Dynamical model of coherent pion production in neutrino-nucleus scattering

We study coherent pion production in neutrino-nucleus scattering in the energy region relevant to neutrino oscillation experiments of current interest. Our approach is based on the combined use of the Sato-Lee model of electroweak pion production on a nucleon and the $\Delta$-hole model of pion-nucleus reactions. Thus we develop a model that describes pion-nucleus scattering and electroweak coherent pion production in a unified manner. Numerical calculations are carried out for the case of the ${}^{12}\mathrm{C}$ target. All the free parameters in our model are fixed by fitting to both total and elastic differential cross sections for $\pi$-${}^{12}\mathrm{C}$ scattering. Then we demonstrate the reliability of our approach by comparing our prediction for coherent pion photoproductions with the data. Finally, we calculate total and differential cross sections for neutrino-induced coherent pion production, and some of the results are compared with recent data from K2K, SciBooNE, and MiniBooNE. We also study the effect of nonlocality of $\Delta$ propagation in the nucleus and compare the elementary amplitudes used in different microscopic calculations.

Figure 1

Total cross sections for ${\pi }^{-}$-${}^{12}\mathrm{C}$ scattering. The solid curve was obtained with our full calculation, while the dashed curve was obtained without the spreading potential. Data are from Ref. [35].

Figure 2

$\pi -{}^{12}$C elastic differential cross sections. The pion kinetic energy in the laboratory frame is indicated in each figure. The solid curve is obtained with our full calculation, while the dashed curve was obtained without the phenomenological terms in Eq. (54). Data are from Ref. [36] for (a), Ref. [37] for (b), and Ref. [35] for (c)–(f).

Figure 3

Differential cross sections for $\gamma +{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ for different incident photon energies (indicated in each panel). Solid lines represent the results of the full calculation. Dashed lines were obtained without the FSI and without the medium effects on the $\Delta$-propagation, while dotted lines were obtained with the medium effects on the $\Delta$ included. Dash-dotted curves correspond to a case in which the pion production operator includes only the $\Delta$ mechanism. For more detailed explanations for the different cases, see the text. Data are from Ref. [38] for (a) and from Ref. [39] for (b)–(d).

Figure 4

$E_{\nu }$ dependence of the total cross section for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ (solid line), $\nu +{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \nu +{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ (dashed line), ${\mathrm{\nu ̄}}_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{+}+{\pi }^{-}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ (dotted line), and $\mathrm{\nu ̄}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \mathrm{\nu ̄}+{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ (dash-dotted line).

Figure 5

Pion momentum distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV; $p_{\pi }$ is the pion momentum in the laboratory frame. Solid, dashed, dotted, and dash-dotted lines follow the same convention as in Fig. 3.

Figure 6

Same as Fig. 5 but for $E_{\nu }=0.5$ GeV.

Figure 7

$Q^2$ spectrum for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 8

Same as Fig. 7 but for the NC process at $E_{\nu }=1$ GeV.

Figure 9

Contributions from the axial-vector (solid) and vector (dashed) currents for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 10

Effect of the nonlocality of $\Delta$ propagation for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$. The ratio $\mathcal{R}$ of the total cross sections was obtained with and without taking account of the nonlocal effect.

Figure 11

The $\cos {\theta }_{\pi }$ distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ and ${\mathrm{\nu ̄}}_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{+}+{\pi }^{-}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV. The position of ${\theta }_{\pi }={35}^{°}$ is indicated by the arrow.

Figure 12

$Q_{\mathrm{rec}}^2$ distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 13

Pion momentum distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 14

Muon momentum distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 15

Muon scattering angle distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV.

Figure 16

Coplanar angle difference ($\Delta \varphi$) distribution for ${\nu }_{\mu }+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to {\mu }^{-}+{\pi }^{+}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ at $E_{\nu }=1$ GeV. The definition of $\Delta \varphi$ is given in Fig. 17.

Figure 17

Graphical definition for the coplanar angle difference ($\Delta \varphi$).

Figure 18

Flux-convoluted $\eta$ distribution for $\nu +{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \nu +{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$ obtained in our full calculation. The neutrino flux is taken from MiniBooNE [41]. Also shown is the Monte Carlo result from MiniBooNE [3] rescaled to match our result at $\eta =0.005$ GeV.

Figure 19

Flux-convoluted ${\pi }^0$-momentum distribution for $\nu +{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \nu +{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$. The neutrino flux is taken from MiniBooNE [41].

Figure 20

Flux-convoluted $\eta$ distribution for $\mathrm{\nu ̄}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \mathrm{\nu ̄}+{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$. The antineutrino flux is taken from MiniBooNE [41].

Figure 21

Flux-convoluted ${\pi }^0$-momentum distribution for $\mathrm{\nu ̄}+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}\to \mathrm{\nu ̄}+{\pi }^0+{}^{12}{\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$. The antineutrino flux is taken from MiniBooNE [41].