Is $1^-+$ Meson a Hybrid?

TL;DR

This study investigates whether the $1^{-+}$ meson is a hybrid state by calculating specific matrix elements on the lattice and comparing them to ordinary mesons, finding no evidence supporting its hybrid nature.

Contribution

The paper provides lattice QCD calculations of matrix elements for the $1^{-+}$ meson and compares them to known mesons, offering insights into its structure and whether it is a hybrid.

Findings

Matrix elements of $1^{-+}$ are comparable to those of $2^{++}$ mesons.

No evidence supports $1^{-+}$ as a hybrid in $car{c}$ and $sar{s}$ regions.

The quantum number $1^{-+}$ can be explained by quark-antiquark center-of-mass motion.

Abstract

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.