# Three-particle distribution in B meson and charm-quark loops in FCNC B decays

## TL;DR

This paper analyzes the nonfactorizable charm-quark loop contributions to FCNC B-meson decays, emphasizing the role of the three-particle Bethe-Salpeter amplitude and its properties in the heavy-quark limit.

## Contribution

It clarifies the properties of the B-meson three-particle Bethe-Salpeter amplitude necessary for accurately describing charm-loop effects in FCNC B-decays.

## Key findings

- Dominant charm-loop contribution described by convolution with B-meson 3BS in double-collinear configuration.
- Emergence of new constraints on distribution amplitudes parametrizing the 3BS.
- Heavy-quark limit simplifies the nonfactorizable charm contribution analysis.

## Abstract

We discuss a nonfactorizable (NF) contribution of a charm loop to the FCNC $B$-decay amplitude given through the three-particle Bethe-Salpeter amplitude (3BS) of the $B$-meson. This 3BS contains one heavy-quark field and two light fields (a light quark and a gluon). Our discussion is aimed at clarifying properties of the $B$-meson 3BS necessary to describe properly charm-loop contributions to the amplitudes of FCNC $B$-decays. We demonstrate that the dominant contribution of nonfactorizable charm to FCNC $B$-decay amplitude is given in the heavy-quark limit by a convolution of some hard kernel and the $B$-meson 3BS in a "double-collinear" light cone (LC) configuration: one of the light degrees of freedom $\phi(x)$, $x^2=0$, lies on the $(+)$-direction of the LC, whereas another light degree of freedom $\phi'(x')$, $x'^2=0$ lies on the $(-)$-direction. We show the emergence of new constraints on the distribution amplitudes which parametrize the 3BS in this double-collinear configuration.