Is the1−+meson a hybrid?

Abstract

We calculate the vacuum to meson matrix elements of the dimension-4 operator $\overline{\ensuremath{\psi}}{\ensuremath{\gamma}}_{4}{\stackrel{\ensuremath{\leftrightarrow}}{D}}_{i}\ensuremath{\psi}$ and dimension-5 operator $\overline{\ensuremath{\psi}}{\ensuremath{\epsilon}}_{ijk}{\ensuremath{\gamma}}_{j}\ensuremath{\psi}{B}_{k}$ of the ${1}^{\ensuremath{-}+}$ 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 ${1}^{\ensuremath{-}+}$ are comparable in size as compared to other known $q\overline{q}$ 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 ${1}^{\ensuremath{-}+}$ mesons in the $c\overline{c}$ and $s\overline{s}$ 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 ${1}^{\ensuremath{-}+}$ are identical up to a proportional constant and it involves a center-of-mass momentum operator of the quark-antiquark pair. This explains why ${1}^{\ensuremath{-}+}$ is an exotic quantum number in the constituent quark model where the center of mass of the $q\overline{q}$ 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 $q\overline{q}$ to recoil against them, it can accommodate such states as ${1}^{\ensuremath{-}+}$. By the same token, hadronic models with additional constituents besides the quarks can also accommodate the $q\overline{q}$ center-of-mass motion. To account for the quantum numbers of these $q\overline{q}$ mesons in QCD and hadron models in the nonrelativistic case, the parity and total angular momentum should be modified to $P=(\ensuremath{-}{)}^{L+l+1}$ and $\stackrel{\ensuremath{\rightarrow}}{J}=\stackrel{\ensuremath{\rightarrow}}{L}+\stackrel{\ensuremath{\rightarrow}}{l}+\stackrel{\ensuremath{\rightarrow}}{S}$, where $L$ is the orbital angular momentum of the $q\overline{q}$ pair in the meson.