1. Constraints on symmetry energy from different collective excitations G. Colò NUSYM Krakow July 2 nd, 2015

2. Outline Introduction and general problem(s). Different nuclear excitations chosen as a probe to extract symmetry energy parameters. Are the result consistent ? Can we disentangle “observable dependence” from “model dependence” ? Another tool: study of the correlations between observables within a model.

3. Co-workers A. Bracco, M. Brenna, P.F. Bortignon, F. Camera, A. Carbone, X. Roca-Maza, L. Trippa, E. Vigezzi, O. Wieland (Università di Milano and INFN, Italy) M. Centelles, X. Viñas (University of Barcelona, Spain) N. Paar, D. Vretenar (University of Zagreb, Croatia) J. Piekarewicz (Florida State University, USA) B.K. Agrawal (SINP, Kolkata, India) L. Cao (NCEPU, Beijing, P.R. China) H. Sagawa (University of Aizu and RIKEN, Japan)

4. Nuclear matter EOS Symmetric matter EOS Symmetry energy S Uncertainties affect The nuclear equation of state and the symmetry energy From the energy per particle as a function of the density we can extract the pressure. For this reason we call E/A the “equation of state” of nuclear matter. In this quantity, the part that depends on the neutron-proton imbalance is poorly known. J = S 0 = S v = a 4 = a τ

5. Isovector modes Neutrons and protons oscillate in opposition of phase. Aim: relate their measurable properties to bulk ones – mainly S. Problems: the nucleus is not a homogeneous system, it has a shell structure, and there is isoscalar/isovector mixing. Aim: relate their measurable properties to bulk ones – mainly S. Problems: the nucleus is not a homogeneous system, it has a shell structure, and there is isoscalar/isovector mixing.

6. Extracting values for the EoS parameters EoS PARAMETER B MEASURABLE QUANTITY A The “points” correspond to calculations using different EDFs, essentially Skyrme forces and RMF Lagrangians. IVGDR PRC 77, 061304(R) (2008) PDR PRC 81, 041301(R) (2010) J = 32.3 ± 1.3; L = 64.8 ± 15.7 Dipole polarizability PRC 88, 024316 ( 2013) (J = 31 ± 2); L = 43 ± 16 IVGQR PRC 87, 034301 (2013) (J = 32 ± 1); L = 37 ± 18 Anti-analog ch.exch. dipole PRC (2015) J = 33.2 ± 1.0; L = 97.3 ± 11.2 NUMBERS in MeV

7. From L to the neutron skin in 208 Pb PDR L = 64.8 ± 15.7 MeV; Δ R np = 0.194 ± 0.024 fm Dipole polarizability L = 43 ± 16 MeV; Δ R np = 0.165 ± 0.026 fm IVGQR L = 37 ± 18 MeV; Δ R np = 0.14 ± 0.03 fm AGDR L = 97.3 ± 11.2 MeV; Δ R np = 0.236 ± 0.018 fm Values of J fully compatible Other quantities compatible if extracted from dipole and quadrupole Charge-exchange AGDR leads to higher values of L and of the skin

8. Self-consistent mean-field and/or EDF Slater determinant 1-body density matrix H eff = T + V eff. If V eff is well designed, the resulting g.s. (minimum) energy can fit experiment at best. Within a time-dependent theory, one can describe oscillations around the minimum. In the harmonic approximation the restoring force is: The linearization of the equation of the motion leads to the well known Random Phase Approximation.

9. Skyrme vs. relativistic functionals attraction short-range repulsion Skyrme effective force In the relativistic (that is, covariant) models the nucleons are described as Dirac particles that exchange effective mesons. There are effective Lagrangians that include free nucleons, free mesons and interactions. Also point coupling versions !

10. EoS PARAMETER B MEASURABLE QUANTITY A 1.reliability of experimental data; 2.understanding the physical meaning of the correlation between observable A and parameter B; 3.possible model dependence (a bias in the model can impact the correlation). Critical analysis ? In the method shown at right, there are three natural critical points.

11. The isovector quadrupole resonance High intensity polarized photon beam on 209 Bi Scattering parallel and perpendicular to the polarization plane Three-parameter fit of the IVGQR energy, width and strength S. Henshaw et al., PRL 93, 122501 (2011). HIγS (10 7 γ/s, ΔE/E≈2-3%)

12. QHO model and the relation IVGQR vs. S Schematic RPA: Bohr-Mottelson formula: We assume: (i) simple density profile; (ii) relationship with S shell gap pot E(ISGQR) = 61 A -1/3, Fermi energy = 37 MeV, S(0.1) = 24 MeV ⇒ E(IVGQR) = 135 A -1/3

13. Systematically varied SAMi and DDME families X. Roca-Maza, G.C., H. Sagawa, Phys. Rev. C 86, 031306(R) (2012). D. Vretenar, T. Nikšić, P. Ring, Phys. Rev. C68, 024310 (2002). All sets have comparable quality. Fits on exp. data (binding energies, radii etc.) are repeated each time by fixing only either m* (SAMi- m) or J (SAMi-J or DDME-x).

14. Model dependence Interestingly, experiment lies in the region where the model dependence is minimal.

15. 1.For the IVGQR one does not see experimental problems, and the reason for the correlation with S is transparent. Model dependence (perhaps accidentally) small.

16. The debated nature of the “pygmy” dipole O.Wieland et al., PRL 102, 092502 (2009) 68 Ni A. Klimkiewicz et al., PRC 76, 051603(R) (2007). Many experiments have identified strength (well) below the GDR region. Is this a “skin mode” possessing some degree of collectivity ? Or does it just have single-particle character ?

17. Pygmy “states” (PDS) in the IV response The PDR collectivity can vary Polarizability gets contribution from it

18. Isoscalar response The states in the PDR region are more prominent in the IS response

19. Transition densities and cross sections IS dominance, in particular at the surface X. Roca-Maza et al., Phys. Rev. C85, 024601 (2012). F.L. Crespi et al., Phys. Rev. Lett., 113, 012501 (2014). Cf. his talk. Experimental data support the relevance of IS surface part and can validate the microscopic t.d.

20. 1.For the IVGQR one does not see experimental problems, and the reason for the correlation is transparent. Model dependence (perhaps accidentally) small. 2.The PDR seems admixed with IS components. In this respect, it does not seem the best candidate to extract S. Despite model dependence of the PDR, no discrepancy with the results for L and skin extracted from the IVGQR.

21. The droplet model and the relation between polarizability and L or skin The droplet model provides an expression for the dipole polarizability: Also, it provides an expression for the neutron skin. Under the hypothesis that (i) JA -1/3 /Q can be treated as a small parameter, (ii) that the density has a simple Fermi profile, and (iii) that J/Q is linearly correlated with L, as displayed by many models, then Conclusion: the droplet model provides a relation between α D, J and L. ALSO IT SHOWS THE EXISTENCE OF A LINEAR RELATIONSHIP BETWEEN α D J AND r np.

22. Results with realistic models

23. 1.For the IVGQR one does not see experimental problems, and the reason for the correlation is transparent. Model dependence (perhaps accidentally) small. 2.The PDR seems admixed with IS components. In this respect, it does not seem the best candidate to extract S. Despite model dependence of the PDR, no discrepancy with the results for L and skin extracted from the IVGQR. 3.The dipole polarizability displays also a trasparent correlation with S. - Cf. the talk by X. Viñas.

24. The AGDR (cf. talk by A. Krasznahorkay) The AGDR is the analogous state of the GDR, in the same way as the IAS is the analogous of the g.s. Anti- ? Perhaps misleading. We expect E1 transitions between AGDR and IAS in the same way as between GDR and g.s. In this respect, we expect sensitivity to the symmetry energy… but the argument should be refined.

25. Explaining the correlation E(AGDR)-E(IAS) vs. neutron skin Z N Using sum rules and schematic RPA, as above: By taking the difference, and doing some mild approxmations related again to (i) density profiles, (ii) the fact that ε -U is small and U is related to V 1, one arrives at a correlation L. Cao et al., PRC (submitted)

26. Sensitivity to the experimental input From [54]: L = 86.1 ± 9.1 MeV; Δ R np = 0.254 ± 0.062 fm From [57]: L = 108.5 ± 35.8 MeV Δ R np = 0.218 ± 0.015 fm [54] A. Krasznahorkay et al. Cf. his talk. [57] J. Yasuda et al. Polarized (p,n) at 296 MeV plus MDA

27. Additional model dependence [54] A. Krasznahorkay et al. Cf. his talk. [57] J. Yasuda et al. Polarized (p,n) at 296 MeV plus MDA Compare with the IVGQR case !

28. 1.For the IVGQR one does not see experimental problems, and the reason for the correlation is transparent. Model dependence (perhaps accidentally) small. 2.The PDR seems admixed with IS components. In this respect, it does not seem the best candidate to extract S. Despite model dependence of the PDR, no discrepancy with the results for L and skin extracted from the IVGQR. 3.The dipole polarizability displays also a trasparent correlation with S. - Cf. the talk by X. Viñas. 4.The AGDR is also correlated with the skin or with L in a transparent way but the model dependence plays a stronger role.

29. Correlations - generalities Let us assume we have fitted a model characterized by a set of parameters p, and that we move around the optimal model (i.e., the χ 2 minimum). It is possible to calculate the covariance between two observables A, B and the Pearson-product correlation coefficient c AB ≈ 0 c AB ≈ 1 is a measure of the correlation within the given model.

30. Correlations – difference between models The isoscalar properties show mutual correlations in both cases (except for the Dirac mass in the case of DDME-min1). On the other hand, it is striking to notice that the mutual correlations among isovector properties is strong in the case of DDME-min1 and does not show up so clearly in the case of SLy5-min. The reason must have to do with the different fitting protocols.

31. Correlations – effect of the fitting protocol When the constraint on a property A included in the fit is relaxed, correlations with other observables B become larger. When a strong constraint is imposed on A, the correlations with other properties become very small. Constraint on neutron EoS almost released In addition, neutron skin fixed !

32. Conclusions We have already a large amount of information concerning symmetry energy parameters and neutron skins extracted from collective excitations like giant resonances. Most of the outcome is consistent ! J looks fine, and L is between 35 MeV and 65 MeV in three cases – except when deduced from AGDR. However, there is room for improvement. Mainy, to understand the model dependence. Correlation analysis can help ! Open issues: pairing, correlations beyond mean-field.