Improved QCD sum rule study of $Z_{c}(3900)$ as a $\bar{D}D^{*}$ molecular state

In the framework of QCD sum rules, we present an improved study of our previous work [Phys. Rev. D {\bf80}, 056004 (2009)] particularly on the $\bar{D}D^{*}$ molecular state to investigate that the possibility of the newly observed $Z_{c}(3900)$ as a $S$-wave $\bar{D}D^{*}$ molecular state. To ensure the quality of QCD sum rule analysis, contributions of up to dimension nine are calculated to test the convergence of operator product expansion (OPE). We find that the two-quark condensate $<\bar{q}q>$ is very large and makes the standard OPE convergence (i.e. the perturbative at least larger than each condensate contribution) happen at very large values of Borel parameters. By releasing the rigid OPE convergence criterion, one could find that the OPE convergence is still under control. We arrive at the numerical result $3.86\pm0.27 {GeV}$ for $\bar{D}D^{*}$, which agrees with the mass of $Z_{c}(3900)$ and could support the explanation of $Z_{c}(3900)$ in terms of a $S$-wave $\bar{D}D^{*}$ molecular state.