Study of the decay pattern of $f_0 (1370)$ as a $\kappa \bar{\kappa }$ molecular state

Abstract

Under the hypothesis that the $f_0(1370)$ is a $\kappa \bar{\kappa}$ molecular state, we calculate the partial widths of its various decay channels, including the two-body decay $K \bar{K}$, $\pi \pi$, $\eta \eta$ and the four-body decay $\rho \rho / \sigma \sigma \to 4 \pi$ and $K \bar{K} \pi \pi$. The coupling of $g_{f_0(1370) \kappa \bar{\kappa}}\approx 13$ 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 $K \bar{K}$, $\pi \pi$ and $4 \pi$ ranked as $\Gamma(K \bar{K }) > \Gamma(4 \pi) \approx \Gamma (\pi \pi) $ with $g_{f_0(1370) \kappa \bar{\kappa}}= 35$ GeV. Around $1.37$ GeV, the decay widths of the two-body channels $K \bar{K}$, $\pi \pi$ and $\eta \eta$ remain stable with variation in $\sqrt{s}$, whereas the decay widths of the four-body channels $4 \pi$ and $K \bar{K }\pi \pi$ increase continuously with $\sqrt{s}$. Most current data are model-dependent and conflicting, particularly regarding the conclusion of $4 \pi$ dominance and the ratio of $K\bar{K}$ to $\pi \pi$ decay widths. The current data can not rule out the $\kappa \bar{\kappa}$ assignment for $f_0(1370)$. Further reliable theoretical and experimental analyses of $f_0(1370)$ are required to reveal its nature.