Study of $B_s^0\to TT\left(a_2\right(1320),K_2^{*}\left(1430\right),f_2\left(1270\right),f_2^{\prime }\left(1525\right))$ in the perturbative QCD approach

In the present study, the calculations of two-body decays $B_s^0\to TT$ [$T$ denotes tensor mesons, $a_2\left(1320\right)$, $K_2^{*}\left(1430\right)$, $f_2\left(1270\right)$, $f_2^{\prime }\left(1525\right)$] in the perturbative QCD approach are presented. The ensuing predictions encompass branching ratios, polarization fractions, and direct $CP$ violations, all elucidated in comprehensive detail. It is discerned that (1) for pure annihilation decay, the longitudinal polarization is around 90.0%, whereas the transverse polarizations manifest comparatively diminutive magnitudes. (2) The direct $CP$ asymmetry is directly proportional to the interference between the tree and penguin contributions. For most of the decays investigated within this discourse, the direct $CP$ asymmetry remains modest in magnitude. (3) There are precisely six distinct categories of Feynman diagrams for $B_s^0\to TT$, because the tensor mesons cannot be produced through the ($V\pm A$) currents or ($S\pm P$) density, thereby prohibiting factorizable emission diagrams. The nonfactorizable and annihilation contributions are ascertained to be pivotal in these decay modes. The calculated branching ratios of our calculation for $B_s^0\to TT$ are at the order of ${10}^{-6}$ and ${10}^{-7}$, which can be tested in the LHCb and Belle II experiments. (4) Mixing exists for the $f_2\left(1270\right)$ and $f_2^{\prime }\left(1525\right)$, just as the $\eta$ and ${\eta }^{\prime }$ mixing, the branching ratios about the mixing angle $\theta$ are given in this work. However, it is different from $f_1\left(1285\right)-f_1\left(1420\right)$, the mixing angle is notably small, thereby resulting in only marginal alterations in the decay branching ratios.