Extraction of $P_{11}$ resonances from $\pi N$ data

We show that two $P_{11}$ nucleon resonance poles near the $\pi \Delta$ threshold, obtained in several analyses, are stable against large variations of parameters within a dynamical coupled-channels analysis based on meson-exchange mechanisms. By also performing an analysis based on a model with a bare nucleon state, we find that this two-pole structure is insensitive to the analytic structure of the amplitude in the region below the $\pi N$ threshold. Our results are $M_{\mathrm{pole}}=({1363}_{-6}^{+9}-i{79}_{-5}^{+3})$ and $({1373}_{-10}^{+12}-i{114}_{-9}^{+14})$ MeV. We also demonstrate that the number of poles in the 1.5-GeV $⩽W⩽$ 2-GeV region could be more than one, depending on how the structure of the single-energy solution of SAID is fitted. For three-pole solutions, our best estimated result of a pole near $N(1710)$ listed by Particle Data Group is $({1829}_{-65}^{+131}-i{192}_{-110}^{+88})$ MeV, which is close to the results of several previous analyses. Our results indicate the need for more accurate $\pi N$ reaction data in the $W>$ 1.6-GeV region for high-precision resonance extractions.

Figure 1

The real (a) and imaginary (b) parts of the on-shell $P_{11}$ amplitudes as a function of the $\pi N$ invariant mass $W$ (MeV). The solid curves are from the JLMS fit; the dashed (dotted) curves are from the 1$N^{*}$-3$p$-H (1$N^{*}$-3$p$-L) fit to the SAID-EDS [5] up to $W=2$ GeV ($W=1.6$ GeV); the open circles are the SAID-EDS [5]. $T$ is unitless in the convention of Ref. [5].

Figure 2

The real (top panels) and imaginary (bottom panels) parts of the $P_{11}$ amplitudes as a function of the $\pi N$ invariant mass $W$ (MeV). The JLMS (solid) results are compared with the results from the 2$N^{*}$-3$p$ (dotted) and 2$N^{*}$-4$p$ (dashed) fits. The points with errors are from the SAID-SES [5]. $T$ is unitless in the convention of Ref. [5].

Figure 3

The real (a) and imaginary (b) parts of the $P_{11}$ amplitudes as a function of the $\pi N$ invariant mass $W$ (MeV). The $2N^{*}$-4$p$ fit (solid) and the $2N^{*}$-4$p$-CMB fit (dashed) are compared with the data. The open circles with errors are from the SAID-SES [5], and the crosses with errors are from the SAID-SES at $W<1.55$ GeV and the CMB solution [3] at $W>$ 1.55 GeV. $T$ is unitless in the convention of Ref. [5].

Figure 4

The real (top panels) and imaginary (bottom panels) parts of the $P_{11}$ amplitudes as a function of the $\pi N$ invariant mass $W$ (MeV). The JLMS fit (solid) and the 1$N_0$1$N^{*}$-3$p$ fit (dashed) are compared with the SAID-SES [5]. $T$ is unitless in the convention of Ref. [5].