Statistical and dynamical fluctuations of Binder ratios in heavy ion collisions

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by  Zhiming Li

Statistical and dynamical fluctuations of Binder ratios in heavy ion collisions
Zhiming Li, Fengbo Xiong, and Yuanfang Wu
Institute of Particle Physics, Central China Normal University, Wuhan 430079, China
Key Laboratory of Quark and Lepton Physics (Central China Normal University),Ministry of Education
Higher moments of net-proton Binder ratio, which is suggested to be a good experimental mea-
surement to locate the QCD critical point, is measured in relativistic heavy ion collisions. We
firstlyestimatetheeffectofstatistical fluctuationsofthethirdandforthorderBinderratios. Then
the dynamical Binder ratio is proposed and investigated in both transport and statistical models.
The energy dependenceof dynamical Binder ratios with different system sizes at RHIC beam scan
energies are presented and discussed.
PACSnumbers: 25.75.Gz,25.75.Nq
2
1
0 I. INTRODUCTION order parameter in current heavy ion experiment [14].
2 It is suggested recently that the Binder-like ratios

n are good identification of critical behavior in relativis-
One of the main goals of currentrelativistic heavy ion
a tic heavy ion collisions[15–17]. In statistical physics, the
J collisions is to map the QCD phase diagram [1]. At Binder ratios are defined as normalized raw moments of
vanishing baryon chemical potential µ = 0, finite tem-
3 B orderparameter. ThethirdandforthorderBinderratios
1 perature Lattice QCD calculations predict that a cross- can be written as
over transition from hadronic phase to the Quark Gluon

] Plasma (QGP) phase will occur around a temperature
h
t of 170 - 190 MeV [2, 3]. QCD based model calculations <M3 >
l- indicatethatthetransitioncouldbeafirstorderatlarge B3 = <M2 >3/2
c µ [4]. The point where the first order phase transition
u B <M4 >
ends is the so-called QCD Critical Point (QCP) [5, 6]. B = , (1)
n 4 <M2 >2
Attempts are being made to locate the QCP both ex-
[
perimentally and theoretically [7]. Lattice QCD calcula-
where M is the order parameter. The Binder ratio is
2 tions at finite µ face numerical challenges in comput-
B a new observable in heavy ion collisions. The differ-
v ing. Thus the location of the QCP are highly uncertain
ence between Binder ratios and the well-known higher
2
intheoretically. Inexperimentalaspect,the RHIC beam
8 moments[14, 18]is thatBinder ratiosare the normalized
energy scan program [8] has been motivated to search
9 higher raw moments of the order parameter while the
2 for the QCP in experiment. By decreasing the collision higher moments use central moments.
9. energy downto a center of mass of 5 GeV, RHIC will be The universality argument indicates that the static
able to vary the baryon potential from µ 0 to 500
0 B ∼ criticalexponentsofthesecondorderphasetransitionare
MeV.
1 determined by the dimensionality and symmetry of the
1 Fluctuations of conserved charges, which behave dif- system. The QCD critical point of deconfinement phase
:
v ferently between the hadronic and QGP phase, are gen- transitionbelongstothesameuniversalityclassasliquid-
i erally considered to be sensitive indicators for the tran- gasphase transitionand the 3D-Ising model [11, 12, 19].
X
sition [9, 10]. The singularity at the QCP, at which the Its universal critical properties are discussed to be valid
r transition is believed to be second order, may cause en- in various of models and relevant to heavy ion colli-
a
hancement of fluctuations if fireballs created by heavy sions [11, 20, 21], in particular the event-by-event fluc-
ioncollisions pass near the criticalpoint during the time tuations of baryon numbers [11].
evolution [11]. It has been shown that near the critical In the calculations of the 3D-Ising model[15], Binder
point, the density-density correlator of baryon number ratios of B and B as a function of temperature show
3 4
follow the same power law behavior as the correlator of a step jump from a lower platform to a higher one near
the sigma field which is associated with the chiral or- the vicinity of critical point by choosing the magnet of
der parameter [11, 12]. Therefore, the baryon number is spin as the order parameter. Therefore, if the formed
considered as an equivalent order parameter of formed system in heavy ion collisions reaches the critical point
system in nuclear collisions. In experiment, net-proton andthefreeze-outcurveisclosetothetransitionline,the
multiplicity distribution is much easier to measure than step function liked behavior of Binder ratios in the Ising
the net-baryon numbers. Theoretical calculations have modelmayserveasagoodprobeofQCPincurrentheavy
shown that in QCD with exact isospin invariance, the ioncollisions,wherecriticalincidentenergyisdifficultto
relevant corrections due to isospin breaking are small assign precisely in priori.
and the net-proton fluctuations can reflect the singular- Inthemeantime,theeffectoftrivialstatisticalfluctu-
ityofthebaryonnumbersusceptibilityasexpectedatthe ations [22] due to insufficient number of particles should
QCP [13]. Hence, the net-proton number is used as the be studied and properly eliminated in higher moment
2
calculations. Therefore, we should discuss the statisti- 1.5 6 AMPT default
cal contributions from the measured fluctuations firstly, AMPT string melting
Therminator
then we couldidentify the dynamicalpart whichis more 5 AMPT default (SK)
1 AMPT string melting (SK)
relevanttothecriticalpointoftheQCDphasetransition. Therminator (SK)
B3 B4 4
In this paper, we firstly investigate the statistical and
dynamicalfluctuationsofnet-protonBinderratiosbyus- 0.5 Au + Au 200 GeV 3
(a) (b)
ingtheAMPTandTHERMINATORmodels. Then,the
energy dependence of dynamical Binder ratios in Au + 00 100 200 300 400 20 100 200 300 400
N N
Au collisions at various RHIC beam scan energies are part part
studied.
FIG. 1: Binder ratios of B3 (a) and B4 (b) as a function
In our analysis, two versions of a multi-phase trans-
of number of participants from AMPT default (solid cir-
port (AMPT) model [23] are used. One is the AMPT cle), AMPT with string melting (open circle), and THER-
defaultandtheotheroneisthe AMPTwithstringmelt- MINATOR (open square) models in Au + Au collisions at
ing. In both versions, the initial conditions are obtained √sNN =200GeV.Thedashedlinesrepresentthecorrespond-
from the heavy ion jet interaction generator (HIJING) ing statistical fluctuations.
model, and then the scattering among partons is given
by the Zhangs parton cascade (ZPC) model. In the
AMPT default model, the partons recombine with their
parent strings when they stop interacting, and the re- ∆3+6µ∆+∆
sulting strings are converted to hadrons using the Lund B3,stat = (∆2+2µ)3/2
stringfragmentationmodel,whereasintheAMPTmodel
withstringmelting,quarkcoalescenceisusedincombin- ∆4+12µ∆2+4∆2+12µ2+2µ
B = , (2)
ing partons into hadrons. The dynamics of the hadronic 4,stat (∆2+2µ)2
matter is described by the ART model. The THERMI-
where ∆ = N N is the average number of net-
NATOR statistical model [24] is a Monte Carlo event p p¯
h i−h i
protons, and µ = ( N + N )/2 is the mean value
generatordesignedforstudying ofparticleproductionin p p¯
h i h i
of protons and antiprotons in the event sample. More
relativistic heavy ion collisions from SPS to LHC ener-
details of the calculation of this formula could be found
gies. It implements thermal models of particle produc-
in the appendix.
tion with single freeze out.
In Fig. 1 (a) and (b), we show the results of B and
3
In order to make our calculations convenient for com- B as a function of number of participants (N ) from
4 part
parisonwiththeRHICbeamenergyscandata,wechoose AMPT default (solid circle), AMPT with string melt-
the midrapidity (y < 0.5) region with transverse mo- ing (open circle), and THERMINATOR (open square)
| |
mentum 0.4 < pT < 0.8 GeV/c. This phase space is models in Au + Au collisions at √sNN = 200 GeV, re-
where the STAR experiment can do the particle identi- spectively. Forcomparison,thestatisticalfluctuationsof
fication for proton numbers with its main tracking de- the Binder ratios,calculated from Eq. (2), are presented
tector - the Time ProjectionChamber [14]. The number as dashed lines. We can see that in both transport and
of events used in this analysis is around 6 million. This statistical models, the statistical fluctuations give main
statistics is needed for the calculation of the dynamical contributions to the Binder ratios. It shows that the
Binderratiosofnet-protontoensurethestatisticalerrors influence of statistical fluctuations are not negligible in
under control. the measurement of net-proton Binder ratios at RHIC
energy.
III. DYNAMICAL NET-PROTON BINDER
II. STATISTICAL FLUCTUATIONS OF BINDER RATIOS
RATIOS
In the THERMINATOR model, it is well-known that
the fluctuations are thermal. From Fig. 1, we observe
In the measurement of the net-proton Binder ratios,
it gives a good agreement with the Skellam statistical
finite number of protons and antiprotons will cause non-
fluctuations. It is difficult to disentangle purely statis-
negligible statistical fluctuations. If the produced pro-
tical effects from thermal fluctuations which follow the
tons and antiprotons are two independent Poisson-like
physicsofa hadronresonancegas. Since neither ofthem
distributions [22, 25], the net-protons then obey a Skel-
is associate with the QCP behavior,we suggestto elimi-
lam (SK) distribution [26].
nate these statistical or thermal fluctuations in order to
According to the definition of Eq. (1), the statistical get the dynamical part.
fluctuations of the net-proton Binder ratios can be di- As shown in section II, the statistical fluctuations of
rectly deduced from the Skellam distribution Binder ratio can be expressed by Eq. (2) given proton
3
0.2
(a) 1 (b) (a) 7600--8700%% (b) AMPT default
0.15 AMPT default 0.2 AMPT default 5400--6500%% 1
AMPT string melting Au + Au 200 GeV 30-40%
20-30%
B3,dyn0.1 Therminator B4,dyn0.5 B3,dyn0.1 1500--15-20%0%% B4,dyn0.5
0.05
0 0
0 0
-0.05
0 100 200 300 400 0 100 200 300 400 10 102 10 102
Npart Npart s s
FIG. 2: The third (a) and forth (b) order dynamical Binder
1.5
ratios as a function of Npart from AMPT default (solid cir- 0.3 (c) AMPT strig melting (d) AMPT strig melting
cle), AMPT with string melting (open circle), and THER-
MINATOR (open square) models in Au + Au collisions at 0.2 1
√sNN =200 GeV. B3,dyn B4,dyn
0.1 0.5
and antiproton obey independent Poisson distributions.
0 0
We define the so-called dynamical Binder ratios as,
10 102 10 102
s s
B3,dyn = B3 B3,stat FIG. 3: Energy dependence of the dynamical B3 (left panel)
B = B −B . (3) and B4 (right panel) at RHIC energies from AMPT default
4,dyn 4 4,stat
− model(upperpanel)andstringmelting(lowerpanel),respec-
tively. The nine different symbols represent nine collision
We suggest to measure these dynamical Binder ratios
sizes, which goes from most peripheral (70-80% central) to
instead of the original definition given by Eq. (1) in rel-
mostcentral(0-5%central)collisions. Thelinesareusedonly
ativistic heavy ion experiment. to guideeyes.
The dynamical net-proton B and B as a function
3 4
of N from AMPT default (solid circle), AMPT with
part
stringmelting(opencircle),andTHERMINATOR(open the critical point is most probably in the intersection of
square) in Au + Au collisions at √sNN = 200 GeV are different size systems between the two platforms.
shown in Fig .2 (a) and (b), respectively. We find that In the upper panels of Fig. 3 (a) and (b), we show
boththethirdandforthorderBinderratiosfromTHER- the energy dependence of the dynamical B and B at
3 4
MINATOR are zero at all centralities. This is because six RHIC energies, 7.7, 9.2, 11.5, 39, 62.4, and 200 GeV
thatTHERMINATORis basedonthe hadronresonance from AMPT default model. The nine different symbols
gasmodelandtheproducednet-protonsinthefinalstate represent nine collision sizes (denoted by centralities in
obey the Skellam distribution [27]. While, in transport experiments). From the figure, we can see that, in all
models,bothdynamicalB3andB4arelargerthanzeroin beamenergies,whencentralitygoesfrommostperipheral
peripheral collisions, then tend to be zero in central col- (70-80%central)tomostcentral(0-5%central)collisions,
lisions. TheresultsfromAMPTstringmeltingarelarger the dynamical Binder ratios decrease and are close to
thanthatfromthedefaultmodel. Thisisduetodifferent zero in the most central collisions. It means that both
mechanismsofhadronizationschemeusedforfinite state dynamical B and B are system size dependent. In the
3 4
particles in different versions of AMPT models. calculations of 3D-Ising model [15], the Binder ratios of
different sizes should intersectto eachother, i.e. there is
a fix point for the system near the critical point. From
IV. ENERGY DEPENDENCE OF DYNAMICAL 7.7 GeV to 200 GeV, we observeno fixed point from the
BINDER RATIOS IN TRANSPORT MODELS AMPT default model. The lower panels of (c) and (d)
for the string melting version give similar results.
Based on the Wolf algorithm [28], the 3D-Ising model Therefore, there is no step function behavior observed
calculations show that the third and forth order Binder in both two versions of the AMPT models. This is un-
ratios exhibit a step function behavior near the critical derstandable since there is no QCD critical mechanism
temperature [15]. In experiment, if the freeze-out curve implemented in these transport models.
is close to the QCD critical point along the transition
line and the criticalfluctuations are survivedin the final
V. SUMMARY AND OUTLOOK
state, one may expect to observe this behavior exper-
imentally. For RHIC beam energy scan program, if we
scanalltheexistingenergiesandobservetwoplatformsof Inthis paper,the statisticalanddynamicalBinderra-
Binderratiosatlowandhighenergyregions,respectively, tios of net-proton are studies in Au + Au collisions at
4
RHIC energies. Using transport and statistical models, N and N , the net-proton will follow a Skellam dis-
p p¯
h i h i
it is shownthat statisticalfluctuations arenot negligible tribution. If we define the net-proton as M = N N ,
p p¯
−
inthemeasurementofhigherBinderratiosinrelativistic then the probability distribution function of M is
heavy ion collisions.
In order to obtain a clean signature which may be re-
f(M; N , N )
latedtotheQCDcriticalpoint,wesuggesttousethedy- h pi h p¯i
namicalBinder ratio in experimentalmeasurement. The N M/2
= e−(hNpi+hNp¯i) h pi I 2 N N ,
dynamical net-protonBinder ratio is found to be zero in (cid:18) Np¯ (cid:19) |M|(cid:18) qh pih p¯i(cid:19)
the THERMINATOR model but larger than zero in pe- h i
ripheral collisions in AMPT model. The energy depen-
dence of dynamical Binder ratios with different system
where I 2 N N is the modified Bessel func-
sdiezfeasulsthoorwsstrninogsmteepltfiunngcmtioodnelbse,hwahviicohriesitchoenrsisintenAtMwPitTh tion of t|hMe|fi(cid:16)rspt khinpdi.h p¯i(cid:17)
expectation that no QCD critical point physics is imple- The nth moment of M, which is defined as Mn =
mented in these models. ∞ Mnf(M; N , N )dM,canbecalculatedfhromithe
Itisinterestingtoseewhetherthestepfunctionbehav- Ra−bo∞ve distributhionpifuhncp¯tiion. We obtain
ior of the dynamical Binder ratios could show up in the
coming high energy collisions at RHIC, SPS, and FAIR
experiments, where the critical incident energy of QCD M = ∆
h i
phase transition may be covered. Our model study can M2 = ∆2+2µ
h i
serve as a background study of the behavior expected M3 = ∆3+6µ∆+∆
from known physics effects for the experimental search h i
M4 = ∆4+12µ∆2+4∆2+12µ2+2µ,
for the QCD critical point. h i
where ∆ = N N is the average number of net-
p p¯
h i−h i
protons, and µ = ( N + N )/2 is the mean value of
p p¯
h i h i
VI. ACKNOWLEDGMENTS proton and antiproton.
By the definitions of the third and forth Binder ra-
We thank Dr. Lizhu Chen and Shusu Shi for valuable tios of Eq. (1), we get the Binder ratios of the Skellam
discussions and remarks. This work is supported in part statistical distribution as
by the NSFC of China with project No. 11005046 and
No. 10835005.
<M3 > ∆3+6µ∆+∆
B = = ,
VII. APPENDIX: BINDER RATIOS FROM 3,stat <M2 >3/2 (∆2+2µ)3/2
SKELLAM DISTRIBUTION <M4 > ∆4+12µ∆2+4∆2+12µ2+2µ
B = = .
4,stat <M2 >2 (∆2+2µ)2
Given the distributions of proton and antiproton are
independent Poisson distributions with mean values are
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