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)

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.

3. Variational calculation of light nuclei with NN interaction
VMC+GFMC
VNNN
Fujita-Miyazawa
C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci.51(2001)
Relativistic
We want to calculate heavy nuclei!!
Pion is a key element

4. RCNP experiment (good resolution)
Giant GT
Not simple
Y. Fujita et al.,
E.Phys.J A13 (2002) 411
H. Fujita et al., PRC

5. The pion (tensor) is important.
S=1 and L=0 or 2
NN interaction
Deuteron (1+)

6. Deuteron and tensor interaction
Pion Tensor spin-spin
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 attraction.
D-wave is spatially shrank by a half.

7. 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 dynamically
Pion is the Nambu-Goldstone particle of the chiral symmetry breaking

8. He was motivated by the BCS theory（１９５８）.
Nobel prize (2008)
He was motivated by the BCS theory（１９５８）.
is the order parameter
is the order parameter
Particle number
Chiral symmetry

9. Nambu-Jona-Lasinio Lagrangian
Chiral transformation
Mean field approximation; Hartree approximation
Fermion gets mass.
The chiral symmetry is spontaneously broken.
Pion appears as a Nambu-Goldstone boson.

10. Weinberg transformation
Chiral sigma model
Y. Ogawa et al. PTP (2004)
Pion is the Nambu boson of chiral symmetry
Linear Sigma Model Lagrangian
Polar coordinate
Weinberg transformation

11. Non-linear sigma model
Lagrangian
r = fp + j
where
M = gsfp M* = M + gs j
mp2 = m2 + l fp
ms2 = m2 +3 l fp
mw = gwfp mw* = mw + gwj ~
Free parameters are and
(Two parameters)

12. Relativistic Chiral Mean Field Model
Wave function for mesons and nucleons p p
Mean field approximation for mesons. h h
Nucleons are moving in the mean
field and occasionally brought
up to high momentum states
due to pion exchange interaction
Bruekner argument

13. Relativistic Brueckner-Hartree-Fock theory
Brockmann-Machleidt (1990)
RBHF
Non-RBHF
relativity
Us~ -400MeV
Uv~ 350MeV
RBHF theory provides a theoretical foundation of RMF model.

14. Density dependent RMF model
Brockmann Toki PRL(1992)

15. Why 2p-2h states are necessary for the tensor interaction?
The spin flipped states are already
occupied by other nucleons.
Pauli forbidden
G.S.
Spin-saturated

16. Energy minimization with respect to meson and nucleon fields
(Mean field equation)

17. Energy
Energy minimization

18. RCMF equation

19. Energy minimization with respect to meson and nucleon fields
(Mean field equation)
(Corrrelation function)

20. Unitary Correlation Operator Method
(UCOM)
short-range correlator
Bare Hamiltonian
Shift operator depending on the relative distance r
H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61

21. Short-range correlator : C
Hamiltonian in UCOM
2-body approximation in the cluster expansion of operator

22. Numerical results (1) 4He 12C 16O Ogawa Toki NP 2009
Adjust binding energy
and size.

23. Pion tensor provides large attraction to 12C
Numerical results 2 O
The difference between 12C and 16O is 3MeV/N.
P1/2 C
The difference comes from low pion spin states (J<3).
This is the Pauli blocking effect.
P3/2
S1/2
Pion tensor provides large attraction to 12C
Pion energy

24. Chiral symmetry Ogawa Toki NP(2009) Nucleon mass is reduced N
by 20% due to sigma. N
Not 45%
We want to work out heavier nuclei for magic number.
Spin-orbit splitting should be worked out systematically.

25. Nuclear matter E/A Hu Ogawa Toki Phys. Rev. 2009 Total Pion 10.2.23

26. Deeply bound pionic atom Predicted to exist
Toki Yamazaki, PL(1988)
Found by GSI
Itahashi, Hayano, Yamazaki..
Z. Phys.(1996), PRL(2004)
Findings: isovector s-wave

27. Halo structure in 11Li Myo Kato Toki Ikeda PRC(2008)
Deuteron wave function
Deuteron-like state is
made by 2p-2h states
in shell model.

28. Tensor interaction Tensor interaction needs 2p-2h excitation of pn
pair.
P1/2 orbit is used for this
Excitation.
This orbit is blocked
When we want to put
two neutrons.
S1/2 orbit is free of this.

29. Conclusion
Pion (tensor) is treated within relativistic chiral mean field model.
We extended RBHF theory for finite nuclei.
Nucleon mass is reduced by 20%
Chiral condensate is similar to the model independent value. (Sigma term~50MeV)
Deeply bound pionic atom seems to verify partial recovery of chiral symmetry.

30. Picture of nucleus proton Snapshot neutron pionic pair 10.2.23