8. 6. 2010nantes‘101 femtoscopic correlations and final state resonance formation r. lednický,...
TRANSCRIPT
8. 6. 2010 Nantes‘10 1
Femtoscopic Correlations and Final State Resonance Formation
R. Lednický, JINR Dubna & IP ASCR Prague
• History
• Assumptions
• Technicalities
• Narrow resonance FSI contributions to π+- K+K- CF’s
• Conclusions
2
History
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
measurement of space-time characteristics R, c ~ fm
Correlation femtoscopy :
of particle production using particle correlations
4
Modern correlation femtoscopy formulated by Kopylov & Podgoretsky
KP’71-75: settled basics of correlation femtoscopyin > 20 papers
• proposed CF= Ncorr /Nuncorr &
• showed that sufficiently smooth momentum spectrum allows one to neglect space-time coherence at small q*
(for non-interacting identical particles)
mixing techniques to construct Nuncorr
• clarified role of space-time characteristics in various models
|∫d4x1d4x2p1p2(x1,x2)...|2 → ∫d4x1d4x2p1p2(x1,x2)|2...
5
QS symmetrization of production amplitude momentum correlations of identical particles are
sensitive to space-time structure of the source
CF=1+(-1)Scos qx p1
p2
x1
x2
q = p1- p2 → {0,2k} x = x1- x2 → {t,r}
nnt , t
, nns , s
2
1
0 |q|
1/R0
total pair spin
2R0
KP’71-75
exp(-ip1x1)
CF → |S-k(r)|2 = | [ e-ikr +(-1)S eikr]/√2 |2
PRF
6
Final State InteractionSimilar to Coulomb distortion of -decay Fermi’34:
e-ikr -k(r) [ e-ikr +f(k)eikr/r ]
eicAc
F=1+ _______ + …kr+krka
Coulomb
s-wavestrong FSIFSI
fcAc(G0+iF0)
}
}
Bohr radius}
Point-likeCoulomb factor k=|q|/2
CF nnpp
Coulomb only
|1+f/r|2
FSI is sensitive to source size r and scattering amplitude fIt complicates CF analysis but makes possible
Femtoscopy with nonidentical particles K, p, .. &
Study relative space-time asymmetries delays, flow
Study “exotic” scattering , K, KK, , p, , ..Coalescence deuterons, ..
|-k(r)|2Migdal, Watson, Sakharov, … Koonin, GKW, ...
Assumptions to derive “Fermi” formula for CF
CF = |-k*(r*)|2
- tFSI ddE tprod
- equal time approximation in PRF
typical momentum transfer
RL, Lyuboshitz’82 eq. time condition |t*| r*2 OK fig.
RL, Lyuboshitz ..’98
+ 00, -p 0n, K+K K0K0, ...& account for coupled channels within the same isomultiplet only:
- two-particle approximation (small freezeout PS density f)~ OK, <f> 1 ? low pt fig.
- smoothness approximation: p qcorrel Remitter Rsource
~ OK in HIC, Rsource2 0.1 fm2 pt
2-slope of direct particles
tFSI (s-wave) = µf0/k* k* = ½q*
hundreds MeV/c
tFSI (resonance in any L-wave) = 2/ hundreds MeV/c
in the production process
8
Phase space density from CFs and spectra
Bertsch’94
May be high phase space density at low pt ?
? Pion condensate or laser
? Multiboson effects on CFsspectra & multiplicities
<f> rises up to SPSLisa ..’05
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)
p1
p2
x1
x2
2r0
W(p1,p2|∫d4x1d4x2 p1p2(x1,x2) Τ(x1,x2;)|2
x1’x2’
≈ ∫d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2
r0 - Source radius
rA - Emitter radiusp1p2(x1,x2)p1p2*(x1’,x2’)
Τ(x1,x2 ;)Τ*(x1’,x2’ ;)
G(x1,p1;x2,p2)
= ∫d4ε1d4ε2 exp(ip1ε1+ip2ε2)
Τ(x1+½ε1,x2 +½ε2;)Τ*(x1-½ε1,x2-½ε2;)
Source function
= ∫d4x1d4x1’d4x2d4x2’
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(qx)]
approx. result: ≈ ∫d4x1d4x2 G(x1,p1;x2,p2) [1+cos(qx)]
= ∫ d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2
11
Effect of nonequal times in pair cmsRL, 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 if0 ~r0 or lower
→
Technicalities – 5: treating the spin & angular dependence
In the following we ssume
write (angular dependence enters only through the angle between the vectors k and r):
Since then L’=L, S’=S=j,=1/2 or 0, one can put m=j and
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
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-side-long 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
• The same for p-wave resonance () FSI contributions to K+K- CF for Gaussian radius of 5 fm
Rpeak(NA49) 0.10
Rpeak(STAR) 0.025
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
27
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
Angular dependence in the *-resonance region (k*=140-160 MeV/c)
r* < 1 fm
r* < 0.5 fm
0-10% 200 GeV Au+AuFASTMC-code