2.2.1. Bare Loop
The perturbative part is a double dispersion integral as follows:
(8)
Π
per
=
-
1
4
π
2
∫
d
s
′
∫
d
s
ρ
(
s
,
s
′
,
q
2
)
(
s
-
p
2
)
(
s
′
-
p
′
2
)
+
subtraction
terms
,
where
ρ
(
s
,
s
′
,
q
2
)
is called spectral density. We aim to evaluate the spectral density with the help of the bare loop diagram in Figure 1. One of the generic methods to calculate this bare loop integral is the Cutkosky method, where the quark propagators of Feynman integrals are replaced by the Dirac delta functions:
(9)
1
q
2
-
m
2
⟶
(
-
2
π
i
)
δ
(
q
2
-
m
2
)
.
Then, using the Cutkosky method we get spectral density as
(10)
ρ
(
s
,
s
′
,
q
2
)
=
2
m
c
(
b
)
N
c
(
-
4
m
c
(
b
)
2
+
q
2
+
s
-
s
′
)
3
λ
1
/
2
(
s
,
s
′
,
q
2
)
(
q
2
+
s
-
s
′
)
,
where
λ
(
a
,
b
,
c
)
=
a
2
+
b
2
+
c
2
-
2
a
c
-
2
b
c
-
2
a
b
and
N
c
=
3
is the color number. Note that, since three
δ
functions of integrand must vanish simultaneously, the physical regions in the
s
-
s
′
plane must satisfy the following inequality:
(11)
-
1
≤
f
(
s
,
s
′
)
=
s
(
q
2
+
s
-
s
′
)
λ
1
/
2
(
m
c
(
b
)
2
,
m
c
(
b
)
2
,
s
)
λ
1
/
2
(
s
,
s
′
,
q
2
)
≤
1
.
2.2.2. Two Gluon Condensates
We consider the two gluon condensate diagrams. Note that we do not include the heavy quarks condensate diagrams, since the heavy quark contributions are exactly reducible to the gluon condensate [26]. Now, as a nonperturbative part, we must add contributions coming from the gluon condensates presented in Figures 2(a), 2(b), 2(c), 2(d), 2(e), and 2(f).
These diagrams are calculated in the Fock-Schwinger fixed-point gauge [27–29], where the vacuum gluon field is as follows:
(12)
A
μ
a
(
k
′
)
=
-
i
2
(
2
π
)
4
G
ρ
μ
a
(
0
)
∂
∂
k
ρ
′
δ
(
4
)
(
k
′
)
,
where
k
′
is the gluon momentum and
A
μ
a
is the gluon field. In addition, the quark-gluon-quark vertex is used as
(13)
Γ
i
j
μ
a
=
i
g
γ
μ
(
λ
a
2
)
i
j
.
We come across the following integrals in calculating the gluon condensate contributions [30, 31]:
(14)
I
0
[
a
,
b
,
c
]
=
∫
d
4
k
(
2
π
)
4
m
m
×
1
×
(
[
(
p
′
+
k
)
2
-
m
c
(
b
)
2
]
c
[
k
2
-
m
c
(
b
)
2
]
a
[
(
p
+
k
)
2
-
m
c
(
b
)
2
]
b
m
m
m
m
m
m
×
[
(
p
′
+
k
)
2
-
m
c
(
b
)
2
]
c
)
-
1
,
I
μ
[
a
,
b
,
c
]
=
∫
d
4
k
(
2
π
)
4
m
m
×
k
μ
×
(
[
(
p
′
+
k
)
2
-
m
c
(
b
)
2
]
c
[
k
2
-
m
c
(
b
)
2
]
a
[
(
p
+
k
)
2
-
m
c
(
b
)
2
]
b
m
m
m
m
i
m
m
×
[
(
p
′
+
k
)
2
-
m
c
(
b
)
2
]
c
)
-
1
,
where
k
is the momentum of the spectator quark
m
c
(
b
)
.
These integrals are calculated by shifting to the Euclidean space-time and using the Schwinger representation for the Euclidean propagator:
(15)
1
(
k
2
+
m
2
)
n
=
1
Γ
(
n
)
∫
0
∞
d
α
α
n
-
1
e
-
α
(
k
2
+
m
2
)
.
This kind of expression is very easy for the Borel transformation since
(16)
B
p
^
2
(
M
2
)
e
-
α
p
2
=
δ
(
1
M
2
-
α
)
,
where
M
is Borel parameter.
We perform integration over the loop momentum and over the two parameters which we use in the exponential representation of propagators [28]. As a final operation we apply double Borel transformations to
p
2
and
p
′
2
. We get the Borel transformed form of the integrals in (14) as
(17)
I
^
0
(
a
,
b
,
c
)
=
i
(
-
1
)
a
+
b
+
c
16
π
2
Γ
(
a
)
Γ
(
b
)
Γ
(
c
)
×
(
M
1
2
)
2
-
a
-
b
(
M
2
2
)
2
-
a
-
c
×
U
0
(
a
+
b
+
c
-
4,1
-
c
-
b
)
,
I
^
0
μ
(
a
,
b
,
c
)
=
I
^
1
(
a
,
b
,
c
)
p
μ
+
I
^
2
(
a
,
b
,
c
)
p
μ
′
,
where
(18)
I
^
1
(
a
,
b
,
c
)
=
i
(
-
1
)
a
+
b
+
c
+
1
16
π
2
Γ
(
a
)
Γ
(
b
)
Γ
(
c
)
×
(
M
1
2
)
2
-
a
-
b
(
M
2
2
)
3
-
a
-
c
×
U
0
(
a
+
b
+
c
-
5,1
-
c
-
b
)
,
I
^
2
(
a
,
b
,
c
)
=
i
(
-
1
)
a
+
b
+
c
+
1
16
π
2
Γ
(
a
)
Γ
(
b
)
Γ
(
c
)
×
(
M
1
2
)
3
-
a
-
b
(
M
2
2
)
2
-
a
-
c
×
U
0
(
a
+
b
+
c
-
5,1
-
c
-
b
)
,
and
M
1
2
and
M
2
2
are the Borel parameters. The function
U
0
(
a
,
b
)
is as follows:
(19)
U
0
(
a
,
b
)
=
∫
0
∞
d
y
(
y
+
M
1
2
+
M
2
2
)
a
y
b
mmi
×
exp
[
-
B
-
1
y
-
B
0
-
B
1
y
]
,
where
(20)
B
-
1
=
1
M
1
2
M
2
2
[
m
c
(
b
)
2
(
M
1
4
+
M
2
4
)
+
M
2
2
M
1
2
(
2
m
c
(
b
)
2
-
q
2
)
]
,
B
0
=
2
m
c
(
b
)
2
M
1
2
M
2
2
[
M
1
2
+
M
2
2
]
,
B
1
=
m
c
(
b
)
2
M
1
2
M
2
2
,
where the circumflex of
I
^
in the equations is used for the results after the double Borel transformation. As a result of the lengthy calculations we obtain the following expressions for the two gluon condensate:
(21)
Π
nonper
=
-
2
π
α
s
〈
G
2
〉
3
m
c
(
b
)
m
c
(
b
)
2
k
×
{
3
m
c
(
b
)
2
[
2
[
I
^
0
(
1,4
,
1
)
+
I
^
0
(
4,1
,
1
)
mmikmkkkm
+
I
^
1
(
1,1
,
4
)
+
2
I
^
1
(
1,4
,
1
)
+
2
I
^
1
(
4,1
,
1
)
]
k
k
k
k
k
k
k
m
k
+
I
^
0
(
1,1
,
4
)
]
ikm
-
I
^
0
(
1,2
,
2
)
+
6
I
^
0
(
1,3
,
1
)
+
I
^
0
(
2,1
,
2
)
-
2
I
^
0
(
2,2
,
1
)
ikm
-
2
I
^
1
(
1,2
,
2
)
+
6
I
^
1
(
1,3
,
1
)
+
2
I
^
1
(
2,1
,
2
)
kim
-
6
I
^
1
(
2,2
,
1
)
+
6
I
^
1
(
3,1
,
1
)
-
3
I
^
2
(
1,1
,
3
)
kim
+
6
I
^
2
(
1,3
,
1
)
}
.
Now, we can compare
g
μ
ν
coefficient of (6) and (7). Our result related to the sum rules for the corresponding form factor is as follows:
(22)
F
(
q
2
)
=
e
m
S
2
/
M
2
e
m
V
2
/
M
′
2
f
V
f
S
m
V
m
S
×
[
1
4
π
2
∫
4
m
c
(
b
)
2
s
0
d
s
∫
4
m
c
(
b
)
2
s
0
′
d
s
′
ρ
(
s
,
s
′
,
q
2
)
θ
m
m
m
∫
4
m
c
(
b
)
2
s
0
1
4
π
2
×
[
1
-
(
f
(
s
,
s
′
)
)
2
]
e
-
s
/
M
2
e
-
s
′
/
M
′
2
+
Π
nonper
]
.
Note that finally we have to set
q
2
=
0
for the real photon.