Quantitative understanding nuclear structure and scattering processes, based on underlying NN interactions.

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1 Microscopic descriptions of nuclear scattering and reaction processes on the basis of chiral EFT M. Kohno Research Center for Nuclear Physics, Osaka, Japan Quantitative understanding nuclear structure and scattering processes, based on underlying NN interactions. Saturation properties: mean field and shell structure LOBT nuclear matter calculations with Ch-EFT NN and 3N forces Three-nucleon forces in chiral effective field theory Many-body forces necessary in effective theory (effects from non-nucleonic degrees of freedom are to be recovered) Microscopic optical model potentials based on the G-matrices of Ch-EFT interactions Characteristic 3NF effects Saturation, enhanced spin-orbit and tensor components in the medium

2 Excerpt from Feynman lecture on physics, Vol. III Chap.15 The independent particle approximation : In atomic nuclei the protons and neutrons interact with each other quite strongly. Even so, the independent particle model can again be used to analyze nuclear structure. It was first discovered experimentally that nuclei were especially stable if they contained certain particular numbers of neutrons-namely 2, 8, 20, 28, 50, 82. Nuclei containing protons in these numbers are also especially stable. Since there was initially no explanation for these numbers they were called the magic numbers of nuclear physics. It is well known that neutrons and protons interact strongly with each other; people were, therefore, quite surprised when it was discovered that an independent particle model predicted a shell structure which came out with the first few magic numbers... Then it was discovered by Maria Mayer, and independently by Jensen and his collaborators, that by taking the independent particle model and adding only a correction for what is called the spin-orbit interaction, one could make an improved model which gave all of the magic numbers..

3 How the saturation and s.-p. structure of nuclei appear, in spite of the NN interaction singular at short distance? Brueckner theory (1950 s) provided a semi-quantitative explanation. The effective interaction namely G-matrix, after short-range correlations are taken into account by the matrix equation G ω = v + v Q G ω, is weak enough to produce mean field. ω H Importance of the Pauli and dispersion effects. Saturation mechanism: the attraction from the tensor correlation becomes weaker at larger densities due to the Pauli blocking. However, no realistic NN potential reproduces an empirical saturation curve in nuclear matter. (Energies and radii of nuclei are not simultaneously reproduced in ab initio calculations.)

4 Nuclear matter saturation curves with various modern NN forces LOBT calculations do not reproduce the empirical saturation point. (Higher order contributions are believed to be small.) Off-shell uncertainties: Coester band [F. Coester et al., Phys. Rev. C1, 769 (1970)] [various modern realistic NN potentials] [low momentum interactions from AV18, and Ch-EFT with three choices of cutoff scale Λ]

5 Nuclear matter saturation curves with various modern NN forces LOBT calculations do not reproduce the empirical saturation point. (Higher order contributions are believed to be small.) Off-shell uncertainties: Coester band [F. Coester et al., Phys. Rev. C1, 769 (1970)] [various modern realistic NN potentials] [low momentum interactions from AV18, and Ch-EFT with three choices of cutoff scale Λ]

6 Missing effects or contributions? Higher orders, relativistic effects, 3NF, medium modification,. Since the 1950 s, 3NF effects have been expected. π For example: Fujita-Miyazawa, P.T.P. 17 (1957) 360. Phenomenological terms and adjustments are included in the studies of 3NF effects in the literature. Variational calculations by Pandharipande et al. ( ) Progress on the description of NN interaction: Ch-EFT Low energy effective theory, respecting the symmetry of QCD and its spontaneous breaking pattern. power counting: expansion in terms of the momentum involved At N 3 LO level, comparable accuracy for reproducing NN scattering data with other modern NN interactions is achieved. NN and NNN are constructed systematically and consistently. π

7 NNLO 3NFs in Ch-EFT c 1 c 3 c 4 c D c E V 3N (2π) = g A 2 i j k 8f π 4 (2π) (1π) V 3N V 3N (σ i q i )(σ j q j ) (q i 2 + m π 2 )(q j 2 + m π 2 ) F ijk αβ τi α τ j β V 3N (ct) αβ = δ αβ 4c 1 m π 2 + 2c 3 q i q j + c 4 ε αβγ τ k γ σ k q i q j F ijk V 3N (1π) = g A c D i j k 8fπ 4 Λ χ σ j q j (q j 2 +m π 2 ) σ i q j τ i τ j V 3N (ct) = c E i j k τ i τ j q i = p i p i 2f π 4 Λ χ c 1 = 0.81 GeV 1, c 3 = 3.4 GeV 1, and c 4 = 3.4 GeV 1 are determined in NN sector. c D and c E are to be determined in many-body systems. Λ χ = 700 MeV, m π = MeV

8 Reduction of 3NF v 123 to density dependent NN v 12(3) ab v 12(3) cd abh v A 123 cdh A h Diagrammatical representation by Holt et al. [J.W. Holt, N. Kaiser, and W. Weise, Phys. Rev. C81, (2010)] c D c D c E Contributions from the two left diagrams tend to cancel. Expand them into partial waves, add them to NN and carry out G matrix calculation. (*) A factor of 1 3 This diagram partly corresponds to the Pauli blocking of the isobar Δ excitation in a conventional picture. is necessary for the calculation of energy.

9 Effective two-body interaction and partial wave expansion Averaging over the third nucleon in nuclear matter( k 3 k F, σ 3, τ 3 ). Example: c 1 term of the Ch-EFT NNLO 3NF. k σ 1 τ 1, k σ 2 τ c 2 V 1 12(3) kσ1 τ 1, kσ 2 τ 2 = c 1g 2 2 A m π k σ f4 1 τ 1, k σ 2 τ 2, k 3 σ 3 τ 3 π k 3,σ 3,τ 3 σ 1 q 1 σ 2 q 2 q m2 π q m2 π τ 1 τ 2 + σ 1 q 1 σ 3 q 3 q m2 π q 2 τ 2 + m2 1 τ 3 + σ 2 q 2 σ 3 q 3 π q m2 π q 2 τ 3 + m2 2 τ 3 π k σ 1 τ 1, k σ 2 τ 2 a, k 3 σ 3 τ 3 + k σ 2 τ 2 a, k 3 σ 3 τ 3, k σ 1 τ 1 a +k 3 σ 3 τ 3, k σ 1 τ 1, k σ 2 τ 2 a Spin and isospin summations two-body central, spin-orbit, and tensor components k 3 -integration expand in partial waves Then, form factors in the form of exp k Λ 6 k Λ 6 are multiplied Different statistical factors for V 12(3) in total and s.p. energy, reflecting the three-nucleon force origin.

10 Caution for the use of the reduced 2NF: different statistical factors in total energy E and s.-p. energy e h. Total energy (in the HF approximation) E = h t h + 1 2! h hh 1 hh v 12 hh A + 3! hh h hh h v 123 hh h A define ab v 12(3) cd A h abh v 123 cdh A = h t h + 1 h hh v 2! hh v 3 12(3) hh Single-particle energy (potential) 1 e h = h t h + h hh v 12 hh A + 2! h h A hh h v 123 hh h A = h t h + hh v v 12(3) hh h A = h t h + hh v h v 3 12(3) hh + hh 1 h v 6 12(3) hh A A

11 Prescription for the G matrix equation including 3NF term G 12 = v v 12(3) + v v 12(3) Q ω H G 12 total energy is given by E = k t k + 1 k U 2 E(k), where U E (k) k kk G 12 kk A s.p. energy e k = k t k + U G k, and the potential in the propagator ω H = e k1 + e k2 t 1 + U G k 1 + t 2 + U G k 2 The factor in front of v 12(3) is different in U G (k): U G (k) kk G v 12(3) 1 + Q k ω H G 12 kk A Note that the difference between U E (k) and U G (k) is typically 5 MeV, and this difference does not much affect G, because of the cancellation in the denominator ω H. OMP potential corresponds to U G k.

12 numerical difference between U E (k) and U G (k) U E (k) k kk G 12 kk A U G (k) kk G v 6 12(3) 1 + Q k ω H G 12 kk A

13 LOBT calculation with NN+ 3NF of Ch-EFT Calculated saturation curves with three choices of cutoff Λ. Results of c D = 0 and c E = 0. Pauli effects are sizable. Tune c D and c E.

14 c D and c E terms provide very similar contributions to the NM Born energy, if c D 4c E c D and c E term contributions [MeV] 20 0 symmetric nuclear matter c D =4.2 c E = 1.0 c D = k F [fm 1 ] E cd ρ = c D ( ρ ρ 2 ) E ce ρ = c E ( ρ 128.3ρ 2 ).

15 each spin- and isospin-channel contributions in LOBT E/A thin curves:λ = 450 MeV, thick curves: Λ = 550 MeV Large repulsion in 3 O and 1 S channels.

16 3NF effects in scattering processes It is important to verify 3NF effects in various nuclear properties: structures and reaction processes. Specific characters: repulsive in the 1 S 0 and 3 P channel and enhancement of the tensor component in 3 S 1 channel. The 3 S- 3 D tensor component is enhanced by about 15%. 3NF effects become significant at and beyond the normal density, but small at low densities. Therefore, 3NF contributions are probably small in nucleon scattering at surface region and also in neutron rich nuclei. Verify characteristic features of the 3NF contributions by considering scattering processes. The real part of the optical model potential becomes shallow. The enhancement of the tensor correlation leads to a larger imaginary potential for particle states. These predictions should be testified by explicitly calculating cross sections corresponding to experimental data.

17 Optical model potential calculated from G-matrices in nuclear matter including 3NF effects Folding G-matrices in nuclear matter parameterized in a 3-range Gaussian form, using LDA, to obtain OM potential. V ST ST r = i=1,3 V 0,i e (r/a i )2 a i : 2.5, 0.9, 0.4 fm for real, 1.8, 0.9, 0.5 fm for imaginary Knock-on exchange term is localized by the Brieva-Rook method. U direct (R; E) = dr τ=p,n ρ τ r g τ direct s; k F, E U exch. (R; E) = τ=p.n dr ρ τ r s 3j 1 k F s 2 k F s j 0 K r s g τ exch. s; k F, E K(r) 1 ħ 2μ E U r V C(r) Density profiles of target nuclei ( 58 Ni and 208 Pb) are provided by Hartree- Fock calculations with Gogny D1S force. s g τ (s; k F, E) R r

18 Folding model calculations by the Kyushu University group: M. Toyokawa, T. Matsumoto, K. Minomo, M. Yahiro, and M. Kohno Diagonal G-matrix elements and non-diagonal G-matrix elements around the diagonal part are satisfactory reproduced by a three-range Gaussian function, together with the original single-particle potential in nuclear matter. No phenomenological adjustment is introduced. p- 208 Pb

19 Folding model calculations by the Kyushu University group: M. Toyokawa, T. Matsumoto, K. Minomo, M. Yahiro, and M. Kohno Because elastic scattering takes place at the low-density nuclear surface, 3NF effects are minor. This explains the success of the past microscopic optical-model-potential description such as those of the Melbourne group [K. Amos et al., Adv. Nucl. Phys. 25 (2000) 275]. p- 58 Ni

20 Comparison of the folding potentials with and without 3NF effects: proton case Real part: repulsive contributions from 3NF and enhanced LS potential Imaginary part: strength becomes larger due to the enhanced tensor force p- 208 Pb potential at E=65 MeV p- 208 Pb potential at E=200 MeV

21 Comparison of the folding potentials with the global potentials by Koning-Delaroche [Nucl. Phys. A713, 231 (2003)] Real part: Different at 200 MeV, LS potential is very similar. Imaginary part: large difference at the inner region, but irrelevant for cross sections. p- 208 Pb potential at E=65 MeV p- 208 Pb potential at E=200 MeV

22 Differential cross sections and polarizations obtained from the folding potential and the Koning-Delaroche global potential p- 208 Pb at E=65 MeV p- 208 Pb at E=185 MeV

23 Neutron elastic scattering on 208 Pb by the folding potentials with and without 3NF effects

24 Neutron elastic scattering on 208 Pb: folding potential vs. Koning-Delaroche potential n- 208 Pb at E=65 MeV n- 208 Pb at E=185 MeV

25 Comparison of the folding potentials with and without 3NF effects: neutron case Real part: repulsive contributions from 3NF, and enhanced LS potential. Imaginary part: strength becomes larger due to the enhanced tensor correlation. n- 208 Pb potential at E=65 MeV n- 208 Pb potential at E=200 MeV

26 Comparison of the folding potentials with the Koning-Delaroche [Nucl. Phys. A713, 231 (2003)] global potentials Real part: Different at 200 MeV, LS potential is similar. Imaginary part: Large difference at the center, but irrelevant for cross sections. n- 208 Pb potential at E=65 MeV n- 208 Pb potential at E=185 MeV

27 4 He- 58 Ni scattering Double folding potential. Target density approximation for the LDA.

28 4 He- 208 Pb scattering Double folding potential. Target density approximation for the LDA works well for the alpha particle.

29 Effects of a chiral three-nucleon force on nucleus-nucleus scattering K. Minomo, M. Toyokawa, K. Ogata, M. Yahiro, and M. Kohno The density at the center of the two interacting nucleons is assumed for the LDA. The larger imaginary potential due to the 3NF effects as well as the repulsive effect in the real part, is favorable to account for experimental data, but not enough. 12 C- 12 C at 85 MeV/nucleon 16 O- 16 O at 70 MeV/nucleon

30 Effects of a chiral three-nucleon force on nucleus-nucleus scattering K. Minomo, M. Toyokawa, K. Ogata, M. Yahiro, and M. Kohno preliminary The cross section is further reduced when mutual excitation effects are taken into account by coupled-channel calculations. 12 C- 12 C at 85 MeV/nucleon 16 O- 16 O at 70 MeV/nucleon Preliminary

31 dρ(r) 0 one-body LS field in the Thomas type U 1 ls l σ r dr Relation to the two-body effective LS force by Scheerbaum [N.P. A257,77(1976)] B S = 2π q Strength of spin-orbit force V 3 O ls (r)j 1 (qr)r 3 dr (q 0.7 fm 1 ), then U 0 ls = 3 B 4 S Relation to the δ-type effective LS force: iw σ 1 + σ 2 δ r HF LS field 1 2 W 1 r dρ r d(ρ r +ρ τ r ) if ρ p r = ρ n r = 1 ρ(r), then 3 W 1 l σ. Thus W B 2 4 r dr S DDHF calculations (Skyrme int., Gogny int., ) use W ph = 120~130 MeV fm 5. dr l σ On the other hand, realistic NN forces predict B S ~90 MeV fm 5.

32 Phenomenological W 0 and Scheerbaum factor B S obtained from nuclear matter calculations [M. Kohno, Phys. Rev. 86, (R) 2012] W 0 [Mev fm 5 ] type spin orbit strength Scheerbaum factor in n.m. k F =1.35 fm 1 k F =1.07 fm 1 SkIII SkM* SLy4 SLy5 SLy7 Skp BSk3 BSk5 BSk7 Gogny D1 Gogny D1S NSC97 AV18 CD Bonn Ch EFT (Julich)

33 Phenomenological W 0 and Scheerbaum factor B S obtained from nuclear matter calculations [M. Kohno, Phys. Rev. 86, (R) 2012] W 0 [Mev fm 5 ] type spin orbit strength Scheerbaum factor in n.m. k F =1.35 fm 1 k F =1.07 fm 1 SkIII SkM* SLy4 SLy5 SLy7 Skp BSk3 BSk5 BSk7 Gogny D1 Gogny D1S NSC97 AV18 CD Bonn Ch EFT (Julich) Ch EFT+3NF with 3N effects

34 Effects of three-nucleon spin-orbit interaction on isotope shifts of Pb nuclei : H. Nakada and T. Inakura, Phys. Rev. C91, (R) (2015) Density-dependent δ-type spin-orbit interaction suggested by the Ch-EFT 3NF effects. Δ r 2 A p Pb APb r p Pb r 2 p Stronger kink behavior is produced.

35 Tensor force including the 3NF effects at normal density Bare diagonal matrix elements with and without 3NF effects. Diagonal matrix elements of lowmomentum tensor interaction.

36 Novel features of nuclear forces and shell evolution in exotic nuclei : T. Otsuka et al., Phys. Rev. Lett. 104 (2010) Otsuka et al. introduced the V MU interaction consisting of simple central and tensor forces, and explain the change of shell structures in exotic nuclei, and demonstrated the important role of tensor force. No core shell model calculations have been advanced, but systematic and extensive applications for medium-mass and open shell nuclei are still not feasible.

37 Resemblance between the tensor components of V MU and Ch-EFT Detailed analyses of the shell-model matrix elements of the Ch-EFT tensor interaction including 3NF are in progress by Yoshida in the Tokyo University. Shell model calculations for Ni isotopes need stronger tensor force by about 30 %. Stronger tensor force is needed to understand the repelling of f7/2 and h11/2 orbits at neutron excess region in Sb isotopes by the tensor force,

38 Summary (1) N 3 LO NN potential in Ch-EFT is as accurate as other modern NN potentials in reproducing scattering data. Saturation curves in the LOBT are naturally similar. Coester band. 3NFs, consistent with the NN sector, are introduced systematically in Ch-EFT. Saturation properties are much improved by including the 3NF. 3NFs are reduced to effective NN force by folding over the third nucleon. Salient characters of the 3NF effects, in addition to the repulsive contribution: A) Spin-orbit force is enhanced, which accounts for the empirical strength. B) Tensor component is enhanced, which brings about some attraction in the 3S1 channel and enhance the imaginary potential for the scattering state.

39 Summary (2) Applications of the G-matrix to scattering problems. The G-matrices, parametrized in a 3-range Gaussian form, are used to construct OMP potentials. No adjustable parameters except for the standard local-density approximation (LDA) and (Brieva-Rook) localization method. Good reproduction of scattering data, up to the incident energy of 200 MeV. It is explicitly shown that the 3NF effects are small in the description of elastic scattering, because the process mainly takes place at the low-density surface. Application to the reaction process such as (p,2p) is in progress. As for the structure: Possibility to explain the needed tensor component in the shellmodel calculations: shell evolution in exotic nuclei. Hint for the density-dependent spin-orbit field to solve the problem of the kink structure in the radius.

40 Comparison of the folding potentials with the cutoff of 450 MeV and 550 MeV E in = 65 MeV, E in = 200 MeV

41 Comparison of the cross sections with the cutoff of 450 MeV and 550 MeV p- 208 Pb 4 He- 208 Pb

42 k Λ Equivalent interaction in restricted (low-mom.) space k H k H = e S 12 He S 12 k H k Q P Q Q k Λ Q P Q Q Λ k Λ k Apply a unitary transformation e s 12 to H to obtain an equivalent Hamiltonian H in a restricted (P) space [Suzuki and Lee, PTP64 (1980)] Unitary transformation e s 12 satisfies a decoupling condition Q H P = P H Q = 0 in two-body space. Singular high-momentum components are eliminated. Eigenvalues in the restricted (P) space do not change. On-shell properties are preserved. Off-shell properties naturally change. Induced many-body forces appear in many-body space.