1. 5. 7. 2011GDRE meeting, Nantes 20111 Femtoscopic Correlations and Final State Resonance Formation R. Lednický @ JINR Dubna & IP ASCR Prague P. Chaloupka and M. Šumbera @ NPI ASCR Řež History Assumptions Narrow resonance FSI contributions to π +  -  K + K - CF’s Conclusions

2. 2 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

3. 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...

4. 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 %

5. Caution: Smoothness approximation is justified for small k<<1/r 0 CF(p 1,p 2 )  ∫d 3 r W P (r,k) |  -k (r)| 2 should be generalized in resonance region k~150 MeV/c  ∫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 The smoothness approximation 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

6. Accounting for the r-k correlation in the emission function Substituting the simple Gaussian emission function: W P (r,k) = (8π 3/2 r 0 3 ) -1 exp(-r 2 /4r 0 2 ) by (  = angle between r and k) : W P (r,k) = (8π 3/2 r 0 3 ) -1 exp(-b 2 r 0 2 k 2 ) exp(-r 2 /4r 0 2 + bkrcos  ) Exponential suppression generated in the resonance region (k ~ 150 MeV/c) by a collective flow: b > 0

7. Accounting for the r-k correlation in the emission function In the case of correlation asymmetry in the out direction: W P (r,k) = (8π 3/2 r 0 3 ) -1 exp(-b 2 r 0 2 k 2 - bk out  out )  exp{-[( r out -  out ) 2 + r side 2 + r long 2 ]/4r 0 2 + bkrcos  } Note the additional suppression of W P (0,k) if  out  0: W P (0,k) ~ exp[-(  out /2r 0 ) 2 ] (~20% suppression if  out  r 0 ) & correlation asymmetry even at r  0: W P (0,k) ~ exp(- bk out  out )  1 - bk out  out

8. r-k correlation in the  *- and  -resonance regions from FASTMC code  = angle between r and k fitted by W P (r,k) ~ exp[-r 2 /4r 0 2 + b krcos  ] r* = 9-12 fm  b = 0.13 r* = 0 – 27 fm  b (K + K - ) = 0.32 – 0.09 r* = 9-12 fm  b = 0.18 r* = 0 – 27 fm  b (π +  - ) = 0.18 – 0.08 π+-π+- K+K-K+K-

9. Approximate resonance FSI contribution In good agreement with generalized smoothness approximation (see a figure later) Exponential suppression by the r-k correlation & out shift exp[-b 2 r 0 2 k 2 - (  out /2r 0 ) 2 - b k out  out ] + correlation asymmetry: ~ 1 - b k out  out to be compared with the correlation asymmetry in the Coulomb region (k  0): ~ 1 + 2k out  out /(k a) ! same sign for oppositely charged particles (a < 0) and b > 0 (resulting from collective flow) ! as indicated by STAR  +  - CF

10. 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 preparation

11.  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 at k* < 0.1 GeV/c: r 0 =4.8±0.7 fm,  out = -5.6±1.0 fm r 0 =[½(r π 2 +r  2 )] ½ of ~5 fm is in agreement with the dominant r π of 7 fm P. Chaloupka, JPG 32 (2006) S537; M. Sumbera, Braz.J.Phys. 37(2007)925

12.      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  0.14 after purity correction 3D-Gaussian fit of 0- 5% CF’s: out-side-long radii of 4-5 fm PLB 557 (2003) 157

13. 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 underestimates (overestimates) measured  by 12 (20) % for r 0 = 5 (4.5) fm Little or no room for direct production !  R peak (NA49)  0.10  0.14  after purity  correction  R peak (STAR)  0.025 ----------- ----- --------------------- r 0 = 5 fm

14. 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 > 0.2 is required for π +  - system k=146 MeV/c, r 0 =5 fm k=126 MeV/c, r 0 =5 fm -----------

15. 15 Summary Assumptions behind femtoscopy theory in HIC seem OK, up to a problem of the r-k correlation in the resonance region  the usual smoothness approximation must be generalized. The effect of narrow resonance FSI scales as inverse emission volume r 0 -3, compared to r 0 -1 or r 0 -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 disfavored by the theoretical uncertainty in case of a strong r-k correlation. The NA49 (K + K - ) & STAR (π +  - ) correlation data from the most central collisions point to a strong r-k correlation, required to leave a room for a direct (thermal) production of near threshold narrow resonances.

16. 16 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,...