Paper 2, Section II, C

(a) A moving particle with rest mass $M$ decays into two particles (photons) with zero rest mass. Derive an expression for $\sin \frac{\theta}{2}$, where $\theta$ is the angle between the spatial momenta of the final state particles, and show that it depends only on $M c^{2}$ and the energies of the massless particles. ( $c$ is the speed of light in vacuum.)

(b) A particle $P$ with rest mass $M$ decays into two particles: a particle $R$ with rest mass $0<m<M$ and another particle with zero rest mass. Using dimensional analysis explain why the speed $v$ of $R$ in the rest frame of $P$ can be expressed as

$v=c f(r), \quad \text { with } \quad r=\frac{m}{M}$

and $f$ a dimensionless function of $r$. Determine the function $f(r)$.

Choose coordinates in the rest frame of $P$ such that $R$ is emitted at $t=0$ from the origin in the $x$-direction. The particle $R$ decays after a time $\tau$, measured in its own rest frame. Determine the spacetime coordinates $(c t, x)$, in the rest frame of $P$, corresponding to this event.

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