Factorization of multiparticle contributions to amplitudes of B-meson weak decays

We show that multiparticle contributions to amplitudes of weak decays of the generic topology (heavy quark hits some intermediate point of the propagator line joining the end-points from which momenta $q$ and $q^{\mathrm{\prime }}$ are emitted) is given in the heavy quark limit and at the leading order in ${\alpha }_s$ by the convolution of (i) hard kernel composed of highly virtual propagators of light degrees of freedom and (ii) the $B$-meson multiparticle wave function, $⟨0\mathrm{\mid }\phi (x)\phi (x_1)\dots \phi (x_n){\phi }_b(0)\phi (x_{n^{\mathrm{\prime }}}^{\mathrm{\prime }})\dots \phi (x_1^{\mathrm{\prime }})\phi (x^{\mathrm{\prime }})\mathrm{\mid }B(p)⟩$, in a {\it double-collinear} light-cone configuration: the coordinates $x,x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ are ordered and aligned along the light-like 4-vector $a_{\mu }$, $a^2=0$, $q_{\mu }\propto a_{\mu }$, while the coordinates $x^{\mathrm{\prime }},x_1^{\mathrm{\prime }},\mathrm{.}\mathrm{.}\mathrm{.},x_{n^{\mathrm{\prime }}}^{\mathrm{\prime }}$ are ordered and aligned along the light-like 4-vector $a_{\mu }^{\mathrm{\prime }}$, $a^{\mathrm{\prime }2}=0$, $q_{\mu }^{\mathrm{\prime }}\propto a_{\mu }^{\mathrm{\prime }}$, $a^{\mathrm{\prime }}a\mathrm{\ne }0$. Corrections to this factorization formula are suppressed by powers of ${\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}\mathrm{/}{M}_B$.