Is the 1- ⁺ meson a hybrid?

We calculate the vacuum to meson matrix elements of the dimension-4 operator ψ̄γ ₄D → iψ and dimension-5 operator ψ̄ε _ijkγ _jψB _k of the 1 ^-+ meson on the lattice and compare them to the corresponding matrix elements of the ordinary mesons to discern if it is a hybrid. For the charmoniums and strange quarkoniums, we find that the matrix elements of 1 ^-+ are comparable in size as compared to other known qq̄ mesons. They are particularly similar to those of the 2 ⁺⁺ meson, since their dimension-4 operators are in the same Lorentz multiplet. Based on these observations, we find no evidence to support the notion that the lowest 1 ^-+ mesons in the cc̄ and ss̄ regions are hybrids. As far as the exotic quantum number is concerned, the nonrelativistic reduction reveals that the leading terms in the dimension-4 and dimension-5 operators of 1 ^-+ are identical up to a proportional constant and it involves a center-of-mass momentum operator of the quark-antiquark pair. This explains why 1 ^-+ is an exotic quantum number in the constituent quark model where the center of mass of the qq̄ is not a dynamical degree of freedom. Since QCD has gluon fields in the context of the flux tube which is appropriate for heavy quarkoniums to allow the valence qq̄ to recoil against them, it can accommodate such states as 1 ^-+. By the same token, hadronic models with additional constituents besides the quarks can also accommodate the qq̄ center-of-mass motion. To account for the quantum numbers of these qq̄ mesons in QCD and hadron models in the nonrelativistic case, the parity and total angular momentum should be modified to P=(-)L ⁺l ⁺1 and J→= L→+l→+S→, where L is the orbital angular momentum of the qq̄ pair in the meson.