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 perturbative QCD part of the correlation functions. Instead of meson-meson or diquark-antidiquark currents, we use diagonalized currents under operator renormalization. We find that differing from conventional mesons $\bar qq$ and baryons $qqq$, a unique feature of the multiquark systems like $\bar{Q}Q\bar{Q}Q$ is the operator mixing or color configuration mixing induced by NLO corrections, which is crucial to understand the color structure of the states. Our 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 the scheme and scale dependence and make Borel platform more distinct, especially for the $\bar{b}b\bar{b}b$ in the $\overline{\rm{MS}}$ scheme. We use currents that have good perturbation convergence in our phenomenological analysis. With the $\overline{\rm{MS}}$ scheme, we get three $J^{PC}=0^{++}$ states, with masses $6.35^{+0.20}_{-0.17}$ GeV, $6.56^{+0.18}_{-0.20}$ GeV and $6.95^{+0.21}_{-0.31}$ GeV, respectively. The first two seem to agree with the broad structure around $6.2\sim6.8$ GeV measured by the LHCb collaboration in the $J/\psi J/\psi$ spectrum, and the third seems to agree with the narrow resonance $X(6900)$. For the $2^{++}$ states we find one with mass $7.03^{+0.22}_{-0.26}$ GeV, which is also close to that of $X(6900)$, and another one around $7.25^{+0.21}_{-0.35}$ GeV, which has good scale dependence but slightly large scheme dependence.