Long distance effects in inclusive rare $B$ decays and phenomenology of $\bar{B}\to X_d \ell^+\ell^-$

Rare inclusive $B$ decays such as $\bar{B}\to X_{s(d)} \ell^+\ell^-$ are interesting probes for physics beyond the Standard Model. Due to the complementarity to their exclusive counterparts, they might shed light on the anomalies currently seen in exclusive $b \to s$ transitions. Distinguishing new-physics effects from the Standard Model requires precise predictions and necessitates the control of long distance effects. In the present work we revisit and improve the description of various long distance effects in inclusive decays such as charmonium and light-quark resonances, nonfactorisable power corrections, and cascade decays. We then apply these results to a state-of-the-art phenomenological study of $\bar{B}\to X_d \ell^+\ell^-$, including also logarithmically enhanced QED corrections and the recently calculated five-body contributions. To fully exploit the new-physics potential of inclusive flavour-changing neutral current decays, the $\bar{B}\to X_d \ell^+\ell^-$ observables should be measured in a dedicated Belle II analysis.