Revisiting nonfactorizable charm-loop effects in exclusive FCNC $B$-decays

We revisit the calculation of nonfactorizable corrections induced by charm-quark loops in exclusive FCNC $B$-decays. For the sake of clarity, we make use of a field theory with scalar particles: this allows us to focus on the conceptual issues and to avoid technical complications related to particle spins in QCD. We perform a straightforward calculation of the appropriate correlation function and show that it requires the knowledge of the full generic three-particle distribution amplitude with non-aligned arguments, $⟨0\mathrm{\mid }\bar{s}(y)G_{\mu \nu }(x)b(0)\mathrm{\mid }B(p)⟩$. Moreover, the dependence of this quantity on the variable $(x-y{)}^2$ is essential for a proper account of the ${\left({\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}{m}_b\mathrm{/}{m}_c^2\right)}^n$ terms in the amplitudes of FCNC $B$-decays.