4.4 Feynman Supergraphs
69
4.4 Feynman Supergraphs
Now, after we have introduced the generating functional (4.75), we can start with
formulation of the superfield Feynman diagram technique, or supergraph technique.
It can be introduced as follows.
One can easily read off from (4.78) that any ¯
-propagator corresponds to
( − m
2
)
−1 , at a chiral vertex each propagator is associated with the factor (−
1
4
¯
D
2
),
and at an antichiral one—with (−
1
4
D
2
). However, each chiral (or antichiral) vertex
involves an integration over d
6 z (or d
6
¯
z). Furthermore, since we deal with the delta
function δ
8
(z 1 − z 2 ), for the sake of unification it is more convenient to represent all
contributions in the form of integrals over d
8 z via the rule
d
6 z(−
1
4
) ¯
D
2
F =
d
8 zF,
with F be some function of superfields. As a result, if all superfields associated to a
given
d
6 z
n -vertex are contracted into propagators, this vertex is associated with
n − 1 (−
1
4
¯
D
2
) factors, and, similarly, any
d
6
¯
z ¯
m -vertex of course, in the case
when all superfields are contracted into propagators—with m − 1 (−
1
4
D
2
)-factors.
And the vertex
d
8 z
m ¯
n , in the same case when all superfields are contracted into
propagators, is associated with m factors and n (−
1
4
D
2
) factors. Here and further we
refer to superfields contracted into propagators as to the quantum ones. The quantum
chiral (antichiral) superfields will be denoted as ( ¯
). We see that the number of D
2 ,
¯
D
2 factors for such vertices is the number of antichiral (chiral) quantum superfields
associated with this vertex. There is no such D
2
, ¯
D
2 -factors arising in propagators of
a non-chiral (e.g. real) superfield, since, as we argued in the Sect. 4.1, the presence of
such factors is motivated by the variational derivative with respect to a chiral superfield. However, only quantum fields, i.e. those ones contracted into propagators, are
associated with D
2 , ¯
D
2 factors at corresponding vertices. External lines do not carry
such factors, and if one, two … n chiral (antichiral) superfields associated with the
vertex are external the number of ¯
D
2 (D
2 ) factors corresponding to this vertex is less
by one, two … n than in the case when all superfields are contracted to propagators.
Then, the propagator ( ¯
¯
) corresponds to the
m D
2
4(−m 2 )
(
m ¯
D
2
4(−m 2 )
) factor,
with the D
2 ( ¯
D
2 ) factor is associated with the propagator itself. Besides of this, the
quantum chiral (antichiral) fields contracted to this propagator are associated to ¯
D
2 ,
D
2 by the rules defined in the beginning of this paragraph.
If we consider the N = 1 SYM theory, its quadratic action being the sum of
the action (4.49) and the simplest gauge-fixing term S g f = −
1
2
tr
d
8 zV
{D
2 , ¯
D
2 }
16
V
corresponding to the Feynman gauge (we consider a more general gauge fixing in
Sect. 4.11) looks like
S = −
1
2
tr
d
8 zV V.
(4.86)
Here tr is matrix trace (remind that the superfield V is Lie-algebra valued). There
is no D factors associated with this propagator but they are associated with vertices.
In a pure N = 1 SYM theory vertices are given by
69
4.4 Feynman Supergraphs
Now, after we have introduced the generating functional (4.75), we can start with
formulation of the superfield Feynman diagram technique, or supergraph technique.
It can be introduced as follows.
One can easily read off from (4.78) that any ¯
-propagator corresponds to
( − m
2
)
−1 , at a chiral vertex each propagator is associated with the factor (−
1
4
¯
D
2
),
and at an antichiral one—with (−
1
4
D
2
). However, each chiral (or antichiral) vertex
involves an integration over d
6 z (or d
6
¯
z). Furthermore, since we deal with the delta
function δ
8
(z 1 − z 2 ), for the sake of unification it is more convenient to represent all
contributions in the form of integrals over d
8 z via the rule
d
6 z(−
1
4
) ¯
D
2
F =
d
8 zF,
with F be some function of superfields. As a result, if all superfields associated to a
given
d
6 z
n -vertex are contracted into propagators, this vertex is associated with
n − 1 (−
1
4
¯
D
2
) factors, and, similarly, any
d
6
¯
z ¯
m -vertex of course, in the case
when all superfields are contracted into propagators—with m − 1 (−
1
4
D
2
)-factors.
And the vertex
d
8 z
m ¯
n , in the same case when all superfields are contracted into
propagators, is associated with m factors and n (−
1
4
D
2
) factors. Here and further we
refer to superfields contracted into propagators as to the quantum ones. The quantum
chiral (antichiral) superfields will be denoted as ( ¯
). We see that the number of D
2 ,
¯
D
2 factors for such vertices is the number of antichiral (chiral) quantum superfields
associated with this vertex. There is no such D
2
, ¯
D
2 -factors arising in propagators of
a non-chiral (e.g. real) superfield, since, as we argued in the Sect. 4.1, the presence of
such factors is motivated by the variational derivative with respect to a chiral superfield. However, only quantum fields, i.e. those ones contracted into propagators, are
associated with D
2 , ¯
D
2 factors at corresponding vertices. External lines do not carry
such factors, and if one, two … n chiral (antichiral) superfields associated with the
vertex are external the number of ¯
D
2 (D
2 ) factors corresponding to this vertex is less
by one, two … n than in the case when all superfields are contracted to propagators.
Then, the propagator ( ¯
¯
) corresponds to the
m D
2
4(−m 2 )
(
m ¯
D
2
4(−m 2 )
) factor,
with the D
2 ( ¯
D
2 ) factor is associated with the propagator itself. Besides of this, the
quantum chiral (antichiral) fields contracted to this propagator are associated to ¯
D
2 ,
D
2 by the rules defined in the beginning of this paragraph.
If we consider the N = 1 SYM theory, its quadratic action being the sum of
the action (4.49) and the simplest gauge-fixing term S g f = −
1
2
tr
d
8 zV
{D
2 , ¯
D
2 }
16
V
corresponding to the Feynman gauge (we consider a more general gauge fixing in
Sect. 4.11) looks like
S = −
1
2
tr
d
8 zV V.
(4.86)
Here tr is matrix trace (remind that the superfield V is Lie-algebra valued). There
is no D factors associated with this propagator but they are associated with vertices.
In a pure N = 1 SYM theory vertices are given by