Extraction of resonances from meson-nucleon reactions

We present a pedagogical study of the commonly employed speed-plot (SP) and time-delay (TD) methods for extracting the resonance parameters from the data of two-particle coupled-channels reactions. Within several exactly solvable models, it is found that these two methods find poles on different Riemann sheets and are not always valid. We then develop an analytic continuation method for extracting nucleon resonances within a dynamical coupled-channel formulation of $\pi N$ and $\gamma N$ reactions. The main focus of this paper is on resolving the complications from the coupling with the unstable $\pi \Delta$, $\rho N$, and $\sigma N$ channels, which decay into $\pi \pi N$ states. By using the results from the considered exactly solvable models, explicit numerical procedures are presented and verified. As a first application of the developed analytic continuation method, we present the nucleon resonances in some partial waves extracted within a recently developed coupled-channels model of $\pi N$ reactions. The results from this realistic $\pi N$ model, which includes $\pi N$, $\eta N$, $\pi \Delta$, $\rho N$, and $\sigma N$ channels, also show that the simple pole parametrization of the resonant propagator using the poles extracted from SP and TD methods works poorly.

Figure 1

(a) The complex momentum $p$-plane and its corresponding complex energy $E$-plane, which has (b) a physical sheet and (c) an unphysical sheet. Their correspondence is indicated by the same number. Solid squares (circles) represent the bound state (resonance) poles.

Figure 2

Poles and zeros of the simplified two-channel Breit-Wigner form (${\mu }_1={\mu }_2$ and $\Delta =0)$ of the $S$-matrix on the complex $E$-plane, which has $\mathit{pp}$, $\mathit{uu}$, $\mathit{up}$, and $\mathit{pu}$ sheets. The open circles on the $\mathit{uu}$-sheet (b) are the resonance pole $E_R$ and its conjugate pole $E_R^{*}$. The open circles on the $\mathit{up}$-sheet (c) are the shadow pole $E_S$ and its conjugate pole $E_S^{*}$. The crosses on the $\mathit{pp}$-sheet (a) are the zero $E_{\mathrm{ZS}}$ and its conjugate $E_{\mathrm{ZS}}^{*}$, which are at the same energies of the shadow poles $E_S$ and $E_S^{*}$. The crosses on the $\mathit{pu}$-sheet (d) are the zero $E_{\mathrm{ZR}}$ and its conjugate $E_{\mathrm{ZR}}^{*}$, which are at the same energies of the resonance poles $E_R$ and $E_R^{*}$.

Figure 3

The poles and zeros of the $S$-matrix shown in Fig. 2 displayed on the complex $t$-plane defined by Eqs. (43) and (44).

Figure 4

The $\mu {\gamma }_{+}^2/M$ dependence of the pole positions on the $\mathit{uu}$-sheet (solid curve) and $\mathit{up}$-sheet (dashed curve) of the simplified two-channel Breit-Wigner form (${\mu }_1={\mu }_2,\Delta =0$) calculated at ${\gamma }_{-}/{\gamma }_{+}=0.5$. The real and imaginary parts of poles are shown in panels (a) and (b), respectively. The solid squares (solid circles) are the results obtained from using the SP (TD) method. Triangles in (b) are the widths of the poles obtained using Eq. (18).

Figure 5

The shift of the singularity (open circle) of the propagator of the two-particle scattering equation [Eq. (69)] as energy $E$ moves from the physical sheet to the unphysical sheet. $C_1^{\text{'}}$ in (a) or $C_1$ in (b) is the integration path for calculating the scattering amplitude with $E$ on the unphysical plane.

Figure 6

λ dependence of the pole position of the one-channel, one-resonance model. The line represents the pole position of the $t$-matrix on the unphysical sheet. Solid squares are the pole positions extracted by using the SP and TD methods.

Figure 7

${\lambda }_2$ dependence of the pole positions on the $\mathit{uu}$-sheet (dash-dotted curve), $\mathit{up}$-sheet (solid curve), and $\mathit{pu}$-sheet (dashed curve) of the two-channel, one-resonance model. The open squares (solid squares) are obtained from using the SP (TD) method.

Figure 8

Pole positions (crosses connected by solid lines) on the $\mathit{uu}$-sheet of the two-channel, two-resonance model. The results from using the SP method are the open squares connected by the dashed lines. The numbers on the figure are the value of ${\tilde{M}}_2$.

Figure 9

Pole positions (crosses connected by solid lines) on the $\mathit{up}$-sheet of the two-channel, two-resonance model. The results from using the TD method are the open squares connected by the dashed lines. The numbers on the figure are the value of ${\tilde{M}}_2$.

Figure 10

Contour $C_2$ for calculating the $\pi \Delta$ self-energy on the unphysical sheet. See the text for explanations of the dashed line and the singularity $p_x$.

Figure 11

Contour $C_3$ for calculating the $\pi N$ self-energy on the unphysical sheet. The dashed curve is the singularity $q_0$ of the propagator in Eq. (102), which depends on the spectator momentum $p$ on the contour $C_2$ of Fig. 10.

Figure 12

$N^{*}$ Green function. The solid (dash-dotted) curve is the real (imaginary) part of exact Green function $G_{N^{*}}(E)=1/[E-M_0-{\Sigma }_{\pi \Delta }(E)]$. They are compared with the dashed (dotted) curve of the real (imaginary) part of the approximate Green function $G_{N^{*}}^{\mathrm{approx}}$.

Figure 13

Comparisons of various resonance propagators: exact propagator $G_{N^{*}}(E)$ (crosses), $G_{N^{*}}^{(0)}(E)$ (dashed curve), $G_{N^{*}}^{(1)}(E)$ (dash-dotted curve), and $G_{N^{*}}^{(2)}(E)$ (solid curve). The real and imaginary parts are shown in panels (a) and (b), respectively.

Figure 14

The resonant propagator $G_{N^{*}}(E)$ (crosses) compared with $G_{N^{*},\mathrm{SP}}(E)$ (solid curve) using the SP poles and $G_{N^{*},\mathrm{TD}}(E)$ (dashed curve) using the TD poles. The real and imaginary parts are shown in panels (a) and (b), respectively.