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8. 6. 2010Nantes‘101 Femtoscopic Correlations and Final State Resonance Formation R. Lednický, JINR Dubna & IP ASCR Prague History Assumptions Technicalities Narrow resonance FSI contributions to π + - K + K - CF’s Conclusions
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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
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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
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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 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...
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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 PRF
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6 Final State Interaction Similar to Coulomb distortion of -decay Fermi’34: e -ikr -k ( r ) [ e -ikr +f( k )e ikr / r ] eicAceicAc 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,...
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Assumptions to derive “Fermi” formula for CF CF = | -k* (r*)| 2 - t FSI d dE t prod - 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 correl R emitter R source ~ OK in HIC, R source 2 0.1 fm 2 p t 2 -slope of direct particles t FSI (s-wave) = µf 0 /k* k * = ½q * hundreds MeV/c t FSI (resonance in any L-wave) = 2/ hundreds MeV/c in the production process
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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
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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
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Smoothness approximation: r A « r 0 (q « p) 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 G (x 1,p 1 ;x 2,p 2 ) | p 1 p 2 ( x 1,x 2 ) | 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 ’ For non-interacting identical spin-0 particles – exact result (p=½(p 1 +p 2 ) ): W(p 1,p 2 ∫ d 4 x 1 d 4 x 2 [G(x 1,p 1 ;x 2,p 2 )+G(x 1,p;x 2,p) cos(q x)] approx. result: ≈ ∫d 4 x 1 d 4 x 2 G(x 1,p 1 ;x 2,p 2 ) [1+cos(q x)] = ∫ d 4 x 1 d 4 x 2 G(x 1,p 1 ;x 2,p 2 ) | p 1 p 2 (x 1,x 2 )| 2
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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 →
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Technicalities – 1: neglecting complex intermediate channels
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Technicalities – 2: spin & isospin equilibration
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Technicalities – 3: equal-time approximation
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Technicalities – 4: simple Gaussian emission functions
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Technicalities – 5: treating the spin & angular dependence
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In the following we ssume write (angular dependence enters only through the angle between the vectors k and r): Since then L’=L, S’=S=j,=1/2 or 0, one can put m=j and
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Technicalities – 6: contribution of the outer region
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Technicalities – 7: projecting pair spin & isospin =π+-=π+- =K + K -
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Technicalities – 8: resonance dominance in the JT-wave
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Technicalities – 9: contribution of the inner region
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Technicalities – 10: volume integral In the single flavor case For s & p-waves it recovers the result of Wigner’55 & Luders’55
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correlations in Au+Au (STAR) Coulomb and strong FSI present * 1530, k*=146 MeV/c, =9.1 MeV No energy dependence seen Centrality dependence observed, quite strong in the * region; 0-10% CF peak value CF-1 0.025 Gaussian fit of 0-10% CF’s: r 0 =6.7±1.0 fm, out = -5.6±1.0 fm
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correlations in Pb+Pb (NA49) Coulomb and strong FSI present 1020, k*=126 MeV/c, =4.3 MeV Centrality dependence observed, particularly strong in the region; 0-5% CF peak value CF-1 0.10 3D-Gaussian fit of 0-5% CF’s: out-side- long radii of 4-5 fm
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Resonance FSI contributions to π + - K + K - CF’s Complete and corresponding inner and outer contributions of p-wave resonance ( *) FSI to π + - CF for two cut parameters 0.4 and 0.8 fm and Gaussian radius of 7 fm The same for p-wave resonance ( ) FSI contributions to K + K - CF for Gaussian radius of 5 fm R peak (NA49) 0.10 R peak (STAR) 0.025
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Peak values of resonance FSI contributions to π + - K + K - CF’s vs cut parameter Complete and corresponding inner and outer contributions of p-wave resonance ( *) FSI to peak value of π + - CF for Gaussian radius of 7 fm The same for p-wave resonance ( ) FSI contributions to K + K - CF for Gaussian radius of 5 fm R peak (STAR) 0.025 R peak (NA49) 0.10
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27 Summary Assumptions behind femtoscopy theory in HIC seem OK, including both short-range s-wave and narrow resonance FSI (? up to a problem of angular dependence in the resonance region) The effect of narrow resonance FSI scales as inverse emission volume r 0 -3, compared to volume r 0 -1 or r 0 -2 scaling of the short-range s-wave FSI, thus being more sensitive to the space-time extent of the source The NA49 and STAR correlation data from the most central collisions seem to leave a little or no room for a direct (thermal) production of narrow resonances
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Angular dependence in the *-resonance region (k*=140-160 MeV/c) r* < 1 fm r* < 0.5 fm 0-10% 200 GeV Au+Au FASTMC-code
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Angular dependence – example parametrization
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