Tensor optimized antisymmetrized molecular dynamics (TOAMD) for - PowerPoint PPT Presentation

Tensor optimized antisymmetrized molecular dynamics (TOAMD) for relativistic nuclear matter Hiroshi Toki （RCNP/Osaka） with Takayuki Myo（Osaka IT） Kiyomi Ikeda（RIKEN） Hisashi Horiuchi（RCNP/Osaka） Tadahiro Suhara（Matsue TC） toki@nakanoshimaosaka 2015.11.19 (17-20)

Tensor Optimized Antisymmetrized Molecular Dynamics (TOAMD) Tensor optimized shell model（TOSM） 1. We include tensor interaction most effectively to shell model 2. Difficult to treat cluster structure + Antisymmetrized molecular dynamics (AMD) 1. Cluster+shell structure is handled on the same footing with effective interaction 2. Difficult to treat bare nucleon-nucleon interaction Study nuclear structure based on nuclear interaction Myo Toki Ikeda Horiuchi Enyo Kimura..

TOAMD wave function (variational wave function) 2. Momentum space Matrix elements Gauss integration(analytical） symmetrization Anti- 3. Anti-symmetrization Tensor correlation Short range correlation 1. Gauss expansion Ψ = (1 + F S )(1 + F D ) Φ ( AMD ) Φ ( AMD ) = A Π i e − ν ( x i − D i ) 2 χ i ( σ ) ξ i ( τ ) D = f D ( r : α )(3 σ 1 ⋅ ˆ r σ 2 ⋅ ˆ r − σ 1 ⋅ σ 2 ) F S = f S ( r : β ) F A ∑ ∑ AMD FVF AMD = g I ( ij ..) F g V g F ( space ) M ( spin ) C ( ij ..) ij .. gauss

TOAMD project 1. T. Myo : S-shell nuclei (demonstrated) Make fundamental programs and establish 
 the TOAMD concept 2. T. Suhara : P-shell nuclei Establish the treatment of shell structure 3. H. Toki, T. Yamada : Nuclear matter Study infinite matter 4. Many collaborations : China, Korea

Nuclear Matter（Relativistic effect） Brockmann Machleidt : PRC42(1990)1965 G Relativistic Brueckner-Hartree-Fock with Bonn-potential V Q ( α ⋅ p + β m + U ) ! ψ = E ! ψ ∑ m ) = m ) kp − pk U k ( ! G ( ! kp p m = m + U S ( ! U = β U S + U V ! m )

Infinite matter（non-relativistic framework： 
 ＝C.M. boost interaction ３ body attraction（Δ） + 3 body repulsion +Boost corrections） Variational chain summation (VCS) C.M. boost effect ＝3 body repulsion Effective mass Relativistic effect Akmal Pandhyaripande Ravenhall : PRC58(1998)1804 ( ) ∏ p correlation function Ψ = ij 1 + F Φ F ij ij

Brueckner-Hartree-Fock type equation TOSM for relativistic matter ∑ Ψ = C 0 0 + 2 p 2 h : α C α α

Numerical results of TOSM and comments 5MeV/A short It is difficult to include 3 body interaction in TOSM (low momentum) (high momentum) 3 body interaction (Fujita-Miyazawa delta term) Tensor Short-range C 0 C α

TOAMD vs TOSM Concept is same, but TOAMD is flexible 2 body term 3 body term 4 body term We can include naturally the 3 body interaction ∑ Ψ TOSM = C 0 0 + 2 p 2 h : α C α α Ψ TOAMD = C 0 0 + F D 0 ∑ = C 0 0 + 2 p 2 h : α 2 p 2 h : α F D 0 α C α D = 1 1 ∑ ∑ VF V ij F Dkl 2 2 i ≠ j k ≠ l = 1 + 1 ∑ ∑ ∑ Dij + V ij F V ij F V ij F Djk Dkl 2 4 i ≠ j i ≠ j ≠ k i ≠ j ≠ k ≠ l

TOAMD for nuclear matter (Three body interaction) Ψ = (1 + F S )(1 + F D ) Φ ( RNM ) ∏ A Φ ( RNM ) = ψ p ( r , s ) ξ p ( t ) p ⎛ ⎞ χ p ( s ) E p + ! ⎜ ⎟ m 1 ψ p ( r , s ) = σ ⋅ p e ipr ⎜ ⎟ m χ p ( s ) 2 ! m V ⎜ ⎟ E p + ! ⎝ ⎠ ∑ D = f D ( r ij )(3(2 m ) 2 γ 5 i γ 5 j − k 2 γ 5 i γ i x γ 5 j γ j x ) τ i ⋅ τ j → 3 σ 1 ⋅ k σ 2 ⋅ k − k 2 σ 1 ⋅ σ 2 F x S = f S ( r ij ) γ i 0 γ j → 1 0 F H = T + V Bonn + U Δ

Formulation is simple（2 body＋3 body..) MC(Metropolis) method for integration 2 body term S RNM = 1 ∑ ∑ RNM F S VF C ( p 1 p 2 : q 1 q 2 ) C µ 1 C µ 2 C µ 3 2 µ 1 µ 2 µ 3 p 1 p 2 : q 1 q 2 ∑ − k 1 2 / k µ 1 2 − k 2 2 / k µ 2 2 − ( p 1 − q 1 − k 1 − k 2 ) 2 / k µ 3 2 M ( p 1 − k 1 | Γ | p 1 − k 1 − k 2 ) M ( p 2 + k 1 | Γ | p 2 + k 1 + k 2 ) e e e k 1 k 2 ⎛ ⎞ E p + m E q + m p 1 p 2 † 1 − σ ⋅ p σ ⋅ q M ( p |1| q ) = χ p ⎟ χ q ⎜ k 1 E p + m E q + m ⎝ ⎠ 2 E p 2 E q p 2 + k 1 ⎛ ⎞ p 1 − k 1 E p + m E q + m k 2 † − σ ⋅ p E p + m + σ ⋅ q M ( p | γ 5 | q ) = χ p ⎟ χ q ⎜ Γ Γ E q + m ⎝ ⎠ 2 E p 2 E q p 1 − k 1 − k 2 p 2 + k 1 + k 2 k 3 C ( p 1 p 2 : q 1 q 2 ) = δ p 1 q 1 δ p 2 q 2 − δ p 1 q 2 δ p 2 q 1 q 2 q 1 ∫ − k 1 2 / k µ 1 2 e − k 1 2 / k µ 2 2 f = dp 2 dk 1 dk 2 f ( p 1 p 2 k 1 k 2 ) θ ( p 1 − k F ) θ ( p 2 − k F ) e dp 1

3 body term MC(Metropolis) method for integration ∑ ∑ S RNM = RNM F S VF C ( p 1 p 2 p 3 : q 1 q 2 q 3 ) C µ 1 C µ 2 C µ 3 µ 1 µ 2 µ 3 p 1 p 2 p 3 : q 1 q 2 q 3 ∑ − k 1 2 / k µ 1 2 − ( p 1 − q 1 − k 1 ) 2 / k µ 2 2 − ( p 3 − q 3 ) 2 / k µ 3 2 M ( p 1 − k 1 | Γ | q 1 ) M ( p 2 + k 1 | Γ | p 1 − q 1 − k 1 + p 2 + k 2 ) e e e k 1 p 1 p 2 p 3 δ p 1 q 1 δ p 1 q 2 δ p 1 q 3 k 1 C ( p 1 p 2 p 3 : q 1 q 2 q 3 ) = δ p 2 q 1 δ p 2 q 2 δ p 2 q 3 p 2 + k 1 p 1 − k 1 p 1 − q 1 − k 1 δ p 3 q 1 δ p 3 q 2 δ p 3 q 3 Γ Γ p 3 − q 3 p 1 − q 1 − k 1 + p 2 + k 2 q 1 q 2 q 3 ∫ − k 1 2 / k µ 1 2 f = dp 2 p 3 dk 1 f ( p 1 p 2 p 3 k 1 ) θ ( p 1 − k F ) θ ( p 2 − k F ) θ ( p 3 − k F ) e dp 1

Non-relativistic nuclear matter (1/m expansion) spin-orbit square c.m. correction spin-orbit Analytical gauss integration and differentiation ⎛ ⎞ E p + m E q + m † 1 − σ ⋅ p σ ⋅ q M ( p |1| q ) = χ p ⎟ χ q ⎜ E p + m E q + m ⎝ ⎠ 2 E p 2 E q ( p + q ) 2 M ( p |1| q ) = 1 − 1 1 ∑ − ε xyz i σ x p y q z m 2 (2 m ) 2 8 xyz ( p 1 + q 1 ) 2 + ( p 2 + q ) 2 M ( p 1 |1| q 1 ) M ( p 2 |1| q 2 ) = 1 − 1 1 ∑ − ε xyz i σ 1 x p 1 y q 1 z m 2 (2 m ) 2 8 xyz 1 1 ∑ ∑ ∑ − ε xyz i σ 2 x p 2 y q 2 z + ε xyz i σ 1 x p 1 y q 1 z ε x ' y ' z ' i σ 2 x ' p 2 y ' q 2 z ' (2 m ) 2 (2 m ) 4 xyz xyz x ' y ' z '

+pion with short and tensor correlation -13MeV Hartree-Fock with sigma+omega exchange Numerical results 5 0 E[MeV] -5 -10 -15 -20 0.6 0.8 1 1.2 1.4 1.6 k(F)[1/fm]

Present status (limitation) E(kin)=18MeV+5MeV+10MeV=33MeV Short Fermi Tensor (Bonn potential) One gaussian -> many gaussians Two body term + (many body term) Two -> Three body interaction σ + ω + π + ( ρ + δ + η ) 1 2 Effective Mass ratio 1.5 0.9 1 0.8 0.5 0.7 C 0 0.6 -0.5 0.5 -1 0.4 -1.5 0.6 0.8 1 1.2 1.4 1.6 0.5 1 1.5 2 k(F)[1/fm] r

1. Reproduce TOSM results by TOAMD 2. Add three body interaction in TOAMD 3. Complete EOS in nuclear matter 4. Hyper-nuclear matter Hu Toki Ogawa TOSM

Conclusion 1. We formulated relativistic nuclear matter using TOAMD 2. We formulated non-relativistic nuclear matter using TOAMD 3. We calculated various terms using Bonn potential 4. We express 3 body term and 3 body interaction 5. We get first (preliminary) results with correlations 6. We shall get relativistic EOS soon using Bonn potential