\input phyzzx

\def\bp{{\bf p}}

%\hsize=6.8in	\hoffset=-0.2in	%\vsize=8.5in	\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#10.\quad
Due November 24, 1998.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
In a linear sigma model, calculate the tree-level decay rate
$\Gamma(\sigma\to\pi\pi)$.

\point
Consider two Dirac fields $\Psi_1(x)$ and $\Psi_2(x)$ coupled
to a real scalar field $\Phi(x)$:
$$
\eqalign{
{\cal L}\ ={}&
\overline\Psi_1(i\!\not\!\partial-m_1)\Psi_1\
    +\ \overline\Psi_2(i\!\not\!\partial-m_2)\Psi_2\
    +\ \half(\partial_\mu\Phi)^2\ -\ \half M^2\Phi^2\cr
&-\ g_1\Phi\overline\Psi_1\Psi_1\ -\ g_2\Phi\overline\Psi_2\Psi_2\,.\cr
}\eqn\ThreeFields
$$
At the tree level, calculate the matrix element, the partial cross-section
and the total cross-section for 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.
\subpar
Hint: Prove and use
$$
\half\sum_{s,s'}\left|\bar u(p',s') u(p,s)\right|^2\
=\ 2(m^2+EE'-\bp\bp').
\eqno\eq
$$

\point
Consider Rutherford scattering of an electron off a static point-like
electric charge.
See problem 4.4 of the Peskin\&Schroeder textbook for details and
follow steps (a), (b) and (c) therein.
\sitem{\rm (d)}
In addition, calculate the scattering cross-section for a relativistic
un\-polarized electron.
\subpar
Hint: Prove and use
$$
\half\sum_{s,s'}\left|\bar u(p',s')\gamma^0 u(p,s)\right|^2\
=\ 2(m^2+EE'+\bp\bp').
\eqno\eq
$$



\bye
