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Isospin relation and SU(3) breaking effects of strong phases in Charmless B decays

Isospin and flavor SU(3) relations in charmless hadronic B decays $B\to ππ, πK$ are investigated in detail with paying attention to the SU(3) symmetry breaking effects in both amplitudes and strong phases. In general, the isospin and the flavor SU(3) structure of the effective Hamiltonian provide several relations among the amplitudes and strong phases. Whereas a global fit to the latest data shows that some relation seems not to be favorable for a consistent explanation to the current data within the standard model (SM). By considering several patterns of SU(3) breaking, the amplitudes and the corresponding strong phases are extracted and compared with the theoretical estimations. It is found that in the case of SU(3) limits and also the case with SU(3) breaking only in amplitudes, the fitting results lead to an unexpected large ratio between two isospin amplitudes $a^{c}_{3/2}/a^{u}_{3/2}$, which is about an order of magnitude larger than the SM prediction. The results are found to be insensitive to the weak phase $γ$. By including SU(3) breaking effects on the strong phases, one is able to obtain a consistent fit to the current data within the SM, which implies that the SU(3) breaking effect on strong phases may play an important role in understanding the observed charmless hadronic B decay modes $B\to ππ$ and $πK$. It is possible to test those breaking effects in the near future from more precise measurements of direct CP violation in B factories.