= Three-body phase space with one massive daughter
{title2=$\Phi_3=\frac{M^2}{256\pi^3}(1-r^2+2r\ln r)$}
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> gives
$$
\Phi_3=\frac1{128\pi^3M^2}\int_{m^2}^{M^2}\frac{(M^2-s)(s-m^2)}s\,ds
=\frac{M^2}{256\pi^3}(1-r^2+2r\ln r).
$$
This 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 $(1-r)^3/3$.
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