Pions in nuclei and tensor force
description
Transcript of 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)
10.2.23 [email protected] 2
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.
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Variational calculation of light nuclei 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 a key elementWe want to calculate heavy nuclei!!
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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
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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
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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
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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
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He was motivated by the BCS theory (1958) .
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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
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Δ is the order parameter is the order parameter
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m
Particle number Chiral symmetry€
εiψ i + Δψ i * = E iψ i
−ε iψ i * + Δ*ψ i = E iψ i
*
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E i = ±(ε i2 + Δ2)1/ 2
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Nambu-Jona-Lasinio Lagrangian
Mean field approximation; Hartree approximation
Fermion gets mass.
The chiral symmetry is spontaneously broken.
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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.
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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
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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
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mσ
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gω (Two parameters)
N
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€
Ψ=Ψ(σ ,ω)⊗Ψ(N)Mean field approximation for mesons.
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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
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σ
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σ
h h
p p
Bruekner argument
Relativistic Chiral Mean Field ModelWave function for mesons and nucleons
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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
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Density dependent RMF model
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Brockmann Toki PRL(1992)
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Why 2p-2h states are necessary for the tensorinteraction?
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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Energy minimization with respect tomeson and nucleon fields
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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
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RCMF equation
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Energy minimization with respect tomeson and nucleon fields
€
δΨ H Ψ
Ψ Ψ= 0
€
δE
δσ= 0
δE
δω= 0
(Mean field equation)
€
δE
δψ i(x)= 0
δE
δCi
= 0
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δE
δbi
= 0 (Corrrelation function)
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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)
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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
€
˜ 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+
+′ =
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Numerical results (1)
4He12C16O
Ogawa TokiNP 2009
Adjust binding energyand size.
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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
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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
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Nuclear matter
Hu Ogawa TokiPhys. Rev. 2009
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ψ ψ
E/A
Total
Pion
€
Σ ~ 50MeV
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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
€
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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Halo structure in 11LiQuickTime˛ Ç∆
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Deuteron-like state ismade by 2p-2h states in shell model.
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Deuteron wave function
Myo Kato Toki Ikeda PRC(2008)
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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
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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.