Relativistic point-coupling models for finite nuclei

Transcription

1 Relativistic point-coupling models for finite nuclei 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 9/23/04, Seattle

2 Outline the relativistic mean-field point-coupling (RMF-PC) approach to nuclear ground states extensions of the model some trade-offs additional nonlinear terms asymmetry energy neutron matter exchange conclusions observables binding energy, radii, surface, single-particle energies, shell structure,... naturalness

3 Motivation Successful models with point couplings Predictive power comparable to Waleckatype relativistic mean-field (RMF) models with meson exchange (QHD) Features 1 m 2 q 2 1 m 2 + q2 m those we don t speak of 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 Fermi theory NJL model Skyrme-Hartree-Fock Effective Field Theories low momentum: derivative expansion should work does work Covariant framework Large scalar and vector potentials: saturation Spin-orbit automatic and of right size!

4 !hart of "e Nuclide#? Density [1/fm^3] z [fm] r [f m] RIA, GSI,... Thanks to my sister

5 E EFT low energy degrees of freedom Strings? even more crazy stuff? QCD DFT Hohenberg-Kohn: full many particle ground state is a unique functional of its density symmetries, power counting, scales, truncation approximations, parameters,... Kohn-Sham equations [ h ] 2m!2 +V int +V ext +V xc " i = # i " i! = " # i 2 i self-consistency relativistic Kohn-Sham scheme L(!,",#,$,...) L(!,!!,%(!!),!& µ!,...)

6 Philosophy of self-consistent point-coupling mean-field models for nuclei Construct an Effective Contact Interaction (or an Energy Functional) between point-like nucleons Introduce approximations and solve the reduced problem selfconsistently numerically Adjust the coupling constants introduced through the interaction (>=9, 2 for pairing): a force is born (biased) There is no best force : the adjustment procedure determines the predictive power for various observables Predict nuclear ground-state observables throughout the known nuclear chart and extrapolate

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

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) 9 parameters L em = ea µ ψ[(1 τ3 )/2]γ µ ψ 1 4 F µνf µν +BCS +! f orce(volume)pairing

9 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

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

11 Different selections of nuclei Sn Pb Neutron Number Neutron Number missing groundstate correlations,... in light nuclei 1.0 N = N = Proton Number PC-F1 Z >= Sn Z <= Zr Z <= Ni Proton Number TJB, D. G. Madland, and P.-G. Reinhard, to be published in NPA

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

13 He [Feynman] said, I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk. (told by Dyson) future-generation energy functional? RMF-PC? Basis (density, k F,...)? Terms? Observables? Parameters?...?

14 asymmetry energy - a 4 = 38 MeV in RMF-FR/RMF-PC from adjustments to finite nuclei: why? - empirical values rather hint at 32 MeV ( SHF) - correlation between surface region of nucleus and a 4 - only one isovector parameter in RMF-FR (2 in RMF-PC)

15 a 4 (MeV) a TV = 1.28 fm 2 a TV = 1.30 a TV = 1.32 a TV = 1.34 a TV = 1.36 a TV = 1.38 a TV = 1.40 a TV = 1.42 a TV = 1.44 a TV = 1.46 a TV = 1.48 a TV = 1.50 no T=1 terms finite nuclei fits V TV =! TV " TV + # TV " 3 TV alpha TV grid, all others kept fixed (PC-F3) (fm -3 )

16 a 4 (MeV) a TV = 1.28 fm 2 a TV = 1.30 a TV = 1.32 a TV = 1.34 a TV = 1.36 a TV = 1.38 a TV = 1.40 a TV = 1.42 a TV = 1.44 a TV = 1.46 a TV = 1.48 a TV = 1.50 alpha TV grid, gamma TV adjusted, all others fixed!" 2 /" 2 4/ (fm -3 ) chisquared of PC-F3 is reference

17 gamma TV adjusted all adjusted!" 2 /" 2 4/100!" 2 /" 2 0 a 4 (MeV) a TV = 1.28 fm 2 a TV = 1.30 a TV = 1.32 a TV = 1.34 a TV = 1.36 a TV = 1.38 a TV = 1.40 a TV = 1.42 a TV = 1.44 a TV = 1.46 a TV = 1.48 a TV = 1.50 a 4 (MeV) a TV = 1.28 fm 2 a TV = 1.30 a TV = 1.32 a TV = 1.34 a TV = 1.36 a TV = 1.38 a TV = 1.40 a TV = 1.42 a TV = 1.44 a TV = 1.46 a TV = 1.48 a TV = (fm -3 ) (fm -3 ) entanglement of isoscalar/isovector channels

18 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,...

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

20 QCD scaling scale the Lagrangian using two scales: f π = 93.5 MeV Λ = 770 MeV pion decay constant QCD mass scale L = c lmn ( ψψ f 2 πλ )l ( π f π ) m ( µ, m π Λ )n f 2 π Λ 2 first application to nuclei in RMF-PC: J. L. Friar, D. G. Madland, and B. W. Lynn, PRC 53 (1996) 3085 = l + n 2 0. 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 best forces to date satisfy QCD scaling

21 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,..., gradients 3 S 5,... 4 S 6, V 6, organizing scheme perspective phenomenology perspective

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

23 E/A [MeV] Friedmann & Pandharipande (1981) PC-F1 Fit-NPC-F1 Fit-NPC-F3 (PC-F1+(TV) 4 ) PC-F1+ (TV) 4 + V 2 (TV) 2 PC-F3 + all mixed T=0/TV terms up to 4th order (3 more than PC-F3) simultaneous adjustment of finite nuclei and neutron matter a 4 = 45 MeV (freedom?!) chisquared = 115 (PC-F1: 98) trends in isotop(n)es OK n [fm -3 ] more to come...

24 Fitting to energies only Sn Pb L6 : + S 2 V 2 L3 : - V 4 + S 5,6 L7 : + V 2 V T Neutron Number N = Proton Number Neutron Number N = 126 PC-F1 PC-F1 oe L6 oe L3 oe L7 oe Proton Number

25 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... )

26 Naturalness? stringent criterion: i) scaled coupling constants ~ 1 ii) max / min < 10 PC-F1: i) fullfilled ii) max / min = 8.92 yet no natural extended set of coupling constants has been obtained i) fullfilled for ~ 80-90% (not all sets have been checked yet) ii) max / min > 18

27 (explicit) Exchange? why bother? - effective interaction but still fermions - correct one-particle limit (no self-interaction error; odd nuclei) - which part of the energy functional is XC? Kohn-Sham functional with exact exchange - Coulomb exchange? - alternate representation: efficient? physical interpretations? rather questions than answers

28 effective toy Lagrangian L =!(i" µ # µ m)! 1 2 (!!) 2 no exchange: Ψ : L : Ψ L =! a " a (i# µ $ µ m)" a 1 2! a ( " a " a ) 2 N=1: the particle feels its own potential with exchange: L =! a " a (i# µ $ µ m)" a 1 2! a ( " a " a ) ! a,b( " a " b )( " b " a ) N=1: free particle - driving out the self-interaction error - sometimes used in functionals for molecules - relevant for odd-even nuclei J. P. Perdew and M. Enzerhof, in Electronic Density Functional Theory, Plenum Press, New York, 1998

29 Fierz Transformations express as ( ψ a Γ i ψ b )( ψ b Γ i ψ a ) c j ( ψ a Γ j ψ a )( ψ b Γ j ψ b ) j (Γ i = 1, γ µ, σ µν, γ 5 γ µ, γ 5 ) Example, ( ψ a ψ b )( ψ b ψ a ) = 1 4 ( ψ a ψ a )( ψ b ψ b ) ( ψ a γ µ ψ a )( ψ b γ µ ψ b ) ( ψ a σ µν ψ a )( ψ b σ µν ψ b ) ( ψ a γ 5 γ µ ψ a )( ψ b γ µ γ 5 ψ b ) ( ψ a γ 5 ψ a )( ψ b γ 5 ψ b )

30 The 4-fermion model L = 1 2 α S( ψψ) α V ( ψγ µ ψ)( ψγ µ ψ) 1 2 α T S( ψ τψ) ( ψ τψ) 1 2 α T V ( ψγ µ τψ) ( ψγ µ τψ) T = 0 T = 1 Φ : L HF : Φ = 1 2 α Sρ S α V ρ V α T Sρ T S α T V ρ T V 2 exchange terms α Sρ Sex α V ρ Vex α T Sρ T Sex α T V ρ T Vex 2 L HF = 1 2 α Sρ S α V ρ V α T Sρ T S α T V ρ T V 2 + T erms(γ 5, γ µ γ 5, σ µν ) α S 7 8 α S 1 2 α V 3 8 α T S 3 2 α T V α V 1 8 α S α V 3 8 α T S α T V α T S 1 8 α S 1 2 α V α T S α T V α T V 1 8 α S α V α T S α T V

31 Hartree: Hartree-Fock: Hartree-Fock α i = α i α i = f( α j ) Hartree 4244,0 1754,8-734, MeV ,8-5713,0 S V TS TV identical observables, different physics: size of isovector coupling constants is strongly representation dependent (H vs. HF) D. G. Madland, TJB, J. A. Maruhn, and P.-G. Reinhard, NPA 741 (2004) 52

32 Coulomb exchange?

33 Sn Neutron Number Pb Neutron Number N=82 N= Proton Number PC-F1 PC-F1 X switched on PC-EX Proton Number Coulomb exchange in Slater Approximation can be absorbed in the nuclear mean-field (similar for radii)

34 Conclusions - extended relativistic point-coupling models provide many enhancements and possibilities - there still exist some trade-offs - a 4 at saturation density is not the problem anymore - the problem appears to be the density dependence and the entanglement of isoscalar and isovector channels - improved simultaneous description of neutron matter and finite nuclei - we might want to keep explicit exchange as an option to consider - more/different observables needed (isovector, density dependence,...)

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