\input phyzzx

\def\Re{\mathop{\rm Re}\nolimits}

\hsize=6.8in	\hoffset=-0.2in
\vsize=8.75in	\overfullrule=-0.25in
\noindent PHY--396~K.\quad
Problem set \#13.\quad
Due December 4, 2003.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider so-called Bhabha scattering $e^-e^+\to e^-e^+$.
In QED, there are two tree-level Feynman diagrams contributing to this
process; their contributions must be added {\sl before} squaring the
amplitude and adding/averaging over spins,
$$
\left|{\cal M}_1+{\cal M}_2\right|^2\
=\ |{\cal M}_1|^2\ +\ |{\cal M}_2|^2\ +\ 2\Re\bigl({\cal M}_1^*{\cal M}_2\bigr)\
\neq\ |{\cal M}_1|^2\ +\ |{\cal M}_2|^2.
\eqno\eq
$$
Calculate the partial cross-section $d\sigma/d\Omega$ for the
Bhabha scattering.
For simplicity, assume $E\gg m_e$ and neglect the electron's mass throughout
your calculation.
%\subpar
You may find it convenient to use Mandelstam's Lorentz-invariant kinematic
variables $s$, $t$ and $u$, see eq.~(5.69) of the Peskin\&Schroeder textbook
for details.
Notice $s+t+u=4m_e^2\approx0$ in the $m_e\approx0$ approximation.
\subpar
The answer to this problem is simple:
$$
\left({d\sigma\over d\Omega}\right)_{\rm c.m.}\
=\ {\alpha^2\over 2s}\,\left[ \left({t\over s}\right)^2\,
    +\,\left({s\over t}\right)^2\,+\,\left({u\over s}+{u\over t}\right)^2\right]
\eqno\eq
$$
but the intermediate steps are quite complicated, so beware.


\bigskip
\point %2
The $Z^0$ (or simply $Z$) particle of the Standard Model
is a massive ($M_Z\approx 91\,$GeV) neutral vector boson.
The Lagrangian for the $Z_\mu^0$ field is
$$
{\cal L}[Z]\ =\ -\coeff14 Z_{\mu\nu}Z^{\mu\nu}\
+\ \half M_Z^2\,Z_\mu Z^\mu\ +\ g' Z_\mu J_0^\mu\ +\ \cdots
\eqn\ZLAG
$$
where $Z_{\mu\nu}=\partial_\mu Z_\nu-\partial_\nu Z_\mu$,
$J_0^\mu$ is the {\it neutral week current} governing the
$Z^0$ coupling to the leptons and quarks
(\cf\ the charged week currents $J_\pm^\mu$ governing the
$W^\pm$ couplings to the fermions) and the `$\cdots$'
stand for additional interaction terms involving the $W^\pm$,
Higgs and EM fields.
The specific form of the neutral current is
$$
J_0^\mu\ = \sum_{\rm fermions}
\overline\Psi (g_V+g_A\gamma^5)\gamma^\mu\Psi
\eqn\NCurrent
$$
where
$$
\vcenter{\openup 1\jot \ialign{%
	$\displaystyle{g_V\ =\ (#)}$,\hfil\quad &
	$\displaystyle{g_A\ =\ #}$\hfil\quad &
	for #\hfil\cr
\sin^2\theta_W-\coeff14 & +\coeff14 &
	charged leptons $e^-$, $\mu^-$ and $\tau^-$,\cr
+\coeff14 & -\coeff14 &
	neutrinos $\nu_e$, $\nu_\mu$ and $\nu_\tau$,\cr
\coeff13 \sin^2\theta_W-\coeff14 & +\coeff14 &
	charge $-{1\over3}e$ quarks $d$, $s$ and $b$,\cr
-\coeff23 \sin^2\theta_W+\coeff14 & -\coeff14 &
	charge $+{2\over3}e$ quarks $u$, $c$ and $t$,\crr
\omit $\displaystyle{
	g'\ =\ {e\over\sin \theta_W\,\cos\theta_W}}\,$,\hfil\quad &
\omit $\sin^2\theta_W\ \approx\ 0.23$.\hfil\cr
}}\eqn\Zparams
$$
The $\theta_W$ here is called the {\sl weak mixing angle};
it is an experimentally
determined parameter of the Standard Model.

Write down the Feynman rules for the $Z$ field and its interactions
with the leptons and the quarks and
calculate the total decay rate of the $Z$ particle and the branching
ratios $B(Z\to e^+e^-)$, $B(Z\to \mu^+\mu^-)$ and
$B(Z\to q\bar q\to{\rm hadrons})$.
Use tree-level approximation.

For your information, the top quark $t$ is heavier than the $Z$
particle while the other five quark flavors and all the leptons
are so much lighter than the $Z$ that you may neglect their
masses altogether.
You should also keep in mind that all quarks (but not the leptons)
come in three colors.
The color degree of freedom is fundamental to QCD but for the purposes
of this calculation, it's just an index taking three values.

\bigskip
\point %3
Finally consider the $e^+e^-\to\mu^+\mu^-$ pair production
in the Standard Model.
There are two tree-level Feynman diagrams contributing to this
process: one involving a virtual photon and the other a virtual $Z$
particle.
In QED, there is only the first diagram --- which was discussed
in class in much detail --- but at high energies available at
the LEP $e^+e^-$ collider both diagrams are equally important.

\spointbegin
Write down the combined tree-level amplitude
${\cal M}(e^+e^-\to\mu^+\mu^-)$.

\spoint %b
Assume both the electrons and the muons to be ultra-relativistic
($E_{\rm c.m.}=O(M_Z)\gg m_\mu,m_e$) and evaluate the amplitude
(a) for all possible particle helicities.
(Use the center-of-mass frame.)\brk
Hint:
$$
(g_V+g_A\gamma^5)\gamma^\alpha\
=\ g_L\,{1-\gamma^5\over2}\,\gamma^\alpha\
+\ g_R\,{1+\gamma^5\over2}\,\gamma^\alpha
$$
where $g_L=g_V-g_A$ and $g_R=g_V+g_A$.

\spoint %c
Finally,
calculate the total cross section $\sigma(e^+e^-\to\mu^+\mu^-)$
and the forward-backward asymmetry
$$
A\ =\ {\sigma(\theta<\pi/2)-\sigma(\theta>\pi/2)\over
	\sigma(\theta<\pi/2)+\sigma(\theta>\pi/2)}
\eqn\Asymmetry
$$
as functions of the total energy $E_{\rm c.m.}$.
For simplicity, approximate $\sin^2\theta_w\approx\coeff14$ and hence
$g_V\approx0$.
\subpar
Note that in QED the tree-level pair production is symmetric with
respect to $\theta\to\pi-\theta$; the asymmetry in the Standard Model
arises from the
interference between the virtual-photon and virtual-$Z$ diagrams.

\bye

