Amplitude analysis of the $B_{(s)} to K^{*0} overline{K}^{*0}$ decays and measurement of the branching fraction of the $B to K^{*0} overline{K}^{*0}$ decay

The $B^0 to K^{*0} overline{K}^{*0}$ and $B^0_s to K^{*0} overline{K}^{*0}$ decays are studied using proton-proton collision data corresponding to an integrated luminosity of 3fb$^{-1}$. An untagged and time-integrated amplitude analysis of $B^0_{(s)} to (K^+pi^-)(K^-pi^+) $ decays in two-body invariant mass regions of 150 MeV$/c^2$ around the $K^{*0}$ mass is performed. A stronger longitudinal polarisation fraction in the ${B^0 to K^{*0} overline{K}^{*0}}$ decay, ${f_L = 0.724 pm 0.051 ,({rm stat}) pm 0.016 ,({rm syst})}$, is observed as compared to ${f_L = 0.240 pm 0.031 ,({rm stat}) pm 0.025 ,({rm syst})}$ in the ${B^0_sto K^{*0} overline{K}^{*0}}$ decay. The ratio of branching fractions of the two decays is measured and used to determine $mathcal{B}(B^0 to K^{*0} overline{K}^{*0}) = (8.0 pm 0.9 ,({rm stat}) pm 0.4 ,({rm syst})) times 10^{-7}$.