We study the decays of the pseudotensor mesons $[ pi_{2}(1670) , K_{2}(1770) , eta_{2}(1645) , eta_{2}(1870) ]$ interpreted as the ground-state nonet of $1^1 D_{2}$ $bar{q}q$ states using interaction Lagrangians which couple them to pseudoscalar, vector, and tensor mesons. While the decays of $pi_2 (1670)$ and $K_2 (1770)$ can be well described, the decays of the isoscalar states $eta_2 (1645)$ and $eta_2 (1870)$ can be brought in agreement with experimental data only if the mixing angle between nonstrange and strange states is surprisingly large (about $-42^circ$, similar to the mixing in the pseudoscalar sector, in which the chiral anomaly is active). Such a large mixing angle is however at odd with all other conventional quark-antiquark nonets: if confirmed, a deeper study of its origin will be needed in the future. Moreover, the $bar{q}q$ assignment of pseudotensor states predicts that the ratio $[ eta_2 (1870) rightarrow a_2 (1320) pi]/[eta_2 (1870) rightarrow f_2 (1270) eta]$ is about $23.5$. This value is in agreement with Barberis et al., ($20.4 pm 6.6$), but disagrees with the recent reanalysis of Anisovich et al., ($1.7 pm 0.4$). Future experimental studies are necessary to understand this puzzle. If Anisovichs value shall be confirmed, a simple nonet of pseudoscalar mesons cannot be able to describe data (different assignments and/or additional state, such as an hybrid state, will be needed). In the end, we also evaluate the decays of a pseudoscalar glueball into the aforementioned conventional $bar{q}q$ states: a sizable decay into $K^ast_2 (1430) K$ and $a_2 (1230) pi$ together with a vanishing decay into pseudoscalar-vector pairs [such as $rho(770) pi$ and $K^ast (892) K$] are expected. This information can be helpful in future studies of glueballs at the ongoing BESIII and at the future PANDA experiments.
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