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\noindent PHY--396~L.\qquad
Problem set \#23.\quad
Due April 21, 2005.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
First, a bit of group theory.
Consider a generic simple non-abelian Lie group $G$ and its
generators $T^a$.
The (quadratic) Casimir operator $C_2=\sum_a T^aT^a$ commutes with
all the generators
and hence for any irreducible representation $(r)$ of the group,
$C_2$ restricted to $(r)$ is simply a unit matrix times a number $C(r)$.
In other words, if $T^a_{(r)}$ is a matrix of the generator $T^a$ in the
representation $(r)$, then $\sum_a T^a_{(r)}T^a_{(r)}=C(r)
\times\bf 1$.
For example, for the isospin group $SU(2)$, the irreps are
characterized by the isospin $I$ and $C(I)=I(I+1)$.

\spointbegin
By symmetry, for any complete representation $(r)$ of the group,
$$
\tr_{(r)}(T^aT^b)\ \equiv\ \tr\left(T^a_{(r)}T^b_{(r)}\right)\
=\ R(r)\delta^{ab}
\eqno\eq
$$
for some coefficient $R(r)$.
Show that for any irreducible representation,
$$
{R(r)\over C(r)}\ =\ {\dim(r)\over\dim(G)}\,.
\eqn\RC
$$
In particular, for the $SU(2)$ group, this formula gives
$R(I)={1\over3}I(I+1)(2I+1)$.

\spoint %b
Suppose the first three generators of $G$ generate an $SU(2)$ subgroup.
Show that if a representation $(r)$ of $G$ decomposes into several
$SU(2)$ multiplets of isospins $I_1,I_2,\ldots,I_n$, then
$$
R(r)\ =\sum_{i=1}^n\,\coeff13 I_i(I_i+1)(2I_i+1).
\eqn\decomp
$$

\spoint %c
Now consider the $SU(N)$ group with an obvious $SU(2)$ subgroup
of matrices acting on the first two components of a complex $N$-vector.
The fundamental representation $(N)$ of the $SU(N)$ decomposes into
one doublet and $(N-2)$ singlets of the $SU(2)$ subgroup,
hence
$$
R(N)\ =\ \half\qquad{\rm and}\quad C(N)\ =\ {N^2-1\over 2N}\,.
\eqno\eq
$$
Show that the adjoint representation of the $SU(N)$ decomposes into
one $SU(2)$ triplet, $2(N-2)$ doublets and $(N-2)^2$ singlets and hence
$$
R({\rm adj})\ =\ C({\rm adj})\ \equiv\ C(G)\ =\ N.
\eqno\eq
$$
Hint: $(N)\times(\overline N)=({\rm adj})+(1)$.

\spoint
The symmetric and the anti-symmetric 2--index tensors form irreducible
representations of the $SU(N)$ group.
Find out the decomposition of these irreps under an $SU(2)\subset SU(N)$
and calculate their respective $R$ factors.


\medskip
\point %2
And now let's apply group theory to a physical process of
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 %3
Next, 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 %4
To convert the annihilation amplitude into a cross-section we need to
sum / average over the colors of all the particles.
As a first step in this direction, it's convenient to write the amplitude as
$$
{\cal M}(j+i\to a+b)\
=\ F\times\{T^a,T^b\}^i_{\,\,j}\ +\ iG\times[T^a,T^b]^i_{\,\,j}
\eqn\Matrix
$$
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 scalars' colors.
Show that
$$
{1\over\dim^2(r)}\sum_{ij}\sum_{ab}|{\cal M}|^2\
=\ {C(r)\over\dim(r)}\times\left( 4C(r)\times |F|^2\,
 +\,C({\rm adj})\times (|G|^2-|F|^2)\right).
\eqn\ColorSum
$$
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.
In this frame, the vector particles' polarizations
$e^\mu_{1,2}=(0,{\bf e}_{1,2})$
are purely spatial and transverse to the vectors momenta $\pm\bf k$.
For simplicity, use planar rather than circular polarizations.

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

\bye
