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\noindent PHY--396~K.\quad
Problem set \#12.\quad
Due November 23, 2004.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
In a Yukawa theory with  $M_s>2m_f$,
the scalar particles are unstable against
 decay into $\rm fermion + antifermion$ pairs.
\subpar
Write down the tree-level matrix element ${\cal M}(S\to f+\bar f)$,
sum $|{\cal M}|^2$ over final particles' spins, and calculate
the total decay rate $S\to f+\bar f$.


\bigskip
\point %2
Consider  a Yukawa theory of
two Dirac fields $\Psi_1(x)$ and $\Psi_2(x)$ coupled
to the same real scalar field $\Phi(x)$:
$$
\eqalign{
{\cal L}\ ={}&
\overline\Psi_1(i\!\not\!\partial-m_1)\Psi_1\
    +\ \overline\Psi_2(i\!\not\!\partial-m_2)\Psi_2\
    +\ \half(\partial_\mu\Phi)^2\ -\ \half M_s^2\Phi^2\cr
&-\ g_1\Phi\overline\Psi_1\Psi_1\ -\ g_2\Phi\overline\Psi_2\Psi_2\,.\cr
}\eqn\SFF
$$
At the tree level, calculate the matrix element, the partial cross-section
and the total cross-section for elastic scattering of one type of a fermion
off the other type, $f_1+f_2\to f_1+f_2$.
Take the initial fermions to be unpolarized and sum over the final
fermion's polarizations.

\bigskip
\point %4
Finally, consider the 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;
In this exercise, we calculate these quantities from the
Fermi theory of weak interactions.
\pointcon
According to the Fermi theory,
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.

\spointbegin
Sum the absolute square of the amplitude~\MuonAmplitude\ over the
final particle spins and average over the initial muon's spin.
Show that altogether,
$$
\half\sum_{\rm all\atop spins}
	\left|\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}\right|^2\
=\ 64G^2_F (p_\mu\cdot p_{\bar\nu})\,(p_e\cdot p_\nu) .
\eqn\MuonAverage
$$

\pointcon
The rest of this exercise is the phase space calculation.
The following lemma is very useful for three-body decays like
$\mu^-\to e^-+\nu_\mu+\bar\nu_e$: 

\spoint %b
Consider a generic three-body decay of some particles of mass~$M_0$
into three particles of respective masses $m_1$, $m_2$, and $m_3$. 
Show that in the rest frame of the original particle the partial decay
rate is given by
$$
d\Gamma\ =\ {1\over 2M_0}\times\overline{|{\cal M}|^2}\times
{d^3\Omega\over 256 \pi^5}\times
dE_1\,dE_2\,dE_3\,\delta(E_1+E_2+E_3-M_0)
\eqn\TPS
$$
where $d^3\Omega$ refers to three angular variables parameterizing
the directions of the three final-state particles relative to some
external frame
but not affecting the angles between the three momenta.
For example, one may use two angles to describe the orientation of the
decay plane (the three momenta are coplanar, $\bp_1+\bp_2+\bp_3=0$)
and one more angle to fix the direction of \eg, $\bp_1$ in that plane.
Altogether, $\int\!d^3\Omega=4\pi\times2\pi=8\pi^2$.
\spointcon
Also show that when $m_1=m_2=m_3=0$, the kinematically allowed range
of the final particles' energies is given by
$$
0\ \le\ E_1,E_2,E_3\ \le \half M_0\qquad{\rm while}\quad
E_1+E_2+E_3\ =\ M_0,
\eqn\MasslessLimits
$$
but for the non-zero masses $m_{1,2,3}$ this range is much more complicated.

\pointcon
The electron and the neutrinos are much lighter then the muon,
so in most decay events all three final-state particles are ultra-relativistic.
This allows us to approximate $m_e\approx m_\nu\approx m_{\bar\nu}\approx0$,
which greatly simplifies the last part of this exercise:

\spoint %c
Integrate the muon's partial decay rate over the final particle energies
and derive first $d\Gamma/dE_e$ and then the total decay rate.


\bye
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bigskip
\point %4
And now consider the pseudoscalar version of the Yukawa theory~\SFF:
Two Dirac fields $\Psi_1(x)$ and $\Psi_2(x)$ coupled
to the same real \undertext{pseudoscalar} field $\phi(x)$:
$$
\eqalign{
{\cal L}\ ={}&
\overline\Psi_1(i\!\not\!\partial-m_1)\Psi_1\
    +\ \overline\Psi_2(i\!\not\!\partial-m_2)\Psi_2\
    +\ \half(\partial_\mu\phi)^2\ -\ \half M_p^2\phi^2\cr
&-\ ig_1\phi\overline\Psi_1\gamma^5\Psi_1\
    -\ ig_2\phi\overline\Psi_2\gamma^5\Psi_2\,.\cr
}\eqn\PFF
$$
\spointbegin
Write down the Feynman rules of this theory.
\spoint %b
Calculate the tree-level matrix element ${\cal M}(f_1+f_2\to f_1+f_2)$
for elastic scattering of two distinct fermions off each other.
\spoint %c
Calculate the partial cross section and the total cross-section for this
scattering.
Assume un-polarized beams of initial fermions and spin-blind detectors
for the final particles.
