Light tetraquark states with exotic quantum numbers $J^{PC}=2^{+-}$

We study the masses of light tetraquark states $ud\bar{u}\bar{d}$ , $us\bar{u}\bar{s}$ and $ss\bar{s}\bar{s}$ with exotic quantum numbers $J^{PC}=2^{+-}$ using the method of QCD sum rules. It is found that there is no tetraquark operator with two Lorentz indices coupling to the $2^{+-}$ quantum numbers. To investigate such tetraquark states, we construct the interpolating tetraquark currents with three Lorentz indices and without derivative operator. We calculate the correlation functions up to dimension 10 condensates, and extract the $2^{+-}$ invariant functions via the projector operator. Our results show that the masses of the $ud\bar{u}\bar{d}$, $us\bar{u}\bar{s}$ and $ss\bar{s}\bar{s}$ tetraquark states with $J^{PC}=2^{+-}$ are about $3.3-3.5 ~\mathrm{GeV}$, $3.5-3.7 ~\mathrm{GeV}$ and $3.67 ~\mathrm{GeV}$, respectively. We further discuss the strong decays of these light tetraquarks into the two-meson and baryon-antibaryon final states, and suggest to search for them in the $\rho\pi, \omega\pi, \phi\pi$, $b_{1}\pi$, $h_{1}\pi$, $K\bar K^\ast, K\bar{K}_{1}$, $\Delta\bar{\Delta}$, $\Sigma^{\ast} \bar{\Sigma }^{\ast}$, $\Xi^{\ast} \bar{\Xi }^{\ast}$, $\Omega\bar{\Omega }$ channels in the future.