BEC from "inside"BEC from "inside"

O.UtyuzhO.Utyuzh

The Andrzej Sołtan Institute for Nuclear Studies (SINS), Warsaw, Poland

* In collaboration with G.Wilk and Z.Wlodarczyk

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 22

High-Energy collisions High-Energy collisions

0

0

K K

K K

0K

p

p

p

p x h

Quantum

statistics

p

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 33

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Quantum Correlations (QS)Quantum Correlations (QS)

1 212 1 2( ) ( )p pA x x 1 22 1( ) ( )p px x

x1

x2

p1

p2

12 1 2

2 1 222

( , )( )

( , ), ref

N p pC Q

N p pp p

2

2 1 2 2 2 11 1 2( , ) ( , ( ,) )Ω

ΩN p p x ρ x xA x dµ ò

BE enhancement

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 44

2 1 22 1 2

2 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p

CorrelationCorrelation functionfunction (1D) – (1D) – sourcesource sizesize

24

2 ( ) 1 ( ) iQxC Q d x x e 12

( )QR

x1

x2

p1

p2

R sourcesize

12

( )QR

R

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q1

R

R.Hunbury Brown and Twiss, Nature 178 (1956) 1046G.Goldhaber, S.Goldhaber, W.Lee and A.Pais, Phys.Rev 120 (1960) 300

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 55

Mod

el

Mod

el

Monte-Carlo event generators (MC)Monte-Carlo event generators (MC)

Assumption 1

Assumption 2

Assumption 4

Assumption 3 3

3

1

...

( )

...ch

dNN dy

P N

d Ndp

MCMC

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 66

Mod

el

Mod

el

Monte-Carlo event generators (MC)Monte-Carlo event generators (MC)

Assumption 1

Assumption 2

Assumption 4

Assumption 3

MCMC

Available Phase-Space

3

1

...

( )

...ch

dNN dy

P N

dNdp

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 77

Monte-Carlo event generators (MC)Monte-Carlo event generators (MC)

2 1 22

2 1 2

( , )( )

( , )

MC

ref

N p pC Q

N p p2 1 2

22 1 2

( , )( )

( , )

MC

ref

N p pC Q

N p p

π+

π-

0π

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

C2(Q

inv)

Qinv

[GeV]

MCDELPHI Data

change MC output to simulate proper behaviour

2( )C Q

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 88

Numerical modeling of BECNumerical modeling of BEC

* L.Lönblad, T.Sjöstrand, Eur.Phys.J. C2 (1998) 165

(a) Momenta shiftingMomenta shifting** 3 2

2 2( )

4

d p Q dQdN Q

E Q m

2

2 20

2

2 20

4

( )4

Q

Q Q

BE

q dq

q m

q dqf Q

q m

( ) 1BEf Q 0Q

π+

π-

0π

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 99

(a) Momenta shiftingMomenta shifting** 3 2

2 2( )

4

d p Q dQdN Q

E Q m

2

2 20

2

2 20

4

( )4

Q

Q Q

BE

q dq

q m

q dqf Q

q m

( ) 1BEf Q 0Q

π+

π-

0π

Numerical modeling of BECNumerical modeling of BEC

* L.Lönblad, T.Sjöstrand, Eur.Phys.J. C2 (1998) 165

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1010

Numerical modeling of BECNumerical modeling of BEC

* K.Fiałkowski,R.Wit,J.Wosiek, Phys.Rev. D57 (1998) 0940013

(b) weighting of eventsweighting of events**

i jW W

i j

{ , , , , }i jE E

in jn

{ , , , , }i jE E

i iWn j jW n

2 21

2

{ ( )} 1

eventi input

NQ R

eP i i

W e

events recounting

1j

i

W

W

for each EEii event one should take

EEjj events

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1111

( ) e!

N

BltzP NN

( ) (1 ) NBEP N

( )N i ii

x { , }

1( )

!N i jP i j i

xN

1

( ) e 1iE

kTin E

ssymmetrizationymmetrization**ssymmetrizationymmetrization**

non-identicalnon-identical VSVS identicalidentical BoltzmannBoltzmann VSVS Bose-EinsteinBose-Einstein

QuantumQuantum statisticsstatisticsQuantumQuantum statisticsstatistics

- K.Zalewski, Nucl. Phys. Proc. Suppl. 74 (1999) 65

- A. Giovannini and H.B.Nielsen, Proc. Of the IV Int. Symp. On Mult. Hadrodyn., Pavia 1973

- S.Pratt, in “Quark-Gluon Plasma”, ed.R.C.Hwa (World Scientific Oubl. Co, Singaoure, 1999), p.700

* - E.M.Purcell, Nature 174 (1956) 1449

GEOMETRICALGEOMETRICAL

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1212

0

0 eE

kT

0

0 eE

kT

cell formationuntil first

failure( ) (1 ) N

BEP N ( ) (1 ) NBEP N

2

2

( )

2( )cell

E

E E

cellg E e

2

2

( )

2( )cell

E

E E

cellg E e

smearing

particle energyin the cells

E-kTP(E) eµE

-kTP(E) eµ

MC particlesproduction example example

model (1D)model (1D)

phasephase spacespace (1D) (1D)

phasephase spacespace (1D) (1D)

phasephase spacespace (1D) (1D)

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1313

model (1D)model (1D)

phasephase spacespace (1D) (1D)

phasephase spacespace (1D) (1D)

phasephase spacespace (1D) (1D)

( ) (1 ) NBEP N

0

0 eE

kT

inputinputinputinputn

cell n1P (n)=

1+ n 1+ n

é ùê úê úê úë û

-10

1( )

-1E

T

P E

e

outputoutput-1

( ; )

1

N

chN k

N

N k kP N k

N N

k

+

æ ö÷ç ÷ç ÷ç ÷çæ ö+ è ø÷ç= ÷ç ÷ç ÷ç æ öè ø ÷ç ÷+ç ÷ç ÷çè ø

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1414

Clan modelClan model**

HadronicSource

Clan1

Clan2

Clan3Ind

epen

den

t p

rod

uct

ion

Ind

epen

den

t p

rod

uct

ion

( )NBP N

( )Possion ClanP N log ( )partP N

corr

elat

edco

rrel

ated

corr

elat

edco

rrel

ated

corr

elat

edco

rrel

ated

* L. Van Hove and A. Giovannini, XVII Int. Symp. On Mult. Dyn., ed. by M.Markitan (World Scientific, Singapore 1987), p. 561

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1515

Clan2

Clan modelClan model**

HadronicSource

Clan1

Clan3Ind

epen

den

t p

rod

uct

ion

Ind

epen

den

t p

rod

uct

ion

( )NBP N

( )Possion ClanP N log ( )partP N

corr

elat

edco

rrel

ated

corr

elat

edco

rrel

ated

corr

elat

edco

rrel

ated

0 5 10 15 20

100

101

102

103

104

P(N

cell)

Ncell

<Ncell

> = 6.28, N = 1.53

<Ncell

> = 6.30, N = 1.57

0 5 10 15 20 25 3010-1

100

101

102

103

104

105

106

<np> = 1.53,

n = 1.02

<np> = 1.57,

n = 1.07

P(n

p)

np

* L. Van Hove and A. Giovannini, XVII Int. Symp. On Mult. Dyn., ed. by M.Markitan (World Scientific, Singapore 1987), p. 561

0 5 10 15 20 25 30 35 40 45 50 551E-6

1E-5

1E-4

1E-3

0,01

0,1

DELPHI e+e-@91.2 GeV <nch

>=20.71, n=6.28

T=3.5 GeV, P0=0.7, =0.3*T; <n

ch>=20.87,

n=6.35

T=3.7 GeV, P0=0.7, =0.1*T; <n

ch>=20.76,

n=6.76

P(n

ch)

nch

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1616

Clan model …Clan model … (MD)(MD)

( )!

cellN

Poisson cellcell

P N eN

Pólya-Aeppli ( PA ) multiplicity distribution

( ) ( )!

partcellnN

Poisson cell Logarithm partcell part

bP N e P n

N n

Negative Binominal ( NB ) multiplicity distribution

Quantum statisticsQuantum statistics**

( ) partn

Geometric partP n b

* J.Finkelstein, Phys. Rev. D37 (1988) 2446 and Ding-wei Huang, Phys. Rev. D58 (1998) 017501

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1717

model (3D)model (3D)

p-Spacep-Space x-Spacex-Space

x·x·p-correlationsp-correlations

1+cos(δx δp)

symetrizationsymetrization

1D-model1D-model ( )

( , )

P N

P E P

3D-model3D-model

,

, ,P P

R RR

under conditionunder condition cos(δx δp)=2 Rand- 1

plane waves

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 1818

Preliminary resultsPreliminary results

ISMD'2005, Kromeriz O.Utyuzh/SINS 19

W T T P0 <nch> σn <npart> <ncell>

45.645.6 3.53.5 0.30.3 1.01.055

0.0.77

10.8310.83 4.294.2922

1.54/1.01.54/1.022

3.23/1.613.23/1.61

91.291.2 3.53.5 0.30.3 1.01.055

0.0.77

20.8820.88 6.376.3700

1.55/1.01.55/1.055

6.31/2.396.31/2.39

182.182.44

3.53.5 0.30.3 1.01.055

0.0.77

41.9741.97 8.988.9855

1.57/1.01.57/1.088

12.60/3.212.60/3.299

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

1.3

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

T=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...) W = 46.5 GeV W = 91.2 GeV W = 182.4 GeV

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

T=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...) W = 46.5 GeV W = 91.2 GeV W = 182.4 GeV

W - dependence

ISMD'2005, Kromeriz O.Utyuzh/SINS 20

T T P0 <nch> σn <npart> <ncell>

3.13.1 0.30.3 0.930.93 0.70.7 23.3423.34 6.6966.696 1.56/1.01.56/1.044

7.12/2.57.12/2.500

3.53.5 0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.93.9 0.30.3 1.171.17 0.70.7 18.8618.86 6.0756.075 1.57/1.01.57/1.077

5.72/2.45.72/2.422

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

T = 3.1 GeV | T = 3.5 GeV > = 0.3*T GeV, P=0.7*exp(...) T = 3.9 GeV |

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

T = 3.1 GeV | T = 3.5 GeV > = 0.3*T GeV, P=0.7*exp(...) T = 3.9 GeV |

T - dependence

ISMD'2005, Kromeriz O.Utyuzh/SINS 21

T T P0 <nch> σn <npart> <ncell>

3.3.55

0.30.3 1.051.05 0.60.6 19.9119.91 5.7145.714 1.41/0.71.41/0.700

6.74/2.46.74/2.466

3.3.55

0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.3.55

0.30.3 1.051.05 0.80.8 22.1522.15 7.3017.301 1.79/1.51.79/1.555

5.89/2.25.89/2.266

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

P0 = 0.6 |

P0 = 0.7 > T = 3.5 GeV, =0.3*T GeV

P0 = 0.8 |

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

P0 = 0.6 |

P0 = 0.7 > T = 3.5 GeV, =0.3*T GeV

P0 = 0.8 |

P0 - dependence

ISMD'2005, Kromeriz O.Utyuzh/SINS 22

- dependence

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

0 = 0.1 |

0 = 0.3 > T = 3.5 GeV, P=0.7*exp(...)

0 = 0.5 |

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

1.3

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

0 = 0.1 |

0 = 0.3 > T = 3.5 GeV, P=0.7*exp(...)

0 = 0.5 |

T T P0 <nch> σn <npart> <ncell>

3.3.55

0.10.1 0.350.35 0.70.7 21.8421.84 6.9276.927 1.57/1.01.57/1.077

6.62/2.46.62/2.444

3.3.55

0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.3.55

0.50.5 1.751.75 0.70.7 19.6519.65 5.8165.816 1.56/1.01.56/1.044

5.99/2.25.99/2.299

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2323

summarysummary

2 steps way to model Bose-Einstein correlations:

- create cells in phase space allocate particles to them (until first failure)

- correlate momenta and positions of particles in the cells according tofunction obtained from symmetrization procedure.

What one can get:

- geometrical distribution in the cells

Negative-Binomial likedistribution in the event

( ) (1 ) NBEP N

-1( ; )

1

N

chN k

N

N k kP N k

N N

k

+

æ ö÷ç ÷ç ÷ç ÷çæ ö+ è ø÷ç= ÷ç ÷ç ÷ç æ öè ø ÷ç ÷+ç ÷ç ÷çè ø

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2424

- energy distribution has a Bose-Einstein form

1

( ) e 1iE

kTin E

… to be continued …

Real MC implementation

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2525

Back-up SlidesBack-up Slides

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2626

- energy distribution has a Bose-Einstein form

- a simple way to include Final State Interaction (FSI) effects:

1

( ) e 1iE

kTin E

e eCiδC -iCψ ( )= A (η) F -iη,1; i( +kr)kr

k r kr

Coulomb Final State Interaction (FSI)

e i kr

Correlate x·p according to instead ofk r2Cψ ( ) 1 cos( )x p

… to be continued …

ISMD'2005, Kromeriz O.Utyuzh/SINS 27

W T T P0 <nch> <nch2>1/2 <npart> <ncell>

91.91.22

3.53.5 0.00.0 0.00.0 0.70.7 22.0322.03 7.0817.081 1.57/1.01.57/1.077

6.69/2.46.69/2.466

91.91.22

3.53.5 0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

, W=91.2 GeVT=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...)

(E-E0)

exp(-1/a (E-E0)2)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

, W=91.2 GeVT=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...)

(E-E0)

exp(-1/a (E-E0)2)

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2828

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 2929

2 1 22 1 2

2 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p

CorrelationCorrelation functionfunction (1D) - chaoticity (1D) - chaoticity

24

2 ( ) 1 ( ) iQxC Q d x x e 12

( )QRλ

x1

x2

p1

p2

12

( )QR

R

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q

2 (0) 2C ¹2 (0) 2C ¹

chaoticitchaoticityy

• resonancesresonances• finalfinal statestate interactionsinteractions• flowsflows• particlesparticles mismisinindificationdification• momentum resolutionmomentum resolution• ......

1 0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3030

High-Energy collisions …High-Energy collisions …

AA BB

1

2s

1

2s

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3131

2 1 22 1 2

2 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p

CorrelationCorrelation functionfunction (1D) (1D)

"+-", "mix", 1 1 1 2( ) ( )N p N p×

resonancesFSI+

e i- kr

1 2 1 2( , ) ( ) ( )ρ x x ρ x ρ x= ×

24

2( ) 1 ( )e iQxC Q d x x 12

( )QR

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3232

0,0 0,5 1,0 1,5 2,00,8

1,0

1,2

1,4

C2(Q

)

Q [GeV]

-- DELPHI e+e-@91.2 GeVP.Abreu et al., DELPHI Collab., Phys. Lett., B286 (1992) 201

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3333

0,00 0,05 0,10 0,15 0,200,0

0,2

0,4

0,6

0,8

1,0

1,2

C2(Q

)

Q [GeV]

p-p NA49 PbPb 158 A GeVH.Appelshauser et al., NA49 Collab., Phys. Lett., B467 (1999) 21

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3434

””... The correlation is determined by the size of the region from ... The correlation is determined by the size of the region from which pions are emitted with roughly the same momenta. which pions are emitted with roughly the same momenta. This has the consequence that for collectively streaming This has the consequence that for collectively streaming matter this region is smaller than the total source due to the matter this region is smaller than the total source due to the strong correlation between the momenta and the emission strong correlation between the momenta and the emission points of the particles.points of the particles.”- H.W.Barz, nucl-th/9808027.”- H.W.Barz, nucl-th/9808027.

””... The chaoticity of the source, i.e., the absence of initial ... The chaoticity of the source, i.e., the absence of initial correlations between the two emitted pions except those correlations between the two emitted pions except those correlations coming from the Bose-Einstein statistics”- correlations coming from the Bose-Einstein statistics”- H.W.Barz, nucl-th/9808027.H.W.Barz, nucl-th/9808027.

ISMD'2005, KromerizISMD'2005, Kromeriz O.Utyuzh/SINSO.Utyuzh/SINS 3535