Three-body phase space with one massive daughter (${\Phi }_3=\frac{M^2}{256{\pi }^3}(1-r^2+2r\ln r)$)

For one daughter of mass $m$ and two massless daughters, set $r=m^2/M^2$, where $M$ is the parent mass. In natural units, integrating over all directions and all kinematically allowed final momenta givesThis follows by splitting the final state into a massless spectator and a two-particle cluster of invariant squared mass $s$, and using $d{\Phi }_3=d{\Phi }_2(M;\sqrt{s},0)d{\Phi }_2(\sqrt{s};m,0)ds/(2\pi )$. The expression counts phase space alone; a physical decay rate must weight it with the squared decay amplitude. Near threshold the dimensionless factor vanishes as $(1-r{)}^3/3$.