Effects of Brown-Rho scaling in nuclear matter, neutron stars and finite nuclei

1. Effects of Brown-Rho scaling in nuclear matter, neutron stars and finite nuclei ★ T.T.S. Kuo ★ Collaborators: H. Dong (StonyBrook), G.E. Brown (StonyBrook) R. Machleidt (Idaho), J.W. Holt (TU Munchen), J.D. Holt (Oak Ridge)

2. Brown-Rho (BR) scaling of in-medium mesons (in medium) ≠ (in free space) ? ? Using one-boson exchange (OBE) models, we have studied effects of BR scaling in • nuclear matter • neutron stars • finite nuclei

3. * • Brown-Rho scaling: in-medium meson mass m is ‘dropped’ relative to m in vacuum ρis nuclear matter density, ρ is that at saturation. • How to determine the Cs ? ? We adopt: fixing Cs by requiring BR-scaled OBE (BonnA and Nijmegen) giving symmetric nuclear matter 0 -3 MeV fm and

4. Symmetric (N=Z) nuclear matter equation of state (EOS): Most theories can not ‘simultaneously’ reproduce its binding energy and saturation density • This difficulty is well known (the ‘Coester’ band). MeV -3 fm

5. Coester band B.-A. Li el at., Phys. Rep. 464, 113 (2008)

6. We calculate nuclear matter EOS using a ring-diagram method: • The pphh ring diagrams are included to all orders. • Each vertex is • In BHF and DBHF, only first-order G-matrix diagram included.

7. EOS with all-order pphh ring diagrams: Ground state energy The transition amplitudes Y are given by RPA equations and it is equivalent to treating nuclear matter as a system of “quasi bosons” (quasi-boson approximation).

8. is used in our nuclear matter calculation. It is obtained by ‘integrating’ out the k >Λ components of , namely • is a smooth (no hard core) potential, and reproduces phase shifts of up to -1 (We use Λ~ 3 fm )

9. Ring-diagram EOSs for N=Z nuclear matter with from CDBonn and BonnA , and Λ= 3 and 3.5 fm Empirical values: MeV -3 fm -1

10. 1 • Linear BR scaling (BR ), not suitable for large ρ • Non-linear BR scaling (BR ) 2

11. Skyrme 3b-forces (TBF) We have 3 calculations for EOS: with BR 1 with BR 2 unscaled - plus TBF

12. Ring-diagram EOSs for N=Z nuclear matter (Λ=3.5 fm ) -1 alone, too soft with BR , too stiff 1 with BR and plus TBF satisfactory 2 Effects of BR scaling ≈ that of Skyrme TBF

13. Ring-diagram EOS for N=Z nuclear matter using ( plus TBF ) with CDBonn, BonnA, Λ =3 and 3.5 fm A common t =2000 MeVfm used for all cases. -1 6 3

14. Can we test EOSs and BRs at high densities ( ρ ≈ 5ρ ) ? 0 • Heavy-ion scattering experiments (e.g. Sn+Sn ) • Neutron stars where ρ≈ 8ρ 132 132 0

15. Experiment constraint for N=Z nuclear matter Danielewicz el at., Science 298, 1592(2002)

16. Comparison with the Friedman-Pandharipande (FP) neutron matter EOS solid lines: FP various symbols: + TBF dotted line: only (CDBonn)

17. Tolman-Oppenheimer-Volkov (TOV) equations for neutron stars: To solve TOV, need EOS for energy density vs pressure. • Neutron star outer crust ( ρ<~3×10 fm ), Nuclei EOS of Baym, Pethick and Sutherland (BPS) • Neutron star core ( >~4×10 M c/km ), Extrapolated polytrope EOS • Ring-diagram EOS used for intermediate region -4 -3 p -4 2 3 

18. Mass-radius trajectories of pure neutron stars • Ring-diagram EOSs, CD-Bonn with and without TBF • Causality limit: the straight line in the upper left core

19. Density profile for Maximum mass Pure Neutron stars • Ring-diagram EOSs, CD-Bonn with and without TBF

20. Pure neutron stars’ moment of inertia • CD-Bonn with and without TBF • Middle solid points are the empirical constraint (Lattimmer-Schutz)  

21. Neutron stars with β-stable ring diagram EOS: • Consider medium including p, n, e, μ • Equilibrium conditions:

22. Proton fraction ( ) of β-stable neutron stars

23. Mass, radius and moment of inertia of β-stable neutron stars 2   Ring and nuclei-crust EOS Top four rows with TBF, bottom without TBF 3b

24. Carbon-14 decay This β-decay has a long half-life T ≈ 5170 yrs 1/2 Tensor force is important for this long life time

25. Tensor forces from π- and ρ-mesons are of opposite signs: m decrease substantially at nuclear matter density m remains relatively constant (Goldstone boson) BR scaling is to decrease the tensor force at finite density ρ π

26. Shell model calculations (2 holes in p-p shell) using LS-coupled wave functions: Gamow-Teller transition matrix element (Talmi 1954) from BonnB with BR-scaled (m , m , m ) ρ ω σ

27. for decay GT 0 at

28. Ericson (1993) scaling: • Leads to non-linear BRS • Calculations with this scaling for m , m , m in progress • Recall BR scaling is with ρ ω σ

29. Summary and outlook: • Effects from BR scaling is important and desirable for nuclear matter saturation, neutron stars and C β-decay. • At densities (<~ρ ), BR scaling is likely linear, but at high densities it is an OPEN question. • BR scaling is similar to Skyrme 14 0

30. Thanks to organizers A. Covello, A. Gargano, L.Coraggio and N. Itako