The nature and line shapes of charmonium in the $e^+e^- \to D\bar{D}$ reactions

Abstract

We explore the nature of sharp resonances with asymmetric line shape observed in cross section data, a general physical phenomenon produced by the interference of continuum background and resonances. A Fano scheme and the coupled-channel T-matrix approach are employed to this aim and their close relationship is present. As a typical example, we point out that the $\psi(3770)$ state observed in the $e^+e^-$ reactions with an anomalous line shape can be explained naturally as a resonance embedded in the $D\bar{D}$ continuum. From a coupled-channels analysis the background of $\psi(3770)$ resonance is found to originate from a pole at $\sqrt{s}=3716.0 \pm$ 30.0~MeV. As a by-product, the broad structure $X(3900)/G(3900)$ seen in the Belle data, is found to be the tail of the $\psi(3770)$ state, distorted by the opening of the $D^*\bar{D} + c.c$ channel and the onset of the $\psi(4040)$ spectral distribution, thus making the assignment as a genuine charmonium state unlikely.