The puzzle of excessive non-$D\bar D$ component of the inclusive $\psi(3770)$ decay and the long-distant contribution

In this letter we suggest that the obvious discrepancy between theoretical prediction on the $\mathrm{non}-D\bar D$ decays of $\psi(3770)$ and data is to be alleviated by taking final state interaction (FSI) into account. By assuming that $\psi(3770)$ overwhelmingly dissociates into $D\bar D$, then the final state interaction induces a secondary process, we calculate the branching ratios of $\psi(3770)\to D\bar D\to J/\psi\eta, \rho\pi, \omega\eta, K^*K$. Our results show that the branching ratio of $\psi(3770)\to \mathrm{non}-D\bar{D}$ can reach up to $\mathcal{B}_{\mathrm{non}-D\bar{D}}^{FSI}=(0.2\sim1.1)%$ while typical parameters $I=0.4$ GeV$^{-2}$ and $\alpha=0.8\sim 1.3$ are adopted. This indicates that the FSI is obviously non-negligible.