Quark model study of the $\pi N\to \pi N$ reactions up to the $N\left(1440\right)$ resonance region

A combined analysis of the reactions ${\pi }^{+}p\to {\pi }^{+}p, {\pi }^{-}p\to {\pi }^{-}p$, and ${\pi }^{-}p\to {\pi }^{0}n$ is carried out with a chiral quark model. The observations are reasonably described from the $\mathrm{\Delta }\left(1232\right)$ resonance region up to the $N\left(1440\right)$ resonance region. Besides the $\mathrm{\Delta }\left(1232\right)P_{33}$, a confirmed role of $N\left(1440\right)P_{11}$ is found in the polarizations of the ${\pi }^{-}p\to {\pi }^{-}p$ and ${\pi }^{-}p\to {\pi }^{0}n$ reactions. It is found that the $N\left(1440\right)N\pi$ and $\mathrm{\Delta }\left(1232\right)N\pi$ couplings are about 1.7 and 4.8 times larger than the expectations from the simple quark model, respectively, which may suggest the unusual property of $N\left(1440\right)P_{11}$ and deficiency of the simple quark model in the description of $N\left(1440\right)P_{11}$ and $\mathrm{\Delta }\left(1232\right)P_{33}$. The $t$- and $u$-channel backgrounds have notable contributions to the ${\pi }^{+}p\to {\pi }^{+}p$ reaction, while in the ${\pi }^{-}p\to {\pi }^{-}p,{\pi }^{0}n$ reactions, the $s$-channel nucleon and $t$- and $u$-channel backgrounds play crucial roles.

Figure 1

Differential cross sections of the ${\pi }^{+}p\to {\pi }^{+}p$ reaction compared with the experimental data (open circles) from Refs. [64, 65, 66, 67, 68] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ center-of-mass (c.m.) energy $W$ (MeV), respectively.

Figure 2

Differential cross sections of the ${\pi }^{-}p\to {\pi }^{-}p$ reaction compared with the experimental data (open circles) from Refs. [64, 65, 66, 67, 68] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 3

Differential cross sections of the reaction ${\pi }^{-}p\to {\pi }^{0}n$ compared with the experimental data (open circles) from Refs. [69, 70, 71, 72, 73, 74] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 4

Polarizations of the ${\pi }^{+}p\to {\pi }^{+}p$ reaction compared with experimental data (open circles) from Refs. [75, 76, 77, 78, 79, 80, 81, 82] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 5

Effects of the main contributors on the differential cross sections of the reaction ${\pi }^{+}p\to {\pi }^{+}p$ at two energy points $P_{\pi }=240$, 427 $\mathrm{MeV}/c$. The experimental data are taken from Refs. [64, 65, 68]. The bold solid curves correspond to the full model result. The predictions by switching off the contributions from $\mathrm{\Delta }\left(1232\right)P_{33}, u$- and $t$-channel backgrounds are indicated explicitly by the legend in the figures.

Figure 6

Predicted total cross sections of the reaction ${\pi }^{+}p\to {\pi }^{+}p$ and ${\pi }^{-}p\to {\pi }^{-}p,{\pi }^{0}n$ compared with experimental data. The data for the ${\pi }^{\pm }p\to {\pi }^{\pm }p$ reactions are obtained from the PDG [4], and the data of ${\pi }^{-}p\to {\pi }^{0}n$ are taken from Refs. [70, 71, 73, 83, 84, 85]. The bold solid curves correspond to the full model result. In the figure, exclusive cross sections for $\mathrm{\Delta }\left(1232\right)P_{33}, N\left(1535\right)S_{11}$, and $N\left(1440\right)P_{11}$ are indicated explicitly by the legends.

Figure 7

Polarizations of the ${\pi }^{-}p\to {\pi }^{-}p$ reaction compared with experimental data (open circles) from Refs. [75, 77, 108, 109, 110] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 8

Polarizations of the ${\pi }^{-}p\to {\pi }^{0}n$ reaction compared with the experimental data (open circles) from Refs. [74, 109, 111, 112, 113] and the solutions (dotted curves) from the GWU group [38]. The bold solid curves correspond to the full model result. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 9

Effects of the main contributors on the differential cross sections (left side) and polarizations (right side) of the reaction ${\pi }^{-}p\to {\pi }^{-}p$. The experimental data are taken from Refs. [64, 65, 66, 67, 68, 77]. The bold solid curves correspond to the full model result. The results by switching off the one of the main contributors in the resonances and nonresonance backgrounds are indicated explicitly by the legend in the figures. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 10

Effects of the main contributors on the differential cross sections (left side) and polarizations (right side) of the reaction ${\pi }^{-}p\to {\pi }^{0}n$. The experimental data are taken from Refs. [70, 72, 114, 115, 116]. The bold solid curves correspond to the full model result. The predictions by switching off the one of the main contributors in the resonances and nonresonance backgrounds are indicated explicitly by the legend in the figures. The first and second numbers in each figure correspond to the incoming $\pi$ momentum $P_{\pi }$ (MeV) and the $\pi N$ c.m. energy $W$ (MeV), respectively.

Figure 11

The $\pi N$ partial amplitudes of $P_{33}$ and $P_{11}$. Solid (dashed) curves give the real (imaginary) parts of amplitudes extracted from our quark model. The filled (open) circles stand for the real (imaginary) parts (solution WI08 [11]) extracted by the GWU group [38].