Analysis of the $X(4475/4710)$, $X(4500/4700)$, $Z_{c}(4600)$, $Z_{\bar{c}\bar{s}}(4600)$ and related tetraquark states with the QCD sum rules

Abstract: In this work, we introduce an explicit P-wave to construct the diquarks $[qc]{\widehat{V}}$, then construct the local four-quark currents to explore the hidden-charm tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$ and $2^{++}$ in the framework of the QCD sum rules at length. Our calculations indicate that the light-flavor $SU(3)$ breaking effects on the tetraquark masses are tiny. The predictions support assigning the $X(4475)$ and $X(4500)$ as the $[uc]{\widehat{V}}[\overline{uc}]{\widehat{V}}-[dc]{\widehat{V}}[\overline{dc}]{\widehat{V}}$ and $[sc]{\widehat{V}}[\overline{sc}]{\widehat{V}}$ tetraquark states with the $J^{PC}=0^{++}$ respectively, and assigning the $Z{c}(4600)$ and $Z_{\bar{c}\bar{s}}(4600)$ as the $[uc]{\widehat{V}}[\overline{dc}]{\widehat{V}}$ and $[qc]{\widehat{V}}[\overline{sc}]{\widehat{V}}$ tetraquark states with the $J^{PC}=1^{+-}$ respectively. On the other hand, there is no room for the $X(4710)$ and $X(4700)$. Combined with previous works, the $X(4475)$, $X(4500)$, $Z_{c}(4600)$ and $Z_{\bar{c}\bar{s}}(4600)$ might have other important Fock components besides the $\widehat{V}\widehat{V}$ type components.