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\noindent PHY--396~K.\quad
Problem set \#12.\quad
Due November 25, 2003.
\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 %3
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.

\bigskip\goodbreak
\point %4
Finally, let us finish the muon decay problem of homework~\#10:
Given the matrix element
$$
\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}
=\ {G_F\over\sqrt{2}}\,
\left[\bar u(\nu_\mu)(1-\gamma^5)\gamma^\alpha u(\mu^-)\right]\times
\left[\bar u(e^-) (1-\gamma^5)\gamma_\alpha v(\bar\nu_e)\right] ,
\eqn\MuonAmplitude
$$
show that
$$
\half\sum_{\rm all\atop spins}
	\left|\bra{e^-,\bar\nu_e,\nu_\mu}{\cal M}\ket{\mu^-}\right|^2\
=\ 64G^2_F (p_\mu\cdot p_{\bar\nu})\,(p_e\cdot p_\nu) .
\eqn\MuonAverage
$$


\bye
