1. 11-14. 09. 2008WPCF'08 Krakow1 Femtoscopic Correlations of Nonidentical Particles R. Lednický, JINR Dubna & IP ASCR Prague History Assumptions Correlation asymmetries Conclusion

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(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. 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: (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. 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 

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 in multiparticle production CF =  |  -k* (r*)| 2  - t FSI  t prod  |k * | = ½|q * |  hundreds MeV/c - 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|  peak  R emitter  R source ~ OK in HIC, R source 2  0.1 fm 2  p t 2 -slope of direct particles

8. 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

9. 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

10. 10 Smoothness approximation: r A « r 0 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 |  p 1 p 2 ( x 1,x 2 ) | 2 G (x 1,p 1 ;x 2,p 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 ’

11. 11 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 →

12. 12 NA49 central Pb+Pb 158 AGeV vs RQMD 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  

13. 13 Correlation asymmetries CF of identical particles sensitive to terms even in k*r* (e.g. through  cos 2k*r*  )  measures only dispersion of the components of relative separation r * = r 1 * - r 2 * in pair cms CF of nonidentical particles sensitive also to terms odd in k*r*  measures also relative space-time asymmetries - shifts  r *  RL, Lyuboshitz, Erazmus, Nouais PLB 373 (1996) 30  Construct CF +x and CF -x with positive and negative k* -projection k x * on a given direction x and study CF-ratio CF +x /CF  x SUBATECH-94-20 (1994)‏

14. 14 Simplified idea of CF asymmetry (valid for Coulomb FSI)‏ x x v v v1v1 v2v2 v1v1 v2v2 k*/  = v 1 -v 2   p p k* x > 0 v  > v p k* x < 0 v  < v p Assume  emitted later than p or closer to the center p   p Longer t int Stronger CF  Shorter t int Weaker CF   CF  CF  LLEN’94 (also Gelderloos et al’94, Cornell et al'96)‏

15. 15 CF-asymmetry for charged particles Asymmetry arises mainly from Coulomb FSI CF  A c (  )  |F(-i ,1,i  )| 2  =(k*a) -1,  =k*r*+k*r* F  1+   = 1+r*/a+k*r*/(k*a)‏ r*  |a| k*  1/r* Bohr radius } ±226 fm for  ± p ±388 fm for  +  ±  CF +x /CF  x  1+2  x*  /a k*  0  x* = x 1 *-x 2 *  r x *  Projection of the relative separation r* in pair cms on the direction x In LCMS ( v z =0) or x || v :  x* =  t (  x - v t  t)‏  CF asymmetry is determined by space and time asymmetries

16. 16 Large lifetimes evaporation or phase transition x || v |  x|  |  t|  CF-asymmetry yields time delay Ghisalberti’95 GANIL Pb+Nb  p+d+X CF + (pd)‏ CF  (pd)‏ CF + /CF  < 1 Deuterons earlier than protons in agreement with coalescence e -t p /  e -t n /   e -t d /(  /2) since t p  t n  t d Two-phase thermodynamic model CF + /CF  < 1 123 1 2 3 Strangeness distillation : K  earlier than K  in baryon rich QGP Ardouin et al.’99

17. 17 ad hoc time shift  t = –10 fm/c CF + /CF  Sensitivity test for ALICE a, fm   84  226  249 CF + /CF   1+2  x*  /a k*  0 Here  x*  = -  v  t   CF-asymmetry scales as -  t  /a Erazmus et al.’95 Delays of several fm/c can be easily detected

18. 18 Usually:  x  and  t  comparable RQMD Pb+Pb   p +X central 158 AGeV :  x  = -5.2 fm  t  = 2.9 fm/c  x*  = -8.5 fm  + p-asymmetry effect 2  x*  /a  -8%  Shift  x  in out direction is due to collective transverse flow RL’99-01  x p  >  x K  >  x   > 0 & higher thermal velocity of lighter particles rtrt y x FF tTtT tt  FF = flow velocity tTtT = transverse thermal velocity tt =  F +  t T = observed transverse velocity  x   r x  =  r t cos   =  r t (  t 2 +  F2 -  t T2 )/(2  t  F )   y   r y  =  r t sin   = 0 mass dependence  z   r z    sinh  = 0 in LCMS & Bjorken long. exp. out side measures edge effect at y CMS  0

19. pion Kaon Proton BW Retiere@LBL’05 Distribution of emission points at a given equal velocity: - Left,  x = 0.73c,  y = 0 - Right,  x = 0.91c,  y = 0 Dash lines: average emission R x   R x (  )  <  R x (K)  <  R x (p)  p x = 0.15 GeV/c p x = 0.3 GeV/c p x = 0.53 GeV/cp x = 1.07 GeV/c p x = 1.01 GeV/cp x = 2.02 GeV/c For a Gaussian density profile with a radius R G and linear flow velocity profile  F (r) =  0 r/ R G RL’04, Akkelin-Sinyukov’96 :  x  = R G  x  0 /[  0 2 +T/m t ] 0.73c0.91c

20. NA49 & STAR out-asymmetries Pb+Pb central 158 AGeV not corrected for ~ 25% impurity r* RQMD scaled by 0.8 Au+Au central  s NN =130 GeV corrected for impurity  Mirror symmetry (~ same mechanism for  and  mesons)‏  RQMD, BW ~ OK  points to strong transverse flow pp pp KK (  t  yields ~ ¼ of CF asymmetry)‏

21. 21 Decreasing R(p t ): x-p correlation usually attributed to collective flow taken for granted femtoscopy the only way to confirm x-p correlations  x 2  -p correlation:yes  x  -p correlation:yes Non-flow possibility hot core surrounded by cool shell important ingredient of Buda-Lund hydro picture Csörgő & Lörstad’96  x 2  -p correlation:yes  x  -p correlation:no  x  = R G  x  0 /[  0 2 +T/m t +  T/T  r ] radial gradient of T  decreasing asymmetry ~1 ? problem M. Lisa’07

22. 22 Summary Assumptions behind femtoscopy theory in HIC seem OK 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 Rather direct evidence for strong transverse flow in HIC at SPS & RHIC comes from nonidentical particle correlation asymmetries (lower thermal velocities of heavier particles lead to mass hierarchy of the shifts arising due to flow)‏ For femtoscopic correlations of nonidentical particles, there is no problem with two-track resolution though, in the case of very different masses a large detector acceptance is required Info on two-particle strong interaction:  &  & p  scattering lengths from HIC at SPS and RHIC. Good perspective at RHIC and LHC

23. 23 Beta decay rate d 6 N/(d 3 p 1 d 3 p 2 ) = d 6 N 0 /(d 3 p 1 d 3 p 2 )  |  -k (r * )| 2  d 5 w = ∑ spins ∫ d 3 p d 3 q d 3 k δ 4 (p 0 -p-q-k) |∫d 3 x Τ(x; spins ) e iqx  -k * (x)| 2 = ∑ spins ∫ d 3 p d 3 q d 3 k δ 4 (p 0 -p-q-k)‏ ∫ d 3 x d 3 x’ Τ(x; spins ) Τ * (x’;spins) e iq(x-x’)  -k * (x)  -k (x’)‏ p 0 (A) = p(A’) - q(ν) - k(e)‏ ∑ spins ∫ d 3 k ∫d 3 x |Τ(x; spins )| 2 |  -k * (x)| 2 Neglecting the limitation due to energy-momentum conservation cf classical sources approximation valid in multiparticle production ∫ d 3 q →

24. 24 Fermi theory of beta decay d 5 w = ∑ spins ∫ d 3 p d 3 q d 3 k δ 4 (p 0 -p-q-k) |∫d 3 x Τ(x; spins ) e iqx  -k * (x)| 2 ≈ ∑ spins ∫ d 3 p d 3 q d 3 k δ 4 (p 0 -p-q-k)  |  -k (x)| 2  |∫d 3 x Τ(x; spins )| 2 ρ

25. 25 Conclusion from beta decay Formally (FSI) correlations in beta decay and multiparticle production are determined by the same (Fermi) function  |  -k (x)| 2  But this factor appears for different reasons in beta decay (a weak r-dependence of  -k (r) within the nucleus volume + point like emission) and in multiparticle production (valid classical particle sources approximation + small source sizes compared to their separation in HIC)‏

26. 26 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

27. 27 Correlation study of particle interaction -  +   &  & p  scattering lengths f 0 from NA49 and STAR NA49 CF(  +   ) vs RQMD with SI scale: f 0  sisca f 0 (= 0.232fm )‏ sisca = 0.6  0.1 compare ~0.8 from S  PT & BNL data E765 K  e  Fits using RL-Lyuboshitz’82 NA49 CF(  ) data prefer | f 0 (  )|  f 0 (NN) ~ 20 fm 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  pp

28. 28 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]