Competitive effects of nuclear deformation and density dependence of ΛN interaction
aa r X i v : . [ nu c l - t h ] S e p APS/123-QED
Competitive effects of nuclear deformation and density dependence of Λ N interactionin B Λ values of hypernuclei M. Isaka , Y. Yamamoto , and Th.A. Rijken Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka, 567-0047, Japan Nishina Center for Accelerator-Based Science, Institute for Physicaland Chemical Research (RIKEN), Wako, Saitama, 351-0198, Japan IMAPP, University of Nijmegen, Nijmegen, The Netherlands (Dated: July 11, 2018)Competitive effects of nuclear deformation and density dependence of Λ N -interaction in Λ bindingenergies B Λ of hypernuclei are studied systematically on the basis of the baryon-baryon interactionmodel ESC including many-body effects. By using the Λ N G-matrix interaction derived from ESC,we perform microscopic calculations of B Λ in Λ hypernuclei within the framework of the antisym-metrized molecular dynamics under the averaged-density approximation. The calculated values of B Λ reproduce experimental data within a few hundred keV in the wide mass regions from 9 to 51. Itis found that competitive effects of nuclear deformation and density dependence of Λ N -interactionwork decisively for fine tuning of B Λ values. PACS numbers: Valid PACS appear here
I. INTRODUCTION
Basic quantities in hypernuclei are Λ binding energies B Λ , from which a potential depth U Λ in nuclear mat-ter can be evaluated. The early success to reproduce the U Λ value was achieved by Nijmegen hard-core models [1],where the most important role was played by the Λ N -Σ N coupling term. Medium and heavy Λ hypernuclei havebeen produced by counter experiments such as ( π + , K + )reactions. Accurate data of B Λ values in ground andexcited states of hypernuclei have been obtained by γ -ray observations and ( e, e ′ K + ) reactions. With the in-crease of experimental information [2], precise interactionmodels have been constructed. In the Nijmegen group,the soft-core models have been developed with contin-uous efforts so as to reproduce reasonably hypernucleardata [3–6]. In the recent versions of the Extended-SoftCore (ESC) models [5, 6], two-meson and meson-pair ex-changes are taken into account explicitly, while these ef-fects are implicitly and roughly described by exchanges of“effective bosons” in one-boson exchange (OBE) models.The latest model ESC08c aims to reproduce consistentlyalmost all features of the S = − − . This version is used also in the present work.Recently, the dependence of B Λ on structures of corenuclei, in particular nuclear deformations, has been dis-cussed in p shell [8], and sd - pf shell hypernuclei [9–11]theoretically. Generally, values of B Λ are related to nu-clear structure in two ways: One is that an increase The fortran code ESC08c2012.f is put on the permanent open-access website, NN-Online facility: http://nn-online.org . of deformation reduces the overlap of the densities be-tween a Λ particle and the core nucleus, which makes B Λ smaller. Such effects are seen in sd - pf shell hypernuclei.In Refs. [9, 11], the antisymmetrized molecular dynam-ics for hypernuclei (HyperAMD) [12, 13] was applied toseveral sd - pf shell hypernuclei such as Ca and
Sc. Itis found that B Λ values in deformed states are decreasedreflecting smaller overlaps.The other effect is due to the density dependence ofthe Λ N effective interaction. In light hypernuclei and/ordilute states like cluster states, the density overlap be-tween a Λ and nucleons is significantly decreased, whichcan affect the B Λ through the density dependence. Forexample, in Be hypernuclei having a 2 α cluster struc-ture with surrounding neutrons, it was discussed thatthe overlap becomes much smaller in the well-pronounced2 α cluster states [8]. When the Λ N effective interactionderived from the G -matrix calculation is designed to de-pend on the nuclear Fermi momentum k F , the smalleroverlap makes the relevant value of k F small, i.e. lessPauli-blocking, resulting in the increase of B Λ . Consid-ering this effect, it is expected that appropriate valuesof k F in finite systems are reduced as overlaps becomesmall with mass numbers, which would affect the massdependence of B Λ .Λ N interactions are related intimately to the recenttopic of heavy neutron stars (NS). The stiff equationof state (EoS) giving the large NS-mass necessitates thestrong three-nucleon repulsion in the high-density region,the existence of which has been established by manyworks [14] in nuclear physics. However, the hyperon mix-ing in neutron-star matter brings about the remarkablesoftening of the EoS, canceling this repulsive effect. Apossible way to solve such a problem is to assume thatstrong repulsions exist universally in three-baryon chan-nels. More specifically, it is assumed that the Λ N N repulsion works in Λ hypernuclei as well as the three-nucleon repulsion. A Λ
N N three-body effect, that isgenerally a hyperonic many-body effect (MBE), has toappear as an additional density dependence of the Λ N effective interaction. It is important to study MBE byanalyzing the experimental data of B Λ systematically.The aim of the present work is to reveal how the den-sity dependence of the Λ N effective interaction affectsthe mass dependence of B Λ . Since the p - sd - pf shell hy-pernuclei have various structures in the ground states,they would affect the values of B Λ through the density-dependence of the Λ N interaction. To investigate it, weuse the HyperAMD combined with Λ N G -matrix inter-action, which successfully describes various structures ofhypernuclei without assumptions on specific clusteringand deformations [12, 13].This paper is organized as follows. In the next sec-tion, the Λ N G-matrix interaction is explained as wellas treatment of MBE. In Sec. III, we explain how todescribe hypernuclei, namely the theoretical frameworkof HyperAMD. In Sec. IV, we show the calculated val-ues of B Λ including MBE, and discuss effects from corestructures on B Λ . Section V summarizes this paper. II. Λ N G-MATRIX INTERACTION
We start from ESC08c(2012), which was used in theanalysis of Λ hypernuclei based on the HyperAMD mostsuccessfully [9]. One should be careful, however, thatthe main conclusion in this work has to be valid qualita-tively also for other realistic interaction models includingΛ N -Σ N coupling terms which lead to strong density de-pendences of the Λ N effective interactions. Hereafter,ESC08c(2012) is denoted as ESC simply. As a modelincluding an additional density dependence due to a hy-peronic MBE, we adopt the model given in Ref. [15].Here, the multi-pomeron exchange repulsion (MPP) isadded into ESC together with the phenomenologicalthree-body attraction (TBA), where both of them arerepresented as density-dependent two-body interactions.Using ESC+MPP+TBA, G-matrix calculations are per-formed with the continuous choice for off-shell single par-ticle potentials: Contributions of MPP and TBA arerenormalized into Λ N G-matrices. The MPP part is givenas V ( N ) MP P ( r ; ρ )= g ( N ) P g NP ρ N − M N − (cid:18) m P √ π (cid:19) exp (cid:18) − m P r (cid:19) , (1)corresponding to triple ( N = 3) and quartic ( N = 4)pomeron exchange. The values of the two-body pomeronstrength g P and the pomeron mass m P are the same asthose in ESC. A scale mass M is taken as the protonmass. The TBA part is assumed as V T BA ( r ; ρ )= V exp( − ( r/ . ) ρ exp( − . ρ ) (1 + P r ) / , (2) TABLE I: Values of parameters in ∆ V Λ N ( k F ; r ) = ( a + bk F + ck F ) exp − ( r/β ) with β = 0 . E E O Oa b − − − − c P r being a space-exchange operator. In Refs. [15, 16],these interactions were assumed to be universal in allbaryonic channels. Namely, the parameters g (3) P , g (4) P and V in hyperonic channels were taken to be thesame as those in nucleon channels, assuring the stiffEoS of hyperon-mixed neutron-star matter. There wereused three sets with different strengths of MPP inRefs. [15, 16]. In the case of the set MPa, for instance,the parameters were taken as g (3) P = 2 . g (4) P = 30 . V = − .
8. In the present analysis, however, sucha choice leads to a too strong density-dependence of theΛ N G-matrix interaction for reproducing the mass de-pendence of B Λ values: In the case of ESC08c(2012),the mass dependence of B Λ values are reproduced ratherwell without the additional MBE. Then, the values of g (3) P and g (4) P may be much smaller than the above valuesso that the additional density dependence is not strong.Here, the parameters are determined so that calculatedresults of B Λ values in the present framework are con-sistent with the experimental data. They are taken as g (3) P = 0 . g (4) P = 0 . V = − .
0: MPP (TBA)is far less repulsive (attractive) than those in the abovecase. In this case, the calculated value of B Λ is 13.0 MeVin O, which is consistent with the observed value (seeTable III). Thus, MBE is represented by MPP+TBA,having only minor effects on the results in this work.Λ N G-matrix interactions V Λ N for ESC are con-structed in nuclear matter with Fermi momentum k F [17]. They are represented in coordinate space andparameterized in a three-range Gaussian form [17], V Λ N ( r ; k F ) = X i =1 ( a i + b i k F + c i k F ) exp ( − r /β i ) . (3)The parameters ( a i , b i , c i ) are determined so as to simu-late the calculated G-matrix for each spin-parity state.The procedures to fit the parameters are given inRef. [17], and the determined parameters for ESC aregiven in Ref. [9].Contributions from MBE (MPP+TBA) to G-matricesare represented by modifying the second-range partsof V Λ N ( k F , r ) for ESC by ∆ V Λ N ( k F , r ) = ( a + bk F + ck F ) exp (cid:8) − ( r/β ) (cid:9) . It should be noted that the valuesof parameters g (3) P , g (4) P and V are connected to the val-ues of a , b and c through this procedure. The values ofparameters are given in Table I.In applications of nuclear matter G-matrix interactions V Λ N ( r ; k F ) to finite systems, a basic problem is how to TABLE II: Values of B Λ in Y and
O calculated withADA and LDA (in MeV). Observed values of B Λ ( B expΛ ) areshifted by 0.54 MeV from those in Refs. [18, 19] as explainedin Sec. IV A. − B calΛ ADA LDA − B expΛ89Λ Y − − − . ± .
10 [18] O − − − . ± .
05 [19] choose k F values in each system: An established man-ner is to use so called local-density and averaged-densityapproximations etc based on physical insight. As thebetter choice to describe Λ single particle (s.p.) states,we adopt an averaged-density approximation (ADA) [17],where the averaged value of k F is defined by k F = (cid:18) π h ρ i (cid:19) / , h ρ i = Z d rρ N ( r ) ρ Λ ( r ) . (4)In the case of local-density approximation (LDA), k F values are obtained from ( ρ N ( r ) + ρ Λ ( r )) / r . We compare ADA and LDA by calculating B Λ values for Y and
O with use of the Λ-nucleus foldingmodel in which Λ N G-matrix interactions V Λ N ( r ; k F ) arefolded into density distributions [17]. For spherical-coresystems, the results calculated with the G-matrix foldingmodel are similar to those with the HyperAMD used inthe following section. In Table II, the result is shown inthe case of using ESC without MBE. It is demonstratedhere that the B Λ values in Y are reproduced nicelyin both cases of ADA and LDA with no adjustable pa-rameter. On the other hand, in
O, the value of B Λ obtained with LDA is found to be smaller than thatwith ADA. Thus, the B Λ values with LDA are similarto (smaller than) those with ADA in heavy (light) sys-tems, and eventually the mass dependence of B Λ valuescan be reproduced better using ADA than LDA. Hence,the ADA is employed in the present work as an approxi-mate way to use nuclear matter G-matrix interactions infinite systems. III. ANALYSIS BASED ON HYPERAMD
In this study, we apply the HyperAMD to p , sd , and pf shell Λ hypernuclei, namely from Li up to
Fe, in orderto describe various structures of these hypernuclei suchas an α clustering and prolate, oblate, and triaxial defor-mations in ground states. Combined with the generatorcoordinate method (GCM), we perform the systematicanalysis of B Λ . A. Hamiltonian and wave function
The Hamiltonian used in this study is H = T N + T Λ − T g + V NN + V C + V Λ N , (5)where T N , T Λ , and T g are the kinetic energies of the nucle-ons, Λ particle, and center-of-mass motion, respectively.We use Gogny D1S [20, 21] as the effective nucleon-nucleon interaction V NN , and the Coulomb interaction V C is approximated by the sum of seven Gaussians. Asfor the Λ N interaction V Λ N , we use the G-matrix inter-action discussed above.The variational wave function of a single Λ hypernu-cleus is described by the parity-projected wave function,Ψ ± = ˆ P ± {A{ ϕ , . . . , ϕ A } ⊗ ϕ Λ } , where ϕ i ∝ e − P σ ν σ (cid:0) r σ − Z iσ (cid:1) ⊗ ( u i χ ↑ + v i χ ↓ ) ⊗ ( p or n ) , (6) ϕ Λ ∝ M X m =1 c m e − P σ ν σ (cid:0) r σ − z mσ (cid:1) ⊗ ( a m χ ↑ + b m χ ↓ ) . (7)Here the s.p. wave packet of a nucleon ϕ i is described bya single Gaussian, while that of Λ, ϕ Λ , is represented by asuperposition of Gaussian wave packets. The variationalparameters are Z i , z m , ν σ , u i , v i , a m , b m , and c m . Inthe actual calculation, the energy variation is performedunder the constraint on the nuclear quadrupole deforma-tion parameters ( β, γ ) in the same way as in Ref. [13].By the frictional cooling method, the variational param-eters in Ψ ± are determined for each set of ( β, γ ), and theresulting wave functions are denoted as Ψ ± ( β, γ ). B. Angular momentum projection and generatorcoordinate method
After the variation, we project out the eigenstate of thetotal angular momentum J for each set of ( β, γ ) (angularmomentum projection; AMP),Ψ J ± MK ( β, γ ) = 2 J + 18 π Z d Ω D J ∗ MK (Ω) R (Ω)Ψ ± ( β, γ ) . (8)The integrals over the three Euler angles Ω are performednumerically. Then the wave functions with differing val-ues of K and ( β, γ ) are superposed (generator coordinatemethod; GCM):Ψ J ± n = X p J X K = − J c npK Ψ J ± MK ( β p , γ p ) . (9)The coefficients c npK are determined by solving theGriffin-Hill-Wheeler equation [13]. C. B Λ and analysis of wave function The B Λ ’s are calculated as the energy differencebetween the ground states of a hypernucleus ( A +1Λ Z ) TABLE III: − B Λ [MeV] calculated with ESC + MBE to-gether with h ρ i [fm − ] and k F [fm − ] defined by Eq.(4), andnuclear quadrupole deformation ( β, γ ) for each hypernucleus.Values in parentheses are calculated with ESC08c(2012) onlyin unit of MeV. Observed values B expΛ are taken from Refs.[2, 18, 19, 22–28]. Values of B expΛ with dagger are also ex-plained in text. β γ h ρ i k F − B calΛ − B expΛ9Λ Li 0.50 2 ◦ − . − . − . ± . Be 0.87 1 ◦ − . − . − . ± . B 0.45 2 ◦ − . − . − . ± . Be 0.57 1 ◦ − . − . − . ± . − . ± . B 0.68 1 ◦ − . − . − . ± . B 0.50 29 ◦ − . − . − . ± . B 0.39 44 ◦ − . − . − . ± . − . ± . C 0.41 34 ◦ − . − . − . ± . C 0.45 60 ◦ − . − . − . ± . C 0.52 22 ◦ − . − . − . ± . N 0.28 60 ◦ − . − . − . ± . O 0.02 – 0.105 1.16 − . − . − . ± . † O 0.30 3 ◦ − . − .
0) –
Ne 0.46 0 ◦ − . − .
1) –
Mg 0.478 21 ◦ − . − .
8) –
Mg 0.36 36 ◦ − . − .
4) –
Si 0.32 53 ◦ − . − . − . ± . † S 0.23 16 ◦ − . − . − . ± . † K 0.01 – 0.136 1.26 − . − .
2) –
Ca 0.03 – 0.136 1.26 − . − . − . ± . † Ca 0.13 12 ◦ − . − .
4) –
K 0.01 – 0.141 1.27 − . − .
1) –
V 0.18 2 ◦ − . − . − . ± . † Fe 0.26 23 ◦ − . − .
3) – and the core nucleus ( A Z ) as B Λ = E ( A Z ; j ± ) − E ( A +1Λ Z ; J ± ), where E ( A Z ; j ± ) and E ( A +1Λ Z ; J ± ) arecalculated by GCM.We also calculate squared overlap between theΨ J ± MK ( β, γ ) and GCM wave function Ψ J ± α , O J ± MKα ( β, γ ) = |h Ψ J ± MK ( β, γ ) | Ψ J ± α i| , (10)which we call the GCM overlap. O J ± MKα ( β, γ ) shows thecontribution of Ψ J ± MK ( β, γ ) to each state J ± , which is use-ful to estimate deformation of each state. In this study,we regard ( β, γ ) corresponding to the maximum value ofthe GCM overlap as nuclear deformation of each state. IV. RESULTS AND DISCUSSIONSA. B Λ in p , sd , and pf shell Λ hypernuclei The calculated values of B Λ for ESC including MBEare summarized in Table III together with the values of k F and h ρ i , and compared with those calculated by us-ing ESC only (in parentheses) and observed values of B Λ ( B expΛ ). Here, the k F values are calculated by Eq.(4) onthe basis of ADA. In Table III, we also show ( β, γ ) whichgives the maximum value of the GCM overlap defined byEq.(10). Recently, in Ref. [28], it has been discussed thatthe B expΛ measured by the ( π + , K + ) experiments are sys-tematically shallower by 0.54 MeV in average than theemulsion data for Li, Be,
B and
C. It indicates thatthe reported binding energy of
C [24] would be shal-lower by 0.54 MeV, which is used for the binding energymeasurements as the reference in the ( π + , K + ) experi-ments. Therefore, in Table III, the values of B expΛ mea-sured by the ( π + , K + ) or ( K − , π − ) experiments (withdagger) are shifted by 0.54 MeV deeper from the val-ues reported by Refs. [2, 18, 19, 22, 26]. In spite of thiscorrection, there still remain calibration ambiguities inthe ( π + , K + ) data. One should be careful for this prob-lem, when the calculated values of B Λ are compared withthese data.Let us discuss the calculated values of B Λ shown inTable III. As mentioned in Sec. II, we determine theparameters of MPP and TBA in Eqs. (1) and (2) so asto reproduce B expΛ in O in the HyperAMD calculationwith ESC + MPP + TBA. It is seen that the B Λ withESC + MPP + TBA reproduces the observed data withinabout 200 keV except for Be,
N and
Si, which isachieved owing to the k F dependence of the Λ N G-matrixinteraction used. As seen in Table III, the k F valuesbecome small with decreasing mass number, which meansthat the Λ N G-matrix interaction becomes attractive.The main origin of the k F dependence is from the Λ N -Σ N coupling terms included in ESC. B. Effects of core deformation
For the fine agreement of B Λ values to the experimen-tal data, it is very important to describe properly thecore structures, in particular nuclear deformations. Re-cently, many authors have been studying deformations ofhypernuclei in p -shell [29–32], sd -shell [9, 10, 29–34], and pf -shell [9, 10, 29] mass regions. In this study, we takeinto account deformations of hypernuclei by performingGCM calculations in which intrinsic wave functions withvarious ( β, γ ) deformations Ψ ± ( β, γ ) are diagonalized.In order to study the importance of core deformationsin the systematic calculations of B Λ values, we performthe GCM calculation by using the spherical wave func-tions Ψ J ± MK ( β = 0 .
0) in Eq.(9) (case (B)), whereas TableIII summarizes the GCM results with various deforma-tions (case (A)). In the case (B), the k F value is deter-mined independently from the case (A) with Ψ J ± MK ( β =0 .
0) by Eq.(4) for each hypernucleus. By using the k F values determined in the case (B), we also perform theGCM calculations with various ( β, γ ) deformations (case(C)). Table IV shows the calculated values of B Λ in thecases (A) - (C) in the typical p -shell hypernuclei B, B, and
C. Comparing the cases (A) and (B), we findthe considerable discrepancy of B Λ , i.e. the B Λ in the FIG. 1: (Color online) (a) Comparison of B Λ between cases(A) (solid) and (B) (dashed). Open circles show observedvalues with mass numbers from A = 9 up to A = 51, whichare taken from Refs. [2, 18, 19, 22–27]. B expΛ measured by( π + , K + ) and ( K − , π − ) reactions are shifted by 0.54 MeVas explained in text. (b) Same as (a), but magnified in the5 ≤ A ≤
20 region. case (B) are shallower than those in the case (A), whichindicates that the B Λ becomes smaller, if the core nucleiare spherical. This is mainly due to the larger k F valuein the case (B) compared with the case (A), which comesfrom the increase of h ρ i in a spherical state (see Eq.(4)).For example, in the case of B, the obtained value of B Λ is 9.7 MeV with k F = 1 .
16 fm − in the case (B), whereas B Λ = 11 . k F = 1 .
07 fm − in the case (A)( cf. B expΛ = 11 . ± .
02 MeV [27]). The same differencebetween the cases (A) and (B) is seen in the other hy-pernuclei, in particular light hypernuclei with
A <
16, asshown in Figure 1 (a) and (b).In Table IV, it is also found that the values of B Λ in thecase (C) are shallower than those in the case (B), whichdeviate much from those in the case (A) and the obser-vations. This is because the deformation of the core nu-clei decreases the overlap between the Λ and core nuclei.Since we use the same k F in the cases (B) and (C), thesmaller overlap with deformation in the case (C) makes B Λ shallower. Therefore, it can be said that the consis-tent descriptions of the deformation and the values of k F determined in deformed states are essential to reproducethe observations. B Λ values are given by the balance oftwo competitive effects: (1) The deformation makes theΛ s.p. energy ( k F value) shallower (smaller). (2) Thesmaller value of k F makes the Λ s.p. energy deeper dueto the density dependence of Λ N interaction. In A > B Λ values smallerbecause the effect (2) is not so remarkable to cancel theeffect (1). On the other hand, in A <
16 region, defor-mations make B Λ values larger due to the effect (2).Let us confirm whether the core deformation is success-fully described under the present AMD framework withthe Gogny D1S interaction. It can be done essentially by comparing E B ( E
2) of the corenuclei with the observations, which are quite sensitive tothe nuclear deformation. For example, in
B, we calcu-late B ( E
2) in B as B ( E
2; 5 / − → / − ) = 16 e fm by performing the GCM calculation with various ( β, γ )deformations following Refs. [35, 36], which is consis-tent with the experimental value B ( E
2; 5 / − → / − ) =14 ± e fm [37]. On the basis of the structure calcula-tion for B, we obtain a very reasonable value of B Λ in B by the addition of a Λ particle. Then, it is confirmedthat our calculations for B Λ are performed in the modelspace to describe core deformations properly.Here, we compare the deformation of hypernuclei withthat predicted by Ref. [30], in which C and
Si arepredicted to be spherical within the framework of rela-tivistic mean-field, whereas the core nuclei C and Siare oblately deformed. It means that the addition ofa Λ particle makes the core nucleus spherical. In thepresent work, we also find the reduction of the core de-formation by the addition of a Λ particle. However,the degree of deformation change is rather small. Thusthese hypernuclei are still deformed as shown in Table III,while ( β, γ ) = (0 . , ◦ ) in C and ( β, γ ) = (0 . , ◦ )in Si. This difference between the present work andRef. [30] mainly comes from the effects by rotational mo-tions, which are included by performing the angular mo-mentum projection (AMP) (see Eq.(8)). In fact, it isalso found that the deformation of
C becomes spheri-cal before performing the AMP [32], which is the sametrend as predicted by Ref. [30]. In the present calcula-tion, not only rotational motions but also configurationmixing and shape fluctuations are taken into account byperforming the AMP and GCM, which can affect the de-formation of hypernuclei.
C. Deviation of B Λ in several hypernuclei We comment on the large deviation of B Λ in Be,
N, and
Si. In Be, it is considered that the GognyD1S force [20, 21] overestimates the size of each α parti-cle of 2 α cluster structure of the core Be due to thezero-range density-dependent term, as pointed out byRef. [38], which would cause the overestimation of B Λ bythe decrease of k F through Eq.(4). It is found that the k F value which reproduces the B expΛ of Be ( k F = 1 .
08 fm − )is much larger than that shown in Table III ( k F = 0 . − ). The smallness of the latter value of k F is due tothe overestimation of the size of α with Gogny D1S. Itis also found that the same phenomenon appears in theΛ hypernuclei with A < α cluster structure byusing Gogny D1S. Therefore, we exclude them from be-ing the subject of the present analysis. In such cases, itwould be necessary to use appropriate effective N N in-teractions instead of Gogny D1S. In
N, the B expΛ mea-sured by the emulsion experiment [25] seems to be devi-ating from those of the neighboring hypernuclei in Figure1(b). This might be due to the difficulties of the analysis TABLE IV: Comparison of B Λ with the cases (A), (B) and (C) in B, B, and
C. Value of k F calculated by Eq.(4) in eachcase is also shown. ( β, γ ) giving the maximum values of the GCM overlap (Eq.(10)) are also shown in cases (A) and (C). B B Ccase (A) case (B) case (C) case (A) case (B) case (C) case (A) case (B) case (C) − B Λ − . − . − . − . − . − . − . − . − . k F β, γ ) (0.50,29 ◦ ) (0.50,29 ◦ ) (0.39,44 ◦ ) (0.39,44 ◦ ) (0.45,60 ◦ ) (0.45,60 ◦ ) − B expΛ − . ± . − . ± . − . ± . − . ± . and smaller numbers of events in the emulsion experi-ments. Therefore, we hope to perform a new analysis ofthe emulsion measurements with large statistic in the fu-ture. In Si, the value of B Λ is underestimated in case(A), whereas that in case (B) (17.3 MeV) is much closerto the experimental value. This might be due to an over-estimation of the core deformation, which is seen in thecomparison of the electric quadrupole moment Q in theground state 5 / + of Si, namely, Q (5 / + , AMD) = 10 efm , whereas Q (5 / + , exp) = 6 . ± . [39]. Sincethe calculated values of k F are almost the same in thecases (A) and (B) (1.23 fm − ), the value of B Λ would bein between the values of these cases, if the deformationof Si is smaller than the present result. D. B Λ and strength of many-body force Finally, we also comment on the relation between B Λ and the strength of MPP and TBA. In the present study,the parameters g (3) P and g (4) P in Eq.(1) ( V in Eq.(2)) aretaken as far smaller (less attractive) than those in in Refs.[15, 16]. They are determined so as to improve the fittingof B Λ values to the experimental data. As seen in TableIII, the calculated values of B Λ with ESC only reproducerather well the experimental ones. Therefore, there re-mains only a small room to introduce MBE on the basisof ESC. On the other hand, in case of MPa [15, 16],the parameters of MPP and TBA in hyperonic chan-nels are taken to be the same as those in nucleon chan-nels assuming the stiff EoS of hyperon mixed neutron-star matter. It is found that values of B Λ are overes-timated if the parameter set of MPa is used combinedwith ESC. For example, B Λ with MPa are 13.0 MeV for C ( cf. B expΛ = 11 . ± .
19 MeV), and 14.2 MeV for
O ( cf. B expΛ = 12 . ± .
05 MeV). This indicates thatthe strength of MPP and TBA in MPa is too strong to reproduce the observations, when MPa is used togetherwith ESC. It is known that two-body Λ N effective inter-actions still have ambiguities, and thus potential depthand k F dependence are different among models. The de-pendence of MBE on two-body Λ N effective interactionmodels will be discussed in following paper. Here, forinstance, the strong MPP such as MPa is shown to beallowable in the case of the latest version of ESC08c. V. SUMMARY
On the basis of the baryon-baryon interaction modelESC including MBE, competitive effects of nuclear de-formation and density dependence of the Λ N interactionare investigated. By using the G-matrix interaction de-rived from ESC, we perform microscopic calculations of B Λ within the framework of HyperAMD with the ADAtreatment for the hypernuclei with 9 ≤ A ≤
59. It isfound that the calculated values of B Λ reproduce theexperimental data within a few hundred keV, when theadditional density dependence by MBE is taken into ac-count. This is achieved by the competition between thenuclear deformation and density dependence of Λ N inter-action. Generally, the overlap between the Λ and nucle-ons varies depending on the degree of core deformation.In the light hypernuclei with A ≤
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