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What is the meaning of the statistical model ? F.B. hep-ph 0410403

F. Becattini, Kielce workshop, October 15 2004. What is the meaning of the statistical model ? F.B. hep-ph 0410403. OUTLINE Introduction Discussion: phase space dominance, triviality and Lagrange multipliers Basic microcanonical formulation Future tests.

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What is the meaning of the statistical model ? F.B. hep-ph 0410403

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  1. F. Becattini, Kielce workshop, October 15 2004 What is the meaning of the statistical model ?F.B. hep-ph 0410403 OUTLINE Introduction Discussion: phase space dominance, triviality and Lagrange multipliers Basic microcanonical formulation Future tests

  2. The statistical model is successful in describing soft observables in Heavy Ion Collisions • M. Gazdzicki, M. Gorenstein • F. B., A. Keranen, J. Manninen • J. Cleymans, H. Satz • P. Braun-Munzinger, J. Stachel, D. Magestro • W. Broniowski, W. Florkowski • J. Letessier, J. Rafelski • K. Redlich, A. Tounsi • A. Panagiotou, C. Ktorides • Nu Xu, M . Kaneta And many more...

  3. The statistical model is even more successful in describing relevant soft observables in elementary collisions • F.B. Z. Phys. C 69 (1996) 485 • F. B., Proc. XXXIII Eln. Workshop Erice, hep-ph 9701275 • F. B., U. Heinz, Z. Phys. C 76 (1997) 269. • F. B., G. Passaleva, Eur. Phys. J. C 23 (2001) 551 F.B., Nucl. Phys. A 702 (2002) 336 F.B., G. Passaleva, Eur. Phys. J. C 23 (2002) 551 Warning! Strangeness phase space is undersaturated

  4. Why? From: L. Mc Lerran, Lectures “The QGP and the CGC”, hep-ph 0311028 • Three kind of answers (criticisms) • The thermodynamical-like behaviour is only mimicked by • the data. It should be rather called “phase space dominance” • The Statistical Model results are somehow trivial due to • large involved multiplicities • The Statistical Model results can be obtained as a by • -product of other models, at least in elementary collisions The temperature is not a real temperature

  5. Phase space dominance VS Discussed in detail inJ. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th 0001044 If |Mif|2has a very weak dependence on kinematical independent variables, e.g.pi·pj,we could somehow recovera pseudo- thermal shape of the multiplicity and pT spectrum function

  6. for large mults IF Where bis such that: Conclusion:b is not a temperature and inclusive particle multiplicities are not sensitive enough to the different integration measure to distinguish between a genuine thermal behaviour and this pseudo-thermal function (phase space dominance) However, there is a quantitative difference!

  7. In principle,|Mif|2may depend on as well as on Phase space dominance is not trivial Example: quite restrictive: again, only one scale a and factorization The thermal-like behaviour can be easily distorted at ANY scale of Multiplicity (just take g(I)=AI2+C or f(am)=(am)5)

  8. Disagreement with “Triviality” arguments e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002 |Mif|2depends onN; Nis large; small fluctuations ofN |Mif|2 is unessential at high Nand therefore the statistical model results are trivially recovered |Mif|2may not depend just on N, also on specific particle content in the channel (through mass, isospin etc.) In analyses of e.g. pp collisions overall multiplicities are not large enough to make fluctuations negligible

  9. “Lagrange multiplier” or what is a temperature? See e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002; U. Heinz, hep-ph 0407360 “Concepts in heavy ion physics” Seem to advocate the idea that the temperature determined with hadron abundances is not a “real temperature”, rather a “Lagrange multiplier constraining maximization of entropy” • This is just a possible definition of temperature • There might be different definitions in small systems (e.g. 1/T=S/E, saddle point for microcanonical partition function, etc.) but ALL OF THEM converge to the same quantity in the thermodynamic limit • A quantitative difference is needed: if you have volume, energy and statistical equilibrium, temperature is a temperature regardless of how the system got there!

  10. Derive the statistical features within other models A. Bialas, Phys. Lett. B 466 (1999) 301 W. Florkowski, Acta Phys. Pol. B 35 (2004) 799 Fluctuation of the string tension may lead to an exponential shape, e.g. of the pT spectrum Occam razor argument From: Delphi collaboration, CERN-PPE 96-120

  11. How to probe a genuine statistical model ? Need to test exclusive channel rates Much more sensitive to the integration measure (V d3p vs d3p/2e) because information is not integrated away Data available at low energy ( s < 10 GeV) Need full microcanonical calculations see e.g. W. Blumel, P. Koch, U. Heinz, Z. Phys. C63 (1994) 637

  12. Basic scheme Clusters: extended massive objects with internal charges Every multihadronic state within the cluster compatible with conservation laws is equally likely

  13. The microcanonical ensembleand its partition function A usual definition reads Pi projector on the cluster’s initial state Can be generalized as | hV >multihadronic state within the cluster canonical: What is the probability of an asymptotic free state| f >? Define

  14. All pf are positive definite as: The cluster is described by the mixture Note: PiPVPiPVPiPV Used in Eur.Phys. J. C 35 (2004) 243 In principle, projection PV should be made on localized field states: In all studies, relativistic quantum field effects are neglected: good approximation forV1/3 > lC (at most 1.4 fm)

  15. F.B., L. Ferroni, Eur. Phys. J. C 35 (2004) 243 F.B., L. Ferroni, hep-ph 0407117, Eur. Phys. J. C in print Full microcanonical ensembleProjection onto an irreducible state P 4-momentum J spin l helicity p parity c C-parity Q abelian charges I, I3 isospin Decompose the projector: The projectorPP,J,l,pcan be written (formally) as an integral over the extended Poincare’ groupIO(1,3)↑ The projectors on 4-momentum, spin-helicity and parity factorize ifP=(M,0)

  16. Other projectors: Integral projection technique already used in the canonical ensemble (Cerulus,Turko, Redlich,Cleymans, et al.) Restricted microcanonical ensemble: only four-momentum and abelian charges

  17. Rate of a multi-hadronic channel {Nj}=(N1,...,NK) For non-identical particles: For identical particles: cluster decomposition Generalization of the expression in M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445 which holds only for large V partitions

  18. Comparison between mC and C hadron multiplicities Q=0cluster,M/V=0.4 GeV/fm3 pp-likecluster,M/V=0.4 GeV/fm3 Baryons Mesons

  19. Summary and Conclusions • Discussion on the statistical model • Temperature, phase space dominance: only quantitative differences are differences. • More quantitative tests of the picture, e.g. on exclusive channels (at low energy) require full microcanonical calculations and Monte Carlo implementations (matching with parton shower) • Microcanonical ensemble sampling algorithm for hadron system accomplished. Ongoing work to include ang. Mom., isospin etc.

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