bec from "inside" o.utyuzh the andrzej sołtan institute for nuclear studies (sins),...
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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
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