30
3 Superfield Description of Three-Dimensional Supersymmetric Theories
i S 1a ( p) =
1
4
d
2
θ 1 d
2
θ 2
d
3 k
(2π) 3 A
α
(− p, θ 1 )A
β
( p, θ 2 ) ×
×
D α1 (D
2
1 + m)
k 2 + m 2 δ 12
(D
2
1 + m)D β2
(k + p) 2 + m 2 δ 12
−
D α1 (D
2
1 + m)D β2
k 2 + m 2
δ 12
D
2
1 + m
(k + p) 2 + m 2 δ 12
.
(3.100)
Integrating by parts some of the spinor derivatives and using the identity D β2 (k, θ 2 )
δ 12 = −D β1 (−k, θ 1 )δ 12 (further, we will omit momentum arguments of spinor supercovariant derivatives), we arrive at
i S 1a ( p) =
1
4
d
2
θ 1 d
2
θ 2
d
3 k
(2π) 3 I (k, p)
×
2(D
2
1 + m)δ 12 D α1 (D
2
1 + m)D β1 δ 12 A
α
(− p, θ 1 )A
β
( p, θ 2 )
+ (D
2
1 + m)δ 12 (D
2
1 + m)D β1 δ 12 (D
α A α )(− p, θ 1 )A
β
( p, θ 2 )
. (3.101)
where
I (k, p) =
1
(k 2 + m 2 )[(k + p) 2 + m 2 ]
.
(3.102)
It is convenient to separate S 1a into two parts, S 1a = S
(1)
1a + S
(2)
1a , where i S
(1)
1a and
i S
(2)
1a are associated to two terms in the large brackets of (3.101). Let us consider
first i S
(1)
1a , which, after transporting D
2 from one of the propagators to other factors,
becomes
i S
(1)
1a ( p) =
1
4
d
2
θ 1 d
2
θ 2
d
3 k
(2π) 3 I (k, p)
×
2mδ 12 D α1 (D
2
1 + m)D β1 δ 12 A
α
(− p, θ 1 )A
β
( p, θ 2 )
+ 2δ 12 D
2
1
D α1 (D
2
1 + m)D β1 δ 12 A
α
(− p, θ 1 )
A
β
( p, θ 2 )
. (3.103)
Now we employ the identity {D α1 , D
2
1 } = 0 which leads to
i S
(1)
1a ( p) =
1
4
d
2
θ 1 d
2
θ 2
d
3 k
(2π) 3 I (k, p)
(3.104)
×
2δ 12 (k
2
+ m
2
)D α1 D β1 δ 12 A
α
(− p, θ 1 )A
β
( p, θ 2 )
+ 2δ 12 (−D
2
1 + m)D α1 D β1 δ 12 (D
2 A
α
(− p, θ 1 ))A
β
( p, θ 2 )
.
The use of the relationship (3.13) now provides