Dynamical coupled-channels study of $\pi N\to \pi \pi N$ reactions

As a step toward performing a complete coupled-channels analysis of the world data of $\pi N,{\gamma }^{*}N\to \pi N,\eta N,\pi \pi N$ reactions, the $\pi N\to \pi \pi N$ reactions are investigated starting with the dynamical coupled-channels model developed in Phys. Rev. C 76, 065201 (2007). The channels included are $\pi N,\eta N$, and $\pi \pi N$ which has $\pi \Delta ,\rho N$, and $\sigma N$ resonant components. The nonresonant amplitudes are generated from solving a set of coupled-channels equations with the meson-baryon potentials defined by effective Lagrangians. The resonant amplitudes are generated from 16 bare excited nucleon $(N^{*})$ states that are dressed by the nonresonant interactions as constrained by the unitarity condition. The data of total cross sections and $\pi N$ and $\pi \pi$ invariant mass distributions of ${\pi }^{+}p\to {\pi }^{+}{\pi }^{+}n,{\pi }^{+}{\pi }^{0}p$ and ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n,{\pi }^{-}{\pi }^{0}p,{\pi }^0{\pi }^{0}n$ reactions from threshold to the invariant mass $W=2$ GeV can be described to a very large extent. We show the importance of the coupled-channels effects and the strong interference among the contributions from the $\pi \Delta ,\sigma N$, and $\rho N$ channels. The large interference between the resonant and nonresonant amplitudes is also demonstrated. Possible future developments are discussed.

Figure 1

Graphical representations of $T_{\pi N,\pi \pi N}$ of Eqs. (3, 4, 5, 6, 7).

Figure 2

The considered $v_{\pi N,\pi \pi N}$.

Figure 3

The total cross sections predicted (solid curves) from the JLMS model compared with the data [31]. The dotted curves are from turning off the amplitude $T_{\pi N,\pi \pi N}^{\mathrm{dir}}$.

Figure 4

The total cross sections predicted (solid curves) from the JLMS model compared with the data [31]. The dotted curves are from turning off the amplitude $T_{\pi N,\pi \pi N}^{\mathrm{dir}}$.

Figure 5

The coupled-channels effects on $\pi N\to \pi \pi N$ reactions. The solid curves are from full calculations, the dotted curves are from keeping only $M^{\text{'}}{B}^{\text{'}}=\mathit{MB}$ in Eq. (36) and in ${\overline{\Gamma }}_{N^{*}\to \mathit{MB}}$ of Eqs. (12) and (13), and the dashed curves are from setting $t_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}=v_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}$. The data are from Ref. [31].

Figure 6

Coupled-channels effects on the invariant mass distributions of ${\pi }^{+}p\to \pi \pi N$ at $W=1.79$ GeV. The solid curves are from full calculations, the dotted curves are from keeping only $M^{\text{'}}{B}^{\text{'}}=\mathit{MB}$ in Eq. (36) and in ${\overline{\Gamma }}_{N^{*}\to \mathit{MB}}$ of Eqs. (12) and (13), and the dashed curves are from setting $t_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}=v_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}$. The data are from Arndt [30], with magnitudes determined by normalizing them to the total $\pi N\to \pi \pi N$ cross sections listed in Ref. [2].

Figure 7

Coupled-channels effects on the invariant mass distributions of ${\pi }^{-}p\to \pi \pi N$ at $W=1.79$ GeV. The solid curves are from full calculations, the dotted curves are from keeping only $M^{\text{'}}{B}^{\text{'}}=\mathit{MB}$ in Eq. (36) and in ${\overline{\Gamma }}_{N^{*}\to \mathit{MB}}$ of Eqs. (12) and (13), and the dashed curves are from setting $t_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}=v_{\mathit{MB},M^{\text{'}}{B}^{\text{'}}}$. The data are from Arndt [30], with magnitudes determined by normalizing them to the total $\pi N\to \pi \pi N$ cross sections listed in Ref. [2].

Figure 8

Contributions from $\pi \Delta$ (dashed) and $\rho N$ (dot-dashed) channels to the invariant mass distributions of ${\pi }^{+}p\to \pi \pi N$ at $W=1.79$ GeV. The data are from Arndt [30], with magnitudes determined by normalizing them to the total $\pi N\to \pi \pi N$ cross sections listed in Ref. [2].

Figure 9

Contributions from $\pi \Delta$ (dashed), $\sigma N$ (dotted), and $\rho N$ (dot-dashed) channels to the invariant mass distributions of ${\pi }^{-}p\to \pi \pi N$ at $W=1.79$ GeV. The data are from Arndt [30], with magnitudes determined by normalizing them to the total $\pi N\to \pi \pi N$ cross sections listed in Ref. [2].

Figure 10

The invariant mass distributions of ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ at $W=1.44$, 1.60, and 1.79 GeV. The solid curves are the full calculations, the dashed curves are from nonresonant amplitudes, and the dotted curves are phase-space normalized to the data. The data are from Arndt [30], with magnitudes determined by normalizing them to the total $\pi N\to \pi \pi N$ cross sections listed in Ref. [2].

Figure 11

The invariant mass distributions of ${\pi }^{-}p\to {\pi }^0{\pi }^{0}n$ at $W=1.40$ GeV. The solid curve is the full calculation, the dashed (dot-dashed) curve is from keeping only the nonresonant (resonant) amplitude in the calculations, and the dotted curve is phase-space normalized to the data. The data are from Ref. [36]. (The data are transformed to $d\sigma /\mathit{dM}$ by using the relation $d\sigma /\mathit{dM}=2Md\sigma /{\mathit{dM}}^2$.)

Figure 12

Total cross sections of ${\pi }^{+}p$ (left) and ${\pi }^{-}p$ (right) reactions calculated using the optical theorem ${\sigma }_{\mathrm{opt}}^{(\mathrm{tot})}=(2\pi /k)F_{\pi N,\pi N}(\theta =0)$ (solid curves), ${\sigma }^{(\mathrm{tot})}={\sigma }_{\pi N,\pi N}+{\sigma }_{\pi N,\pi \eta }+{\sigma }_{\pi N,\pi \pi N}$ (with no $T_{\pi N,\pi \pi N}^{\mathrm{dir}}$) (dotted curves), and ${\sigma }^{(\mathrm{tot})}={\sigma }_{\pi N,\pi N}+{\sigma }_{\pi N,\pi \eta }+{\sigma }_{\pi N,\pi \pi N}$ (dashed curves). Solid and dotted curves are not distinguishable. For a clear comparison of the curves, only a few data points are shown. The data are from Refs. [29, 37].

Figure 13

Graphical illustration of Eq. (A1) for calculating the Δ self-energy in the $\pi \Delta$ propagator.