Spectrum of $[8]_{[c\bar{s}]} \otimes [8]_{[q \bar{q^\prime}]}$ systems with quantum numbers $J^{P}=0^\pm$ and $1^\pm$

Inspired by the recent experimental progress on the $T_{c\bar{s}0}^a(2900)^{0/++}$, the fully open tetraquark spectrum with the configuration of $[8]_{[c\bar{s}]} \otimes [8]_{[q \bar{q^\prime}]}$ is systematically investigated by the QCD sum rules. In this article, we concentrate on the quantum numbers $J^{P}=0^{+}/0^{-}/1^{+}/1^{-}$. Firstly, we construct four scalar currents ($J^{P}=0^+$) in the form of $[8]_{[c\bar{s}]} \otimes [8]_{[u\bar{d}]}$ type tetraquark structure and perform the analysis of the QCD sum rules, where we consider the leading order contributions up to dimension 11 in the operator product expansion and retain contributions linear in the strange quark mass $m_s$. Our results, $M_{0^{+}}^A = 2.91_{-0.20}^{+0.20}~\text{GeV}$ and $M_{0^{+}}^C = 2.98_{-0.21}^{+0.20}~\text{GeV}$, are consistent with the experimentally discovered $T_{c\bar{s}0}^a(2900)^{++}$ within error margins. Thus, our calculation supports classifying $T_{c\bar{s}0}^a(2900)^{++}$ as a tetraquark state with the $[8]_{[c\bar{s}]} \otimes [8]_{u \bar{d}}$ color configuration. Moreover, on the basis of $J^P=0^+$, we also study tetraquark states with quantum numbers $J^P=0^-$, $1^+$, and $1^-$, predicting two new hadronic states awaiting experimental verification for each quantum number. Their masses are as follows: $M_{0^{-}}^{A}=3.45_{-0.10}^{+0.12}~\text{GeV}$, $M_{0^{-}}^{D}=3.20_{-0.19}^{+0.17}~\text{GeV}$, $M_{1^{+}}^{A}=2.95_{-0.19}^{+0.19}~\text{GeV}$, $M_{1^{+}}^{C}={2.95_{-0.19}^{+0.19}}~\text{GeV}$, $M_{1^{-}}^{A}={3.23_{-0.10}^{+0.11}}~\text{GeV}$ and $M_{1^{-}}^{D}={3.23_{-0.11}^{+0.09}}~\text{GeV}$. With advances in experimental techniques and accumulation of new data, these predicted results are hoped to be confirmed in future experiments.