Color halo scenario of charmonium-like hybrids

The internal structures of $J^{PC}=1^{--}, (0,1,2)^{-+}$ charmonium-like hybrids are investigated under lattice QCD in the quenched approximation. We define the Bethe-Salpeter wave function $\Phi_n(r)$ in the Coulomb gauge as the matrix element of a spatially extended hybrid-like operator $\bar{c}{c}g$ between the vacuum and $n$-th state for each $J^{PC}$ with $r$ being the spatial separation between a localized $\bar{c}c$ 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 $\bar{c}c$ 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 $Y(4260)$ and suggests LHCb and BelleII to search for $(0,1,2)^{-+}$ charmonium-like hybrids in $\chi_{c0,1,2}\eta$ and $J/\psi \omega (\phi)$ final states.