Physics of Elementary Particles and Atomic Nuclei. Theory

# Multiplicity correlations in the model with string clusters in pp collisions
at LHC energies

###### Abstract

In the framework of the model with string fusion and the formation of string
clusters the correlations between multiplicities in two separated rapidity
windows in pp collisions at LHC energies were studied and the results obtained
were compared with data from the ALICE collaboration at CERN.

The simulation is carried out within the framework of a Monte Carlo
implementation of the colour quark-gluon string model. String fusion effects
are taken into account by implementing a finite lattice in the plane of the
impact parameter.

The dependence of the correlation coefficient between multiplicities in two
rapidity observation windows on the distance between these windows is
calculated for four values of their width and three values of initial energy.

It is shown that the model with string clusters describes the main features of
the behavior of the correlation coefficient: its increase with increasing
initial energy, decrease with increasing rapidity distance between observation
windows, and nonlinear dependence on the width of the rapidity window.

a Saint Petersburg State University, 7/9 Universitetskaya nab., St.
Petersburg, 199034 Russia.

PACS: 12.40.$-$y; 25.75.$-$q

## Introduction

It is known that the study of correlations and fluctuations of various
observables in multiparticle production processes at high energies gives
information about the initial stage of hadronic interaction corresponding to
the maximum density of formed quark–gluon matter.

This study was carried out within the framework of quark-gluon (colour) string
models [1, 2] — a phenomenological model that is used to quantitatively
describe soft range of the strong interaction with low momentum transfer,
where perturbative QCD calculations are impossible. The concept of strings
represented by colour flux tubes, which is qualitatively substantiated in the
QCD framework, is used in this approach [3, 4].

This model assumes that interaction occurs in two stages. At the first stage
tubes filled with a colour gluon field (strings) are formed as a result of
colour reconnection processes. At the second stage observable hadrons are
formed as a result of hadronization of these strings.

In the case of high string density in the transverse plane, for example, with
nucleus-nucleus collisions and/or ultra-high energies of the LHC, it is
necessary take into account the interaction between the strings [5, 6]. For
the case of interaction of heavy nuclei, M.A. Brown and C. Pajares proposed
[7, 8] the model of fusion (percolation) of primary strings before their
fragmentation as a way to take into account the processes of interaction
between them.

Later, the simple discrete version of this model was proposed to simplify the
consideration of string fusion processes. In this simple model the transverse
plane is divided into cells with an area on the order of the cross-sectional
area of the string (the cross-sectional radius of the string is $r_{str}\sim
0,2\div 0,3$ fm). If the centres of two strings are in the same cell, we
assume that they fuse.

## Correlation coefficients and strongly intensive variables

The study of so-called forward-backward (FB) correlations between observed
quantities for particles in two rapidity intervals separated by an interval
has been proposed as one of the tools for studying string fusion processes.

Usually these two rapidity intervals (observation windows) are chosen
symmetrically, one in the front and the other in the back hemisphere of the
reaction. As observable quantities, $F$ and $B$, we usually choose the
multiplicity (number) of particles, $n_{F}$ and $n_{B}$, with rapidity within
the selected intervals, and the average transverse momentum of these particles
in a given event, $p_{t}^{F}$ and $p_{t}^{B}$,

$p_{t}^{F}=\frac{1}{n_{F}}\sum_{i=1}^{n_{F}}\left|\textbf{p}_{t}^{i}\right|,\
p_{t}^{B}=\frac{1}{n_{B}}\sum_{i=1}^{n_{B}}\left|\textbf{p}_{t}^{i}\right|.$
(1)

It was proposed [12] to study 3 types of correlations: $n$ – $n$, correlations
between multiplicities of charged particles, $p_{t}$ – $p_{t}$, correlations
between the average transverse momenta and $p_{t}$ – $n$, correlations between
the average transverse momentum in one rapidity window and the multiplicity of
charged particles in another rapidity window.

Unfortunately, the coefficient including multiplicity depends on so-called
volume fluctuations, which arise due to fluctuations in the volume of the
interaction region (for example, the number of strings formed) from event to
event. One of the possible ways to exclude the contribution of “volume”
fluctuations and obtaining information about the properties of objects formed
at the initial stage of strong interaction is the use of strongly intensive
variables [13] — such variables, the value of which does not depend either on
the volume of interaction or on its fluctuations.

General methods for constructing such observables were studied in [13]. In
particular, it was shown that within a certain class of statistical models
considered in the work, the value

$\Sigma(A,B)\equiv\frac{\left\langle A\right\rangle\,\omega_{B}+\left\langle
B\right\rangle\,\omega_{A}-2\,\mathrm{cov}(A,B)}{\left\langle
A\right\rangle+\left\langle B\right\rangle}\ ,$ (2)

composed of any two extensive quantities $A$ and $B$ is a strongly intensive
variable. In this formula $\left\langle A\right\rangle$ and $\left\langle
B\right\rangle$ — average values of $A$ and $B$, $\omega_{A}$ and $\omega_{B}$
\- their scaled variances:

$\omega_{A}\equiv\frac{D_{A}}{\left\langle
A\right\rangle}=\frac{\left\langle{A^{2}}\right\rangle-\left\langle{A}\right\rangle^{2}}{\left\langle
A\right\rangle},\omega_{B}\equiv\frac{D_{B}}{\left\langle
B\right\rangle}=\frac{\left\langle{B^{2}}\right\rangle-\left\langle{B}\right\rangle^{2}}{\left\langle
B\right\rangle}\ ,$ (3)

$\mathrm{cov}(A,B)$ — covariance of this variables:

$\mathrm{cov}(A,B)=\left\langle{AB}\right\rangle-\left\langle{A}\right\rangle\left\langle{B}\right\rangle.$
(4)

Later it was proposed [14] to use the values of the number of particles
$n_{F}$ and $n_{B}$, registered in this event in two separated rapidity
intervals of these particles in the observation windows usually called forward
and backward, as these extensive variables $A$ and $B$.

We will restrict ourselves to the case of symmetric reactions and the equal in
width, $\delta y_{F}=\delta y_{B}=\delta y$, symmetrically located relative to
$y=0$ rapidity observation intervals. In this case

$\left\langle n_{F}\right\rangle=\left\langle n_{B}\right\rangle=\left\langle
n\right\rangle,\ D_{n_{F}}=D_{n_{B}}=D_{n},\
\omega_{n_{F}}=\omega_{n_{B}}=\omega_{n}=\frac{D_{n}}{\left\langle
n\right\rangle}.$ (5)

Then (2) can be rewritten in a simpler form:

$\Sigma\left[n_{F},\ n_{B}\right]=\frac{\left\langle
n^{2}\right\rangle-\left\langle n_{F}\ n_{B}\right\rangle}{\left\langle
n\right\rangle}.$ (6)

For the case of a symmetric reaction and symmetrically located windows of
equal width, the following relationship can be obtained between the
correlation coefficient $b_{nn}$ and the strongly intensive variable
$\Sigma\left[n_{F},n_{B}\right]$:

$b_{nn}(\Delta y)=1-\frac{\Sigma[n_{F},n_{B}](\Delta y)}{\omega_{n}}.$ (7)

The formula (7) was used to analyse the dependence of the correlation
coefficient $b_{nn}$ on the rapidity distance between observation windows.

## Calculation of variables $\Sigma\left[n_{F},\ n_{B}\right]$ and $b_{nn}$ on
the lattice

Within the framework of a model with a finite lattice on a transverse plane,
it was shown [15] that $\Sigma$ is equal to the sum of the variables
$\Sigma_{\eta}$ introduced for a string cluster in one cell with some
weighting coefficients $\alpha(\eta)$.

$\Sigma[n_{F},n_{B}]=\sum_{\eta=1}^{\infty}\alpha(\eta)\Sigma_{\eta}(\mu_{F},\mu_{B}),\
\Sigma_{\eta}(\mu_{F},\mu_{B})=1+\left\langle{\mu}\right\rangle_{\eta}[J^{\eta}_{FF}-J^{\eta}_{FB}].$
(8)

The weight coefficients are the average fraction of particles formed as a
result of cluster decay. $J_{FF}^{\eta}$ and $J_{FB}^{\eta}$ are expressed
through the standard introduced single-particle distribution function of
particles (hadrons) $\lambda_{\eta}(y)$ formed during the fragmentation of
such a string and the two-particle (pair) correlation function
$\Lambda_{\eta}(y_{1},y_{2})$, characterizing the correlations between
particles formed from the decay of a given string [15]:

$J^{\eta}_{FB}\equiv\frac{1}{\left\langle{\mu_{F}}\right\rangle_{\eta}\left\langle{\mu_{B}}\right\rangle_{\eta}}\int_{\delta
y_{F}}dy_{1}\int_{\delta y_{B}}dy_{2}\
\lambda_{\eta}(y_{1})\lambda_{\eta}(y_{2})\Lambda_{\eta}(y_{1},y_{2}),$ (9)
$J^{\eta}_{FF}\equiv\frac{1}{\left\langle{\mu_{F}}\right\rangle^{2}_{\eta}}\int_{\delta
y_{F}}dy_{1}\int_{\delta y_{F}}dy_{2}\
\lambda_{\eta}(y_{1})\lambda_{\eta}(y_{2})\Lambda_{\eta}(y_{1},y_{2}).$ (10)

Similarly, we can obtain the following expression for the scaled variance of
the multiplicity $\omega_{n}$ in terms of $\omega_{\mu}^{\eta}$ introduced for
the string cluster

$\omega_{n}=\sum_{\eta=1}^{\infty}\alpha(\eta)\,\omega_{\mu}^{\eta}+\sum_{C_{\eta}}P(C_{\eta})\frac{\left\langle
n\right\rangle_{C_{\eta}}^{2}}{\left\langle n\right\rangle}-\left\langle
n\right\rangle,\ \omega_{n_{i}}^{\eta_{i}}=1+\left\langle
n_{i}\right\rangle_{\eta_{i}}J_{FF}^{\eta_{i}}.$ (11)

Here $\left\langle n_{i}\right\rangle_{\eta_{i}}=\mu_{0}\delta
y\sqrt{\eta_{i}}$ is the average multiplicity of charged particles produced
during the decay of a cluster of $\eta_{i}$ strings in $i$ -th cell in the
rapidity interval $\delta y$ [16].

If the observation windows $\delta y_{F}$ and $\delta y_{B}$ are chosen in the
central rapidity region so all the resulting strings contribute to both of
these rapidity intervals at once, then, due to the locality of the strong
interaction in the space of rapidities and the resulting homogeneity of the
distribution of particles from string fragmentation by rapidity, translational
invariance with respect to rapidity takes place. This approximation works well
at LHC energies, when the strings contribute over a wide range of rapidities.
If the two-particle (pair) correlation function $\Lambda_{\eta}(\Delta y)$ was
chosen in its simplest form

$\Lambda_{\eta}(\Delta y)=\Lambda_{0}^{\eta}e^{-\frac{|\Delta
y|}{y_{corr}^{(\eta)}}},$ (12)

then, within the framework of the assumptions described above, the integrals
$J_{FF}^{\eta}$ and $J_{FB}^{\eta}$ can be calculated explicitly:

$J_{FF}^{\eta}=\frac{2\Lambda_{0}^{\eta}}{(\delta
y)^{2}}y_{corr}^{{(\eta)}}\left(\delta
y-y_{corr}^{(\eta)}\left(1-e^{-\frac{\delta
y}{y_{corr}^{{(\eta)}}}}\right)\right),$ (13)
$J_{FB}^{\eta}=\frac{\Lambda_{0}^{\eta}(y_{corr}^{{(\eta)}})^{2}}{(\delta
y)^{2}}e^{-\frac{\Delta y}{y_{corr}^{{(\eta)}}}}\left(e^{-\frac{\delta
y}{y_{corr}^{{(\eta)}}}}+e^{\frac{\delta y}{y_{corr}^{{(\eta)}}}}-2\right).$
(14)

According to [17] we assumed the following dependence of the string cluster
correlation function parameters on $\eta$:

$y_{corr}^{(\eta)}=\frac{y^{(1)}}{\sqrt{\eta}},\ \Lambda_{0}^{(\eta)}=const.$
(15)

## Parameters selection

Previously [18, 19] the value $\mu_{0}=0.7$ was chosen to describe the
multiplicity of all particles with impulses $0<p_{t}<\infty$. The experimental
distribution over $p_{t}$ gives that it will be 58% particles in the interval
$0.3<p_{t}<1.5$ GeV, which gives $\mu_{0}(0.3<p_{t}<1.5)=0.7\cdot 0.58=0.41$,
which was not taken into account earlier.

The parameters (15) characterizing clusters with different numbers of fused
strings were determined by comparison with experimental data from the ALICE
collaboration [20]: $y_{corr}^{(1)}=2.7,\ \Lambda_{0}^{(1)}=0.8$.

The results for the strongly intensive variable $\Sigma\left(n_{F},\
n_{B}\right)$ depend on the product $\mu_{0}\Lambda_{0}^{(1)}$, therefore,
simultaneously with the change in $\mu_{0}$ as a result of taking into account
the distribution of particles over $p_{t}$, it is also necessary to change the
value of the parameter $\Lambda_{0}^{(1)}$ so that the product remains the
same. We adjust the parameters as follows:

$\mu_{0}(0.3<p_{t}<1.5)=0.41,\ y_{corr}^{(1)}=2.7,\ \Lambda_{0}^{(1)}=1.36.$
(16)

The (16) parameters were used for further calculations.

## Results

To calculate the value $\Sigma\left[n_{F},n_{B}\right]$ included in (7), its
expression in terms of $\Sigma_{\eta}$ (8) was used. To calculate the scaled
variance $\omega_{n}$ on the lattice, the formula (11) was used. Thus, the
calculation requires only the values of $\eta_{i}$ — the number of strings in
each cell of the lattice.

To calculate the weighting coefficients $\alpha(k)$, we first simulated the
distribution of the primary strings in the transverse planes taking into
account real pp-collision conditions using the Monte Carlo algorithm developed
in [18, 19] according to the methodology proposed in the work [21]. Next, the
process of fusion of primary strings and forming string clusters was simulated
introducing a finite lattice in the impact parameter plane.

We used the following parameter values at the modelling the distribution of
strings in the plane of the impact parameter:

$\Delta=0.2\,,\alpha^{\prime}=0.05\ GeV^{-2},\,\gamma_{pp}=1.035\
GeV^{-2},\,R_{pp}^{2}=3.3\ GeV^{-2}\ ,\,C=1.5.$ (17)

|  |
---|---|---
Figure 1: Correlation coefficient $b_{nn}$ as a function of the rapidity
distance between windows $y_{gap}$ for energies 0.9 TeV (left), 2.76 TeV
(center), 7 TeV (right). Lines — the result of MC calculations for windows
with width $\delta y=0.2,0.4,0.6,0.8$ (from bottom to top). Points —
experimental data obtained by the ALICE collaboration [20] by analysing data
on the yields of charged particles with transverse momenta in the range
0.3-1.5 GeV/$c$ in pp collisions for window widths of 0.2 ($\bullet$) , 0.4
($\blacksquare$), 0.6 ($\blacktriangle$), 0.8 ($\blacktriangledown$).

Fig.1 shows the dependence of the correlation coefficient $b_{nn}$ on the
rapidity distance between windows $y_{gap}$ for energies 0.9, 2.76, 7 TeV.

The studied model with the formation of string clusters qualitatively
describes the behaviour of $b_{nn}$: a decrease in $b_{nn}$ with increasing
distance between windows $y_{gap}$, an increase in $b_{nn}$ with increasing
energy, a nonlinear dependence of $b_{nn}$ from the window width $\delta y$.

However, we see that the model under study does not provide an ideal numerical
description of the absolute value of the correlation coefficient $b_{nn}$. A
possible reason is that with the selected parameter values (17) our model
gives a large value of the scaled variance of the total number of initial
strings in the event, which leads to a large value of the scaled variance of
the number of charged particles. Taking into account the processes of string
merging significantly reduces the value of the scaled variance of the number
of charged particles compared to the model of identical non-interacting
strings. Numerically, this reduction allows us to obtain accurate experimental
values of $b_{nn}$ only for an initial energy of 2.76 TeV. At an energy of 0.9
TeV this decrease turns out to be insufficient, and at an energy of 7 TeV, on
the contrary, it is excessive. To solve this problem, it seems that a revision
of the Regge parameters (17) is necessary. We leave a systematic model
analysis of this issue for future research.

## Conclusion

The previously developed [18, 19] Monte Carlo algorithm was applied to
calculate the correlation coefficient between multiplicities $b_{nn}$ (7) of
charged particles within the framework of a model with string fusion on a
lattice.

It is shown that the model with string clusters describes the main features of
the behaviour of the correlation coefficient $b_{nn}$: its increase with
increasing initial energy, decrease with increasing rapidity distance between
observation windows, and nonlinear dependence on the width of the rapidity
window. However, it is not possible to obtain an accurate quantitative
description of the experimental data for all initial energies.

To solve this problem, we need to revise the Regge parameters (17), which are
used in our model to find the configurations of the initial strings before
taking into account the processes of their fusion. In this case, it is
necessary to provide a simultaneous description of a large set of experimental
data at different energies — total, elastic and diffraction cross sections,
multiplicities, correlations and fluctuations between various observables.
This represents a laborious and important task, which we consider as a subject
for our future research.

## Acknowledgments

Supported by Saint Petersburg State University, project ID: 94031112.

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