Measurements of $\bar{\nu}_{\mu}$ and $\bar{\nu}_{\mu} + \nu_{\mu}$ charged-current cross-sections without detected pions nor protons on water and hydrocarbon at mean antineutrino energy of 0.86 GeV

We report measurements of the flux-integrated $\bar{\nu}_\mu$ and $\bar{\nu}_\mu+\nu_\mu$ charged-current cross-sections on water and hydrocarbon targets using the T2K anti-neutrino beam, with a mean neutrino energy of 0.86 GeV. The signal is defined as the (anti-)neutrino charged-current interaction with one induced $\mu^\pm$ and no detected charged pion nor proton. These measurements are performed using a new WAGASCI module recently added to the T2K setup in combination with the INGRID Proton module. The phase space of muons is restricted to the high-detection efficiency region, $p_{\mu}>400~{\rm MeV}/c$ and $\theta_{\mu}<30^{\circ}$, in the laboratory frame. Absence of pions and protons in the detectable phase space of"$p_{\pi}>200~{\rm MeV}/c$ and $\theta_{\pi}<70^{\circ}$", and"$p_{\rm p}>600~{\rm MeV}/c$ and $\theta_{\rm p}<70^{\circ}$"is required. In this paper, both of the $\bar{\nu}_\mu$ cross-sections and $\bar{\nu}_\mu+\nu_\mu$ cross-sections on water and hydrocarbon targets, and their ratios are provided by using D'Agostini unfolding method. The results of the integrated $\bar{\nu}_\mu$ cross-section measurements over this phase space are $\sigma_{\rm H_{2}O}\,=\,(1.082\pm0.068(\rm stat.)^{+0.145}_{-0.128}(\rm syst.)) \times 10^{-39}~{\rm cm^{2}/nucleon}$, $\sigma_{\rm CH}\,=\,(1.096\pm0.054(\rm stat.)^{+0.132}_{-0.117}(\rm syst.)) \times 10^{-39}~{\rm cm^{2}/nucleon}$, and $\sigma_{\rm H_{2}O}/\sigma_{\rm CH} = 0.987\pm0.078(\rm stat.)^{+0.093}_{-0.090}(\rm syst.)$. The $\bar{\nu}_\mu+\nu_\mu$ cross-section is $\sigma_{\rm H_{2}O} = (1.155\pm0.064(\rm stat.)^{+0.148}_{-0.129}(\rm syst.)) \times 10^{-39}~{\rm cm^{2}/nucleon}$, $\sigma_{\rm CH}\,=\,(1.159\pm0.049(\rm stat.)^{+0.129}_{-0.115}(\rm syst.)) \times 10^{-39}~{\rm cm^{2}/nucleon}$, and $\sigma_{\rm H_{2}O}/\sigma_{\rm CH}\,=\,0.996\pm0.069(\rm stat.)^{+0.083}_{-0.078}(\rm syst.)$.


Introduction
The Tokai-to-Kamioka (T2K) experiment [1] is a long baseline neutrino oscillation experiment in Japan. Using either the ν µ or the ν µ beam produced at the J-PARC accelerator complex, both electron (anti-)neutrino appearance and muon (anti-)neutrino disappearance are measured at the far-detector, Super-Kamiokande (SK). T2K aims to make precision measurements of neutrino oscillation parameters, including a search for CP violation in the leptonic sector by precisely measuring the (anti-)neutrino oscillation. In these measurements, the neutrino event rate at SK is constrained by the cross-section and neutrino flux measured in the near-detector, ND280. The ND280 includes two Fine-Grained Detectors, FGD1 and FGD2 [2], used as a target for neutrino interactions and as a tracking device. The FGD1 interaction target is made up of plastic scintillators, and FGD2 consists of water and plastic scintillator targets, while SK is a water-target detector. Uncertainties in the modeling of neutrino-nucleus interactions due to the difference in the target at the near and the fardetector constitute an additional source of systematic uncertainties in the T2K oscillation analysis. In addition, unknown nuclear effects like the so called 2-particle-2-hole (2p2h) process with large uncertainties motivate testing the interaction model at multiple neutrino energies. The neutrino interaction model is used to extrapolate the neutrino beam energy distributions and interactions at the near-detector to the far-detector. Indeed, the T2K offaxis near-detector angular acceptance is more limited than the far-detector. Moreover, the near-detector event rate also includes significant interactions on materials other than the far-detector target. The interaction model is tuned from the near detector measurement and its parameterization can be incomplete. Therefore, testing the interaction model with different target materials and at various ranges of the neutrino energies are essential to improve the T2K oscillation analysis.
In the T2K experiment, the neutrino beam is directed 2.5 degrees off-axis with respect to the SK direction to ensure that the detector sees a narrow-band neutrino beam with a † also at INFN-Laboratori Nazionali di Legnaro ‡ also at J-PARC, Tokai, Japan § affiliated member at Kavli IPMU (WPI), the University of Tokyo, Japan ¶ also at National Research Nuclear University "MEPhI" and Moscow Institute of Physics and Technology, Moscow, Russia deceased ** also at JINR, Dubna, Russia † † also at Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP) ‡ ‡ also at BMCC/CUNY, Science Department, New York, New York, U.S.A.

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peak energy at 0.6 GeV, which maximizes the oscillation probability. In this energy range, neutrino interactions with nucleons are dominated by charged-current quasi-elastic (CCQE) and charged-current resonant-pion production (CC-Resonant). The neutrino energies from incoming CCQE interactions are reconstructed from the outgoing charged lepton kinematics. However, if multi-nucleon interactions or pion absorption occur in the nucleus, 2p2h and CC-resonant interactions may be misidentified as CCQE interactions because only a muon-like track may be observed in the final state. Furthermore, the reconstructed neutrino energy spectrum could be distorted. For this reason, in modern experiments, signals are classified by final-state particles, such as protons and pions. For example, CC0π (chargedcurrent interactions with no pions in the final state) cross-sections are measured instead of measuring CCQE cross-sections making them less dependent on nuclear models.
So far, T2K has published two results of neutrino cross-sections on water at a mean neutrino energy of 0.6 GeV: CC-resonant π + production cross-section using FGD2 [3] and CC0π cross-section using a dedicated water target in the ND280 detector, called the PØD [4]. CC-inclusive neutrino cross-sections using the INGRID Water Module, which consists of 80% water and 20% plastic scintillators, with a mean neutrino energy of 1.5 GeV [5] have also been measured. However, there has been only one publication of CC0π anti-neutrino cross-sections on water using PØD [6] with a neutrino energy peak at 0.6 GeV. In this article, we measure CC0π0p (CC0pi without detected protons) cross-sections on water and hydrocarbon in antineutrino beam mode by using a new neutrino detector called the WAGASCI module [7], and other T2K detectors, the Proton Module [8] and the INGRID module [9] with a mean neutrino energy of 0.86 GeV at an off-axis angle of 1.5 degrees. As described in Sec. 2.2, the WAGASCI module and the INGRID Water Module are basically the same except for the detector position and electronics. In the future, we will use both detectors to measure neutrino cross-sections at an off-axis angle 1.5 degrees.
Hereafter, we will describe the experimental apparatus, the Monte Carlo simulations, the datasets, the event selections, the analysis method, the systematic uncertainties, and the results.

Neutrino Beam
The accelerator complex J-PARC in Tokai (Japan) is composed of a linear accelerator (LINAC), a rapid cycling synchrotron (RCS), and the main ring (MR). The 30 GeV proton beam is extracted from the MR every 2.48 s. The beam spill consists of eight bunches with 581 ns interval. The protons impinge onto a graphite target fixed in the most upstream electromagnetic horn. Produced charged hadrons are focused by three electromagnetic horns into a 96 m-long decay volume where they decay preferentially producing ν µ (ν µ ) and µ + (µ − ). By changing the polarity of the horns, the beam mode can be switched between the neutrino mode and anti-neutrino mode. In this article, the data are collected in the anti-neutrino mode with a beam power of about 470 kW.

Detector Configuration
We use two detectors with different interaction targets, the WAGASCI module (water) and the Proton Module (hydrocarbon). The INGRID module is located at the most downstream 6/29 position as shown in Fig. 1, and is used as a muon detector. These detectors are located at an off-axis angle of 1.5 degrees in the T2K near-detector hall since August 2017. They are exposed to neutrinos with a higher energy distribution than the ND280 detector, since the off-axis angle is smaller than the ND280 angle of 2.5 degrees. A typical event display is shown in Fig. 2.
The WAGASCI module is a neutrino detector with 0.6 tons of water and 1280 plastic scintillator bars. The total fraction of water target in the fiducial volume is 80%, and is higher than the one in other T2K detectors (PØD and FGD2) [2]. The type of scintillator bar (3 × 25 × 1020 cm 3 ) and wavelength-shifting (WLS) fiber (Kuraray, Y-11(200)) used in the WAGASCI module is the same as that used in the INGRID Water Module [5]. The readout electronics are newly developed with a Silicon PM Integrated Read-Out Chip (SPIROC) which is a 36-channel auto-triggered front-end ASIC. The WAGASCI module consists of 16 scintillator tracking planes in total, and each tracking plane consists of 40 scintillators positioned perpendicularly to the neutrino-beam axis (plane scintillator) and another 40 scintillators positioned in parallel to the beam with a grid structure (grid scintillator), as shown in Fig. 3. Figure 4 shows the schematic view of the scintillators from the x-and y-directions, where the definition of the coordinate system is shown in Fig. 2.
The Proton Module is a fully active tracking detector. It consists of 34 tracking planes, where each tracking plane is an array of two types of 32 scintillator bars, as shown in Fig. 5. Two types of scintillators, SciBar type (13 × 25 × 1203 cm 3 ) and INGRID type (10 × 50 × 1203 cm 3 ), are used, and their chemical composition is the same as that of the WAGASCI type scintillator bar. The six veto planes surrounding the tracking planes are used to track the charged particles coming from outside the Proton Module. The tracking planes also serve as the neutrino-interaction target. The target mass in the fiducial volume is 303 kg in total which corresponds to 98% of the total target mass. More detailed information about the Proton Module can be found in Ref. [8].
The INGRID module has a sandwich structure comprising nine iron plates and eleven tracking planes which are surrounded by veto planes, as shown in Fig. 6. The tracking planes are formed by two scintillator layers each of which is composed of 24 scintillator bars oriented perpendicularly to one another. The thicknesses of each iron plate and scintillator bar are 6.5 cm and 1.0 cm, respectively. More detailed information about the INGRID module can be found in Ref. [9].
In all three detectors, the scintillation light emitted from the scintillator bar is collected by a WLS fiber, and it is detected by a Multi-Pixel Photon Counter (MPPC) [10]. To digitize and record the integrated charge and hit timing of 1280 channels, the SPIROC2D electronics [11] are used for the WAGASCI module, and the Trip-t electronics [12] are used for the Proton Module and the INGRID module. For each beam bunch, the threshold is set to 2.5 p.e. (photon equivalent) to exclude accidental dark noise from MPPCs.

Monte Carlo Simulation
To estimate backgrounds, neutrino fluxes and signal detection efficiencies, a set of Monte Carlo (MC) simulations is used as follows: • JNUBEAM [13]   The beam axis corresponds to the z-axis. The muon angle is defined as the scattering angle with respect to z-axis.
Software settings for the simulation are the same as those used in [5]. The neutrino energy spectra at the WAGASCI and Proton Module positions predicted by JNUBEAM, with hadronic processes tuned from the NA61/SHINE measurements [16], are shown in Fig. 7. The mean neutrino energy is 0.86 GeV, and the peak is at 0.66 GeV with 1σ spread of +0.40 −0.25 GeV. The flux-integrated CC cross-sections per nucleon predicted by NEUT are summarized in Table 1. To compare predicted neutrino cross-sections in Sec. 7, an alternative event generator, GENIE [17] (2.12.8), is also used. In both generators, a Relativistic Fermi-Gas (RFG) model [18] is used, but the Bodek-Ritchie modifications [19,20] are implemented in GENIE. In NEUT, random-phase approximation (RPA) [21], and multi-nucleon (2p2h) 8/29  interactions [22] are considered. In addition, they use the Rein-Sehgal model [23,24] for the single-meson production, the Berger-Sehgal model [25] for the coherent-pion production, and Glück-Reya-Vogt-1998 (GRV98) [26] parton distributions with Bodek-Yang modifications [27,28] for the deep-inelastic scattering. NEUT is also used for the T2K neutrino oscillation analysis, and more details can be found in Ref. [29].

Datasets and Event Selections
In defined as the charged-current interaction with no detected pions nor protons. This signal is characterized by a muon-like track produced inside the detector. The cross-section is calculated for signal events both from ν µ interactions (ν µ cross-section) and ν µ + ν µ interactions (ν µ + ν µ cross-section), as described in Sec. 5.2.
The selections applied to the two detectors are similar to those in a previous analysis [5], where cross-sections on water and hydrocarbon targets were measured. The selection criteria in this analysis are briefly described below.

Selections for the WAGASCI module
A scintillator channel having an ADC charge greater than 2.5 p.e. is defined as a "hit". Based on a cellular automaton algorithm [30], these hits are fitted by a line (track reconstruction). The two-dimensional tracks are reconstructed in each detector from more than two hits in a beam bunch, and then at least one track in the WAGASCI module and the Proton Module is required to be matched with a reconstructed track in the INGRID module to select a muon-like track. Three-dimensional tracks are searched for among pairs of two-dimensional XZ tracks and YZ tracks. After the reconstruction of the three-dimensional tracks, the upstream point of the longest track is defined as a neutrino interaction vertex.
Subsequently, in order to reduce non-beam backgrounds such as cosmic rays, the event timing for a vertex is required to be within 100 ns from the expected beam-bunch timing (beam-timing cut). In addition, to reduce the beam-induced backgrounds mainly from neutrino interactions in the walls of the detector hall, two cuts are applied. First, if the most upstream point of a reconstructed track is in the first or second plane of the parallel scintillators, then that event is excluded. Second, if a vertex is in the outer region of the fiducial volume (FV), then that event is excluded. The FV is defined as the central area of the WAGASCI module with dimensions of 70 cm (in x-coordinate) × 70 cm (in y-coordinate) × 21 cm (in z-coordinate).
Since the WAGASCI module lies closer to the INGRID module than the Proton Module, the angular acceptance by the INGRID module is larger. In order to obtain a similar angular acceptance to that of the Proton Module, an extrapolation of the reconstructed track from the WAGASCI module is required to reach an imaginary INGRID module. The imaginary INGRID module is set as shown in Fig. 8 so that the distance between the downstream edge of the Proton Module and the upstream edge of the INGRID module (1034.5 cm) is almost the same as that between the downstream edge of the WAGASCI module and the upstream edge of the imaginary INGRID module (1035.5 cm). For signal interactions, a single muon-like track is expected in the final state. To reduce the multi-track backgrounds from other neutrino interactions, events having more than one track are excluded.
The number of selected events and the background fraction in the WAGASCI module are summarized in Table 2. The last cut of the reconstructed track angle is due to the final selection acceptance, and it is described in Sec. 4.3. The neutrino energy, muon momentum, and angular distributions of the selected events predicted by the MC simulation are shown in Fig. 9. The left panel of Fig. 10 shows the angular distribution of the reconstructed single muon-like track for events passing the one-track extraction in the WAGASCI module.

Selections for the Proton Module
Selection criteria for the Proton Module basically use the same method as those for the WAGASCI module, except for the two-dimensional track matching. Since the WAGASCI module is located between the Proton Module and the INGRID module, two-dimensional tracks in the Proton Module are required to be matched to both the WAGASCI module and the INGRID module.
The number of selected events and the background fraction in the Proton Module are summarized in Table 3. The neutrino energy, muon momentum, and angular distributions of the selected events predicted by the MC simulation are shown in Fig. 11. The right panel of Fig. 10 shows the angular distribution of the reconstructed single muon-like track for events passing the one-track extraction in the Proton Module.   of induced muons are restricted to the high-detection efficiency-region, θ µ < 30 • and p µ > 400 MeV/c, in the laboratory frame. According to this restriction, the charged-current events are classified into six bins based on the muon angles, as summarized in Table 4. Although the signal is CC0π0p with a muon angle smaller than 30 degrees, the selected events for crosssection calculations also include two bins for multi-track samples (labelled as CCother) and higher angle samples (labelled as single track 30 • -180 • CC0π0p). In addition, detectable 13/29     space is defined allowing no pions nor protons in these regions. Detection efficiencies for each bin are summarized in Table 5.

Cross-Section Extraction
In this paper, the following notations are used: • X reco j represents the j-th reconstructed single-track angle bin. The analysis method is almost the same as that used in Ref. [5], and detailed information can be found in that reference. 16

Calculation formula
The CC0π0p flux-integrated differential cross-sections are calculated as follows: where N sel is the number of selected events, Φ is the integrated ν µ (ν µ + ν µ ) flux, T is the number of target nucleons, and ε is the signal-selection efficiency. The N BG is the number of expected backgrounds, and N BG WM is estimated not only by the MC simulation but also by the calculated cross-section on the hydrocarbon target to take into account the contribution from the plastic scintillators in the WAGASCI module. Quantities, Φ and T , are summarized in Table 6. The U ij is an unfolding matrix which is iteratively calculated based on the D'Agostini method [31]. To avoid any dependence of U ij on the input neutrino interaction simulation the number of iterations is not truncated but rather ran through to convergence such that the result is effectively unregularized (more details are presented in Sec. 7). In the unfolding procedure, we choose a flat prior, and define the number of iterations as 1500. The subscripts of WM and PM represent the WAGASCI module and the Proton Module, respectively, and those of H 2 O and CH represent target materials. All of the backgrounds are estimated by the MC simulation, except for interactions on WAGASCI plastic scintillators. They constitute one of main background sources for σ H2O , since about 20% of the fiducial volume of the WAGASCI module is occupied by plastic scintillators. They are calculated by normalizing from the number of selected events in the Proton Module.

ν µ cross-sections and ν µ + ν µ cross-sections
As shown in Tables 2 and 3, the ν µ CC interactions are the dominant background and are irreducible in our ν µ event selection since we cannot determine the charge of the outgoing muon. In order to be less model-dependent, we also measured a combined ν µ + ν µ cross-section, since this measurement does not rely on model assumptions to subtract the ν µ background.
The event selection, the number of selected events (N sel j ), and the number of target nucleons (T ) are common to the ν µ and ν µ + ν µ cross-section measurements. Differences between these measurements are summarized in Table 7. 17/29 Table 7: Summary of differences between the ν µ and ν µ + ν µ cross-section measurements.

Uncertainties
Evaluation methods for each uncertainty are almost the same as those considered in Ref. [5], and detailed information can be found in that reference.

Systematic uncertainties from neutrino flux uncertainties
The uncertainty on the neutrino flux is estimated according to knowledge of hadron interactions and the J-PARC beamline. For systematic uncertainties on the cross-section extraction, effects on the number of background events (N BG ), integrated flux (Φ), detection efficiency (ε), and the unfolding matrix (U ij ) are considered. Events generated in the MC simulation are varied based on the estimated flux uncertainties in bins of the true neutrino energies, and correlations among them. The variation of the cross-section is calculated by using 10,000 toy samples accordingly. The ±1σ range of the distribution is taken as the systematic uncertainty.
The uncertainties from neutrino flux on the integrated cross-section for CC0π0p with a muon angle smaller than 30 degrees are expected to be about 10% for σ H2O and σ CH , and they give the dominant contributions to the total uncertainty. On the other hand, the uncertainties for the cross-section ratio (σ H2O /σ CH ) are about 0.5%, since most of the parameters are strongly correlated and the uncertainties cancel.

Systematic uncertainties from neutrino-interaction model
Uncertainties on the neutrino-interaction model are estimated based on the understanding of the model applied to the MC-event generator. Each parameter related to this analysis is varied to cover model uncertainties, and the propagation to the extracted cross-sections is calculated. The parameters with their default values and 1σ variations are summarized in Table 8. When the uncertainty is calculated, no correlation is assumed between different target nuclei for the Fermi momentum (P f ), binding energy (E b ), 2p2h, CC coherent parameters, and nucleon final state interactions (FSI). Full correlation between the different targets is assumed for the other parameters.
The uncertainties due to the neutrino-interaction model are dominated by effects from CCQE and 2p2h interactions, and nucleon FSI, followed by pion production (M Res A and C A5 ) and Fermi momentum (P f ). The CCQE and 2p2h interactions have uncertainties that are 2% larger than other categories and have the largest effect on the detection-efficiency estimation, since they dominate the CC0π0p signal and then largely distort the prior distribution. Nucleon FSI mainly affect the number of backgrounds via ν µ interactions, since more nucleons often exist in the final state of ν µ interactions than that of ν µ interactions. Hence, this effect becomes smaller for the ν µ + ν µ cross-section measurement. 18/29

Systematic uncertainties from detector response
Uncertainties on the detector response are estimated based on measurements during the detector construction, commissioning data taking with cosmic muons, and operation with 19/29 the anti-neutrino beam. Effects on the number of selected events are estimated according to the uncertainty on the detector response, and the systematic uncertainty on the cross-section measurement is estimated by applying fluctuations to the measured number of selected events. In order to apply fluctuations to the number of selected events, no correlation between the WAGASCI module and the Proton Module is assumed, except for the beam-related backgrounds which should be common between the two detectors. Correlations between each bin of reconstructed tracks are considered. The target mass, MPPC noise, scintillator crosstalk, reconstruction efficiency, event pileup, beam-related backgrounds, and event selections are considered as sources of uncertainty. Uncertainties from the event selection are estimated from the difference between data and simulation in the variation of the number of selected events for each selection criterion.

Total uncertainty
Total uncertainties are summarized in Appendix A. For the cross-section measurements of CC0π0p with a muon angle smaller than 30 degrees, the total uncertainty on the absolute cross-sections, σ H2O and σ CH , is dominated by the neutrino-flux uncertainty, while that on the cross-section ratio, σ H2O /σ CH , is dominated by statistical errors and errors on the detector response.  Figure 14 shows correlation matrices including all uncertainties for ν µ (top) and ν µ + ν µ (bottom) cross sections. Figure 15 shows the distributions of the measured differential cross-sections for CC0π0p with a muon angle smaller than 30 degrees, with their uncertainties and expectations from NEUT from NEUT and GENIE. Considering the number of degrees of freedom is eight, the calculated χ 2 values suggest that the measured cross-sections agree well with neutrino-interaction models implemented in those generators.

Conclusion
In this paper, we report measurements of (anti-)neutrino cross-sections on water and hydrocarbon targets with the WAGASCI module and the Proton Module using the T2K      muons, pions, and protons. The differential cross-sections and integrated cross-sections for the ν µ only and ν µ + ν µ fluxes are measured. The results agree with the current neutrinointeraction models used in the T2K oscillation analysis within their statistical and systematic uncertainties. 23/29