From scattering amplitudes to cross sections

Use natural units and metric $(+---)$. An S-matrix compares incoming and outgoing states that behave as separated stable particles far before and after a localized collision. We use states with $\langle p|q\rangle=2E_{\mathbf p}(2\pi)^3\delta^{(3)}(\mathbf p-\mathbf q)$. The connected matrix element defines $\mathcal M$ after removing the overall energy-momentum delta function and one factor of $i$. Delta-normalized states are calculation tools; experimental beams are wave packets.

LSZ reduction relates scattering amplitudes to time-ordered correlation functions by extracting external one-particle poles and their residues. Its usual massive-particle form assumes suitable asymptotic states with isolated stable-particle poles. It is not a blanket rule that any field correlator defines any desired particle. Unstable resonances are not exact asymptotic states, and massless gauge theories introduce infrared issues that require inclusive observables or appropriately dressed states. Our massive scalar example avoids those complications.

The invariant flux factor normalizes by how often the incoming beams encounter each other. The final-state phase-space measure counts allowed momenta with relativistic normalization and imposes total four-momentum conservation. These ingredients can suppress a rate near threshold even when the amplitude is nonzero. Conversely, a large number of accessible final configurations can enhance a rate without changing one individual transition’s amplitude.

For two-body scattering in the center-of-momentum frame, the remaining differential measure is a solid angle and a ratio of outgoing to incoming momentum magnitudes. For distinguishable spinless final particles the formula below applies directly. A cross section has dimensions of area, or energy to the power minus two in natural units. Checking that dimension is a fast way to catch a missing flux or energy factor when the amplitude itself is dimensionless.

Two identical final scalars at opposite momenta do not form two different outcomes when their labels are exchanged. Integrating the labeled angular formula over the full sphere counts that pair twice. One remedy is a factor $1/2!$ over the full sphere; another is an appropriate half-domain without that factor. Applying both would undercount. The usual two-beam cross section does not acquire an extra initial-state factorial merely because the incoming particles are identical.

To compare with data, combine a cross section with integrated luminosity and efficiencies. An ideal expected count is the luminosity times the accepted cross section, not the bare amplitude squared. Real analyses also address backgrounds, resolution, selection, radiative corrections, and uncertainties. A clean tree-level scalar prediction is a baseline calculation with stated accuracy, rather than a detector simulation or an all-orders claim about nature.