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

Abstract

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 ${\bar{B}}_d^0\to {\pi }^{+}{\pi }^{-}$ is very small, but is noticeable for ${\bar{B}}_d^0\to {\pi }^0{\pi }^0$.

Latex, 22 pages, 3 figures, a few comments added, version to appear in Phys. Rev. D

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• CP Violation in the B System and Relations to $K\to \pi \nu \bar{\nu }$ Decays
• Exploring Light-Cone Sum Rules for Pion and Kaon Form Factors
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