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.