QCD prediction for the non-$D\bar D$ annihilation decay of $\psi(3770)$

To clarify the marked difference between BES and CLEO measurements on the non-$D\bar D$ decays of the $\psi(3770)$, a $1^{3}D_{1}$-dominated charmonium, we calculate the annihilation decay of $\psi(3770)$ in NRQCD. By introducing the color-octet contributions, the results are free from infrared divergences. The color-octet matrix elements are estimated by solving the evolution equations. The S-D mixing effect is found to be very small. With $m_{c}=1.5\pm0.1\textrm{GeV}$ our result is $\Gamma(\psi(3770)\to \textrm{light hadrons})=467^{-187}_{+338}\textrm{KeV}$. For $m_c=1.4$ GeV, together with the observed hadronic transitions and E1 transitions, the non-$D\bar D$ decay branching ratio of $\psi(3770)$ could reach about 5%. Our results do not favor either of the results of BES and CLEO collaborations, and further experimental tests are urged.