110
14. QCD I: Introduction, Tree Graph Predictions, and Jets
where c 1 , c 2 and c 3 label the colour degree of freedom of the quark,
antiquark and gluon respectively, and the sum on the index c has
been indicated explicitly. The χ’s are the colour wavefunctions of
the quark and antiquark, and are represented by three-component
column vectors; a convenient choice is
( )
( )
(
)
1
0
0
χ(r) = ( 0 ) , χ(b) = ( 1 ) , χ(g) = ( 0 )
(14.84)
0
0
1
by analogy with the spin wavefunctions of SU(2). The cross section
is obtained by forming the modulus squared of (14.83) and summing
over the colour labels c i :
∑
(λ )
(λ d ) lm
a c (c )χ
∗
c rs
3
r (c 1 )
χ s (c 2 )χ
∗
l (c 2 )
χ m (c 1 )a
∗
2
2
d (c 3 )
c,c1,c2,c3
(14.85)
where summation is understood on the matrix indices on the χ’s and
λ’s, which have been indicated explicitly. In this form the expression is very similar to the spin summations considered in chapter 8
(cf equation (8.62)). We proceed to evaluate it as follows:
(i) Show that
∑
χ s (c 2 )χ
∗
l (c 2 ) = δ sl .
c2
(ii) Assuming the analogous result
∑
a c (c 3 )a
∗
d (c 3 ) = δ cd
c3
show that (14.85) becomes
∑ 8 (
)
λ c λ c
,
2 2
c=1
rr
where the (implied) sum on r runs from 1 to 3.
ˆ
(iii) The expression
λc λc
c
is just the Casimir operator C 2 (see
2 2
section M.5 in appendix M) for SU(3) in the fundamental representation 3, which from (M.67) has the value C F 1 3 , where 1 3 is the
unit 3 × 3 matrix, and C F = 4/3. Hence show that the colour factor
for (14.73) is 4.
Note that if we averaged over the colours of the initial quark, or
considered one particular colour, the colour factor would be C F .