Inspired by the states $X(2370)$ and $X(2600)$ reported by the BESIII Collaboration, we systematically investigate the mass spectra of light compact tetraquark states with configurations $ud\overline{u}\overline{d}$, $us\overline{u}\overline{s}$, and $ss\overline{s}\overline{s}$ in the $J^{PC}=0^{-+}$ and $2^{-+}$ channels using the QCD sum rules approach. To effectively describe these tetraquark states, we construct appropriate interpolating tetraquark currents featuring three Lorentz indices while avoiding derivative operators. Through meticulous calculations of correlation functions up to dimension 10 condensates, we extract the mass spectra for both $0^{-+}$ and $2^{-+}$ states by employing the projection operator technique. Our results indicate that the masses of light tetraquarks span 1.5–2.5  GeV for $0^{-+}$ states and 2.4–2.7 GeV for $2^{-+}$ states. Notably, our analysis suggests that the $X(2370)$ state could be interpreted as a $0^{-+}us\overline{u}\overline{s}$ or $ss\overline{s}\overline{s}$ tetraquark state, while the $X(2600)$ state is likely to be a $2^{-+}us\overline{u}\overline{s}$ tetraquark. These intriguing findings warrant further detailed investigation in future studies to better understand the nature of these states.

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• QCD phenomenology
• Hadrons