Gogny’s pairing forces in covariant density functional theory ISTANBUL-06 Bruyères-le-Châtel, Dec. 10, 2015 Technical University Munich Excellence Cluster “Universe” Peking University Peter Ring 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Dogmata in nuclear physics: A quantum field theory has to be renormalizable Forces have to be „phase-shift equivalent“ Nuclei are non-relativistic systems DFT has to be derived from a Hamiltonian DFT has to be local … 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Dogmata in nuclear physics: A quantum field theory has to be renormalizable Forces have to be „phase-shift equivalent“ Nuclei are non-relativistic systems DFT has to be derived from a Hamiltonian DFT has to be local … 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Dogmata in nuclear physics: A quantum field theory has to be renormalizable Forces have to be „phase-shift equivalent“ Nuclei are non-relativistic systems DFT has to be derived from a Hamiltonian DFT has to be local … Egido-poles ! 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Dogmata in nuclear physics: A quantum field theory has to be renormalizable Forces have to be „phase-shift equivalent“ Nuclei are relativistic systems DFT has to be derived from a Hamiltonian DFT has to be local … 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel My conclusions: Nuclei are relativistic systems (Covariant DFT) The concept of one Hamiltonian for mean field and pairing does not work in relativistic systems Pairing is a non-relativistic property We use Gogny-pairing together with a relativistic DFT There is a separable pairing force equivalent to Gogny Finite range is very important for pairing 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Density functional theory in nuclei ● The nuclear chart is two-dimensional: isospin (p/n) ● Magic numbers need large spin-orbit: spin (↑↓) ● Most of the nuclei have open shells: pairing (u,v) ● Nuclei are relativistic systems: components (f,g) We deal with spinors of dimension 16 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Covariant DFT is based on the Walecka model Dürr and Teller, Phys.Rev 101 (1956) Walecka, Phys.Rev. C83 (1974) Boguta and Bodmer, Nucl.Phys. A292 (1977) The nuclear fields are obtained by coupling the nucleons through the exchange of effective mesons through an effective Lagrangian. (J,T)=(0+,0) (J,T)=(1-,0) (J,T)=(1-,1) sigma-meson: attractive scalar field omega-meson: short-range repulsive rho-meson: isovector field 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Covariant DFT is based on the Walecka model Dürr and Teller, Phys.Rev 101 (1956) Walecka, Phys.Rev. C83 (1974) Boguta and Bodmer, Nucl.Phys. A292 (1977) This model has only 4 parameters (J,T)=(0+,0) (J,T)=(1-,0) (J,T)=(1-,1) gσ mσ gρ gω We need in addition density dependence: gm(ρ) and pairing We do not need: t3 with strong repulsion spin-orbit time-odd terms 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Relativistic Kohn-Sham equations: S(r) ≈ -400 MeV V(r) ≈ 350 MeV scalar potential: vector potential: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel scalar potential: S(r) ≈ -400 MeV V(r) ≈ 350 MeV vector potential: Fermi sea Dirac sea 2m* ≈ 1100 MeV V-S ≈ 750 MeV V+S ≈ 50 MeV 2m ≈ 1900 MeV continuum 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? In medium QCD-sum rules relate the scalar condensate and the quark density to the scalar and vector self-energies of the nucleon in the medium: 1) In QCD we have very large scalar and vector fields Cohen, Furnstahl, Griegel, PRC 67 (1992) 961 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? 2) Similar fields in the Walecka model: Walecka, Ann.Phys. (1974) The EoS in the σω-model depends on two parameters Gσ and Gω. They are determined by the density ρ0 and the binding energy E/A at saturation 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? 2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV Walecka, Ann.Phys. (1974) The EoS in the σω-model depends on two parameters Gσ and Gω. They are determined by the density ρ0 and the binding energy E/A at saturation 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? 2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV Walecka, Ann.Phys. (1974) The EoS in the σω-model depends on two parameters Gσ and Gω. They are determined by the density ρ0 and the binding energy E/A at saturation This gives the proper spin-orbit splitting in finite nuclei 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? Large fields V ≈ 350 MeV, S ≈ – 400 MeV Large spin-orbit splittings in nuclei (a factor -40) Success of relativistic Brueckner calculations Success of intermediate energy proton scattering Relativistic saturation mechanism Consistent treatment of time-odd fields Natural explanation of pseudospin symmetry s Connection to underlying theories ? Use as many symmetries as possible in phenomenology Ch. Fuchs No three-body forces Coester-line 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? Large fields V ≈ 350 MeV, S ≈ – 400 MeV Large spin-orbit splittings in nuclei Success of relativistic Brueckner calculations A relativistic saturation mechanism: No parameter t3 The σ-field is the origin of attraction. Its source is the scalar density: In the non-rel. case, Hartree with Yukawa forces would lead to collapse 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? Large fields V ≈ 350 MeV, S ≈ – 400 MeV Large spin-orbit splittings in nuclei Success of relativistic Brueckner calculations A relativistic saturation mechanism: Consistent description of time-odd fields: + - nuclear magnetism scalar potential vector potential (time-like) vector potential (space-like) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Moments of inertia in rotating nuclei: Skyrme RMF NL1 Dobaczewski, Dudek, PRC (1995) Afanasjev, P.R. PRC (1996) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? 1) Relativistic kinematic is not important in nucl. structure 2) Large fields V ≈ 350 MeV , S ≈ – 400 MeV 3) Large spin-orbit splittings in nuclei 4) Success of relativistic Brueckner calculations 5) Relativistic saturation mechanism 6) Consistent treatment of time-odd fields 7) Natural explanation of pseudospin symmetry 9) Success of intermediate energy proton scattering 10) Use as many symmetries as possible in phenomenology 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Pairing in a relativistic quantum field theory: H. Kucharek et al, ZPA 1991 Leads to Relativistic Hartree-(Fock)-Bogoliubov Theory normal density anomaleous density (pairing tensor) where: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel neglect retardation π,δ,η = + 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Pairing in nuclear matter: RMF+BCS Gap equation: e.g. ω-meson: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

The pairing gap at the Fermi surface maximal pairing at the Fermi surface: -------------------------------------------------- potential ΔF(MeV) kF(fm-1) Bonn A 2.80 0.76 Bonn B 2.84 0.76 Bonn C 2.83 0.76 Gogny D1S 2.78 0.80 free NN-forces, which reproduce the phase shift in the 1S0 channel give pairing similar to the Gogny force 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Contributions of the various mesons in the Bonn-potential to pairing: M. Serra, A. Rummel, P. R., PRC 65 (2002) 014304 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel σ-ω model All relativistic forces, overestimate nuclear pairing by a factor 3, because of the very large cut off in momentum space 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Pairing matrix elements in the σ-ω ´model vpp(k,p=0.8) mσ = 520 MeV mσ = 390 MeV k[fm-1] H. Kucharek, P. R., Z. Phys. A 1991 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Relativistic Hartree-Bogoliubov therefore we neglect total scalar vector time-like vector spacelike M. Serra, P. R., PRC 65 (2002) 064324 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Hybride model: RMF + Gogny pairing Gonzales-Llarena, Egido et al, PLB 379, 13 (1996) pairing in superdeformed bands: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Separable ansatz fitted to Gogny force perfect agreement for nuclear matter P(k) Gaussian form P(k) obtained by mapping 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

In finite nuclei: comparison with Gogny: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

dynamic results for the new force: comparison with Gogny: comparison with experiment: Y. Tian, Z.Y. Ma, P.R. (2007) A. Ansari PLB 623, 37 (2005) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Comparison between Gogny- and δ-pairing: 244Pb Tian, Ma, P.R. PLB 676, 44 (2009) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Influence of pairing on fission barriers: 240Pu Shell effects are washed out with increasing pairing Lalazissis et al PLB 689, 72 (2010) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Influence of the pairing window on fission barriers: Barrier height as a function of the ground state gap For Gogny pairing and for δ-pairing with various Pairing windows. Lalazissis et al PLB 689, 72 (2010) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Influence of the pairing window on fission barriers: For each pairing window the strength is adjusted such that the ground state gap ∆gs stays the same 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Summary and outlook: Nuclei are relativistic systems: Covariant DFT No Hamiltonian common for pairing and bulk properties Pairing is a non-relativistic phenomenon RHB with Gogny-pairing is very successful A simple separable form for Gogny-pairing Applications for fission barriers: strong dependence on pairing correlations strong dependence on the pairing window for δ-pairing Finite range pairing increases predictive power Next problem: implement separable pairing into Skyrme calculations finite range pairing + zero range mean field 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel M. Serra† (TUM, Munich, Tokyo Univ.) H. Kucharak (TUM, Munich) J. Koenig (TUM, Munich) J. L. Egido (UAM, Madrid) L. Robledo (UAM, Madrid) A.V. Afanasjev (Mississippi State) Y. Tian (CIAE, Beijing) Z.Y. Ma (CIAE, Beijing) S. Karatzikos (Thessaloniki) G.A. Lalazissis (Thessaloniki) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

large gap caused by the repulsive part of the force: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Contributions of the various mesons in the Bonn-potential to pairing: M. Serra, A. Rummel, P. R., PRC 65 (2002) 014304 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Wave functions of the Cooper pair in r-space: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Wave functions of the Cooper pair in momentum space: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Influence of the repulsive core in Bonn-pot.: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Coherence length: M. Serra, A. Rummel, P. R., PRC 65 (2002) 014304 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Density functional: Density functional 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel normal deformed bands in the rare earth region A.V. Afanasjev et al., PRC 62, 054306 (2000) 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Relativistic Kohn-Sham equations: V0(r) Vi (r) time-like space-like S(r) scalar potential: vector potential: The space-like parts V break time reversal symmetry 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Why covariant ? 2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV The EoS in the σω-model depends on two parameters Gσ and Gω. They are determined by the density ρ0 and the binding energy E/A at saturation This gives the proper spin-orbit splitting σω - model J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Pairing in a relativistic quantum field theory: Lagrangian with fermions and mesons : quantization leads to a Hamiltonian 2 Greens functions: 2 equations of motion: elimination of meson fields: relativistic 2-body interaction: neglect retardation 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Relativistic Hartree-Bogoliubov (RHB) theory: Gorkov factorization: direct term exchange term pairing term Relativistic Hartree-Bogoliubov equations: quasiparticle energy pairing field Dirac hamiltonian quasiparticle wave function 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel H. Kucharek et al, ZPA 1991

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel effective pairing forces: nucl. matter: seniority force, constant G zero range; δ-force pairing part of Gogny D1S Gonzales-Llarena et al, PLB 379, 13 (1996) Gogny equivalent separable force: Tian, Ma, P.R. PLB 676, 44 (2009) finite nuclei: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Simplified pairing energy functional: Y. Tian, Z.Y. Ma, P.R. Gogny: separable: 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel Content Covariant density functional theory Pairing in a relativistic quantum field theory BCS-theory with meson-exchange forces Rel. Hartree-Bogoliubov (RHB) with Gogny pairing Applications with zero- and finite-range pairing forces A separable version of Gogny‘s pairing force Fission barriers with zero and finite range pairing 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

Density functional Theory (DFT) Motivation: proton number Z Density functional Theory (DFT) Shell model Coupled cluster Ab initio neutron number N 16:58 First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel