Charmless $B\to PPP$ Decays: the Fully-Antisymmetric Final State

Abstract

Under flavor $SU(3)$ symmetry (SU(3)${}_F$), the final-state particles in $B\to PPP$ 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 22 $B\to PPP$ decays that can be in an FA state in terms of both 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 $B\to PPP$ 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.

Comments: 13 pages, 4 tables, minor typos corrected, to appear in Phys. Rev. D