Investigating triply heavy tetraquark states through QCD sum rules

We apply the method of QCD sum rules to study the $QQ\bar{Q}\bar{q}$ and $QQ\bar{Q}\bar{s}$ tetraquark states, where $Q=c,b$ and $q=u,d$, with the quantum number $J^P=0^{+}$. We consider the contributions of vacuum condensates up to dimension-9 in the operator product expansion, and use the energy scale formula $\mu =\sqrt{M_X^2-(i{\mathbb{M}}_c+j{\mathbb{M}}_b{)}^2}-k{\mathbb{M}}_s$ to determine the optimal energy scales for the QCD spectral densities. Our results indicate that triply charm tetraquark states $cc\bar{c}\bar{q}$ and $cc\bar{c}\bar{s}$ have masses in the ranges of $5.38-5.84\text{GeV}$ and $5.66-6.16\text{GeV}$, respectively. In the bottom sector, triply bottom tetraquark states $bb\bar{b}\bar{q}$ and $bb\bar{b}\bar{s}$ have masses in the ranges of $14.89-15.55\text{GeV}$ and $14.95-15.66\text{GeV}$, respectively. This study could help distinguish these states in upcoming high-energy nuclear and particle experiments.