Phenomenological study of two-meson couplings of ${\Theta }^{+}$

We evaluate two-meson couplings of ${\Theta }^{+}$, using experimental information of nucleon resonances decaying into $\pi \pi N$ channels, in which the two pions are in scalar- and vector-type correlations. We examine two assignments of spin and parity of $J^P=1/2^{+}$ and $3/2^{-}$, for which the experimental spectra of known resonances with exotic baryons are properly reproduced by an octet-antidecuplet representation mixing scheme. With the obtained coupling constants, total cross sections of the reactions ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ and $K^{+}p\to {\pi }^{+}{\Theta }^{+}$ are calculated. Substantial interference of two terms may occur in the reaction processes for the $J^P=1/2^{+}$ case, whereas the interference effect is rather small for the $3/2^{-}$ case.

Figure 1

Feynman diagram for the three-body decay of the $N^{*}$ resonance.

Figure 2

Three-body decay of the $N^{*}$ resonance with insertion of the vector meson propagator.

Figure 3

(Color online) Numerical results for the coupling constants with $J^P=1/2^{+}$. The two choices of the relative phase between coupling constants are marked as “phase 1” and “phase 2”. Allowed regions of the coupling constants are shown by the vertical bar. Horizontal bars represent the results obtained with the averaged values, which are absent for the vector case. Horizontal dashed lines show the upper limits of the coupling constants derived from the self-energy $\left|\mathrm{Re}\Sigma \right|<200$ MeV.

Figure 4

(Color online) Numerical results for the coupling constants with $J^P=3/2^{-}$. The two choices of the relative phase between coupling constants are marked as “phase 1” and “phase 2”. Allowed regions of the coupling constants are shown by the vertical bar. Horizontal dashed lines show the upper limit of the coupling constants derived from the self-energy $\left|\mathrm{Re}\Sigma \right|<200$ MeV.

Figure 5

Feynman diagrams for the meson-induced reactions for ${\Theta }^{+}$ production.

Figure 6

Feynman diagram for the meson-induced reaction for ${\Theta }^{+}$ production with a vector meson propagator.

Figure 7

(Color online) Total cross sections for the $J^P=1/2^{+}$ case with $g^s=0.47$ and $g^v=0.47$. The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 8

(Color online) Total cross sections for the $J^P=1/2^{+}$ case with $g^s=0.47$ and $g^v=-0.47$. The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 9

(Color online) Total cross sections for the $J^P=1/2^{+}$ case with $g^s=0.47$ and $g^v=-0.08$, when the most destructive interference for ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ takes place. Note that the vertical scale is different in the two panels. The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 10

(Color online) Total cross section for the $J^P=3/2^{-}$ case with $g^s=0.2$ and $g^v=0.4$, when the most destructive interference for ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ takes place. The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 11

(Color online) Total cross sections for the $J^P=1/2^{+}$ case with $g^s=0.47$ and $g^v=-0.08$, including a hadronic form factor (51). The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 12

(Color online) Total cross sections for the $J^P=3/2^{-}$ case with $g^s=0.2$ and $g^v=0.4$, including a hadronic form factor (51). The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 13

(Color online) The total cross section of ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ at $P_{\mathrm{lab}}=1920$ MeV and the real part of the self-energy of ${\Theta }^{+}$ as functions of the factor F defined in Eqs. (53) and (58) for $J^P=1/2^{+}$ (left) and $J^P=3/2^{-}$ (right). Solid, dashed, and dash-dotted vertical lines show the upper limit of the cross section given by the KEK experiment [16], the limit of the coupling constant, and the point where $\mathrm{Re}\Sigma =-100$ MeV, respectively.

Figure 14

(Color online) Angular dependence of the differential cross section for ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ at $P_{\mathrm{lab}}~1920$ MeV (left) and for $K^{+}p\to {\pi }^{+}{\Theta }^{+}$ at $P_{\mathrm{lab}}~1200$ MeV (right) for the $J^P=1/2^{+}$ case with $g^s=1.59$ and $g^v=-0.27$, including a hadronic form factor (51). The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 15

(Color online) Angular dependence of the differential cross section for ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ at $P_{\mathrm{lab}}~1920$ MeV (left) and for $K^{+}p\to {\pi }^{+}{\Theta }^{+}$ at $P_{\mathrm{lab}}~1200$ MeV (right) for the $J^P=3/2^{-}$ case with $g^s=0.106$ and $g^v=0.212$, including a hadronic form factor (51). The thick line shows the result with full amplitude. Dash-dotted and dashed lines are the contributions from s and $v$ terms, respectively.

Figure 16

Born terms for the reaction: u-channel diagram for $K^{+}p\to {\pi }^{+}{\Theta }^{+}$ (left); s-channel diagram for ${\pi }^{-}p\to K^{-}{\Theta }^{+}$ (right).

Figure 17

(Color online) Total cross section for the $J^P=1/2^{+}$ case with $g^s=1.59$ and $g^v=-0.27$, including a hadronic form factor (51) and Born terms. The thick line shows the result with full amplitude. Dash-dotted, dashed, and thin lines are the contributions from $s,v$, and Born terms, respectively. The result without Born terms is shown by the long-dashed lines.