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, $\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|B(p)\rangle$. Moreover, the dependence of this quantity on the variable $(x-y)^2$ is essential for a proper account of the $\left(\Lambda_{\rm QCD}m_b/m_c^2\right)^n$ terms in the amplitudes of FCNC $B$-decays.