126
4 Four-Dimensional Superfield Supersymmetry
and for (4.299)
S V = tr
d 8 z(( ¯
e gV e −gV ) − ¯
) = tr
d 8 z(g ¯
[V, ,] +
g 2
2
¯
[V, [V, ,]] + . . .). (4.303)
The propagators of chiral superfields in this case are the standard ones given by (4.78),
taken at m = 0 (we note that namely massless chiral superfields were considered in
the Sect. 4.8 when the chiral effective potential in N = 1 SYM theory coupled to a
chiral matter was evaluated, and here we restrict out consideration to the massless
case as well), and multiplied by delta symbols corresponding to algebraic indices.
The diagram technique derived now is very suitable for calculations in the sector
of background (anti)chiral superfields only and for obtaining the divergences.
Let us consider an example. The N = 2 SYM theory with a chiral matter is
described by the action (see e.g. [2, 101]):
S =
1
64g 2
d
6 z tr W
α W α + tr
d
8 z ¯
e
gV
e
−gV
+
+
n
i=1
ig(
d
6 z Q i ˜
Q i + h.c.) +
d
8 z ¯ ˜
Q i e
−gV ˜
Q i +
d
8 z ¯
Q i e
gV Q i
.
(4.304)
Here is the Lie-algebra-valued chiral superfield, and Q
i
, ˜
Q i are chiral superfields
transformed under mutually conjugated representations of Lie algebra. They are
often called matter hypermultiplets. In the particular case n = 1, where we have
only one superfield Q and one superfield ˜
Q, this action describes the N = 4 SYM
theory (the extended supersymmetry manifests itself through the so-called hidden
supersymmetry transformations relating mutually the chiral and real superfields, see
e.g. [23]).
We note that there is another equivalent formulation of the N = 4 SYM theory
characterized instead of , Q and ˜
Q, by three Lie-algebra-valued chiral superfields
i , with i = 1 . . . 3, with the action (see [23]):
S =
1
64g 2
d
6 z tr W
α W α + tr
d
8 z
3
i=1
¯
i e
gV
i e
−gV
+
+
g
3!
i jk tr
(
d
6 z i j k + h.c.)
.
(4.305)
Let us consider the structure of one-loop divergences in the theory (4.304). For the
sake of the simplicity we choose the Feynman gauge ξ = 1 in which the propagator
is given by (4.297), therefore all tadpole diagrams given in [2] evidently vanish.
The contributions to the two-point function of the field are given by Fig. 4.15.
Here and further, the thin line is for the propagator of the Lie-algebra valued
chiral superfield , the thick one—of hypermultiplets Q, ˜
Q, the wavy one—of the
4 Four-Dimensional Superfield Supersymmetry
and for (4.299)
S V = tr
d 8 z(( ¯
e gV e −gV ) − ¯
) = tr
d 8 z(g ¯
[V, ,] +
g 2
2
¯
[V, [V, ,]] + . . .). (4.303)
The propagators of chiral superfields in this case are the standard ones given by (4.78),
taken at m = 0 (we note that namely massless chiral superfields were considered in
the Sect. 4.8 when the chiral effective potential in N = 1 SYM theory coupled to a
chiral matter was evaluated, and here we restrict out consideration to the massless
case as well), and multiplied by delta symbols corresponding to algebraic indices.
The diagram technique derived now is very suitable for calculations in the sector
of background (anti)chiral superfields only and for obtaining the divergences.
Let us consider an example. The N = 2 SYM theory with a chiral matter is
described by the action (see e.g. [2, 101]):
S =
1
64g 2
d
6 z tr W
α W α + tr
d
8 z ¯
e
gV
e
−gV
+
+
n
i=1
ig(
d
6 z Q i ˜
Q i + h.c.) +
d
8 z ¯ ˜
Q i e
−gV ˜
Q i +
d
8 z ¯
Q i e
gV Q i
.
(4.304)
Here is the Lie-algebra-valued chiral superfield, and Q
i
, ˜
Q i are chiral superfields
transformed under mutually conjugated representations of Lie algebra. They are
often called matter hypermultiplets. In the particular case n = 1, where we have
only one superfield Q and one superfield ˜
Q, this action describes the N = 4 SYM
theory (the extended supersymmetry manifests itself through the so-called hidden
supersymmetry transformations relating mutually the chiral and real superfields, see
e.g. [23]).
We note that there is another equivalent formulation of the N = 4 SYM theory
characterized instead of , Q and ˜
Q, by three Lie-algebra-valued chiral superfields
i , with i = 1 . . . 3, with the action (see [23]):
S =
1
64g 2
d
6 z tr W
α W α + tr
d
8 z
3
i=1
¯
i e
gV
i e
−gV
+
+
g
3!
i jk tr
(
d
6 z i j k + h.c.)
.
(4.305)
Let us consider the structure of one-loop divergences in the theory (4.304). For the
sake of the simplicity we choose the Feynman gauge ξ = 1 in which the propagator
is given by (4.297), therefore all tadpole diagrams given in [2] evidently vanish.
The contributions to the two-point function of the field are given by Fig. 4.15.
Here and further, the thin line is for the propagator of the Lie-algebra valued
chiral superfield , the thick one—of hypermultiplets Q, ˜
Q, the wavy one—of the