Transcript Probing hadron-production Processes by Using New

Probing hadron-production processes by using new
statistical methods to analyze data
LIU Qin
Department of Physics, CCNU, 430079 Wuhan, China
I.
Introduction and Motivation
II.
Fluctuations
III. Correlations
IV. Concluding Remarks
I. Introduction and Motivation
This talk is a brief summary of the following two papers:
Fluctuation studies at the subnuclear level of matter:
Evidence for stability, stationarity, and scaling,
Phys. Rev. D 69, 054026 (2004).
(LIU Qin and MENG Ta-chung)
Direct Evidence for the Validity of Hurst’s Empirical
Law in Hadron Production Processes,
arXiv: hep-ph/0404016v2
(MENG Ta-chung and LIU Qin)
What we want to present in the talk are the results obtained
in a series of preconception-free data-analyses
The purpose of these analyses is to extract useful
information on the reaction mechanism(s) of hadronproduction processes, directly from experimental data.
The method we used to perform such analyses are
borrowed from other sciences:
Mandelbrot’s approach in Economics
Hurst’s R/S-analysis in Marine Sciences
We see a great need for preconception-free data-analyses!
The currently popular ways of describing
haron-production processes are based:
either on a “Three-Step Scenario”
in terms of parton-momentum
distributions, pQCD, and partonfragmentation functions;
or on a “Two-Component Picture”
in which the fluctuations in every
experimental distribution are
separated by hand into a “pure
statistical part” and a part which is
considered to be physical.
Under this assumption, the method
of Bialas and Peschanski is used to
calculate factorial moments.
Step 1
Step 2
Step 3
One common characteristic of these conventional approaches is:
They describe details about the reaction mechanisms.
For example:
how quarks interact with one another;
whether they form multiquark states, etc.
The price one has to pay for such detailed information is:
Large number of inputs (assumptions, adjustable parameters).
Hence, a rather natural question is:
Do we really need so much detailed information as input, if
we only wish to know the key features of such hadronproduction processes?
II. Fluctuations
Bachelier’s Contribution
Since 1900, there has long been a tradition among economists:
Price changes in speculative markets behave like random walks.
The paradigm of such an approach is:
Bachelier’s Gaussian hypothesis (in 1900), which is based on
two assumptions:
Price changes, z (t  T )  z (t ) , are independent random
variables;
⑵ The changes are approximately Gaussian.
⑴


Mathematical basis: Central limit theorem
Fatal defect: Empirical data are not Gaussian!
Mandelbrot’s Contribution
A radically new approach (1963) to the theory of random walks.
An important observation:
The variances of the empirical distributions of price
changes,
LM (t , T )  ln z(t  T )  ln z(t )
can behave as if they were infinite;
Immediate result:
The Gaussian distribution (in Bachelier’s approach) should
be replaced by a family of limiting distributions called Stable
distributions which contain Gaussian as the only member with
finite population variance.


Mathematical basis: Generalized central limit theorem
Main advantage: Empirical data conform best to the nonGaussian members of stable distributions!
Fluctuations in Subnuclear Reactions
we introduce the quantity:
dN
dN
L( ,  )  ln
(   )  ln
( )
d
d
JACEE 1
In analogy with Mandelbrot’s LM (t , T )
We assume:
The L( ,  )' s are identically distributed
random variables;
We examine:
 the resulting distributions by using the
JACEE data as an example
We show that the obtained distributions are:
 Stable
 Stationary
 Scale invariant
JACEE 2
A few words on the kinematics
To study the space-time properties of such fluctuations, we
introduce, in analogy with rapidity,
y
1 E  p//
ln
2 E  p//
1
2
  ln
p  p//
p  p//
a quantity l , which we call “locality” :
1 t  x//
l  ln
2 t  x//
1 r  x//
  ln
2 r  x//
The uncertainty principles lead, in particular, to:
 ~ const.
Results: comparison with data
— Stability test
A non-degenerate random variable X is stable, if and only if for
all m  1, there exist constants cm  m1/ with   (0,2) and d m  R
such that:
d
Sm  X 1  X 2    X m  cm X  d m
where X i ' s are independent, identical copies of X.
Let:
and
X  L( ,  )
d
Sm  L1  L2    Lm  cm L  d m
1
m
d
c (Sm  d m )  L
Here, {L1 , L2 ,, Lm} : stands for a m-dimensional random variable
the components of which can be considered as independent.
JACEE2
1
m
d
c (Sm  d m )  L
The striking agreement
between both sides of the
above equation shows that
there are such constants
for which the data
obtained from both sides
coincide.
The two sets of L( ,  )
obtained from the two
JACEE events are indeed
stable random variables.
JACEE1
— Stationarity test
J1
J2
Stationarity expresses the invariance
principle with respect to time.
Hence in hadron-production processes, the
property of stationarity manifests itself in
the sense that whether the L( ,  )
obtained from the  -distribution
measured at different times (or timeintervals) have the same statistical
properties.
What we can do at present is to compare
between the two JACEE events.
What we see is they are very much the
same!
The fact that: these two events occurred at
different times; in reactions at different
energies; by using different projectiles and
targets, makes the observed similarity
particularly striking!
— Scale Invariance Test
Scaling expresses invariance with
respect to change in the unit of the
quantity with which we do
measurements
One possible way of testing scaling
is to apply the method proposed by
Mandelbrot:
Divide the entire data range
into equal-size samples;
Evaluate the corresponding
variance of each sample;
Plot the frequency distribution
of these variances and examine
their power-law behavior.
In doing so, our result is not as impressive as Mandelbrot’s,
because our data sample is much smaller!
JACEE 2
JACEE 1
4
5
In order to amend this deficiency, we
propose to evaluate the running sample
variance:
1 n
S ( ) 
{L(i ,  )  Ln ( ,  )}2

n  1 i 1
2
n
of JACEE-data and plot their frequency
distributions.
The straight-line structure and thus the
scale invariance property is evident.
The power-law behavior of JACEE events
is in sharp contrast to that of a standard
Gaussian variable.
Combined with the result that L( ,  )
is stable, we are led to the conclusion:
It is not only stable but also nonGaussian!
III.Correlations
In order to extract more information about the reaction
mechanism, we now take an even closer look at the
m-dimensional random variable, {L1, L2 ,, Lm} , and ask:
Is there global statistical dependence between the
components Li (,  )' s ?
In terms of mathematical statistics, the question is:
Does the joint distribution of the above mentioned
m-dimensional random variable have non-zero
correlation coefficients?
i. Method
The method we propose to check the existence of such
statistical dependence is the Hurst’s rescaled range
analysis (also known as R/S analysis)
It is a robust and universal method for testing the presence
of global statistical dependence of many records in Nature
The reason why we choose this method is because of the
fact that the main statistical technique to treat very global
statistical dependence, spectral analysis, performs poorly
on records which are far from being Gaussian
R/S analysis
  t0 , 
Step 1. Average influx:
1 t0 
 (t )


t t 0
t0 
Step 2. Accumulated departure:
X (t0 , t , )  { (u)    t0 , }
u t 0
Step 3. Range:
R(t0 , )  max X (t0 , t , )  min X (t0 , t , )
Step 4. Standard deviation:
Step 5. Hurst’s empirical law:
0t 
0t 
 1 t0 

2
S (t0 , )    { (t )    t0 , } 
  t t0

1
2
R
(t0 , ) ~  H (t0 ) , H  (0, 1)
S
R/S intensity: J=H-1/2
H=1/2 thus J=0: absence of global statistical dependence
H>1/2 thus J>0: persistence
H<1/2 thus J<0: antipersistence
  t0 , 
1 t0 
 (t )


dN
1
ln
 yi ,Y 
dy
Y
t t 0
t0 t
yi Y
 ln
y  yi
yi  y
X (t0 , t , )  { (u)    t0 , }
X ( yi , y, Y )  {ln
R (t0 , ) 
R( yi , Y ) 
u t 0
u  yi
max X (t0 , t , )  min X (t0 , t , )
0t 
R
(t0 , ) ~  H (t0 ) ,
S
dN
dN
(u)  ln
 yi ,Y }
dy
dy
max X ( yi , y, Y )  min X ( yi , y, Y )
0t 
 1 t0 

2
S (t0 , )    { (t )    t0 , } 
  t t0

dN
( y)
dy
0 y Y
0 y Y
1
2
S ( yi , Y ) 
1

Y

yi Y
 {ln
y  yi

dN
dN
( y )   ln
 yi ,Y }2 
dy
dy

R
( yi , Y ) ~ Y H ( y i )
S
1
2
ii. Results
Validity of Hurst’s law, and scaling
behavior, with universal features of
H=0.9 for both JACEE events
Hurst exponent H ( yi ) is independent of
the selected starting point yi .
Existence of global statistical
dependence and thus the existence of
global structure in the set {ln dN d ( )}
Self-affine property of the records
with fractal dimension
where DT  DG  1.1
is divider dimension
DT  1 H
D  2  H is capacity dimension
G
IV. Concluding Remarks
The fact that non-Gaussian stable distributions which are
stationary and scale-invariant describe the existing data
remarkably well calls for further attention.
It would be very helpful to have a comparison with data taken
at other energies and/or for other collision processes.
The validity of Hurst’s empirical law with the same exponent
(H=0.9>0.5) for the two JACEE events is not only another
example for the existence of universal features in the complex
system of produced hadrons, but also implies the existence of
global statistical dependence and thus the existence of global
structure between the different parts of the system.
The fact that the extremely robust quantity such as the
frequency distribution of running sample variance and the
rescaled range R/S obey universal power-laws which are
independent of the colliding energy, independent of the
colliding objects, and independent of the size of the
rapidity intervals, strongly suggests that the system under
consideration has no intrinsic scale in space-time.
Since none of the above-mentioned features can be directly
related to the basis of the conventional picture, it is not
clear whether, (and if yes, how and why) these striking
empirical regularities can be understood in terms of the
conventional theory, including QCD.