Lorentz-invariant phase-space measure ($d{\Phi }_n(P)=(2\pi {)}^4{\delta }^4(P-{\sum }_i{p}_i){\prod }_i\frac{d^3{p}_i}{(2\pi {)}^{3}2E_i}$)

The on-shell measure $d^{3}p/(2E)$ is Lorentz invariant. The product measure together with the four-dimensional Dirac delta distribution enforces four-momentum conservation. For equal masses in two-body scattering, $d{\Phi }_2=\sqrt{1-4m^2/s}d\Omega /(32{\pi }^2)$.

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

When the full Lorentz-invariant phase-space measure treats $n$ identical final particles as labeled, each physical final configuration is counted $n!$ times. The unlabelled relativistic scattering cross-section therefore includes $1/n!$. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.