= Three-body threshold phase space
{title2=$\Phi_3\propto Q^2$}
For three massive daughters with a smooth nonzero <decay amplitude> at the <decay threshold>, the integrated <Lorentz-invariant phase space> vanishes as $Q^2$, where $Q$ is the parent's <mass> minus the sum of daughter <masses>. The spatial <Dirac delta function> leaves six independent <momentum> coordinates. Rescaling each by $Q^{1/2}$ contributes $Q^3$, while the <conservation of energy> <Dirac delta function> contributes $Q^{-1}$. The factors $1/(2E)$ are smooth nonzero constants at the <decay threshold>. Thus the <decay width> also scales as $Q^2$ when the <decay amplitude> has a nonzero constant term. A vanishing <decay amplitude>, a massless daughter, or a threshold singularity can change this law. In three spatial dimensions the same argument gives $Q^{(3N-5)/2}$ for $N$ massive daughters.
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