1. 28.09.2013NTHEP, Crimea, September 23-29, 2013I1 Correlation Femtoscopy – status & prospects History QS correlations FSI correlations Correlation study of particle interaction Correlation asymmetries Summary In memory of V.L.Lyuboshitz (19.3.1937-4.5.2013) R. Lednický @ JINR Dubna & IP ASCR Prague

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 2 x Goldhaber, Lee & Pais GGLP’60: enhanced  +  +,  -  - vs  +  - at small opening angles – interpreted as BE enhancement depending on fireball radius R 0 R 0 = 0.75 fm p p  2  + 2  - n  0

5. 5 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...

6. 6 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 -ik*r* +(-1) S e ik*r* ]/√2 | 2  PRF

7. 7 Assumptions to derive KP formula CF - 1   cos q  x  - two-particle approximation (small freeze-out PS density f ) - smoothness approximation: R emitter  R source   |  p|    |q|  peak - incoherent or independent emission ~ OK,  1 ? low p t fig. ~ OK in HIC, R source 2  0.1 fm 2  p t 2 -slope of direct particles 2  and 3  CF data ~ consistent with KP formulae: CF 3 (123) = 1+|F(12)| 2 +|F(23)| 2 +|F(31)| 2 +2Re[F(12)F(23)F(31)] CF 2 (12) = 1+|F(12)| 2, F(q)| =  e iqx  - neglect of FSI OK for photons, ~ OK for pions up to Coulomb repulsion

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. “General” parameterization at |q|  0 Particles on mass shell & azimuthal symmetry  5 variables: q = {q x, q y, q z }  {q out, q side, q long }, pair velocity v = {v x,0,v z } R x 2 =½  (  x-v x  t) 2 , R y 2 =½  (  y) 2 , R z 2 =½  (  z-v z  t) 2  q 0 = qp/p 0  qv = q x v x + q z v z y  side x  out  transverse pair velocity v t z  long  beam Podgoretsky’83, Bertsch, Pratt’95; so called out-side-long parameterization Interferometry or correlation radii:  cos q  x  =1-½  (q  x) 2  +..  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 ) Grassberger’77 RL’78 Csorgo, Pratt’91: LCMS v z = 0

10. 10 Examples of NA49 & STAR data 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 radii KK STAR  Coulomb corrected

11. 11 Probing source shape and emission duration Static Gaussian model with space and time dispersions R  2, R || 2,  2 R x 2 = R  2 +v  2  2  R y 2 = R  2  R z 2 = R || 2 +v || 2  2 Emission duration  2 = (R x 2 - R y 2 )/v  2  (degree) R side 2 fm 2 If elliptic shape also in transverse plane  R y  R side oscillates with pair azimuth  R side (  =90°) small R side  =0°) large z A B Out-of reaction plane In reaction plane In-planeCircular Out-of plane KP (71-75) …

12. Probing source dynamics - expansion Dispersion of emitter velocities & limited emission momenta (T)  x-p correlation: interference dominated by pions from nearby emitters  Interferometry radii decrease with pair velocity  Interference probes only a part of the source Resonances GKP’71.. Strings Bowler’85.. Hydro P t =160 MeV/cP t =380 MeV/c R out R side R out R side Collective transverse flow  F  R side  R/(1+m t  F2 /T) ½ during proper freeze-out (evolution) time   R long  (T/m t ) ½  /coshy Pratt, Csörgö, Zimanyi’90 Makhlin-Sinyukov’87 } 1 in LCMS ….. Bertch, Gong, Tohyama’88 Hama, Padula’88 Mayer, Schnedermann, Heinz’92 Pratt’84,86 Kolehmainen, Gyulassy’86 Longitudinal boost invariant expansion

13. pion Kaon Proton , , Flow & Radii in Blast Wave model ← Emission points at a given tr. velocity p x = 0.15 GeV/c0.3 GeV/c p x = 0.53 GeV/c1.07 GeV/c p x = 1.01 GeV/c2.02 GeV/c For a Gaussian density profile with a radius R G and linear flow velocity profile  F (r) =  0 r/ R G : 0.73c0.91c R z 2   2  (T/m t ) R x 2 =  x’ 2  -2v x  x’t’  +v x 2  t’ 2  R z   = evolution time R x   = emission duration R y 2 =  y’ 2  R y 2 = R G 2 / [1+  0 2 m t /T] R x, R y   0 = tr. flow velocity p t –spectra  T = temperature  t’ 2    (  -  ) 2   (  ) 2 BW: Retiere@LBL’05  x  = R G  x  0 /[  0 2 +T/m t ]   x(  )  <  x(K)  <  x(p) 

14. BW fit of Au-Au 200 GeV T=106 ± 1 MeV = 0.571 ± 0.004 c = 0.540 ± 0.004 c R InPlane = 11.1 ± 0.2 fm R OutOfPlane = 12.1 ± 0.2 fm Life time (  ) = 8.4 ± 0.2 fm/c Emission duration = 1.9 ± 0.2 fm/c  2 /dof = 120 / 86 Retiere@LBL’05 R β z ≈ z/τ β x ≈ β 0 (r/R)

15. Femto-puzzle I Contradiction with transport and simple hydro calcul. - - small space-time scales - - their weak energy dep. - - R out /R side ~ 1 Basically solved due to the initial flow increasing with energy (likely related to the increase of the initial energy density and partonic energy fraction)

16. 16 Femtoscopy Puzzle I basically solved due to initial flow appearing in realistic IC

17. Femtoscopic signature of QGP onset 3D 1-fluid Hydrodynamics Rischke & Gyulassy, NPA 608, 479 (1996) With 1 st order Phase transition Initial energy density  0 Long-standing signature of QGP onset: increase in , R OUT /R SIDE due to the Phase transition hoped-for “turn on” as QGP threshold in  0 is reached  decreases with decreasing Latent heat & increasing tr. Flow (high  0 or initial tr. Flow)

18. 18 Cassing – Bratkovskaya: Parton-Hadron-String-Dynamics Perspectives at FAIR/NICA energies

19. 19 Femto-puzzle II No signal of a bump in R out near the QGP threshold (expected at AGS- SPS energies) !? likely solved due to a dramatic decrease of partonic phase with decreasing energy

20. Femtoscopy of Pb+Pb at LHC ALICE arXiv:1012.4035 All radii increase with N ch from RHIC to LHC (not from SPS to RHIC)! Multiplicity scaling of the correlation volume  universal freeze-out density Freeze-out time  f from R long =  f (T/m t ) 1/2 The LHC fireball: - hotter - lives longer & - expands to a larger size

21. Ridge effect Dense matter (collective flows) also in pp collisions at LHC (for high N ch ) ? -  p t  increases with n ch and particle mass - BE CF vs n ch and p t points to expansion at high n ch - Ridge effect observed in angular correlations at high n ch R(k t )  at large N ch  expansion CMS

22. 22 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, LL,...

23. Assumptions to derive “Fermi-like” formula CF =  |  -k* (r*)| 2  t FSI (s-wave) = µf 0 /k*  |k * | = ½|q * |  hundreds MeV/c - same as for KP formula in case of pure QS & - equal time approximation in PRF  typical momentum transfer in production RL, Lyuboshitz’82  eq. time condition |t*|   r* 2 OK (usually, to several % even for pions) fig. RL, Lyuboshitz..’98 same isomultiplet only:  +     0  0,  - p   0 n, K + K   K 0 K 0,... & account for coupled channels within the - t FSI  d  dE >  t prod

24. 24 Effect of nonequal times in pair cms RL, Lyuboshitz SJNP 35 (82) 770; RL nucl-th/0501065 Applicability condition of equal-time approximation: |t*|  m 1,2 r* 2 r 0 =2 fm  0 =2 fm/c r 0 =2 fm v=0.1  OK for heavy particles & small k*  OK within 5% even for pions if  0 ~r 0 or lower → |k*t*|  m 1,2 r*

25. Resonance contribution to CF vs r-k correl. 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 is required for π +  - system; however, BW  b ~ 0.2;  likely eq.-time approx. not valid for π +  - at k*~ 150 MeV/c  * (k=146 MeV/c), r 0 =5 fm  (k=126 MeV/c), r 0 =5 fm -----------

26. FSI effect on CF of neutral kaons STAR data on CF(K s K s ) Goal: no Coulomb. But R may go up by ~1 fm if neglected FSI in = 1.09  0.22 R = 4.66  0.46 fm 5.86  0.67 fm KK (~50% K s K s )  f 0 (980) & a 0 (980) RL-Lyuboshitz’82 couplings from Martin or Achasov t Achasov’01,03 Martin’77 no FSI Lyuboshitz-Podgoretsky’79: K s K s from KK also show BE enhancement R ↓ M t  Universal expansion !

27. 27 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  

28. 28 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 & m t scaling Goal: No Coulomb suppression as in pp CF & Wang-Pratt’99 Stronger sensitivity to R =0.5  0.2 R=4.5  0.7 fm Scattering lengths, fm: 2.31 1.78 Effective radii, fm: 3.04 3.22 singlet triplet AGSSPS STAR R=3.1  0.3  0.2 fm

29. 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

30. 30 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 But R( p  ) < R( p  ) ? Residual correlations  pp

31. Correlation study of particle interaction - -  scattering lengths f 0 from NA49 correlation data Fit using RL-Lyuboshitz (82) with fixed Pair Purity =0.16 from feed-down and PID Data prefer |f 0 | « f 0 (NN) ~ 20 fm  -  CF=1+ [  CF FSI +  S  S (-1) S exp(-r 0 2 Q 2 )]  0 = ¼(1-P 2 )  1 = ¼(3+P 2 ) P=Polariz.=0  CF FSI = 2  0 [½|f 0 (k)/r 0 | 2 (1-d 0 0 /(2r 0  )) +2Re(f 0 (k)/(r 0  ))F 1 (r 0 Q) - 2Im(f 0 (k)/r 0 )F 2 (r 0 Q)] f S (k)=(1/f 0 S +½d 0 S k 2 - ik) -1 k=Q/2 F 1 (z)=  0 z dx exp(x 2 -z 2 )/z F 2 (z)=[1-exp(-z 2 )]/z

32. CF=N{1+ [  CF FSI -½exp(-r 0 2 Q 2 )]} N r0r0 f00f00 d00d00 fm B

33. 33 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

34. 34 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 

35. 35 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

36. 36 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

37. 37 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

38. 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)

39. 39 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

40. 40 Summary Assumptions behind femtoscopy theory in HIC seem OK at k  0. At k > ~ 100 MeV/c, the usual smoothness and equal-time approximations may not be valid. 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 correlations LHC femtoscopic data vs n ch and m t point to a universal fireball freeze- out density settled at RHIC energies and indicate collective expansion in high-multiplicity pp-collisions Absence of femtoscopic signals of QGP 1-st order PT (onset and CEP) at √s NN < 10 GeV is likely due to a dramatic decrease of partonic phase with decreasing energy  dedicated high statistics, precise experiments (like planned at NICA and FAIR) are required Info on two-particle strong interaction:  &  & p  scattering lengths from HIC at SPS and RHIC. Good perspective at RHIC & LHC (a problem of residual correlations is to be solved).

41. AGS  SPS  RHIC:  radii STAR Au+Au at 200 AGeV0-5% central Pb+Pb or Au+Au Clear centrality & m t dependence Weak energy dependence R ↑ with centrality &  with m t only R long slightly ↑ with energy  R side  R/(1+m t  F2 /T) ½ R long   (T/m t ) ½  tr. collective flow velocity  F  Evolution (freeze-out) time 

42. 42 hadronization initial state pre-equilibrium QGP and hydrodynamic expansion hadronic phase and freeze-out Expected evolution of HI collision vs RHIC data dN/dt 1 fm/c5 fm/c10 fm/c50 fm/c time Kinetic freeze out Chemical freeze out RHIC side & out radii:   2 fm/c R long & radii vs reaction plane:   10 fm/c Bass’02

43. 43 Even stronger effect of KK-bar FSI on K s K s correlations in pp-collisions at LHC ALICE: PLB 717 (2012) 151 e.g. for k t < 0.85 GeV/c, N ch =1-11 the neglect of FSI increases by ~100% and R inv by ~40% = 0.64  0.07  1.36  0.15 > 1 ! R inv = 0.96  0.04  1.35  0.07 fm

44. 44 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]

45. 45 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

46. 46 Coalescence: deuterons.. E d d 3 N/d 3 p d = B 2 E p d 3 N/d 3 p p E n d 3 N/d 3 p n p p  p n  ½p d WF in continuous pn spectrum  -k* (r*)  WF in discrete pn spectrum  b (r*) Coalescence factor: B 2 = (2  ) 3 (m p m n /m d ) -1  t  |  b (r*)| 2  ~ R -3 Triplet fraction = ¾  unpolarized Ns Usually: n  p Much stronger energy dependence of B 2 ~ R -3 than expected from pion and proton interferometry radii B2B2 R(pp) ~ 4 fm from AGS to SPS Lyuboshitz (88)..