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

The $B^0\to K^{\ast 0}{\bar{K}}^{\ast 0}$ 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 $B_{(s)}^0\to (K^{+}{\pi }^{-})(K^{-}{\pi }^{+})$ decays in two-body invariant mass regions of 150 MeV$\mathrm{/}{c}^2$ around the $K^{\ast 0}$ mass is performed. A stronger longitudinal polarisation fraction in the ${B^0 \to K^{*0} \overline{K}^{*0}}$decay,${f_L = 0.724 \pm 0.051 ,({\rm stat}) \pm 0.016 ,({\rm syst})}$,isobservedascomparedto${f_L = 0.240 \pm 0.031 ,({\rm stat}) \pm 0.025 ,({\rm syst})}$inthe${B^0_s\to K^{*0} \overline{K}^{*0}}$ decay. The ratio of branching fractions of the two decays is measured and used to determine $$\mathcal{B}$(B^0 \to K^{*0} \overline{K}^{*0}) = (8.0 \pm 0.9 ,({\rm stat}) \pm 0.4 ,({\rm syst})) \times 10^{-7}$.