$SU(3)$ and isospin breaking effects on $B\to PPP$ amplitudes

Several modes of $B$ decays into three pseudoscalar octet mesons $PPP$ are measured. These decays provide useful information for $B$ decays in the standard model (SM). Some powerful tools in analyzing $B$ decays are flavor $SU(3)$ and isospin symmetries. Such analyses are usually hampered by $SU(3)$ breaking effects due to a relatively large strange quark mass which breaks $SU(3)$ symmetry down to isospin symmetry. The isospin symmetry also breaks down when the up and down quark mass difference is nonzero. It is, therefore, interesting to find relations which are not sensitive to $SU(3)$ and isospin breaking effects. We find that the relations among several fully symmetric $B\to PPP$ decay amplitudes are not affected by first-order $SU(3)$ breaking effects due to a nonzero strange quark mass, and also some of them are not affected by first isospin breaking effects. These relations, therefore, hold to good precisions. The measurements for these relations can provide important information about $B$ decays in the SM.