Measurement of the branching fractions and longitudinal polarisations of $B^0_{(s)} \to K^{*0} \kern 0.18em \overline{\kern -0.18em K}{}^{*0}$ decays

A time- and flavour-integrated amplitude analysis of $B^0$ and $B^0_s$ decays to the $(K^+π^-)(K^-π^+)$ final state in the $K^*(892)^0 \kern 0.18em \overline{\kern -0.18em K}{}^{*}(892)^0$ region is presented, using $pp$ collision data recorded with the LHCb detector in 2011--2018, corresponding to an integrated luminosity of $9\,\text{fb}^{-1}$. The branching fractions of the $B^0$ and $B^0_s$ decays are measured relative to the $B^0 \to D^-π^+$ and $B^0_s \to D^-_s π^+$ modes, respectively. The corresponding longitudinal polarisation fractions are found to be $f_L^{d} = 0.600 \pm 0.022 \pm 0.017$ and $f_L^{s} = 0.159 \pm 0.010 \pm 0.007$, where the uncertainties are statistical and systematic, respectively. The theory-motivated $L_{K^{*0} \kern 0.18em \overline{\kern -0.18em K}{}^{*0}}$ observable is found to be $L_{K^{*0} \kern 0.18em \overline{\kern -0.18em K}{}^{*0}} = 4.92 \pm 0.55 \pm 0.47 \pm 0.02 \pm 0.10$, where the uncertainties are statistical, systematic, due to uncertainty of external mass and lifetime measurements, and due to knowledge of the fragmentation fraction ratio, respectively. This confirms the previously reported tension between experimental determinations and theoretical predictions of longitudinal polarisation in $B \to VV$ decays.