1. Scott PrattMichigan State University Femtoscopy: Theory ____________________________________________________ Scott Pratt, Michigan State University

2. Scott PrattMichigan State University Deriving the Fundamental Formula

3. Scott PrattMichigan State University Deriving the Fundamental Formula Step 1: Define the source function

4. Scott PrattMichigan State University Deriving the Fundamental Formula Step 2: Write 2-particle probability = probability relative momentum q and separation x evolves to q asymptotically

5. Scott PrattMichigan State University Deriving… Identical particles Smoothness approximation

6. Scott PrattMichigan State University With final-state interactions Smoothness approximation Approximate (in frame of pair),

7. Scott PrattMichigan State University Deriving… Summary Assumptions Identical Particles 1.Symmetrize pairwise 2.Independent emission 3.Smoothness Strong/Coulomb 1.Independent emission 2.Ignore time difference for evolution 3.Smoothness * * * Tested *

8. Scott PrattMichigan State University Femtoscopy – Theory Measures phase space cloud for fixed velocity Overall source can be larger Inversion depends on |  (q,r)| 2

9. Scott PrattMichigan State University Hadronic Interferometry – Theory Theories predict S P (r)  C(P,q) Correlations provide stringent test of space-time evolution

10. Scott PrattMichigan State University Using Identical Particles Examples: , KK, … Easy to invert 3-dimensional information R out, R side, R long are functions of P

11. Scott PrattMichigan State University Identical Particles: Measuring Lifetime Has been studied for , KK, pp, nn Source function S(p,r,t) is 7-dimensional – requires one dimension of common sense

12. Scott PrattMichigan State University Strong Interactions Peak height determined by scattering length or resonance width Examples: pp, p , nn, p , Kp, p , d , … d  Correlations  E (MeV) G. Verde / MSU Miniball Group

13. Scott PrattMichigan State University Coulomb Interactions Can be calculated classically for larger fragments Kim et al., PRC45 p. 387 (92)

14. Scott PrattMichigan State University Proton-proton Correlations Deconvoluting C(q) provides detailed source shape S.Panitkin and D.Brown, PRC61 021901 (2000)

15. Scott PrattMichigan State University Measuring shape without identical particles

16. Scott PrattMichigan State University Example: pK + correlations Gaussian Sources: R x =R y =4, R z =8 fm

17. Scott PrattMichigan State University Detailed Shape Information Standard formalism: Defining, Using identities for Y lm s, Simple correspondence! Danielewicz and Brown

18. Scott PrattMichigan State University Moments L=0 L=1, M=1 L=2, M=0,2 L=3, M=1,3 Angle-integrated shape Lednicky offsets Shape (R out /R side, R long /R side ) Boomerang distortion

19. Scott PrattMichigan State University Blast Wave Model (z  -z) C L+M=even (q) = 0 (y  -y) Imag C L,M = 0 S.P. and S.Petriconi, PRC 2003

20. Scott PrattMichigan State University Liquid-Gas Phase Transition Definition of Gas: “Expands to fill available volume” Liquid = Evaporation  Long lifetimes Gas = Explosion  Short lifetimes

21. Scott PrattMichigan State University Change to Explosive Behavior (GAS) at ~ 50 AMeV

22. Scott PrattMichigan State University Experimental Signatures Dramatic change in nn correlations  ~ 500 fm/c  ~ 50 fm/c

23. Scott PrattMichigan State University Phase Transition at RHIC Transparency complicates the problem For complete stopping, times could be ~ 100 fm/c For Bjorken, strong first-order EOS leads to  ~ 20 fm/c

24. Scott PrattMichigan State University Phase Transition at RHIC? Stiffer EOS -> Smaller source sizes Data demonstrate no latent heat or significant softness

25. Scott PrattMichigan State University THE HBT PUZZLE AT RHIC To fit data: a) Stiff (but not too stiff) EOS b) Reduce emissivity from surface c) Not that much different than SPS

26. Scott PrattMichigan State University Phase space density Any method to extract R inv is sufficient

27. Scott PrattMichigan State University Phase space density rises until threshold of chemical equilibrium   ~ 80 MeV at break-up

28. Scott PrattMichigan State University HBT and Entropy Entropy can be determined from average Phase space density determined from: correlations (  ) coalescence (KK ,pp  d) thermal models…

29. Scott PrattMichigan State University Entropy for 130 GeV Au+Au at  = 1 fm/c S.Pal and S.P., PLB 2003 hydro  Bjorken

30. Scott PrattMichigan State University Summary Correlations CRUCIAL for determining Pressure Entropy Reaction Dynamics