Strangeonium-like hybrids on the lattice *

The strangeonium-like $s\bar{s}g$ hybrids are investigated from lattice QCD in the quenched approximation. In Coulomb gauge, spatially extended operators constructed for $1^{--}$ and $(0,1,2)^{-+}$ states with color octet $s\bar{s}$ component being separated chromomagnetic field strength by spatial distances $r$, whose matrix elements between vacuum corresponding interpreted as Bethe-Salpeter (BS) wave functions. each of $(1,2)^{-+}$ channels, masses BS functions reliably derived. $1^{-+}$ ground state mass is around 2.1-2.2 GeV, that $2^{-+}$ 2.3-2.4 while first excited roughly 1.4 GeV higher. This splitting much larger than expectation phenomenological flux-tube model or constituent gluon hybrids, which usually a few hundred MeV. respect to $r$ show clear radial nodal structures non-relativistic two-body system, imply meaningful dynamical variable these motivate halo picture surrounded gluonic degrees freedom. channel, properties lowest two comply those $\phi(1020)$ $\phi(1680)$. We have not obtained convincing information relevant $\phi(2170)$ yet, however, we argue whether conventional meson hybrid within scenario, ratio partial decay widths $\Gamma(\phi \eta)$ $\Gamma (\phi \eta')$ observed BESIII can be understood mechanism hadronic transition along $\eta-\eta'$ mixing.