66
13. Local Non-Abelian (Gauge) Symmetries
where t is some 4-vector. This certainly fixes ∈ μ , and enables us to calculate
(13.122), but of course now two further difficulties have appeared: namely, the
physical results seem to depend on t μ ; and have we not lost Lorentz covariance,
because the theory involves a special 4-vector t μ ?
Setting these questions aside for the moment, we can calculate (13.122)
using the conditions (13.123) and (13.124), finding (problem 13.11)
P μν = −g μν − [t
2 k μ k ν − k · t(k μ t ν + k
2
ν t μ )]/(k · t) .
(13.125)
But only the first term on the right-hand side of (13.125) is to be seen in
(13.121). A crucial quantity is clearly
U μν (k, t) ≡ −g μν − P μν
= [t
2 k μ k ν − k · t(k μ t ν + k ν t μ )]/(k · t)
2 .
(13.126)
We note that whereas
k
μ P μν = k
ν P μν = 0
(13.127)
(from the condition k · ∈ = 0), the same is not true of k
μ U μν – in fact,
k
μ U μν = −k ν
(13.128)
where we have used k
2 = 0. It follows that U μν may be regarded as including
polarization states for which ∈ · k = 0. In physical terms, therefore, a gluon
appearing internally in a Feynman graph has to be regarded as existing in more
than just the two polarization states available to an external gluon (cf section
7.3.1). U μν characterizes the contribution of these unphysical polarization
states.
The discrepancy between (13.121) and (13.119) is then
∫
2Im M
(4)
= M
(2) [U
μ1μ2
(2)
ν1ν2
q¯ q→q¯ q
μ1ν1
(k 1 , t 1 )]M μ2ν2 [U
(k 2 , t 2 )]dρ 2 , (13.129)
together with similar terms involving one P and one U . It follows that these
unwanted contributions will, in fact, vanish if
μ
k
1
1 M
(2)
μ1ν1 = 0,
(13.130)
and similarly for k 2 . This will also ensure that amplitudes are independent of
t μ .
Condition (13.130) is apparently the same as the U(1) gauge invariance
requirement of (8.165), already recalled in the previous section. As discussed
there, it can be interpreted here also as expressing gauge invariance in the
non-Abelian case, working to this given order in perturbation theory. Indeed,
the diagrams of figure 13.11 are essentially ‘crossed’ versions of those in figure
13.5. However, there is one crucial difference here. In figure 13.5, both the
X’s were physical, their polarizations satisfying the condition ∈ · k = 0. In
figure 13.11, by contrast, neither of the gluons, in the discrepant contribution
/
13. Local Non-Abelian (Gauge) Symmetries
where t is some 4-vector. This certainly fixes ∈ μ , and enables us to calculate
(13.122), but of course now two further difficulties have appeared: namely, the
physical results seem to depend on t μ ; and have we not lost Lorentz covariance,
because the theory involves a special 4-vector t μ ?
Setting these questions aside for the moment, we can calculate (13.122)
using the conditions (13.123) and (13.124), finding (problem 13.11)
P μν = −g μν − [t
2 k μ k ν − k · t(k μ t ν + k
2
ν t μ )]/(k · t) .
(13.125)
But only the first term on the right-hand side of (13.125) is to be seen in
(13.121). A crucial quantity is clearly
U μν (k, t) ≡ −g μν − P μν
= [t
2 k μ k ν − k · t(k μ t ν + k ν t μ )]/(k · t)
2 .
(13.126)
We note that whereas
k
μ P μν = k
ν P μν = 0
(13.127)
(from the condition k · ∈ = 0), the same is not true of k
μ U μν – in fact,
k
μ U μν = −k ν
(13.128)
where we have used k
2 = 0. It follows that U μν may be regarded as including
polarization states for which ∈ · k = 0. In physical terms, therefore, a gluon
appearing internally in a Feynman graph has to be regarded as existing in more
than just the two polarization states available to an external gluon (cf section
7.3.1). U μν characterizes the contribution of these unphysical polarization
states.
The discrepancy between (13.121) and (13.119) is then
∫
2Im M
(4)
= M
(2) [U
μ1μ2
(2)
ν1ν2
q¯ q→q¯ q
μ1ν1
(k 1 , t 1 )]M μ2ν2 [U
(k 2 , t 2 )]dρ 2 , (13.129)
together with similar terms involving one P and one U . It follows that these
unwanted contributions will, in fact, vanish if
μ
k
1
1 M
(2)
μ1ν1 = 0,
(13.130)
and similarly for k 2 . This will also ensure that amplitudes are independent of
t μ .
Condition (13.130) is apparently the same as the U(1) gauge invariance
requirement of (8.165), already recalled in the previous section. As discussed
there, it can be interpreted here also as expressing gauge invariance in the
non-Abelian case, working to this given order in perturbation theory. Indeed,
the diagrams of figure 13.11 are essentially ‘crossed’ versions of those in figure
13.5. However, there is one crucial difference here. In figure 13.5, both the
X’s were physical, their polarizations satisfying the condition ∈ · k = 0. In
figure 13.11, by contrast, neither of the gluons, in the discrepant contribution
/