4.6 Effective Action in Superfield Theories. Superfield Proper-Time Technique
83
called the Schwinger kernel. Of course, the U depends on background superfields.
It follows from this definition that U satisfies the equation:
i
∂U
∂s
= −U .
(4.123)
The is supposed to have a form of a power series in derivatives, and U by the
definition satisfies the initial condition
U (x, x
)| s=0 = δ
4
(x − x
).
In a usual case (especially, in the gravity theories) the U is represented in the form
of infinite power series in parameter s, called the proper time, as [36]
U = −
i
(4πs) 2 exp(
i
4s
(x − x
)
2
)
∞
n=0
a n (is)
n
.
(4.124)
The (ultraviolet) divergences correspond to lower orders of this expansion, one must
note that the ultraviolet limit corresponds to s → 0, infrared one—to s → ∞. Coefficients a n depend on background fields and their derivatives. We note that if background fields are put to zero, we arrive at
U
(0)
(x, x
; s) = e
is
δ
4
(x − x
) = −
i
(4πs) 2 exp(
i
4s
(x − x
)
2
), (4.125)
and the U
(0)
(x, x
; s) satisfies the condition
i
∞
0
dsU
(0)
(x, x
; s) =
1
δ
4
(x − x
).
(4.126)
The approach in the case of superfield theories is quite analogous. However, in the
superfield case using of the proper time approach is characterized by an essential
advantage. Indeed, in this case it is more convenient to expand Schwinger kernel
U (x, x
; s) not in an infinite power series in s but in a power series in spinor supercovariant derivatives which is finite due to anticommutation properties of spinor derivatives (we already know that the similar situation takes place in three-dimensional
superfield theories).
Actually, in most cases the operator in superfield theories looks like
= + A
α
10 D α + A 01 ˙
α
¯
D
˙
α
+ . . . ≡ + ˜
,
(4.127)
with ˜
is a some background dependent operator. In typical cases it contains only
even orders in spinor derivatives, here we consider just this case, and A nm ’s are
background dependent coefficients. We introduce the superfield Schwinger kernel
(cf. [27, 33]):
83
called the Schwinger kernel. Of course, the U depends on background superfields.
It follows from this definition that U satisfies the equation:
i
∂U
∂s
= −U .
(4.123)
The is supposed to have a form of a power series in derivatives, and U by the
definition satisfies the initial condition
U (x, x
)| s=0 = δ
4
(x − x
).
In a usual case (especially, in the gravity theories) the U is represented in the form
of infinite power series in parameter s, called the proper time, as [36]
U = −
i
(4πs) 2 exp(
i
4s
(x − x
)
2
)
∞
n=0
a n (is)
n
.
(4.124)
The (ultraviolet) divergences correspond to lower orders of this expansion, one must
note that the ultraviolet limit corresponds to s → 0, infrared one—to s → ∞. Coefficients a n depend on background fields and their derivatives. We note that if background fields are put to zero, we arrive at
U
(0)
(x, x
; s) = e
is
δ
4
(x − x
) = −
i
(4πs) 2 exp(
i
4s
(x − x
)
2
), (4.125)
and the U
(0)
(x, x
; s) satisfies the condition
i
∞
0
dsU
(0)
(x, x
; s) =
1
δ
4
(x − x
).
(4.126)
The approach in the case of superfield theories is quite analogous. However, in the
superfield case using of the proper time approach is characterized by an essential
advantage. Indeed, in this case it is more convenient to expand Schwinger kernel
U (x, x
; s) not in an infinite power series in s but in a power series in spinor supercovariant derivatives which is finite due to anticommutation properties of spinor derivatives (we already know that the similar situation takes place in three-dimensional
superfield theories).
Actually, in most cases the operator in superfield theories looks like
= + A
α
10 D α + A 01 ˙
α
¯
D
˙
α
+ . . . ≡ + ˜
,
(4.127)
with ˜
is a some background dependent operator. In typical cases it contains only
even orders in spinor derivatives, here we consider just this case, and A nm ’s are
background dependent coefficients. We introduce the superfield Schwinger kernel
(cf. [27, 33]):