29. 2. 2012 RBRC Hyperon Workshop, BNL2012 1

Femtoscopic Correlations and Final State Interactions

R. Lednický @ JINR Dubna & IP ASCR Prague

• History

• Assumptions

• Correlation study of strong interaction

• 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 F(k,Z,R) in β-decay

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

Z=83 (Bi)β-

β+

R=84 2 fm

k MeV/c

Modern correlation femtoscopy formulated by Kopylov & Podgoretsky

KP’71-75: settled basics of correlation femtoscopyin > 20 papers (for non-interacting identical particles)

• proposed CF= Ncorr /Nuncorr &

• argued that sufficiently smooth momentum spectrum allows one to neglect space-time coherence at small q*

smoothness approximation:

mixing techniques to construct Nuncorr

• clarified role of space-time production characteristics: shape & time source picture from various q-projections

|∫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

usually OK

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

- 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

to several %

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. p1≈ p2 : ≈ ∫d4x1d4x2 G(x1,p1;x2,p2) [1+cos(qx)]

= ∫ d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2

10

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

→

Equal time & smoothness approx. “Fermi” formula

∫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) and n = r/r

Smoothness approx. 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

CF(p1,p2) ∫d3r WP(r,k) |-k(r)|2

Caution: Smoothness approximation is justified for small k << 1/r0

It should be generalized in the region k > ~100 MeV/c

Resonance contribution vs r-k correlation parameter b

Rpeak(STAR) ----------- 0.025

Rpeak(NA49) ---------- 0.10 0.14

Smoothness assumption:WP(r,½(k-kn)) WP(r,-kn) WP(r,k) Exact

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

CF suppressed by a factor WP(0,k) ~ exp[-b2r02k2]

To leave a room for a direct production b > 0.3 (0.15) is required for π+- (K+K-) system

*(k=146 MeV/c), r0=5 fm (k=126 MeV/c), r0=5 fm-----------

4-6.02.2006 R. Lednický dwstp'06 13

Examples of present data: NA49 & STAR

3-dim fit: CF=1+exp(-Rx2qx

2 –Ry2qy

2 -Rz

2qz2

-2Rxz2qx qz)

z x y

Correlation strength or chaoticity

NA49

Interferometry or correlation (Gaussian) radii

KK STAR Coulomb corrected

Gaussian source function (~ OK)

14

mt scaling of the invariant Gaussian radius universal transverse flow

π, K, p, Λ STAR (200 AGeV Au+Au) radii show mt scaling expected in hydrodynamics

ππ

pΛ

pΛ

KsKs

15

A. Kisiel … THERMINATOR hydro-like freezeout + resonances

Non-Gaussian r*-tails

16

Non-Gaussian r*-tails

CF = |-k* (r*)|2Be careful when comparing like-sign (QS+FSI) and unlike-sign (FSI) correlations different sensitivity to r*-distribution tails

QS & strong FSI: non-Gaussian r*-tail influences only first few bins in Q=2k* and its effect is mainly

absorbed in suppression parameter Coulomb FSI: sensitive to r*-tail up to r* ~ |a| (Bohr radius)

|a|=|z1z2e2|-1

fm K p KK pp388 249 223 110 58

to analyze CF’s of charged particles, instead of simple Gaussian r*-distribution use those simulated within realistic models (like transport codes)

-k*(r*) Ac [1 + r*(1+cos*)/a + f(0)/r*] at k* 0 and r* << |a|

-k*(r*)| 1 at r* >> |a|

17

NA49 central Pb+Pb 158 AGeV vs RQMD: FSI theory OKLong tails in RQMD: r* = 21 fm for r* < 50 fm

29 fm for r* < 500 fm

Fit CF=Norm [Purity RQMD(r* Scaler*)+1-Purity]

Scale=0.76 Scale=0.92 Scale=0.83

RQMD overestimates r* by 10-20% at SPS cf ~ OK at AGS worse at RHIC

p

18

Correlation study of particle interaction

-

+ scattering length f0 from NA49 CF

Fit CF(+) by RQMD with SI scale: f0 sisca f0

input f0

input = 0.232 fm

sisca = 0.60.1 Compare with

~0.8 from SPT & BNL E765

K e

+

CF=Norm [Purity RQMD(r* Scaler*)+1-Purity]

4-6.02.2006 R. Lednický dwstp'06 19

p CFs at AGS & SPS & STAR

Fit using RL-Lyuboshitz’82 with consistent with estimated impurityR~ 3-4 fm consistent with the radius from pp CF

Goal: No Coulomb suppression as in pp CF & Gaussian SF more reliable &Wang-Pratt’99 Stronger sensitivity to the correlation radius R

=0.50.2R=4.50.7 fm

Scattering lengths (f0S), fm: 2.31 1.78

Effective radii (d0S), fm: 3.04 3.22

S = singlet triplet

AGS SPS STAR

R=3.10.30.2 fm

R=1.50.10.3 fm

Pair purity problem for CF(p)@STAR PairPurity ~ 15%Assuming no correlation for misidentified particles and particles from weak decays

Fit using RL-Lyuboshitz’82 (for np)

but, there can be residualcorrelations for particles fromweak decays requiring knowledgeof , p, , , p, , correlations

21

Correlation study of particle interaction

-

Spin-averaged p scattering length f0 from STARFits using RL-Lyuboshitz’82

STAR CF(p) data point to

Ref0(p) < Ref0(pp) 0

Imf0(p) ~ Imf0(pp) ~ 1 fm

But R(p) < R(p) ? Residual correlations

pp

Correlation study of particle interaction

-

-

scattering lengths f0 from NA49 correlation data

Fit using RL-Lyuboshitz (82) with fixed =0.16 from feed-down and PIDData prefer |f0| f0(NN) ~ 20 fm

-

23

interaction potential from LEP CF = Norm (1 e-R2Q2)

=0.620.09R=0.110.02 fm

=0.540.10R=0.110.03 fm

=0.600.07R=0.100.02 fm

Pure QS:

= ½(1+P2) < 0.3Feed-down & PID: ~ 0.5 Polarization < 0.3 }

String picture: lstring~ 2mt/~2 fm ~1 fm

Rz (T/mt)½ ~ 0.3 fm R > Rz /3 ~ 0.17 fm

QS fit yields too low R & too big

FSI potential core RL (02)

=0.6 fixedR=0.290.03 fm

NSC97eneglectedSpin-orbit &Tensor parts

- R ~ OK but - pot. tuning ? - smooth. appr. ?

PLB 475 (00) 395

CF at LEP dominated by ! Direct core signal

Summary

• Assumptions behind femtoscopy theory in HIC seem OK at k 0. At k > ~ 100 MeV/c, the r-k correlation requires a generalization of the usual smoothness approximation.

• Wealth of data on correlations of various particle species (,K0,p,,) is available & gives unique space-time info on production characteristics including collective flows.

• Info on two-particle strong interaction: & & p scattering lengths comes from HIC data at SPS and RHIC. Good perspective at RHIC and LHC (a problem of residual correlations is to be solved).

• An evidence on potential core from LEP (however, a small source size questions the smoothness approximation).

25

Grassberger’77: fire sausage

Dispersion of emitter velocities & limited emission momenta (T) x-p correlation: correlation dominated by pions from nearby emitters

besides geometry, femtoscopy probes source dynamics - expansion

References related to resonance formation in final state:

R. Lednicky, V.L. Lyuboshitz, SJNP 35 (1982) 770R. Lednicky, V.L. Lyuboshitz, V.V. Lyuboshitz, Phys.At.Nucl. 61 (1998) 2050S. Pratt, S. Petriconi, PRC 68 (2003) 054901S. Petriconi, PhD Thesis, MSU, 2003S. Bekele, R. Lednicky, Braz.J.Phys. 37 (2007) 994B. Kerbikov, R. Lednicky, L.V. Malinina, P. Chaloupka, M. Sumbera, arXiv:0907.061v2 B. Kerbikov, L.V. Malinina, PRC 81 (2010) 034901 R. Lednicky, Phys. Part. Nucl. Lett. 8 (2011) 965R. Lednicky, P. Chaloupka, M. Sumbera, in preparation

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 overestimates measured * by a factor 4 (3) for r0 = 5 (5.5) fm factor 3 (2) if account for out -6 fm

• The same for p-wave resonance () FSI contributions to K+K- CF FSI contribution overestimates measured by 20% for r0 = 4.5 fm

• Little or no room for direct production when neglecting r-k correlation!

Rpeak(NA49) 0.10 0.14after purity correction

Rpeak(STAR) 0.025 ----------- -----

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

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

r0 = 5 fm