Large SU(3) breaking effects and CP violation in $B^{+}$ decays into three charged SU(3) octet pseudoscalar mesons

Abstract

TThe LHCb collaboration has recently reported evidence for non-zero CP asymmetries in $B^{+}$ decays into ${\pi }^{+}{K}^{+}{K}^{-},{\pi }^{+}{\pi }^{+}{\pi }^{-},K^{+}{K}^{+}{K}^{-}$ and $K^{+}{\pi }^{+}{\pi }^{-}$. The branching ratios for these decays have also been measured with different values ranging from $5\times 10^{-6}$ to $51\times 10^{-6}$. If flavor $SU(3)$ symmetry is a good symmetry for $B$ decays, in the case that the dominant amplitude is momentum independent it is expected that branching ratios $Br$ and CP violating rate differences ${\Delta }_{CP}=\Gamma -\overline{\Gamma }$ satisfy, $Br({\pi }^{+}{\pi }^{+}{\pi }^{-})=2Br({\pi }^{+}{K}^{+}{K}^{-})$, $Br(K^{+}{K}^{+}{K}^{-})=2Br(K^{+}{\pi }^{+}{\pi }^{-})$, and ${\Delta }_{CP}({\pi }^{+}{\pi }^{+}{\pi }^{-})=2{\Delta }_{CP}({\pi }^{+}{K}^{+}{K}^{-})=-{\Delta }_{CP}(K^{+}{K}^{+}{K}^{-})=-2{\Delta }_{CP}(K^{+}{\pi }^{+}{\pi }^{-})$. The experimental data do not exhibit the expected pattern for the branching ratios. The rate differences for $B^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $B^{+}\to K^{+}{K}^{+}{K}^{-}$ satisfy the relation between $\Delta S=0$ and $\Delta S=1$ well, but the other two do not, with the CP asymmetries having different signs than expected. In this work we study how to including momentum dependent and also $SU(3)$ breaking effects on these decays to explain experimental data. We find that only including lowest order derivative terms, in the $SU(3)$ limit, the decay patterns cannot be explained. Large $SU(3)$ breaking effects are needed to explain the data.

Review Tex, 20 pages, no figure. Materials extensively extended to include momentum dependent decay amplitudes

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