STATISTICAL MODELSSTATISTICAL MODELSin high energy collisionsin high energy collisions

OUTLINEOUTLINE

o IntroductionIntroductiono Formulation for full microcanonical ensembleFormulation for full microcanonical ensembleo Discussion on “triviality”Discussion on “triviality”o Numerical methods and comparison micro-canNumerical methods and comparison micro-can

F. Becattini, BNL, May 10 2004

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

statistical lawstatistical law

The statistical lawThe statistical lawEvery multihadronic state within the cluster compatible with Every multihadronic state within the cluster compatible with

conservation laws is equally likelyconservation laws is equally likely

Statistical equilibriumStatistical equilibrium

Set of multi-hadronic states having the energy-momentum, angular Set of multi-hadronic states having the energy-momentum, angular momentum, parity and charges of the cluster =momentum, parity and charges of the cluster = the microcanonical the microcanonical ensembleensemble

Hadrons and resonances treated as free states (Hagedorn, on the Hadrons and resonances treated as free states (Hagedorn, on the basis of BDM paper)basis of BDM paper) – ideal hadron-resonance gas– ideal hadron-resonance gas

Cluster has a spacial extension (like in the MIT bag model and unlike in Cluster has a spacial extension (like in the MIT bag model and unlike in HERWIG MC)HERWIG MC)

Statistical model can be considered a model for the decays of MIT bagsStatistical model can be considered a model for the decays of MIT bags

What is the origin of equilibrium ?What is the origin of equilibrium ? Collisions among Collisions among formed hadronsformed hadrons in a slowly in a slowly

expanding fireballexpanding fireball ((scattscatt << <<expexp)) until until

decoupling like in the Hot Big Bang theorydecoupling like in the Hot Big Bang theory (thermalization) (thermalization)

Quantum evolution leads to a uniform superposition Quantum evolution leads to a uniform superposition over the multihadronic states within the cluster and, over the multihadronic states within the cluster and, as a consequence, equiprobability of observation as a consequence, equiprobability of observation when measurement is madewhen measurement is made

),(),()(1

lim0

pqdpqfdttfT

T

T

A temptative reformulationA temptative reformulation (F.B., L. Ferroni, “Statistical hadronization and hadronic microcanonical ensemble I” (F.B., L. Ferroni, “Statistical hadronization and hadronic microcanonical ensemble I”

hep-ph 0307061, to appear in Eur. Phys. J. C)hep-ph 0307061, to appear in Eur. Phys. J. C)

Vh

Vf f

i

fV

hhViVV

hhViVVV

hhWiWffWWf

fhhhhfhhhhΓ

V

VVVV

withP

PP

2

''''

''

Vh

ViV hhΩΓ P

| i >| i > cluster’s initial statecluster’s initial state

| h| hVV > > multihadronic state multihadronic state within within thethe

clustercluster

Starting from the end:Starting from the end:

2iWfΓ f

Full microcanonical ensembleFull microcanonical ensemble

QPPPPP3,,,, IIJPi

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

The projector can be decomposed as:The projector can be decomposed as:

IΠ,z

,,, )()()(dim2

1 P z

iizzJP gUgDgd

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

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

Basis for microcanonical calculationBasis for microcanonical calculation

IΠ,z

,,, )()()(dim2

1 P z

iizzJP gUgDgd

)ˆexp())(T(

2

)Π(I)R()R(R)12())(T(e

)(2

1 P *4

4,,,

xPixU

UUDdJxUxd JxiP

JP

The projectors on 4-momentum,The projectors on 4-momentum,angular momentum and parity angular momentum and parity factorize iffactorize if P=(M,P=(M,00))

])ˆ(exp[)2(

1P

CI2

1P

)()(1)(2 P *,

3

33

QQQ id

gUgDdgI

MM

II

III

Other projectors:Other projectors:

Integral projection technique Integral projection technique already used in the canonical already used in the canonical ensemble.ensemble.

A recursive method for the A recursive method for the canonical ensemble recently canonical ensemble recently used by S. Pratt et al.used by S. Pratt et al.

(Phys. Rev.C68 (2003) 024904 and ref. (Phys. Rev.C68 (2003) 024904 and ref. therein)therein)

““Restricted” microcanonical ensembleRestricted” microcanonical ensemble

Neglecting angular momentum, parity and isospin Neglecting angular momentum, parity and isospin conservation: summing over all conservation: summing over all J, J, , I, I, I, I33 with equal with equal

weightsweights

Usual definition of microcanonical ensembleUsual definition of microcanonical ensembleM. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445

fWWffWWfiWfΓ PIIJi

IIJf Q,

,,,,

2

,,,,

PPconst33

QQQ ,ˆ4

, )ˆ(P iP PP

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

N

iiN

j j

Nj

N

N

Nk

k pPpdpdN

JVΩΓΓ

j

j

jN

jNjN1

431

33 !

)12(

)2(

It can be shown that, It can be shown that, for non-identical particles:for non-identical particles:

Usually found in literatureUsually found in literature (e.g. K. Werner, J. Aichelin Phys. Rev. C 52 (1995) 1584)(e.g. K. Werner, J. Aichelin Phys. Rev. C 52 (1995) 1584)

In relativistic quantum field theory, confined states are NOTIn relativistic quantum field theory, confined states are NOTeigenstates of properly defined particle number operator.eigenstates of properly defined particle number operator.The above expression holds provided that The above expression holds provided that VV1/31/3 > > ComptCompt

For pions: For pions: ComptCompt = 1.4 fm = 1.4 fm

Generalized expression: phase space Generalized expression: phase space volume as avolume as a cluster expansioncluster expansion

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

Partitions

The leading term is theThe leading term is the {N{Njj}} for the classical Boltzmann statisticsfor the classical Boltzmann statistics

Subleading terms enhance phase space volume for identical bosonsSubleading terms enhance phase space volume for identical bosonsand suppress it for identical fermions. They disappear in the limit and suppress it for identical fermions. They disappear in the limit VV

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

Phase space and Fermi golden rulePhase space and Fermi golden rule

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(

Different measures (proper phase spaceDifferent measures (proper phase space dd33x dx d33pp vs invariant momentum vs invariant momentum SpaceSpace dd33p/2p/2 ) leading to different averages) leading to different averages

Statistical phase space modelStatistical phase space model predictspredicts definite ratios betweendefinite ratios between different channelsdifferent channels

Statistical phase space model has built-in quantum statistics effects Statistical phase space model has built-in quantum statistics effects (BEC) due to the finite volume(BEC) due to the finite volume

VS

They try to demonstrate that the same results of the They try to demonstrate that the same results of the statisticalstatistical

model can be obtained starting from different assumptionsmodel can be obtained starting from different assumptions

Is statistical population trivial?Is statistical population trivial?

,,,,, 213

212

22

21 mmmm

N

iiif

N

N

j j

Nj

NN pPMpdpd

N

JBR

j

j1

423

1

13

3 22!

)12(

)2(

1

,,,,,, 312121 IIII II

J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th 0001044J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th 0001044D. Rischke, Nucl. Phys. A698 (2001) 153, talk at QM2001D. Rischke, Nucl. Phys. A698 (2001) 153, talk at QM2001V. 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

Relativistic invariant: depends onRelativistic invariant: depends onas well as on as well as on

Correct, but not trivialCorrect, but not trivial

The peculiar prediction of the statistical modelThe peculiar prediction of the statistical model

which can be easily spoiled by most which can be easily spoiled by most |M|Mifif||22 if channel if channel constants depend on particle contentconstants depend on particle content

jM

jN

j

j

Ω

Ω

BR

BR

M

N

ExampleExample

Quite restrictive: only a single scale Quite restrictive: only a single scale and factorization and factorization

N

iii

Nif IhmfMMM

1

342)()()(

jM

jN

j

j

Ω

Ω

BR

BR

M

N

Average multiplicities for large M:Average multiplicities for large M:

where where is such that is such that

The actual production pattern The actual production pattern maymay be similar to the prediction be similar to the prediction of the statistical model, though this is not trivial of the statistical model, though this is not trivial

(e.g. if(e.g. if f f is a steep function of the mass) is a steep function of the mass)

can indeed fake a temperaturecan indeed fake a temperature

)exp()()()2(

))(12( 2233

3

jjjj

j mppdmfIhMJ

n

j

jjjj mppdmfIh

J0)exp()()(

)2(

)12(1 223

3

3

““Triviality” argument advocated in nucl-th 0210070Triviality” argument advocated in nucl-th 0210070

|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

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

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

ALL CONSERVATION LAWS MUST BEALL CONSERVATION LAWS MUST BEIMPLEMENTED IMPLEMENTED (W. Blumel et al., Z. Phys. C63 (1994) 637)(W. Blumel et al., Z. Phys. C63 (1994) 637)

NEED MICROCANONICAL CALCULATIONSNEED MICROCANONICAL CALCULATIONS

Size (Mass, Volume)

Microcanonical ensemble. All conservation laws includingMicrocanonical ensemble. All conservation laws includingenergy-momentum (angular momentum, parity), chargesenergy-momentum (angular momentum, parity), chargesenforcedenforced..

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 conservedCanonical ensemble. Energy and momentum conservedon average, charges exactly. Temperature is introducedon 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 conservedGrand-canonical ensemble. Also charges are conservedon average. Chemical potentials are introducedon average. Chemical potentials are introduced

Difficulty of computing

Microcanonical ensemble calculationMicrocanonical ensemble calculation(angular momentum, parities, isospin neglected)(angular momentum, parities, isospin neglected)

(F.B., L. Ferroni, talk given in ISMD 2003 and “Statistical hadronization and hadronic microcanonical ensemble II”, in (F.B., L. Ferroni, talk given in ISMD 2003 and “Statistical hadronization and hadronic microcanonical ensemble II”, in prep.)prep.)

jj jjj

jj jjjj

NNN

NNNN

Ω

ΩO

OqQ

qQ

,}{

,}{}{

Main difficulty: size Main difficulty: size 271 light-flavoured species in the hadron-resonance gas271 light-flavoured species in the hadron-resonance gasgive rise to a huge number of channelsgive rise to a huge number of channels {N {Njj}}

Analytical integration impossibleAnalytical integration impossible Compute averages numerically via numerical integrationsCompute averages numerically via numerical integrations ofof{Nj}{Nj}

Monte-Carlo methodsMonte-Carlo methods

Importance sampling Monte-CarloImportance sampling Monte-Carlo Random sampling of channels from a known and quick-to-Random sampling of channels from a known and quick-to-

sample distribution sample distribution as close as possible to the target as close as possible to the target distributiondistribution Nj}Nj}

Calculate average as (for Calculate average as (for MM samples): samples):

Metropolis algorithmMetropolis algorithm (suitable for event generation)(suitable for event generation)

Random walk in the channel space governed by gain-loss Random walk in the channel space governed by gain-loss equations. At equilibrium, points of the walk are samples of equations. At equilibrium, points of the walk are samples of

the distributionthe distribution (K. Werner, J. Aichelin, Phys. Rev. C 52 (1995) 1584)(K. Werner, J. Aichelin, Phys. Rev. C 52 (1995) 1584)

M

k

M

kk

kjN

kjN

kjN

kjN

Ω

ΩO

O

1

1

Speeding up the calculation to affordable Speeding up the calculation to affordable computing timescomputing times

Use as Use as the grand-canonical correspondant of the grand-canonical correspondant of Nj} Nj}

i.e. the multi-poissonian distributioni.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 computation in terms of efficiency in the importance

sampling method and reducing the relaxation time sampling method and reducing the relaxation time in the Metropolis algorithmin the Metropolis algorithm

This method can be made even more effective for LARGE This method can be made even more effective for LARGE systems and opens the way to make fast event generators systems and opens the way to make fast event generators

for the statistical modelfor the statistical model

averagescanonicalgrand with)exp(

! j

jj

j

N

zzN

zΠ

j

kjN

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

Comparison between Comparison between C and C hadron C and C hadron multiplicity distributionsmultiplicity distributions

Inequivalence between C and Inequivalence between C and C in the thermodynamic limitC in the thermodynamic limit

QQ=0=0 cluster, cluster, M/V=0.4 M/V=0.4 GeV/fmGeV/fm33 pp-likepp-like cluster, cluster, M/V=0.4 M/V=0.4 GeV/fmGeV/fm33

ConclusionsConclusions

A formulation of the statistical model A formulation of the statistical model suitable for small systemssuitable for small systems More stringent tests: exclusive channels ? More stringent tests: exclusive channels ?

Involves full microcanonical calculationsInvolves full microcanonical calculations Microcanonical ensemble calculations Microcanonical ensemble calculations

techniques. Fast and reliable. techniques. Fast and reliable. Future: release of an event generator based Future: release of an event generator based

on Metropolis algorithmon Metropolis algorithm