Muon Distance and Mandelstam Variables
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3.18 (a) A pion at rest decays into a muon and a neutrino (pi^- --> mu^- + V_mu). On the average, how far will the muon travel (in vacuum) before disintegrating?
(b) The length of the muon track in Figure 1.7 is about 0.6 mm (the photograph has been enlarged). How do you explain this?
3.23 For elastic scattering of identical particles, A + A --> A + A, show that the Mandelstam variables (Problem 3.22) become
s = 4(p^2 + m^2c^2)/c^2
t = -2p^2(1 - cos theta)/c^2
u = -2p^2(1 + cos theta)/c^2
where p is the CM momentum of the incident particle, and theta is the scattering angle.
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Question 3.18 rm
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(a)
Let us set $c=1$ for the ease of derivations, and put it back in the formula for the answer,
Let X-axis be directed along the direction of muon's velocity. Then, if the pion is at rest, the neutrino moves along the X-axis as well, but in the opposite direction, and the 4-momenta of the three particles can be written as
$$
p_pi = m_pi(1,0,0,0),
p_mu = m_mugamma(1,beta,0,0)
p_nu = nu(1,-1,0,0)
$$
(recall, for use below that $1-beta^2 = gamma^{-2}$).
The conservation of energy-momentum says $p_pi = p_mu + p_nu$, so that
$nu = m_mugammabeta$ and
$$
m_pi ...
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