= Solution
In a <relativistic two-body decay>, <four-momentum conservation> in the parent's <rest frame> requires $\mathbf p_1^*=-\mathbf p_2^*$ and
$$
M=\sqrt{p^{*2}+m_1^2}+\sqrt{p^{*2}+m_2^2}.
$$
For fixed daughter <masses>, this fixes a unique $p^*$, independent of direction. In the <Lorentz-invariant phase space> integral, the <Dirac delta distribution> enforcing <conservation of energy> therefore performs the radial <momentum> integration. Only the <solid angle> remains to be integrated; the resulting $p^*$ can be taken outside that integral, even if the <matrix element> depends strongly on direction.
For three or more final particles, <four-momentum conservation> generally leaves continuously variable daughter <energies> and <momenta>, so the <decay rate> includes integrations over them, or equivalently over independent <invariant masses> in a <Dalitz plot>. Integrating over a variable daughter <invariant mass> is also necessary when an unstable daughter is treated with a finite <decay width> rather than a fixed <mass>.
Back to article page
