Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon Forces in Vacuum and in the medium” Trento, Italy July 11 (11-15), 20 11

Outline 1. Monopole problem in the shell model 2. Shell evolution in exotic nuclei 3. Solution by three-body force Introduction to talks by J. Holt, A. Schwenk and T. Suzuki

Spectra of Ca isotopes calculated by most updated NN interaction microscopically obtained By Y. Tsunoda and N. Tsunoda N3LO V low-k with  =2.0 fm -1 2 nd and 3 rd order Q-box 4hw and 6hw s.p.e. used Present GXPH1A KB3G GXPF1A KB3G for comparison

Two-body matrix elements (TBME) may be calculated to a rather good accuracy 40 Ca core is not very stable yet -> 0+ energy lowered

48 Ca

Monopole component of the NN interaction Averaged over possible orientations As N or Z is changed to a large extent in exotic nuclei, the shell structure is changed (evolved) by Strasbourg group made a major contribution in initiating systematic use of the monopole interaction. (Poves and Zuker, Phys. Rep. 70, 235 (1981)) can be ~ 10 in exotic nuclei -> effect quite relevant to neutron-rich exotic nuclei n j’ : # of particles in j’ Linearity: Shift

T=1 monopole interactions in the pf shell j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A What’s this ? Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

T=1 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) j = j’ Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

T=0 monopole interaction The correction is opposite !

T=0 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force (  +  exchange) f-fp-p f-p “Local pattern”  tensor force

T=0 monopole interactions in the pf shell GXPF1A shell-model int. G-matrix (H.-Jensen) Tensor force (  +  exchange) Tensor component is subtracted Correction is attractive

Outline 1. Monopole problem in the shell model 2. Shell evolution in exotic nuclei 3. Solution by three-body force

Treatment of tensor force by V low k and Q box (3 rd order) Monopole component of tensor interactions in pf shell short-range correlation by V low k in-medium correction with intermediate states (> 10 hw, 3 rd order) Bare (AV8’) only for comparison

From a comparison to shell model interactions which can reproduce many experimental data in sd and pf shells, an extremely simple interaction “Monopole-based Universal Interaction V MU has been introduced. The same interaction for all nuclei range = 1 fm

monopole component of tensor force in nuclear medium monopole component of tensor force in free space almost equal ? Partly work by Tsunoda, O, Tsukiyama, H.-Jensen (2011)

Systematic description of monopole properties of exotic nuclei can be obtained by an extremely simple interaction as Parameters are fixed for all nuclei monopole component of tensor force in nuclear medium monopole component of tensor force in free space almost equal ?

Shell evolution due to proton-neutron tensor + central forces Changes of single-particle properties due to these nuclear forces

ls splitting smaller exotic nucleus with neutron skin proton neutron  d  /dr stable nucleus T=1 NN interaction more relevant to ls splitting change

From RIA Physics White Paper

Neutron single-particle energies at N=20 for Z=8~20 Z 8 14 Z These single-particle energies are “normal” f 7/2 -p 3/2 2~3 MeV N=20 gap ~ 6 MeV energy (MeV) d 5/2 s 1/2 d 3/ solid line : full VMU (central + tensor) dashed line : central only p 3/2 low Tensor force makes changes more dramatic. PRL 104, (2010) more exotic

Increase of 2+ excitation energy Neutron number 2+ level (MeV)

Outline 1. Monopole problem in the shell model 2. Shell evolution in exotic nuclei 3. Solution by three-body force

Proton number  Neutron number  Nuclear Chart - Left Lower Part - Why is the drip line of Oxygen so near ?

Single-Particle Energy for Oxygen isotopes Utsuno, O., Mizusaki, Honma, Phys. Rev. C 60, (1999) G-matrix+ core-pol. : Kuo, Brown SDPF-M - G-matrix + fit - V low-k : Bogner, Schwenk, Kuo trend by microscopic eff. int. by phenomenological eff. int. USD-B Brown and Richter, Phys. Rev. C 74, (2006)

What is the origin of the repulsive modification of T=1 monopole matrix elements ? A solution within bare 2-body interaction is very unlikely (considering efforts made so far) Zuker, Phys. Rev. Lett. 90, (2003)  3-body interaction The same puzzle as in the pf shell

The clue : Fujita-Miyazawa 3N mechanism  -hole excitation)  particle m=1232 MeV S=3/2, I=3/2  N NN   Miyazawa, 2007

Renormalization of NN interaction due to  excitation in the intermediate state  T=1 attraction between NN effectively Modification to bare NN interaction (for NN scattering)

Pauli blocking effect on the renormalization of single-particle energy Pauli blocking effect on the renormalization of single-particle energy Pauli Forbidden  The effect is suppressed m  m m’ Renormalization of single particle energy due to  -hole excitation  more binding (attractive) m  m m’ single particle states Another valence particle in state m’

Pauli forbidden (from previous page) This Pauli effect is included automatically by the exchange term.  m m m’  m m Inclusion of Pauli blocking

Most important message with Fujita-Miyazawa 3NF + Renormalization of single particle energy m  m m’ same Monopole part of Fujita-Miyazawa 3-body force Pauli blocking m m m’  Effective monopole repulsive interaction

(i)  -hole excitation in a conventional way  -hole dominant role in determining oxygen drip line (ii)EFT with  (iii) EFT incl. contact terms (N 2 LO) -> J.Holt, A. Schwenk, T. Suzuki

Ground-state energies of oxygen isotopes Drip line NN force + 3N-induced NN force (Fujita-Miyazawa force) O, Suzuki, Holt, O, Schwenk, Akaishi, PRL 105 (2010)

 (Effective) two-body interaction constant change of single-particle energy If the origin is “forgotten”, This is what happened in “microscopic theories”, leading to wrong drip line. N N N N N N N N or What was wrong with “microscopic theories” ? Observed in NN scattering present picture

For neutron matter : states below Fermi level attractive repulsive k k k k Brown and Green, Nucl.Phys. A137, 1 (1969 Fritsch, Kaiser and Weise, Nucl. Phys. A750, 259 (2005); Tolos, Friman and Schwenk, Nucl.Phys. A806}, 105 (2008); Hebeler and Schwenk, arXiv: [nucl-th]

For valence neutrons: states outside the core Attractive (single-particle energy renormalization) repulsive (valence neutron interaction)

Major monopole forces are due to + V  =1 fm Quick Summary More from J. Holt, A. Schwenk and T. Suzuki + FM 3NF basic binding (T=0), repulsive (T=1) except for j=j’ variation of shell structure limit of existence, shell structure at far stability

Casablanca mechanism Rick Victor This love is reduced by the presence of Rick This love is reduced by the presence of Victor repulsion Love = attractive force* *This equation has no proof.

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The central force is modeled by a Gaussian function V = V 0 exp( -(r/  ) 2 ) (S,T dependences) with V 0 = -166 MeV,  =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) relative strength Can we explain the difference between f-f/p-p and f-p ?

Spin-orbit splitting Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) h  4h  3h  2h  1h 

density saturation + short-range NN interaction + spin-orbit splitting  Mayer-Jensen’s magic number with rather constant gaps (except for gradual A dependence) robust feature -> nuclear forces not included in the above can change it -> tensor force

Magic numbers may change due to spin-isospin nuclear forces Tensor force produces unique and sizable effect Tensor and central forces -> Weinberg-type model Brief history on our studies on tensor force

V T = (           2) Y (2    Z(r) contributes only to S=1 states relative motion  meson (~  +  ) : minor (~1/4) cancellation  meson : primary source    Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92) Tensor Interaction by pion exchange Yukawa

How does the tensor force work ? Spin of each nucleon is parallel, because the total spin must be S=1 The potential has the following dependence on the angle  with respect to the total spin S. V ~ Y 2,0 ~ 1 – 3 cos 2  attraction  =0 repulsion  =  /2  S relative coordinate

Monopole effects due to the tensor force - An intuitive picture - wave function of relative motion large relative momentum small relative momentum attractive repulsive spin of nucleon TO et al., Phys. Rev. Lett. 95, (2005) j > = l + ½, j < = l – ½

wave function when two nucleons interact k1k1 k2k2 k1k1 k2k2 - approx. by linear motion - k = k 1 – k 2, K = k 1 + k 2 large relative momentum k strong damping wave function of relative coordinate k1k1 k2k2 wave function of relative coordinate small relative momentum k loose damping k1k1 k2k2

j > = l + ½ j < = l – ½ j’ < proton neutron j’ > General rule of monopole interaction of the tensor force Identity for tensor monopole interaction (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T TO. et al., Phys. Rev. Lett. 95, (2005)

The central force is modeled by a Gaussian function V = V 0 exp( -(r/  ) 2 ) (S,T dependences) with V 0 = -166 MeV,  =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) relative strength Can we explain the difference between f-f/p-p and f-p ?

GXPF1 G-matrix (H.-Jensen) Central (Gaussian) - Reflecting radial overlap - Tensor force (  +  exchange) T=0 monopole interactions in the pf shell f-fp-p f-p

S. Weinberg, PLB 251, 288 (1990) Tensor force is explicit Central force: strongly renormalized finite range (Gaussian) In nuclei  +  exchange Similarity to Chiral Perturbation of QCD

Central part changes as the cut-off  changes T=0 T=1 j-j’ Monopole int. (MeV) Tensor (reminder)

from Dickhoff Ratio to naïve single-particle mo del Measured spectroscopic factors short-range + in-medium corrections Tensor force remains almost unchanged ! Higher order effects due to the tensor force yield renormalization of central forces.

Multipole component of tensor forces - diagonal matrix elements -

Actual potential Depends on quantum numbers of the 2-nucleon system (Spin S, total angular momentum J, Isospin T) Very different from Coulomb, for instance 1S01S0 Spin singlet (S=0) 2S+1=1 L = 0 (S) J = 0 From a book by R. Tamagaki (in Japanese)

Tensor potential Question : Can the tensor force survive after -treatment of short-range correlations (hard-core) -treatment of core polarization

Test by experiments

h 11/2 g 7/2 proton single-particle levels change driven by neutrons in 1h 11/2 Z =51 isotopes No mean field theory, (Skyrme, Gogny, RMF) explained this before. h 11/2 - h 11/2 repulsive h 11/2 - g 7/2 attractive An example with 51 Sb isotopes  +  meson exchange tensor force (splitting increased by ~ 2 MeV)

Z =51 (= ) isotopes No mean field theory, (Skyrme, Gogny, RMF) explained this before. h 11/2 - h 11/2 h 11/2 - g 7/2 attractive repulsive An example with 51 Sb isotopes with VMU interaction tensor force in VMU (splitting increased by ~ 2 MeV) change driven by neutrons in 1h 11/2 g 7/2

Consistent with recent experiment One of the Day 1 experiments at RIBF by Nakamura et al. - Position of p 3/2

Monte Carlo Shell Model (MCSM) results have been obtained by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space. Expansion of the territory Effective N=20 gap between sd and pf shells Ca O Ne Mg SDPF-M (1999) WBB (1990) ~2MeV ~5MeV Neyens et al Mg Tripathi et al Na Dombradi et al Ne Terry et al Ne

From Grawe, EPJA25, 357 g 9/2 occupied E (MeV) N Central Gaussian + Tensor solid line: full VMU effect dotted line: central only Crossing here is consistent with exp. on Cu isotopes Proton single-particles levels of Ni isotopes shaded area : effect of tensor force

solid line : full VMU (central + tensor) dashed line : central only shaded area : effect of tensor force Shell structure of a key nucleus 100 Sn Exp. d5/2 and g7/2 should be close Seweryniak et al. Phys. Rev. Lett. 99, (2007) Gryzywacz et al. Zr Sn

Potential Energy Surface s 1/2 Z=28 gap is reduced also proton neutron f 7/2 d 3/2 d 5/2 full Tensor force removed from cross-shell interaction Strong oblate Deformation ? Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008) exp Si Si Other calculations show a variety of shapes. 42 Si: B. Bastin, S. Grévy et al., PRL 99 (2007) Si isotopes SM calc. by Utsuno et al.

Spectroscopic factors obtained by (e,e’p) on 48 Ca and the tensor force Collaboration with Utsuno and Suzuki

Spectroscopic factor for 1p removal from 48 Ca  d 5/2 deep hole state – More fragmentation Distribution of strength – quenching factor 0.7 is needed (as usual). – Agreement between experiment and theory for both position and strength (e,e’p): Kramer et al., NP A679, 267 (2001) Same interaction as the one for 42 Si

What happens, if the tensor force is taken away ? no tensor in the cross shell part s 1/2 d 3/2 d 5/2 with full tensor force

Summary 2. The tensor force remain ~unchanged by the treatments of short-range correlations and in-medium correction. This feature is very unique. 3. (e, e’p) data on 48 Ca suggests the importance of the tensor force, which is consistent with exotic feature of 42 Si. Direct reactions with RI beam should play important roles in exploring structure of exotic nuclei driven by nuclear forces. 1. Changes of shell structure and magic numbers in exotic nuclei are a good probe to see effects of nuclear forces. Such changes are largely due to tensor force, as have been described by VMU. Transfer reactions have made important contributions. 4. Fujita-Miyazawa 3N force can be the next subject for the shell evolution.

Summary 1.Monopole interactions : effects magnified in neutron-rich nuclei 2.Tensor force combined with central force : a unified description particularly for proton-neutron monopole correlation. -> N=20 Island of inversion, 42 Si, ７８ Ni, 100 Sn, Sb, 132 Sn, Z=64,… Tensor force in nuclear medium is very similar to the bare one. This central force may be a challenge for microscopic theories. 3. Fujita-Miyazawa 3-body force produces repulsive effective interaction between valence neutrons in general. The spacings between neutron single-particle levels can become wider as N increases, and new magic numbers may arise. Examples are shown for O and Ca isotopes with visible effects. shell quenching 4.Structure change on top of the shell evolution -> diagonalization with super computer

Collaborators T. Suzuki Nihon U. M. Honma Aizu Y. Utsuno JAEA N. Tsunoda Tokyo K.Tsukiyama Tokyo M. H.-Jensen Oslo A.Schwenk Darmstadt J.Holt ORNL K.Akaishi RIKEN

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SPE : GXPF1 f7:-8.62 f5: p3: p1: Ca ground-state energy cont’d

Ca 2+ level systematics 2 + of 48 Ca rises by 3N becomes about right by using GXPF1A SPE N=32, 34 higher 2 + levels

Spin quenching factor MeV GXPF1 spe : GXPF1 48 Ca M1 excitation 10

Summary The tensor force is similar to bare  +  meson exchange. 3. Fujita-Miyazawa 3N force generates a repulsive T=1 monopole force robustly for valence nucleons on top on the core, resolving long-standing difficulty of shell-model interaction. FM force should produce density-dependent repulsive effective interaction, which may give new direction to mean field models. 1. Shell structure and magic numbers do change from Mayer- Jensen’s in exotic nuclei, due to the nuclear forces. 2. Such changes can be predicted/explained by a simple interaction consisting of central + tensor forces body force affects crucially oxygen drip line and new magic numbers (N=14, 16), as well as properties of Ca isotopes. This feature will continue to heavier nuclei.

Dominant monopole forces are due to + V  =1 fm Summary-2 + FM 3NF basic binding variation of shell structure limit of existence

古典力学での三体問題と三体力 （有効三体力） ここでの三体力は、二体力＋超多体問題を回避するための “ 有効 ” 相互作用. 正真正銘の三体力は存在するか？ GPS の位置をこの方程式を数値的に 解いても正確には求まらない。 GPS の役割を果たさない！ それは地球が変形するから（もちろん相対 論効果もあるが） 酒井（英）氏より拝借

Ground-state energies of oxygen isotopes Drip line NN force + 3N-induced NN force (Fujita-Miyazawa force)

Other diagram included T=1 interaction between valence particles  Pauli blocking Particle in the inert core Related effect was discussed by Frisch, Kaiser and Weise for neutron matter See also Nishizaki, Takatsuka and Hiura PTP 92, 93 (1994)  Compressibility of neutron-rich matter

 -hole excitation may be crucial to neutron matter property Chiral Perturbation incl.  Frisch, Kaiser and Weise

D, E terms fitted to E(3H) and radius(4He) NN for smooth cutoff V low k (n_exp=4) from N 3 LO(500) Conventional calculation with πNΔ coupling  exchange with radial cut-off at 0.5 fm, ΔE =293 MeV f_{πNΔ ｝ /f_{πNN} = \sqrt{9/2} A.M. Green, Rep. Prog. Phys. 39, 1109 (1976)

Ground-state energies of oxygen isotopes Drip line NN force + 3N-induced NN force  m m m’ (Fujita-Miyazawa force)

Collaborators T. Suzuki Nihon U. M. Honma Aizu Y. Utsuno JAEA N. Tsunoda Tokyo K.Tsukiyama Tokyo M. H.-Jensen Oslo A.Schwenk TRIUMF/Darmstadt J.Holt ORNL K.Akaishi RIKEN

Systematic description of monopole properties of exotic nuclei can be obtained by an extremely simple interaction as Can this be really the same as the bare tensor interaction ?

Hartree-Fock energies by Skyrme Hartree-Fock The shell structure remain rather unchanged -- orbitals shifting together -- change of potential depth ~ Woods-Saxon. Neutron Single-Particle Ene ｒ gies at N=20

Other effects from neutron-skin, continuum etc., may arise, particularly near drip line. We need to know more precisely nuclear and hadronic forces, in order to understand exotic nuclei. -> EFT, Lattice QCD particularly for 3-body force although FM force is well under control -> Fully ab initio calculations of single-particle levels by Coupled Cluster calculation, etc. will be of great interest

(i)  -hole excitation in a conventional way  -hole dominant role in determining oxygen drip line (ii)EFT with  (iii) EFT incl. contact terms (N 2 LO)

Spin quenching factor MeV GXPF1 spe : GXPF1 48 Ca M1 excitation 10

Monopole part of the NN interaction averaged over J As N or Z is changed to a large extent in exotic nuclei, the following quantity plays a crucial role. Strasbourg group made a major contribution in initiating systematic use of the monopole interaction. (Poves and Zuker, Phys. Rep. 70, 235 (1981)) Linearity : Shift  e(a) = V ab * n b n b : # of particles in orbit b can be ~ 10 for g9/2, for instance -> effect quite relevant to neutron-rich exotic nuclei

Theoretical calculations (i)  -hole excitation in a conventional way (ii) EFT with  included (iii) EFT incl. contact terms (N 2 LO)  -hole dominant role in determining oxygen drip line Phenomenological int.

What are the major monopole components in the shell-model effective interaction which are successful in systematic description over many nuclei ? Example taken from pf-shell GXPF1A interaction * G-matrix obtained from Bonn-C potential + 3 rd order Q box + empirical refinement by  2 -fit to experimental data *) M. Honma et al., PRC65 (2002) (R)