Beta transition rates and EOS for massive stars

(New Nuclear Theory: TOAMD)

Hiroshi Toki （RCNP/Osaka） with

!

toki@monash 2016.2.2 (2.1~2.6)

Takayuki Myo（Osaka IT） Kiyomi Ikeda（RIKEN）

Hisashi Horiuchi（RCNP/Osaka） Tadahiro Suhara（Matsue TC）

Toshio Suzuki (Nihon) Ken’ichi Nomoto (WPI/Tokyo)

Hong Shen (Nankai) Jinniu Hu (Nankai)

My desire: 1. Provide best NP results for astrophysics 2. EOS table: H. Shen, H. Toki, K. Oyamatsu, K. SumiyoshiNucl. Phys. A637 (1998) 435

3. Beta transition rate for nuclear URCA: H. Toki, T. Suzuki, K. Nomoto, S. Jones, R. Hirschi, Phys. Rev. C88 (2013) 015806 !

Content: 1. Beta transition rates for URCA nuclei 2. EOS table for supernova and neutron star (URCA) 3. New theory using nuclear interaction (TOAMD)

Beta transition rates in sd-shell nuclei

Several works: 1. T.Kajino, E.Shiino, H.Toki, A.Brown, H.Wildenthal, NP (1988) 2. T.Oda et al, Atomic and nuclear data table (1994) 3. H.Toki, T.Suzuki, K.Nomoto, S.Jones, R.Hirschi, PR (2013) 4. T.Suzuki, H.Toki, K.Nomoto, APJ (2016)

These works are all based on the shell modelfor sd-shell nuclei by A. Brown and H. Wildenthal

Nuclear shell structure

54Fe (Most stable)sd − shell

pf − shell(Suzuki)

HΨ = EΨ

H = [T +U ]ii=1

A

∑ + Viji≠ j

A

∑Ψ = AC

C∑ (s)n1(d3/2 )

n2 (d5/2 )n3

No. of parameters

ε j

sd V sd

3

63

[T +U ]ψ j = ε jψ j

(s)n1(d3/2 )n2 (d5/2 )

n3 V (s)n1(d3/2 )n2 (d5/2 )

n3 = C sd∑ V sd

Phenomenologicalsd − shell

Level scheme

!gA = 0.77gA

f στ i Beta transition rate

URCA process for rapid cooling of star(A,Z )+ e− → (A,Z −1)+ν(A,Z −1)→ (A,Z )+ e− +ν

25Na − 25Mg 23Na − 23Ne

sd-shell model calculationToki-Suzuki table

Oda table

Jones calculation

25Na − 25Mg

23Na − 23Ne

Suzuki, Toki, NomotoAstro J. (2016)

other beta rates

neutron star cooling

p + e− → n +νn→ p + e− +ν

URCA process for

npn(k)1

kFp kF

n

µn = µp + µe(Beta equilibrium)

Momentum conservation

kFp = kF

e

(Charge neutrality)

kFp + kF

e > kFn

(2kFp )3 > (kF

n )3

ρ p

ρ> 19

8ρ p > ρn

EOS with the RMF theory (Shen EOS)

Relativistic Mean Field Theory (RMF)

mean-field approximation: meson field operators are replaced by their expectation values !no-sea approximation: contributions from the negative-energy Dirac sea are ignored

Applications flavor SU(2) flavor SU(3)

infinite matter: nuclear matter strange hadronic matter

finite system: nuclei hypernuclei

, ,σ ω ρ

Shen Toki Oyamatsu Sumiyoshi Nucl. Phys. A637 (1998) 435

Relativistic Mean Field Theory (Phenomenological model)

2 2 3 42 3

2 23

2

[ ]1 1 1 12 2 3 41 1 1 ( )4 2 41 1 ...4 2

aa

a a a a

L i M g g g

m g g

W W m c

R R m

µ µ µµ σ ω µ ρ µ

µµ σ

µυ µ µµυ ω µ µ

µυ µµυ ρ µ

ψ γ σ γ ω γ τ ρ ψ

σ σ σ σ σ

ω ω ω ω

ρ ρ

= ∂ − − − −

+ ∂ ∂ − − −

− + +

− + +

Lagrangian

Lagrangian Equations Mean-Field Approximation

Calculate everything such as , , ...p sε

TM1 parameter set

EOS

6 parameters

L. S. Geng, H. Toki, J. Meng, Prog. Theor. Phys. 113 (2005) 785

( )2theo expt1

2.1

ni i

iM M

nσ =

−=

=

∑

2157 nuclei

Comparison with nuclear physics data

Shen EOS H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Prog. Theor. Phys. 100, 1013 (1998) H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Astrophys. J. Suppl. 197, 20 (2011)

ρc ∼ 2.2ρ⊙ρ p ∼ 0.08ρ

URCA process does not occur for neutron star cooling

ρ p

ρ> 19(URCA condition)

LS PSU (non-relativistic) soft EOSH.Shen G.Shen (relativistic) hard EOS

npn(k)1

kFp kF

n

npn(k)1

kFp kF

n

Brueckner-Hartree-Fock TOAMD

!kp +!ke =!kn becomes possible!!

URCA becomes possible in neutron star cooling

New theory with tensor and short range correlations

toki@numazupion 17

Deuteron (1+)NN interaction

S=1 and L=0 or 2

�

π

Ψ J=1 =ψ SY0χS=1 +ψ D Y2χS=1[ ]1

AV8

ψ S

ψ D

18

Variational calculation of light nuclei with NN interaction

C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci.51(2001)

�

ΨVπ ΨΨVNN Ψ

~ 80%

�

Ψ = φ(r12)φ(r23)...φ(rij )

VMC+GFMC

VNNN

Fujita-Miyazawa

Relativistic

Pion is keyHeavy nuclei (Super model)

19

Pion is important in nucleus• 80% of attraction is due to pion • Tensor interaction is particularly important

and difficult to handle (50%)

�

! σ 1 ⋅! q ! σ 2 ⋅! q

mπ2 + q2 =

13

q2

mπ2 + q2 S12( ˆ q ) +

13

q2

mπ2 + q2

! σ 1 ⋅! σ 2

=13

q2

mπ2 + q2 S12( ˆ q ) +

13

1− mπ2

mπ2 + q2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ! σ 1 ⋅! σ 2

Pion Tensor spin-spin

high momentum low momentum

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 project1. T. Myo : S-shell nuclei (He3, He4) 　　Make fundamental programs and establish the TOAMD concept (almost finished) 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）G

VQ

Brockmann Machleidt : PRC42(1990)1965

(α ⋅ p + βm +U ) !ψ = E !ψ

U = βUS +UV !m = m +US ( !m)

Uk ( !m) = kpp∑ G( !m) kp − pk

Relativistic Brueckner-Hartree-Fock with Bonn-potential

npn(k)1

kFp kF

n

Infinite matter（non-relativistic framework： 3 body repulsion +Boost corrections）

Akmal Pandhyaripande Ravenhall : PRC58(1998)1804

Relativistic effectEffective mass 　　　＝3 body repulsion

C.M. boost effect 　　　＝C.M. boost interaction

Ψ = ij 1+ Fij( )∏ Φ

+ ３ body attraction（Δ）

Fijp correlation function

Variational chain summation (VCS)

TOSM for relativistic matter

Ψ = C0 0 + Cαα∑ 2p2h :α

Brueckner-Hartree-Fock type equation

Numerical results of TOSM and comments

5MeV/A short

It is difficult to include 3 body interaction in TOSM

C0 Cα(low momentum)(high momentum)

3 body interaction (Fujita-Miyazawa delta term)

Tensor Short-range

TOAMD for nuclear matter

Ψ = (1+ FS )(1+ FD )Φ(RNM )

Φ(RNM ) = ψ p (r, s)ξ p (t)p

A∏

ψ p (r, s) =Ep + !m2 !m

χ p (s)

σ ⋅ pEp + !m

χ p (s)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

1Veipr

FD = fD (rij )(3(2m)2γ 5iγ 5 j − k

2 γ 5iγ ixγ 5 j

x∑ γ j

x )τ i ⋅τ j

FS = fS (rij )γ i0γ j

0

→ 3σ 1 ⋅ kσ 2 ⋅ k − k2σ 1 ⋅σ 2

→1

H = T +VBonn +UΔ(Three body interaction)

RNM FSVFS RNM = 12

C(p1p2 :q1q2 )p1p2:q1q2∑ Cµ1

Cµ2Cµ3

µ1µ2µ3∑

e−k12 /kµ1

2

k1k2∑ e−k2

2 /kµ22

e−( p1−q1−k1−k2 )2 /kµ3

2

M (p1 − k1 |Γ | p1 − k1 − k2 )M (p2 + k1 |Γ | p2 + k1 + k2 )

M (p |1 | q) =Ep +m2Ep

Eq +m2Eq

χ p† 1− σ ⋅ p

Ep +mσ ⋅qEq +m

⎛

⎝⎜⎞

⎠⎟χq

M (p |γ 5 | q) =Ep +m2Ep

Eq +m2Eq

χ p† − σ ⋅ p

Ep +m+ σ ⋅qEq +m

⎛

⎝⎜⎞

⎠⎟χq

MC(Metropolis) method for integration

Formulation is simple（2 body＋3 body..)

C(p1p2 :q1q2 ) = δ p1q1δ p2q2

−δ p1q2δ p2q1

p1

q1 q2

p2

k3

k2

k1

Γ Γp1 − k1

p1 − k1 − k2

p2 + k1

p2 + k1 + k2

2 body term (we can use Feynmann rule)

f = dp1∫ dp2dk1dk2 f (p1p2k1k2 )θ(p1 − kF )θ(p2 − kF )e−k1

2 /kµ12

e−k22 /kµ 2

2

Present status (preliminary)σ +ω +π +δ +η + (ρ)One gaussian -> many gaussiansTwo body term -> many body termTwo -> Three body interaction

(Bonn potential)

-1.5-1

-0.5 0

0.5 1

1.5 2

0.5 1 1.5 2

C

r

-30-20-10

0 10 20 30

0 0.1 0.2 0.3

E [M

eV]

density [fm-3]

0 10 20 30 40 50 60

0 0.1 0.2 0.3

E [M

eV]

density [fm-3]

N=Z N only

D

S

HF

S

S+D

HF S

S+D

Nuclear matter Neutron matter

M (p→ n) ~ (1− n(kn ))n(kp )URCA

M (n→ p) ~ (1− n(kp ))n(kn )

TOSM=Two body version of TOAMD

possible!!

Conclusion 1. Nuclear URCA process (sd-shell) 2. Fine mesh table ̶> T. Suzuki talk 3. URCA in neutron stars: RMF NO!! 4. Many body theory with correlation YES!! 5. BHF should be extended to TOAMD 6. We are close to get numbers with TOAMD 7. URCA in neutron star cooling will be possible 8. EOS using TOAMD will come soon