Nantes‘101 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

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(k,Z,R) in β-decay =  |  -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 proposed CF= N corr /N uncorr & showed that sufficiently smooth momentum spectrum allows one to neglect space-time coherence at small q* (for non-interacting identical particles)‏ mixing techniques to construct N uncorr clarified role of space-time characteristics in various models |∫ 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 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 |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 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,...

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 OK fig. 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 fig. - 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

8 Phase space density from CFs and spectra Bertsch’94 May be high phase space density at low p t ?  ? Pion condensate or laser ? Multiboson effects on CFs spectra & multiplicities rises up to SPS Lisa..’05

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

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. result: ≈ ∫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

11 Effect of nonequal times in pair cms RL, Lyuboshitz SJNP 35 (82) 770; RL nucl-th/ 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 →

Technicalities – 1: neglecting complex intermediate channels

Technicalities – 2: spin & isospin equilibration

Technicalities – 3: equal-time approximation

Technicalities – 4: simple Gaussian emission functions

Technicalities – 5: treating the spin & angular dependence

In the following we ssume write (angular dependence enters only through the angle  between the vectors k and r): Since then L’=L, S’=S=j,=1/2 or 0, one can put m=j and

Technicalities – 6: contribution of the outer region

Technicalities – 7: projecting pair spin & isospin =π+-=π+-  =K + K -

Technicalities – 8: resonance dominance in the JT-wave

Technicalities – 9: contribution of the inner region

Technicalities – 10: volume integral In the single flavor case For s & p-waves it recovers the result of Wigner’55 & Luders’55

 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  Gaussian fit of 0-10% CF’s: r 0 =6.7±1.0 fm,  out = -5.6±1.0 fm

     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  D-Gaussian fit of 0-5% CF’s: out-side- long radii of 4-5 fm

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 7 fm The same for p-wave resonance (  ) FSI contributions to K + K - CF for Gaussian radius of 5 fm  R peak (NA49)  0.10  R peak (STAR)  0.025

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 7 fm The same for p-wave resonance (  ) FSI contributions to K + K - CF for Gaussian radius of 5 fm  R peak (STAR)   R peak (NA49)  0.10

27 Summary Assumptions behind femtoscopy theory in HIC seem OK, including both short-range s-wave and narrow resonance FSI (? up to a problem of angular dependence in the resonance region) The effect of narrow resonance FSI scales as inverse emission volume r 0 -3, compared to volume 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 NA49 and STAR correlation data from the most central collisions seem to leave a little or no room for a direct (thermal) production of narrow resonances

Angular dependence in the  *-resonance region (k*= MeV/c) r* < 1 fm r* < 0.5 fm 0-10% 200 GeV Au+Au FASTMC-code

Angular dependence – example parametrization