Reanalysis of the $X(3915)$, $X(4500)$ and $X(4700)$ with QCD sum rules

Abstract

In this article, we study the $C{\gamma }_5\otimes {\gamma }_{5}C$ type and $C\otimes C$ type scalar $cs\bar{c}\bar{s}$ tetraquark states with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension 10 in a consistent way. The ground state masses $M_{C{\gamma }_5\otimes {\gamma }_{5}C}=3.89\pm 0.05\mathrm{GeV}$ and $M_{C\otimes C}=5.48\pm 0.10\mathrm{GeV}$ support assigning the $X(3915)$ to be the ground state $C{\gamma }_5\otimes {\gamma }_{5}C$ type tetraquark state with $J^{PC}=0^{++}$, but do not support assigning the $X(4700)$ to be the ground state $C\otimes C$ type $cs\bar{c}\bar{s}$ tetraquark state with $J^{PC}=0^{++}$. Then we tentatively assign the $X(3915)$ and $X(4500)$ to be the 1S and 2S $C{\gamma }_5\otimes {\gamma }_{5}C$ type scalar $cs\bar{c}\bar{s}$ tetraquark states respectively, and obtain the 1S mass $M_{1S}=3.85_{-0.17}^{+0.18}\mathrm{GeV}$ and 2S mass $M_{2S}=4.35_{-0.11}^{+0.10}\mathrm{GeV}$ from the QCD sum rules, which support assigning the $X(3915)$ to be the 1S $C{\gamma }_5\otimes {\gamma }_{5}C$ type tetraquark state, but do not support assigning the $X(4500)$ to be the 2S $C{\gamma }_5\otimes {\gamma }_{5}C$ type tetraquark state.

Comments: 17 pages, 6 figures