STATISTICAL MODELS in high energy collisions OUTLINE oIntroduction oFormulation for full microcanonical ensemble oDiscussion on “triviality” oNumerical methods and comparison micro-can F. Becattini, BNL, May

Hadronization : formation of extended massive regions (clusters or fireballs) emitting hadrons according to a pure statistical law

The statistical law Every multihadronic state within the cluster compatible with conservation laws is equally likely Statistical equilibrium  Set of multi-hadronic states having the energy-momentum, angular momentum, parity and charges of the cluster = the microcanonical ensemble  Hadrons and resonances treated as free states (Hagedorn, on the basis of BDM paper) – ideal hadron-resonance gas  Cluster has a spacial extension (like in the MIT bag model and unlike in HERWIG MC) Statistical model can be considered a model for the decays of MIT bags

What is the origin of equilibrium ?  Collisions among formed hadrons in a slowly expanding fireball (  scatt <<  exp ) until decoupling like in the Hot Big Bang theory (thermalization)  Quantum evolution leads to a uniform superposition over the multihadronic states within the cluster and, as a consequence, equiprobability of observation when measurement is made

A temptative reformulation (F.B., L. Ferroni, “Statistical hadronization and hadronic microcanonical ensemble I” hep-ph , to appear in Eur. Phys. J. C) | i > cluster’s initial state | h V > multihadronic state within the cluster Starting from the end:

Full microcanonical ensemble P 4-momentum J spin helicity helicity  parity  C-parity Qabelian charges I, I 3 isospin The projector can be decomposed as: The projector P P,J,  can be written as an integral over the extended Poincare’ group IO(1,3) ↑ The projector P P,J,  can be written as an integral over the extended Poincare’ group IO(1,3) ↑ Basis for microcanonical calculation

The projectors on 4-momentum, angular momentum and parity factorize if P=(M,0) Other projectors: Integral projection technique already used in the canonical ensemble. Integral projection technique already used in the canonical ensemble. A recursive method for the canonical ensemble recently used by S. Pratt et al. A recursive method for the canonical ensemble recently used by S. Pratt et al. (Phys. Rev.C68 (2003) and ref. therein) (Phys. Rev.C68 (2003) and ref. therein)

“Restricted” microcanonical ensemble Neglecting angular momentum, parity and isospin conservation: summing over all J, , I, I 3 with equal weights Usual definition of microcanonical ensemble M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445

Rate of a multi-hadronic channel {N j }=(N 1,...,N K ) It can be shown that, for non-identical particles: Usually found in literature (e.g. K. Werner, J. Aichelin Phys. Rev. C 52 (1995) 1584) In relativistic quantum field theory, confined states are NOT eigenstates of properly defined particle number operator. The above expression holds provided that V 1/3 > Compt For pions: Compt = 1.4 fm

Generalized expression: phase space volume as a cluster expansion Partitions  The leading term is the  {N j } for the classical Boltzmann statistics  Subleading terms enhance phase space volume for identical bosons and suppress it for identical fermions. They disappear in the limit V   Generalization of the expression in M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445 which holds only for large V

Phase space and Fermi golden rule  Different measures (proper phase space d 3 x d 3 p vs invariant momentum Space d 3 p/2  ) leading to different averages  Statistical phase space model predicts definite ratios between different channels  Statistical phase space model has built-in quantum statistics effects (BEC) due to the finite volume VS

They try to demonstrate that the same results of the statistical model can be obtained starting from different assumptions Is statistical population trivial?  J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th  D. Rischke, Nucl. Phys. A698 (2001) 153, talk at QM2001  V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th , talk given at QM2002 Relativistic invariant: depends on as well as on Correct, but not trivial

The peculiar prediction of the statistical model which can be easily spoiled by most |M if | 2 if channel constants depend on particle content Example Quite restrictive: only a single scale  and factorization

Average multiplicities for large M: where  is such that The actual production pattern may be similar to the prediction of the statistical model, though this is not trivial (e.g. if f is a steep function of the mass)  can indeed fake a temperature

“Triviality” argument advocated in nucl-th  |M if | 2 depends on N; N is large; small fluctuations of N  |M if | 2 is unessential at high N and therefore the statistical model results are trivially recovered |M if | 2 is unessential at high N and therefore the statistical model results are trivially recovered 1.|M if | 2 may not depend just on N, also on specific particle content in the channel (through mass, isospin etc.) 2.In analyses of e.g. pp collisions overall multiplicities are not large enough to make fluctuations negligible Verify statistical model with exclusive channels BR’s! Verify statistical model with exclusive channels BR’s! (e.g.  pp annihilation at rest) ALL CONSERVATION LAWS MUST BE IMPLEMENTED (W. Blumel et al., Z. Phys. C63 (1994) 637) NEED MICROCANONICAL CALCULATIONS

Size (Mass, Volume) Microcanonical ensemble. All conservation laws including energy-momentum (angular momentum, parity), charges enforced. V > 20 fm 3, M > 10 GeV (F. Liu et al., Phys. Rev. C 68 (2003) ) F. B., L. Ferroni, talk in ISMD 2003) Canonical ensemble. Energy and momentum conserved on average, charges exactly. Temperature is introduced V > 100 fm 3, M > 50 GeV (A. Keranen, F.B., Phys. Rev. C 65 (2002) ) Grand-canonical ensemble. Also charges are conserved on average. Chemical potentials are introduced Difficulty of computing

Microcanonical ensemble calculation (angular momentum, parities, isospin neglected) (F.B., L. Ferroni, talk given in ISMD 2003 and “Statistical hadronization and hadronic microcanonical ensemble II”, in prep.) Main difficulty: size 271 light-flavoured species in the hadron-resonance gas give rise to a huge number of channels {N j }  Analytical integration impossible  Compute averages numerically via numerical integrations of  {Nj} of  {Nj} Monte-Carlo methods

 Importance sampling Monte-Carlo Random sampling of channels from a known and quick-to- sample distribution  as close as possible to the target distribution   Nj} Random sampling of channels from a known and quick-to- sample distribution  as close as possible to the target distribution   Nj} Calculate average as (for M samples):  Metropolis algorithm (suitable for event generation) Random walk in the channel space governed by gain-loss equations. At equilibrium, points of the walk are samples of the distribution (K. Werner, J. Aichelin, Phys. Rev. C 52 (1995) 1584) Random walk in the channel space governed by gain-loss equations. At equilibrium, points of the walk are samples of the distribution (K. Werner, J. Aichelin, Phys. Rev. C 52 (1995) 1584)

Speeding up the calculation to affordable computing times Use as  the grand-canonical correspondant of   Nj} Use as  the grand-canonical correspondant of   Nj} i.e. the multi-poissonian distribution This greatly enhances the performance of the This greatly enhances the performance of the computation in terms of efficiency in the importance sampling method and reducing the relaxation time in the Metropolis algorithm This method can be made even more effective for LARGE systems and opens the way to make fast event generators for the statistical model

Comparison between  C and C hadron multiplicities Q=0 cluster, M/V=0.4 GeV/fm 3 Mesons Baryons pp-like cluster, M/V=0.4 GeV/fm 3

Comparison between  C and C hadron multiplicity distributions Inequivalence between C and  C in the thermodynamic limit Q=0 cluster, M/V=0.4 GeV/fm 3 pp-like cluster, M/V=0.4 GeV/fm 3

Conclusions  A formulation of the statistical model suitable for small systems  More stringent tests: exclusive channels ? Involves full microcanonical calculations  Microcanonical ensemble calculations techniques. Fast and reliable.  Future: release of an event generator based on Metropolis algorithm