2 Gauge Theories and the Standard Model
19
Thus L[φ, D μ φ] is indeed invariant under gauge transformations. But, at this
stage, the gauge fields V A
μ appear as external fields that do not propagate. In order to
construct a gauge-invariant kinetic energy term for the gauge fields V A
μ , we consider
[D μ , D ν ]φ = ig{∂ μ V ν − ∂ ν V μ + ig[V μ , V ν ]}φ ≡ igF μν φ ,
(2.20)
which is equivalent to
F
A
μν = ∂ μ V
A
ν − ∂ ν V
A
μ − gC ABC V
B
μ V
C
ν .
(2.21)
From Eqs. (2.8), (2.18) and (2.20) it follows that the transformation properties of
F A
μν are those of a tensor of the adjoint representation
F
μν = U F μν U
−1 .
(2.22)
The complete Yang–Mills lagrangian, which is invariant under gauge transformations, can be written in the form
L YM = −
1
2
T rF μν F
μν
+ L[φ, D μ φ] = −
1
4
A
F
A
μν F
Aμν
+ L[φ, D μ φ] .
(2.23)
Note that the kinetic energy term is an operator of dimension 4. Thus if L is
renormalizable, also L YM is renormalizable. In fact it is the most general gauge
invariant and renormalizable lagrangian density. If we give up renormalizability then
more gauge invariant higher dimension terms could be added. It is already clear at
this stage that no mass term for gauge bosons of the form m 2 V μ V μ is allowed by
gauge invariance.
For an abelian theory, as for example QED, the gauge transformation reduces to
U [θ(x)] = exp[ieQθ(x)], where Q is the charge generator. The associated gauge
field (the photon), according to Eq. (2.15), transforms as
V
μ = V μ − ∂ μ θ(x) .
(2.24)
and the familiar gauge transformation by addition of a 4-gradient of a scalar function
is recovered. The QED lagrangian density is given by:
L = −
1
4
F
μν F μν +
ψ
¯
ψ(iD / − m ψ )ψ .
(2.25)
Here D / = D μ γ μ , where γ μ are the Dirac matrices and the covariant derivative is
given in terms of the photon field A μ and the charge operator Q by:
D μ = ∂ μ + ieA μ Q
(2.26)