%Title: Gluonic Penguins in $B\to\pi\pi$ from QCD Light-Cone Sum Rules
%Author: Alexander Khodjamirian,Thomas Mannel, Piotr Urban
%Published: * Phys.Rev. * ** D67 ** (2003) 054027.
%hep-ph/0210378
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TTP02-16
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\begin{center}
{\Large\bf Gluonic Penguins in $B \to \pi \pi$ \\
from QCD Light-Cone Sum Rules}
\vskip 1.5true cm
{\large\bf
Alexander Khodjamirian\,$^{a,*)}$,
Thomas Mannel\,$^{a}$, Piotr Urban\,$^{a,b}$}
\\[1cm]~
\vskip 0.5true cm
{\it $^a$ Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe, \\ D-76128 Karlsruhe, Germany\\
$^b$ Henryk Niewodnicza\'nski Institute of Nuclear Physics\\
Kawiory 26a, 30-055 Krak\'ow, Poland }
\end{center}
\vskip 1.0true cm
\begin{abstract}
\noindent
The $B\to \pi\pi$ hadronic matrix element
of the chromomagnetic dipole operator $O_{8g}$
(gluonic penguin) is calculated using the QCD
light-cone sum rule approach. The resulting sum rule for
$\langle \pi\pi |O_{8g}|B\rangle $
contains, in addition to the $O(\alpha_s)$ part induced by
hard gluon exchanges, a contribution due to
soft gluons.
We find that in the limit $m_b\to \infty$ the soft-gluon contribution
is suppressed as a second power of $1/m_b$
with respect to the leading-order factorizable
$B\to \pi\pi$ amplitude, whereas the hard-gluon contribution has
only an $\alpha_s$ suppression. Nevertheless,
at finite $m_b$, soft
and hard effects of the gluonic penguin
in $B\to \pi\pi$ are of the same order.
Our result indicates that soft contributions
are indispensable for an accurate counting
of nonfactorizable effects in charmless $B$ decays.
On the phenomenological side we predict that the
impact of gluonic penguins on $\bar{B}^0_d\to \pi^+\pi^-$
is very small,
but is noticeable for $\bar{B}^0_d\to \pi^0\pi^0$.
\end{abstract}
%{\em Keywords:} B-meson decays; Quantum Chromodynamics; Sum rules \\
\vspace*{\fill}
\noindent $^{*)}${\small \it On leave from
Yerevan Physics Institute, 375036 Yerevan, Armenia } \\
\newpage
\section{Introduction}
Intensive experimental studies of
$B\to \pi\pi$ and other charmless two-body hadronic
$B$ decays are currently under way at $B$ factories ~\cite{Bpipiexp}.
The major goal is to complement the already observed
CP-violation in the $B\to J/\psi K$ channel
by measuring other CP-violating effects
and quantifying them
in terms of the angles $\alpha$ and $\gamma$
of the CKM unitarity triangle (for a recent comprehensive review,
see, e.g. Ref.~\cite{Fleischer}). The extraction of these fundamental
parameters from experimental data on $B\to \pi\pi, K\pi, K\bar{K}$
demands reliable theoretical estimates of
hadronic matrix elements of the
effective weak Hamiltonian between the initial $B$ and the final
two-light-meson states. These matrix elements are not yet
accessible in lattice QCD, and thus other QCD methods of their calculation
are actively being developed.
The factorization ansatz in hadronic two-body
$B$ decays has recently been put on a more solid ground.
It has been shown \cite{BBNS}, to $O(\alpha_s)$,
that in the limit $m_b\to \infty$
the amplitude for $B\to \pi\pi$ factorizes
into a product of the $B\to \pi$ form factor
and the pion decay constant.
The nonfactorizable effects due to hard gluon exchanges
are directly calculated combining perturbative QCD
with certain nonperturbative inputs. The latter include the
$B\to \pi$ form factor and the light-cone distribution
amplitudes (DA) of pions and $B$ meson.
According to QCD factorization,
various soft nonfactorizable contributions
to $B\to \pi\pi$, such as exchanges of
low virtuality gluons between the ``emission'' pion
and the remaining $B\to \pi$ part of the process,
vanish in the $m_b \to \infty$ limit.
An open phenomenological
problem for QCD factorization is to obtain quantitative estimates of
soft nonfactorizable effects at a finite, physical $b$ quark mass.
Among these effects,
the contributions of current-current operators
in the penguin topology (e.g., the ``charming penguins'') could be
important, as was argued in Ref.~\cite{charmpeng}.
Recently, a new method to calculate the $B\to \pi\pi$ hadronic
matrix elements from QCD
light-cone sum rules (LCSR) \cite{lcsr,BF1,CZ} has been suggested
by one of us \cite{AK}.
The main advantage of LCSR is that
the hadronic matrix elements of heavy-to-light
transitions can be calculated
including simultaneously soft and hard effects.
Thus, the $B\to \pi$ form factor
including both soft (end-point) contributions and hard-gluon exchanges
was obtained from LCSR \cite{Bpi,KRWY,BBB,BallZ}
providing the main input for the factorizable $B\to \pi\pi$ amplitude.
Furthermore, in Ref.~\cite{AK} it was shown that with LCSR
one achieves a quantitative control over $1/m_b$
effects of soft nonfactorizable gluons in $B\to \pi\pi$.
The soft-gluon effect in the matrix element of the current-current
operator in the emission topology
turns out to be of the same size as the
hard-gluon nonfactorizable contributions obtained from
QCD factorization.
It is important therefore to investigate other
nonfactorizable effects, related to the penguin and dipole
operators and/or to non-emission topologies for the current-current operators.
Especially interesting is to compare the size of soft- and hard-gluon
contributions.
To continue a systematic study in this direction
we apply in this paper the LCSR method
to the $B\to \pi\pi$ hadronic matrix element of
the chromomagnetic dipole operator
$O_{8g}$ (gluonic penguin).
Apart from being phenomenologically
interesting by itself, the $\langle \pi\pi| O_{8g}| B\rangle$
matrix element provides a very useful
study case for the LCSR approach. Indeed, as we shall see, the
hard-gluon contribution is given by one-loop diagrams
and is therefore relatively simple.
The soft-gluon effect corresponding to a tree-level diagram
is also easily calculable.
One is therefore able to estimate both hard and soft
gluonic-penguin contributions using one and the same method and input.
The paper is organized as follows. In section 2
we introduce the relevant three-point correlation function
and calculate it using operator-product expansion
(OPE) near the light-cone and taking into
account both hard- ($O(\alpha_s)$) and
soft-gluon contributions.
In section 3, following the procedure described in Ref.~\cite{AK}
we match the result of this calculation to
the dispersion relations in $\pi$ and $B$ channels
combined with quark-hadron duality, and obtain the
sum rule for the $\langle \pi\pi|O_{8g}|B\rangle$
matrix element. In section 4 we investigate the $m_b\to \infty$ limit
of the sum rule. Section 5 contains the numerical analysis.
Furthermore, in section 6 our prediction is discussed
from the phenomenological point of view and compared, in section 7,
with the corresponding result of QCD factorization.
We conclude in section 8. The appendix contains
some useful formulae: the decompositions of
the vacuum-to-pion matrix elements
in terms of the light-cone DA of various twist.
\section{Correlation function}
Following Ref.~\cite{AK} we start with defining a
vacuum-pion correlation function:
\be
F_\alpha (p,q,k)=
-\int d^4x\, e^{-i(p-q)x} \int d^4y\, e^{i(p-k)y}
\langle\, 0 \mid T\{j^{(\pi)}_{\alpha 5}(y)O_{8g}(0)j^{(B)}_5(x)\}\mid \pi^-(q)
\rangle\,,
\label{corr}
\ee
where the quark currents $j^{(\pi)}_{\alpha5}=\bar{u}\gamma_\alpha \gamma_5 d $
and $j^{(B)}_5 = m_b \bar{b}i \gamma_5 d$ interpolate
$\pi$ and $B$ mesons, respectively, and
the quark-gluon chromomagnetic dipole operator
$O_{8g}$ relevant for $B\to \pi\pi $ has the following
standard expression:
\be
O_{8g}=\frac{m_b}{8\pi^2}\bar{d}\sigma^{\mu\nu}(1+\gamma_5)
\frac{\lambda^a}2 g_sG^a_{\mu\nu} b\,.
\label{peng}
\ee
This operator enters the $\Delta B =1 $ effective weak Hamiltonian
\be
H_W= \frac{G_F}{\sqrt{2}}
\left\{ \lambda_u\left[\left( c_1(\mu)+ \frac{c_2(\mu)}{3}\right)O_1(\mu) +
2c_2(\mu)\widetilde{O}_1(\mu)\right]
+ ... +\lambda_t c_{8g}(\mu)O_{8g}(\mu)\right\}\,,
\label{H}
\ee
where $\lambda_q=V_{qb}V^*_{qd}$, $q=u,t$ are the combinations of CKM parameters,
and we have singled out the relevant current-current operators:
\be
O_1=(\bar{d}\Gamma_\mu u)(\bar{u}\Gamma^\mu b)\,,~~~
\widetilde{O}_1=(\bar{d}\Gamma_\mu \frac{\lambda^a}2u)(\bar{u}\Gamma^\mu
\frac{\lambda^a}2 b)\,,
\label{o1}
\ee
denoting by ellipses all remaining operators which
we do not consider in this paper.
In Eqs.~(\ref{peng})-(\ref{o1}),
$\mbox{Tr}(\lambda^a\lambda^b) = 2\delta^{ab}$, $c_{1,2,8g}$ are the Wilson
coefficients, $\mu\sim m_b$ is the normalization scale and $\Gamma_\mu=\gamma_\mu(1-\gamma_5)$.
Furthermore, the operator $O_2=(\bar{u}\Gamma_\mu u)(\bar{d}\Gamma^\mu b)\,$
has been Fierz transformed: $O_2= O_1/3+ 2\widetilde{O}_1$.
The correlator (\ref{corr}) is a function of
three independent momenta, which are chosen to be $q$, $p-k$, $k$,
and, for simplicity, $p^2=k^2=0$.
As explained in Ref.~\cite{AK}, the momentum $k$ plays an
auxiliary role and will not appear in the sum rule for the hadronic matrix element.
The pion is on shell, $q^2=m_\pi^2$.
We will work in the chiral limit
and set $m_\pi=0$ everywhere, except in the enhanced
combination $\mu_\pi=m_\pi^2/(m_u+m_d)$.
The Lorentz-decomposition of the correlation function (\ref{corr})
contains four invariant amplitudes:
\be
F_\alpha=
(p-k)_\alpha F
+ q_\alpha \widetilde{F}_1
+ k_\alpha\widetilde{F}_2
+ \epsilon_{\alpha\beta\lambda\rho}q^\beta p^\lambda k^\rho
\widetilde{F}_3\,,
\label{decompos}
\ee
depending
on three kinematical invariants $(p-k)^2$, $(p-q)^2$ and
$P^2=(p-q-k)^2$. In what follows only the amplitude $F$ is needed.
In the region of large virtualities: $(p-k)^2,(p-q)^2,P^2<0$,
$|(p-k)^2|,|(p-q)^2|,|P^2|\gg \Lambda_{QCD}^2$,
the correlation function (\ref{corr}) is calculated
using light-cone OPE, in a form of the sum of short-distance coefficient functions (hard amplitudes) convoluted with the pion DA of
growing twist and multiplicity.
Only a few first terms of this expansion are
usually retained, the components with higher twist/multiplicity are
neglected being suppressed by inverse powers of large virtualities.
The gluon emitted from the vertex $O_8$ in the correlation function
contributes in two different ways. First, a highly virtual (hard) gluon
is absorbed by one of the quark lines at a short light-cone separation
from the emission point. The corresponding one-loop diagrams
(see Fig.~1) are then calculated perturbatively, as
an $O(\alpha_s)$ part of the
hard amplitude. The second possibility is that the emitted gluon
together with the quark-antiquark pair forms
the quark-antiquark-gluon DA of the pion (see Fig.~2a). We will call
these low-virtuality gluons {\em soft}.
Thus, in the correlation function the hard- and soft-gluon
contributions belong to different terms of the light-cone OPE
and can be calculated separately.
In what follows, we take into account
all $O(\alpha_s)$ hard-gluon effects of twist 2 and 3, and, in addition
the effects of quark-antiquark-gluon DA of twist 3 and 4 in zeroth order
in $\alpha_s$.
This approximation corresponds to the accuracy of the
light-cone expansion adopted for the
LCSR calculation of the $B\to \pi$ form factor.
The higher twist terms
corresponding to the soft-gluon contributions are
subleading in the light-cone OPE, being suppressed
by inverse powers of large virtualities.
However, in the correlation function (\ref{corr}) the lowest twist
contribution is of $O(\alpha_s)$, therefore both hard and soft contributions
can be equally important. We may again refer to the LCSR for
the $B\to \pi$ form factor where the $O(\alpha_s)$ twist 2 and
the zeroth-order in $\alpha_s$ twist 4 contributions are
at the same level numerically.
In the following three subsections the
calculation of the various contributions to the correlation function
(\ref{corr}) is presented.
\subsection{Hard gluon contribution}
We begin with calculating the leading $O(\alpha_s)$ contribution
to the correlator (\ref{corr}) generated by hard gluons.
To obtain the relevant diagrams
one has to contract the $b$-quark fields
and, in addition, the $d$- and $\bar{d}-$quark
fields from the operators $j^{(\pi)}_{\alpha5}$
and $O_{8g}$, respectively, inserting
the free-quark propagators for both contractions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.9\textwidth]{figpert.ps}
\end{center}
\caption{
{\it Diagrams corresponding to the perturbative $O(\alpha_s)$ contributions
to the correlation function (\ref{corr}).
Solid, dashed, wavy lines and ovals represent quarks, gluons,
external currents and pseudoscalar meson DA, respectively.
Thick points indicate the gluonic penguin vertex. }
\label{fig1}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The remaining on-shell $\bar{u} $ and $d$
fields form the quark-antiquark DA of the pion.
The decomposition of the relevant vacuum-pion
matrix elements in terms of DA of the lowest twists
2 and 3 is presented in the appendix. Twist 4 components
are neglected, because the $O(\alpha_s)$ twist-4 effects
are beyond the accuracy of our calculation.
The gluon emitted by $O_{8g}$ is absorbed by one of the four quark lines
leading to the four diagrams shown in Fig.~1.
Note that we disregard the possibility to
contract the $\bar{d}$ quark from $O_{8g}$
with the $d$ quark from $j_5^{(B)}$. That type of contraction
leads to ``penguin-annihilation'' diagrams where
the heavy-light loop is connected by a single gluon with the
pion part of the correlation function. An additional gluon
has to be added in this case to obey color conservation, making
this contribution either $O(\alpha_s^2)$,
with both gluons hard, or $O(\alpha_s)$, with one hard gluon and one
soft gluon, the latter entering the pion DA. Both effects are
beyond the adopted approximation for the correlation function
and presumably very small.
The one-loop diagrams are calculated employing dimensional
regularization.
Among other tools we used the FORM program \cite{FORM}.
The invariant amplitude $F$ is obtained by
taking the coefficient at
$(p-k)_\alpha$.
The result for $F$ is ultraviolet divergent; however,
this divergence does not play any role in the resulting sum rule
because the $1/\epsilon$ terms vanish after Borel transformations.
On the other hand, we find that
the amplitude $F$ does not contain infrared-collinear
divergences. This is in accordance with the fact that the perturbative
expansion for the correlation function starts, in
twists 2 and 3, with $O(\alpha_s)$.
If one takes into account an additional perturbative gluon correction
in the correlation function, the resulting two-loop diagrams
will yield infrared-collinear divergences which
should be absorbed by the evolution of the pion DA, as in
the case of the $O(\alpha_s)$ correction to the
LCSR for the $B\to \pi$ form factor \cite{KRWY,BBB}.
According to the procedure of
the sum rule derivation explained in Ref.~\cite{AK}
we have to retain in the final answer for the amplitude $F$
only those terms which, in the limit of large $|P^2|\sim m_B^2$,
have a nonvanishing double imaginary part
in the variables $(p-k)^2$ and $ (p-q)^2 $, taken in the
duality region $\;0<(p-k)^2 K+- pi0 and B --> K0 pi0, and evidence for B --> pi+ pi-,''
Phys.\ Rev.\ Lett.\ {\bf 85} (2000) 515;
%%CITATION = PRLTA,85,515.%%
%\bibitem{Abe:2001nq}
%\cite{Aubert:2002tv}
%\bibitem{Aubert:2002tv}
B.~Aubert [BABAR Collaboration],
%``Measurements of branching fractions and CP-violating asymmetries in B0 $\to$ pi+ pi-, K+ pi-, K+ K- decays,''
hep-ex/0205082;\\
%%CITATION = HEP-EX 0205082;%%
%\cite{Casey:2002yd}
%\bibitem{Casey:2002yd}
B.~C.~Casey {\it et al.} [Belle Collaboration],
%``Charmless hadronic two-body B meson decays,''
arXiv:hep-ex/0207090.
%%CITATION = HEP-EX 0207090;%%
%\cite{Fleischer:2002ys}
\bibitem{Fleischer}
R.~Fleischer,
%``CP violation in the B system and relations to K $\to$ pi nu anti-nu
%decays,''
hep-ph/0207108.
%%CITATION = HEP-PH 0207108;%%
\bibitem{BBNS}
%\cite{Beneke:1999br}
%\bibitem{Beneke:1999br}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``{QCD} factorization for B $\to$ pi pi decays: Strong phases and CP violation in the heavy quark limit,''
Phys.\ Rev.\ Lett.\ {\bf 83} (1999) 1914;
%[hep-ph/9905312].
%%CITATION = HEP-PH 9905312;%%
%\cite{Beneke:2001ev}
%M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``QCD factorization in B $\to$ pi K, pi pi decays and extraction of Wolfenstein parameters,''
Nucl.\ Phys.\ B {\bf 606} (2001) 245.
%[arXiv:hep-ph/0104110].
%%CITATION = HEP-PH 0104110;%%
%\cite{Ciuchini:2001gv}
\bibitem{charmpeng}
M.~Ciuchini, E.~Franco, G.~Martinelli, M.~Pierini and L.~Silvestrini,
%``Charming penguins strike back,''
Phys.\ Lett.\ B {\bf 515} (2001) 33.
%[arXiv:hep-ph/0104126].
%%CITATION = HEP-PH 0104126;%%
\bibitem{lcsr}
I.~I.~Balitsky, V.~M.~Braun and A.~V.~Kolesnichenko,
%``Radiative Decay Sigma+ $\to$ P Gamma In Quantum Chromodynamics,''
Nucl.\ Phys.\ {\bf B312} (1989) 509.
%%CITATION = NUPHA,B312,509;%%
\bibitem{BF1}
V.~M.~Braun and I.~E.~Filyanov,
%``QCD Sum Rules In Exclusive Kinematics And Pion Wave Function,''
Z.\ Phys.\ {\bf C44} (1989) 157.
%%CITATION = ZEPYA,C44,157;%%
\bibitem{CZ}
V.~L.~Chernyak and I.~R.~Zhitnitsky,
%``B Meson Exclusive Decays Into Baryons,''
Nucl.\ Phys.\ {\bf B345} (1990) 137.
%%CITATION = NUPHA,B345,137;%%
%\cite{Khodjamirian:2000mi}
\bibitem{AK}
A.~Khodjamirian,
%``B $\to$ pi pi decay in QCD,''
Nucl.\ Phys.\ {\bf B605} (2001) 558.
%%CITATION = HEP-PH 0012271;%%
\bibitem{Bpi}
V.~M.~Belyaev, A.~Khodjamirian and R.~R\"uckl,
%``QCD calculation of the B $\to$ pi, K form-factors,''
Z.\ Phys.\ {\bf C60} (1993) 349,
%[hep-ph/9305348].
%%CITATION = HEP-PH 9305348;%%
%\cite{Belyaev:1994zk}
%\bibitem{Belyaev:1994zk}
V.~M.~Belyaev, V.~M.~Braun, A.~Khodjamirian and R.~Ruckl,
%``D* D pi and B* B pi couplings in QCD,''
Phys.\ Rev.\ D {\bf 51} (1995) 6177.
%[arXiv:hep-ph/9410280].
%%CITATION = HEP-PH 9410280;%%
\bibitem{KRWY}
A.~Khodjamirian, R.~R\"uckl, S.~Weinzierl and O.~Yakovlev,
%``Perturbative QCD correction to the B --> pi transition form factor,''
Phys.\ Lett.\ {\bf B410} (1997) 275.
%[hep-ph/9706303].
%%CITATION = HEP-PH 9706303;%%
%\bibitem{Bagan}
\bibitem{BBB}
E.~Bagan, P.~Ball and V.~M.~Braun,
%``Radiative corrections to the decay B --> pi e nu and
%the heavy quark limit,''
Phys.\ Lett.\ {\bf B417} (1998) 154.
%[hep-ph/9709243].
%%CITATION = HEP-PH 9709243;%%
%\cite{Ball:2001fp}
\bibitem{BallZ}
P.~Ball and R.~Zwicky,
%``Improved analysis of B $\to$ pi e nu from QCD sum rules on the light-cone,''
JHEP {\bf 0110} (2001) 019.
%[arXiv:hep-ph/0110115].
%%CITATION = HEP-PH 0110115;%%
\bibitem{FORM}
%\cite{Vermaseren:2000nd}
%\bibitem{Vermaseren:2000nd}
J.~A.~Vermaseren,
%``New features of FORM,''
%arXiv:
math-ph/0010025.
%%CITATION = MATH-PH 0010025;%%
%\cite{Braun:1999uj}
\bibitem{BKM}
V.~M.~Braun, A.~Khodjamirian and M.~Maul,
%``Pion form factor in QCD at intermediate momentum transfers,''
Phys.\ Rev.\ D {\bf 61} (2000) 073004.
%[arXiv:hep-ph/9907495].
%%CITATION = HEP-PH 9907495;%%
\bibitem{SVZ}
%\cite{Shifman:bx}
%\bibitem{Shifman:bx}
M.~A.~Shifman, A.~I.~Vainshtein and V.~I.~Zakharov,
%``QCD And Resonance Physics. Sum Rules,''
Nucl.\ Phys.\ B {\bf 147} (1979) 385,
%%CITATION = NUPHA,B147,385;%%
%\cite{Shifman:by}
%\bibitem{Shifman:by}
%M.~A.~Shifman, A.~I.~Vainshtein and V.~I.~Zakharov,
%``QCD And Resonance Physics: Applications,''
%Nucl.\ Phys.\ B {\bf 147} (1979)
448.
%%CITATION = NUPHA,B147,448;%%
\bibitem{KR}
%\cite{Khodjamirian:1998ji}
A.~Khodjamirian and R.~Ruckl,
%``{QCD} sum rules for exclusive decays of heavy mesons,''
in {\em Heavy Flavors}, 2nd edition, eds., A.J. Buras and M. Lindner,
World Scientific (1998), p. 345, hep-ph/9801443.
%%CITATION = HEP-PH 9801443;%%
%\cite{Khodjamirian:1998vk}
\bibitem{KRW}
A.~Khodjamirian, R.~Ruckl and C.~W.~Winhart,
%``The scalar B $\to$ pi and D $\to$ pi form factors in {QCD},''
Phys.\ Rev.\ D {\bf 58} (1998) 054013.
%[arXiv:hep-ph/9802412].
%%CITATION = HEP-PH 9802412;%%
\bibitem{ElKLuke}
%\bibitem{El-Khadra:2002wp}
A.~X.~El-Khadra and M.~Luke,
%``The mass of the b quark,''
hep-ph/0208114.
%%CITATION = HEP-PH 0208114;%%
\bibitem{delta}
V.L.~Chernyak, A.R.~Zhitnitsky and I.R.~Zhitnitsky;
Sov. J. Nucl. Phys. {\bf 38} (1983) 645;\\
%%CITATION = SJNCA,38,645;%%
V.A.~Novikov, M.A.~Shifman, A.I.~Vainshtein, M.B.~Voloshin and V.I.~Zakharov,
Nucl. Phys. {\bf B237} (1984) 525.
%%CITATION = NUPHA,B237,525;%%
%\cite{Bijnens:2002mg}
\bibitem{BK}
J.~Bijnens and A.~Khodjamirian,
%``Exploring light-cone sum rules for pion and kaon form factors,''
hep-ph/0206252, to be published in Eur.~Phys.~J. C.
%%CITATION = HEP-PH 0206252;%%
\bibitem{BF}
V.~M.~Braun and I.~E.~Filyanov,
%``Conformal Invariance And Pion Wave Functions Of Nonleading Twist,''
Z.\ Phys.\ {\bf C48} (1990) 239;\\
%%CITATION = ZEPYA,C48,239;%%
%\bibitem{BallDA}
P.~Ball,
%``Theoretical update of pseudoscalar meson distribution amplitudes of
%higher twist: The nonsinglet case,''
JHEP {\bf 01} (1999) 010.
%, hep-ph/9812375
%\cite{Zhitnitsky:dd}
\bibitem{ZZC}
A.~R.~Zhitnitsky, I.~R.~Zhitnitsky and V.~L.~Chernyak,
%``QCD Sum Rules And Properties Of Wave Functions Of Nonleading Twist. (In Russian),''
Sov.\ J.\ Nucl.\ Phys.\ {\bf 41} (1985) 284.
%[Yad.\ Fiz.\ {\bf 41} (1985) 445].
%%CITATION = SJNCA,41,284;%%
%\cite{Buras:2000ra}:
\bibitem{BurasSilv}
A.~J.~Buras and L.~Silvestrini,
%``Non-leptonic two-body B decays beyond factorization,''
Nucl.\ Phys.\ {\bf B569} (2000) 3.
%[hep-ph/9812392].
%%CITATION = HEP-PH 9812392;%%
%\cite{Buchalla:1996vs}:
\bibitem{Hrev}
G.~Buchalla, A.~J.~Buras and M.~E.~Lautenbacher,
%``Weak Decays Beyond Leading Logarithms,''
Rev.\ Mod.\ Phys.\ {\bf 68} (1996) 1125.
%[hep-ph/9512380].
%%CITATION = HEP-PH 9512380;%%
\bibitem{pqcd}
%\cite{Keum:2000wi}
%\bibitem{Keum:2000wi}
Y.~Y.~Keum, H.~n.~Li and A.~I.~Sanda,
%``Penguin enhancement and B $\to$ K pi decays in perturbative QCD,''
Phys.\ Rev.\ D {\bf 63} (2001) 054008,\\
%[arXiv:hep-ph/0004173].
%%CITATION = HEP-PH 0004173;%%
for a recent review see
%\cite{Li:2002uy}
%\bibitem{Li:2002uy}
H.~n.~Li,
%``PQCD approach to exclusive B decays,''
hep-ph/0210198.
%%CITATION = HEP-PH 0210198;%%
\end{thebibliography}
\end{document}