$D$-wave excited $cs\bar{c}\bar{s}$ tetraquark states with $J^{PC}=1^{++}$ and $1^{+-}$

Abstract

We study the mass spectra of D-wave excited $cs\bar{c}\bar{s}$ tetraquark states with $J^{PC}=1^{++}$ and $1^{+-}$ in both symmetric $6_{cs}\otimes {\bar{6}}_{\bar{c}\bar{s}}$ and antisymmetric ${\bar{3}}_{cs}\otimes 3_{\bar{c}\bar{s}}$ color configurations by using the QCD sum rule method. We construct the D-wave diquark-antidiquark type of $cs\bar{c}\bar{s}$ tetraquark interpolating currents in various excitation structures with $(L_{\lambda },L_{\rho }\{l_{{\rho }_1},l_{{\rho }_2}\})=(2,0\{0,0\}),(1,1\{1,0\}),(1,1\{0,1\}),(0,2\{1,1\}),(0,2\{2,0\}),(0,2\{0,2\})$. Our results support to interpret the recently observed $X(4685)$ resonance as a D-wave $cs\bar{c}\bar{s}$ tetraquark state with $J^{PC}=1^{++}$ in the $(2,0\{0,0\})$ or $(0,2\{2,0\})$ excitation mode, although some other possible excitation structures cannot be excluded exhaustively within theoretical errors. Moreover, our results provide a mass relation $6_{\rho \rho }<3_{\lambda \lambda }<3_{\lambda \rho }<3_{\rho \rho }$ and $6_{\rho \rho }<3_{\lambda \lambda }<6_{\lambda \lambda }<3_{\rho \rho }$ for the positive and negative $\mathbb{C}$-parity D-wave $cs\bar{c}\bar{s}$ tetraquarks, respectively. We suggest searching for these possible D-wave $cs\bar{c}\bar{s}$ tetraquarks in both the hidden-charm channels $J/\psi \varphi$, ${\eta }_c\varphi$ and open-charm channels such as $D_s{\bar{D}}_s^{\ast }$, $D_s{\bar{D}}_{s1}^{\ast }$ and so on.

Comments: 31 pages, 5 figures. Accepted by Chinese Physics C