CC0π contamination is α A ¼ 1.02.

Control sample B
The second control sample is a subset of the CC-Other sample, obtained requiring, besides the muon, the presence of two TPC tracks tagged as positively charged pions. Events with three or more TPC tracks in addition to the muon track are rejected as they are high-energy, multipletrack events which are less representative of the actual backgrounds. Table IV lists the topological composition of the control sample B. Figure 9 shows the data and Monte Carlo comparison for the muon momentum and angle and for the pion momentum in this control sample.  This control sample is used to constrain the event contamination from CC-Other due to multiple pion events. From this control sample we extract the normalization value α B ¼ 0.94.

Control sample C
Similarly to the previous sample, this is a subset of the CC-Other sample, obtained by additionally requiring that at least one TPC track is tagged as an electron or positron. This control sample is used to constrain the event contamination from CC-Other due to neutral pion events. To reject misidentified protons, positron tracks are required to have a momentum smaller than 0.4 GeV=c. The absence of π þ tracks in the TPC is required in order to avoid overlap with the control sample B. The number of TPC tracks in addition to the muon track is required to be exactly two since with this requirement the shape of the control sample in Monte Carlo is found to be more similar to the actual background. Table IV provides the topological composition of control sample C. Figure 10 shows the comparison between data and Monte Carlo for the muon momentum and angle for this sample. The normalization value obtained for this control sample C is α C ¼ 0.99.

G. Systematic error
The systematic error can be split into three categories related to the flux, detector and modeling of interactions. A detailed description of all the systematic errors can be found in Ref. [64].
Uncertainty in the neutrino flux prediction arises from the hadron production model, proton beam profile, horn current, horn alignment, and other factors. For each source of uncertainty, the underlying parameters are varied to evaluate the effect on the flux prediction. The average effect of this systematic error varies in the range 10-15% along the different differential measurements.
The detector systematic errors are estimated by comparing the simulation predictions and dedicated data control samples. This list of detector systematic errors includes track efficiency in the FGD and TPC, particle identification, ECAL pion rejection, charge identification and momentum scale and resolution. The uncertainties caused by simultaneous events (pile-up), tracks coming from outside of the detector (sand muons), OFV events and secondary interactions (SIs) of pions and nucleons in the detector are also evaluated. On average, the largest contribution from the detector systematics are the SIs, while for low energies the charge misidentification is dominant.
A set of systematic parameters characterizes the uncertainties on the predictions of the NEUT generator. These uncertainties are propagated through the analyses to estimate the impact on the background and signal predictions, as well as the effect of the final-state interactions. A number of those parameters are normalization uncertainties for the different interaction modes simulated by NEUT (energy dependent for the dominant modes at the T2K neutrino energy spectrum). Other parameters describe uncertainties on the values of the axial mass (using separate parameters for CCQE and resonant interactions), the binding energy, and the Fermi momentum. An additional systematic parameter covers the difference between the predictions obtained with the default relativistic Fermi gas model used by NEUT and a spectral function describing the momentum and energy of nucleons inside the nucleus [65]. The modeling uncertainties are constrained by fits to external neutrino and pion scattering data (see Ref. [64] for more details).
Detector, beam and cross section model uncertainties were propagated in the selected sample. The beam and the cross-section model uncertainties are propagated by weighting the events according to true particle kinematics including the neutrino. Detector uncertainties are propagated event by event according to the observable on which the systematic depends; the algorithm depends on the systematic type. The propagation of the detector uncertainties was described in detail in Ref. [64].
All systematic uncertainties were propagated using a sample of toy experiments generated using the nominal values of each uncertainty and taking into account their correlation. Each toy experiment is treated as data, i.e., the cross section is determined for each toy, and the results were used to calculate a covariance matrix defined as where, for each source of uncertainty, labeled by s, 2000 pseudoexperiments are performed, giving a new differential cross section σ ðs n Þ each time, and the nominal cross section in bin i is given by σ nominal i . As an example of the effect of these systematics we show their impact in the double differential cross section measurement on the muon momentum and cosine of the angle; see Fig. 11. The systematic error contributions to this particular observable is 15.4% from the beam flux uncertainty, 8.2% from the detector uncertainty and 8.7% from the cross-section model uncertainties.

H. Phase space
The acceptance of the detector is limited in angle and momentum both for pions and muons. It is necessary to find suitable restrictions to identify the phase space where the observables can be unfolded without introducing large model dependencies. Complex kinematical observables (i.e., Q 2 , E ν and the Adler angles) depend nontrivially on the ranges of angle and momentum of the selected particles. We performed the phase-space optimization independently for pions and muons. The reconstruction efficiency has been studied both for the subsample of pions reconstructed in the TPC and the subsample of pions identified by the Michel electron tag. The resulting phase space for the reconstructed quantities is then associated with the true phase space contributing to the measured cross section.
The phase space for the muon observables is restricted to cos θ μ > 0.2 and p μ > 0.2 GeV=c. The same acceptance restrictions are applied to the pion observables: cos θ π > 0.2 and p π > 0.2 GeV=c for charged pions with a TPC segment. In the cases when the pion is tagged with a Michel electron no pion phase-space restriction is required. The bins in muon angle and momentum have been selected to ensure a large efficiency per bin while maintaining a large number of bins. Table V summarizes the phase-space restrictions applied for the differential cross section measurements presented in the next section.

IV. RESULTS
The differential cross sections are extracted using the unfolding method proposed by D'Agostini [41]. The background prediction is subtracted from the data after they are weighted by the corresponding sideband normalization (α k ). To assess the robustness of the method against potential biases, several tests were done using the nominal Observable cos θ μ > 0 cos θ μ > 0.2 p μ > 0.2 GeV=c cos θ π > 0.2 cos θ π > 0 p π > 0.2 GeV=c Michel electron Monte Carlo to unfold pseudoexperimental data produced with different Monte Carlo simulations obtained by changing the parameters and the models used to describe signal and background. This study enables an understanding of the impact of the control samples and the optimal number of iterations needed for the unfolding procedure. These tests show an optimal result with only one iteration. The binning for each observed variable was chosen by taking into account the available statistics and the resolution of the reconstructed variables calculated using Monte Carlo simulation. Figure 12 shows the flux-integrated cross section for d 2 σ=dp μ d cos θ μ . The rightmost bins are truncated and contain entries from 2 to 15 GeV=c. The unfolded double differential cross sections as a function of the muon kinematics are well reproduced by the Monte Carlo given the large errors of this measurement, except for some bins at high angles and momentum between 1.2 and 2.0 GeV=c. Figure 11 shows the covariance matrix including statistical and systematic errors.
Using the fill sample, including the Michel electron tag, and integrating over the T2K off-axis neutrino beam spectrum, peaking at 0.6 GeV, the flux-averaged total cross section is σ ¼ð11.76 AE 0.44ðstatÞAE2.39ðsystÞÞ × 10 −40 cm 2 nucleon −1 while the corresponding value predicted by NEUT is 12.25 × 10 −40 cm 2 =nucleon. The total cross section measurement depends strongly on the model assumptions for the extrapolation to the full phase space and it is provided only as a reference. Figure 13 shows the unfolded dσ=dQ 2 flux-integrated cross section, measured in the restricted phase space of cos θ μ > 0.2, p μ > 0.2 GeV=c and cos θ π > 0.2, p π > 0.2 GeV=c. There is a significant difference in the shape of experimental results and the predictions. The pronounced model excess at low Q 2 might be an indication of deficiencies in the nuclear model. Figure 14 shows the dσ=dp π flux-integrated cross section, measured in the restricted phase space of cos θ μ > 0.2, p μ > 0.2 GeV=c and cos θ π > 0.2. Simulations overshoot data over the whole momentum range. NEUT shows a good agreement above 0.7 GeV=c. A similar model excess for low-momentum pions has been observed in other experiments such as MiniBooNE [9] and MINERνA [10][11][12].
The dσ=dθ π flux-integrated cross section is shown in Fig. 15. The θ π -dependent cross section is measured in the restricted phase space cos θ μ > 0.2, p μ > 0.2 GeV=c for the muon and cos θ π > 0, p π > 0.2 for the pion. Consistently with the dσ=dp π cross section above, the measured differential cross section as a function of the pion angle also shows a disagreement with the predictions. Figure 16 shows the dσ=dθ πμ flux-integrated cross section, measured in the restricted phase space cos θ μ > 0.2, p μ > 0.2 GeV=c for the muon and cosθ π > 0.2, p π > 0.2 GeV=c for the pion. Figure 17 shows the dσ=dϕ Adler flux-integrated cross section, measured in the restricted phase space of cosθ μ > 0.2, p μ > 0.2 GeV=c and cosθ π > 0.2, p π > 0.2 GeV=c.   The shape of the distribution is reasonably described by NEUT except for those values in between 0.8 and 2.8 rad.
The region with the largest data deficit is around ϕ Adler ≃ 1.5 similar to the deficit observed in ANL data around π=2 for charged pions [1] and around the same value for neutral pions in MINERνA [66]. A significant difference of the ANL measurement compared to T2K is the use of a deuterium target where both the Fermi momentum and the FSIs are reduced with respect to the CH target. The ratio of the integrated cross section for positive ϕ Adler angles over the negative ϕ Adler angles, similar to the left-right asymmetry measured in MINERνA [66], gives a value of 1.08 AE 0.10.N E U Ta n dGENIE generators predict a value equal to 1. Both generators' predictions show an unexpected dependency on the ϕ Adler angle(seeFig.17), most probably caused by the effect of intranuclear cascades (FSIs) on the reconstruction of the Adler reference system [40]. The experimental results are consistently below the NEUT prediction for negative values of cos θ Adler ; see Fig. 18. A negative cos θ Adler corresponds to lowmomentum pions (≤0.3 GeV=c). This observation is consistent with the predicted excess observed at low pion momentum; see Fig. 14.
While the Monte Carlo reproduces reasonably well the muon observables, the predictions for the pion observables are larger than data. The difference between the two is the inclusion of Michel electron tags for the muon-only observables. The difference might be an indication of a biased estimation of the Michel electron tagging efficiency in Monte Carlo, but final-state interaction modeling or the model prediction for the pion momentum could also contribute to the observed disagreement. Even if the numbers of events were similar, there are significant shape differences in most of the observables investigated.

V. CONCLUSIONS
The analysis presented in this paper describes the CC1π þ cross section measurement on CH realized in ND280, the off-axis near detector of the T2K experiment.
Using NEUT as the default MC generator we observed a purity of the CC1π þ signal of 61.5%. The main contamination in the sample is due to unidentified CC-Other events. Three control samples have been investigated in order to subtract the background using data instead of applying the Monte Carlo purity correction, with the aim to reduce the model dependency.
We have presented differential cross section measurements using a set of observables that will be most useful for comparison with neutrino interaction models. One example is the use of Adler angles, which were measured before in light targets [1,2] and free protons [3,4] in old bubble chamber experiments and more recently on hydrocarbon targets [66] by the MINERνA experiment at higher energy. This is, together with the recent MINERνA result, the first time those angles have been measured in interactions of neutrinos on heavy nuclei.
The largest contribution to the measurement error overall is the uncertainty on the flux, while the largest contribution from detector systematics comes from pion secondary interactions and, at low energies, the TPC charge misidentification. Uncertainties in the cross section model are the second largest contribution to the uncertainties, which serves as a reminder of the importance of cross section measurements.
From the differential cross section measurements presented we highlight the following: (1) We observed a good description of the data for the CC1π þ topological channel in all the muon kinematics observables. These distributions use inclusively all pions, including the low-energy pions identified by Michel electron tagging. (2) The shape of the predicted Q 2 distribution shows large discrepancies with data all over the available Q 2 space being more pronounced for Q 2 ≤ 0.3 GeV 2 =c 2 . (3) We observed, in general, that the models predict larger cross sections for the angular pion observables. Only pions with momentum above 0.2 GeV=c, which have been identified as tracks in the TPC, are included. The discrepancy is more pronounced for low-momentum pions and are almost independent of the value of the θ π and θ μπ angles. might be caused either by a distorted pion momentum spectrum or by deficiencies in the efficiency predictions. We have also computed the flux-averaged cross section value: σ ¼ð11.76AE0.44ðstatÞAE2.39ðsystÞÞ×10 −40 cm 2 nucleon −1 : To obtain this value the full CC1π þ candidate sample was considered, including pions identified by the Michel electron tag. From this result we extrapolated to the full phase space, including regions where the detector efficiency is small or even null: this result is strongly dependent on model assumptions and should be used with care.