2 Gauge Theories and the Standard Model
21
in an abelian theory like QED. In QED the photon is coupled to all electrically
charged particles but itself is neutral. In QCD the gluons are coloured hence selfcoupled. This is reflected in the fact that in QED F μν is linear in the gauge field,
so that the term F 2
μν in the lagrangian is a pure kinetic term, while in QCD F A
μν is
quadratic in the gauge field so that in F A2
μν we find cubic and quartic vertices beyond
the kinetic term. Also instructive is to consider the case of scalar QED:
L = −
1
4
F
μν F μν + (D μ φ)
† (D
μ φ) − m
2 (φ
† φ)
(2.32)
For Q = 1 we have:
(D μ φ)
† (D
μ φ) = (∂ μ φ)
† (∂
μ φ) + ieA μ [(∂
μ φ)
† φ − φ
† (∂
μ φ)] + e
2 A μ A
μ φ
† φ
(2.33)
We see that for a charged boson in QED, given that the kinetic term for bosons is
quadratic in the derivative, there is a two-gauge vertex of order e 2 . Thus in QCD the
3-gluon vertex is there because the gluon is coloured and the 4-gluon vertex because
the gluon is a boson.
2.6 Quantization of a Gauge Theory
The lagrangian density L Y M in Eq. (2.23) fully describes the theory at the classical
level. The formulation of the theory at the quantum level requires that a procedure of
quantization, of regularization and, finally, of renormalization is also specified. To
start with, the formulation of Feynman rules is not straightforward. A first problem,
common to all gauge theories, including the abelian case of QED, can be realized
by observing that the free equation of motion for V A
μ , as obtained from Eqs. ((2.21),
(2.23)), is given by
[∂
2 g μν − ∂ μ ∂ ν ]V
Aν
= 0
(2.34)
Normally the propagator of the gauge field should be determined by the inverse of
the operator [∂ 2 g μν − ∂ μ ∂ ν ] which, however, has no inverse, being a projector over
the transverse gauge vector states. This difficulty is removed by fixing a particular
gauge. If one chooses a covariant gauge condition ∂ μ V A
μ = 0 then a gauge fixing
term of the form
GF = −
1
2λ
A
|∂
μ V
A
μ |
2
(2.35)
21
in an abelian theory like QED. In QED the photon is coupled to all electrically
charged particles but itself is neutral. In QCD the gluons are coloured hence selfcoupled. This is reflected in the fact that in QED F μν is linear in the gauge field,
so that the term F 2
μν in the lagrangian is a pure kinetic term, while in QCD F A
μν is
quadratic in the gauge field so that in F A2
μν we find cubic and quartic vertices beyond
the kinetic term. Also instructive is to consider the case of scalar QED:
L = −
1
4
F
μν F μν + (D μ φ)
† (D
μ φ) − m
2 (φ
† φ)
(2.32)
For Q = 1 we have:
(D μ φ)
† (D
μ φ) = (∂ μ φ)
† (∂
μ φ) + ieA μ [(∂
μ φ)
† φ − φ
† (∂
μ φ)] + e
2 A μ A
μ φ
† φ
(2.33)
We see that for a charged boson in QED, given that the kinetic term for bosons is
quadratic in the derivative, there is a two-gauge vertex of order e 2 . Thus in QCD the
3-gluon vertex is there because the gluon is coloured and the 4-gluon vertex because
the gluon is a boson.
2.6 Quantization of a Gauge Theory
The lagrangian density L Y M in Eq. (2.23) fully describes the theory at the classical
level. The formulation of the theory at the quantum level requires that a procedure of
quantization, of regularization and, finally, of renormalization is also specified. To
start with, the formulation of Feynman rules is not straightforward. A first problem,
common to all gauge theories, including the abelian case of QED, can be realized
by observing that the free equation of motion for V A
μ , as obtained from Eqs. ((2.21),
(2.23)), is given by
[∂
2 g μν − ∂ μ ∂ ν ]V
Aν
= 0
(2.34)
Normally the propagator of the gauge field should be determined by the inverse of
the operator [∂ 2 g μν − ∂ μ ∂ ν ] which, however, has no inverse, being a projector over
the transverse gauge vector states. This difficulty is removed by fixing a particular
gauge. If one chooses a covariant gauge condition ∂ μ V A
μ = 0 then a gauge fixing
term of the form
GF = −
1
2λ
A
|∂
μ V
A
μ |
2
(2.35)