Evidence of selfaffine target fragmentation process in 197AuAgBr interactions at 10'7 A GeV

1
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

2
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.

3
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?ß?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 lt3 mm and velocity ßlt0.3
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.
4
1. Introduction
In the analysis of intermittency most of the
studies are performed in the one-dimensional
space only, but the real process occurs in three
dimensions. So one-dimensional analysis is not
sufficient enough to make any comment on the
complete dynamical fluctuations pattern.
According to Ochs (PL B247(1990) 101), in a
lower-dimensional 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 self-similar 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).
5
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 dqaq/(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 self-similar cascading is not
consistent with particle creation during one
phase but instead requires a non-thermal phase
transition.
6
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 ?q(aq1)/q. The condition for
non-thermal phase transition may occur when the
function ?q has a minimum value at
qqc(Peschanski, NP B327(1989) 144, PL B410(1991)
323, Bialas and Zaeeswski, PL B238(1990) 413).
Among the two different regions qltqc and qgtqc,
numerous small fluctuations dominate the region
qltqc, but in the region qgtqc, 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 ?q may be a manifestation
of the fact that the liquid and the dust phase
coexist.
7
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).
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)
Where dx1dx2 is the size of a two-dimensional
cell, nm is the multiplicity in the mth cell,
ltnmgt is the average multiplicity of all events in
the mth cell, M' is the number of two-dimensional
cells into which the considered phase-space has
been divided. To fix dx1, dx2 and M' we
consider a two-dimensional region ?x1?x2 and
divide it into subcells with widths
9
3. Method of study
in the x1 and x2 directions where M1?M2 and
M'M1M2. Here M1 and M2 are the scale
factors that satisfy the equation
Where the parameter H (0ltH1) (called Hurst
exponent) characterizes the self-affine property
of dynamical fluctuations. The scaling behavior
that we were looking into has the form
10
3. Method of study
The power aq(gt0) 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 H1, the
fluctuation pattern is called self-similar. If
scaling behavior is found for Hlt1, 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 non-integral 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 lt 1. When the elementary bins of width dy?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.
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3. Method of study
Putting the smaller bin at the last place of the
region and doing average with only the first N
bins, ltFq(dy)gt becomes
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(cos?) and X(f) are used instead of cos? and f.
The cumulative variable X(x) is given by the
relation as follows
12
3. Method of study
where x1 and x2 are two extreme points of the
distribution ?(x). The variable X(x) varies
between 0.0 and 1.0 keeping ?(X(x)) almost
constant.   To probe the anisotropic structure
of phase space we have calculated factorial
moments for the qth order (q3,4,5,6) with the
varying values of Hurst exponent. The partition
numbers along Xcos? and Xf directions are chosen
as Mf 3, 4, , 30, and Mcos? given by We have
not considered the first two data points
corresponding to Mf 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.
13
4. Experimental results
We have plotted the natural logarithm of average
value of factorial moments (lnltFqgt) along Y axis
and the natural logarithm of ?Xcos?.?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 ?2 per degrees of
freedom (DOF) for each linear fit. Interestingly
best linear behavior is revealed at H0.7 and not
at H1 for each order of moment for the data set.
The plots of lnltFqgt against ?Xcos?.?X? at H0.7,
and 1.0 for different order of moment are shown
in figure 1, and 2, respectively. Table 1
represents the value of ?2 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 ?2 per
DOF is smaller at H0.7 for different order of
moment. So the dynamical fluctuation pattern in
197Au-AgBr interactions is not self-similar but
self-affine.
14
4. Experimental results
Fig.1

Fig.2
15
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.
16
4. Experimental results
The variation of anomalous fractal dimension
dq(dqaq/(q-1)) with the order of moment q under
the self-affine scaling scenario (H0.7) 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
197Au-AgBr interactions.
17
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 for
the multifractal Bernoulli fluctuations. In the
above relation, Dq is the generalized dimensions
which is related intermittent indices aq as
Dq1-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
197Au-AgBr collisions at 10.7 A GeV in cos? and f
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.
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4. Experimental results
Fig.4, The generalized dimension Dq versus
lnq/(q-1) at H0.7 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
19
4. Experimental results
Finally we discuss the property of non-thermal
phase transition in the emission of target
fragments in 197Au-AgBr interactions.
Figure 5 presents the dependence of ?q on the
order q. From the plot it is seen that a slight
minimum of ?q is appeared at q4, which may
indicate the coexistence of two different phase,
i.e. the "liquid" and "dust" phases.  
Fig.5
20
Conclusions
From the present study of the 10.7 A GeV
197Au-AgBr interactions, it may be concluded
that 1). The effect of the intermittency is
observed and the best power law behavior is
exhibited at H0.7 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 ?q is observed at q4, which suggested
that there is a coexistence of the liquid and the
dust phases.
Thank you !