132
4 Four-Dimensional Superfield Supersymmetry
Hence the Feynman gauge is the most convenient one. The propagator of the v field
in this gauge is
v
I
(z 1 )v
J
(z 2 ) = i
−1
δ
I J
δ
8
(z 1 − z 2 ).
(4.325)
The interaction part of the total action of v includes both covariant generalizations
of the usual vertices presented in (4.293) and new vertices involving the background
strengths W α , ¯
W ˙
α manifestly (these expressions are analogues of those ones used in
[66, 106], where, however, other conventions are adopted):
S int = tr
d
8 z
1
2
v(W
α
D α + ¯
W ˙
α D
˙
α
)v +
1
16
gv{D
α v, ¯
D
2
D α v} −
−
1
48
g[[D
α v, v], v]W α −
1
64
g
2
[v, D
α v] ¯
D
2
[v, D α v] −
−
1
48
g
2 ¯
D
2
D
α v[v, [v, D α v]] −
1
192
g
2
[[[D
α v, v], v], v]W α
+ . . . . (4.326)
We note that the terms proportional to W α in the triple- and higher-order vertices in
v arose from the anticommutation of gauge covariant derivatives in (4.313) between
themselves, with using of the expression (4.318).
We also can prove the following important relation (cf. [66]):
D
α ¯
D
2
D α −
1
2
{D
2
, ¯
D
2
} − W
α
D α −
1
2
(D
α W α ) = −8( + W
α
D α + ¯
W α ¯
D
˙
α
);
D
2 ¯
D
2
D
2
= 16( + ¯
W ˙
α
¯
D
˙
α
+
1
2
( ¯
D ˙
α
¯
W
˙
α
))D
2
≡ 16 − D
2
;
¯
D
2
D
2 ¯
D
2
= 16( + W
α
D α +
1
2
(D
α W α )) ¯
D
2
≡ 16 − ¯
D
2
.
(4.327)
Because of the gauge symmetry, one needs to introduce ghosts. Since the form
of the gauge transformations and the gauge fixing action is very similar to the usual
superfield case evaluated at zero background fields in the previous subsection, with
only difference consisting in the covariant chirality of matter and ghost fields instead
of the usual one, the action of ghosts in this case is also analogous to the zero
background case with only difference consisting in the covariant chirality of the
ghosts c, c
and the covariant antichirality of the ghosts ¯
c, ¯
c
:
S gh = tr
d
8 z( ¯
c
− c
)L gv/2 (c + ¯
c + cthL gv/2 (c − ¯
c)),
(4.328)
which after expansion in the power series gives:
S gh = tr
d
8 z( ¯
c
c + c
¯
c +
1
2
( ¯
c
− c
)[v, c + ¯
c] +
1
12
(c
− ¯
c
)[v, [v, c − ¯
c]] + . . .).
(4.329)
4 Four-Dimensional Superfield Supersymmetry
Hence the Feynman gauge is the most convenient one. The propagator of the v field
in this gauge is
v
I
(z 1 )v
J
(z 2 ) = i
−1
δ
I J
δ
8
(z 1 − z 2 ).
(4.325)
The interaction part of the total action of v includes both covariant generalizations
of the usual vertices presented in (4.293) and new vertices involving the background
strengths W α , ¯
W ˙
α manifestly (these expressions are analogues of those ones used in
[66, 106], where, however, other conventions are adopted):
S int = tr
d
8 z
1
2
v(W
α
D α + ¯
W ˙
α D
˙
α
)v +
1
16
gv{D
α v, ¯
D
2
D α v} −
−
1
48
g[[D
α v, v], v]W α −
1
64
g
2
[v, D
α v] ¯
D
2
[v, D α v] −
−
1
48
g
2 ¯
D
2
D
α v[v, [v, D α v]] −
1
192
g
2
[[[D
α v, v], v], v]W α
+ . . . . (4.326)
We note that the terms proportional to W α in the triple- and higher-order vertices in
v arose from the anticommutation of gauge covariant derivatives in (4.313) between
themselves, with using of the expression (4.318).
We also can prove the following important relation (cf. [66]):
D
α ¯
D
2
D α −
1
2
{D
2
, ¯
D
2
} − W
α
D α −
1
2
(D
α W α ) = −8( + W
α
D α + ¯
W α ¯
D
˙
α
);
D
2 ¯
D
2
D
2
= 16( + ¯
W ˙
α
¯
D
˙
α
+
1
2
( ¯
D ˙
α
¯
W
˙
α
))D
2
≡ 16 − D
2
;
¯
D
2
D
2 ¯
D
2
= 16( + W
α
D α +
1
2
(D
α W α )) ¯
D
2
≡ 16 − ¯
D
2
.
(4.327)
Because of the gauge symmetry, one needs to introduce ghosts. Since the form
of the gauge transformations and the gauge fixing action is very similar to the usual
superfield case evaluated at zero background fields in the previous subsection, with
only difference consisting in the covariant chirality of matter and ghost fields instead
of the usual one, the action of ghosts in this case is also analogous to the zero
background case with only difference consisting in the covariant chirality of the
ghosts c, c
and the covariant antichirality of the ghosts ¯
c, ¯
c
:
S gh = tr
d
8 z( ¯
c
− c
)L gv/2 (c + ¯
c + cthL gv/2 (c − ¯
c)),
(4.328)
which after expansion in the power series gives:
S gh = tr
d
8 z( ¯
c
c + c
¯
c +
1
2
( ¯
c
− c
)[v, c + ¯
c] +
1
12
(c
− ¯
c
)[v, [v, c − ¯
c]] + . . .).
(4.329)