${\overline{D}}_s^{(*)}{D}^{(*)}$ molecular state with $J^P=1^{+}$

In this paper, we construct ${\overline{D}}_s^{(*)}{D}^{(*)}$-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 ${\eta }_c{K}^{*}$, $J/\psi K$, $\overline{D}{D}_s^{*}$, and ${\overline{D}}^{*}{D}_s$ via three-point sum rules. The numerical values are $m_{Z_{(\pm )cs}}=3.99_{-0.14}^{+0.17}\mathrm{GeV}$, ${\lambda }_{Z_{(\pm )cs}}=2.07_{-0.16}^{+0.28}\times {10}^{-2}{\mathrm{GeV}}^5$, and ${\mu }_{Z_{(\pm )cs}}=0.18_{-0.09}^{+0.16}{\mu }_N$ with ${\mu }_N$ the nucleon magneton, ${\mathrm{\Gamma }}_{Z_{(+)cs}}=17.47_{-8.08}^{+12.70}$, and ${\mathrm{\Gamma }}_{Z_{(-)cs}}=13.86_{-6.51}^{+10.37}$. The masses are in agreement with the recently measured value of $Z_{cs}(3985)$ by the BESIII Collaboration, $m_{Z_{cs}}^{\exp }=({3982.5}_{-2.6}^{+1.8}\pm 2.1)\mathrm{MeV}$. The widths are compatible with the experimental value ${\mathrm{\Gamma }}_{Z_{cs}}^{\exp }=({12.8}_{-4.4}^{+5.3}\pm 3.0)\mathrm{MeV}$. 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.

Figure 1

(a) denotes the various condensates as functions of $M_B^2$ with $\sqrt{s_{Z_{cs}}^0}=4.5\mathrm{GeV}$; (b) represents $R{P}_0$ and $R{H}_0$ varying with $M_B^2$ at $\sqrt{s_{Z_{cs}}^0}=4.5\mathrm{GeV}$.

Figure 2

The dependence of the mass $m_{Z_{cs}}$ and pole residue ${\lambda }_{Z_{cs}}$ on the Borel parameter $M_B^2$ with $\sqrt{s_{Z_{cs}}^0}=4.5\mathrm{GeV}$ (dot-dashed line), $\sqrt{s_{Z_{cs}}^0}=4.6\mathrm{GeV}$ (real line), and $\sqrt{s_{Z_{cs}}^0}=4.6\mathrm{GeV}$ (dashed line).