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

Dmitri Melikhov  D. V. Skobeltsyn Institute of Nuclear Physics, M. V.
Lomonosov Moscow State University, 119991, Moscow, Russia;
Joint Institute for Nuclear Research, 141980 Dubna, Russia;
and Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria

###### 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
$\phi^{\prime}(x^{\prime})$, $x^{\prime 2}=0$ lies along the $(-)$-direction.
We show the emergence of new constraints on the distribution amplitudes which
parametrize the 3BS in this double-collinear configuration.

##  1 Introduction

Charming loops in rare flavour-changing neutral current (FCNC) decays of the
$B$-meson have impact on the $B$-decay observables neubert  and provide an
unpleasant noise for the studies of possible new physics effects (see, e.g.,
ciuchini2022 ; ciuchini2020 ; ciuchini2021 ; diego2021 ; matias2022 ;
gubernari2022 ; hurth2022 ; stangl2022 ).

A number of theoretical analyses of nonfactorizable (NF) charming loops in
FCNC $B$-decays have been published: In voloshin , an effective gluon-photon
local operator describing the charm-quark loop was 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
were 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 configuration $\langle
0|\bar{s}(x)G_{\mu\nu}(ux)b(0)|B(p)\rangle$, $x^{2}=0$ japan ; Braun2017 .
This approximation was later used for the analysis of other FCNC $B$-decays
gubernari2020 .

The collinear LC configuration was known to provide the dominant 3BS
contribution to meson tree-level form factors braun1994 ; offen2007 , in
particular, to form factors of semileptonic (SL) $B$-decay induced by $b\to u$
weak charged current (CC). It was tempting to use the collinear 3BS for the
description of FCNC $B$-decays. However, the 3BS contribution to the CC
$B$-decay and to the FCNC $B$-decay have a qualitative difference. To
demonstrate this 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 the case of charming loops 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. In mk2018 ; m2019 ; m2022  it was demonstrated that
the FCNC $B$-decay amplitude is dominated not by the collinear configuration
of 3BS, but rather by a double-collinear configuration $\langle
0|\bar{q}(x)G_{\mu\nu}(x^{\prime})b(0)|B(p)\rangle$, $x^{2}=0$, $x^{\prime
2}=0$, but $(x-x^{\prime})^{2}\neq 0$. The first application of a double-
collinear 3BS to FCNC $B$-decays was presented in wang2022 .

The noncollinear configuration of the $B$-meson 3BS leads to the appearance of
new Lorentz structures of the type $(x_{\mu}p_{\nu}-x_{\mu}p_{\nu})/(xp)$ and
$(x^{\prime}_{\mu}p_{\nu}-x^{\prime}_{\mu}p_{\nu})/(x^{\prime}p)$. These
Lorentz structures contain kinematical singularities at $1/(xp)$ and
$1/(x^{\prime}p)$ which should not be singularities of the physical 3BS. This
requirement and the continuity of the 3BS at the point $x^{2}=0$, $x^{\prime
2}=0$, $xp=0$, and $x^{\prime}p=0$ yield constraints on the corresponding
distribution amplitudes (DAs). Moreover, these constraints turn out to be
valid also for those DAs which appear in the collinear 3BS. To avoid
complications related to the spinorial structure of the 3BS in QCD, Sec. 2
demonstrates the derivation of these constraints in the case of field theory
with scalar “quarks”. A generalization to QCD is straightforward.

In Sec. 3 we analyze Feynman diagrams of the type corresponding to charming
loops in FCNC $B$-decays (i.e., those diagrams in which the heavy field hits
the middle point of the line along which light degrees of freedom propagate;
hereafter referred to as FCNC-type diagrams) and show that to the leading
order in the heavy-quark (HQ) expansion, the amplitude is given by the
convolution of the hard kernel composed of propagators of the light degrees of
freedom and the $B$-meson 3BS in the double-collinear configuration. We also
derive the exact expression for the $B$-decay amplitudes of FCNC-type using
the $\alpha$-representation and show that the exact amplitudes and the double-
collinear approximation for these amplitudes coincide in the heavy-quark
limit.

##  2 Properties of the 3BS wave function of the $B$-meson

Leaving aside technical details related to the spins of the constituent fields
and of the bound state, and considering heavy $B$-meson as bound state of
scalar fields, one of which, $\phi_{b}(x)$, is heavy, the 3BS of the $B$-meson
may be parametrized as follows (m2022  and refs therein):

$\displaystyle\langle
0|\phi(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle$ $\displaystyle=$
$\displaystyle\int D(\omega,\omega^{\prime})e^{-ipx\omega-
ipx^{\prime}\omega^{\prime}}\Psi(\omega,\omega^{\prime},x^{2},x^{\prime
2},(x-x^{\prime})^{2}),$ (2.1)

where

$\displaystyle D(\omega,\omega^{\prime})\equiv
d\omega\,d\omega^{\prime}\,\theta(\omega)\theta(\omega^{\prime})\theta(1-\omega-\omega^{\prime}).$
(2.2)

For $\Psi(\omega,\omega^{\prime},x^{2},x^{\prime 2},(x-x^{\prime})^{2})$ one
can write down Taylor expansion in its variables $x^{2}$, $x^{\prime 2}$,
$(x-x^{\prime})^{2}$:

$\displaystyle\Psi(\omega,\omega^{\prime},x^{2},x^{\prime
2},(x-x^{\prime})^{2})$ $\displaystyle=$
$\displaystyle\Psi_{0}(\omega,\omega^{\prime})+x^{2}\Psi_{12}(\omega,\omega^{\prime})+x^{\prime
2}\Psi_{23}(\omega,\omega^{\prime})+(x-x^{\prime})^{2}\Psi_{13}(\omega,\omega^{\prime})+\dots,$
(2.3)

where $\dots$ stand for higher powers of $x^{2},x^{\prime
2},(x-x^{\prime})^{2}$. The distribution amplitudes (DAs) $\Psi_{i}$ have
support in the region $0<\omega$, $0<\omega^{\prime}$,
$\omega+\omega^{\prime}<1$, and peak at small values
$(\omega,\omega^{\prime})=O(\Lambda_{\rm QCD}/M_{B})$, reflecting the fact
that the heavy $b$-quark carries almost the full momentum of the heavy
$B$-meson, whereas the light degrees of freedom carry its small fraction
$O(\Lambda_{\rm QCD}/M_{B})$. (In practical model calculations, the DAs may be
nonzero in a narrower region, e.g. $0<\omega$, $0<\omega$,
$\omega+\omega^{\prime}<2\omega_{0}<1$.)

Interesting constraints on the DAs emerge in those cases when the 3BS has a
nontrivial Lorentz structure. We shall consider a more complicated 3BS, when
one of the light scalar fields is replaced by the gauge field, $\phi(x)\to
G_{\mu\nu}(x)$:111In QCD, the analogous 3BS is $\langle
0|\bar{q}(x^{\prime})G_{\mu\nu}(x)b(0)|B(p)\rangle$. This amplitude is not
gauge-invarinat as it contains field operators at different locations. Gauge-
invariant 3BS is constructed by introducing the Wilson lines as follows
$\langle
0|\bar{q}(x^{\prime})U(x^{\prime},x)G_{\mu\nu}(x)U(x,0))b(0)|B(p)\rangle$ with
$U(x,y)=P\exp(i\int_{y}^{x}A_{\mu}(z)dz_{\mu})$.

$\displaystyle\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle.$ (2.4)

Our discussion will be technically rather simple but will allow a direct
generalization to QCD, in which case more Lorentz structures emerge.

###  2.1 Collinear 3BS of $B$-meson

In the literature, much attention has been given to the collinear 3BS, where
the coordinates of the light degrees of freedom are proportional to each other
offen2007 :

$\displaystyle\langle
0|G_{\mu\nu}(ux)\phi_{b}(0)\phi^{\prime}(x)|B(p))\rangle=\int
D(\omega,\omega^{\prime})\,e^{-i(u\omega+\omega^{\prime})px}\left(\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{xp}\right)\left[X_{A}(\omega,\omega^{\prime})+O(x^{2})\right],\;0<u<1.$
(2.5)

For $x^{2}=0$ and $xp\to 0$, all components of the 4-vector $x_{\mu}$ vanish
in the rest frame of the $B$-meson ($p_{+}=p_{-}$, $p_{\perp}=0$):

$\displaystyle xp=x_{+}p_{-}+x_{-}p_{+}-x_{\perp}p_{\perp}=0\Longrightarrow
x_{+}=-x_{-};$ $\displaystyle
x^{2}=2x_{+}x_{-}-x_{\perp}^{2}=-2x_{+}^{2}-x_{\perp}^{2}=0\Longrightarrow
x_{+}=-x_{-}=0,x_{\perp}=0.$

So, if (2.5) is defined only for $x^{2}=0$, one can introduce the new variable
$\tau$, $x_{\mu}=\tau n_{\mu}$, $n^{2}=0$ ($n_{\mu}$ lies, e.g., along the
$(+)$ direction of the light cone). Then the Lorentz structure
$(x_{\mu}p_{\nu}-x_{\nu}p_{\mu})/xp$ has no singularity at $\tau\to 0$ and
therefore the amplitude (2.5) is finite at $xp\to 0$ Braun2017 .

However, if one considers the collinear 3BS (2.5) also at $x^{2}\neq 0$, then
the singularity at $xp\to 0$ in the Lorentz structure emerges. The 3BS (2.5)
as the function of two arguments $x^{2}$ and $xp$ should be continuous and
finite at $x^{2}=0$ and $xp=0$, so one has to require that the kinematical
singularity of the Lorentz structure should not be the singularity of the 3BS
(2.5). Expanding the exponential in (2.5) and requiring the absence of a
singularity at $xp\to 0$ in the r.h.s. of Eq. (2.5) leads to a well-known
constraint [see Eq. (5.5) of Braun2017 ]:

$\displaystyle\int D(\omega,\omega^{\prime})X_{A}(\omega,\omega^{\prime})=0.$
(2.6)

###  2.2 Generalization to a noncollinear kinematics

For a proper description of the nonfactorizable charm in FCNC $B$-decay, 3BS
of the $B$-meson in a noncollinear configuration is necessary (see mk2018 ;
m2019 ; m2022 ; wang2022  and the demonstration in the next Section). We
therefore consider the generalization of the amplitude (2.5) and allow the
gluon field $G_{\mu\nu}(x)$ and the light-quark field
$\phi^{\prime}(x^{\prime})$ to have arbitrary different non-collinear
coordinates. Then a more general decomposition emerges:222We do not display
here further structures like
$(x_{\mu}x^{\prime}_{\nu}-x^{\prime}_{\mu}x_{\nu})/((xp)\,(x^{\prime}p))$. The
way one should properly treat the corresponding DAs will become obvious.

$\displaystyle\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle$
$\displaystyle=$ $\displaystyle\int D(\omega,\omega^{\prime})\,e^{-i\omega
xp-i\omega^{\prime}x^{\prime}p}$ (2.7)
$\displaystyle\times\bigg{\\{}\left(\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{x\,p}\right)X^{(1)}_{A}(\omega,\omega^{\prime})+\left(\frac{x^{\prime}_{\mu}p_{\nu}-x^{\prime}_{\nu}p_{\mu}}{x^{\prime}\,p}\right)X^{(2)}_{A}(\omega,\omega^{\prime})+\dots\bigg{\\}}.$

The amplitude (2.7) contains factors $1/xp$ and $1/x^{\prime}p$ in the Lorentz
structures. If (2.7) is defined for $x^{2}=0$ and $x^{\prime 2}=0$ only, one
can introduce two new variables $\tau$ and $\tau^{\prime}$ such that
$x_{\mu}=\tau n_{\mu}$, $x^{\prime}_{\mu}=\tau^{\prime}n^{\prime}_{\mu}$,
$n^{2}=0$ and $n^{\prime 2}=0$, but $n^{\prime}n\neq 0$ (e.g. $n_{\mu}$ lies
along the $(+)$-direction of the light cone, and $n^{\prime}_{\mu}$ lies along
the $(-)$-direction, see also wang2022 ). Then the Lorentz structure
$(x_{\mu}p_{\nu}-x_{\nu}p_{\mu})/xp$ has no singularity at $\tau\to 0$, the
Lorentz structure
$(x^{\prime}_{\mu}p_{\nu}-x^{\prime}_{\nu}p_{\mu})/x^{\prime}p$ has no
singularity at $\tau^{\prime}\to 0$, and therefore the amplitude (2.7) has no
singularity at $xp\to 0$ and $x^{\prime}p\to 0$.

However, the noncollinear 3BS (2.7) should be a regular continuous function of
its arguments at the point $x^{2}=0$, $x^{\prime 2}=0$, $xp=0$, and
$x^{\prime}p=0$ independently of the way one approaches this point. If one
first takes the limit $xp\to 0$ and $x^{\prime}p\to 0$ keeping $x^{2}\neq 0$
and $x^{\prime 2}\neq 0$, then the singularities at $xp\to 0$ and
$x^{\prime}p\to 0$ in the Lorentz structures emerge. To compensate these
singularities, the DAs should satisfy the following constraints [obtained by
expanding the exponential in (2.7)]

$\displaystyle\int\limits_{0}^{1-\omega^{\prime}}d\omega
X^{(1)}_{A}(\omega,\omega^{\prime})=0\quad\forall\,\omega^{\prime}\quad{\rm
and}\quad\int\limits_{0}^{1-\omega}d\omega^{\prime}X^{(2)}_{A}(\omega,\omega^{\prime})=0\quad\forall\,\omega.$
(2.8)

For $X_{A}$ parametrizing the 3BS in the collinear limit, $x=ux^{\prime}$, one
finds the relation

$\displaystyle
X_{A}(\omega,\omega^{\prime})=X^{(1)}_{A}(\omega,\omega^{\prime})+X^{(2)}_{A}(\omega,\omega^{\prime})$
(2.9)

Obviously, the condition (2.6) follows from (2.8). However, Eq. (2.8) shows
that each of the parts of $X_{A}(\omega,\omega^{\prime})$ should satisfy more
restrictive constraints.

For further use we introduce the primitives:

$\displaystyle\overline{X}^{(1)}_{A}(\omega,\omega^{\prime})=\int\limits_{0}^{\omega}d\underline{\omega}{X}^{(1)}_{A}(\underline{\omega},\omega^{\prime}),\qquad$
$\displaystyle{X}^{(1)}_{A}(\omega,\omega^{\prime})=\partial_{\omega}\overline{X}^{(1)}_{A}(\omega,\omega^{\prime})$
(2.10)
$\displaystyle\overline{X}^{(2)}_{A}(\omega,\omega^{\prime})=\int\limits_{0}^{\omega^{\prime}}d\underline{\omega}^{\prime}{X}^{(2)}_{A}(\omega,\underline{\omega}^{\prime}),\qquad$
$\displaystyle{X}^{(2)}_{A}(\omega,\omega^{\prime})=\partial_{\omega^{\prime}}\overline{X}^{(2)}_{A}(\omega,\omega^{\prime}).$
(2.11)

By virtue of (2.8), we obtain the following contraints on the primitives

$\displaystyle\overline{X}^{(1)}_{A}(\omega=0,\omega^{\prime})=\overline{X}^{(1)}_{A}(\omega=1-\omega^{\prime},\omega^{\prime})=0\quad\forall\omega^{\prime},$
(2.12)
$\displaystyle\overline{X}^{(2)}_{A}(\omega,\omega^{\prime}=0)=\overline{X}^{(2)}_{A}(\omega,\omega^{\prime}=1-\omega)=0\quad\,\forall\omega.$
(2.13)

These relations mean that the primitives should vanish on the boundary of the
DA’s-support region.

Performing parts integration in $\omega$ or $\omega^{\prime}$, and taking into
account that the surface terms vanish due to (2.12) and (2.13), we may rewrite
(2.7) in the form that does not contain factors $1/xp$ and
$1/x^{\prime}p$:333Noteworthy, the functions of the type $X_{A}$ extensively
used in the literature Braun2017 ; hidr ; gubernari2020  do not satisfy (2.8).
In this case, the surface terms do not vanish and the 3BS given by Eqs. (2.7)
and (2.14) are not equivalent to each other.

$\displaystyle\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle$
$\displaystyle=$ $\displaystyle i\int D(\omega,\omega^{\prime})\,e^{-i\omega
xp-i\omega^{\prime}x^{\prime}p}$ (2.14)
$\displaystyle\times\bigg{\\{}\left(x_{\mu}p_{\nu}-x_{\nu}p_{\mu}\right)\overline{X}^{(1)}_{A}(\omega,\omega^{\prime})+\left(x^{\prime}_{\mu}p_{\nu}-x^{\prime}_{\nu}p_{\mu}\right)\overline{X}^{(2)}_{A}(\omega,\omega^{\prime})+\dots\bigg{\\}}.$

Let us summarize the material of this Section: We required that the
noncollinear 3BS is a continuous function of its arguments at $x^{2}=0$,
$x^{\prime 2}=0$, $xp=0$ and $x^{\prime}p=0$ and has a finite value at this
point. Then, the kinematical singularities in the Lorentz structures at $xp\to
0$ and $x^{\prime}p\to 0$ should be compensated by certain properties of the
DAs: namely, for the DA of the $X_{A}$-type, its primitives should vanish on
the boundary of $X_{A}$-support region. This property also guarantees that two
forms of the 3BS, Eqs. (2.7) and (2.14), are equivalent and may be safely
applied to the calculation of the amplitudes of $B$-meson decays.

##  3  $B$-decay in the kinematics of charming loops in FCNC decays

In this Section we focus on 3BS contributions involving charming loops in FCNC
$B$-decays and show that to the leading order in the HQ expansion the
amplitude is given by the convolution of the hard kernel and the $B$-meson 3BS
in the noncollinear kinematical configuration.

###  3.1 The dominant contribution to the FCNC $B$-decay amplitude involving
charming loops

As noticed in m2022  (see Appendix A for details) the kinematics of FCNC
$B$-decay amplitudes involving charming loops is equivalent to the 3BS
correction to the $B$-decay form factor with the difference that the heavy
field, $\phi_{b}(0)$, is in the middle of the quark line.

Fig. 1:  The generic 3BS amplitude of the form factor topology. If the field
$\phi_{b}$ is heavy, while other fields $\phi$ and $\phi^{\prime}$ are light,
the diagram corresponds to the topology of charming loops in FCNC $B$-decay
amplitude.

Placing the heavy-quark field at zero, $\phi_{b}(0)$, we have [Fig. 1]:

$\displaystyle
A(p|q,q^{\prime})=\int\frac{dxdx^{\prime}dkdk^{\prime}}{(\mu^{2}-k^{2})(m^{2}-k^{\prime
2})}e^{iqx+ikx+iq^{\prime}x^{\prime}-ik^{\prime}x^{\prime}}\langle
0|\phi(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle,$ (3.15)

where $\phi$ and $\phi^{\prime}$ are light degrees of freedom (in practical
calculations, the gluon and the light quark). We are going to derive the
leading-order behaviour of this amplitude.

####  3.1.1 The $x$-vertex

Let us discuss the $x$-vertex and introduce $\kappa=k+q$, the momentum carried
by the constituent field $\phi(x)$:

$\displaystyle\int\frac{dxd\kappa}{\mu^{2}-(\kappa-q)^{2}}e^{i\kappa x}\langle
0|\phi(x)...|B(p)\rangle.$ (3.16)

We consider the case $q^{2}=q^{\prime 2}=0$ and work in the rest frame of the
$B$-meson and take the axes such that momentum $q$ is along $(+)$-axis,
whereas momentum $q^{\prime}$ is along the $(-)$-axis. [One can formulate this
in a covariant form by introducing vectors $n_{\mu}$ and $\bar{n}_{\mu}$
Braun2017 ; wang2022 .]

Due to the properties of the $B$-meson 3BS, the vector $\kappa$ is soft, i.e.
all its components are $\kappa_{\mu}\sim O(\Lambda_{\rm QCD})$. The component
$q_{+}$ is large, $q_{+}\sim M_{B}$, and the propagator is highly virtual,
$\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}\sim\Lambda_{\rm
QCD}M_{B}$.

Let us expand the field operator $\phi(x)$ near $x=0$ (this would correspond
to considering a tower of local operators of the increasing dimension). The
expansion in powers of $x_{-}$ and $x_{\perp}$ leads to a well behaved Taylor
series as

$\displaystyle
x_{-}e^{i\kappa_{+}x_{-}}\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\to\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\partial_{\kappa_{+}}e^{i\kappa_{+}x_{-}}$
$\displaystyle\to
e^{i\kappa_{+}x_{-}}\partial_{\kappa_{+}}\left(\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\right)\to
e^{i\kappa_{+}x_{-}}\frac{\kappa_{-}}{(\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2})^{2}}.$
(3.17)

Since $\kappa_{-}=O(\Lambda_{\rm QCD})$ and the virtuality of the propagator
is $O(\Lambda_{\rm QCD}M_{B})$, any term $(x_{-})^{n}$ is suppressed by a
factor $(1/M_{B})^{n}$ compared to the term $(x_{-})^{0}$. The same property
holds for $(x_{\perp})^{n}$.

However, for powers of the variable $x_{+}$ the situation is different:

$\displaystyle
x_{+}e^{i\kappa_{-}x_{+}}\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\to\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\partial_{\kappa_{-}}e^{i\kappa_{-}x_{+}}$
$\displaystyle\to\partial_{\kappa_{-}}\left(\frac{1}{\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2}}\right)e^{i\kappa_{-}x_{+}}\to
e^{i\kappa_{-}x_{+}}\frac{q_{+}}{(\mu^{2}-2\kappa_{-}(\kappa_{+}-q_{+})+\kappa_{\perp}^{2})^{2}}.$
(3.18)

Since $q_{+}\sim M_{B}$, all powers of $x_{+}^{n}$ in the expansion of
$\phi(x_{+})$ near $x_{+}=0$ have the same order of magnitude. The Taylor
expansion of $\phi(x_{+})$ near $x_{+}=0$ leads to no hierarchy in the
corresponding expansion of the $B$-decay amplitude, and we need to keep the
full $x_{+}$ dependence of the operator $\phi(x_{+})$ on the light cone
($x^{2}=0$). This result is valid for two phenomenologically interesting
cases: (i) the charm-quark loop contribution to the FCNC amplitude,
$\mu^{2}\sim m_{c}^{2}=O(\Lambda_{\rm QCD}m_{b})$, and (ii) the light-quark
loop contribution to the FCNC amplitude, $\mu$ being the light-quark mass.

So, the leading term of the expansion of the $B$-decay amplitude related to
the $x$-vertex corresponds to the expansion near $x_{-}=0,x_{\perp}=0$ and has
the form

$\displaystyle\int
dx_{+}dx_{-}dx_{\perp}d\kappa_{+}d\kappa_{-}d\kappa_{\perp}\frac{1}{\mu^{2}-2(\kappa_{+}-q_{+})k_{-}+k_{\perp}^{2}}e^{i\kappa_{+}x_{-}+i\kappa_{-}x_{+}-ik_{\perp}x_{\perp}}\langle
0|...\phi(x_{+})...|B(p)\rangle,$ (3.19)

The $x_{-}$ and $x_{\perp}$ integrals here may be taken and lead to
$\delta(\kappa_{\perp})\delta(\kappa_{+})$. Integrating these
$\delta$-functions, we obtain for the part of the amplitude related to the
$x$-vertex (we denote $\tau=x_{+}$, and recall that $q$ has only the
$(+)$-component):

$\displaystyle\int d\tau
d\kappa_{-}\frac{1}{\mu^{2}+2q_{+}\kappa_{-}}e^{i\kappa_{-}\tau}\langle
0|\phi(\tau)...|B(p)\rangle.$ (3.20)

####  3.1.2 The $x^{\prime}$-vertex

We can perform a similar analysis of the $x^{\prime}$ vertex. The crucial
difference is that now $q_{-}\sim M_{B}$ is the only nonzero component of the
vector $q^{\prime}$. The propagator has the form
$m^{2}-2\kappa^{\prime}_{-}(\kappa^{\prime}_{-}-q_{-})+{\kappa^{\prime}_{\perp}}^{2}\sim\Lambda_{\rm
QCD}M_{B}$. Obviously, we can perform Taylor expansion of
$\phi^{\prime}(x^{\prime})$ near $x^{\prime}_{+}=0$ and $x^{\prime}_{\perp}=0$
but have to keep its full dependence on the variable $x_{-}$. Taking into
account this property and denoting $\tau^{\prime}=x^{\prime}_{-}$, the
dominant contribution of the $x^{\prime}$-vertex reads ($q_{-}\sim M_{B}$ is
the only nonzero component of the vector $q^{\prime}$)

$\displaystyle\int
d\tau^{\prime}d\kappa^{\prime}_{+}\frac{1}{m^{2}+2q^{\prime}_{-}\kappa^{\prime}_{+}}e^{i\kappa^{\prime}_{+}\tau^{\prime}}\langle
0|...\phi^{\prime}(\tau^{\prime})...|B(p)\rangle.$ (3.21)

####  3.1.3 The amplitude of the FCNC $B$-decay

Making use of the leading contributions of the $x$\- and $x^{\prime}$-vertices
we obtain the leading contribution to the $B$-decay amplitude in the form, see
also wang2022 :

$\displaystyle A(p|q,q^{\prime})=\int d\tau
d\kappa_{-}\frac{1}{\mu^{2}+2q_{+}\kappa_{-}}e^{i\kappa_{-}\tau}\int
d\tau^{\prime}d\kappa^{\prime}_{+}\frac{1}{m^{2}+2q^{\prime}_{-}\kappa^{\prime}_{+}}e^{i\kappa^{\prime}_{+}\tau^{\prime}}\langle
0|\phi(\tau)\phi_{b}(0)\phi^{\prime}(\tau^{\prime})|B(p)\rangle.$ (3.22)

This relation is the factorization theorem that represents the dominant
contribution to the FCNC amplitude as the convolution of the hard kernel
composed from the propagators of the light degrees of freedom and the 3BS in
the kinematical configuration which may be called “double collinear”: the
upper and the lower parts of the diagram are aligned along the different
light-cone directions.

####  3.1.4 On the collinear light-cone 3DA in FCNC $B$-decay

We would like to emphasize that the 3BS in a collinear light-cone
configuration

$\displaystyle\langle 0|\phi(x)\phi_{b}(0)\phi^{\prime}(\lambda
x)|B(p)\rangle,\quad x^{2}=0,\quad 0<\lambda<1,$ (3.23)

is not related to the dominant contribution to the amplitude of FCNC
$B$-decay. So, we do not find justification of the statement of hidr  that the
dominant contribution to the FCNC amplitude may be calculated via the
collinear light-cone 3DA of the $B$-meson (3.23).

####  3.1.5 Multi-particle BS contributions to amplitudes of $B$-decays

Our analysis may be generalized to other contributions to $B$-decay amplitudes
of the type shown in Fig. 2.

Fig. 2:  An example of a multi-particle BS contribution to the $B$-decay
amplitude. The field $\phi_{b}$ is heavy, while all other fields, $\phi$,
$\phi^{\prime}$, …, are light. The dominant contribution comes from the double
collinear LC configuration: $x_{1}=u_{1}x$, …, $x_{n}=u_{n}x$,
$0<u_{n}<\ldots<u_{1}<1$, $x^{2}=0$;
$x^{\prime}_{1}=u^{\prime}_{1}x^{\prime}$, …,
$x^{\prime}_{n^{\prime}}=u_{n^{\prime}}x^{\prime}$,
$0<u^{\prime}_{n^{\prime}}<\ldots<u^{\prime}_{1}<1$, $x^{\prime 2}=0$, whereas
$xx^{\prime}\neq 0$, i.e., the set of collinear field coordinates in the upper
part of the diagram is not aligned with the set of collinear field coordinates
in the lower part of the diagram.

The corresponding diagram involves a multi-particle BS of the $B$-meson:

$\displaystyle\langle
0|\phi(x)\phi_{1}(x_{1})\ldots\phi_{n}(x_{n})\phi_{b}(0)\phi^{\prime}_{n^{\prime}}(x^{\prime}_{n^{\prime}})\ldots\phi_{1}^{\prime}(x_{1}^{\prime})\phi^{\prime}(\tau^{\prime})|B(p)\rangle,$
(3.24)

where all fields except for $\phi_{b}$ are light. Combining the analysis
presented above and the discussion of Sect. 3 of m2022 , one can easily prove
by induction that the dominant contribution to the $B$-decay amplitude comes
from the double-collinear light-cone configuration
[$a_{\mu}=(a_{+},a_{-},a_{\perp})$]:

$\displaystyle x=(\tau,0,0),\quad$ $\displaystyle x_{1}=(\tau
u_{1},0,0),\quad\,\,\ldots,\quad x_{n}=(\tau u_{n},0,0),\qquad
0<u_{n}<\ldots<u_{1}<1,$ (3.25) $\displaystyle
x^{\prime}=(0,\tau^{\prime},0),\quad$ $\displaystyle
x^{\prime}_{1}=(0,\tau^{\prime}u^{\prime}_{1},0),\quad\ldots,\quad
x^{\prime}_{n^{\prime}}=(0,\tau^{\prime}u^{\prime}_{n^{\prime}},0),\,\quad
0<u^{\prime}_{n^{\prime}}<\ldots<u^{\prime}_{1}<1.$

The coordinates $x,x_{1},\ldots,x_{n}$ are ordered and lie on the $(+)$-axis
of the LC, and the coordinates
$x^{\prime},x^{\prime}_{1},\ldots,x^{\prime}_{n^{\prime}}$ are ordered and lie
on the $(-)$-axis of the LC. That is why we refer to this configuration as to
the double collinear light-cone configuration.

####  3.1.6 Collinear vs Double-collinear 3DAs

Here we would like to emphasize the differences between the collinear and the
double-collinear 3DAs. Let us get back to Eq. (2.7). Again introduce LC
variables in the $B$-meson rest frame and consider two vectors
$n_{\mu}=(\sqrt{2},0,0)$ and $n^{\prime}_{\mu}=(0,\sqrt{2},0)$ such that
$n^{2}=0$, $n^{\prime 2}=0$, $nn^{\prime}=2$,
$v_{\mu}=\frac{1}{2}(n_{\mu}+n^{\prime}_{\mu})$, $nv=n^{\prime}v=1$.

$\bullet$ In a collinear LC configuration, $x^{2}=0$, $x^{\prime 2}=0$,
$x^{\prime}_{\mu}=ux_{\mu}$, we can choose $x_{\mu}$ and $x^{\prime}_{\mu}$
along $n_{\mu}$: $x_{\mu}=\tau n_{\mu}$,
$x^{\prime}_{\mu}=\tau^{\prime}n^{\prime}_{\mu}$. Then two Lorentz structures
in (2.7) reduce to one Lorentz structure (we use relation
$v=(n+n^{\prime})/2$):

$\displaystyle\langle
0|G_{\mu\nu}(n\tau)\phi_{b}(0)\phi(n\tau^{\prime})|B(p)\rangle=\int
D(\omega,\omega^{\prime})e^{-iM_{B}(\omega\tau+\omega^{\prime}\tau^{\prime})}\frac{1}{2}\left(n_{\mu}n^{\prime}_{\nu}-n_{\nu}n^{\prime}_{\mu}\right)\left(X^{(1)}_{A}+X^{(2)}_{A}\right),$
(3.26)

where $X_{A}\equiv X^{(1)}_{A}+X^{(2)}_{A}$ is the collinear LC 3DA.

$\bullet$ We now turn to the double-collinear LC configuration. Since the 3BS
(2.7) is antisymmetric in $\mu\leftrightarrow\nu$, both Lorentz structures in
(2.7) are again reduced to the same Lorentz structure as in (3.26) with
however a different 3DA:

$\displaystyle\langle
0|G_{\mu\nu}(n\tau)\phi_{b}(0)\phi(n^{\prime}\tau^{\prime})|B(p)\rangle=\int
D(\omega,\omega^{\prime})e^{-iM_{B}(\omega\tau+\omega^{\prime}\tau^{\prime}}\frac{1}{2}\left(n_{\mu}n^{\prime}_{\nu}-n_{\nu}n^{\prime}_{\mu}\right)\left(X^{(1)}_{A}-X^{(2)}_{A}\right).$
(3.27)

So, we conclude: Although the 3BS of Eq. (2.7) both in the collinear and the
double-collinear configurations contain one and the same Lorentz structure,
the corresponding 3DAs in the collinear and the double-collinear
configurations are different and in general are independent of each other, see
also wang2022 .

###  3.2 Gauge field as one of the light fields: double-collinear
approximation

Let us now turn to the situation with more Lorentz structures. We have to take
into account that the kinematics is double-collinear and the new Lorentz
structures and new DAs emerge compared to collinear kinematics considered in
Braun2017 . Still, some features established for the collinear 3DA survive
also in this more complicated case. For instance, let us consider the Lorentz
structures containing $xp$ and $x^{\prime}p$ in the denominator

$\displaystyle\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{x\,p}\quad{\rm
and}\quad\frac{x^{\prime}_{\mu}p_{\nu}-x^{\prime}_{\nu}p_{\mu}}{x^{\prime}p}.$
(3.28)

If we keep the “large” components (i.e. $x_{\mu}$ along the $(+)$-axis and
$x^{\prime}_{\mu}$ along the $(-)$-axis), then powers of $\tau$ and
$\tau^{\prime}$ in the numerator and the denominator cancel. In the end, the
$\tau$\- and $\tau^{\prime}$-integrations may be easily performed.

To be more specific, we consider the case when the light field $\phi(x)$ is
replaced by the vector gauge field $G_{\mu\nu}(x)$ and discuss one structure
in the $B$-meson 3BS

$\displaystyle\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi(x^{\prime})|B(p)\rangle=\int
D(\omega,\omega^{\prime})e^{-i\omega
xp-i\omega^{\prime}x^{\prime}p}\left(\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{x\,p}\right)X_{A}(\omega,\omega^{\prime})$
(3.29)

The $B$-decay amplitude induced by this 3BS is anti-symmetric in two indiices
$\mu$ and $\nu$ and thus contains one form factor:

$\displaystyle A_{\mu\nu}(p|q,q^{\prime})$ $\displaystyle=$
$\displaystyle(q_{\mu}q^{\prime}_{\nu}-q_{\nu}q^{\prime}_{\mu})F(q^{2},q^{\prime
2})$ (3.30) $\displaystyle=$ $\displaystyle\int\frac{dxdx^{\prime}d\kappa
d\kappa^{\prime}}{(\mu^{2}-(\kappa-q)^{2})(m^{2}-(\kappa^{\prime}-q^{\prime})^{2})}e^{i\kappa
x+i\kappa^{\prime}x^{\prime}}\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle,$

Taking into account the derivation of the previous subsection, the leading
contribution may be written as

$\displaystyle A_{\mu\nu}(p|q,q^{\prime})$ $\displaystyle=$ $\displaystyle\int
d\tau d\kappa_{-}\frac{1}{\mu^{2}+2q_{+}\kappa_{-}}e^{i\kappa_{-}\tau}\int
d\tau^{\prime}d\kappa^{\prime}_{+}\frac{1}{m^{2}+2q^{\prime}_{-}\kappa^{\prime}_{+}}e^{i\kappa^{\prime}_{+}\tau^{\prime}}\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{x\,p}$
(3.31) $\displaystyle\times\int
D(\omega,\omega^{\prime})\,X_{A}(\omega,\omega^{\prime})e^{-i\omega
p_{-}\tau-i\omega^{\prime}p_{+}\tau^{\prime}}.$

Obviously, the main contribution to the amplitude comes from the following
regions:

  * •

In the upper part of the diagram, $x_{\mu}$ has nonzero $(+)$-component,
$q_{\mu}$ has nonzero $(+)$-component, $\kappa_{\mu}$ has nonzero
$(-)$-component, and $xp=\tau p_{+}$.

  * •

In the lower part of the diagram, the situation is opposite:
$x^{\prime}_{\mu}$ has nonzero $(-)$-component, $q^{\prime}_{\mu}$ has nonzero
$(-)$-component, $\kappa^{\prime}_{\mu}$ has nonzero $(+)$-component, and
$x^{\prime}p=\tau^{\prime}p_{-}$.

Essentially, we have two independent one-dimensional configurations. These
configurations talk to each other via the $\omega$ and $\omega^{\prime}$
dependence of the 3DA $X_{A}(\omega,\omega^{\prime})$.

In the double-collinear configuration, the 4-vector $x$ has only one nonzero
component, $x_{+}$, and the combination $x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ is
linear in $x_{+}$. Therefore the combination

$\displaystyle\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{x\,p}$ (3.32)

is non-singular for $x_{+}\to 0$. [The same property holds e.g. for the
structure ${(x_{\mu}\gamma_{\nu}-x_{\nu}\gamma_{\mu})}/{x\,p}$ for spinor
fields.]

Taking into account the nonzero components, for calculating the form factor
$F(q^{2},q^{\prime 2})$ we may set $\mu=(+)$ and $\nu=(-)$, or $\mu=(-)$ and
$\nu=(+)$. In the double-collinear configuration, we find

$\displaystyle\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{xp}\to 1\qquad\mbox{ for
}\mu=(+),\nu=(-),\mbox{ and}$
$\displaystyle\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{xp}\to-1\qquad\mbox{ for
}\mu=(-),\nu=(+).$ (3.33)

In the end, we obtain the same result as for the scalar form factor of the
previous Section:

$\displaystyle q^{\prime}q\,F(q^{2},q^{\prime 2})=\int d\tau
d\kappa_{-}\frac{1}{\mu^{2}+2q_{+}\kappa_{-}}e^{i\kappa_{-}\tau}\int
d\tau^{\prime}d\kappa^{\prime}_{+}\frac{1}{m^{2}+2q^{\prime}_{-}\kappa^{\prime}_{+}}e^{i\kappa^{\prime}_{+}\tau^{\prime}}\int
D(\omega,\omega^{\prime})\,e^{-i\omega
p_{-}\tau-i\omega^{\prime}p_{+}\tau^{\prime}}\,X_{A}(\omega,\omega^{\prime}).$

###  3.3 Gauge field as one of the light fields: the $\alpha$-representation

Another possibility to handle the $1/xp$ factor without making approximations
and without performing parts integration is to make use of the $\alpha$
-representation

$\displaystyle\frac{1}{ixp}=\int_{0}^{\infty}e^{-iapx-\epsilon a}da.$ (3.35)

This representation leads to a shift of the expressions containing $\omega$ in
the denominator of the quark propagator and enables us to take explicitly the
$\alpha$-integration in the amplitude (3.30). So, we do not need to make the
parts integration in $\omega$. In this way we could avoid making
approximations and considering the collinear kinematics.

Taking into account that the quadratic form in the denominator is the quark
propagator, we reconstruct the imaginary part as follows

$\displaystyle\int\limits_{0}^{\infty}\frac{da}{(a-a_{+})(a-a_{-})-i0}=\frac{1}{a_{+}-a-}\log\left({-\frac{a_{-}}{a_{+}}}\right)+i\pi\frac{1}{a_{+}-a-},\quad
a_{-}<0,\quad a_{+}>0\quad\mbox{for }0<\omega<1.$ (3.36)

Let us discuss the contribution to the form factor from the “upper part of the
diagram” including the propagator of the quark with mass $\mu$; the “lower”
part of the diagram is treated precisely the same way. Important is that we do
not need to make any approximations and the calculation is fully covariant. By
virtue of the $\alpha$-representation, the contribution of the 3DA structure
(3.29) to the form factor (3.22) takes the form (we explicitly write only the
upper part of the diagram containing the vertex $x$ that emits the momentum
$q$, the quark propagator $1/(\mu^{2}-k^{2}-i0)$ and the relevant part of the
$B$-meson 3BS):

$\displaystyle A_{\mu\nu}(p|q,q^{\prime})=\int dxdke^{iqx+ikx-i\omega
px}\frac{x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}{ixp}\frac{1}{\mu^{2}-k^{2}-i0}d\omega
X_{A}(\omega,\omega^{\prime})\\{\ldots\\}.$ (3.37)

We substitute $1/px$ as the $\alpha$-integral and representing the $x_{\beta}$
as the $q_{\beta}$-derivative of $\exp(iqx)$. This $q_{\beta}$ derivative may
be taken out of the integral bringing us to the following expression:

$\displaystyle
A_{\mu\nu}(p|q,q^{\prime})=\left(p_{\nu}\frac{\partial}{\partial
q_{\mu}}-p_{\mu}\frac{\partial}{\partial
q_{\nu}}\right)\int\limits_{0}^{\infty}da\,dx\,dk\,e^{iqx+ikx-i\omega px-
iapx-\epsilon a}\frac{1}{\mu^{2}-k^{2}-i0}d\omega
X_{A}(\omega,\omega^{\prime}\\{\ldots\\}$ (3.38)

Taking $x$\- and $k$-integrations yields

$\displaystyle
A_{\mu\nu}(p|q,q^{\prime})=\left(p_{\nu}\frac{\partial}{\partial
q_{\mu}}-p_{\mu}\frac{\partial}{\partial
q_{\nu}}\right)\int\limits_{0}^{\infty}dae^{-\epsilon
a}\frac{1}{\mu^{2}-(q-p(a+\omega))^{2}-i0}d\omega
X_{A}(\omega,\omega^{\prime})\\{\ldots\\}$ (3.39)

The $q$-derivative may be written as

$\displaystyle\frac{\partial}{\partial
q_{\beta}}\left(\frac{1}{\mu^{2}-(q-p(a+\omega))^{2}-i0}\right)$
$\displaystyle=$
$\displaystyle-2\frac{\partial}{\partial\mu^{2}}\left(\frac{1}{\mu^{2}-(q-p(a+\omega))^{2}-i0\cdots}\right)\left(q_{\beta}-p_{\beta}(a+\omega)\right)$
(3.40) $\displaystyle\to$
$\displaystyle-2q_{\beta}\frac{\partial}{\partial\mu^{2}}\left(\frac{1}{\mu^{2}-(q-p(a+\omega))^{2}-i0\cdots}\right)$

since the term proportional to $p_{\beta}$ does not contribute to the anti-
symmetric amplitude $A_{\mu\nu}$.

The next steps in our handling of Eq. (3.39) are as follows:
The $a$-integral converges so we may take the limit $\epsilon\to 0$ and set
$\exp(-\epsilon a)\to 1$.
Isolating the factor $M_{B}^{2}$ in the denominator, the $a$-integral in Eq.
(3.39) is equal to (3.36) with

$\displaystyle a_{+}=\frac{-1+\sqrt{1+4\bar{\mu}^{2}}+2\omega}{2}>0,\quad
a_{-}=\frac{-1-\sqrt{1+4\bar{\mu}^{2}}+2\omega}{2}<0,\quad{\rm for}\quad
0<\omega<1,$ (3.41)

where $\bar{\mu}^{2}\equiv\mu^{2}/M_{B}^{2}$ and
$a_{+}-a_{-}=\sqrt{1+4\bar{\mu}^{2}}$. For further use we denote the real part
of (3.36) as $K(\omega,\bar{\mu}^{2})$:

$\displaystyle
K(\omega,\bar{\mu}^{2})=\frac{1}{a_{+}-a-}\log\left({-\frac{a_{-}}{a_{+}}}\right),$
(3.42)

with $a_{\pm}$ given by (3.41). Notice a useful relation

$\displaystyle\partial_{\omega}K(\omega,\bar{\mu}^{2})=\frac{1}{\bar{\mu}^{2}+\omega(1-\omega)}$
(3.43)

Adding the contribution of the “lower” part of the diagram yields

$\displaystyle
A_{\mu\nu}(p|q,q^{\prime})\sim\left(q^{\prime}_{\nu}q_{\nu}-q_{\nu}q^{\prime}_{\mu}\right)\frac{\partial}{\partial\bar{\mu}^{2}}F_{s}(\bar{\mu}^{2},\bar{m}^{2}),$
(3.44)

with (recall that we consider $q^{2}=q^{\prime 2}=0$)

$\displaystyle F_{s}(\bar{\mu}^{2},\bar{m}^{2})$ $\displaystyle=$
$\displaystyle\int
D(\omega,\omega^{\prime})\,X_{A}(\omega,\omega^{\prime})K(\omega,\bar{\mu}^{2})\frac{1}{\bar{m}^{2}+\omega^{\prime}(1+\omega^{\prime})}$
(3.45) $\displaystyle+i\pi\frac{1}{\sqrt{1+4\bar{\mu}^{2}}}\int
D(\omega,\omega^{\prime})\,X_{A}(\omega,\omega^{\prime})\frac{1}{\bar{m}^{2}+\omega^{\prime}(1+\omega^{\prime})}.$

Here $\bar{m}^{2}\equiv m^{2}/M_{B}^{2}$.

To compare the amplitude in the double-collinear approximation, Eq. (3.2), and
the exact amplitude, Eq. (3.44), in the HQ limit ($M_{B}\to\infty$, $m$ and
$\mu$ fixed, $\omega\sim\Lambda_{\rm QCD}/M_{B}$), we make use of the
following expansions

$\displaystyle-\partial_{\bar{\mu}^{2}}K(\omega,\bar{\mu}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{\omega}-(1+2\log(\omega)-2i\pi)-3\omega+O(\omega^{2}),$
(3.46) $\displaystyle\frac{1}{\bar{\mu}^{2}+\omega(1-\omega)}$
$\displaystyle=$ $\displaystyle\frac{1}{\omega}+1+\omega+O(\omega^{2}).$
(3.47)

Taking into account that any 3DA is peaked in region
($\omega,\omega^{\prime})\sim\Lambda_{\rm QCD}/m_{b}$ , we observe the
following important features of the amplitude in the HQ limit:

  * (a)

The amplitude in the “double-collinear” approximation, Eqs. (3.2) and (3.47)
and the exact amplitude, Eqs. (3.44) and (3.46), have one and the same leading
behaviour in the HQ limit.

  * (b)

The imaginary part of the exact amplitude, see Eqs. (3.44) and (3.45), is
parametrically suppressed compared to its real part. The amplitude in the
“double-collinear” approximation (3.2) does not have any imaginary part.

  * (c)

The “strong” imaginary part of the exact $B\to\gamma\gamma$ amplitude, Eq.
(3.45), gained due to soft gluon interactions, is unphysical as there are no
appropriate hadron intermediate states which might lead to the appearance of
such imaginary part at $q^{2}=0$ and $q^{\prime 2}=0$. Noteworthy, the
imaginary part vanishes identically if $X_{A}$ satisfies the constraint (2.8);
the constraint on $X_{A}$ given by Eq. (2.6) is not sufficient to guarantee
the absence of the unphysical imaginary part.

##  4 Conclusions

We studied a generic $B$-decay amplitude of the FCNC-type — i.e., an amplitude
given by diagrams in which the heavy field hits the middle point of the line
along which light degrees of freedom propagate — and obtained the following
results:

(i) As already demonstrated in the literature mk2018 ; m2019 ; m2022 , the
leading contribution to the amplitude of a $B$-decay of FCNC-type, $B\to
j(q)j^{\prime}(q^{\prime})$, is given by the convolution formula of the hard
kernel composed of propagators of the light degrees of freedom and the 3BS of
the $B$-meson

$\displaystyle\langle
0|\phi(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle$ (4.48)

in the following configuration:

$\displaystyle x^{2}=0;\qquad x^{\prime 2}=0,\qquad xx^{\prime}\neq 0.$ (4.49)

We have now formulated this result as a factorization theorem by a direct
analysis of Feynman diagrams: Considering the FCNC-type $B$-decay into two
real photons ($q^{2}=q^{\prime 2}=0$) in the rest frame of the $B$-meson and
choosing the momenta $q$ and $q^{\prime}$ along the $(+)$ and $(-)$ axes of
the light cone, respectively, we have shown that the dominant contribution to
the amplitude in the heavy-quark limit comes from the “double-collinear”
configuration

$\displaystyle\langle
0|\phi(x_{+})\phi_{b}(0)\phi^{\prime}(x^{\prime}_{-})|B(p)\rangle.$ (4.50)

Eq. (3.22) represents the factorization formula for the amplitudes of FCNC-
type; corrections to the contribution given by Eq. (3.22) are suppressed by
powers of the heavy-quark mass.

(ii) The $B$-meson 3BS in a collinear LC configuration

$\displaystyle\langle 0|\phi(x)\phi_{b}(0)\phi^{\prime}(\lambda
x)|B(p)\rangle,\qquad x^{2}=0,\qquad 0<\lambda<1,$ (4.51)

does not appear in the convolution formula (3.22) for the leading 3BS
contribution to the FCNC $B$-decay amplitude. This means that the calculation
of the FCNC $B$-decay amplitude as the convolution of the hard kernel and the
collinear 3BS as is done e.g. in hidr  does not provide a well-defined
starting point: corrections to this approximation are not suppressed by any
large parameter.

This makes an essential difference between the 3DA contributions to charming
loops in FCNC $B$-decays and to SL $B$-decays: The 3DA contribution to the SL
form factor factorizes into the convolution of a hard kernel and the collinear
LC 3BS offen2007 , corrections to this configuration being suppressed by
$1/m_{b}$.

(iii) The generic 3BS involves more Lorentz structures and 3DAs than the
collinear 3BS. In particular, it involves structures of the type

$\displaystyle\langle
0|G_{\mu\nu}(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle$
$\displaystyle=$ $\displaystyle\int D(\omega,\omega^{\prime})\;e^{-ipx\omega-
ipx^{\prime}\omega^{\prime}}$ (4.52)
$\displaystyle\times\left\\{\left(\frac{p_{\mu}x_{\nu}-p_{\nu}x_{\mu}}{xp}\right)X^{(1)}_{A}(\omega,\omega^{\prime})+\left(\frac{p_{\mu}x^{\prime}_{\nu}-p_{\nu}x^{\prime}_{\mu}}{x^{\prime}p}\right)X^{(2)}_{A}(\omega,\omega^{\prime})\right\\}$

with $D(\omega,\omega^{\prime})=d\omega
d\omega^{\prime}\theta(\omega)\theta(\omega)\theta(1-\omega-\omega^{\prime})$.
These Lorentz structures contain kinematical singularities in $1/xp$ or
$1/x^{\prime}p$. Such singularities are unphysical. Requiring that the 3DA is
a continuous and finite function of its variables at $x^{2}=0$, $x^{\prime
2}=0$, $xp=0$ and $x^{\prime}p=0$, we conclude that the kinematical
singularities should vanish due to the properties of the 3DAs $X^{(1)}_{A}$
and $X^{(2)}_{A}$, leading to the following constraints:

$\displaystyle\int\limits_{0}^{1-\omega}d\omega
X^{(1)}_{A}(\omega,\omega^{\prime})=0\quad\forall\omega^{\prime};\qquad\int\limits_{0}^{1-\omega^{\prime}}d\omega^{\prime}X^{(2)}_{A}(\omega,\omega^{\prime})=0\quad\forall\omega.$
(4.53)

In other words, the primitives
$\overline{X}^{(1)}_{A}(\omega,\omega^{\prime})$ and
$\overline{X}^{(2)}_{A}(\omega,\omega^{\prime})$ [Eqs. (2.10) and (2.11)]
should vanish on the boundaies of the $X_{A}$-suppport area:

$\displaystyle\overline{X}^{(1)}_{A}(\omega=0,\omega^{\prime})=\overline{X}^{(1)}_{A}(1-\omega^{\prime},\omega^{\prime})=0\quad\forall\omega^{\prime},$
$\displaystyle\overline{X}^{(2)}_{A}(\omega,\omega^{\prime}=0)=\overline{X}^{(2)}_{A}(\omega,1-\omega)=0\quad\forall\omega.$
(4.54)

Constraints of this type emerge for all 3DAs which parametrize the Lorents
structures $x_{\mu}/xp$, $x^{\prime}_{\mu}/x^{\prime}p$,
$x_{\mu}x_{\nu}/(xp)^{2}$, $x_{\mu}x^{\prime}_{\nu}/(xp)(x^{\prime}p)$ etc in
the 3BS of the $B$-meson. These constraints should be taken into account when
building models for the 3DAs.

(iv) The $B$-meson 3BS of the $X_{A}$-type of Eq. (4.52) in the collinear LC
configuration $x_{\mu}=\tau n_{\mu}$, $x^{\prime}_{\mu}=\tau^{\prime}n_{\mu}$,
$n^{2}=0$, Eq. (3.26), and in the double-collinear LC configuration,
$x_{\mu}=\tau n_{\mu}$, $x^{\prime}_{\mu}=\tau^{\prime}n^{\prime}_{\mu}$,
$n^{2}=n^{\prime 2}=0$, $n^{\prime}n=2$, Eq. (3.27), may be parametrized via
one and the same Lorentz structure
$n_{\mu}n^{\prime}_{\nu}-n_{\nu}n^{\prime}_{\mu}$. However, the corresponding
3DAs are different and in general independent of each other: the collinear LC
3DA is equal to $X^{(1)}_{A}+X^{(2)}_{A}$, whereas the double-collinear LC 3DA
is equal to $X^{(1)}_{A}-X^{(2)}_{A}$.

Acknowledgments. I am grateful to I. Belov, A. Berezhnoy, M. Ferre, E. Kou, O.
Nachtmann, and H. Sazdjian for discussions and to Q. Qin, Y.-L. Shen, C. Wang
and Y.-M. Wang for comments on their paper wang2022 . I am particularly
indebted to Yu-Ming Wang for his illuminating remarks.

##  Appendix A Nonfactorizable charming loop in FCNC $B$-decay amplitude

Nonfactorizable charm-loop contribution to the amplitude of FCNC decay is
given by the diagram of Fig. 3. The corresponding analytic expression reads
m2022

$\displaystyle A_{\rm FCNC}(p|q,q^{\prime})=\frac{G_{F}}{\sqrt{2}}\int
d\kappa^{\prime}dx^{\prime}e^{i\kappa^{\prime}x^{\prime}}\frac{1}{m^{2}-(\kappa^{\prime}-q^{\prime})^{2}-i0}d\kappa
dxe^{i\kappa x}\Gamma_{cc}(\kappa,q)\langle
0|\phi(x)\phi_{b}(0)\phi^{\prime}(x^{\prime})|B(p)\rangle,$ (A.55)

where the expression for the charm-quark loop in the case of scalar “quarks”
has the form

$\displaystyle\Gamma_{cc}(\kappa,q)=\frac{1}{8\pi^{2}}\int\limits_{0}^{1}du\int\limits_{0}^{1-u}dv\frac{1}{m_{c}^{2}-2uv\kappa
q-\kappa^{2}u(1-u)-q^{2}v(1-v)-i0}.$ (A.56)

Fig. 3:  Diagram describing $A_{\rm FCNC}$, a nonfactorizable charming-loop
corrections to the amplitude of FCNC $B$-decay.

The amplitude (A.55) corresponds to the generic amplitude describing 3BS
correction to the form factor (3.15) with the replacement of the usual quark
propagator by an “effective” propagator describing the charm-quark triangle
$\Gamma_{cc}(\kappa,q)$

$\displaystyle\frac{1}{\mu^{2}-(\kappa-q)^{2}-i0}\to\Gamma_{cc}(\kappa,q).$
(A.57)

Important is that $\Gamma_{cc}(\kappa,q)$, similar to the usual propagator, is
a quadratic function of its momentum variables. Therefore the consideration
presented in the text for the amplitude (3.15) can be directly applied to the
amplitude (A.55).

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