Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 12C (Dynamics and Relativity)

The four-momentum of a massive particle iswhile for a massless particleFor $P_1=P_2+P_3$, invariance givesIn the rest frame of particle 1, $P_1=(m_{1}c,0)$, sowith $E_1=m_1{c}^2$. Equal and opposite final momenta are real exactly whenthe threshold condition for a relativistic two-body decay.

At step $n$, the parent and daughter rest masses areThe decay is allowed when $M\ge m$, which holds because $M=\lambda m$ with $\lambda >1$. Hence all $N$ stages occur:The initial rest mass is $NM+m_0$ and the final rest mass is $Nm+m_0$. Thusof rest mass is converted into kinetic energy, corresponding to energy $Nm(\lambda -1)c^2$.

Momentum conservation is $P_A+P_B=P_C+P_D$. The diagonal bilinears are $P_a^2=m_a^2{c}^2$. The six off-diagonal ones areandThese express every bilinear invariant through the Mandelstam variables and masses. Expanding $s+t+u$ and using momentum conservation cancels all mixed products, leaving