4.11 Supergauge Theories
125
The vertices can be read off from (4.293), (4.294). The propagators look like
V
I
(z 1 )V
J
(z 2 ) = iδ
I J 1
−
1
8
D
α ¯
D
2 D α + ξ
{D
2
, ¯
D
2
}
16
δ
8
(z 1 − z 2 ); (4.296)
¯
c
I
(z 1 )c
J
(z 2 ) = =c
I
(z 1 ) ¯
c
J
(z 2 ) = −iδ
I J 1
δ
8
(z 1 − z 2 ).
We note that ghosts are fermions, hence any ghost loop corresponds to minus sign.
Then, D-factors are associated with vertices containing ghosts just by the same rule
as with vertices containing any chiral superfields. We note that if we choose ξ = 1
(the Feynman gauge) the propagator of gauge superfield takes the simplest form
V
I
(z 1 )V
J
(z 2 ) = iδ
I J 1
δ
8
(z 1 − z 2 ).
(4.297)
Note that its sign is opposite to the sign of the propagator of a chiral superfield. This
difference of signs plays an important role for some cancellations of divergences,
see e.g. [23].
The action of a chiral superfield coupled to a gauge one looks like
S =
d
8 z ¯
i (e
gV
)
i
j
j
,
(4.298)
if = { i } is transformed under some representation of the gauge group (i.e. it is
an isospinor), or
S =
d
8 ztr( ¯
e
gV
e
−gV
),
(4.299)
if =
A T
A is Lie-algebra-valued. Note that under gauge transformations (4.277)
the is transformed as
→ e
−ig
(4.300)
for an isospinor case and as
→ e
−ig
e
ig
(4.301)
for a Lie-algebra-valued chiral superfield case. The , ¯
are Lie-algebra-valued
parameters in both cases. The vertices can be easily obtained by expanding interaction
parts of these actions into power series: for (4.298) we have
S V =
d
8 z[ ¯
i (e
gV
)
i
j
j
− i ¯
i
] =
d
8 z
∞
n=1
1
n!
¯
i ((gV )
n
)
i
j
j
, (4.302)
125
The vertices can be read off from (4.293), (4.294). The propagators look like
V
I
(z 1 )V
J
(z 2 ) = iδ
I J 1
−
1
8
D
α ¯
D
2 D α + ξ
{D
2
, ¯
D
2
}
16
δ
8
(z 1 − z 2 ); (4.296)
¯
c
I
(z 1 )c
J
(z 2 ) = =c
I
(z 1 ) ¯
c
J
(z 2 ) = −iδ
I J 1
δ
8
(z 1 − z 2 ).
We note that ghosts are fermions, hence any ghost loop corresponds to minus sign.
Then, D-factors are associated with vertices containing ghosts just by the same rule
as with vertices containing any chiral superfields. We note that if we choose ξ = 1
(the Feynman gauge) the propagator of gauge superfield takes the simplest form
V
I
(z 1 )V
J
(z 2 ) = iδ
I J 1
δ
8
(z 1 − z 2 ).
(4.297)
Note that its sign is opposite to the sign of the propagator of a chiral superfield. This
difference of signs plays an important role for some cancellations of divergences,
see e.g. [23].
The action of a chiral superfield coupled to a gauge one looks like
S =
d
8 z ¯
i (e
gV
)
i
j
j
,
(4.298)
if = { i } is transformed under some representation of the gauge group (i.e. it is
an isospinor), or
S =
d
8 ztr( ¯
e
gV
e
−gV
),
(4.299)
if =
A T
A is Lie-algebra-valued. Note that under gauge transformations (4.277)
the is transformed as
→ e
−ig
(4.300)
for an isospinor case and as
→ e
−ig
e
ig
(4.301)
for a Lie-algebra-valued chiral superfield case. The , ¯
are Lie-algebra-valued
parameters in both cases. The vertices can be easily obtained by expanding interaction
parts of these actions into power series: for (4.298) we have
S V =
d
8 z[ ¯
i (e
gV
)
i
j
j
− i ¯
i
] =
d
8 z
∞
n=1
1
n!
¯
i ((gV )
n
)
i
j
j
, (4.302)