pions in nuclei and tensor force

10.2.23 [email protected] 1

Pions in nuclei and tensor force

Hiroshi Toki (RCNP, Osaka)in collaboration with

Yoko Ogawa (RCNP, Osaka)Jinniu Hu (RCNP, Osaka)

Takayuki Myo (Osaka Inst. Tech.)Kiyomi Ikeda (RIKEN)


Pion is important !! In Nuclear Physics

Yukawa introduced pion as mediator of nuclear interaction for nuclei. (1934)

Nuclear Physics started by shell model with strong spin-orbit interaction.

　 (1949: Meyer-Jensen: Phenomenological)The pion had not played the central role in

nuclear physics until recent years.


Variational calculation of light nuclei with NN interaction

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

€

ΨVπ Ψ

Ψ VNN Ψ~ 80%

€

Ψ=φ(r12)φ(r23)...φ(rij )

VMC+GFMC

VNNN

Fujita-Miyazawa

Relativistic

Pion is a key elementWe want to calculate heavy nuclei!!


RCNP experiment (good 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


The pion (tensor) is important.

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Deuteron (1+)QuickTime˛ Ç∆

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NN interaction S=1 and L=0 or 2

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π


Deuteron and tensor interaction

Central interaction has strong repulsion.Tensor interaction is strong in 3S1 channel.S-wave function has a dip.D-wave component is only 6%.Tensor attraction provides 80% of entire at

traction.D-wave is spatially shrank by a half.

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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

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

Pion Tensor spin-spin


Chiral symmetry (Nambu:1960)

Chiral symmetry is the key symmetry to connect real world with QCD physics

Chiral model is very powerful in generating various hadronic states

Nucleon gets mass dynamicallyPion is the Nambu-Goldstone particle of the c

hiral symmetry breaking




He was motivated by the BCS theory （１９５８） .

€

E p = ±( p2 + m2)1/ 2

Nobel prize (2008)

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rσ ⋅

rp ψ R + mψ L = E pψ R

−r σ ⋅

r p ψ L + mψ R = E pψ L

€

Δ is the order parameter is the order parameter

€

m

Particle number Chiral symmetry€

εiψ i + Δψ i * = E iψ i

−ε iψ i * + Δ*ψ i = E iψ i

*

€

E i = ±(ε i2 + Δ2)1/ 2


Nambu-Jona-Lasinio Lagrangian

Mean field approximation; Hartree approximation

Fermion gets mass.

The chiral symmetry is spontaneously broken.

€

ψ ψ → ψ ψ cos(2α ) +ψ iγ 5ψsin(2α )

ψ iγ 5ψ →ψ iγ 5ψ cos(2α ) −ψ ψsin(2α )

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ψ → e iαγ 5ψChiral transformation

Pion appears as a Nambu-Goldstone boson.


Chiral sigma model

Chiral sigma model

Linear Sigma Model Lagrangian

Polar coordinate

Weinberg transformation

Y. Ogawa et al. PTP (2004)

Pion is the Nambu boson of chiral symmetry


Non-linear sigma modelNon-linear sigma model

Lagrangian = fπ + φ

whereM = gσfπ M* = M + gσ φ

mπ2 = 2 + fπ mσ

2 = 2 +3 fπ

m = gfπ m

= m + gφ

~ ~

Free parameters are and

€

mσ

€

gω (Two parameters)

N


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Ψ=Ψ(σ ,ω)⊗Ψ(N)Mean field approximation for mesons.

€

Ψ(N) = C0 RMF + Ci

i

∑ 2p − 2hi

Nucleons are moving in the mean field and occasionally broughtup to high momentum states

due to pion exchange interaction

€

σ

€

σ

h h

p p

Bruekner argument

Relativistic Chiral Mean Field ModelWave function for mesons and nucleons


Relativistic Brueckner-Hartree-Fock theory

Brockmann-Machleidt (1990)



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G = V + VQ

eG

Us~ -400MeVUv~ 350MeV

€

π

€

πrelativity

RBHF theory provides a theoretical foundation of RMF model.

RBHF

Non-RBHF

€


Density dependent RMF model







Brockmann Toki PRL(1992)


Why 2p-2h states are necessary for the tensorinteraction?

G.S.

Spin-saturated

The spin flipped states are alreadyoccupied by other nucleons.

Pauli forbidden

€

σ1σ 2[ ]2⋅Y2(r)


Energy minimization with respect tomeson and nucleon fields

€

δΨ H Ψ

Ψ Ψ= 0

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δE

δσ= 0

δE

δω= 0

(Mean field equation)

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δE

δψ i(x)= 0

δE

δCi

= 0




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Energy

Energy minimization










RCMF equation






Energy minimization with respect tomeson and nucleon fields

€

δΨ H Ψ

Ψ Ψ= 0

€

δE

δσ= 0

δE

δω= 0

(Mean field equation)

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δE

δψ i(x)= 0

δE

δCi

= 0


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δE

δbi

= 0 (Corrrelation function)


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 † 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)


Short-range correlator : C

† 1 1

( ) ( )r rC p C p

R r R r+ +

=′ ′

†C lC l=r r

† ( )C r C R r+= †12 12C S C S=†C sC s=r r

Hamiltonian in UCOM

2-body approximation in the cluster expansion of operator€

H = T + V = Ti

i

∑ − TC .M + V (rij

i< j

∑ )

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C+HC = ˜ T + ˜ V

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˜ T = T + ΔT

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ΔT = uij

i< j

∑

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˜ V = V (R+

i< j

∑ (rij ))

( ( ))( )

( )

s R rR r

s r+

+′ =


Numerical results (1)

4He12C16O

Ogawa TokiNP 2009

Adjust binding energyand size.


Numerical results 2

The difference between 12C and 16O is 3MeV/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 to 12C


Chiral symmetry

Nucleon mass is reducedby 20% due to sigma.

We want to work out heavier nuclei for magic number.Spin-orbit splitting should be worked out systematically.

Ogawa TokiNP(2009)

Not 45%

N


Nuclear matter

Hu Ogawa TokiPhys. Rev. 2009

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ψ ψ

E/A

Total

Pion

€

Σ ~ 50MeV


Deeply bound pionic atom

Toki Yamazaki, PL(1988)

Predicted to exist

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

Findings: isovector s-wave

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b1

b1(ρ )=1− 0.37

ρ

ρ 0

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fπ2mπ

2 = −2mq ψ ψ

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ψ ψ

ψ ψ=1− 0.37

ρ

ρ 0

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b1 ∝1

fπ2


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Deuteron-like state ismade by 2p-2h states in shell model.



Deuteron wave function

Myo Kato Toki Ikeda PRC(2008)




Tensor interaction needs2p-2h excitation of pnpair.

P1/2 orbit is used for thisExcitation.

This orbit is blocked When we want to put two neutrons.

S1/2 orbit is free of this.

Tensor interaction


ConclusionPion (tensor) is treated within relativistic chi

ral mean field model.We extended RBHF theory for finite nuclei.Nucleon mass is reduced by 20%Chiral condensate is similar to the model in

dependent value. (Sigma term~50MeV)Deeply bound pionic atom seems to verify p

artial recovery of chiral symmetry.


Picture of nucleus

proton

neutron

pionic pair

Snapshot

pions in nuclei and tensor force

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