76
4 Four-Dimensional Superfield Supersymmetry
It turns out that the technique for solving these problems is quite analogous to that
one used in standard field theory. The first problem can be solved on the base of the
superficial degree of divergence. The natural way for solving the second one consists
in introducing superfield counterterms which are quite analogous to standard ones.
First of all let us generalize the procedure to find the superficial degree of divergence (see e.g. [71]) for superfield theories. To do this, let us consider the most
natural example, that is, the N = 1 SYM theory coupled to a chiral matter with the
Wess-Zumino self-interaction [33]. For all other models, the consideration is quite
analogous. The action of the theory is
S =
d
8 z ¯
i (e
gV
)
i
j
j
+ (
d
6 z(
1
2
m i j i j +
λ i jk
3!
i j k ) + h.c.) − (4.106)
− tr
1
16g 2
d
8 z(e
−gV D
α e
gV
) ¯
D
2
(e
−gV D α e
gV
) +
+
d
8 ztr
¯
c
c − ¯
cc
+
1
2
g( ¯
c
− c
)[V, c + ¯
c] + . . .
.
The dots are for the higher-order couplings involving ghosts. The triple vertices in
this theory are (cf. [66])
λ i jk
3!
d
6 z i j k + h.c.; g
d
8 z ¯
i V
A
(T
A
)
i
j
j
;
g
16
tr
d
8 z( ¯
D
2 D
α V )[V, D α V ];
g
2
tr
d
8 z( ¯
c
− c
)[V, c + ¯
c]. (4.107)
The i, j play the role of matrix indices since i is an isospinor, and the gauge
superfield V ≡ V
A T
A takes values in the Lie algebra as well as the ghosts do.
It follows from a direct inspection of the purely gauge sector that all SYM selfinteraction vertices involve exactly two chiral and two antichiral derivatives. In the
previous section, we have already proved that all corrections should be proportional
to one integral over d
4
θ.
As usual, the superficial degree of divergence (SDD) of the given Feynman
(super)graph is the order of the integral over internal momenta for the corresponding contribution, or, as is the same, is a degree of homogeneity of the (super)graph
in momenta, considered after performing D-algebra transformations [33]. The only
difference of the SDD in the superfield case is the additional impact from D-factors.
It is easy to see that contributions to the SDD are generated by momentum
depending factors in propagators and vertices (as usual, any internal momentum
k yields contribution 1), loop integrations, or, in other words, by manifest momentum dependence which is associated with propagators and loop integration, and by
D-factors which are associated with propagators and vertices (note that due to identities D
2 ¯
D
2 D
2
= 16D
2 , {D α , ¯
D ˙
α } = 2i∂ α ˙
α , one chiral derivative combined with
an antichiral one can be converted to one momentum, therefore any D-factor contributes to the SDD with 1/2). If some spinor derivatives are not converted to internal
momenta, the SDD from the supergraph evidently decreases.
4 Four-Dimensional Superfield Supersymmetry
It turns out that the technique for solving these problems is quite analogous to that
one used in standard field theory. The first problem can be solved on the base of the
superficial degree of divergence. The natural way for solving the second one consists
in introducing superfield counterterms which are quite analogous to standard ones.
First of all let us generalize the procedure to find the superficial degree of divergence (see e.g. [71]) for superfield theories. To do this, let us consider the most
natural example, that is, the N = 1 SYM theory coupled to a chiral matter with the
Wess-Zumino self-interaction [33]. For all other models, the consideration is quite
analogous. The action of the theory is
S =
d
8 z ¯
i (e
gV
)
i
j
j
+ (
d
6 z(
1
2
m i j i j +
λ i jk
3!
i j k ) + h.c.) − (4.106)
− tr
1
16g 2
d
8 z(e
−gV D
α e
gV
) ¯
D
2
(e
−gV D α e
gV
) +
+
d
8 ztr
¯
c
c − ¯
cc
+
1
2
g( ¯
c
− c
)[V, c + ¯
c] + . . .
.
The dots are for the higher-order couplings involving ghosts. The triple vertices in
this theory are (cf. [66])
λ i jk
3!
d
6 z i j k + h.c.; g
d
8 z ¯
i V
A
(T
A
)
i
j
j
;
g
16
tr
d
8 z( ¯
D
2 D
α V )[V, D α V ];
g
2
tr
d
8 z( ¯
c
− c
)[V, c + ¯
c]. (4.107)
The i, j play the role of matrix indices since i is an isospinor, and the gauge
superfield V ≡ V
A T
A takes values in the Lie algebra as well as the ghosts do.
It follows from a direct inspection of the purely gauge sector that all SYM selfinteraction vertices involve exactly two chiral and two antichiral derivatives. In the
previous section, we have already proved that all corrections should be proportional
to one integral over d
4
θ.
As usual, the superficial degree of divergence (SDD) of the given Feynman
(super)graph is the order of the integral over internal momenta for the corresponding contribution, or, as is the same, is a degree of homogeneity of the (super)graph
in momenta, considered after performing D-algebra transformations [33]. The only
difference of the SDD in the superfield case is the additional impact from D-factors.
It is easy to see that contributions to the SDD are generated by momentum
depending factors in propagators and vertices (as usual, any internal momentum
k yields contribution 1), loop integrations, or, in other words, by manifest momentum dependence which is associated with propagators and loop integration, and by
D-factors which are associated with propagators and vertices (note that due to identities D
2 ¯
D
2 D
2
= 16D
2 , {D α , ¯
D ˙
α } = 2i∂ α ˙
α , one chiral derivative combined with
an antichiral one can be converted to one momentum, therefore any D-factor contributes to the SDD with 1/2). If some spinor derivatives are not converted to internal
momenta, the SDD from the supergraph evidently decreases.