Extraction of electromagnetic transition form factors for nucleon resonances within a dynamical coupled-channels model

We explain the application of a recently developed analytic continuation method to extract the electromagnetic transition form factors for the nucleon resonances ($N^{*}$) within a dynamical coupled-channel model of meson-baryon reactions. Illustrative results of the obtained $N^{*}\to \gamma N$ transition form factors, defined at the resonance pole positions on the complex energy plane, for the well-isolated $P_{33}$ and $D_{13}$ and the complicated $P_{11}$ resonances are presented. A formula was developed to give a unified representation of the effects due to the first two $P_{11}$ poles, which are near the $\pi \Delta$ threshold, but are on different Riemann sheets. We also find that a simple formula, with its parameters determined in the Laurent expansions of the $\pi N\to \pi N$ and $\gamma N\to \pi N$ amplitudes, can reproduce to a very large extent the exact solutions of the considered model at energies near the real parts of the extracted resonance positions. We discuss the important differences between our approach, which is consistent with the earlier formulations of resonances, and the phenomenological approaches using the Breit-Wigner parametrization of resonant amplitudes to fit the data.

Figure 1

The shift of the on-shell momentum (open circle/solid circle) of the two-particle Green function Eq. (7) as energy $E$ moves from a real value above/below the threshold energy to a complex value with negative imaginary part. The $C_1$ (dotted line) is the integration path for calculating Eqs. (4, 5, 6) amplitude for $E$ on the unphysical Riemann sheet.

Figure 2

Contour $C_2$ for calculating Eqs. (4, 5, 6) for $E$ on the unphysical Riemann sheet with the unstable particle propagators, such as Eq. (9) for the $\pi \Delta$ channel. See the text for the explanations of the dashed line and the singularity $p_x$.

Figure 3

Contour $C_3$ for calculating the $\Delta$ self-energy Eq. (10) on the unphysical Riemann sheet. Dashed curve is the singularity $q_0$ of the propagator in Eq. (13), which depends on the spectator momentum $p$ on the contour $C_2$ of Fig. 2.

Figure 4

The contour (solid curve) for calculating electromagnetic matrix element. $p_0$ and $p_x$ are the singularities shown in Fig. 2. The dashed-dot curve is the singularity of the pion-exchange $\gamma N\to \pi \Delta$ matrix element at $E=(1357,-76i)$ MeV.

Figure 5

The real (left) and the imaginary (right) parts of the half-off-shell matrix-elements of the nonresonant potential $v_{\gamma N\to \pi \Delta }$ for $P_{11}$ partial wave at $E=(1357,-76i)$, as functions of the real part [$\mathrm{Re}(p)$] of off-shell momentum. The solid (dashed) curve is from the calculation along the path $C_2$ ($C_2^{\prime }$) shown in Fig. 4.

Figure 6

Energy dependence of (a) the $\pi N$ amplitude and (b) the $\gamma \pi$ $M_{1+}(3/2)$ and (c) $E_{1+}(3/2)$ amplitudes of the $P_{33}$ channel. The solid circle (triangle) shows the real (imaginary) part of the amplitude calculated using Eq. (15). The solid (dashed) curve shows the real (imaginary) part of the amplitude of the EBC-DCC model.

Figure 7

Energy dependence of the (a) $\pi N$ amplitude and (b) the $\gamma \pi$ $M_{2-}(1/2p)$ and (c) $E_{2-}(1/2p)$ amplitudes of the $D_{13}$ channel. The solid circle (triangle) shows the real (imaginary) part of the amplitude calculated using Eq. (15). The solid (dashed) curve shows the real (imaginary) part of the amplitude of the EBC-DCC model.

Figure 8

Energy dependence of (a) the $P_{11}\pi N$ scattering and (b) the $M_{1-}(1/2p)$ $\gamma \pi$ amplitude. The solid circle (triangle) shows the real (imaginary) part of the amplitude calculated using Eq. (45). The solid (dashed) curve shows the real (imaginary) part of the amplitude of the EBC-DCC model.

Figure 9

The magnetic $N$-$\Delta$(1232) transition form factor $G_M^{*}(Q^2)$ defined in Ref. [7]. $F_D=1/(1+Q^2/b^2){}^2$ with $b^2=0.71$ (GeV/c)${}^2$. Left: the solid circles (solid triangles) are the real (imaginary) parts of our results. Right: The data points are from recent analyses [18, 38, 39, 40] using the Breit-Wigner parametrization. Open squares are our results at $W=1232$ Mev of the Breit-Wigner position, as discussed in the text.

Figure 10

The extracted $\gamma N\to N^{*}[D_{13}(1521-58i)]$ form factors: (a) $A_{1/2}(Q^2)$, (b) $A_{3/2}(Q^2)$, and (c) $S_{1/2}(Q^2)$. The solid circles (solid triangles) are their real (imaginary) parts.

Figure 11

The extracted $\gamma N\to N^{*}$ form factors $A_{1/2}(Q^2)$ for three $P_{11}$ resonance poles; (a) $P_{11}(1357-76i)$, (b) $P_{11}(1364-105i)$, and (c) $P_{11}(1820-248i)$. They are on different Riemann sheets as listed in Table . The solid circles (solid triangles) are their real (imaginary) parts.

Figure 12

The form factor $A_{1/2}(Q^2)$ of the $P_{11}$ resonance near the $\pi \Delta$ threshold calculated from using the unified representation Eq. (45) and the residues of two $P_{11}$ poles at $(1357-76i)$ MeV and $(1364-105i)$ MeV listed in Table . See text for the procedures for this calculation. Solid circle (solid triangle) shows the real (imaginary) part of the helicity amplitude. Solid square (connected by the dot-dashed line) shows the contribution of the bare form factor.