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\noindent PHY--396~K/L.\qquad
Problem set \#23.\quad
Due April 25, 2001.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider the quark-antiquark pair production in {\sl Quantum ChromoDynamics} (QCD).
Specifically, let us focus on the $u\bar u\to d\bar d$ process so there is only
one tree-level diagram contributing to this process.
Draw this diagram and calculate the amplitude, then sum/average the $|{\cal M}|^2$
over both spins and {\it colors} of the final/initial particles and calculate
the total cross section.
For simplicity, you may neglect the quark masses.
\pointcon
Note that the $u\bar u\to d\bar d$ pair production in QCD is very similar to the
$e^-e^+\to\mu^-\mu^+$ pair production in QED, so the only new aspect of this
problem is summing over the colors.

\medskip
\point
Now consider a scalar analogue of QCD or more generally a theory of Yang--Mills
fields $A_\mu^a$ and complex scalars $\Phi_i$ in some representation $(r)$ of the
Gauge group $G$.
\spointbegin
Write down the Lagrangian and the Feynman rules of this theory.
\pointcon
Next, consider the annihilation process $\Phi+\Phi^*\to 2$~gauge bosons.
At the tree level, there are four Feynman diagrams contributing to this process.
\spoint %b
Draw the diagrams and write down the tree-level annihilation amplitude.
\pointcon
As discussed in class, amplitudes involving the non-abelian gauge fields
satisfy a weak form of the Ward identity:
{\sl On-shell Amplitudes involving {\bf a} longitudinally polarized gauge boson
vanish, provided all other gauge bosons are transversely polarized.}
In other words,
$$
\displaylines{
{\cal M}\ 
\equiv\ e_1^{\mu_1}e_2^{\mu_2}\cdots e_n^{\mu_n}\,
{\cal M}_{\mu_1\mu_2\cdots\mu_n}({\rm momenta})\ =\ 0\cr
{\rm when}\quad e_1^\mu\propto k_1^\mu\quad{\rm but}\quad
e_2^\nu k_{2\nu}\ =\ \cdots\ =\ e_n^\nu k_{n\nu}\ =\ 0.\cr
}$$
\spoint %c
Verify this identity for the scalar annihilation amplitude.

\medskip
\point %3
Finally, let us calculate the annihilation cross-section in Scalar QCD.
Generally, the tree-level annihilation amplitude (\cf\ the previous problem)
can be written in the form
$$
{\cal M}\ =\ F\{T^a,T^b\}^i_{\,\,j}\ +\ iG[T^a,T^b]^i_{\,\,j}
\eqn\Matrix
$$
where $j$ is the `color' index of the scalar particle
belonging to some representation $(r)$ of the gauge group $G$,
$i$ is the color index of the scalar anti-particle belonging to the
conjugate representation $(\bar r)$,
and $a$ and $b$
are the color indices of the gauge bosons belonging to the adjoint representation.

\spointbegin
Show that the annihilation amplitude indeed has form~\Matrix\ and write down
the coefficients $F$ and $G$ as explicit functions of the particles momenta
and polarizations.

\spoint %b
Next, let us sum the $|{\cal M}|^2$ over the gauge boson's colors and average over the
scalar colors.
Show that
$$
{1\over\dim^2(r)}\sum_{ij}\sum_{ab}|{\cal M}|^2\
=\ {C(r)\over\dim(r)}\times\left( 4C(r)\,|F|^2\,+\,C({\rm adj})\,(|G|^2-|F|^2)\right).
\eqn\ColorSum
$$
In particular, for scalars in the fundamental representation of the $SU(N)$ gauge group,
$$
{1\over N^2}\sum_{ij}\sum_{ab}|{\cal M}|^2\
=\ {N^2-1\over 2N^2}\left( {N^2-2\over N}\,|F|^2\,+\,N|G|^2\right) .
\eqno\eq
$$

\spoint %c
Evaluate $F$ and $G$ in the center of mass frame.
Take the gauge boson's polarization vectors $e^\mu_{1,2}$ to be purely spatial and transverse
(to the vector particle's momenta).

\spoint %d
Finally,
calculate the (polarized, partial) cross-section for the annihilation process.

\bye
