R. Lednický wpcf'051 Femtoscopy in HIC. Theory R. Lednický, JINR Dubna & IP ASCR Prague History QS correlations FSI correlations Correlation asymmetries Summary

2 History GGLP’60: observed enhanced  +  +,     vs  +   measurement of space-time characteristics R, c  ~ fm KP’71-75: settled basics of correlation femtoscopy proposed CF= N corr /N uncorr & mixing techniques to construct N uncorr clarified role of space-time characteristics in various production models noted an analogy Grishin,KP’71 & differences KP’75 with HBT effect in Astronomy (see also Shuryak’73, Cocconi’74) Correlation femtoscopy : in > 20 papers at small opening angles – interpreted as BE enhancement of particle production using particle correlations

3 Formal analogy of photon correlations in astronomy and particle physics Grishin, Kopylov, Podgoretsky’71: for conceptual case of 2 monochromatic sources and 2 detectors R  and d  are distance vectors between sources and detectors projected in the plane perpendicular to the emission direction Correlation ~ cos(  R  d  /L) “…study of energy correlation allows one to get information about the source lifetime, and study of angular correlations – about its spatial structure. The latter circumstance is used to measure stellar sizes with the help of the Hanbury Brown & Twiss interferometer.” correlation takes the same form both in astronomy and particle physics: L >> R , d  is distance between the emitters and detectors

The analogy triggered misunderstandings: Shuryak’73: “The interest to correlations of identical quanta is due Cocconi’74: “The method proposed is equivalent to that used … “For a stationary source of the measurement of star radii.” ! Correlation(q) = cos(q d) Grassberger’77 (ISMD): case the opposite happens” “While.. interference builds up mostly.. near the detectors.. in our ! Same mistake: many others.. astronomy and is the basis of Hanbury Brown and Twiss method structure of the source of quanta. This idea originates from radio to the fact that their magnitude is connected with the space and time is the standard one |q d|   1” condition for interference (such as a star) the … by radio astronomers to study angular dimensions of radio sources”

5 QS symmetrization of production amplitude  momentum correlations in particle physics CF=1+(-1) S  cos q  x  p 1 p 2 x1x1 x 2 q = p 1 - p 2,  x = x 1 - x 2 nn t,  t , nn s,  s |q| 1/R 0 total pair spin 2R 0 KP’75: different from Astronomy where the momentum correlations are absent due to “infinite” star lifetimes

6 Intensity interferometry of classical electromagnetic fields in Astronomy HBT‘56  product of single-detector currents cf conceptual quanta measurement  two-photon counts p1p1 p2p2 x1x1 x2x2 x3x3 x4x4 star detectors-antennas tuned to mean frequency    Correlation ~  cos  p  x 34   |  p|  -1 |  x 34 | Space-time correlation measurement in Astronomy  source momentum picture  |  p|  =  |  |   star angular radius  |  |  orthogonal to momentum correlation measurement in particle physics  source space-time picture  |  x|  KP’75  no info on star lifetime Sov.Phys. JETP 42 (75) 211 & longitudinal size no explicit dependence on star space-time size

7  momentum correlation (GGLP,KP) measurements are impossible in Astronomy due to extremely large stellar space-time dimensions  space-time correlation (HBT) measurements can be realized also in Laboratory: while Phillips, Kleiman, Davis’67: linewidth measurement froma mercurury discharge lamp 900 MHz  t nsec Goldberger,Lewis,Watson’63-66Intensity-correlation spectroscopy Measuring phase of x-ray scattering amplitude & spectral line shape and width Fetter’65 Glauber’65

8 GGLP’60 data plotted as CF R 0 ~1 fm p p  2  + 2  - n  0

9 Examples of present data: NA49 & STAR 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

10 “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 ; often called cartesian or BP’95 parameterization Interferometry 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

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

12 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

13 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) …

14 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  t  R side  R/(1+m t  t 2 /T) ½ Longitudinal boost invariant expansion 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

15 AGS  SPS  RHIC:  radii STAR Au+Au at 200 AGeV0-5% central Pb+Pb or Au+Au Clear centrality dependenceWeak energy dependence

16 AGS  SPS  RHIC:  radii vs p t R long : increases smoothly & points to short evolution time  ~ 8-10 fm/c R side, R out : change little & point to strong transverse flow  t ~ & short emission duration  ~ 2 fm/c Central Au+Au or Pb+Pb

17 Interferometry wrt reaction plane STAR data:  oscillations like for a static out-of-plane source stronger then Hydro & RQMD  Short evolution time Out-of-planeCircularIn-plane Time Typical hydro evolution STAR’04 Au+Au 200 GeV 20-30%     &    

18 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

19 Puzzle ? 3D Hydro 2+1D Hydro 1+1D Hydro+UrQMD (resonances ?) But comparing 1+1D H+UrQMD with 2+1D Hydro  kinetic evolution at small p t & increases R side ~ conserves R out,R long  Good prospect for 3D Hydro Hydro assuming ideal fluid explains strong collective (  ) flows at RHIC but not the interferometry results + hadron transport Bass, Dumitru,.. Huovinen, Kolb,.. Hirano, Nara,.. ? not enough  t + ? initial  t

Why ~ conservation of spectra & radii? qx i ’  qx i +q(p 1 +p 2 )T/(E 1 +E 2 ) = qx i Sinyukov, Akkelin, Hama’02 : free streaming also in real conditions and thus initial interferometry radii t i ’= t i +T, x i ’ = x i + v i T, v i  v =(p 1 +p 2 )/(E 1 +E 2 ) Based on the fact that the known analytical solution of nonrelativistic BE with spherically symmetric initial conditions coincides with free streaming one may assume the kinetic evolution close to ~ conserving initial spectra and ~ justify hydro motivated freezeout parametrizations Csizmadia, Csörgö, Lukács’98

21 Checks with kinetic model Amelin, RL, Malinina, Pocheptsov, Sinyukov’05: System cools & expands but initial Boltzmann momentum distribution & interferomety radii are conserved due to developed collective flow   ~  ~ tens fm   =  = 0 in static model

22 Hydro motivated parametrizations Kniege’05 BlastWave: Schnedermann, Sollfrank, Heinz’93 Retiere, Lisa’04

23 BW fit of Au-Au 200 GeV T=106 ± 1 MeV = ± c = ± 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

24 Other parametrizations Buda-Lund: Csanad, Csörgö, Lörstad’04 Similar to BW but T(x) &  (x) hot core ~200 MeV surrounded by cool ~100 MeV shell Describes all data: spectra, radii, v 2 (  ) Krakow: Broniowski, Florkowski’01 Describes spectra, radii but R long Single freezeout model + Hubble-like flow + resonances Kiev-Nantes: Borysova, Sinyukov, Erazmus, Karpenko’05 closed freezeout hypersurface Generalizes BW using hydro motivated Additional surface emission introduces x-t correlation  helps to desribe R out at smaller flow velocity volume emission surface emission ? may account for initial  t Fit points to initial  t of ~ 0.3

25 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 CF =  |  -k (r)| 2  - t FSI  t prod  |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 fig. RL, Lyuboshitz..’98 same isomultiplet only:  +     0  0,  - p   0 n, K + K   K 0 K 0,... & account for coupled channels within the

27 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 Note:  v   ~ 0.8  CF FSI (  0  0 )

28 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 t  Achasov’01,03  Martin’77  no FSI Lyuboshitz-Podgoretsky’79: K s K s from KK also show BE enhancement

29 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  

30 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 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: Effective radii, fm: singlet triplet AGSSPS STAR R=3.1  0.3  0.2 fm

31 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

32 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

33 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 

34 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

35 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

36 pion Kaon Proton BW 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 flow velocity profile  F (r) =  0 r/ R G RL’04, Akkelin-Sinyukov’96 :  x  = R G  x  0 /[  0 2 +T/m t ]

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)

38 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 correlations Weak energy dependence of correlation radii contradicts to 2+1D hydro & transport calculations which strongly overestimate out&long radii at RHIC. However, a good perspective seems to be for 3D hydro ?+  F initial & transport A number of succesful hydro motivated parametrizations give useful hints for microscopic models (but fit   true  ) Info on two-particle strong interaction:  &  & p  scattering lengths from HIC at SPS and RHIC. Good perspective at RHIC and LHC

39 Apologize for skipping Coalescence (new d, d data from NA49 ) Beyond Gaussian form RL, Podgoretsky,..Csörgö.. Chung.. Imaging technique Brown, Danielewicz,.. Multiple FSI effects Wong, Zhang,..; Kapusta, Li; Cramer,.. Spin correlations Alexander, Lipkin; RL, Lyuboshitz ……

40 Kniege’05

41 collective flow chaotic source motion x-p correlation yes no x 2 -p correlation yes yes T eff  with m yes yes R  with m t yes yes  x   0 yes no CF asymmetry yes yes if  t   0