A Monte Carlo Study of Multiplicity Fluctuations in Pb-Pb Collisions at LHC Energies

0.96 MB·English

by  Ramni Gupta

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A Monte Carlo Study of Multiplicity Fluctuations
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e in Pb-Pb Collisions at LHC Energies
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u Ramni Gupta∗
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[ Department of Physics & Electronics, University of Jammu, Jammu, India

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v January 16, 2015
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Abstract
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. With large volumes of data available from LHC, it has become possible
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to study the multiplicity distributions for the various possible behaviours of
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the multiparticle production in collisions of relativistic heavy ion collisions,
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1 where a system of dense and hot partons has been created. In this context
: it is important and interesting as well to check how well the Monte Carlo
v
generators can describe the properties or the behaviour of multiparticle pro-
i
X duction processes. One such possible behaviour is the self-similarity in the
particle production, which can be studied with the intermittency studies and
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a further with chaoticity/erraticity, in the heavy ion collisions. We analyse the
behaviour of erraticity index in central Pb-Pb collisions at centre of mass
energy of 2.76 TeV per nucleon using the AMPT monte carlo event gener-
ator, following the recent proposal by R.C. Hwa and C.B. Yang, concerning
thelocalmultiplicityfluctuationstudyasasignatureofcriticalhadronization
in heavy-ion collisions. We report the values of erraticity index for the two
versions of the model with default settings and their dependence on the size
of the phase space region. Results presented here may serve as a reference
sample for the experimental data from heavy ion collisions at these energies.
∗email:[email protected]
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1 Introduction
Dynamics of the initial processes, that is the distributions and the nature of inter-
actions of quarks and gluons, in the heavy ion collisions affect the final distribution
of the particles produced [1]. Of the various distributions, multiplicity distributions
playfundamentalroleinextractingfirsthandinformationontheunderlyingparticle
production mechanism. If QGP is formed at these energies the QGP-hadron phase
transition is expected to be accompanied by large local fluctuations in the number
of produced particles in the regions of phase space [2]. Thus the study of fluctua-
tions in the multiplicity is an important tool to understand the dynamics of initial
processes and consequently the processes of strong interactions, phase transition
and also to understand correlations of QGP formation [3].
A comprehensive theoretical model which can explain and give answers to all
the complexities of the physics involved at high energy and densities, as is created
intheheavyioncollisions, isstillnotavailable. Asuccessfulmodelfocussedonone
aspectoftheproblemmaynotsaymuchabouttheotheraspects,butatleastshould
not contradict what is observed. The measures which are studied in the present
work rely on the large bin multiplicities. At LHC energies multiplicities are high
and it is possible to have detailed study of the local properties in (η,φ) space for
narrow p bins and thus to explore the dynamical properties of the system created
T
in the heavy ion collisions. Thus as an initial attempt to understand the nature of
global properties, as manifested in local fluctuations, here we develop and test the
methodologyandeffectivenessoftheanalysis,analysingsimulatedeventsforPb-Pb
√
collisions at s = 2.76 TeV using A Multi-Phase Transport (AMPT) model.
NN
Study of charged particle multiplicity fluctuations is one of the sensitive probes
to learn about the properties of the system produced in the heavy ion collisions.
Factorial moments are one of the convenient tools for studying fluctuations in the
particle production. The concept of factorial moments was first used by A. Bialas
and R. Peschanski [4] to explain unexpectedly large local fluctuations in high mul-
tiplicity events recorded by the JACEE Collaboratin. Advantage of studying fluctu-
ations using factorial moments is that these filter out statistical fluctuations. The
normalised factorial moment F is defined as
q
(cid:104)n!/(n−q)!(cid:105)
F (δd)= (1)
q (cid:104)n(cid:105)q
where n is the number of particles in a bin of size δd in a d-dimensional space
of observables and (cid:104)...(cid:105) is either vertical or horizontal averaging. q is the order of
the moment and is a positive integer ≥2. Then a power-law behaviour
F (δ)∝δ−ϕq (2)
q
over a range of small δ is referred to as intermittency. In terms of the number
of bins M ∝1/δ, Eq. 2 may be written as
F (M)∝Mϕq (3)
q
where ϕ is the intermittency index, a positive number.
q
2
Even if the scaling behaviour in Eq. 2 is not satisfied, to a high degree of
accuracy, F satisfies the power law behaviour
q
F ∝Fβq (4)
q 2
This is referred to as F-scaling. In attempts to quantify systems undergoing
second order phase transition, in Ginzburg-Landau (GL) theory [5], it is observed
that
β =(q−1)ν, ν =1.304, (5)
q
the scaling exponent, ν, is essentially idependent of the details of the GL pa-
rameters.
Factorial moments (F ’s) do not fully account for the fluctuations that the
q
system exhibits. Vertically averaged horizontal moments, can gauge the spatial
fluctuations, neglecting the event space fluctuations. On the other hand, horizon-
tallyaveragedverticalmomentsloseinformationaboutspatialfluctuationsandonly
measure the fluctuations from event-to-event. Erraticity analysis introduced in [6],
where one finds moments of factorial moment distribution, takes into account the
spatialaswellastheeventspacefluctuations. Itmeasuresfluctuationsofthespatial
patterns and quantifies this in terms of an index named as erraticity index (µ ). In
q
a recent work [7], µ is observed to be a measure sensitive to the dynamics of the
q
particle production mechanism and hence to the different classes of quark-hadron
phase transition.
In erraticity analysis, event factorial moments are studied, defined for an eth
event as
fe(M)
Fe(M)= q (6)
q [fe(M)]q
1
wherein,
fe(M)=(cid:104)n (n −1)......(n −q+1)(cid:105) (7)
q m m m h
where n ≥ q is the bin multiplicity of the mth bin, and (cid:104)...(cid:105) is the average
m h
over all bins such that for M2 cells
M2
1 (cid:88)
fe(M)= n (n −1)...(n −q+1) (8)
q M2 m m m
m=1
Now if Fe(M) fluctuates from event-to-event, then the deviation of Fe(M) from
q q
(cid:104)Fe(M)(cid:105) ((cid:104)...(cid:105) is for averaging over all events) for each event can be quantified
q v v
usingpth ordermomentsofthenormalisedqth orderfactorial(horizontal)moments
that can be defined as
C (M)=(cid:104)φp(M)(cid:105) (9)
p,q q v
where p is a positive real number and
[Fe(M)]p
φp(M)= q (10)
q (cid:104)F (M)(cid:105)p
q v
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10 1
10 2 (aS)M 10 2 (b)
counts/Nevents1100 43 DF dn / NdPT111000 543
10 5 10 6
0 1000 2000 3000 4000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Multiplicity PT(GeV/c)
0.005 0.004
(c) (d)
0.004 0.0038
0.0036
dn/N dη00..000023 dn/Ndφ0.0034
0.0032
0.001 0.003
0 1 0.5 0 0.5 1 0 1 2 3 4 5 6
η φ
Figure 1: (a) Multiplicity distribution (b) p (c) η and (d) φ distributions of
T √
charged particles generated in Pb-Pb collisions at s =2.76 TeV using the
NN
DF and SM AMPT.
Pb Pb s = 2.76 TeV
10 ALICNEN Data
2) CMS Data
>/Npart 8 DSMF
<
)/(η 6
d
/Nch 4
d
(
2
0 50 100 150 200 250 300 350 400
<N >
part
Figure2: dN /dη versusN plotforDFandtheSMAMPT,comparedwith
ch part
the ALICE and CMS data
To search for M-independent property of C (M) one looks for a power-law be-
p,q
haviour of C (M) in M,
p,q
C (M)∝Mψq(p) (11)
p,q
this is referred to as erraticity [6]. If ψ (p) is found to have a linear dependence on
q
p, then erraticity index µ can be defined as
q
dψ (p)
µ = q (12)
q dp
in the linear region so that it is independent of both M and p. µ is a number
q
thatcharacterizesthefluctuationsofspatialpatternsfromevevt-to-event. µ isob-
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served [7] to be an effective measure to distinguish different criticality classes, viz.,
critical, quasicritical, pseudocritical and non-critical, having low value for critical
4
p Default String Melting
T
window <N > <N >
0.2≤p ≤0.3 285.2 434.8
T
0.3≤p ≤0.4 279.2 355.5
T
0.4≤p ≤0.5 243.7 271.6
T
0.6≤p ≤0.7 163.3 155.5
T
0.9≤p ≤1.0 80.5 66.1
T
Table 1: Average Multiplicity of the Simulated Data sets analyzed in different
p windows
T
DF AMPT eta phi phase space hheettaapphhiiXX
EEnnttrriieess 227777
MMeeaann xx 00..44999999
MMeeaann yy 00..55003399
RRMMSS xx 00..22886688
2 RRMMSS yy 00..22887733
1.5
1
0.5
01
0.8
0.6 1
X(ϕ)0.4 0.2 00 0.2 0.4 X(η0).6 0.8
0.2≤pT≤0.3GeV/c
Figure 3: (X(η),X(φ)) phase space of an event with M = 32 in DF and SM
case
hadronizationcomparedtothosehavingrandomhadronization. Toagoodapproxi-
mation,itisobserved[7]thatforthemodelwithcontractionowingtoconfinement,
µ (criticalandquasicriticalcase)=1.87±0.84andformodelswithoutcontraction
4
µ (pseudocriticalandnoncritical)=4.65±0.06. Thesemodelvaluesaresuggestive
4
of the significance of erraticity index to characterize dynamical processes.
2 Data Analysed
Charged particles in |η| ≤ 0.8 and full azimuth, generated using two versions of
√
A MultiPhase Transport (AMPT) model [8, 9], in Pb-Pb collisions at s =
NN
2.76 TeV are analysed. AMPT model is a hybrid model that includes both initial
partonic and the final hadronic state interactions and transition between these two
phases. Thismodeladdressesthenon-equilibriummanybodydynamics. Depending
onthewaythepartonshadronizetherearetwoversions,default(DF)andthestring
melting (SM). In the DF version partons are recombined with their parent strings
when they stop interacting and the resulting strings are converted to hadrons using
LundStringFragmentationmodel. WhereasintheSMversion,aquarkcoalescence
model is used to obtain hadrons from the partons.
We have generated 23424 DF and 19669 SM events with impact parameter,
b ≤ 5, using the model parameters, a = 2.2, b = 0.5, µ = 1.8 and α = 0.47 .
5
102 (cid:9) (cid:9)(cid:9)00(cid:9)..(23a ) ≤≤D ppFT ≤≤ 00..33 GGeeVV//cc 102 (cid:9)(cid:9) (b) SM
0.4 ≤ pT ≤ 0.5 GeV/c
0.6 ≤ pT ≤ 0.7 GeV/c
0.9 ≤ pT ≤ 1.0 GeV/c
T
10 10
> >
n n
< <
1 1
10 1 10 1
10 10
M M
Figure 4: Average bin multiplicity dependence on M for the five p bins, in
T
case of DF and the SM AMPT model
Multiplicity,p ,pseudorapidityandφdistributionsofthesimulatedeventsisshown
T
intheFigure1. Chargedparticlesgeneratedinthe|η|≤0.8andfullazimuthhaving
p ≤ 1.0 GeV/c in the small p bins of width 0.1 GeV/c are studied for the local
T T
multiplicity fluctuations in the spatial patterns. Five p bins are considered in the
T
present analysis, as tabulated in the Table 1, along with the average multiplicity of
the generated charged particles in the respective p bins.
T
ThoughAMPTdoesnotcontainthedynamicsofcollectiveinteractionsthatare
responsible for critical behaviour, but it is a good model to test the effectiveness
of the methodology of analysis for finding observable signal of quark-hadron phase
transition(intermittencyanalysis)andthequantitativemeasureofcriticalbehaviour
of the system (erraticity analysis) at LHC energies.
3 Analysis and Observations
For an‘eth’event, theqth ordereventfactorial moment(Fe(M))asdefined inEq.
q
(6) are determined so as to obtain a simple characterization of the spatial patterns
in two dimensional (η,φ) spacein narrow p windows. However we first obtain flat
T
single particle density distribution using cumulative variable X(η) and X(φ) [10],
which are defined as
(cid:82)y
ρ(y)dy
X(y)= ymin (13)
(cid:82)ymaxρ(y)dy
ymin
here y is η or φ, y and y denote respectively the minimum and maximum
min max
values of y interval considered. η and φ is mapped to X(η) and X(φ) between 0
and 1 such that ρ(y) is the single particle η or φ density. (X(η),X(φ)) unit square
of an event in a selected p window, is binned into a square matrix with M2 bins
T
wherethemaximumvaluethatM cantakedependsonthemultiplicityinthe∆p
T
interval and the order parameter, so that the important part of the M dependence
is captured.
6
(a) q = 2 (DF) 104 (b) q = 4 (DF)
103 M=2
MM==48 103
102 M=16
M=32 102
10
10
1 1
0 0.5 1 F21.5 2 10 2 10 1 1 F4 10
(c) q = 2 (SM) 104 (d) q = 4 (SM)
103
103
102
102
10
10
1 1
0 0.5 1 F 1.5 2 10 2 1 0 1 1 F 10
2 4
Figure 5: P(Fe) distributions for order moment q = 2 and q = 4 for DF and
q
SM (0.3≤p ≤0.4). M values in multiples of 2 are shown only.
T
Togiveavisualizationofthebinninginthe(η,φ)spaceinnarrowp bin,alego
T
plot for an arbitrary event from DF AMPT data, in 0.2 ≤ p ≤ 0.3 window, with
T
M = 32, is shown in Fig. 3.
AsvalueofMandp increases, the(η,φ)spacebecomesempty, asisobserved
T
from Figure 4 which shows the dependence of the average bin multiplicity ((cid:104)n(cid:105)) on
M. Because of the denominator in Eq. (6), a cluster of particles with multiplicity
n ≥ q in an event would produce a large value for Fe(M) for that event. On the
q
other hand, if the particles are evenly distributed, Fe(M) would be smaller. Thus
q
the spatial pattern of the event structure should be revealed in the distribution of
Fe(M) after collecting all events.
q
Study of the event factorial moment distributions, (P(F )) reveal that the dis-
q
tributionsbecomewiderasMincreasesanddeveloplongtailsathigherq, especially
at higher M values. Further the peaks of the distributions shift towards left with
increase in M leading to decrease in (cid:104)Fe(cid:105) (referred to as F hereafter) with M
q q
whereas the upper tails move towards right, at higher q values as M increases. It
meansthatinsmallbinstheaveragebinmultiplicity(cid:104)n(cid:105)issosmallthatwhenthere
is a spike of particles in one such bin with n ≥ q, the non-vanishing numerator in
Eq. (6) results in a large value for Fe(M) for that eth event.
q
Dependence of F on M can be studied in log-log plots as shown in Fig. 6 for
q
various p cuts. From the plots, it is observed that F ’s decrease as the bin size
T q
decreasesorinotherwords, asM valueincreases. WeobserveforboththeDFand
SM in AMPT that relationship between F (M) and M is inverse of that in the Eq.
q
(3); that is
FqAMPT(M)∝Mϕ−q , ϕ−q <0 (14)
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e<F>q1 (a) 0.2 ≤ pT ≤ 0.3 GeV/c e <F>q1 (b) 0.3 ≤ pT ≤ 0.4 GeV/c e> <Fq1 (c) 0.4 ≤ pT ≤ 0.5 GeV/c
DF SM
q = 2 q = 2
q = 3 q = 3
10 1 q = 4 10 1 q = 4 10 1
q = 5 q = 5
10 M(η,φ) 10 M(η,φ) 10 M(η,φ)
e> <Fq1 (d) 0.6 ≤ pT ≤ 0.7 GeV/c e <F>q1 (e) 0.9 ≤ pT ≤ 1.0 GeV/c
10 1
10 1 10 2
10 M(η,φ) 10 M(η,φ)
Figure 6: M Dependence of F , for DF as well as SM.
q
p ν ν
T − −
(Default) (String Melting)
0.2≤p ≤0.3 1.738±0.008 1.753±0.004
T
0.3≤p ≤0.4 1.774±0.007 1.793±0.005
T
0.4≤p ≤0.5 1.758±0.006 1.755±0.006
T
0.6≤p ≤0.7 1.824±0.008 1.869±0.016
T
0.9≤p ≤1.0 1.778±0.013 1.781±0.011
T
Table2: ScalingexponentsfornegativeintermittencyintheDefaultandString
Melting versions of the AMPT Model
Hence, with negative ϕ− it is found that the charged particles generated by the
q
default and the string melting version of the AMPT model exhibit negative inter-
mittency.
Eq. (14) suggests that FAMPT(M) → 0 at large M and q, implying that in
q
AMPT there are too few rare high-multiplicity spikes anywhere in phase space. Eq.
(14)isaquantificationofthephenomenonexemplifiedbyFig. 3foroneevent,and
is a mathematical characterization after averaging over many events. This same
behaviour was observed in [7] for the events belonging to the non-critical class.
We plot F versus F in Fig. 7 to check F-scaling. For each set of p bins
q 2 T
linear fit has been performed to determine the value, β , the slope, as exemplified
q
by the straight lines in Fig. 7 (a). The dependence of β on (q−1) is shown in
q
Fig. 8, which exhibits good linearity in the log-log plots. Thus we obtain a scaling
exponent,denotedhereasν . InTable2aregiventhevalueofthenegativescaling
−
exponentfordifferentp windowsandforbothversionsoftheAMPTmodelstudied
T
here. Since the scaling that is there in Eq. 3 is different to that we observe here for
AMPTdata, thusthescalingexponentobtainedherecannotbecomparedwiththe
ν =1.304 for the second order phase transition, as obtained from the GL theory.
Since large fluctuations result in the high F tails of P(F ), as exemplified in
q q
Fig. 5 (b) and (d), it is advantageous to put more weight on the high F side in
q
8
e<F>q (a) 0.2 ≤ pT ≤ 0.3 GeV/c e<F>q (b) 0.3 ≤ pT ≤ 0.4 GeV/c e><Fq (c) 0.4 ≤ pT ≤ 0.5 GeV/c
1 1 1
DF SM
10 1 q = 3 10 1 q = 3 10 1
q = 4 q = 4
q = 5 q = 5
F2 1 F2 1 F2 1
e<F>q1 (d) 0.6 ≤ pT ≤ 0.7 GeV/c e<F>q1 (e) 0.9 ≤ pT ≤ 1.0 GeV/c
10 1 10 1
10 2 10 2
F2 1 F2 1
Figure 7: F-Scaling for DF as well as SM mode of AMPT
β10 00000.....23469(cid:9) ≤≤≤≤≤ (cid:9) ppppp(cid:9)(cid:9)TTTTT (cid:9)≤≤≤≤≤(a 00001) .....D33570 FGGGGG eeeeeVVVVV/////ccccc β10 (cid:9) (cid:9)(cid:9)(b) SM
3 3
1 2 3 4 5 6 1 2 3 4 5 6
q 1 q 1
Figure 8: β versus (q−1) plot for determination of the scaling exponents.
q
averaging over P(F ). That is just what the double moment C (M) does. We
q p,q
have determined C (M) for q = 2, 3, 4, 5 and p = 1.0, 1.25, 1.5, 1.75 and 2.0.
p,q
The number of bins, M, takes on values from 2 to the maximum value possible
whilehavingreasonable(cid:104)n (cid:105)suchthatF (cid:54)=0. TocheckwhetherC (M)follows
m q p,q
the scaling behaviour with M, C is plotted against M. Fig. 9 (a) to (d) shows
p,q
respectively, for q = 2, 3, 4 and 5, the C versus M plot in the log-log scale
p,q
for the window 0.6 ≤ p ≤ 0.7 GeV/c, and for various values of p between 1 and
T
2. As expected for all values of q, for p = 1.0, the C = 1. For p > 1.0, C
p,q p,q
increases with M and q values. Similar calculations are also done for the other p
T
windows. InthehighM regionlinearfitsareperformedforeachq andpvaluesoas
to determine ψ (p). We see in Fig. 10 that for 0.6≤p ≤0.7 ψ (p) depends on
q T q
p linearly for each q. Thus the erraticity indices defined in Eq. (12) are determined.
Similar plots are obtained for the other p windows also and the values of µ are
T q
given in Table 3. It can be seen from the table that as p value increases, the
T
erraticity indices increase for both versions of the AMPT model.
Comparing the µ values for the DF and SM data within the same window and
q
9
Cp,q 0.6 ≤ pT ≤ 0.7 GeV/c (a) q =2 Cp,q10 0.6 ≤ pT ≤ 0.7 GeV/c (a) q =3
DF SM
p = 1.0 p = 1.0
p = 1.25 p = 1.25
p = 1.5 p = 1.5
p = 1.75 p = 1.75
p = 2.0 p = 2.0
1 1
10 M 10 M
Cp,q 0.6 ≤ pT ≤ 0.7 GeV/c (a) q =4 Cp,q104 0.6 ≤ pT ≤ 0.7 GeV/c (a) q =5
102
102
10
1 1
10 M 10 M
Figure 9: M dependence of C for the p window 0.6 ≤ p ≤ 0.7 GeV/c in
p,q T T
case of DF and SM versions of the AMPT model
for the same values of q, it is observed that µ has higher values for the DF version
q
in comparison to SM for the p windows below 0.6 GeV/c. That phenomenon
T
is related to the average multiplicities of the two versions reversing their relative
magnitudesathigherp . HoweveritistobenotedfromFig.10thatthedependence
T
of ψ (p) on p is better distinguishable for the two versions of the AMPT for only q
q
=4. Coincidentally, as observedin[7], µ seemsto be agoodmeasuretocompare
4
the erraticity indices of the different systems and data sets at these energies.
We observe that the values of µDF and µSM for all p windows are larger
4 4 T
than those obtained for the critical data set in [7], on the same side as the non-
critical case. We have found ϕ− to be negative because P(F ) broadens, as M
q
increases, with (cid:104)F e(cid:105) shifting to the lower region of Fe, thus resulting in negative
q q
intermittency. We did notice that the upper tails move to the right, suggesting
the presence of some degree of clustering. To emphasize that part of P(F ) we
q
have taken higher p-power moments of φ (M), which suppress the lower side of
q
F while boosting the upper side. The scaling properties of C (M) therefore
q p,q
deemphasize what leads to negative intermittency. Thus the erraticity indices µ
q
reveal a different aspect of the fluctuation patterns than the scaling indices ν−.
Our study here has revealed interesting properties of scale-invariant fluctuations
that should be compared to the real data.
4 Summary
Fluctuations in the spatial patterns of charged particles and their event-by-event
fluctuations, as are present in the events generated using the default and string
meltingversionofAMultiPhaseTransport(AMPT)modelarestudied. Thisisafirst
attempt to study intermittency and erraticity at such high energies. It is observed
10