%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 \documentclass[12pt]{article} \usepackage{a4wide} \usepackage{graphicx} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} %%%%%%%% Numbering in the Appendices: %%%%%%%%%%%%%%%%%%% \newcounter{saveeqn} \newcounter{App} %\setcounter{App}{0} \newcommand{\app}{% \stepcounter{App}% \setcounter{saveeqn}{\value{equation}}% \setcounter{equation}{0}% \renewcommand{\theequation}{\Alph{App}\arabic{equation}} } \newcommand{\appende}{% \setcounter{equation}{\value{saveeqn}}% \renewcommand{\theequation}{\arabic{equation}} } \begin{document} %%%%%%%%%%%%%%% \begin{flushright} TTP02-16 \end{flushright} \vspace{0.5cm} \thispagestyle{empty} \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 $O(\alpha_s)$. 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