# Presentation: 875376

## STATISTICAL MODELSin high energy collisions

- OUTLINE
- Introduction
- Formulation for full microcanonical ensemble
- Discussion on “triviality”
- Numerical methods and comparison micro-can

*F. Becattini, BNL, May 10 2004*

**Notes:**
In this talk I will focus on some general aspects of the statistical hadronization model. 
I will try to give a short overview of the model, then to discuss how to make microcanonical
Calculations which are needed to test this model againts more refined observables through
Event generators. I will also address the issue of the triviality of results and I will finally
Describe some numerical results that we have recently obtained in the micro-can comparison.

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

**Notes:**
This is a picture put forward by Fermi in the 50’s ....and developed by Hagedorn in the 60’s. Today we imagine that  an arbitrary number of clusters is formed, possibly as a result of a parton cascade or any other dynamical underlying process.

## 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*

**Notes:**
This law can be summarized in two words: statistical equilibrium. So, making calculation in this approach
Amounts to deal with the set of...
Key features of the model is....

## 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 0307061, to appear in Eur. Phys. J. C)

- A temptative reformulation (F.B., L. Ferroni, “Statistical hadronization and hadronic microcanonical ensemble I” hep-ph 0307061, to appear in Eur. Phys. J. C)

*| i >*       cluster’s initial state

*| h**V** >*     multihadronic state *within *the
- cluster

- Starting from the end:

**Notes:**
The latter concept can be formulated more precisely by assuming a special form of an effective
 transition operator. The aim is keeping the volume in the business.

## Full microcanonical ensemble

*P* 	4-momentum
*J *	spin
** 	helicity
** 	parity
*** *	C-parity
**Q**	abelian 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)↑

- 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.
-      A recursive method for the canonical ensemble recently used by S. Pratt et al.
-      (Phys. Rev.C68 (2003) 024904 and ref. therein)

**Notes:**
Ricorda di dire che neglecting ang mom and parity amounts to retain the first
Factor only and this gives rise to a delta

## “Restricted” microcanonical ensemble

- “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{Nj}=(N1,...,NK)

- 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 V1/3 > Compt
- For pions: Compt = 1.4 fm

## Generalized expression: phase space volume as a cluster expansion

- 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 *

**Notes:**
This is an effect which is owing to finite volume. It is most clearly seen in the
Large V limit where all subleading terms go to zero.

## 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**

**Notes:**
There are strinking similarities between the N-particle phase space and Fermi golden
Rule and in fact there have been some people who have argued that observed statistical features
Can be interpreted in terms of constancy of matrix element. However, it must be pointed out that...
More importantly,....

## Is statistical population trivial?

- model can be obtained starting from different assumptions

- J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th 0001044
- D. Rischke, Nucl. Phys. A698 (2001) 153, talk at QM2001
- V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002

- Relativistic invariant: depends on
- as well as on

*Correct, but not trivial*

**Notes:**
There are strinking similarities between the N-particle phase space and Fermi golden
Rule and in fact there have been some people who have argued that observed statistical features
Can be interpreted in terms of constancy of matrix element. However, it must be pointed out that...
More importantly,....

## The peculiar prediction of the statistical model

which can be easily spoiled by most |Mif|2 if channel constants depend on particle content

- 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

- 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 0210070

|Mif|2 depends on N; N is large; small fluctuations of N  
       |Mif|2 is unessential at high N and therefore the statistical model results are trivially recovered

*|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* may 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

-   Verify statistical model with exclusive channels BR’s!
- (e.g. pp annihilation at rest)

- ALL CONSERVATION LAWS MUST BEIMPLEMENTED (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 fm3, M > 10 GeV  (F. Liu et al., Phys. Rev. C 68 (2003) 024905)
- F. B., L. Ferroni, talk in ISMD 2003)

- Canonical ensemble. Energy and momentum conserved
- on average, charges exactly. Temperature is introduced

- V > 100 fm3, M > 50 GeV  (A. Keranen, F.B., Phys. Rev. C 65 (2002) 044901)

- 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.)

- 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}*

- 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}
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) 

- 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}*
- 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)

## Speeding up the calculation to affordable 
computing times

           Use as   the grand-canonical correspondant of Nj}  
i.e. the multi-poissonian distribution

    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

- computing times
-            Use as **  the grand-canonical correspondant of *****Nj}  *
- i.e. the multi-poissonian distribution
-     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/fm3

- Mesons

- Baryons

**pp-like** cluster, *M/V=0.4 *GeV/fm3

## 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/fm3

**pp-like** cluster, *M/V=0.4 *GeV/fm3

## 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