B d , s 0 → f 1 f 1 decays with f 1 ( 1285 ) − f 1 ( 1420 ) mixing in the perturbative QCD approach

We investigate the ${B}_{d,s}^{0}\\ensuremath{\\rightarrow}{f}_{1}{f}_{1}$ decays in the framework of perturbative QCD (PQCD) approach with a referenced value ${\\ensuremath{\\phi}}_{{f}_{1}}\\ensuremath{\\sim}24\\ifmmode^\\circ\\else\\textdegree\\fi{}$. Here, ${f}_{1}$ denotes the axial-vector mesons ${f}_{1}(1285)$ and ${f}_{1}(1420)$ with mixing angle ${\\ensuremath{\\phi}}_{{f}_{1}}$ in the quark-flavor basis. The observables such as branching ratios, direct $CP$ violations, and polarization fractions of the ${B}_{s}^{0}\\ensuremath{\\rightarrow}{f}_{1}{f}_{1}$ decays are predicted for the first time. We find that (i) the almost pure penguin modes ${B}_{s}^{0}\\ensuremath{\\rightarrow}{f}_{1}{f}_{1}$ have large branching ratios in the order of ${10}^{\\ensuremath{-}6}\\ensuremath{\\sim}{10}^{\\ensuremath{-}5}$ due to the Cabibbo-Kobayashi-Maskawa enhancement and generally constructive interferences between the amplitudes of ${B}_{s}^{0}\\ensuremath{\\rightarrow}{f}_{n}{f}_{s}$ and ${B}_{s}^{0}\\ensuremath{\\rightarrow}{f}_{s}{f}_{s}$ with ${f}_{n}$ and ${f}_{s}$ being the quark-flavor states of ${f}_{1}$ mesons. (ii) The observables receive important contributions from the weak annihilation diagrams in the PQCD approach. In particular, without the annihilation contributions, the ${B}_{s}^{0}\\ensuremath{\\rightarrow}{f}_{1}(1420){f}_{1}(1420)$ branching ratio will decrease about 81% and its longitudinal polarization fraction will reduce around 43%. And (iii) the dependence of the ${B}_{d,s}^{0}\\ensuremath{\\rightarrow}{f}_{1}{f}_{1}$ decay rates on ${\\ensuremath{\\phi}}_{{f}_{1}}$ exhibits some interesting line shapes, whose confirmations would be helpful to constrain the determination of ${\\ensuremath{\\phi}}_{{f}_{1}}$ inversely. All the PQCD predictions await for the (near) future examinations at Large Hadron Collider beauty and/or Belle-II experiments to further understand the properties of the axial-vector mesons and the perturbative dynamics released from the considered decay modes.