Striking empirical regularities in hadron-production processes: Can they be understood in terms of QCD?

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Striking empirical regularities in
hadron-production processes Can they be
understood in terms of QCD?

LIU Qin Department of Physics, CCNU, 430079
Wuhan, China
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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 Hursts
  empirical law in hadron production processes,
• arXiv hep-ph/0404016v2
• (MENG Ta-chung and LIU Qin)

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What we want to present in these two talks 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
• Mandelbrots approach in Economics
• Hursts R/S-analysis in Marine Sciences.

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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
  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.

Step 1
Step 2
Step 3
Under this assumption, the method of Bialas and
Peschanski is used to calculate factorial
moments.
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• 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 hadron-production processes?
  

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II. Fluctuations

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Bacheliers Contribution
Bachelier, a then young French student, was the
first who used the idea of random walk to study
fluctuations. It was in year 1900, five years
earlier than Einsteins. Bacheliers Gaussian
hypothesis says

• ? Price changes, , are
  independent random variables
• ? The changes are approximately Gaussian.
• Mathematical basis Central limit theorem
• Fatal defect Empirical data are not Gaussian!

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Mandelbrots Contribution
A radically new approach in 1963

• An important observation
• The variances of the empirical
  distributions of price changes,
• can behave as if they were infinite
• an immediate result
• The Gaussian distribution (in
  Bacheliers 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!

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Fluctuations in Subnuclear Reactions
In analogy with Mandelbrots , we
introduce the quantity
JACEE 1

• 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 2
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A few words on kinematics
To study the space-time properties of such
fluctuations, we introduce, in analogy with
rapidity,
a quantity , which we call locality
The uncertainty principles lead, in particular,
to
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• Results comparison with data
• Stability test

• 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

stands for a m-dimensional random variable
Here,
the components of which can be considered as
independent.
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• 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.

JACEE1
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JACEE2
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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
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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.

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In doing so, our result is not as impressive as
Mandelbrots, because our data sample is much
smaller!
JACEE 2
JACEE 1
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• 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!

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1. Correlations

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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 ?

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?

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i. Method

• The method we propose to check the existence of
  such statistical dependence is the Hursts
  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

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R/S analysis

• Step 1. Average influx
• Step 2. Accumulated departure
• Step 3. Range
• Step 4. Standard deviation
• Step 5. Hursts empirical law

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• R/S intensity JH-1/2
• H1/2 thus J0 absence of global
  statistical dependence
• Hgt1/2 thus Jgt0 persistence
• Hlt1/2 thus Jlt0 antipersistence

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ii. Results

• The validity of Hursts law, also scaling
  behavior, with universal features of H0.9 for
  both JACEE events
• Hurst exponent is independent of
• the selected starting point .
• There exists global statistical dependence and
  thus global structure in the set
• .
• Self-affine property of the records
• with fractal dimension
• where
• is divider
  dimension
• is capacity
  dimension

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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 Hursts empirical law with the
  same exponent (H0.9gt0.5) 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.

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• 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.