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					                            Sir Syed Ahmad Khan                   di Ettore Majorana


Some Aspects of Dynamical Fluctuations
in Heavy Ion collisions at AGS Energies

         Collaborators:- Prof. Shafiq Ahmad and M. Ashraf T
     The Aligarh Muslim University, Aligarh, India
    International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                Plan of Presentations

Introduction
Experimental Details
Mathematical Analysis
Results and Discussions
 Conclusions

International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                                   Introduction
 The holy grail of Ultra-relativistic heavy ion collisions (URHIC) is the
 discovery and characterisation of the Quark-gluon Plasma (QGP) in the
 laboratory.
 QGP: a state of matter in which                  QGP study: very active program
quarks and gluons are no longer                   at International leveal – both in
confined to volumes of hadronic                   theoretical and experiment sectors.
dimensions.


Nuclei approaching to collide     Hot regime              QGP           Cooling and particle production




          International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
A schematic phase diagram of strongly interacting matter, showing phase
transition between hadronic matter and QGP as a function of temperature
and baryonic chemical potential.




                                                         QGP and
                                                         its signatures




                                              Fig. taken out from a ref. depicted in Ph. D. Thesis
                                              Submitted by M. Ayaz Ahmad at AMU, Aligarh (2008).

        International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
          Facilities to hunt the QGP and its signatures
                                                                              The Future
                                                                              Experiment
 -What were the expectations                    Little Bang                   CBM: Fixed Target
 in the beginning?                                                            U235 @ 2-45 GeV/A
                                                                 2009-10
                                                                 LHC: Collider
                                                      2000       Pb+Pb @5.5TeV/A
                                                  RHIC: Collider
                                  1996            Au+Au @ 200GeV/A

                                  SPS: Fixed Target
                                  Pb at 158GeV/A
               1991               (Ec.m.=17.3GeV)             - What have we learned so far?
               AGS: Fixed Target Au                           - What are the prospects for the
               at 11.7 GeV/A                                  future?
1984           (Ec.m.=4.86GeV)

Bevalac:
Fixed Target
Au at 1GeV/A
                      International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                           Event-by-event Fluctuations
                                      At the critical point of QGP transition: large
                                      fluctuations are expected in several quantities.
WMAP Data


                                                  Fluctuations are following types:
                                                       •Multiplicity fluctuations
                                                       •Temperature or <pT> fluctuation
                                                       •Fluctuation in particle ratio
                                                       •Long range correlations
                                                       •Net charge fluctuations
                                                       •Balance functions
WMAP and the COBE DMR                                  •Disoriented chiral condensations
(Differential microwave radiator)
results show tiny variations (< 10-5K)
in the Cosmic Background Radiation
temperature. This reflects small
density fluctuations in the early
Universe => galaxy formation.
       Nobel Prize 2006: Mather & Smoot
Dec 11, 2007   International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                                 DAE Nuclear Physics Symposium, Sambalpur                          6
The study of fluctuations in rapidity density as a possible signature for
QGP, has gained much eagerness during last couple of years.

We have studied the dynamical fluctuations, in terms of the scaled factorial
moments, Fq, which describes the genuine phenomena of multiparticle
production in one and two dimensional phase space.

A modest attempt has been made to study the behaviour of intermittency,
anomalous fractal dimensions and Renyi dimensions using scaled factorial
moments (SFMs).


We have also discussed Levy stable theory. The experimental results have
been compared with ultrarelativistic quantum molecular dynamics (UrQMD)
and uncorrelated Monte Carlo (Mc-RAND) models.


     International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                            Experimental Details
For present investigations, FUJI type nuclear emulsion has been used, with a
beam of 28Si nuclei at 14.6A GeV/c at Alternating Gradient Synchrophasotron
(AGS) of Brookhaven National Laboratory (BNL), NewYork, USA.




28Si   projectile at 14.6A GeV
   Input informations
   Total data = 2000, and <NS> for 28Si-Em data is 21.09  0.17
   To minimize statistical fluctuations events with NS  8 have been used,
   which is 1251 and their <NS> is 21.72  0.17.
   For present data min is –2.309 and max is 7.129
 More details about present experiments published by M. Ayaz Ahmad in the
 IJMPE Vol.16, (2007), 2241, and References there in.
       International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                            Mathematical Analysis
First we have changed the values of pseudo-rapidity in all the spaces (, , and
,) in a new normalized “cumulative” variables X() and X() proposed by
Bialas and Gazdzicki, which is given by the following relation:
                                    max
 X ( ) 
            
                   ( ) d  /       ( ) d 
                                     min
                                                                , will be change into 0-1

             min




    On the basis of bin averaging the normalized scaled factorial moments
 of the order of q is defined in vertical form as:

                                                                                 nm 
                                      d
                                  M                                          Md     q
                       1/ M   d
                                  n            q
                                                m
                                                                      1
                                                            Fq ()  d
                                                              V
                                                                             1  n  q
  FqH ( )                     m 1
                                                                    M       m
                      ( n  / M )          d       q                              m




      International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
 Using normalized scaled factorial moments an increasing trend in
   fluctuations with decreasing bin size is a representation of an
   intermittent behaviour, which leads to a power law expressed by:

                                             q
                        Fq (X )  X                   (X  0)

                                                   q
                         Fq (X )  M                    ( M  0)




      International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                            Results and Discussions

 Dynamical fluctuations in one dimension (1D):
     3.0                                                        3.0


     2.8                                                        2.8
                                                                               Space
     The SFMsare capable of measuring
     2.6          -Space                  q=6                   2.6
                                                                              Experimental Data                   q=6
                Experimental Data                               2.4
                 fluctuations
the large-scale Experimental Data and provide
     2.4                                                                      UrQMD Data
                                                                2.2           McRAND Data
     2.2
information about the pattern of
                Fig. 1(b)                                       2.0           Fig. 1(a)
     2.0




                                                        <lnFq()>
                                          q=5
dynamical fluctuations.                                         1.8                                               q=5
     1.8
 <lnF q()>




                                                                1.6
     1.6

                                                                1.4
    The pseudorapidity interval, is
     1.4                                                        1.2
successively divided into M = 30 bins q=4
     1.2                                  and                   1.0
                                                                                                                  q=4
     1.0                                                        0.8
the results of calculated values of ln<Fq>                      0.6
                                                                                                                  q=3
                                          q=3
      function of ln M in the  and  -
as a 0.8
     0.6
                                                                0.4                                               q=2
spaces respectively for each orderq=2of                         0.2
     0.4                                                        0.0
moments are shown in Fig. 1(a & b)
     0.2                                                       -0.2
along with the corresponding UrQMD
     0.0                                                              0.5   1.0   1.5     2.0   2.5   3.0   3.5     4.0
    -0.2
predictions. 1.0 1.5 2.0 2.5 3.0 3.5 4.0                                                  ln M 
         0.5
                           ln M

              International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                                                           3.0

The experimental result is not                            2.8
                                                           2.6            Space
                                                                         Experimental Data                   q=6
                                                           2.4
    reproduced by the independent                          2.2
                                                                         UrQMD Data
                                                                         McRAND Data
    emission mode of the particle.                         2.0           Fig. 1(a)




                                                   <lnFq()>
                                                           1.8                                               q=5
                                                           1.6
                                                           1.4
This gives an indication for the absence                  1.2
                                                           1.0
    of statistical contribution in the                     0.8
                                                                                                             q=4
                                                                                                             q=3
    experimental data.                                     0.6
                                                           0.4                                               q=2
                                                           0.2
                                                           0.0
                                                          -0.2
                                                                 0.5   1.0   1.5     2.0   2.5   3.0   3.5     4.0
                                                                                     ln M

The     flat    behaviour in MC events is
expected for independent emission of particle.




        International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
     Dynamical fluctuations in two dimension (2D)

In order to see the existence of
                                                               4
intermittency signal in two-dimension,                                           - Space
we started with a rectangle in ()                                            Experimental Data
                                                                               Mc-RAND Data                     q=6
space.                                                         3                Fig. 1 (c)




                                                 <lnFq()>
The rectangle was divided into                                 2
                                                                                                                q=5

M, bins each of size ()
=(/M) ( /M).                                                                                              q=4
                                                               1                                                q=3
                                                                                                                q=2
Fig. 1 (c) shows the plots of ln <Fq>
                                                               0
as a function of ln MM in the
interactions of 28Si-Em collisions at                              0.5   1.0     1.5    2.0   2.5   3.0   3.5     4.0
14.6A GeV/c for the order q = 2-6.                                                  ln (MM)




       International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
A linear rise of SFMs is seen from the figures and a larger q is reported in
given Table
  Data Set         2              3              4              5              6            Phase
                                                                                              Space/Models
                                                                                                 -space
 28
  Si-Em       0.089 0.003    0.187 0.002    0.214 0.003    0.551 0.007    0.911 0.008
 14.6     A   0.095  0.002   0.211  0.003   0.293 0.002    0.680  0.008   1.043  0.009      -space
 GeV/c        0.220  0.004   0.301  0.003   0.506  0.008   0.901  0.008   1.373  0.007     -space
              0.084  0.004   0.182 0.003    0.314 0.008    0.679 0.008    1.109 0.007      UrQMD
              0.005  0.002   0.049  0.004   0.050 0.008    0.053  0.007   0.054  0.004    M- Carlo




A stronger intermittency effect is observed in 2D, where as a weak signal is
seen in one-dimension (1D).


Study of Intermittency in one dimension published in Nucl. Phys. A 780, (2006), 206



         International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
         Levy stable law and anomalous fractal dimension, dq
The anomalous dimensions, dq, in terms of q,
                                                            0.32
                                                            0.30
is expressed by the following relation:                     0.28          
                                                            0.26          

             d q   q (q  1) 1
                                                                          
                                                            0.24
                                                                          UrQMD
                                                            0.22        Fig. 2
                                                            0.20




                                                       dq
The variation of dq with the order of the                   0.18

moments, q, is depicted in Fig. 2 for ,  and              0.16

 -spaces respectively. The UrQMD results
                                                            0.14
                                                            0.12
shown by the solid spheres in  and  spaces                0.10
are also presented in Fig. 2.                               0.08
                                                            0.06
                                                                   2     3        4     5    6
                                                                                  q


The increasing trend of dq with q reflects a self-similar cascade mechanism for the
production of final particles. Thus, the behaviour of anomalous fractal dimension,
dq, does not favour the origin of any exotic phenomenon. These observations are
in agreement with the results from lepton–hadron and hadron–hadron collisions.


         International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
    The Levy stability index parameter, , is considered to be useful in describing the
    intermittency in multiparticle production at high-energy collisions. The Levy
    index helps in classifying the intermittency regimes due to different kinds of
    phase transitions during the cascading process.
                                                            12
                            
    q  2 .(q  q) /(2  2)                               10
                                                                             experimental
                                                                             Theoretical
                                                                               Fig.3
       q   2 . q(q  1) / 2                                  8

                                                                6




                                                        q/2
It is seen that both the Eqns. are valid for                    4
q  0 and in limit  2 passes to Eqn. in
the whole range of q is                                         2

                                                                0   Nucl. Phys. A 789, (2007), 298
 q  2 .[(q 2 )  / 2  q] /(2   2)                         -2       0       2
                                                                                      q 4   6        8


The values of the intermittency indices (q /2) as a function of q are plotted in Fig.3.
The value of the Levy index, , for our data obtained from the fits of Eqn. ,which is
shown in Fig.3 is found to be 1.507  0.006. This value of  lies in the Levy stable
region (0    2) and it is consistent with the Levy stable theory.

         International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
                      Generalized fractal dimension Dq

            q                                       1.00
Dq  1              or   Dq  1  d q               0.98
           (q  1)                                   0.96
                                                     0.94
                                                     0.92
                                                     0.90
The decreasing pattern of Dq with the                0.88
                                                     0.86
order of moments, q clearly gives an                 0.84




                                                Dq
agreement with the multifractal                      0.82
                                                     0.80
cascade mechanism.                                   0.78          
                                                                   
                                                     0.76
                                                                   
                                                     0.74          UrQMD
This behaviour of Dq does not favour                 0.72
                                                                 Fig.4
the existence of second order phase                  0.70
                                                     0.68
transition. Similar results have been                0.66
reported in various experiments.                            2       3       4      5       6
                                                                           q

Therefore, the observed scaled factorial
moment analysis reveals a self-similarity
characteristic in multiparticle production

       International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
    Non-thermal phase transitions                       0.70

The possibility of observing a non-                    0.65
                                                                             
thermal phase transition can be obtained                0.60                 
                                                                             
by calculating the relevant parameter, q,              0.55
                                                                             UrQMD
using the relation:                                     0.50




                                                   q
         q  ( q  1) / q                             0.45

                                                        0.40

If a non-thermal phase transition is                    0.35
                                                                     Fig.5
present, then q, should have a                         0.30
minimum at certain value of q = qc,                     0.25
where qc is some minimum point in the
                                                               2       3       4      5     6
distribution.                                                                 q

In order to study the existence of the non-thermal phase transition, the variation of
q as a function of q is shown in Fig. 5 for the variables ,  and  respectively.
 However, no clear dip is observed in q vs q plot for certain value of q in one
 dimension as reported by other workers. Thus our data do not support a clear
 evidence for the existence of non-thermal phase-transition.

        International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
             Conclusions
 On the basis of the result presented, the following conclusions may be drawn.
 A generalized power law behaviour of scaled factorial moments in ,  and 
   spaces gives an evidence of self-similar structure in multiparticle production.
 A weak intermittency effect has been observed experimentally in one dimension,
  whereas a strong signal of intermittency behaviour is seen in a two dimensional
  phase space.
 The behaviour of anomalous fractal dimension, dq, with q does not favour the
  origin of any exotic phenomenon.
 The decreasing trend of Dq with increasing order of moments, q indicates that
   multiparticle production is due to a self-similar cascade process in one and two
   dimensional phase spaces.
 No clear evidence for the existence of non-thermal phase transition is observed
  from the study of q vs q graph in one as well as two dimensional phase-space.
 The value of Levy index, , found in the Levy stability analysis is 1.507  0.006,
   which is consistent with the Levy stable region (0    2) .
 Comparison of experimental results with the data generated using the ultra-relativistic
   quantum molecular dynamics (UrQMD) model reproduces similar pattern in most of
   the results.

          International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009
International School of Sub Nuclear Physics at ERICE, from Aug. 29- Sep .07, 2009

				
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