Nucleon resonances within a dynamical coupled-channels model of $\pi N$ and $\gamma N$ reactions

The nucleon resonances are investigated within a dynamical coupled-channels model of $\pi N$ and $\gamma N$ reactions up to the invariant mass $W=2$ GeV. The meson-baryon ($MB$) channels included in the calculations are $MB=$ $\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$, and $\pi \pi N$ that has $\pi \Delta$, $\rho N$, and $\sigma N$ resonant components. The meson-baryon amplitudes $T_{M^{\prime }{B}^{\prime },MB}\left(W\right)$ are calculated from solving a set of coupled-channels integral equations defined by an interaction Hamiltonian consisting of (a) meson-exchange interactions $v_{M^{\prime }{B}^{\prime },MB}$ derived from phenomenological Lagrangian and (b) vertex interactions $N^{*}\to MB$ for describing the transition of a bare excited nucleon state $N^{*}$ to a meson-baryon channel $MB$.  The parameters of $v_{M^{\prime }{B}^{\prime },MB}$ are mainly constrained by the fit to the data of $\pi N\to \pi N$ in the low-energy region up to $W=1.4$ GeV. The bare masses of $N^{*}$ and the $N^{*}\to MB$ parameters are then determined in simultaneous fits to the data of $\pi N\to \pi N$ up to $W=2.3$ GeV and those of $\pi N\to \eta N,K\Lambda ,K\Sigma$ and $\gamma N\to \pi N,\eta N,K\Lambda ,K\Sigma$ up to $W=2.1$ GeV. The pole positions and residues of nucleon resonances are extracted by analytically continuing the meson-baryon amplitudes $T_{M^{\prime }{B}^{\prime },MB}\left(W\right)$ to the complex Riemann energy surface. From the extracted residues, we have determined the $N^{*}\to \pi N,\gamma N,\eta N,K\Lambda ,K\Sigma$ transition amplitudes at resonance poles. We compare the resonance pole positions from our analysis with those given by the Particle Data Group and the recent coupled-channels analyses by the Jülich and Bonn-Gatchina groups. Four results agree well only for the first $N^{*}$ in each spin-parity-isospin $(J^P,I)$ channel. For higher mass states, the number of states and their resonance positions from four results do not agree well. We discuss the possible sources of the discrepancies and the need of additional data from new hadron facilities such as the Japan Proton Accelerator Research Complex.

Figure 1

Meson-exchange mechanisms.

Figure 2

$Z$-diagram mechanisms.

Figure 3

Dressed $\gamma N\to N^{*}$ vertex defined by Eq. (18).

Figure 4

Partial-wave amplitudes of $\pi N$ scattering with isospin $I=1/2$. Upper (lower) panels are for real (imaginary) parts of the amplitudes. Red solid curves, current results; green dotted curves, results from our previous analysis [2].

Figure 5

Partial-wave amplitudes of $\pi N$ scattering with isospin $I=3/2$. Upper (lower) panels are for real (imaginary) parts of the amplitudes. Red solid curves, current results; green dotted curves, results from our previous analysis [2].

Figure 6

Differential cross section $d\sigma /d\Omega$ of $\pi N\to \pi N$. Red solid curves, current results; green dotted curves, results from our previous analysis [2]. The corresponding values of $W$ are indicated in each panel. The same applies to the rest of figures.

Figure 7

Target polarization $P$ of $\pi N\to \pi N$. Red solid curves, current results; green dotted curves, results from our previous analysis [2].

Figure 8

$d\sigma /d\Omega$ of $\gamma p\to {\pi }^{0}p$.

Figure 9

$d\sigma /d\Omega$ of $\gamma p\to {\pi }^{0}p$ (continued).

Figure 10

$\Sigma$ of $\gamma p\to {\pi }^{0}p$.

Figure 11

$\Sigma$ of $\gamma p\to {\pi }^{0}p$ (continued).

Figure 12

$P$ of $\gamma p\to {\pi }^{0}p$.

Figure 13

$T$ of $\gamma p\to {\pi }^{0}p$.

Figure 14

$\widehat{E}$ of $\gamma p\to {\pi }^{0}p$.

Figure 15

$G$ and $H$ of $\gamma p\to {\pi }^{0}p$.

Figure 16

$d\sigma /d\Omega$ of $\gamma p\to {\pi }^{+}n$.

Figure 17

$d\sigma /d\Omega$ of $\gamma p\to {\pi }^{+}n$ (continued).

Figure 18

$\Sigma$ of $\gamma p\to {\pi }^{+}n$.

Figure 19

$\Sigma$ of $\gamma p\to {\pi }^{+}n$ (continued).

Figure 20

$P$ of $\gamma p\to {\pi }^{+}n$.

Figure 21

$T$ of $\gamma p\to {\pi }^{+}n$.

Figure 22

$\widehat{E}$ of $\gamma p\to {\pi }^{+}n$.

Figure 23

$G$ and $H$ of $\gamma p\to {\pi }^{+}n$.

Figure 24

$d\sigma /d\Omega$ of ${\pi }^{-}p\to \eta n$.

Figure 25

$d\sigma /d\Omega$ of $\gamma p\to \eta p$.

Figure 26

$d\sigma /d\Omega$ of $\gamma p\to \eta p$ (continued).

Figure 27

$\Sigma$ of $\gamma p\to \eta p$.

Figure 28

$T$ of $\gamma p\to \eta p$.

Figure 29

$d\sigma /d\Omega$ of ${\pi }^{+}p\to K^{+}{\Sigma }^{+}$.

Figure 30

$P$ of ${\pi }^{+}p\to K^{+}{\Sigma }^{+}$.

Figure 31

$\beta$ of ${\pi }^{+}p\to K^{+}{\Sigma }^{+}$.

Figure 32

$d\sigma /d\Omega$ of ${\pi }^{-}p\to K^0{\Lambda }^0$.

Figure 33

$P$ of ${\pi }^{-}p\to K^0{\Lambda }^0$.

Figure 34

$\beta$ of ${\pi }^{-}p\to K^0{\Lambda }^0$.

Figure 35

$d\sigma /d\Omega$ of ${\pi }^{-}p\to K^0{\Sigma }^0$.

Figure 36

$P$ of ${\pi }^{-}p\to K^0{\Sigma }^0$.

Figure 37

$d\sigma /d\Omega$ of $\gamma p\to K^{+}\Lambda$.

Figure 38

$d\sigma /d\Omega$ of $\gamma p\to K^{+}\Lambda$. (continued)

Figure 39

$P$, $\Sigma$, and $T$ of $\gamma p\to K^{+}\Lambda$.

Figure 40

$C_{x^{\prime }}$, $C_{z^{\prime }}$, $O_{x^{\prime }}$, and $O_{z^{\prime }}$ of $\gamma p\to K^{+}\Lambda$.

Figure 41

$d\sigma /d\Omega$ of $\gamma p\to K^{+}{\Sigma }^0$.

Figure 42

$P$ and $\Sigma$ of $\gamma p\to K^{+}{\Sigma }^0$.

Figure 43

$C_{x^{\prime }}$ and $C_{z^{\prime }}$ of $\gamma p\to K^{+}{\Sigma }^0$.

Figure 44

$d\sigma /d\Omega$ of $\gamma p\to K^0{\Sigma }^{+}$.

Figure 45

$P$ and $\Sigma$ of $\gamma p\to K^0{\Sigma }^{+}$.

Figure 46

$N^{*}$ spectrum with the isospin $I=1/2$ determined by ANL-Osaka (AO) collaboration. For each $N^{*}$ state, $\mathrm{Re}\left(M_R\right)$ together with the $\mathrm{Re}\left(M_R\right)\pm \mathrm{Im}\left(M_R\right)$ band is plotted. The results are compared with four- and three-star states listed by the PDG [50], as well as the results from Jülich (J) (model A in Ref. [20]) and Bonn-Gatchina (BG) [23] groups. The spin and parity of states are denoted as $J^P$ with $P=\pm$ and the associated $\pi N$ partial wave.

Figure 47

$N^{*}$ spectrum with the isospin $I=3/2$ determined by ANL-Osaka (AO) collaboration. For each $N^{*}$ state, $\mathrm{Re}\left(M_R\right)$ together with the $\mathrm{Re}\left(M_R\right)\pm \mathrm{Im}\left(M_R\right)$ band is plotted. The results are compared with four- and three-star states listed by the PDG [50], as well as the results from Jülich (J) (model A in Ref. [20]) and Bonn-Gatchina (BG) [23] groups. The spin and parity of states are denoted as $J^P$ with $P=\pm$ and the associated $\pi N$ partial wave.