Azimuthal angle correlations at large rapidities: revisiting density variation mechanism
Eur. Phys. J. C
Azimuthal angle correlations at large rapidities: revisiting density variation mechanism
E. Gotsman 2
E. Levin 0 1 2
0 Centro CientíficoTecnológico de Valparaíso , Avda. Espana 1680, Casilla 110V, Valparaiso , Chile
1 Departemento de Física, Universidad Técnica Federico Santa María , Valparaiso , Chile
2 Department of Particle Physics, Raymond and Beverly Sackler Faculty of Exact Science, School of Physics and Astronomy, Tel Aviv University , 69978 Tel Aviv , Israel
We discuss the angular correlation present in hadronhadron collisions at large rapidity difference (α¯ S y12 1). We find that in the CGC/saturation approach the largest contribution stems from the density variation mechanism. Our principal results are that the odd Fourier harmonics (v2n+1) decrease substantially as a function of y12, while the even harmonics (v2n ) increase considerably with the growth of y12. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Correlations in the momentum representation . . . . . 3 Single inclusive production in a one parton shower . . 3.1 BFKL Pomeron: the simplest approach for a one parton shower . . . . . . . . . . . . . . . . . . . 3.2 General estimates . . . . . . . . . . . . . . . . . 4 Double inclusive cross section for two parton shower production . . . . . . . . . . . . . . . . . . . . . . . 4.1 The simplest diagram . . . . . . . . . . . . . . . 4.2 The CGC/saturation approach . . . . . . . . . . 5 Azimuthal angle correlations . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . Appendix A: BFKL Pomeron in the mixed representation Appendix B: Calculation of the integrals for the contribution of the simplest diagram . . . . . . . . . . . . . Appendix C: Integration over dipole sizes in the CGC/saturation approach . . . . . . . . . . . . . . . Appendix D: Integration over dipole sizes in the angle correlation function . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

Contents
1 Introduction
In this paper we address the problem of the azimuthal
angle correlations of two hadrons with transverse momenta
pT 1 and pT 2 and rapidities y1 and y2, at large values of
y12 ≡ y1 − y1 1/α¯ S . Our main theoretical
assumption is that these correlations stem from interactions in
the initial state. We are aware that, unlike rapidity
correlations which at large rapidities originate from the initial
state interactions due to causality reasons [
1
], a
substantial part of these correlations could be due to the
interactions in the final states [
2–4
]. On the other hand, it
has been demonstrated that at small rapidity difference
α¯ S y12 < 1 the interactions in the initial state [
5–14
]
yield the value of the correlations, which describe the
major part of the experimentally observed correlations [
15–
37
].
In this paper we concentrate our efforts on calculating the
long range rapidity part of angular correlations with large
value of the rapidity difference y12. All previous
calculations assumed that α¯ S y12 < 1[
5–14
]. It turns out that in this
kinematic region, the main source of the azimuthal angle
correlations is the Bose–Einstein correlations of identical
gluons, corresponding to the interference diagram in the
production of two partonic showers. Intuitively, we expect that
the correlations in the process, where two different gluons
are produced from two different partonic showers, should
not depend on the difference of rapidities (y12), nor on the
values of y1 and y2. Using the AGK cutting rules [38]1 one
1 In the framework of perturbative QCD for the inclusive cross sections,
the AGK cutting rules were discussed and proven in Refs. [
44–52
].
However, in Ref. [
47
] it was shown that the AGK cutting rules are
violated for double inclusive production. This violation is intimately
related to the enhanced diagrams [
47–49,52
] and to the production of
gluon from the triple Pomeron vertex. It reflects the fact that different
(y1, pT1)
lines denote gluons. Figure 1c shows the example of a more
complicated structure of the partonic cascades, than the exchange of the BFKL
Pomeron. The color of the lines indicates the parton shower
can prove that the two gluon correlations can be calculated
using the Mueller diagrams [39] of Fig. 1.
The diagrams of Fig. 1 lead to correlations which do not
depend on y1 and y2, but only for α¯ S y12 1. For large y12
the contributions of Fig. 1 decrease. The main goal of this
paper is to find the contributions which survive at large y12
(α¯ S y12 1).
At large y12, we have to take into account the emission
gluons, with rapidities y2 < yi < y1, which transform
the Mueller diagram of Fig. 1b to the more general
diagrams of Fig. 2. The general feature of Fig. 1b is that the
lower Pomerons carry momenta QT + p12 and −QT − p12
with p12 = pT 1 − pT 2. QT denotes the momentum along
the BFKL Pomeron. After integration over QT , we obtain
p12 ∼ 1/ Rh , where Rh is the size of the target (projectile),
which has a nonperturbative origin. Roughly speaking, the
correlation function turns out to be proportional to G ( p12),
where G denotes the nonperturbative form factor of the
target or projectile [
12
]. This conclusion stems from the value of
the typical QT for the BFKL Pomeron, which is determined
by the size of the largest dipoles in the Pomeron. Figure 2
does not have these features. We will show that the azimuthal
angle correlations originate from the integration over QT (see
Fig. 2), due to the structure of the vertices of emission of the
Footnote 1 continued
cuts of the triple BFKL Pomeron vertex with the produced gluon lead
to different contributions. We do not consider such diagrams.
gluons with pT 1 and pT 2, which have contributions
proportional to ( pT 1 · QT )n ( pT 2 · QT )n . Recall that these kinds
of vertices are the only possibilities to obtain angular
correlations in the classical Regge analysis [
53
]. This mechanism
for azimuthal angular correlations was suggested in Ref. [
54
]
(see also Refs. [
10, 55–57
]), and in the review of Ref. [57] it
was called the density variation mechanism.
The paper is organized as follows. In the next section
we discuss the contribution of the diagram of Fig. 2 in the
momentum representation. In the remainder of the paper,
we will use the mixed representation: the dipole sizes and
momentum transferred (QT ), which will be introduced in
Sect. 3 and appendix A. Section 4 is devoted to the
discussion of the single inclusive production in the Color Glass
Condensate (CGC)/saturation approach. The double
inclusive production is considered in Sect. 4, in which the rapidity
dependence of the master diagram of Fig. 2 will be
calculated. In Sect. 5, we estimate the angular correlation function
and Fourier harmonics vn , and we present our prediction for
the dependence of vn on the difference of rapidities (y12). In
Sect. 6 we draw our conclusions and outline problems for
future investigation.
2 Correlations in the momentum representation
The double inclusive cross section of Fig. 2 takes the
following form:
d2σ
(Fig. 2)
dy1d2 pT 1 dy2d2 pT 2
2CF αS 2
= (2π )2
−QT )2 × N QT φHG −kT + QT , kT ; Y − y1
where φHG kT , −k + QT , as well as all other functions φ
of this type, are the correlation functions, which at QT = 0
give the probability to find a gluon with transverse
momentum kT in the hadron (nucleus) of the projectile (target).
φ kT , −k + QT ; kT , −kT + QT describes the interaction
of two gluons with momenta kT and kT , which scatter at
momentum transferred QT . N QT is a pure
phenomenological form factor that describes the probability to find two
Pomerons in the projectile or target, with transferred moment
QT and −QT . CF = Nc2 − 1 /2Nc where Nc is the
number of colors. The Lipatov vertex μ (kT , pT 1) has the
following form:
1
μ (kT , pT 1) = pT 1
2
Using Eq. (2.2) we obtain
2 ν −k + QT , pT 1
ν k − QT − QT , pT 1
kT2 pT 1,μ − kT,μ pT 1 .
2
(2.2)
−k + QT
kT − pT 1 − QT − QT
1
= pT 1
2
1
= pT 2
2
2
2
2
2
+
−k + pT 1 − QT
kT − QT − QT
2 μ −kT − pT 2 + QT , pT 2
μ kT − pT 2 − QT , pT 2
−kT − pT 2 + QT
which in Eq. (2.1) is absorbed in the phenomenological form
factors N (QT ) and N (QT ). Therefore, the typical QT and
QT turn out to be of the order of the soft scale μsoft, which
is much smaller that the other typical momenta in Eq. (2.1),
assuming that PT 1 and PT 2 are larger than μsoft. Introducing
(2.4)
2
μsoft =
d2 Q
T N QT
(2 π )2
we can neglect QT and QT in the BFKL Pomeron Green
functions and rewrite Eq. (2.1) in the form
d2kT d2kT
(2π )2 (2π )2 d(22πQ)T2 kT2 (kT
−QT )2 × φHG
−kT , kT ; Y − y1 × φHG kT
−QT , −kT + QT ; Y − y1
× ν − k
+QT , pT 1
ν k − QT , pT 1 × φ − kT ,
−kT + QT ; kt + pT 2, −kT − p2T − QT ; y12
×φ − kT + pT 1, kT − pT 1 − QT ; kT + QT , kT
− pT 2; y12 × φHG kT + QT , −kT − QT ; y2
G
×φH
−kT + QT , kT ; y2 ×
μ(−kT
− pT 2, pT 2) μ kT − pT 2, pT 2 ,
(2.5)
2
2
2
1
= pT 1
2
1
= pT 2
2
with Eq. (2.3), which takes the following form:
2 ν −k + QT , pT 1
ν k − QT , pT 1
−k + QT
kT − pT 1 − QT
+
−k + pT 1 − QT
y1 and y2, and its expression has the following form:
The most economical way of calculating the diagram
of Fig. 2, is to use the mixed representation of the BFKL
Pomeron Green function, G r , R, QT , Y , where r and R
are the sizes of two interacting dipoles, QT denotes the
momentum transferred by the Pomeron, and Y the
rapidity between the two dipoles. This Green function is well
known [
42,43
], and for the completeness of presentation we
discuss it in Appendix A, referring to Refs. [
42,43,58–60
]
for all details.
3 Single inclusive production in a one parton shower
3.1 BFKL Pomeron: the simplest approach for a one parton
shower
The single inclusive cross section resulting from the one
BFKL Pomeron is well known, and it is equal to
d2σ 2CF αS
dyd2 pT = (2π)2
(d22πk)T2 φHG kT , QT = 0; Y − y
×φHG kT − pT , QT = 0; y
ν kT , pT
ν −kT , pT .
The relation between the parton densities φ and the Green
function of the BFKL Pomeron has been given in Ref. [
46
]:
N BFKL (r, r1; y, QT = 0)
αS
= 2
d2kT
1 − eikT ·r
φHG kT , QT = 0; y
kT2
(3.2)
where N BFKL (r, r1; Y ) is given by Eq. (A.1) or by Eq. (A.9),
in the high energy limit. Equation (3.2) can be rewritten as
follows:
We have
d2r eikT ·r ∇r2 N BFKL (r, r1; y, QT = 0) . (3.3)
kT2
kT − pT
pT2
2
.
ν kT , pT
ν −kT , pT
=
Substituting Eq. (3.2) and also Eq. (3.4) into Eq. (3.1) we
obtain [
46
]
d2σ 8CF 1
dy d2 pT = αS (2π )2 pT2
× (r, r1; Y − y, QT = 0) ∇r2 NtBrFKL
d2r ei pT ·r ∇r2 NpBrFKL
× (r, r2; y, QT = 0)
where Npr and Ntr denote the probability to find a dipole in
the projectile and target, respectively. r1 and r2 are the typical
dipoles sizes in the projectile and target.
As can be seen from Eq. (2.1) we need to generalize
Eq. (3.5) for the case QT = 0. Equation (3.1) has to be
replaced by
d2σ
dyd2 pT (QT = 0)
2CF αS
= (2π )2
(d22πk)T2 φHG kT , QT , Y − y
× φHG kT
− pT , QT ; y
ν kT , pT
ν −kT + QT , pT .
1
= 2
1
pT2
2
Taking into account Eq. (3.2) for QT = 0 and
ν kT , pT
ν −kT + QT , pT
2
−y, QT ) NtBrFKL (r, r2; y, QT ) .
d2r ei pT ·r NpBrFKL (r, r1; Y
(3.8)
3.2 General estimates
It should be stressed that the single inclusive production has
the form of Eqs. (3.5) and (3.8) for the general structure of the
single parton shower, as was shown in Ref. [
46
]. For example,
for the process shown in Fig. 1c. We need only to substitute
NtGr (r, r2; y, QT ) for 2NtBrFKL (r, r2; y, QT ) where
2NtBrFKL (r, r2; y, QT ) →
NtGr (r, r2; y, QT )
= 2Ntr (r, r2; y, QT ) −
d2 QT Ntr r, r2; y, QT
−QT Ntr r, r2; y, QT ;
Ntr (r, r2; y, QT ) is a solution to the nonlinear evolution
equation. For the case of inclusive production, we can
considerably simplify the estimates noting that
r2 Qs2(y) 1
∇r2 Ntr (r, r2; y, QT ) −−−−−−−→ NtBrFKL (r, r2; y, QT )
r2 Qs2(y) 1
∇r2 Ntr (r, r2; y, QT ) −−−−−−−→ 0,
where Qs (y) denotes the saturation momentum.
In other words, the main contribution to inclusive produc
tion comes from the vicinity of the saturation scale, where
r 2 Q2
s ≈ 1. Fortunately, the behavior of N in this kinematic
region is determined by the linear BFKL evolution
equation [
62–67
] and has the following form [68]:
Ntr (r, r2; y, QT = 0) ∝
r 2 Qs2(y)
1−γcr
with Q2
s
= (1/r22) exp
= (1/r22)eκ y
1
ω γ = 2 + i ν = γcr y
where γcr = 0.37.
From Eq. (A.8) we see that, for QT = 0, the scattering
amplitude decreases at Q4T r 2 r22 1. Therefore, we need
to consider rather small values of QT : Q4T r 2 r 2
2 ≤ 1. The
product of vertices that determines the amplitude has two
terms (see Eq. (A.5)) which are proportional to r 2/r22 iν
and to Q4T r 2 r22 I ν . Therefore, the maximum of ∇r2 N can
be reached if r 2/r22eκ y ∼ 1 and Q4T r 2 r22eκ y ∼ 1 and the
amplitude then has the following form:
Ntr (r, r2; y, QT ) ∝ c1
+ c2 Q4T r 2 r22 eκ y 1−γcr .
r 2 eκ y
r22
The first term does not depend on QT and, therefore, the
upper limit of the integral over QT , goes up to QmTax 2 ≈
1/(r r2). The second term, both for Q2T r r2 < e− 21 κ y and
for Q2T r r2 > e− 21 κ y , turns out to be small. Indeed, in the
first region the amplitude is small, while in the second region
we are deep in the saturation domain where ∇r2 N → 0.
Hence, we expect that in the integral over QT , the first term
gives a larger contribution than the second term, and we will
only keep this contribution in our estimates.
4 Double inclusive cross section for two parton shower
production
4.1 The simplest diagram
In this section we calculate the simplest diagram of Fig. 2.
We need to integrate the product of two BFKL Pomerons
over QT (see Eq. (2.5)):
From Eq. (2.5) in the momentum representation, we see that
r1 = r1 (r2 = r2), however, they are close to each other,
being determined by the same momentum kT . We assume that
pT 1 < kT , since kT ∼ Qs (Y − y1) μsoft. Considering
r1 ≈ r1 r2 ≈ r2 we will show that in the integral over
QT , the typical QT ∼ 1/r2. In other words, the dependence
of QT is determined by the largest of interacting dipoles.
From Eq. (A.8) we see that, for large QT , when r12 Q2
T
1 and r22 Q2T 1, the integrand is proportional to 1/Q4T
and converges. The main region of interest is r22 Q2T 1
and r12 Q2T 1. In this kinematic region for the vertices
Vν1 r1, QT and Vν2 r1, QT , we can use Eq. (A.6), while
the conjugated vertices are still in the regime of Eq. (A.8).
Eq. (4.1) then takes the form
I = 26i(ν1+ν2)
−26ν1 ν2 π
The appearance of the pole ν1 = −ν2 indicates that the
contribution from this kinematic region is large.
Closing the contour of integration on ν2 over the pole, we
obtain
Actually, the double inclusive cross section depends on
∇2 N as we argued in the previous section. Repeating the
procedure for
Using the method of steepest descent, to integrate over ν1,
we obtain the following contribution:
d2σ
d y1d2 pT 1 dy2d2 pT 2
2
In Eq. (4.8) we have neglected the terms which are pro
portional to Q2T (see Eq. (2.5)), since, as we have argued, the
typical QT are small, and because these terms do not lead to
additional correlations in the azimuthal angles. In Appendix
(4.4)
4.2 The CGC/saturation approach
B we calculate this integral and obtain the final expression
for the double inclusive cross section:
The integral over kT in Eq. (B.6) has an infrared singularity
with a cutoff at pT 2, since we assume that pT 2 is the smallest
momentum. This reflects the principal feature of the BFKL
Pomeron parton cascade, which has diffusion, both in the
region of small and large transverse momenta. On the other
hand, we know that the CGC/saturation approach suppressed
the diffusion in the small momenta [
44
], providing the
natural cutoff for the infrared divergency. We expect that such a
cutoff will be the value of the smallest saturation momenta:
Qs (Y − y1) or Qs (y2), which will replace one of the pT2 2 in
the dominator of Eq. (4.9). Therefore, we anticipate that for
a realistic structure of the one parton shower cascade, (see
Fig. 1c for example), the contribution for the double inclusive
cross section will be different.
We need to specify the behavior of the scattering ampli
tude in the vicinity of the saturation scale. We have discussed
the basic formulas [
68
] of Eq. (3.11), but for integration over
the dipole sizes we need to know the size of this region. The
scattering amplitude can be written in the form
where ω(γ , 0) is given by Eq. (A.2), replacing 21 + i ν ≡ γ
and ξ = ln r12/r22 . The saturation scale is determined by
the line on which the amplitude is a constant (C), of the order
one. This leads to the following equation for the saturation
scale [
62,68
]:
ω (γcr , 0) Y − (1 − γcr )ξs = 0;
ωγ (γ , 0) Y − ξs = 0,
(4.11)
which results in the value of γcr given by the equation
ω (γcr , 0)
= ωγ (γ , 0)
and gives γcr = 0.37, with the equation for the saturation
momentum:
ξs ≡ ln Qs2 r22
= κY = 1ω −(γγcrc)r Y.
(4.12)
(4.13)
Fig. 3 a − N (τ ) = − ∇2
N (τ ) = − 4 τ 1 d τ ddτ N (τ )
τ dτ
versus τ for the behavior of the
scattering amplitude deep in the
saturation domain[
72
]. b The
example of a more complicated
structure of the partonic
cascades than the exchange of
the BFKL Pomeron, which are
shown in Fig. 2. The color of the
lines indicates the parton shower
0.8
Q’
(y1, pT1 )
(y2, pT2 )
Expanding the phase ω (γ , 0) Y − (1 − γ )ξ in the vicinity
ξ = ξ − ξs and γ = γ − γcr we obtain
N (r1, r2; Y )
= C
−i∞
+i∞ dγ
2π
= r12 Qs2 1−γcr C
r12 Qs2 1−γcr
π e− (4DξY)2 .
DY
d γ e 21 ωγγ (γ,0)Y ( γ )2+ γ ξ
2πi
(4.14)
At first sight, Eq. (4.14) shows that the amplitude has a
maximum at τ = r12 Qs2 = 1. However, this is not correct.
Equation (4.14) gives the correct behavior for τ < 1, while
for τ > 1 we need to take into account the interaction of the
BFKL Pomerons and the nonlinear evolution, generated by
these interactions. The general result of this evolution is the
fact that the amplitude depends on one variable [
69–72
] τ ,
i.e. N (τ ) (as it shows geometric scaling behavior). The peak
at τ = 1 appears in
∇r21 N (r1, r2; Y ) = 4 Qs2(Y )
1 d d
τ
τ dτ dτ
N (τ ) .
(4.15)
From Eq. (4.15) we can conclude that the width of the
distribution in r12 is of the order of Qs2, but it depends crucially
on the model for the Pomeron interaction. In Fig. 3a we plot
this value for the behavior of the scattering amplitude deep
in the saturation domain (see Ref. [
72
]).
This approach is not correct for τ → 1 and −∇2 N = 1.58
at τ = 1, but it starts to be small at τ > 2, which could be
large enough to trust the formulas of Ref. [
72
]. At least such a
conclusion can be justified considering the fit of the DIS data
in the saturation model of Refs. [
73,74
], which is based on
the idea of Ref. [75], and which has the correct behavior of
the scattering amplitude, both deep in the saturation domain,
and near τ = 1. Hence, we expect that ∇2 N decreases faster
than we can see from Eq. (4.14). Bearing these conclusions
in mind, we will calculate the contribution of Fig. 2, keeping
all N in Eq. (4.8) in the vicinities of the saturation scales, by
replacing 0∞ dτ (−∇2 N ) = − 01 dτ (−∇2 N ).
We will show in the following that we cannot integrate
over the dipole sizes, so that all six Pomerons will be in the
vicinity of the saturation scale. At least two of the Pomerons
occur either deep in the saturation domain, or in the
perturbative QCD region. We believe that the largest contribution
stems from the exchange of two Pomerons between rapidities
y1 and y2 (see Fig. 3b), which are in the perturbative QCD
region. Unfortunately, we cannot use the AGK cutting rules
[
38
], which state that these Pomerons will not be affected
by the Pomeron interaction, and the contributions of these
interactions (see the red Pomeron in Fig. 3b) are canceled.
Indeed, it has been proven that for the double inclusive
production [
47
] they are not applicable in perturbative QCD. On
the other hand, these Pomerons carry transverse momentum
QT , unlike the others in the diagram, which is larger than
the saturation scale Qs (y2); hence, their contributions are
suppressed in comparison with the other Pomerons in Fig. 2.
In addition our choice leads to the natural matching with the
region α¯ S y12 < 1.
The integration over QT will produce the same result as
Eq. (4.7), as in the previous section. In Appendix C we
discuss making estimates for the integrals over the dipole sizes
which lead for pT 1 Qs2 (y1) to the following cross
section:
d2σ
dy1d2 pT 1 (QT = 0; Eq. (3.5))
8CF 1
= αS (2π )2 p2
T
×∇r2 NtBrFKL (r, r2; y, QT = 0)
= αS (2π )2 p12 C2 (4γ¯ 2)2 exp −
8CF
T
d2r ei pT ·r ∇r2 NpBrFKL (r, r1; Y − y, QT = 0)
ln2 Qs2 (Y − y) /Q2 (y)
4D y
pp W 13 TeV
Fig. 4 The ratio of Eq. (4.17) at W=13 TeV versus y12, assuming that
the experiment has a symmetric pattern with Y − y1 = y2 = 21 (Y − y12).
The dotted line in a is for the estimates for the y12 dependence of the
Bose–Einstein contribution at small y12 [
11,76
]. a and b The estimates
in the leading order of perturbative QCD with α¯ S = 0.25. In c we take
BFKL = 0.25 and Qs2(Y ) ∝ exp (λY ) with λ = 0.25. These numbers
correspond to the BFKL phenomenology
×
× exp
× exp
One can see that Eq. (4.17) demonstrates the additional
suppression due to the infrared cutoff at Qs (y2) instead of at pT 2,
as taken in the calculation of the simplest diagram. The factor
exp (2 BFKL y12) reflects the fact that two BFKL Pomerons
between rapidities y1 and y2 are taken in the perturbative
QCD region. It should be stressed that we can only trust
our estimates for values of y12 at which the exchange of the
BFKL Pomeron with rapidity y12 give a contribution smaller
than C. This condition means that
1
(2 D y12)
e2 BFKL y12 < C.
(4.18)
Taking BFKL = 0.25 and Qs2(Y ) ∝ exp (λY ) with λ =
0.25 (these values correspond to the BFKL phenomenology)
we see that the l.h.s. of Eq. (4.18) is smaller than 0.15 for
y12 ≤ 7. Therefore, we can trust our estimates shown in
Fig. 4 for C > 0.15. We take C = 0.3, which leads to
the contribution of the shadowing corrections of the order of
30%.
The two last factors in Eq. (4.17) stem from the perturbative QCD nature of two Pomerons in Eq. (C.8) (see Eq. (4.16)).
In Fig. 4 we plot the ratio R as a function of y12 for y12 ≤ 7
(see Eq. (4.18). One can see that the ratio increases for large
y12.
5 Azimuthal angle correlations
The azimuthal angle correlations stem from terms (QT · ri )n
in the vertices (see Eqs. (A.5), (A.6)). Indeed, after
integrating over ri these terms transform to expressions of the
folm1 m2
, which lead
lowing type [
54
]: QT · pT 1 QT · pT 2
to the term ( pT 1 · pT 2)m . We have illustrated in Eqs. (A.5)
and (A.6) how these originate from the general form of
the BFKL Pomeron vertices in the coordinate
representation. From Eqs. (A.5) and (A.6) only terms proportional to
n
QT · ri with even n appear in the expansion. Therefore,
the azimuthal angle (φ) correlation function contains only
terms cos2n (φ), and it is invariant with respect to φ → π −φ.
In other words, vn with odd n turn out to be zero. Hence,
we have the first prediction: the value vn with odd n should
decrease with y12, and their dependence should follow the
dotted lines in Fig. 4a.
We return to Eq. (4.1) and integrate over QT ,
collecting terms that depend on the angles between QT and ri ,
which we have neglected in the previous section. As we have
learned, the typical values of QT ∝ 1/r2 ∼ 1/r2 where r2
and r2 are larger than r1 and r1. In other words, we showed
that the main contributions stem from the kinematic regions
r12 Qs2 (Y − y1) ∼ 1 ( r12 Qs2 (Y − y1) ∼ 1) and r22 Qs2 (y2) ∼
1 ( r22 Qs2 (y2) ∼ 1). Assuming that Qs (Y − y1) Qs (y2)
we conclude that r1(r1) r2(r2). The typical QT is
determined by the largest dipoles and, therefore, we expect QT ≈
1/r2(1/r2), as has been demonstrated above. Bearing these
estimates in mind, we can replace vertices Vν1 r1, QT
and Vν2 r1, QT in Eq. (4.1) by Eq. (A.6) in which we
put QT = 1/r2 and QT = 1/r2, respectively. Taking into
account that r1/r2 1(r1/r2 1), we obtain
Vν1 r1, QT
Vν2 r1, QT
1
+ 28
1
+ 28
QT · r1
r 2 −iν2
26
QT · r1
4
−
4
At first sight Eq. (5.1) should enter two angles between QT
and r1 and r1, respectively. However, in the integrand for
integration over ri (see Eq. (B.1)) it depends only on one
vector pT 1. Therefore, after integration over all angles, we
find that the angle φ in Eq. (5.1) is the angle between QT
and pT 1.
For vertices Vν∗1 r2, QT and Vν∗2 r2, QT in Eq. (4.1)
we use Eq. (A.8). Finally, we need to evaluate the integral
IQ = −16 ν1 ν2
QT d QT Vν1 r1, QT Vν2 r1, QT
× Q2T −i(ν1+ν2) cos2 Q212TQrT22 · r2 cos QT · r2
r1 =r1=1/Qs(Y−y1)
Eq. (5.1)
(5.2)
with better accuracy that we did in Sect. 5.1, keeping the
dependence on the angle between QT and r2. Note that
the factor cos QT · r2 comes from exp i QT · r2 in
Eq. (4.8). Taking this integral we substitute for the terms
in parentheses in Eq. (5.1), QT  = 1/r22(1/r22).
The integral is equal to
IQ = 26i(ν1+ν2)
−27ν1 ν2
where n = QT /QT , and φ2 is the angle between n and
n2 = r2/r2. In Eq. (5.3) the terms in (. . . ) (. . . ) stem from
the expansion with respect to r12/r22 1. However, for the
terms in {. . . } there are no such small parameters, and we
expand the function of φ2 in a Fourier series.
Integrating over n one obtains
3 r14
+ 212 r 4 (n1 · n2)4 +
2
2
n1 · n2
4
(5.4)
where ϕ is the angle between pT 1 and pT 2. vn is calculated
from vn,n ( pT 1, pT 2),
1. vn ( pT ) =
2. vn ( pT ) =
vn,n ( pT , pT );
vn,n pT , pTRef
vn,n pTRef , pTRef
.
Equations (5.8)1 and (5.8)2 depict two methods of how
the values of vn have been extracted from the
experimentally measured vn,n ( pT 1, pT 2), where pRef denotes
T
the momentum of the reference trigger. These two
definitions are equivalent if vn,n ( pT 1, pT 2) can be factorized as
vn,n ( pT 1, pT 2) = vn ( pT 1) vn ( pT 2). In this paper we use
the definition in Eq. (5.8)1.
Introducing the angular correlation function as
C ( pT , φ) ≡
we obtain
vn,n =
In Eq. (4.17) we have calculated the part of C ( pT , φ)
which does not depend on φ, which coincides with C ( pT ,
φ = 0) = R of Eq. (4.17) for Qs (Y − y1) Qs (y2). To
calculate the contribution to C , which depends on φ, we need
to take the separate integrals over ν1 and ν2, since the terms,
which are proportional to cos2 (φ) and cos4 (φ) do not have
a pole at ν1 = −ν2 (see Eq. (5.5)). These integrations lead
to the following extra factor in C ( pT , φ) − C ( pT , φ = 0):
C ( pT , φ) − C ( pT , φ = 0)
pT2 pT2
∝ R Qs2 (Y − y1) Qs2 (Y − y1)
R = 2 ξ 2
1
(2 D y12)3 exp −2ξ 2/ (4 D y12)
C ( pT , φ = 0) ;
(5.11)
where ξ = ln Qs2 (Y − y1) /Qs2 (y2) . We took factors
proportional to pT from the expression for A (kT , pT 1) and
A(4) (kT , pT 1) putting pT 1 = pT 2 = pT . To find the
final correlation function and v2,2 and v4,4, we need to
collect all numerical factors that come from A (kT , pT 1),
A(4) (kT , pT 1) and Eq. (5.6), and to integrate over φ, as given
in Eq. (5.9).
Note that in the case of symmetric kinematics, where
Y − y1 = y2 = 21 (Y − y12), ξ = 0 and Eq. (D.7
vanishes. In this case, we have to use Eq. (A.5) instead
of Eq. (A.6), keeping track of the corrections, which are
(5.8)
(5.9)
(5.10)
proportional to νi . As a result, we can consider ξ = 0
in Eq. (D.7), but we need to replace the factor ξ 2 by
1.
at α¯ S y12
Equations (4.17) and (D.7) contain numerical
uncertainties, which stem both from the values of the soft
parameters μ˜ soft and μsoft, as well as the values of the saturation
scale at low energies, and from the integration in Eqs. (C.3)
and (C.5), which were taken neglecting contribution from
the region τ < 1. On the other hand, the contribution to
the double inclusive cross sections of the diagram of Fig. 2
1 coincides with the contribution of Fig. 1b,
d2σ (Fig. 2)
dy1 dy2d2 pT 1d2 pT 2
α¯ S y12→ 1
−−−−−−→
d2σ (Fig. 1b)
dy1 dy2d2 pT 1d2 pT 2
(5.12)
Therefore, to obtain the realistic estimate we use the
following procedure of matching:
where v2 ( pT = 5 GeV) Fig. 1b and v4 ( pT = 5 GeV) Fig. 1b
are taken from Ref. [
14
] where the estimates were performed
based on the model for soft interaction which describes all
features of the soft interaction at high energy and provides
an interface with the hard processes.
Figure 5 shows the pT and y dependence of the v2 and
v4 using Eq. (D.8) for normalization. In addition we take
BFKL = 0.25 and Qs2(y) ∝ exp (λ y) with λ = 0.25. These
values correspond to the BFKL Pomeron phenomenology.
We believe that this figure illustrates the scale of rapidity
dependence and will be instructive for future experimental
observations.
6 Conclusions
In this paper we generalize the interference diagram that
described the Bose–Einstein correlation for small rapidity
difference α¯ S y12 1, to include the emission of the
gluons with rapidities (yi ) between y1 and y2 (y1 , yi < y2).
We calculate the resulting diagram in the CGC/saturation
approach and make two observations, which we consider as
the main result of this paper. The first one is a substantial
decrease of the odd Fourier harmonics v2n+1 as a function of
the rapidity difference y12 ( see Fig. 4c). The second result is
that even Fourier harmonics v2n have a rather strong
dependence on y12, showing a considerable increase in the region
of large y12 (see Fig. 5). We believe that our calculations,
which have been performed both for the simplest diagrams
Fig. 5 vn versus pT (a, c) and versus y (b, d) at W = 13 TeV
assuming that the experiment has a symmetric pattern with Y − y1 = y2 =
21 (Y − y12). In all these figures we use Eq. (D.8) for normalization and
and for the CGC/saturation approach, will be instructive for
further development of the approach especially in the part
that is related to the integration of the momenta transferred
by the BFKL Pomerons.
We demonstrated in this paper the general origin of the
density variation mechanism, whose nature does not depend
on the technique that has been used. This mechanism has
to be taken into account, since it leads to the values of the
Fourier harmonics that are large and of the order of vn, which
have been observed experimentally.
We hope that the paper will be useful in the
clarification of the origin of the angular correlation, especially for
hadron–hadron scattering at high energy. We firmly believe
that the experimental observation of both phenomena, the
sharp decrease of vn with odd n and the substantial increase
of vn with even n as a function of y12, will be a strong
argument for the CGC/saturation nature of the angular
correlations.
Acknowledgements We thank our colleagues at Tel Aviv university
and UTFSM for encouraging discussions. Our special thanks go to
Carlos Contreras, Alex Kovner and Misha Lublinsky for elucidating
discussions on the subject of this paper. This research was supported by
the BSF Grant 2012124, by Proyecto Basal FB 0821(Chile), Fondecyt
(Chile) Grant 1140842, and by CONICYT grant PIA ACT1406.
pp W 13 TeV
0
1
2
3
4
5
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: BFKL Pomeron in the mixed representation
In this appendix we discuss the BFKL Pomeron Green
function G r , R, QT , Y in the mixed representation, where r
and R are the sizes of two interacting dipoles, QT denotes
the momentum transferred by the Pomeron, and Y the
rapidity between the two dipoles. This Green function has the
following form [
42,43,58–60
]:
G r , R, QT ; Y
r R
∞
= 16 n=−∞ −∞
×Vν,n r , QT
∞
dν
1
ν2 + 41 (n − 1)2
ν2 + 41 (n + 1)
Vν∗,n R, QT eω(ν,n) Y
(A.1)
where
ω (ν, n) = 2 α¯ SRe ψ
ω (ν, 0) = 2 α¯ SRe ψ
where ψ (z) is the Euler ψ function (see Ref. [
61
]
formulas 8.36) and BFKL = α¯ S4 ln 2, D = α¯ S14ζ (3), ξ =
ln r12/r22 .
Each term in Eq. (A.1) has a very simple structure, being
the typical contribution of the Reggepole exchange: the
product of two vertices and Reggepole propagator. From
Eq. (A.2) one can see that at large Y the main contribution
comes from the term with n = 0, and in what follows we will
concentrate on this particular term. The vertices with n = 0
have been determined in Refs. [
42,43,58–60
], and they have
an elegant form in the complex number representation for the
point on the two dimensional plane, viz.,
For r (x , y) : ρ = x + i y; ρ∗ = x − i y;
For QT (Qx , Q y ) : q = Qx + i Q y ; q∗ = Qx − i Q y .
(A.3)
Using this notation the vertices have the following structure:
Vν r , QT
=
Q2T iν
× J−iν
− Jiν
At QT → 0 this vertex takes the form
QT r 1 r2 −iν
26iν Vν r, QT −−−−−→ 26
⎛ (ν + i) 8(QT · r)4 − 8(QT · r)2 Q2T r2 + 5 21 Q4T r4 + (2i + ν)Q4T r4)
× ⎝ 642(ν + 2i)(1 − iν)2
+
i(ν + i) (2(QT · r)2 − Q2T r2
32(1 − iν)2
⎞
+ 1⎠ + Q2 iν Q2r2 iν
26
⎛ (ν − 2i) 8(QT · r)4 − 8(QT · r)2 Q2T r2 + 5 21 Q4T r4 )
× ⎝ 212((2 + iν)(1 + iν))2
+
2(1 + iν) (2(QT · r)2 − Q2T r2
26(1 + iν)2
For small values of ν (which are related to the region of
large α¯ SY 1), Eq. (A.5) can be simplified and reduced to
the form
26iν Vν r , QT
QT r 1
−−−−−→
−
−
−
Using
2(QT · r )2 − Q2T r 2
25
dν
1
ν2 + 1/4 2
Y 1;ν 1
−−−−−−→ 2 r1 r2
−α¯ S14ζ (3)ν2 Y
eω(ν,0)Y 2
dν exp ((α¯ S 4 ln 2
r22 iν
r12
= r1 r2
D2πY exp
ξ 2
BFKLY − 4DY
r22 iν
r12
(A.6)
(A.7)
4 i ν
.
QT r
(A.8)
(A.9)
(A.5)
where
BFKL and D are defined in Eq. (A.2).
In the derivation of Eq. (A.9) we neglected the
contri−iν
butions that are proportional to Q4T2r1222 r12 , since this
contribution will be the same as in Eq. (A.9), but with
ξ = ln Q4T2r1222 r12 1. To integrate over ν, we use the
method of steepest descent, and the expansion of ω (ν, 0) at
small ν (diffusion approximation; see the second equation in
Eq. (A.2)).
N (r1, r2; Y ) denotes the imaginary part of the
dipoledipole sacttering amplitude at QT = 0, which is related to
the cross section. One can check that Eq. (A.9) has the correct
dimension.
Appendix B: Calculation of the integrals for the
contribution of the simplest diagram
In this appendix we discuss the integrations in Eq. (2.5). The
integral over QT has been considered in Sect. 4.1 and it has
the form of Eq. (4.7). The extra eir1·QT give an additional
numerical factor, replacing 25 by 27 in Eq. (4.7). To integrate
over kT and kT we replace
! dφi e−i pT 1·r˜1 δ(2) r1 + r1 − r˜1 − r˜1
→ (2π )4
kT dkT J0 (kT r1) J0 kT
+ pT 1 r˜1 J0 (kT r˜1) J0 (kT r˜1) .
0
r˜2
0
∞
Now we can take the integrals over ri bearing in mind
Eq. (4.7) and
N pr (ri , Y − y1) =
dν
2π
2 2 21 +iνi eω(νi ,0)(Y −y1).
μsoft ri
The integrals over r˜1and r˜1 have the following form (see
Ref. [
61
], Eq. 6.511(6)):
1
J0 (kT r˜1) dr˜1 = r˜2 J0 (kr˜2) + 2 πr˜2 J1 (kr˜2) H0 (kr˜2)
− J0 (kr˜2) H1 (kr˜2)
=
Using Eq. (B.2) we obtain
r˜2
1
k
if kr˜2
if kr˜2
Collecting Eqs. (B.2), (B.3) and (B.4) we see that the main
contribution stems from the region kr˜2 1 and the integral
over kT has the form
r˜2r˜ 2
2
pT1
We see that the integrals over r1 and r1 lead to r1 ∼ 1/Qs (Y −
y1) and r1 ∼ 1/Qs (Y − y1). The same holds for the integrals
over r2 and r2, leading to r2 ∼ 1/Qs (y2) and r2 ∼ 1/Qs (y2).
Assuming that Qs (Y − y1) > Qs (y2) we conclude that ri
and ri are much smaller than r2 and r2. Replacing
∇r21 Npr (r1; Y − y1) ei QT ·r˜ 1 ∇r˜21 Npr r˜1; Y − y1
r12 Qs2 (Y − y1) γ¯ r12 Qs2 (Y − y1) γ¯ (C.2)
(C.1)
1 1
= Qs 1 + 2γ¯
where γ¯ = 1 − γcr , we see from Eq. (C.1) that integration
over r takes the form
We need to estimate the diagram of Fig. 1a (see Eq. (2.7)). This diagram can be rewritten as (C.8)
Examining Eq. (3.5), one can see that in the general case
when Y − y1 = y1 and Y − y2 = y2 all four Pomerons cannot
be in the vicinity of the saturation scale. Actually we have
two kinematic regions which give the maximal contributions
(assuming Qs (Y − y1) > Qs2 (y1)):
1. r 2 Qs2 (Y − y1) ≈ 1 but r 2 Qs2 (y1) → Qs2 (Y − y1) /
Qs2 (y1) 1;
2. r 2 Qs2 (y1) ≈ 1 but r 2 Qs2 (y1) → Qs2 (y1) /Qs2 (Y − y1)
1.
In region 1 the upper Pomeron is in the vicinity of the
saturation scale, while the lower Pomeron is in the perturbative
QCD region. In region 2 the lower Pomeron is in the vicinity
of the saturation scale, and the upper Pomeron is deep inside
the saturation domain. As we have discussed (see Fig. 3a)
∇2 N decreases in the saturation region much faster than in
the perturbation QCD region and, therefore, we assume that
the kinematic region 1 gives the largest contribution. Hence,
for pT 1 Qs2 (y1) we obtain Eq. (4.16).
Appendix D: Integration over dipole sizes in the angle
correlation function
In this appendix we collect tedious integration over the dipole
sizes in Eq. (5.6).
Each term in this equation can be factorized as a product
of two functions which depend on r 1i and on r 2i. Bearing this
feature in mind we calculate each term going to the
momentum representation using Eq. (C.1). We obtain a product of
functions of kT . Each of these functions has the following
general form:
j
d2r eikT ·r ! rμi F (r )
i=1
= (−i ∇kT ) j
= 2π (−i ∇kT ) j
d2r eikT ·r F (r )
d2r J0 (kT r ) F (r ) .
(D.1)
Using Eq. (C.5) for pT 2
be reduced to
1
(1 + 2γ¯ )2
×
ln τ
τ
0
1
dτ
2
1 F2
= 3.50.
Qs (y2) the integral over kT can
Finally, collecting all numerical coefficients, we obtain
where the constant C is the value of the amplitude at τ = 1.
This contribution is proportional to
∝
e2 BFKL y12 /Qs2 (Y − y1)
for pT 1 Qs (Y − y1) and pT 2
Qs2 (Y − y1) > Qs2 (y2).
Qs (y2). Note that
As we have seen the dependence on ri stems from the
integration over QT or, in other words, from (IQ ). In IQ the
dependence on r1 and r1 can be extracted explicitly, leading
to F (r ) ∝ 1/r . Hence the momentum image for Eq. (D.1)
has a simple form:
The expressions for A and B can be written in a
general form. Assuming that both pT 1 and pT 2 are smaller
than Qs (y2), we can expand the answer, taking into account
only terms that are proportional to pT2 1/ kT2 and pT2 2/ kT2. We
obtain
j
i=1
=
d2r eikT ·r ! rμi F (r ) = 2π (−i ∇kT )
(D.2)
.
For j = 2 and j = 4, which we need to calculate considering
Eq. (5.6), we have
(−i ∇kT )
(−i ∇kT )
.
Note that, for integration over r1, Eq. (D.2) takes the form
j
d2r1 ei(kT +pT1)·r1 ! r1,μi F (r1)
= 2π
−i ∇kT +pT1
i=1
j
1
(kT + pT 1)2
.
2
The term r12 (n1 · n2)2
+ r12 n1 · n2
can be rewritten as
r1,μ r1,ν + r1,μ r1,ν r2,μ r2,ν and in the momentum
representation it is
A (kT , pT 1) =
B (kT , pT 1) =
3 pT2 1
4 kT8
1
8 kT4
The integrations over r2 and r2 differ from the integrations
over r1 and r1, due to the extra factor 1/r22, which comes
from the integration over QT in Eqs. (4.2) and (4.3). Since
r22 ≈ 1/Qs2(y2) we replace it by 1/r22 = Qs2 (y2). In the case
the integral over kT takes the same form as the integral over
kT , leading to the following expression, which is proportional
to cos2 (φ), where φ is the angle between pT 1 and pT 2:
d2σ
⎧
dφ ⎪⎪⎪⎨
+ pT 1)i −
= A
pT 1,i pT 1,i
2
pT 1
3 δii
kT5 kT,i kT,i − kT3
1
2 2
kT + pT 1 + 2 cos (φ) kT pT 1
3
2 2
kT + pT 1 + 2 cos (φ) kT pT 1
5 (kT + pT 1)i (kT
δii
⎟
2 2 3 ⎟⎟ kT
kT + pT 1 + 2 cos (φ) kT pT 1 ⎠
+ B δii .
⎞
⎫
⎪
1 ⎪⎪⎬
⎪⎪⎪⎭
(D.4)
(D.7)
(D.8)
1. A. Dumitru , F. Gelis , L. McLerran , R. Venugopalan , Nucl. Phys. A 810 , 91 ( 2008 ). arXiv: 0804 .3858 [hepph]
2. E.V. Shuryak , Phys. Rev. C 76 , 047901 ( 2007 ). arXiv: 0706 .3531 [nuclth]
3. S.A. Voloshin , Phys. Lett. B 632 , 490 ( 2006 ). arXiv:nuclth/0312065
4. S. Gavin , L. McLerran , G. Moschelli, Phys. Rev. C 79 , 051902 ( 2009 ). arXiv: 0806 .4718 [nuclth]
5. K. Dusling , R. Venugopalan , Phys. Rev. D 87 ( 9 ), 094034 ( 2013 ). arXiv: 1302 .7018 [hepph] (reference therein)
6. A. Kovner , M. Lublinsky , Phys. Rev. D 83 , 034017 ( 2011 ). arXiv: 1012 .3398 [hepph]
7. Y.V. Kovchegov , D.E. Wertepny , Nucl. Phys. A 906 , 50 ( 2013 ). arXiv: 1212 .1195 [hepph]
8. T. Altinoluk , N. Armesto , G. Beuf , A. Kovner , M. Lublinsky , Phys. Lett. B 752 , 113 ( 2016 ). arXiv: 1509 .03223 [hepph]
9. T. Altinoluk , N. Armesto , G. Beuf , A. Kovner , M. Lublinsky , Phys. Lett. B 751 , 448 ( 2015 ). arXiv: 1503 .07126 [hepph]
10. E. Gotsman , E. Levin, U. Maor , S. Tapia, Phys. Rev. D 93 ( 7 ), 074029 ( 2016 ). arXiv: 1603 .02143 [hepph]
11. E. Gotsman , E. Levin, U. Maor, Phys. Rev. D 95 ( 3 ), 034005 ( 2017 ). arXiv: 1604 .04461 [hepph]
12. E. Gotsman , E. Levin, Phys. Rev. D 95 ( 1 ), 014034 ( 2017 ). arXiv: 1611 .01653 [hepph]
13. A. Kovner , M. Lublinsky , V. Skokov , Phys. Rev. D 96 , 016010 ( 2017 ). arXiv: 1612 .07790 [hepph]
14. E. Gotsman , E. Levin , I. Potashnikova , Eur. Phys. J. C 77 , 632 ( 2017 ). arXiv: 1706 .07617 [hepph]
15. V. Khachatryan et al. [CMS Collaboration], Phys. Rev. Lett . 116 , 172302 ( 2016 ). arXiv: 1510 .03068 [nuclex]
16. V. Khachatryan et al. [CMS Collaboration], JHEP 1009 , 091 ( 2010 ). arXiv: 1009 .4122 [hepex]
17. J. Adams et al. [STAR Collaboration], Phys. Rev. Lett . 95 , 152301 ( 2005 ). arXiv:nuclex/0501016
18. B. Alver et al. [PHOBOS Collaboration], Phys. Rev. Lett . 104 , 062301 ( 2010 ). arXiv: 0903 .2811 [nuclex]
19. H. Agakishiev et al. [ STAR Collaboration], Measurements of Dihadron Correlations Relative to the Event Plane in Au+Au Collisions at √sN N = 200 GeV . arXiv: 1010 .0690 [nuclex]
20. S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 718 , 795 ( 2013 ). arXiv: 1210 .5482 [nuclex]
21. V. Khachatryan et al. [CMS Collaboration], JHEP 1009 , 091 ( 2010 ). arXiv: 1009 .4122 [hepex]
22. S. Chatrchyan et al. [CMS Collaboration], JHEP 1402 , 088 ( 2014 ). arXiv:1312 . 1845 [nuclex]
23. S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. C 89 ( 4 ), 044906 ( 2014 ). arXiv: 1310 .8651 [nuclex]
24. S. Chatrchyan et al. [CMS Collaboration], Eur. Phys. J. C 72 , 2012 ( 2012 ). arXiv: 1201 .3158 [nuclex]
25. J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett . 117 , 182301 ( 2016 ). arXiv: 1604 .07663 [nuclex]
26. J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett . 116 ( 13 ), 132302 ( 2016 ). arXiv: 1602 .01119 [nuclex]
27. L. Milano [ALICE Collaboration], Nucl. Phys. A 931 , 1017 ( 2014 ). arXiv: 1407 .5808 [hepex]
28. Y. Zhou [ALICE Collaboration], J. Phys. Conf. Ser . 509 , 012029 ( 2014 ). arXiv: 1309 .3237 [nuclex]
29. B.B. Abelev et al. [ALICE Collaboration], Phys. Rev. C 90 ( 5 ), 054901 ( 2014 ). arXiv: 1406 .2474 [nuclex]
30. B.B. Abelev et al. [ALICE Collaboration], Phys. Lett. B 726 , 164 ( 2013 ). arXiv: 1307 .3237 [nuclex]
31. B. Abelev et al. [ALICE Collaboration], Phys. Lett. B 719 , 29 ( 2013 ). arXiv:1212 . 2001 [nuclex]
32. M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. C 96 , 024908 ( 2017 ). arXiv: 1609 .06213 [nuclex]
33. G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett . 116 , 172301 ( 2016 ). arXiv: 1509 .04776 [hepex]
34. G. Aad et al. [ATLAS Collaboration] Phys. Rev. C 90 ( 4 ), 044906 ( 2014 ). arXiv: 1409 .1792 [hepex]
35. B. Wosiek [ATLAS Collaboration], Ann. Phys. 352 , 117 ( 2015 )
36. G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 725 , 60 ( 2013 ). arXiv:1303 . 2084 [hepex]
37. B. Wosiek [ATLAS Collaboration], Phys. Rev. C 86 , 014907 ( 2012 ). arXiv: 1203 .3087 [hepex]
38. V.A. Abramovsky , V.N. Gribov , O.V. Kancheli , Yad. Fiz. 18 , 595 ( 1973 ) [Sov . J. Nucl. Phys . 18 , 308 ( 1974 )]
39. A.H. Mueller , Phys. Rev. D 2 , 2963 ( 1970 )
40. E.A. Kuraev , L.N. Lipatov , F.S. Fadin , Sov. Phys. JETP 45 , 199 ( 1977 )
41. Ya Ya. Balitsky , L.N. Lipatov , Sov. J. Nucl. Phys . 28 , 22 ( 1978 )
42. L.N. Lipatov , Phys. Rep . 286 , 131 ( 1997 )
43. L.N. Lipatov , Sov. Phys. JETP 63 , 904 ( 1986 ) (references therein)
44. Yuri V. Kovchegov , Eugene Levin, “ Quantum Choromodynamics at High Energies”, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, 2012 )
45. Y.V. Kovchegov , Phys. Rev. D 64 , 114016 ( 2001 ). arXiv:hepph/0107256 [Erratumibid. D 68 , 039901 ( 2003 )]
46. Y.V. Kovchegov , K. Tuchin , Phys. Rev. D 65 , 074026 ( 2002 ). [arXiv:hepph/0111362]
47. J. JalilianMarian , Y.V. Kovchegov , Phys. Rev. D 70 , 114017 ( 2004 ). arXiv:hepph/0405266 [Erratumibid. D 71 , 079901 ( 2005 )]
48. M.A. Braun , Eur. Phys. J. C 48 , 501 ( 2006 ). arXiv:hepph/0603060
49. M.A. Braun , Eur. Phys. J. C 55 , 377 ( 2008 ). arXiv: 0801 .0493 [hepph]
50. C. Marquet , Nucl. Phys. B 705 , 319 ( 2005 ). arXiv:hepph/0409023
51. A. Kovner , M. Lublinsky , JHEP 0611 , 083 ( 2006 ). arXiv:hepph/0609227
52. E. Levin , A. Prygarin , Phys. Rev. C 78 , 065202 ( 2008 ). arXiv: 0804 .4747 [hepph]
53. K.G. Boreskov , A.B. Kaidalov , O.V. Kancheli , Eur. Phys. J. C 58 , 445 ( 2008 ). arXiv: 0809 .0625 [hepph]
54. E. Levin , A.H. Rezaeian , Phys. Rev. D 84 , 034031 ( 2011 ). arXiv: 1105 .3275 [hepph]
55. Y. Hagiwara , Y. Hatta , B.W. Xiao , F. Yuan , Phys. Lett. B 771 , 374 ( 2017 ). arXiv: 1701 .04254 [hepph]
56. L. McLerran , V. Skokov , Nucl. Phys. A 947 , 142 ( 2016 ). arXiv: 1510 .08072 [hepph]
57. A. Kovner , M. Lublinsky , Int. J. Mod. Phys. E 22 , 1330001 ( 2013 ). arXiv:1211 . 1928 [ hepph] (references therein)
58. H. Navelet , R.B. Peschanski , Nucl. Phys. B 507 , 35 ( 1997 ). arXiv:hepph/9703238
59. H. Navelet , R.B. Peschanski , Phys. Rev. Lett . 82 , 1370 ( 1999 ). arXiv:hepph/9809474
60. H. Navelet , R.B. Peschanski , Nucl. Phys. B 634 , 291 ( 2002 ). arXiv:hepph/0201285
61. I. Gradstein , I. Ryzhik , Table of Integrals, Series, and Products, 5th edn. (Academic Press, London, 1994 )
62. L.V. Gribov , E.M. Levin , M.G. Ryskin , Phys. Rep . 100 , 1 ( 1983 )
63. A.H. Mueller , J. Qiu , Nucl. Phys. B 268 , 427 ( 1986 )
64. L. McLerran , R. Venugopalan , Phys. Rev. D 49 ( 2233 ), 3352 ( 1994 )
65. L. McLerran , R. Venugopalan , Phys. Rev. D 50 , 2225 ( 1994 )
66. L. McLerran , R. Venugopalan , Phys. Rev. D 53 , 458 ( 1996 )
67. L. McLerran , R. Venugopalan , Phys. Rev. D 59 , 09400 ( 1999 )
68. A.H. Mueller , D.N. Triantafyllopoulos , Nucl. Phys. B 640 , 331 ( 2002 ). arXiv:hepph/0205167
69. E. Iancu, K. Itakura , L. McLerran , Nucl. Phys. A 708 , 327 ( 2002 ). arXiv:hepph/0203137
70. A.M. Stasto , K.J. GolecBiernat , J. Kwiecinski, Phys. Rev. Lett . 86 , 596 ( 2001 ). arXiv:hepph/0007192
71. J. Bartels , E. Levin, Nucl. Phys. B 387 , 617 ( 1992 )
72. E. Levin, K. Tuchin , Nucl. Phys. B 573 , 833 ( 2000 ). arXiv:hepph/9908317
73. C. Contreras , E. Levin, R. Meneses , I. Potashnikova , Phys. Rev. D 94 ( 11 ), 114028 ( 2016 ). arXiv: 1607 .00832 [hepph]
74. C. Contreras , E. Levin , I. Potashnikova , Nucl. Phys. A 948 , 1 ( 2016 ). arXiv: 1508 .02544 [hepph]
75. E. Iancu, K. Itakura , S. Munier, Phys. Lett. B 590 , 199 ( 2004 ). arXiv:hepph/0310338
76. E. Gotsman , E. Levin, Phys. Rev. D 96 ( 7 ), 074011 ( 2017 ). arXiv: 1705 .07406 [hepph]