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

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 $51\times 10^{-6}$\mathrm{.}Ifflavor$SU(3)$symmetryisagoodsymmetryfor$B$ 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,$Br(\pi^+\pi^+\pi^-) = 2Br(\pi^+ K^+ K^-)$,$Br(K^+K^+K^-) = 2 Br(K^+\pi^+\pi^-)$,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$B^+\to K^+ K^+ K^-$satisfytherelationbetween$\Delta S =0$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.