Meson-Hybrid Mixing in $J^{PC}=1^{++}$ Heavy Quarkonium from QCD Sum-Rules

Abstract

We explore conventional meson-hybrid mixing in $J^{PC}=1^{++}$ heavy quarkonium using QCD Laplace sum-rules. We calculate the cross-correlator between a heavy conventional meson current and heavy hybrid current within the operator product expansion, including terms proportional to the four- and six-dimensional gluon condensates and the six-dimensional quark condensate. Using experimentally determined hadron masses, we construct models of the $1^{++}$ charmonium and bottomonium mass spectra. These models are used to investigate which resonances couple to both currents and thus exhibit conventional meson-hybrid mixing. In the charmonium sector, we find almost no conventional meson-hybrid mixing in the ${\chi }_{c1}(1P)$, minimal mixing in the $X(3872)$, and significant mixing in both the $X(4140)$ and $X(4274)$. In the bottomonium sector, we find minimal conventional meson-hybrid mixing in the ${\chi }_{b1}(1P)$ and significant mixing in both the ${\chi }_{b1}(2P)$ and ${\chi }_{b1}(3P)$.

Comments: Updated citations and added figures in latest version