Impact of final state interactions on neutrino-nucleon pion production cross sections extracted from neutrino-deuteron reaction data

TL;DR

This paper investigates how final state interactions affect neutrino-induced pion production data from deuteron reactions, providing corrected cross sections to improve neutrino-nucleus models for oscillation experiments.

Contribution

It introduces a theoretical model incorporating FSI effects to accurately extract elementary neutrino-nucleon pion production cross sections from deuteron data.

Findings

FSI significantly reduce the spectra over the quasi-free peak region.

A recipe is developed to extract elementary process information from deuteron data.

Corrected cross sections improve neutrino-nucleus reaction models.

Abstract

The current and near-future neutrino oscillation experiments require significantly improved neutrino-nucleus reaction models. Neutrino-nucleon pion production data play a crucial role to validate corresponding elementary amplitudes that go into such neutrino-nucleus models. Thus the currently available data extracted from charged-current neutrino-deuteron reaction data ($\nu_\mu d\to \mu^-\pi NN$) must be corrected for nuclear effects such as the Fermi motion and final state interactions (FSI). We study $\nu_\mu d\to \mu^-\pi NN$ with a theoretical model including the impulse mechanism supplemented by FSI from $NN$ and $\pi N$ rescatterings. An analysis of the spectator momentum distributions reveals that the FSI effects significantly reduce the spectra over the quasi-free peak region, and leads to a useful recipe to extract information of elementary $\nu_\mu N\to \mu^-\pi N$ processes using $\nu_\mu d\to \mu^-\pi NN$ data, with the important FSI corrections taken into account. We provide $\nu_\mu N\to \mu^-\pi N$ total cross sections by correcting the deuterium bubble chamber data for the FSI and Fermi motion. The results will bring a significant improvement on neutrino-nucleus reaction models for the near-future neutrino-oscillation experiments.

Tables

Table 1. Table 1: Parameters for N α fit ​ ( E ν , p s ) superscript subscript 𝑁 𝛼 fit subscript 𝐸 𝜈 subscript 𝑝 𝑠 N_{\alpha}^{\rm fit}(E_{\nu},p_{s}) defined in Eqs. ( 8 )-( 9 ).

$\alpha$	${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}p$	${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{+}n$	${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{0}p$
$a_{\alpha }$	$0.9498$	$0.9348$	$0.9930$
$b_{\alpha }$	$-0.5632$	$-0.7111$	$-1.035$
$c_{\alpha }$	$0.2788$	$0.2211$	$0.3142$
$d_{\alpha }$	$-0.6381$	$-0.7832$	$-1.047$
$e_{\alpha }$	$0.3171$	$0.2574$	$0.3188$
$f_{\alpha }$	$-0.5496$	$-0.3170$	$-25.18$
$g_{\alpha }$	$0.0087$	$-1.387$	$-67.88$
$h_{\alpha }$	$-0.2983$	$0.3030$	$378.5$

Table 2. Table 2: Total cross sections for ν μ ​ p → μ − ​ p ​ π + → subscript 𝜈 𝜇 𝑝 superscript 𝜇 𝑝 superscript 𝜋 \nu_{\mu}p\to\mu^{-}p\pi^{+} from correcting the reanalyzed ANL data.

$E_{\nu }^{\min }$ (GeV)	$E_{\nu }^{\max }$ (GeV)	$\sigma$ (${10}^{-38}{\mathrm{cm}}^2$)	$\delta \sigma$ (${10}^{-38}{\mathrm{cm}}^2$)
0.3	0.6	0.065	0.010
0.6	0.8	0.312	0.036
0.8	1.0	0.542	0.065
1.0	1.5	0.531	0.047
1.5	2.2	0.821	0.125

Table 3. Table 4: Total cross sections for ν μ ​ n → μ − ​ n ​ π + → subscript 𝜈 𝜇 𝑛 superscript 𝜇 𝑛 superscript 𝜋 \nu_{\mu}n\to\mu^{-}n\pi^{+} from correcting the reanalyzed ANL data.

$E_{\nu }^{\min }$ (GeV)	$E_{\nu }^{\max }$ (GeV)	$\sigma$ (${10}^{-38}{\mathrm{cm}}^2$)	$\delta \sigma$ (${10}^{-38}{\mathrm{cm}}^2$)
0.3	0.5	0.021	0.008
0.5	0.7	0.078	0.014
0.7	0.8	0.160	0.035
0.8	1.0	0.169	0.029
1.0	1.2	0.151	0.026
1.2	1.3	0.124	0.043
1.3	1.5	0.237	0.053

Table 4. Table 6: Total cross sections for ν μ ​ n → μ − ​ p ​ π 0 → subscript 𝜈 𝜇 𝑛 superscript 𝜇 𝑝 superscript 𝜋 0 \nu_{\mu}n\to\mu^{-}p\pi^{0} from correcting the reanalyzed ANL data.

$E_{\nu }^{\min }$ (GeV)	$E_{\nu }^{\max }$ (GeV)	$\sigma$ (${10}^{-38}{\mathrm{cm}}^2$)	$\delta \sigma$ (${10}^{-38}{\mathrm{cm}}^2$)
0.3	0.5	0.007	0.004
0.5	0.7	0.046	0.011
0.7	0.8	0.128	0.030
0.8	1.0	0.205	0.032
1.0	1.2	0.114	0.022
1.2	1.3	0.240	0.065
1.3	1.5	0.268	0.059

Equations

$$\frac{d\sigma }{d{\Omega }_{l^{\prime }}d{E}_{l^{\prime }}}$$

$$W^{\mu \nu }$$

$$⟨f|J_{\mathrm{imp}}^{\mu }|i⟩$$

$$⟨f|J_{NN}^{\mu }|i⟩$$

$$⟨f|J_{\pi N}^{\mu }|i⟩$$

$$\frac{d{\tilde{\sigma }}_{\alpha }(E_{\nu })}{d{p}_s}=p_s^2\int d{\Omega }_{p_s}{\sigma }_{\alpha }({\tilde{E}}_{\nu })|{\Psi }_d({\vec{p}}_s){|}^2,$$

$$\frac{d{\sigma }_{\nu d}(E_{\nu })}{d{p}_s}=N_{\alpha }(E_{\nu },p_s)\frac{d{\tilde{\sigma }}_{\alpha }(E_{\nu })}{d{p}_s}.$$

$$N_{\alpha }^{\mathrm{fit}}(E_{\nu },p_s)=A_{\alpha }({\hat{E}}_{\nu })+B_{\alpha }({\hat{E}}_{\nu }){\hat{p}}_s,$$

$$A_{\alpha }(x)$$

$${\bar{N}}_{\alpha }(E_{\nu }){\sigma }_{\alpha }(E_{\nu })=\frac{1}{A}{\int }_0^{p_s^{\max }}d{p}_s\frac{d{\sigma }_{\nu d}(E_{\nu })}{d{p}_s},$$

Full text

††preprint: LFTC-18-15/36, J-PARC-TH-0139

The current frontier of neutrino-oscillation experiments, such as the T2K t2k and the DUNE dune , is to determine the charge-parity violating phase ($\delta_{\rm CP}$) and the neutrino mass hierarchy as the primary objective. In this era of precision neutrino experiments, we must improve the current situation that the uncertainty in our knowledge of neutrino-nucleus cross sections in the few-GeV neutrino energy region is one of the largest sources of systematic uncertainty in extracting the oscillation parameters from the data white ; mahn .

A reliable model for the elementary neutrino-nucleon reactions is a key ingredient in neutrino-nucleus interaction generators such as NEUT neut , GENIE genie , NuWro nuwro , and GiBUU gibuu , to be used in neutrino-oscillation analyses. Many microscopic models microscopic for the neutrino-nucleon single pion productions ($\nu N\to l\pi N$) have been developed with different dynamical contents nu_model_pi ; SUL ; dcc_nu ; see Ref. nu-review for an overview of these microscopic models, and Ref. pi_comp for a detailed comparison. The common procedure in developing all the models is to adjust the dominant $\Delta(1232)$-excitation mechanism to fit the total cross section data anl ; bnl of $\nu_{\mu}p\to\mu^{-}\pi^{+}p$, $\nu_{\mu}n\to\mu^{-}\pi^{+}n$, and $\nu_{\mu}n\to\mu^{-}\pi^{0}p$. These currently available data for $\nu_{\mu}N\to\mu^{-}\pi N$ were extracted from neutrino-deuteron reaction ($\nu_{\mu}d\to\mu^{-}\pi NN$) data, assuming the quasifree mechanism [Fig. 1(a)]. In order to reduce the systematic uncertainty of the neutrino-oscillation parameters, an urgent task is to clarify the effects of the final state interactions (FSI), such as the nucleon and pion rescattering processes [Fig. 1(b,c)], and then correct the extracted $\nu_{\mu}N\to\mu^{-}\pi N$ cross sections.

In this paper, we analyze the spectator momentum distributions of $\nu_{\mu}d\to\mu^{-}\pi NN$, the data of which would be obtainable from a possible future neutrino-deuteron experiment int_ws . Taking account of the important FSI corrections, we then find a useful recipe for extracting the information of $\nu_{\mu}N\to\mu^{-}\pi N$ processes from the deuteron-target data. We will also provide improved $\nu_{\mu}N\to\mu^{-}\pi N$ total cross sections by correcting the bubble chamber data reanalysis1 ; reanalysis2 , which are free from the flux uncertainty, for the FSI and Fermi motion effects. The improved data will enable one to develop a more accurate $\nu_{\mu}N\to\mu^{-}\pi N$ model to be used in extracting the neutrino properties from the oscillation experiments of the precision era.

The first attempt towards understanding the FSI effects on $\nu_{\mu}d\to\mu^{-}\pi NN$ differential cross sections has been made recently wsl15 , and sizable FSI effects were found in the quasifree $\Delta(1232)$-production region. However, their analysis focused on this particular kinematical region, and thus understanding the FSI corrections on the $\nu_{\mu}N\to\mu^{-}\pi N$ data anl ; bnl is beyond their scope. Our calculation will cover the whole phase space of $\nu_{\mu}d\to\mu^{-}\pi NN$ with the Monte Carlo phase-space integral.

Our $\nu_{\mu}d\to\mu^{-}\pi NN$ reaction model consists of the relevant mechanisms depicted in Fig. 1: (a) impulse mechanism; (b) $NN\to NN$ FSI mechanism; (c) $\pi N\to\pi N$ FSI mechanism. We use elementary weak pion production amplitudes and $\pi N\to\pi N$ scattering amplitudes generated from a dynamical coupled-channels (DCC) model dcc_nu ; nu-review ; knls13 ; knls16 . The DCC model has been developed for a unified description of the hadronic and electroweak reactions on the single nucleon in the nucleon resonance region: $\pi N,\gamma^{(*)}N\to X$ and $\nu_{l}N\to l^{-}X$ with $X=\pi N,\pi\pi N,\eta N,K\Lambda,K\Sigma$. The model has been extensively tested by a large amount of data ($\sim$27,000 data points) on $\pi N,\gamma N\to\pi N,\eta N,K\Lambda$, $K\Sigma$ knls13 ; knls16 , and also by data on electron-induced reactions dcc_nu for W\ \raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\ 2 GeV ($W$: invariant mass of the hadron system) and $Q^{2}\leq$ 3 GeV2. The DCC elementary amplitudes are particularly suited for describing the neutrino-deuteron reactions that include loop diagrams of hadronic rescatterings. This is because the DCC model possesses unique features, which the other microscopic models do not, such as: (i) a consistent description of the weak pion productions and $\pi N$ reactions satisfying two-body as well as three-body unitarity requirements; (ii) off-shell amplitudes are available by construction. The latter feature is crucial for embedding the elementary amplitudes into the deuteron reaction model in a manner consistent with the multiple scattering theory. The deuteron wave function and $NN$ scattering amplitudes are generated from a realistic $NN$ potential; here we employ the CD-Bonn potential cdbonn .

We have already studied $\gamma d\to\pi NN$ gd-pinn ; gd-pinn2 and $\gamma d\to\eta pn$ gd-epn using a similar DCC-based model for meson photoproductions and have shown that significant FSI effects bring model predictions into a good agreement with the data. These results validate the predictive power of our approach, and allow us to estimate the FSI effects on the neutrino-induced pion productions with a good level of reliability.

The cross sections for charged-current neutrino-deuteron reactions in the laboratory frame are given as

[TABLE]

where $G_{F}$ and $V_{ud}$ are the Fermi coupling constant and CKM matrix element, respectively. The lepton tensor $L^{\mu\nu}$ is given with the initial neutrino ($p_{l}$) and the final lepton ($p_{l^{\prime}}$) momenta. The hadron tensor is defined by

[TABLE]

where $p_{i}$ ($p_{f}$) is the total four-momentum of the initial (final) hadron state and $q=p_{l}-p_{l^{\prime}}$; the average (sum) over the initial (final) hadron states is denoted by $\sum_{\bar{i}}$ ($\sum_{f}$). The hadron current matrix element, $\langle f|J^{\mu}|i\rangle$, includes the impulse, $NN$ FSI, and $\pi N$ FSI mechanisms which are given more explicitly with the particle labels and momenta defined in Fig. 1 as

[TABLE]

[TABLE]

and the exchange terms that can be obtained from Eqs. (3)-(5) by flipping the overall sign and interchanging all subscripts 1 and 2 for the nucleons in the intermediate and final $\pi NN$ states. Here, the deuteron state with spin projection $s_{d}$ is denoted as $|{\Psi_{d}(s_{d})}\rangle$; $|{N(\bm{p},s,t)}\rangle$ the nucleon state with momentum $\bm{p}$ and spin and isospin projections $s$ and $t$; $|{W^{+}(\bm{q},\lambda)}\rangle$ the $W^{+}$ boson state with momentum $\bm{q}$ and polarization $\lambda$; $|{\pi(\bm{k},t_{\pi})}\rangle$ the pion state with momentum $\bm{k}$ and the isospin projection $t_{\pi}$. The total energy in the laboratory frame is denoted by $E$, and the energy of a particle $x$ with the mass $m_{x}$ is $E_{x}(k)=\sqrt{m_{x}^{2}+k^{2}}$. The two-body invariant masses $M_{\pi N_{1}}$ and $M_{N_{1}N_{2}}$ are determined by the final states while $W=\sqrt{[E-E_{N}(-\bm{l})]^{2}-(\bm{l}+\bm{q})^{2}}$ is used for the intermediate two-body invariant mass.

The elementary amplitudes $\langle{\pi N}|t_{\pi N,W^{+}N}|{W^{+}N^{\prime}}\rangle$ of the weak pion production dcc_nu and $\langle{\pi N}|t_{\pi N,\pi N}|{\pi^{\prime}N^{\prime}}\rangle$ of the pion-nucleon scattering knls13 ; knls16 in Eqs. (3)-(5) are first generated from the DCC model in the two-body CM frame, and then boosted to the laboratory frame. The same frame-transformation procedure is also needed to calculate the matrix element $\langle{NN}|t_{NN,NN}|{NN}\rangle$ of $NN$ scattering cdbonn in Eq. (4). The formulas for calculating these matrix elements in the laboratory frame are given in Appendix A in Ref.  gd-pinn .

As in the bubble chamber experiments anl ; bnl , we look into the spectator momentum ($p_{s}$) distribution in $\nu_{\mu}d\to\mu^{-}\pi N^{\prime}N_{s}$ ($N_{s}$: spectator). The cross section for $\nu_{\mu}p\to\mu^{-}\pi^{+}p$ ($\nu_{\mu}n\to\mu^{-}\pi^{+}n$) can be extracted from the $p_{n}$ ($p_{p}$) distribution in $\nu_{\mu}d\to\mu^{-}\pi^{+}pn$ because, in a low-$p_{n}$ ($p_{p}$) region, the quasifree pion production on the proton (neutron) is expected to dominate while the neutron (proton) would hardly contribute. If this expectation is right, the spectator momentum distribution ($d\sigma_{\nu_{\mu}d}/dp_{s}$) calculated with the impulse approximation (IA) should be approximated accurately with the $\nu_{\mu}N\to\mu^{-}\pi N^{\prime}(\equiv\alpha)$ cross section convoluted with the deuteron wave function ($\Psi_{d}$) as

[TABLE]

where the total cross section $\sigma_{\alpha}$ is calculated with the same $\nu_{\mu}N\to\mu^{-}\pi N^{\prime}$ amplitudes implemented in the $\nu_{\mu}d\to\mu^{-}\pi NN$ model. The boosted neutrino energy $\tilde{E}_{\nu}$ is obtained from $E_{\nu}$ by the same Lorentz transformation that boosts the struck nucleon with the momentum $-\vec{p}_{s}$ to its rest.

Indeed, when the spectator in $\nu_{\mu}d\to\mu^{-}\pi^{+}pn$ is the neutron as in Fig. 2(left) where $E_{\nu}=0.5$ GeV, the convoluted $\nu_{\mu}p\to\mu^{-}\pi^{+}p$ cross section (black solid curves) agrees almost perfectly with $d\sigma^{\rm IA}_{\nu_{\mu}d}/dp_{n}$ (green triangles). On the other hand, when the spectator is the proton as in Fig. 2(right), the convoluted $\nu_{\mu}n\to\mu^{-}\pi^{+}n$ cross section undershoots $d\sigma^{\rm IA}_{\nu_{\mu}d}/dp_{p}$ in the quasifree peak region. As $p_{p}$ increases, the difference between them becomes significantly larger. This difference is due to the contribution from the $\nu_{\mu}p\to\mu^{-}\pi^{+}p$ mechanism which is responsible for $\sim 87$% of the $\nu_{\mu}d\to\mu^{-}\pi^{+}pn$ total cross section from the IA calculation at this neutrino energy.

The $NN$ FSI significantly reduces $d\sigma_{\nu_{\mu}d}/dp_{s}$ for $\nu_{\mu}d\to\mu^{-}\pi^{+}pn$, especially around the quasifree peak in the low-$p_{s}$ region, as can be seen in the differences between the blue diamonds and green triangles in Fig. 2. On the other hand, the spectator momentum distribution for $\nu_{\mu}d\to\mu^{-}\pi^{0}pp$ is hardly changed by the FSI and, thus, is in good agreement with the convoluted $\nu_{\mu}n\to\mu^{-}\pi^{0}p$ cross section. The distinct differences between the $\pi^{+}$ and $\pi^{0}$ productions in the FSI effects is due to the fact that the deuteron ($pp$) bound state can (cannot) be formed in the $\pi^{+}$ ($\pi^{0}$) production.

For a more quantitative study of the above observations on $d\sigma_{\nu_{\mu}d}/dp_{s}$, we define a coefficient $N_{\alpha}(E_{\nu},p_{s})$ by

[TABLE]

The predicted $N_{\alpha}(E_{\nu},p_{s})$ are shown in Fig. 3. A deviation from $N_{\alpha}(E_{\nu},p_{s})\simeq 1$ indicates the contributions from neutrino reactions on the other nucleon and/or FSI. Within the IA (green triangles), the quasifree $\nu_{\mu}p\to\mu^{-}\pi^{+}p$ dominates in the low-$p_{n}$ (upper panel), while the quasifree $\nu_{\mu}n\to\mu^{-}\pi^{+}n$ dominance in the low-$p_{p}$ (lower panel) is prevented by the stronger $\nu_{\mu}p\to\mu^{-}\pi^{+}p$ channel and thus $N_{\alpha}(E_{\nu},p_{s})$ quickly deviates from one.

Figure 3 also shows that the total FSI effects reduce the $p_{n}$ distribution, depending on $p_{n}$, by 10-20% (5-10%) for $E_{\nu}=0.5$ (1.5) GeV. The reduction of the $p_{p}$ distribution is even larger and, near the quasifree peak, it is $\sim 2$ times larger than the reduction of the $p_{n}$ spectra. We also find that the FSI effect depends strongly on the neutrino energy $E_{\nu}$. The $NN$ FSI effects, which are large at small $NN$ relative energies, become smaller as $E_{\nu}$ increases. The $\pi N$ FSI effects become as large as the $NN$ FSI effect for E_{\nu}\ \raise 1.29167pt\hbox{>\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\ 1 GeV around the quasifree peak of the $p_{p}$ spectrum where the $\nu_{\mu}n\to\mu^{-}\pi^{+}n$ mechanism is dominant.

Once $N_{\alpha}(E_{\nu},p_{s})$ is provided in a simple phenomenological formula, one can easily extract $d\tilde{\sigma}_{\alpha}/dp_{s}$ of Eq. (6) from $d\sigma_{\nu d}(E_{\nu})/dp_{s}$ data using Eq. (7) with the FSI effects taken into account. Thus such a formula offers a useful recipe: one can determine a $\nu_{\mu}N\to\mu^{-}\pi N^{\prime}$ model by fitting the extracted $d\tilde{\sigma}_{\alpha}/dp_{s}$. Also, $N_{\alpha}(E_{\nu},p_{s})$ is expected to have a small model dependence from using our particular DCC-based model, because it is given by the ratio of $d\sigma_{\nu d}/dp_{s}$ to $d\tilde{\sigma}_{\alpha}/dp_{s}$ both of which are calculated with the same DCC elementary $\nu_{\mu}N\to\mu^{-}\pi N^{\prime}$ amplitudes. Regarding the functional form, we find it convenient to use

[TABLE]

with $\hat{E}_{\nu}\equiv E_{\nu}/(1\,{\rm GeV})$, $\hat{p}_{s}\equiv p_{s}/(1\,{\rm GeV})$, and

[TABLE]

We fit the parameters ($a_{\alpha}$–$h_{\alpha}$) to the numerically computed $N_{\alpha}(E_{\nu},p_{s})$ at several neutrino energies between $E_{\nu}=0.4$ and 2 GeV and in the $p_{s}\leq 50$ MeV region where the cross sections are dominated by the quasifree process aside from the FSI effects. The obtained parameters are presented in Table 1. The fit functions are shown in Fig. 3 by the solid lines in comparison with the original numerical results.

Our finding on the FSI effects requires a modification of the flux-corrected bubble chamber data reanalysis1 ; reanalysis2 . The data for $d\sigma_{\nu d}/dp_{s}$ from the $\nu d$ experiments anl ; bnl are unavailable by now. Therefore, we use the $\nu d$ cross section in the quasifree peak region, which includes a large portion of the total events, to estimate the corrections due to the FSI effects. We integrate Eq. (7) with respect to $p_{s}$ up to $p_{s}^{\rm max}=90$ MeV, and then introduce an effective coefficient $\bar{N}_{\alpha}(E_{\nu})$ to satisfy the following equation:

[TABLE]

with $A\equiv\int_{0}^{p_{s}^{\rm max}}d^{3}p_{s}|\Psi_{d}(\vec{p}_{s})|^{2}$. The effective coefficient $\bar{N}_{\alpha}(E_{\nu})$ can be interpreted as an average of $N_{\alpha}(E_{\nu},p_{s})$, weighted with the nucleon momentum distribution in the deuteron, over $p_{s}$. We may identify $\bar{N}_{\alpha}(E_{\nu})\sigma_{\alpha}(E_{\nu})$ as the reanalyzed bubble chamber data ($\sigma^{\rm data}_{\alpha}(E_{\nu})$) reanalysis1 ; reanalysis2 , because both of them can be regarded as the effective $\nu N$ cross sections extracted from $\sigma_{\nu d}(E_{\nu})$ without correcting for the FSI and the Fermi motion. Because the factor $\bar{N}_{\alpha}(E_{\nu})$ distorts the true $\nu N$ cross sections ($\sigma_{\alpha}(E_{\nu})$) by including the FSI and Fermi motion effects, the corrected data are given by $[\bar{N}_{\alpha}(E_{\nu})]^{-1}\sigma^{\rm data}_{\alpha}(E_{\nu})$. The corrected data are shown in Fig. 4 suppl in comparison with the original ones reanalysis2 to which no $W$ cut has been applied. The correction is larger for smaller $E_{\nu}$ and enhances the cross sections by scaling factors of 1.05–1.12, 1.10–1.27, and 1.01–1.02 for $\nu_{\mu}p\to\mu^{-}\pi^{+}p$, $\nu_{\mu}n\to\mu^{-}\pi^{+}n$, and $\nu_{\mu}n\to\mu^{-}\pi^{0}p$, respectively.

In summary, we have studied for the first time $\nu_{\mu}d\to\mu^{-}\pi NN$ over the whole phase space with a reaction model including the FSI mechanisms. The FSI is found to significantly reduce the spectator momentum distributions, depending on the proton or neutron momentum, $\pi^{+}/\pi^{0}$ production, and neutrino energy. We have proposed a recipe to determine an elementary $\nu_{\mu}N\to\mu^{-}\pi N$ model by using data for $d\sigma_{\nu d}(E_{\nu})/dp_{s}$ which could be obtained from possible future deuteron-target experiments. We also presented the $\nu_{\mu}N\to\mu^{-}\pi N$ total cross sections by correcting the flux-corrected bubble chamber data reanalysis2 for the FSI and Fermi motion. Because the bubble chamber data are currently the most important information for studying the elementary neutrino-induced pion production mechanisms, the corrected data pave the way to implementing a significantly improved pion production mechanism into a neutrino-nucleus reaction model for the near-future neutrino-oscillation experiments. An extension of the present analysis to differential cross sections such as $W$ and $Q^{2}$ dependences will be presented elsewhere.

Appendix A Supplemental material

The following tables present numerical values for the total cross sections ($\sigma$) of $\nu_{\mu}N\to\mu^{-}N\pi$ and their errors ($\delta\sigma$) in each neutrino energy bin specified by the range $[E^{\rm min}_{\nu},E^{\rm max}_{\nu}]$. The cross sections are obtained by correcting the reanalyzed ANL and BNL data (no $W$ cut) reanalysis2 , which are free from the flux uncertainty, for the final state interactions and the Fermi motion effects. In each bin, the correction factor is calculated at the central value of $[E^{\rm min}_{\nu},E^{\rm max}_{\nu}]$.