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Multiplicity fluctuations in high energy hadronic and nuclear collisions

Explore fluctuations in elementary interactions and system variables in cosmic ray experiments and nuclear collisions. Investigate the impact of inelasticity and parameter q on system fluctuations. Study multiplicity distributions and fluctuating temperatures in relation to dynamical fluctuations.

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Multiplicity fluctuations in high energy hadronic and nuclear collisions

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  1. Multiplicity fluctuations in high energy hadronic and nuclear collisions M. Rybczyński(a), G. Wilk(b) and Z. Włodarczyk(a) (a)Świętokrzyska Academy, Kielce, Poland (b) Soltan Institute for Nuclear Studies, Warsaw, Poland XIII ISVHECRI – NESTOR Institute Pylos, Grece, 6-12 September 2004

  2. (*) What counts most in the cosmic ray experiments in which such showers are observed? (*) Cross-section of elementary interactions (or, rather, of hA and AA interactions): (E) (*) Inelasticity K(E) defined as fraction of the actual energy used to produce secondaries and therefore lost for the subsequent interaction: (*) We shall add here that important are also possible fluctuations in these variables and argue that they can be deduced from the measurements of multiplicities (Problem that experimentally (,K)are interrelated will not be discussed, see: SWWW, JPhG18 (1992) 1281)

  3. Historical example: (*) observation of deviation from the expected exponential behaviour (*) successfully intrepreted (*) in terms of cross-section fluctuations: (*) can be also fitted by: (*) immediate conjecture: q fluctuations present in the system Depth distributions of starting points of cascades in Pamir lead chamber Cosmic ray experiment (WW, NPB (Proc.Suppl.) A75 (1999) 191 (*) WW, PRD50 (1994) 2318

  4. q – measure of fluctuations (*) Parameter q is known in the literature as measure of nonextensivity in the Tsallis statistics based on Tsallis entropy (a): Sq = - (1-piq)/(1-q) => -  pi ln pi forq1 (*) It can be shown to be a measure of fluctuations existing in the system (b): • WW, Physica A305 (2002) 227 • WW, PRL 84 (2000) 2770

  5. Inelasticity from UA5 and similar data..... q=1 q>1 NUWW PRD67 (2003) 114002

  6. (*) Input: s, T, Ncharged (*) Fitted parameter: q, q-inelasticity q NUWW PRD67 (2003) 114002 q • (*) Inelasticity K:= fraction of the total energy s, which goes into • observed secondaries produced in the central region of reaction • very important quantity in cosmic ray research and statistical models

  7. NUWW PRD67 (2003) 114002

  8. NUWW PRD67 (2003) 114002

  9. NUWW PRD67 (2003) 114002

  10. NUWW PRD67 (2003) 114002

  11. Possible meaning of parameter q in rapidity distributions (*) From fits to rapidity distribution data one gets systematically q>1 with some energy dependence (*) What is now behid this q? (*) y-distributions  ‘partition temperature’ TK s/N (*) q  fluctuating Tfluctuating N   (*) Conjecture: q-1 should measure amount of fluctuation in P(N) (*) It does so, indeed, see Fig. where data on q obtained from fits are superimposed with data on parameter k in Negative Binomial Distribution! NUWW PRD67 (2003) 114002

  12. Negative-Binomial Distribution generating function: average and variance: k =  Poisson distribution k = - N binomial distribution

  13. Parameter q as measure of dynamical fluctuations in P(N) (*) Experiment: P(N) is adequately described by NBD depending on <N> and k (k1) affecting its width: (*) If 1/k is understood as measure of fluctuations of <N> then with (*) one expects: q=1+1/k what indeed is observed (P.Carruthers,C.C.Shih, Int.J.Phys. A4 (1989)5587)

  14. Multiplicity is important ... Notice: there is remarkable linear relation between <Ncharged> and the corresponding cross section for pp and ppcollisions (cf. also: NP. in NC 63A (1981) 129 or Yokomi, PRL 36 (1976) 924) V3/2 Fluctuations of multiplicity and  should also be related ......

  15. Multiplicity Distributions: (UA5, DELPHI, NA35) Kodama et al.. SS (central) 200GeV e+e- 90GeV Delphi UA5 200GeV <n> = 21.1; 21.2; 20.8 D2 = <n2>-<n>2 = 112.7; 41.4; 25.7  Deviation from Poisson: 1/k 1/k = [D2-<n>]/<n2> = 0.21; 0.045; 0.011

  16. (*) Experiment: P(N) is adequately described by NBD depending on <N> and k (k1) affecting its width: (*) If 1/k is understood as measure of fluctuations of <N> then with (*) one expects: q=1+1/k what indeed is observed Parameter q as measure of dynamical fluctuations in P(N)

  17. Recent example from AA – (1) (MWW, APP B35 (2004) 819) With increasing centrality fluctuations of the multiplicity become weaker and the respective multiplicity distributions approach Poissonian form. ??? Perhaps: smaller NW smaller volume of interaction V smaller total heat capacity C greater q=1+1/C  greater 1/k = q-1 Dependence of the NBD parameter 1/k on the number of participants for NA49 and PHENIX data

  18. Recent example from AA – (2) (MWW, APP B35 (2004) 819) It can be shown that ( Wróblewski law ) ( for p/e=1/3) • in this case • q1.59 It (over)saturates therefore the limit imposed from Tsallis statistics: q1.5 . For q=1.5 one has: 0.33  0.28 (in WL) or 1/3  0.23 (in EoS) Dependence of the NBD parameter 1/k on the number of participants for NA49 and PHENIX data

  19. Potentially very important result from AA collisions concerning fluctuations (MRW, nucl-th/0407012) for AA collisions the usual superposition model deos not work when applied to fluctuations (signal for the phase transition to Quark-Gluon-Plasma phase of matter?...)

  20. Limitations on fluctuation... Notice: q  1.5 limit, if applied here, leads to saturation of fluctuations at energies s  33.32 TeV or ELAB 0.5 1018 eV i.e., in the UHECR energy range where effects of the GZK cut starts to be important It is important for any analysis connected with GZK to know the fluctuation pattern - it can be decisive factor here!

  21. Summary (*) Inelasticity K and cross-section  seem still be main parameters influencing development of the cosmic ray cascades (*) In some recent analysis presenting cross section obtained from cosmic ray data it is not clear whether it was accounted for that in any single CR experiment K and  are measured in junction (*) Some data call for proper accounting of fluctuations, which can be most economically described by changing exp[ -x/ ]  expq[ -x/ ] = [1-(1-q) x/ ]1/(1-q) with q being new parameter (reaction and energy dependent) (*) Fluctuations in  , K and multiplicity can substantially change the predicted(expected?) development of all kinds of CR cascades (*) The single parameter q seems to summarily account for all new effects, which can have different (mostly unknown yet) sources

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