Eur. Phys. J. C (2016) 76 :142 Page 5 of 21 142
10
uus
( p) =−
i
3
ε
ila
ε
ijk
ε
lmn
ε
i
l
a
ε
i
j
k
ε
l
m
n
×
d
4
xe
ip·x
γ
5
γ
μ
CC
T
a
a
(−x )Cγ
ν
γ
5
×{2Tr[γ
μ
U
kk
(x )γ
ν
CU
T
jj
(x )C] Tr[γ
5
C
nn
(x )γ
5
CS
T
mm
(x )C]
+4Tr[γ
μ
S
kk
(x )γ
ν
CU
T
jj
(x )C] Tr[γ
5
C
nn
(x )γ
5
CU
T
mm
(x )C]
−4Tr[γ
μ
U
kk
(x )γ
ν
CU
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CS
T
jm
(x )C]
−4Tr[γ
μ
U
kk
(x )γ
ν
CS
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CU
T
jm
(x )C]
−4Tr[γ
μ
S
kk
(x )γ
ν
CU
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CU
T
jm
(x )C]},
(27)
10
uss
( p) =−
i
3
ε
ila
ε
ijk
ε
lmn
ε
i
l
a
ε
i
j
k
ε
l
m
n
×
d
4
xe
ip·x
γ
5
γ
μ
CC
T
a
a
(−x )Cγ
ν
γ
5
×{2Tr[γ
μ
S
kk
(x )γ
ν
CS
T
jj
(x )C] Tr[γ
5
C
nn
(x )γ
5
CU
T
mm
(x )C]
+4Tr[γ
μ
U
kk
(x )γ
ν
CS
T
jj
(x )C] Tr[γ
5
C
nn
(x )γ
5
CS
T
mm
(x )C]
−4Tr[γ
μ
S
kk
(x )γ
ν
CS
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CU
T
jm
(x )C]
−4Tr[γ
μ
S
kk
(x )γ
ν
CU
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CS
T
jm
(x )C]
−4Tr[γ
μ
U
kk
(x )γ
ν
CS
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CS
T
jm
(x )C]},
(28)
10
sss
( p) =−i ε
ila
ε
ijk
ε
lmn
ε
i
l
a
ε
i
j
k
ε
l
m
n
×
d
4
xe
ip·x
γ
5
γ
μ
CC
T
a
a
(−x )Cγ
ν
γ
5
×{2Tr[γ
μ
S
kk
(x )γ
ν
CS
T
jj
(x )C] Tr[γ
5
C
nn
(x )γ
5
CS
T
mm
(x )C]
−4Tr[γ
μ
S
kk
(x )γ
ν
CS
T
mj
(x )Cγ
5
C
nn
(x )γ
5
CS
T
jm
(x )C]}, (29)
where U
ij
(x), S
ij
(x), and C
ij
(x) are the full u, s, and c quark
propagators, respectively,
U
ij
(x ) =
iδ
ij
x
2π
2
x
4
−
δ
ij
¯qq
12
−
δ
ij
x
2
¯qg
s
σ Gq
192
−
ig
s
G
a
αβ
t
a
ij
(xσ
αβ
+ σ
αβ
x)
32π
2
x
2
−
1
8
¯q
j
σ
μν
q
i
σ
μν
+···,
S
ij
(x ) =
iδ
ij
x
2π
2
x
4
−
δ
ij
m
s
4π
2
x
2
−
δ
ij
¯ss
12
+
iδ
ij
xm
s
¯ss
48
−
δ
ij
x
2
¯sg
s
σ Gs
192
+
iδ
ij
x
2
xm
s
¯sg
s
σ Gs
1152
−
ig
s
G
a
αβ
t
a
ij
(xσ
αβ
+ σ
αβ
x)
32π
2
x
2
−
1
8
¯s
j
σ
μν
s
i
σ
μν
+···, (30)
C
ij
(x ) =
i
(2π)
4
d
4
ke
−ik·x
×
δ
ij
k − m
c
−
g
s
G
n
αβ
t
n
ij
4
σ
αβ
(k + m
c
) + (k + m
c
)σ
αβ
(k
2
− m
2
c
)
2
−
g
2
s
(t
a
t
b
)
ij
G
a
αβ
G
b
μν
( f
αβμν
+ f
αμβν
+ f
αμνβ
)
4(k
2
− m
2
c
)
5
+···
,
f
αβμν
= (k + m
c
)γ
α
(k + m
c
)γ
β
(k + m
c
)γ
μ
(k + m
c
)γ
ν
(k + m
c
),
(31)
and t
n
=
λ
n
2
, λ
n
is the Gell-Mann matrix [42], then compute
the integrals both in the coordinate and momentum spaces
to obtain the correlation functions
j
L
j
H
q
1
q
2
q
3
( p), therefore the
QCD spectral densities ρ
j
L
j
H
,1
q
1
q
2
q
3
(s) and ρ
j
L
j
H
,0
q
1
q
2
q
3
(s) at the quark
level through the dispersion relation,
Im
j
L
j
H
q
1
q
2
q
3
(s)
π
= p ρ
j
L
j
H
,1
q
1
q
2
q
3
(s) + m
c
ρ
j
L
j
H
,0
q
1
q
2
q
3
(s). (32)
The explicit expressions of ρ
j
L
j
H
,1
q
1
q
2
q
3
(s) and ρ
j
L
j
H
,0
q
1
q
2
q
3
(s) are
given in the appendix. In Eq. (30), we retain the term
¯q
j
σ
μν
q
i
(¯s
j
σ
μν
s
i
) comes from the Fierz re-arrangement
of q
i
¯q
j
(s
i
¯s
j
) to absorb the gluons emitted from other
quark lines to form ¯q
j
g
s
G
a
αβ
t
a
mn
σ
μν
q
i
(¯s
j
g
s
G
a
αβ
t
a
mn
σ
μν
s
i
)
to extract the mixed condensate ¯qg
s
σ Gq (¯sg
s
σ Gs). A
number of terms involving the mixed condensates ¯qg
s
σ Gq
and ¯sg
s
σ Gs appear and play an important role in the QCD
sum rules.
Once the analytical QCD spectral densities ρ
j
L
j
H
,1
q
1
q
2
q
3
(s)
and ρ
j
L
j
H
,0
q
1
q
2
q
3
(s) are obtained, we can take the quark–hadron
duality below the continuum thresholds s
0
and introduce the
weight function exp
−
s
T
2
to obtain the following QCD
sum rules:
2M
P,−
λ
−
P
2
exp
−
M
2
P,−
T
2
=
s
0
4m
2
c
ds
√
sρ
j
L
j
H
,1
q
1
q
2
q
3
(s) + m
c
ρ
j
L
j
H
,0
q
1
q
2
q
3
(s)
exp
−
s
T
2
,
(33)
2M
P,+
λ
+
P
2
exp
−
M
2
P,+
T
2
=
s
0
4m
2
c
ds
√
sρ
j
L
j
H
,1
q
1
q
2
q
3
(s) − m
c
ρ
j
L
j
H
,0
q
1
q
2
q
3
(s)
exp
−
s
T
2
,
(34)
where we take into account the contributions of the terms
D
0
, D
3
, D
5
, D
6
, D
8
, D
9
, and D
10
,
D
0
= perturbative terms,
D
3
∝¯qq, ¯ss,
D
5
∝¯qg
s
σ Gq, ¯sg
s
σ Gs,
D
6
∝¯qq
2
, ¯qq¯ss, ¯ss
2
,
D
8
∝¯qq¯qg
s
σ Gq, ¯ss¯qg
s
σ Gq, ¯qq¯sg
s
σ Gs,
¯ss¯sg
s
σ Gs,
D
9
∝¯qq
3
, ¯qq¯ss
2
, ¯qq
2
¯ss, ¯ss
3
,
D
10
∝¯qg
s
σ Gq
2
, ¯qg
s
σ Gq¯sg
s
σ Gs, ¯sg
s
σ Gs
2
.
(35)
In this article, we carry out the operator product expansion
to the vacuum condensates up to dimension 10, and assume
123