3.5 Effective Action of the Three-Dimensional Superfield Theories …
33
where is a scalar superfield. We will use again the loop expansion methodology.
To do it, we make a shift in the field :
→ +
√
φ,
(3.114)
where, is now a background (super)field, and φ is a quantum one. As a result, the
classical action (3.113) takes the form
S[, φ] = S[] +
d
5 z
1
2
φ[D
2
− V
(()]φ
−
3/2 1
3!
V
(()φ
3
−
2 1
4!
V
(I V )
(()φ
4
+ . . . ,
(3.115)
where dots are for higher orders in φ which are irrelevant in the two-loop approximation. Following the definitions given in the Chap. 2, we see that the corresponding
effective action [] is defined by the expression
exp
i
[]
=
Dφ exp
i
S[, φ]
.
(3.116)
The general structure of the effective action can be cast in a form of the derivative
expansion:
[] =
d
5 z K (() +
d
5 z F(D α , D
2
; ) + . . . ,
(3.117)
where the K (() is the term which we call the Kählerian effective potential by analogy with four-dimensional studies (see the next chapter), depending only on the
superfield but not on its derivatives, and the F is called auxiliary fields effective
potential whose key property is its vanishing in the case when all derivatives of the
superfields are equal to zero (these definitions have been firstly introduced in [27]
for the four-dimensional superfield theories), and dots are for terms involving spacetime derivatives of superfields. It is easy to see that F is at least of the second order
in the auxiliary field of the scalar supermultiplet. It can be explicitly written as
F(D α , D
2
; ) = F 2 (()D
α
D α + . . . ,
(3.118)
where the F 2 (() is a function of only but not of its derivatives, and the dots
correspond to terms with four or more supercovariant derivatives. It is clear that
this definition does not require to impose the condition D α = 0 which naively can
be treated as a supercovariant analogue of the usual requirement for a background
superfield to be constant, but known to imply in strong conceptual difficulties (see
33
where is a scalar superfield. We will use again the loop expansion methodology.
To do it, we make a shift in the field :
→ +
√
φ,
(3.114)
where, is now a background (super)field, and φ is a quantum one. As a result, the
classical action (3.113) takes the form
S[, φ] = S[] +
d
5 z
1
2
φ[D
2
− V
(()]φ
−
3/2 1
3!
V
(()φ
3
−
2 1
4!
V
(I V )
(()φ
4
+ . . . ,
(3.115)
where dots are for higher orders in φ which are irrelevant in the two-loop approximation. Following the definitions given in the Chap. 2, we see that the corresponding
effective action [] is defined by the expression
exp
i
[]
=
Dφ exp
i
S[, φ]
.
(3.116)
The general structure of the effective action can be cast in a form of the derivative
expansion:
[] =
d
5 z K (() +
d
5 z F(D α , D
2
; ) + . . . ,
(3.117)
where the K (() is the term which we call the Kählerian effective potential by analogy with four-dimensional studies (see the next chapter), depending only on the
superfield but not on its derivatives, and the F is called auxiliary fields effective
potential whose key property is its vanishing in the case when all derivatives of the
superfields are equal to zero (these definitions have been firstly introduced in [27]
for the four-dimensional superfield theories), and dots are for terms involving spacetime derivatives of superfields. It is easy to see that F is at least of the second order
in the auxiliary field of the scalar supermultiplet. It can be explicitly written as
F(D α , D
2
; ) = F 2 (()D
α
D α + . . . ,
(3.118)
where the F 2 (() is a function of only but not of its derivatives, and the dots
correspond to terms with four or more supercovariant derivatives. It is clear that
this definition does not require to impose the condition D α = 0 which naively can
be treated as a supercovariant analogue of the usual requirement for a background
superfield to be constant, but known to imply in strong conceptual difficulties (see