Gluonic Penguins in $B\to\pi\pi$ from QCD Light-Cone Sum Rules

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 $<\pi\pi |O_{8g}|B>$ 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 $\bar{B}^0_d\to \pi^0\pi^0$.