Ab-initio no-core shell model study of 10−14 B isotopes with realistic NN interactions
aa r X i v : . [ nu c l - t h ] S e p Ab-initio no-core shell model study of − B isotopes with realistic NN interactions Priyanka Choudhary ∗ , Praveen C. Srivastava † and Petr Navr´atil ‡ Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada (Dated: September 23, 2020)We report a comprehensive study of − B isotopes within the ab-initio no-core shell model(NCSM) using realistic nucleon-nucleon ( NN ) interactions. In particular, we have applied the insidenon-local outside Yukawa (INOY) interaction to study energy spectra, electromagnetic propertiesand point-proton radii of the boron isotopes. The NCSM results with the charge-dependent Bonn2000 (CDB2K), the chiral next-to-next-to-next-to-leading order (N LO) and optimized next-to-next-to-leading order (N LO opt ) interactions are also reported. We have reached basis sizes up to N max = 10 for B, N max = 8 for , , B and N max = 6 for B with m-scheme dimensions up to1.7 billion. We also compare the NCSM calculations with the phenomenological YSOX interactionusing the shell model to test the predictive power of the ab-initio nuclear theory. Overall, ourNCSM results are consistent with the available experimental data. The experimental ground statespin 3 + of B has been reproduced using the INOY NN interaction. Typically, the 3 N interactionis required to correctly reproduce the aforementioned state. PACS numbers: 21.60.Cs, 21.30.Fe, 21.10.Dr, 27.20.+n
I. INTRODUCTION
In nuclear physics, our focus is to describe the nuclearstructure including the exotic behaviour of atomic nucleithroughout the nuclear chart. Conventional shell model[1–6], where interactions are assumed to exist only amongthe valence nucleons in a particular model space is un-able to determine the drip line [7, 8], cluster [9] and halo[10] structures. The study of interactions derived fromfirst principles has been a challenging area of researchover the past decades. These fundamental interactionsare determined from either meson-exchange theory orQuantum chromodynamics (QCD) [11]. QCD is non-perturbative in low-energy regime which makes analyticsolutions difficult. This difficulty is overcome by chiral ef-fective field theory ( χ EFT) [12–15]. Chiral perturbationtheory ( χ PT) [16] within χ EFT provides a connectionbetween QCD and the hadronic system.Progress has been made in the development of differ-ent many-body modern ab-initio approaches [17–19], oneof them being the NCSM [20–31].
Ab-initio methods aremore fundamental compared to the nuclear shell model.The aim of this paper is to explain the nuclear struc-ture of boron isotopes with realistic NN interactions asthe only input. The well-bound stable B have posed achallenge to the microscopic nuclear theory in particularconcerning the reproduction of its ground-state spin [32].The boron isotopes have been investigated in the pastusing the shell model [33, 34]. Shell model Hamiltonianconstructed from a monopole-based universal interaction(V MU ) in full psd model space including (0 − ~ Ω ex- ∗ [email protected] † Corresponding author: [email protected] ‡ [email protected] citations has been used for a systematic study of boronisotopes [33]. This phenomenological effective interactionis obtained by fitting experimental data, thus, it at leastpartly includes three-body effects. So it is able to repro-duce spin of the ground state (g.s.) of B. This V MU based Hamiltonian, however, fails to describe the dripline nucleus B. Tensor-optimized shell model (TOSM)[34] has been applied to study B using effective barenucleon-nucleon ( NN ) interaction Argonne V8 ′ (AV8 ′ )[35]. The g.s. obtained with AV8 ′ interaction is 1 + ,which, in experiment, is the first excited state of B.AV8 ′ eff interaction, which is a modification of tensorand spin-orbit forces of AV8 ′ interaction, gives correctg.s. spin and low-lying spectra, indicating that the tensorforces affect the level ordering. TOSM with Minnesota(MN) effective interaction [36] without tensor force alsogives correct g.s. spin but a smaller g.s. radius comparedto the experimental result, which affects the nuclear sat-uration property, thus providing the small level density.In Refs. [37–39], the structure of B was studiedwithin the NCSM, using accurate charge dependent NN potentials up to the 4 th order of χ PT in basis spaces( N max) of up to 10 ~ Ω. Using the NN interactions aloneled to an incorrect g.s. of B. By including the chiralthree-nucleon interaction (3 N ), the g.s. was correctly re-produced as 3 + [37, 39]. The ab-initio NCSM study of B with the chiral N LO (next-to-next-to-leading order) NN interaction [40] including three-body forces has beendone in Ref. [41], where it was shown that the g.s. energyand spin depends on the chiral order. To correctly repro-duce the 3 + as an experimental g.s., 3 N force with theN LO NN interaction is needed. In Ref. [42], N LO opt interaction was employed in the NCSM calculation for B up to N max = 10 (10 ~ Ω) to calculate ground andlow-lying excited states. This study reported 1 + as a g.s.instead of 3 + . Realistic shell model calculations includ-ing contributions of a chiral three-body force [N LO NN + N LO 3 N potential] for B is reported in Ref. [43].These results are consistent with the NCSM results withthe same interaction. The NCSM with CDB2K potential( N max = 8) and AV8’ ( N max = 6) predict 1 + as g.s.of B [32, 44]. Green’s function Monte Carlo (GFMC)approach with AV8’ and AV18 has also been employedto investigate the g.s. of B [45] and similarly predictsthe 1 + as the ground state with these NN forces.In Ref. [46], the Daejeon16 and JISP16 (J-matrix in-verse scattering potential) NN interactions were appliedto p -shell nuclei. For B, excitation energies of 1 + statewith respect to 3 + state of 0.5(1) MeV and 0.9(2.4) MeVwere reported with Daejeon16 and JISP16 NN interac-tions, respectively. This means both these NN interac-tions reproduce correct g.s. without adding 3 N forcesbut the ordering could not be confirmed on the accountof uncertainty in the energy result obtained from JISP16interaction.In recent years, several experimental techniques havebeen used to measure nuclear charge radius for neutron-rich nuclei towards the drip line [47]. These then serveas a test of the predictive power of ab-initio calculation.Charge radii inform us about the breakdown of the con-ventional shell gaps and the evolution of new shell gaps.One of the reasons behind the disappearance of the shellgap is the presence of the halo structure. Tanihata etal. [48] have measured interaction cross-sections ( σ I ) for , − B using radioactive nuclear beams at the LawrenceBerkeley Laboratory. In this experiment, the interactionnuclear radii and the effective root-mean-square (rms)radii of nucleon distribution have been deduced from σ I .Point-proton radii of − B are also measured from thecharge-changing cross-section ( σ cc ) at GSI, Darmstadt[49]. Further, the proton radii were extracted from afinite-range Glauber model analysis of the σ cc . The mea-surement shows the existence of a thick neutron surfacein B [50]. A recent experiment on nitrogen chain es-tablishes the neutron skin and signature of the N = 14shell gap by measuring proton-radii of − N isotopes.In the present work, we perform systematic NCSM cal-culations for − B isotopes using INOY [51], N LO [52],CDB2K [53] and N LO opt [42] NN interactions. For thefirst time, we report NCSM structure results with theINOY interaction for these isotopes. We have reached ba-sis sizes up to N max = 10 for B, N max = 8 for , , Band N max = 6 for B with m-scheme dimensions upto 1.7 billion. Apart from energy spectra, we have alsocalculated electromagnetic properties and point-protonradii. In addition, we compare shell model results ofenergy levels and nuclear observables obtained with theYSOX interaction [33] with present ab-initio results.The paper is organized as follows: In section II, wedescribe the NCSM formalism. In section III, we brieflyreview the NN interactions used in our calculations. Wepresent the NCSM results of the energy spectra and com-pare them to those obtained with the shell model YSOXinteraction in section IV. In section V, electromagneticproperties of − B are reported. In section VI, we dis- cuss point-proton radii of − B. Finally, we summarizethe paper in section VII.
II. NO-CORE SHELL MODEL FORMALISM
In NCSM [27, 29], all nucleons are treated as active,which means there is no assumption of an inert core,unlike in standard shell model. The nucleus is describedas a system of A non-relativistic nucleons which interactby realistic NN or NN + 3 N interactions.In the present work, we have considered only realistic NN interactions between the nucleons. The Hamiltonianfor the A nucleon system is then given by H A = T rel + V = 1 A A X i III. REALISTIC NN AND SHELL MODELINTERACTIONS In the present work, apart from the INOY interaction[51, 68, 69], we also report results with the CDB2K [53,70–72], N LO [11, 52] and N LO opt [42, 73] interactions. The Inside Non-Local Outside Yukawa (INOY) inter-action [51, 68, 69] has a local character (Yukawa tail) atlong distances ( r ≥ r < V fullll ′ ( r, r ′ ) = W ll ′ ( r, r ′ ) + δ ( r − r ′ ) F cutll ′ ( r ) V Y ukawall ′ ( r ) , (4)where, the cut-off function is defined as: F cutll ′ ( r ) = ( − e − [ α ll ′ ( r − R ll ′ )] f or r ≥ R ll ′ , f or r ≤ R ll ′ , and W ll ′ ( r, r ′ ) and V Y ukawall ′ ( r ) are the non-local partand the Yukawa tail (the same as in AV18 potential [74]),respectively. The parameters α ll ′ and R ll ′ have the val-ues 1.0 fm − and 2.0 fm, respectively. Because of thenon-local character in the INOY interaction, three-bodyforce effects are in part absorbed by nonlocal terms, e.g.,it produces correct binding energy of the three-nucleonsystem ( H and He) without adding three-body forcesexplicitly.The Charge-Dependent Bonn 2000 potential (CDB2K)is a meson exchange based potential [53, 70–72]. It in-cludes all the mesons with masses below the nucleonmass, i.e. π ± , , η , ρ ± , and ω as an exchange particlebetween nucleons. The η has a vanishing coupling con-stant and as such, can be ignored. This potential alsoincludes two scalar-isoscalar σ (or ǫ ) bosons. Charge de-pendence of nuclear forces, which is investigated by theBonn full model based on charge independence break-ing (difference between proton-proton/neutron-neutronand proton-neutron interaction; pion mass splitting) andcharge symmetry breaking (difference between proton-proton and neutron-neutron interaction; nucleon masssplitting) in all partial waves with J ≤ 4, is also repro-duced. 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 im-portant role in nuclear structure calculations, is affectedby imposing locality on the Feynman amplitudes. So,non-local Feynman amplitudes are used in the CDB2Kpotential. This momentum-space dependent potentialfits proton-proton data with χ per datum of 1.01 andthe neutron-proton data with χ /datum = 1.02 below350 MeV, where χ is the square of theoretical error overthe experimental error.Chiral perturbation theory is a perturbative expansionin Q/ Λ χ , where Q ≪ Λ χ ≈ NN potential [11, 52] at fourth order(next-to-next-to-next-to-leading order; N LO) of χ PT inthe momentum-space. In χ PT, two class of contribu-tions determine the NN amplitude: Contact terms andpion-exchane diagrams. The N LO interaction contains24 contact terms, whose parameters contribute to thefit of partial waves of NN scattering with angular mo-mentum L ≤ 2. Charge dependence is also included upto next-to-leading order of the isospin-violation scheme.The N LO has two charge-dependent contacts. Thus,the total number of contact terms is 26. The N LO hasone pion-exchange (OPE) as well as two pion-exchange(TPE) contributions. Contributions of three pion ex-change in the N LO, however, are negligible. OPE andTPE depend on the axial-vector coupling constant g A (1.29), the pion decay constant f π (92.4 MeV) and eightlow-energy constants (LEC). Three of them ( c , c and c ) are varied in the fitting process and other are fixed.All constants are determined from the NN data. With atotal of 29 parameters, the N LO yields χ /datum ≈ NN data for this order iscomparable to the high-precision phenomenological AV18potential [74].The N LO opt [42, 73] is a softer interaction and as such,the OLS or SRG renormalization is not needed. Thisinteraction was derived from χ EFT at the N LO order.For the optimization of the LECs, Practical OptimizationUsing No Derivatives algorithm (POUNDERs) was used.In particular, the optimisation is performed for the pion-nucleon ( π N) couplings ( c , c , c ) and 11 partial wavecontact parameters C and ˜ C . The N LO opt interactionreproduces reasonably well experimental binding energiesand radii of A = 3, 4 nuclei.For comparison, we have also performed shell modelcalculations with the phenomenological YSOX interac-tion [33] developed by the Tokyo group. In the YSOXinteraction, He is assumed as a core and interactionstake place in the psd valence space. Single-particle ener-gies are e p / = 1.05 MeV, e p / = 5.30 MeV, e d / = 8.01MeV, e s / = 2.11 MeV and e d / = 10.11 MeV. Thereare 516 two-body matrix elements (TBMEs) in this in-teraction.NCSM calculations presented in this paper have beenperformed with the pAntoine code [75–77]. We have usedKSHELL code [78] for the shell model calculation withthe YSOX interaction [33]. Recently, we have reportedNCSM results for N, O and F isotopes in Refs. [79, 80]performed in an analogous way. IV. RESULTS AND DISCUSSIONS The dimensions corresponding to different N max forboron isotopes are shown in Table I. We can see that theyincrease rapidly with N max and the mass number. Inthe present work, we were able to perform NCSM calcu-lations up to N max = 10 for B, N max = 8 for , , Band N max = 6 for B. First, we investigate the depen-dence on the HO frequency ( ~ Ω) for various N max bases,typically up to the next to the largest accesible for com-putational reasons. The optimal HO frequency used tocalculate the entire energy spectrum is found from theg.s. energy minimum in the largest N max space. Fig. 1 shows variation of g.s. energy of B for different ba-sis spaces as a function of HO frequencies for the fourinteractions that we employ. Overall, we observe a de-crease of the g.s. energy dependece on the frequencyat higher N max as expected. Let us re-iterate that theN LO opt calculations are variational while those with theOLS renormalized interactions are not. TABLE I: Dimensions in m-scheme for boron isotopes corre-sponding to different N max . The dimensions up to which wehave reached are shown in blue. N max B B B B B0 84 62 28 5 482 1 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × 10 1 . × . × . × . × . × We note that minima of the g.s. energy are at the samefrequency for both N max = 6 and 8 for the INOY inter-action. Thus, we expect to obtain the minimum at thesame frequency also for N max = 10. Optimal frequencyvalues for the INOY, CDB2K, N LO and N LO opt inter-actions are at ~ Ω = 20 MeV, 14 MeV, 12 MeV and 22MeV, respectively. Only for those values we performedthe N max = 10 calculations. We have determined theoptimal frequencies for other boron isotopes as shownin Fig. 2 corresponding to INOY and N LO opt interac-tions. Similarly, we have obtained optimal frequenciesfor CDB2K and N LO interactions.The NCSM results of low-lying states for boron iso-topes corresponding to the INOY interaction in the basisspaces 0 ~ Ω to highest N max, and for the other interac-tions in the highest N max are shown in Figs. 3-4. Fromthe figures, we can see how the energy states approachthe experimental values. Along with the NCSM results,we have also reported shell model results correspondingto YSOX interaction. All results are compared with ex-perimental data. We have calculated only natural paritystates for each nucleus. A. Energy spectra for , , B Experimentally, the g.s. of B is 3 + and the first ex-cited state 1 + lies 0.718 MeV above the g.s. For theINOY interaction, we obtain the correct g.s. 3 + as seenin the energy spectrum shown in the top panel of Fig. 3.The difference between 3 + and 1 + states decreases as N max increases and for N max = 10, the difference is1.250 MeV. Previously, the NCSM results using CDB2Kinteraction have been reported for N max = 8 [44]. Inthe present paper, we have extended the basis size from N max = 8 to 10 to further improve convergence. Overall,the present results are consistent with those of Ref. [44].The CDB2K interaction is unable to reproduce the cor-rect g.s. 3 + . For comparison, we have also studied NCSM − − − − − − B [INOY] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT − − − − − B [CDB2K] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT − − − B [N LO] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT − − − B [N LO opt ] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT FIG. 1: Ground state energy of B as a function of HO frequency for N max = 2 to 10 with the INOY, CDB2K, N LO andN LO opt interactions. Experimental g.s. energy is shown by the horizontal line. results with N LO and N LO opt interactions for N max= 10. These interactions predict 1 + as the g.s. con-trary to the experimental result, albeit the difference be-tween 3 + and 1 + states is very small (0.035 MeV) for theN LO opt interaction. We note that the calculated 3 +1 cor-responding to CDB2K and N LO interactions is respec-tively, 1.069 MeV and 1.594 MeV above the 1 +1 state. Wecan also see that the INOY interaction predicts the cor-rect ordering of 3 + -1 + -0 + -1 + -2 + states contrary to thephenomenological YSOX interaction.As seen in the second panel of Fig. 3, the INOY interac-tion fails to predict correct g.s. 1 + for B, while CDB2K,N LO and N LO opt interactions are able to predict theg.s. correctly. At the same time, it is clear that thedifference between 1 + and 2 + states decreases with in-creasing N max for INOY interaction. So, we expect thatfor larger N max, the g.s. would be 1 + also for the INOYinteraction. Using CDB2K and N LO interactions, theNCSM results are too compressed compared to experi-mental results. In particular, the 0 + state is too low. The N LO opt interaction gives the correct order of theenergy levels up to 3 +1 with lower energy values than theexperimentally obtained energies.For B, we have reached only N max = 6 space, dueto huge dimension of Hamiltonian matrix involved inthe calculation. All interactions provide the correct g.s.as 2 − . Experimentally, 1 − and 3 − states are tentative,which are confirmed with the CDB2K and N LO interac-tions. These states are also confirmed with YSOX inter-action. For the INOY interaction, the order of states 1 − ,3 − and 2 − , 4 − is reversed in comparison to the (tenta-tive) experimental data. The energy difference between2 − and 1 − states is larger for all ab-initio interactionscompared to that obtained in experiment. B. Energy spectra for , B For B, we employed HO frequencies of 20 MeV, 16MeV and 24 MeV for the INOY, CDB2K and N LO opt interaction, respectively. For N LO interaction, optimal − − − − B [INOY] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT 16 18 20 22 24 26 − − − − B [N LO opt ] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT − − − B [INOY] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT 16 18 20 22 24 26 − − − − B [N LO opt ] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT 14 16 18 20 22 − − − − B [INOY] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT 16 18 20 22 24 26 − − − − B [N LO opt ] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6 N max = 8EXPT − − − − B [INOY] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6EXPT 16 18 20 22 24 26 − − − − B [N LO opt ] ¯ h Ω (MeV) G r o und s t a t ee n e r g y ( M e V ) N max = 2 N max = 4 N max = 6EXPT FIG. 2: Ground state energy of , , , B as a function of HO frequency for different N max with the INOY and N LO opt interactions. h Ω 2¯ h Ω 4¯ h Ω 6¯ h Ω 8¯ h Ω 10¯ h Ω EXPT 10¯ h Ω 10¯ h Ω 10¯ h Ω YSOX0 . . . . . . . . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + B INOY [¯ h Ω = 20 MeV] N LO [¯ h Ω = 12 MeV] CDB2K [¯ h Ω = 14 MeV] N LO opt [¯ h Ω = 22 MeV] E x c i t a t i o n e n e r g y ( M e V ) h Ω 2¯ h Ω 4¯ h Ω 6¯ h Ω 8¯ h Ω EXPT 8¯ h Ω 8¯ h Ω 8¯ h Ω YSOX0 . . . . . . . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + B INOY [¯ h Ω = 20 MeV] N LO [¯ h Ω = 14 MeV] CDB2K [¯ h Ω = 16 MeV] N LO opt [¯ h Ω = 22 MeV] E x c i t a t i o n e n e r g y ( M e V ) h Ω 4¯ h Ω 6¯ h Ω EXPT 6¯ h Ω 6¯ h Ω 6¯ h Ω YSOX012345 − − − − − − − − − − − − − − − − − − − − − − − − − − (1 − )(3 − )2 − (4 − ) 2 − − − − − − − − − − B INOY [¯ h Ω = 18 MeV] CDB2K [¯ h Ω = 14 MeV] N LO [¯ h Ω = 14 MeV] N LO opt [¯ h Ω = 20 MeV] E x c i t a t i o n e n e r g y ( M e V ) FIG. 3: Comparison of theoretical and experimental energy spectra of , , B isotopes. The NCSM results are reportedwith the INOY, CDB2K, N LO and N2LO opt interactions at their optimal HO frequencies. Shell model results with the YSOXinteraction is also shown. h Ω 2¯ h Ω 4¯ h Ω 6¯ h Ω 8¯ h Ω EXPT 8¯ h Ω 8¯ h Ω 8¯ h Ω YSOX0 . . . . . . . . . . . / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − (3 / − )5 / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − B INOY [¯ h Ω = 20 MeV] [¯ h Ω = 15 MeV] N LO [¯ h Ω = 16 MeV] CDB2K [¯ h Ω = 24 MeV] N LO opt E x c i t a t i o n e n e r g y ( M e V ) h Ω 2¯ h Ω 4¯ h Ω 6¯ h Ω 8¯ h Ω EXPT 6¯ h Ω 6¯ h Ω 6¯ h Ω YSOX02468101214 / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − B INOY [¯ h Ω = 18 MeV] CDB2K [¯ h Ω = 16 MeV] N LO [¯ h Ω = 14 MeV] N LO opt [¯ h Ω = 22 MeV] E x c i t a t i o n e n e r g y ( M e V ) FIG. 4: Comparison of theoretical and experimental energy spectra of , B isotopes. The NCSM results are reported withthe INOY, CDB2K, N LO and N2LO opt interactions at their optimal HO frequencies. Shell model results with the YSOXinteraction is also shown. frequency is taken to be 15 MeV from Ref. [37]. The 3 / − state is the experimental g.s. of B. Our NCSM calcu-lations reproduce the correct g.s. with all four interac-tions. We get correct excited states up to ∼ LO. The experimental g.s.energy of the 3 / − state is -76.205 MeV. With the INOYinteraction, we obtain the energy of -74.9 MeV for thisstate, fairly close to the experimental value. For N LOinteraction, 3 / − and 1 / − states are almost degenerate,while the INOY gives a splitting close to experimental.This splitting depends on the strength of the spin-orbitinteraction, which is apparently the largest for the INOYinteraction. We note that the energy gap between the states 7 / − and 5 / − obtained using the INOY interac-tion is very large compared to the experimental value.This could be because the optimal HO frequency is cho-sen with respect to the g.s. which is then used to predictthe whole energy spectrum. It is possible that a fasterconvergence of the excited states could be achieved witha different optimal frequency.Our NCSM calculations have been performed up to N max = 8 for B, for which we obtain correct g.s. withall interactions. The energy difference between theoreti-cal and experimental excited states is rather large, whichmakes it difficult to use the present calculations for as-signing experimentally unknown spin and parity to the − . − . − . . . . . . . B [INOY] ¯ h Ω (MeV) B ( M ; + → + )( µ N ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT B [INOY] ¯ h Ω (MeV) B ( E ; + → + )( e f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT − . − . − . . . . . . . B [CDB2K] ¯ h Ω (MeV) B ( M ; + → + )( µ N ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT B [CDB2K] ¯ h Ω (MeV) B ( E ; + → + )( e f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT − . − . − . . . . . . . B [N LO] ¯ h Ω (MeV) B ( M ; + → + )( µ N ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT B [N LO] ¯ h Ω (MeV) B ( E ; + → + )( e f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT FIG. 5: Variation of B(M1:2 +1 → +1 ) and B(E2:3 +1 → +1 ) for B with HO frequency for N max = 2 to 10, corresponding tothe INOY, N LO and CDB2K interactions. Experimental values are shown by horizontal line with uncertainty. TABLE II: Electromagnetic observables of − B corresponding to the largest N max at their optimal HO frequencies.Quadrupole moments, magnetic moments, g.s. energies, E M µ N ),MeV, e fm and µ N respectively. Experimental values are taken from Refs. [81, 82]. YSOX results are also shown forcomparison. B EXPT INOY CDB2K N LO N LO opt YSOXQ(3 + ) 0.0845(2) 0.061 0.071 0.077 0.067 0.073 µ (3 + ) 1.8004636(8) 1.836 1.852 1.856 1.838 1.806E g.s. (3 + ) -64.751 -63.433 -54.979 -53.225 -54.181 -65.144 B ( E 2; 3 +1 → +1 ) 1.777(9) 0.911 2.091 2.686 1.482 0.757 B ( M 1; 2 +1 → +1 ) 0.00047(27) 0.0007 0.002 0.003 0.0001 0.004 B EXPT INOY CDB2K N LO N LO opt YSOXQ(3 / − ) 0.04059(10) 0.027 0.030 0.031 0.029 0.043 µ (3 / − ) 2.688378(1) 2.371 2.537 2.622 2.366 2.501E g.s. (3 / − ) -76.205 -74.926 -66.034 -62.915 -59.993 -76.686 B ( E 2; 7 / − → / − ) 1.83(44) 0.814 1.258 1.478 1.032 3.118 B ( M 1; 3 / − → / − ) 0.519(18) 0.708 0.976 1.051 0.766 0.835 B EXPT INOY CDB2K N LO N LO opt YSOXQ(1 + ) 0.0132(3) 0.009 0.009 0.010 0.010 0.014 µ (1 + ) 1.003(1) 0.561 0.134 0.022 0.282 0.737E g.s. (1 + ) -79.575 -78.304 -69.350 -68.062 -61.226 -79.264 B ( M 1; 1 +1 → +1 ) NA 0.047 0.078 0.086 0.066 0.026 B ( M 1; 2 +1 → +1 ) 0.251(36) 0.125 0.197 0.339 0.170 0.204 B EXPT INOY CDB2K N LO N LO opt YSOXQ(3 / − ) 0.0365(8) 0.025 0.029 0.031 0.028 0.042 µ (3 / − ) 3.1778(5) 2.844 2.815 2.830 2.781 2.959E g.s. (3 / − ) -84.454 -85.205 -75.856 -74.716 -65.624 -84.185 B ( E 2; 5 / − → / − ) NA 1.800 2.281 2.721 1.990 0.787 B ( M 1; 3 / − → / − ) NA 0.984 1.035 1.065 0.982 0.729 B EXPT INOY CDB2K N LO N LO opt YSOXQ(2 − ) 0.0297(8) 0.016 0.025 0.025 0.004 0.026 µ (2 − ) 1.185(5) 0.778 0.926 0.914 0.550 0.614E g.s. (2 − ) -85.422 -82.002 -76.929 -77.549 -51.413 -84.454 B ( M 1; 2 − → − ) NA 2.579 2.457 2.436 2.755 2.656 excited states. V. ELECTROMAGNETIC PROPERTIES Table II contains quadrupole moments ( Q ), mag-netic moments ( µ ), g.s. energies (E g.s. ), reduced elec-tric quadrupole transition probabilities ( B ( E B ( M B is -64.751MeV. The INOY interaction underbinds the B nucleusby 1.32 MeV while YSOX interaction overbinds this by0.39 MeV. The other used realistic interactions underesti-mate the experimental binding energy more significantly.The g.s. Q and µ moments of , B are in a reason-able agreement with experiment for all interactions. Onthe other hand, the calculated B ( E 2; 3 +1 → +1 ) value for B varies substantially. Similarly, we find interactiondependence and stronger disagreements with experimentfor the , , B g.s. moments. We predict several B ( E 2) and B ( M 1) values for − B which are not yet measuredexperimentally. In Fig. 5, we show B ( M 1; 2 +1 → +1 ) and B ( E 2; 3 +1 → +1 ) transition strengths corresponding todifferent N max and ~ Ω for B with the INOY, CDB2Kand N LO interactions. B ( M 1; 2 +1 → +1 ) curves becomeflat, which means they become independent of N max and ~ Ω. So, the convergence of the B ( M 1) result is obtainedat smaller ~ Ω and lower N max. As discussed, e.g., inRefs. [30, 31], it is a big task to compute the E B ( E 2) value varieseven for large value of the N max parameter. The best B ( E 2) value is then taken where these curves becomeflat, although clearly we have not reached convergencewithin the model spaces used in this work.The quadrupole and magnetic moments of the studiedisotopes are summarized in Fig. 6. Overall, the experi-mental trends are well reproduced for both observablesalthough the NCSM calculations systematically underpredict the experimental quadrupole moments.1 10 11 12 13 140 . . . . . 08 A Q g . s . ( b a r n ) INOYCDB2KN3LON2LO opt YSOXEXPT 10 11 12 13 140123 A µ g . s . ( µ N ) INOYCDB2KN3LON2LO opt YSOXEXPT FIG. 6: Ground state quadrupole and magnetic moment dependencies on the mass number of the studied boron isotopes.NCSM results obtained at the largest accessible N max space with the optimal frequency are shown. Experimental values aretaken from Ref. [82]. 10 11 12 13 14 − − − 60 A E g . s . ( M e V ) INOYCDB2KN3LON2LO opt YSOXEXPTExtrapolated[N2LO opt ] FIG. 7: Dependence of the calculated g.s. energies on Aof boron isotopes with INOY, CDB2K, N LO, N LO opt ,YSOX interactions and compared with experimental energies.NCSM results obtained at the largest accessible N max spacewith the optimal frequency are shown. In Fig. 7, the dependence of the calculated g.s. ener-gies on the mass number of boron isotopes is plotted withINOY, CDB2K, N LO, N LO opt , YSOX interactions andcompared with experimental energies. NCSM results ob-tained at the largest accessible N max space with the opti-mal frequency are shown. From Fig. 7, we can concludethat INOY interaction provides better description for g.s.energy than other used ab initio interactions.For the N LO opt interaction, we have extrapolatedthe g.s. energy using an exponential fitting function E g.s. ( N max ) = a exp ( − bN max ) + E g.s. ( ∞ ) with E g.s. ( ∞ )the value of g.s. energy at N max → ∞ . In particular,we have used last three N max points in the extrapola-tion procedure. For B, no meaningful extrapolationwas possible. VI. POINT-PROTON RADII In Table III, we have presented point-proton radii ( r p )using NCSM with INOY, CDB2K and N LO interactionsat their optimal frequencies along with experimentallyobserved radii [50]. The INOY interaction considerablyunderestimates the radii. For , B, the CDB2K andN LO interactions produce better results, with the for-mer slightly underestimating and the latter slightly over-estimating the radii. For − B, the radii are underesti-mated for all interactions. TABLE III: Calculated point-proton radii ( r p ) of − B withINOY, CDB2K and N LO interactions at highest N max cor-responding to their optimal HO frequencies. Experimentalpoint-proton radii are taken from Ref. [50]. The point-protonradii are given in fm. r p EXPT INOY CDB2K N LO B 2.32(5) 2.03 2.27 2.38 B 2.21(2) 1.97 2.15 2.24 B 2.31(7) 1.96 2.13 2.23 B 2.48(3) 1.98 2.10 2.20 B 2.50(2) 1.99 2.18 2.20 In Fig. 8, we present the variation of B r p withfrequency and N max for INOY, CDB2K and N LO in-teraction. With the enlargement of basis size N max, the2 . . B [INOY] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT . . . . . B [CDB2K] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT . . . . . B [N LO] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6 N max = 8 N max = 10EXPT FIG. 8: Variation of r p of B with HO frequency for N max = 2 to 10, corresponding to the INOY, N LO and CDB2Kinteractions. The horizontal line shows the experimental value with the vertical bars representing uncertainty. dependence of r p on frequencies decreases. The curves of r p corresponding to different N max intersect each-otherapproximately at the same point. We take this cross-ing point as an estimate of the converged radius [46, 83].In particular, we consider the intersection point of thecurves at the highest successive N max as an estimate ofthe converged radius. In this way, we obtain B point-proton radii for INOY, CDB2K and N LO interactions2.14, 2.30 and 2.36 fm, respectively.Similarly, we have shown variation of r p with frequencyand N max for other isotopes corresponding to INOY in-teraction in Fig. 9. Obtained r p values for B, B, Band B are 2.00, 1.99, 1.95 and 1.99 fm, respectively.However, even with this determination of the radii, theexperimental trend is not reproduced.We can conclude that the CDB2K and N LO interac-tions give radii which are much closer to experimentalvalue 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. Wehave obtained different optimal frequencies for the en-ergy spectra and the point-proton radii. Similar findingswere reported for C using Daejeon16 and JISP16 inter-actions in Ref. [46]. VII. CONCLUSIONS In this work, we have applied ab-initio no-core shellmodel to obtain spectroscopic properties of boron iso-topes using INOY, CDB2K, N LO and N LO opt nucleon-nucleon interactions. We have calculated low-lying spec-tra and other observables with all four interactions and,in addition, compared the NCSM results with shell modelusing YSOX valence-space effective interaction. We wereable to correctly reproduce the g.s. spin of B only withthe INOY NN interaction. Overall, the INOY interac-tion reproduced quite reasonably g.s. energies of all the3 ✽ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽ ✷✵ ✷✷ ✷✹ ✷✻ ✷✽✶✿✺✷✷✿✺✸ (cid:0)(cid:0)❇❬■◆❖❨❪✖❤✡ ✭▼❡❱✮r ♣✁❢♠✂ ✄☎❛① ❂ ✆✄☎❛① ❂ ✝✄☎❛① ❂ ✞✄☎❛① ❂ ✟❊❳P❚ . . B [INOY] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6 N max = 8EXPT . . B [INOY] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6 N max = 8EXPT . . B [INOY] ¯ h Ω (MeV) r p ( f m ) N max = 2 N max = 4 N max = 6EXPT FIG. 9: Variation of r p of , , , B with HO frequency for different N max, corresponding to the INOY interaction. Thehorizontal line shows the experimental value with the vertical bars representing uncertainty. studied isotopes, − B.Considering electromagnetic properties, we have ob-tained fast convergence for M E M LO interactions give radii which are much closer to ex-perimental value than the radii obtained with the INOYinteraction.The present study confirms that non-locality in the NN interaction can account for some of the many-nucleonforce effects. The non-local NN interaction like INOYcan provide a quite reasonable description of ground-state energies, excitation spectra and selected electro- magnetic properties, e.g., magnetic moments and M N interaction,in particular 3 N interaction with non-local regulators, isessential for a correct simultaneous description of nuclearbinding and nuclear size [39, 84, 85]. ACKNOWLEDGMENTS We would like to thank Christian Forss´en for makingavailable the pAntoine code. We thank Toshio Suzukifor the YSOX interaction. 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