Probing the Equation of State of Nuclear Matter with Two-Particle Correlations

1. Probing the equation of state of nuclear matter with two-particle correlations Zbigniew Chajęcki Western Michigan University for the HiRA group

2. Outline • Equation of state of nuclear matter at low energies • transport theory (BUU) • p-p correlations • NSCL 03045 Experiment • particle emission chronology • (neutron and proton emission times and symmetry energy) • Summary Z. Chajecki - WPCF 2017

3. EOS and Symmetry Energy adapted from M.B. Tsang, Prog. Part.Nucl.Phys. 66, 400 (2011) Brown, Phys. Rev. Lett. 85, 5296 (2001) E/A (,) = E/A (,0) + d2S() d = (n- p)/ (n+ p) = (N-Z)/A Stiff • Both astrophysical and laboratory observables can constrain the EoS indirectly • What are the observables? • At what densities or asymmetries do these constraints apply? • What are the accuracies or model dependencies of these constraints? Soft Super soft Z. Chajecki - WPCF 2017

4. Modeling heavy ion collisions • Our tool: Transport models • BUU – Boltzmann-Uehling-Uhlenbeck • QMD – Quantum Molecular Dynamics • AMD – Antisymmetrized Molecular Dynamics • Simulates the time-dependent evolution of the collision • Main ingredients • Nucleons in mean-field • Symmetry energy • Momentum-dependent nuclear interaction • In-medium cross section effective mass • Cluster production Danielewicz, Acta. Phys. Pol. B 33, 45 (2002) Danielewicz, Bertsch, NPA533 (1991) 712 Z. Chajecki - WPCF 2017

5. What we hope to learn? (Double-)ratios Isospin diffusion Spectra Femtoscopy Flow pBUUTransport model ingredients Our approach: Use different isotopes (fix Z of your initial system and vary N) Z. Chajecki - WPCF 2017

6. Proton femtoscopy (p,p) correlation function (p,p) correlation function S-wave interraction S(r) S(r) S-wave interraction r1/2 Coulomb Coulomb r r uncorrelated uncorrelated 0 0 |q| = 0.5 |p1 - p2| |q| = 0.5 |p1 - p2| Theoretical CF: Koonin-Pratt equation p1 S.E. Koonin, PLB70 (1977) 43 S.Pratt et al., PRC42 (1990) 2646 x1 r … 2-particle wave function … source function x2 p2 few fm Z. Chajecki - WPCF 2017

7. Our experiment Z. Chajecki - WPCF 2017

8. NSCL experiment 05045: HiRA + 4 detector = High Resolution Array beam • 4π detector => impact parameter + reaction plane • HiRA => light charge particle correlations (angular coverage 20-60º in LAB, • 63 cm from target (= ball center)) Reaction systems: 40Ca + 40Ca @ 80 MeV/u 48Ca + 48Ca @ 80 MeV/u Z. Chajecki - WPCF 2017

9. NSCL experiments 05045: HiRA + 4 detector 4x CsI(Tl) 4cm Si-E 500 mm Si-DE 65mm = High Resolution Array Particle identification 4He 3He t d p Z. Chajecki - WPCF 2017

10. Experimental correlation functions C(q) Low PT : [150,350] MeV High PT : [350,700] MeV Measured correlation functions depend on rapidity and the transverse momentum of the pair Next step:extract the sizes Z. Chajecki - WPCF 2017

11. Gaussian S(r)? C(q) Two ways of characterizing the size of the p-p source S(r) - Gaussian shape Imaged S(r) (Brown, Danielewicz) Koonin-Pratt Equation Brown, Danielewicz, PLB398 (1997) 252 Danielewicz, Pratt, PLB618 (2005) 60 Z. Chajecki - WPCF 2017

12. Fits to the data C(q) Both methods give consistent fits Two ways of characterizing the size of the p-p source S(r) - Gaussian shape Imaged S(r) (Brown, Danielewicz) Koonin-Pratt Equation Brown, Danielewicz, PLB398 (1997) 252 Danielewicz, Pratt, PLB618 (2005) 60 Z. Chajecki - WPCF 2017

13. Fits to the data r1/2 Source distribution : S(r) x103 Correlation function C(Q) Z. Chajecki - WPCF 2017

14. Fits to the data Correlation function C(Q) rapidity & momentum dependence Z. Chajecki - WPCF 2017

15. Fit results Small rapidity:reflect the participant zone of the reaction Large rapidity:reflect the expanding, fragmenting and evaporating projectile-like residues Higher velocity protons are more strongly correlated than their lower velocity counterparts, consistent with emission from expanding and cooling sources Sensitivity to the initial size Z. Chajecki - WPCF 2017

16. Comparing data to theory (pBUU) Proton femtoscopy in 48Ca+48Ca ..@ 80 AMeV 6 5 4 3 Importance of cluster production and in-medium cross section Henzl et al., Phys.Rev. C85 (2012) 014606 Z. Chajecki - WPCF 2017

17. Comparing data to theory (pBUU) . BUU Pararameters • Rostock in-medium reduction • Importance of the clusters • No dependence on symmetry energy – not shown BUU does reasonably well Except at larger rapidities - Spectator source Where evaporation and secondary decays are important! Z. Chajecki - WPCF 2017

18. Comparing data to theory (pBUU) . Assuming the decay occurs from a spherical source with RG and the timescale of the decay is t Z. Chajecki - WPCF 2017

19. Averaged emission time of particles in transport theory Z. Chajecki - WPCF 2017

20. Emission of p’s and n’s: Sensitivity to SymEn adapted from M.B. Tsang, Prog. Part.Nucl.Phys. 66, 400 (2011) Brown, Phys. Rev. Lett. 85, 5296 (2001) 52Ca 48Ca Stiff Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft Soft EoS Super soft Soft Stiff EoS (γ=2) Soft EoS (γ=0.5) p’s and n’s emitted at similar time fasteremission times p’s emittedafter n’s later emission times Stiff Z. Chajecki - WPCF 2017

21. Emission of p’s and n’s: Sensitivity to SymEn 52Ca 48Ca Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft EoS Soft EoS (γ=0.5) Stiff EoS (γ=2) p’s emitted after n’s later emission times p’s and n’s emitted at similar time fasteremission times Z. Chajecki - WPCF 2017

22. Possible emission configurations (stiff EOS) n n n n p p p p Catching up Catching up qx<0 qx>0 Moving away Moving away qx<0 qx>0 (n,p) correlation function Dq=0.5(pp –pn)=(qx, qy=0,qz=0); r=(x, y=0,z=0) qx<0 qx>0 S(x) x 0 q = 0.5(pp - pn) Z. Chajecki - WPCF 2017

23. Emission of p’s and n’s 52Ca 48Ca Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft EoS Soft EoS (γ=0.5) Stiff EoS (γ=2) p’s emitted after n’s later emission times p’s and n’s emitted at similar time fasteremission times Z. Chajecki - WPCF 2017

24. Sensitivity to particle emission (soft EOS) n n p p Moving away Catching up qx<0 qx>0 (n,p) correlation function qx<0 qx>0 S(x) x 0 Dq=0.5(pp –pn)=(qx, qy=0,qz=0); r=(x, y=0,z=0) qx = 0.5(px,p - px,n) Z. Chajecki - WPCF 2017

25. Relating asymmetry in the CF to space-time asymmetry Stiff EoS Soft EoS (n,p) correlation function qx<0 qx>0 S(x) <x> x 0 qx = 0.5(px,p - px,n) Classically, average separation b/t protons and neutrons Not expected if n,p emitted from the same source (no n-p differential flow) =0 Protons emitted later Voloshin et al., PRL 79:4766-4769,1997Lednicky et al., PLB 373:30-34,1996 Z. Chajecki - WPCF 2017

26. Summary • Two particle correlations provide a unique probe to study the space-time extend of the source • add constrains on the in-medium cross-section • importance of the clusters, symmetry energy • validate theoretical models • The average relative emission time of n’s and p’s potentially sensitive to the symmetry energy and can be “measured” with two particle correlations • Transport models • Predictions are model dependent • Collaboration between theorists and experimentalists beneficial for both sides Z. Chajecki - WPCF 2017

27. Thank you HiRA collaborators Z. Chajecki - WPCF 2017