1. Kondo effect in charm/bottom nuclei International workshop on J-PARC hadron physics in 2016@KEK, Tokai, 2-4 Mar. 2016 Shigehiro Yasui Tokyo Institute of Technology arXiv:1602.00227 [hep-ph]

2. 1. Hadron-nucleon interaction 1. Multi-flavor nucleus Nucleus + Strangeness 2. Hadron in medium 3. Nuclear structure Impurity physics

3. 1. Hadron-nucleon interaction 1. Multi-flavor nucleus Nucleus + Charm 2. Hadron in medium 3. Nuclear structure Mass hierarchy (m c >>Λ QCD ) ~1.2 GeV ~0.2 GeV Impurity physics ??? S. H. Lee’s talk D. Suenaga’s talk

4. Nucleus D, D* q c 1. Multi-flavor nucleus - Binding energy - Dispersion relation - Spectral function - Reaction - Restoration of χSB - Nuclear structure - etc. Kondo effect : quantum impurity physics Charm nucleus mass spectroscopy

5. Contents 1. Multi-flavor nucleus 2. Kondo effect 2.1 What’s Kondo effect? 2.2 Kondo effect in atomic nuclei (mean-field + RPA) 3. Conclusion F. Takano, T. Ogawa, Prog. Theor. Phys. 35, 343 (1966) A. Yoshimori, A. Sakurai, Suppl. Prog. Theor. Phys. 46, 162 (1970) M. Eto, Y. V. Narazov, Phys. Rev. B 64, 085322 (2001) bulk systemsfinite systems This work Mean-field approach to Kondo effect

6. 2. Kondo effect J UN K ONDO (1930-)

7. 2.1 What’s Kondo effect? Original Work: J. Kondo (Prog. Theor. Phys. 32, 37 (1964)) Log T/T K (quantum) T 2 (classical) T K : Kondo temperature Impurity atom with spin ½ with T a ・ T a interaction electron metal T 1,..., T n^2-1 : generators of SU(n) (n=2 for spin ½) impurity spin electron spin “Kondo bound state”

8. 3 2 1Fermi surface (particle-hole creation) Loop effect Non-Abelian int. (SU(n) symmetry) Original Work: J. Kondo (Prog. Theor. Phys. 32, 37 (1964)) 2.1 What’s Kondo effect? Heavy impurity impurity fermion

9. 2.1 What’s Kondo effect? Original Work: J. Kondo (Prog. Theor. Phys. 32, 37 (1964)) Scattering amplitude at tree and one-loop levels k, l, i, j=↑, ↓ in SU(n) (n=2 for spin ½) T 1,..., T n^2-1 : generators of SU(n) Log E divergence for infrared limit (E→0) for any small coupling E: energy from Fermi surface impurity atom electron → Logarithmic divergence in resistance of electron ++ electron hole k l i j l j l j k i k i (T a ) kl (T a ) ij interaction

10. 2.1 What’s Kondo effect? nucleon D, B meson “Isospin Kondo effect” u,d,s quark c, b quark “Color (QCD) Kondo effect” ① nuclear matter ② nucleon-hole loop ③ isospin SU(2) symmetry ① quark matter ② quark-hole loop ③ color SU(3) symmetry Application to nuclear/quark matter NJL-type: S.Y., K.Sudoh, Phys. Rev. C88, 015201 (2013) QCD: K. Hattori, K. Itakura, S. Ozaki, S. Y., Phys. Rev. D92, 065003 (2015) (τ a ) kl (τ a ) ij interaction(λ a ) kl (λ a ) ij interaction Cf. “Magnetic catalysis QCD Kondo effect” S. Ozaki, K. Itakura, Y. Kuramoto, arXiv:1509:06966 [hep-ph]

11. 2.1 What’s Kondo effect? Application to nuclear/quark matter nucleon D, B meson “Isospin Kondo effect” u,d,s quark c, b quark “Color (QCD) Kondo effect” NJL-type: S.Y., K.Sudoh, Phys. Rev. C88, 015201 (2013) QCD: K. Hattori, K. Itakura, S. Ozaki, S. Y., Phys. Rev. D92, 065003 (2015) What’s about in atomic nuclei? (finite size nuclear systems) ① nuclear matter ② nucleon-hole loop ③ isospin SU(2) symmetry ① quark matter ② quark-hole loop ③ color SU(3) symmetry (τ a ) kl (τ a ) ij interaction(λ a ) kl (λ a ) ij interaction Cf. “Magnetic catalysis QCD Kondo effect” S. Ozaki, K. Itakura, Y. Kuramoto, arXiv:1509:06966 [hep-ph]

12. 2.2 Kondo effect in atomic nuclei D meson Simple model for atomic nucleus Purpose: the ground state energy? k = 123N nucleon sector (N states) ↑/↓ for proton/neutron D sector εkεk kinetic term Kondo (isospin-flipping) interaction c kσ : annihilation operator for nucleon in level k and isospin σ T +, T -, T 3 : isospin operator for D meson HKHK à la Lipkin model Cf. π-exchange interaction: S.Y., K.Sudoh, Phys. Rev. D80, 034008 (2009)

13. 2.2 Kondo effect in atomic nuclei Exact solution Case of nucleon #=1 singlet w.f. linear algebraic equation solution !! ε ε-3Ng/2 : “Kondo bound state” (superposed state of nucleons and D meson) k’ = 123N Simple case: ε k =ε ε k = 123N

14. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Step 2. Introduce Lagrange multiplier λ Step 4. Variation by λ and Δ Step 3. Apply mean-field (Δ) approx. “gap” color singlet Step 1. Introduce auxiliary fermion fields f σ (SU(2)) auxiliary fermion # constraint fermion # 0, 2 fermion # 1 physical space How does it work?

15. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Step 2. Introduce Lagrange multiplier λ Step 4. Variation by λ and Δ Step 3. Apply mean-field (Δ) approx. “gap” color singlet Step 1. Introduce auxiliary fermion fields f σ (SU(2)) auxiliary fermion # constraint fermion # 0, 2 fermion # 1 physical space

16. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Simple case: ε k =ε N↑N↑ N↑N↑ D↑D↑ N↓N↓ N↓N↓ D↓D↓ N↑N↑ N↑N↑ D↑D↑ N↓N↓ N↓N↓ D↓D↓ Diagonalization (next page) MatrixBasis ε k = 123N Step 3 Attraction by N-D mixing (Δ) ↓ New ground state

17. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Simple case: ε k =ε Diagonalized matrix New basis lowest energy (↑) lowest energy (↓) ε k = 123N Step 3 Linear combination of (coherent) nucleon and D meson

18. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Simple case: ε k =ε Diagonalized matrix Variation by lowest energy (↓) lowest energy (↑) ε k = 123N Step 3 Step 4 “ Kondo bound state ” Binding energy (MF): Ng different form exact solution 3Ng/2?

19. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Simple case: ε k =ε ① Mean-value of auxiliary fermion number is one. ε k = 123N ② Fluctuation effect (random-phase approximation; RPA) Binding energy (MF+RPA): 1.378Ng close to exact solution 3Ng/2=1.5Ng!! zero-point energy in H.O. potential

20. 2.2 Kondo effect in atomic nuclei What’s about more general cases? Mean-field (+RPA) approach Simple case: ε k =ε ③ Violation of isospin symmetry (application) ε+δε ε k = 123N ε ↑ =ε+δε, ε ↓ =ε Δ=0 at δε=4Ng E MF =ε at δε=4Ng Δ δε 4Ng0

21. 3. Conclusion 2. We apply the mean-field + RPA approach to the ground state in a simple model in success. 1. We study the ground state of charm nucleus (D meson) with isospin Kondo effect. Future works: 1. Application to realistic D meson-nucleon interaction. 2. Effect by nuclear shell effects/collective modes. 3. Observables for experiments. 4. etc. 3. This method can be applied to realistic nuclear structure with charm/bottom hadron. Isospin-dependent interaction x Nuclear system with D meson = “Kondo bound state”

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