62
4 Four-Dimensional Superfield Supersymmetry
via some gauge transformation. Indeed, if we consider the gauge parameter (z)
whose component structure is
(z) = i(
1
2
C(x) + θ
α
χ α (x) + θ
2 M(x)),
(4.54)
with ¯
C = C, we identically cancel the above-mentioned lower components of the real
scalar superfield V . The gauge, in which the θ-expansion of the V (4.24) starts with
i( ¯
θσ
a
θ)A a (x), is called the Wess-Zumino gauge, and it implies that V
n
= 0 for any
n ≥ 3. However, this gauge, although it allows to remove the nonpolynomiality of
the action which seems to be very useful in the non-Abelian case, is used rarely since
it breaks the superfield structure being thus very inconvenient within the superfield
approach. Some attempts to conciliate this gauge with the superfield methodology
have been carried out in [39].
4.2.3 Non-Abelian Gauge Theories
Now, let us generalize the gauge theory for the non-Abelian case, see e.g. [33]. The
key idea consists in using the action (4.49) where the superfield strength (4.50) is
promoted to a non-Abelian case. Indeed, let us suggest that it has the form
W α = − ¯
D
2
(e
−gV D α e
gV
)
(4.55)
Here we suggest that V is a non-Abelian, Lie-algebra valued real scalar superfield,
V = V
A T
A , where T
A are the Hermitian generators of some Lie group, the most
popular examples of the gauge groups are U (N ) and SU (N ). It is clear that in the
Abelian case this strength reduces to the expression (4.50) multiplied by the coupling
g.
Let us require the theory to be invariant under the following gauge transformations:
e
gV
→ e
−ig ¯
e
gV e
ig
,
(4.56)
where =
A T
A is a Lie-algebra-valued chiral parameter of the gauge transformation, and ¯
is an antichiral one. It is clear that the superfield strength W α , being
Lie-algebra-valued, in this case is not invariant but transformed in a covariant manner:
W α → e
−ig W α e
ig
.
(4.57)
The action of the non-Abelian gauge theory can be obtained by a straightforward
promotion of the action (4.49) to the non-Abelian case (cf. [66]):
4 Four-Dimensional Superfield Supersymmetry
via some gauge transformation. Indeed, if we consider the gauge parameter (z)
whose component structure is
(z) = i(
1
2
C(x) + θ
α
χ α (x) + θ
2 M(x)),
(4.54)
with ¯
C = C, we identically cancel the above-mentioned lower components of the real
scalar superfield V . The gauge, in which the θ-expansion of the V (4.24) starts with
i( ¯
θσ
a
θ)A a (x), is called the Wess-Zumino gauge, and it implies that V
n
= 0 for any
n ≥ 3. However, this gauge, although it allows to remove the nonpolynomiality of
the action which seems to be very useful in the non-Abelian case, is used rarely since
it breaks the superfield structure being thus very inconvenient within the superfield
approach. Some attempts to conciliate this gauge with the superfield methodology
have been carried out in [39].
4.2.3 Non-Abelian Gauge Theories
Now, let us generalize the gauge theory for the non-Abelian case, see e.g. [33]. The
key idea consists in using the action (4.49) where the superfield strength (4.50) is
promoted to a non-Abelian case. Indeed, let us suggest that it has the form
W α = − ¯
D
2
(e
−gV D α e
gV
)
(4.55)
Here we suggest that V is a non-Abelian, Lie-algebra valued real scalar superfield,
V = V
A T
A , where T
A are the Hermitian generators of some Lie group, the most
popular examples of the gauge groups are U (N ) and SU (N ). It is clear that in the
Abelian case this strength reduces to the expression (4.50) multiplied by the coupling
g.
Let us require the theory to be invariant under the following gauge transformations:
e
gV
→ e
−ig ¯
e
gV e
ig
,
(4.56)
where =
A T
A is a Lie-algebra-valued chiral parameter of the gauge transformation, and ¯
is an antichiral one. It is clear that the superfield strength W α , being
Lie-algebra-valued, in this case is not invariant but transformed in a covariant manner:
W α → e
−ig W α e
ig
.
(4.57)
The action of the non-Abelian gauge theory can be obtained by a straightforward
promotion of the action (4.49) to the non-Abelian case (cf. [66]):