Hybrid quarkonium masses up to the order of ${\cal O}(1/m_Q)$

Abstract

We have calculated the binding energy of the hybrid quarkonium up to the order of ${\cal O}(1/m_Q)$ and found their decay constants scale like $m_Q^{5/4}$ as $m_Q\to \infty$. The $0^{--}$ and $0^{++}$ hybrid quarkonium is exactly degenerate in the limit of $m_Q\to\infty$ while the ${\cal O}(1/m_Q)$ correction renders the $0^{--}$ mass lower than that for $0^{++}$. The $1^{-+}$ and $1^{+-}$ hybrid is nearly degenerate and lies 0.7 GeV lower than $0^{--}$. With $m_b=4.8$GeV the mass of $1^{-+}$ $b\bar b g$ state is $(10.75\pm 0.10)$, which is the lightest exotic hybrid quarkonium.