1. Probing two-particle sources in HIC Giuseppe Verde, NSCL/Michigan State University HIC03, Montreal, 25-28 June, 2003 1+R(E * ) E * (MeV) p-p d-   - 6 Li Outline p-p correlation functions: physics information content Imaging Complex particle correlations (d-  ), effects of collective flow Conclusions

2. HIC at intermediate energies Projectile Target Pre-equilibrium Compression Expansion Fragmentation Secondary decays Short time scales Long time scales Probe nuclear equation of state (EoS) Find pace-time probes of the reaction:… Taking “photographs” Volume, density, shape, lifetime of fragmenting system Probe reaction models (transport, dynamics/EoS)

3. Intensity interferometry: from large scales... Star R d << R Static systems: exploring the geometry (size, R) HBT: R. Hanbury Brown, R.Q. Twiss, Phil. Mag., Ser. 7 45 (1954) 663 … to subatomic physic scales ( , K-K, , p-p, n-n, IMF-IMF, …) Fast evolving systems: 10 -23 -10 -15 sec: geometry changing in time Detectors Nuclear reaction d >> R R+V  G. Goldhaber et al., PR 120 (1960) 300

4. Measuring correlation functions 112 Sn R(q) probes space-time properties of source 1+R(q) q (MeV/c) p-p d-  6 Li 2.19 6 Li 4.31 LASSA (IU, MSU, WU) 124 Sn E/A=50 MeV/u

5. Koonin-Pratt Eqn and Source function S.E. Koonin, PLB70 (1977) 43 S.Pratt et al., PRC42 (1990) 2646 = Source function Probability distribution of emitting a pair separated by when last particle is emitted If (not simultaneous)Space-time ambiguity in Directional correlations to reduce space-time ambiguity… … only if r0r0

6. Very-Long-Lived emitting sources Directional correlation functions insensitive proton Detectors … such as secondary decays, evaporation, … Source elongated up to

7. Angle-averaged correlation functions Angle-averaging over Spherical symmetric Gaussian profiles extensively used r 0 =3.4 fm 4.2 5.9 14 N+ 197 Au E/A=75 MeV  ~25 o Gaussian spherical sources “Common wisdom”… R(20 MeV/c) Size

8. Low q region not accessible experimentally: probing only fast source Fast and slow emitting sources in HIC FastSlow Proton emission: Fast: Pre-equilibrium Slow: Evaporation, Secondary Decays FastSlow Contribution from:

9. Size of two-proton sources Width (not height!!) of peak at 20 MeV/c measures uniquely the size of the source G. Verde at al., PRC65, 054609 (2002) Slope~ 2.7 MeV/c/fm C(q)= Size (r 0 ) Width – FWHM (MeV/c)

10. Fast and slow d-  sources 1+R(q) S(r) (fm -3 ) r (fm)q (MeV/c) d-  sourced-  correlation 2 1 Peak width (MeV/c) Size (fm) Peak 2 Peak 1 Size of fast source from width of peak 2 6 Li (2.19) 6 Li (4.31) Peak 1 dominated by detector resolution

11. q (MeV/c) 1+R(q) p-p correlations: physics information content Y total =Pre-eq. + Sec. Decays Y fast + Y slow Peak Height Relative contribution from fast pre-eq. source Y fast /Y total Peak width (shape) Size (shape) of two-proton fast source S(r) G. Verde et al., PRC65, 054609 (2002) Shape analysis required! G. Verde at al., PRC65, 054609 (2002)

12. Imaging two-proton sources D.A. Brown, P. Danielewicz PRC57 (1998) 2474, PRC64 (2001) 014902 G. Verde et al., PRC65 (2002) 054609 Source Function Imaging = Inverting KP Eqn KP Eqn Model independent and multi-dimensional approach All the points deviating from 1 contain information about S(r), not only C(q=20 MeV/c)

13. Imaging: profile of the short-lived dynamical source size from r 1/2 relative contribution from long-  emissions: 14 N+ 197 Au E/A=75 MeV  ave ~25 o Imaging 3.4 4.2 5.9 r 0 (fm)= Zero-lifetime Gaussian sources

14. r 1/2 weakly sensitive to P sum : size of fast dynamical sources Long-lifetime contributions 1-f strongly depend on P sum Properties of two-proton sources 7 5 4 2.5 3.1 2.9 r 1/2 (fm)  f  Source SizesLong-lived contributions

15. Imaging p-p correlations Relative contributions from FAST and SLOW emitting sources Constraints on secondary decays Size of emitting sources – from peak width (shape), not from peak height! Measure densities Profile of dynamical two-proton emitting source

16. Test of transport theories Constraints contributions from secondary decays with f-value Imaging analysis Height of the peak not reproduced Long-lived emissions not handled correctly Ar+Sc BUU  in-med BUU  free G. Verde et al., Phys. Rev. C 67, 034606 (2003)

17. Source shape: probing transport models Shape of BUU source probes probes details about  NN r (fm) S(r) (fm -3 ) Imaging BUU free NN BUU red NN Model Ar+Sc, E/A=120 MeV Reduced  NN favored G. Verde et al., Phys. Rev. C 67, 034606 (2003)

18. Peak height sensitive to V asy (  0 ): Shorter emission times for asy-stiff? Peak height not reliable (long-lifetime decays)  0 V asy (MeV) Asy-soft Asy-stiff Lie-Wen Chen et al., nucl-th/0211002, Nov 2002 1+R(q) q (MeV/c) Asy-stiff Asy-soft IBUU: Isospin effects in p-p correlations 52 Ca+ 48 Ca, 80 MeV/u

19. IBUU: Source shape and Asy-EOS Shape of peak sensitive to Asy-EOS Asy-soft: more extended source, longer proton emission times Measure at q<15 MeV/c required!! Asy-stiff r 1/2 ~3.6 fm Asy-soft r 1/2 ~4.4 fm r (MeV/c) S(r) (a.u.) 1+R(q) q (MeV/c) Asy-stiff Asy-soft Sources Lie-Wen Chen et al., nucl-th/0211002, Nov 2002 52 Ca+ 48 Ca, 80 MeV/u P>500 MeV/c

20. Isospin effects in Two-proton sources Central collisionsSources Need more statistics and higher resolution (future experiments): explore the shape up to q<8 MeV/c Protons from secondary decays: more in 112 Sn+ 112 Sn Preliminary LASSA

21. Two-proton correlations in 112 Sn+ 112 Sn and 124 Sn+ 124 Sn q (MeV/c) 1+R(q) E 1,E 2 >60 MeVE 1,E 2 <50 MeV 124 Sn+ 124 Sn 112 Sn+ 112 Sn 124 Sn+ 124 Sn 112 Sn+ 112 Sn Fast protonsSlow protons

22. Complex particle correlations 1+R(E * ) E * (MeV) p-p d-   - 6 Li Densities, fast/slow contributions, source profiles and test of reaction models

23. d-  in 112 Sn+ 124 Sn reactions q (MeV/c) 1+R(q) Central 112 Sn+ 124 Sn, E/A=50 MeV S(r) (fm -3 ) r (fm) p-p Size~5.5+-0.5 d-  Size~3.5+-0.5 Sources Good news: d-  can probe long-lived emitting source Warning! Height of peak 2 overpredicted LASSA Size

24. Collective motion requires special considerations Reduction of source size Shape of correlation functions between complex particles (d-  ) strongly distorted.

25. Source size reduction Only thermal motion Detectors Thermal + Collective motion Position-momentum correlations Size reduction enhanced with heavier particles

26. Collective motion distortions Yields Coincidence Event mixing E rel (MeV) 1+R(E rel ) Correlation Event b Event a G. Verde et al., in prep.

27. Effective temperature correction q (MeV/c) 1+R(q) q (MeV/c) 1+R(q) No Flow Flow Nuclear part of correlation function needs correction Data reproduced for T eff =5 MeV KP eq. G. Verde et al., in prep.

28. Size correction: p-p vs d-  S(r) (fm -3 ) r (fm) d-  p-p Before correction Source sizes p-p 5.5  0.2 fm d-  0.5 fm After correction Source sizes p-p 7.5  0.5 fm d-  1 fm Differences p-p vs d-  reduced p-p and d-  probe different sources Central 112 Sn+ 124 Sn, E/A=50 MeV

29. Conclusions Important physics information from imaging of p-p size (from width/shape of correlation peak), contributions from dynamical/equilibrium emissions, profiles of dynamical sources (probes of microscopic models, BUU, IBUU, …) Extension to more complex particle correlations (d-  ) Effects of collective flow need special consideration (sizes, shape of nuclear resonance peaks) Two-particle correlations can provide “snapshots” of emitting sources… …and we actually need them!

30. Acknowledgements D.A. Brown, LLNL P. Danielewicz, C.K. Gelbke, T.X. Liu, X.D. Liu, W.G. Lynch, W.P. Tan, M.B. Tsang, A.Wagner, H.S. Xu, NSCL/MSU B. Davin, Y. Larochelle, R.T. de Souza, IU R.J. Charity, L.G. Sobotka, WU