Study of the decay pattern of $f_0(1370)$ as a $\hat{I}º\overline{\hat{I}º}$ molecular state

Abstract

Under the hypothesis that the $f_0(1370)$ is a $\hat{I}º\bar{\hat{I}}º$ molecular state, we calculate the partial widths of its various decay channels, including the two-body decay $K\bar{K}$, $\ddot{I}€\ddot{I}€$, $\hat{I}\cdot \hat{I}\cdot$ and the four-body decay $\ddot{I}\ddot{I}/\ddot{I}ƒ\ddot{I}ƒ\to 4\ddot{I}€$ and $K\bar{K}\ddot{I}€\ddot{I}€$. The coupling of $g_{f_0(1370)\hat{I}º\bar{\hat{I}}º}\approx 13$ GeV estimated from the Weinberg criterion yields a width of $f_0(1370)$ significantly smaller than the experimental data. By adjusting this coupling to $25\sim 40$ GeV, the total width of $f_0(1370)$ can be fitted to the measured value $200\sim 500$ MeV. At the center-of-mass energy $\sqrt{s}=1.37$ GeV, the channels that mainly contribute to the total width are $K\bar{K}$, $\ddot{I}€\ddot{I}€$ and $4\ddot{I}€$ ranked as $\hat{I}“(K\bar{K})>\hat{I}“(4\ddot{I}€)\approx \hat{I}“(\ddot{I}€\ddot{I}€)$ with $g_{f_0(1370)\hat{I}º\bar{\hat{I}}º}=35$ GeV. Around $1.37$ GeV, the decay widths of the two-body channels $K\bar{K}$, $\ddot{I}€\ddot{I}€$ and $\hat{I}\cdot \hat{I}\cdot$ remain stable with variation in $\sqrt{s}$, whereas the decay widths of the four-body channels $4\ddot{I}€$ and $K\bar{K}\ddot{I}€\ddot{I}€$ increase continuously with $\sqrt{s}$. Most current data are model-dependent and conflicting, particularly regarding the conclusion of $4\ddot{I}€$ dominance and the ratio of $K\bar{K}$ to $\ddot{I}€\ddot{I}€$ decay widths. The current data can not rule out the $\hat{I}º\bar{\hat{I}}º$ assignment for $f_0(1370)$. Further reliable theoretical and experimental analyses of $f_0(1370)$ are required to reveal its nature.

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