Dynamical coupled-channels model for neutrino-induced meson productions in resonance region

A dynamical coupled-channels (DCC) model for neutrino-nucleon reactions in the resonance region is developed. Starting from the DCC model that we have previously developed through an analysis of $\pi N,\gamma N\to \pi N,\eta N,K\mathrm{\Lambda },K\mathrm{\Sigma }$ reaction data for $W\le 2.1\mathrm{GeV}$, we extend the model of the vector current to $Q^2\le 3.0(\mathrm{GeV}/c{)}^2$ by analyzing electron-induced reaction data for both proton and neutron targets. We derive axial-current matrix elements that are related to the $\pi N$ interactions of the DCC model through the partially conserved axial current (PCAC) relation. Consequently, the interference pattern between resonant and nonresonant amplitudes is uniquely determined. We calculate cross sections for neutrino-induced meson productions, and compare them with available data. Our result for the single-pion production reasonably agrees with the data. We also make a comparison with the double-pion production data. Our model is the first DCC model that can give the double-pion production cross sections in the resonance region. We also make comparison of our result with other existing models to reveal an importance of testing the models in the light of PCAC and electron reaction data. The DCC model developed here will be a useful input for constructing a neutrino-nucleus reaction model and a neutrino event generator for analyses of neutrino experiments.

Figure 1

Momentum variables and scattering angles for a neutrino-induced single meson production reaction.

Figure 2

Diagrammatic representation of the Lippmann-Schwinger equation. The corresponding equation numbers in the text are indicated on the right. The symbols $T$, $t$, and $V$ in the figures correspond to $T^{(\pm )}(W)$ and $t_{\text{nonres}}^{(\pm )}(W)$ in Eq. (21), and $V^{(\pm )}(W)$ in Eq. (23), respectively. The third line represents the dressed $N^{*}$ decay vertex, $⟨{\chi }^{(-)}|\mathrm{\Gamma }|N^{*}⟩$, contained in Eq. (26). The double lines represent $N^{*}$-propagators and the fourth line corresponds to Eq. (27).

Figure 3

Kinematical region covered by available single pion electroproduction data from the CLAS Collaboration [43, 44, 45, 46, 47, 48, 49, 50]. The red triangle [blue cross] points indicate the kinematical points where data for $p(e,e^{\prime }{\pi }^0)p$ [$p(e,e^{\prime }{\pi }^{+})n$] are available. At the green square points, both $p(e,e^{\prime }{\pi }^0)p$ and $p(e,e^{\prime }{\pi }^{+})n$ data are available.

Figure 4

The virtual photon cross section $d{\sigma }_T/d{\mathrm{\Omega }}_{\pi }^{*}+\epsilon d{\sigma }_L/d{\mathrm{\Omega }}_{\pi }^{*}$ ($\mu \mathrm{b}/\mathrm{sr}$) at $Q^2=0.40(\mathrm{GeV}/c{)}^2$ for $p(e,e^{\prime }{\pi }^0)p$ (left) and $p(e,e^{\prime }{\pi }^{+})n$ (right) from the DCC model. The number in each panel indicates $W$ (MeV). The data are from Ref. [43] for $p(e,e^{\prime }{\pi }^0)p$, and Ref. [45] for $p(e,e^{\prime }{\pi }^{+})n$.

Figure 5

The virtual photon cross section $d{\sigma }_T/d{\mathrm{\Omega }}_{\pi }^{*}+\epsilon d{\sigma }_L/d{\mathrm{\Omega }}_{\pi }^{*}$ ($\mu \mathrm{b}/\mathrm{sr}$) at $Q^2=1.76(\mathrm{GeV}/c{)}^2$. The data are in the range, $1.72\le Q^2\le 1.80(\mathrm{GeV}/c{)}^2$, and are from Ref. [43] for $p(e,e^{\prime }{\pi }^0)p$, and Refs. [46, 47] for $p(e,e^{\prime }{\pi }^{+})n$. The other features are the same as those in Fig. 4.

Figure 6

The virtual photon cross section $d{\sigma }_T/d{\mathrm{\Omega }}_{\pi }^{*}+\epsilon d{\sigma }_L/d{\mathrm{\Omega }}_{\pi }^{*}$ ($\mu \mathrm{b}/\mathrm{sr}$) at $Q^2=2.95(\mathrm{GeV}/c{)}^2$. The data are in the range, $2.91\le Q^2\le 3.00(\mathrm{GeV}/c{)}^2$, and are from Ref. [48] for $p(e,e^{\prime }{\pi }^0)p$ and Refs. [46, 47] for $p(e,e^{\prime }{\pi }^{+})n$. The other features are the same as those in Fig. 4.

Figure 7

Comparison of DCC-based calculation with data for inclusive electron-proton scattering at $E_e=5.498\mathrm{GeV}$. The red solid curves are for inclusive cross sections while the magenta dashed-curves are for contributions from the $\pi N$ final states. The range of $Q^2$ and the electron scattering angle (${\theta }_{e^{\prime }}$) are indicated in each panel. The data are from Ref. [54].

Figure 8

Unpolarized differential cross sections, $d\sigma /d{\mathrm{\Omega }}_{\pi }^{*}$ ($\mu \mathrm{b}/\mathrm{sr}$), for $\gamma n\to {\pi }^{-}p$. The data are from Refs. [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78].

Figure 9

Unpolarized differential cross sections $d\sigma /d{\mathrm{\Omega }}_{\pi }^{*}$ ($\mu \mathrm{b}/\mathrm{sr}$), for $\gamma n\to {\pi }^{0}n$. The data are from Refs. [79, 80, 81, 82].

Figure 10

Comparison of DCC-based calculation with data for differential cross sections per nucleon of inclusive electron-deuteron scattering at $E_e=4.628\mathrm{GeV}$. The red solid curves are for inclusive cross sections while the magenta dashed-curves are for contributions from the $\pi N$ final states. The range of $Q^2$ and the electron scattering angle (${\theta }_{e^{\prime }}$) are indicated in each panel. The data are from Ref. [88].

Figure 11

The structure function $F_2^{\mathrm{CC}}$ at $Q^2=0$. $F_2^{\mathrm{CC}}$ calculated with the axial current amplitude (lines) are compared with those from $\pi N$ cross sections via the PCAC relation (points). The upper (lower) line and point are for inclusive ($\pi N$) final state(s). The left (right) panel is for CC $\nu \text{−}\mathrm{proton}/\mathrm{CC}$ $\overline{\nu }$-neutron (CC $\nu \text{−}\mathrm{neutron}/\mathrm{CC}$ $\overline{\nu }$-proton) process.

Figure 12

Total cross sections for the CC ${\nu }_{\mu }p$ (left) and ${\nu }_{\mu }n$ (right) reactions.

Figure 13

The same as Fig. 12 but in log scale.

Figure 14

Various mechanisms contributing to ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}p$ (left) and ${\nu }_{\mu }n\to {\mu }^{-}\pi N$ (right).

Figure 15

Various mechanisms contributing to ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}{\pi }^{0}p$ (left) and ${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{+}{\pi }^{-}p$ (right).

Figure 16

Comparison of the DCC-based calculation (red solid curves) with data for ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}p$ (left), ${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{0}p$ (middle), and ${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{+}n$ (right). The DCC calculation with $0.8\times g_{AN\mathrm{\Delta }(1232)}^{\mathrm{PCAC}}$ is also shown (magenta dashed curve). ANL (BNL) data are from Ref. [12] ([13]).

Figure 17

Left: DCC-based calculation for NC neutrino-induced single pion productions. Middle: Comparison of the DCC model and ANL data [90] for $\nu n\to \nu p{\pi }^{-}$. Right: Comparison of the DCC model and BNL data [91] for $\nu n\to {\mu }^{-}{K}^{+}\mathrm{\Lambda }$.

Figure 18

The flux-averaged ($0.5\mathrm{GeV}\le E_{\nu }\le 6\mathrm{GeV}$) $d\sigma /d{Q}^2$ for ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}p$. Left: The DCC-based calculation (red solid curve) is compared with the ANL data [12]. The DCC calculation with $0.8\times g_{AN\mathrm{\Delta }(1232)}^{\mathrm{PCAC}}$ is also shown (magenta dashed curve). Contributions from high $W$ ($>1.4\mathrm{GeV}$) are cut in the calculation in order to be consistent with the data. Right: The DCC-based calculation (red solid curve) is compared with the BNL data [13].

Figure 19

Comparison of the DCC-based calculation with data for ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}{\pi }^{0}p$ (left), ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}{\pi }^{+}n$ (middle), and ${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{+}{\pi }^{-}p$ (right). ANL (BNL) data are from Ref. [93] ([13]).

Figure 20

Contour plots of $d^2\sigma /dWd{Q}^2$ for ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}p$ (left) and ${\nu }_{\mu }n\to {\mu }^{-}\pi N$ (right) at $E_{\nu }=2\mathrm{GeV}$.

Figure 21

Contour plots of $d^2\sigma /dWd{Q}^2$ for ${\nu }_{\mu }p\to {\mu }^{-}{\pi }^{+}{\pi }^{0}p$ (left) and ${\nu }_{\mu }n\to {\mu }^{-}{\pi }^{+}{\pi }^{-}p$ (right) at $E_{\nu }=2\mathrm{GeV}$.

Figure 22

$F_2^{\mathrm{CC}}$ at $Q^2=0$ from contribution of the single pion production only. The DCC model is compared with the LPP model due to Lalakulich et al. [18] and the Rein-Sehgal (RS) model [35, 36]. The left (right) panel is for the CC ${\nu }_{\mu }p$ (${\nu }_{\mu }n$) reaction.

Figure 23

The inclusive structure function $F_2^{\mathrm{CC}}$ at $Q^2=0$. The other features are the same as those in Fig. 22.