4.11 Supergauge Theories
135
W
2 f (DW, ¯
D ¯
W ), with f (DW, ¯
D ¯
W ) is a some function of covariant derivatives of
superfield strengths, reducing to a non-zero constant when these derivatives are equal
to zero.
We start with the definition based on the well-known zeta function regularization
procedure (see e.g. [109] for a review on this methodology):
ln det = −ζ
(0),
(4.335)
where the zeta function corresponding to the operator ˆ
K = e
t , with t is a proper
time (see [36] for the general review on the proper time methodology) is defined as
ζ(s) =
1
(s)
∞
0
dtt
s−1 K (t),
(4.336)
and K (t) is a functional trace of ˆ
K . Explicitly, it looks like
K (t) =
d
8 z lim
z→z
e
t
δ
8
(z − z
).
(4.337)
Alternatively, one can use straightforwardly the Schwinger-De Witt representation
which allows to write
ln det =
∞
0
dt
t
tre
t
≡
∞
0
dt
t
K (t).
(4.338)
Here we follow the methodology developed in [74] and use the notations adopted
there. In our case, for the effective action given by (4.334), the K (t) reads as
K (t) =
d
8 z lim
z→z
e
t (D
a D a +W
α D α + ¯
W ˙
α
¯
D
˙
α +||
2 )
δ
8
(z − z
).
(4.339)
Then, we use the Fourier representation of the complete delta function δ
8
(z − z
),
both for its bosonic and fermionic parts:
δ
8
(z − z
) =
d
4 k
(2π) 4 e
ik(x−x
)
d
4
e
i
α (θ α −θ
α ) e
i ¯
˙
α ( ¯
θ
˙
α − ¯
θ
˙
α )
.
(4.340)
To simplify the calculations, we factorize out the chiral matter superfields (which are
considered as constants within this calculation since we disregard all their derivatives), writing K (t) = e
t||
2
˜
K (t). Then, we introduce the key point of the methodology proposed in [74]: the K (t) (4.339), being a kernel of the operator, is actually
nothing other as the operator acting on the delta function, which further must be
applied to some other function. Since the delta function is expanded into the Fourier
series through (4.340), one can verify that when the operator whose kernel is given
by (4.339) acts on an arbitrary function, the following objects will emerge:
135
W
2 f (DW, ¯
D ¯
W ), with f (DW, ¯
D ¯
W ) is a some function of covariant derivatives of
superfield strengths, reducing to a non-zero constant when these derivatives are equal
to zero.
We start with the definition based on the well-known zeta function regularization
procedure (see e.g. [109] for a review on this methodology):
ln det = −ζ
(0),
(4.335)
where the zeta function corresponding to the operator ˆ
K = e
t , with t is a proper
time (see [36] for the general review on the proper time methodology) is defined as
ζ(s) =
1
(s)
∞
0
dtt
s−1 K (t),
(4.336)
and K (t) is a functional trace of ˆ
K . Explicitly, it looks like
K (t) =
d
8 z lim
z→z
e
t
δ
8
(z − z
).
(4.337)
Alternatively, one can use straightforwardly the Schwinger-De Witt representation
which allows to write
ln det =
∞
0
dt
t
tre
t
≡
∞
0
dt
t
K (t).
(4.338)
Here we follow the methodology developed in [74] and use the notations adopted
there. In our case, for the effective action given by (4.334), the K (t) reads as
K (t) =
d
8 z lim
z→z
e
t (D
a D a +W
α D α + ¯
W ˙
α
¯
D
˙
α +||
2 )
δ
8
(z − z
).
(4.339)
Then, we use the Fourier representation of the complete delta function δ
8
(z − z
),
both for its bosonic and fermionic parts:
δ
8
(z − z
) =
d
4 k
(2π) 4 e
ik(x−x
)
d
4
e
i
α (θ α −θ
α ) e
i ¯
˙
α ( ¯
θ
˙
α − ¯
θ
˙
α )
.
(4.340)
To simplify the calculations, we factorize out the chiral matter superfields (which are
considered as constants within this calculation since we disregard all their derivatives), writing K (t) = e
t||
2
˜
K (t). Then, we introduce the key point of the methodology proposed in [74]: the K (t) (4.339), being a kernel of the operator, is actually
nothing other as the operator acting on the delta function, which further must be
applied to some other function. Since the delta function is expanded into the Fourier
series through (4.340), one can verify that when the operator whose kernel is given
by (4.339) acts on an arbitrary function, the following objects will emerge: