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

Abstract

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 $q'$ 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, $\langle 0|\phi(x)\phi(x_1)\dots\phi(x_n)\phi_b(0)\phi(x'_{n'})\dots\phi(x'_1)\phi(x')|B(p)\rangle$, in a {\it double-collinear} light-cone configuration: the coordinates $x,x_1,...,x_n$ are ordered and aligned along the light-like 4-vector $a_\mu$, $a^2=0$, $q_\mu\propto a_\mu$, while the coordinates $x',x_1',...,x'_{n'}$ are ordered and aligned along the light-like 4-vector $a'_\mu$, $a'^2=0$, $q'_\mu\propto a'_\mu$, $a'a\ne 0$. Corrections to this factorization formula are suppressed by powers of $\Lambda_{\rm QCD}/M_B$.