On the Δt time scalein Bose-Einstein and Fermi-Dirac correlations
Gideon Alexander and Erez Reinherz-Aronis
Tel-Aviv University
OUTLINE
1. A brief introduction
3. A closer look at Δt
2. R1D from Z0 decays
4. Δt as the particle emission time
5. It’s consequence to heavy ions
arXiv: 0910.0138 [hep-ph]
6. Comments and remarks
WPCF 2009
G. Alexander, E. Reinherz-Aronis
Bose-Einstein Correlation (BEC)-Reminder
1-Dimension analysisof two identical bosons
2 1, 22 1 2
1 1 2 2
( )( , )
( ) ( )
p pC p p
p p
22221
22 4)( BBB mMppQ
Correlation function:
GGLP variable:
2 21 2
22 2( ) 1 DR QC Q e 1
1nnR nR
G. Alexander, E. Reinherz-Aronis
Taken from WA98 collaboration (2007)arXiv 0709.2477
π± π± BEC from heavy nuclei – dependence on A
Note: only few BEC of K-pairs
R vs. the target A1/3
1 / 31DR ( m ) ( 0.91 0.02 ) A fm
G. Alexander, E. Reinherz-Aronis
The extension to Fermi-Dirac Correlation for di-fermions
2. The phase space density approach (Pauli Exclusion principle)
Like in the BEC analysis one considers the density of the identical baryon pairs as Q 0
2 2 2 2s 2 R Q a 2 R Q
1,2 1,2| | 1 e and | | 1 e 2 22
1,2R Q| | 1 0.5e
ΛΛ FDC Three referencesamples
Aleph
P.L. B475 (00) 395
No need forCoulomb correction!
1 .ΛΛ Spin-Spin correlation (Alexander & Lipkin P.L. B352 (1995) 162)
Does not need a reference sample nor Coulomb correction
G. Alexander, E. Reinherz-Aronis
R1D (m) from BEC and FDC analysesof the Z0 hadronic decays at LEP
G. Alexander, E. Reinherz-Aronis
R1D (m) from BEC and FDC analysesof the Z0 hadronic decays at LEP
G. Alexander, E. Reinherz-Aronis
R(m) derived from the Heisenberg uncertainty relations[G.Alexander, I.Cohen, E.Levin, Phys. Lett. B452 (99) 159, G.Alexander, Phys.Lett. B506 (2001) 45]
The two bosons are near threshold in their CMS, i.e. non-relativistic
2c
p R c vR mvR Rp
2
2 / /p
E t t p m t p m tm
1D
hc/ /Δt c ΔtR (m)= =
m m
It was further assumed that:
1) Δt
2) ΔE
3) RΔR
Represents the interaction strength i.e.~10-24 sec for S.I.
Depends on the kinetic energy i.e. potential energy is small
Note: For m≠0
G. Alexander, E. Reinherz-Aronis
R1D (m) from BEC and FDC analysesof the Z0 hadronic decays at LEP
Uncertainty relations
1 /DR c t m
For S.I. QCD potential
r
crV S
3
4
fmGeV /7.0
)/87.12ln(9
2
rS
24(1 0.5) 10 st -D = ± ´
G. Alexander, E. Reinherz-Aronis
Application of the Bjorken-Gottfried relation to R(mT)
There exists a linear relation qµ=λxµ between the 4-momentum and the time-space which implies λ=mT/τ
where τ=(t2-z2)0.5 is the longitudinal proper time
mT [GeV]
Bialas et al. P.R. D62 (00) 114007 ;
Bialas et al. Acta Phys. Polon. B32 (01) 2901
2 20T T
Transverse mass
m m p
0: 0T TJust note m m when P
G. Alexander, E. Reinherz-Aronis
Question: Is this m dependence of R unique to the Z decays ?WA98 Collaboration (2007); arXiv:0709.2477
Central Pb +Pb collisions at 158 GeV/A
Δt=(1.28±0.04)x10-22 sec
1/ 2
1
(2.75 0.04)( )
( )D
Const fmGeVR m
m m GeV
1st time BEC of di-deuteron
G. Alexander, E. Reinherz-Aronis
Δt=10 -24 sec
Δt=10 -19 sec
Δt=10 -12 sec
π K p,Λ
Q: Is Δt a measure of the interaction strengthof the two identical outgoing particles?
R1D as a function of m and Δt
Unfortunately so far no systematic
BEC or FDC of WI particles
have been measured !
Aggarwal et al. (WA98) PRL 93 (04) 022301
γγ BEC in Pb + Pb at 158 GeV/A
π
100<KT <200 MeV
200<KT <300 MeV
1
1
5.4 0.8 0.9
5.8 0.8 1.2
ID
IID
R fm
R fm
Aggarwal et al. (WA98) arXiv:0709.2477
R ) long
long T
T
c Δt(m ≈
m
[G. Alexander, P.L. B506 (2001) 45]
Δtlong=(1.61±0.05)x10-22 sec
G. Alexander, E. Reinherz-Aronis
Thus we assign Δt as a measure of the particle emission time
1D
c tR m
m
0 25 24: 10 10 sec ZFor Z decay to
Z0 Hadronic decay
G. Alexander, E. Reinherz-Aronis
As a next steplet us relate the R dependence on A
13
1DR m aA
a=0.91 fm
with
1D
c tR m
m
G. Alexander, E. Reinherz-Aronis
22
32
m at A
c
To get the particle emission time which depends on the
surface area of the nucleus
Question: Does it have an energy dependence?
NA49 Collaboration, P.R. C77 (2008) 064908 found that
3
20, 30, 40, 80, 158 /
( )D
In Pb Pb collisions at and GeV A
the R m values change very little if at all
G. Alexander, E. Reinherz-Aronis
Particle emission time Δt vs the Atomic number A
Δt(A=1)=Δt(proton)=2.4x10-24 sec for a=1 fm[Data from: WA80, WA98, STAR Collaborations and from Chacon et. al,. (1991)]
22
32
m at A
c
a=1.2 fm
a=0.8 fm
a=1 fm
From measured Rcalculate
2
2
( )m R mt
c
and insert
G. Alexander, E. Reinherz-Aronis
Comments and remarks
2) One should have more BEC and FDC analyzes like that of WA98
3) In heavy ion collisions Heisenberg derived formulae are applied
1) Δt is attributed to the particles’ emission time
4) A simple merge of the 1D Heisenberg derived formula with R vs A
22/
23 23 2 2; 1 , 10 10 secfor A and
m at A
ca fm t is to
5) Consistent with the data
7) An extension of this work from 1D to 3D should be explored
6) The Δt dependence on A2/3 e.g. Fireball shell model ?
G. Alexander, E. Reinherz-Aronis
Experimental opportunity may be offered by the SLHC pp Z0Z0 + X data
Comments and remarks (con’t)
8) FDC of directly produced e±e±/µ±µ± are unlikely to be measured
9) An attractive possibility may be the BEC study of Z0Z0 pairs
a) It is a BEC of weak interacting particles (what is its Δt value?)
b) R determination of a very high mass boson
c) No Coulomb correction
Because :
7) BEC and/or FDC for WI particles are of interest to measure
G. Alexander, E. Reinherz-Aronis
Pythia MC study of ppZ0Z0 at 14 TeV
Monte Carlopp->ZZ->4ℓ (ℓ= e/μ)Pythia 6.403, CTEQ 6L1
Generated events (study sample)~69000Equivalent to 1200 fb-1
Expected generated events (100 1/fb) 5600
Major selection cuts:pT > 20 GeVAt least 1 μ
2 lepton pairs
Invariant mass cut (for both)74 GeV <Mz<110 GeV
Pythia includes the Z’s width
Detection efficiency after detector simulation
~25%
Fraction of the ZZ->4ℓ (ℓ=e or μ) decay ~0.36%Fraction of Z->Jets Z->2ℓ (ℓ= e or μ) decay ~9.32%
G. Alexander, E. Reinherz-Aronis
Pythia MC study of ppZ0Z0 at 14 TeV
One parameter fit: R1D=0.019 ± 0.006 fm24
110 sec ( ) 0.025D ZFor t one expects R m fm
G. Alexander, E. Reinherz-Aronis
Last remark
MANY THANKS FOR YOUR ATTENTION
0 /e e Z qq hadrons
G. Alexander, E. Reinherz-Aronis
Backup slides
G. Alexander, E. Reinherz-Aronis
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
[A simple minded approach , G. Alexander. RPP 66 (03) 481]
G. Alexander, E. Reinherz-Aronis
Baryon production in the Lund Model
See e.g., Delgado, Gustafson, Lönnblad, (LUND group)
Eur. Phys. J.C. 52 (2007) 113