Probing hadron-production processes by using new Probing hadron-production processes by using new

statistical methods to analyze datastatistical methods to analyze data

LIU Qin

Department of Physics, CCNU, 430079 Wuhan, China

I. Introduction and Motivation 

II. Fluctuations 

III. Correlations 

IV. Concluding Remarks

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)

I. Introduction and Motivation

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 hadron-production 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

either on a “Three-Step Scenario” in terms of parton-momentum distributions, pQCD, and parton-fragmentation 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.

We see a great need for preconception-free data-analyses!

Step 1 Step 2 Step 3

The currently popular ways of describing haron-production processes are based:

Under this assumption, the method of Bialas and Peschanski is used to calculate factorial moments.

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 Do we really need so much detailed information as input, if

we only wish to know the key features of such hadron-productiwe only wish to know the key features of such hadron-production processes?on processes?

II. Fluctuations

Bachelier’s Contribution

⑴ Price changes, , are independent random variables;

⑵ The changes are approximately Gaussian.

Mathematical basis: Central limit theorem Fatal defect: Empirical data are not Gaussian!

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:

)()( tzTtz

Mandelbrot’s Contribution

An important observation: The variances of the empirical distributions of price change

s, 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 non- Gaussian members of stable distributions!

A radically new approach (1963) to the theory of random walks.

)(ln)(ln),( tzTtzTtLM

Fluctuations in Subnuclear Reactions

we introduce the quantity:

)(ln)(ln),(

d

dN

d

dNL

We assume:

The 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 1

JACEE 2

In analogy with Mandelbrot’s ),( TtLM

sL )',(

A few words on the kinematics

To study the space-time properties of such fluctuations, we introduce, in analogy with rapidity,

l

//

//ln2

1

xt

xtl

//

//ln2

1

xr

xr

//

//ln2

1

pE

pEy

//

//ln2

1

pp

pp

.~ const

a quantity , which we call “locality” :

The uncertainty principles lead, in particular, to:

A non-degenerate random variable X is stable, if and only if for all , there exist constants with and such that:

where are independent, identical copies of X.

Let:

and

1m /1mcm )2,0( Rdm

mm

d

mm dXcXXXS 21

sX i '

),( LX

mm

d

mm dLcLLLS 21LdSc

d

mmm )(1

Results: comparison with data

— Stability test

},,,{ 21 mLLL : stands for a m-dimensional random variableHere,the components of which can be considered as independent.

JACEE2

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

obtained from the two JACEE events are indeed stable random variables.

LdScd

mmm )(1

),( L

JACEE1

— Stationarity test

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 obtained from the -distribution measured at different times (or time-intervals) 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!

J1 J2),( L

— 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 1 JACEE 2

45

In order to amend this deficiency, we propose to evaluate the running sample variance:

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 is stable, we are led to the conclusion:

It is not only stable but also non-Gaussian!

n

inin LL

nS

1

22 )},(),({1

1)(

),( L

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, , and ask:

Is there global statistical dependence between the components ?sLi )',(

},,,{ 21 mLLL

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

Step 1.Step 1. Average influx:

Step 2.Step 2. Accumulated departure:

Step 3.Step 3. Range:

Step 4.Step 4. Standard deviation:

Step 5.Step 5. Hurst’s empirical law:

0

0

0)(

1,

t

ttt t

0

0

0})({),,( ,0

t

tututtX

),,(min),,(max),( 00

00

0

ttXttXtRtt

2

1

2,0 })({

1),(

0

0

0

t

t

tt

ttS

,~),( )(0

0tHtS

R )1,0(H

R/S intensityR/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

0

0

0)(

1,

t

ttt t

tt

tututtX

0

0

0})({),,( ,0

2

1

2,0 })({

1),(

0

0

0

t

t

tt

ttS

),,(min),,(max

),(

00

00

0

ttXttX

tR

tt

Yy

yyYy

i

i

iy

dy

dN

Ydy

dN)(ln

1ln ,

yy

yuYyi

i

i

idy

dNu

dy

dNYyyX }ln)({ln),,( ,

),,(min),,(max

),(

00YyyXYyyX

YyR

iYy

iYy

i

2

1

2, }ln)({ln

1

),(

Yy

Yy

yy

i

i

i

idy

dNy

dy

dN

Y

YyS

)(~),( iyHi YYy

S

R,~),( )(0

0tHtS

R

ii. ResultsValidity of Hurst’s law, and scaling behavior, with universal features of H=0.9 for both JACEE events

Hurst exponent is independent of the selected starting point .

Existence of global statistical dependence and thus the existence of global structure in the set

Self-affine property of the records with fractal dimension where is divider dimension is capacity dimension

)( iyH

iy

)}({ln ddN

1.1 GT DDHDT 1

HDG 2

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.