Rare $b\to s\ell^+\ell^-$ decays provide some of the most sensitive tests of the Standard Model and require precise and systematically improvable hadronic input from lattice QCD.
For the phenomenologically important channel $B\to K^*\ell^+\ell^-$ this entails a first-principles treatment of a resonant $K\pi$ final state together with controlled heavy-quark dynamics.
We present the status of a new exploratory lattice calculation that combines a variational determination of finite-volume $K\pi$ states with the $1+J\to2$ finite-volume formalism to access the relevant matrix elements.
The computation is carried out on an RBC/UKQCD domain-wall fermion ensemble with $a^{-1} \approx 2.7\,\mathrm{GeV}$ and employs a dual heavy-quark strategy, using both a relativistic heavy-quark action tuned to the physical $b$ mass and domain-wall heavy masses extrapolating from charm.
All correlation functions are computed using (stochastic) distillation, providing a versatile setup that supports a broad range of heavy-to-light transitions into resonant final states.
We show first two-point results for the $K^*\leftrightarrow K\pi$ system and discuss the accessible kinematic region, which allows for a controlled study at high $q^2$.
The outlook for extending the calculation to lower $q^2$ and for incorporating effects from charmonium resonances is outlined.