Analysis of the $D\bar{D}^*K$ system with QCD sum rules

In this article, we construct the color singlet-singlet-singlet interpolating current with $I\left(J^P\right)=\frac{3}{2}\left(1^-\right)$ to study the $D\bar{D}^*K$ system through QCD sum rules approach. In calculations, we consider the contributions of the vacuum condensates up to dimension-16 and employ the formula $\mu=\sqrt{M_{X/Y/Z}^{2}-\left(2{\mathbb{M}}_{c}\right)^{2}}$ to choose the optimal energy scale of the QCD spectral density. The numerical result $M_Z=4.71_{-0.11}^{+0.19}\,\rm{GeV}$ indicates that there exists a resonance state $Z$ lying above the $D\bar{D}^*K$ threshold to saturate the QCD sum rules. This resonance state $Z$ may be found by focusing on the channel $J/\psi \pi K$ of the decay $B\longrightarrow J/\psi \pi \pi K$ in the future.