NLO results with operator mixing for fully heavy tetraquarks in QCD sum rules

We study the mass spectra of $\bar{Q}Q\bar{Q}Q(Q=c,b)$ systems in QCD sum rules with the complete next-to-leading order (NLO) contribution to the pertabative QCD part of the correlation functions. Instead of meson-meson or diquark-diquark currents, we use diagonalized currents under operator renormalization.
Numerical results show that the NLO corrections are very important for the $\bar{Q}Q\bar{Q}Q$ system, because they not only give significant contributions but also reduce parameter dependence and makes Borel platform more distinct, especially for the $\bar{b}b\bar{b}b$ in the $\overline{MS}$ scheme. We find that the operator mixing induced by NLO corrections is crucial to understand the color structure of the states. We use currents that have good perturbative convergence in our phenomenological analysis.
We get three $J^{PC}=0^{++}$ states, with masses ${6.35}_{-0.17}^{+0.20}$ GeV, ${6.56}_{-0.20}^{+0.18}$ GeV and ${6.95}_{-0.31}^{+0.21}$ GeV, respectively. The first two seem to agree with the broad structure around 6.2∼6.8 GeV measured by the LHCb collaboration in the $J/\psi J/\psi$ spectrum, and the third seems to agree with the narrow resonance X(6900). ... mehrFor the 2${}^{++}$ states we find one with mass ${7.03}_{-0.26}^{+0.22}$ GeV, which is also close to that of X(6900), and another one around 7.3 GeV but with larger uncertainties.