Open questions related to Bose-Einstein correlations in e + e _ hadrons G. Alexander Tel-Aviv University OUTLINE ISMD 2003 1. Introduction 3. Fermi-Dirac.

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

Baryon production in the Lund Model

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

rz dependence on mT in ee Zohadrons

Z0 hadrons (DELPHI preliminary)

z TT

hΔtr (m )≈c

m

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: