Cross Sections and Decay Rates

Core Idea

Cross sections and decay rates are the measurable quantities extracted from S-matrix elements. The differential cross section is proportional to |M|^2 times the phase space available to the final-state particles. Decay rates follow the same structure but for a single initial particle at rest. Fermi's golden rule is the non-relativistic limit of these formulas.

Explainer

The S-matrix gives probability amplitudes, but experiments measure cross sections (for scattering) and decay rates (for unstable particles). Converting amplitudes to observables requires squaring the amplitude, summing over unobserved final-state quantum numbers (spins, colors), averaging over initial-state quantum numbers, and integrating over the phase space of the final-state particles. The differential cross section for 2 -> n scattering is d sigma = (1 / 4E_a E_b |v_a - v_b|) |M|^2 d(LIPS_n), where LIPS_n is the n-body Lorentz-invariant phase space.

Phase space measures the density of available final states. For n final-state particles, it is d(LIPS_n) = product over final particles of [d^3p_i / ((2pi)³ 2E_i)] times (2pi)⁴ delta^4(p_initial - sum p_i). The delta function enforces energy-momentum conservation, which constrains the final momenta. For a 2 -> 2 process in the center-of-mass frame, the phase space reduces to an integral over the scattering angle, giving d sigma/d Omega = |M|^2 / (64 pi² s), where s is the center-of-mass energy squared.

Decay rates have the same structure but with a single initial particle. For a particle of mass M at rest decaying into n particles, Gamma = (1/2M) integral |M|^2 d(LIPS_n). The lifetime is tau = 1/Gamma_total. The total width Gamma_total has a direct physical interpretation: it determines the width of the resonance peak in the invariant mass distribution via the Breit-Wigner formula, sigma ~ 1/[(s - M²)² + M² Gamma²]. A short-lived particle has a broad resonance; a long-lived particle has a narrow one. This is the energy-time uncertainty relation made precise.

These formulas connect the theoretical output of quantum field theory (the amplitude M computed from Feynman diagrams) to the experimental input (measured cross sections and lifetimes). The separation into dynamics (|M|^2) and kinematics (phase space) is clean and universal. The same phase-space formulas apply regardless of the underlying theory -- QED, QCD, or the full Standard Model. All the theory-specific physics is encoded in the invariant amplitude M.