# Centrality dependence of forward-backward multiplicity correlation
in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV

Yu-Liang Yan1, Dai-Mei Zhou2, Bao-Guo Dong1,3, Xiao-Mei Li1, Hai-Liang Ma1,
Ben-Hao Sa1,2,4111Corresponding author<EMAIL_ADDRESS>1 China Institute of
Atomic Energy, P.O. Box 275(18), Beijing 102413, China
2 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079,
China
3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion
Accelerator of Lanzhou, Lanzhou 730000,China
4 China Center of Advanced Science and Technology, World Laboratory, P. O. Box
8730 Beijing 100080, China

###### Abstract

We have studied the centrality dependence of charged particle forward-backward
multiplicity correlation strength in Au+Au collisions at $\sqrt{s_{NN}}$=200
GeV with a parton and hadron cascade model, PACIAE, based on PYTHIA. The
calculated results are compared with the STAR data. The experimentally
observed correlation strength characters: (1) the approximately flat
pseudorapidity dependence in central collisions and (2) the monotonous
decrease with decreasing centrality are well reproduced. However the
theoretical results are larger than the STAR data for the peripheral
collisions. A discussion is given for the comparison among the different
models and STAR data. A prediction for the forward-backward multiplicity
correlation in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV is also given.

###### pacs:

24.10.Lx, 24.60.Ky, 25.75.Gz

##  I INTRODUCTION

The study of fluctuations and correlations has been suggested as a useful
means for revealing the mechanism of particle production and Quark-Gluon-
Plasma (QGP) formation in relativistic heavy ion collisions hwa2 ; naya .
Correlations and fluctuations of the thermodynamic quantities and/or the
produced particle distributions may be significantly altered when the system
undergoes phase transition from hadronic matter to quark-gluon matter because
of the very different degrees of freedom between two matters.

The experimental study of fluctuations and correlations becomes a hot topic in
relativistic heavy ion collisions with the availability of high multiplicity
event-by-event measurements at the CERN-SPS and BNL-RHIC. An abundant
experimental data have been reported star2 ; phen2 ; phob ; star  where a lot
of new physics arise and are urgent to be studied. A lot of theoretical
investigations have been reported as well paja ; hwa1 ; brog ; yan ; fu ; bzda
; konc ; yan2 .

Recently STAR collaboration have measured the charged particle forward-
backward (FB) multiplicity correlation strength $b$ in a given centrality bin
size at different centralities from the most central to peripheral Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV star . The outstanding features of STAR
data are:

  1. 1.

In most central collisions, the correlation strength is approximately
independent of the distance between the centers of forward and backward
pseudorapidity bins, $\Delta\eta$.

  2. 2.

The correlation strength monotonously decreases with decreasing centrality.

  3. 3.

In the peripheral collisions, the correlation strength approaches to an
exponential function of $\Delta\eta$.

A lot of theoretical interests fu ; bzda ; konc ; yan2  has been stimulated.

The wounded nucleon model was used in bzda  to study the correlation strength,
the first two characters of STAR data were reproduced, but the third one was
not. In Ref. konc , the Glauber Monte Carlo code (GMC) with a “toy” wounded-
nucleon model and the Hadron-String Dynamics (HSD) transport approach have
been used to analyze the STAR data. They used three different centrality
determinations: the impact parameter $b_{i}$, the number of participant
(wounded) nucleons $N_{part}$, and the charged particle multiplicity $N_{ch}$
in midrapidity $|\eta|<1$. The first two characters of STAR data can be
reproduced by the $N_{part}$ and $N_{ch}$ centrality determinations, while the
third character can not.

We have used a parton and hadron cascade model, PACIAE, to investigate the
centrality bin size dependence of charged particle multiplicity correlation in
most central (5, 0-5, and 0-10%) Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200
GeV yan2 . It turned out that the correlation strength increases with
increasing bin size. In yan2  we follow cape  defining the charged particle FB
multiplicity correlation strength $b$ as

$\ b=\frac{\langle n_{f}n_{b}\rangle-\langle n_{f}\rangle\langle
n_{b}\rangle}{\langle n_{f}^{2}\rangle-\langle
n_{f}\rangle^{2}}=\frac{{\rm{cov}}(n_{f},n_{b})}{{\rm{var}}(n_{f})},$ (1)

where $n_{f}$ and $n_{b}$ are, respectively, the number of charged particles
in forward and backward pseudorapidity bins defined relatively and
symmetrically to a given pseudorapidity $\eta$. $\langle n_{f}\rangle$ refers
to the mean value of $n_{f}$ for instance. cov($n_{f}$,$n_{b}$) and
var($n_{f}$) are the FB multiplicity covariance and forward multiplicity
variance, respectively.

In this paper we use the PACIAE model to study the charged particle FB
multiplicity correlation strength $b$ in a given centrality bin size at
different centralities from the most central to peripheral Au+Au collisions at
$\sqrt{s_{NN}}$=200 GeV. A discussion is given for the comparison among the
models and STAR data as well as the STAR’s convention of different centrality
determination for different measured $\Delta\eta$ points. A prediction for the
forward-backward multiplicity correlation in Pb+Pb collisions at
$\sqrt{s_{NN}}$=5500 GeV is also given.

##  II THE PACIAE MODEL

The parton and hadron cascade model, PACIAE sa , is based on PYTHIA soj2
which is a model for hadron-hadron ($hh$) collisions. The PACIAE model is
composed of four stages: the parton initialization, parton evolution
(rescattering), hadronization, and hadron evolution (rescattering).

1\. PARTON INITIALIZATION: In the PACIAE model, a nucleus-nucleus collision is
decomposed into the nucleon-nucleon collisions based on the collision
geometry. A nucleon-nucleon ($NN$) collision is described with the PYTHIA
model, where a $NN$ (hadron-hadron, $hh$) collision is decomposed into the
parton-parton collisions. The hard parton-parton collision is described by the
lowest-leading-order (LO) pQCD parton-parton cross section comb  with
modification of parton distribution function in the nucleon. And the soft
parton-parton interaction is considered empirically. The semihard, between
hard and soft, QCD $2\rightarrow 2$ processes are also involved in the PYTHIA
(PACIAE) model. Because of the initial- and final-state QCD radiation added to
the parton-parton collision processes, the PYTHIA (PACIAE) model generates a
partonic multijet event for a $NN$ ($hh$) collision. That is followed, in the
PYTHIA model, by the string-based fragmentation scheme (Lund string model
and/or Independent Fragmentation model), thus a hadronic final state is
reached for a $NN$ ($hh$) collision. However, in the PACIAE model the above
fragmentation is switched off temporarily, so the result is a partonic
multijet event (composed of quark pairs, diquark pairs and gluons) instead of
a hadronic final state. If the diquarks (anti-diquarks) are split forcibly
into quarks (anti-quarks) randomly, the consequence of a $NN$ ($hh$) collision
is its initial partonic state composed of quarks, anti-quarks, and gluons.

2\. PARTON EVOLUTION: The next stage in the PACIAE model is the parton
evolution (parton rescattering). Here the $2\rightarrow 2$ LO-pQCD
differential cross sections comb  are employed. The differential cross section
of a subprocess $ij\rightarrow kl$ reads

$\frac{d\sigma_{ij\rightarrow
kl}}{d\hat{t}}=K\frac{\pi\alpha_{s}^{2}}{\hat{s}}\sum_{ij\rightarrow kl},$ (2)

where the factor $K$ is introduced considering the higher order pQCD and non-
perturbative QCD corrections as usual, $\alpha_{s}$ stands for the strong
(running) coupling constant. Taking the process $q_{1}q_{2}\rightarrow
q_{1}q_{2}$ as an example one has

$\sum_{q_{1}q_{2}\rightarrow
q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{\hat{t}^{2}},$ (3)

where $\hat{s}$, $\hat{t}$, and $\hat{u}$ are the Mandelstam variables. Since
it diverges at $\hat{t}$=0 one has to regularize it with the parton color
screen mass $\mu$

$\sum_{q_{1}q_{2}\rightarrow
q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{(\hat{t}-\mu^{2})^{2}}.$
(4)

The total cross section of the parton collision, $i+j$, then reads

$\sigma_{ij}(\hat{s})=\sum_{k,l}\int_{-\hat{s}}^{0}d\hat{t}\frac{d\sigma_{ij\to
kl}}{d\hat{t}}.$ (5)

With the total and differential cross sections above the parton evolution
(parton rescattering) can be simulated by the Monte Carlo method.

3\. HADRONIZATION: The parton evolution stage is followed by the hadronization
at the moment of partonic freeze-out (no any more parton collision). In the
PACIAE model, the Lund string fragmentation model and phenomenological
coalescence model are supplied for the hadronization of partons after
rescattering. The Lund string fragmentation model is adopted in this paper. We
refer to sa  for the details of the hadronization stage.

4\. HADRON EVOLUTION: After hadronization the rescattering among produced
hadrons is dealt with the usual two-body collision model. Only the
rescattering among
$\pi,k,p,n,\rho(\omega),\Delta,\Lambda,\Sigma,\Xi,\Omega,J/\Psi$ and their
antiparticles are considered in the calculations. An isospin averaged
parametrization formula is used for the $hh$ cross section koch ; bald . We
also provide an option for the constant total, elastic, and inelastic cross
sections ($\sigma_{\rm{tot}}^{NN}=40$ mb, $\sigma_{\rm{tot}}^{\pi N}=25$ mb,
$\sigma_{\rm{tot}}^{kN}=35$ mb, $\sigma_{\rm{tot}}^{\pi\pi}=10$ mb) and the
assumed ratio of inelastic to total cross section of 0.85. More details of
hadronic rescattering can see sa1 .

##  III CALCULATIONS AND RESULTS

In our calculations the default values given in the PYTHIA model are adopted
for all model parameters except the parameters $K$ and $b_{s}$ (in the Lund
string fragmentation function). $K$=3 is assumed and $b_{s}$=6 is tuned to the
PHOBOS data of charged particle multiplicity in 0-6% most central Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV phob2 , as shown in Tab. 1.

Table 1: Total charged particle multiplicity in three $\eta$ fiducial ranges in 0-6% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. | $N_{\rm{ch}}(|\eta|<4.7)$ | $N_{\rm{ch}}(|\eta|<5.4)$ |  $N_{\rm{ch}}$(total)
---|---|---|---
PHOBOSa | 4810 $\pm$ 240 | 4960 $\pm$ 250 | 5060 $\pm$ 250
PACIAE | 4819 | 4983 | 5100
a The experimental data are taken from phob2 .

In the theoretical calculation it is convenient to define the centrality by
impact parameter $b_{i}$. The mapping relation between centrality definition
in theory and experiment

$b_{i}=\sqrt{g}b_{i}^{\rm{max}},\qquad b_{i}^{\rm{max}}=R_{A}+R_{B},$ (6)

is introduced sa2 . In the above equation, $g$ stands for the geometrical
(total) cross section percentage (or charged multiplicity percentage) used in
the experimental determination of centrality. $R_{A}=1.12A^{1/3}+0.45$ fm is
the radius of nucleus $A$. We have also used the centrality determination
based on the charged multiplicity $N_{ch}$ in the central pseudorapidity
window $|\eta|<1$ to study the FB multiplicity correlation strength as a
function of $\Delta\eta$.

Figure 1: (Color online) The charged particle pseudorapidity distribution for
the specified centrality bin in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV.
The solid and open symbols are the PHOBOS data phob2  and PACIAE results,
respectively.

We compare the calculated charged particle pseudorapidity distribution in
specified centrality bin in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV with
the corresponding PHOBOS data phob2  in Fig. 1. Here the theoretical
centrality are defined by impact parameter $b_{i}$ via Eq.(6). One sees that
the PHOBOS data are well reproduced.

Figure 2: (Color online) The FB multiplicity correlation strength $b$ in Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV for centrality determination of (a)
impact parameter $b_{i}$ and (b) charged particle multiplicity $N_{ch}$
($|\eta|<1$). The open symbols with line and solid symbols are the PACIAE
results and STAR data star , respectively. Figure 3: (Color online) The FB
multiplicity correlation strength $b$ calculated by different models are
compared with the STAR datastar  for (a) 0-10% and (b) 40-50% central Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV.

The PACIAE model results of FB multiplicity correlation strength are shown in
Fig. 2 (a) and (b) for the centrality determinations of the impact parameter
$b_{i}$ and charged multiplicity $N_{ch}$ ($|\eta|<1$), respectively. One sees
in Fig. 2 (a) that the theoretical results can reproduce the STAR data for
0-10% most central collisions. However the theoretical results are all higher
than the corresponding STAR data for the 10-20%, 20-30%, 30-40%, and 40-50%
most central collisions and those theoretical results are closed to each
other. That is because the FB multiplicity correlation in Au+Au collisions is
mainly the statistical correlation steaming from the multiplicity fluctuation
yan2 . And the multiplicity fluctuation in those centralities defined by
impact parameter are similar to each other.

Figure 2 (b) gives the FB correlation strength calculated with the charged
particle multiplicity ($|\eta|<1$) centrality determination together with the
STAR data. We see in this panel that the STAR data are well reproduced by the
PACIAE model for the 0-10%, 10-20%, and 20-30% central collisions. However,
the PACIAE results are higher than the corresponding STAR data for 30-40%, and
40-50% central collisions. The correlation strength approaches an exponential
function of $\Delta\eta$ observed by STAR in 40-50% central collisions can not
be reproduced especially.

In Fig. 3 we compare the FB multiplicity correlation strength as a function of
$\Delta\eta$ calculated by the wounded nucleon model bzda , GMC code with a
“toy” wounded-nucleon model, HSD transport approach konc , and PACIAE model
with the STAR data for the 0-10% (panel(a)) and 40-50% central collisions
(panel (b)). We see in Fig. 3 (a) that all of the four models can nearly
reproduce the STAR data for 0-10% central collisions and the PACIAE model is
somewhat better than others. However all of the model results are higher than
the STAR data for the 40-50% central collisions, therefore can not reproduce
$b$ as an exponential function of $\Delta\eta$ observed by STAR as shown in
Fig. 3 (b).

Figure 4: (Color online) The same as Fig. 2 (b) but the PACIAE results are
calculated with STAR’s convention of different centrality determination for
different measured $\Delta\eta$ points (see text for the details).

We have noted the mentions in star  that “To avoid a bias in the FB
correlation measurements, care was taken to use different pseudorapidity
selection for the centrality determination which is also based on
multiplicity. Therefore, the centrality determination for the FB correlation
strength for $\Delta\eta$=0.2, 0.4, and 0.6 is based on the multiplicity in
$0.5<|\eta|<1.0$, while for $\Delta\eta$=1.2, 1.4, 1.6, and 1.8, the
centrality is obtained from $|\eta|<0.5$. For $\Delta\eta$=0.8 and 1.0, the
sum of multiplicities from $|\eta|<0.3$ and $0.8<|\eta|<1.0$ is used for the
centrality determination.” So we follow STAR’s centrality determination
convention to repeat all of the PACIAE calculations and draw Fig. 4 with these
results. Comparing Fig. 4 with Fig. 2 (b) we see this complicated centrality
determination does not improve but even worsens the agreement between
experiment and theory. That means STAR’s centrality determination convention
may be needed for the experimental measurement of FB correlation strength but
not for the theoretical calculation. In this kind of theoretical calculations,
each definite centrality curve in Fig. 4 is composed of $b$ calculated at
different $\Delta\eta$ with different centrality determination, such kind of
curve is not reasonable in theoretical physics.

Figure 5: (Color online) The PACIAE calculated FB multiplicity correlation
strength $b$ in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV with centrality
determination of charged particle multiplicity $N_{ch}$ ($|\eta|<1$).

Figure 5 gives the PACIAE model prediction for the charged particle FB
multiplicity correlation strength in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500
GeV. We can see that the FB multiplicity correlation strength in Pb+Pb
collisions is smaller than in Au+Au collisions. One reason may be that the
multiplicity in Pb+Pb collisions at LHC energy is much larger than in Au+Au
collisions at RHIC energy (the $N_{ch}/d\eta$ at mid-rapidity, $|\eta|<$0.5,
is about 600 in 0-10% most central Au+Au collisions at RHIC energy but it is
around 1200 in 0-10% most central Pb+Pb collisions at LHC energy sa3 ). This
observation is similar to the report that the elliptic flow parameter $v_{2}$
in Pb+Pb collisions at LHC energy is significantly smaller than in Au+Au
collisions at RHIC energy eyyu . That is attributed to the fact that the hard
process is more influential at the LHC energy than RHIC energy in eyyu .
Whether the competition between the hard and soft processes is also the reason
of the FB multiplicity correlation decreasing from RHIC energy to LHC energy
is beyond this paper scope, and it would be studied later.

##  IV CONCLUSION

We have used a parton and hadron cascade model, PACIAE, to study the
centrality dependence of charged particle FB multiplicity correlation strength
in 0-10%, 10-20%, 20-30%, 30-40%, and 40-50% central Au+Au collisions at
$\sqrt{s_{NN}}$=200 GeV. For the 0-10%, 10-20%, and 20-30% central collisions,
the STAR data are well reproduced. The STAR observed characters of (1) $b$ as
a function of $\Delta\eta$ is approximately flat for central collisions and
(2) $b$ decreases with decreasing centrality are reproduced as well. However
the PACIAE results are higher than the STAR data for the 30-40% and 40-50%
central collisions and can not obtain $b$ as an exponential function of
$\Delta\eta$ for the 40-50% central collisions, especially.

It turned out that the PACIAE model is somewhat better than the wounded
nucleon model, the GMC code with a “toy” wounded-nucleon model, or the HSD
transport model in comparing with the STAR correlation data. However all the
models can not reproduce $b$ as an exponential function of $\Delta\eta$ in the
40-50% central collisions observed by STAR. That should be studied further.

The PACIAE calculations are repeated using the STAR’s centrality determination
convention mentioned above. The results not improve but even worsens the
agreement between theory and experiment. This means STAR’s centrality
determination convention may be needed for the experimental measurement but
not for the theoretical calculations. Because in such theoretical
calculations, each definite centrality curve is composed of $b$ calculated at
different $\Delta\eta$ with different centrality determination, this kind of
curve is not reasonable in theoretical physics.

A prediction for the charged particle FB multiplicity correlation strength in
0-10% Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV is also given. The charged
particle FB multiplicity correlation strength in Pb+Pb collisions at LHC
energy is much smaller than in Au+Au collisions at RHIC energy. The further
study is out of present paper scope and has to be investigated in another
paper.

ACKNOWLEDGMENT

Finally, the financial support from NSFC (10635020, 10605040, 10705012,
10475032, 10975062, and 10875174) in China is acknowledged.

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