50
13. Local Non-Abelian (Gauge) Symmetries
μ
μ
μ
where A 1 , A 2 , . . . A 8 are eight gauge fields which are called gluons. The coupling is denoted by ‘g s ’ in anticipation of the application to strong interactions
via QCD.
The infinitesimal version of (13.52) is (cf (13.13))
ψ
′ = (1 + ig s λ · η(x)/2)ψ
(13.54)
where ‘1’ stands for the unit matrix in the three-dimensional space of components of the triplet ψ. As in (13.14), it is clear that ∂
μ ψ
′ will involve an
‘unwanted’ term ∂
μ
η(x). By contrast, the desired covariant derivative D
μ ψ
should transform according to
D
′μ ψ
′ = (1 + ig s λ · η(x)/2)D
μ ψ
(13.55)
without the ∂
μ
η(x) term. Problem 13.6 verifies that this is fulfilled by having
the gauge fields transform by
A
′μ
a = A
μ
a − ∂
μ η a (x) − g s f
μ
abc η b (x)A c .
(13.56)
Comparing (13.56) with (12.80) we can identify the term in f abc as telling us
that the 8 fields A
μ
a transform as an SU(3) octet, the η’s now depending on
x, of course. This is the adjoint, or regular representation of SU(3), as we
have now come to expect for gauge fields. However, the ∂
μ η a (x) piece spoils
this simple transformation property under local transformations. But it is
just what is needed to cancel the corresponding ∂
μ
η(x) term in ∂
μ ψ
′ , leaving
D
μ ψ transforming as a proper triplet via (13.55). The finite version of (13.56)
can be derived as in section 13.1 for SU(2), but we shall not need the result
here.
As in the SU(2) case, the free Dirac equation for an SU(3)-triplet ψ,
(iγ μ ∂
μ
− m)ψ = 0,
(13.57)
can be ‘promoted’ into one which is covariant under local SU(3) transformations by replacing ∂
μ by D
μ of (13.53), leading to
(i ∂ − m)ψ = g s λ/2 · Aψ
(13.58)
(compare (13.39)). This leads immediately to the one gluon emission amplitude (see figure 13.2)
∫
−
¯
ig s ψ f λ/2γ
μ ψ i · A μ d
4 x
(13.59)
as already suggested in section 12.3.1: the SU(3) current of (12.133) – but
this time in colour space – is ‘dotted’ with the gauge field. The Feynman rule
for figure 13.2 is therefore
−ig s λ a /2 γ
μ .
(13.60)
/
/
13. Local Non-Abelian (Gauge) Symmetries
μ
μ
μ
where A 1 , A 2 , . . . A 8 are eight gauge fields which are called gluons. The coupling is denoted by ‘g s ’ in anticipation of the application to strong interactions
via QCD.
The infinitesimal version of (13.52) is (cf (13.13))
ψ
′ = (1 + ig s λ · η(x)/2)ψ
(13.54)
where ‘1’ stands for the unit matrix in the three-dimensional space of components of the triplet ψ. As in (13.14), it is clear that ∂
μ ψ
′ will involve an
‘unwanted’ term ∂
μ
η(x). By contrast, the desired covariant derivative D
μ ψ
should transform according to
D
′μ ψ
′ = (1 + ig s λ · η(x)/2)D
μ ψ
(13.55)
without the ∂
μ
η(x) term. Problem 13.6 verifies that this is fulfilled by having
the gauge fields transform by
A
′μ
a = A
μ
a − ∂
μ η a (x) − g s f
μ
abc η b (x)A c .
(13.56)
Comparing (13.56) with (12.80) we can identify the term in f abc as telling us
that the 8 fields A
μ
a transform as an SU(3) octet, the η’s now depending on
x, of course. This is the adjoint, or regular representation of SU(3), as we
have now come to expect for gauge fields. However, the ∂
μ η a (x) piece spoils
this simple transformation property under local transformations. But it is
just what is needed to cancel the corresponding ∂
μ
η(x) term in ∂
μ ψ
′ , leaving
D
μ ψ transforming as a proper triplet via (13.55). The finite version of (13.56)
can be derived as in section 13.1 for SU(2), but we shall not need the result
here.
As in the SU(2) case, the free Dirac equation for an SU(3)-triplet ψ,
(iγ μ ∂
μ
− m)ψ = 0,
(13.57)
can be ‘promoted’ into one which is covariant under local SU(3) transformations by replacing ∂
μ by D
μ of (13.53), leading to
(i ∂ − m)ψ = g s λ/2 · Aψ
(13.58)
(compare (13.39)). This leads immediately to the one gluon emission amplitude (see figure 13.2)
∫
−
¯
ig s ψ f λ/2γ
μ ψ i · A μ d
4 x
(13.59)
as already suggested in section 12.3.1: the SU(3) current of (12.133) – but
this time in colour space – is ‘dotted’ with the gauge field. The Feynman rule
for figure 13.2 is therefore
−ig s λ a /2 γ
μ .
(13.60)
/
/