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Femtoscopic Correlations and Final State Interactions R. Lednický @ JINR Dubna & IP ASCR Prague

Femtoscopic Correlations and Final State Interactions R. Lednický @ JINR Dubna & IP ASCR Prague. History Assumptions Correlation study of strong interaction Conclusions. History. Correlation femtoscopy :. measurement of space-time characteristics R, c  ~ fm.

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Femtoscopic Correlations and Final State Interactions R. Lednický @ JINR Dubna & IP ASCR Prague

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  1. RBRC Hyperon Workshop, BNL2012 Femtoscopic Correlations and Final State InteractionsR. Lednický@ JINR Dubna & IP ASCR Prague • History • Assumptions • Correlation study of strong interaction • Conclusions

  2. History 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

  3. Fermi function F(k,Z,R) in β-decay F = |-k(r)|2~ (kR)-(Z/137)2 Z=83 (Bi)‏ β- 2 fm 4 R=8 β+ 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= Ncorr /Nuncorr& mixing techniques to construct Nuncorr •argued that sufficientlysmooth momentum spectrum allows one to neglect space-time coherence at small q* smoothness approximation: |∫d4x1d4x2p1p2(x1,x2)...|2 →∫d4x1d4x2p1p2(x1,x2)|2... •clarified role of space-time production characteristics: shape & time source picture from various q-projections

  5. QS symmetrization of production amplitudemomentum correlations of identical particles are sensitive to space-time structure of the source KP’71-75 total pair spin CF=1+(-1)Scos qx exp(-ip1x1)‏ p1 2 x1 ,nns,s x2 1/R0 1 p2 2R0 nnt,t PRF q =p1- p2 → {0,2k} x = x1- x2 → {t,r} |q| 0 CF →|S-k(r)|2  =| [ e-ikr +(-1)S eikr]/√2 |2 

  6. Final State Interaction |-k(r)|2 Similar to Coulomb distortion of -decay Fermi’34: Migdal, Watson, Sakharov, … Koonin, GKW, ... fcAc(G0+iF0)‏ s-wave strong FSI } FSI nn e-ikr -k(r)  [ e-ikr +f(k)eikr/r ] CF pp Coulomb |1+f/r|2 kr+kr F=1+ _______ + … eicAc ka } } Bohr radius Coulomb only Point-like Coulomb factor k=|q|/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, .. & Coalescence deuterons, .. Study “exotic” scattering,K, KK,, p,, .. Study relative space-time asymmetriesdelays, flow

  7. 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

  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(p1,p2|Τ(p1,p2;)|2

  9. Smoothness approximation: rA« r0 (q « p) W(p1,p2|∫d4x1d4x2p1p2(x1,x2) Τ(x1,x2;)|2 =∫d4x1d4x1’d4x2d4x2’ r0 - Source radius rA - Emitter radius p1p2(x1,x2)p1p2*(x1’,x2’)‏ Τ(x1,x2 ;)Τ*(x1’,x2’ ;)‏ p1 x1 ≈ ∫d4x1d4x2G(x1,p1;x2,p2) |p1p2(x1,x2)|2 x1’ x2’ Source function G(x1,p1;x2,p2) = ∫d4ε1d4ε2 exp(ip1ε1+ip2ε2)‏ Τ(x1+½ε1,x2 +½ε2;)Τ*(x1-½ε1,x2-½ε2;) x2 p2 2r0 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. p1≈ p2: ≈ ∫d4x1d4x2 G(x1,p1;x2,p2) [1+cos(qx)] = ∫ d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2

  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 r0=2 fm 0=2 fm/c r0=2 fm v=0.1  OK for heavy particles  OK within 5% even for pions if 0 ~ r0 or lower

  11. Equal time & smoothness approx.  “Fermi” formula CF(p1,p2)∫d3r WP(r,k) |-k(r)|2 Caution: Smoothness approximation is justified for small k << 1/r0 ∫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 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 It should be generalized in the region k > ~100 MeV/c

  12. Resonance contribution vs r-k correlation parameter b 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 ----------- Rpeak(NA49) ----------  0.10 0.14 Smoothness assumption: WP(r,½(k-kn))  WP(r,-kn)  WP(r,k) Exact Rpeak(STAR) -----------  0.025  

  13. Gaussian source function (~ OK) R. Lednický dwstp'06 Examples of present data: NA49 & STAR 3-dim fit:CF=1+exp(-Rx2qx2 –Ry2qy2-Rz2qz2-2Rxz2qx qz) Correlation strength or chaoticity Interferometry or correlation (Gaussian) radii STAR KK NA49 Coulomb corrected z x y

  14. mt scaling of the invariant Gaussian radius  universal transverse flow π, K, p, ΛSTAR (200 AGeV Au+Au) radii show mt scaling expected in hydrodynamics ππ KsKs pΛ pΛ

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

  16. Non-Gaussian r*-tails Be careful when comparing like-sign (QS+FSI) and unlike-sign (FSI) correlations  different sensitivity to r*-distribution tails CF = |-k* (r*)|2 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 pp 388 249 223 110 58 -k*(r*)  Ac [1 + r*(1+cos*)/a+ f(0)/r*] at k* 0 and r* << |a| -k*(r*)|  1 at r* >> |a| • to analyze CF’s of charged particles, instead of simple • Gaussian r*-distribution use those simulated within • realistic models (like transport codes)

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

  18. Correlation study of particle interaction CF=Norm[Purity RQMD(r*Scaler*)+1-Purity] +scattering length f0 from NA49 CF + Fit CF(+) by RQMD with SI scale: f0siscaf0input f0input = 0.232 fm - sisca = 0.60.1 Compare with ~0.8 from SPT & BNL E765 K  e

  19. R. Lednický dwstp'06 Goal: No Coulomb suppression as in pp CF & Gaussian SF more reliable & p CFs at AGS & SPS & STAR Wang-Pratt’99 Stronger sensitivity to the correlation radius R S = singlet triplet Fit using RL-Lyuboshitz’82 with Scattering lengths (f0S), fm:2.31 1.78  consistent with estimated impurity Effective radii (d0S), fm:3.04 3.22 R~ 3-4 fm consistent with the radius from pp CF STAR AGS SPS =0.50.2 R=4.50.7 fm 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. Correlation study of particle interaction Spin-averaged pscattering lengthf0 from STAR Fits using RL-Lyuboshitz’82 pp STAR CF(p) data point to Ref0(p) < Ref0(pp)  0 Imf0(p) ~ Imf0(pp) ~ 1 fm But R(p) < R(p) ? Residual correlations  -

  22. 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 PID Data prefer|f0| f0(NN) ~ 20 fm  - - - 

  23. interaction potential from LEP CF= Norm (1e-R2Q2) Pure QS: PLB 475 (00) 395 Feed-down & PID:~ 0.5 } = ½(1+P2) < 0.3 Polarization < 0.3 =0.620.09 R=0.110.02 fm String picture: lstring~ 2mt/~2 fm  ~1 fm Rz(T/mt)½ ~ 0.3 fmR > Rz /3 ~ 0.17 fm =0.540.10 R=0.110.03 fm  QS fit yields too low R & too big  CF at LEP dominated by ! Direct core signal =0.600.07 R=0.100.02 fm FSI potential coreRL (02) NSC97e neglected Spin-orbit & Tensor parts  • R ~ OK but • pot. tuning ? • smooth. appr. ? =0.6 fixed R=0.290.03 fm

  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. 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 r0 = 5 fm • 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 overestimatesmeasured * 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(STAR)  0.025 ----------- ----- --------------------- • Rpeak(NA49) • 0.10  0.14 • after purity • correction ----------- -----

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