What is the meaning of the What is the meaning of the statistical model ?statistical model ?

F.B. hep-ph 0410403F.B. hep-ph 0410403

OUTLINEOUTLINE

o IntroductionIntroductiono Discussion: phase space dominance, triviality Discussion: phase space dominance, triviality

and Lagrange multipliersand Lagrange multiplierso Basic microcanonical formulationBasic microcanonical formulationo Future testsFuture tests

F. Becattini, Kielce workshop, October 15 2004

The statistical model is successful in describing The statistical model is successful in describing soft observables in Heavy Ion Collisionssoft observables in Heavy Ion Collisions

M. Gazdzicki, M. GorensteinM. Gazdzicki, M. Gorenstein F. B., A. Keranen, J. ManninenF. B., A. Keranen, J. Manninen J. Cleymans, H. SatzJ. Cleymans, H. Satz P. Braun-Munzinger, J. Stachel, D. MagestroP. Braun-Munzinger, J. Stachel, D. Magestro W. Broniowski, W. FlorkowskiW. Broniowski, W. Florkowski J. Letessier, J. RafelskiJ. Letessier, J. Rafelski K. Redlich, A. TounsiK. Redlich, A. Tounsi A. Panagiotou, C. KtoridesA. Panagiotou, C. Ktorides Nu Xu, M . KanetaNu Xu, M . Kaneta

And many more...And many more...

The statistical model is even more The statistical model is even more successful in describing relevant soft successful in describing relevant soft observables in elementary collisionsobservables in elementary collisions

F.B. Z. Phys. C 69 (1996) 485F.B. Z. Phys. C 69 (1996) 485 F. B., Proc. XXXIII Eln. Workshop Erice, hep-ph 9701275F. B., Proc. XXXIII Eln. Workshop Erice, hep-ph 9701275 F. B., U. Heinz, Z. Phys. C 76 (1997) 269.F. B., U. Heinz, Z. Phys. C 76 (1997) 269. F. B., G. Passaleva, Eur. Phys. J. C 23 (2001) 551F. B., G. Passaleva, Eur. Phys. J. C 23 (2001) 551

F.B., Nucl. Phys. A 702 (2002) 336F.B., Nucl. Phys. A 702 (2002) 336 F.B., G. Passaleva, Eur. Phys. J. C 23 (2002) 551F.B., G. Passaleva, Eur. Phys. J. C 23 (2002) 551

Warning! Warning! Strangeness phase space is undersaturatedStrangeness phase space is undersaturated

Why?Why?

From:From: L. Mc Lerran, Lectures “The QGP and the CGC”, hep-ph 0311028 L. Mc Lerran, Lectures “The QGP and the CGC”, hep-ph 0311028

Three kind of answers (criticisms)Three kind of answers (criticisms)

The thermodynamical-like behaviour is only mimicked byThe thermodynamical-like behaviour is only mimicked bythe data. It should be rather called the data. It should be rather called “phase space dominance”“phase space dominance” The Statistical Model results are somehow trivial due to The Statistical Model results are somehow trivial due to large involved multiplicitieslarge involved multiplicities The Statistical Model results can be obtained as a byThe Statistical Model results can be obtained as a by-product of other models, at least in elementary collisions-product of other models, at least in elementary collisions

The temperature is not a real temperatureThe temperature is not a real temperature

Phase space dominancePhase space dominance

N

iiif

N

N

j j

Nj

Nf

N

iiN

j j

Nj

N

N

N

pPMpdpd

N

JΓ

pPpdpdN

JVΩ

j

j

j

1

423

1

13

3

1

431

33

22!

)12(

)2(

1

!

)12(

)2(

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

VS

If If |M|Mifif||22 has a very weak dependence on kinematical independenthas a very weak dependence on kinematical independentvariables, e.g.variables, e.g. ppii·· ppjj ,we could somehow recover,we could somehow recover a pseudo-a pseudo-thermal shape of the multiplicity and thermal shape of the multiplicity and ppTT spectrum function spectrum function

NifM

2)exp(

2)2(

)12( 223

3 jj

j mppdJ

n

IFIF

for large multsfor large mults

j

jj mp

pdJM 0)exp(

2)2(

)12( 223

3

)exp()2(

)12( 2233 j

jj mppd

VJn

Where Where is such that:is such that:

Conclusion:Conclusion: is not a temperature and inclusive particle is not a temperature and inclusive particlemultiplicities are not sensitive enough to the different integrationmultiplicities are not sensitive enough to the different integrationmeasure to distinguish between a genuine thermal behaviourmeasure to distinguish between a genuine thermal behaviourand this pseudo-thermal function (phase space dominance)and this pseudo-thermal function (phase space dominance)

However, there is a quantitative difference!However, there is a quantitative difference!

Example:Example:

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

Phase space dominance is not trivialPhase space dominance is not trivial

N

iii

Nif IgmfMM

1

32)()()(

,,,,, 213

212

22

21 mmmm

,,,,,, 312121 IIII IIIn principle,In principle, |M|Mifif||22 may depend onmay depend onas well as onas well as on

)exp(2

)()()2(

))(12( 223

3

3

jjjj

j mppd

mfIgMJ

n

The thermal-like behaviour can be easily distorted at ANY scale ofThe thermal-like behaviour can be easily distorted at ANY scale ofMultiplicity (just take Multiplicity (just take g(I)=AIg(I)=AI22+C+C or or f(f(m)=(m)=(m)m)55))

Disagreement with “Triviality” argumentsDisagreement with “Triviality” arguments

e.g.e.g.V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002

1.1. |M|Mifif||22 may not depend just on may not depend just on NN, , also on specific also on specific particle content in the channel (through mass, isospin particle content in the channel (through mass, isospin etc.)etc.)

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

|M|Mifif||22 depends ondepends on NN; ; NN is large; small fluctuations ofis large; small fluctuations of NN |M|Mifif||22 is unessential at high is unessential at high NN and therefore the statistical and therefore the statistical model results are trivially recoveredmodel results are trivially recovered

““Lagrange multiplier” or Lagrange multiplier” or what is a temperature?what is a temperature?

See e.g. See e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002;V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002; U. Heinz, hep-ph 0407360 “Concepts in heavy ion physics”U. Heinz, hep-ph 0407360 “Concepts in heavy ion physics”

Seem to advocate the idea that the temperature determined with hadron Seem to advocate the idea that the temperature determined with hadron abundances is not a abundances is not a “real temperature”“real temperature”, rather a , rather a “Lagrange multiplier “Lagrange multiplier constraining maximization of entropy”constraining maximization of entropy”

This is just a possible definition of temperatureThis is just a possible definition of temperature There might be different definitions in There might be different definitions in smallsmall systems (e.g. systems (e.g. 1/T=1/T=S/S/EE, ,

saddle point for microcanonical partition function, etc.) but ALL OF saddle point for microcanonical partition function, etc.) but ALL OF THEM converge to the same quantity in the thermodynamic limitTHEM converge to the same quantity in the thermodynamic limit

A quantitative difference is needed: if you have volume, energy and A quantitative difference is needed: if you have volume, energy and statistical equilibrium, temperature is a temperature regardless of how statistical equilibrium, temperature is a temperature regardless of how the system got there!the system got there!

Derive the statistical features within Derive the statistical features within other modelsother models

A. Bialas, Phys. Lett. B 466 (1999) 301A. Bialas, Phys. Lett. B 466 (1999) 301 W. Florkowski, Acta Phys. Pol. B 35 (2004) 799W. Florkowski, Acta Phys. Pol. B 35 (2004) 799

Fluctuation of the string tension may lead to an Fluctuation of the string tension may lead to an exponential shape, e.g. of the pexponential shape, e.g. of the pTT spectrum spectrum

Occam razor argumentOccam razor argument

From: Delphi collaboration, CERN-PPE 96-120From: Delphi collaboration, CERN-PPE 96-120

Need to test exclusive channel ratesNeed to test exclusive channel rates

Much more sensitive to the integration measure (Much more sensitive to the integration measure (V dV d33pp vs vs dd33p/2p/2) because information is not integrated away) because information is not integrated away

jM

jN

j

j

Ω

Ω

BR

BR

M

N

How to probe a genuine statistical How to probe a genuine statistical model ?model ?

Data available at low energy ( Data available at low energy ( s < 10 GeV)s < 10 GeV)Need Need fullfull microcanonical calculations microcanonical calculationssee e.g. W. Blumel, P. Koch, U. Heinz, Z. Phys. C63 (1994) 637see e.g. W. Blumel, P. Koch, U. Heinz, Z. Phys. C63 (1994) 637

Basic schemeBasic scheme

Clusters: Clusters: extended massive extended massive objects with internal chargesobjects with internal charges

Every multihadronic state within the cluster Every multihadronic state within the cluster compatiblecompatible

with conservation laws is equally likelywith conservation laws is equally likely

The microcanonical ensembleThe microcanonical ensemble and its partition functionand its partition function

states

statePPΩ )(4

ΩhhpVh

ViVViiVif

f P)PP(tr)PPP(tr

Vh

VVViVif hhffp Pwith PPP

PPii projector on the projector on the

cluster’s initial statecluster’s initial state

| h| hVV > > multihadronic state multihadronic state

within within the clusterthe cluster

A usual definition readsA usual definition reads

Can be generalized asCan be generalized as Vih

V hhΩV

P

What is the probability of an asymptotic free stateWhat is the probability of an asymptotic free state | f >| f > ??

DefineDefine

canonical:canonical: Vh

V hThV

)/Hexp(

iViih

VVi

V

hhW PPPPP

Vh

ViiVi hfff2

PPPP

The cluster is described by the mixtureThe cluster is described by the mixture

All All ppff are positive definite as: are positive definite as:

In principle, projection In principle, projection PPVV should be made on localized field should be made on localized field states:states:

)(withP 1 xx V

particleV D

In all studies, relativistic quantum field effects are neglected: In all studies, relativistic quantum field effects are neglected: good approximation forgood approximation for VV1/31/3 > > C C (at most 1.4 fm)(at most 1.4 fm)

Note:Note:PPiiPPVVPPii PPVVPPiiPPVVUsed in Eur.Phys. J. C 35 (2004) 243Used in Eur.Phys. J. C 35 (2004) 243

Full microcanonical ensembleFull microcanonical ensembleProjection onto an irreducible stateProjection onto an irreducible state

QPPPPP3,,,, IIJPi

PP 4-momentum4-momentumJ J spinspin helicityhelicity parityparity C-parityC-parityQQ abelian chargesabelian chargesI, II, I33 isospin isospin

Decompose the projector:Decompose the projector:

IΠ,z

,,, )()()(dim2

1 P z

iizzJP gUgDgd

The projectorThe projector PPP,J,P,J, can be written (formally) as an integral over the can be written (formally) as an integral over the

extended Poincare’ groupextended Poincare’ group IO(1,3)IO(1,3)↑↑

2

)Π(I)R()R(R)12()ˆexp(e

)(2

1 P *4

4,,,

UUDdJxPixd JxiP

JP

The projectors on 4-momentum,The projectors on 4-momentum,spin-helicity and parity spin-helicity and parity factorize iffactorize if P=(M,0)P=(M,0)

F.B., L. Ferroni, Eur. Phys. J. C 35 (2004) 243F.B., L. Ferroni, Eur. Phys. J. C 35 (2004) 243F.B., L. Ferroni, hep-ph 0407117, Eur. Phys. J. C in printF.B., L. Ferroni, hep-ph 0407117, Eur. Phys. J. C in print

])ˆ(exp[)2(

1P

CI2

1P

)()(1)(2 P *,

3

33

QQQ id

gUgDdgI

MM

II

III

Other projectors:Other projectors:

Integral Integral projection projection technique technique already used already used in the in the canonical canonical ensembleensemble

(Cerulus,Turko, (Cerulus,Turko, Redlich,Cleymans, Redlich,Cleymans, et al.)et al.)

Restricted microcanonical ensemble: Restricted microcanonical ensemble: only four-momentum and abelian chargesonly four-momentum and abelian charges

QQQ ,ˆ4

, )ˆ(P PPP

Rate of a multi-hadronic channel Rate of a multi-hadronic channel {N{Njj}=(N}=(N11,...,N,...,NKK))

N

iiN

N

iiN

N

N pPpdpdJV

ΩΓjjN

1

431

3

13

)12()2(

ForFor non-identical particles: non-identical particles:

V

ici

n

in

N

nnjj

N

nnj

j

H

ln

hN

nn

hj

Hj

HN

fNN

lll

l

l

l

j

j

j

j

j

j

j

j

jl

jn

j

j

j

jn

jjj

j

ixdFhnNhH

F

hn

JPPpdpdΩ

)(3

13

11

1}{

1

431

3

exp)2(

1

!

)12(1

ppx

For identical particles: cluster decompositionFor identical particles: cluster decomposition

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

Comparison between Comparison between C and C hadron C and C hadron multiplicitiesmultiplicities

QQ=0=0 cluster,cluster, M/V=0.4 M/V=0.4 GeV/fmGeV/fm33

MesonsMesons BaryonsBaryonspp-likepp-like cluster,cluster, M/V=0.4 M/V=0.4 GeV/fmGeV/fm33

Summary and ConclusionsSummary and Conclusions Discussion on the statistical modelDiscussion on the statistical model Temperature, phase space dominance: only Temperature, phase space dominance: only

quantitative differences are differences.quantitative differences are differences. More quantitative tests of the picture, e.g. on More quantitative tests of the picture, e.g. on

exclusive channels (at low energy) require full exclusive channels (at low energy) require full microcanonical calculations and Monte Carlo microcanonical calculations and Monte Carlo implementations (matching with parton shower)implementations (matching with parton shower)

Microcanonical ensemble sampling algorithm for Microcanonical ensemble sampling algorithm for hadron system accomplished. Ongoing work to hadron system accomplished. Ongoing work to include ang. Mom., isospin etc.include ang. Mom., isospin etc.