Problems
111
(b) The colour part for the triple gluon vertex g 1 → g 2 + g 3 is
∑
a
∗
d (c 2 )a
∗
e (c 3 )f dec a c (c 1 ).
c,d,e
Show that the modulus squared of this, averaged over the initial
gluon colours and summed over the final gluon colours, is
1 ∑
f dec f dec ,
8
c,d,e
where each of c, d, e runs from 1 to 8. Deduce using (12.84) that
this expression can be written as
(
)
1 ∑ ∑ (8) (8)
G G
,
8
d
d
e
d
ee
(8)
where G d , (d = 1 . . . 8) are the 8×8 matrices representing the generators of SU(3) in the 8-dimensional (adjoint) representation (see
(8) (8)
section 12.2). The expression ( d G d G d ) is the SU(3) Casimir
ˆ
operator C 2 in the adjoint representation, which from (M.67) has
the value C A 1 8 , where 1 8 is the 8 × 8 unit matrix, and C A = 3.
Hence show that the (averaged, summed) triple gluon vertex colour
factor is C A = 3.
(c) The colour part of the g → q + q v
¯ ertex is
λ
χ
∗
c
r (c 3 )( ) rs χ s (c 2 )a c (c 1 ).
2
Show that the modulus squared of this, averaged over the initial
gluon colours and summed over the final quark colours is
(
)
1 ∑ λ c λ c
1
= .
8
2 2
c
rr
2
This number is usually denoted by T R .
14.6 Verify equation (14.60).
14.7 Verify equation (14.72).
14.8 Verify that expression (14.68) becomes the factor in large parentheses in
equation (14.73), when expressed in terms of the x i ’s.