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\noindent PHY--396~K.\quad
Problem set \#9.\quad
Due November 4, 2003.
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This homework is primarily a reading assignment.
Specifically, you should study section \S4.5 of the {\sl Peskin\&Schroeder}
textbook about scattering cross sections and unstable particles' decay rates
and their relations to the  transition matrix elements
$\cal M\rm(initial\ particles\,\to\,final\ particles)$.

In class, I shall explain where such matrix elements come from and
we shall spend the rest of the semester learning how to calculate them in
perturbation theory.
Unfortunately, I do not have enough class-time to explain how these matrix
elements govern particles' scattering and decay processes, hence this
assignment.

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In addition to the reading assignment, I ask you to do a simple exercise
as a practical application of the material you read:

In Fall of 2000, CERN lab in Geneva shut down
LEP~II --- the second-stage Large Electron--Positron collider ---
to make room for the LHC proton-antiproton collider
(presently in construction).
Just before its final shutdown, LEP~II saw a few events that could
be the first evidence of the Higgs particle being produced in $e^+e^-$
collisions --- but unfortunately not enough
of these events to decide whether they really involved the Higgs particle.
The experimental discovery of the Higgs and the measurement of its mass
is therefore postponed until the next experiment at Fermilab or LHC
can do the job.

This exercise is a (part of a)
theoretical calculation of the Higgs production rate at
an electron-positron collider such as LEP.
Specifically, consider the
$$
e^+\ +\ e^-\ \to\ H^0\ +\ Z^0
\eqn\eeHZ
$$
process in which the positron and the electron annihilate into the Higgs
particle (spin=0) and the $Z^0$ particle (spin=1).
The $Z^0$ particle (one of the three mediators of the weak force, the other two
being $W^\pm$) has mass $M_Z\approx 91~{\rm GeV}/c^2$; the mass of the Higgs
particle is currently unknown but is believed to be heavier --- but not too
much heavier --- than $M_Z$;
the initial electron and positron are so ultra-relativistic that we may treat
them as massless, $M_e\approx0$.

The Standard Model of weak and electromagnetic forces (which we shall study
in some detail next semester) yields a simple formula for the transition
process~\eeHZ.
Or rather, {\sl to the leading order of the perturbation theory} there is a simple
formula for the matrix element,
$$
{\cal M}(e^++e^-\to H^0+Z^0)\
=\ {e^2 M_Z\over 4 \sin^2\theta_W}\,{1\over s-M_Z^2}\times
\epsilon_\mu(Z^0)\,\bar v(e^+)(4\sin^2\theta_W-1+\gamma^5)\gamma^\mu u(e^-)
\eqn\MatEl
$$
where
$$
s\ =\ (p_{e^+}+p_{e^-})^2\ =\ (p_Z+p_H)^2\ =\ E_{\rm c.m.}^2 ,
\eqn\MandelstamS
$$
{}$\epsilon_\mu(Z^0)$ is the polarization vector of the final $Z^0$
vector particle, $\bar v(e^+)$ and $u(e^-)$ are the polarization spinors
of the initial positron and electron,
$\theta_W$ is the electroweak mixing angle, $\sin^2\theta_W\approx0.233$
and the electric charge $e$ is in the rationalized $\hbar=c=1$ units,
$\alpha_{EM}=e^2/(4\pi)\approx1/137$.

Later this semester, we shall learn how to average the $|{\cal M}|^2$
over the spin states of the initial electron and positron and sum over the
spin states of the final $Z^0$ particle, but for the purpose of this exercise,
let me simply give you the result:
$$
\eqalign{
\coeff14\sum_{\rm all\atop spins}\,\left|{\cal M}(e^++e^-\to H^0+Z^0)\right|^2\
={}\ &
A\left({e^2M_Z\over s-M_Z^2}\right)^2 \cr
&\times\left( (p_{e^+}\cdot p_{e^-})\,+\,{2\over M_Z^2}
	(p_Z\cdot p_{e^+})(p_Z\cdot p_{e^-})\right) \cr
}\eqn\spinsum
$$
where
$$
A\ =\ {1+(1-4\sin^2\theta_W)^2\over(4\sin^2\theta_w)^2}\ \approx\ 1.15.
\eqno\eq
$$
Note the Lorentz invariance of the right hand side of eq.~\spinsum.

Your task is to calculate the total cross section for the process~\eeHZ\
as a function of the net energy $E_{\rm c.m.}$ in the center-of-mass frame
\foot{%
    The ``center of mass'' frame is the frame of zero net 3--momentum
    ${\bf P}_{\rm net}$ of the initial (or final) particles.
    Strictly speaking, for relativistic particles this frame
    should be properly called the ``center of energy frame'',
    but everybody calls it the ``center of mass frame'' anyway.
    }
(see eq.~\MandelstamS\ for the Lorentz-invariant formula)
and angular dependence of the partial cross section $d\sigma/d\Omega$ on
the directions of the final Higgs and $Z_0$ particles.

\bye
