QCD sum rule predictions on gluonic tetraquark states with $J^{PC}=0^{+-},0^{--}$ and $1^{\pm \pm}$

Abstract

In this work, we present a systematic calculation of the mass spectrum for tetraquark hybrid states, focusing on the $8_{[c\bar{c}]}\otimes 8_{[G]}\otimes 8_{[c\bar{c}]}$ color configuration, within the framework of QCD sum rules. As an extension of our previous work on $0^{++}$ and $0^{-+}$ states, we now construct 18 distinct interpolating currents with $J^{PC} = 0^{+-}$, $0^{--}$, and $1^{\pm\pm}$. Using operator product expansion (OPE) techniques and including nonperturbative contributions up to dimension six, we obtain key results: for the $0^{+-}$, $1^{--}$, and $1^{-+}$ states, the predicted masses lie in the range of $7.2-7.3$ GeV, while the $1^{+-}$ and $1^{++}$ states have slightly lower masses, between 6.9 and 7.1 GeV. These predictions provide strong support for the possible existence of an $8_{[c\bar{c}]}\otimes 8_{[G]}\otimes 8_{[c\bar{c}]}$ component within the di-$J/ψ$ structure reported by LHCb. Moreover, our analogous calculations for tetrabottom hybrid states yield mass ranges of $19.4-19.5$ GeV (for $0^{+-}$, $1^{--}$, and $1^{-+}$) and $19.2-19.3$ GeV (for $1^{+-}$ and $1^{++}$), offering crucial references for future searches.

Figures (11)

• Figure 1: The leading-order Feynman diagrams for the tetracharm hybrid states are shown, with their corresponding contributions to the spectral density given in Eq. \ref{['Pi-MB']}. Only representative diagrams for each type are displayed; topologically equivalent ones are omitted for clarity. Diagrams I and II represent the perturbative contribution and the effects of the two-gluon condensate, respectively. Diagrams III and IV illustrate contributions from the three-gluon condensate.
• Figure 2: (color).The numerical analysis results for current operator $j^{0^{+-}}$ are presented graphically: Figure (a) displays the variation of pole contribution ratio $R^{\text{PC}}_{0^{+-}}$ and OPE convergence ratio $R^{\text{cond}}_{0^{+-}}$ with respect to Borel parameter $M_B^2$ at the central value of continuum threshold $s_0$, while Figure (b) illustrates the evolution of state mass $M_{\text{H}}^{0^{+-}}$ versus $M_B^2$, where three distinct curves correspond to parameter choices of $s_0=7.8^{2}\, \text{GeV}^{2}$ (lower bound), $8.0^{2}\, \text{GeV}^{2}$ (central value), and $8.2^{2}\, \text{GeV}^{2}$ (upper bound). The valid Borel working window, demarcated by vertical gray dashed lines in the figures, is established relative to the central $s_0$ value.
• Figure 3: (color). Following the same caption format as Fig. \ref{['fig2']} but now addressing the current $j_{A}^{1^{--}}$.
• Figure 4: (color). Following the same caption convention as Fig. \ref{['fig2']} but now pertaining to the current $j_{C}^{1^{--}}$.
• Figure 5: (color). Following the same caption convention as Fig. \ref{['fig2']} but now pertaining to the current $j_{B}^{1^{+-}}$.