Babichev, L. F.

Khmialeuski, A. N.

English

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c© 2010 Proceedings of the F&ANS-2010 Conference-School
Study the signals of QGP using Monte Carlo
for Pb+Pb for Energy from 500 to 2000 GeV
Babichev L.F.∗ and Khmialeuski A.N.†
Joint Institute for Power and Nuclear Research - Sosny,
National Academy of Sciences of Belarus
Acad. Krasina Str. 99, 220109 Minsk, Belarus
The heavy-ion collisions (Pb+Pb) for LHC and RHIC energy are simulated by
Monte-Carlo generator HIJING and HIDJET++ for ssudy signals of QGP. With help
the factorial moments is studied posibility phase transition from QGP to hadrons. A
comparison of Monte Carlo generators HIJING and HIDJET++ for modeling such
phenomena as QGP-hadrons phase transition is conducted.
1. Introduction
The collision of high-energy heavy ions with atoms and atomic nucleus begins to manifest
various collective phenomena. For example, the fluctuations of the multiplicity of secondary
particles increase with the growing of initial energy of collisions. The final distribution of particles
in the phase space has great depends from the dynamics of initial processes, i.e. the distribution
and nature of the interactions of quarks and gluons. The study of such fluctuations can provide
the necessary information about the dynamics of initial processes and, consequently, on the
properties of strong interactions.
In collisions characterized by sufficiently high temperatures and/or energy density in accor-
dance with the predictions of quantum chromodynamics should form a new superdense state of
matter — quark-gluon plasma (QGP). The most promising for such purposes are the heavy ion
collisions. Such experiments were carried out at RHIC (Au collisions) and curently are conducted
at the LHC (Pb collisions). But detection of the QGP formation is a more difficult. Firstly,
there is a large background due to the strong interaction between nucleons. Secondly, the dura-
tion of the QGP evolution of the nuclear system is a small fraction of the total evolution time.
If the QGP was formed in a collision, then there should be observed a phase transition (PT)
QGP→hadrons. Such PT can be defined by the study of the particles multiplicities fluctuations.
The availability of sufficiently large fluctuations can be a indication of PT and correspondingly
∗E-mail: babichev@sosny.bas-net.by
†E-mail: akhmelevskiy@sosny.bas-net.by
152 Proceedings of the F&ANS-2010 Conference-School, (2010) 152 - 162
Study the signals of QGP using Monte Carlo for . . . 153
QGP formation.
In this paper we used Monte Carlo generators for simulation of the relativistic heavy ions
collisions to investigated of the multiplicity fluctuations of secondary particles. We used dif-
ference factorial moments for analysis of such fluctuations and for the determination of QGP
signals.
2. Monte Carlo generators
To carry out model calculations, we needed generators include the possibility to simulate
heavy ion collisions at high energies. Also generators should have been simulate by various
fluctuations. We have selected the following Monte Carlo generators:
• HIJING (Heavy Ion Jet INteraction Generator) is Monte Carlo generator with special
emphasis on the role of minijets in pp, pA and AA reactions at collider energies [1, 2].
HIJING basically is designed to simulate multiple jets and particle production in pp, pA
or AA collisions.
Based on a pQCD-inspired model, multiple minijet production is combined together with
Lund-type model for soft interactions. Within this model, triggering on large PT jet pro-
duction automatically biases toward enhanced minijet production. Binary approximation
and Glauber geometry for multiple interaction are used to simulate pA and AA collisions.
A parametrized parton distribution function inside a nucleus is used to take into account
parton shadowing. Jet quenching is modeled by an assumed energy loss dE/dz of par-
tons traversing the produced dense matter. A simplest color configuration is assumed for
the multiple jet system and Lund jet fragmentation model is used for the hadronization.
PYTHIA subroutines and Lund jet fragmentation scheme are used for calculations.
The program is only valid for collisions with c.m. energy (
√
s) above 4 GeV/n. For central
Pb+Pb collisions, some arrays have to be extended
√
s = 10 TeV/n.
Currently the HIJING Monte Carlo model is updated with the latest parton distributions
functions (PDF) and new set of the parameters in the two-component mini-jet model that
controls total p+ p cross section and the central pseudorapity density [3].
• HYDJET++ (HYDrodynamics plus JETs) is the event generator to simulate relativistic
heavy ion AA collisions as a superposition of the soft, hydro-type state and the hard state
resulting from multi-parton fragmentation [4]. It is designed for studying of multi-particle
production in the wide energy range of heavy ion experimental facilities: from FAIR and
NICA to RHIC and LHC.
A heavy ion event in HYDJET++ is a superposition of the soft, hydro-type state and
the hard, multi-parton state. Both states are treated independently. HYDJET++ is
Proceedings of the F&ANS-2010 Conference-School, 2010
154 Babichev L.F., Khmialeuski A.N.
the development and continuation of HYDJET MC model. The main program is writ-
ten in the object-oriented C++ language under the ROOT environment. The hard part
of HYDJET++ is identical to the hard part of Fortran-written HYDJET (version 1.5).
The routine for generation of single hard NN collision, generator PYQUEN, modies the
“standard“ jet event obtained with the generator of hadron-hadron interactions PYTHIA
6.4. The event-by-event simulation procedure in PYQUEN includes the generation of
initial parton spectra with PYTHIA and production vertexes at given impact parameter;
rescatteringby-rescattering simulation of the parton path length in a dense zone and radia-
tive and collisional energy loss; nal hadronization according to the Lund string model for
hard partons and in-medium emitted gluons. Then the full hard part of the event includes
PYQUEN multi-jets generated according to the binomial distribution around its mean
value. In order to take into account the effect of nuclear shadowing on parton distribution
functions, the impact parameter dependent parameterization obtained in the framework
of Glauber-Gribov theory is used.
The soft part of HYDJET++ event is the ”thermal” hadronic state generated on the
chemical and thermal freeze-out hypersurfaces represented by a parameterization of rel-
ativistic hydrodynamics with preset freeze-out conditions. Fast soft hadron simulation
procedure includes the generation of the 4-momentum of a hadron in the rest frame of a
liquid element in accordance with the equilibrium distribution function; the generation of
spatial position of a liquid element and its local 4-velocity in accordance with phase space
and the character of motion of the uid; the standard von Neumann rejection/acceptance
procedure to account for the difference between the true and generated probabilities; boost
of the hadron 4-momentum in the center mass frame of the event; and nally the two- and
three-body decays of resonances with the branching ratios taken from the SHARE particle
decay table.
Although HYDJET++ is optimized for very high energies of RHIC and LHC colliders, in
practice it can also be used for studying of multiparticle production in
√
s a wider energy
range of other heavy ion experimental facilities, down to
√
s ∼ 10 GeV per nucleon pair.
HYDJET++ is only applicable for symmetric AA collisions of heavy (A & 40) ions at high
energies (c.m.s. energy
√
s & 10 GeV per nucleon pair). The results obtained for very
peripheral collisions (with impact parameter of the order of two nucleus radii, b ∼ 2RA)
and for very forward rapidities may be not adequate.
3. The simulation of heavy ion collisions at high energies
To simulate the heavy ions interaction processes we use Monte Carlo generator HIJING
and HYDJET++. But as for HYDJET++ we do not collected sufficient statistics, then further
Proceedings of the F&ANS-2010 Conference-School, 2010
Study the signals of QGP using Monte Carlo for . . . 155
FIG. 1. Distribution of the multiplicity of secondary particles for the pseudorapidity η for√
s = 500, 700, 900, 1100, 1300 GeV (with HIJING)
calculations were based on data obtained with HIJING.
We calculate the lead ions (Pb) collisions at high energies. We simulated N = 105 collisions
for center of mass energies in range
√
s from 500 to 1400 GeV (N = 104 from 1400 to 2000 GeV).
The highest energy density obtained in central collisions in which there exists the greatest
probability of QGP formation. The probability of such events depends on the impact parameter.
To simulate the distributions of particle collisions on impact parameter we have used Glauber
model [7].
The highest energy density obtained in central collisions in which there exists the greatest
probability of QGP formation. To study the most likely scenario of QGP formation we were
selected the most central events, representing 5% of the total number of events.
Based on these data we calculated multiplicity distribution of charged secondary particles
on pseudorapidity η (FIG. 1), azimuthal angle ∆φ (FIG. 2) and transverse momentum pt (FIG.
3).
FIG. 2. Distribution azimuthal angle of emission particle with respect to the reaction plane ∆φ for
energy 500, 700, 900, 1100, 1300 GeV
FIG. 3. Transverse momentum distribution for different energy (from 500 to 1400 GeV).
4. The calculation of the factorial moments and scaling exponent
Based on the distributions from figures 1, 2 and 3, it is possible to analyze the fluctuations
of the secondary charged particles multiplicity. As it was shown in [8, 9], a sufficiently large
fluctuations can serve as a criterion for the existence of a phase transition from QGP to hadrons.
Convenient means for determining such fluctuations are the factorial moments (FM) [5]. In [5]
it was shown that this method can help filter out statistical fluctuations and reveal the presence
of fluctuations of the dynamic nature.
But the usage of factorial moments is a good means for detecting fluctuations in processes
which have not very large average multiplicity of the event, for example, in e−e+, pp, as well as in
collisions between ions of light elements. However, the average multiplicity in the event of heavy-
ion collisions is high enough. In this case, the usual factorial moments are not effective. A new
Proceedings of the F&ANS-2010 Conference-School, 2010
156 Babichev L.F., Khmialeuski A.N.
method for studying the particles distribution in events with very high multiplicity was proposed
in [10]. The idea was to use the so-called factorial moments of the multiplicity difference, wich
are multiplicity difference correlators (MDC). They are hybrid of usual moments and wavelets
and have the form
Fq(δ,∆) =
∑
m
m(m− 1)...(m− q + 1)Qm(δ,∆)(∑
m
mQm(δ,∆)
)q (1)
where m = m(δ,∆) = |n1 − n2| is difference between the multiplicities in the two bins located
at the distance ∆ from each other, Qm(δ,∆) is the multiplicities difference distribution.
Typically, for numerical calculations the factorial moments are averaged over several bins
(see, [5] or review [11]) and are calculated as follows. Interval under consideration (for example
pseudorapidity ∆η) is divided into M equal parts. Size of bin is defined as δη = ∆η/M , where
∆η is considered interval of the phase space, M is the number of bins in this interval. To
calculate MDC we using the formula:
Fq =M q−1
〈
N∑
i=1
mi(mi − 1)...(mi − q + 1)
〉
〈
N∑
i=1
mi
〉q (2)
where mi = |n1 − n2| is difference between the multiplisities of two bins of size δ, separated by
a distance ∆, (i = 1, . . .M), M is number of bins on the interval.
Averaging 〈. . . 〉 is the ensemble of events. By varying the number of bins M , and hence
the size of the bin δη, we obtain the dependence of Fq(δη). Thus we calculated MDC of (2), for
q = 2, . . . 7. The examples of such dependencies are shown on figures 4, ?? and ??.
FIG. 4. Dependence ln(Fq(δη)) on − ln(δη) for
√
s = 500 GeV, q = 1 . . . 7 for η
FIG. 5. Example of dependence ln(Fq) on ln(F2), q = 1 . . . 7
In the series of works [5] it was shown that if there are fluctuations of the multiplicity par-
ticles, which are not statistical, it should be observed as power dependence of factorial moments
of the size of the bin:
Fq(δ) ∝ δ−αq (3)
Proceedings of the F&ANS-2010 Conference-School, 2010
Study the signals of QGP using Monte Carlo for . . . 157
where δ is the bin size. The dependence of this type is called intermittency, where αq is index of
intermittency. In [5] it was shown that this dependence should be observed for the MDC. Power
dependence of (3) is a criterion of dynamic fluctuations of multiplicity [11–14]. As was shown in
[5, 17], if the system is characterized by the phenomenon of intermittency that may be observed
the scaling behavior.
Fq(δ) ∝ F−βq2 (4)
where coefficients βq can be obtained by using the following expression:
βq = (q − 1)γ (5)
Here γ is difference universal scaling exponent. As the value of this magnitude can be tested on
the presence in the phase transition from QGP into hadrons. Thus, the scaling exponent is the
criterion by which to test whether the system has undergone a phase transition or not.
In the paper [10] it was shown that the presence of intermittency and the scaling are
characteristic of the MDC. It within Ginzburg-Landau theory has been calculated the scaling
exponent for the second order phase transition, a value of which is 1.09 ± 0.02. Consequently,
if the system is observed scaling behavior of (4) and the value of the difference of the scaling
exponent γ is approximately equal to 1.1, then we can say that the system has undergone a
phase transition of the second kind. In addition, was shown that if the system is not a phase
transition occurs, then the value of the scaling exponent will be equal to 1.33± 0.02.
Figure 5 shows example of the dependence ln(Fq) of ln(F2). This dependence is approx-
imated by straight lines, which are also shown in the figure. Slope coefficients β are obtained
from lines on the figure. The value of the difference of the scaling exponents γ are calculated
by the formula (5) from slope coefficients β. The values of the scaling exponents obtained for
different energies are presented in the table 1. There is lower value of that exponent means
larger fluctuations.
If we compare our data with MDC obtained in the theory of Ginzburg-Landau phase tran-
sition [10], we can say that the value of the scaling exponent γ indicates that in collisions may
be seen the first order phase transition. But in these calculations existents rather large errors,
which would make accurate conclusions. We have slightly modified the method of calculation.
We were taken for calculations is not the whole interval δη, and that part where intermittensy
most pronounced. These results are displayed in the table 2. From Data from the table we can
conclude that on the possible existence of the first order phase transition above 1500 Gev. For
energy from 1500 to 1700 GeV is the possible existence of the second order phase transition.
Proceedings of the F&ANS-2010 Conference-School, 2010
158 Babichev L.F., Khmialeuski A.N.
Table 1. The value of the difference of the scaling exponents for different energies of interaction (Pb +
Pb)√
s, γη on [0; 5,0], γφ on [0; 3,0], γp⊥ on [0; 2,0],
GeV ∆ = 0, 42 ∆ = 0, 19 ∆ = 0, 21
500 1,21 ± 0,03 0,86 ± 0,03 1,22 ± 0,16
600 1,22 ± 0,03 0,86 ± 0,06 1,22 ± 0,16
700 1,23 ± 0,03 0,86 ± 0,07 1,22 ± 0,17
800 1,24 ± 0,03 0,86 ± 0,08 1,21 ± 0,20
900 1,25 ± 0,04 0,85 ± 0,08 1,21 ± 0,20
1000 1,28 ± 0,05 0,84 ± 0,11 1,20 ± 0,32
1100 1,26 ± 0,04 0,85 ± 0,10 1,21 ± 0,25
1200 1,27 ± 0,04 0,84 ± 0.10 1.24 ± 0.29
1300 1,28 ± 0,05 0.85 ± 0.11 1.20 ± 0.33
1400 1,29 ± 0,05 0.85 ± 0.10 1.20 ± 0.32
5. Conclusion
As part of this work it was carried out the Monte Carlo simulation of lead ions colli-
sions for different energies and phase variables and also it was calculated scaling exponents of
MDC γ. Comparing the results for γ with the values calculated within the framework of the
Ginzburg-Landau phase transitions general theory, we can say that the fluctuatians are grow
wirh increasing collision energy. The simulation of collisions of heavy ions to indicate on the
possible existence of the first order phase transition above 1500 Gev. For energy from 1500 to
1700 GeV is the possible existence of the second order phase transition.
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Proceedings of the F&ANS-2010 Conference-School, 2010
Study the signals of QGP using Monte Carlo for . . . 159
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Proceedings of the F&ANS-2010 Conference-School, 2010


Study the signals of QGP using Monte Carlo for Pb+Pb for Energy from 500 to 2000 GeV

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Study the signals of QGP using Monte Carlo for Pb+Pb for Energy from 500 to 2000 GeV

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