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

3

Fermi function(k,Z,R) in β-decay

= |-k(r)|2 ~ (kR)-(Z/137)2

Z=83 (Bi)β-

β+

R=84 2 fm

k MeV/c

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

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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 – 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

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 – 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

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

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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

Angular dependence – example parametrization