Charged-Current Neutrino-Induced Single-Pion Production in the Superscaling Approach and the Relativistic Distorted-Wave Impulse Approximation

1. Introduction

Neutrino oscillation experiments constitute a fundamental tool for exploring the properties of neutrinos, such as the possible violation of charge-parity symmetry or their mass hierarchy [1,2,3]. The interaction of the neutrino with the nucleus is a key piece in the reconstruction of the neutrino energy in oscillation experiments, a fundamental step in the determination of neutrino oscillation parameters and a source of major uncertainties in the analysis. As such, multiple theoretical and experimental efforts have been developed for this purpose. In these experiments, multiple channels are involved. For example, in the neutrino energy range of hundreds of MeV to a few GeV, the dominant interaction of the neutrino with the nucleus is the quasielastic (QE) process in which a single nucleon is knocked-out from the nucleus. In spite of this process being heavily relevant in the few GeV region, in experimental inclusive analyses, where all reaction channels are considered, other processes also give crucial contributions to the cross section, like two-particle two-hole interaction, the excitation of the nucleonic resonances, or deep-inelastic scattering (DIS). Hence, a reduction in the nuclear-medium uncertainties associated with all these processes, including neutrino-induced pion production, is needed to improve cross-section measurements and neutrino oscillation analyses.

Most experiments – MiniBooNE [4], MicroBooNE [5], T2K [6], MINERvA[7], NO$\nu$A [8] and future ones like HyperK [9] or DUNE [10] – operate on different energy ranges, such as 0.5-1 GeV (MicroBooNE, T2K, MiniBooNE) or around 10 GeV (DUNE, ArgoNEUT[11]). The importance of each channel depends on the energy in which the experiment operates: quasielastic scattering dominates at a neutrino energy of $0.5-1$ GeV, while pion production and other inelastic processes become dominant at higher energies. Diverse experimental efforts have been devoted to the study of these different reaction mechanisms. This is the case of charged-current neutrino interactions with only one pion detected in the final state of the process, together with the scattered lepton. These so-called CC1$\pi$ events can be modeled as the excitation of a nucleonic resonance followed by its decay into nucleons, pions and other mesons, non-resonant pion production processes or intranuclear pion rescattering effects. These studies also allow us to compare and validate theoretical models for the description of neutrino-induced pion production.

On the theoretical side, several groups [12,13,14,15,16,17,18,19,20,21,22] have studied neutrino-induced pion production on nuclei, providing different descriptions of the initial nuclear state, the pion production from a bound nucleon, and the subsequent pion-nucleon interaction within the residual nucleus. Most of the initial studies were based on Fermi gas approaches of non-interacting nucleons [23,24], but recently more sophisticated descriptions have been developed, incorporating Random Phase Approximation (RPA) calculations [18,25], the plane-wave impulse approximation (PWIA) together with the use of realistic spectral functions [26] or relativistic distorted-wave impulse approximation (RDWIA) approaches [27].

All previous nuclear models describing lepton-induced single-pion production (SPP) on the nucleus are based on the impulse approximation, i.e., they consider that the virtual boson couples only one nucleon in the nucleus. The description of the elementary boson-nucleon-pion vertex is done differently depending on the model considered. For example, the Hybrid model [28,29,30] is used in the RDWIA approach [31], and the ANL-Osaka Dynamical Coupled-Channels model [32,33,34,35,36] (DCC in what follows) has been recently implemented in the SuSAv2 framework [37]. The DCC has also been used in the extended factorization scheme [26], which was incorporated in the ACHILLES event generator [38]; the Hybrid was implemented in the NuWro event generator using a local Fermi gas as nuclear model [22]. A different approach for neutrino-induced single pion production is the MK model [39], which is used in the NEUT event generator [40].

In a recent work [37,41], the superscaling model SuSAv2, developed for the charged-current quasielastic neutrino-nucleus cross section, was extended to the full inelastic regime (SuSAv2-inelastic model). In this framework, the DCC model has been implemented [37] to give an accurate description of neutrino-induced one- and two-pion production, among other meson-production mechanisms in the resonance region. This model has been widely tested for photon, electron and neutrino scattering off single nucleons, providing information about the single-nucleon inelastic structure functions ($W_{1-5}$) [32,33,34,35,36]. This model accounts for the structure functions associated with the whole inclusive contribution and also the functions connected to lepton-induced SPP. In what follows, we refer to them as inclusive-DCC and $\pi$-DCC, respectively. The inclusive-DCC takes into account two- and three-body meson-baryon final states ($\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$ and $\pi \pi N$) in a couple channel approach, while the $\pi$-DCC accounts only for a particular SPP channel. Then, using $\pi$-DCC inelastic structure functions, we can distinguish between channels with a single ${\pi }^{+}$, ${\pi }^0$, or ${\pi }^{-}$ in the final state.

The Hybrid model, presented in [29], aims at describing lepton-induced SPP in a broader energy region, so it could be used without phase-space restrictions in neutrino-oscillation experiments. It is based on the low-energy model of [28] and a high-energy Regge-based model, and can be used for SPP induced by electrons and charged- and neutral-current neutrinos. The Hybrid model was later incorporated in the RDWIA nuclear framework, so it could be used to model cross sections for incoherent SPP on the nucleus [21,30,31,42]. It has been compared to MiniBooNE, MINERvA and T2K pion-detected cross section data. The tension observed in these studies between bubble-chamber and MINERvA data was shown in [31], using the RDWIA and also comparing with the relativistic plane-wave impulse-approximation (RPWIA).

All these approaches are described in Sect. 2, where the theoretical formalisms for the SuSAv2-inelastic and RDWIA frameworks are summarized. In Sect. 3, we show the comparison between the predictions of the different theoretical models and the CC1$\pi$ experimental data provided by MiniBooNE (Sect. 3.0.1), MINERvA (Sect. 3.0.2) and T2K (Sect. 3.0.3), that operate at diverse kinematics and with different targets. In Sect. 4, we draw our conclusions.

2. Formalism

The superscaling approach (SuSA) is based on the scaling properties exhibited by inclusive electron scattering where the QE scattering cross section can be written, under certain conditions, as a term containing the single-nucleon cross section times a scaling function (f) that embodies the nuclear dynamics. The analysis of inclusive electron scattering data [43] has shown that for transferred momentum (q) values around 300 MeV/c or higher, the scaling function does not depend on q (scaling of 1st kind) nor on the nuclear species (scaling of 2nd kind) and can therefore be expressed in terms of a single variable $\psi$, the so-called scaling variable. A more detailed description of superscaling can be found in [43,44,45,46,47,48,49,50]. This approach has also been successfully applied to inclusive charged-current quasielastic (CCQE) neutrino scattering and, most recently, to the full inelastic regime for both electron and neutrino reactions. The corresponding model for the quasielastic region (SuSAv2-QE) is based on a set of QE scaling functions extracted from the relativistic mean field (RMF) theory [51].

The SuSAv2-inelastic model is an extension of the SuSAv2-QE approach to the inelastic regime [52,53]. The double differential cross section for lepton-nucleus scattering with respect to the transferred energy $\omega$ and the scattered lepton solid angle $\Omega$ can be written in the general form [44]

$${\displaystyle \frac{d\sigma }{d\Omega d\omega }}={\sigma }_0\sum _K{v}_K{R}^K,$$

(1)

where ${\sigma }_0$ is an elementary cross section (the Mott cross section in the case of electron scattering), $v_K$ are lepton kinematic Rosenbluth factors and $R^K$ are the nuclear response functions, containing all the information about nuclear dynamics and the inner nucleon structure functions. The summed index K is associated to different components of the nuclear tensor with respect to the direction of the transferred momentum $\mathbf{q}$.

The inclusive nuclear responses, after integrating over the possible values of the invariant mass $W_X$ of the hadronic final states, depend only on the energy and momentum transferred to the nucleus.

The invariant mass for lepton-nucleus reactions, $W_X$, can be determined via the 4-momentum of the final hadronic system $P_X$, which is defined by 4-momentum conservation as:

$$k_i+P_A=k_f+P_{A-1}+P_X,$$

(2)

where $k_i$ and $k_f$ are the initial and final leptons 4-momenta, and $P_A$ and $P_{A-1}$ are the 4-momenta of the target and residual nucleus, respectively. Hence,

$$W_X^2=P_X^2.$$

(3)

In the case of the SPP on nucleons, one has

$$W_X^2={(P_N+P_{\pi })}^2={(Q+P_i)}^2=P_X^2,$$

(4)

being $P_N$, $P_{\pi }$ and $P_i$ the 4-momentum of the final nucleon, produced pion and initial nucleon, respectively.

In the SuSAv2-inelastic model [37,41,52,53], the nuclear responses are given by

$$\begin{array}{cc}\hfill & R_K^{inel}(q,\omega )=N{\displaystyle \frac{2T_F{m}_N^3}{k_F^{3}q}}\hfill \\ \hfill & \times {\int }_{{\mu }_X^{min}}^{{\mu }_X^{max}}d{\mu }_X{\mu }_X{f}^{model}\left({\psi }_X\right)G_K^{inel}(q,\omega ,W_X),\hfill \end{array}$$

(5)

being ${\mu }_X\equiv W_X/m_N$ the dimensionless invariant mass, N the number of nucleons from the target involved in the reaction, $k_F$ the Fermi momentum and $T_F\equiv \sqrt{m_N^2+k_F^2}-m_N$ the Fermi kinetic energy. Thus, the inelastic nuclear responses are defined as the integral over all possible final hadronic states of the single-nucleon inelastic hadronic tensor $G_K^{inel}$ (characterized by the inelastic structure functions $W_{1-5}$) times the inelastic scaling function $f^{model}$ evaluated in a given nuclear model. The latter is written in terms of ${\psi }_X\equiv {\psi }_X(q,\omega ,W_X)$, which is the extension of the QE scaling variable $\psi$ to the inelastic regime and now depends on the final state invariant mass $W_X$. The limits of the integral (${\mu }_X^{min/max}$) can be adjusted depending on the kinematics considered for a particular experimental or theoretical analysis and the limitations of the single-nucleon model used to characterize the hadronic tensor. For example, in the case of the DCC model, there is a limitation up to an invariant mass of 2.1 GeV and $Q^2$ below 3 GeV² [32]. In general, the limits of this integral either in the SuSAv2 model or in a RFG framework can be defined as ${\mu }_X^{min}=1+m_{\pi }/m_N$ and ${\mu }_X^{max}=1+\omega /m_N-E_S$, being $m_{\pi }$ the pion mass and $E_S$ the separation energy for a bound nucleon [54].

In Figure 1, we show the single-nucleon structure functions of the DCC model, displaying both $\pi$-DCC and inclusive-DCC contributions. As expected, both curves match below the two-pion production threshold, in the region dominated by the $\Delta$ resonance, but as the invariant mass increases, the single-pion results ($\pi$-DCC) are below the inclusive ones that also take into account effects beyond SPP as mentioned above.

In Figure 1, we show the single-nucleon structure functions of the DCC model, displaying both $\pi$-DCC and inclusive-DCC contributions. As expected, both curves match below the two-pion production threshold, in the region dominated by the $\Delta$ resonance, but as the invariant mass increases, the single-pion results ($\pi$-DCC) are below the inclusive ones that also take into account effects beyond SPP as mentioned above.

Unlike the SuSAv2 model, in the RDWIA approach, the cross section describing the SPP process gives information not only about final-state leptons but also about final-state hadrons (pion and nucleon). It reads [27]:

$${\displaystyle \frac{d^8\sigma }{d{E}_{f}d{\Omega }_{f}d{E}_{\pi }d{\Omega }_{\pi }d{\Omega }_N}}=\mathcal{F}{\displaystyle \frac{k_f{E}_f{p}_N^2{E}_{\pi }{k}_{\pi }}{{\left(2\pi \right)}^8{f}_{rec}}}{l}_{\mu \nu }{h}^{\mu \nu },$$

(6)

where

$$f_{rec}={\displaystyle \frac{p_N}{E_N}}\left(1+{\displaystyle \frac{E_N}{E_{A-1}}}\left|1+{\displaystyle \frac{{\mathbf{p}}_N·({\mathbf{k}}_{\pi }-\mathbf{q})}{p_N^2}}\right|\right).$$

(7)

This expression is valid for SPP induced by electromagnetic as well as weak-neutral and weak-charged current interactions. It is written in terms of the lepton, outgoing pion and outgoing nucleon variables. The factor $\mathcal{F}$ depends on the particular process under study and includes the boson propagator and the coupling constants at the leptonic vertex. The expression involves the leptonic tensor ($l_{\mu \nu }$) and the hadronic tensor ($h^{\mu \nu }$).

The hadronic tensor is defined as

$$h^{\mu \nu }=\sum _{\kappa }\sum _{m_j,s_N}{\left[J^{\mu }(\kappa ,m_j,s_N)\right]}^{*}{J}^{\nu }(\kappa ,m_j,s_N).$$

(8)

The outer sum runs over bound states labelled by $\kappa$. The total angular momentum of the bound state is $j=\left|\kappa \right|-1/2$ and the orbital angular momentum is $l=j\pm 1/2$ for $\kappa =\pm \left|\kappa \right|$. Here $m_j$ is the third component of the angular momentum, and $s_N$ is associated with the spin projection of the outgoing nucleon. We have suppressed here and in the following the explicit dependence on the four-momenta of the exchanged boson, the nucleon and the pion. The tensor is written in terms of the matrix elements of the hadronic current ($J^{\mu }$), which can be expressed as the Fourier transform of a current density

$$J^{\mu }(\kappa ,m_j,s_N)=\int {\mathrm{d}}^3\mathbf{r}{e}^{i\mathbf{q}·\mathbf{r}}{\mathcal{J}}^{\mu }(\mathbf{r},\kappa ,m_j,s_N).$$

(9)

In the relativistic impulse approximation it has the following structure

$${\mathcal{J}}^{\mu }\left(\mathbf{r}\right)\sim {\varphi }_{\pi }^{*}\left(\mathbf{r}\right){\overline{\psi }}_N(\mathbf{r},s_N){\widehat{\mathcal{O}}}_{1\pi }^{\mu }\psi (\mathbf{r},\kappa ,m_j).$$

(10)

Here ${\psi }_N$ and $\psi$ are, respectively, scattering and bound-state nucleon wavefunctions represented by four-component spinors, ${\varphi }_{\pi }$ is the pion wavefunction, and ${\widehat{\mathcal{O}}}_{1\pi }$ is a bilinear operator. More details of the spinors and the operator can be found in [29,30,31]. In particular, the calculations for charged pion production presented here are identical to those of [31], and ${\pi }^0$ production results were included in [55].

Results in this work are obtained using the following approximations: the pion is treated as a plane wave, and the ‘local approximation’ is invoked. The latter means that momenta that enter in the operator are fixed to their asymptotic values. With these approximations the calculation of the current simplifies as explained in [31]. Recently, results without the local approximation and an analysis of the effects of invoking it were presented in [21]. The outgoing nucleon wavefunctions are obtained with the energy-dependent RMF (EDRMF) potential from [27]. In this approach, initial and final state potentials are identical for low nucleon energies, which leads naturally to Pauli blocking, and the conservation of the Dirac current. At higher nucleon kinetic energies the potentials soften, thereby avoiding unsound behavior caused by using an energy-independent potential [56].

In the following section, we compare the predictions of the models described above for charged-current neutrino- and antineutrino-induced SPP off the nucleon and on the nucleus.

3. Results

In this section, we compare the predictions of the SuSAv2-DCC and EDRMF-Hybrid models with data on charged current neutrino-induced SPP. Our analysis focuses on channels that result in a single pion in the final state, where a significant contribution arises from the $\Delta$ resonance. The SuSAv2-inelastic model with $\pi$-DCC structure functions allows to separate the different SPP channels, i.e., ${\pi }^{+}$, ${\pi }^0$, or ${\pi }^{-}$. Similarly, the SPP channels are computed separately in the EDRMF-Hybrid model.

Before comparison to nuclear target data, we present in Figure 2, the total cross sections for SPP off the nucleon for all neutrino and antineutrino channels obtained from the Hybrid and DCC models. For $W_X<1.4$ GeV, where the main contribution comes from the $\Delta$ resonance, the prediction of the DCC model is higher than that of the Hybrid model, except for the ${\pi }^0$ channel. On the other hand, without this limitation in the invariant mass, the Hybrid model predicts larger total cross sections with the exception of the ${\pi }^{+}$ channel.

In Table 1 and Table 2, the flux-folded CC1$\pi$ total cross sections for the different experiments are shown. We consider both neutrino and antineutrino fluxes for MINERvA and the neutrino fluxes for T2K and MiniBooNE to analyze the flux-folded total cross sections for different neutrino- and antineutrino-induced SPP off free nucleons. This table shows the cross sections for $W_X<1.4$ GeV, where the $\Delta$ resonance dominates, and for $W_X<1.8$ GeV, which is a common threshold used in the MINER$\nu$A analyses. In the case of MINER$\nu$A, we generally observe a notable difference between the total cross section from the DCC and Hybrid models, matching the differences shown in Figure 2 at higher energies. For T2K and MiniBooNE, the models differ significantly for neutrino-induced ${\pi }^{+}$ production, which is also observed at energies around and below 1 GeV in the curves of Figure 2 for that particular channel. The models tend to be more similar for the other pion production channels.

In the following subsections, we show the comparison between the SuSAv2 $\pi$-DCC and the EDRMF-Hybrid predictions and with CC1$\pi$ neutrino-nucleus data.

3.0.1. MiniBooNE

The MiniBooNE experiment uses mineral oil, ${\mathrm{CH}}_2$, as a target. Figure 3 shows the associated neutrino flux for CC1$\pi$ studies. The flux ranges from 0 to 3 GeV, although only values between 0.5 and 2 GeV are considered for the experimental analyses [57,58]. The flux peaks around $E_{\nu }\sim 0.5\mathrm{GeV}$.

Figure 4 presents the CC1${\pi }^0$ single-differential cross sections as a function of the muon momentum and the cosine of the muon scattering angle. In this channel, both the final-state muon and ${\pi }^0$ are detected. The SuSAv2 $\pi$-DCC model in general underestimates the data. The model prediction does not include coherent pion production, though according to [18] its contribution is negligible in this case. In a similar way, intra-nuclear cascade effects (e.g., a ${\pi }^{+}$ converting into a ${\pi }^0$), are not included in the model which can explain part of the observed underestimation [30,59] Another reason for this discrepancy can be related to the difference between DCC and Hybrid, producing the latter larger results for CC1${\pi }^0$, unlike other channels, as observed in Figure 2. However, at the relatively low energies of MiniBooNE, the differences between DCC and Hybrid are not expected to be as significant as for other higher-energy experiments, such as MINERvA, as will be discussed later. In general, the contribution given by SuSAv2 $\pi$-DCC accounts for roughly half of the total observed CC1${\pi }^0$ MiniBooNE result.

The following figures (Figure 5, Figure 6, Figure 7 and Figure 8) show results for CC1${\pi }^{+}$ cross sections as a function of the muon kinetic energy, using the same flux shown in Figure 3. In this case there is an experimental cut in the phase space, so only processes with $W^{exp}<1.35$ GeV are considered, being $W^{exp}\equiv \sqrt{m_N^2+2m_N\omega -\left|Q^2\right|}$. In Figure 5, the single-differential cross section is presented. Here, various pion production channels contribute, namely ${\pi }^{+}$ production from neutrino interactions with both neutrons and protons from carbon. The contribution of protons from the hydrogen component (${\mathrm{H}}_2$) of the target is also considered. The DCC model accounts for most of the cross section strength, but contributions from coherent and pion rescattering processes are missing. On the other hand, the EDRMF-Hybrid results are similar at these kinematics to the SuSAv2-DCC ones, being the latter approximately 5% higher, which is consistent with the results in Figure 2.

Figure 6 shows the double-differential cross section in the very forward angular region. At low muon kinetic energies, the absence of the additional coherent and pion rescattering effects could lead to a more pronounced underestimation of the data [18]. These discrepancies between theory and data diminish as the muon kinetic energy increases, observing a good agreement with data for the EDRMF model at large $T_{\mu }$ values and a slight overestimation when compared with SuSAv2.

In Figure 7 and Figure 8, we present the charged-current single ${\pi }^{+}$ (CC1${\pi }^{+}$) double-differential cross section as a function of the cosine of the muon scattering angle, averaged over bins of muon kinetic energy. In general, we observe good agreement with data for both models except at very forward angles and low-intermediate $T_{\mu }$ where the models underestimate the data; at large $T_{\mu }$ the data are overestimated. The agreement with data is slightly better for the EDRMF-Hybrid.

3.0.2. MINERvA

In this experiment, the target material is hydrocarbon ($\mathrm{CH}$). As shown in Figure 9, the flux for the different pion production channels peaks at approximately 3.5 GeV and extends up to 20 GeV, although it becomes negligible above 7 GeV. This energy range is significantly larger than that of the MiniBooNE experiment discussed previously.

In Figure 10, the MINERvA flux-folded CC1$\pi$ single-differential cross sections in terms of the muon momentum and scattering angle for different neutrino- and antineutrino-induced pion production reactions on single-nucleons are analyzed to compare differences between DCC and Hybrid before including these approached in the corresponding nuclear models. We observe larger cross sections for DCC in the ${\pi }^{+}$ case, while for the ${\pi }^{-}$ and ${\pi }^0$ channels the Hybrid provides larger results. These analyses are useful to understand the following results, when the models are compared with data for pion production on the nucleus. Note that for the results shown in Figure 10 and in Figure 11, Figure 12 and Figure 13, we have considered the following experimental restrictions: for ${\pi }^{+}$, 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.4 GeV; for antineutrino ${\pi }^0$, 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.8 GeV; for neutrino ${\pi }^0$, 1.5 $<E_{\nu }/$GeV < 20, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV; and for ${\pi }^{-}$, 1.5 $<E_{\nu }/$GeV < 10, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV. The restrictions to relatively low values of the invariant mass imply a more prominent contribution of the $\Delta$ resonance, while the restrictions to forward angles imply focusing, on average, on smaller values of the transferred energy and momentum, where nuclear effects are more important.

In Figure 11, we show results for CC neutrino-induced ${\pi }^{+}$ and antineutrino-induced ${\pi }^0$ production. For the neutrino CC1${\pi }^{+}$ channel, two experimental datasets are shown: one corresponding to single-pion events and another that includes events with one or more pions ($n\pi$). We observe a general good agreement with data for this channel using both models, though some overestimation is obtained for scattering angles between 5 and 10 deg.

In the antineutrino CC1${\pi }^0$ channel, the SuSAv2-DCC results exhibit a similar lack of strength as observed for MiniBooNE in Figure 4, producing the EDRMF-Hybrid approach better agreement with data. This is in accordance with the DCC vs. Hybrid differences observed in Figure 2 and Figure 10 for the free-nucleon case.

In the neutrino CC1${\pi }^0$ channel, Figure 12, we observe an important underestimation of the data. In this case, also the EDRMF-Hybrid model underpredict the data. As mentioned for MiniBooNE, Figure 4, other processes not included in this analysis, such as pion rescattering effects, could help to explain these discrepancies. Nevertheless, it is worth mentioning that, for the antineutrino case, the EDRMF-Hybrid model produces a reasonable agreement with data at these kinematics. As discussed in [55], this is peculiar since ¹²C is a symmetric nucleus. Based on isospin symmetry, the leading order difference between the neutrino and antineutrino case is then the change of sign of the vector-axial interference contribution. The inconsistency in the description of these data may hence point to large uncertainties in the axial nucleon-to-resonance transition form factors in the second resonance region [14,63].

Finally, for the antineutrino CC1${\pi }^{-}$ channel, Figure 13, we observe that the EDRMF-Hybrid agrees with the data reasonably well while SuSAv2-DCC underestimate them. It can be ascribed to the differences between DCC and Hybrid shown in Figure 2 and Figure 10 for the free-nucleon case.

In general, when comparing the ratios between DCC and Hybrid results in Figure 10 for scattering off single-nucleons at MINERvA kinematics with the SuSAv2-DCC and EDRMF-Hybrid ones in Figure 11, Figure 12 and Figure 13, the cross section per nucleon for a bound nucleon compared to a free one is reduced in both models. This reduction is found to be stronger in the SuSAv2 model than in the EDRMF. ¹

3.0.3. T2K

In this section, the results for CC1${\pi }^{+}$ at T2K kinematics are shown. The target of the T2K experiment is $C_8{H}_8$. The normalized neutrino flux is shown in Figure 14. It peaks at around 0.6 GeV, covering an energy range similar to the MiniBooNE flux (Sect. 3.0.1). In this case, the restriction $W^{exp}<2.0$ GeV is applied, so contributions from processes beyond the $\Delta$ region are not removed as was the case for the MiniBooNE data. However, the main contribution to the ${\pi }^{+}$ production cross section remains the delta resonance, especially since the T2K flux peaks at relatively low energy.

Figure 10. MINER$\nu$A flux-folded CC1$\pi$ single-differential cross sections in terms of the muon momentum (top panels) and scattering angle (bottom panels) for different neutrino- and antineutrino-induced pion production scattering off single nucleons. (Left) ${\pi }^{+}$ processes for 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.4 GeV; (Center Top) $\overline{\nu }{\pi }^0$ processes for 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.8 GeV; (Center Bottom) $\nu {\pi }^0$ processes for 1.5 $<E_{\nu }/$GeV < 20, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV; and (Right) ${\pi }^{-}$ processes for 1.5 $<E_{\nu }/$GeV < 10, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV.

Figure 10. MINER$\nu$A flux-folded CC1$\pi$ single-differential cross sections in terms of the muon momentum (top panels) and scattering angle (bottom panels) for different neutrino- and antineutrino-induced pion production scattering off single nucleons. (Left) ${\pi }^{+}$ processes for 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.4 GeV; (Center Top) $\overline{\nu }{\pi }^0$ processes for 1.5 $<E_{\nu }/$GeV < 10 and $W^{exp}<$ 1.8 GeV; (Center Bottom) $\nu {\pi }^0$ processes for 1.5 $<E_{\nu }/$GeV < 20, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV; and (Right) ${\pi }^{-}$ processes for 1.5 $<E_{\nu }/$GeV < 10, ${\theta }_{\mu }<$ 25 deg. and $W^{exp}<$ 1.8 GeV.

Figure 11. MINERvA flux-averaged CC${\pi }^{+}$${\nu }_{\mu }$-CH (left) and CC1${\pi }^0$${\overline{\nu }}_{\mu }$-CH (right) differential cross sections as a function of the muon momentum (top) and the scattering angle (bottom) for multiple- (black data) and single-pion production (red data). The kinematical restrictions are (left) $W^{exp}<$ 1.4 GeV, (right) $W^{exp}<$ 1.8 GeV and 1.5 $<E_{\nu }/$GeV < 10 for both cases. The experimental data points are taken from [60,64] .

Figure 11. MINERvA flux-averaged CC${\pi }^{+}$${\nu }_{\mu }$-CH (left) and CC1${\pi }^0$${\overline{\nu }}_{\mu }$-CH (right) differential cross sections as a function of the muon momentum (top) and the scattering angle (bottom) for multiple- (black data) and single-pion production (red data). The kinematical restrictions are (left) $W^{exp}<$ 1.4 GeV, (right) $W^{exp}<$ 1.8 GeV and 1.5 $<E_{\nu }/$GeV < 10 for both cases. The experimental data points are taken from [60,64] .

Figure 12. MINERvA flux-averaged CC1${\pi }^0$${\nu }_{\mu }$-CH (left) differential cross sections as a function of the muon momentum (left) and the scattering angle (right). The kinematical restrictions are 1.5 $<E_{\nu }/$GeV< 20, ${\theta }_{\mu }<25$ deg. and $W^{exp}<$ 1.8 GeV. The experimental data points are taken from [61].

Figure 12. MINERvA flux-averaged CC1${\pi }^0$${\nu }_{\mu }$-CH (left) differential cross sections as a function of the muon momentum (left) and the scattering angle (right). The kinematical restrictions are 1.5 $<E_{\nu }/$GeV< 20, ${\theta }_{\mu }<25$ deg. and $W^{exp}<$ 1.8 GeV. The experimental data points are taken from [61].

Figure 13. MINERvA flux-averaged CC1${\pi }^{-}$${\overline{\nu }}_{\mu }$-CH (left) differential cross sections as a function of the muon momentum (left) and the scattering angle (right). The kinematical restrictions are 1.5 $<E_{\nu }/$ GeV < 10, ${\theta }_{\mu }<25$ deg. and $W^{exp}<$ 1.8 GeV. The experimental data points are taken from [62].

Figure 13. MINERvA flux-averaged CC1${\pi }^{-}$${\overline{\nu }}_{\mu }$-CH (left) differential cross sections as a function of the muon momentum (left) and the scattering angle (right). The kinematical restrictions are 1.5 $<E_{\nu }/$ GeV < 10, ${\theta }_{\mu }<25$ deg. and $W^{exp}<$ 1.8 GeV. The experimental data points are taken from [62].

Figure 14. Normalized neutrino flux from T2K [65].

Figure 14. Normalized neutrino flux from T2K [65].

Figure 15 shows the flux-averaged ${\nu }_{\mu }$ CC1${\pi }^{+}$ double-differential cross section in bins of the muon scattering angle. The results are consistent with those shown for MiniBooNE. The predictions from EDRMF and SuSAv2 are close to the data and similar between them, with EDRMF producing smaller cross sections in general, which is consistent with the DCC and Hybrid differences observed in Figure 2. It is worth mentioning that, as in the previous cases, coherent pion production and pion rescattering effects are not taken into account by the models.

4. Conclusions

This paper presents an extensive comparison of two different theoretical approaches with neutrino- and antineutrino-nucleus CC1$\pi$ cross-section measurements. The single-pion production off the nucleon is described with the DCC [32,33,34,35,36] and the Hybrid [29,30] models. The DCC is incorporated into the SuSAv2-inelastic nuclear framework [37,52,53], while the Hybrid is integrated into an RDWIA approach; in particular, we use the EDRMF model to treat the distortion of the final nucleon [27], while the pion is described as a plane wave.

In general, we found significant discrepancies between the two model predictions for all channels. In spite of the level of agreement observed in some figures, none of the models is able to provide a satisfactory description of all datasets.

When comparing our results with those of other theoretical approaches [18,30] and Monte Carlo event generators [59,60,61,62], we find similar pion-production contributions to other theoretical models, while generators predict larger cross sections for ${\pi }^0$ production results. In this context, the Monte Carlo intranuclear cascade effects (mainly pion rescattering effects in the nuclear medium, leading to pion charge exchanged) could improve our agreement with data, as observed in the Supplemental Material of [59], where these final-state interactions increase the CC1${\pi }^0$ results. Note also the important discrepancies observed in this work for neutrino and antineutrino reactions in the ${\pi }^0$ channel between models and data, which may point to the importance of improvements in the modeling of axial contributions for CC1$\pi$ interactions.

It is interesting that variations at the nucleon level, i.e., differences between DCC and Hybrid [Figure 2 and Figure 10], are generally more relevant than those introduced by the nuclear model, i.e. differences due to the use of SuSAv2 or EDRMF. A slight general increase of the EDRMF results with respect to the SuSAv2 ones has also been observed, in particular at MINERvA kinematics. However, the ratio of cross sections per bound nucleon compared to the free nucleon is generally slightly smaller for SuSAv2 compared to the EDRMF, in particular in MINERvA kinematics.

The limits $Q^2<3$ GeV²$/c^2$ and $W_x<2.1$ GeV in the DCC model have little impact on the results presented here due to the experimental constraints. Nevertheless, it could be explored in the future via detailed studies with higher-energy neutrino experiments. Other recent and forthcoming datasets from different experiments, such as NOvA, T2K, MINERvA, and MicroBooNE [66,67,68,69], or on different targets, such as water [70] or argon [59], will also be analyzed in future works.

On the other hand, the recent implementation of superscaling and RDWIA models in neutrino event generators [71,72,73,74], such as GENIE or NEUT, mostly done for the QE and 2p2h regimes, will be extended in the future for the pion production and the full inelastic regime. This could allow us to apply intranuclear cascade effects (e.g., rescattering processes that can convert a ${\pi }^{+}$ into a ${\pi }^0$) to these analyses, with the possibility of having a more accurate description of final-state interaction effects, thus improving the agreement with data. The addition of other approaches for the single-nucleon inelastic structure functions, such as the MK model [39] will also be considered. Finally, the use of different nuclear potentials in the RDWIA framework will also be explored in further studies in a similar way to previous works [56,75], also bearing in mind more exclusive processes, i.e. cross-section measurements in terms of the kinematics of final-state pions and nucleons, which can improve nuclear model selection.