\input phyzzx

\def\bp{{\bf p}}

%\hsize=6.8in	\hoffset=-0.2in	%\vsize=8.5in	\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#9.\quad
Due November 17, 1998.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
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\)$.

\pointcon
According to the Fermi theory of weak interactions,
the matrix element for muon decay is
$$
\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}
=\ {G_F\over\sqrt{2}}\,
\left[\bar u(\nu_\mu){1-\gamma^5\over2}\gamma^\alpha u(\mu^-)\right]\times
\left[\bar v(\bar\nu_e) {1-\gamma^5\over2}\gamma_\alpha u(e^-)\right] .
\eqn\MuonAmplitude
$$
The modern Standard Model of particle interactions produces essentially the
same answer in the leading order of the perturbation theory.
(This order is usually called {\it tree-order} or tree-level because it
involves Feynman diagrams of tree topology).

\pointcon
Later in this course, we shall learn how to spin-sum $|{\cal M}|^2$ for amplitudes
such as~\MuonAmplitude,
but for this exercise, let me simply give you the answer:
Summing over spin states of the
outgoing particles and averaging over the spin states
of the initial muon gives
$$
\half\sum_{\rm all\atop spins}
	\left|\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}\right|^2\
=\ 64G^2_F (p_\mu\cdot p_{\bar\nu})\,(p_e\cdot p_\nu) .
\eqn\MuonAverage
$$
Note Lorentz invariance of this formula.

\pointcon
Your task is to 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
$$
\bp_e\ +\ \bp_\nu\ +\ \bp_{\bar\nu}\ =\ {\bf0},\qquad
E_e\ +\ E_\nu\ +\ E_{\bar\nu}\ =\ M_\mu\,\approx\,105.66\,\rm MeV .
\eqn\Constraint
$$
Note that 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.

\medskip
\point
Now consider a quantum field theory of a complex scalar field with a Lagrangian
$$
{\cal L}\ =\ \partial\Phi^*\cdot\partial\Phi\ -\ M^2|\Phi|^2\
-\ \coeff14\lambda\,|\Phi|^4.
$$
\spointbegin
Derive the Feynman rules for this theory.
\spoint
Calculate to lowest non-trivial order in $\lambda$
the elastic scattering cross-section for one particle and one
anti-particle.

\bye
