We study the mass spectra of $$ \overline{Q}Q\overline{Q}Q $$ Q ¯ Q 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 $$ \overline{q}q $$ q ¯ q and baryons qqq, a unique feature of the multiquark systems like $$ \overline{Q}Q\overline{Q}Q $$ Q ¯ Q 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 $$ \overline{Q}Q\overline{Q}Q $$ Q ¯ Q 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 $$ \overline{b}b\overline{b}b $$ b ¯ b b ¯ b in the $$ \overline{\textrm{MS}} $$ MS ¯ scheme. We use currents that have good perturbation convergence in our phenomenological analysis. With the $$ \overline{\textrm{MS}} $$ MS ¯ scheme, we get three JPC = 0++ states, with masses $$ {6.35}_{-0.17}^{+0.20} $$ 6.35 − 0.17 + 0.20 GeV, $$ {6.56}_{-0.20}^{+0.18} $$ 6.56 − 0.20 + 0.18 GeV and $$ {6.95}_{-0.35}^{+0.21} $$ 6.95 − 0.35 + 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/ψJ/ψ spectrum, and the third seems to agree with the narrow resonance X(6900). For the 2++ states we find one with mass $$ {7.03}_{-0.26}^{+0.22} $$ 7.03 − 0.26 + 0.22 GeV, which is also close to that of X(6900), and another one around $$ {7.25}_{-0.35}^{+0.21} $$ 7.25 − 0.35 + 0.21 GeV, which has good scale dependence but slightly large scheme dependence.

We study the triply heavy baryons in the QCD sum rules by performing the first calculation of the next-to-leading order (NLO) contribution to the perturbative QCD part of the correlation functions. Compared with the leading order (LO) result, the NLO contribution is found to be very important to the . This is because the NLO not only results in a large correction but also reduces the parameter dependence, making the Borel platform more distinct, especially for the in the scheme, where the platform appears only at NLO but not at LO. Particularly, owing to the inclusion of the NLO contribution, the renormalization schemes ( and On-Shell) dependence and the scale dependence are significantly reduced. Consequently, after including the NLO contribution to the perturbative part in the QCD sum rules, the masses are estimated to be GeV for and GeV for , where the results are obtained at with errors including those from the variation of the renormalization scale μ in the range . A careful study of the μ dependence in a wider range is further performed, which shows that the LO results are very sensitive to the choice of μ whereas the NLO results are considerably better. In addition to the result, a more stable value, (4.75-4.80) GeV, for the mass is found in the range of , which should be viewed as a more relevant prediction in our NLO approach because of dependence.

We study the mass spectra of QQ QQ (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 QQ QQ system, because they not only give significant contributions but also reduce parameter dependence and makes Borel platform more distinct, especially for the bb bb in the 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 P C = 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 ∼ 6.8 GeV measured by the LHCb collaboration in the J/ψJ/ψ 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 7.3 GeV but with larger uncertainties.