Past exam of the particle and nuclear physics course of the University of Cambridge / 2012 / Paper 1 / 1 / b

Neglect atomic binding energy and take the target electron at rest. Use natural units temporarily. The final state contains three electrons and one Positron, so its threshold invariant mass is $4m_e$. For incident total energy $E$,The corresponding incident kinetic energy is $6m_e{c}^2=3.066\mathrm{MeV}$. At threshold the products have no relative momentum in the centre-of-momentum frame; their common laboratory motion requires more energy than merely the pair's rest energy.