Disentangling the Dynamical Origin of $P_{11}$ Nucleon Resonances

We show that two almost degenerate poles near the $\pi \Delta$ threshold and the next higher mass pole in the $P_{11}$ partial wave of $\pi N$ scattering evolve from a single bare state through its coupling with $\pi N$, $\eta N$, and $\pi \pi N$ reaction channels. This finding provides new information on understanding the dynamical origins of the Roper $N^{*}(1440)$ and $N^{*}(1710)$ resonances listed by Particle Data Group. Our results for the resonance poles in other $\pi N$ partial waves are also presented.

Figure 1

(left) Trajectories of the evolution of $P_{11}$ resonance poles $A(1357,76)$, $B(1364,105)$, and $C(1820,248)$ from a bare $N^{*}$ with 1763 MeV, as the couplings of the bare $N^{*}$ with the meson-baryon reaction channels are varied from zero to the full strengths of the JLMS model. See text for detailed explanations. Brunch cuts for all channels are denoted as dashed lines. The branch points, $E_{\mathrm{b}.\mathrm{p}.}$, for unstable channels are determined by $E_{\mathrm{b}.\mathrm{p}.}-E_M(k)-E_B(k)-{\Sigma }_{MB}(k,E_{\mathrm{b}.\mathrm{p}.})=0$ of the their propagators (described in the text) evaluated at the spectator momentum $k=0$. With the parameters [14] used in JLMS model, we find that $E_{\mathrm{b}.\mathrm{p}.}(\mathrm{MeV})=(1365.40,-32.46)$, (1704.08, $-74.98$), (1907.57, $-323.62$) for $\pi \Delta$, $\rho N$, and $\sigma N$, respectively. (right) Three-dimensional depiction of the behavior of $|\det [D(E)]{|}^2$ of the $P_{11}$ $N^{*}$ propagator (in arbitrary units) as a function of complex $E$.