Localized form of Fock terms in nuclear covariant density functional theory

Transcription

1 Nuclear Structure and Astrophysical Applications July 8-12, 213, ECT*, Trento, Italy Localized form of Fock terms in nuclear covariant density functional theory Haozhao Liang ùíî RIKEN Nishina Center, Japan Peking University, China July 9, 213

2 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances and more? 4 Summary and Perspectives

3 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances and more? 4 Summary and Perspectives

4 Nuclear charge-exchange excitations Nuclear charge-exchange excitations β-decay charge-exchange reactions These excitations are important to understand spin and isospin properties of the in-medium nuclear interaction Osterfeld:1992 effective nucleon-nucleon tensor forces Bai:21 neutron skin thickness Krasznahorkay:1999, Vretenar:23, Yako:26 β-decay rates of nuclei in r-process path Engel:1999, Borzov:26 inclusive neutrino-nucleus cross sections Kolbe:23, Vogel:26, Paar:28 ββ-decay rates Ejiri:2, Avignone:28 isospin corrections for superallowed β deacys Towner & Hardy:21, HL:29 They play crucial roles in many important issues in nuclear physics, astrophysics, and particle physics.

5 Roles in nuclear physics To investigate the spin and isospin properties of nuclear interaction in medium Gamow-Teller and spin-dipole resonances R - ( 1 / M e V ) H F e x p. e x p. 9 Z r G T R 2 P K O E ( M e V ) HL, Giai, Meng, PRL 11, (28); HL, Zhao, Meng, PRC 85, 6432 (212) Data: Wakasa et al., PRC 55, 299 (1997); 84, (211)

6 Roles in nuclear astrophysics To investigate rapid neutron capture (r-) and rapid proton capture (rp-) processes Nuclear β-decay and electron-capture rates P r o t o n N u m b e r r - p r o c e s s p a t h N e u t r o n N u m b e r (a ) P r o t o n N u m b e r N e u t r o n N u m b e r E E E + 1 E E - 2 (b ) s s s Niu, Niu, HL, Long, Nikšić, Vretenar, Meng, PLB 723, 172 (213); Niu, Niu, Liu, HL, Guo, PRC 87, 5133(R) (213) Data: Audi:23 with NNDC update; Nishimura et al., PRL 16, 5252 (211)

7 Roles in particle physics To testify the unitarity of Cabibbo-Kobayashi-Maskawa matrix: V ud Eronen s talk Superallowed β decays & theoretical isospin corrections V u d P D G L i a n g e t a l. S a t u l a e t a l Y e a r HL, Giai, Meng, PRC 79, (29); Satu la et al., PRL 16, (211); PRC 86, (212) Professor William Marciano: Issue needs complete quantitative resolution at INPC21 plenary talk.

8 Localized form of Fock terms in nuclear covariant density functional theory

9 Covariant density functional theory RH theory The covariant version of DFT takes into account Lorentz symmetry. Vretenar s talk unification of the time-even and time-odd components stringent restrictions on the number of parameters CDFT in Hartree level (RH/RMF theory) has received wide attention due to its successful description of lots of nuclear phenomena. Serot:1986, Ring:1996, Vretenar:25, Meng:26, Paar:27, Nikšić:211 spin-orbit splittings EoS in symmetric and asymmetric nuclear matter ground-state properties of finite spherical and deformed nuclei collective rotational and vibrational excitations low-lying spectra of transitional nuclei involving quantum phase transitions...

10 Covariant density functional theory RH theory The covariant version of DFT takes into account Lorentz symmetry. Vretenar s talk unification of the time-even and time-odd components stringent restrictions on the number of parameters CDFT in Hartree level (RH/RMF theory) has received wide attention due to its successful description of lots of nuclear phenomena. Serot:1986, Ring:1996, Vretenar:25, Meng:26, Paar:27, Nikšić:211 spin-orbit splittings EoS in symmetric and asymmetric nuclear matter ground-state properties of finite spherical and deformed nuclei collective rotational and vibrational excitations low-lying spectra of transitional nuclei involving quantum phase transitions... Common problems in the isovector channels [Reinhard s talk] Difficult to disentangle the isovector-scalar (δ) and isovector-vector (ρ) channels, unless a tuning is performed based on selected microscopic calculations. Roca-Maza:211 Nuclear spin-isospin resonances, e.g., GTR and SDR, cannot be described in a fully self-consistent way.

11 RH+RPA for spin-isospin resonances RH+RPA for spin-isospin resonances De Conti:1998, 2, Vretenar: 23, Ma:24, Paar:24, Nik sić:25 example: Gamow-Teller resonance (GTR) in 28 Pb ( S = 1, L =, J π = 1 + ) a. add π-meson b. fit g

12 Covariant density functional theory RHF theory CDFT in Hartree-Fock level (RHF theory) several attempts to include the Fock term in the relativistic framework Bouyssy:1985,1987, Bernardos:1993, Marcos:24 DDRHF theory achieved quantitative descriptions of binding energies and radii Long, Giai, Meng, PLB 64, 15 (26); Long, Sagawa, Giai, Meng, PRC 76, (27); Long, Sagawa, Meng, Giai, EPL 82, 121 (28); Long, Ring, Giai, Meng, PRC 81, 2438 (21); Ebran, Khan, Peña Arteaga, Vretenar, PRC 83, (211) effective mass splitting in asymmetric nuclear matter can be described naturally Long, Giai, Meng, PLB 64, 15 (26) improvement on the descriptions of nuclear shell structures and their evolutions Long, Sagawa, Giai, Meng, PRC 76, (27); Long, Sagawa, Meng, Giai, EPL 82, 121 (28); Long, Nakatsukasa, Sagawa, Meng, Nakada, Zhang, PLB 68, 428 (29) nuclear spin-isospin resonances can be described in a fully self-consistent way HL, Giai, Meng, PRL 11, (28); HL, Giai, Meng, PRC 79, (29); HL, Zhao, Meng, PRC 85, 6432 (212)

13 RHF+RPA for Gamow-Teller resonances Gamow-Teller resonances in 48 Ca, 9 Zr, and 28 Pb H F e x p. 4 8 C a G T R e x p. 9 Z r G T R R - ( 1 / M e V ) e x p. P K O 1 R - ( 1 / M e V ) H F e x p. P K O E ( M e V ) E ( M e V ) GTR excitation energies can be reproduced in a fully self-consistent way. 2 1 HL, Giai, Meng, PRL 11, (28)

14 Physical mechanisms of GTR RH+RPA no contribution from isoscalar mesons (σ, ω), because exchange terms are missing. π-meson is dominant in this resonance. g has to be refitted to reproduce the experimental data RHF+RPA isoscalar mesons (σ, ω) play an essential role via the exchange terms. π-meson plays a minor role. g = 1/3 is kept for self-consistency. HL, Giai, Meng, PRL 11, (28)

15 Physical mechanisms of GTR The characteristics of GT and SD excitations are understood with the delicate balance between the σ- and ω-meson fields via the exchange terms To find a CEDF based on only local potentials, yet keeping the merits of the exchange terms HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212)

16 To construct RH functionals from RHF scheme Possible/promising solution: construct RH functionals from RHF scheme take the constraints introduced by exchange terms of RHF scheme into account We start from an important observation: RHF functional PKO2 [Long:28] well describes the neutron-proton Dirac mass splitting in asymmetric nuclear matter and nuclear spin-isospin resonances only includes σ-, ω-, ρ-mesons, but not π-meson masses of mesons are heavy zero-range approximation is reasonable Fierz transformation: Fock terms local Hartree terms With such RH functional thus obtained, to investigate proton-neutron Dirac mass splitting in neutron matter Gamow-Teller and spin-dipole resonances Goal(s) To verify whether the important effects of exchange terms can be maintained by the mapping from RHF functional to RH functional.

17 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances and more? 4 Summary and Perspectives

18 Relativistic Hartree-Fock theory Covariant density functional theory RHF theory Effective Lagrangian density Bouyssy:1987, Long:26 ( L = [iγ ψ µ µ M g σ σ γ µ g ω ω µ + g ρ τ ρ µ + e 1 τ )] 3 A µ ψ µ σ µ σ 1 2 m2 σσ Ωµν Ω µν m2 ωω µ ω µ 1 4 R µν R µν m2 ρ ρ µ ρ µ 1 4 F µν F µν (1) System Hamiltonian H = T = L φ i φi L (2) Ground-state trial wave function Φ = a c a (3) Energy functional of the system E = Φ H Φ = E k + E D σ + E D ω + E D ρ + E D A +E E σ + E E ω + E E ρ + E E A (4)

19 Zero-range reduction Zero-range reduction Yukawa propagators of the mesons for m i q, D i (r, r ) = 1 e m i r r 4π r r, D i(q) = 1 m 2 i + q 2 (5) D i (q) 1 m 2 i q2 m 4 i + D i (r, r ) 1 m 2 i δ(r r ) (6) within the zero-order approximation. Zero-range reduction of meson-nucleon couplings α HF S = g σ 2, α HF mσ 2 V = g ω 2, α HF mω 2 tv = g ρ 2, (7) mρ 2

20 Fierz transformation Fierz transformation (I) Sixteen Dirac matrices form a complete system O S = 1, O V = γ µ, O T = σ µν, O PS = γ 5, O PV = γ 5 γ µ so that any one can be expressed as a linear superposition of variants with a changed sequence of spinors, (āo i d)( co i b) = k c ik (āo k b)( co k d), (8) with the coefficients c ik in the so-called Fierz table Fierz:1937, Okun:1982, Sulaksono:23 S 1 4 S V T PS PV V T PS PV (9) For the isospin coefficients, δ qa q d δ qc q b = 1 2 [ δqa q b δ qc q d + q a τ q b q c τ q d ], (1a) q a τ q d q c τ q b = 1 2 [ 3δqa q b δ qc q d q a τ q b q c τ q d ]. (1b)

21 Fierz transformation Fierz transformation (II) Fierz transformation: from α HF to α H α H S = αhf S 4 8 αhf V 12 8 αhf tv (11a) α H ts = 1 8 αhf S 4 8 αhf V αhf tv (11b) α H V = 1 8 αhf S αhf V αhf tv (11c) α H tv = 1 8 αhf S αt H = 1 16 αhf S αtt H = 1 16 αhf S α H PS = 1 8 αhf S αhf V αhf V αhf tv αhf tv (11d) (11e) (11f) (11g) α H tps = 1 8 αhf S αhf V 4 8 αhf tv (11h) α H PV = αhf S αhf V αhf tv (11i) α H tpv = αhf S αhf V 2 8 αhf tv (11j)

22 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances and more? 4 Summary and Perspectives

23 Coupling strengths in different channels Nucleon-meson coupling strengths of PKO2 g Starting point: nucleon-meson coupling strengths g σ, g ω, and g ρ of PKO2 P K O ( f m -3 ) g P K O ( f m -3 ) g P K O ( f m -3 ).4.5 Long, Sagawa, Meng, Giai, EPL 82, 121 (28) Zero-range reduction α HF S = g σ 2, α HF mσ 2 V = g ω 2, α HF mω 2 tv = g ρ 2, mρ 2 Fierz transformation α H j = k c jk α HF k,

24 Coupling strengths in different channels RHF equivalent zero-range coupling strengths (I) The RHF equivalent parametrization derived from PKO2 is called PKO2-H S V P K O 2 -H D D -M E 2 S ( f m 2 ) P K O 2 -H D D -M E 2 V ( f m 2 ) 1 t S ( f m 2 ) ( f m -3 ) ts P K O 2 -H t V ( f m 2 ) ( f m -3 ) tv P K O 2 -H D D -M E ( f m -3 ) ( f m -3 ) αs H and αh V are consistent with those of DD-ME2 Lalazissis:25 αts H appears proton-neutron Dirac mass splitting in asymmetric nuclear matter is modified for another delicate balance between ts and tv channels α H tv

25 Dirac mass splitting Dirac mass and its isospin splitting M * D /M P K O 2 - H p r o to n n e u tr o n ( f m -3 ) M * D /M ( M * D, p - M * D, n ) / M.1 D B H F. 5 P K O 2 -H P K O ( f m -3 ) HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212); DBHF: van Dalen, et al., EPJA 31, 29 (27) It is found that MD,p > M D,n. The splitting behavior in neutron matter is quantitatively consistent with the prediction of the Dirac-Brueckner-Hartree-Fock (DBHF) calculations.

26 Dirac mass splitting Dirac mass and its isospin splitting M * D /M P K O 2 - H p r o to n n e u tr o n ( f m -3 ) M * D /M ( M * D, p - M * D, n ) / M.1 D B H F. 5 P K O 2 -H P K O ( f m -3 ) HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212); DBHF: van Dalen, et al., EPJA 31, 29 (27) It is found that MD,p > M D,n. The splitting behavior in neutron matter is quantitatively consistent with the prediction of the Dirac-Brueckner-Hartree-Fock (DBHF) calculations. The constraints introduced by the Fock terms of the RHF scheme into the ts channel of the present density functional are straight forward and quite robust.

27 Spin-isospin resonances RHF equivalent zero-range coupling strengths (II) T P K O 2 -H 1 8 P S P K O 2 -H 1.5 P V P K O 2 -H T ( f m 2 ) 1..5 P S ( f m 2 ) P V ( f m 2 ) ( f m -3 ) ( f m -3 ) ( f m -3 ) tt P K O 2 -H 1 8 tp S P K O 2 -H 1.5 tp V P K O 2 -H t T ( f m 2 ) 1..5 t P S ( f m 2 ) t P V ( f m 2 ) ( f m -3 ) ( f m -3 ) ( f m -3 ) α(t)t H, αh (t)ps, and αh (t)pv are explicitly determined by exchange effects of RHF scheme (t)ps and (t)pv channels vanish in ground-state descriptions due to the parity conservation, but crucial for spin-isospin resonances

28 Spin-isospin resonances Particle-hole residual interactions in charge-exchange channel Particle-hole (ph) residual interactions ts channel: V ts (1, 2) = αts[γ H τ] 1 [γ τ] 2 δ(r 1 r 2 ), (12a) tv channel: V tv (1, 2) = αtv H [γ γ µ τ] 1 [γ γ µ τ] 2 δ(r 1 r 2 ), (12b) tt channel: V tt (1, 2) = αtt[γ H σ µν τ] 1 [γ σ µν τ] 2 δ(r 1 r 2 ), (12c) tps channel: V tps (1, 2) = αtps[γ H γ 5 τ] 1 [γ γ 5 τ] 2 δ(r 1 r 2 ), (12d) tpv channel: V tpv (1, 2) = αtpv H [γ γ 5 γ µ τ] 1 [γ γ 5 γ µ τ] 2 δ(r 1 r 2 ). (12e) For the charge-exchange spin-flip modes, the tt and tpv channels are expected to play the dominant roles, where the operator [σ τ] [σ τ] is sandwiched by the large components of wave functions.

29 Spin-isospin resonances Gamow-Teller resonances R - ( 1 / M e V ) R - ( 1 / M e V ) e x p t. 4 8 C a G T R u n p e r. P K O 2 P K O 2 - H E ( M e V ) 2 8 P b G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) R - ( 1 / M e V ) Z r G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) GTR excitation energies can be well reproduced by the ph residual interactions of PKO2-H. The present results are similar as those by the original RHF+RPA. The difference is due to the zero-range approximation. RHF+RPA: HL, Giai, Meng, PRL 11, (28)

30 Spin-isospin resonances Gamow-Teller resonances in 28 Pb R - ( 1 / M e V ) P b G T R P K O 2 - H P K O 2 D D - M E 2 w i t h M i g d a l D D - M E 2 w / o M i g d a l e x p t E ( M e V ) Strength distribution of the GTR in 28 Pb. HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212) It is fitted for DD-ME2 by adjusting the Migdal term. RHF+RPA: HL, Giai, Meng, PRL 11, (28) RH+RPA: Paar, Nikšić, Vretenar, Ring, PRC 69, 5433 (24)

31 Spin-isospin resonances Physical mechanisms of GTR d i a g o n a l m a t r i x e l e m e n t s ( M e V ) t o t a l σ ω ρ t o t a l t S t V t T t P S t P V 2 8 P b G T R ( ν 1 h 1 1 /2 ) - 1 ( π 1 h 9 /2 ) 1 t o t a l ρ M i g d a l π P V P K O 2 P K O 2 -H D D -M E 2 In RHF+RPA (PKO2), σ- and ω-mesons play the most important role in determining the properties of GTR via the exchange terms. In the RHF equivalent RPA (PKO2-H), tt and tpv channels are most important, their coupling strengths are intrinsically determined by Fierz transformation. In conventional RH+RPA (DD-ME2), the free π residual interaction is attractive and the Migdal term is repulsive by fitting to experimental data.

32 Spin-isospin resonances Physical mechanisms of GTR by PKO2-H d i a g o n a l m a t r i x e l e m e n t s ( M e V ) t o t a l σ ω ρ t o t a l t S t V t T t P S t P V 2 8 P b G T R ( ν 1 h 1 1 /2 ) - 1 ( π 1 h 9 /2 ) 1 t o t a l ρ M i g d a l π P V P K O 2 P K O 2 -H D D -M E 2 The dominant ph interactions are the operator [σ τ] [σ τ] sandwiched by large components of w.f., i.e., [σ ij τ] [σ ij τ] 2αtT H and [γ 5γ i τ] [γ 5 γ i τ] αtpv H The net contribution is then proportional to 2αtT H αtpv H = 1 ( ) α H 3 S + αv H Total ph strengths are determined by the delicate balance between S and V channels.

33 Spin-isospin resonances SDR in localized RHF equivalent RPA R - ( f m 2 / M e V ) Z r S D R e x p t. ( a ) e x p t. R H ( b ) s u m R H F -lo c 3 e x p t. ( c ) 2 1 R H F E ( M e V ) For RHF-loc, not only the total strengths but also the individual contribution from different spin-parity J π components are almost identical to RHF. The energy hierarchy E(2 ) < E(1 ) < E( ) can be obtained naturally in the present RPA calculations in the local scheme. HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212) exp: Yako, Sagawa, Sakai, PRC 74, 5133(R) (26)

34 Spin-isospin resonances SDR in localized RHF equivalent RPA R - ( f m 2 / M e V ) Z r S D R e x p t. ( a ) e x p t. R H ( b ) s u m R H F -lo c 3 e x p t. ( c ) 2 1 R H F E ( M e V ) For RHF-loc, not only the total strengths but also the individual contribution from different spin-parity J π components are almost identical to RHF. The energy hierarchy E(2 ) < E(1 ) < E( ) can be obtained naturally in the present RPA calculations in the local scheme. HL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212) exp: Yako, Sagawa, Sakai, PRC 74, 5133(R) (26) The constraints introduced by the Fock terms of the RHF scheme into the ph residual interactions are straight forward and robust.

35 and more? Coulomb exchange term Slater approximation for Coulomb exchange term (Relativistic LDA) Engel & Dreizler, DFT: an Advanced Course E RLDA Cex = 3 4 ( ) 1/3 3 e 2 π Relative deviations E Cex = (E LDA Cex d 3 rρ 4/3 p [ ECex exact )/ECex exact (3π 2 ρ p ) 2/3 ]. M 2 E C e x ( % ) C a N i Z r R L D A N R L D A A Gu, HL, Long, Giai, Meng, PRC 87, 4131(R) (213) S n P b The exact Coulomb exchange energies can be reproduced by RLDA within 5% from drip line to drip line. The relativistic corrections improve the agreement with the exact results by 3 5%.

36 and more? CDFT + FAM (I) To include the deformation degree of freedom Peña Arteaga:28,29 CDFT + finite amplitude method (FAM): feasibility? advantages? FAM: Nakatsukasa:27 R ( 1 3 f m 4 / M e V ) (a ) e x p t. 2 8 P b I S G M R m - F A M ( n o s e a ) m - F A M ( f u l l ) i - F A M D D -P C E ( M e V ) R ( 1 3 f m 4 / M e V ) (a ) 2 8 P b I S G M R i - F A M ( n o r e a r. ) m - F A M ( n o r e a r. ) m - F A M ( f u l l ) D D -P C E ( M e V ) HL, Nakatsukasa, Niu, Meng, PRC 87, 5431 (213) The effects of the Dirac sea can be automatically taken into account in the coordinate-space representation. The rearrangement terms can be implicitly calculated without extra computational costs.

37 and more? CDFT + FAM (II) FAM for the charge-exchange channels involving proton-neutron mixing e x p t. 4 8 C a I A S R P A i - F A M P b I A S R P A i - F A M e x p t. R - ( 1 / M e V ) 3 2 R - ( 1 / M e V ) D D -P C E ( M e V ) 5 D D -P C E ( M e V ) Isobaric analog states in 48 Ca and 28 Pb by i-fam and conventional RPA.

38 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances and more? 4 Summary and Perspectives

39 Summary and Perspectives Summary A new method is proposed to take into account the Fock terms in local covariant density functionals. The advantages of existing RH functionals can be maintained, while the problems in the isovector channel can be solved. The neutron-proton Dirac mass splitting in asymmetric nuclear matter is in a very good agreement with the prediction of DBHF. The properties of GTR and SDR can be reproduced in a natural way. Perspectives This opens a new door for the development of nuclear local covariant density functionals with proper isoscalar and isovector properties in the future.

40 Acknowledgments In collaboration with Huaiqiang Gu Jianyou Guo Quan Liu Wenhui Long Jie Meng Takashi Nakatsukasa Tamara Nikšić Yifei Niu Zhongming Niu Peter Ring Xavier Roca-Maza Nguyen Van Giai Dario Vretenar Pengwei Zhao Lanzhou University, China Anhui University, China Anhui University, China Lanzhou University, China Peking University, China RIKEN, Japan University of Zagreb, Croatia China Academy of Engineering Physics, China Anhui University, China Technische Universität München, Germany INFN-Milano, Italy IPN-Orsay, France University of Zagreb, Croatia Peking University, China