# Deep Reinforcement Learning for Foreign Exchange Trading

1st Yun-Cheng Tsai  School of Big Data Management
Soochow University
Taipei, Taiwan
<EMAIL_ADDRESS>2nd Chun-Chieh Wang  Department of Computer Science
and Information Engineering
National Taipei University
Taipei, Taiwan

###### Abstract

Reinforcement learning can interact with the environment and is suitable for
applications in decision control systems. Therefore, we used the reinforcement
learning method to establish a foreign exchange transaction, avoiding the
long-standing problem of unstable trends in deep learning predictions. In the
system design, we optimized the Sure-Fire statistical arbitrage policy, set
three different actions, encoded the continuous price over a period of time
into a heat-map view of the Gramian Angular Field (GAF) and compared the Deep
Q Learning (DQN) and Proximal Policy Optimization (PPO) algorithms. To test
feasibility, we analysed three currency pairs, namely EUR/USD, GBP/USD, and
AUD/USD. We trained the data in units of four hours from 1 August 2018 to 30
November 2018 and tested model performance using data between 1 December 2018
and 31 December 2018. The test results of the various models indicated that
favourable investment performance is achieved as long as the model is able to
handle complex and random processes and the state is able to describe the
environment, validating the feasibility of reinforcement learning in the
development of trading strategies.

###### Index Terms:

Gramian Angular Field (GAF), Deep Q Learning (DQN), Proximal Policy
Optimization (PPO), Reinforcement Learning, Foreign Exchange Trading

The corresponding author is supported in part by the Ministry of Science and
Technology of Taiwan under grant 107-2218-E-002-031. The first author is
supported in part by the Ministry of Science and Technology of Taiwan under
grant 107-2218-E-002-065.

##  I Introduction

We plan to use deep-enhanced learning to mimic how humans make decisions,
using the state of the current environment to execute actions and obtain
rewards from the environment. Moreover, people’s actions impact the
environment, causing the situation to enter a new state. To check the
feasibility of this approach, we adopted the method of training four-hour
units of EUR/USD, GBP/USD, and AUD/USD data between 1 August 2018 and 30
November 2018. We then applied the trained data to the period between 1
December 2018 and 31 December 2018 to validate the system performance.

##  II Preliminary

In the paper, we adopted the Sure-Fire arbitrage strategy, which is a variant
of the Martingale. It involves increasing bets after every loss so that the
first win recovers all previous losses plus a small profit. After entering the
market and initiating trading, the investor uses the margin between the stop-
loss and stop-gain prices as the raised margin. As long as the price
fluctuates within the increased margin and touches on the risen price, the
Sure-Fire Strategy urges investors to continue growing the stakes until they
surpass the margin to profit.

Figure 1: Sure-Fire arbitrage strategy 1.

Figure 2: Sure-Fire arbitrage strategy 2.

Figure 3: Sure-Fire arbitrage strategy 3.

The most prominent shortcoming of Martingale is the lack of stakes or funds
when the favorable odds are low. Therefore, we applied reinforcement learning
to optimize the Sure-Fire Strategy. Data train to obtain the trading behavior
with the minimum number of raises to achieve the maximum winning odds. A
detailed transactions are illustrated from Fig. 3 to Fig. 3. In Sure-Fire
arbitrage strategy 1, we purchase one unit at any price and set a stop-gain
price of $+k$ and a stop-loss price of $-2k$. At the same time, we select an
amount with a difference of $-k$ to the buy price and $+k$ to the stop-loss
price and set a backhand limit order for three units. Backhand refers to
engaging in the opposite behavior. The backhand of buying is selling, and the
backhand of selling is buying. A limit order refers to the automatic
acquisition of corresponding units. In Sure-Fire arbitrage strategy 2, we
place an additional backhand limit order, where the buy price is $+k$ to the
selling price, and $-k$ to the stop-loss price when a limit order is
triggered, and three units successfully sold backhand. We set the stop-gain
point as the difference of $+k$ and the stop-loss point as the difference of
$-2k$, after which an additional six units buy. In Sure-Fire arbitrage
strategy 3, the limit order triggers in the third transaction. The final price
exceeded the stop-gain amount of the first transaction, the stop-loss price of
the second transaction, and the stop-gain price of the third transaction. In
this instance, the purchase is complete. The calculation in the right block
shows that the profit is $+1k$.

###  II-A Deep Q Network (DQN)

The Deep Q Network (DQN) is a deep, reinforcement learning framework
[hausknecht2015deep]. For the selection of actions, if the greedy method alone
is applied to select the action with the highest expected return every round,
then the chance of selecting other actions would be lost. Therefore, DQN
adopts the $\epsilon$-greedy exploration method. Specifically, each time an
action is selected, the system provides a small probability ($\epsilon$) for
exploration, in which the best action abandoned and other new actions
executed. The probability of exploration increases concurrently with the
$\epsilon$ value, making the agent more likely to try a new state or action.
This process improves the learning effects. The disadvantage is that the
convergence time increases exponentially with the duration of exploration.

###  II-B Proximal Policy Optimization (PPO)

Proximal policy optimization (PPO) is a modified version of the policy
gradient method. It used to address the refresh rate problem of policy
functions. The algorithms used in the reinforcement learning method can be
categorized into dynamic programming (DP), the Monte Carlo method and
temporal-difference (TD) [1]. The policy gradient method uses the Monte Carlo
method, wherein samples collected for learning. The advantage of this method
is that it can apply to unknown environments. The policy gradient method
abandons value functions. Instead, it directly uses rewards to update the
policy function, which outputs the probability of taking specific action in an
individual state. Thus, it can effectively facilitate decision making for non-
discrete actions [2].

###  II-C Gramian Angular Field (GAF)

Foreign exchange prices can be viewed as a time series. Therefore, it is
necessary to consider prices over time rather than adopting a set of opening
and closing prices as a single state. The paper uses the Gramian Angular Field
method, which is a time series coding method consisting of many Gram matrices.

Figure 4: GAP results: Heat-map and numerical line chart.

A Gram matrix is a useful tool in linear algebra and geometry. It is often
used to calculate the linear correlation of a set of vectors
[tsai2019encoding]. In Fig. 4, the left image is a heat-map view of the GAF,
and the right image is the corresponding numerical line chart. The chart shows
high values.

##  III Methodology

Our reinforcement learning trading system designs as follows:

  1. 1.

State Design: States are derived from an agent’s observations of the
environment. They are used to describe or represent environments. The agents
then perform actions corresponding to the perceived state. Therefore, defining
the state is key to learning performance. In the paper, we use the sliding
window method on the “opening price, highest point, lowest point, and closing
price” of the foreign exchange state in units of four hours to obtain sets of
12. After GAF encoding, the data are inputted as a state with dimension
$12\times 12\times 4$.

  2. 2.

Action Design: The purpose of the paper is to optimize the Sure-Fire policy.
There are two optimization objectives. The first is to reduce overall buy-ins
and the second is to set favorable stop-loss and stop-gain points. The former
is to avoid the lack of funding while the latter is to prevent the fluctuation
of prices and unnecessary placement of investments. After pragmatic
consideration, three discrete actions are developed:

     * •

Upper limit to additional buy-ins after entering the market: $\\{1,2,3\\}$.

     * •

First buy or sell: $\\{BUY,SELL\\}$.

     * •

Stop-gain: $\\{20,25,30\\}$.

  3. 3.

Reward Design: Reward is defined as follows:

$\textup{Reward}=\textup{Profit}\times\textup{Discount},$

where profit is the net income of the current transaction and the discount is

$1.0-0.1\times\textup{(number of additional buy-ins)}.$

This calculation is a variant of the system quality number (SQN). SQNs are one
of the indicators used to score trading strategies. The formula is:

$(\textup{expected profit}/\textup{standard deviation})\times$
$(\textup{square root of the number of transactions}).$

The discount factor is a variable that decreases concurrently with the
increase in transactions. The reason for adding a discount factor to profit is
to inform the agent that the reward decreased concurrently with the increase
in the number of additional buy-ins. That is, risk increases concurrently with
the number of additional buy-ins. Therefore, excessive buy-ins should be
avoided.

Oue experiment environment designs as follows:

  1. 1.

Experiment Data: we used the EUR/USD, GBP/USD, and AUD/USD exchange data in
units of four hours between 1 August 2018 and 31 December 2018 as a reference
for the environment design and performance calculation of the system. The
period between 1 August 2018 and 30 November 2018 is the training period for
the agent. In this period, the agent repeatedly applied the data to learn,
eventually obtaining an optimal investment strategy. The period between 1
December 2018 and 31 December 2018 is the agent’s performance evaluation
period. The agent uses a trading strategy to form decisions. The system
accumulates the rewards obtained during the evaluation period, which served as
a reference for performance.

  2. 2.

Trade Environment Settings: EUR/USD, GBP/USD, and AUD/USD are adopted as
investment targets. The smallest unit is 1 pip (0.00001 of the price). The
default earnings for an action is set to at least 20 pips, which would be
greater than the slip price and transaction fee. Therefore, the slip price and
transaction fees are not taken into account. Whenever a transaction is tested,
the closing price is used as the data point rather than considering whether
the transactions are triggered at a high point or low point in the data (that
is, of the four values for each day-opening prices, high point, low point, and
closing price-the final one is chosen). Time scales smaller than one day
rarely contained stop-loss and stop-gain points.

  3. 3.

Experiment Model Design: a constant agent is added to the experiment model,
serving as a reference value for performance. The input of the three different
algorithms is a GAF-encoded time series of dimension $12\times 12\times 4$. A
CNN is used as the policy network or Q-network.

##  IV Experiment Results and Discussions

Model | Algorithm | Currencies
---|---|---
CEU | Constant | EUR/USD
DEU | DQN | EUR/USD
PEU | PPO | EUR/USD
CGU | Constant | GBP/USD
DGU | DQN | GBP/USD
PGU | PPO | GBP/USD
CAU | Constant | AUD/USD
DAU | DQN | EUR/USD
PAU | PPO | EUR/USD
TABLE I:  Model Code Information

In Table II, PEU and DEU profits are lower than those of CEU. The accumulated
return trajectories for PEU and DEU are almost identical, with the sole
differences being the slope. The lines for accumulated returns of PEU and DEU
are smoother than CEU. The max drawdown is also much lower, which shows that
net profits for PEU and DEU are lower only because no wrong decisions made in
the more conservative markup models. The max drawdown of PEU is almost half
that of CEU. A more economical risk-orientated strategy for stable returns
very possibly discovered for PEU and DEU. The returns for the three are about
the same, but the PGU profit factor is much higher than that of the other two,
and the PGU max drawdown is the lowest of the three. It can see that PGU using
the PPO trade algorithm is the most consistently profitable. Interestingly
enough, the trading decisions made in DGU are almost all reached later than
those in PGU. PGU more clearly captured profit maximization properties than
DGU during the training period, resulting in a situation in which PGU did not
experience the same delays as DGU. PAU made trading decisions that maximized
profitability, with every evaluation being far better than that of CAU. Though
the max drawdown of PAU is approximately 1.25 times that of DAU, returns are
almost 1.5 times those of DAU, while the profit factor remained the highest of
the three. Besides the unexpectedly high PGU earnings, there is not much out
of the ordinary in the accumulated returns line graphs for DAU and CAU. DAU
continually made sound trading decisions, and its max drawdown is the lowest
of the three.

Model Code | Net Profit | Profit Factor | Max Draw-down
---|---|---|---
CEU | 753 | 3.67 | -5.18%
DEU | 486 | 3.09 | -3.07%
PEU | 615 | 3.82 | -2.92%
CGU | 753 | 3.24 | -5.58%
DGU | 717 | 3.10 | -4.63%
PGU | 763 | 4.47 | -4.33%
CAU | 351 | 1.62 | -14.53%
DAU | 402 | 2.32 | -5.97%
PAU | 597 | 2.85 | -7.62%
TABLE II:  Comparing DQN model with PPO model trading performance. Using
EUR/USD 4 hours data from 2018/12/01 to 2018/12/31 at 1300 episode.

##  V Conclusion

We used a GAF-encoded time series as the state, two different algorithms (DQN
and PPO), and three different currency pairs (EUR/USD, GBP/USD, AUD/USD) to
create six different trading models. From the above comparisons that using
PPO’s reinforcement learning algorithm to optimize forex trading strategies is
quite feasible, with PPO performance being better than DQN. The results of the
various models indicated that favorable investment performance achieved as
long as the model can handle complex and random processes, and the state can
describe the environment, validating the feasibility of reinforcement learning
in the development of trading strategies. We found that the most challenging
aspect is the definition of reward.

## References

  * [1] Richard S Sutton and Andrew G Barto.  Reinforcement learning: An introduction.  MIT press, 2018.
  * [2] Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour.  Policy gradient methods for reinforcement learning with function approximation.  In Advances in neural information processing systems, pages 1057–1063, 2000.


# Rapidity window dependence of ridge correlations in the glasma

Donghai Zhang  Key Laboratory of Quark and Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
Yeyin Zhao  Key Laboratory of Quark and Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China
Mingmei Xu<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics
(MOE) and Institute of Particle Physics, Central China Normal University,
Wuhan 430079, China  Xue Pan  School of Electronic Engineering, Chengdu
Technological University, Chengdu 611730, China  Yuanfang Wu
<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China

###### Abstract

We study ridge correlations of the glasma in pp collisions at
$\sqrt{s_{\mathrm{NN}}}=7$ TeV by using the color glass condensate (CGC)
formalism. The azimuthal collimation at long range rapidity is intrinsic to
glasma dynamics and is reproduced here. When rapidity window enlarges, ridge
correlations in two dimensional $\Delta y$-$\Delta\phi$ distribution and one
dimensional $\Delta\phi$ distribution at long range rapidity gap are enhanced.
The enhancements are demonstrated to be the contributions of source gluons.
The quantum evolution of the gluons presents unique correlation patterns in
differential correlation function. These characters of two gluon correlations
open a way of testing the production mechanism from experimental measurements.

##  I Introduction

The long range rapidity ridge correlation in small systems is one of the most
unexpected discoveries CMS-pp-JHEP-2010 ; CMS-pPb-PLB-2013 ; CMS-pPb-PbPb-
PLB-2013 ; ATLAS-pPb-PRL-2013 ; ALICE-pPb-PLB-2013 ; ALICE-pPb-PLB-2016 ;
ridge-RHIC-small-1 ; ridge-RHIC-small-2 . The structure of ridge in two
dimensional $\Delta\eta$-$\Delta\phi$ correlations is twofold: a long ridge in
$\Delta\eta$ direction and two peaks in $\Delta\phi$ direction. The two peaks
at $\Delta\phi=0$ and $\pi$ are also termed the collimation production,
revealing the collectivity at long range rapidity in small systems. The
physics underlying the collectivity at long range rapidity is open yet and
under extensive studies.

Final state interactions, especially the hydrodynamic description, combined
with a variety of initial conditions, attribute the long range ridge to the
fluid nature pp-hydro1 ; pp-hydro2 ; pp-hydro3 ; pp-hydro4 ; pp-hydro5 ; pp-
hydro6 . The transverse expansion of the fluid driven by the pressure gradient
can explain the azimuthal anisotropy of final state particles. The
difficulties of hydrodynamic description are that it can not reproduce the
multi-particle cumulant $c_{2}\\{4\\}$ hydro-cumu-1 ; hydro-cumu-2  and the
elliptic flow of heavy flavor particles hydro-flow-hf-1 ; hydro-flow-hf-2 . In
these respects, CGC calculations reproduce the data well cgc-c2 ; cgc-heavy .
Only initial correlations from CGC can explain the long range ridge yield very
well, too cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III . The trouble of CGC is that it
can not generate the right ordering of Fourier harmonics $v_{n}$ in the three
system sizes of pAu, dAu and HeAu cgc-vn-ordering . In a word, the two
mechanisms, i.e. hydrodynamics and CGC, can only describe some of the data in
small systems, respectively. Therefore, they both need further studies.

CGC describes the initial state of a colliding hadron or nucleus. The gluon
density grows rapidly and gets saturated as the collision energy increases or
as Feynman variable $x$ decreases. At saturation the small-$x$ gluons have
large occupation numbers (of order $1/\alpha_{s}$), so their mutual
interactions can be treated as the classical field with a strong field
strength (of order $1/\sqrt{\alpha_{s}}$). Thus two colliding hadrons or
nuclei at very high energies are two sheets of colored glass approaching one
another. After the collision, strong longitudinal color electric and magnetic
fields are formed in the region between the nuclei, which is called glasma.
The glasma fields are localized in the transverse scale (of order $1/Q_{s}$)
that are smaller than the nucleon size, which gives the picture of glasma flux
tube. The strong longitudinal color fields in glasma flux tubes are
approximately boost invariant and multi-particle productions in the small $x$
region naturally generate long range longitudinal correlations cgc-NPA-2008 ;
cgc-NPA-2010 .

The correlations between gluons in high energy scatterings are caused by the
quantum evolution, which is illustrated in Fig. 1(a). The short lines
represent valence quarks, which radiate gluons in the form of gluon cascades.
Besides the gluon radiation, when the gluon density gets large, recombinations
of radiated gluons (non-linear effects) are also important and shown in the
evolution. They result in the saturation phenomenon of the gluon distribution
at $k_{\perp}\ll Q_{s}$, while the gluon distribution at $k_{\perp}\gg Q_{s}$
follows a behavior of linear approximation. Therefore, $Q_{s}$, called
saturation momentum, as a scale separating linear and non-linear behavior, is
the only parameter which can determine the physics in a given collision. This
indicates an equivalence between nuclei and protons. The universality of
hadronic interactions at high energies in the CGC effective field theory
succeeds in explaining long range azimuthal correlations in pp, pPb
systematically cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III  and may account for the
similarity between PbPb and pPb with the same multiplicity Dusling-PRL-2018 .

The CGC framework succeeds in explaining not only the multiplicity and
transverse momentum dependence, but also the $\Delta\eta$ acceptance
dependence of the long range azimuthal correlations in the LHC pPb data at
5.02 TeV cgc-PRD-III . With a common set of parameters, ref cgc-PRD-III  gets
good fits to the CMS, ALICE and ATLAS data of different acceptances
simultaneously. The overall agreement with data is a spectacular achievement
of CGC. Because they considers the combined effect of rapidity acceptance and
normalizations of different LHC experiments, an explicit dependence on the
rapidity acceptance alone is not presented yet.

In high energy limit, both rapidity and transverse momentum of a gluon are
related to its Feynman $x$. For a right moving projectile, the relation reads

$x=\frac{\mathrm{p_{\perp}}}{\sqrt{s}}e^{y},$ (1)

with $y$ representing rapidity, $\mathrm{p_{\perp}}$ transverse momentum and
$\sqrt{s}$ center-of-mass energy. The dependence on rapidity is much more
sensitive due to the exponential function. At $\sqrt{s}=7$ TeV and
intermediate $\mathrm{p_{\perp}}$, e.g. 2 GeV$/$c, the rapidity region
$y\leqslant 1.2$ corresponds to $x\lesssim 10^{-3}$, while $y\leqslant 3.5$
corresponds to $x\lesssim 10^{-2}$. It means that gluons at central rapidity
region ($|y|\leqslant 1.2$) reflect properties of the small-$x$ ($x\lesssim
10^{-3}$), as the green band shows in Fig. 1(b). In that rapidity region
quantum evolutions are essential. In contrast, gluons at middle rapidity
regions ($1.2<|y|\leqslant 3.5$), as the blue bands show in Fig. 1(b), present
features of moderate-$x$ ($10^{-3}<x<10^{-2}$) degree of freedom. Within this
region, radiated gluons still dominate, but contributions of color sources
begin to show up. The large-$x$ ($x\gtrsim 10^{-2}$) degrees of freedom, i.e.
gluons at large rapidity ($|y|>3.5$) , act as sources and are referred to as
source gluons, which are denoted by red bands in Fig. 1(b).

It has been demonstrated that, the strong correlation between radiated gluons
and source gluons can explain long range rapidity correlations zhao-1 ; zhao-2
. However, ridge correlations in experiments are usually measured within
certain rapidity or pseudorapidity windows. For example, the pseudorapidity
windows of ALICE and CMS are [-0.9, 0.9] and [-2.4, 2.4], respectively. A
quantitative comparison of results from different experiments requires a study
of the rapidity window dependence of ridge correlations.

On the theoretical side, different rapidity windows are expected to reveal
gluon dynamics at different $x$ values. Fig. 2 demonstrates that different
rapidity windows ($Y_{\rm W}$ for short) pick pairs of different $x$ when a
given rapidity gap like $\Delta y=2$ is being concerned. A narrow rapidity
window only include contributions of radiated gluons, while a wider rapidity
window can further include correlations between radiated gluons and source
gluons. So, rapidity window dependence of long range ridge correlations is
sensitive to the quantum evolution of gluon saturation dynamics.

Considering the above reasons, it would be valuable to study the rapidity
window dependence of the ridge correlations systematically. This paper is
organized as follows. In section II, the definition of correlation function
and some related formulae of single- and double-gluon inclusive production in
CGC framework are given. The formulae in this manuscript follow those in Ref.
cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III  and are identical with those at gluon
level without fragmentation functions. The new aspects of analysis method here
lie in the exploration of contributions of different $x$ degrees of freedom.
Results of correlations are shown and discussed in section III, where the
sensitivity to rapidity windows are carefully compared. Section IV gives the
summary and discussion.

Figure 1: (a) The quantum evolution of gluons in the right moving projectile.
The black axis represents the longitudinal momentum fraction $x$ of partons in
the projectile, and the blue axis roughly indicates corresponding rapidity
$y$. (b) The quantitative correspondence between $x$ regions and rapidity
regions for the case of $\mathrm{p_{\perp}}=$ 2 GeV$/$c in the collision of
$\sqrt{s}=7$ TeV.  Figure 2: A schematic plot for pairs with rapidity
separation $\Delta y=2$. Rapidity window ($Y_{\rm W}$) of [-1, 1] only
includes pairs with $y_{\rm p}=-1$ and $y_{\rm q}=1$, as the green curve
shows. Rapidity window of [-2, 2] counts more pairs including pairs with
$y_{\rm p}=0$ and $y_{\rm q}=\pm 2$, as blue curves show. Rapidity window of
[-4, 4] further includes pairs with $y_{\rm p}=\pm 2$ and $y_{\rm q}=\pm 4$,
as red curves show.

##  II Two-gluon $\Delta y$-$\Delta\phi$ correlations from high energy QCD
evolution

In a high energy collision, both the projectile and the target are regarded as
high parton density sources. When they pass through each other, strong
longitudinal color electric and magnetic fields are formed. The framework that
describes the physics of high parton densities and strong color fields is the
CGC effective field theory (CGC EFT) Gribov-1983 ; Iancu-2003 ; Weigert-2005 ;
Gelis-2010 . The effective degrees of freedom in this framework are color
sources $\rho$ at large $x$ and classical gauge fields $\mathcal{A}_{\mu}$ at
small $x$, as Fig.1(a) shows. The classical gauge field $\mathcal{A}_{\mu}$
can be obtained by numerically solving Yang-Mills equations with a given
configuration of color source. For a given initial configuration of color
source, fields in the nuclear wave functions evolve with Feynman $x$, which is
described by the Jalilian-Marian-Iancu-McLerran-Weigert-Kovner (JIMWLK)
renormalization group equations Jalilian-Marian-1997 ; Jalilian-Marian-PRD ;
Iancu-2001 . In mean field approximation and large-$N_{c}$ limit, the JIMWLK
equation is reduced to the Balitsky-Kovchegov (BK) equation Balitsky1996 ;
Balitsky1999 ; Kovchegov1999 , which describes the quantum evolution in Fig.
1(a).

Supposing two gluons are produced with transverse momentum
$\mathbf{p}_{\perp}$, $\mathbf{q}_{\perp}$ and rapidity $y_{\rm p}$, $y_{\rm
q}$, the correlation function is defined as,

$\displaystyle C(\mathbf{p}_{\bot},y_{\rm p};\mathbf{q}_{\bot},y_{\rm q})$
$\displaystyle=$
$\displaystyle\frac{\frac{dN_{2}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}}{\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}\frac{dN_{1}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}-1=\frac{\frac{dN_{\mathrm{2}}^{\rm corr}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}}{\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}\frac{dN_{1}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}},$ (2)

where $\frac{dN_{2}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}}$,
$\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}$ are the double- and single-
gluon inclusive productions and $\frac{dN_{2}^{\rm
corr}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}}$
is the correlated double-gluon production. Subscripts “p” and “q” are used to
mark the two gluons.

In the framework of CGC EFT, for a given collision, the observable under the
leading-log approximation is factorized as cgc-NPA-2008 ,

$\displaystyle\left\langle\mathcal{O}\right\rangle_{\mathrm{LLog}}$
$\displaystyle=$
$\displaystyle\int[D\rho_{1}][D\rho_{2}]W[\rho_{1}]W[\rho_{2}]\mathcal{O}[\rho_{1},\rho_{2}]_{\mathrm{LO}},$
(3)

where $\mathcal{O}[\rho_{1},\rho_{2}]_{\mathrm{LO}}$ is leading-order single-
or double- gluon inclusive distribution for a fixed distribution of color
sources, and the integration denotes an average over different distributions
of color sources with the weight functional $W[\rho_{1,2}]$. In general,
$W[\rho_{1,2}]$ encodes all possible color charge configurations of the
projectile and target, and obeys the JIMWLK renormalization group equations
Jalilian-Marian-1997 ; Jalilian-Marian-PRD ; Iancu-2001 . All the quantum
evolution of the projectile/target is absorbed into the distribution
$W[\rho_{1,2}]$.

The averaging over color sources can be done under the McLerran-Venugopalan
(MV) model with a Gaussian weight functional. According to Ref. cgc-PRD-I ;
cgc-PRD-II ; cgc-PRD-III , the correlated two-gluon production can be
expressed by unintegrated gluon distributions (uGD) as,

$\displaystyle\frac{dN_{\mathrm{2}}^{\mathrm{corr}}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}=\frac{C_{2}}{\mathbf{p}^{2}_{\bot}\mathbf{q}^{2}_{\bot}}\left[\int\frac{d^{2}\mathbf{k}_{\bot}}{(2\pi)^{2}}(D_{1}+D_{2})+\sum_{j=\pm}\left[D_{3}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})+\frac{1}{2}D_{4}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})\right]\right],$
(4)

where
$C_{2}=\frac{\alpha_{s}(\rm{p}_{\bot})\alpha_{s}(\rm{q}_{\bot})N_{c}^{2}S_{\bot}}{\pi^{8}(N_{c}^{2}-1)^{3}}$,
and

$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle\Phi_{A_{1}}^{2}(y_{\rm
p},\mathbf{k}_{\bot})\Phi_{A_{2}}(y_{\rm
p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot})\left[\Phi_{A_{2}}(y_{\rm
q},\mathbf{q}_{\bot}+\mathbf{k}_{\bot})+\Phi_{A_{2}}(y_{\rm
q},\mathbf{q}_{\bot}-\mathbf{k}_{\bot})\right],$ $\displaystyle D_{2}$
$\displaystyle=$ $\displaystyle\Phi_{A_{2}}^{2}(y_{\rm
q},\mathbf{k}_{\bot})\Phi_{A_{1}}(y_{\rm
p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot})[\Phi_{A_{1}}(y_{\rm
q},\mathbf{q}_{\bot}+\mathbf{k}_{\bot})+\Phi_{A_{1}}(y_{\rm
q},\mathbf{q}_{\bot}-\mathbf{k}_{\bot})].$ (5)

Here, $\Phi_{A_{1(2)}}(y,\mathbf{k}_{\bot})$ denotes the uGD of projectile
$A_{1}$ or target $A_{2}$. In Eq.(4),

$D_{3}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})=\delta^{2}(\mathbf{p}_{\bot}+j\mathbf{q}_{\bot})\left[\mathcal{I}_{1}^{2}+\mathcal{I}_{2}^{2}+2\mathcal{I}_{3}^{2}\right],$
(6)

with

$\displaystyle\mathcal{I}_{1}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2})^{2}}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(7) $\displaystyle\mathcal{I}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{|\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot}|^{2}}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(8) $\displaystyle\mathcal{I}_{3}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2})|\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot}|}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(9)

and

$\displaystyle D_{4}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A1}(y_{\rm p},\mathbf{k}_{2\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{2\bot})$ (10) $\displaystyle\times$
$\displaystyle\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2}){(\mathbf{k}_{2\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{2\bot}^{2})}+(\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot})\cdot(\mathbf{k}_{2\bot}\times\mathbf{p}_{\bot})}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}}$
$\displaystyle\times$ $\displaystyle\frac{(\mathbf{k}_{1\bot}\cdot
j\mathbf{q}_{\bot}-\mathbf{k}_{1\bot}^{2}){(\mathbf{k}_{2\bot}\cdot
j\mathbf{q}_{\bot}-\mathbf{k}_{2\bot}^{2})}+(\mathbf{k}_{1\bot}\times\mathbf{q}_{\bot})\cdot(\mathbf{k}_{2\bot}\times\mathbf{q}_{\bot})}{\mathbf{k}_{2\bot}^{2}(j\mathbf{q}_{\bot}-\mathbf{k}_{1\bot})^{2}},$

where
$\mathbf{k}_{2\bot}\equiv\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}+j\mathbf{q}_{\bot}$.

The single-gluon inclusive production reads

$\displaystyle\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}$
$\displaystyle=$ $\displaystyle\frac{\alpha_{s}({\rm
p}_{\bot})N_{c}S_{\bot}}{\pi^{4}(N_{c}^{2}-1)}\frac{1}{\mathbf{p}_{\bot}^{2}}\int\frac{d\mathbf{k}_{\bot}^{2}}{(2\pi)^{2}}\Phi_{A}(y_{\rm
p},\mathbf{k}_{\bot})\Phi_{A}(y_{\rm p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot}).$

Based on correlation function $C(\mathbf{p}_{\bot},y_{\rm
p};\mathbf{q}_{\bot},y_{\rm q})$, the associated yield per trigger is defined
as

$\displaystyle Y(\Delta\phi,\Delta
y)=\frac{1}{N_{\rm{Trig}}}\frac{d^{2}N_{\rm{Assoc}}}{d\Delta\phi d\Delta y},$
(12)

where

$\displaystyle\frac{d^{2}N_{\rm{Assoc}}}{d\Delta\phi d\Delta y}$
$\displaystyle=$ $\displaystyle\int_{y^{\rm min}}^{y^{\rm max}}dy_{\rm
p}\int_{y^{\rm min}}^{y^{\rm max}}dy_{\rm q}\delta(y_{\rm q}-y_{\rm p}-\Delta
y)\int_{0}^{2\pi}d\phi_{\rm p}\int_{0}^{2\pi}d\phi_{\rm q}\delta(\phi_{\rm
q}-\phi_{\rm p}-\Delta\phi)$ (13) $\displaystyle\times\int^{{\rm
p}^{\rm{max}}_{\bot}}_{{\rm p}^{\rm{min}}_{\bot}}\frac{d{\rm
p}^{2}_{\bot}}{2}\int^{{\rm q}^{\rm{max}}_{\bot}}_{{\rm
q}^{\rm{min}}_{\bot}}\frac{d{\rm
q}^{2}_{\bot}}{2}\frac{dN^{\rm{corr}}_{\mathrm{2}}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}},$

and

$\displaystyle N_{\rm{Trig}}$ $\displaystyle=$
$\displaystyle\iiint_{\mathrm{Acceptance}}dyd^{2}\mathbf{p}_{\bot}\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}.$ (14)

The associated yield per trigger $Y(\Delta\phi,\Delta y)$ describes
correlations of two gluons with rapidity separation $\Delta y$ and azimuthal
separation $\Delta\phi$ in given transverse momentum intervals
($\mathrm{p^{min}_{\bot}}$, $\mathrm{p^{max}_{\bot}}$),
($\mathrm{q^{min}_{\bot}}$, $\mathrm{q^{max}_{\bot}}$) and rapidity window
$(y^{\rm min},y^{\rm max})$. In the following we do not distinguish between
trigger gluons and associated gluons, but we still use the name “associated
yield per trigger” for simplicity.

The framework is valid to leading-logarithmic accuracy in $x$ and momentum
${\rm p}_{\bot}$, ${\rm q}_{\bot}\mathrm{\gtrsim}Q_{s}$. The important
ingredient in the above expressions is uGD ($\Phi$), which can be obtained by
solving the BK equation with running coupling corrections with a given initial
condition. To avoid repetition, details can be found in Ref. cgc-NPA-2010  and
our previous paper zhao-1 ; zhao-2 . For pp collision at $7\ \mathrm{TeV}$,
$Q^{2}_{s0}$ (with $Q_{s0}$ the initial value of $Q_{s}$ at $x_{0}$) is chosen
to be $0.168\ \mathrm{GeV^{2}}$ cgc-PRD-I .

The solution of the BK equation is reliable when $x\leqslant 0.01$. For larger
values of $x$, the BK description breaks down. A phenomenological
extrapolation for the unintegrated gluon distribution has the form

$\Phi(x,\mathbf{k}_{\bot})=(\frac{1-x}{1-x_{0}})^{\beta}\Phi(x_{0},\mathbf{k}_{\bot}),\mbox{
for }x>x_{0},$ (15)

where $x_{0}=0.01$ and the parameter $\beta=4$cgc-NPA-2010 .

Both the experimental data CMS-pp-JHEP-2010 ; ALICE-pPb-PLB-2016  and the CGC
cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III  show that the correlation at near side
gets strongest only in an intermediate $\mathrm{p_{\bot}}$ interval,
approximately $1<\mathrm{p_{\bot}}<3\ \mathrm{GeV/c}$. A calculation from the
CGC points out that the correlation function gets a maximum when the
transverse momenta are close to $\mathrm{2Q_{sp}}\sim 2\ \mathrm{GeV/c}$ for
minimum-bias pp collisions, where $\mathrm{Q_{sp}}$ denotes the saturation
momentum of proton zhao-1 . To obtain the strongest correlation, the
correlation function is integrated within
$1\leq\mathrm{p_{\bot}}(\mathrm{q_{\bot}})\leq 3\ \mathrm{GeV/c}$ here.

##  III Rapidity window dependence of the ridge correlations

|  |
---|---|---
Figure 3: The associated yield per trigger on $\Delta y$-$\Delta\phi$ plane
for 7 TeV pp collisions with transverse momentum integrated within $1\leq
p_{\perp}(q_{\perp})\leq 3$GeV$/$c and rapidity integrated in (a) $-0.9\leq
y_{\rm p}(y_{\rm q})\leq 0.9$, (b) $-2.4\leq y_{\rm p}(y_{\rm q})\leq 2.4$ and
(c) $-4.2\leq y_{\rm p}(y_{\rm q})\leq 4.2$, respectively.

Ridge correlations are usually illustrated by two dimensional $\Delta
y$-$\Delta\phi$ distribution and one dimensional $\Delta\phi$ distribution. In
this paper we start with the calculation of the $\Delta y$-$\Delta\phi$
distribution, i.e. the associated yield per trigger in Eq. (12), which is
widely measured in CMS, ALICE and ATLAS experiments.

In order to study the rapidity window dependence of the long range ridge
correlations, three rapidity windows, i.e. $[-0.9,0.9],[-2.4,2.4]$ and
$[-4.2,4.2]$, are used. The correlations in pp collisions at $\sqrt{s}$ = 7
TeV with transverse momentum integrated within
$1\leq\mathrm{p_{\bot}}(\mathrm{q_{\bot}})\leq 3\ \mathrm{GeV/c}$ are plotted
on $\Delta y$-$\Delta\phi$ plane in Fig. 3.

The rapidity window $[-0.9,0.9]$ is chosen to be the same with the ALICE
acceptance. As Fig. 3(a) shows, the correlations have two moderate peaks of
equal height at $\Delta\phi=0$ and $\pi$ for a fixed $\Delta y$. For a fixed
$\Delta\phi$, the correlations show a downward trend as $|\Delta y|$
increases.

As rapidity window extends to $[-2.4,2.4]$, i.e. the CMS acceptance, the peaks
at $\Delta\phi=0$ and $\pi$ get pronounced. The double ridges, i.e. two raised
lines along $\Delta y$ direction at $\Delta\phi=0$ and $\pi$ on the surface
plot, are clearly seen in Fig. 3(b). The range of $\Delta y$ is only plotted
to $\pm 2$ so that a direct comparison can be made with Fig. 3(a). For longer
range ($|\Delta y|>2$) the ridges extend continuously which are not shown in
the plot.

When rapidity window extends to $[-4.2,4.2]$, the surface in Fig. 3(c) is
overall uplifted compared to Fig. 3(a) and 3(b). The correlations at long
range rapidity are uplifted more, as the color at large $|\Delta y|$ changes
from blue in Fig. 3(a) to light blue in Fig. 3(b) to green in Fig. 3(c). When
it comes to the $\Delta\phi$ direction, the amplitude of $\Delta\phi$
distribution, both at short range and long range rapidity gap, increases
significantly with increasing rapidity window.

Fig. 3 demonstrates an enhancement of correlations at long range rapidity gap
for larger $Y_{\rm W}$. As $Y_{\rm W}$ increases, rapidity correlations at
fixed $\Delta\phi$ change from a steep to a relatively flat shape, indicating
that long range rapidity correlations increase with rapidity window. On the
other hand, the amplitude of $\Delta\phi$ distribution at fixed $\Delta y$
also increases with rapidity window.

The downward trend in $\Delta y$ direction is gentle in the ALICE measurement
of the associated yield per trigger in pPb collisions at a center-of-mass
energy of 5.02 TeV ALICE-pPb-PLB-2013 . Compared to the ALICE results, the
downward trend here is much more steep. When rapidity window extends to
$[-2.4,2.4]$, ridges that can extend to $|\Delta y|=4$ are not as flat as
those obtained in CMS measurements, either CMS-pp-JHEP-2010 . It shows that
glasma graphs have significant short range rapidity correlations zhao-2 .

Of particular interest in studies of ridge is the long range (large $|\Delta
y|$) structure of two-particle correlation functions. In order to get a better
view of the ridge correlation at long range rapidity and its enhancement, we
plot one dimensional $\Delta\phi$ distribution at fixed long range rapidity
gap in Fig. 4(a). For $Y_{\rm W}$ of $[-0.9,0.9]$, the $\Delta\phi$
distribution at $\Delta y=1.8$ (1.8 units are the maximum rapidity gap for
that rapidity window) is straight, as the black curve shows. In fact, the two
peaks are still there. They are not visible in the plot just because the
amplitude is small. For $Y_{\rm W}$ of $[-2.4,2.4]$, the $\Delta\phi$
distribution at $\Delta y=2$ has a larger amplitude and the peaks at 0 and
$\pi$ is clearly seen, as the blue curve shows. As rapidity window gets larger
to $[-4.2,4.2]$, the peaks get more pronounced and the amplitude of the
distribution grows larger, as the red curve shows.

|
---|---
Figure 4: (a) The $\Delta\phi$ distribution at a fixed long range rapidity
gap. (b) The $\Delta\phi$ distribution with a constant background subtracted
by the ZYAM method.

According to the zero-yield-at-minimum (ZYAM) method note , the integrated
associated yield is defined as the area under the peak of the $\Delta\phi$
distribution above a constant background. The $\Delta\phi$ distribution with a
constant background subtracted by the ZYAM method is shown in Fig. 4(b). ZYAM
subtraction scheme makes the signals cleaner. The biggest amplitude is the red
curve, corresponding to the widest rapidity window. The area under the peaks
for the three curves qualitatively shows that the integrated associated yield
grows drastically with rapidity window.

The two peak structure of $\Delta\phi$ distribution at long range rapidity for
the three rapidity windows confirms an intrinsic collimation production in the
CGC formalism. Quantitative calculations about the near side ridge yield in pp
and pPb collisions are consistent with CMS, ATLAS and ALICE data cgc-PRD-I ;
cgc-PRD-II ; cgc-PRD-III . It means that small $x$ dynamics can explain the
ridge yield. The enhancement at wider rapidity window, demonstrated in Fig.
4(b) of this paper, indicates that large $x$ gluons, i.e. source gluons, make
significant contributions to the ridge yield.

The ridge structure is twofold. The azimuthal collimation in CGC comes from
Cauchy-Schwartz inequality Dumitru-PLB-2011 , which states that the azimuthal
correlations get maximum when the two transverse momenta are parallel or
antiparallel. The ridge at long range rapidity is due to the contribution of
color sources. Beyond that, we also demonstrate that source gluons enhance the
azimuthal collimation, indicating an interplay between longitudinal and
transverse directions.

The rapidity window of track reconstruction from the ALICE detector is very
different from the CMS and ATLAS. Whether the data from the three experiments
at the LHC shows the increasing trend with rapidity window as shown in Fig.
4(b) is hard to conclude at present. As mentioned before, long range angular
correlations are sensitive to centrality class and $\mathrm{p_{\bot}}$ window.
In this paper we show that long range angular correlations are also sensitive
to rapidity window. In order to see the trend with rapidity window, the
correlations should be compared at the same centrality and the same
$\mathrm{p_{\bot}}$ window. The ALICE collaboration reports long range angular
correlations in a centrality bin that is different from CMS and ATLAS CMS-pPb-
PLB-2013 ; ALICE-pPb-PLB-2013 ; ATLAS-pPb-PRL-2013 , which makes a direct
comparison between the three experiments impossible at the moment.

Enlarging the rapidity window is one way of seeing the contributions of source
gluons. Obviously, the integration over a given rapidity window includes
different combinations of gluons. In order to further clarify the
contributions of source gluons, a more direct observable is the differential
correlation function $C(\mathbf{p}_{\bot},y_{\rm p};\mathbf{q}_{\bot},y_{\rm
q})$, i.e. Eq.(2) with specific $y_{\rm p}$ and $y_{\rm q}$, which is shown in
Fig.5.

Figure 5: The differential correlation function at
$\mathrm{p_{\bot}}=\mathrm{q_{\bot}}=2.0$ GeV/c with $\phi_{\rm p}=\phi_{\rm
q}=0$. Black, blue and red curves denote $y_{\rm p}=0$, 2 and 3.5,
respectively.

A similar figure has already been shown in ref zhao-1 . The motivation of
presenting it here again is to understand the integrated correlations by the
differential correlations. The enhanced correlations originating from source
gluons is reflected more directly by the rapidity location dependence of the
differential correlation function.

The differential correlation function in Fig. 5 shows some interesting
correlation patterns. In the case of $y_{\rm p}=0$, i.e. the trigger particle
is at small $x$, the correlation function is almost flat and shows a rising
trend when $\Delta y$ exceeds 3 as the black curve shows. The solid, dash and
dash-point lines denote $x_{\rm q}<10^{-3}$, $10^{-3}<x_{\rm q}<10^{-2}$ and
$x_{\rm q}>10^{-2}$, labeling the results of the small, moderate and large-$x$
degrees of freedom of the associated particle, respectively. The point
connecting the dash line and dash-point line represents $x_{0}=0.01$. Since
quantitative results of the extrapolation Eq. (15) is unreliable, we only take
the dash-point lines that are near the connection points as large-$x_{\rm q}$.

A rapidity difference of only 2 units is sensitive to the small $x$ evolution,
denoted by the solid lines in Fig. 5. A wider region is sensitive to the
moderate and large $x$ degrees of freedom. The black curve indicates that the
correlation strength between radiated gluons (r for short), moderate-$x$
gluons (m for short) and source gluons (s for short) has

$C_{\rm rr}\approx C_{\rm rm}<C_{\rm rs},$ (16)

where each subscript represents a kind of gluon. It shows that correlations
between radiated gluons are not as strong as those with source gluons.

Similarly, the rising trend of the blue curve in Fig. 5 indicates

$C_{\rm mr}<C_{\rm mm}<C_{\rm ms},$ (17)

and the sharp rise of the red curve indicates

$C_{\rm sr}\ll C_{\rm sm}\ll C_{\rm ss}.$ (18)

$C_{\rm rr}$, $C_{\rm rm}$ and $C_{\rm ms}$ represents the green, blue and red
curves in Fig. 2, respectively. The strong correlations between the
moderate-$x$ gluon and the source gluon, i.e. $C_{\rm ms}$, make the ridge
yield enhanced significantly, as the red curve in Fig. 4(b) shows.

The ordering in Eqs. (16)-(18) demonstrates that correlations with source
gluons are the strongest for each type of trigger gluon. For trigger gluons at
non-central rapidity, like the case of $y_{\rm p}=2$ and 3.5, correlations
with small-x, moderate-x and large-x gluons show a continuously increasing
trend. The strongest correlations exist between source gluons. These ordering
patterns, especially the continuously rising trend of correlations between
trigger particle at non-central rapidity and associated particle at other
rapidities demonstrate characteristic features of CGC.

##  IV Summary and discussion

According to the relation of Feynman $x$ to rapidity $y$, different rapidity
regions reveal gluon dynamics at different $x$ and thus the rapidity window
dependence of long range ridge correlations is sensitive to the quantum
evolution of gluon saturation dynamics.

The ridge structure is twofold, i.e. the longitudinal and transverse
structures. Two dimensional $\Delta y$-$\Delta\phi$ distributions illustrate
that ridge structures in both directions have rapidity window dependence. As
the rapidity window enlarges, the longitudinal structure has a rising and
flattening trend at long range rapidity gap. It indicates an enhancement of
long range rapidity correlations. On the other hand, the angular distribution
at long range rapidity gap shows larger amplitudes as rapidity window
enlarges. It indicates the enhancement of azimuthal correlations and the
resulting ridge yields. In summary, ridge correlations in both longitudinal
and transverse directions get stronger as the rapidity window enlarges.

Within the CGC framework, the azimuthal collimation comes from Cauchy-Schwartz
inequality which requires parallel or antiparallel of two gluon transverse
momenta to get the maximum correlations. The long range rapidity correlations
is due to the contribution of color sources. As the rapidity window enlarges,
source gluons are included and begin to play a role. The enhancement in both
longitudinal and transverse correlations demonstrates that source gluons
enhance the long range rapidity correlations and azimuthal collimation.

While the origin of the ridge in small systems is inconclusive, it is already
widely known that the color flux tube picture naturally explains the long
range rapidity correlation and angular collimation within the CGC framework.
This paper notices that the rapidity of gluons is related to the quantum
evolution and the inclusion of large rapidity, or equivalently the source
gluons, will enhance the long range rapidity ridge correlations significantly.

So far the correlation patterns calculated are for gluons. It is shown in Ref.
Dusling-PRL-2012  that fragmentation functions only have a major impact on
transverse momentum dependence. The correlation patterns as a function of
rapidities would remain in the final state.

Since the quantum evolution of the CGC gives characteristic rapidity location
dependence of the differential correlation, differential measurements are more
direct for probing the CGC and are highly needed in the future measurement.
The correlation patterns of one particle at non-central rapidities with the
other particle at other rapidities can provide sensitive tests of this
picture. Similar suggestions have been made in Ref. Dumitru-PLB-2011 .

The work of Ref.Schenke-PRL-2016  where IP-glasma model is combined with Lund
string fragmentation demonstrates similar ridge structures. Flatter ridges
along rapidity direction shown there are a result of an exactly boost
invariant distribution implemented. The similarity in ridge structures
indicates that the long range rapidity ridge correlation in high energy
hadronic collisions is intrinsic to glasma dynamics.

Ultra-long range rapidity ridge, up to nearly 8 units of pseudorapidity
separation, has been observed in a preliminary result of ALICE experiment
ultra-long . The picture of classical fields has a restricted range of
validity. In a larger range of rapidity, quantum fluctuations become
important, and the description in terms of classical fields breaks down. For
large rapidity, one has less theoretical control within this framework. The
contribution of quarks should be considered.

## Acknowledgement

This work is supported in part by the NSFC of China under Grant No. U1732271.

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