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We use the Laplace/Borel sum rules (LSR) and the finite energy/local dualitysum rules (FESR) to investigate the non-strange $ud\bar u\bar d$ andhidden-strange $us\bar u\bar s$ tetraquark states with exotic quantum numbers$J^{PC}=0^{+-}$ . We systematically construct all eight possible tetraquarkcurrents in this channel without covariant derivative operator. Our analysesshow that the $ud\bar u\bar d$ systems have good behaviour of sum rulestability and expansion series convergence in both the LSR and FESR analyses,while the LSR for the $us\bar u\bar s$ states do not associate with convergentOPE series in the stability regions and only the FESR can provide validresults. We give the mass predictions $1.43\pm0.09$ GeV and $1.54\pm0.12$ GeVfor the $ud\bar u\bar d$ and $us\bar u\bar s$ tetraquark states, respectively.Our results indicate that the $0^{+-}$ isovector $us\bar u\bar s$ tetraquarkmay only decay via weak interaction mechanism, e.g. $X_{us\bar{u}\bar{s}}\toK\pi\pi$, since its strong decays are forbidden by kinematics and the symmetryconstraints on the exotic quantum numbers. It is predicted to be very narrow,if it does exist. The $0^{+-}$ isoscalar $us\bar u\bar s$ tetraquark is alsopredicted to be not very wide because its dominate decay mode$X_{us\bar{u}\bar{s}}\to\phi\pi\pi$ is in $P$-wave.

Exotic tetraquark states with $J^{PC}=0^{+-}$