# Forward-backward multiplicity and momentum correlations in $pp$ and $pPb$
collisions at the LHC energies

Joyati Mondal<EMAIL_ADDRESS>Nuclear and Particle Physics Research
Centre, Department of Physics,
Jadavpur University, Kolkata - 700032, India  Hirak Koley  Nuclear and
Particle Physics Research Centre, Department of Physics,
Jadavpur University, Kolkata - 700032, India  Somnath Kar
<EMAIL_ADDRESS>University Department of Physics, Kolhan University,
Chaibasa - 833201, India  Nuclear and Particle Physics Research Centre,
Department of Physics,
Jadavpur University, Kolkata - 700032, India  Premomoy Ghosh  Nuclear and
Particle Physics Research Centre, Department of Physics,
Jadavpur University, Kolkata - 700032, India  Argha Deb  Nuclear and Particle
Physics Research Centre, Department of Physics,
Jadavpur University, Kolkata - 700032, India  School of Studies in
Environmental Radiation &
Archaeological Sciences, Jadavpur University, Kolkata - 700032, India  Mitali
Mondal<EMAIL_ADDRESS>Nuclear and Particle Physics Research Centre,
Department of Physics,
Jadavpur University, Kolkata - 700032, India  School of Studies in
Environmental Radiation &
Archaeological Sciences, Jadavpur University, Kolkata - 700032, India

###### Abstract

Correlations and fluctuations between produced particles in an ultra-
relativistic nuclear collision remain one of the successor to understand the
basics of the particle production mechanism. More differential tools like
Forward-Backward (FB) correlations between particles from two different phase-
space further strengthened our cognizance. We have studied the strength of FB
correlations in terms of charged particle multiplicity and summed transverse
momentum for proton-proton ($pp$) and proton-lead ($pPb$) collisions at the
centre-of-mass energies $\sqrt{s}$ = 13 TeV and $\sqrt{s_{\rm NN}}$ = 5.02 TeV
respectively for the EPOS3 simulated events with hydrodynamical evolution of
produced particles. Furthermore, the correlation strengths are separately
obtained for the particles coming from the core and the corona. FB correlation
strengths are examined as a function of psedorapidity gap ($\eta_{gap}$),
psedorapidity window-width ($\delta\eta$), centre-of-mass energy ($\sqrt{s}$),
minimum transverse momentum ($p_{Tmin}$) and different multiplicity classes
following standard kinematical cuts used by the ALICE and the ATLAS
experiments at the LHC for all three EPOS3 event samples. EPOS3 model shows a
similar trend of FB multiplicity and momentum correlation strengths for both
$pp$ & $pPb$ systems, though the correlation strengths are found to be larger
for $pPb$ system than $pp$ system. Moreover, $\delta\eta$-weighted average of
FB correlation strengths as a function of different center-of-mass energies
for $pp$ collisions delineates a tendency of saturation at very high energies.

††preprint: APS/123-QED

##  I Introduction

The formation of a hot dense medium of quasi-free quarks and gluons, known as
the Quark-Gluon Plasma, in relativistic heavy-ion collisions at the
Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC)
unlatch the path to understand the early Universe as well as to testify the
theory of the strong interactions between quarks mediated by gluons [1, 2, 3].
The relativistic viscous hydrodynamic calculations [4, 5] have been found to
be most successful in explaining the properties of the produced hot and dense
matter in heavy-ion collisions and demonstrate the space-time evolution of the
medium through observables such as harmonic flow ($v_{n}$) [6, 7, 8] which
represent the translation of initial state spatial inhomogeneities to the
final state momentum anisotropies.

In heavy-ion collisions, the initial energy density fluctuates strongly event
to event which leads to the fluctuations of the space-time evolution of the
produced medium in the final state. Owing to the viscous hydrodynamics, such
density fluctuations are manifested as anisotropic harmonic flow. The large
initial state fluctuations effectuate the observed long-range correlations
(LRC) between final state particles which are observed as correlation between
multiplicity densities in different pseudorapidity ($\eta$)-windows [9].
Another aspect of longitudinal multiplicity correlations is the short-range
correlations (SRC), localized over a smaller range of $\eta$, manifested in
the single-jet, mini-jets, resonance decays etc. Forward-Backward (FB)
correlations between charged particle multiplicities or transverse momenta in
two symmetrically separated $\eta$-windows about the collision vertex promote
us to differentiate between LRC and SRC components [10] and to study the
dynamics of particle production mechanism in high-energy hadron or nuclear
collisions.

Positive FB multiplicity correlation strength was first observed in $p\bar{p}$
collisions at $\sqrt{s}$ = 540 GeV at CERN SPS collider [12] and it has been
then rigorously inspected in $p\bar{p}$ collisions at ISR energies from
$\sqrt{s}$ = 200 GeV to 900 GeV [13, 14]. Later, FB correlations were also
examined in $pp$ and $p\bar{p}$ collisions over wider range of collision
energies [15, 16, 17]. No significant FB multiplicity correlation has been
reported in $e^{+}e^{-}$ collisions [18] whereas a very weak correlation
strength was observed in $e^{+}e^{-}$ annihilation [19, 20]. CLAN structure
had been used for better understanding of observed stronger positive FB
correlation in $pp$ and $p\bar{p}$ collisions compared to weak correlation in
$e^{+}e^{-}$ annihilation [21]. There are also significant positive FB
correlation values reported in different collision systems e.g. $pp$, $pA$ and
$AA$ collisions [9, 22, 23, 24, 25]. The ATLAS [24] and the ALICE [25]
collaborations at the LHC reported strong FB correlations in $pp$ collisions
at $\sqrt{s}$ = 0.9, 2.76 and 7 TeV which contradicts STAR collaboration
findings of weak correlation [23].

To explain experimental results, numerous theoretical models have been put
forth. In Dual Parton Model (DPM) [11] a Pomeron exchange between colliding
hadrons was initially thought of as an inelastic scattering, later, the idea
of many Pomeron exchanges was implemented. DPM projected that particles
created in two well selected rapidity intervals would have a significant long-
range correlation [26]. The Quark Gluon String Model (QGSM) [27, 28] is
similar to DPM with some essential differences. In this model new objects,
quark-gluon strings are formed which fragment into hadrons and resonances. The
QGSM model successfully described ALICE data and concluded that the
multistring processes due to multi-Pomeron exchanges were the main contributor
to the FB correlations. The String Fusion Model (SFM) [29, 30] incorporates
the string fusion phenomenon and is based on the Parton model of strong
interactions. The SFM framework is used to investigate the characteristics of
the strongly intensive variable that characterizes correlations between the
number of particles in two separated rapidity intervals in $pp$ interactions
at LHC energy [31]. Using various dynamics of the string interaction
assumptions, correlations between multiplicities and average transverse
momentum are accomplished in the percolating colour strings picture [32, 33].
A string percolation process in $pp$ collisions was used to study the FB
correlations [32] and observed an approximately constant FB correlation over a
substantial range of rapidity window.

The correlations between mean transverse momentum and multiplicity of charged
particles in $pp$ and $p\bar{p}$ collisions at $\sqrt{s}$ from 17 GeV to 7 TeV
are studied using a modified multi-pomeron exchange model in which string
collectivity has been included in an effective way [34]. In $pp$, $pPb$, and
$PbPb$ collisions at LHC energies it is explored using a dipole based Monte
Carlo String Fusion Model [35]. According to the Color Glass Condensate model
[36, 37, 38] long-range rapidity correlations continue throughout the
development of the quark–gluon plasma that result from the collision. Using a
model that regards strings as independent identical emitters, the forward-
backward (FB) charged particle multiplicity correlations between windows
spaced apart in rapidity and azimuth are investigated in Ref. [39].
Theoretical background of long-range correlations in heavy-ion collisions has
been studied using a Monte-Carlo method in Ref. [40].

The high multiplicity data of $pp$ and $pPb$ collisions at the LHC and $dAu$
collisions at the RHIC show some collective-like features resembling the
heavy-ion collision [41, 42, 43, 44, 45, 46, 47, 48]. The two-particle
correlation studies in high multiplicity $pp/pPb$ collisions showed the heavy-
ion signature; “the ridge” which triggered many theoretical
discussion/interpretations on the origin of it. Recent studies [49, 50] show
that the hydrodynamical modelling which remains successful in explaining many
features of heavy-ion collisions, is also found to be applicable in small
collision systems. We discussed in our previous article [51] on how EPOS3
model [52] with hydrodynamical evolution of produced particles (referred as
“with hydro” in rest of the texts) successfully reproduced many features of
small collision systems at the LHC energies [53]. Furthermore, we investigated
the FB multiplicity and momentum correlations using EPOS3 model by switching
ON/OFF hydrodynamical evolution of produced particles which does not affect
much the final outcomes. Studies using different models show that the FB
correlation strength is found to be decreasing with increasing nuclear size
upon the selected $\eta$-windows and with increasing collision energy for a
fixed collision system [54, 55]. It has also been proposed that instead of the
contribution coming from particle production in initial stages of collisions,
subsequent stage could modify the behaviour of FB correlations and hadron
nucleus collision is expected to give more information on the whole scenario.
Keeping this in mind, a comparative analysis of $pp$ and $pPb$ systems has
been performed to improve our current understanding of FB phenomena. We have
inspected FB multiplicity and momentum correlation in $pp$ and $pPb$
collisions at the centre-of-mass energy $\sqrt{s}$ = 13 TeV and $\sqrt{s_{\rm
NN}}$ = 5.02 TeV respectively using EPOS3 simulated with hydro events. In
order to clarify the functions of each component of the model that contribute
to the outcomes, we further divided the model into core and corona approaches.
The energy density of the strings in the core is sufficient to activate the
hydrodynamically evolving QGP description. In the corona, hadron creation from
nucleon-nucleon collisions is viewed as an independent phenomenon [52].

The rest of the paper is organized as follows: the definition of FB
multiplicity and momentum correlation coefficients are introduced in Section
II. Section III describes briefly the basic principles of EPOS3 model and
sample size of generated events. Choice of EPOS3 simulated events and FB
windows are illustrated in Section IV. In Section V, the dependence of FB
correlation strength by varying $\eta_{gap}$, $\delta\eta$, $p_{T_{min}}$ and
different multiplicity classes have been discussed in details. More
importantly, the behaviour of $\delta\eta$-weighted average of FB multiplicity
and momentum correlation strengths as a function of centre-of-mass energy
using EPOS3 simulated $pp$ events have been studied for the first time.
Finally, conclusions are drawn in Section VI.

##  II Forward-backward Charged Particle Multiplicity and Momentum Correlation
Coefficient

Forward-backward correlations are measured between different observables in
separated $\eta$-intervals, namely n–n (the correlation between charged-
particle multiplicities), $p_{T}$–$p_{T}$ (the correlation between mean or
summed transverse momenta of charged particles) and $p_{T}$–n (the correlation
between mean or summed transverse momenta in one pseudorapidity interval and
the multiplicity of charged particles in another pseudorapidity interval)
[56]. Two $\eta$-intervals, one from the forward and another from the backward
window, are symmetrically chosen around the collision centre. The detailed
window construction has already been shown and discussed in Ref. [51].

A linear relationship between average charged particle multiplicity in
backward window ($\langle N_{b}\rangle_{N_{f}}$) and the charged particle
multiplicity in the forward window ($N_{f}$) has been reported and discussed
in Ref. [13, 14]:

$\langle N_{b}\rangle_{N_{f}}=a+b_{corr}(mult)N_{f}$ (1)

Here, FB multiplicity correlation strength is characterized by
$b_{corr}(mult)$. Considering the linear relation between ($\langle
N_{b}\rangle_{N_{f}}$) and $N_{f}$, $b_{corr}(mult)$ can be determined using
the following Pearson Correlation Coefficient:

$b_{corr}(mult)=\frac{\langle N_{f}N_{b}\rangle-\langle N_{f}\rangle\langle
N_{b}\rangle}{\langle N_{f}^{2}\rangle-\langle N_{f}\rangle^{2}}$ (2)

The measurement of FB multiplicity correlation coefficient $b_{corr}(mult)$ is
defiled by the so-called “volume fluctuations”, which arises due to the event-
by-event fluctuations of the number of the participating nucleons [57, 58].
Hence, we have considered an intensive observable like the sum of the absolute
transverse momentum of charged-particles within the selected $\eta$ windows to
reduce the contribution of volume fluctuations. Similar to the multiplicity
correlation we have estimated FB momentum correlation coefficient,
$b_{corr}(\Sigma p_{T})$ using the following formula:

$\small b_{corr}(\Sigma p_{T})=\frac{\langle\Sigma p_{T_{f}}\Sigma
p_{T_{b}}\rangle-\langle\Sigma p_{T_{f}}\rangle\langle\Sigma
p_{T_{b}}\rangle}{\langle(\Sigma p_{T_{f}})^{2}\rangle-\langle\Sigma
p_{T_{f}}\rangle^{2}}$ (3)

Here, $\Sigma p_{T_{f}}$ ($\Sigma p_{T_{b}}$) denotes the event averaged
transverse momenta of charged-particles in forward (backward) window.

An intuitive observable, $\delta\eta$-weighted average of FB multiplicity and
momentum correlation strength has been introduced for the first time and is
defined as follows:

$\small<b_{corr}(mult/\Sigma
p_{T})>_{\delta\eta}=\frac{\Sigma_{i}b_{corr}(mult/\Sigma
p_{T})_{i}\delta\eta_{i}}{\Sigma_{i}\delta\eta_{i}}$ (4)

The behaviour of such observable has been studied as a function of the centre-
of-mass energy in $pp$ collisions taking into account of our earlier
measurements in smaller collision system [51].

Figure 1: Charged-particle invariant yields as a function of $p_{\rm T}$ in
$pp$ collisions at $\sqrt{s}$ = 13 TeV (top) and in $pPb$ collisions at
$\sqrt{s_{\rm NN}}$ = 5.02 TeV (bottom) compared to ALICE data [61, 62].

Figure 2: (Average) Pseudorapidity density of charged particles in ($pp$
collisions at $\sqrt{s}$ = 13 TeV (top)) $pPb$ collisions at $\sqrt{s_{\rm
NN}}$ = 5.02 TeV (bottom) compared to ALICE data [61, 63].

##  III EPOS3 Model

The EPOS3 model [52] is based on Gribov-Regge multiple scattering theory. In
this approach an individual scattering is labeled as a ‘Pomeron’. A pomeron
creates a parton ladder which may be considered as longitudinal flux tube
carrying the transverse momentum from the initial hard scatterings [59]. In a
collision, many elementary parton-parton hard scatterings form a large number
of flux tubes that expand and are fragmented into string segments. Higher
string density forms the so-called “core” which undergoes a three-dimensional
(3D)+1 viscous hydrodynamical evolution expecting no jet parton escape and
hadronizes via usual Cooper-Frye formalism at a “hadronization temperature”,
TH. Another part of lower string density forms the so-called “corona” where we
can expect the escape of jet partons. Such string segments having high
transverse momentum that are close to the surface leave the bulk matter and
hadronize (including jet hadrons) via the Schwinger mechanism. The phase
transition from parton to hadron follows a realistic equation of state which
is compatible with the lattice gauge results with subsequent hadronic cascade
using UrQMD model [60].

Using EPOS3 model, we have generated 3 million minimum-bias $pp$ events at
$\sqrt{s}$ = 13 TeV and $pPb$ events at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. On top
of the minimum bias analysis, a more differential approach has been introduced
by taking particles coming from either core or corona and we have varied
certain model parameter in order to achieve it. We have measured FB
multiplicity and momentum correlations for the EPOS3 generated events with all
charged particles and particles from core and corona.

To validate the generated event samples of different centre-of-mass energies,
we have compared minimum bias EPOS3 simulated events with ALICE data [61, 62,
63]. Fig. 1 shows that the invariant yields of charged-particles as a function
of $p_{\rm T}$ as measured by ALICE experiment in $pp$ collisions at
$\sqrt{s}$ = 13 TeV (top) and in $pPb$ collisions at $\sqrt{s_{\rm NN}}$ =
5.02 TeV (bottom) have been successfully reproduced by the EPOS3 simulated
events at the chosen energies. Average pseudorapidity density and
pseudorapidity density of charged-particles has been plotted in Fig. 2 for
EPOS3 simulated $pp$ events at $\sqrt{s}$ = 13 TeV (top) and pPb events at
$\sqrt{s_{\rm NN}}$ = 5.02 TeV (bottom) respectively. Compared results reflect
that EPOS3 simulated events agreed well with the experimental measurements in
the chosen kinematic intervals [61, 63].

##  IV Events & FB Window Selection

Events are selected with a minimum of two charged particles in the chosen
kinematic interval. The whole analyses for the $pp$ and $pPb$ events have been
carried out following ALICE [25] and ATLAS [24] kinematics. By the word ALICE
kinematics we mean the cuts on the kinematic variables $p_{\rm T}$ and $\eta$
as 0.3 $<p_{\rm T}<$1.5 GeV/c and $|\eta|<$ 0.8 respectively. Similarly for
the ATLAS kinematics we use $p_{\rm T}>$ 0.1 GeV/c and $|\eta|<$ 2.5. Only
caveat is those cuts were used for lower centre-of-mass energies for $pp$
collisions by the ALICE and the ATLAS Collaborations.

##  V Results & Discussions

We have calculated and plotted in Fig. 3 the average backward multiplicity
($\langle N_{b}\rangle_{N_{f}}$) for each fixed value of forward multiplicity
$N_{f}$ for window width $\delta\eta=$ 0.6 and $\eta_{gap}=$ 0.4 for EPOS3
simulated $pp$ events at $\sqrt{s}$ = 13 TeV (left panel) and $pPb$ events at
$\sqrt{s_{\rm NN}}$ = 5.02 TeV (right panel). From the scatter plots we can
see a linear relationship between $\langle N_{b}\rangle_{N_{f}}$ and $N_{f}$.
A linear fit has been displayed by the red lines in both panels. Slope of
these lines actually quantify the correlation strength between multiplicities
in F $\&$ B windows.

Figure 3: Variation of $\langle N_{b}\rangle_{N_{f}}$ with $N_{f}$ for FB
window width $\delta\eta$ = 0.6 and $\eta_{gap}$ = 0.4 for EPOS3 generated
$pp$ events at $\sqrt{s}$ = 13 TeV (left panel) and $pPb$ events (right panel)
at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. A linear fit has been performed (red line)
for both the systems.

We have, therefore, applied Pearson Correlation Coefficient formula described
in Eq. 2 to compute FB multiplicity correlation strengths. To eliminate the
incorporated volume fluctuations in FB multiplicity correlation, we have
evaluated FB momentum correlation coefficient, $b_{corr}(\Sigma p_{\rm T})$
using Eq. 3 for the EPOS3 simulated $pp$ events at $\sqrt{s}$ = 13 TeV and
$pPb$ events at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.

###  V.1 Dependence on the gap between
FB windows ($\eta_{gap}$)

The variation of FB multiplicity and momentum correlation coefficients with
$\eta_{gap}$ for four different window widths ($\delta\eta$ = 0.2, 0.4, 0.6 &
0.8) have been shown in Fig. 4 and Fig. 5 respectively for EPOS3 simulated all
charged particles, only-core and only-corona particles in $pp$ collisions at
$\sqrt{s}$ = 13 TeV (top panel) and $pPb$ collisions at $\sqrt{s_{\rm NN}}$ =
5.02 TeV (bottom panel).

Figure 4: FB multiplicity correlation strength, $b_{corr}$(mult) as a function
of $\eta_{gap}$ for $\delta\eta$ = 0.2, 0.4, 0.6 and 0.8 for EPOS3 generated
all charged particles and particles from core & corona in $pp$ collisions at
$\sqrt{s}$ = 13 TeV (top panel) and $pPb$ collisions (bottom panel) at
$\sqrt{s_{\rm NN}}$ = 5.02 TeV.  Figure 5: Forward-backward summed-$p_{\rm T}$
correlation as a function of $\eta_{gap}$ for four window widths $\delta\eta$
= 0.2, 0.4, 0.6 and 0.8 for EPOS3 generated all charged particles and
particles from core & corona in $pp$ and $pPb$ collisions at $\sqrt{s}$ = 13
TeV (top panel) and $\sqrt{s_{\rm NN}}$ = 5.02 TeV (bottom panel)
respectively.

We have compared all three cases for window widths $\delta\eta$ = 0.2 & 0.4
(left panel) and $\delta\eta$ = 0.6 & 0.8 (right panel) for both $pp$ and
$pPb$ events. We observed that for a fixed window width, the FB correlation
strengths decrease slowly with the increasing $\eta_{gap}$ and increase with
increasing $\delta\eta$ at a fixed $\eta_{gap}$ which resemble the trend at
lower centre-of-mass energies in $pp$ collisions as described in our earlier
study [51].

Quantitatively we found that the correlation strengths are larger for $pPb$
collisions than $pp$ collisions for all the chosen $\eta_{gap}$ and
$\delta\eta$ combinations. The asymmetric nature of $pPb$ collisions where the
proton collides with a nucleus having a larger number of sources compared to
$pp$ collisions, results in a larger initial-state parton density in the lead
nucleus compared to the proton. Such asymmetric collisions could have larger
fluctuations in the final state which may contribute to stronger forward-
backward correlation strength in $pPb$ collisions w.r.t $pp$ collisions.

Interestingly, we have noticed that for $pPb$ events the correlation strengths
decrease faster with increasing $\eta_{gap}$ as compared to $pp$ events. The
SRC component depends strongly on collision system and it is asymmetric
between forward and backward windows in $pPb$ collisions while the LRC
component is nearly symmetric in all collision systems [9]. Thus, the faster
dilution of SRC component at large $\eta_{gap}$ between forward and backward
regions could be the reason behind the faster decrease of correlation strength
in asymmetric $pPb$ collisions w.r.t symmetric $pp$ collisions.

The dominance of correlation strength due to only-core particles is clearly
visible over only-corona particles. Since the particles from corona are mostly
dominated by jets or minijet partons, the paucity of LRC component results in
smaller correlation strength for particles from corona over core at large
$\eta_{\rm gap}$ for both the collision systems.

###  V.2 Dependence on the width of FB windows ($\delta\eta$)

The $\delta\eta$-dependence of FB multiplicity and momentum correlation
coefficients for contiguous ($\eta_{gap}$ = 0), symmetrical windows with
respect to collision centre are shown in Fig. 6 and Fig. 7 for $\sqrt{s}$ = 13
TeV in $pp$ collisions and $\sqrt{s_{\rm NN}}$ = 5.02 TeV in $pPb$ collisions
using EPOS3 simulated with hydro events.

Figure 6: FB multiplicity correlation strength, $b_{corr}$(mult) as a function
of $\delta\eta$ for $\eta_{gap}$ = 0 using EPOS3 generated $pp$ and $pPb$
events at $\sqrt{s}$ = 13 TeV and $\sqrt{s_{\rm NN}}$ = 5.02 TeV respectively
for all charged particles and particles from core & corona.

We have studied and presented the multiplicity and momentum correlation
coefficients for the EPOS3 generated event samples with all charged particles
and particles from core & corona. We observed that the only-core and only-
corona cases underestimate both the correlation strengths for minimum-bias
event sample. The correlation coefficients increase non-linearly with
$\delta\eta$ for both $pp$ and $pPb$ events though the values are higher in
case of $pPb$ events which may be due to fact as described in V.1. The results
are found to be similar to our earlier measurements qualitatively [51],
featuring the dominance of SRC component for the non-linear growth of the FB
correlation strengths. As discussed and explained in V.1, here also we have
found that the correlation strengths w.r.t $\delta\eta$ are larger for only-
core particles than only-corona particles.

Figure 7: Contributions of $b_{corr}(\Sigma p_{T})$ on $\delta\eta$ for
$\eta_{gap}$ = 0 for EPOS3 generated $pp$ and $pPb$ events.

###  V.3 Dependence on collision energy ($\sqrt{s}$)

We have examined the behaviour of $\delta\eta$-weighted average of FB
multiplicity and momentum correlation strengths as a function of centre-of-
mass energy using EPOS3 simulated $pp$ event samples. Such an unconventional
measurement is still not available experimentally. Hence, to compare our
findings in a systematic way we have evaluated $\delta\eta$-weighted average
for the available experimental $b_{corr}(mult)$ and $b_{corr}(\Sigma p_{\rm
T})$ values for the ALICE [25] and the ATLAS [24] data. Fig. 8 and Fig. 9 show
the $\delta\eta$-weighted average of FB multiplicity and momentum correlation
as a function of centre-of-mass energy following ALICE and ATLAS kinematics.

In the left panel of Fig. 8 and Fig. 9, we observed that initially $\langle
b_{corr}(mult)\rangle_{\delta\eta}$ and $\langle b_{corr}(\Sigma p_{\rm
T})\rangle_{\delta\eta}$ values increase rapidly with increasing $\sqrt{s}$
upto 2.76 TeV, then grows moderately upto $\sqrt{s}$ = 7 TeV for both EPOS3
simulated events and experimental data. For comparison, we have incorporated
the results from other available theoretical models which also show a similar
trend for the $\delta\eta$-weighted average of FB correlations [28, 39].

It is also very interesting to find that for EPOS3 simulated events, the
$\delta\eta$-weighted average of FB correlation strengths as a function of
$\sqrt{s}$ lean towards saturation approximately beyond $\sqrt{s}$ = 7 TeV,
where there is no available experimental data for such a measurement. The
results from QGSM model although do not show such a strong saturation effect
at higher energy.

To gain a more nuanced understanding of the fascinating behavior observed in
the $\delta\eta$-weighted average of FB correlation strengths, we computed
this metric separately for the particles coming either from core or corona. We
have found that the observed saturation at higher centre-of-mass energy
predominantly due to the saturation for the only-core particles whereas the
only-corona particles show increasing trend.

Figure 8: Comparison of $\delta\eta$-weighted average FB multiplicity
correlations, ($\langle b_{corr}(mult)\rangle_{\delta\eta}$) as a function of
$\sqrt{s}$ for the EPOS3 simulated $pp$ events (all charged particles, core
and corona) with derived ALICE (left), ATLAS (right) data and theoretical
models (left).

In EPOS3 model, the corona is dominated by the high-$p_{T}$ particles whereas
the core contains particles which undergo 3+1D hydro mimicking the formation
of QGP-like medium [64].

The exchange of multiple pomerons between colliding particles [65] in a
collision remains the primary sources of fluctuations producing multiple
particles in a correlated way. The multiplicity of produced particle and their
transverse momentum are thus very much influenced by the initial conditions of
a collision and in particluar are much more apparent in small collision
systems like $pp$ or $pA$ where final state effects are less. In the CGC
framework [66, 67] it has been argued that at small $x\sim p_{T}/\sqrt{s}$,
gluon density first grows and then gets saturated with increase in energy
which results in moderate increase in charge particle multiplicity density in
$pp$ or $pA$ collisions as the beam energy increases [68]. Since the
observable like bcorr (mult or $\Sigma p_{\rm T}$) is an extensive quantity,
it might show such saturation effect mainly because fluctuations associated
with the number of sources get saturated [65]. Henceforth, the EPOS3 model
based FB correlation analysis at higher center-of-mass energy encourages
further experimental study.

Figure 9: Comparison of $\delta\eta$-weighted average FB summed-$p_{\rm T}$
correlations, ($\langle b_{corr}(\Sigma p_{\rm T})\rangle_{\delta\eta}$) as a
function of $\sqrt{s}$ for EPOS3 simulated $pp$ events (all charged particles,
core and corona) following ALICE kinematics (left) and with derived ATLAS
(right) data.

###  V.4 Dependence on the minimum transverse momentum ($p_{\rm T_{min}}$)

The variation of FB multiplicity and momentum correlations with minimum
transverse momentum of charged particles ($p_{\rm T_{min}}$) have been shown
in Fig. 10 and Fig. 11 for both EPOS3 generated with hydro $pp$ events at
$\sqrt{s}$ = 13 TeV and $pPb$ events at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
following ATLAS kinematics [24]. We calculated the values of $b_{corr}$ at
seven different levels of minimum transverse momentum ($p_{\rm T_{min}}$),
specifically at $p_{\rm T_{min}}$ = 0.1, 0.2, 0.3, 0.5, 1.0, 1.5, and 2.0 GeV.
These calculations were performed for symmetric forward-backward (FB) windows
without any separation.

Figure 10: Forward-backward multiplicity correlations as a function of $p_{\rm
T_{min}}$ for window width $\delta\eta$ = 0.5 for EPOS3 simulated $pp$ and
$pPb$ events.

The multiplicity and momentum correlations strengths decrease rapidly with the
increase of $p_{\rm T_{min}}$ values for both $pp$ and $pPb$ events confirming
similar trend at lower center-of-mass energies [51]. With the increase of
$p_{\rm T_{min}}$, domination of LRC component decreases resulting weaker FB
multiplicity correlation strength suggesting the transition from soft process
to hard processes with increasing transverse momentum of the produced
particles. As discussed in V.1, here also the correlation strengths are found
to be greater for $pPb$ collisions than $pp$ collisions.

Figure 11: Forward-backward summed-$p_{\rm T}$ correlation as a function of
$p_{\rm T_{min}}$ for window width $\delta\eta$ = 0.5 for EPOS3 generated $pp$
and $pPb$ events.

###  V.5 Multiplicity dependent $b_{corr}(\Sigma p_{\rm T}$)

Figure 12: Forward-backward summed-$p_{\rm T}$ correlations as a function of
$\eta_{gap}$ for window width $\delta\eta$ = 0.5 in different multiplicity
ranges for EPOS3 simulated $pp$ events at $\sqrt{s}$ = 13 TeV. Figure 13:
Forward-backward summed-$p_{\rm T}$ correlations as a function of $\eta_{gap}$
for window width $\delta\eta$ = 0.5 in different multiplicity ranges for EPOS3
simulated $pPb$ events at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.

In addition to the study using minimum-bias EPOS3 events, we have exploited a
multiplicity-dependent summed-$p_{\rm T}$ FB correlations study as well. Fig.
12 and Fig. 13 show FB momentum correlation as a function of $\eta_{gap}$ for
$\delta\eta$ = 0.5 in three multiplicity ranges estimated following ATLAS
kinematics [24] using EPOS3 simulated $pp$ events at $\sqrt{s}$ = 13 TeV and
$pPb$ events at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. Red, blue and green points
correspond to non-overlapping multiplicity regions; Low- (2 $<\rm N_{ch}<$
30), Mid- (30 $<\rm N_{ch}<$ 60) and High-multiplicity (60 $<\rm N_{ch}<$ 90)
regions respectively. We have kept the multiplicity ranges same for both $pp$
and $pPb$ events for better understanding. We can see the similar decrease of
correlation strength with increasing $\eta_{gap}$ and for a fixed $\eta_{gap}$
value, $b_{corr}(\Sigma p_{\rm T})$ also decreases with increasing
multiplicity which may be due to the fact that fusion of strings into core in
high multiplicity EPOS3 events lowers the FB correlation strength [51].

##  VI Summary and Conclusions

We have performed a rigorous study on FB multiplicity and momentum correlation
in $pp$ and $pPb$ collisions at the centre-of-mass energies, $\sqrt{s}$ = 13
TeV and $\sqrt{s_{\rm NN}}$ = 5.02 TeV respectively at the LHC using EPOS3
simulated events with all charged particles and particles from core & corona.
We have investigated the behaviour of FB correlation strengths on gap between
two pseudorapidity windows ($\eta_{gap}$), window width ($\delta\eta$),
minimum transverse momentum ($p_{Tmin}$) and different multiplicity classes.
Many LHC findings confirm a strong analogy between the small collision
systems, $pp$ and $pPb$ particularly in terms of particles correlations and
fluctuations [42, 43, 69]. We have also noticed here that general trends of
both FB correlation strengths are similar in case of both EPOS3 simulated with
hydro $pp$ and $pPb$ collisions irrespective of energy difference. We have
found following results:

  * •

The linear relationship between $\langle N_{b}\rangle_{N_{f}}$ and $N_{f}$ is
verified for EPOS3 generated both $pp$ and $pPb$ events and is more steeper
for the $pPb$ events than $pp$ events.

  * •

Our model based study fairly describes two general features of both FB
correlation coefficients in two different collision systems ($pp$ and $pPb$)
that it decreases slowly with the increase of gap between two selected
$\eta$-windows irrespective of window widths and increases non-linearly with
window width for a fixed separation between $\eta$-windows.

  * •

Rapid decrease in the values of both $b_{corr}(mult)$ and
$b_{corr}(\sum{p_{T}})$ with the small increase of minimum transverse momentum
$p_{Tmin}$ has been observed for both $pp$ and $pPb$ events.

  * •

Multiplicity dependent summed-$p_{T}$ correlation study also reveals that with
the increase of multiplicity the value of $b_{corr}(\sum p_{T})$ decreases at
a fixed $\eta_{gap}$ for both $pp$ and $pPb$ events.

All these facts resemble our previous assessment of FB correlations in $pp$
collisions at three comparatively lower centre-of-mass energies, $\sqrt{s}$ =
0.9, 2.76 & 7 TeV [51]. The observed larger FB multiplicity and momentum
correlation strength in $pPb$ collisions w.r.t $pp$ collisions could be due to
the fact that the initial asymmetry of the $pPb$ collisions and the large
system size w.r.t the $pp$ collisions may enhance the event-by-event
fluctuations which in turn may increase the FB correlation.

The most interesting result from our present study is the behaviour of
$\delta\eta$-weighted average of FB multiplicity and momentum correlation
strengths as a function of centre-of-mass energy ($\sqrt{s}$ = 0.9, 2.76, 7 &
13 TeV) using EPOS3 simulated $pp$ events. The increase of both the
correlation strengths ($b_{corr}(mult)$ & $b_{corr}(\Sigma p_{\rm T})$) with
$\sqrt{s}$ is clearly visible and interestingly we observed that it tends to
saturate at very high energy. We have incorporated different theoretical model
studies to compare our results and the correlation strengths are found to be
following the similar trend as of EPOS3 simulated events. To investigate the
possible reason behind such an interesting observation, we have calculated the
correlation strengths for the EPOS3 generated events with all charged
particles and particles from core & corona. We found that for the particles
from corona, the $\delta\eta$-weighted average of FB correlations do not show
any saturation trend whereas for particles form core are perfectly exhibited
the trend. We may infer that it may be due to the dominance of gluon-
saturation effect at such higher centre-of-mass energies.

Our analyses uncover the fact that the FB multiplicity and momentum
correlation as a function of $\eta_{gap}$, $\delta\eta$, $p_{Tmin}$ and
different multiplicity classes in EPOS3 simulated $pPb$ events qualitatively
resemble the outcome of EPOS3 simulated $pp$ events though the values of
correlation coefficients are higher for $pPb$ events than those for $pp$
events. Overall, we may conclude that the systematic study on FB correlations
in different dimensions using the hybrid Monte-Carlo model EPOS3 [52] adds
more valuable information to understand the existing experimental results as
well as encourages more experimental measurements at higher centre-of-mass
energies and in different collision systems.

## Acknowledgements

The authors are thankful to Dr. Klaus Warner for providing us with the EPOS3
model. The authors are thankful to the members of the grid computing team of
VECC and cluster computing team of dept. of Physics, Jadavpur University for
providing uninterrupted facility for event generation and analyses. We also
gratefully acknowledge the financial help from the DST-GOI under the scheme
“Mega facilities in basic science research” [Sanction Order No.
SR/MF/PS-02/2021-Jadavpur (E-37128) dated 31.12.2021]. One of the authors (JM)
acknowledges DST-INDIA for providing fellowship under INSPIRE Scheme.

## References

  * [1] E. Shuryak, Rev. Mod. Phys. 89, 035001 (2017)
  * [2] P. Braun-Munzinger, V. Koch, T. Schaefer et al., Phys. Rep. 621, 76 (2016)
  * [3] K. Fukushima and T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011)
  * [4] C. Gale, S. Jeon, and B. Schenke, Int. J. Mod. Phys. A28, 1340011 (2013)
  * [5] U. Heinz and R. Snellings, Annu. Rev. Nucl. Part. Sci. 63, 123 (2013)
  * [6] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. Lett. 106, 042301(2011)
  * [7] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C85, 024901(2012)
  * [8] G. Aad et al. (ATLAS Collaboration), Phys. Rev. C86, 014907 (2012)
  * [9] M. Aaboud et al. (ATLAS Collaboration), Phys. Rev. C95, 064914 (2017)
  * [10] A. Capella and A. Krzywicki, Phys. Rev. D18, 4120 (1978)
  * [11] A. Capella, U. Sukhatme, C.-I. Tan, and J. T. T. Van, Phys. Rep. 236, 225 (1994)
  * [12] K. Alpgard et al. (UA5 Collaboration), Phys. Lett. B123, 361 (1983)
  * [13] G. J. Alner et al. (UA5 Collaboration), Phys. Rep. 154, 247 (1987)
  * [14] R. E. Ansorge et al. (UA5 Collaboration), Z. Phys. C37, 191 (1988)
  * [15] L. V. Bravina et al., Sov. J. Nucl. Phys. 50, 245 (1989)
  * [16] V. V. Aivazyan et al. (NA22 Collaboration), Z. Phys. C42, 533 (1989)
  * [17] T. Alexopoulos et al. (E735 Collaboration), Phys. Lett. B353, 155 (1995)
  * [18] M. Derrick et al. (HRS Collaboration), Phys. Rev. D34, 3304 (1986)
  * [19] W. Braunschweig et al. (TASSO Collaboration), Z. Phys. C45, 193 (1989)
  * [20] R. Akers et al. (OPAL Collaboration), Phys. Lett. B320, 417 (1994)
  * [21] A. Giovannini and R. Ugoccioni, Phys. Rev. D66, 034001 (2002)
  * [22] B. B. Back et al. (PHOBOS Collaboration), Phys. Rev. C74, 011901 (2006)
  * [23] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 103, 172301 (2009)
  * [24] G. Aad et al. (ATLAS Collaboration), J. High Energy Phys. 07 019 (2012)
  * [25] J. Adam et al. (ALICE Collaboration), J. High Energy Phys. 05 097 (2015)
  * [26] A. Capella and J. T. T. Van, Z. Phys. C18, 85 (1983)
  * [27] A. B. Kaidalov, Phys. Lett. 116B, 459 (1982); Phys. At. Nucl. 66, 1994 (2003)
  * [28] L. V. Bravina, J. Bleibel, and E. E. Zabrodin, Phys. Lett. B787, 146 (2018)
  * [29] N. S. Amelin , M. A. Braun, C. Pajares, Physics Letters B306 (1993) 312; Z. Phys. C63 (1994) 507; N. S. Amelin, N. Armesto, M. A. Braun, E. G. Ferreiro, and C. Pajares, Phys. Rev. Lett. 73, 2813 (1994)
  * [30] M. A. Braun, C. Pajares, Nucl. Phys. B390, 542 (1993)
  * [31] V. V. Vechernin and S. N. Belokurova 2020 J. Phys.: Conf. Ser. 1690 012088; E. Andronov & V. V. Vechernin, Euro. Phys. J. A55, 14 (2019)
  * [32] P. Brogueira, J. Dias de Deus, C. Pajares, Phys. Lett. B675, 308 (2009)
  * [33] M. A. Braun, R. S. Kolevatov, C. Pajares, and V. V. Vechernin, Eur. Phys. J. C32, 535 (2004)
  * [34] E. O. Bodnya, V. N. Kovalenko, A. M. Puchkov and G. A. Feofilov AIP Conference Proceedings 1606, 273 (2014)
  * [35] V. N. Kovalenko, Physics of Particles and Nuclei volume 48, 945 (2017)
  * [36] L. McLerran, Nuclear Physics A699, 73-81 (2002)
  * [37] N. Armesto, L. McLerran, C. Pajares, Nucl. Phys. A781 (2007) 201
  * [38] T. Lappi, L. McLerran, Nucle. Phys. A832, 330 (2010)
  * [39] V. Vechernin, Nuclear Physics A939, 21 (2015); Proceedings of Science, QFTHEP, 055 (2013)
  * [40] V. V. Vechernin, R. S. Kolevatov, Phys. Atom. Nucl. 70, 1797 (2007) ; Phys. Atom. Nucl. 70, 1809 (2007)
  * [41] V. Khachatryan et al. (CMS Collaboration), Phys. Rev. Lett. 116, 172302 (2016)
  * [42] V. Khachatryan et al. (CMS Collaboration), Phys. Lett. B765, 193 (2017)
  * [43] B. Abelev et al. (ALICE Collaboration), Phys. Lett. B719, 29 (2013)
  * [44] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B718, 795 (2013)
  * [45] G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 110, 182302 (2013)
  * [46] A. Adare et al. (PHENIX Collaboration), Phys. Rev. Lett. 114, 192301 (2015)
  * [47] V. Khachatryan et al. (CMS Collaboration), J. of High Energy Phys. 09, 091 (2010)
  * [48] G. Aad et al. (ATLAS Collaboration) Phys. Rev. Lett. 116, 172301 (2016)
  * [49] K. Werner, I. .Karpenko and T. Pierog, Phys. Rev. Lett. 106, 122004 (2011)
  * [50] P. Bozek, Phys. Rev. C85, 014911 (2012)
  * [51] M. Mondal et al., Phys. Rev. D102, 014033 (2020)
  * [52] K. Werner, B. Guiot, Iu. Karpenko, and T. Pierog, Phys. Rev. C89, 064903 (2014)
  * [53] S. Kar et al., J. Phys. G: Nucl. Part. Phys. 45, 125103 (2018)
  * [54] Yi-An Li et al., Phys. Rev. C104, 044906 (2021)
  * [55] N. Armesto, M. A. Braun, C. Pajares, Phys. Rev. C75 054902 (2007)
  * [56] B. Alessandro et al. (ALICE Collaboration), J. Phys. G 32, 1295 (2006)
  * [57] Konchakovski, V.P. Hauer, M. Torrieri et al., Phys. Rev. C79, 034910 (2009)
  * [58] A. Bzdak, Phys. Rev. C80, 024906 (2009)
  * [59] H. J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, and K. Werner, Phys. Rep. 350, 93 (2001)
  * [60] M. Bleicher et al., J. Phys. G25, 1859 (1999); H. Petersen, J. Steinheimer et al., Phys. Rev. C78, 044901 (2008)
  * [61] J. Adam et al. (ALICE Collaboration), Phys. Lett. B753, 319-329 (2016)
  * [62] B. Abelev et al. (ALICE Collaboration), Eur. Phys. J. C74, 3054, (2014)
  * [63] B. Abelev et al. (ALICE Collaboration) Phys. Rev. Lett. 110, 032301 (2013)
  * [64] J. Aichelin, K. Werner, J. Phys. G: Nucl. Part. Phys. 37, 094006 (2010)
  * [65] Kovalenko, V., Feofilov, G., Puchkov, A., Valiev, F., Universe 8, 246 (2022)
  * [66] Larry McLerran, arXiv:0804.1736v1 [hep-ph]
  * [67] F. Gelis, J. Phys. G34, S421 (2007)
  * [68] J. Adam et al. (ALICE Collaboration) Phys. Rev. Lett. 116, 222302 (2016)
  * [69] V. Khachatryan et al. (CMS Collaboration), Phys. Lett. B742, 200-224 (2015)