Color suppressed contribution toB¯d0→π0π0

Abstract

The decay modes of the type $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ are dynamically different. For the case ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ there is a substantial factorized contribution which dominates. In contrast, the decay mode ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ has a small factorized contribution, being proportional to a small Wilson coefficient combination. However, for the decay mode ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ there is a sizeable nonfactorizable (color suppressed) contribution due to soft (long distance) interactions, which dominate the amplitude. We estimate the branching ratio for the mode ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ in the heavy quark limit for the $b$ quark. In order to estimate color suppressed contributions we treat the energetic light ($u$, $d$, $s$) quark within a variant of Large Energy Effective Theory combined with a recent extension of chiral quark models in terms of model- dependent gluon condensates. We find that our calculated color suppressed amplitude is suppressed by a factor of order ${\ensuremath{\Lambda}}_{\mathrm{QCD}}/{m}_{b}$ with respect to the factorizable amplitude, as it should according to QCD-factorization. Further, for reasonable values of the constituent quark mass and the gluon condensate, the calculated nonfactorizable amplitude for ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ can easily accommodate the experimental value. Unfortunately, the color suppressed amplitude is very sensitive to the values of these model-dependent parameters. Therefore fine-tuning is necessary in order to obtain an amplitude compatible with the experimental result for ${\overline{B}}_{d}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$. A possible link to the triangle anomaly is discussed.