Theoretical study of the $d^{*}\left(2380\right)\to d\pi \pi$ decay width

The decay widths of the $d^{*}\to d{\pi }^0{\pi }^0$ and $d^{*}\to d{\pi }^{+}{\pi }^{-}$ processes are explicitly calculated in terms of our chiral quark model. By using the experimental ratios of cross sections between various decay channels, the partial widths of the $d^{*}\to pn{\pi }^0{\pi }^0,d^{*}\to pn{\pi }^{+}{\pi }^{-},d^{*}\to pp{\pi }^0{\pi }^{-}$, and $d^{*}\to nn{\pi }^{+}{\pi }^0$ channels are also extracted. Further including the estimated partial width for the $d^{*}\to pn$ process, the total width of the $d^{*}$ resonance is obtained. In the first step of the practical calculation, the effect of the dynamical structure on the width of $d^{*}$ is studied in the single $\mathrm{\Delta }\mathrm{\Delta }$ channel approximation. It is found that the width is reduced by a few tens of MeV, in comparison with the one obtained by considering the effect of the kinematics only. This presents the importance of such an effect from the dynamical structure. However, the obtained width with the single $\mathrm{\Delta }\mathrm{\Delta }$ channel wave function is still too large to explain the data. It implies that the $d^{*}$ resonance will not consist of the $\mathrm{\Delta }\mathrm{\Delta }$ structure only, and instead there should be enough room for other structures such as the hidden-color (CC) component. Thus, in the second step, the width of $d^{*}$ is further evaluated by using a wave function obtained in the coupled $\mathrm{\Delta }\mathrm{\Delta }$ and CC channel calculation in the framework of the resonating group method (RGM). It is shown that the resultant total width for $d^{*}$ is about 69 MeV, which is compatible with the experimental observation of about 75 MeV and justifies our assertion that the $d^{*}$ resonance is a hexaquark-dominated exotic state.

Figure 1

Four possible emission paths in the decay of the $d^{*}$ resonance composed of the $\mathrm{\Delta }\mathrm{\Delta }$ structure only. Two pions with momenta of ${\vec{k}}_1$ and ${\vec{k}}_2$ are emitted from one of the three quarks in two $\mathrm{\Delta }\mathrm{s}$, respectively.

Figure 2

Relative wave functions in the $S$ wave in the extended chiral SU(3) quark model: (a) for deuteron, (b) for ${\chi }_{\mathrm{\Delta }\mathrm{\Delta }}$ in the single $\mathrm{\Delta }\mathrm{\Delta }$ channel case for $d^{*}\left(2380\right)$, and (c) for ${\chi }_{\mathrm{\Delta }\mathrm{\Delta }}$ in the coupled $\mathrm{\Delta }\mathrm{\Delta }$ and CC channel case for $d^{*}\left(2380\right)$.