Line-shape and poles of the $\psi(3770)$

We study the non-Breit-Wigner line-shape of the $\psi(3770)$ resonance, predominantly a $1\ ^{3}D_{1}$ $\bar{c}c$ state, using an unitarized effective Lagrangian approach, including the one-loop effects of the nearby thresholds $D^+D^-$ and $D^0\bar{D}^0$. A fit of the theoretical result to the total cross-section $e^{+}e^{-}\rightarrow D\bar{D}$ is performed, leading to a good description of data ($\chi^{2}/d.o.f.\sim 1.03$). The partial cross sections $e^{+}e^{-}\rightarrow D^0\bar{D}^0$ and $e^{+}e^{-}\rightarrow D^+D^-$ turn out to be separately in good agreement with the experiment. We find a pole at $3777-i12$ MeV, that is within the Particle Data Group (PDG) mass and width estimation for this state. Quite remarkably, we find an additional, dynamically generated, companion pole at $3741-i19$ MeV, which is responsible for the deformation on the lower energy side of the line-shape. The width for the leptonic decay $\psi(3770)\rightarrow e^{+}e^{-}$ is 112 eV, hence smaller than the PDG fit of $262\pm 18$ eV, yet in agreement with a recent experimental study.