Possible molecular states of $\bar D^{*}K^{*}$ ($ D^{*}K^{*}$) and the new exotic states $X_0(2900)$ and $X_1(2900)$ ($T^a_{cs0}(2900)^0$ and $T^a_{cs0}(2900)^{++}$)

Two iso-singlet hadron states $X_0(2900)$ and $X_1(2900)$ with $J=0$ and 1 respectively, discovered by the LHCb collaboration in 2020, were identified as molecular bound states of $\bar D^*K^*$. Recently two structures $T^a_{cs0}(2900)^0$ and $T^a_{cs0}(2900)^{++}$ have been observed at the hadron spectra, one would suspect if they also are molecular states of $D^*$ and $K^*$. As long as they were of the molecular structures of $D^*K^*$, the hadron states must be in an iso-vector, namely $T^a_{cs0}(2900)^0$ and $T^a_{cs0}(2900)^{++}$ were $I_3=-1, 1$ components of the iso-vector. If it is the case, the corresponding $T^a_{cs0}(2900)^+$ of ($I=1,I_3=0$) and $T^{'a}_{cs0}(2900)^{+}$ of $I=0,I_3=0$ so far evade experimental observation, but should be found by the future experiments. To testify this ansatz, in this paper we study the possible molecular structures of $\bar D^{*}K^{*}$ and $D^{*}K^{*}$ within the Bethe-Salpeter (B-S) framework. With reasonable input parameters it is found that $\bar D^{*}K^{*}$ iso-scalar systems with $J^P=0^+$ and $1^+$ are solutions. The result supports the ansatz of $X_0(2900)$ ($X_1(2900)$) being molecular states of $\bar D^*K^{*}$. Whereas for the system of $ D^{*}K^{*}$ with $I=1$ the corresponding B-S equation has no solution. Thus we can draw a clear conclusion that $T^a_{cs0}(2900)^0$ and $T^a_{cs0}(2900)^{++}$ should not be bound states of $ D^{*}$ and $K^{*}$. The two structures observed by the LHCb collaboration may be caused by dynamics, such as the well-recognized triangle anomalies or other mechanisms.
Comment: 14 pages, 2 figures and 5 tables. Accepted by PRD. arXiv admin note: text overlap with arXiv:2112.14142