36
3 Superfield Description of Three-Dimensional Supersymmetric Theories
in the space-time, the Moyal product of these superfields reduces to the usual one
(in higher loops, this is not so—the noncommutative deformation of the two-loop
effective potential is discussed in [45]). It is interesting to note that this result, up to a
constant multiplier, can be obtained without calculations. Indeed, it follows already
from (3.121) that the one-loop effective potential is a function of V
(() only. Since
our result is naturally finite (as we already noted, in three-dimensional theories the
one-loop results diverge only for theories with exotic effective dynamics, e.g. with
propagators ∝
1
√
k 2 ), it cannot depend on any arbitrary parameter like the cutoff scale
μ. Therefore, by dimensional reasons, the only form of the effective potential is
a
V
(()
2 , where a is a some number.
We note that in principle one could elaborate the expression (3.121), for the
constant background, by a much more straightforward way. Indeed, if one denotes
−V
= , the one-loop effective action becomes
(1)
=
i
2
Tr ln[D
2
+ ], which is
a function of . Then, one can consider
d
(1)
d
=
i
2
Tr
1
D 2 +
,
(3.131)
which, after finding the inverse operator together with the Fourier transform, becomes
d
(1)
d
= −
i
2
d
5 z
d
3 k
(2π) 3
D
2
−
k 2 + 2 δ
5
(z − z
)| z=z .
(3.132)
The D-algebra is trivial, and after Wick rotation we have
d
(1)
d
= −
1
2
d
5 z
d
3 k
(2π) 3
1
k 2 + 2 =
d
5 z
8π
,
(3.133)
Integrating this equation with respect to , we arrive at the expression (3.130)
obtained above. Nevertheless, we remind that the proper-time method we formulated can be applied for a wide class of superfield theory models, and its significance
is not exhausted by a scalar superfield theory considered here. Further in this section
we discuss the case of contributions depending on derivatives of background superfields.
Now, let us go to two loops. Applying the expansion given in Eqs. (3.115)–(3.119),
we can find that the expression for the two-loop effective action
(2) looks like,
(2) = −i
Dφ exp
i
2
φ[D 2 − V (()]φ
1
2
1
3!
V (()φ 3
2
−
1
4!
V (I V ) (()φ 4
.
(3.134)
The two-loop contributions are schematically represented by two supergraphs given
by Fig. 3.2.
3 Superfield Description of Three-Dimensional Supersymmetric Theories
in the space-time, the Moyal product of these superfields reduces to the usual one
(in higher loops, this is not so—the noncommutative deformation of the two-loop
effective potential is discussed in [45]). It is interesting to note that this result, up to a
constant multiplier, can be obtained without calculations. Indeed, it follows already
from (3.121) that the one-loop effective potential is a function of V
(() only. Since
our result is naturally finite (as we already noted, in three-dimensional theories the
one-loop results diverge only for theories with exotic effective dynamics, e.g. with
propagators ∝
1
√
k 2 ), it cannot depend on any arbitrary parameter like the cutoff scale
μ. Therefore, by dimensional reasons, the only form of the effective potential is
a
V
(()
2 , where a is a some number.
We note that in principle one could elaborate the expression (3.121), for the
constant background, by a much more straightforward way. Indeed, if one denotes
−V
= , the one-loop effective action becomes
(1)
=
i
2
Tr ln[D
2
+ ], which is
a function of . Then, one can consider
d
(1)
d
=
i
2
Tr
1
D 2 +
,
(3.131)
which, after finding the inverse operator together with the Fourier transform, becomes
d
(1)
d
= −
i
2
d
5 z
d
3 k
(2π) 3
D
2
−
k 2 + 2 δ
5
(z − z
)| z=z .
(3.132)
The D-algebra is trivial, and after Wick rotation we have
d
(1)
d
= −
1
2
d
5 z
d
3 k
(2π) 3
1
k 2 + 2 =
d
5 z
8π
,
(3.133)
Integrating this equation with respect to , we arrive at the expression (3.130)
obtained above. Nevertheless, we remind that the proper-time method we formulated can be applied for a wide class of superfield theory models, and its significance
is not exhausted by a scalar superfield theory considered here. Further in this section
we discuss the case of contributions depending on derivatives of background superfields.
Now, let us go to two loops. Applying the expansion given in Eqs. (3.115)–(3.119),
we can find that the expression for the two-loop effective action
(2) looks like,
(2) = −i
Dφ exp
i
2
φ[D 2 − V (()]φ
1
2
1
3!
V (()φ 3
2
−
1
4!
V (I V ) (()φ 4
.
(3.134)
The two-loop contributions are schematically represented by two supergraphs given
by Fig. 3.2.