\input phyzzx

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\noindent PHY--396~K/L.\quad
Problem set \#12.\quad
Due January 24, 2001.
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Consider muon decay, $\mu^-\to e^-\bar\nu_e\nu_\mu$.
Since neutrinos are hard to detect experimentally, the readily measurable quantities
for this process
are the total muon decay rate $\Gamma_\mu=1/\tau_\mu$ and the energy distribution of
electrons produced by decaying muons;
the latter is known to have a maximum at the highest kinematically allowed value
of $E_e\)$.

According to the Fermi theory of weak interactions,
the matrix element for muon decay is
$$
\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}
=\ {G_F\over\sqrt{2}}\,
\left[\bar u(\nu_\mu)(1-\gamma^5)\gamma^\alpha u(\mu^-)\right]\times
\left[\bar u(e^-) (1-\gamma^5)\gamma_\alpha v(\bar\nu_e)\right] .
\eqn\MuonAmplitude
$$
The modern Standard Model of particle interactions produces essentially the
same answer at the tree level of the perturbation theory.

\pointbegin
Show that
$$
\half\sum_{\rm all\atop spins}
	\left|\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}\right|^2\
=\ 64G^2_F (p_\mu\cdot p_{\bar\nu})\,(p_e\cdot p_\nu) .
\eqn\MuonAverage
$$

\point %2
Now calculate the $d\Gamma/d E_e$ and the $\Gamma_{\rm tot}$ in the
muon's rest frame.
This is a straightforward but non-trivial exercise because of three particles in
the final state, with momenta subject to constraints
$$
\bp_e\ +\ \bp_\nu\ +\ \bp_{\bar\nu}\ =\ {\bf0},\qquad
E_e\ +\ E_\nu\ +\ E_{\bar\nu}\ =\ M_\mu\,\approx\,105.66\,\rm MeV .
\eqn\Constraint
$$
Fortunately, the neutrinos are massless while the electron may be approximated
as massless because in most decay events the electron's
energy $E_e=O(M_\mu)\gg m_e$.
You are advised to take this approximation $m_e\approx0$ as it simplifies
the calculation quite a bit.

\bye
