Probing the photon polarization in $B \to K^*\gamma$ with conversion

We re-examine the possibility to measure the photon polarization in $B \to K^*\gamma$ decays, via decays in which the photon subsequently undergoes nuclear conversion to a lepton pair. We obtain compact expressions for the full decay-plus-conversion amplitude. With these results we show that interference between the $B \to (K^*\to K\pi)\gamma$ decay and the $\gamma N \to \ell^+\ell^-N$ conversion permits both the ratio and relative weak phase between the left- and right-handed photon amplitudes to be probed by an angular observable, constructed from the final state dilepton, kaon and pion kinematic configuration. Exploiting this technique will be experimentally challenging. However, we present special kinematic cuts that enhance the statistical power of this technique by an $\mathcal{O}(1)$ factor. We verify this effect and extract pertinent angular kinematic distributions with dedicated numerical simulations.