Fine structure of nuclear spin-dipole excitations in covariant density functional theory

1 o3iø(œ April 12 16, 2012, Huzhou, China Fine structure of nuclear spin-dipole excitations in covariant density functional theory ùíî (Haozhao Liang) ŒÆÔnÆ 2012 c 4 13 F ÜŠöµ Š # Ç!Nguyen Van Giai Ç!ë+ Æ

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

Nuclear spin-isospin resonances Nuclear charge-exchange excitations β-decay charge-exchange reactions These excitations play important roles in spin and isospin properties of the in-medium nuclear interaction neutron skin thickness Krasznahorkay:1999PRL, Vretenar:2003PRL, Yako:2006PRC β-decay rates of nuclei in r-process path Engel:1999PRC, Borzov:2006NPA ββ-decay rates Ejiri:2000PhysRep, Avignone:2008RMP inclusive neutrino-nucleus cross sections Kolbe:2003JPG, Vogel:2006NPA isospin corrections for superallowed β deacys Towner & Hardy:2010RPP, HZL:2009PRC... Nuclear spin-isospin resonances become one of the central topics in nuclear physics, particle physics, and astrophysics.

SD excitations and resolving different J π components Spin-dipole (SD) excitations with S = 1 and L = 1 have attracted more and more attentions due to their connection with neutron-skin thickness Krasznahorkay:1999PRL cross section of neutrino-nucleus scattering Kolbe:2003JPG, Vogel:2006NPA ββ-decay rates Ejiri:2000PhysRep Different from Gamow-Teller (GT) excitations having single J π = 1 + component, SD excitations are composed of three collective components with spin-parity J π = 0, 1 and 2. Resolving the J π = 0, 1, 2 components is crucial to understand strengths of the nucleon-nucleon effective tensor interactions Bai:2010PRL,2011PRC multipole-dependent effects on the neutrino-nucleus scattering Lazauskas:2007NPA and the 0νββ decays Šimkovic:2008PRC, Fang:2011PRC

Microscopic theories for SD excitations Shell models (A 60) Caurier:2005RMP Random Phase Approximation (RPA) based on density functional theories traditional (non-relativistic) density functional Auerbach:1984PRC, Bai:2010RPL,2011PRC covariant (relativistic) density functional RHF+RPA Excellent agreement with the GT resonances data was obtained without any readjustment of the covariant density functional. HZL, Giai, Meng, PRL 101, 122502 (2008)

In this work In this work Self-consistent RPA approach based on the RHF theory will be applied to investigate the SD excitations. Results will be compared to the fine structure provided by the most up-to-date experiments in 16 O. RH+RPA results will be also shown to present the effects of the exchange terms.

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

Relativistic Hartree-Fock theory Covariant density functional theory RHF theory Effective Lagrangian density Bouyssy:1987PRC, Long:2006PLB ( 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 f ] π γ 5 γ µ µ π τ 2 m π + 1 2 µ σ µ σ 1 2 m2 σσ 2 1 4 Ωµν Ω µν + 1 2 m2 ωω µ ω µ 1 4 R µν R µν + 1 2 m2 ρ ρ µ ρ µ + 1 2 µ π µ π 1 2 m2 π π π 1 4 F µν F µν (1) ψ Energy functional of the system E = Φ 0 H Φ 0 = 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 π + E E A (2)

Random Phase Approximation Random Phase Approximation RPA equations Ring & Schuck:1980 A B B A X Y = ω ν X Y (3) where the matrix elements of particle-hole residual interactions read A = (E A E a )δ AB δ ab (E α E a )δ αβ δ ab + f Af b V f B f a f a f B f A f b V f β f a f a f β, f α f b V f B f a f a f B f α f b V f β f a f a f β B = f Af B V f b f a f a f b f A f β V f b f a f a f b f α f B V f b f a f a f b f α f β V f b f a f a f b (4a) (4b) Particle-hole (ph) residual interactions in self-consistent RPA derived from the second derivative of the energy functional with rearrangement terms, if the meson-nucleon couplings are density-dependent

RHF+RPA RHF+RPA in charge-exchange channel Particle-hole residual interactions σ-meson: V σ (1, 2) = [g σ γ 0 ] 1 [g σ γ 0 ] 2 D σ (1, 2) (5a) ω-meson: V ω (1, 2) = [g ω γ 0 γ µ ] 1 [g ω γ 0 γ µ ] 2 D ω (1, 2) (5b) ρ-meson: V ρ (1, 2) = [g ρ γ 0 γ µ τ] 1 [g ρ γ 0 γ µ τ] 2 D ρ (1, 2) (5c) pseudovector π-n coupling: V π (1, 2) = [ f π m π τγ 0 γ 5 γ k k ] 1 [ f π m π τγ 0 γ 5 γ l l ] 2 D π (1, 2) (5d) zero-range counter-term of π-meson: V πδ (1, 2) = g [ f π m π τγ 0 γ 5 γ] 1 [ f π m π τγ 0 γ 5 γ] 2 δ(r 1 r 2 ), g = 1/3 (5e) π-meson is included naturally. g = 1/3 in the zero-range counter-term of π-meson is maintained for the sake of self-consistency.

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

RHF+RPA for Gamow-Teller resonances Gamow-Teller resonances in 48 Ca, 90 Zr, and 208 Pb 12 14 10 12 8 10 6 8 4 6 4 2 2 0 0 5 10 15 20 25 30 0 0 5 10 15 20 25 30 60 50 40 30 GTR excitation energies can be reproduced in a fully self-consistent way. 20 10 0 0 5 10 15 20 25 30 HZL, Giai, Meng, PRL 101, 122502 (2008)

Physical mechanisms of GTR 60 50 40 30 20 10 0 0 5 10 15 20 25 30 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. 60 50 40 30 20 10 0 0 5 10 15 20 25 30 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. HZL, Giai, Meng, PRL 101, 122502 (2008)

Spin-dipole resonances Main peak can be reproduced by RHF+RPA exp. Yako:2006PRC Energy hierarchy RHF+RPA: E(2 ) < E(1 ) < E(0 ) agree with SHF+RPA Fracasso:2007PRC RH+RPA: E(2 ) < E(0 ) < E(1 ) Separating experimentally the different components from the total transition strength would be helpful to evaluate the theoretical predictive power, e.g., SDR in 208 Pb Wakasa:2010arXiv and 16 O Wakasa:2011PRC

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

Fine structure of GT and SD excitations in 16 O A most recent 16 O( p, n) 16 F experiment Wakasa et al., PRC 84, 014614 (2011) SHF+RPA calculations: Bai et al., PRC 84, 044329 (2011)

Spin-dipole excitations by RHF+RPA HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th] SDR in T channel by RHF+RPA, the lowest RPA state as the reference of E x. exp. Wasaka:2011PRC In general, the 0, 1, and 2 excitations are well reproduced. The 0 1, 1 1, and 2 1 triplets are found at E x 0 MeV. The shoulder at E x 6 MeV and giant resonance at E x 7.5 MeV are nicely reproduced. In particular, the shoulder state cannot be described by shell model calculations.

Spin-dipole excitations by RHF+RPA HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th] SDR in T channel by RHF+RPA, the lowest RPA state as the reference of E x. exp. Wasaka:2011PRC The broad resonances at E x 9.5 and 11 13 MeV are understood as the mixture of the J π = 1 and 2 excitations, and the former one is dominant by 2 component, whereas the latter one is dominant by 1 component. The 0 resonances are predicted to be fragmented at 12 18 MeV with the peak at E x 14.5 MeV.

SD excitations by RHF+RPA and RH+RPA RH+RPA results The general pattern of 2 excitations are similar to that of RHF+RPA calculations, except the peak at E x 9.5 MeV is missing. The 1 resonances are predicted at 12 15 MeV, somehow too high in energy by comparing to data. The 0 resonances are predicted to be centralized at 10 12 MeV, but not seen in experiments yet. Conclusion By comparing with the experimental date, it is found that the self-consistent RHF+RPA calculations are more favored.

Unperturbed and collective excitations R - (fm 2 /M e V ) R - (fm 2 /M e V ) R - (fm 2 /M e V ) 0.8 0.6 0.4 0.2 0.0 2.0 1.5 1.0 0.5 P K O 1 H a r tr e e -F o c k R P A J = 0 - J = 1-0.0 5 4 J = 2-3 2 1 0-5 0 5 1 0 1 5 2 0 E x ( M e V ) D D - M E 2 H a r tr e e R P A J = 0 - J = 1 - J = 2-0 5 1 0 1 5 2 0 E x ( M e V ) HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th]

Diagonal matrix elements of ph interactions 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 ) 2 1 0-1 -2 6 4 2 0-2 3 2 1 0-1 -2 J = 0-1 /2 πs 1 /2 J = 1-1 /2 πs 1 /2 J = 2-1 /2 πd 5 /2 P K O 1 3 /2 πd 3 /2 3 /2 πd 5 /2 3 /2 πd 5 /2 3 /2 πd 3 /2 3 /2 πs 1 /2 J = 0-1 /2 πs 1 /2 J = 1-1 /2 πs 1 /2 J = 2-1 /2 πd 5 /2 D D -M E 2 3 /2 πd 3 /2 3 /2 πd 5 /2 3 /2 πd 5 /2 t o t a l σ + ω ρ π P V π Z R 3 /2 πd 3 /2 3 /2 πs 1 /2 HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th]

Spin-dipole excitations in T + channel by RHF+RPA HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th] SDR in T + channel by RHF+RPA, the lowest RPA state as the reference of E x. exp. Hicks:1991PRC The 0 1, 1 1, and 2 1 triplets are found at E x = 0 2 MeV. The main resonance at E x 8.3 MeV is nicely reproduced, where the 2 component is predicted as the dominant component. A shoulder structure at E x 6.5 MeV is mainly formed by the 2 excitations.

Spin-dipole excitations in T + channel by RHF+RPA HZL, Zhao, Meng, arxiv:1201.0071 [nucl-th] SDR in T + channel by RHF+RPA, the lowest RPA state as the reference of E x. exp. Hicks:1991PRC Giant resonances at E x = 11 15 MeV are also reproduced. It is predicted that these resonances mainly consist of 1 and 2 excitations, and the 1 component is the dominant component. The 0 resonances are predicted to be fragmented at E x = 13 17 MeV.

Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Random Phase Approximation RHF+RPA 3 GT and SD Resonances 4 Fine structure of SD excitations in 16 O 5 Summary

Summary SD excitations have been investigated with the fully self-consistent RPA based on the covariant density functional theory. The fine structure of SD excitations in the most up-to-date 16 O( p, n) 16 F experiment is excellently reproduced without any readjustment in the functional. The characteristics of SD excitations are understood with the delicate balance between the σ- and ω-meson fields via the exchange terms. The fine structure of SD excitations for 16 O(n, p) 16 N channel has also been predicted for future experiments.