Ab-initio no-core shell model study of ^10-14B isotopes with realistic NN interactions

In nuclear physics, our focus is to describe the nuclear structure including the exotic behaviour of atomic nuclei throughout the nuclear chart. Conventional shell model SM ; p ; sd ; fp ; O.Sorlin ; T.Otsuka , where interactions are assumed to exist only among the valence nucleons in a particular model space is unable to determine the drip line dripline1 ; dripline2 , cluster cluster and halo halo structures. The study of interactions derived from first principles has been a challenging area of research over the past decades. These fundamental interactions are determined from either meson-exchange theory or Quantum chromodynamics (QCD) QCD . QCD is non-perturbative in low-energy regime which makes analytic solutions difficult. This difficulty is overcome by chiral effective field theory ($\chi$EFT) RMP ; EFT1 ; EFT2 ; EFT3 . Chiral perturbation theory ($\chi$PT) PT within $\chi$EFT provides a connection between QCD and the hadronic system.

Progress has been made in the development of different many-body modern ab-initio approaches ab-initio ; HHMS2015 ; SHKH2019 , one of them being the NCSM Phys.Rev.C502841(1994) ; Phys.Rev.C542986(1996) ; Phys.Rev.C573119(1998) ; PRL84 ; 1999 ; PRC62 ; 2000 ; 2009 ; MVS2009 ; nocore_review ; stetcu1 ; stetcu2 . Ab-initio methods are more fundamental compared to the nuclear shell model. The aim of this paper is to explain the nuclear structure of boron isotopes with realistic NN interactions as the only input. The well-bound stable ¹⁰B have posed a challenge to the microscopic nuclear theory in particular concerning the reproduction of its ground-state spin av8'nocore . The boron isotopes have been investigated in the past using the shell model monopole ; TOSM . Shell model Hamiltonian constructed from a monopole-based universal interaction (V_MU) in full $psd$ model space including $(0{-}3)\hbar\Omega$ excitations has been used for a systematic study of boron isotopes monopole . This phenomenological effective interaction is obtained by fitting experimental data, thus, it at least partly includes three-body effects. So it is able to reproduce spin of the ground state (g.s.) of ¹⁰B. This V_MU based Hamiltonian, however, fails to describe the drip line nucleus ¹⁹B. Tensor-optimized shell model (TOSM) TOSM has been applied to study ¹⁰B using effective bare nucleon-nucleon (NN) interaction Argonne V8${}^{{}^{\prime}}$ (AV8${}^{{}^{\prime}}$) av8' . The g.s. obtained with AV8${}^{{}^{\prime}}$ interaction is $1^{+}$, which, in experiment, is the first excited state of ¹⁰B. AV8${}^{{}^{\prime}}_{eff}$ interaction, which is a modification of tensor and spin-orbit forces of AV8${}^{{}^{\prime}}$ interaction, gives correct g.s. spin and low-lying spectra, indicating that the tensor forces affect the level ordering. TOSM with Minnesota (MN) effective interaction MN without tensor force also gives correct g.s. spin but a smaller g.s. radius compared to the experimental result, which affects the nuclear saturation property, thus providing the small level density.

In Refs. Nmax8 ; 10B_PRL ; Nmax10 , the structure of ¹⁰B was studied within the NCSM, using accurate charge dependent NN potentials up to the $4^{th}$ order of $\chi$PT in basis spaces ($N_{\mbox{max}}$) of up to $10\hbar\Omega$. Using the NN interactions alone led to an incorrect g.s. of ¹⁰B. By including the chiral three-nucleon interaction (3N), the g.s. was correctly reproduced as $3^{+}$ 10B_PRL ; Nmax10 . The ab-initio NCSM study of ¹⁰B with the chiral N²LO (next-to-next-to-leading order) NN interaction EKM2015 including three-body forces has been done in Ref. N2LO , where it was shown that the g.s. energy and spin depends on the chiral order. To correctly reproduce the $3^{+}$ as an experimental g.s., 3N force with the N²LO NN interaction is needed. In Ref. N2LOopt , N²LO_opt interaction was employed in the NCSM calculation for ¹⁰B up to $N_{\mbox{max}}$ = 10 (10$\hbar\Omega$) to calculate ground and low-lying excited states. This study reported $1^{+}$ as a g.s. instead of $3^{+}$. Realistic shell model calculations including contributions of a chiral three-body force [N³LO NN + N²LO 3N potential] for ¹⁰B is reported in Ref. MBPT . These results are consistent with the NCSM results with the same interaction. The NCSM with CDB2K potential ($N_{\mbox{max}}$ = 8) and AV8’ ($N_{\mbox{max}}$ = 6) predict $1^{+}$ as g.s. of ¹⁰B CDB2Knocore ; av8'nocore . Green’s function Monte Carlo (GFMC) approach with AV8’ and AV18 has also been employed to investigate the g.s. of ¹⁰B GFMC and similarly predicts the $1^{+}$ as the ground state with these NN forces.

In Ref. PLB , the Daejeon16 and JISP16 (J-matrix inverse scattering potential) NN interactions were applied to $p$-shell nuclei. For ¹⁰B, excitation energies of $1^{+}$ state with respect to $3^{+}$ state of 0.5(1) MeV and 0.9(2.4) MeV were reported with Daejeon16 and JISP16 NN interactions, respectively. This means both these NN interactions reproduce correct g.s. without adding 3N forces but the ordering could not be confirmed on the account of uncertainty in the energy result obtained from JISP16 interaction.

In recent years, several experimental techniques have been used to measure nuclear charge radius for neutron-rich nuclei towards the drip line nature_Ca . These then serve as a test of the predictive power of ab-initio calculation. Charge radii inform us about the breakdown of the conventional shell gaps and the evolution of new shell gaps. One of the reasons behind the disappearance of the shell gap is the presence of the halo structure. Tanihata et al. RMSradii have measured interaction cross-sections ($\sigma_{I}$) for ^8,12-15B using radioactive nuclear beams at the Lawrence Berkeley Laboratory. In this experiment, the interaction nuclear radii and the effective root-mean-square (rms) radii of nucleon distribution have been deduced from $\sigma_{I}$. Point-proton radii of ^12-17B are also measured from the charge-changing cross-section ($\sigma_{cc}$) at GSI, Darmstadt N=14 . Further, the proton radii were extracted from a finite-range Glauber model analysis of the $\sigma_{cc}$. The measurement shows the existence of a thick neutron surface in ¹⁷B chargechanging . A recent experiment on nitrogen chain establishes the neutron skin and signature of the $N=14$ shell gap by measuring proton-radii of ^17-22N isotopes.

In the present work, we perform systematic NCSM calculations for ^10-14B isotopes using INOY INOY , N³LO Machleidt , CDB2K CDBonn and N²LO_opt N2LOopt NN interactions. For the first time, we report NCSM structure results with the INOY interaction for these isotopes. We have reached basis sizes up to $N_{\mbox{max}}$ = 10 for ¹⁰B, $N_{\mbox{max}}$ = 8 for ^11,12,13B and $N_{\mbox{max}}$ = 6 for ¹⁴B with m-scheme dimensions up to 1.7 billion. Apart from energy spectra, we have also calculated electromagnetic properties and point-proton radii. In addition, we compare shell model results of energy levels and nuclear observables obtained with the YSOX interaction monopole with present ab-initio results.

The paper is organized as follows: In section II, we describe the NCSM formalism. In section III, we briefly review the NN interactions used in our calculations. We present the NCSM results of the energy spectra and compare them to those obtained with the shell model YSOX interaction in section IV. In section V, electromagnetic properties of ^10-14B are reported. In section VI, we discuss point-proton radii of ^10-14B. Finally, we summarize the paper in section VII.

In NCSM 2009 ; nocore_review , all nucleons are treated as active, which means there is no assumption of an inert core, unlike in standard shell model. The nucleus is described as a system of $A$ non-relativistic nucleons which interact by realistic NN or NN + 3N interactions.

In the present work, we have considered only realistic NN interactions between the nucleons. The Hamiltonian for the $A$ nucleon system is then given by

where T_rel is the relative kinetic energy, $m$ is the mass of nucleon and V${}^{NN}_{ij}$ is the realistic NN interaction that contains both nuclear and electromagnetic (Coulomb) parts.

In the NCSM, translational invariance as well as angular momentum and parity of the nuclear system are conserved. The many-body wave function is cast into an expansion over a complete set of antisymmetric $A$-nucleon harmonic oscillator (HO) basis states containing up to $N_{\rm max}$ - HO excitations above the lowest possible configuration.

We use a truncated HO basis while the realistic NN interactions act in the full space. Unless the potential is soft like, e.g., the N²LO_opt, we need to derive an effective interaction to facilitate the convergence. Two renormalization methods based on similarity transformations have been applied in the NCSM, the Okubo-Lee-Suzuki (OLS) scheme Prog.Theor.Phys.12 ; Prog.Theor.Phys. ; Prog.Theor.Phys.68 ; Prog.Theor.Phys.92 and more recently the Similarity Renormalization Group (SRG) Bogner2007 . The latter has the advantage in being more systematic and in the fact that renormalized potentials are phaseshift equivalent. The three-body induced terms, however, cannot be neglected. Those, in turn, are difficult to converge for potentials that generate strong short-range correlations, such as the CDB2K Jurgenson2011 . The OLS method is applied directly in the HO basis and results in an $A$- and $N_{\rm max}$-dependent effective interaction, i.e, the calculation is not variational. The three-body induced terms are less important. It has been observed that the method works particularly well for the INOY interaction Forssen2005 ; Caurier2006 ; Forssen2009 ; Forssen2013 . Consequently, in this work we apply the OLS method for the INOY, CDB2K and, for a consitent comparison also for the N³LO NN interaction. For the latter, the SRG method is, however, more appropriate Jurgenson2011 ; Jurgenson2013 . The softer N²LO_opt NN interaction is not renormalized.

To facilitate the derivation of the OLS effective interaction, we add centre-of-mass (c.m.) HO Hamiltonian to equation (1) which makes the Hamiltonian dependent on the HO frequency.

where

The intrinsic properties of the system are not affected by the addition of HO c.m. Hamiltonian due to translational invariance of the Hamiltonian (1).

Thus, we obtain a modified Hamiltonian:

We divide the A nucleon large HO basis space into two parts: one is the finite active space ($P$) which contains all states up to $N_{\mbox{max}}$, and, the other is the excluded space ($Q=1-P$). NCSM calculations are performed in the truncated $P$ space. The two-body OLS effective is derived by applying the Hamiltonian (2) to two nucleons and performing the unitary transformation in the HO basis 2009 ; nocore_review . Eventually, the second term in the brackets in (2) is replaced by the effective interaction.

Finally, we subtract the c.m. Hamiltonian $H_{\mbox{c.m.}}$ and include the Lawson projection term Lawson to shift the spurious c.m. excitations.

An extension of the NCSM that provides a unified description of both bound and unbound states is the no-core shell model with continuum (NCSMC) approach NCSMC . It has been successfully applied, e.g., to explain the parity inversion phenomenon in ¹¹Be 11Be . It has not been applied to boron isotopes yet although NCSMC calculations for ^10,11B are now in progress.

In the present work, apart from the INOY interaction INOY ; nonlocal ; Doleschall , we also report results with the CDB2K offshell ; Phys.Rep. ; Adv.Nucl.Phys. ; CDBonn , N³LO QCD ; Machleidt and N²LO_opt optimizedn2lo ; N2LOopt interactions.

The Inside Non-Local Outside Yukawa (INOY) interaction INOY ; nonlocal ; Doleschall has a local character (Yukawa tail) at long distances ($r\geq 3$ fm) and a non-local one at short distances ($r\,{<}\,3$ fm), where the non-local part is due to the internal structure of the nucleon. As it is constructed in coordinate space, the range of locality and non-locality is explicitly controllable. This interaction has the form:

where, the cut-off function is defined as:

and $W_{ll^{{}^{\prime}}}(r,r^{{}^{\prime}})$ and $V_{ll^{{}^{\prime}}}^{Yukawa}(r)$ are the non-local part and the Yukawa tail (the same as in AV18 potential av18 ), respectively. The parameters $\alpha_{ll^{\prime}}$ and $R_{ll^{\prime}}$ have the values 1.0 fm^-1 and 2.0 fm, respectively. Because of the non-local character in the INOY interaction, three-body force effects are in part absorbed by nonlocal terms, e.g., it produces correct binding energy of the three-nucleon system (³H and ³He) without adding three-body forces explicitly.

The Charge-Dependent Bonn 2000 potential (CDB2K) is a meson exchange based potential offshell ; Phys.Rep. ; Adv.Nucl.Phys. ; CDBonn . It includes all the mesons with masses below the nucleon mass, i.e. $\pi^{\pm,0}$, $\eta$, $\rho^{\pm,0}$ and $\omega$ as an exchange particle between nucleons. The $\eta$ has a vanishing coupling constant and as such, can be ignored. This potential also includes two scalar-isoscalar $\sigma$ (or $\epsilon$) bosons. Charge dependence of nuclear forces, which is investigated by the Bonn full model based on charge independence breaking (difference between proton-proton/neutron-neutron and proton-neutron interaction; pion mass splitting) and charge symmetry breaking (difference between proton-proton and neutron-neutron interaction; nucleon mass splitting) in all partial waves with $J\leq 4$, is also reproduced. The potential is represented in terms of the one-boson-exchange (OBE) covariant Feynman amplitudes. The off-shell behavior of the potential, which plays an important role in nuclear structure calculations, is affected by imposing locality on the Feynman amplitudes. So, non-local Feynman amplitudes are used in the CDB2K potential. This momentum-space dependent potential fits proton-proton data with $\chi^{2}$ per datum of 1.01 and the neutron-proton data with $\chi^{2}$/datum $=$ 1.02 below 350 MeV, where $\chi^{2}$ is the square of theoretical error over the experimental error.

Chiral perturbation theory is a perturbative expansion in $Q/\Lambda_{\chi}$, where $Q\ll\Lambda_{\chi}\approx$ 1 GeV. Entem and Machleidt constructed the NN potential QCD ; Machleidt at fourth order (next-to-next-to-next-to-leading order; N³LO) of $\chi$PT in the momentum-space. In $\chi$PT, two class of contributions determine the NN amplitude: Contact terms and pion-exchane diagrams. The N³LO interaction contains 24 contact terms, whose parameters contribute to the fit of partial waves of NN scattering with angular momentum $L\leq 2$. Charge dependence is also included up to next-to-leading order of the isospin-violation scheme. The N³LO has two charge-dependent contacts. Thus, the total number of contact terms is 26. The N³LO has one pion-exchange (OPE) as well as two pion-exchange (TPE) contributions. Contributions of three pion exchange in the N³LO, however, are negligible. OPE and TPE depend on the axial-vector coupling constant $g_{A}$ (1.29), the pion decay constant $f_{\pi}$ (92.4 MeV) and eight low-energy constants (LEC). Three of them ($c_{2}$, $c_{3}$ and $c_{4}$) are varied in the fitting process and other are fixed. All constants are determined from the NN data. With a total of 29 parameters, the N³LO yields $\chi^{2}$/datum $\approx$ 1 up to 290 MeV for the fit of neutron-proton data. The accuracy in the reproduction of NN data for this order is comparable to the high-precision phenomenological AV18 potential av18 .

The N²LO_opt N2LOopt ; optimizedn2lo is a softer interaction and as such, the OLS or SRG renormalization is not needed. This interaction was derived from $\chi$EFT at the N²LO order. For the optimization of the LECs, Practical Optimization Using No Derivatives algorithm (POUNDERs) was used. In particular, the optimisation is performed for the pion-nucleon ($\pi$N) couplings ($c_{1}$, $c_{3}$, $c_{4}$) and 11 partial wave contact parameters $C$ and $\tilde{C}$. The N²LO_opt interaction reproduces reasonably well experimental binding energies and radii of $A$ = 3, 4 nuclei.

For comparison, we have also performed shell model calculations with the phenomenological YSOX interaction monopole developed by the Tokyo group. In the YSOX interaction, ⁴He is assumed as a core and interactions take place in the $psd$ valence space. Single-particle energies are $e_{p_{3/2}}$ = 1.05 MeV, $e_{p_{1/2}}$ = 5.30 MeV, $e_{d_{5/2}}$ = 8.01 MeV, $e_{s_{1/2}}$ = 2.11 MeV and $e_{d_{3/2}}$ = 10.11 MeV. There are 516 two-body matrix elements (TBMEs) in this interaction.

NCSM calculations presented in this paper have been performed with the pAntoine code pAntoine1 ; pAntoine11 ; pAntoine2 . We have used KSHELL code KSHELL2019 for the shell model calculation with the YSOX interaction monopole . Recently, we have reported NCSM results for N, O and F isotopes in Refs. arch1 ; arch2 performed in an analogous way.

The dimensions corresponding to different $N_{\mbox{max}}$ for boron isotopes are shown in Table 1. We can see that they increase rapidly with $N_{\mbox{max}}$ and the mass number. In the present work, we were able to perform NCSM calculations up to $N_{\mbox{max}}$ = 10 for ¹⁰B, $N_{\mbox{max}}$ = 8 for ^11,12,13B and $N_{\mbox{max}}$ = 6 for ¹⁴B. First, we investigate the dependence on the HO frequency ($\hbar\Omega$) for various $N_{\mbox{max}}$ bases, typically up to the next to the largest accesible for computational reasons. The optimal HO frequency used to calculate the entire energy spectrum is found from the g.s. energy minimum in the largest $N_{\mbox{max}}$ space. Fig. 1 shows variation of g.s. energy of ¹⁰B for different basis spaces as a function of HO frequencies for the four interactions that we employ. Overall, we observe a decrease of the g.s. energy dependece on the frequency at higher $N_{\mbox{max}}$ as expected. Let us re-iterate that the N²LO_opt calculations are variational while those with the OLS renormalized interactions are not.

We note that minima of the g.s. energy are at the same frequency for both $N_{\mbox{max}}$ = 6 and 8 for the INOY interaction. Thus, we expect to obtain the minimum at the same frequency also for $N_{\mbox{max}}$ = 10. Optimal frequency values for the INOY, CDB2K, N³LO and N²LO_opt interactions are at $\hbar\Omega$ = 20 MeV, 14 MeV, 12 MeV and 22 MeV, respectively. Only for those values we performed the $N_{\mbox{max}}$ = 10 calculations. We have determined the optimal frequencies for other boron isotopes as shown in Fig. 2 corresponding to INOY and N²LO_opt interactions. Similarly, we have obtained optimal frequencies for CDB2K and N³LO interactions.

The NCSM results of low-lying states for boron isotopes corresponding to the INOY interaction in the basis spaces 0$\hbar\Omega$ to highest $N_{\mbox{max}}$, and for the other interactions in the highest $N_{\mbox{max}}$ are shown in Figs. 3-4. From the figures, we can see how the energy states approach the experimental values. Along with the NCSM results, we have also reported shell model results corresponding to YSOX interaction. All results are compared with experimental data. We have calculated only natural parity states for each nucleus.

Table 2 contains quadrupole moments ($Q$), magnetic moments ($\mu$), g.s. energies (E_g.s.), reduced electric quadrupole transition probabilities ($B(E2)$) and reduced magnetic dipole transition probabilities ($B(M1)$). Only one-body electromagnetic operators were considered. The experimental binding energy of ¹⁰B is -64.751 MeV. The INOY interaction underbinds the ¹⁰B nucleus by 1.32 MeV while YSOX interaction overbinds this by 0.39 MeV. The other used realistic interactions underestimate the experimental binding energy more significantly. The g.s. $Q$ and $\mu$ moments of ^10,11B are in a reasonable agreement with experiment for all interactions. On the other hand, the calculated $B(E2;3_{1}^{+}\rightarrow 1_{1}^{+})$ value for ¹⁰B varies substantially. Similarly, we find interaction dependence and stronger disagreements with experiment for the ^12,13,14B g.s. moments. We predict several $B(E2)$ and $B(M1)$ values for ^12-14B which are not yet measured experimentally.

In Fig. 5, we show $B(M1;2_{1}^{+}\rightarrow 3_{1}^{+})$ and $B(E2;3_{1}^{+}\rightarrow 1_{1}^{+})$ transition strengths corresponding to different $N_{\mbox{max}}$ and $\hbar\Omega$ for ¹⁰B with the INOY, CDB2K and N³LO interactions. $B(M1;2_{1}^{+}\rightarrow 3_{1}^{+})$ curves become flat, which means they become independent of $N_{\mbox{max}}$ and $\hbar\Omega$. So, the convergence of the $B(M1)$ result is obtained at smaller $\hbar\Omega$ and lower $N_{\mbox{max}}$. As discussed, e.g., in Refs. stetcu1 ; stetcu2 , it is a big task to compute the $E2$ transition operator, as it depends on the long-range correlations in the nucleus i.e. the tails of nuclear wave functions. From Fig. 5, we can see that $B(E2)$ value varies even for large value of the $N_{\mbox{max}}$ parameter. The best $B(E2)$ value is then taken where these curves become flat, although clearly we have not reached convergence within the model spaces used in this work.

The quadrupole and magnetic moments of the studied isotopes are summarized in Fig. 6. Overall, the experimental trends are well reproduced for both observables although the NCSM calculations systematically under predict the experimental quadrupole moments.

In Fig. 7, the dependence of the calculated g.s. energies on the mass number of boron isotopes is plotted with INOY, CDB2K, N³LO, N²LO_opt, YSOX interactions and compared with experimental energies. NCSM results obtained at the largest accessible $N_{\rm max}$ space with the optimal frequency are shown. From Fig. 7, we can conclude that INOY interaction provides better description for g.s. energy than other used ab initio interactions.

For the N²LO_opt interaction, we have extrapolated the g.s. energy using an exponential fitting function $E_{g.s.}(N_{max})=a~{}exp(-bN_{max})+E_{g.s.}(\infty)$ with $E_{g.s.}(\infty)$ the value of g.s. energy at $N_{max}$ $\rightarrow$ $\infty$. In particular, we have used last three $N_{max}$ points in the extrapolation procedure. For ¹⁴B, no meaningful extrapolation was possible.

In Table 3, we have presented point-proton radii (r_p) using NCSM with INOY, CDB2K and N³LO interactions at their optimal frequencies along with experimentally observed radii chargechanging . The INOY interaction considerably underestimates the radii. For ^10,11B, the CDB2K and N³LO interactions produce better results, with the former slightly underestimating and the latter slightly overestimating the radii. For ^12-14B, the radii are underestimated for all interactions.

In Fig. 8, we present the variation of ¹⁰B $r_{p}$ with frequency and $N_{\mbox{max}}$ for INOY, CDB2K and N³LO interaction. With the enlargement of basis size $N_{\mbox{max}}$, the dependence of $r_{p}$ on frequencies decreases. The curves of $r_{p}$ corresponding to different $N_{\mbox{max}}$ intersect each-other approximately at the same point. We take this crossing point as an estimate of the converged radius PLB ; Phys.Rev.C90034305(2014) . In particular, we consider the intersection point of the curves at the highest successive $N_{\mbox{max}}$ as an estimate of the converged radius. In this way, we obtain ¹⁰B point-proton radii for INOY, CDB2K and N³LO interactions 2.14, 2.30 and 2.36 fm, respectively.

Similarly, we have shown variation of $r_{p}$ with frequency and $N_{\mbox{max}}$ for other isotopes corresponding to INOY interaction in Fig. 9. Obtained $r_{p}$ values for ¹¹B,¹²B,¹³B and ¹⁴B are 2.00, 1.99, 1.95 and 1.99 fm, respectively. However, even with this determination of the radii, the experimental trend is not reproduced.

We can conclude that the CDB2K and N³LO interactions give radii which are much closer to experimental value than the radii obtained with the INOY interaction. To some extent this is not surprising given the fact that those interactions underbind the studied isotopes. We have obtained different optimal frequencies for the energy spectra and the point-proton radii. Similar findings were reported for ¹²C using Daejeon16 and JISP16 interactions in Ref. PLB .

In this work, we have applied ab-initio no-core shell model to obtain spectroscopic properties of boron isotopes using INOY, CDB2K, N³LO and N²LO_opt nucleon-nucleon interactions. We have calculated low-lying spectra and other observables with all four interactions and, in addition, compared the NCSM results with shell model using YSOX valence-space effective interaction. We were able to correctly reproduce the g.s. spin of ¹⁰B only with the INOY NN interaction. Overall, the INOY interaction reproduced quite reasonably g.s. energies of all the studied isotopes, ^10-14B.

Considering electromagnetic properties, we have obtained fast convergence for $M1$ values, whereas, converging $E2$ observables is a computational challenge. The INOY interaction again appears to do better than the other interactions in the reproduction of the $M1$ observables for all isotopes.

Concerning proton radii, we find that optimal frequency obtained from the minima of the g.s. energy curves and that obtained from the intersection of radii curves could be different. In this case, the CDB2K and N³LO interactions give radii which are much closer to experimental value than the radii obtained with the INOY interaction.

The present study confirms that non-locality in the NN interaction can account for some of the many-nucleon force effects. The non-local NN interaction like INOY can provide a quite reasonable description of ground-state energies, excitation spectra and selected electromagnetic properties, e.g., magnetic moments and $M1$ transitions. However, the description of nuclear radii and consequently of the density remains unsatisfactory. Recent studies show that the inclusion of the 3N interaction, in particular 3N interaction with non-local regulators, is essential for a correct simultaneous description of nuclear binding and nuclear size Nmax10 ; N2LOsat ; NN3Nlnl .

We would like to thank Christian Forssén for making available the pAntoine code. We thank Toshio Suzuki for the YSOX interaction. P.C. acknowledges financial support from MHRD (Government of India) for her PhD thesis work. P.C.S. acknowledges a research grant from SERB (India), CRG/2019/000556. P.N. acknowledges support from the NSERC Grant No. SAPIN-2016-00033. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada.