# Strangeonium-like hybrids on the lattice

• The strangeonium-like $s\bar{s}g$ hybrids are investigated from lattice QCD in the quenched approximation. In the Coulomb gauge, spatially extended operators are constructed for $1^{--}$ and $(0,1,2)^{-+}$ states with the color octet $s\bar{s}$ component being separated from the chromomagnetic field strength by the spatial distance $r$, whose matrix elements between the vacuum and the corresponding states are interpreted as Bethe-Salpeter (BS) wave functions. In each of the $(1,2)^{-+}$ channels, the masses and the BS wave functions are reliably derived. The $1^{-+}$ ground state mass is approximately 2.1-2.2 GeV, and that of $2^{-+}$ is approximately 2.3-2.4 GeV, whereas the mass of the first excited state is approximately 1.4 GeV higher. This mass splitting is much larger compared to that expected based on the phenomenological flux-tube model or constituent gluon model for hybrids, which is usually a few hundred MeV. The BS wave functions with respect to $r$exhibit clear radial nodal structures of a non-relativistic two-body system, which imply that $r$ is a meaningful dynamical variable for these hybrids and motivate a color halo picture of hybrids, in which the color octet $s\bar{s}$ is surrounded by gluonic degrees of freedom. In the $1^{--}$ channel, the properties of the lowest two states are consistent with those of $\phi(1020)$ and $\phi(1680)$. We did not obtain convincing information with respect to $\phi(2170)$. However, we argue that regardless of whether $\phi(2170)$ is a conventional $s\bar{s}$ meson or a $s\bar{s}g$ hybrid in the color halo scenario, the ratio of partial decay widths $\Gamma(\phi \eta)$ and $\Gamma (\phi \eta')$ observed by BESIII can be understood based on the mechanism of hadronic transition of a strangeonium-like meson in addition to $\eta-\eta'$ mixing.
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Yunheng Ma, Ying Chen, Ming Gong and Zhaofeng Liu. Strangeonium-like hybrids on the lattice[J]. Chinese Physics C. doi: 10.1088/1674-1137/abc241
Yunheng Ma, Ying Chen, Ming Gong and Zhaofeng Liu. Strangeonium-like hybrids on the lattice[J]. Chinese Physics C.
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• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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## Strangeonium-like hybrids on the lattice

###### Corresponding author: Ying Chen, cheny@ihep.ac.cn
• 1. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
• 2. School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China

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

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