Eur. Phys. J. C (2014) 74:3084 Page 3 of 12 3084
p
2
x
2
p
1
x
1
k
1
k
2
Fig. 1 Mueller–Navelet jet production process
dσ
dy
J
1
dy
J
2
d|
k
J
1
|d|
k
J
2
|dφ
J
1
dφ
J
2
=
1
(2π)
2
C
0
+
∞
n=1
2 cos(nφ) C
n
, (3)
where φ = φ
J
1
− φ
J
2
− π and the cross section C
0
and the
other coefficients C
n
are given by
C
n
≡
2π
0
dφ
J
1
2π
0
dφ
J
2
cos[n(φ
J
1
− φ
J
2
− π)]
×
dσ
dy
J
1
dy
J
2
d|
k
J
1
|d|
k
J
2
|dφ
J
1
dφ
J
2
=
x
J
1
x
J
2
|
k
J
1
||
k
J
2
|
+∞
−∞
dν
x
J
1
x
J
2
s
s
0
¯α
s
(μ
R
)χ(n,ν)
×α
2
s
(μ
R
)c
1
(n,ν,|
k
J
1
|, x
J
1
)c
2
(n,ν,|
k
J
2
|, x
J
2
)
×
1 + α
s
(μ
R
)
c
(1)
1
(n,ν,|
k
J
1
|, x
J
1
)
c
1
(n,ν,|
k
J
1
|, x
J
1
)
+
c
(1)
2
(n,ν,|
k
J
2
|, x
J
2
)
c
2
(n,ν,|
k
J
2
|, x
J
2
)
+¯α
2
s
(μ
R
) ln
x
J
1
x
J
2
s
s
0
×
¯χ(n,ν)+
β
0
8C
A
χ(n,ν)
− χ(n,ν)
+
10
3
+ ln
μ
4
R
k
2
J
1
k
2
J
2
. (4)
Here ¯α
s
(μ
R
) ≡ α
s
(μ
R
)N
c
/π, with N
c
the number of colors,
β
0
=
11
3
N
c
−
2
3
n
f
(5)
is the first coefficient of the QCD β-function,
χ
(
n,ν
)
= 2ψ
(
1
)
− ψ
n
2
+
1
2
+iν
−ψ
n
2
+
1
2
−iν
(6)
is the LO BFKL characteristic function,
c
1
(n,ν,|
k|, x) = 2
C
F
C
A
(
k
2
)
iν−1/2
⎛
⎝
C
A
C
F
f
g
(x,μ
F
)
+
a=q, ¯q
f
q
(x,μ
F
)
⎞
⎠
(7)
and
c
2
(n,ν,|
k|, x) =
c
1
(n,ν,|
k|, x)
∗
, (8)
are the LO jet vertices in the ν-representation. The remaining
objects are related with the NLO corrections of the BFKL
kernel ( ¯χ(n,ν)) and of the jet vertices in the small-cone
approximation (c
(1)
1,2
(n,ν,|
k
J
2
|, x
J
2
))intheν-representation.
Their expressions are given in Eqs. (23), (36) and (37) of
Ref. [38].
The representation (4) is valid both in the leading loga-
rithm approximation (LLA), which means resummation of
leading energy logarithms, all terms
(
α
s
ln
(
s
))
n
, and in the
next-to-leading approximation (NLA), which means resum-
mation of all terms α
s
(
α
s
ln
(
s
))
n
. The scale s
0
is artificial.
It is introduced in the BFKL approach at the time to perform
the Mellin transform from the s-space to the complex angu-
lar momentum plane and cancels in the full expression, up to
terms beyond the NLA.
Equation (4) represents just one of infinitely many repre-
sentations of the coefficients C
n
. One can consider alterna-
tive representations, aiming at catching some of the unknown
next-to-NLA corrections. Introducing for the sake of brevity
the definitions
Y = ln
x
J
1
x
J
2
s
|
k
J
1
||
k
J
2
|
, Y
0
= ln
s
0
|
k
J
1
||
k
J
2
|
,
the representations we will use in this work are the following:
123
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