Nuclear Energy Density Functional from Chiral Two- and Three-Nucleon Interactions N. Kaiser EMMI seminar, 3.May 2012 Introduction: nuclear energy density functional Tool: (improved) density-matrix expansion Chiral two- and three-nucleon interactions Diagrammatic calculation of energy density functional Results for isospin-symmetric nuclear systems Isovector part of nuclear energy density functional Publications: J. Holt, N. Kaiser, W. Weise, Eur. Phys. J. A47 (2011) 128; Eur. Phys. J. A48 (2012) 36. N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Introduction Nuclear energy density functional: many-body method of choice for systematic calculation of medium-mass and heavy nuclei Non-relativistic (parametrized) Skyrme functionals and relativistic mean-field models ( σ + ω -mesons) are widely and successfully used RMF: strong Lorentz scalar and vector mean-fields of opposite sign act coherently to generate nuclear spin-orbit interaction Complementary approach: constrain form of a predictive energy density functional and its couplings by many-body perturbation theory and the underlying two- and three-nucleon interactions Switch from hard-core NN-potentials to low-momentum interactions: with V low − k nuclear many-body problem becomes more perturbative Non-local Fock contributions to energy: approximate them by functionals expressed in terms of local densities and currents only Key ingredient: Density-matrix expansion Negele and Vautherin, Phys. Rev. C5 (1972) 1472 Gebremariam, Bogner, Duguet, Nucl. Phys. A851 (2011) 17: used N 2 LO chiral NN-potential + Skyrme → got small but systematic reduction of χ 2 This work: Improved chiral NN-potential N 3 LO + chiral 3N-interaction N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Improved density-matrix expansion Improved density-matrix expansion via phase-space averaging: Gebremariam, Duguet and Bogner, Phys. Rev. C82 (2010) 14305 a a j 1 ( ak f ) − a r − � r + � = 3 ρ � τ − 3 f − 1 � � � � � j 1 ( ak f ) 5 ρ k 2 � Ψ † ∇ 2 ρ � � � Ψ α ak f 2 k f α 2 2 4 α + 3 i j 1 ( ak f ) � a × � J ) + . . . σ · ( � 2 ak f f / 3 π 2 local nucleon density, � ρ = 2 k 3 J = � α i � σ × � α Ψ † ∇ Ψ α spin-orbit density Few % accuracy for Fock contrib. from central and tensor interactions a × � J ) of Negele-Vautherin DME makes 50 % error Spin-dependent part ( � Fourier-transform: ”medium insertion” for inhomogenous nuclear system p | ) + π 2 � � d 3 r e − i � q · r � � p ,� q ) = � θ ( k f − | � k f δ ′ ( k f − | � p | ) − 2 δ ( k f − | � p | ) Γ( � 4 k 4 f − 3 π 2 � � � τ − 3 f − 1 5 ρ k 2 � δ ( k f − | � p | ) � p × � J ) ∇ 2 ρ σ · ( � × 4 k 4 4 f generalizes step-function θ ( k f − | � p | ) for infinite nuclear matter N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Improved density-matrix expansion Comparison of density-matrix expansions: central interaction N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Improved density-matrix expansion Comparison of density-matrix expansions: tensor interaction INM: quadrupolar deformation of local Fermi-moment. distribution neglected N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Nuclear energy density functional Energy density functional for N = Z even-even nuclei: �� 1 k 2 � � τ − 3 f E [ ρ, τ,� J ] = ρ ¯ E ( ρ ) + 5 ρ k 2 4 M 3 + F τ ( ρ ) 2 M − f ∇ ρ ) 2 F ∇ ( ρ ) + � J 2 F J ( ρ ) ∇ ρ · � J F so ( ρ ) + � + ( � J 2 term effective nucleon mass M ∗ ( ρ ) , surface term, spin-orbit coupling, � Relation to slope of single-particle potential at Fermi surface: ∂ U ( p , k f ) = − k f 1 � F τ ( ρ ) = 3 π 2 f 1 ( k f ) � 2 k f ∂ p � p = k f i.e. same effective nucleon mass M ∗ ( ρ ) as in Fermi-liquid theory ∇ ρ ) 2 emerges directly from interaction Decomposition: for F d ( ρ ) , factor ( � ∂ F τ ( ρ ) F ∇ ( ρ ) = 1 + F d ( ρ ) 4 ∂ρ For zero-range Skyrme force: improved density-matrix expansion and Negele-Vautherin DME give identical results (quadratic p -dependence) Differences expected for long-range 1 π - and 2 π -exchange interaction N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Chiral NN and 3N interactions Preferred 2-body interact.: universal low-momentum NN-potential V low − k Partial wave matrix elements, explicit spin-isospin operators better suited Easier tractable substitute for V low − k : Chiral N 3 LOW potential, Λ= 414MeV Finite-range part of N 3 LOW: one- and two-pion exchange of the form V ( π ) V C ( q ) + � τ 2 W C ( q ) + V S ( q ) + � τ 2 W S ( q ) � � = τ 1 · � τ 1 · � � σ 1 · � σ 2 NN V T ( q ) + � τ 2 W T ( q ) q � q � � σ 1 · � σ 2 · � + τ 1 · � � V SO ( q ) + � τ 2 W SO ( q ) i ( � q × � p ) , � � σ 2 ) · ( � + τ 1 · � σ 1 + � dependence only on momentum transfer q , no quadratic spin-orbit comp. 60 0.3 0 60 50 finite-range central potentials finite-range spin-spin potentials V T (q) 0.2 40 40 W SO (q) 0.1 -0.5 20 30 W S (q) W C (q) 0 20 -4 ] -4 ] 0 -2 ] 3 GeV -1 W T (q) 3 GeV -0.1 -2 ] [GeV 10 [GeV -20 -0.2 0 [10 [10 -1.5 -40 V S (q) -0.3 V SO (q) -10 -60 V C (q) -0.4 -20 -2 finite-range tensor potentials finite-range spin-orbit potentials -80 -30 -0.5 -100 -40 -0.6 -2.5 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 q [MeV] q [MeV] q [MeV] q [MeV] Short-range part: 24 contact terms up 4th power of momenta, C ST , C j , D j determined in fits to NN-phase shifts and deuteron ( → Machleidt’s code) N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Two-body contributions at 1st order p 1 ,� q ) Γ( � p 2 , − � q ) Finite-range pieces: Hartree-Fock, employing Γ � � 1 E ( ρ ) = ρ 2 V C ( 0 ) − 3 ρ ¯ dx x 2 ( 1 − x ) 2 ( 2 + x )[ V C ( q ) + 3 V S ( q ) + q 2 V T ( q ) + ... ] 2 0 � 1 k f F τ ( ρ ) = dx ( x − 2 x 3 )[ V C ( q ) + 3 V S ( q ) + q 2 V T ( q ) + 3 W comb ( q )] 2 π 2 0 F d ( ρ ) = 1 4 V ′′ C ( 0 ) � 1 F so ( ρ ) = 1 2 V SO ( 0 ) + dx x 3 [ V SO ( 2 xk f ) + 3 W SO ( 2 xk f )] 0 � 1 3 F J ( ρ ) = dx � ( 2 x 3 − x )[ V C ( q ) − V S ( q )] − x 3 q 2 V T ( q ) + 3 W comb ( q ) � 8 k 2 f 0 Short-range pieces: 8 ( C S − C T ) + 3 ρ k 2 20 ( C 2 − C 1 − 3 C 3 − C 6 ) + 9 ρ k 4 E ( ρ ) = 3 ρ f f ¯ 140 ( D 2 − 4 D 1 + ... ) 4 ( C 2 − C 1 − 3 C 3 − C 6 ) + ρ k 2 F τ ( ρ ) = ρ f 4 ( D 2 − 4 D 1 − 12 D 5 − 4 D 11 ) 32 ( 16 C 1 − C 2 − 3 C 4 − C 7 ) + k 2 F d ( ρ ) = 1 f 48 ( 9 D 3 + 6 D 4 − 9 D 7 − 6 D 8 + ... ) 8 C 5 + k 2 F so ( ρ ) = 3 f 6 ( 2 D 9 + D 10 ) N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Three-body contributions at 1st order Leading order chiral 3N-interaction: contact + 1 π -exchange + 2 π -exch. LECs c E = − 0 . 625 , c D = 2 . 06 fitted to binding energies of 3 H and 4 He 3-body correlations in inhomogeneous nuclear many-body systems: factorized density-matrices in p -space Γ( � p 1 ,� q 1 ) Γ( � p 2 ,� q 2 ) Γ( � p 3 , − � q 1 − � q 2 ) c E - and c D -terms: c E k 6 f E ( ρ ) = − ¯ 12 π 4 f 4 π Λ χ � u 6 g A c D m 6 3 − 3 u 4 + u 2 8 + u 3 arctan 2 u − 1 + 12 u 2 � E ( ρ ) = ¯ π ln ( 1 + 4 u 2 ) ( 2 π f π ) 4 Λ χ 4 32 F τ ( ρ ) = 2 g A c D m 4 � ( 1 + 2 u 2 ) ln ( 1 + 4 u 2 ) − 4 u 2 � π ( 4 π f π ) 4 Λ χ � 1 g A c D m π 2 u � F d ( ρ ) = 2 u ln ( 1 + 4 u 2 ) − ( 4 f π ) 4 π 2 Λ χ 1 + 4 u 2 3 g A c D m π u = k f � 2 u − 1 1 � F J ( ρ ) = 4 u 3 ln ( 1 + 4 u 2 ) u + , ( 4 f π ) 4 π 2 Λ χ m π N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Three-body contributions at 1st order 2 π -exchange Hartree diagram prop. to c 1 = − 0 . 76 , c 3 = − 4 . 78 (GeV − 1 ) g 2 A m 6 � ( 12 c 1 − 10 c 3 ) u 3 arctan 2 u − 4 3 c 3 u 6 + 6 ( c 3 − c 1 ) u 4 E ( ρ ) ¯ π = ( 2 π f π ) 4 � 1 4 ( 2 c 3 − 3 c 1 ) + 3 u 2 � � +( 3 c 1 − 2 c 3 ) u 2 + 2 ( 3 c 3 − 4 c 1 ) ln ( 1 + 4 u 2 ) � 2 3 g 2 A m π F so ( ρ ) u ( 4 c 1 − 3 c 3 ) − 4 c 3 u = ( 8 π ) 2 f 4 π � 4 u ( c 3 − c 1 ) + 3 c 3 − 4 c 1 u = k f � � ln ( 1 + 4 u 2 ) + , 2 u 3 m π 3-body spin-orbit coupling originally suggested by Fujita and Miyazawa Most tedious to evaluate: 2 π -exchange Fock diagram, c 4 = 3 . 96 GeV − 1 N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Results for isospin-symmetric nuclear systems Energy per particle: for 2-body part V low − k ≃ V N 3 LOW 20 energy per particle 10 0 E( ρ ) [MeV] -10 -20 2-body 3-body total E pot -30 E pot +E kin V low-k -40 0 0.05 0.1 0.15 0.2 -3 ] ρ [fm Improved description: treat 2-body interaction to second order etc. Effective nucleon mass M ∗ ( ρ 0 ) : in phenomenological reasonable range 1 12 effective nucleon mass 2-body 10 3-body 0.9 total 2 ] V low-k F τ ( ρ ) [MeVfm 8 * ( ρ ) / M 0.8 6 M 4 0.7 2 0 0.6 0 0.2 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 -3 ] -3 ] ρ [fm ρ [fm N. Kaiser Nuclear Energy Density Functional from 2N +3N Chiral Interactions
Recommend
More recommend
