#  Non-factorizable charming loops in FCNC B decays
vs B-decay semileptonic form factors

Dmitri Melikhova,b,c aD. V. Skobeltsyn Institute of Nuclear Physics, M. V.
Lomonosov Moscow State University, 119991 Moscow, Russia
bJoint Institute for Nuclear Research, 141980 Dubna, Russia
cFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria

###### Abstract

We compare a non-factorizable charming-loop correction to an exclusive FCNC
$B$-decay given in terms of the 3-particle Bethe-Salpeter amplitude (3BS) of
the $B$-meson, $\langle 0|\bar{q}(x)G_{\mu\nu}(z)b(0)|B(p)\rangle$, with the
corresponding correction to the $B$-meson semileptonic form factor. In spite
of certain similarities, these two corrections are shown to have substantial
differences: The form factor correction is dominated by the collinear light-
cone configuration of 3BS: $z_{\mu}=ux_{\mu}$, $0<u<1$, $x^{2}=0$. In
contrast, the FCNC amplitude is dominated by a different configuration with
non-collinear arguments: $x^{2}=0$, $z^{2}=0$, but $(x-z)^{2}\neq 0$ (i.e.,
$z_{\mu}\neq ux_{\mu}$).

##  1 Introduction

Charming loops in rare FCNC (flavour-changing neutral current) decays of the
$B$-meson have impact on the $B$-decay observables neubert  providing an
unpleasant noise for the studies of possible new physics effects (see, e.g.,
recent discussions ciuchini2020 ; ciuchini2021 ; diego2021 ; matias2022  and
refs therein).

A number of theoretical analyses of non-factorizable (NF) charming loops in
FCNC $B$-decays has been published. We mention here those directly related to
the discussion of this paper: in voloshin , an effective gluon-photon local
operator describing the charm-quark loop has been calculated as an expansion
in inverse charm-quark mass $m_{c}$ and applied to inclusive $B\to
X_{s}\gamma$ decays (see also ligeti ; buchalla ); in khod1997 , NF
corrections in $B\to K^{*}\gamma$ using local operator product expansion (OPE)
have been studied; NF corrections induced by the local photon-gluon operator
have been calculated in zwicky1 ; zwicky2  in terms of the light-cone (LC)
3-particle antiquark-quark-gluon Bethe-Salpeter amplitude (3BS) of
$K^{*}$-meson braun ; ball1 ; ball2  with two field operators having equal
coordinates, $\langle 0|\bar{s}(0)G_{\mu\nu}(0)u(x)|K^{*}(p)\rangle$,
$x^{2}=0$. However, local OPE for the charm-quark loop in FCNC $B$ decays
leads to a power series in $\Lambda_{\rm QCD}m_{b}/m_{c}^{2}$; numerically
this parameter is close to one. To sum up $O(\Lambda_{\rm
QCD}m_{b}/m_{c}^{2})^{n}$ corrections, Ref. hidr  obtained a nonlocal photon-
gluon operator describing the charm-quark loop and evaluated its effect making
use of 3BS of the $B$-meson in a collinear LC approximation $\langle
0|\bar{s}(x)G_{\mu\nu}(ux)b(0)|B(p)\rangle$, $x^{2}=0$. This approximation was
used later for the analysis of other FCNC $B$-decays gubernari2020 .

The collinear LC configuration is known to provide the dominant 3BS
contribution to meson form factors braun1994 ; offen2007 , in particular, to
form factors of semileptonic (SL) $B$-decay induced by the tree-level $b\to u$
weak charged current (CC). So it may seem attractive to express also the FCNC
$B$-decay amplitude via this collinear LC 3BS of the $B$-meson.

However, the 3BS contribution to the CC $B$-decay and to the FCNC $B$-decay
have a qualitative difference: Let us consider the $B$-decay in the $B$-meson
rest frame. In CC $B$-decays, the $b$-quark emits a fast light $u$-quark which
is later hit by a soft gluon and thus keeps moving in the same space
direction. In FCNC $B$-decays, a fast light $s$-quark and a pair of fast
$c$-quarks emitted by the $b$-quark move in the opposite space directions. We
shall demonstrate that, as the consequences of this difference, the $B$-meson
CC weak form factor is dominated by a collinear LC configuration $\langle
0|\bar{q}(x)G_{\mu\nu}(ux)b(0)|B(p)\rangle$, $x^{2}=0$ japan , whereas the
FCNC $B$-decay amplitude is dominated by a non-collinear configuration
$\langle 0|\bar{q}(x)G_{\mu\nu}(z)b(0)|B(p)\rangle$, $x^{2}=0$, $z^{2}=0$, but
$(x-z)^{2}\neq 0$ mk2018 ; m2019 . The first application of a NC 3BS to FCNC
$B$-decays was presented in wang2022 .

We study the general properties of the 3BS contributions to the amplitudes of
$B$-decays and formulate the conditions necessary for the dominance of the
amplitude by a collinear 3BS configuration. We perform the analysis using
field theory with scalar quarks/gluons which is free of technical
complications and allows one to focus on the conceptual issues; the
generalization of our analysis to QCD is straightforward. Section 2
demonstrates the technical similarities between the CC and the FCNC amplitudes
and their equivalence to the generic diagram of the form factor topology, and
Section 3 studies the conditions under which this generic diagram is dominated
by a collinear 3BS configuration. As follows from this analysis, large $O(1)$
corrections to the collinear LC 3BS contribution should emerge in the
amplitudes of FCNC $B$-decays. Chapter 4 studies in detail the FCNC $B$-decay
amplitude, including the cases of the light $u$-quark and the $c$-quark in the
triangle loop, adopting for the latter case the counting scheme $\Lambda_{\rm
QCD}m_{b}/m_{c}^{2}\simeq 1$ paz . The origin of large $O(1)$ corrections to
the collinear LC approximation is identified: namely, we show that a non-
collinear 3BS configuration (both $x$ and $z$ on the light cone, but on
different axes: $x$ on the $(+)$ axis, and $z$ on the $(-)$ axis or vice versa
paz2010 ) give parametrically unsuppressed contributions compared to the
collinear LC 3BS contribution. So, the full dependence of 3BS of the $B$-meson
on the variable $(x-z)^{2}$ is necessary to sum properly the $(\Lambda_{\rm
QCD}m_{b}/m_{c}^{2})^{n}$ corrections in FCNC $B$-decays.

##  2 The amplitudes of $B$-meson CC decay vs FCNC decay

In this Section we show that the 3BS contributions to the CC and to the FCNC
amplitudes may be reduced to the diagram of a generic form factor topology
with an (essential) difference of the location of the heavy-quark field in
this diagram.

###  2.1 The amplitude of semileptonic $B$-meson decay induced by weak charged
current

To exemplify the essential part of our analysis, we neglect the spins of the
$B$-meson constituents (quarks and gluons are treated as scalar fields) as
well as the Lorentz structure of the weak currents. So, instead of the full
QCD amplitude describing a semileptonic (SL) $B$-meson weak decay induced by a
charged current,

$\displaystyle A^{\nu}_{\rm SL}(p|q,q^{\prime})=i\int
dx_{1}dx_{3}e^{iqx_{1}+iq^{\prime}x_{3}}\langle 0|T\left\\{\bar{u}(x_{3}){\cal
O}u(x_{3}),\bar{u}(x_{1})\gamma^{\nu}(1-\gamma_{5})b(x_{1})\right\\}|B_{u}(p)\rangle,$
(2.1)

$\cal O$ is a Dirac matrix, we consider the amplitude

$\displaystyle A_{\rm SL}(p|q,q^{\prime})=i\int
dx_{1}dx_{3}e^{iqx_{1}+iq^{\prime}x_{3}}\langle
0|T\left\\{u^{\dagger}(x_{3})u(x_{3}),u^{\dagger}(x_{1})b(x_{1})\right\\}|B_{u}(p)\rangle$
(2.2)

with scalar “quarks” and “gluons”. Here $q$ is the momentum emitted by the
weak vertex, and $q^{\prime}$ is the momentum emitted by the interpolating
current of the outgoing state, see Fig. 1. The $u$ and the $b$ quark fields
are the Heisenberg operators with respect to the strong interaction. We will
be interested in the part of the amplitude (2.2) that emerges at the first
order of the expansion in the strong coupling and contains the gluon field
$G$. The corresponding Feynman graph is shown in Fig. 1. After some
manipulations (and making use of the Fock-Schwinger gauge, see braun1994 ;
offen2007 ) the relevant to us part of this amplitude may be written as

$\displaystyle A_{\rm SL}(p|q,q^{\prime})$ $\displaystyle=$ $\displaystyle\int
dx_{1}dx_{2}dx_{3}dkdk^{\prime}\,e^{iqx_{1}-ik(x_{2}-x_{1})}\,e^{-ik^{\prime}(x_{3}-x_{2})+iq^{\prime}x_{3}}$
(2.3)
$\displaystyle\;\;\times\frac{1}{m_{u}^{2}-k^{2}}\frac{1}{m_{u}^{2}-k^{\prime
2}}\langle 0|u^{\dagger}(x_{3})G(x_{2})b(x_{1})|B_{s}(p)\rangle.$

Making use of the transformation properties of field operators under
translations, we may shift the coordinate of the gluon field to zero and
introduce the new variables through the relations $k=\kappa_{1}-q$ and
$k^{\prime}=q^{\prime}-\kappa_{3}$, $x_{1}-x_{2}\to x_{1}$ and $x_{3}-x_{2}\to
x_{3}$. After that, we perform the $x_{2}$-integration that leads to the
momentum conservation $\delta(p-q-q^{\prime})$, and the amplitude (2.3) takes
the form

$\displaystyle A_{\rm SL}(p|q,q^{\prime})$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta(p-q-q^{\prime})\overline{A}_{\rm
SL}(p|q,q^{\prime}),$ (2.4)

with

$\displaystyle\overline{A}_{\rm SL}(p|q,q^{\prime})$ $\displaystyle=$
$\displaystyle\int
dx_{1}dx_{3}d\kappa_{1}d\kappa_{3}\,e^{i\kappa_{1}x_{1}+i\kappa_{3}x_{3}}\frac{1}{m_{u}^{2}-(\kappa_{1}-q)^{2}}\frac{1}{m_{u}^{2}-(q^{\prime}-\kappa_{3})^{2}}\langle
0|u^{\dagger}(x_{3})G(0)b(x_{1})|B_{s}(p)\rangle.$ (2.5)

Fig. 1:  Momentum notations in the 3BS contribution to the form factor
describing weak $b\to u$ semileptonic $B$-decay.

###  2.2 Non-factorizable part of the amplitude of FCNC $B$-decay

The amplitude of FCNC $B_{s}$-decay such as, e.g.
$B_{s}\to\gamma^{*}\gamma^{*}$ decay, is given by the following expression
mnk2018 :

$A^{\nu\mu}_{\rm FCNC}(p|q,q^{\prime})=\int
dx\exp(iqx)dx_{3}\exp(iq^{\prime}x_{3})\langle
0|T\left\\{\bar{c}(x)\gamma^{\nu}c(x),\bar{s}(x_{3})\gamma^{\mu}s(x_{3})\right\\}|B_{s})p)\rangle.$
(2.6)

Here the $c$ and $s$ quark operators are the Heisenberg operators with respect
to strong and weak interactions. Notice that in the case of an FCNC $B$-decay,
the $b$-quark field is not contained in the current operators under the
$T$-product, but comes into the came at order $G_{F}$ in weak interaction. The
part of the amplitude describing the non-factorizable contribution of the
charming loop emerges at order $G_{F}$ in weak interaction and at the first
order in strong interaction. The contribution of the charming loop is given
through the gluon field $G_{\mu\nu}$ lm , so the Feynman diagram describing
the FCNC amplitude (2.6) is represented by Fig. 2.

Fig. 2: Feynman diagram describing the 3BS contribution to a NF amplitude of
FCNC $B$-decay. The crossed propagator line means that the propagator is
replaced by the triangle charming loop $\Gamma_{cc}(\omega_{1}p,q)$.

We omit all complications related to Lorentz and spinor structure details and
consider scalar quark fields $b$, $c$, and $s$, and a scalar gluon field $G$.
We then come to the following expression for the non-factorizable part of the
amplitude of FCNC $B$-decay, see Fig. 2:

$\displaystyle A_{\rm
FCNC}(p|q,q^{\prime})=(2\pi^{4})\delta(p-q-q^{\prime})\overline{A}_{\rm
FCNC}(p|q,q^{\prime})$ (2.7)

with

$\displaystyle\overline{A}_{\rm
FCNC}(p|q,q^{\prime})=\frac{G_{F}}{\sqrt{2}}\int
dx_{1}dx_{3}d\kappa_{1}d\kappa_{3}e^{ix_{1}\kappa_{1}+i\kappa_{3}x_{3}}\Gamma_{cc}(\kappa_{1},q)\frac{d\kappa_{3}}{m_{s}^{2}-(q^{\prime}-\kappa_{3})^{2}}\,\langle
0|s^{\dagger}(x_{3})G(x_{1})b(0)|B_{s}(p)\rangle.$ (2.8)

Here $\Gamma_{cc}(\kappa_{1},q)$ is the charm-quark triangle diagram which may
be written as a double integral in Feynman parameters (see a detailed
discussion in hidr ; mk2018 ; m2019 ; mnk2018 ; lm )

$\displaystyle\Gamma_{cc}(\kappa_{1},q)=\frac{1}{8\pi^{2}}\int\limits_{0}^{1}du\int\limits_{0}^{1}dv\frac{\theta(u+v<1)}{m_{c}^{2}-uv(\kappa_{1}-q)^{2}-u(1-u-v)\kappa_{1}^{2}-v(1-v-u)q^{2}}.$
(2.9)

Important for us is that the quantity is a quadratic function in momentum
variables. So, the FCNC amplitude is similar to the SL amplitude, with the
light-quark propagator replaced by an “effective” propagator
$\Gamma_{cc}(\kappa_{1},q)$

$\displaystyle\frac{1}{m_{u}^{2}-(\kappa_{1}-q)^{2}}\to\Gamma_{cc}(\kappa_{1},q).$
(2.10)

The main difference between the SL and the FCNC amplitudes arises from the
fact that the heavy $b$-quark in a SL decay amplitude is attached to the end-
point of the line connecting $x_{1}$ and $x_{3}$, along which the energetic
light quarks are propagating, while in the FCNC the heavy field is attached to
the middle point of the line connecting $x_{1}$ and $x_{3}$. We shall see that
this features of the SL and FCNC $B$-decays are responsible for a qualitative
difference between the configurations of the 3BS of the $B$-meson, that
provide the dominant contributions in SL $B$ decays and in FCNC $B$-decays.

To demonstrate this difference, the next Section considers the general
amplitude of the form-factor topology and figures out the properties necessary
for the dominance of the collinear 3BS configuration.

##  3 3BS contribution to a generic amplitude of the form factor topology

Let us consider the generic form factor amplitude $A(p|q,q^{\prime})$ shown in
Fig. 3.

Fig. 3:  3BS contribution to the generic amplitude of the form factor
topology.

As we have mentioned above, both $A_{\rm FCNC}$ and $A_{\rm SL}$ are reduced
to this amplitude, such that each of the FCNC and the SL amplitudes is
characterized by a specific (and different) content of the heavy and the light
fields $\varphi_{1,2,3}$: in FCNC decays, the field $\varphi_{2}$ is heavy,
while $\varphi_{1,3}$ are light; in SL decays, $\varphi_{1}$ is heavy, while
$\varphi_{2,3}$ are light. If one considers the case of the 3BS correction to
the form factor of a light meson, all fields $\varphi_{1,2,3}$ are light
degrees of freedom (light quarks or gluons).

The properties of the set of meson constituents $\varphi_{1,2,3}$ (i.e., which
of these fields are heavy and which are light) are reflected in the properties
of the amplitude $\langle
0|\varphi_{1}(x_{1})\varphi_{2}(x_{2})\varphi_{3}(x_{3})|B(p)\rangle$. The
goal of our analysis in this Section is to figure out the kinematical
configuration of the constituent fields that dominate the amplitude of Fig. 3.

The analytic expression corresponding to the diagram of Fig. 3 has the form

$\displaystyle
A(p\,|q,q^{\prime})=\int\frac{dx_{1}\,dx_{2}\,dx_{3}\,dk\,dk^{\prime}}{(\mu^{2}-k^{2})(m^{2}-k^{\prime
2})}e^{iqx_{1}-ik(x_{2}-x_{1})-ik^{\prime}(x_{3}-x_{2})+iq^{\prime}x_{3}}\langle
0|\varphi_{1}(x_{1})\varphi_{2}(x_{2})\varphi_{3}(x_{3})|B(p)\rangle.$ (3.11)

As already mentioned above, the amplitude contains $\delta(p-q-q^{\prime})$
which may be isolated by making use of the transformation properties of the
field operators $\varphi_{1,2,3}$ under translations: one can set one of the
arguments of the field operators equal zero, and integrate over one of the
coordinate differences. For the moment we will not make use of this property
and will keep all three arguments $x_{1,2,3}$ nonzero.

By introducing the Feynman parameter $v$ to combine two propagators in a
single propagator squared, and after redefinitions of the variables

$\displaystyle\tilde{k}=k-v\kappa,$ (3.12) $\displaystyle
x_{2}=x_{1}(1-v)+x_{3}v+z_{2},$ (3.13)

the amplitude takes the form convenient for a further analysis

$\displaystyle A(p|q,q^{\prime})$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}dv\int
d\kappa\,d\tilde{k}\,dx_{1}\,dz_{2}\,dx_{3}\
\frac{e^{ix_{1}(\tilde{k}+q)+ix_{3}(q^{\prime}-\tilde{k})+i\kappa
z_{2}}}{\left[m^{2}(1-v)+\mu^{2}v-\tilde{k}^{2}-v(1-v)\kappa^{2}\right]^{2}}$
(3.14) $\displaystyle\times\langle
0|\varphi_{1}(x_{1})\varphi_{2}(x_{1}(1-v)+x_{3}v+z_{2})\varphi_{3}(x_{3})|B(p)\rangle,$

Here $\kappa$ is the momentum transferred in the central point $x_{2}$ and the
variable $z_{2}$ measures the deviation of the configuration
$x_{1},x_{2},x_{3}$ from the straight line joining the end points $x_{1}$ and
$x_{3}$.

To proceed further, one can attempt to expand $\langle
0|\varphi_{1}(x_{1})\varphi_{2}(x_{1}(1-v)+x_{3}v+z_{2})\varphi_{3}(x_{3})|B(p)\rangle$
in powers of $z_{2}$ and obtain in this way a tower of collinear operators of
the increasing dimension containing derivatives of $\varphi_{2}$. The
expansion in powers of $z_{2}$ corresponds to expanding the denominator in
powers of $\kappa^{2}$. Such expansion is meaningful if $\kappa$ is soft
compared to the virtualities of the propagators $D(x_{1}-x_{2})$ and
$D(x_{2}-x_{3})$. Obviously, the amplitude is dominated by a collinear
configuration of the 3BS only if the momentum transferred in the central
vertex is soft compared to the virtualities of the particle propagators along
the line $x_{1}$-$x_{3}$.

There are phenomenologically relevant cases where the collinear 3BS
configuration indeed dominates the amplitude $A(p|q,q^{\prime})$:

  * $\bullet$

QCD radiative correction to the $B\to j_{1}j_{2}$ weak form factor. In this
case the field $\varphi_{1}$ is heavy, whereas $\varphi_{2}$ and $\varphi_{3}$
are light (the gluon and the light quark, respectively). If $q^{2},q^{\prime
2}\ll M_{B}^{2}$, $\kappa^{2}=O(\Lambda_{\rm QCD}^{2})$, the virtualities of
the particles propagating along the segments $x_{1}-x_{2}$ and $x_{2}-x_{3}$
are $O(m_{b}\Lambda_{\rm QCD})$, and $\kappa^{2}\simeq\Lambda_{\rm QCD}^{2}$.
The expansion in $\kappa^{2}$ seems meaningful and the amplitude is dominated
by a collinear light-cone (LC) configuration $x_{2}=x_{1}\,(1-v)+x_{3}\,v$,
$x_{1}^{2}\simeq 0$ and $x_{3}^{2}\simeq 0$.

  * $\bullet$

QCD radiative correction to $M\to j_{1}j_{2}$ form factor, $M$ is a light
meson. In this case, all three fields $\varphi_{1,2,3}$ are light
($\varphi_{1}$ and $\varphi_{3}$ are the light quarks and $\varphi_{2}$ is the
gluon). To make the OPE convergent, we cannot set $q^{2}=q^{\prime 2}=0$, but
must keep $q^{2}\simeq q^{\prime 2}\ll-1\,{GeV}^{2}$. In this case both quark
propagators are highly virtual, $O(q^{2},q^{\prime 2})$, the momentum transfer
$\kappa$ is soft, $\kappa^{2}=O(\Lambda_{\rm QCD}^{2})$, and the collinear 3BS
configuration dominates the amplitude.

Unfortunately, the non-factorizable correction to the FCNC amplitude of the
$B$-meson decay does not fall into this class of processes: $\varphi_{1}$ and
$\varphi_{3}$ are light degrees of freedom (the gluon and the light quark,
respectively), whereas $\varphi_{2}$ is a heavy quark which carries almost the
full momentum of the $B$-meson, $\kappa_{2}^{2}\sim M_{B}^{2}$. The momentum
$\kappa_{2}$ is thus by far not soft compared to the virtualities of the
particles along the line $x_{1}$-$x_{3}$, and the expansion around the
collinear 3BS configuration does not converge: Expanding in powers of
$\kappa_{2}^{2}$ leads to a series in which all terms have the same order of
magnitude.

In the next Section, we look in more detail what kind of expansion of the
amplitude arises in this case. In particular, we show that a non-collinear 3BS
configuration with $(x_{1}-x_{2})^{2}=0$, $(x_{2}-x_{3})^{2}=0$, but
$(x_{1}-x_{3})^{2}\neq 0$ dominates the FCNC amplitude.

##  4 The amplitude of FCNC $B$-decay

We start with briefly recalling the general properties of the 3-particle
(antiquark-quark-gluon) BS wave function of the meson and then obtain the
$B$-decay FCNC amplitude in terms of this BS wave function. To simplify the
analytical expressions, we will consider the case $q^{2}=q^{\prime 2}=0$.

###  4.1 Parametrization of the 3-particle BS amplitude

The 3-particle BS amplitude may be written in the form (see e.g. braun1994 ;
ball2 ; japan )111 Refs. braun1994 ; ball2 ; japan  provide the expansion of
3BS for collinear arguments $x_{1}=vx_{3}$. A generalization to the case of
non-collinear arguments presented here is straightforward.

$\displaystyle\langle
0|G(x_{1})b(x_{2})s^{\dagger}(x_{3})|B_{s}(p)\rangle=\int D\omega
e^{-i\omega_{1}x_{1}p-i\omega_{2}x_{2}p-i\omega_{3}x_{3}p}\left[\psi_{0}(\omega)+\psi_{12}(\omega)x_{12}^{2}+\psi_{13}(\omega)x_{13}^{2}+\psi_{23}(\omega)x_{23}^{2}+\dots\right],$

where

$\displaystyle x_{ij}^{2}=(x_{i}-x_{j})^{2},\quad D\omega\equiv
d\omega_{1}d\omega_{2}d\omega_{3}\delta(1-\omega_{1}-\omega_{2}-\omega_{3}),\quad\psi_{i}(\omega)\equiv\psi_{i}(\omega_{1},\omega_{2},\omega_{3}).$
(4.16)

Since the $b$-quark is heavy, all functions $\psi_{i}(\omega)$ in the
amplitide (4.1) have support in the end-point regions

$\displaystyle\omega_{2}\sim 1-O(\Lambda_{\rm QCD}/m_{b}),\quad$
$\displaystyle\omega_{1,3}\sim O(\Lambda_{\rm QCD}/m_{b}).$ (4.17)

One of the arguments of the field operators may be set to zero by using the
transformation properties of the field operators under translations, so we set
the coordinate of the heavy $b$-quark to zero $x_{2}=0$ (in this way
$\delta(p-q-q^{\prime})$-function describing the momentum conservation has
been singled out from $A_{\rm FCNC}$). As the next step, we insert this
expression in the general formula for the amplitude (3.11) which takes the
following form

$\displaystyle\overline{A}(p\,|q,q^{\prime})=\int
dx_{1}\,dx_{3}\,d\kappa_{1}\,d\kappa_{3}e^{i\kappa_{1}x_{1}+i\kappa_{3}x_{3}}\Gamma_{cc}(\kappa_{1},q)D_{s}(\kappa_{3}-q^{\prime})\Psi_{p}(x_{1},x_{3}),$
(4.18)

where

$\displaystyle\Psi_{p}(x_{1},x_{3})\equiv\langle
0|G(x_{1})b(0)s^{\dagger}(x_{3})|B(p)\rangle$ (4.19)

and $D_{s}(k)=1/(m_{s}^{2}-k^{2}-i0)$. We will not introduce the Feynman
parameter $v$ to combine the propagators as was done in Eq. (3.14), but
evaluate the amplitude directly. Following the results of the previous
section, we are going to demonstrate directly that the FCNC amplitude is
indeed not dominated by the collinear field configuration.

###  4.2 The contribution of the $\psi_{0}$ term in 3BS (neglecting all powers
of $x_{ij}^{2}$).

The term $\propto\psi_{0}(\omega)$ in 3BS does not contain $x_{ij}^{2}$ so its
contribution to $A(q,p)$ is calculated easily: the integrals over $x_{1,3}$
give the $\delta(\kappa_{1}-\omega_{1}p)$ and $\delta(\kappa_{3}-\omega_{3}p)$
so one ends up with the following expression (see Fig. 4):

$\displaystyle
A_{\psi_{0}}(q,p)=\int\limits_{0}^{1}d\omega_{1}\int\limits_{0}^{1-\omega_{1}}d\omega_{3}\,\psi_{0}(\omega_{1},\omega_{3})\Gamma_{cc}\left(\omega_{1}p,q\right)D_{s}\left((q^{\prime}-\omega_{3}p)^{2}\right).$
(4.20)

Fig. 4:  Momenta values in the 3BS contribution to the amplitude of FCNC
$B$-decay. The crossed propagator line means that the propagator is replaced
by the triangle charming loop $\Gamma_{cc}(\omega_{1}p,q)$

####  4.2.1 $s$-quark propagator

The denominator of the $s$-quark propagator takes the form

$\displaystyle m_{s}^{2}-(\omega_{3}p-q^{\prime})^{2}=m_{s}^{2}-\lambda
q^{2}-(1-\omega_{3})q^{\prime 2}+(1-\omega_{3})\omega_{3}M_{B}^{2}.$

For $q^{2}=q^{\prime 2}=0$, in the region $\omega_{3}\sim\Lambda_{\rm
QCD}/m_{b}$ that dominates the integral, the $s$-quark is highly virtual: its
momentum squared is $O(\Lambda_{\rm QCD}m_{b})$.

####  4.2.2 Charm-quark loop

The analytic expression for the triangle charming loop was already given in
(2.9). We now rewrite it in a slightly different way by introducing a new
variable $0<\tau<1$, $v=(1-u)\tau$:

$\displaystyle\Gamma_{cc}(\kappa_{1},q)=\frac{1}{8\pi^{2}}\int\limits_{0}^{1}du(1-u)\int\limits_{0}^{1}d\tau\frac{1}{m_{c}^{2}-u(1-u)[\tau(\kappa_{1}-q)^{2}+\kappa_{1}^{2}(1-\tau)]-q^{2}(1-u)^{2}\tau(1-\tau)}.$
(4.21)

To shorten the formulas, we set hereafter $q^{2}=0$. This expression appears
under the convolution Eq. (4.18) with the 3BS of the $B$-meson. Then the
$\omega_{1}$-integral is peaked near $\omega_{1}\sim\Lambda_{\rm QCD}/m_{b}$
so the gluon is soft: $\kappa_{1}=\omega_{1}p$ and $\kappa_{1}^{2}\sim
O(\Lambda_{\rm QCD}^{2})\ll m_{c}^{2}$. In (4.21), $\kappa_{1}^{2}$ may be
neglected compared to $m_{c}^{2}$, and we find

$\displaystyle\Gamma_{cc}(\kappa_{1},q)=\frac{1}{8\pi^{2}}\int\limits_{0}^{1}du(1-u)\int\limits_{0}^{1}d\tau\frac{1}{m_{c}^{2}-u(1-u)\tau(\kappa_{1}-q)^{2}}.$
(4.22)

Taking the $\tau$-integral one comes to a relation very similar to the one
obtained in hidr  (up to the appropriate changes related to the spins of the
constituents and the Lorentz structures of the currents).

Let us make two remarks:

  * •

The obtained expression Eq. (4.20) for the amplitude $A_{\psi_{0}}(q,p)$ does
not “feel” the relative location of the arguments $x_{i}$. In particular,
$\psi_{0}$ is precisely the same function that parameterizes, e.g., the
collinear LC configuration discussed in hidr . Moreover, up to technical
complications related to the spins of the $B$-meson constituents and the
Lorentz structure of the currents, the obtained expression corresponds to the
approximation considered in hidr . So one may say that the $A_{\psi_{0}}$
approximation to the FCNC amplitude corresponds to the contribution of the
collinear LC 3BS of the $B$-meson.

  * •

We have shown in the previous Section, that the collinear approximation does
not dominate the FCNC amplitude, and expanding the amplitude near the
collinear configuration should lead to sizeable $O(1)$ corrections to the
collinear approximation. The 3BS feels the relative location of the arguments
$x_{i}$ only starting with the terms $x_{ij}^{2}$. So, based on the general
argument, one expects sizeable corrections, of the order of unity, coming from
powers of $x_{ij}^{2}$. We shall see that indeed large $O(1)$ corrections
emerge from all powers of $(x-z)^{2}$, whereas terms containing powers of
$x^{2}$ and $z^{2}$ lead to the suppressed contributions.

###  4.3 Contributions induced by $x_{ij}^{2}$ terms in the 3-particle BS
amplitude

We now turn to the calculation of the contributions of $x_{ij}^{2}$ terms in
the 3BS amplitude.

In the problem under consideration one encounters two heavy quark scales,
$m_{c}$ and $m_{b}$, such that $\Lambda_{\rm QCD}\ll m_{c}\ll m_{b}$. Taking
into account the real values of the quark masses one encounters a new
parameter of order of unity:

$\displaystyle\Lambda_{\rm QCD}m_{b}/m_{c}^{2}\simeq 1.$ (4.23)

One needs therefore to sum all powers of the parameter $\Lambda_{\rm
QCD}m_{b}/m_{c}^{2}$. We shall see that this task is related to a summation of
all corrections of the order $(x_{1}-x_{3})^{2n}$.

$\bullet$ Let us start with the $x_{1\mu}$ term under the integral. It may be
written as

$\displaystyle
x_{1\mu}e^{i\kappa_{1}x_{1}}\Gamma_{cc}(\kappa_{1},q)d\kappa_{1}=-i\frac{\partial}{\partial\kappa_{1}^{\mu}}e^{i\kappa_{1}x_{1}}\Gamma_{cc}(\kappa_{1},q)d\kappa_{1}=ie^{i\kappa_{1}x_{1}}\frac{\partial}{\partial\kappa_{1}^{\mu}}\Gamma_{cc}(\kappa_{1},q)\to
i\frac{\partial}{\partial\kappa_{1}^{\mu}}\Gamma_{cc}(\kappa_{1},q)|_{\kappa_{1}=\omega_{1}p}.$
(4.24)

where we have performed the parts integration. The $x_{1}$-dependence then
remains only in the exponential factors, and $x_{1}$-integartion may be taken
and leads to $\delta(\kappa_{1}-\omega_{1}p)$.

$\bullet$ Similarly, the $x_{3\mu}$ term under the integral may be handled
leading to $\delta(\kappa_{3}-\omega_{3}p)$:

$\displaystyle
x_{3\mu}e^{i\kappa_{3}x_{3}}D_{s}(\kappa_{3}-q^{\prime})d\kappa_{3}=-i\frac{\partial}{\partial\kappa_{3}^{\mu}}e^{i\kappa_{3}x_{3}}D_{s}(\kappa_{3}-q^{\prime})d\kappa_{3}=ie^{i\kappa_{3}x_{3}}\frac{\partial}{\partial\kappa_{3}^{\mu}}D_{s}(\kappa_{3}-q^{\prime})\to
i\frac{\partial}{\partial\kappa_{3}^{\mu}}D_{s}(\kappa_{3}-q^{\prime})|_{\kappa_{3}=\omega_{3}p}.$
(4.25)

Making use of these formulas we obtain the factors in the integrands that
describe the relative contributions to $A_{\psi_{ij}}(p,q)$ of the
$x_{ij}^{2}\psi_{ij}$ terms compared to the $\psi_{0}$ contribution to
$A_{\psi_{0}}(p,q)$ (i.e., the factors given below are to be compared with
$\psi_{o}$):

$\displaystyle x_{1}^{2}\Lambda_{\rm QCD}^{2}\psi_{12}:$
$\displaystyle\Lambda_{\rm
QCD}^{2}\psi_{12}\frac{8u^{2}(1-u)^{2}(\kappa_{1}-q\tau)^{2}}{(\left[m_{c}^{2}-u(1-u)[(\kappa_{1}-q)^{2}\tau+\kappa_{1}^{2}(1-\tau)]\right]^{2}}\big{|}_{\kappa_{1}=\omega_{1}p}\sim\frac{\Lambda_{\rm
QCD}^{2}\omega_{1}M_{B}^{2}}{m_{c}^{4}}\psi_{12}\sim\frac{\Lambda_{\rm
QCD}^{3}m_{b}}{m_{c}^{4}}\psi_{12};$ $\displaystyle x_{3}^{2}\Lambda_{\rm
QCD}^{2}\psi_{23}:$ $\displaystyle\Lambda_{\rm
QCD}^{2}\psi_{23}\frac{8(\kappa_{3}-q^{\prime})^{2}}{\left[m_{s}^{2}-(\kappa_{3}-q^{\prime})^{2}\right]^{2}}\big{|}_{\kappa_{1}=\omega_{1}p}\sim\frac{\Lambda_{\rm
QCD}^{2}\omega_{3}M_{B}^{2}}{(\omega_{3}M_{B}^{2})^{2}}\psi_{23}\sim\frac{\Lambda_{\rm
QCD}}{m_{b}}\psi_{23};$ $\displaystyle x_{1}x_{3}\Lambda_{\rm
QCD}^{2}\psi_{13}:$
$\displaystyle\frac{2u(1-u)(\kappa_{1}^{\mu}-q^{\mu}\tau)}{m_{c}^{2}-u(1-u)[(\kappa_{1}-q)^{2}\tau+\kappa_{1}^{2}(1-\tau)]}\frac{2(\kappa_{3}-q^{\prime})^{\mu}}{m_{s}^{2}-(\kappa_{3}-q^{\prime})^{2}}\bigg{|}_{\kappa_{1,3}=\omega_{1,3}p}\sim\Lambda_{\rm
QCD}\frac{q^{\prime}q}{m_{c}^{2}m_{b}}\psi_{13}\sim\frac{\Lambda_{\rm
QCD}m_{b}}{m_{c}^{2}}\psi_{13}.$

Taking into account the adopted scaling, $\Lambda\ll m_{c}\ll m_{b}$, and
$\Lambda m_{b}/m_{c}^{2}\simeq 1$, we see that the factors describing the
relative contributions of powers of $x_{1}^{2}$ and $x_{3}^{2}$ are small
(i.e. the dominant contribution comes from the region where $x_{1}$ and
$x_{3}$ are on the light cone, $x_{1}^{2}=0$ and $x_{3}^{2}=0$), whereas, as
expected from the general arguments, each term of the form
$(x_{1}-x_{3})^{2n}$ in $\Psi_{p}(x_{1},x_{3})$ leads to the contribution to
$A(p,q)$ of the same order as $A_{\psi_{0}}(p,q)$. So, the knowledge of the
full functional dependence of $\Psi_{p}(x_{1},x_{3})$ on the variable
$(x_{1}-x_{3})^{2}$ is essential for a proper resummation of large
$\Lambda_{\rm QCD}m_{b}/m_{c}^{2}$ corrections to the amplitude $A(p,q)$ of
FCNC $B$-decay.

$\bullet$ We would like to remark on what happens if one considers the
contribution of the light $u$-quark instead of the $c$-quark in the loop:
First, in this case one cannot set $q^{2}=0$ but has to keep $q^{2}\leq-1$
GeV2 to validate the perturbative calculation of the $u$-quark loop. Second,
the scaling relations change

$\displaystyle x_{1}^{2}\Lambda_{\rm QCD}^{2}\psi_{12}:$
$\displaystyle\sim\frac{\Lambda_{\rm QCD}}{m_{b}}\psi_{12};$ (4.29)
$\displaystyle x_{3}^{2}\Lambda_{\rm QCD}^{2}\psi_{23}:$
$\displaystyle\sim\frac{\Lambda_{\rm QCD}}{m_{b}}\psi_{23};$ (4.30)
$\displaystyle x_{1}x_{3}\Lambda_{\rm QCD}^{2}\psi_{13}:$
$\displaystyle\sim\psi_{13}.$ (4.31)

Obviously, the statement that all $(x_{1}-x_{3})^{2n}$ terms provide the
$O(1)$ contributions applies to both cases of the $c$ and the $u$-quarks in
the triangle diagram.

$\bullet$ Finally, let us emphasize the following feature of the $B\to
j_{1}j_{2}$ amplitude: If in Eq. (4.3) we keep the leading term only and
neglect all corrections $O(\Lambda_{\rm QCD}/m_{b})$, then the contributions
of $\psi_{13}$ simplifies considerably and takes the form similar to the
contribution of $\psi_{0}$ (4.20). Moreover, the contribution of the full 3BS
Eq. (4.1), including all powers of $(x_{1}-x_{3})^{2n}$
($\psi^{(1)}_{13}\equiv\psi_{13}$),

$\displaystyle\Psi_{p}(x_{1},x_{3})=\int
d\omega_{1}d\omega_{3}e^{-i\omega_{1}x_{1}p-i\omega_{3}x_{3}p}\left[\psi_{0}(\omega_{1},\omega_{3})+\sum\limits_{n=1}\psi^{(n)}_{13}(\omega_{1},\omega_{3})x_{13}^{2n}+O(x_{1}^{2},x_{3}^{2})\right],$
(4.32)

to a FCNC $B\to j_{1}j_{2}$ amplitude may be written with $O(\Lambda_{\rm
QCD}/m_{b})$ accuracy in a simple form wang2022private :

$\displaystyle
A(q,p)=\int\limits_{0}^{1}d\omega_{1}\int\limits_{0}^{1-\omega_{1}}d\omega_{3}\,\psi_{\rm
eff}(\omega_{1},\omega_{3})\Gamma_{cc}\left(\omega_{1}p,q\right)D_{s}\left((q^{\prime}-\omega_{3}p)^{2}\right),$
(4.33)

with

$\displaystyle\psi_{\rm
eff}(\omega_{1},\omega_{3})=\psi_{0}(\omega_{1},\omega_{3})+\frac{4}{M_{B}^{2}}\frac{\partial^{2}}{\partial\omega_{1}\partial\omega_{3}}\psi_{13}(\omega_{1},\omega_{3})+\dots.$
(4.34)

Here the dots stand for the contributions of higher functions
$\psi_{13}^{(n)}$, $n\geq 2$. The expression (4.34) as well as the
contributions of $\psi_{13}^{(n)}$ may be easily obtained making use of the
relation wang2022private :

$\displaystyle x_{1}x_{3}=\frac{2}{M_{B}^{2}}x_{1}p\,x_{3}p.$ (4.35)

The latter relation is valid to $O(\Lambda_{\rm QCD}/m_{b})$ accuracy as soon
as $x_{1}^{2}$ and $x_{3}^{2}$ are near the LC, $x_{1}^{2}=O(1/\Lambda_{\rm
QCD}m_{b})$ and $x_{3}^{2}=O(1/\Lambda_{\rm QCD}m_{b})$.

As said above, in general, the invariant amplitudes appearing in 3BS are
functions of 5 independent variables: $x_{1}p$, $x_{3}p$, $x_{1}^{2}$,
$x_{3}^{2}$, $(x_{1}-x_{3})^{2}$. However, if $x_{1}^{2}=0$ and $x_{3}^{2}=0$,
the variable $(x_{1}-x_{3})^{2}$ is not an independent variable anymore and is
reduced to the combination of the variables $x_{1}p$ and $x_{3}p$, Eq. (4.35).

##  5 Conclusions

We performed a detailed comparison of the contributions coming from 3-particle
BS amplitude of the $B$-meson

$\displaystyle\langle 0|\bar{s}(x)G(z)b(0)|B(p)\rangle$ (5.36)

to (i) $B$-meson weak decay form factor that describes the CC $B$-decay
amplitude and to (ii) non-factorizable part of FCNC $B$-decay amplitude. Both
amplitudes are related to the same diagram of the form factor topology with,
however, a different location of the heavy-quark field: In the CC amplitude,
the heavy $b$-quark is located at the end point of the line along which fast
light quarks propagate in the diagram, whereas in the FCNC amplitude the heavy
$b$-quark hits the middle of this line. As the result, the dominant
contributions to these two amplitudes come from different 3BS configurations:

  * •

The dominant contribution to a CC amplitude comes from the collinear
configuration when both vertices $x$ and $z$ lie along the same light-cone
direction $z_{\mu}=ux_{\mu}$, $x^{2}=0$ and $z^{2}=0$. Therefore, a collinear
3BS $\langle 0|\bar{s}(x)G(ux)b(0)|B(p)\rangle$ is sufficient for the
calculation of the 3BS correction to the form factor.

  * •

The dominant contribution to a FCNC amplitude comes from a different
configuration when both vertices $x$ and $z$ lie on the light cone, but along
the different light-cone directions such that $z^{2}=x^{2}=0$, but $xz\neq 0$.
Therefore, a non-collinear 3BS of the $B$-meson, $\langle
0|\bar{s}(x)G(z)b(0)|B(p)\rangle$, with $x^{2}=z^{2}=0$, but $x-z)^{2}\neq 0$,
is necessary for a reliable calculation of the 3BS correction to the FCNC form
factor.

We point out that a simple physics picture lies beyond these results: One
considers the $B$-meson rest frame; in this rest frame the $b$-quark is almost
at rest.

In a CC decay, the $b$ quark decays in a fast lepton pair with momentum $q$
and a fast light quark which moves, say, along the $+$ light-cone direction.
At the point $z$ it is hit by a soft gluon and continues to move practically
along the same direction before it reaches the point $x$ where it emits the
momentum $q^{\prime}$. So we come to the well-known result that the 3BS
correction to the $B$-decay CC form factor is dominated by the collinear
light-cone configuration $x^{2}=0$, $z^{2}=0$, and $z_{\mu}=ux_{\mu}$.

In a FCNC amplitude the situation is different: a resting $b$-quark emits a
fast $s$-quark in one space direction and a fast pair of charmed
quarks222Obviously, the picture does not change if $c$-quark in the triangle
is replaced by $u$-quark. in an opposite space direction. If we translate this
into light-cone directions and assign the direction of the $s$ quark as $(+)$,
then the $c$-quark pair moves along the $(-)$ LC direction. At point $x$ the
$s$-quark emits the momentum $q^{\prime}$. The point $x$ thus lies on the LC
along its $(-)$ direction. The fast $c$-quark pair is hit by the soft gluon at
the point $z$ and continues to move up to the point $z^{\prime}$ where it
emits the momentum $q$. Both $z$ and $z^{\prime}$ lie along the $(+)$ LC
direction. We see that $x^{2}=z^{2}=0$, but, in general, $(x-z)^{2}\neq 0$.

Notice, that this argument does not say yet that all
$O\left((x-z)^{2n}\right)$ terms in 3BS of the $B$-meson lead to $O(1)$
contributions compared to the contribution of the $\psi_{0}$ term.

The analysis of Section 4 showed that for the case of the $c$-quark in the
loop, the dominant contributions to the FCNC amplitude come from the region
$x^{2}=z^{2}=0$, and $\Lambda_{\rm QCD}^{2}(x-z)^{2}\simeq{\Lambda_{\rm
QCD}m_{b}}/{m_{c}^{2}}=O(1)$. In this case, all $O\left((x-z)^{2n}\right)$
terms in the 3BS of the $B$-meson lead to $O(1)$ contributions compared to the
contribution of the $\psi_{0}$ term. The same result holds for the light quark
in the triangle instead of the $c$-quark.

In conclusion, let us recall that the idea of going from local OPE for the
FCNC amplitude to a non-local OPE was motivated by the necessity to sum up
large $(\Lambda_{\rm QCD}m_{b}/m_{c}^{2})^{n}$ corrections to the FCNC
amplitude. However, keeping only the collinear LC part of the 3BS of the
$B$-meson and neglecting all terms of the order $(x-z)^{2n}$ leads to the
resummation of a part of these large $(\Lambda_{\rm QCD}m_{b}/m_{c}^{2})^{n}$
corrections, whereas another source of the corrections of the same order of
magnitude remains unaccounted. So, the full dependence of 3BS of the $B$-meson
on the variable $(x-z)^{2}$ is necessary to sum properly the $(\Lambda_{\rm
QCD}m_{b}/m_{c}^{2})^{n}$ corrections.

Acknowledgments. I am grateful to I. Belov, A. Berezhnoy, M. Ferre, E. Kou, O.
Nachtmann, and H. Sazdjian for fruitful discussions, to E. Thommes and G.
Wolschin for their kind hospitality, and to the AvH-Stiftung for support. My
special thanks are due to Q. Qin, Y.-L. Shen, C. Wang and Y.-M. Wang for
illuminating comments concerning their paper wang2022 .

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