Nuclear Force from Lattice QCD. [1] Why nuclear force now? [2] NN force from lattice QCD [3] BB and BM forces from lattice QCD [4] Summary and Future

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1 Nuclear Force from Lattice QCD [1] Why nuclear force now? [2] NN force from lattice QCD [3] BB and BM forces from lattice QCD [4] Summary and Future Tetsuo Hatsuda (Univ. Tokyo) NFQCD, Feb 17, 2010

2 The nuclear force is a basis for understanding Structure of ordinary and hyper nuclei Structure of neutron stars Ignition of Type II supernovae

3 NN phase shifts mid-range attraction short-range repulsion deuteron mid-range attraction short-range repulsion Nijmegen partial-wave analysis, Stoks et al., Phys.Rev. C48 (1993) 792

4 Phenomenological NN potentials (~40 parameters to fit 5000 phase shift data) One-pion exchange by Yukawa (1935) repulsive core 2p, 3p,... (s, r, w,...) p Multi-pions by Taketani et al. (1951) Repulsive core by Jastrow (1951)

5 Phys. Rev. 81 (1951) 165 Repulsive core : history Phys. Rev. 106 (1957) 1366 ω-meson

6 So I got up in the question period and I said, Maybe the reason is that inside the nuclear force of attraction, which holds nuclei together, there's a very strong short-range force of repulsion, like a little hard sphere inside this attractive Jell-O. I'll never forget, Oppenheimer got up, he liked to needle the young fellows and he said, very dryly, "Thank you so much for, we are grateful for every tiny scrap of help we can get. But I ignored his needle and pursued my idea, and actually calculated the scattering of neutrons by protons. I showed that it fit the data very well. Oppenheimer read my paper for the Physical Review and took back his criticisms. This work became a permanent element of the literature of physics.

7 From Phenomenology to First Principle Y. Nambu, Quarks : Frontiers in Elementary Particle Physics, World Scientific (1985) Even now, it is impossible to completely describe nuclear forces beginning with a fundamental equation. But since we know that nucleons themselves are not elementary, this is like asking if one can exactly deduce the characteristics of a very complex molecule starting from Schroedinger equation, a practically impossible task.

8 space Lattice QCD x x L a quarks q(n) Euclidean time gluons U μ (n) O well defined statistical system (finite a and L) gauge invariant fully non-perturbative Monte Carlo simulations

9 Supercomputer peak performance Keisoku (Kobe) Roadrunner (LANL) Jaguar (ORNL) First LQCD simulation First LQCD calculation? Ukawa (2009)

10 QCD Vacuum and Hadron Masses from Lattice QCD quark nucleon Chiral condensate Dynamical quark mass Hadron masses

11 Chiral condensate in lattice QCD Dirac operator: QCD Banks-Casher relation : free Dynamical overlap fermion, 2-flavor S. Hashimoto (JLQCD Coll.) arxiv: [hep-lat]

12 Dynamical quark-mass in lattice QCD Landau gauge, Improved KS fermion, (2+1)-flavor P. O. Bowman et al., Phys. Rev. D71 (2005)

13 Hadron masses in lattice QCD 3% accuracy at present Improved-Wilson, (2+1)-flavor, L=2.9 fm, a=0.09 fm, 32 3 x 64, (p, K, W) inputs, m ud (min.)= 3.5MeV m p (min.) = 156 MeV PACS-CS Coll., Phys. Rev. D79 (2009)

14 Nambu-Bethe-Salpeter Amplitude and Nuclear Forces in Lattice QCD How to extract the NN force at low energies in lattice QCD? Y. Nambu, Force Potentials in Quantum Field Theory, Prog. Theor. Phys. 5 (1950) 614. K. Nishijima, Formulation of Field Theories for Composite Particles, Phys. Rev. 111 (1958) 995.

15 Nambu-Bethe-Salpeter Amplitude and Nuclear Forces in Lattice QCD How to extract the NN force at low energies in lattice QCD? Y. Nambu, T. Hatsuda, Y. Ikeda, N. Ishii (Tokyo) Force Potentials in Quantum Field Theory, Prog. Theor. Phys. 5 (1950) 614. S. Aoki, T. Doi, T. Inoue, K. Murano, K. Sasaki (Tsukuba) K. Nishijima, Formulation of Field Theories for Composite H. Nemura (Tohoku) Particles, Phys. Rev. 111 (1958) 995.

16 Equal-time NBS amplitude φ (r) in lattice QCD x J y + all possible combinations y J y space Imaginary time φ (r > R) phase shift : φ (r < R) potential : Luscher, Nucl. Phys. B354 (1991) 531 Ishii, Aoki & Hatsuda, PRL 99 (2007)

17 Central NN potentials from lattice QCD wave function NN potential Quenched QCD (m π =530MeV, L=4.4 fm) Full QCD (m π =570MeV, L=2.9 fm) r [fm] Ishii, Aoki & Hatsuda, PRL 99 (2007) Ishii, Aoki & Hatsuda, arxive [hep-lat]

18 Systematic procedure to define the NN potential in lattice QCD Full details: Aoki, Hatsuda & Ishii, [hep-lat], PTP 123 (2010) (i) Choose your favorite operator: e.g. observables do not depend on the choice yet the local operator is useful Nishijima, Haag, Zimmermann (1958) (ii) Measure the NBS amplitude: (iii) Define the non-local potential: (iv) Velocity expansion : (v) LO LO NLO NNLO Okubo-Marshak (1958), Tamagaki-Watari (1967) Calculate observables : phase shifts, binding energies etc

19 Key channels in NN scattering ( 2s+1 L J ) LO LO NLO NNLO 1 S 0 Central force nuclear BCS pairing Bohr, Mottelson & Pines, Phys. Rev. 110 (1958) 3 S 1-3 D 1 Tensor force deuteron binding Pandharipande et al., Phys. Rev. C54 (1996) 3 P 2-3 F 2 LS force neutron superfluidity in neutron stars Tamagaki, Prog. Theor. Phys. 44 (1970) Density profile of the deuteron with S z =±1

20 Frequently Asked Questions [Q1] Operator dependence of the latttice potential [Q2] Energy dependence of the lattice potential [A1] (N(x), U(x,x )) is a combination to define observables remember, QM : (Φ, U) ~ (Φ, U ) observables QFT : (asymptotic field, vertices) observables EFT : (choice of field, vertices) observables local operator = a convenient choice for reduction formula [A2] U(x,x ) is E-independent by construction non-locality can be determined order by order in velocity expansion (c.f. ChPT)

21 LO potential (central & tensor) in lattice QCD For J P = 1 +, φ > comprises S-wave and D-wave, φ = φ S + φ D where, φ S = P φ = ( 1/24 ) R O R φ φ D = Q φ = ( 1 P ) φ Therefore, we have 2-component Schrödinger eq. S-wave: P ( T + V C + V T S 12 ) φ = E P φ D-wave: Q ( T + V C + V T S 12 ) φ = E Q φ

22 Central & tensor potentials : V C (r) & V T (r) Aoki, Hatsuda & Ishii, [hep-lat] PTP 123 (2010) quenched QCD E ~ 0 MeV

23 Central & tensor potentials : V C (r) & V T (r) Aoki, Hatsuda & Ishii, [hep-lat] PTP 123 (2010) quenched QCD quenched E ~ QCD 0 MeV E ~ 0 MeV V c (r 0) ~ (log r) β /r 2, V T (r 0) 0 from operator product expansion (Aoki, Balog & Weisz, arxiv: )

24 Central & tensor potentials : V C (r) & V T (r) Aoki, Hatsuda & Ishii, [hep-lat] PTP 123 (2010) quenched QCD quenched E ~ QCD 0 MeV E ~ 0 MeV quenched QCD E ~ 0 MeV fit function V c (r 0) ~ (log r) β /r 2, V T (r 0) 0 Rapid quark-mass dependence of V T (r) from operator product expansion Evidence of the one-pion-exchange (Aoki, Balog & Weisz, arxiv: )

25 One boson exchange model : V C (r) & V T (r) From Machleidt, Adv. Nucl.Phys.vol.19

26 velocity dependence of the potential (NNLO) PBC (E~0 MeV) APBC (E~46 MeV)

27 velocity dependence of the potential (NNLO) NNLO PBC (E~0 MeV) APBC (E~46 MeV) 1 S 0 3 S 1-3 D 1 NNLO can be determined from φ (r) for different E NNLO is small at least up to E cm ~ 46 MeV (T lab ~ 100 MeV) quenched QCD m π = 529 MeV Murano et al. (HAL QCD Coll.)

28 Phase shift δ 0 (k) from V c (r) in 1 S 0 (m π = 0 MeV) m π = 380 MeV m π = 529 MeV m π = 731 MeV quenched QCD 1 S 0

29 V C (r) and V T (r) in full QCD (m π =570MeV, L=2.9 fm) Quenched QCD Full QCD

30 Phase shift from V(r) in full QCD 1 S 0 3 S 1 1 S 0 3 S 1

31 unitary regime NN scattering lengths in full QCD BS wave func. E Luscher s formula (CP-PACS method, 2005) Overall attraction Still far from unitary regime a 0 [fm] 1 S 0 3 S 1 1 S 0 3 S 1 [MeV] Kuramashi Plot [hep-lat/ ]

32 YN and YY interactions in lattice QCD no phase shifts available for YN and YY scatterings J-PARC (Tokai, Japan) plenty of hyper-nucleus data will soon be available at J-PARC predictions from lattice QCD difference between NN and YN?

33 Hyperon Core of Neutron Stars Radius ~ 10 km Mass ~ solar mass Central density ~ kg/cm 3 Hyperon matter Schaffner-Bielich, NP A804 (2008). Solid Crust Neutron Liquid Hyperon Matter?

34 M BB [GeV] De Swart, Nagels, Rijken, Verhoeven (1971)

35 S=-1 system: LN interaction (I=1/2) in full QCD Full QCD m π =701 MeV Nemura et al. (HAL QCD Coll.)

36 unitary regime NN and ΛN Scattering lengths in full QCD ΛN NN

37 S=-2 system: XN interaction (I=1) J-PARC DAY-1 exp. : 12 C(K -,K + ) 12 Be X Nemura, Ishii, Aoki, T.H., Phys.Lett. B673, 136 (2009) quenched QCD m π =510 MeV 1. Repulsive core + attractive well 2. Large spin dependence 3. Overall attraction

38 BB interaction What makes the difference among these six?

39 1 S 0 Pauli principle at work 3 S x32, full (CP-PACS/JLQCD config.) a=0.12 fm, L=2fm SU(3) limit: m π =m K =835 MeV, m B =1745 MeV Inoue et al. (HAL QCD Coll.)

40 BB phase shift from BB potential in full QCD 3-flavor full QCD CP-PACS/JLQCD configurations a=0.12fm, L=2.0fm m π,k =835 MeV Inoue (HAL QCD) Oka-Yazaki

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44 V C [MeV] V C [MeV] S-wave N-K + (us bar ) interaction δ 0 [deg] I=0 I=0 (lat) I=1(lat) I=1 r [fm] I=1(exp) I=0(exp) p lab [MeV] r [fm] (2+1)-flavor full QCD CP-PACS/JLQCD configurations a=0.12 fm, L=1.93 fm m π =871 MeV Ikeda et al. (HAL QCD Coll.)

45 S-wave N-η c (cc bar ) interaction Quenched QCD 32 3 x48, a = fm Kawanai & Sasaki (2010)

46 Summary and Future Nuclear forces from LQCD (HAL QCD strategy) BS amplitude YN, YY potentials observables exact few body calculations Full QCD with m π =140 MeV is our ultimate goal current : PACS-CS config. (N f =2+1) with L=2.9fm & m p = MeV in 1-2 years: PACS-CS config. (N f =2+1) with L=5.8fm & m p = 140 MeV in 5 years: new full QCD config. on 10 Pflops machine at Kobe (2012-) Current and Future targets of HAL QCD tensor force and π-n coupling LS force 3B forces light nuclei from LQCD potentials B-M interactions r [fm] r [fm]

47 Some Recent References NN force in quenched QCD: Ishii, Aoki & T.H., Phys. Rev. Lett. 99 (2007) [nucl-th/ ]. Theoretical foundation of the HAL formalism: Aoki, T.H. & Ishii, Prog. Theo. Phys 123 (2010) [arxive: [hep-lat]]. YN force in quenched QCD: Nemura, Ishii, Aoki & T.H., Phys. Lett. B673 (2009) 136 [arxiv: [nucl-th]]. NN force in full QCD: Ishii, Aoki & T.H. (for PACS-CS Coll.), LATTICE08 Proc. arxiv: [hep-lat] YN force in full QCD: Nemura, Ishii, Aoki & T.H. (for PACS-CS Coll.), LATTICE08 Proc. arxiv: [hep-lat]