Invariant scattering amplitude ($⟨f|S-1|i⟩=i(2\pi {)}^4{\delta }^{(4)}(P_f-P_i)\mathcal{M}$)

With relativistically normalized external states, $\mathcal{M}$ is the reduced S-matrix element after removing the four-momentum conservation distribution. A relativistic scattering cross-section combines $|\mathcal{M}{|}^2$ with the Lorentz-invariant phase-space measure and the invariant flux factor. Identical-leg factorials in Feynman vertices and the identical final-state symmetry factor in the event integral have different roles.