Determination of the resonant parameters of excited vector strangenia with e^(+)e^(-)→ηФdata

We determine the resonant parameters of the vector statesϕ(1680)andϕ(2170)by performing a combined fit to the e+e−→ηϕcross sections from the threshold to 2.85GeV measured by the BaBar,Belle,BESIII,and CMD-3 experiments.The mass(1678_(−3)^(+5)±7)MeV/c^(2) and width(156±5±9)MeV are obtained forϕ(1680),and the mass(2169±5±6)MeV/c^(2) and width(96_(−14)^(+17)±9)MeV are obtained forϕ(2170).The statistical significance ofϕ(2170)is 7.2σ.Depending on the interference betweenϕ(1680),ϕ(2170),and a non-resonantηϕamplitude in the nominal fit,we obtain four solutions andΓe+e−ϕ(1680)⋅B[ϕ(1680)→ηϕ]=(79±4±16),(127±5±12),(65_(−4)^(+5)±13)or(215_(−5)^(+8)±11)eV,andΓ_(ϕ(2170))^(e+e−)⋅B[ϕ(2170)→ηϕ]=(0.56_(−0.02)^(+0.03)±0.07),(0.36_(−0.03)^(+0.05)±0.07),(38±1±5)or(41±2±6)eV.We also search for the production of X(1750)→ηϕ,and the significance is only 2.0σ.We then determine the upper limit ofΓ_(X(1750))^(e+e−)⋅B[X(1750)→ηϕ]at the 90%confidence level.

Strangeonium-like hybrids on the lattice

The strangeonium-like ss^-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 ss^-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 ss^-is surrounded by gluonic degrees of freedom.In the 1^--channel,the properties of the lowest two states are consistent with those ofФ(1020)andФ(1680).We did not obtain convincing information with respect toФ(2170).However,we argue that regardless of whetherФ(2170)is a conventional ss^-meson or a ss^-g hybrid in the color halo scenario,the ratio of partial decay widthsΓ(Фη)andΓ(Фη')observed by BESIII can be understood based on the mechanism of hadronic transition of a strangeonium-like meson in addition toη-η'mixing.