Extended Brueckner-Hartree-Fock theory in many body system - Importance of pion in nuclei -

11.7.22 toki@genkenpion 1

Extended Brueckner-Hartree-Fock theory in many body system

- Importance of pion in nuclei -

Hiroshi Toki (RCNP, KEK)In collaboration with

Yoko Ogawa (RCNP)

Jinniu Hu (RCNP)

Kaori Horii (RCNP)

Takayuki Myo (Osaka Inst. of Technology)

Kiyomi Ikeda (RIKEN)


Pion is important in Nuclear Physics ！

• Yukawa （１９３４） predicted pion as a mediator of nuclear interaction to form nucleus

• Meyer-Jansen （１９４９） introduced shell model ー beginning of Nuclear Physics

• Nambu （１９６０） introduced the chiral symmetry and its breaking produced mass and the pion as pseudo-scalar particle


Shell model (Meyer-Jensen)

• Phenomenological

• Strong spin-orbit interaction added by hand

• Magic number

• 2,8,20,28,50,82

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2

8

20

40

70

Harmonicoscillator


The importance of pion is clear in deuteron





Deuteron (1+)QuickTime˛ Ç∆

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NN interaction S=1 and L=0 or 2

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π




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Variational calculation of few body system with NN interaction

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

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ΨVπ Ψ

Ψ VNN Ψ~ 80%

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Ψ=φ(r12)φ(r23)...φ(rij )

VMC+GFMC

VNNN

Fujita-Miyazawa

Relativistic

Pion is keyHeavy nuclei (Super model)


Pion is important in nucleus

• ８０％ of attraction is due to pion

• Tensor interaction is particularly important

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rσ 1 ⋅

r q

r σ 2 ⋅

r q =

1

3q2S12( ˆ q ) +

1

3

r σ 1 ⋅

r σ 2q

2

Pion Tensor spin-spin

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S12( ˆ q ) = 24π Y2( ˆ q ) σ 1σ 2[ ]2[ ]0


Halo structure in 11LiQuickTime˛ Ç∆






Deuteron structure in shell model is producedby 2p-2h states



Deuteron wave function

Myo Kato Toki Ikeda PRC(2008)

(Tanihata..)

11.7.22 toki@genkenpion 9Pairing-blocking : 　　 K.Kato,T.Yamada,K.Ikeda,PTP101(‘99)119, 　 Masui,S.Aoyama,TM,K.Kato,K.Ikeda,NPA673('00)207. TM,S.Aoyama,K.Kato,K.Ikeda,PTP108('02)133, H.Sagawa,B.A.Brown,H.Esbensen,PLB309('93)1.


Tensor optimized shell model （ TOSM ）

Myo, Toki, Ikeda, Kato, Sugimoto, PTP 117 (2006)

0p-0h + 2p-2h

Energy variation

(size parameter)

Gc


11Li G.S. properties 　 (S2n=0.31 MeV)

Tensor+Pairing

Simon et al.

P(s2)

Rm

E(s2)-E(p2) 2.1 1.4 　 0.5 -0.1 [MeV]

Pairing interaction couples (0p)2 and (1s)2 states.


Unitary Correlation Operator Method

H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61

corr. uncorr. SM, HF, FMDCΨ = ⋅Φ ←

{ }12exp( ), ( ) ( )ij r r

i j

C i g g p s r s r p<

= − = +∑

short-range correlator

( ( ))( )

( )

s R rR r

s r+

+′ =

† 1 (Unitary trans.)C C−=

rp p pΩ= +r r r

Bare Hamiltonian

† : Hermitian generatorg g=

Shift operator depending on the relative distance r

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C+HCΦ = EΦ

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HΨ = EΨ

(UCOM)

† ( )C r C R r+=


4He with UCOM

He : (0 ) with -parameterA As b


TOSM+UCOM with AV8’

T

VLS

E

VC

VT

Few bodyCalculation

(Kamda et al)

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Ψ =C0 0 + Cα 2 p2h :αα∑

(Myo Toki Ikeda)






Y0 Y2+

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D S12 S ≠ 0


TOSM should be used for nuclear many body problem2p-2h excitation is essential for treatment of pion

G.S.

Spin-saturated

The spin flipped states are alreadyoccupied by other nucleons.

Pauli forbidden

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σ1σ 2[ ]2⋅Y2(r)



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HF state cannot handle the tensor interaction



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δΨ H Ψ

Ψ Ψ= 0






Total energy









Energy variation




Variation with HF state







We can solve finite nuclei by solving the above equation.


EBHF equation





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∂∂ψ b

* (x)0 Veff 0

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Tψ b (x) + d3x'ψ d* (x')V (x,x ') ψ bψ d[ ]A∫

d∑ + C0

2 ∂ 0 ˆ V αμ

∂ψ b* (x)

1

E − βν ˆ H αμαμ ,βν∑ βν ˆ V 0

+ C0

20 ˆ V αμ

αμ ,βν ,α 'μ ',β 'ν '∑

1

E − α 'μ ' ˆ H αμ

∂ α 'μ ' ˜ H β 'ν '

∂ψ b* (x)

1

E − βν ˆ H β 'ν 'βν ˆ V 0 = ε bψ b (x)

Similar to BHF equation


Feshbach projection method









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P = 0 0

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Q = 2p2h :αα∑ 2p2h :α

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Veff = C0

2V + C0

20

αβ∑ V α

1

E − β H αβ V 0

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Comparison to BHF theory

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Veff = C0

2V + C0

20

αβ∑ V α

1

E − β H αβ V 0

= C0

2V − 0

α∑ V α

1

Eα

α V 0 + 0αβ∑ V α

1

Eα

α V β1

Eβ

β V 0 + .. ⎡

⎣ ⎢

⎤

⎦ ⎥

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E − β H α ≈ −Eα δαβ − β V α

•There appears |C0|2.•|C0|2 is not normalized to 1.

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C0

2+ Cα

α∑

2= 1

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G = V − VQ

eG = V − V

Q

eV + V

Q

eV

Q

eV − ...






Nucl. Phys (2011)

Direct terms only


Relativistic chiral mean field model

The difference between 12C and 16O is 1.5 MeV/N.

The difference comes from low pion spin states (J<3).This is the Pauli blocking effect.

P3/2

P1/2

C

O

S1/2

Pion energy Pion tensor provides large attraction for 12C

OC

Pion contribution Individual contribution

O

C


Renaissance in Nuclear Physics by pion

• We have a EBHF theory with pion• Unification of hadron and nuclear physics• Pion is the main player ー Pion renaissance• High momentum components are produced by

pion （ Tensor interaction ）ー Nuclear structure renaissance

• Nuclear Physics is truly interesting!!




Monden




Relativistic Brueckner-Hartree-Fock theoryBrockmann-Machleidt (1990)

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G = V + VQ

eG

Us~ -400MeVUv~ 350MeV

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π

€

πQuickTime˛ Ç∆


relativity

RBHF

Non-RBHF

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ρ


Important experimental data


Deeply bound pionic atom

Toki Yamazaki, PL(1988)

Prediction

Found by (d,3He) @ GSIItahashi, Hayano, Yamazaki..Z. Phys.(1996), PRL(2004)

Physics : isovector s-wave

€

b1

b1(ρ )=1− 0.37

ρ

ρ 0

€

fπ2mπ

2 = −2mq ψ ψ

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ψ ψρ

ψ ψ=1− 0.37

ρ

ρ 0

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b1 ∝1

fπ2




Suzuki, Hayano, Yamazaki..PRL(2004)

Optical model analysisfor the deeply bound state.


Nuclear structure caused by pion

• ２ p− ２ h excitation is ２０％• High momentum components• Low momentum components

（ Shell model ） are reduced

by ２０ %


RCNP experiment (high resolution)

Y. Fujita et al.,E.Phys.J A13 (2002) 411

H. Fujita et al., PRC

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Ψf στ Ψ i

Not simpleGiant GT


16O (p,d) Ep = 198 MeVΘd = 10°

16O (p,d) Ep = 295 MeVΘd = 10°

16O (p,d) Ep = 392 MeVΘd = 10°

15OLevel scheme

1/2

-0.0

- -+ +Ong, Tanihata et al


Relative Cross Section

a) J. L. Snelgrove et al., PR187, 1246(1969) b) J.K.P. Lee et al., NP A106, 357(1968) c) G.R. Smith et al., PRC30, 593(1984)

Ex[MeV] Jπ

gs 0.0 1/2-

1 5.2 1/2+, 5/2+ 2 6.2 3/2-

3 6.8 3/2+, 5/2+

4 7.3 7/2+

5 7.6 1/2+

6 8.3 3/2+

7 8.9 5/2+,1/2+,(1/2)- 8 9.5 (3/2)+,5/2- 9 10.5 (9/2+),(3/2-),(3/2)+

10 11.6 5/2-

11 11.9 5/2-, 5/2-

• 13.6 5/2+

13.7 3/2-

1 23

45

67 8 9 10

11

12gs

c)a)

Incident proton energy [MeV]

rela

tive

cros

s se

ctio

n

σ1 /σgs

σ2 /σgs

σ4

/σgsσ6

/σgsσ12 /σgs

This workb)


Centrifugal potential ([email protected]) pushes away the L=2 wave function

The property of tensor interaction

Extended Brueckner-Hartree-Fock theory in many body system - Importance of pion in nuclei -

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