1. 29. 2. 2012RBRC Hyperon Workshop, BNL20121 Femtoscopic Correlations and Final State Interactions R. Lednický @ JINR Dubna & IP ASCR Prague History Assumptions Correlation study of strong interaction Conclusions

2. 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. 3 Fermi function F(k,Z,R) in β-decay F =  |  -k (r)| 2  ~ (kR) -(Z/137) 2 Z=83 (Bi)‏ β-β- β+β+ R=8 4 2 fm k MeV/c

4. 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= N corr /N uncorr & argued that sufficiently smooth momentum spectrum allows one to neglect space-time coherence at small q* smoothness approximation: mixing techniques to construct N uncorr clarified role of space-time production characteristics: shape & time source picture from various q-projections |∫ d 4 x 1 d 4 x 2  p 1 p 2 ( x 1,x 2 )... | 2 → ∫ d 4 x 1 d 4 x 2  p 1 p 2 ( x 1,x 2 )| 2...

5. 5 QS symmetrization of production amplitude  momentum correlations of identical particles are sensitive to space-time structure of the source CF=1+(-1) S  cos q  x  p 1 p 2 x1x1 x 2 q = p 1 - p 2 → {0,2k}  x = x 1 - x 2 → {t,r} nn t,  t , nn s,  s 2 1 0 |q| 1/R 0 total pair spin 2R 0 KP’71-75 exp(-ip 1 x 1 )‏ CF →  |  S -k ( r )| 2  =  | [ e -ikr +(-1) S e ikr ]/√2 | 2  PRF

6. 6 Final State Interaction Similar to Coulomb distortion of  -decay Fermi’34: e -ikr   -k ( r )  [ e -ikr +f( k )e ikr / r ] eicAceicAc F=1+ _______ + … kr+kr kaka Coulomb s-wave strong FSI FSI f c  A c  (G 0 +iF 0 )‏ } } Bohr radius } Point-like Coulomb factor k=|q|/2 CF nn pp Coulomb only  | 1+f/r| 2   FSI is sensitive to source size r and scattering amplitude f It 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)| 2  Migdal, Watson, Sakharov, … Koonin, GKW,...

7. Assumptions to derive “Fermi” formula for CF CF =  |  -k* (r*)| 2  - t FSI  d  dE  t prod - equal time approximation in PRF  typical momentum transfer RL, Lyuboshitz’82  eq. time condition |t*|   r* 2 usually OK RL, Lyuboshitz..’98  +     0  0,  - p   0 n, K + K   K 0 K 0,... & account for coupled channels within the same isomultiplet only: - two-particle approximation (small freezeout PS density f )‏ ~ OK,  1 ? low p t - smoothness approximation:  p    q  correl  R emitter  R source ~ OK in HIC, R source 2  0.1 fm 2  p t 2 -slope of direct particles t FSI (s-wave) = µf 0 /k*  k * = ½q *  hundreds MeV/c t FSI (resonance in any L-wave) = 2/     hundreds MeV/c in the production process to several %

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(p 1, p 2   | Τ(p 1,p 2 ;  ) | 2

9. Smoothness approximation: r A « r 0 (q «  p) p 1 p 2 x1x1 x 2 2r 0 W(p 1, p 2   |∫ d 4 x 1 d 4 x 2  p 1 p 2 ( x 1,x 2 ) Τ(x 1,x 2 ;  ) | 2 x1’x1’ x2’x2’ ≈ ∫ d 4 x 1 d 4 x 2 G (x 1,p 1 ;x 2,p 2 ) |  p 1 p 2 ( x 1,x 2 ) | 2 r 0 - Source radius r A - Emitter radius  p1p2 (x 1,x 2 )  p1p2 *(x 1 ’,x 2 ’)‏ Τ(x 1,x 2 ;  )Τ*(x 1 ’,x 2 ’ ;  )‏ G(x 1,p 1 ;x 2,p 2 ) =   ∫ d 4 ε 1 d 4 ε 2 exp(ip 1 ε 1 +ip 2 ε 2 )‏ Τ ( x 1 +½ε 1,x 2 +½ε 2 ;  )Τ * (x 1 -½ε 1,x 2 -½ε 2 ;  ) Source function =   ∫d 4 x 1 d 4 x 1 ’d 4 x 2 d 4 x 2 ’ For non-interacting identical spin-0 particles – exact result (p=½(p 1 +p 2 ) ):‏ W(p 1,p 2  ∫ d 4 x 1 d 4 x 2 [G(x 1,p 1 ;x 2,p 2 )+G(x 1,p;x 2,p) cos(q  x)] approx. p 1 ≈ p 2 : ≈ ∫d 4 x 1 d 4 x 2 G(x 1,p 1 ;x 2,p 2 ) [1+cos(q  x)] = ∫ d 4 x 1 d 4 x 2 G(x 1,p 1 ;x 2,p 2 ) |  p 1 p 2 (x 1,x 2 )| 2

10. 10 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 r 0 =2 fm  0 =2 fm/c r 0 =2 fm v=0.1  OK for heavy particles  OK within 5% even for pions if  0 ~ r 0 or lower →

11. Equal time & smoothness approx.  “Fermi” formula  ∫d 3 r {W P (r,k) + W P (r,½(k-kn)) 2Re[exp(ikr)  -k (r)] +W P (r,-kn) |  -k (r)| 2 } where  -k (r) = exp(-ikr)+  -k (r) and n = r/r Smoothness approx. W P (r,½(k-kn))  W P (r,-kn)  W P (r,k) is valid if one can neglect the k-dependence of W P (r,k ), e.g. for k << 1/r 0 CF(p 1,p 2 )  ∫d 3 r W P (r,k) |  -k (r)| 2 Caution: Smoothness approximation is justified for small k << 1/r 0 It should be generalized in the region k > ~100 MeV/c

12. Resonance contribution vs r-k correlation parameter b  R peak (STAR) -----------  0.025  R peak (NA49) ----------  0.10  0.14  Smoothness assumption: W P (r,½(k-kn))  W P (r,-kn)  W P (r,k) Exact W P (r,k) ~ exp[-r 2 /4r 0 2 + bkrcos  ];  = angle between r and k CF suppressed by a factor W P (0,k) ~ exp[-b 2 r 0 2 k 2 ] To leave a room for a direct production  b > 0.3 (0.15) is required for π +  - (K + K - ) system  * (k=146 MeV/c), r 0 =5 fm  (k=126 MeV/c), r 0 =5 fm -----------

13. 4-6.02.2006R. Lednický dwstp'0613 Examples of present data: NA49 & STAR 3-dim fit: CF=1+ exp(-R x 2 q x 2 –R y 2 q y 2 -R z 2 q z 2 -2R xz 2 q x q z ) zxy Correlation strength or chaoticity NA49 Interferometry or correlation (Gaussian) radii KK STAR  Coulomb corrected Gaussian source function (~ OK)

14. 14 m t scaling of the invariant Gaussian radius  universal transverse flow π, K, p, Λ STAR (200 AGeV Au+Au) radii show m t scaling expected in hydrodynamics ππ pΛpΛ pΛpΛ K s K s

15. 15 A. Kisiel … THERMINATOR hydro-like freezeout + resonances Non-Gaussian r*-tails

16. 16 Non-Gaussian r*-tails CF =  |  -k* (r*)| 2  Be 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|=|z 1 z 2 e 2  | -1 fm   K  p KK pp 388 249 223 110 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 * )   A c [1 + r * (1+cos  * )/a + f(0)/r * ] at k *  0 and r * << |a|  -k* (r * )|  1 at r * >> |a|

17. 17 NA49 central Pb+Pb 158 AGeV vs RQMD: FSI theory OK Long tails in RQMD:  r*  = 21 fm for r* < 50 fm 29 fm for r* < 500 fm Fit CF=Norm [ Purity RQMD(r*  Scale  r*)+1-Purity] Scale=0.76Scale=0.92 Scale=0.83  RQMD overestimates r* by 10-20% at SPS cf ~ OK at AGS worse at RHIC  p  

18. 18 Correlation study of particle interaction -  +   scattering length f 0 from NA49 CF Fit CF(  +   ) by RQMD with SI scale: f 0  sisca f 0 input f 0 input = 0.232 fm sisca = 0.6  0.1 Compare with ~0.8 from S  PT & BNL E765 K  e  ++ CF=Norm [ Purity RQMD(r*  Scale  r*)+1-Purity]

19. 4-6.02.2006R. Lednický dwstp'0619 p  CFs at AGS & SPS & STAR Fit using RL-Lyuboshitz’82 with consistent with estimated impurity R~ 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 =0.5  0.2 R=4.5  0.7 fm Scattering lengths (f 0 S ), fm: 2.31 1.78 Effective radii (d 0 S ), fm: 3.04 3.22 S = singlet triplet AGSSPS STAR R=3.1  0.3  0.2 fm R=1.5  0.1  0.3 fm

20. 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 residual correlations for particles from weak decays requiring knowledge of , p , , , p , ,  correlations

21. 21 Correlation study of particle interaction - Spin-averaged p  scattering length f 0 from STAR Fits using RL-Lyuboshitz’82 STAR CF( p  ) data point to Re f 0 ( p  ) < Re f 0 ( pp )  0 Im f 0 ( p  ) ~ Im f 0 ( pp ) ~ 1 fm But R( p  ) < R( p  ) ? Residual correlations  pp

22. Correlation study of particle interaction - -  scattering lengths f 0 from NA49 correlation data Fit using RL-Lyuboshitz (82) with fixed =0.16 from feed-down and PID Data prefer | f 0 |  f 0 (NN) ~ 20 fm  - 

23. 23  interaction potential from LEP CF = Norm (1   e -R 2 Q 2 )  =0.62  0.09 R=0.11  0.02 fm  =0.54  0.10 R=0.11  0.03 fm  =0.60  0.07 R=0.10  0.02 fm Pure QS:  = ½ (1+P 2 ) < 0.3 Feed-down & PID: ~ 0.5 Polarization < 0.3 } String picture: l string ~ 2m  t /  ~2 fm   ~1 fm R z   (T/m  t ) ½ ~ 0.3 fm  R > R z /  3 ~ 0.17 fm  QS fit yields too low R & too big  FSI potential core RL (02) =0.6 fixed R=0.29  0.03 fm NSC97e neglected Spin-orbit & Tensor parts  - R ~ OK but - pot. tuning ? - smooth. appr. ? PLB 475 (00) 395   CF at LEP dominated by ! Direct core signal

24. 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 (  ,K  0,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. 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

26. 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, Phys. Part. Nucl. Lett. 8 (2011) 965 R. Lednicky, P. Chaloupka, M. Sumbera, in preparation

27. 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 r 0 = 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 r 0 = 4.5 fm Little or no room for direct production when neglecting r-k correlation!  R peak (NA49)  0.10  0.14  after purity  correction  R peak (STAR)  0.025 ----------- ----- --------------------- r 0 = 5 fm