Dynamical coupled-channels analysis of ${}^1\mathrm{H}$($e,e^{\prime }\pi$)$N$ reactions

We have performed a dynamical coupled-channels analysis of available $p(e,e^{\prime }\pi )N$ data in the region of $W⩽1.6$ GeV and $Q^2⩽1.45$ (GeV/$c$)${}^2$. The channels included are ${\gamma }^{*}N$, $\pi N$, $\eta N$, and $\pi \pi N$ that has $\pi \Delta$, $\rho N$, and $\sigma N$ components. With the hadronic parameters of the model determined in our previous investigations of $\pi N\to \pi N$, $\pi \pi N$ reactions, we have found that the available data in the considered $W⩽1.6$ GeV region can be fitted well by only adjusting the bare ${\gamma }^{*}N\to N^{*}$ helicity amplitudes for the lowest $N^{*}$ states in $P_{33}$, $P_{11}$, $S_{11}$, and $D_{13}$ partial waves. The sensitivity of the resulting parameters to the amount of data included in the analysis is investigated. The importance of coupled-channels effect on the $p(e,e^{\prime }\pi )N$ cross sections is demonstrated. The meson cloud effect, as required by the unitarity conditions, on the ${\gamma }^{*}N\to N^{*}$ form factors are also examined. Necessary future developments, both experimentally and theoretically, are discussed.

Figure 1

Graphical illustration of the contribution to the $\pi N$ intermediate state to the dressed ${\gamma }^{*}N\to N^{*}$ vertex defined by Eq. (4).

Figure 2

Fit to $p(e,e^{\prime }{\pi }^0)p$ structure functions at $Q^2=0.4$ (GeV/$c){}^2$. Here $\theta \equiv {\theta }_{\pi }^{*}$. The solid curves are the results of Fit1, the dashed curves are of Fit2, and the dotted curves are of Fit3. (See text for the description of each fit.) The data are taken from Refs. [10, 11].

Figure 3

Fit to $p(e,e^{\prime }{\pi }^0)p$ structure functions at $Q^2=0.9$ (GeV/$c){}^2$. Here $\theta \equiv {\theta }_{\pi }^{*}$. The data are taken from Ref. [10].

Figure 4

Fit to $p(e,e^{\prime }{\pi }^0)p$ structure functions at $Q^2=1.45$ (GeV/$c){}^2$. Here $\theta \equiv {\theta }_{\pi }^{*}$. The data are taken from Ref. [10].

Figure 5

$G_M^{*}$ normalized by the dipole factor $G_D=[1+Q^2/0.71$ (GeV/$c){}^2]{}^{-2}$. The solid black circles at $Q^2>0$ are our fit to the values extracted from previous analyses (those values are taken from Ref. [4]). The triangle at $Q^2=0$ is from our photoproduction analysis [4].

Figure 6

The ${\gamma }^{*}N\to \Delta (1232)$ form factors. Solid points are from Fit1; dashed curves are the meson cloud contribution. Open circles and triangles at $Q^2=0.4$ (GeV/$c){}^2$ are from Fit2 and Fit3, respectively. The three points are almost overlapped in $G_M^{*}$. The solid point at $Q^2=0$ is obtained from the photoproduction reaction analysis in Ref. [4].

Figure 7

Extracted helicity amplitudes for $S_{11}$ at $W=1535$ MeV (upper panels), $P_{11}$ at $W=1440$ MeV (middle panels), and $D_{13}$ at $W=1520$ MeV (lower panels). The meaning of each point and curve is the same as in Fig. 6.

Figure 8

Structure functions of $p(e,e^{\prime }{\pi }^{+})n$ at $Q^2=0.4$ (GeV/$c$)${}^2$. Here $\theta \equiv {\theta }_{\pi }^{*}$. The solid curves are the results of Fit1, the dashed curves are of Fit2, and the dotted curves are of Fit3. (See text for the description of each fit.) As for the ${\sigma }_{{\mathit{LT}}^{\text{'}}}$, results at $W=1.14,1.22,1.3,1.38,1.5,1.58$ GeV (from left to right of the bottom row) are shown, in which the data are available. The data in the figure are taken from Refs. [12, 13].

Figure 9

Contribution of the $S_{31}$ and $P_{13}$ helicity amplitudes at $Q^2=0.4$ (GeV/$c){}^2$. The left (right) panels are the structure functions of $p(e,e^{\prime }{\pi }^0)p$ [$p(e,e^{\prime }{\pi }^{+})n$] reaction at $W=1.3$ GeV ($W=1.29$ GeV). Solid (dashed) curves are the results with (without) nonzero $S_{31}$ and $P_{13}$ bare helicity amplitudes. The parameters of Fit2 are used for the $S_{11},P_{11},P_{33}$, and $D_{13}$ helicity amplitudes in both curves. The data are from Refs. [10, 11, 12, 13].

Figure 10

Coupled-channels effect on the integrated structure functions ${\sigma }_T(W)$ and ${\sigma }_L(W)$ for $Q^2=0.4,0.9,1.45$ (GeV/$c){}^2$ for $p(e,e^{\prime }{\pi }^0)p$ reactions. The solid curves are the full results calculated with the bare helicity amplitudes of Fit1. The dashed curves are the same as solid curves but only the $\pi N$ loop is taken in the $M^{\text{'}}{B}^{\text{'}}$ summation in Eqs. (2) and (4).

Figure 11

Coupled-channels effect on the fivefold differential cross sections ${\Gamma }_{\gamma }^{-1}[d{\sigma }^5/(d{E}_{e^{\prime }}d{\Omega }_{e^{\prime }}d{\Omega }_{\pi }^{*})]$ of $p(e,e^{\prime }{\pi }^0)p$ (upper panels) and $p(e,e^{\prime }{\pi }^{+})n$ (lower panels) at $Q^2=0.4$ (GeV/$c){}^2$. Here $\theta \equiv {\theta }_{\pi }^{*}$ and $\varphi \equiv {\varphi }_{\pi }^{*}$. The solid curves are the full results calculated with the bare helicity amplitudes of Fit1. The dashed curves are the same as the solid curves but only the $\pi N$ loop is taken in the $M^{\text{'}}{B}^{\text{'}}$ summation in Eqs. (2) and (4). The data are taken from Ref. [14].