Gluonic penguin contributions inB→ππfrom QCD light-cone sum rules

Abstract

The $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}\ensuremath{\pi}$ hadronic matrix element of the chromomagnetic dipole operator ${O}_{8g}$ (gluonic penguin operator) is calculated using the QCD light-cone sum rule approach. The resulting sum rule for $〈\ensuremath{\pi}\ensuremath{\pi}|{O}_{8g}|B〉$ contains, in addition to the $O({\ensuremath{\alpha}}_{s})$ part induced by hard-gluon exchanges, a contribution due to soft gluons. We find that in the limit ${m}_{b}\ensuremath{\rightarrow}\ensuremath{\infty}$ the soft-gluon contribution is suppressed as a second power of ${1/m}_{b}$ with respect to the leading-order factorizable $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}\ensuremath{\pi}$ amplitude, whereas the hard-gluon contribution has only an ${\ensuremath{\alpha}}_{s}$ suppression. Nevertheless, at finite ${m}_{b},$ the soft and hard effects of the gluonic penguin operator in $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}\ensuremath{\pi}$ 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 penguin operators on ${B}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is very small, but is noticeable for ${B}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}.$