3.5 Effective Action of the Three-Dimensional Superfield Theories …
39
d 0 + d 2 ( + D
2
) + d
α
1 (D α ) + = 0,
−id 1γ ∂
γ α
+ d
α
1 + d 2 D
α
= 0,
d 0 + d 2 + 1 = 0.
(3.147)
The system (3.146), after some simplifications, implies in the following equation:
b 1γ
ω − +
1
2
D
β
D β
(ω − ) 2 − − (D 2 )
C
αγ
− i∂
αγ
= 0.
(3.148)
Since b 1γ = 0 (otherwise the solution is trivial), the ω’s can be found requiring the
2 × 2 matrix
γ α defined as
γ α
=
ω − +
1
2
D
β
D β
(ω − ) 2 − − (D 2 )
C
αγ
− i∂
αγ
(3.149)
to have zero determinant. The corresponding equation for ω is solvable in principle,
but its evaluation is technically very difficult, and we will not obtain the solution
here. The corresponding results would be extremely complicated.
Let us also briefly discuss the calculation of the one-loop effective potential via the
more traditional method, that is, via summation of the supergraphs. In the simplest
case of the purely scalar superfield theory, we can start with the expression (3.121)
and expand it in the power series in the background field = −V
(():
(1)
=
i
2
Tr ln[D
2
+ ] =
i
2
Tr ln(D
2
) +
i
2
Tr ln[1 + D
2
−1
] =
=
i
2
Tr ln(D
2
) +
i
2
Tr
∞
n=1
(−1)
n+1
n
[
D
2
]
n
,
(3.150)
where we can construct corresponding supergraphs and calculate the sum, arriving
at the same result (3.130). However, in the case of the scalar superfield coupled to
the gauge one the situation is much more delicate. Let us consider for example the
scalar QED whose action is
S m =
d
5 z
1
2
(D
α
+ iA
α
)(D α ¯
− i A α ¯
) +
1
2g 2 W
α W α
. (3.151)
First of all, one can find that the triple gauge-matter vertices in the one-loop diagrams
arise within the following fragments given by Fig. 3.3.
In this figure, the factor D
β D
α
+ ξ D
α D
β originates from the propagator (there is
no essential difference between the QED and Chern-Simons theories in this context
since this factor commutes with D
2 ). The indices α and β are the indices of gauge
fields contracted into this propagator. Let us, for the sake of the convenience, refer to
a vertex as to the “proper” one if it contains three and more quantum fields since such
a vertex cannot be absorbed into redefinition of the propagator in the framework of
39
d 0 + d 2 ( + D
2
) + d
α
1 (D α ) + = 0,
−id 1γ ∂
γ α
+ d
α
1 + d 2 D
α
= 0,
d 0 + d 2 + 1 = 0.
(3.147)
The system (3.146), after some simplifications, implies in the following equation:
b 1γ
ω − +
1
2
D
β
D β
(ω − ) 2 − − (D 2 )
C
αγ
− i∂
αγ
= 0.
(3.148)
Since b 1γ = 0 (otherwise the solution is trivial), the ω’s can be found requiring the
2 × 2 matrix
γ α defined as
γ α
=
ω − +
1
2
D
β
D β
(ω − ) 2 − − (D 2 )
C
αγ
− i∂
αγ
(3.149)
to have zero determinant. The corresponding equation for ω is solvable in principle,
but its evaluation is technically very difficult, and we will not obtain the solution
here. The corresponding results would be extremely complicated.
Let us also briefly discuss the calculation of the one-loop effective potential via the
more traditional method, that is, via summation of the supergraphs. In the simplest
case of the purely scalar superfield theory, we can start with the expression (3.121)
and expand it in the power series in the background field = −V
(():
(1)
=
i
2
Tr ln[D
2
+ ] =
i
2
Tr ln(D
2
) +
i
2
Tr ln[1 + D
2
−1
] =
=
i
2
Tr ln(D
2
) +
i
2
Tr
∞
n=1
(−1)
n+1
n
[
D
2
]
n
,
(3.150)
where we can construct corresponding supergraphs and calculate the sum, arriving
at the same result (3.130). However, in the case of the scalar superfield coupled to
the gauge one the situation is much more delicate. Let us consider for example the
scalar QED whose action is
S m =
d
5 z
1
2
(D
α
+ iA
α
)(D α ¯
− i A α ¯
) +
1
2g 2 W
α W α
. (3.151)
First of all, one can find that the triple gauge-matter vertices in the one-loop diagrams
arise within the following fragments given by Fig. 3.3.
In this figure, the factor D
β D
α
+ ξ D
α D
β originates from the propagator (there is
no essential difference between the QED and Chern-Simons theories in this context
since this factor commutes with D
2 ). The indices α and β are the indices of gauge
fields contracted into this propagator. Let us, for the sake of the convenience, refer to
a vertex as to the “proper” one if it contains three and more quantum fields since such
a vertex cannot be absorbed into redefinition of the propagator in the framework of