Correlation femtoscopy R Lednick JINR Dubna IP ASCR
Correlation femtoscopy R. Lednický, JINR Dubna & IP ASCR Prague • • • History QS correlations & Multiboson effects FSI correlations Correlation asymmetries Spin correlations Summary 4 -6. 02. 2006 R. Lednický dwstp'06 1
Reviews, books M. I. Podgoretsky, Sov. J. Part. Nucl. 20 (1989) 266; ЭЧАЯ 20 (1989) 628 D. H. Boal et al. , Rev. Mod. Phys. 62 (1990) 553 U. A. Wiedemann, U. Heinz, Phys. Rep. 319 (1999) 145 T. Csorgo, Heavy Ion Phys. 15 (2002) 1 R. Lednicky, Phys. Atom. Nucl. 67 (2004) 72 M. Lisa et al. , Ann. Rev. Nucl. Part. Sci. 55 (2005) 357 R. M. Weiner, B-E correlations and subatomic interference, John Wiley & sons, LTD 4 -6. 02. 2006 R. Lednický dwstp'06 2
History Correlation femtoscopy : measurement of space-time characteristics R, c�~ fm of particle production using particle correlations GGLP’ 60: observed enhanced + + , vs + at small opening angles – interpreted as BE enhancement KP’ 71 -75: settled basics of correlation femtoscopy in > 20 papers • proposed CF= Ncorr /Nuncorr & mixing techniques to construct Nuncorr • 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) 3
QS symmetrization of production amplitude momentum correlations in particle physics KP’ 75: different from Astronomy where the momentum correlations are absent due to “infinite” star lifetimes x 1 p 1 CF=1+(-1)S cos q x 2 2 R 0 �� , nns , �� s 1/R 0 p 2 1 nnt , �� t q = p 1 - p 2 , �x = x 1 - x 2 0 4 -6. 02. 2006 total pair spin R. Lednický dwstp'06 |q| 4
Intensity interferometry of classical electromagnetic fields in Astronomy HBT’ 56 product of single-detector currents cf conceptual quanta measurement two-photon counts p 1 Correlation ~ � cos �p�x 34� x 1 x 3 x 2 star p 2 x 4 detectors-antennas tuned to mean frequency ��� �|�p|�-1 no explicit dependence on star space-time size |�x 34| Space-time correlation measurement in Astronomy � source momentum picture �|�p|�=��|��|��star angular radius �|� �no info on star lifetime KP’ 75 orthogonal to & longitudinal size Sov. Phys. JETP 42 (75) 211 momentum correlation measurement in particle physics � source space-time picture �|�x|� 5
momentum correlation (GGLP, KP) measurements are impossible in Astronomy due to extremely large stellar space-time dimensions while space-time correlation (HBT) measurements can be realized also in Laboratory: Intensity-correlation spectroscopy Goldberger, Lewis, Watson’ 63 -66 Measuring phase of x-ray scattering amplitude Fetter’ 65 & spectral line shape and width Glauber’ 65 Phillips, Kleiman, Davis’ 67: linewidth measurement from a mercurury discharge lamp 900 MHz 4 -6. 02. 2006 R. Lednický dwstp'06 �t nsec 6
Michelson cf HBT interferometers �IA+B� ~ �|�iexp[i( i+kix. A)] + �jexp[i(�j+kjx. B)] = 2 N+ 2 Re�iexp[ki(x. A-x. B)] filter �IA+B� = �IA+IB�[1+Re�(x. A-x. B)] A B Fourier transform �exp[ik(x. A-x. B)]�(k)d 4 k Field intensity in antenna A: IA ~ | �iexp[i( i+kix. A)] |2 filter ~ N+ 2 �i<j cos[(�i- �j)+(ki-kj)x. A] Product of intensities averaged over �’s: �IA IB�=�IA��IB�[1+(2/N 2)�i<jcos(�kij�x. AB)] =�IA��IB�[1+|�(x. A-x. B)|2] Actually measured product of electric currents after filters (0<|�i-�j|<�� ) integrated in a time F���� T ST=∫dt �JA JB�~ (Ne 2/��) T |�(0, x. A-x. B)|2 1/2 ~ (2�� Ne (T/�� Requirednormalized ST /r. m. s. (Sto. T)r. m. s. (S > 1 �TT) > /Ne)F 2)/��F
HBT paraboloid mirrors focusing the light from a star on photomultipliers 4 -6. 02. 2006 R. Lednický dwstp'06 8
R. Hanbury Brown and R. Q. Twiss: (1954) A new type of interferometer for use in radio astronomy (1956) Correlation between photons in two coherent beams of light (1956) A test of a new type of stellar interferometer on Sirius (1956) The question of correlations between photons in coherent light rays + E. M. Purcell (1956) J. Perina (1984): Quantum statistics of linear and nonlinear phenomena L. Mandel and E. Wolf (1995): Optical coherence and quantum optics 4 -6. 02. 2006 R. Lednický dwstp'06 9
HBT measurement of the angular size of Sirius Ne ~ 108 e/sec, �f ~ 1013 Hz, �f. F ~ 5 -45 MHz Required T ~ (2��/Ne)2/��F ~ hours �ST �/�ST 2 -�ST � 2 �½ Normalized to 1 at d=0 4 -6. 02. 2006 R. Lednický dwstp'06 10
Coincidence measurements A. Adam, L. Janossy, P. Varga (1955) Required time 1011 years E. Brannen, H. I. S. Ferguson (1956) F. T. Arechi, E. Gatti, A. Sona (1966) B. L. Morgan, L. Mandel (1966) 103 years hours compare to HBT technique HBT (1956) minutes-hours E. M. Purcell: Brown and Twiss did not count individual photoelectrons and coincidences, and were able to work with a primary photoelectric current some 104 times greater than that of Brannen and Ferguson. … This only adds lustre to the achievement of Brown and Twiss. 4 -6. 02. 2006 R. Lednický dwstp'06 11
Formal analogy of photon correlations in astronomy and particle physics Grishin, Kopylov, Podgoretsky’ 71: for conceptual case of 2 monochromatic sources and 2 detectors correlation takes the same form both in astronomy and particle physics: Correlation ~ cos(�R�d�/L) R�and d� are distance vectors between sources and detectors projected in the plane perpendicular to the emission direction L >> R�, d� is distance between the emitters and detectors “…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. ” 12
The analogy triggered misunderstandings: Shuryak’ 73: “The interest to correlations of identical quanta is due to the fact that their magnitude is connected with the space and time structure of the source of quanta. This idea originates from radio astronomy and is the basis of Hanbury Brown and Twiss method of the measurement of star radii. ” Cocconi’ 74: “The method proposed is equivalent to that used … … by radio astronomers to study angular dimensions of radio sources” “While. . interference builds up mostly. . near the detectors. . in our case the opposite happens” Grassberger’ 77 (ISMD): “For a stationary source (such as a star) the condition for interference is the standard one |q d| �� 1” ! Correlation(q) = cos(q d) ! Same mistake: many others. . 13
GGLP effect often called HBT, though: • HBT did not count quanta – they measured the product of currents ( field intensities) from two antennas – intensity interferometry useless technique for correlation femtoscopy • Being of classical origin (Superposition Principle), HBT effect would survive when h 0 and quantum interference vanished • Even if quanta measurement were done in Astronomy, it would be orthogonal to that of GGLP 4 -6. 02. 2006 R. Lednický dwstp'06 14
GGLP’ 60 data plotted as CF p p � 2�+ 2�- n� 0 R 0~1 fm 4 -6. 02. 2006 R. Lednický dwstp'06 15
3 -dim fit: CF=1+�exp(-Rx 2 qx 2 –Ry 2 qy 2 -Rz 2 qz 2 -2 Rxz 2 qx qz) Correlation strength or chaoticity Interferometry or correlation radii Examples of present data: NA 49 & STAR NA 49 z 4 -6. 02. 2006 STAR �� KK x Coulomb corrected y R. Lednický dwstp'06 16
“General” parameterization at |q| 0 Particles on mass shell & azimuthal symmetry � 5 variables: q = {qx , qy , qz} �{qout , qside , qlong}, pair velocity v = {vx, 0, vz} q 0 = qp/p 0 �qv = qxvx+ qzvz y �side Grassberger’ 77 RL’ 78 x �out ��transverse pair velocity vt z �long ��beam �cos q�x�=1 -½�(q�x)2�+. . �exp(-Rx 2 qx 2 -Ry 2 qy 2 -Rz 2 qz 2 2 q q ) 2 R xz x z Interferometry or correlation radii: Rx 2 =½ �(�x-vx�t)2 �, Ry 2 =½ �(�y)2 �, Rz 2 =½ �(�zvz�t)2 � Podgoretsky’ 83; often called cartesian or BP’ 95 parameterization
Formalism of independent one-particle sources �x|�A�= (2�)-4 �d 4�u. A(�) exp[-i �(x-x. A)] ��|x�= exp(i �x) �|�A�= �d 4 x ��|x��x|�A�= u. A(�) exp(i �x. A) Momentum (femtoscopic) correlations: Ampl(p) = �p|�A�= u. A(p) exp(i px. A) Ampl(p 1, p 2) = 2 -1/ 2 [u. A(p 1)u. A(p 2) exp(i p 1 x. A+i p 2 x. B) + 1 �� 2] Corr(p 1, p 2) = 2 Re{exp(i q�x) u. A(p 1)u. B(p 2)u. A*(p 2)u. B*(p 1) x 2 2 -1 x [|u. A(p 1)u. B(p 2)| +|u. A(p 2)u. B(p 1)| ] } � cos[(p 1 -p 2)(x. A-x. B)] Space-time (spectroscopic) correlations: Ampl(x) = �x|�A� ~ exp[i p. A(x. A-x)] for ~ monochrom. Ampl(x , x ) ~ exp{i p (x -xsource )+i p (x -x )] + 3 �� 4} 3 4 A A 3 B B 4 Corr(x 3, x 4) ~ cos[(p. A-p. B )(x 3 -x 4)] ! No explicit dependence on x. A, x. B
Femtoscopy through Emission function G(p, x) One particle: E d 3 N/d 3 p = ��|T�(p)|2 = �d 4 x’ exp[-i p(x-x’)] ����(x)��*(x’) = �d 4 x G(p, x) x, x’ � x=½(x+x’), �=x-x’ G(p, x) = partial Fourier transform of space-time density matrix ����(x)��*(x’) Two id. pions: E 1 E 2 d 6 N/d 3 p 1 d 3 p 2 = �d 4 x 1 d 4 x 2 [G(p 1, x 1; p 2, x 2)+ G(p, x 1; p, x 2)cos(q�x)] p = ½(p 1+p 2) q = p 1 -p 2 �x = x 1 -x 2 Corr(p 1, p 2) = �d 4 x 1 d 4 x 2 G(p, x 1; p, x 2) cos(q�x) / �d 4 x 1 d 4 x 2 G(p 1, x 1; p 2, x 2 ��cos(q�x)� � exp(- �i Ri 2 qi 2 - � 2 q 02) if G(p 1, x 1; p 2, x 2)= G(p 1, x 1)G(p 2, x 2) G(p, x) ~ exp(- �i xi 2/2 Ri 2 - x 02/2� 2) 4 -6. 02. 2006 R. Lednický dwstp'06 19
Assumptions to derive KP formula CF - 1 = �cos q�x� - two-particle approximation (small freeze-out PS density f) ~ OK, <f> �� 1 ? low pt fig. - smoothness approximation: Remitter ��Rsource ��|�p|����|q|�pe ~ OK in HIC, Rsource 2 �� 0. 1 fm 2 �pt 2 -slope of direct particles - neglect of FSI OK for photons, ~ OK for pions up to Coulomb repulsion - incoherent or independent emission 2� and 3� CF data consistent with KP formulae: CF 3(123) = 1+|F(12)|2+|F(23)|2+|F(31)|2+2 Re[F(12)F(23)F(31)] CF 2(12) = 1+|F(12)|2 , F(q) = �eiqx� 20
Phase space density from CFs and spectra Bertsch’ 94 Lisa. . ’ 05 <f> rises up to SPS May be high phase space density at low pt ? � ? Pion condensate or laser ? Multiboson effects on CFs spectra & multiplicities 21
Multiboson effects �Coherent emission: pion laser, DCC … �Correlation strength � < 1 due to coherence Fowler-Weiner’ 77 But: impurity, Long-Lived Sources (LLS), . . Deutschman’ 78 RL-Podgoretsky’ 79 LLS effect Heinz-Zhang’ 97 � 3� CF normalized to 2�CFs: get rid of But: problem with 3�Coulomb & extrapolation to Q 3=0 �Coherence modification of FSI effect on 2�CFs Akkelin. . ’ 00 But: requires precise measurement at low Q �Chaotic emission: Podgoretsky’ 85, Zajc’ 87, Pratt’ 93. . See RL et al. PRC 61 (00) 034901 & refs therein & Heinz. . AP 288 (01) 325 Increasing PSD: Widening of n� distribution: Narrowing of spectrum: Widening of CFs: rare gas � BE condensate Poisson � BE �/(2 r 0�) < � width = 1/r 0 width �� �= 1 �� 0 at fixed n
3 data on chaotic fraction Construct ratio r 3 in which LLS contributions to C 3 = CF 3 -1 and C 2 = CF 2 -1 cancel out Heinz-Zhang’ 97 r 3 =[C 3(123) – C 2(12) – C 2(23) – C 2(31) ]/[C 2(12) C 2(23) C 2(31) ]½ Interpolate to r 3(Q 3=0), Q 3 = (Q 122+ Q 232+ Q 312)½ Periph Mid-centr Centr Full chaoticity + ½r 3 STAR’ 03 ½r 3(0) =�½(3 -2�)/(2 -�)¾ � 2 Within large (systematic) errors STAR data is consistent with full chaoticity 4 -6. 02. 2006 R. Lednický dwstp'06 23
Multiboson effects on n & spectra Measure of PSD: �=�/(r 0�+½)3 � 1 Rare gas Width=� Condensate Width=�/(2 r 0�)½ �� BE ~ �n Poisson ~ �n/n!
Multiboson effects on CFs CFn(0) fixed n CF(q) inclusive CF(q) semi-inclusive n�nmax Intercept stays at 2 Width logarithmically increases with PSD n Intercept drops with n faster for softer pions �n�=33. 5 2 nmax 60 120 undershoot 4 -6. 02. 2006 R. Lednický dwstp'06 25
Probing source shape and emission duration If elliptic shape also in transverse plane �Ry�Rside oscillates with pair azimuth f Rside (f=90°) small B z A Rside 2 fm 2 Static Gaussian model with Rx 2 = R� 2 +v� 2�� 2 space and time dispersions �Ry 2 = R� 2 � Emission duration R� 2, R||2, �� 2 Rz 2 = R||2 +v||2�� 2 = (Rx 2 - Ry 2)/v� 2 In reaction plane Rside (f=0°) large Out-of reaction plane R. Lednický dwstp'06 alp fo-tu. O 4 -6. 02. 2006 f (degree) Circular 26 In-plane
Probing source dynamics - expansion Dispersion of emitter velocities & limited emission momenta (T) � x-p correlation: interference dominated by pions from nearby emitters Resonances GKP’ 71. �Interference probes only a part of the source Strings Bowler’ 85. . �Interferometry radii decrease with pair velocity Hydro Pratt’ 84, 86 Pt=160 Me. V/c Rout Pt=380 Me. V/c Rside Rout Rside Kolehmainen, Gyulassy’ 86 Makhlin-Sinyukov’ 87 Bertch, Gong, Tohyama’ 88 Hama, Padula’ 88 Pratt, Csörgö, Zimanyi’ 90 Mayer, Schnedermann, Heinz’ 92 …. . Collective transverse flow �F �Rside�R/(1+mt �F 2/T)½ 1 in LCMS } Longitudinal boost invariant expansion during proper freeze-out (evolution) time �Rlong�(T/mt)½�/coshy 4 -6. 02. 2006 27 � R. Lednický dwstp'06
Longitudinal boost-invariant expansion el. sources of lifetime �produced at t=z=0 uniformly distr. in rapidity �and decaying according to thermal law exp(-E*/T) t= �cosh(�) z= �sinh(�) E= mt cosh(y) pz= mt sinh(y) E*= mt cosh(y- �) In LCMS: pair rapidity y=0 so G ~ exp(-E*/T)= exp(-mt cosh�/T) �exp(-mt/T) exp[-� 2 / 2(T/mt)] � �� 2��(T/mt) 2= �(z-�z�)2���z’ 2� Ry 2= �y’ 2� Rx 2= �(x’-vxt’)2� Rz 2= �(�sinh(�))2�= �� 2��(sinh(�))2�� � � 2� (T/mt) 2 2 �x’ �-2 vx�x’t’�+vx �t’ ���(�-���)2��(��)2 Rz � ���= evolution time Rx � ��= emission duration if �x’t’�=0 & �x’ 2�= �y’ 2�
Transverse expansion Thermal law & gaussian tr. density profile exp(-r 2/ 2 r 02) & linear tr. flow velocity profile �F(r) = � 0 Fr / r 0 Nonrelativistic case: �t. T 2 = �F 2 + �t 2 - 2 �F�t cos� x = r cos� (out) y = r sin� (side) �t = tr. velocity �t. T = tr. thermal velocity G ~ exp(-�t. T 2 mt / 2 T) exp(-r 2/ 2 r 02) = exp[ - (� 0 F 2 r 2/r 02 + �t 2 - 2 � 0 F�t )x/r 0) mt / 2 T - r 2/ 2 r 02] � �y�= 0 �x�= r 0 �t � 0 F / [� 0 F 2+T/mt] Ry 2 = �y’ 2�= �x’ 2�= r 02 / [1+ � 0 F 2 mt /T] Note: for a box-like profile (r < R) � �x’ 2�< �y’ 2� 4 -6. 02. 2006 R. Lednický dwstp'06 29
AGS SPS RHIC: radii Clear centrality dependence Weak energy dependence STAR Au+Au at 200 AGe. V 0 -5% central Pb+Pb or Au+Au 30
AGS SPS RHIC: radii vs pt Rlong: increases smoothly & points to short evolution time �~ 8 -10 fm/c Rside , Rout : change little & point to strong transverse flow � 0 F ~ 0. 4 -0. 6 & short emission duration ��~ 2 fm/c Central Au+Au or Pb+Pb 31
Interferometry wrt reaction plane STAR’ 04 Au+Au 200 Ge. V 20 -30% + + & Typical hydro evolution Out-of-plane Circular In-plane Time STAR data: �oscillations like for a static out-of-plane source stronger then Hydro & RQMD � Short evolution time 32
Expected evolution of HI collision vs RHIC data initial state QGP and hydrodynamic expansion pre-equilibrium hadronic phase and freeze-out Bass’ 02 hadronization Kinetic freeze out Chemical freeze out d. N/dt RHIC side & out radii: Rlong & radii vs reaction plane: 1 fm/c 4 -6. 02. 2006 �� 10 fm/c 5 fm/c R. Lednický ��� 2 fm/c dwstp'06 10 fm/c 50 fm/c time 33
Puzzle ? Hydro assuming ideal fluid explains strong collective (�) flows at RHIC but not the interferometry results But comparing 1+1 D H+Ur. QMD with 2+1 D Hydro �kinetic evolution ~ conserves Rout, Rlong & increases Rside at small pt (resonances ? ) �Good prospect for 3 D Hydro + hadron transport + ? initial �F Bass, Dumitru, . . 1+1 D Hydro+Ur. QMD Huovinen, Kolb, . . 2+1 D Hydro Hirano, Nara, . . 3 D Hydro ? not enough �F 34
Why ~ conservation of spectra & radii? Sinyukov, Akkelin, Hama’ 02: Based on the fact that the known analytical solution of nonrelativistic BE with spherically symmetric initial conditions coincides with free streaming ti’= ti +T, xi’ = xi + vi T , vi v =(p 1+p 2)/(E 1+E 2) one may assume the kinetic evolution close to free streaming also in real conditions and thus ~ conserving initial spectra and Csizmadia, Csörgö, Lukács’ 98 initial interferometry radii qxi’ qxi +q(p 1+p 2)T/(E 1+E 2) = qxi ~ justify hydro motivated freezeout parametrizations
Checks with kinetic model Amelin, RL, Malinina, Pocheptsov, Sinyukov’ 05: System cools & expands but initial Boltzmann momentum distribution & interferomety radii are conserved due ���~ �~ tens fm � ��= �= 0 in static model to developed collective flow 36
Hydro motivated parametrizations Blast. Wave: Schnedermann, Sollfrank, Heinz’ 93 Retiere, Lisa’ 04 Kniege’ 05 37
BW fit of Au-Au 200 Ge. V Retiere@LBL’ 05 T=106 ± 1 Me. V <b. In. Plane> = 0. 571 ± 0. 004 c <b. Out. Of. Plane> = 0. 540 ± 0. 004 c RIn. Plane = 11. 1 ± 0. 2 fm ROut. Of. Plane = 12. 1 ± 0. 2 fm Life time (t) = 8. 4 ± 0. 2 fm/c Emission duration = 1. 9 ± 0. 2 fm/c c 2/dof = 120 / 86 38
Other parametrizations Buda-Lund: Csanad, Csörgö, Lörstad’ 04 Similar to BW but T(x) & �(x) hot core ~200 Me. V surrounded by cool ~100 Me. V shell Describes all data: spectra, radii, v 2(�) Krakow: Broniowski, Florkowski’ 01 Single freezeout model + Hubble-like flow + resonances Describes spectra, radii but Rlong ? may account for initial �F Kiev-Nantes: Borysova, Sinyukov, Erazmus, Karpenko’ 05 Generalizes BW using hydro motivated closed freezeout hypersurface Additional surface emission introduces x-t correlation �helps to desribe Rout at smaller flow velocity Fit points to initial � 0 F of ~ 0. 3 volume emission surface emission 39
Final State Interaction Similar to Coulomb distortion of �-decay Fermi’ 34: �|�-k(r)|2� CF nn pp fc�Ac�(G 0+i. F 0) } Migdal, Watson, Sakharov, … Koonin, GKW, . . . s-wave FSI strong FSI } } e-ikr � �-k(r) � [ e-ikr +f(k)eikr/r ] Coulomb �|1+f/r|2� kr+kr+ … _______ F=1+ ka ei�c�Ac Bohr radius Point-like k=|q|/2 Coulomb factor �FSI is sensitive to source size r and scattering amplitude f It complicates CF analysis but makes possible � Femtoscopy with nonidentical particles �K, �p, . . & Coulomb only Coalescence deuterons, . . �Study “exotic” scattering ��, �K, KK, ��, p�, ��, . . �Study relative space-time asymmetries delays, flow 4 -6. 02. 2006 R. Lednický dwstp'06 40
Assumptions to derive “Fermi” formula CF = �|�-k(r)|2 � - same as for KP formula in case of pure QS & - equal time approximation in PRF RL, Lyuboshitz’ 82 � eq. time condition |t*| ���r*2 OK fig. - t. FSI ��tprod � |k*| = ½|q*| �� hundreds Me. V/c » typical momentum RL, Lyuboshitz. . ’ 98 transfer in production & account for coupled channels within the same isomultiplet only: �+���� 0� 0, �-p �� 0 n, K+K��K 0 K 0, . .
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 =2 fm v=0. 1 �CFFSI(� 0� 0) 0 � OK for heavy particles �OK within 5% even for pions if ���� 0 ~r 0 or lower Note: �v��~ 0. 8 4 -6. 02. 2006 R. Lednický dwstp'06 42
FSI effect on CF of neutral kaons Lyuboshitz-Podgoretsky’ 79: Ks. Ks from KK also show BE enhancement STAR data on CF(Ks. Ks) Goal: no Coulomb. But R may go up by ~1 fm if neglected FSI in KK (~50% Ks. Ks)�f 0(980) & a 0(980) RL-Lyuboshitz’ 82 couplings from �Martin’ 77 �Achasov’ 01, 03 �no FSI l = 1. 09 0. 22 R = 4. 66 0. 46 fm 5. 86 0. 67 fm 4 -6. 02. 2006 R. Lednický dwstp'06 43 t
NA 49 central Pb+Pb 158 AGe. V 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] �RQMD overestimates r* by 10 -20% at SPS cf ~ OK at AGS worse at RHIC Scale=0. 76 Scale=0. 92 Scale=0. 83 ��p�� 4 -6. 02. 2006 R. Lednický dwstp'06 44
p CFs at AGS & SPS & STAR Goal: No Coulomb suppression as in pp CF & Wang-Pratt’ 99 Stronger sensitivity to R singlet triplet Scattering lengths, fm: 2. 31 1. 78 Effective radii, fm: 3. 04 3. 22 Fit using RL-Lyuboshitz’ 82 with �consistent with estimated impurity R~ 3 -4 fm consistent with the radius from pp CF AGS SPS �=0. 5� 0. 2 R=4. 5� 0. 7 fm 4 -6. 02. 2006 STAR R=3. 1� 0. 3� 0. 2 fm R. Lednický dwstp'06 45
Correlation study of particle interaction �+��& p�scattering lengths f 0 from NA 49 and STAR Fits using RL-Lyuboshitz’ 82 STAR CF(p�) data point to Ref 0(p�) < Ref 0(pp) � 0 Imf 0(p�) ~ Imf 0(pp) ~ 1 fm pp � NA 49 CF(�+��) vs RQMD with SI scale: f 00. 1 �sisca f 0 (=0. 232 fm ) sisca = 0. 6� compare ~0. 8 from S�PT & BNL data E 765 K �e��� NA 49 CF(�� ) data prefer |f 0(�� )| ��f 0(NN) ~ 20 fm 4 -6. 02. 2006 R. Lednický dwstp'06 - 46
Correlation study of particle interaction CF=Norm [Purity RQMD(r* �Scale�r*)+1 -Purity] �+�� scattering length f 0 from NA 49 CF �+�� Fit CF(�+��) by RQMD with SI scale: f 0 �sisca f 0 input -= 0. 232 fm sisca = 0. 6� 0. 1 Compare with ~0. 8 from S�PT & BNL E 765 K �e��� 47
interaction potential from LEP 2 2 Pure QS: CF = Norm (1� �e-R Q ) Feed-down & PID: �~ 0. 5 2) < 0. 3 } �� = ½� (1+P Polarization < 0. 3 PLB 475 (00) 395 �=0. 62� 0. 09 R=0. 11� 0. 02 fm String picture: lstring~ 2 m�t/�~2 fm ��~1 fm Rz��(T/m�t)½ ~ 0. 3 fm � R > Rz /� 3 ~ 0. 17 fm �=0. 54� 0. 10 R=0. 11� 0. 03 fm �QS fit yields too low R & too big � ���CF at LEP dominated by ! Direct FSI potential core RL (02) core signal NSC 97 e neglected Spin-orbit & Tensor parts � 4 -6. 02. 2006 R. Lednický �=0. 60� 0. 07 R=0. 10� 0. 02 fm �=0. 6 fixed R=0. 29� 0. 03 fm dwstp'06 R OK but potential tuning required 48
Correlation asymmetries CF of identical particles sensitive to terms even in k*r* (e. g. through �cos 2 k*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 4 -6. 02. 2006 R. Lednický dwstp'06 49
Simplified idea of CF asymmetry (valid for Coulomb FSI) Assume �emitted later than p or closer to the center v 1 v p v 2 k*/�= v 1 -v 2 v 1 � p 4 -6. 02. 2006 x CF� k*x > 0 v�> vp � v v 2 � Longer tint Stronger CF� x CF� k*x < 0 v�< vp R. Lednický � p � Shorter tint Weaker CF� p dwstp'06 50
CF-asymmetry for charged particles Asymmetry arises mainly from Coulomb FSI CF �Ac(�) �|F(�=(k*a)-1, �=k*r*+k*r* 2� i� , 1, i� )| r*��|a| F � 1+ ��= 1+r*/a+k*r*/(k*a) } k*� 1/r* k* � 0 Bohr radius ± 226 fm for �±p ± 388 fm for �+�± � CF+x/CF�x � 1+2 �� x*�/a �x* = x 1*-x 2*�rx* �Projection of the relative separation r* in pair cms on the direction x In LCMS (vz=0) or x || v: �x* = �t(�x - vt�t) �CF asymmetry is determined by space and time asymmetries 4 -6. 02. 2006 R. Lednický dwstp'06 51
Large lifetimes evaporation or phase transition x || v |�x| ��|�t| �CF-asymmetry yields time delay GANIL Pb+Nb �p+d+X Ghisalberti (95) Strangeness distillation: K� earlier than K� in baryon rich QGP Two-phase thermodynamic model Ardouin et al. (99) CF+(pd) CF�(pd) 1 2 3 1 CF+/CF�< 1 2 CF+/CF�< 1 3 Deuterons earlier than protons in agreement with coalescence e-tp/�e-tn/��e-td/(�/2) since tp�tn �td 4 -6. 02. 2006 R. Lednický dwstp'06 52
ad hoc time shift t = – 10 fm/c Sensitivity test for ALICE Erazmus et al. (95) CF+/CF� a, fm � � 249 k* � 0 CF+/CF�� 1+2 ��x*�/a Here ��x*�= - ��v�t� � � 226 CF-asymmetry scales as - ��t�/a � 84 4 -6. 02. 2006 R. Lednický dwstp'06 Delays of several fm/c can be easily detected 53
Usually: x and t comparable RQMD Pb+Pb ��p +X central 158 AGe. V : ��x�= -5. 2 fm ��t�= 2. 9 fm/c �+p-asymmetry effect 2��x*�/a �-8%�� � x*�= -8. 5 fm Shift ��x�in out direction is due to collective transverse flow & higher thermal velocity of lighter particles �xp�> �x. K�> �x��> 0 RL’ 99 -01 x �t. T out �F= flow velocity �t. T= transverse thermal velocity �t= �F + �t. T = observed transverse velocity side �t y �F � � x���rx�= �rt cos��= �rt (�t 2+�F 2 - �t. T 2)/(2�t�F) � mass dependence �y���ry�= �rt sin��= 0 rt �z���r ����sinh��=in 0 LCMS & Bjorken long. exp. z measures edge effect at y. CMS � 0 4 -6. 02. 2006 R. Lednický dwstp'06 54
pion px = 0. 15 Ge. V/c px = 0. 3 Ge. V/c Kaon px = 0. 53 Ge. V/c px = 1. 07 Ge. V/c Proton px = 1. 01 Ge. V/c px = 2. 02 Ge. V/c BW Retiere@LBL’ 05 Distribution of emission points at a given equal velocity: - Left, bx = 0. 73 c, by = 0 - Right, bx = 0. 91 c, by = 0 Dash lines: average emission Rx ��Rx(p)�< �Rx(K)�< �Rx(p)� For a Gaussian density profile with a radius RG and flow velocity profile �F (r) = � 0 r/ RG RL’ 04, Akkelin-Sinyukov’ 96 : �x�= RG x � 0 /[� 02+T/mt]
NA 49 & STAR out-asymmetries Pb+Pb central 158 AGe. V Au+Au central �s. NN=130 Ge. V not corrected for ~ 25% impurity r* RQMD scaled by 0. 8 corrected for impurity �p �p �K � Mirror symmetry (~ same mechanism for �and �mesons) � RQMD, BW ~ OK � points to strong transverse flow (��t�yields ~ ¼ of CF asymmetry)
Spin correlations , tt, . . � n 2 p � �� n 1 p Decay asymmetry parameters: � 1 = � 2 = �(��p��) = 0. 642 �� Joint angular distribution of �decay analyzers n 1 and n 2 is determined by: polarization vectors Pi= ��i� correlation tensor Tik = �� 1 k�� 2 k� 16�²W(n 1, n 2) = 1+ � 1 P 1 n 1+ � 2 P 2 n 2+ � 1� 2�ik. Tik n 1 in 2 k Distribution of correlation x = n 1�n 2 = cos� 12 is determined by Sp. T = �s. Sp. Tsinglet + �t. Sp. Ttriplet = -3 �s+ �t = 4 �t-3 �s and �t are singlet and triplet fractions, �s+ �t = 1: Alexander-Lipkin (95), RL (99) 4 -6. 02. 2006 W(x) = ½[1+ ½ 3� 1� 2 Sp. T x] R. Lednický dwstp'06 57
spin correlations at LEP �ALEPH distributions of correlation x=n 1�n 2= cos� 12 of directions of decay protons Slopes ~ Sp. T = 4 �t-3 � �t = t/� x = cos� 12 Q � 0 �New femtoscopy tool: �t = t/ � 0 triplet state forbidden at Q=0 Noninteracting unpolarized �s �Check two-particle QM coherence: violation of Bell-type inequality RL-Lyuboshitz (01) Sp. T� 1� �t�½ Bell-type inequality = s t t=¾(1�e-r 02 Q 2) s=¼(1�e-r 02 Q 2) r 0=0. 14� 0. 09 fm 58
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+1 D hydro & transport calculations which strongly overestimate out&long radii at RHIC. However, a good perspective seems to be for 3 D hydro ? + Finitial & 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 • Promising results from Spin correlations
Apologize for skipping • • • Coalescence (new d, d data from NA 49) 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 …… 4 -6. 02. 2006 R. Lednický dwstp'06 60
Kniege’ 05
Hydro wrt reaction plane � Heinz, Kolb, hep-ph/0111075 Though Hydro transforms out-of plane source into in-plane one, the expansion dynamics leads to qualitatively similar �dependence as for the static out-of plane source Quantitative differences: • Rs too small, Ro, l too big • oscill. amplit. too small 4 -6. 02. 2006 R. Lednický dwstp'06 62
Finite-size effects �r*�~ 10 fm but ��~30 -40 fm and �’ �~900 fm Ur. QMD: p. Ni � 2 at 24 Ge. V � ~1% �’, ~19% � ML ~ r*2/[1+(r*/r 0)2 a]2 b short-distance parametrization ML � � �’ and � contributions well fitted based on exponential decay law �’ 63
DIRAC CF: CF=N{��|�-k*(r*)|2�SLS +(1 - �)}[1+s Q] SLS determined by: N � r 0 fm a b f� f�’ s f�= 17� 6% �’, f�’ �, f� ML(r 0, a, b), f. ML=1 -f�-f�’ G 2 fm ML G 3 fm � �’ 64
Tails in RQMD: �r*�= 21 fm for r* < 50 fm 29 fm for r* < 500 fm �=0. 89 �r*�=16 fm �=0. 94 �r*�=24. 4 fm �=0. 93 �r*�=24 fm > �=0. 91 �r*�=22. 9 fm �+�+ 2 -G fit Strong FSI on �+��=0. 81 �r*�=18. 4 fm �Strong FSI important for �+�� 1 -G fit: �(�+�+) � 0. 8, �r*� 25% � � 2 -G fit: �+�+ ��+��r*�QS < �r*�Coul 4 -6. 02. 2006 R. Lednický dwstp'06 �=0. 76 �r*�=18. 1 fm 1 -G fit 65
Femtoscopy with nonidentical particles CF = �|�-k* (r*)|2� Be careful when comparing QS (�+�+. . ) and FSI correlations (�+��. . ) � different sensitivity to r*-distribution tails �QS & strong FSI: non-Gaussian r*-tail influences only first few bins in Q=2 k* and its effect is mainly absorbed in suppression parameter � �Coulomb FSI: sensitive to r*-tail up to r* ~ Bohr radius |a|=|z 1 z 2 e 2�|-1 �� �K �p KK pp fm 388 249 223 110 58 �In Gaussian fits one may expect r 0(�+�+) < r 0(�+��) �Use realistic models like transport codes 4 -6. 02. 2006 R. Lednický dwstp'06 66
Coalescence: deuterons. . WF in continuous pn spectrum �-k*(r*) � WF in discrete pn spectrum �b(r*) Edd 3 N/d 3 pd = B 2 Epd 3 N/d 3 pp End 3 N/d 3 pn pp�pn�½pd Coalescence factor: B 2 = (2�)3(mpmn/md)-1�t�|�b(r*)|2� ~ R-3 Triplet fraction = ¾ � unpolarized Ns Lyuboshitz (88). . B 2 Usually: n � p Much stronger energy dependence of B 2 ~ R-3 than expected from pion and proton interferometry radii R(pp) ~ 4 fm from AGS to SPS 4 -6. 02. 2006 R. Lednický dwstp'06 67
collective flow x 2 -p correlation Teff �with m R �with mt yes yes x-p correlation ��x�� 0 CF asymmetry yes yes 4 -6. 02. 2006 R. Lednický dwstp'06 chaotic source motion yes yes no no yes if ��t�� 0 68
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