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Exploring Correlation Femtoscopy in Physics: Space-Time Measurements and Intensity Interferometry

This article delves into the history and principles of correlation femtoscopy, focusing on space-time characteristics measurement using particle correlations. Key topics include correlation asymmetries, spin correlations, and multiboson effects. The text compares correlation femtoscopy in particle physics with measurements in astronomy, highlighting the difference in momentum correlations. It discusses the implications of intensity interferometry, along with relevant techniques and models. The HBT effect and its applications in measuring angular sizes are also covered.

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Exploring Correlation Femtoscopy in Physics: Space-Time Measurements and Intensity Interferometry

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  1. Correlation femtoscopy R. Lednický, JINR Dubna & IP ASCR Prague • History • QS correlations & Multiboson effects • FSI correlations • Correlation asymmetries • Spin correlations • Summary R. Lednický dwstp'06

  2. 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 R. Lednický dwstp'06

  3. 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 analogyGrishin,KP’71 & differencesKP’75 with HBT effect in Astronomy (see also Shuryak’73, Cocconi’74)

  4. QS symmetrization of production amplitudemomentum correlations in particle physics KP’75: different from Astronomy where the momentum correlations are absent total pair spin due to “infinite”star lifetimes CF=1+(-1)Scos qx p1 2 x1 ,nns,s x2 1/R0 1 p2 2R0 nnt,t q =p1- p2 , x = x1- x2 |q| 0 R. Lednický dwstp'06

  5. Intensity interferometry of classical electromagnetic fields in AstronomyHBT’56  product of single-detector currentscf conceptual quanta measurement  two-photon counts p1 Correlation ~  cos px34 x1 x3 no explicit dependence |p|-1 p2 x4 on star space-time size x2 detectors-antennas tuned to mean frequency  star |x34| Space-time correlation measurement in Astronomy  source momentum picture |p|=||  star angular radius ||  no info on star lifetime KP’75 orthogonalto & longitudinal size Sov.Phys. JETP 42 (75) 211 momentum correlation measurement in particle physics  source space-time picture |x|

  6. 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 R. Lednický dwstp'06 t nsec

  7. IA+B ~ |iexp[i(i+kixA)] + jexp[i(j+kjxB)]|2 = 2N+ 2Reiexp[ki(xA-xB)] MichelsoncfHBT interferometers filter IA+B = IA+IB[1+Re(xA-xB)] Fourier transform exp[ik(xA-xB)](k)d4k Field intensity in antenna A: IA ~ | iexp[i(i+kixA)] |2 ~ N+ 2 i<jcos[(i- j)+(ki-kj)xA] filter Product of intensities averaged over ’s: IA IB=IAIB[1+(2/N2)i<jcos(kijxAB)] =IAIB [1+|(xA-xB)|2] Actually measured product of electric currents afterfilters (0<|i-j|<F) integrated in a time T ST=∫dt JA JB~ (Ne2/) T |(0,xA-xB)|2 normalized to r.m.s.(ST) ~ Ne (T/F)1/2 A B Required ST /r.m.s.(ST) > 1  T > (2 /Ne)2/F

  8. HBT paraboloid mirrors focusing the light from a star on photomultipliers R. Lednický dwstp'06

  9. 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 R. Lednický dwstp'06

  10. HBT measurement of the angular size of Sirius Ne ~ 108e/sec, f ~ 1013 Hz, fF ~ 5-45MHz Required T ~ (2 /Ne)2/F ~ hours  ST / ST2- ST 2 ½ Normalized to 1 at d=0 R. Lednický dwstp'06

  11. Coincidence measurements Required time A. Adam, L. Janossy, P. Varga (1955) 1011 years E. Brannen, H.I.S. Ferguson (1956) 103 years F.T. Arechi, E. Gatti, A. Sona (1966) hours B.L. Morgan, L. Mandel (1966) 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. R. Lednický dwstp'06

  12. Formal analogy of photon correlationsin 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andd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.”

  13. 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 andtime 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 ..

  14. 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 R. Lednický dwstp'06

  15. GGLP’60 data plotted as CF p p  2+ 2 - n0 R0~1 fm R. Lednický dwstp'06

  16. 3-dim fit: CF=1+exp(-Rx2qx2 –Ry2qy2-Rz2qz2-2Rxz2qxqz) Examples of present data: NA49 & STAR Correlation strength or chaoticity Interferometry or correlation radii STAR  KK NA49 Coulomb corrected z x y R. Lednický dwstp'06

  17. “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} q0 = qp/p0 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(-Rx2qx2 -Ry2qy2-Rz2qz2-2Rxz2qx qz) Interferometry or correlation radii: Rx2 =½  (x-vxt)2 , Ry2 =½  (y)2 , Rz2 =½  (z-vzt)2  Podgoretsky’83;often called cartesian or BP’95 parameterization

  18. Formalism of independent one-particle sources x|A = (2)-4 d4 uA() exp[-i (x-xA)] |x = exp(i x) |A =  d4x |xx|A = uA() exp(i xA) Momentum (femtoscopic) correlations: Ampl(p) = p|A = uA(p) exp(i pxA) Ampl(p1,p2) = 2-1/ 2 [uA(p1)uA(p2) exp(i p1xA+i p2xB) + 1  2] Corr(p1,p2) = 2Re{exp(i qx) uA(p1)uB(p2)uA*(p2)uB*(p1) x x [|uA(p1)uB(p2)| 2 +|uA(p2)uB(p1)|2]-1 }  cos[(p1-p2)(xA-xB)] Space-time (spectroscopic) correlations: Ampl(x) = x|A ~ exp[i pA(xA-x)] for ~ monochrom. source Ampl(x3,x4) ~ exp{i pA(xA-x3)+i pB (xB-x4)] + 3  4} Corr(x3,x4) ~ cos[(pA-pB )(x3-x4)]! No explicit dependence on xA, xB

  19. Femtoscopy through Emission function G(p,x) One particle: E d3N/d3p = |T(p)|2 =  d4x d4x’ exp[-i p(x-x’)] (x)*(x’) =  d4x 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: E1E2d6N/d3p1d3p2 =  d4x1d4x2 [G(p1,x1;p2,x2)+ G(p,x1;p,x2)cos(qx)] p = ½(p1+p2) q = p1-p2 x = x1-x2 Corr(p1,p2) =  d4x1d4x2G(p,x1;p,x2) cos(qx) /  d4x1d4x2G(p1,x1;p2,x2) cos(qx)  exp(- i Ri2qi2 - 2q02) if G(p1,x1;p2,x2)= G(p1,x1)G(p2,x2) G(p,x) ~ exp(- i xi2/2Ri2- x02/22) R. Lednický dwstp'06

  20. Assumptions to derive KP formula CF - 1 = cos qx - two-particle approximation (small freeze-out PS density f) ~ OK, <f>  1 ? lowptfig. - smoothness approximation: Remitter  Rsource |p|  |q|peak ~ OK in HIC, Rsource20.1 fm2  pt2-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: CF3(123) = 1+|F(12)|2+|F(23)|2+|F(31)|2+2Re[F(12)F(23)F(31)] CF2(12) = 1+|F(12)|2 , F(q) = eiqx

  21. 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

  22. Coherent emission:pion laser, DCC … Multiboson effects  Correlation strength  < 1 due to coherenceFowler-Weiner’77 But: impurity, Long-Lived Sources (LLS), .. Deutschman’78 RL-Podgoretsky’79  3 CF normalized to 2 CFs: get rid of LLS effect Heinz-Zhang’97 But: problem with 3 Coulomb & extrapolation to Q3=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 rare gas BEcondensate IncreasingPSD: Widening of n distribution: Poisson BE Narrowing of spectrum:  /(2r0) <  Widening of CFs: width = 1/r0  = 1 width     0 at fixed n

  23. 3 data on chaotic fraction  Construct ratio r3 in which LLS contributions to C3 = CF3-1 and C2= CF2-1 cancel out Heinz-Zhang’97 r3=[C3(123) – C2(12) – C2(23) – C2(31) ]/[C2(12) C2(23) C2(31) ]½ Interpolate to r3(Q3=0), Q3 = (Q122+ Q232+ Q312)½ ½r3 Periph Mid-centr Centr STAR’03 Full chaoticity p+ p-  ½r3(0) =½(3-2)/(2-)¾ 2 Within large (systematic) errors STAR data is consistent with full chaoticity R. Lednický dwstp'06

  24. Multiboson effects on n & spectra Measure of PSD:=/(r0+½)3  1 Rare gas Width= Condensate Width=/(2r0)½ BE ~ n Poisson ~ n/n!

  25. Multiboson effects on CFs CFn(0) fixed n CF(q) semi-inclusive nnmax CF(q) inclusive Intercept stays at 2 Width logarithmically increases with PSD n Intercept drops with n faster for softer pions n =33.5 120 60 2 nmax undershoot R. Lednický dwstp'06

  26. f (degree) KP (71-75) … Probing source shape and emission duration Static Gaussian model with space and time dispersions R2, R||2, 2 Rx2 = R2 +v22  Ry2 = R2 Rz2 = R||2 +v||22 Emission duration 2 = (Rx2- Ry2)/v2 If elliptic shape also in transverse plane  RyRsideoscillates with pair azimuth f Rside2 fm2 Out-of plane Circular In-plane Rside(f=90°) small Out-of reaction plane A Rside (f=0°) large In reaction plane z B R. Lednický dwstp'06

  27. 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 Kolehmainen, Gyulassy’86 Makhlin-Sinyukov’87 Pt=160MeV/c Pt=380 MeV/c Bertch, Gong, Tohyama’88 Hama, Padula’88 Pratt, Csörgö, Zimanyi’90 Rout Rside Mayer, Schnedermann, Heinz’92 Rout Rside ….. Collective transverse flow F RsideR/(1+mt F2/T)½ Longitudinal boost invariant expansion during proper freeze-out (evolution) time  1 in LCMS }  Rlong(T/mt)½/coshy R. Lednický dwstp'06

  28. 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) Rz2= (z-z)2  z’2 Ry2= y’2Rx2= (x’-vxt’)2 Rz2= (sinh())2 = 2(sinh())2  2 (T/mt) Rx2= x’2-2vxx’t’+vx2t’2t’2  (-)2  ()2 Rz  = evolution time Rx  = emission duration if x’t’=0 & x’2= y’2

  29. Transverse expansion Thermal law & gaussian tr. density profile exp(-r2/ 2r02) & linear tr. flow velocity profile F(r) = 0Fr / r0 Nonrelativistic case: tT2 = F2 + t2 - 2 Ft cos x = rcos (out)y = r sin (side) t= tr. velocity tT= tr. thermal velocity G ~ exp(-tT2 mt / 2T) exp(-r2/ 2r02) = exp[ - (0F2 r2/r02 + t2 - 2 0Ft )x/r0) mt / 2T - r2/ 2r02]  y = 0 x = r0 t0F / [0F2+T/mt] Ry2 = y’2 = x’2 = r02 / [1+ 0F2 mt /T] Note: for a box-like profile (r < R)  x’2 < y’2 R. Lednický dwstp'06

  30. AGSSPSRHIC: radii Clear centrality dependence Weak energy dependence STAR Au+Au at 200 AGeV 0-5% central Pb+Pb or Au+Au

  31. AGSSPSRHIC: radii vs pt Central Au+Au or Pb+Pb Rlong:increases smoothly & points to short evolution time  ~ 8-10 fm/c Rside,Rout: change little & point to strong transverse flow 0F~ 0.4-0.6 & short emission duration  ~ 2 fm/c

  32. Interferometry wrt reaction plane STAR’04 Au+Au 200 GeV 20-30% Typical hydro evolution p+p+ & p-p- 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

  33. Expected evolution of HI collision vs RHIC data Bass’02 QGP and hydrodynamic expansion hadronic phase and freeze-out initial state pre-equilibrium hadronization Kinetic freeze out dN/dt Chemical freeze out RHIC side & out radii: 2 fm/c Rlong & radii vs reaction plane: 10 fm/c 1 fm/c 5 fm/c 10 fm/c 50 fm/c time R. Lednický dwstp'06

  34. Hydro assuming ideal fluid explains strong collective () flows at RHIC but not the interferometryresults Puzzle ? But comparing Bass, Dumitru, .. 1+1D Hydro+UrQMD 1+1D H+UrQMD Huovinen, Kolb, .. 2+1D Hydro with 2+1D Hydro Hirano, Nara, .. 3D Hydro  kinetic evolution ? not enough F ~ conserves Rout,Rlong & increases Rside at small pt (resonances ?)  Good prospect for 3D Hydro + hadron transport + ? initial F

  35. Why ~ conservation of spectra & radii? Sinyukov, Akkelin, Hama’02: Based on the fact that the known analytical solution of nonrelativistic BE with sphericallysymmetric initial conditions coincides with free streaming ti’= ti +T, xi’= xi+ vi T , vi v =(p1+p2)/(E1+E2) one may assume the kinetic evolution close to free streaming alsoin realconditionsand thus ~ conserving initial spectra and Csizmadia, Csörgö, Lukács’98 initial interferometry radii qxi’ qxi +q(p1+p2)T/(E1+E2) = qxi ~ justify hydro motivated freezeout parametrizations

  36. 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

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

  38. BW fit of Au-Au 200 GeV Retiere@LBL’05 T=106 ± 1 MeV <bInPlane> = 0.571 ± 0.004 c <bOutOfPlane> = 0.540 ± 0.004 c RInPlane = 11.1 ± 0.2 fm ROutOfPlane = 12.1 ± 0.2 fm Life time (t) = 8.4 ± 0.2 fm/c Emission duration = 1.9 ± 0.2 fm/c c2/dof = 120 / 86

  39. Buda-Lund: Csanad, Csörgö, Lörstad’04 Other parametrizations Similar to BW but T(x) & (x) hot core ~200 MeV surrounded by cool ~100 MeV shell Describes all data: spectra, radii, v2() 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, volume emission Erazmus, Karpenko’05 Generalizes BW using hydro motivated closed freezeout hypersurface Additional surface emission introduces x-t correlation  helps to desribe Rout surface emission at smaller flow velocity Fit points to initial 0F of ~ 0.3

  40. |-k(r)|2 Similar to Coulomb distortion of -decay Fermi’34: Final State Interaction Migdal, Watson, Sakharov, … Koonin, GKW, ... fcAc(G0+iF0) s-wave strong FSI FSI } nn e-ikr  -k(r)  [ e-ikr +f(k)eikr/r ] CF pp Coulomb |1+f/r|2 kr+kr F=1+ _______ + … eicAc ka } } Bohr radius Coulomb only Point-like Coulomb factor k=|q|/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, .. & Coalescence deuterons, .. Study “exotic” scattering,K, KK,, p,, .. Study relative space-time asymmetriesdelays, flow R. Lednický dwstp'06

  41. 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 OKfig. - tFSI tprod  |k*|= ½|q*| hundreds MeV/c • typical momentum RL, Lyuboshitz ..’98 transfer in production & account for coupled channels within the same isomultiplet only: + 00, -p  0n, K+K K0K0, ...

  42. Effect of nonequal times in pair cms RL, Lyuboshitz SJNP 35 (82) 770; RLnucl-th/0501065 Applicability condition of equal-time approximation: |t*| r*2 r0=2 fm 0=2 fm/c r0=2 fm v=0.1 CFFSI(00)  OK for heavy particles OK within 5% even for pions if 0 ~r0 or lower Note: v ~ 0.8 R. Lednický dwstp'06

  43. Lyuboshitz-Podgoretsky’79: KsKs from KK also show FSI effect on CF of neutral kaons Goal: no Coulomb. But R may go up by ~1 fm if neglected FSI in BE enhancement KK (~50% KsKs) f0(980) & a0(980) STAR data on CF(KsKs) 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 R. Lednický dwstp'06 t

  44. Long tails in RQMD: r* = 21 fm for r* < 50 fm NA49 central Pb+Pb 158 AGeV vs RQMD 29 fm for r* < 500 fm Fit CF=Norm[Purity RQMD(r* Scaler*)+1-Purity] RQMD overestimatesr* by 10-20% at SPS cf ~ OK at AGS worse at RHIC Scale=0.76 Scale=0.92 Scale=0.83 p R. Lednický dwstp'06

  45. Goal: No Coulomb suppression as in pp CF & p CFs at AGS & SPS & STAR Wang-Pratt’99 Stronger sensitivity to R singlet triplet Scattering lengths, fm: 2.31 1.78 Fit using RL-Lyuboshitz’82 with Effective radii, fm: 3.04 3.22  consistent with estimated impurity R~ 3-4 fm consistent with the radius from pp CF STAR AGS SPS =0.50.2 R=4.50.7 fm R=3.10.30.2 fm R. Lednický dwstp'06

  46. Correlation study of particle interaction +&  & pscattering lengthsf0 from NA49 and STAR Fits using RL-Lyuboshitz’82 pp STAR CF(p) data point to Ref0(p) < Ref0(pp)  0 Imf0(p) ~ Imf0(pp) ~ 1 fm  NA49 CF(+) vs RQMD with SI scale:f0siscaf0 (=0.232fm) - sisca = 0.60.1 compare ~0.8 from SPT & BNL data E765 K  e NA49 CF() data prefer |f0()| f0(NN) ~ 20 fm R. Lednický dwstp'06

  47. Correlation study of particle interaction CF=Norm[Purity RQMD(r* Scaler*)+1-Purity] +scattering length f0 from NA49 CF + Fit CF(+) by RQMD with SI scale: f0siscaf0input f0input = 0.232 fm - sisca = 0.60.1 Compare with ~0.8 from SPT & BNL E765 K  e

  48. CF = Norm (1e-R2Q2) Pure QS: interaction potential from LEP PLB 475 (00) 395 Feed-down & PID:~ 0.5 } = ½(1+P2) < 0.3 Polarization < 0.3 =0.620.09 R=0.110.02 fm String picture: lstring~ 2m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 lowR & too big  CF at LEP dominated by ! Direct core signal =0.600.07 R=0.100.02 fm FSI potential core RL (02) NSC97e neglected Spin-orbit & Tensor parts  R OK but potential tuning required =0.6 fixed R=0.290.03 fm R. Lednický dwstp'06

  49. Correlation asymmetries CF of identical particles sensitive to terms evenin k*r* (e.g. through cos 2k*r*)  measures only dispersion of the components of relative separation r* = r1*- r2*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 negativek*-projectionk*x on a given directionx and study CF-ratio CF+x/CFx R. Lednický dwstp'06

  50. Simplified idea of CF asymmetry(valid for Coulomb FSI) Assume emitted later than p or closer to the center  x v Longer tint Stronger CF v1  CF  k*x > 0 v > vp p p v2 k*/= v1-v2 x Shorter tint Weaker CF v CF  v1 k*x < 0 v < vp p  v2 p R. Lednický dwstp'06

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