Some new developments in relativistic point-coupling models

Transcription

1 Some new developments in relativistic point-coupling models T. J. Buervenich 1, D. G. Madland 1, J. A. Maruhn 2, and P.-G. Reinhard 3 1 Los Alamos National Laboratory 2 University of Frankfurt 3 University of Erlangen-Nuernberg INT03 workshop on Theories of Nuclear Forces & Nuclear Systems December 3, 2003

2 Outline The relativistic mean-field point-coupling (RMF-PC) approach to nuclear ground states Current issues, puzzles, questions,... Status Extensions of the model additional observables nonlinear terms binding energy, radii, surface, single-particle energies, shell structure,... for the whole nuclear chart further (on top of the mean field): excitations, collective states,... naturalness

3 Motivation Successful models with point couplings Fermi theory NJL model Skyrme-Hartree-Fock Effective Field Theories 1 m 2 q 2 1 m 2 + q2 m Predictive power comparable to Waleckatype relativistic mean-field (RMF) models with meson exchange low momentum: derivative expansion should work does work Features Hartree and Hartree-Fock Link to relativistic meson models Link to nonrelativistic Skyrme-Hartree-Fock point-coupling models Test of power counting in finite nuclei EFT / DFT covariant framework: large scalar and vector potentials: saturation spin-orbit for free!

4 Density [1/fm^3] Chart of the nuclides? bubbles? z [fm] r [fm]

5 Philosophy of self-consistent mean-field models for nuclei (to set the stage) Construct an Effective Interaction (or an Energy Functional) between point-like nucleons Introduce approximations and solve the reduced problem numerically Adjust the coupling constants introduced through the interaction (6 to10, 2 for pairing): a force is born (biased, thus there are many forces) There is no best force : the adjustment procedure determines the predictive power for various observables Predict nuclear ground-state observables throughout the nuclear chart and extrapolate

6 The relativistic mean-field (RMF) model nucleons interact with each other through the exchange of various mesons (scalar, vector, isovector-vector,...) historic view covariant Lagrangian Hartree approximation (no exchange terms) mean-field (field operators can be replaced by their expectation values) no-sea (vacuum polarization is ignored) RMF approximates the exact density functional of strongly interacting fermions [ Hartree + exchange-correlation functional] - Effective Field Theory modern view interaction due to mean meson fields or point-like (contact) interactions and derivatives

7 Building blocks in principle all possible Lorentz invariants (isoscalar, isovector) should be there Phenomenology, Symmetries, Approximations: large scalar and vector potentials density dependence derivatives (~ finite range) isovector channel (!!) 2, (!" µ!) 2 (!!) 3, (!!) 4, (!" µ!) 4 (! µ "") 2, (! µ "# $ ") 2 (!" µ #!) 2, ($ µ!" % #!) 2 Coulomb force pairing

8 The RMF-PC Lagrangian L = L free + L 4f + L hot + L der + L em L free = ψ(iγµ µ m)ψ L 4f = 1 2 α S( ψψ)( ψψ) 1 2 α V( ψγ µ ψ)( ψγ µ ψ) 1 2 α TV( ψ τγ µ ψ) ( ψ τγ µ ψ) L hot = 1 3 β S( ψψ) γ S( ψψ) γ V[( ψγ µ ψ)( ψγ µ ψ)] 2 L der = 1 2 δ S( ν ψψ)( ν ψψ) 1 2 δ V( ν ψγµ ψ)( ν ψγ µ ψ) 1 2 δ TV( ν ψ τγµ ψ) ( ν ψ τγ µ ψ) L em = ea µ ψ[(1 τ3 )/2]γ µ ψ 1 4 F µνf µν +BCS +! f orce(volume)pairing

9 The assumption of a local δ-like interaction leads to a contribution to E of the form E pair = 1 d 3 r G( r) χ 2 ( r) 4 with the pairing density BCS pairing χ( r) = 2 k>0 u k v k ψ k ( r) 2. standard BCS formalism proton-proton / neutron-neutron pairing only δ-force pairing G( r) = constant = V 0. Epair δ = V 0 d 3 r χ 2 ( r). 4 The pairing potential q = V 0 2 χ q is mainly located inside the nucleus. + cutoff

10 Adjustment of forces binding energy [all forces] diffraction radius [NL-Z2, PC-F1, SkI3/4] surface thickness [NL-Z2, PC-F1, SkI3/4] rms radius [NL-Z2, NL3, PC-F1, SkI3/4, SkP, SLy6] neutron radius [NL3] spin-orbit splitting [Skyrme forces] isotope shift in lead [SkI3/4] nuclear matter [NL3] neutron matter [SLy6] form factor chisquared adjustment to magic and doubly-magic nuclei

11 Adjusting the parameters of the RMF model 16 O 40 Ca 48 Ca 56 Ni 58 Ni 88 Sr observable error E B % R dms 0.5 % σ 1.5 % rrms ch 0.5 % p 5 MeV n 5 MeV Zr 100 Sn 112 Sn 120 Sn 124 Sn 132 Sn 136 Xe 144 Sm 202 Pb 208 Pb 214 Pb NL-Z2 PC-F1 magic and doubly-magic (spherical) nuclei are chosen adjustment to both binding energy and form factor pairing strengths are adjusted simultaneously with the mean-field parameters

12 prediction adjusted surface thickness rms radius diffraction radius binding energy

13 Current puzzles nuclear matter bulk properties differ in RMF and Skyrme- Hartree-Fock (SHF) wrong trends in binding energies asymmetry energy appears to be too large ( ~ 38 MeV) compared to empirical values of ~ 32 MeV surface thicknesses are too small ( ~ 5 %) compressibility does not seem to be determined by groundstate observables axial fission barriers in RMF and SHF differ by up to factor of two for superheavies (trends already visible in actinides)...

14 5 PC-F1 (RMF) SLy6 (SHF) E/A E/A [MeV] ,17 0,14 PC-F1 NL-Z2 SLy6 SkI4! [fm -3 ] 0, PC-F1 NL-Z2 SLy6 SkI4 asym PC-F1 NL-Z2 SLy6 SkI4

15 Sn Pb T=0: attraction T=1: repulsion PC-F1 PC-LA PL-40 NL3!!! Neutron Number Neutron Number 1.0 N = N = 126 T=0: attraction T=1: attraction Coulomb: repulsion Proton Number Proton Number

16 120 SHE 118 axial symmetry reflection asymmetry TJB, M. Bender, J. A. Maruhn, P.- G. Reinhard, to appear in PRC E [MeV] NL-Z2 SLy

17 Some observations and findings both the isovector and the isoscalar channel need further adjustment in both channels the density dependence is not optimal energies and form factor compete in the adjustment procedure as do light and heavy systems isovector channel possibly absorbs mismatch of the isoscalar channel RMF-PC

18 Bulk properties of nuclear matter modified fitting protocols Type of adjustment ρ 0 [fm 3 ] E/A [MeV] K [MeV] m /m a 4 [MeV] {Z Z N i} {Z Z Zr} {Z Z Sn} {Z Z Zr or Z = P b} {Z Z Sn} no T=1 terms a 4! = 34 MeV only E - error % only E - error 0.5 MeV σ error 25 %

19 Different selections of nuclei Sn Pb Neutron Number N = Proton Number Neutron Number N = 126 PC-F1 Z >= Sn Z <= Zr Z <= Ni Proton Number

20 Sn Pb Neutron Number N = Proton Number Neutron Number N = 126 PC-F1 asym!= 34 MeV Energies only - % - 25 % Proton Number

21 Where to go from here? an accurate and well adjusted mean-field is desirable for ground-states and all correlations on top of it ( pairing, ground-state correlations, excited states,... ) drip lines, superheavy nuclei,... guidance for important / physical terms? extend / modify the current models force for the complete nucleus - force only for energies / for the geometry /...

22 QCD scaling scale the Lagrangian using two scales: f π = 93.5 MeV Λ = 770 MeV pion decay constant QCD mass scale = l + n 2 0. L = c lmn ( ψψ f 2 πλ )l ( π f π ) m ( µ, m π Λ )n f 2 π Λ 2 i) c lmn of order unity (natural) if of physical significance (S. Weinberg) ii) in principle this should involve a complete set of Lorentz invariants {1, γ µ, γ 5, γ 5 γ µ, σ µν } and the same coupled to isospin iii) maybe only a subset is needed with QCD scaling so far successful for the best forces

23 RMF-PC force PC-F1 C. Const. Magnitude Dim. Order c lmn α S MeV 2 Λ β S MeV 5 Λ γ S MeV 8 Λ δ S MeV 4 Λ α V MeV 2 Λ γ V MeV 8 Λ δ V MeV 4 Λ α T V MeV 2 Λ δ T V MeV 4 Λ TJB, D. G. Madland, J. A. Maruhn, and P.-G. Reinhard, PRC 65 (2002)

24 Extended relativistic point-coupling models add new terms and adjust them with the standard adjustment protocols but with various algorithms complement adjustment protocols by additional observables ( single-particle energies, spin-orbit and/or pseudo-spin splittings, energies / radii / neutron radii,... ) guidance: power counting (QCD scaling)... TJB, D. G. Madland, J. A. Maruhn, and P.-G. Reinhard, in preparation compare also to work by Furnstahl et al., M. A. Huertas,...

25 nearest minimum is found Bevington (downhill) PC-F1 PC-F for each new force: downhill plus MC runs with max. of 30 walkers and slow cooling many walkers (diffusion Monte Carlo) large region of the parameter space is explored Simulated annealing (Monte Carlo) PC-F PC-F1 PC-F~

26 Extending RMF-PC mixed terms contribute to two potentials simultaneously L =! 2 S +! 2 V +! 2 S! 2 V V S = 2! S + 2! 2 V! S = (2 + 2! 2 V)! S V V = 2! V + 2! 2 S! V = (2 + 2! 2 S)! V (isoscalar / isovector) scalar / vector potentials become interdependent various effects on energies and the form factor (geometry of the nucleus) density dependent coupling constants

27 Symbolic notation L = L free + L 4f + L hot + L der + L em, L free = ψ(iγ µ µ m)ψ, L 4f = 1 2 α S( ψψ)( ψψ) 1 2 α V( ψγ µ ψ)( ψγ µ ψ) 1 2 α TS( ψ τψ) ( ψ τψ) 1 2 α TV( ψ τγ µ ψ) ( ψ τγ µ ψ), L hot = 1 3 β S( ψψ) γ S( ψψ) γ V[( ψγ µ ψ)( ψγ µ ψ)] 2, L der = 1 2 δ ν 1 S( ν ψψ)( ψψ) 2 δ V( ν ψγµ ψ)( ν ψγ µ ψ) 1 2 δ TS( ν ψ τψ) ( ν 1 ψ τψ) 2 δ TV( ν ψ τγµ ψ) ( ν ψ τγ µ ψ), L em = ea µ ψ[(1 τ3 )/2]γ µ ψ 1 4 F µνf µν. We can rewrite it using symbolic notation: (1) L = S 2 + V 2 + S 2 T + V 2 T + S3 + S 4 + V 4 + derivative terms + Coulomb force (2)

28 inverse Order in Λ (yet) no tensor terms, no isovector-scalar terms 0 S 2, V 2, V T 2,... 1 S 3, SV 2, SV T 2,... 2 S 4, V 4, S 2 V 2, S 2 V T 2, V 2 VT 2,... 3 S 5,... 4 S 6, V 6, organizing scheme perspective phenomenology perspective

29 L(1) = L S 3 L(2) = L(1) + S 6 + V 6 L(2 ) = L + S 5 + S 6 + V 6 L(3) = L(1) V 4 + S 6 L(3 ) = L V 4 + S 5 + S 6 L(3 ) = L V 4 + S 5 L(4) = L + S 5 L(5) = L + S 5 + S 6 L(6) = L(1) + S 2 V 2 L(6 ) = L + S 2 V 2 L(7) = L(1) + V 2 VT 2 L(7 ) = L + V 2 VT 2 L(8) = L(1) + S 2 V 2 + V 2 VT 2 L(8 ) = L + S 2 V 2 + V 2 VT 2 L(9) = L(1) + S 2 V 2 + S 2 VT 2 + V 2 VT 2 L(9 ) = L + S 2 V 2 + S 2 VT 2 + V 2 VT 2 L(10) = L + SV 2 + SVT 2 L(11) = L V 4 + SV 2 + SVT 2 L(12) = L V 4 S 4 + SV 2 + SVT 2 L(13) = L(1) + S 2 V 2 + S 2 VT 2 + V 2 VT 2 + S6 L(13 ) = L + S 2 V 2 + S 2 VT 2 + V 2 VT 2 + S5 + S 6 L(14) = L(1) + S 2 V 2 + S 2 VT 2 + V 2 VT 2 + S6 + V 6 L(14 ) = L + S 2 V 2 + S 2 VT 2 + V 2 VT 2 + S5 + S 6 + V 6 (1) Extensions - 3rd and 4th order mixed terms - mixings include isoscalar/ isovector and scalar/vector terms - 5th and 6th order scalar and vector (6th only) terms - no isovector-scalar terms since nuclear ground-state observables only determine the sum of (linear) isovectorvector and isovector-scalar terms

30 force # cc (mf) # w.d. χ 2 dof χ 2 pt χ 2 tot χ 2 BE χ 2 ff adjustment PC-F MC+B PC-F ISO L(3 ) B (sp1 ) L(4) B (sp1 ) L(5) B (sp1 ) L(6 ) B (sp1) L(6 ) B+C (sp1 ) L(6 ) MC 1 L(7 ) B+C (sp1 ) L(7 ) B (sp1) L(8 ) B (sp1) L(8 ) B+C (sp1p L(9 ) B (sp1 ) L(9 ) MC 1 L(10) B (sp1) L(10 ) B (sp1 L(11) B (sp1) L(11) B (sp1 ) L(13 ) B (sp1 ) L(14 ) B (sp1 )

31 Sn Pb Neutron Number N = Proton Number Neutron Number N = 126 PC-F1 Energies only - % L(6 ) Proton Number excellent for tin! (but never only consider one chain alone... )

32 Fitting to energies only Sn Pb L6 : + S 2 V 2 L3 : - V 4 + S 5, L7 : + V 2 V T Neutron Number Neutron Number 1.0 N = N = Proton Number PC-F1 PC-F1 oe L6 oe L3 oe L7 oe Proton Number improvements so far result from increased compatibility of energy and form factor

33 Neutron matter E/A [MeV] Friedmann & Pandharipande (1981) PC-F1 NL-Z2 SLy6 SkI3 SkM* wrong curvature: generic feature of RMF forces SLy6 has been adjusted to the neutron matter EOS n [fm -3 ] density dependence of asymmetry energy neutron star poperties

34 E/A [MeV] Friedmann & Pandharipande (1981) PC-F1 (99) a 4 = 38 MeV 3 PC-F3 (99) a 4 = 35 MeV - V T L(13 ) (78) a 4 = 32 MeV - SV/SV T /VV T PC fit to energies only a 4 = 32 MeV n [fm -3 ] neutron matter EOS not determined by nuclear ground-state observables extended models may provide enough freedom to simultaneously describe neutron matter and finite nuclei some of them: better chisquared at the cost of smaller (and too small) a 4 possible next steps: adjustment to both nuclei and neutron matter calculations / isovectorsensitive observables

35 with a few additional parameters enhancements are possible (yet) no dramatic improvements have been obtained so far form factor vs. binding energies neutron matter description may be possible isovector properties are still an issue new freedom demands more (new / different) observables density dependence - powers of k F vs. powers of density? isovector-scalar terms in nuclear ground states (effective mass splittings, ls-, pseudo-spin splittings)?