Decay threshold ($M\ge {\sum }_i{m}_i$)

A parent of rest mass $M$ can decay to on-shell daughters of rest masses $m_i$ only if $M\ge {\sum }_i{m}_i$. In the parent rest frame, each daughter energy is at least its rest energy, so four-momentum conservation proves necessity. The excess energy controls the available Lorentz-invariant phase space. A two-body decay threshold is the two-daughter special case; spin and other selection rules can still forbid an energetically allowed channel.