Multiplicity Dependence of z-Scaling in AA Collisions at RHIC

Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

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I. Zborovsky* and M.V. Tokarev**

*Nuclear Physics Institute Řež near PragueCzech Republic

**Veksler and Baldin Laboratory of High Energies

JINR, Dubna Russia

Multiplicity Dependence of z-Scalingin AA Collisions at RHIC


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Contents Principles and symmetries: self-similarity, locality, fractality z-Scaling in inclusive reactions Generalized z-scaling Multiplicity dependence of z-scaling in AA collisions at RHIC Conclusions


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Principles & Symmetries

Motivation: Search for phenomenological description of

production cross sections aiming to grasp main principles which influence the particle production at small scales.

Self-similarity. Locality. Fractality.

There exists special symmetry inherent to them: Symmetry with respect to structural degrees of freedom. (The space-time structural relativity)


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Self-similarity Principle:

Dropping of certain quantities out of physical

picture of the interaction. Construction of self-similarity parameters as simple combinations of suitable physical quantities.

Reynolds number in Reynolds number in hydrodynamics hydrodynamics R=UR=U//U-velocity of the fluid U-velocity of the fluid

-density of the fluid-density of the fluid

-viscosity of the fluid-viscosity of the fluid

Point explosion:Point explosion: =r(Et=r(Et22//

r-radius of the front wave r-radius of the front wave

E-energy of the explosionE-energy of the explosiont-elapsed timet-elapsed time

-density of the environment-density of the environment


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Self-similarity in Inclusive Reactions

Production of an inclusive particle depends on:

1. Reaction characteristics (A1, A2, s)

2. Particle characteristics (mi, Ei, i)

3. Structural and dynamical characteristics of the interaction (dN/d

Search for the solution dzdσ

Nσ1

ψ(z)in

depending on a single self-similarity parameter z

Solution:


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Locality in Inclusive Reactions

Gross features of the single particle distributions

are expressed in terms of the constituent sub-process

(x (x 11MM11) + (x) + (x22MM2 2 ) ) m + (x m + (x 11MM11+x+x22MM22+m+m2 2 ) )

The sub-process is subject to the energy-momentun conservation written as follows (x1P1+x2P2 -p)2 = (x1M1+x2M2+m2 )2

MM11+M+M22 m + X m + X

V.S. Stavinsky 1982


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Fractality of Hadron Matter

Extended objects like hadrons and nuclei are considered to have fractal properties with respect to increasing resolution concerning the parton content involved.

(Objects consisting of “subtle nets” of quarks, anti-quarks and gluons).

Assumption of fractality: Self-similarity of parton sub-structure does not exhaust with increasing resolution.


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Fractality at Small Scales


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Fractal character of the scaling variable

z=z0-1The scaling variable

consisting of a finite part z0 and a divergent factor -1.

For a given production process, -1 - characterizes resolutionat which the underlying collision of constituents can be

singled out of this process.

1,2 - anomalous fractal dimensions of the colliding objects with respect to their constituent sub-structure.

is relative number of all initial configurations containing the constituents which carry the momenta x1P1 and x2P2.

21 )1()1(),( 2121 xxxx

is a fractal measure

1)(z


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Momentum fractions x1 and x2

Principle of minimal resolution: For a given inclusive reaction, the fractions x1 and x2 are determined to minimize the resolution -1 of the fractal measure z=z0-1 with respect to all constituent sub-processes in which the inclusive particle can be created.

21 )1()1(),( 2121 xxxx

with the condition (x1P1+x2P2 -p)2 = (x1M1+x2M2+m2 )2 .

This corresponds to the maximum of


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Structure of x1 and x2

1

2

1

2

11

21

21 )(

)(

s

pE

PPpP zp

iii χλx Principle of minimal resolution:

)χ(χ)λ(λ)χ(λ)χ(λ 21212211 (x1M1) + (x2M2 ) m + (x1M1+x2M2+m2 )

2

2

2

2

22

21

12 )(

)(

s

pE

PPpP zp

Uii

2

1U)1)(1( 21

212

1

2


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z-Scaling hypothesis

Production cross sections of particles with large transverse momenta in relativistic collisions of hadrons and nuclei depend in a self-similar way on the scaling variable:

1

0

1/2

Ω|dN/dη

sz )( 22211

2/1 mxMxMmsss

2

2211

2

2211)()( PPsPPs

3

31

in0in dpσd

EJ)σ|(dN/dη

πsψ(z),

dzdσ

Nσ1

ψ(z)

iiixxx ,)1()1( 2121


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Aditivity of fractal dimensions A= A

The property is connected with factorization of the resolution -1

in the fractal measure z=z0-1 for small values of x2 = xA .

-momentum fraction of the interacting nucleus expressed in units of the nucleon mass.

22 Axx

)1()1(

)/1()1(

21

21

1

1

xx

AxxA

Relative number of parton configurationsin a single nucleon interaction regime (x2<A-1).


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Generalized z-Scaling Gross features of the single particle distributions

are expressed in terms of the constituent sub-process

(x (x 11MM11) + (x) + (x22MM2 2 ) ) m/m/yyaa + (x + (x 11MM11+x+x22MM22+m+m2 2 //yybb) )

(x (x 11PP11+x+x22PP2 2 –p/–p/yyaa))22 = (x = (x 11MM11+x+x22MM22+m+m22//yybb))22

MM11+M+M22 m + X m + X

Ws

z2/1

2/1

s - transverse kinetic energy of the sub-process consumed on production of m & m2 W - relative number of all configurations of the system which can lead to production of m & m2

Scaling variable:


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Variable & Entropy SWs

z2/1

Ωc)dη/ (dN0

W 21 )1()1()1()1( 21 xxyy ba

WS lnStatistical entropy Thermodynamical entropy

const. RlnVlnTcV Sconst. ])x(1)x(1)y(1)y(1ln[)dη/dN(lnc 2δ

21δ

1ba0

εε S

The quantities c and dN/dη|0 have physical meaning of “specific heat” and “temperature” of medium, respectively. Entropy of medium decreases with increasing resolution Ω-1 . βψ(z))(z'ψ'

z/βz'

Max. entropy S = Max. number of configurations W(ya,yb,x1,x2)

with the condition: (x1P1+x2P2–p/ya)2 = (x1M1+x2M2+m2/yb)22

10

zz


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Structure of x1 and x2

b

j

a

ji yPP

mM

yPP

pP 1)(

1)(

)(

21

2

21

iiii 22

Uii

2

1

1 2

u

uU

222111 xx

)()()()( 21212211

Maximal entropy:

10,/ 12

Kin.limit: 1

11

u 2121

21

1

uuuu

u

Symmetry: Space-time structural relativity...

spatial resolution


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Scaling variable Ws

z2/1

mMMsyT aa )( 2211

2

2211

2

2211)()( PPsPPs

iiix

- transverse kinetic energy of the sub-process consumed on production of m & m2

ba TTs 2/1

22211 )( mMMsyT bb

2/1s

)()()()( 21212211


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Sub-process illustrationm

m2

M1 M2 x1

x2

ya

yb

Diagram:

1c

0

1/2

Ω)|(dN/dη

sz

21 )1()1()1()1( 21 xxyy ba

(x (x 11MM11) + (x) + (x22MM2 2 ) ) m/m/yyaa + (x + (x 11MM11+x+x22MM22+m+m2 2 //yybb) )

(x (x 11PP11+x+x22PP2 2 –p/–p/yyaa))22 = (x = (x 11MM11+x+x22MM22+m+m22//yybb))22

Larger = smaller y = larger energy losses in the final state


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Properties of the scaling function (z) in pp collisions

Energy independence for s1/2>20 GeV Angular independence in a wide range of Multiplicity independence for various

multiplicity selection criteria Power law (z) z

for large z

universality in pA collisions (


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Charged hadrons in pp collisions

1c

0

1/2

Ω)|(dN/dη

sz 21 )1()1()1()1( 21

xxyy ba

Energy independence of z scaling


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Negative pions in pp collisions



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Negative pions in pp collisions Angular independence of (z)

p+p-+p+++ m2=m(++)-m(p)


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Negative kaons in pp collisions



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Negative kaons in pp collisionsAngular independence of (z)

p+pK+p+p+K+ m2=m(K+)


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Antiprotons in pp collisions



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Antiprotons in pp collisions Angular independence of (z)

p+pp+p+p+p m2=m(p)


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K0s in pp collisions at RHIC

Multiplicity independence of z scaling


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Λ production in pp collisions at RHIC

Multiplicity independence of z scaling


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Summary of z-scaling in pp collisions

• Energy, angular and multiplicity independence of (z) in pp collisions for h, , K, P- ,K0

S , Λ• Specific heat for the pp system: c=0.25• Proton anomalous fractal dimension: =0.5 • Fragmentation anomalous dimension is constant with dN/d • increases with particle mass: ()=0.2, (K)=0.3, (P)=0.35, (Λ)=0.4


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Charged hadronsin peripheral AuAu collisions at

RHIC

• Energy independence of (z) in peripheral AA • Same shape of (z) for peripheral AA & pp• Specific heat: c(AA)=0.09<c(pp)=0.25• Same in peripheral AA & pp


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Charged hadronsin central AuAu collisions

at RHIC

• Energy independence of (z) in central AA • Energy dependence of in central AA • Specific heat: c(AA)=0.09<c(pp)=0.25• increases with centrality in AA (increase of energy losses with centrality)


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Charged hadrons in AuAu collisions

at 200 & 130 GeV at RHIC

• Energy independence of (z) in AA • Same shape of (z) in AA & pp (solid line) Energy dependence of in AA Multiplicity dependence of in AA • Specific heat: c(AA)=0.09<c(pp)=0.25


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Charged hadrons in AuAu collisions

at 62 GeV at RHIC

• Compatibility of STAR & PHOBOS data • Same shape of (z) in AA & pp Energy dependence of in AA Multiplicity dependence of in AA • Specific heat: c(AA)=0.09<c(pp)=0.25


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Charged hadrons in CuCu collisions

at 200 & 62 GeV at RHIC

• Compatibility of STAR & PHOBOS data • Same shape of (z) in AA & pp (solid line) Energy dependence of in AA Multiplicity dependence of in AA • Specific heat: c(CuCu)=0.09<c(pp)=0.25• A-independence of • A=A (additivity of fractal dimensions)


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Charged hadrons in dAu collisions

at 200 GeV at RHIC

• does not depend on centrality in dAu as in pp collisions (no extra losses in this system)• specific heat c increases in dAu system: c(dAu)>c(AuAu)


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Negative pions in AuAu collisions

at 200 GeV at RHIC

STAR and PHENIX data confirm universal shape of (z) for pion production in AuAu & pp


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Positive pions in AuAu collisions

at 200 GeV at RHIC

• Same energy and multiplicity dependence of for pions as for charged particles • Specific heat c(AA)=0.09 is same for pions as for charged particles • Same shape of (z) in AA & pp


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Negative kaons in AuAu collisions

at 200 & 130 GeV

• Same shape of (z) in AA & pp (solid line) Energy dependence of in AA Multiplicity dependence of in AA • Specific heat: c(AA)=0.09<c(pp)=0.25• A=A (additivity of fractal dimensions)


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Positive kaons in AuAu collisions

at 200 & 130 GeV

Similar results as for K


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Ks0 and K*(892) in AuAu

collisions


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Antiprotons in AuAu collisions at 200 & 130 GeV

• Energy independence of (z) in AA • Multiplicity independence of (z) in AA • Nuclear effects in the shape of (z) for small z with respect to pp (solid line)• is same as in pp - independence on dN/d• Specific heat: c(AA)=0.09<c(pp)=0.25


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Ξ+ and Ξ in AuAu collisions


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Direct photons in AuAu collisions

• data prefer 0 - direct formation of in the sub-process with no (or small) energy losses• but errors bars are too large to make strong conclusion on


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Summary Z-scaling in inclusive particle production at high

energies reflects self-similarity, locality and fractality of hadron interactions at constituent level.

The scaling function (z) and scaling variable z are expressed via measurable quantities (inclusive cross sections, particle density, kinematical variables).

The scaling includes multiplicity, energy and angular independence of (z) in pp and pA collisions.

General features of the scaling are found to be valid for particle (h,,K,anti-p) production in A-A collisions at RHIC energies.


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Summary (cont.) Parameters and are interpreted as anomalous

fractal dimensions of the colliding and produced objects, respectively.

Relation between thermodynamical characteristics (entropy, specific heat ) and the quantities W and c entering the z definition was established.

Increase of the fractal dimension with centrality in AA collisions reflects strong energy losses in fragmentation of the scattered and recoil

constituents in the final state. Obtained results are of interest for verification of

z scaling and search for new physics at large multiplicities and high pT at RHIC and LHC energies....

Documents

Multiplicity Dependence of z-Scaling in AA Collisions at RHIC