A new $B$-flavour anomaly in $B_{d,s}\to K^{*0}\bar{K}^{*0}$: anatomy and interpretation

In the context of the recently measured non-leptonic decays $B_{d}\to K^{*0}\bar{K}^{*0}$ and $B_{s}\to K^{*0}\bar{K}^{*0}$ we analyse the anatomy of the $L_{VV}$ observable that compares the longitudinal components of $B_s \to VV$ and $B_d \to VV$ decays. This observable is cleaner than the longitudinal polarisation fraction as it is afflicted only at subleading order in a $1/m_b$ expansion by the theoretical uncertainties arising in the transverse components entering the polarisation fraction. Focusing on the particular case of $B_{d}\to K^{*0}\bar{K}^{*0}$ and $B_{s}\to K^{*0}\bar{K}^{*0}$, we discuss the main sources of hadronic uncertainty in the SM. We find for the SM prediction $L_{K^*\bar{K}^*}=19.5^{+9.3}_{-6.8}$, which implies a $2.6\sigma$ tension with respect to the most recent data, pointing to a deficit in the $b \to s$ transition of the non-leptonic decay versus the corresponding $b \to d$ transition. We discuss possible New Physics explanations for this deviation, first at the level of the Weak Effective Theory and we identify that the two Wilson coefficients ${\cal C}_{4}$ and ${\cal C}_{8g}$ can play a central role in explaining this anomaly. Finally, we briefly explore two different simplified New Physics models which can explain the anomaly through a contribution either in ${\cal C}_4$ (Kaluza-Klein gluon) or in ${\cal C}_{8g}$, with a significant amount of fine tuning, but possible connections to the $b \to s \ell \ell$ anomalies.