Calculations of branching ratios and longitudinal polarization fractions in B 0 → K * 0 K ¯ * 0 and B s 0 → K * 0 K ¯ * 0 decays

In this paper, I have investigated the two-body pure penguin decays of [Formula: see text] and [Formula: see text], that these were collected with the LHCb detector, and the branching fractions were measured to be [Formula: see text] and [Formula: see text]. I have studied these decay modes within the framework of the factorization approach by using three different energy scales; while in calculating the amplitude of these decays, the multiplications of four vectors of the polarization vectors, such as [Formula: see text], [Formula: see text], and [Formula: see text] appear. These parameters have been calculated based on helicity amplitudes [Formula: see text]; the results obtained for branching fractions in the μ = m b /2 scale are [Formula: see text] and [Formula: see text]. I have also estimated the fractions of longitudinal polarizations in [Formula: see text] and [Formula: see text] decay modes by considering the transverse amplitudes [Formula: see text]; at the μ = m b /2 scale the results are obtained to be [Formula: see text] and [Formula: see text], which are comparable to the experimental results, 0.724 ± 0.051 ± 0.016 and 0.694 ± 0.016 ± 0.012, respectively, reported by the LHCb collaboration.