multiplicity dependence of z-scaling in aa collisions at rhic
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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)
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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:
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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.
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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).
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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:
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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 (
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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Negative pions in pp collisions
Energy independence of z scaling
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
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Negative kaons in pp collisions
Energy independence of z scaling
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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
Energy independence of z scaling
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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)
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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)
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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)
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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)
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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.
Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006
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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....