$B \rightarrow TT$ decays in the QCD factorization approach

In this study, the nonleptonic two-body $B$ decays into two tensor mesons (including $a_2(1320)$, $K^*_2(1430)$, $f_2(1270)$, $f^\prime_2(1525)$, denoted generically as $T$) are investigated in the QCD factorization approach. The branching ratios, longitudinal polarization fractions, and CP asymmetries are predicted systematically. It is found that, from the perspective of central values, the branching ratios for the decays $B \rightarrow \{a_2 f^\prime_2, f_2 f^\prime_2\}$, $\{a_2f_2, f_2 f_2\}$ and $\{f_2 K^*_2, K^*_2f^\prime_2\}$ are at the order of $10^{-8}$, $10^{-7}$ and $10^{-6}$, respectively. The branching ratios for the decays into $\{a_2 a_2, K^*_2 K^*_2\}$ and $a_2 K^*_2$ are at the order of $10^{-8} - 10^{-7}$ and $10^{-7} - 10^{-6}$, respectively. The longitudinal polarization fractions are approximately $0.7-1$. Especially, for the $\bar{B}^0/B^0 \rightarrow \{K^{*-}_2 K^{*+}_2, f^\prime_2 f^\prime_2\}$ decays, the longitudinal polarization fractions are equal to $1$. The CP asymmetry for the $B^\pm \rightarrow a^\pm_2 f_2$ modes is most significant, roughly $-22\%$. The CP asymmetries for the decays $B \rightarrow \{a_2 f^\prime_2, f_2 f^\prime_2, f^\prime_2 f^\prime_2\}$ and $\bar{B}^0/B^0 \rightarrow K^{*-}_2 K^{*+}_2$ are equal to zero. Our results may be tested by more precise experiments in the future.