$\bar{D}^{(*)}_{s}D^{(*)}$ molecular state with $J^{P}=1^{+}$

In this paper, we construct $\bar{D}^{(*)}_{s}D^{(*)}$-molecule-type interpolating currents $J_{(\pm)\mu}(x)$ with $J^{P}=1^{+}$, calculate the corresponding mass and magnetic moment using the QCD sum rule method and its extension in the weak electromagnetic field, and study the processes of $Z_{(\pm)cs}$ to $\eta_{c}K^{*}$, $J/\psi K$, $\bar{D}D^{*}_{s}$, and $\bar{D}^{*}D_{s}$ via three-point sum rules. The numerical values are $m_{Z_{(\pm)cs}}=3.99^{+0.17}_{-0.14}~\mbox{GeV}$, and $\lambda_{Z_{(\pm)cs}}=2.07^{+0.28}_{-0.16}\times10^{-2}~\mbox{GeV}^{5}$, $\mu_{Z_{(\pm)cs}}=0.18^{+0.16}_{-0.09}~\mu_{N}$ with $\mu_{N}$ the nucleon magneton, $\Gamma_{Z_{(+)cs}}=17.47^{+12.70}_{-8.08}$, and $\Gamma_{Z_{(-)cs}}=13.86^{+10.37}_{-6.51}$. The masses are in agreement with the recently measured value of $Z_{cs}(3985)$ by the BESIII Collaboration, $m^{exp}_{Z_{cs}}=(3982.5^{+1.8}_{-2.6}\pm2.1)~\mbox{MeV}$. The widths are compatible with the experimental value, $\Gamma^{exp}_{Z_{cs}}=(12.8^{+5.3}_{-4.4}\pm3.0)~\mbox{MeV}$. The magnetic moment and the various decay modes can help us to determine the inner structure of $Z_{cs}(3985)$ when being confronted with experimental data in the future.