4.II.18A

A pion of rest mass $M$ decays at rest into a muon of rest mass $m<M$ and a neutrino of zero rest mass. What is the speed $u$ of the muon?

In the pion rest frame $S$, the muon moves in the $y$-direction. A moving observer, in a frame $S^{\prime}$ with axes parallel to those in the pion rest frame, wishes to take measurements of the decay along the $x$-axis, and notes that the pion has speed $v$ with respect to the $x$-axis. Write down the four-dimensional Lorentz transformation relating $S^{\prime}$ to $S$ and determine the momentum of the muon in $S^{\prime}$. Hence show that in $S^{\prime}$ the direction of motion of the muon makes an angle $\theta$ with respect to the $y$-axis, where

$\tan \theta=\frac{M^{2}+m^{2}}{M^{2}-m^{2}} \frac{v}{\left(c^{2}-v^{2}\right)^{1 / 2}} .$

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