\begin{document}$J^{PC} = 1^{--}, (0,1,2)^{-+}$\end{document} charmonium-like hybrids are investigated under lattice QCD in the quenched approximation. We define the Bethe-Salpeter wave function (\begin{document}$ \Phi_n(r) $\end{document}) in the Coulomb gauge as the matrix element of a spatially extended hybrid-like operator (\begin{document}$ \bar{c}{c}g $\end{document}) between the vacuum and n-th state for each \begin{document}$ J^{PC} $\end{document}, with r being the spatial separation between a localized \begin{document}$ \bar{c}c $\end{document} component and the chromomagnetic strength tensor. These wave functions exhibit some similarities for states with the aforementioned different quantum numbers, and their r-behaviors (no node for the ground states and one node for the first excited states) imply that r can be a meaningful dynamical variable for these states. Additionally, the mass splittings of the ground states and first excited states of charmonium-like hybrids in these channels are obtained for the first time to be approximately 1.2-1.4 GeV. These results do not support the flux-tube description of heavy-quarkonium-like hybrids in the Born-Oppenheimer approximation. In contrast, a charmonium-like hybrid can be viewed as a “color halo” charmonium for which a relatively localized color octet \begin{document}$ \bar{c}c $\end{document} is surrounded by gluonic degrees of freedom, which can readily decay into a charmonium state along with one or more light hadrons. The color halo picture is compatible with the decay properties of \begin{document}$ Y(4260) $\end{document} and suggests LHCb and BelleII to search for \begin{document}$ (0,1,2)^{-+} $\end{document} charmonium-like hybrids in \begin{document}$ \chi_{c0,1,2}\eta $\end{document} and \begin{document}$ J/\psi \omega (\phi) $\end{document} final states."> Color halo scenario of charmonium-like hybrids -
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