Fixed-target threshold for production of equal-mass particles ($E_{\min }=\frac{N^2-2}{2}m{c}^2$)

An incident particle of rest mass $m$ and energy $E$ striking a stationary identical target has total squared invariant energy $W^2=2m{c}^2(E+m{c}^2)$. Producing $N\ge 2$ final particles of the same rest mass requires $W\ge Nm{c}^2$, giving the threshold. Equality means no final kinetic energy in the centre-of-momentum frame. From the Lorentz factor, the corresponding incident speed is at least $cN\sqrt{N^2-4}/(N^2-2)$.