F. Becattini, Kielce workshop, October 15 2004

What is the meaning of the statistical model ?

F.B. hep-ph 0410403

OUTLINE

o o o o Introduction Discussion: phase space dominance, triviality and Lagrange multipliers Basic microcanonical formulation Future tests

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...

   

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

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

Phase space dominance

Ω

 

j



V

( 2 

N

) 3

N



j

( 2

J j

 1 )

N j N j

!



d

3

p

1 

d

3

p N

 4  

P



i N

  1

p i

 

Γ f

 1 ( 2  ) 3

N



j

( 2

J j

 1 )

N j N j

!



d

3 2 

p

1 1 

d

3 2 

p N N M if

2  4  

P



i N

  1

p i

 

VS

Discussed in detail in nucl-th 0001044 J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, If

|M if | 2

has a very weak dependence on kinematical independent variables, e.g.

p i · p j

,we could somehow recover a pseudo thermal shape of the multiplicity and

p T

spectrum function

IF

M if

2  

N

for large mults

n j

 ( 2

J j

( 2   1 )  ) 3 

d

3

p

2  exp(  

p

2 

m

2

j

) Where  is such that:

M

    

j

( 2

J j

( 2   1 ) ) 3  

d

3 2 

p

exp(  

p

2 

m

2

j

)  0 Conclusion:  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!

n j

 ( 2

J j

( 2   1 )

V

) 3 

d

3

p

exp(  

p

2 

m

2

j

)

Phase space dominance is not trivial

In principle, as well as on

|M if | 2 I

1 , may depend on

I

2 ,  ,

I

1 

I

2 ,

I

1 

m

1 2

I

3 , ,

m

2 2  , ,  , 2

m

12 , 2

m

13 ,  Example:

M if

2  (  3

M

)

N i N

  1

f

( 

m i

)

g

(

I i

) quite restrictive: again, only one scale  and factorization

n j

 ( 2

J j

 1 )(  3

M

) ( 2  ) 3

g

(

I j

)

f

( 

m j

) 

d

3 2 

p

exp(  

p

2 

m

2

j

) The thermal-like behaviour can be easily distorted at ANY scale of Multiplicity (just take

g(I)=AI 2 +C

or

f(



m)=(



m) 5

)

Disagreement with “Triviality” arguments e.g.

V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002

|M if | 2 |M if | 2

depends on

N

;

N

is large; small fluctuations of is unessential at high

N

model results are trivially recovered

N

 and therefore the statistical

1.

2.

|M if | etc.) 2

may not depend just on

N

,

also on specific particle content in the channel (through mass, isospin

In analyses of e.g. pp collisions overall multiplicities are not large enough to make fluctuations negligible

“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!

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 p T spectrum

Occam razor argument

From: Delphi collaboration, CERN-PPE 96-120

How to probe a genuine statistical model ?

Need to test exclusive channel rates

BR

 

j BR

 

j Ω



Ω

Much more sensitive to the integration measure ( V d 3 p d 3 p/2  ) because information is not integrated away vs 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

Basic scheme

Clusters:

extended massive objects with internal charges

Every multihadronic state within the cluster compatible with conservation laws is equally likely

The microcanonical ensemble

and its partition function A usual definition reads Can be generalized as canonical: 

h V h V Ω

 

states

 4 (

P



P state

)

Ω

 

h V h V

P

i h V P i | h V

exp(  H /

T

)

h V >

projector on the cluster’s initial state multihadronic state

within

the cluster What is the probability of an asymptotic free state

| f >

?

Define

p f



f

P

i

P

V

P

i f

with P

V

 

h V h V h V



f p f

 tr ( P

i

P

V

P

i

)  tr ( P

i

P

V

)  

h V h V

P

i h V



Ω

All

p f

are positive definite as:

f

P

i

P

V

P

i f

 

h V f

P

i h V

2 The cluster is described by the mixture

W

 

h V

P

i h V h V

P

i

 P

i

P

V

P

i

Note: P i P V P i  P V P i P V Used in Eur.Phys. J. C 35 (2004) 243 In principle, projection

P V

states: should be made on localized field P

V

1 

particle

 

V D

   with   

x

 (

x

) In all studies, relativistic quantum field effects are neglected: good approximation for

V 1/3 >

l

C

(at most 1.4 fm)

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 ensemble

Decompose the projector: P

i

 P

P

, Projection onto an irreducible state

P

4-momentum

J

, l ,  P  P

I

,

I

3 P

Q

J

l  

Q

spin helicity parity C-parity abelian charges

I, I 3

isospin The projector

P P,J,

l, can be written (formally) as an integral over the extended Poincare’ group IO(1,3) ↑ P

P

,

J

, l ,   1 2 z   Π, I dim  

d

 (

g z

)

D

  (

g z

)

i i U

(

g z

)

The projectors on 4-momentum, spin-helicity and parity factorize if

P=(M,0)

P

P

,

J

, l ,   1 (2  ) 4 

d

4

x

e

iP



x

exp( 

i



x

) ( 2

J

 1 ) 

d

R

D J

( R ) l l *

U

( R ) I  

U

( Π ) 2

Other projectors: P

I

,

I

3  (2

I

 1) 

dg D I

(

g

)

I

3 *

U I

3 (

g

) P  P

Q

  1 2  I   C  ( 2  1 )

M

    

d M

 exp[

i

(

Q

 )   ] Integral projection technique already used in the canonical ensemble (Cerulus,Turko, Redlich,Cleymans, et al.) Restricted microcanonical ensemble: only four-momentum and abelian charges P

P

,

Q

  4 ( 

P

)  ,

Q

Rate of a multi-hadronic channel

{N

j

}=(N

1

,...,N

K

)

For

non-identical particles: Γ



Ω

 

j



N V

( 2  ) 3

N i N

  1 ( 2

J i

 1 ) 

d

3

p

1 

d

3

p N

 4   

P



i N

  1

p i

   For identical particles: cluster decomposition

Ω

 

j

 

d

3

p

1 

d

3

p N

 4 

P



P f

 

j

{  }

h n j

 

N j



H n N



j

 1

j n j j

( 2

J h n j j h n j

 1 )

H

!

j j l H



j

 1

F n l j H j



j n N



j

 1

h n j N j



j n N



j

 1

n j h n j F n l



i l n l

  1 1 ( 2  ) 3 

V d

3

x

exp 

i

x

 

p

i l



p

c l

(

i l

)   Generalization of the expression in M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445 which holds only for large

V

partitions

Comparison between



C and C hadron multiplicities

Q

=0

M/V=0.4

GeV/fm 3 3 Mesons Baryons

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.