Dynamical coupled-channels model of $\pi N$ scattering in the $W⩽2$ GeV nucleon resonance region

As a first step to analyze the electromagnetic meson production reactions in the nucleon resonance region, the parameters of the hadronic interactions of a dynamical coupled-channels model, developed in Physics Reports 439, 193 (2007), are determined by fitting the $\pi N$-scattering data. The channels included in the calculations are $\pi N,\eta N$, and $\pi \pi N$, which has $\pi \Delta ,\rho N$, and $\sigma N$ resonant components. The nonresonant meson-baryon interactions of the model are derived from a set of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states in each of $S,P,D$, and $F$ partial waves are included to generate the resonant amplitudes in the fits. The parameters of the model are first determined by fitting as much as possible the empirical $\pi N$ elastic-scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by directly comparing the predicted differential cross section and target polarization asymmetry with the original data of the elastic ${\pi }^{\pm }p\to {\pi }^{\pm }p$ and charge-exchange ${\pi }^{-}p\to {\pi }^{0}n$ processes. The predicted total cross sections of $\pi N$ reactions and $\pi N\to \eta N$ reactions are also in good agreement with the data. Applications of the constructed model in analyzing the electromagnetic meson production data as well as the future developments are discussed.

Figure 1

Graphical representation of Eqs. (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21).

Figure 2

Mechanisms for $v_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}$ of Eq. (7): $v^s$ direct $s$ channel, $v^u$ crossed $u$ channel, $v^t$ one-particle-exchange $t$ channel, $v^c$ contact interactions.

Figure 3

One-particle-exchange interactions $Z_{\pi \Delta ,\pi \Delta }^{(E)}(E),Z_{\rho N,\pi \Delta }^{(E)}$, and $Z_{\sigma N,\pi \Delta }^{(E)}$ of Eq. (7).

Figure 4

Real parts of the calculated $\pi N$ partial-wave amplitudes [Eq. (5)] of isospin $T=1/2$ are compared with the energy independent solutions of Ref. [33].

Figure 5

Imaginary parts of the calculated $\pi N$ partial wave amplitudes [Eq. (5)] of isospin $T=1/2$ are compared with the energy independent solutions of Ref. [33].

Figure 6

Real parts of the calculated $\pi N$ partial-wave amplitudes [Eq. (5)] of isospin $T=3/2$ are compared with the energy independent solutions of Ref. [33].

Figure 7

Imaginary parts of the calculated $\pi N$ partial-wave amplitudes [Eq. (5)] of isospin $T=3/2$ are compared with the energy independent solutions of Ref. [33].

Figure 8

Differential cross section for several different center-of-mass energies. Solid red curve corresponds to our model, whereas blue dashed lines correspond to the SP06 solution of SAID [33]. All data have been obtained through the SAID online applications. Ref. [33].

Figure 9

Differential cross section for several different center-of-mass energies. Similar description as for Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].

Figure 10

Target polarization asymmetry, $P$, for several different center-of-mass energies. Similar description as for Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].

Figure 11

Target polarization asymmetry, $P$, for several different center-of-mass energies. Description as for Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].

Figure 12

Target polarization asymmetry, $P$, for several different center-of-mass energies. Description as for Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].

Figure 13

The predicted total cross sections of the ${\pi }^{+}p\to X$ (solid curve) and ${\pi }^{+}p\to {\pi }^{+}p$ (dashed curve) reactions are compared with the data. Squares and triangles are the corresponding data from Ref. [40].

Figure 14

(Left) The predicted total cross sections of the ${\pi }^{-}p\to X$ (solid curve) and ${\pi }^{-}p\to {\pi }^{-}p+{\pi }^{0}n$ (dashed curve) reactions are compared with the data. Open squares are the data on ${\pi }^{-}p\to X$ from Ref. [40], open triangles are obtained by adding the ${\pi }^{-}p\to {\pi }^{-}p$ and ${\pi }^{-}p\to {\pi }^{0}n$ data obtained from Ref. [40] and SAID database [41], respectively. (Right) Show how the predicted contributions from each channel are added up to the predicted total cross sections of the ${\pi }^{-}p\to X$.

Figure 15

The predicted total cross sections of ${\pi }^{-}p\to \eta n$ reaction, see text for details, are compared with the data [42, 43].