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

TL;DR

This paper proves a leading-order factorization formula for multiparticle contributions to $B$-meson weak-decay amplitudes with a generic topology in the heavy-quark limit. The authors show that these amplitudes can be written as a convolution of a hard kernel, built from highly virtual light-propagator propagators, with a $B$-meson multiparticle light-cone wave function in a double-collinear configuration, where the $(+)$-directed and $(-)$-directed light fields are ordered along their respective light-cone directions. The analysis covers the 3-particle case and extends to arbitrary multiparticle insertions in both the upper and lower parts of the propagator line, yielding explicit expressions for the amplitudes in the semileptonic and generic form-factor topologies. Radiative corrections are discussed, with I-type corrections aligning with conventional SL-factorization results, and II-type corrections requiring a soft-function treatment that will demand further RG analysis of multiparticle soft functions. The work provides a foundational factorization framework for complex $B$-decay amplitudes and clarifies the dominant double-collinear structure that governs multiparticle contributions at leading order in $α_s$.

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 $α_s$ 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, $\langle 0|φ(x)φ(x_1)\dotsφ(x_n)φ_b(0)φ(x'_{n'})\dotsφ(x'_1)φ(x')|B(p)\rangle$, 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 $a_μ$, $a^2=0$, $q_μ\propto a_μ$, while the coordinates $x',x_1',...,x'_{n'}$ are ordered and aligned along the light-like 4-vector $a'_μ$, $a'^2=0$, $q'_μ\propto a'_μ$, $a'a\ne 0$. Corrections to this factorization formula are suppressed by powers of $Λ_{\rm QCD}/M_B$.

Figures (4)

• Figure 1: A diagram of the generic form factor topology: the heavy $b$-quark field which we place at $x=0$ hits the intermediate point of the propagator line between $x$ and $x'$. The part of the propagator line between $x=0$ and $x$, at which the momentum $q$ is emitted, contains $n$ intermediate points coupled to the light fields $\phi(x_1)...\phi(x_n)$. The part of the propagator line between $x=0$ and $x'$, at which the momentum $q'$ is emitted, contains $n'$ intermediate points coupled to the light fields $\phi(x'_n)...\phi(x'_1)$. The corresponding amplitude is denoted as $A^{(n,n')}(q,q')$.
• Figure 2: The double-collinear light-cone field configuration of the wave function of Eq. (\ref{['BSresult']}): The field coordinates $x,x_1,...,x_n,0$ are ordered along the $(+)$ axis of the light cone; The field coordinates $x',x'_1,...,x'_{n'},0$ are ordered along the $(-)$ axis of the light cone. In the reference frame where $q=(q_+,0,0)$ and $q'=(0,q'_-,0)$, this very field configuration gives the dominant contribution to the amplitude of the generic weak form factor topology.
• Figure 3: Three-particle contribution to the amplitude of the generic weak form factor topology: the heavy field $\phi_b$ hits the intermediate point of the (vertical) line along which light degrees of freedom propagate. Except for $\phi_b$, all other fields in the multiparticle wave function are light fields.
• Figure 4: (a) A typical diagram of the SL topology: the heavy field $\phi_b(0)$ and the light field $\phi(x)$ hit, respectively, the upper and the lower end points of the vertical line along which light degrees of freedom propagate, with $n$ intermediate points corresponding to $n$ light fields $\phi(x_1)...\phi(x_n)$. The fields $\phi(x)$, $\phi(x_1),...,\phi(x_n)$ as well as the propagating light degrees of freedom may be different (e.g. light quarks and gluons). (b) Coordinate, momentum and Feynman parameter notations for the amplitude of figure (a).