Turn an invariant amplitude into an event yield

Use $\hbar=c=1$, the positive-time metric, and relativistically normalized asymptotic states. For two incoming on-shell particles, the invariant flux denominator is four times the square root of $(p_a\cdot p_b)^2-m_a^2m_b^2$. In the center-of-momentum frame this becomes $4p_i^*\sqrt s$. The corresponding differential cross section is the squared amplitude times our invariant phase-space measure divided by that flux. The dimensions then give an area, represented as inverse energy squared in natural units.

The ordinary difference of beam velocities is not a Lorentz-invariant relative speed. In the center-of-momentum frame it can be used with the matching energy factors to reproduce the invariant formula, but it can exceed one without describing the speed of a single object. Avoid interpreting that flux factor as a signal velocity. For particles in hadrons, a partonic cross section also needs a convolution with momentum-fraction distributions before becoming the hadronic prediction. The beam luminosity alone does not remove that theoretical distinction.

For unpolarized incoming particles of spins $s_a,s_b$, average over their equally populated spin states by dividing the summed squared amplitude by $(2s_a+1)(2s_b+1)$. Sum over unobserved final spin states. A polarized beam instead uses its actual spin density matrix, so the unpolarized average can be inappropriate. Massless gauge particles have only their physical helicities; counting fictitious extra polarizations as incident states would produce a wrong normalization. Color averaging, when relevant, is another separately declared internal-state average.

Identical final particles create a counting issue even after the amplitude has the required exchange symmetry. Integrating a full labeled phase space covers permutations of the same physical configuration, so divide by the factorial for each identical group. An explicitly nonduplicating ordered region is an alternative. Do not divide again simply because the incoming particles are identical: their prepared beam flux has already specified the collision normalization. Also keep combinatorial factors in an amplitude diagram distinct from a final-state phase-space symmetry factor.

Integrated luminosity counts collision exposure, while cross section measures an interaction's rate per that exposure. Their product is an expected produced count, not a guaranteed integer observation. A branching fraction restricts a produced particle to a specific final channel, and efficiency describes what fraction of that channel is selected. If a quoted cross section already refers to the exclusive channel, multiplying by its branching fraction again would undercount. Likewise, a fiducial cross section includes a declared truth-level region that should not receive a second acceptance extrapolation silently.

Observed counts include statistical fluctuations and background processes. Inferring a cross section generally requires a response model, luminosity calibration, background estimate, and uncertainties. Efficiency can vary across phase space, so a single number is only adequate when its averaging is appropriate for the signal distribution. Convert units explicitly: an inverse-picobarn luminosity must multiply a picobarn cross section to give a dimensionless count. To convert a natural-unit result, restore the squared factor of hbar c rather than treating inverse GeV squared as a barn.