%Title: QCD Light-Cone Sum Rule Estimate of Charming Penguin Contributions in B -> pi pi
%Author: A. Khodjamirian, Th. Mannel, B. Melic
%Published: * Phys.Lett. * ** B571 ** (2003) 75-84; Phys.Lett. ** B572 ** (2003) 171-180 (Published twice in journal) .
%hep-ph/0304179
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TTP03-12 \\
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{\Large \bf
QCD Light-Cone Sum Rule Estimate \\
of Charming Penguin Contributions
in $B \rightarrow \pi\pi$}
\vskip 1.5true cm
{\large\bf
A.~Khodjamirian\,$^{*)}$,
Th.~Mannel\,, B.~Meli\'c\,$^{**)}$}
\\[1cm]~
\vskip 0.3true cm
{\it Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe,\\ D-76128 Karlsruhe, Germany } \\
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%\maketitle
\vspace*{1cm}
\begin{abstract}
\noindent
Employing the QCD light-cone sum rule approach
we calculate the $B \rightarrow \pi\pi$
hadronic matrix element of the current-current operator
with $c$ quarks in the penguin topology (``charming penguin'').
The dominant contribution to the sum rule is
due to the $c$-quark loop at short distances and is of $O(\alpha_s)$ with respect
to the factorizable $B\to \pi\pi$ amplitude. The effects of soft gluons
are suppressed at least by $O(\alpha_s m_b^{-2})$.
Our result indicates that sizable nonperturbative effects generated by
charming penguins at finite $m_b$
are absent. The same is valid for the penguin contractions of
the current-current operators with light quarks.
\end{abstract}
\vspace*{\fill}
\noindent $^{*)}${\small \it On leave from
Yerevan Physics Institute, 375036 Yerevan, Armenia} \\
\noindent $^{**)}${\small \it On leave from
Rudjer Bo\v skovi\'c Institute, Zagreb, Croatia} \\
\newpage
%\section{Introduction}
{\bf 1.} Charmless two-body hadronic $B$ decays, such as $B\to \pi\pi$,
are promising sources of information on CP-violation
in the $b$-flavour sector. The task of unfolding
CKM phases from the decay observables is challenged by our
limited ability to calculate the relevant hadronic interactions.
One has to resort to approximate
methods based on the expansion in the inverse
$b$-quark mass. The long-distance effects
are then parametrized in terms of process-independent characteristics:
heavy-light form factors, hadronic decay constants and
light-cone distribution
amplitudes (DA). For example, in the
QCD factorization approach \cite{BBNS} the exclusive $B$-decay amplitudes
in the $m_b \to\infty$ limit are expressed in terms of the factorizable part
and calculable $O(\alpha_s)$ nonfactorizable corrections.
For phenomenological applications it is important to investigate
the subleading effects in the decay amplitudes suppressed by inverse powers of $m_b$.
Especially interesting are ``soft'' nonfactorizable effects, involving
low-virtuality gluons and quarks, not necessarily
accompanied by an $\alpha_s$-suppression.
Quantitative estimates of nonfactorizable
contributions, including the power-suppressed ones can be obtained \cite{AK}
using the method of QCD light-cone sum rules (LCSR) \cite{lcsr}.
In particular, the $O(1/m_b)$ soft-gluon corrections have been
calculated \cite{AK} for the $B\to \pi\pi$ matrix elements
with the emission topology.
Furthermore, the LCSR estimate
of the chromomagnetic dipole operator (gluonic penguin)
contribution to $B\to \pi\pi$
was obtained in \cite{KMU}. In this case the soft-gluon
contribution, being suppressed as $1/m_b^2$,
at finite $m_b$ is of the magnitude of the $O(\alpha_s)$
hard-gluon part. However, the nonfactorizable corrections are nonuniversal,
depending on the effective operators and quark topologies
involved in a given $B$ decay channel,
therefore these corrections have to be investigated one by one.
Among the most intriguing effects in charmless $B$ decays are
the so called ``charming penguins''.
The $c$-quark pair emitted in the $b\to c \bar{c} d (s) $ decay
propagates in the environment of the light spectator cloud and
annihilates to gluons, the latter being absorbed in the final
charmless state. In this, so called BSS-mechanism \cite{BSS}
the intermediate $c \bar{c}$ loop generates
an imaginary part, contributing to the final-state strong rescattering phase.
In QCD factorization
approach \cite{BBNS}, charming penguins are typically
small, being a part of the $O(\alpha_s)$ nonfactorizable correction
to the $B\to \pi\pi$ amplitude.
On the other hand, fits of two-body charmless $B$ decays do not exclude
substantial $O(1/m_b)$ nonperturbative effects of the charming-penguin type
\cite{Rome}.
It is therefore important to investigate by independent
methods the effects generated by $c$-quark loops in charmless $B$ decays.
In this letter we report on the LCSR
estimate of the penguin topology contributions
to the $B\to \pi\pi$ amplitude. We start with calculating
the $c$-quark part of this effect (charming penguin).
Later on, we extend the sum rule analysis to
the penguin contractions of the $u$-quark current-current operators. \\
{\bf 2.} As a study case we choose the $\bar{B}^0\to\pi^+\pi^-$ channel.
The decay amplitude is given by the hadronic matrix element
$\langle \pi^+\pi^- |H_{\rm eff}|\bar{B}^0\rangle $
of the effective weak Hamiltonian \cite{BBL}
\begin{eqnarray}
H_{\rm eff} &=& \frac{G_F}{\sqrt{2}} \Bigg \{ \sum_{p=u,c}
V_{pb}V^{\ast}_{pd} \left(C_1 {\cal O}_1^p +
C_2 {\cal O}_2^p \right)- V_{tb} V^{\ast}_{td}
\left(\sum_{i=3}^{10} C_i {\cal O}_i +C_{8g} {\cal O}_{8g}\right)\Bigg \}\,,
\label{eq:heff}
\end{eqnarray}
where
$
{\cal O}_1^p = (\overline{d}\Gamma_{\mu}p)(\overline{p}\Gamma^{\mu} b)
$
and
$
{\cal O}_2^p = (\overline{p} \Gamma_{\mu} p)(\overline{d} \Gamma^{\mu}
b)$
are the current-current operators
($p = u,c$ and $\Gamma_\mu=\gamma_\mu(1-\gamma_5)$),
${\cal O}_{3-10}$ are the penguin
operators, and ${\cal O}_{8g}$ is the chromomagnetic dipole operator.
Each operator entering Eq.~(\ref{eq:heff}) contributes to the
$B\to \pi\pi$ decay amplitude
with a number of different contractions of the quark lines
(topologies) \cite{BS}.
In what follows we will mainly concentrate on the operator
$ {\cal O}_1^c$. For convenience we decompose this operator
$
{\cal {O}}_1^c = \frac13{\cal {O}}_2^c + 2 {\cal \tilde{O}}_2^c
$,
extracting the colour-octet part
$
{\cal \tilde{O}}_2^c = (\overline{c} \Gamma_{\mu} \frac{\lambda^a}{2} c)
(\overline{d} \Gamma^{\mu} \frac{\lambda^a}{2} b) \, .
$
To derive LCSR for the $B\to \pi\pi$ hadronic
matrix element of ${\cal {O}}_1^c$
we follow the procedure which is described in detail in
\cite{AK,KMU}. One starts from introducing the correlation function:
\begin{eqnarray}
F_{\alpha}^{(\tilde{{\cal O}}_2^c)} =
i^2 \int \!d^4 x \,e^{-i(p-q)x}\!\int d^4 y \,e^{i(p-k)y} \langle 0 |
T \{ j_{\alpha 5}^{(\pi)}(y) \tilde{{\cal O}}_2^c(0) j_5^{(B)}(x) \} | \pi^-(q)
\rangle
\nonumber
\\
=(p-k)_{\alpha} F(s_1,s_2,P^2) + ...\,,
\label{eq:corr0}
\end{eqnarray}
where
$
j_{\alpha 5}^{(\pi)} = \overline{u} \gamma_{\alpha} \gamma_5 d
$
and
$
j_5^{(B)} = i m_b \overline{b} \gamma_5 d
$
are the quark currents
interpolating pion and $B$ meson, respectively.
In the above, ellipses denote the Lorentz-structures which are not used.
Only the colour-octet part of ${\cal O}_1^c$ needs to be taken into account.
The contributions of ${\cal O}_2^c$ contain at least a two-gluon annihilation
of the color-neutral $c$-quark pair and have to be considered within
higher-order corrections.
The correlation function (\ref{eq:corr0}) depends on
the artificial four-momentum $k$ allowing one to avoid overlaps
of the $b$-flavoured and light-quark states in the
dispersion relation. We also choose $p^2=k^2=0$ for simplicity and adopt
the chiral limit $q^2=m_\pi^2=0$.
Thus, the invariant amplitude $F$ is a function
of three variables $s_1=(p-k)^2$, $s_2= (p-q)^2$ and
$P^2=(p-q-k)^2 $.
The next step is to calculate $F(s_1,s_2,P^2)$ at large spacelike $s_1,s_2,P^2$
employing the operator product expansion (OPE) near the light-cone.
The corresponding diagrams of $O(\alpha_s)$
are shown in Fig.~\ref{fig:hard}. They contain a $c$-quark loop, which
involves a well known function of $m_c^2$.
The divergence of the quark loop is absorbed in the
renormalization of the QCD penguin operators ${\cal O}_{3-6}$.
The finite contribution of the loop is formally of the next-to-leading
order and depends on the renormalization scale $\mu$. This dependence
is compensated by the Wilson coefficients
of the QCD penguin operators so that the remaining $\mu$ dependence is
weak, since it is of $O(\alpha_s ^2 \ln \mu)$.
Furthermore, the finite piece of the loop contains scheme-dependent
constants; we use the NDR scheme \cite{BBL}.
%
\begin{figure}
\begin{center}
\includegraphics*[width=4.5cm]{fig1a.eps} \qquad
\includegraphics*[width=4.5cm]{fig1b.eps}\qquad
\end{center}
\hspace{4.7cm} (a) \hspace{5.1cm}(b)
\caption{{\em Diagrams corresponding to
the $O(\alpha_s)$ penguin contractions
in the correlation function (\protect{\ref{eq:corr0}}).
Only the diagrams contributing to the sum rule are shown.
The square denotes
the four-quark operator ${\tilde{\cal O}}_2^c$.
}}
\label{fig:hard}
\end{figure}
%
The sum rule is derived from the dispersion relations for $F$ in
the variables $s_{1}$ and $s_2$. Therefore the calculation of
the two-loop diagrams in
Fig.~\ref{fig:hard}
can be considerably simplified if one starts
from their imaginary parts obtained by employing the Cutkosky rule
and replacing the propagators by $\delta$-functions.
For the diagram in Fig.~\ref{fig:hard}a it is sufficient
to calculate its imaginary part in $s_1$.
Furthermore, according to the procedure
explained in \cite{AK} we have to consider the dispersion relation in
$s_1$ in the quark-hadron duality interval,
$0\!<\!s_1\!<\!s_0^\pi\ll m_B^2$ and simultaneously at $ |P^2|\sim
m_B^2$. Hence, one can safely neglect all $O(s_1/P^2)$
contributions in $\mbox{Im}_{s_1}F(s_1,s_2,P^2)$. In this approximation
we obtain for the diagram in Fig.~\ref{fig:hard}a:
\begin{eqnarray}
Im_{s_1} F_{a}^{(\tilde{{\cal O}}_2^c)}(s_1,s_2,P^2) &=&
- \frac{\alpha_s C_F f_{\pi} m_b^2}{8 \pi^2}
\int\limits_0^1 \frac{du}{m_b^2 - u s_2} \int\limits_0^1 dz \, I(zuP^2,m_c^2)
\nonumber \\
& & \hspace*{-2cm}
\times P^2\Bigg \{ z(1-z) \varphi_{\pi}(u) + (1-z) \frac{\mu_{\pi}}{2 m_b}
\left [ \left (2 z + \frac{s_2}{P^2} \right )u \varphi_p(u)
\right .
\nonumber \\
& & \hspace*{-1.5cm} \left . +
\left ( 2 z - \frac{s_2}{P^2} \right )
\left ( \frac{\varphi_{\sigma}(u)}{3}- \frac{u\varphi_{\sigma}^{\prime}(u)}{6} \right )
\right ]
\Bigg \} +O\left (\frac{s_1}{P^2}\right )\,.
\label{diaga}
\end{eqnarray}
In the above, $\varphi_\pi$ and $\varphi_p$, $\varphi_\sigma$ are the
pion DA of twist 2 and
3, respectively, the latter are normalized by
$\mu_\pi=m_\pi^2/(m_u+m_d)$
nonvanishing in the chiral limit;
we use the same standard definitions as in \cite{KMU}. Finally,
$\varphi_\sigma^{\prime}(u)=d\varphi_\sigma(u)/du$ and $I(...)$ is the $c$-quark loop function in the NDR scheme:
\begin{eqnarray}
I(l^2,m_c^2) = \frac{1}{6} \left ( \ln \left (\frac{m_c^2}{\mu^2}\right ) + 1
\right ) + \int\limits_0^1 dx\, x (1-x) \ln \left [1 - \frac{x (1-x) l^2}{m_c^2} \right ]\,.
\end{eqnarray}
Similarly, for the diagram in Fig.~\ref{fig:hard}b it is sufficient
to calculate its imaginary part in $s_2$. The result reads:
\begin{eqnarray}
Im_{s_2} F_{b}^{(\tilde{{\cal O}}_2^c)}(s_1,s_2,P^2) &=& \frac{\alpha_s C_F f_{\pi} m_b^2 }{16\pi^2}
\left (\frac{s_2 -m_b^2}{s_2} \right )^2
\int\limits_0^1 \frac{du}{ \overline{u} P^2 + u s_1} \int\limits_0^1 dz I_c(-uz(s_2-m_b^2))
\nonumber \\
& & \hspace*{-2cm}
\times \Bigg \{ (P^2-s_1-s_2) z \, \varphi_{\pi}(u)
+ \frac{\mu_{\pi}}{2 m_b} (P^2-s_1+3s_2)\varphi_p(u) \nonumber \\
& & \hspace*{-1.5cm} - \frac{\mu_{\pi}}{12 m_b} \Bigg [
2 \left ( \frac{P^2-s_1-s_2}{u} + 2 \frac{P^2-s_1}
{\overline{u}P^2 + u s_1}(P^2-s_1- s_2) \right ) \varphi_{\sigma}(u) \nonumber \\
& & \hspace*{-1.5cm}
+ (3 P^2 - 3 s_1 + s_2) \varphi_{\sigma}^{\prime}(u) \Bigg ]
\Bigg \}\,.
\label{diagb}
\end{eqnarray}
The remaining diagrams not shown in Fig.~\ref{fig:hard},
with gluons attached to the virtual $b$ and $d$ lines,
do not contribute to the sum rule because their double imaginary
parts vanish inside the duality regions
$0\!<\!s_1\!