Threshold energy

Threshold energy is the minimum incoming-particle energy required for a specified final state to be kinematically allowed in a collision, consistent with conservation of four-momentum. The calculation is cleanest in the centre-of-momentum (CM) frame: at threshold, all final-state particles emerge at rest relative to one another, so the total CM energy must equal the sum of their rest energies. Using the Lorentz-invariant total four-momentum, the CM energy squared is s = (Σ p^μ)·(Σ p_μ), and the threshold condition is √s ≥ Σ m_i c². In the lab frame — where one particle is typically at rest and the other carries all the kinetic energy — the threshold is higher, because some incoming kinetic energy is "wasted" on giving the final state the momentum required to balance the projectile's momentum.

For pair production from a single photon (γ → e⁺ + e⁻ in vacuum), four-momentum conservation forbids the process at any energy: the photon's four-momentum is null (p² = 0), the pair's combined four-momentum is timelike (p² ≥ (2m_e c)² > 0), and a Lorentz-invariant scalar cannot equal both zero and a positive number. A third body — typically a nucleus — is required to carry off momentum, lowering the threshold to E_γ ≥ 2m_e c² ≈ 1.022 MeV in the limit of an infinitely heavy nucleus. Particle physics extends the same logic up the energy ladder: hadron production thresholds, antiproton creation (E_p ≈ 6m_p c² ≈ 5.6 GeV at fixed-target lab kinematics), Higgs production at the LHC. Every threshold calculation is the same exercise in finding the minimum √s consistent with the final-state rest-energy budget.