Transcript Progress in correlation femtoscopy

Femtoscopic Correlations and
Final State Resonance Formation
R. Lednický, JINR Dubna & IP ASCR Prague
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History
Assumptions
Technicalities
Narrow resonance FSI contributions to
π+-  K+K- CF’s
• Conclusions
8. 6. 2010
Nantes‘10
1
History
Correlation femtoscopy :
measurement of space-time characteristics R, c ~ fm
of particle production using particle correlations
Fermi’34: e± Nucleus Coulomb FSI in β-decay
modifies the relative momentum (k) distribution
→ Fermi (correlation) function F(k,Z,R) is
sensitive to Nucleus radius R if charge Z » 1
2
Fermi function(k,Z,R) in β-decay
= |-k(r)|2 ~ (kR)-(Z/137)
β-
2
Z=83 (Bi)
4 2 fm
R=8
β+
k MeV/c
3
Modern correlation femtoscopy
formulated by Kopylov & Podgoretsky
KP’71-75: settled basics of correlation femtoscopy
in > 20 papers
(for non-interacting identical particles)
• proposed CF= Ncorr /Nuncorr &
mixing techniques to construct Nuncorr
• showed that sufficiently smooth momentum spectrum
allows one to neglect space-time coherence at small q*
|∫d4x1d4x2p1p2(x1,x2)...|2 → ∫d4x1d4x2p1p2(x1,x2)|2...
• clarified role of space-time characteristics in various models
4
QS symmetrization of production amplitude
 momentum correlations of identical particles are
sensitive to space-time structure of the source
KP’71-75
total pair spin
exp(-ip1x1)
p1
CF=1+(-1)Scos qx
2
, nns , s
x1
x2
1/R0
p2
2R0
1
nnt , t
PRF
q = p1- p2 → {0,2k}
x = x1- x2 → {t,r}
0
CF →  |S-k(r)|2  =  | [ e-ikr +(-1)S eikr]/√2 |2 
|q|
5
Final State Interaction
Similar to Coulomb distortion of -decay Fermi’34: |-k(r)|2
nn
CF
pp
fcAc(G0+iF0)
}
Migdal, Watson, Sakharov, … Koonin, GKW, ...
s-wave
FSI
strong FSI
}
}
e-ikr  -k(r)  [ e-ikr +f(k)eikr/r ]
Coulomb
|1+f/r|2
kr+kr + …
_______
F=1+
ka
eicAc
Bohr radius
Point-like
k=|q|/2
Coulomb factor
 FSI is sensitive to source size r and scattering amplitude f
It complicates CF analysis but makes possible
 Femtoscopy with nonidentical particles K, p, .. &
Coulomb only
Coalescence deuterons, ..
 Study “exotic” scattering , K, KK, , p, , ..
 Study relative space-time asymmetries delays, flow
6
Assumptions to derive “Fermi” formula for CF
CF =  |-k*(r*)|2 
- two-particle approximation (small freezeout PS density f)
~ OK, <f>  1 ? low pt fig.
- smoothness approximation: p >> qcorrel  Remitter  Rsource
~ OK in HIC, Rsource2 >> 0.1 fm2  pt2-slope of direct particles
- equal time approximation in PRF
OK fig.
RL, Lyuboshitz’82  eq. time condition |t*|  r*2
- tFSI = d/dE >> tprod
tFSI (s-wave) = µf0/k*  k* = ½q*  hundreds MeV/c
 typical momentum transfer
in the production process
RL, Lyuboshitz ..’98
& account for coupled channels within the same isomultiplet only:
+ 00, -p  0n, K+K K0K0, ...
tFSI (resonance in any L-wave) = 2/    hundreds MeV/c
Phase space density from CFs and spectra
Bertsch’94
Lisa ..’05
<f> rises up to SPS
May be high phase space
density at low pt ?

? Pion condensate or laser
? Multiboson effects on CFs
spectra & multiplicities
8
BS-amplitude 
For free particles relate p to x through Fourier transform:
Then for interacting particles:
Product of plane waves -> BS-amplitude  :
Production probability W(p1,p2=|Τ(p1,p2;)|2
Smoothness approximation: rA « r0 (q « p)
W(p1,p2 = |∫d4x1d4x2 p1p2(x1,x2) Τ(x1,x2;)|2
= ∫d4x1d4x1’d4x2d4x2’
p1p2(x1,x2)p1p2*(x1’,x2’)
r0 - Source radius
rA - Emitter radius
x1
x2’
x2
2r0
x1’
p1
Τ(x1,x2 ;)Τ*(x1’,x2’ ;)
≈ ∫d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2
Source function G(x1,p1;x2,p2)
= ∫d4ε1d4ε2 exp(ip1ε1+ip2ε2)
p2
Τ(x1+½ε1,x2 +½ε2;)Τ*(x1-½ε1,x2-½ε2;)
For non-interacting identical spin-0 particles – exact result (p=½(p1+p2) ):
W(p1,p2 = ∫ d4x1d4x2 [G(x1,p1;x2,p2)+G(x1,p;x2,p) cos(qx)]
approx. result: ≈ ∫d4x1d4x2 G(x1,p1;x2,p2) [1+cos(qx)]
= ∫ d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2
Effect of nonequal times in pair cms
RL, Lyuboshitz SJNP 35 (82) 770; RL nucl-th/0501065
→
Applicability condition of equal-time approximation: |t*|  r*2
r0=2 fm 0=2 fm/c
r0=2 fm v=0.1

OK for heavy
particles
 OK within 5%
even for pions if
=0 ~r0 or lower
11
Technicalities – 1: neglecting complex intermediate channels
Technicalities – 2: spin & isospin equilibration
Technicalities – 3: equal-time approximation
Technicalities – 4: simple Gaussian emission functions
Technicalities – 5: treating the spin & angular dependence
Technicalities – 5: treating the spin & angular dependence
In the following we ssume
Since then L’=L, S’=S=j,=1/2 or 0, one can put m=j and
write (angular dependence enters only through the angle  between the vectors k and r):
Technicalities – 6: contribution of the outer region
Technicalities – 7: projecting pair spin & isospin
=π+=K+K-
Technicalities – 8: resonance dominance in the JT-wave
Technicalities – 9: contribution of the inner region
Technicalities – 10: volume integral
In the single flavor case
For s & p-waves it recovers the result of Wigner’55 & Luders’55
 correlations in Au+Au (STAR)
• Coulomb and strong FSI present
*1530, k*=146 MeV/c, =9.1 MeV
• No energy dependence seen
• Centrality dependence observed,
quite strong in the * region;
0-10% CF peak value CF-1  0.025
• Gaussian fit of 0-10% CF’s:
r0=6.7±1.0 fm, out = -5.6±1.0 fm
K+ K correlations in Pb+Pb (NA49)
• Coulomb and strong
FSI present
1020, k*=126 MeV/c,
=4.3 MeV
• Centrality dependence
observed, particularly
strong in the  region;
0-5% CF peak value
CF-1  0.10
•
3D-Gaussian fit of
0-5% CF’s: out-sidelong radii of 4-5 fm
Resonance FSI contributions to π+-  K+K- CF’s
• Complete and corresponding
inner and outer contributions
of p-wave resonance (*)
FSI to π+- CF for two cut
parameters 0.4 and 0.8 fm
and Gaussian radius of 7 fm
Rpeak(STAR)
 0.025
• The same for p-wave
resonance () FSI
contributions to K+K- CF for
Gaussian radius of 5 fm
Rpeak(NA49)
 0.10
Peak values of resonance FSI contributions
to π+-  K+K- CF’s vs cut parameter 
• Complete and corresponding
inner and outer contributions
of p-wave resonance (*)
FSI to peak value of π+CF for Gaussian radius of
7 fm
• The same for p-wave
resonance () FSI
contributions to K+K- CF for
Gaussian radius of 5 fm
Rpeak(STAR)  0.025
Rpeak(NA49)  0.10
Summary
• Assumptions behind femtoscopy theory in HIC seem OK,
including both short-range s-wave and narrow resonance
FSI (? up to a problem of angular dependence in the
resonance region)
• The effect of narrow resonance FSI scales as inverse
emission volume r0-3, compared to volume r0-1 or r0-2
scaling of the short-range s-wave FSI, thus being more
sensitive to the space-time extent of the source
• The NA49 and STAR correlation data from the most
central collisions seem to leave a little or no room for a
direct (thermal) production of narrow resonances
27
Angular dependence in the *-resonance region
(k*=140-160 MeV/c)
r* < 1 fm
0-10% 200 GeV Au+Au
FASTMC-code
r* < 0.5 fm
Angular dependence – example parametrization