On the color suppressed contribution to $\bar{B_{d}^0} \rightarrow \, \pi^0 \pi^{0}

TL;DR

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Contribution

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Abstract

The decay modes of the type $B \rightarrow \pi \, \pi $ are dynamically different. For the case $\bar{B_{d}^0} \rightarrow \, \pi^+ \pi^- $ there is a substantial factorized contribution which dominates. In contrast, the decay mode $\bar{B_{d}^0} \rightarrow \, \pi^0 \pi^{0} $ has a small factorized contribution, being proportional to a small Wilson coefficient combination. However, for the decay mode $\bar{B_{d}^0} \rightarrow \, \pi^0 \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 $\bar{B_{d}^0} \rightarrow \, \pi^0 \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 $\Lambda_{QCD}/m_b$ 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 $\bar{B_{d}^0} \rightarrow \, \pi^0 \pi^{0} $ can easily accomodate 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 $\bar{B_{d}^0} \rightarrow \, \pi^0 \pi^{0} $. A possible link to the triangle anomaly is discussed.