32
3 Superfield Description of Three-Dimensional Supersymmetric Theories
The complete two-point vertex function for the A α field is the sum of (3.109) and
(3.110) and therefore reads
i S 1 ( p) = −
1
2
d
2
θ
d
3 k
(2π) 3
1
(k 2 + m 2 )[(k + p) 2 + m 2 ]
(k γβ + mC γβ )
×
(D
2 A
γ
(− p, θ))A
β
( p, θ) +
1
2
D
γ D
α A α (− p, θ)A
β
( p, θ)
. (3.111)
One can observe that the linear divergences presented in S 1a and S 1b are cancelled in the above result, and the logarithmic ones vanish by symmetry reasons,
therefore this contribution is finite. However, this finiteness is based on the gauge
symmetry rather than on the supersymmetry. Indeed, after integration over the internal momentum we find that this expression is proportional to the gauge invariant
expression
i S 1 ( p) =
d
2
θ f ( p)(W
α
(− p)W α ( p) + 2mW
α
( p)A α ( p)),
(3.112)
where
f ( p) =
d
3 k
(2π) 3
1
(k 2 + m 2 )[(k + p) 2 + m 2 ]
.
This expression, at p → 0 (notice that f ( p)| p→0 =
1
8π|m|
), reproduces the expression for the quadratic Maxwell-Chern-Simons action. As a result, even if we consider
the spinor A
α superfield as a purely external one, it acquires a nontrivial dynamics
due to the one-loop correction. This is a key effect which also occurs in the supersymmetric C P
N −1 model explicitly studied in [44] in the commutative case, and in
[15] in the noncommutative one. We note that the only difference taking place in the
noncommutative case consists in the modification of the factor f ( p), see (3.158).
3.5 Effective Action of the Three-Dimensional Superfield
Theories and the Proper-Time Method
In this section we develop a prescription for calculating the superfield effective action
within the three-dimensional superfield formalism. Here we adopt the formalism
developed in Chap. 2, for superfield theories, and follow the approach proposed in
[45].
Our starting point is the three-dimensional superfield theory described by the
action (see e.g. [23]):
S[] =
d
5 z
1
2
D
2
− V (()
,
(3.113)
3 Superfield Description of Three-Dimensional Supersymmetric Theories
The complete two-point vertex function for the A α field is the sum of (3.109) and
(3.110) and therefore reads
i S 1 ( p) = −
1
2
d
2
θ
d
3 k
(2π) 3
1
(k 2 + m 2 )[(k + p) 2 + m 2 ]
(k γβ + mC γβ )
×
(D
2 A
γ
(− p, θ))A
β
( p, θ) +
1
2
D
γ D
α A α (− p, θ)A
β
( p, θ)
. (3.111)
One can observe that the linear divergences presented in S 1a and S 1b are cancelled in the above result, and the logarithmic ones vanish by symmetry reasons,
therefore this contribution is finite. However, this finiteness is based on the gauge
symmetry rather than on the supersymmetry. Indeed, after integration over the internal momentum we find that this expression is proportional to the gauge invariant
expression
i S 1 ( p) =
d
2
θ f ( p)(W
α
(− p)W α ( p) + 2mW
α
( p)A α ( p)),
(3.112)
where
f ( p) =
d
3 k
(2π) 3
1
(k 2 + m 2 )[(k + p) 2 + m 2 ]
.
This expression, at p → 0 (notice that f ( p)| p→0 =
1
8π|m|
), reproduces the expression for the quadratic Maxwell-Chern-Simons action. As a result, even if we consider
the spinor A
α superfield as a purely external one, it acquires a nontrivial dynamics
due to the one-loop correction. This is a key effect which also occurs in the supersymmetric C P
N −1 model explicitly studied in [44] in the commutative case, and in
[15] in the noncommutative one. We note that the only difference taking place in the
noncommutative case consists in the modification of the factor f ( p), see (3.158).
3.5 Effective Action of the Three-Dimensional Superfield
Theories and the Proper-Time Method
In this section we develop a prescription for calculating the superfield effective action
within the three-dimensional superfield formalism. Here we adopt the formalism
developed in Chap. 2, for superfield theories, and follow the approach proposed in
[45].
Our starting point is the three-dimensional superfield theory described by the
action (see e.g. [23]):
S[] =
d
5 z
1
2
D
2
− V (()
,
(3.113)