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Striking empirical regularities in hadron-production pStriking empirical regularities in hadron-production p
rocesses: rocesses:
Can they be understood in terms of QCD?Can they be understood in terms of QCD?
LIU Qin
Department of Physics, CCNU, 430079 Wuhan, China
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 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: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!
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 Einstein’s.
Bachelier’s Gaussian hypothesis says:)()( tzTtz
Mandelbrot’s Contribution
An important observationAn important observation:: The variances of the empirical distributions of price change
s, can behave as if they were infinite;
an immediate resultan 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 in 1963:
)(ln)(ln),( tzTtzTtLM
Fluctuations in Subnuclear Reactions
In analogy with Mandelbrot’s ,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
),( TtLM
sL )',(
A few words on 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.
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
JACEE2
— 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. ResultsThe validity of Hurst’s law, also scaling behavior, with universal features of H=0.9 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
)( iyH
iy
)}({ln ddN
1.1 GT DD
HDT 1HDG 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.