Transcript 相对论重离子碰撞实验的现状及展望

Evidence of self-affine target fragmentation
process in 197Au-AgBr interactions at 10.7 A
GeV
D.H. Zhang, F. Wang, J.X. Cheng, B. Cheng, Q. Wang, H.Q. Zhang, R. Xu
Institute of Modern Physics, Shanxi Normal University
Linfen 041004, China
Sept. 2, 2008
1、Introduction
2、Experimental details
3、Method of Study
4、Experimental Results
5、Conclusions
1. Introduction
In high energy interactions, the study of non-statistical fluctuations have entered into
a new era since Bialas and Peschanski(NP B273(1986) 703) introduced an attractive
methodology to study non-statistical fluctuations in multiparticle production. They
suggested that the scaled factorial moment Fq has a growth following a power law with
decreasing phase space interval size and this feature signals the onset of intermittency in
the context of high energy interactions. This scaled factorial moment method has the
feature that it can measure the non-statistical fluctuations avoiding the statistical noise.
Up to now, most of the analysis have been carried out in the relativistic produced
particles with the common belief that these particles are the most informative about the
reaction dynamics and thus could be effective in revealing the underlying physics of
relativistic nucleus-nucleus collisions. However, the physics of nucleus-nucleus
collisions at high energies is not yet conclusive and therefore all the available probes
need to thoroughly exploited towards meaningful analysis of experimental data.
1. Introduction
In relativistic heavy ion induced nuclear emulsion interactions, the target
fragmentation produces highly ionizing particles responsible for heavy tracks which are
subdivided into gray and black tracks. The gray tracks are the medium-energy (30-400
MeV) knocked-out target protons (or recoiled protons) with range 3 mm and velocity
0.3β0.7. They are supposed to carry some information about the interaction dynamics
because the time scale of the emission of these particles is of the same order (10-22s) as
that of the produced particles. The general belief about these recoiled protons is that
they are the low energy part of the internuclear cascade formed in high energy
interactions. The black tracks with range <3 mm and velocity β<0.3 are attributed to
evaporation from highly excited nuclei in the thermodynamically equilibrium state. In
the rest system of the target nucleus, the emission direction of the evaporated particles
is distributed isotropically.
1. Introduction
In the analysis of intermittency most of the studies are performed in the onedimensional space only, but the real process occurs in three dimensions. So onedimensional analysis is not sufficient enough to make any comment on the complete
dynamical fluctuations pattern. According to Ochs (PL B247(1990) 101), in a lowerdimensional projection the fluctuations will be reduced by the averaging process. In
two-dimensional analysis generally the phase space are divided equally in both
directions assuming that the phase spaces are isotropic in nature. Consequently selfsimilar fluctuations are expected. It may happen that the fluctuations are anisotropic and
the scaling behavior is different in different directions giving rise to self-affine scaling.
So far only a few works have been reported where the evidence of self-affine
multiparticle production is indicated by the data(Ghosh et al., EPJ A14(2002) 77, PR
C66(2002) 047901, JP G29(2003) 983, IJMP E13(2004) 1179, MPL A22(2007) 1759,
Wang et al., PL B410(1997) 323, Wu and Liu, PRL 70(1993) 3197).
1. Introduction
In most of the earlier works on intermittency, best linear fits were drawn in the total
bin range from some pre-conceived ideas. Actually, the plots are not perfectly linear in
the whole bin range, rather nice linear behavior is apparent in selective bin ranges. So it
would be better to investigate intermittency in those bin ranges.
The intermittency pattern cannot only suggest the dynamical nature of fluctuation but
also reveals the inner fractal structure of the fluctuation codimensions dq (Ochs, PL
B247 (1990) 101, Bialas and Gazdzicki, PL B252(1990) 483), which are related to the
intermittency indices aq as dq=aq/(q-1). Unique dq for a different order of moments
suggests monofractality whereas order dependence of dq signals the presence of
multifractality. Multifractality may be due to self-similar cascading, whereas
monofractality is associated with thermal transitions. Now, there is a feeling that selfsimilar cascading is not consistent with particle creation during one phase but instead
requires a non-thermal phase transition.
1. Introduction
According to Peschanski (PL B410(1991) 323) if the dynamics of intermittency is
due to self-similar cascading, then there is a possibility of observing a non-thermal
phase transition. The signals of non-thermal phase transition can be studied with the
help of the parameter λq=(aq+1)/q. The condition for non-thermal phase transition may
occur when the function λq has a minimum value at q=qc(Peschanski, NP B327(1989)
144, PL B410(1991) 323, Bialas and Zaeeswski, PL B238(1990) 413). Among the two
different regions q<qc and q>qc, numerous small fluctuations dominate the region q<qc,
but in the region q>qc, dominance of small number of very large fluctuations occur.
This situation resembles a mixture of a "liquid" of many small fluctuations and a
"dust“ consisting of a few grains of very large density. The minimum of the function λq
may be a manifestation of the fact that the liquid and the dust phase coexist.
2. Experimental details
Stacks of NIKFI BR-2 nuclear emulsion plates were horizontally exposed to a 197Au
beam at 10.7 A GeV at BNL AGS. BA2000 microscopes with a 100 oil immersion
objective and 10 ocular lenses were used to scan the plates. The tracks are picked up at
a distance of 5mm from the edge of the plates and are carefully followed until they
either interacted with emulsion nuclei or escaped from the plates. Interactions which are
within 30μm from the top or bottom surface of the emulsion plates are not considered
for final analysis. All the primary tracks are followed back to ensure that the events
chosen do not include interactions from the secondary tracks of other interactions.
When they are observed to do so the corresponding events are removed from the sample.
To ensure that the targets in the emulsion are silver or bromine nuclei, we have chosen
only the events with at least eight heavy ionizing tracks of particles (Nh8).
3. Method of study
We adopted a procedure to study the self-affine scaling behavior of factorial moments,
where the size of the elementary phase-space cells can vary continuously. In two
dimension if the two phase space variables are x1 and x2, factorial moment of order q
may be defined as (Bialas and Peschanski, NP B273(1986) 703)
1 M  nm (nm  1)    (nm  q  1)
Fq (x1x2 ) 


M m1
 nm  q
Where δx1δx2 is the size of a two-dimensional cell, nm is the multiplicity in the mth cell,
<nm> is the average multiplicity of all events in the mth cell, M' is the number of twodimensional cells into which the considered phase-space has been divided.
To fix δx1, δx2 and M' we consider a two-dimensional region Δx1Δx2 and divide it into
subcells with widths
3. Method of study
x1
x1 
M1
x2
x2 
M2
in the x1 and x2 directions where M1M2 and M'=M1·M2.
Here M1 and M2 are the scale factors that satisfy the equation
M1  M 2H
Where the parameter H (0<H≤1) (called Hurst exponent) characterizes the self-affine
property of dynamical fluctuations. The scaling behavior that we were looking into has
the form
Fq (x1x2 )  (x1x2 )
 aq
3. Method of study
The power aq(>0) is a constant at any positive integer q and it is called intermittency
exponent which measures the strength of intermittency. If such a scaling behavior is
found for H=1, the fluctuation pattern is called self-similar. If scaling behavior is found
for H<1, the fluctuation is called self-affine.
It is clear that the scale factors M1 and M2 cannot be an integer simultaneously, so
that the size of the elementary phase space cell would be continuously varying value.
The following method has been adopted for performing the analysis with nonintegral value of scale factor (M'). For simplicity, we considered one-dimensional space
(y) and let
M' = N + a
Where N is an integer and 0 ≤ a < 1. When the elementary bins of width δy=y/M' are
used as the “scale” to “measure” the region y, N of them are obtained and a smaller bin
of width ay/M' is left.
3. Method of study
Putting the smaller bin at the last place of the region and doing average with only the
first N bins, <Fq(δy)> becomes
1 Nev 1 N  nmi (nmi  1)    (nmi  q  1) 
 Fq (y) 


Nev i N m1
 nm  q
Where nmi is the multiplicity in mth cell of the ith event, and M' can be any positive real
number and it can vary continuously.
Our work are performed in two-dimensional emission-azimuthal angle space. As the
shape of the single particle distribution influences the scaling behavior of the factorial
moments, the “cumulative” variables X(cosθ) and X(φ) are used instead of cosθ and φ.
The cumulative variable X(x) is given by the relation as follows:
x
x2
x1
x1
X ( x)    ( x)dx
  ( x)dx
3. Method of study
where x1 and x2 are two extreme points of the distribution ρ(x). The variable X(x) varies
between 0.0 and 1.0 keeping ρ(X(x)) almost constant.
To probe the anisotropic structure of phase space we have calculated factorial
moments for the qth order (q=3,4,5,6) with the varying values of Hurst exponent. The
partition numbers along Xcosθ and Xφ directions are chosen as Mφ= 3, 4, … , 30, and
Mcosθ given by
M cos  MH
We have not considered the first two data points corresponding to Mφ= 1, 2 to reduce
the effect of momentum conservation (Liu, et al., ZP C73(1997) 535) which tends to
spread the particles in opposite directions and thus reduce the value of the factorial
moments. This effect becomes weaker as M increases.
4. Experimental results
We have plotted the natural logarithm of average value of factorial moments (ln<Fq>)
along Y axis and the natural logarithm of Xcosθ.X along X axis for 197Au-AgBr
interactions at 10.7 A GeV for different Hurst exponent values(0.3, 0.4, 0.5, 0.6, 0.7, 0.8,
0.9, and 1.0). For each case linear behavior is observed in two or three regions. In order
to find the partitioning condition at which the scaling behavior is best revealed, we have
performed linear fit in first region, and have estimated the χ2 per degrees of freedom
(DOF) for each linear fit. Interestingly best linear behavior is revealed at H=0.7 and not
at H=1 for each order of moment for the data set. The plots of ln<Fq> against
Xcosθ.X at H=0.7, and 1.0 for different order of moment are shown in figure 1, and 2,
respectively. Table 1 represents the value of χ2 per DOF and the intermittency exponent
for 197Au-AgBr interactions for different values of H and order of moment. From the
table it is seen that χ2 per DOF is smaller at H=0.7 for different order of moment. So the
dynamical fluctuation pattern in 197Au-AgBr interactions is not self-similar but selfaffine.
4. Experimental results
Fig.1
Fig.2
4. Experimental results
The power-law behavior of the scaled factorial moments implies the existence of
some kind of fractal pattern (Hua, 1990) in the dynamics of the particles produced in
their final state. Therefore, it is natural to study the fractal nature of target fragments in
197Au-AgBr interactions under the self-affine scaling scenario.
4. Experimental results
The variation of anomalous
fractal dimension dq(dq=aq/(q-1))
with the order of moment q under
the self-affine scaling scenario
(H=0.7) is presented in right figure.
From the plot it is seen that dq
increases linearly with the order q
for the data set, which suggests the
presence of multifractality of
emission target fragments in 197AuAgBr interactions.
4. Experimental results
Recently Bershadskii (PR C59(1998) 364) showed that the constant specific heat
approximation is also applicable to the multifractal data of multiparticle production
process. Starting from the definition of Gq-moment, he derived the following relation
ln q
Dq  (a  c)  c
q 1
for the multifractal Bernoulli fluctuations. In the above relation, Dq is the generalized
dimensions which is related intermittent indices aq as Dq=1-aq/(q-1), a is some other
constant and constant c can be interpreted as the multifractal specific heat of the
system. We have determined the multifractal specific heat for our data. Fig.4 shows
the plot of Dq obtained from Fq-moment analysis as a function of lnq/(q-1) for 197AuAgBr collisions at 10.7 A GeV in cosθ and φ two-dimensional phase space. Straight
line is the linear fit to the data, indicating good agreement between our data and
multifractal Bernoulli representation. The slope of the fitted line, which gives the
value of the multifractal specific heat c for our data, is 0.520.05.
4. Experimental results
Fig.4, The generalized dimension Dq
versus lnq/(q-1) at H=0.7 in 197Au-AgBr
interactions at 10.7 A GeV. Straight line
is the linear fit to the data, indicating
good agreement between our data and
multifractal Bernoulli representation.
The slope of the fitted line, which gives
the value of the multifractal specific heat
c for our data, is 0.520.05.
Fig. 4
4. Experimental results
Finally we discuss the property of nonthermal phase transition in the emission of
target fragments in 197Au-AgBr interactions.
Figure 5 presents the dependence of λq
on the order q. From the plot it is seen that
a slight minimum of λq is appeared at q=4,
which may indicate the coexistence of two
different phase, i.e. the "liquid" and "dust"
phases.
Fig.5
Conclusions
From the present study of the 10.7 A GeV
concluded that:
197Au-AgBr
interactions, it may be
1). The effect of the intermittency is observed and the best power law behavior is
exhibited at H=0.7 which suggested the dynamical fluctuation pattern in 197Au-AgBr
interactions is not self-similar but self-affine.
2). The anomalous fractal dimensions of the intermittency is found to increase with the
increase of the order of moments, which suggests the presence of multifractality of
emission of target fragments in 197Au-AgBr interactions.
3). A slight minimum value of λq is observed at q=4, which suggested that there is a
coexistence of the liquid and the dust phases.
Thank you !