Multiplicity Dependence of z-Scaling
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Multiplicity Dependence of z-Scaling in AA Collisions at RHIC. I. Zborovsky * and M.V. Tokarev** * Nuclear Physics Institute Ř e ž near Prague Czech Republic ** Veksler and Baldin Laboratory of High Energies JINR, Dubna Russia. Contents. Principles and symmetries:

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Multiplicity Dependence of z-Scaling in AA Collisions at RHIC

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Multiplicity dependence of z scaling in aa collisions at rhic

Multiplicity Dependence of z-Scaling

in AA Collisions at RHIC

I. Zborovsky* and M.V. Tokarev***Nuclear Physics Institute Řež near PragueCzech Republic **Veksler and Baldin Laboratory of High Energies JINR, Dubna Russia

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


Contents

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

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


Principles symmetries

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)

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


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.

Self-similarity Principle:

Point explosion:

  • P=r(Et2/r)-1/5

    r-radius of the front wave

    E-energy of the explosion

    t-elapsed time

    r-density of the environment

Reynolds number in

hydrodynamics

  • R=Ur/m

    U-velocity of the fluid

    r-density of the fluid

    m-viscosity of the fluid

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


Self similarity in inclusive reactions

Self-similarity in InclusiveReactions

Production of an inclusive particle dependson:

  • Reaction characteristics (A1, A2,s)

  • Particle characteristics (mi, Ei, i)

  • Structural and dynamical characteristics of the interaction (d, e, ..dN/dh..)

Solution:

Search for the solution

z

depending on a single self-similarity parameter

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


Locality in inclusive reactions

Gross features of the single particle distributions

are expressed in terms of the constituent sub-process

Locality in Inclusive Reactions

V.S. Stavinsky 1982

  • M1+M2Þ m + X

  • (x 1M1) + (x2M2 ) Þ m + (x 1M1+x2M2+m2 )

  • The sub-process is subject to the energy-momentun conservation written as follows

  • (x1P1+x2P2 -p)2 = (x1M1+x2M2+m2 )2

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


Fractality of hadron matter

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.

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


Fractality at small scales

Fractality at Small Scales

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


Fractal character of the scaling variable

Fractal character of the scaling variable

z=z0-1

The scaling variable

is a fractal measure

consisting of a finite partz0and a divergent factor -1.

is relative number of all initial

configurations containing the constituents which carry the momenta x1P1 and x2P2.

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

For a given production process, -1 - characterizes resolution

at which the underlying collision of constituents can be singled out of this process.

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


Momentum fractions x 1 and x 2

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 -1of the fractal

measure z=z0-1 with respect to all constituent sub-processes

in which the inclusive particle can be created.

This corresponds to the maximum of

  • with the condition

  • (x1P1+x2P2 -p)2 = (x1M1+x2M2+m2 )2 .

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


Structure of x 1 and x 2

Structure of x1 and x2

Principle of minimal resolution:

  • (x1M1) + (x2M2 ) Þ m + (x1M1+x2M2+m2 )

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


Z scaling hypothesis

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:

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


Aditivity of fractal dimensions d a a d

Aditivity of fractal dimensions dA= A d

The property is connected with factorization of the resolution -1

in the fractal measurez=z0-1for small values of x2 = xA .

  • momentum fraction of the interacting nucleus

  • expressed in units of the nucleon mass.

Relative number of parton configurations

in a single nucleon interaction regime (x2<A-1).

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


Generalized z scaling

Gross features of the single particle distributions

are expressed in terms of the constituent sub-process

Generalized z-Scaling

  • M1+M2Þ m + X

  • (x 1M1) + (x2M2 ) Þ (m/ya) + (x 1M1+x2M2+m2 /yb)

(x 1P1+x2P2 –p/ya)2 = (x 1M1+x2M2+m2/yb)2

Scaling variable:

- 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

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


Variable entropy s

Variable & Entropy S

Statistical entropy

Thermodynamical entropy

  • The quantities cand dN/dη|0have physical meaning of

    “specific heat” and “temperature” of medium, respectively.

  • Entropy of medium decreases with increasing resolution Ω-1 .

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

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

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


Structure of x 1 and x 21

Structure of x1 and x2

Maximal entropy:

x - spatial resolution

Kin.limit:

Symmetry: Space-time structural relativity...

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


Scaling variable

Scaling variable

- transverse kinetic energy of the sub-process consumed

on production of m & m2

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


Sub process illustration

Sub-process illustration

Diagram:

  • (x 1M1) + (x2M2 ) Þ (m/ya) + (x 1M1+x2M2+m2 /yb)

(x 1P1+x2P2 –p/ya)2 = (x 1M1+x2M2+m2/yb)2

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

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


Properties of the scaling function y z in pp collisions

Properties of the scaling function y(z) in pp collisions

  • Energy independence for s1/2>20GeV

  • Angular independence in a wide range of h

  • Multiplicity independence for various multiplicity selection criteria

  • Power law y(z) ~z-b for large z

  • A-universality in pA collisions (dA=Ad)

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


Charged hadrons in pp collisions

Charged hadrons in pp collisions

Energy independence of z scaling

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


Negative pions in pp collisions

Negative pions in pp collisions

Energy independence of z scaling

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


Negative pions in pp collisions angular independence of z

Negative pions in pp collisions Angular independence of (z)

p+p-+p+++

m2=m(++)-m(p)

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


Negative kaons in pp collisions

Negative kaons in pp collisions

Energy independence of z scaling

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


Negative kaons in pp collisions angular independence of z

Negative kaons in pp collisionsAngular independence of (z)

m2=m(K+)

p+pK+p+p+K+

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


Antiprotons in pp collisions

Antiprotons in pp collisions

Energy independence of z scaling

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


Antiprotons in pp collisions angular independence of z

Antiprotons in pp collisions Angular independence of (z)

m2=m(p)

p+pp+p+p+p

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


K 0 s in pp collisions at rhic

K0s in pp collisions at RHIC

Multiplicity independence of z scaling

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


Production in pp collisions at rhic

Λ production in pp collisions at RHIC

Multiplicity independence of z scaling

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


Summary of z scaling in pp collisions

Summary of z-scaling in pp collisions

  • Energy, angular and multiplicity independence of (z)

  • in pp collisions for h, , K, P- ,K0S , 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

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


Charged hadrons in peripheral auau collisions at rhic

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

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


Charged hadrons in central auau collisions at rhic

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)

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


Charged hadrons in auau collisions at 200 130 gev at rhic

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

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


Charged hadrons in auau collisions at 62 gev at rhic

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

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


Charged hadrons in cucu collisions at 200 62 gev at rhic

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)

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


Charged hadrons in dau collisions at 200 gev at rhic

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)

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


Negative pions in auau collisions at 200 gev at rhic

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

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


Positive pions in auau collisions at 200 gev at rhic

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

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


Negative kaons in auau collisions at 200 130 gev

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)

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


Positive kaons in auau collisions at 200 130 gev

Positive kaons in AuAu collisions at 200 & 130 GeV

Similar results as for K

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


K s 0 and k 892 in auau collisions

Ks0 and K*(892) in AuAu collisions

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


Antiprotons in auau collisions at 200 130 gev

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

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


And in auau collisions

-

Ξ+ and Ξ- in AuAu collisions

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


Direct photons in auau collisions

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 

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


Summary

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.

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


Summary cont

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....

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


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