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 ρ is nuclear matter density, ρ is that at saturation. How to determine the C s ? ? We adopt: fixing C s by requiring BR-scaled OBE (BonnA and Nijmegen) giving symmetric nuclear matter -3 * 0 MeV and fm

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). -3 MeV 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 (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 fm -3

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

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

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

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. 6 3

14. Can we test EOSs and BRs at high densities ( ρ ≈ 5 ρ ) ? Heavy-ion scattering experiments (e.g. Sn + Sn ) Neutron stars where ρ ≈ 8 ρ 0 132 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  -423 p -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. Ring-diagram EOSs, CD-Bonn with and without TBF Density profile for Maximum mass Pure Neutron stars

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. Ring and nuclei-crust EOS Top four rows with TBF, bottom without TBF POTENTIALS M [M ] R [km] I [M km ] CDBonn 1.80 8.94 60.51 Nijmegen 1.76 8.92 57.84 BonnA 1.81 8.86 61.09 Argonne V18 1.82 9.10 62.10 CDBonn (V =0) 1.24 7.26 24.30 Mass, radius and moment of inertia of β -stable neutron stars 3b  2

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

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 ) ρωσ

27. ρ/ρ ρ/ρ x y a b c M 0 0.844 0.537 0.359 0.168 0.918 -0.615 0.25 0.825 0.564 0.286 0.196 0.938 -0.422 0.50 0.801 0.599 0.215 0.224 0.951 -0.233 0.75 0.771 0.637 0.154 0.250 0.956 -0.065 1.00 0.737 0.675 0.103 0.273 0.956 0.074 fordecay 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 ρωσ

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 0 14

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