2 Effective Action and Loop Expansion: General Formulation
9
the most important of them are, first, the case of background superfields constant
in space-time, second, the case of chiral background superfields only. Further we
consider some examples.
Let us turn again to (2.6). We see that each quantum superfield is accompanied by
−1/2 , and each vertex—by
−1 (which provides a
n/2−1 S
(n)
[]φ
n
form of the vertex). An arbitrary (super)graph with P propagators and V vertices contains 2P quantum superfields (indeed, each propagator is formed by
contraction of two superfields). Therefore if this (super)graph contain vertices
S
(n 1 )
[]φ
n 1 , S
(n 2 )
[]φ
n 2 , . . . , S
(n V )
[]φ
n V , its power in is
V
i=1 (
n i
2
− 1) =
1
2
V
i=1 n i − V . However,
V
i=1 n i is just the number of quantum fields associated
with all vertices which is equal to 2P. Therefore the contribution described by this
(super)graph has the power of equal to P − V = L − 1, with L is number of loops.
But any expression of the form (2.6), by the definition, is a contribution to
, hence a
contribution from L-loop (super)graph to is proportional to
L . Hence we proved
that the order in from an arbitrary (super)graph is just the number of loops in it,
and the expansion in powers of is called the loop expansion. As a result we see that
loop corrections can be calculated on the base of a special (super)field technique.
In the next chapters of this review we will apply this methodology through
evaluating the effective action in the superfield formalism both in three- and fourdimensional superspaces, calculating explicitly one- and two-loop contributions to
the effective actions of different superfield theory models.
9
the most important of them are, first, the case of background superfields constant
in space-time, second, the case of chiral background superfields only. Further we
consider some examples.
Let us turn again to (2.6). We see that each quantum superfield is accompanied by
−1/2 , and each vertex—by
−1 (which provides a
n/2−1 S
(n)
[]φ
n
form of the vertex). An arbitrary (super)graph with P propagators and V vertices contains 2P quantum superfields (indeed, each propagator is formed by
contraction of two superfields). Therefore if this (super)graph contain vertices
S
(n 1 )
[]φ
n 1 , S
(n 2 )
[]φ
n 2 , . . . , S
(n V )
[]φ
n V , its power in is
V
i=1 (
n i
2
− 1) =
1
2
V
i=1 n i − V . However,
V
i=1 n i is just the number of quantum fields associated
with all vertices which is equal to 2P. Therefore the contribution described by this
(super)graph has the power of equal to P − V = L − 1, with L is number of loops.
But any expression of the form (2.6), by the definition, is a contribution to
, hence a
contribution from L-loop (super)graph to is proportional to
L . Hence we proved
that the order in from an arbitrary (super)graph is just the number of loops in it,
and the expansion in powers of is called the loop expansion. As a result we see that
loop corrections can be calculated on the base of a special (super)field technique.
In the next chapters of this review we will apply this methodology through
evaluating the effective action in the superfield formalism both in three- and fourdimensional superspaces, calculating explicitly one- and two-loop contributions to
the effective actions of different superfield theory models.