Double and single pion photoproduction within a dynamical coupled-channels model

Within a dynamical coupled-channels model that has already been fixed by analyzing the data of the $\pi N\to \pi N$ and $\gamma N\to \pi N$ reactions, we present the predicted double pion photoproduction cross sections up to the second resonance region, $W<1.7$ GeV. The roles played by the different mechanisms within our model in determining both the single and double pion photoproduction reactions are analyzed, focusing on the effects attributable to the direct $\gamma N\to \pi \pi N$ mechanism, the interplay between the resonant and nonresonant amplitudes, and the coupled-channels effects. The model parameters that can be determined most effectively in the combined studies of both the single and double pion photoproduction data are identified for future studies.

Figure 1

Graphical representations of $T_{\gamma N,\pi \pi N}$ of Eqs. (1, 2, 3, 4, 5).

Figure 2

Diagrams considered for $v_{\gamma N,\pi \pi N}$.

Figure 3

Near threshold behavior of the total cross section for $\gamma p\to \pi \pi N$: (a) $\gamma p\to {\pi }^{+}{\pi }^{-}p$, (b) $\gamma p\to {\pi }^0{\pi }^{0}p$, and (c) $\gamma p\to {\pi }^{+}{\pi }^{0}n$. The red solid curve represents the full results predicted from our current model, and the blue dashed curves represent the results without the $T_{\gamma N,\pi \pi N}^{\mathrm{dir}}$ contribution. The data are taken from Refs. [7, 8, 9, 10, 11, 12, 13].

Figure 4

Total cross sections of the double and single pion photoproduction reactions up to $W=1.7$ GeV: (a) $\gamma p\to {\pi }^{+}{\pi }^{-}p$, (b) $\gamma p\to {\pi }^0{\pi }^{0}p$, (c) $\gamma p\to {\pi }^{+}{\pi }^{0}n$, (d) $\gamma p\to {\pi }^{0}p$, and (e) $\gamma p\to {\pi }^{+}n$. The red solid curve represents the full result predicted from our current model, and the blue dashed curve in panels (a)–(c) represents the result without $T_{\gamma N,\pi \pi N}^{\mathrm{dir}}$ contribution. The band is generated by allowing a $25\%$ variation in the value of the $\pi N\Delta$ coupling constant $g_{\pi N\Delta }$ used in the electromagnetic amplitudes. The data of the double and single pion photoproduction reactions are taken from Refs. [7, 8, 9, 10, 11, 12, 13] and Refs. [21], respectively.

Figure 5

Contributions of each reaction process described in Eqs. (2, 3, 4, 5) to the total cross sections. (Black solid) $\gamma N\to \pi \Delta$ contribution ($T_{\gamma N,\pi \pi N}^{\pi \Delta }$), (red dashed)$\gamma N\to \sigma N$ contribution ($T_{\gamma N,\pi \pi N}^{\sigma N}$), (green dotted)$\gamma N\to \rho N$ contribution ($T_{\gamma N,\pi \pi N}^{\rho N}$), (blue dashed-dotted) the direct contribution ($T_{\gamma N,\pi \pi N}^{\mathrm{dir}}$). (Top row) Full results of each contribution, (middle row) results with the replacement of $T_{\gamma N,\mathit{MB}}\to t_{\gamma N,\mathit{MB}}^R$; (bottom row) results with the replacement of $T_{\gamma N,\mathit{MB}}\to t_{\gamma N,\mathit{MB}}$. The data are taken from Refs. [7, 8, 9, 10, 11, 12, 13].

Figure 6

Invariant mass distributions of $\gamma p\to {\pi }^{+}{\pi }^{-}p$ at $W=1420,1520$ MeV: (left) $({\pi }^{-}p)$, (middle) $({\pi }^{+}p)$, (right) $({\pi }^{+}{\pi }^{-})$. The red solid curve represents the full result, and the blue dashed curve represents the phase space distribution. The magnitude of both curves is normalized to the data. The data are taken from Ref. [14].

Figure 7

Invariant mass distributions of $\gamma p\to {\pi }^0{\pi }^{0}p$ at $W=1420,1520$ MeV: (left) $({\pi }^{0}p)$, (right) $({\pi }^0{\pi }^0)$. The red solid curve represents the full result, and the blue dashed curve represents the phase space distribution. The magnitude of both curves is normalized to the data. The data are taken from Ref. [9]. The energy bins of the data are 20–30 MeV around the central $W$ shown in the panels.

Figure 8

Invariant mass distributions of $\gamma p\to {\pi }^{+}{\pi }^{0}n$ at $W=1420,1520$ MeV: (left) $({\pi }^{0}n)$, (middle) $({\pi }^{+}n)$, (right) $({\pi }^{+}{\pi }^0)$. The red solid curve represents the full result, and the blue dashed curve represents the phase space distributions. The magnitude of both curves is normalized to the data. The data are taken from Ref. [10]. The energy bins of the data are 20–30 MeV around the central $W$ shown in the panels.

Figure 9

Coupled-channels effects associated with electromagnetic interactions. The red solid curve represents full results, the green dotted curve corresponds to taking only the diagonal element in the $M^{\text{'}}{B}^{\text{'}}$ summation in Eqs. (7) and (9), and the blue dashed curve is obtained by further making a replacement of $t_{\gamma N,\mathit{MB}}\to v_{\gamma N,\mathit{MB}}$ and ${\overline{\Gamma }}_{\gamma N\to N^{*}}\to {\Gamma }_{\gamma N\to N^{*}}$. The data are taken from Refs. [7, 8, 9, 10, 11, 12, 13].

Figure 10

Panels (a)–(e) depict the total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{0}p$, and panels (f)–(j) show total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{+}n$. Each band is obtained by allowing a $50\%$ variation of the helicity amplitudes for the $A_{1/2}$ of $S_{11}(1535)$ (solid blue) and $A_{1/2}$ of $S_{31}(1620)$ (oblique-lined red) listed in Table . The data are taken from Ref. [21].

Figure 11

Total cross sections [panels (a), (d), and (g)] $\gamma p\to {\pi }^{+}{\pi }^{-}p$, [panels (b), (e), and (h)] $\gamma p\to {\pi }^0{\pi }^{0}p$, and [panels (c), (f), and (i)] $\gamma p\to {\pi }^{+}{\pi }^{0}n$. The different bands are generated by allowing $\pm 50\%$ variations of the helicity amplitudes listed in Table . The data are taken from Refs. [7, 8, 9, 10, 11, 12, 13].

Figure 12

Panels (a)–(e) depict the total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{0}p$, and panels (f)–(j) show total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{+}n$. Each band is obtained by allowing a $50\%$ variation of the helicity amplitudes for the $A_{3/2}$ of $P_{33}(1232)$ (solid brown), $A_{1/2}$ of $P_{33}(1232)$ (oblique-lined red), and $A_{1/2}$ of $P_{11}(1440)$ (solid black) listed in Table . The data are taken from Ref. [21].

Figure 13

Panels (a)–(e) depict the total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{0}p$, and panels (f)–(j) show total cross section, differential cross sections, and photon asymmetry for $\gamma p\to {\pi }^{+}n$. Each band is obtained by allowing a $50\%$ variation of the helicity amplitudes for the $A_{3/2}$ of $D_{13}(1520)$ (solid brown), $A_{3/2}$ of $F_{15}(1680)$ (oblique-lined red), and $A_{3/2}$ of $D_{33}(1700)$ (solid black) listed in Table . The data are taken from Ref. [21].