Is $1^-+$ Meson a Hybrid?

We calculate the vacuum to meson matrix elements of the dimension-4 operator $\bar{\psi}\gamma_4\nblr_i \psi$ and dimension-5 operator $\bar{\psi}\eps\gamma_j\psi B_k$ 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 $q\bar{q}$ 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 $c\bar{c}$ and $s\bar{s}$ regions are hybrids. As for the exotic quantum number is concerned, the non-relativistic 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 $q\bar{q}$ 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 $q\bar{q}$ 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 $q\bar{q}$ center-of-mass motion. To account for the quantum numbers of these $q\bar{q}$ mesons in QCD and hadron models in the non-relativistic case, the parity and total angular momentum should be modified to $P = (-)^{L + l +1}$ and $\vec{J} = \vec{L} + \vec{l} + \vec{S}$, where $L$ is the orbital angular momentum of the $q\bar{q}$ pair in the meson.