femtoscopic correlations and final state resonance formation r lednick jinr dubna ip ascr prague n.
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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. History. Correlation femtoscopy :.

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femtoscopic correlations and final state resonance formation r lednick jinr dubna ip ascr prague
WPCF Kiev‘10 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

Correlation femtoscopy :

measurement of space-time characteristics R, c ~ fm

Fermi’34:e± NucleusCoulomb FSI in β-decay modifies the relative momentum (k) distribution → Fermi (correlation) function F(k,Z,R) is sensitive to Nucleusradius R if charge Z » 1

of particle production using particle correlations

fermi function f k z r in decay
Fermi function F(k,Z,R) in β-decay

F = |-k(r)|2~ (kR)-(Z/137)2

Z=83 (Bi)‏


2 fm




k MeV/c

modern correlation femtoscopy formulated by kopylov podgoretsky
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 sufficientlysmooth 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

assumptions to derive fermi formula for cf
Assumptions to derive “Fermi” formula for CF

CF =  |-k*(r*)|2

- two-particle approximation (small freezeout PS density f)‏

~ OK, <f>  1 ? lowpt

- smoothness approximation: pqcorrel Remitter Rsource

~ OK in HIC, Rsource20.1 fm2

 pt2-slope of direct particles

usually OK

- equal time approximation in PRF

RL, Lyuboshitz’82  eq. time condition |t*| r*2

to several %

- tFSI = dd/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

caution smoothness approximation is justified for small k 1 r 0
Caution: Smoothness approximation is justified for small k<<1/r0

∫d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2 ∫d3r WP(r,k) |-k(r)|2

∫d3r {WP(r,k) + WP(r,½(k-kn)) 2Re[exp(ikr)-k(r)]

+WP(r,-kn) |-k(r)|2 }

where-k(r) = exp(-ikr)+-k(r) andn = r/r

Thesmoothness approximation

WP(r,½(k-kn))  WP(r,-kn)  WP(r,k)

is valid if one can neglect the k-dependence of

WP(r,k), e.g. for k << 1/r0


Technicalities – 2: treating the spin & angular dependence

Assuming that one of the two particles is spinless and the other one is

either spin-1/2 or spin-1/2, we have:

Angular dependence enters only through the angle  between the vectors k and r in the argument of the Wigner (rotation) d-functions.


Technicalities – 3: contribution of the outer region

For r > d = the range of the strong interaction potential,

the radial functions:


Technicalities – 8: simple Gaussian emission functions

In the following we use the simple one-parameter Gaussian

emission function:

WP(r,k) = (8π3/2r03)-1 exp(-r2/4r02)

and account for the k-dependence in the form

( = angle between r and k)

WP(r,k) = (8π3/2r03)-1 exp(-b2r02k2) exp(-r2/4r02 + bkrcos)

Note the additional suppression of WP(0,k) if out  0:

WP(0,k) ~exp[-(out/2r0)2] (~30% suppression for π+-)


References related to resonance formation in final state:

R. Lednicky, V.L. Lyuboshitz, SJNP 35 (1982) 770

R. Lednicky, V.L. Lyuboshitz, V.V. Lyuboshitz, Phys.At.Nucl. 61 (1998) 2050

S. Pratt, S. Petriconi, PRC 68 (2003) 054901

S. Petriconi, PhD Thesis, MSU ,2003

S. Bekele, R. Lednicky, Braz.J.Phys. 37 (2007) 994

B. Kerbikov, R. Lednicky, L.V. Malinina, P. Chaloupka, M. Sumbera, arXiv:0907.061v2 B. Kerbikov, L.V. Malinina, PRC 81 (2010) 034901

R. Lednicky, P. Chaloupka, M. Sumbera, in preperation

p x correlations in au au star
p-X correlations in Au+Au (STAR)

P. Chaloupka, JPG 32 (2006) S537; M. Sumbera, Braz.J.Phys. 37(2007)925

  • Coulomb and strong FSI presentX*1530, k*=146 MeV/c, =9.1 MeV
  • No energy dependence seen
  • Centrality dependence observed, quite strong in the X* region; 0-10% CF peak value CF-1  0.025
  • Gaussian fit of 0-10% CF’s at k* < 0.1 GeV/c: r0=4.8±0.7 fm, out = -5.6±1.0 fm

r0 =[½(rπ2+r2)]½ of ~5 fm is in agreement with the dominant rπ of 7 fm

k k correlations in pb pb na49
K+ K-correlations in Pb+Pb (NA49)

PLB 557 (2003) 157

  • 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 k k cf s
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 5 fm  * FSI contribution strongly overestimated
  • The same for p-wave resonance () FSI contributions to K+K- CF   FSI contribution well described but no room for direct production !


 0.025

----------- -----


 0.10

----------- -----

peak values of resonance fsi contributions to k k cf s vs cut parameter
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 5 fm
  • The same for p-wave resonance () FSI contributions to K+K- CF

 for  < r0, CF is practically -independent

 at ~1 fm, CFCF(inner)


 0.025

----- ---------------


 0.10


angular dependence in the resonance region k 140 160 mev c
Angular dependence in the *-resonance region (k*=140-160 MeV/c)

0-10% 200 GeV Au+Au


from L. Malinina

r* < 1 fm

r* < 0.5 fm

 = angle between r and k

WP(r,k) ~ exp[-r2/4r02 + b krcos]

b  0.5

angular dependence example parametrization
Angular dependence – example parametrization

WP(r,k) ~ exp[-r2/4r02 + bkrcos];  = angle between r and k


k=146 MeV/c, r0=5 fm

Suppressing WP(0,k)

by a factor of


Smoothness assumption:

WP(r,½(k-kn))  WP(r,-kn)  WP(r,k)



--------------  0.025

b=0.3 0.4

Similar suppression

of resonance

contribution to CF



Accounting for angular dependence:FASTMC-code (implying smoothness assumption)  close to the STAR peak but too low at small k* (L.M.) OK at small k* but much higher peak (P. Ch.)

b < 0.4

Indicating that the

parametrization of

the angular dependence

~ exp(b krcos), b0.5

is oversimplified


--------------  0.025

  • Assumptions behind femtoscopy theory in HIC seem OK, including both short-range s-wave and narrow resonance FSI up to a problem of the angular dependence in the resonance region.
  • The effect of narrow resonance FSI scales as inverse emission volume r0-3, compared to 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 higher sensitivity may be however disfavoured by the theoretical uncertainty in the case of a strong angular asymmetry.
  • The NA49 (? STAR) correlation data from the most central collisions seem to leave a little or no room for a direct (thermal) production of near threshold narrow resonances.

QS symmetrization of production amplitudemomentum correlations of identical particles are sensitive to space-time structure of the source


total pair spin

CF=1+(-1)Scos qx













q =p1- p2 → {0,2k}

x = x1- x2 → {t,r}



CF →|S-k(r)|2  =| [ e-ikr +(-1)S eikr]/√2 |2 

f inal s tate i nteraction
Final State Interaction


Similar to Coulomb distortion of -decay Fermi’34:

Migdal, Watson, Sakharov, … Koonin, GKW, ...



strong FSI




e-ikr -k(r)  [ e-ikr +f(k)eikr/r ]






F=1+ _______ + …





Bohr radius

Coulomb only


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, .. &

Coalescence deuterons, ..

Study “exotic” scattering,K, KK,, p,, ..

Study relative space-time asymmetriesdelays, flow

bs amplitude
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 r a r 0 q p
Smoothness approximation: rA« r0 (q « p)

W(p1,p2|∫d4x1d4x2p1p2(x1,x2) Τ(x1,x2;)|2


r0 - Source radius

rA - Emitter radius


Τ(x1,x2 ;)Τ*(x1’,x2’ ;)‏



≈ ∫d4x1d4x2G(x1,p1;x2,p2) |p1p2(x1,x2)|2



Source function


= ∫d4ε1d4ε2 exp(ip1ε1+ip2ε2)‏

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
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


 OK within 5%

even for pions if

0 ~r0 or lower


Technicalities – 7: volume integral I(,k)

In the single flavor case & no Coulomb FSI

For s & p-waves it recovers the results of Wigner’55 & Luders’55