Charmless B- PPP decays: The fully antisymmetric final state

Under flavor SU(3) symmetry (SU(3)F), the final-state particles in B - PPP decays (P is a pseudoscalar meson) are treated as identical, and the PPP must be in a fully-symmetric (FS) state, a fully-antisymmetric (FA) state, or in one of four mixed states. In this paper, we present the formalism for the FA states. We write the amplitudes for the 22B - PPP decays that can be in an FA state in terms of both the SU(3)F reduced matrix elements and diagrams. This shows the equivalence of diagrams and SU(3)F. We also give 15 relations among the amplitudes in the SU(3)F limit as well as the additional four that appear when the diagrams E=A=PA are neglected. We present sets of B - PPP decays that can be used to extract gamma using the FA amplitudes. The value(s) of gamma found in this way can be compared with the value(s) found using the FS states.