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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 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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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. Fractality at Small Scales Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. Charged hadrons in pp collisions Energy independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  21. Negative pions in pp collisions Energy independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  22. 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

  23. Negative kaons in pp collisions Energy independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  24. 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

  25. Antiprotons in pp collisions Energy independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  26. 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

  27. K0s in pp collisions at RHIC Multiplicity independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  28. Λ production in pp collisions at RHIC Multiplicity independence of z scaling Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. Ks0 and K*(892) in AuAu collisions Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  41. 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

  42. - Ξ+ and Ξ- in AuAu collisions Workshop on Ultrarelativistic Heavy Ion Physics, March 9-13, Dubna 2006

  43. 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

  44. 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

  45. 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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