2 Gauge Theories and the Standard Model
17
of dimension 4, because the action S is given by the integral of L over d 4 x and
is dimensionless in natural units: ¯
h = c = 1). Once this condition is added to
the specification of a gauge group and of the matter field content the gauge theory
lagrangian density is completely specified. We shall see the precise rules to write
down the lagrangian of a gauge theory in the next Section.
2.4 The Formalism of Gauge Theories
In this Section we summarize the definition and the structure of a gauge Yang–Mills
theory [11]. We will list here the general rules for constructing such a theory. Then
these results will be applied to the SM.
Consider a lagrangian density L[φ, ∂ μ φ] which is invariant under a D dimensional continuous group of transformations:
φ
(x) = U(θ
A )φ(x)
(A = 1, 2, . . . , D) .
(2.8)
with:
U(θ
A ) = exp [ig
A
θ
A T
A
] ∼ 1 + ig
A
θ
A T
A
+ . . . ,
(2.9)
The quantities θ A are numerical parameters, like angles in the particular case of a
rotation group in some internal space. The approximate expression on the right is
valid for θ A infinitesimal. Then, g is the coupling constant and T A are the generators
of the group of transformations (2.8) in the (in general reducible) representation of
the fields φ. Here we restrict ourselves to the case of internal symmetries, so that T A
are matrices that are independent of the space-time coordinates and the arguments
of the fields φ and φ in Eq. (2.8) is the same. If U is unitary, then the generators T A
are Hermitian, but this need not be the case in general (though it is true for the SM).
Similarly if U is a group of matrices with unit determinant, then the traces of T A
vanish: tr(T A ) = 0. The generators T A are normalized in such a way that for the
lowest dimensional non-trivial representation of the group (we use t A to denote
the generators in this particular representation) we have
tr(t
A t
B ) =
1
2
δ
AB .
(2.10)
The generators satisfy the commutation relations
[T
A , T
B
] = iC ABC T
C .
(2.11)
17
of dimension 4, because the action S is given by the integral of L over d 4 x and
is dimensionless in natural units: ¯
h = c = 1). Once this condition is added to
the specification of a gauge group and of the matter field content the gauge theory
lagrangian density is completely specified. We shall see the precise rules to write
down the lagrangian of a gauge theory in the next Section.
2.4 The Formalism of Gauge Theories
In this Section we summarize the definition and the structure of a gauge Yang–Mills
theory [11]. We will list here the general rules for constructing such a theory. Then
these results will be applied to the SM.
Consider a lagrangian density L[φ, ∂ μ φ] which is invariant under a D dimensional continuous group of transformations:
φ
(x) = U(θ
A )φ(x)
(A = 1, 2, . . . , D) .
(2.8)
with:
U(θ
A ) = exp [ig
A
θ
A T
A
] ∼ 1 + ig
A
θ
A T
A
+ . . . ,
(2.9)
The quantities θ A are numerical parameters, like angles in the particular case of a
rotation group in some internal space. The approximate expression on the right is
valid for θ A infinitesimal. Then, g is the coupling constant and T A are the generators
of the group of transformations (2.8) in the (in general reducible) representation of
the fields φ. Here we restrict ourselves to the case of internal symmetries, so that T A
are matrices that are independent of the space-time coordinates and the arguments
of the fields φ and φ in Eq. (2.8) is the same. If U is unitary, then the generators T A
are Hermitian, but this need not be the case in general (though it is true for the SM).
Similarly if U is a group of matrices with unit determinant, then the traces of T A
vanish: tr(T A ) = 0. The generators T A are normalized in such a way that for the
lowest dimensional non-trivial representation of the group (we use t A to denote
the generators in this particular representation) we have
tr(t
A t
B ) =
1
2
δ
AB .
(2.10)
The generators satisfy the commutation relations
[T
A , T
B
] = iC ABC T
C .
(2.11)