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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 $π^+ K^+ K^-,\; π^+π^+π^-,\; K^+ K^+ K^- $ and $K^+π^+π^-$. 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 $Δ_{CP} = Γ- \overlineΓ$ satisfy, $Br(π^+π^+π^-) = 2Br(π^+ K^+ K^-)$, $Br(K^+K^+K^-) = 2 Br(K^+π^+π^-)$, and $Δ_{CP}(π^+π^+π^-) = 2Δ_{CP}(π^+ K^+K^-) = - Δ_{CP}(K^+K^+K^-) = -2Δ_{CP}(K^+π^+π^-)$. The experimental data do not exhibit the expected pattern for the branching ratios. The rate differences for $B^+\to π^+π^+π^-$ and $B^+\to K^+ K^+ K^-$ satisfy the relation between $ΔS =0$ and $ΔS=1$ 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.