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Open questions related to Bose-Einstein
correlations in e+e_ hadrons
G. AlexanderTel-Aviv University
OUTLINE
ISMD 2003
1. Introduction
3. Fermi-Dirac correlations
2. Emitter size vs. ECM
4. Emitter size vs. mass
5. 2-D analysis and TM
6. Bose condensates and BEC?
7. Generalized BEC
8. Summary and conclusions
Bose-Einstein Correlation (BEC) 1-Dimension analysis
1
2121
2
2211
2122 )()(
)(
dp
d
dp
d
dpdp
d
pp
ppC
22221
22 4)( BBB mMppQ
Correlation function:.
GGLP variable
2222
2 2 2( ) 1 QrC Q e
Hadron emitter radius in e+e_ vs. ECM
• HBT effect in the 50’s measured stellar objects dimensions.
In heavy ions, an early compilation of r vs. the projectile A
can be described by
What about the e+e_hadrons BEC dimension vs. ECM ?
AA Collisions
(projectile) 1/3A
A-A Collisions
fmAr 3/12.1[Chacon, P.R. C43 (1991) 2670]
r versus No. of jets and multiplicity
e+e_ Z0 hadrons
2 20.974 0.340 10 ( ); 0.735 0.306 10G ch G chR n fm n
3-jets
2-jets
An approach to the r dependence on ECM
via factorial cumulant moments and hadron sources
22 1122
KeC Qr 1.
2 .Cumulant dependence on number of sources
Dealt with by several authors, among them:P. Lipa & B. Bushbeck, P.L. B223 (1989) 465B. Bushbeck, H.C. Eggers & P. Lipa, P.L. B481 (2000) 187G. Alexander & E. Sarkisyan, P.L. B487 (2000) 215
Dilution factor, Sources and Cumulants
1 Sqq
Sq KDK
11 / qSq SK
Assumption: Pion-pair correlation exist only if both of the same source
Dilution factor Dq
For identical sources
Emitter size vs. hadron sources
22
1)(2SrQ
SS eQC
0
2 2
1[1)(]
S
SS
rdQQC
2 2 2 211
2 2 2 2 2 1( ) 1 1 1 1SQ r Q rS SSC Q e K D K D e
Consider 2-pion BEC from S sources
S SS 1 1
2 1 1
λ λ1r = r ⇒ same sources=S r
D λ λ
•Note: in AA collisions, S vs. A is hard to estimate !
eff
SA
SAD
andfmrArIf
3/112
113/1
2.1
2.1
r(e+e_ h) dependence on ECM
Neglect for simplicity four and more jets so that:
e+e_ q + q + gluon hadrons
)(ln)ln()( 2210 jetjetjetq EaEaaEn
)()( 10 jetqjetg EnRREn
The measured average multiplicity ]Boutemeur, Fortschr. Phys. 50 (01)[
q1 CM q2= CM gluonE =E /2; E E /2- E
gluonE is determined from the total averaged charge multiplicity
The gluon energy
Take:
-
Estimate Dilution Factor
e+e_ hadron Emitter Radius
Source dilution approach taking λS= λ1 normalized at 40 GeV
Words of warning
The experiments use different methods for:
* Selection of data and cuts
* Choice of reference sample
* Fitting procedure
The extension to Fermi-Dirac correlation
1. Spin-Spin Correlation Functions for e.g.
22
10
1
1
22
10
0
0
13/)()(
3/)(2)(
13/)()(
)(2)(
Qr
Qr
eQFQF
QFQC
eQFQF
QFQC
- -ΛΛ⇒ pπpπ
),cos( *2
*1
* ppy *2* )1(1/ ydydN S
Two Methods
; S=S1+S2
The extension to Fermi-Dirac Correlation (cont’d)
2. The phase space density approach
Like in the BEC analysis one considers the density of identical baryon pairs as Q 0
2222
1||1|| 22,1
22,1
QraQrs eande 222222
5.01)[1(31]4
1|| 2
2,1QrQrQr eee
Aleph
pp
Three referencesamples
r(m) from BEC and FDC analyses
Uncertainty relations
mtcr / sec10 24twith
QCD potential
r
crV S
3
4
fmGeV /7.0
)/87.12ln(9
2
rS
Z0hadrons
r(m) derived from the Heisenberg uncertainty relations[G.Alexander, I.Cohen E.Levin, Phys. Lett. B452 (99) 159]
*The two bosons are at threshold in their CMS, i.e. non-relativistic
p
crmvrvrcrp
2
tmptmptm
ptE //2
2
hc/ h/Δt c hΔtr(m)= =
m m *Here we assume that:
1) Δt
2) ΔE
3) rΔr
essentially independent of the mass and is ~10-24 sec
depends on the kinetic energy i.e. potential energy small
A challenge to the Lund string model
bAehhqqM 221 |)......(|
A leading model for multi-hadron production
Expects in its rudimental form ∂ r/ ∂ m>0
1-Dimension string “Toy” model
_
s
⇒
⇒
_ _
uu qq
q s q
Energy density of the hadron emitter
3.exp
2
4
3
h
h
r
m
2/33
2/5
mode )(2
3
tc
mhl
Z0 hadrons
[Dashed lines for sec[10)3.032.1( 24t
2-Dimensional BEC analysis
[1)]1( )(2
2222TTzz QrQr
Tz eQQC 2-dimension Correlation Function:
Transverse mass:
2,2
22,1
2
2
1TTT pmpmm
Longitudinal Center of Mass System
Uncertainty relations applied to )( Tz mr
crprvrp zzzzzz 2z
z p
cr
2
1
2
1
22,
2.
2,
2,
2
i i zTiziyixi pmpppmE
22
22
1 ,
2
,
2
1 2,
2
, )(2
21
TT
zTT
i Ti
zTii
Ti
zTi mm
pmm
m
pm
m
pm
T
zTz m
pmEQAs
2
2:0 tm
ptE
T
z2
z T
T
c hΔtr (m )≈
m
1(
2(
3(
G.A., P.L. B506 (2001) 45
r(mT) in heavy ion collisions
fmrand
fmArFromBEC
Z
A
58.0
2.1
0
3/1
]U. Heinz, Ann.Rev.Nucl.Part.Sci. 49(99)529[
0: : 0.352 /z TFrom BEC in e e Z h r m 0
z zr (S+Pb)/r (Z )=2/0.352=5.7
1.
2. 6.12x0.85
207321.2
2r
rr 1/31/3
Z
PbS
0
Bose Condensates – Brief reminder [A. Einstein (Sitzber. Kgl. Preuss. Akad. Wiss. 1924/5)]
* In a Condensate: All atoms are in the same zero energy state
* E.A.Cornell, W.Ketterle, C.E.Wieman (Nobel 2001) discovered in 1995 rubidium (Rb), sodium (Na), lithium (Li) condensates
* How ? By cooling down below a TB (500nK – 2000nK) dilute bosonic atoms
Any relation between
Condensates and Boson produced in HE reactions?
Try: Inter-Atomic Separation and the dimension extracted from BEC
Inter-Atomic separation in Bose Condensates
* In Bose condensates, when T/TB<< 1, the atomic density is:
[G.A. Phys. Lett. B506(01)45]
2/3
23 2
612.2
h
mkT
V
NdBE
* The de Broglie wave length is:
378.1/2
3/1
2/12
dBBEdB dmkT
h
* Consider two condensates with masses m1 and m2 in the same temperature T0 (<< TB1, TB2) :
2/1
0
2
378.1
2)(
kTm
hmd
iiBE
BE 1 2
BE 2 1
d (m ) m=
d (m ) m
rBEC(m) formula from Bose condensates
* Inter-atomic separation:
2/1
0
2
2378.1
1
mkT
hdBE
* Replace: tEkT /0 to get
BE BEC
c hΔt hΔtd = ≈ c =r (m)
1.378 m m
* However there are obvious differences between condensates and hadrons produced in HE reactions, e.g .
1 (Condensates in thermal equilibrium, hadrons in HE reactions? 2 (Condensates in coherent state, hadrons only partly
Note: dBE = inter-atomic separation NOT the condensate dimension!
Isospin invariance and generalized BEC (GBEC)
* In analogue to the Generalized Pauli exclusion principle one may consider a Generalized BEC where I-spin is included
demanding an over-all symmetric state .
* This possibility was considered by several authors among them Bowler (87), Suzuki (87) and Weiner (2000). Specific GBEC
relations were worked out by Alexander & Lipkin (99) in the case that the multi-hadron final states emerge from an I=0 state .
* The cases where hadrons emerge from an I=0 state is quit frequent. For example, in hadronic decays of
Multi-gluon decays of and into odd numbers of pions/J .,,0 bbccssZ
Relations between the 2-pion systems in the GBEC
Xeven XPXP [)(])3/1([)(])3/2( 0
000
X
even XPXP [)(])3/1([)(])3/2( 000
0
Conclusions (if GBEC is valid)
BEC effect in the inassametheissystem0
0Avoid the use of the system as a reference sample
00 0] ( ) [ ] ( ) [even
X X
P X P X
Summary and Conclusions
*In spite of the fact that BEC is studied over 40 years, absent are systematic studies covering different reactions over a wide energy range
*In addition, a standardization of the analysis methods and reference samples would allow more meaningful interpretation of the
experimental results ____________________
*r(Ecm) is rather well described by a simple approach to hadron-jet sources
*This approach however seems not to be sufficient to account for the dr/dnch seen in the Zohadrons
Summary and Conclusions (cont’d)* r(m), as determined from BEC and FDC analyses on the Zo , follows roughly the expectation derived from the Heisenberg relations as well as that extracted from a QCD potential.
Needs to be measured also in other reactions !
* The dependence dr/dm < 0 poses a challenge to hadron production models including the Lund one
*Above all, the energy density of about 100 GeV/fm3 affixed to the baryon emitter, awakes doubt on the r interpretation
as an emitter radius ____________________
* Generalized BEC has not so far been experimentally verified. If confirmed then it has a considerable effect on the analyses of resonances and on the choice of reference samples
____________________
Summary and Conclusions (cont’d)
* The r(mT) extracted from the 2-D BEC analysis behaves similarly to the r(m) derived from the 1-D analyses and both can be described in terms of the Heisenberg uncertainty relations
* To note is that r(mT) is proportional to (mT)-1/2 in A-A reactions
as is also the case in e+e- collisions even though the latter one is free of nuclear effects!!
* As for Bose condensates , atomBE md /1 is the inter-atomic and NOT the Bose condensate dimension !
Are the behavior of dr/dm, the energy density and the meaning of dBE telling us that we should re-examine what does r measure??
* Final question: