%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 \documentclass[12pt]{article} \usepackage[dvips,final]{graphicx} \usepackage{epsfig} \usepackage{amsmath} \setlength{\evensidemargin}{-3mm} \setlength{\oddsidemargin}{4.0mm} \textwidth 450pt \textheight 24cm \topmargin -40pt %\renewcommand{\baselinestretch}{1.2} \begin{document} \begin{flushright} TTP03-12 \\ \end{flushright} \vspace{0.5cm} \thispagestyle{empty} \begin{center} {\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 } \\ \end{center} %\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\!