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 $\ddot{I}{€}^{+}{K}^{+}{K}^{-},\ddot{I}{€}^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-},K^{+}{K}^{+}{K}^{-}$ and $K^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-}$. 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 $\hat{I}{”}_{CP}=\hat{I}“-\overline{\hat{I}}“$ satisfy, $Br(\ddot{I}{€}^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-})=2Br(\ddot{I}{€}^{+}{K}^{+}{K}^{-})$, $Br(K^{+}{K}^{+}{K}^{-})=2Br(K^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-})$, and $\hat{I}{”}_{CP}(\ddot{I}{€}^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-})=2\hat{I}{”}_{CP}(\ddot{I}{€}^{+}{K}^{+}{K}^{-})=-\hat{I}{”}_{CP}(K^{+}{K}^{+}{K}^{-})=-2\hat{I}{”}_{CP}(K^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-})$. The experimental data do not exhibit the expected pattern for the branching ratios. The rate differences for $B^{+}\to \ddot{I}{€}^{+}\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $B^{+}\to K^{+}{K}^{+}{K}^{-}$ satisfy the relation between $\hat{I}”S=0$ and $\hat{I}”S=1$ 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