*2.8. Monopole Interaction of the Two-Body Spin-Orbit Force*

It is a natural question what effect can be expected from the two-body spin-orbit force of the *NN* interaction. This force can be well described by the M3Y interaction, and the monopole effects of the two-body spin-orbit force were described in detail in [15], particularly in its supplementary document. Although the monopole effects of this force contributes to the spin-orbit splitting [15], the effect is much weaker than the tensor force in most cases, as also discussed in the article by Utsuno in this volume.

An interesting case is found in the coupling between an *s* orbit and *p*3/2,1/2 orbits. There is no monopole effect from the tensor force, if an *s* orbit is involved. Instead, the *s*-*p* coupling due to the two-body spin-orbit force can be exceptionally strong as intuitively stressed in [15]. Figure 8 shows that the possible significant change in the neutron 2*p*3/2-

2*p*1/2 gap between 35Si and 37S is explained to a good extent by the shell evolution due to the two-body spin-orbit force.

**Figure 8.** (**a**) Neutron 2*p*3/2-2*p*1/2 splitting for N = 21 isotones. The symbols are the centroids for 37S [42], 39Ar [50], and 41Ca [50,51]. The horizontal bars are the energy differences between relevant highest peaks [50,52]. Shell evolution predictions are shown by blue closed symbols and the solid line connecting them. The loose binding effect for 35Si is included in the open circle. The calculation with Woods–Saxon potential with parameters adjusted are shown by the yellowish shaded area [50]. (**b**) Neutron 2*p*1/2 single-particle energy (blue solid line) by a Woods–Saxon potential [12] for varying depth parameter, V. The linear dependence of the deeply bound region is linearly extrapolated (blue dashed line) and is compared to the curved dependence that results from the proximity of the continuum. The dashed line is for the 2*p*3/2 orbit, and the loose-binding contribution to the present splitting appears to be 0.06 MeV against 1.5 MeV splitting itself. Taken from Figure 8 of Supplementary Material of [15].

#### *2.9. Monopole Interaction from the Three-Nucleon Force*

The three-nucleon force (3NF) is currently of intense interest (see, for instance, a review [53]). Among various aspects, we showed [54] the characteristic feature of the monopole interaction of the effective *NN* interaction derived from the Fujita–Miyazawa 3NF [55]. Figure 9a displays the effect of the Δ excitation in nucleon–nucleon interaction. The Δ-hole excitation from the inert core changes the SPE of the orbit *j* as shown in Figure 9b, where *m* is one of the magnetic substates of the orbit *j*, and *m* means any state. This diagram renormalizes the SPE, and observed SPE should include this contribution. If there is a valence nucleon in the state *m* as in Figure 9c, the process in Figure 9b is Pauli-forbidden. However, in the shell–model and other nuclear-structure calculations, the SPE containing the effect of Figure 9b is used. One has to somehow incorporate the Pauli effect of Figure 9c, and a solution is the introduction of the process in Figure 9d. In this process, the state *m* doubly appears in the intermediate state, but one can evaluate the Pauli effect by including Figure 9b,d consistently. This is a usual mathematical trick and enables us to correctly treat the Pauli principle within the simple framework. Figure 9d is equivalent to Figure 9e, which is nothing but the Fujita–Miyazawa 3NF, where the state *m* appears in double. Similar treatment is carried out in the chiral Effective Field Theory (EFT) framework. Figure 9f corresponds to Figure 9e, but the violation of the Pauli principle is slightly hidden, because of a vertex in the middle (depicted by a square) instead of the Δ-hole excitation.

In this argument, the 3NF produces a repulsive monopole *NN* interaction in the valence space, after the summation over the hole states of the inert core (see Figure 9 bottom right), which corresponds to the normal ordering in other works.

The plot Figure 9 top right indicates an example of the repulsive effect on the groundstate energy of oxygen isotopes, locating the oxygen dripline at the right place or solving the oxygen anomaly [54]. This is rather strong repulsive monopole interaction, which is a consequence of the inert core. This means that the present case is irrelevant to the no-core shell model or other many-body approaches without the inert core (e.g., Green's Function Monte Carlo calculation [56]). This feature has caused some confusions in the past, but the difference is clear. The present repulsive monopole effect is much stronger than the other effects of the 3NF [57], and the latter will be better clarified by further developments of the chiral EFT for 3NF in the future. I note that the repulsive *T* = 1 *NN* effect was empirically noticed by Talmi in the 1960s [3].

**Figure 9.** Schematic illustration of the three-nucleon force (3NF). **Left**: The diagrams (**a**–**e**) show how Δ-hole excitation effects are incorporated in accordance with Pauli principles, with the final form shown in (**e**), as described in the text. The diagrams in (**f**–**h**) represent three contributions from 3NF obtained in the chiral Effective Field Theory. **Top right:** the ground-state energy of oxygen isotopes, calculated with and without the 3NF and observed experimentally. **Bottom right**: the intuitive explanation of the diagrams in (**d,e**) of the left panel with the 16O inert core. Based on Figures 3 and 4 of [54].

## *2.10. Short Summary of This Section*

The shell evolution phenomena are seen in many isotopic and isotonic chains and sometimes result in the formation of new magic gaps or the vanishing of old ones. Figure 2b displays the emergence of such new magic numbers *N* = 16, 32, and 34, whereas the lowering of some 2<sup>+</sup> levels can mean the weakening of some magic numbers. More changes may appear in the future studies. Thus, the characteristic monopole features of the central, tensor, two-body *LS*, and 3NF-based *NN* interactions and the resulting shell evolution are among the emerging concepts of the nuclear structure. Interestingly, these findings are neither isolated nor limited to particular aspects but are related to other aspects of the nuclear structure. We now move on to such a case.

## **3. Type-II Shell Evolution and Shape Coexistence**
