Pictures Paths Particles Processes

214 November 7, 2013
Also, in the annihilation of an on-shell quark-antiquark pair, we have seen that
= 0 . (8.38)
Taking all this into account, we see that in the now newly available Feynman
diagram for q¯q → gg :
a
p q
1
1
n j
p
2
b
the gluon propagator effectively reduces to just its g µν term, and the corresponding
expression reads
g µν
M 3 = i¯hgg 3 v(p 1 )γ µ u(p 2 )
(p 1 + p 2 ) 2 Y (q 1, ɛ 1 ; q 2 , ɛ 2 ; −q 1 − q 2 , ν) h jk n(T n ) a b .
(8.39)
with summation over the colour n implied. Note that the lowering of indices
in the h symbol does not have any significance, I do it simply to make the
typography looks nicer. Putting the handlebar on gluon 1 as before, we get
Y (q 1 , q 1 ; q 2 , ɛ 2 , −q 1 − q 2 , ν) = ( ∆(q 2 ) νλ − ∆(q 1 + q 2 ) νλ) ɛ 2λ
so that
q 2
k
= −(p 1 + p 2 ) ν (p 1 + p 2 · ɛ 2 ) + (p 1 + p 2 ) 2 ɛ 2 ν , (8.40)
M 3 ⌋ = i¯hgg 3 v(p 1 )/ɛ 2 u(p 2 ) h jk n(T n ) a b . (8.41)
The total handlebarred amplitude thus becomes
M 1+2+3 ⌋ = i¯hg v(p 1 )/ɛ 2 u(p 2 ) ( g 3 h jk nT n − g[T j , T k ] ) ab
(8.42)
The colour current will therefore be conserved if we choose
and
g 3 = g (8.43)
[T j , T k ] = h jk n T n . (8.44)
Note that since the matrices T are hermitean, the constants h must be purely
imaginary 8 . Moreover, we can compute them, using Eq.(8.3), as
h jkl = 2Tr ( T j T k T l − T l T k T j) . (8.45)
8 It is customary to write [T j , T k ] = i f jk n T n . The f’s are then called the structure
constants, and the set of T matrices are then the generators of the Lie algebra of the group
SU(N). The i is then combined with the overall i of the vertex to give a Feynman rule without
any i. This is of course a matter of taste.