Energy spectra in $p$-shell $\Lambda$ hypernuclei and $^{19}_{\Lambda}\textrm{F}$ and spin-dependent $\Lambda N$ interactions

Energy spectra of $0s$-orbit $\Lambda$ states in $p$-shell $\Lambda$ hypernuclei ($^{A}_\Lambda Z$) and those in $^{19}_{\Lambda}\textrm{F}$ are studied with the microscopic cluster model and antisymmetrized molecular dynamics using the $G$-matrix effective $\Lambda N$ ($\Lambda NG$) interactions. Spin-dependent terms of the ESC08a version of the $\Lambda NG$ interactions are tested and phenomenologically tuned to reproduce observed energy spectra in $p$-shell $^{A}_\Lambda Z$. Spin-dependent contributions of the $\Lambda N$ interactions to spin-doublet splitting and excitation energies are discussed. Energy spectra for unobserved excited states in $p$-shell $^{A}_\Lambda Z$ and $^{19}_{\Lambda}\textrm{F}$ are predicted with the modified $\Lambda NG$ interactions.


I. INTRODUCTION
In this decade, experimental and theoretical studies of hypernuclei have remarkably progressed. For Λ hypernuclei, experimental studies with high-resolution γ-ray measurements have been extensively performed to provide detailed information of energy spectra in the p-shell region [1][2][3]. The γ-ray spectroscopic study of sd-shell A Λ Z has just started, and the observed spectra in 19 Λ F have been reported [4]. The observed energy spectra are useful information for study of ΛN interactions. In particular, energy splittings between the spin-doublet J > = I + 1/2 and J < = I − 1/2 states with the Λ-spin coupling in parallel and anti-parallel to the core nuclear spin I are sensitive probes to figure out spin dependences of the ΛN interactions in Λ hypernuclei. Based on compilation of the precise data updated recently, it is time to comprehensively understand the energy spectra in p-shell A Λ Z with theoretical study. Moreover, it is able to test spin dependences of the ΛN interactions in comparison of calculated spectra with observed data.
In the structure calculations of Λ hypernuclei, Y N interactions developed based on the meson-theoretical models by the Nijmegen group have been widely used. After many trials and continuous improvements, new versions (ESC08 series) of the extended-soft-core (ESC) model of the Y N interactions have been proposed [45][46][47]. The spin-independent part of the ESC08(a,b) ΛN interaction was tested and found to be reasonable in description of Λ binding energies in a wide mass number region [42,43,47]. However, in spin dependences of the ESC08(a,b), problems were found in reproducing the observed spin-doublet splitting energies [47]. It is a demanded issue to test the spin dependences of the ESC08 with systematic investigation of energy spectra in p-shell A Λ Z and consider possible modification of the spin-dependent ΛN interactions in comparison with the experimental data.
It is known from observed Λ binding energies that the ΛN interactions are weak compared with the N N interactions. Moreover, from observed energy spectra, the spin-dependent ΛN interactions have been found to be rather weak compared with the spin-independent ΛN interactions, and therefore, they may give perturbative contributions to structures of A Λ Z. It means that the Λ particle in A Λ Z, an impurity embedded in nuclear system, is regarded as a spectator probing the ΛN interactions. In particular, energy spectra of (0s) Λ states can be a good probe that detects rather directly the spindependent ΛN interactions through the spin-dependent mean-filed potential for the 0s-orbit Λ determined by nuclear spin structure in core nuclei. In order to describe detailed energy spectra and understand properties of the spin-dependent ΛN interactions, one needs a reliable structure model which can properly describe nuclear structures, particularly, nuclear spin configurations. Furthermore, for systematic studies of p-shell A Λ Z, it is also demanded to describe cluster structures in light-mass pshell nuclei. Cluster models can respond to the latter demand, but in general they are not sufficient in describing nuclear spin configurations in med-p-shell nuclei because cluster breakings are not taken into account in the model. Shell models are useful to investigate detailed spin configurations, but it is not suitable to deal with remarkable clustering as well as nuclear deformations because of limitation of the model space. The antisymmetrized molecular dynamics (AMD) model [48][49][50][51] is one of use-ful tools for systematic study of p-shell and sd-shell nuclei because it can describe cluster and spin structures in the ground and excited states of general nuclei. A version of the AMD model, variation after projection called AMD+VAP, has been applied to various p-shell nuclei including odd-odd nuclei and proved to be successful in describing nuclear spin properties such as µ moments, M 1, and GT transitions [51][52][53][54]. The HAMD, which is another version of the AMD applied to Λ hypernuclei by one of the authors (M. I.) and his collaborators [39][40][41][42][43], is also a promising approach for study of spin-dependent ΛN interactions though its application is still limited.
The aim of the present work is to investigate energy spectra in p-shell A Λ Z and discuss spin dependences of the ΛN interactions with microscopic structure model calculations. In the previous works by one of the authors (Y. K-E.), spin-averaged energy spectra of lowlying (0s) Λ states have been investigated by applying the cluster model for core nuclei and a single-channel potential model for a Λ particle with the spin-independent effective ΛN interactions [55]. In order to calculate energy spectra of A Λ Z with spin-dependent effective ΛN interactions, here we apply the AMD+VAP model in addition to the microscopic cluster model. The spindependent ΛN interactions are perturbatively treated in the AMD+VAP calculation. Comparing the calculated spin-doublet splitting energies with observed data in p-shell A Λ Z, we test spin dependences of the G-matrix effective ΛN (ΛN G) interactions of the ESC08a model [45][46][47]. A modification of the ESC08a ΛN G interaction is proposed by phenomenological tuning of the spindependent terms to adjust available data of energy spectra. Using the modified ΛN G interactions, the spindependent contributions of the ΛN interactions to energy spectra are investigated. In addition, theoretical spectra for unknown excited states in p-shell A Λ Z and 19 Λ F are predicted.
This paper is organized as follows. In the next section, we explain the framework of the present calculation. The effective N N and ΛN interactions are explained in Sec. III. Structure properties of core nuclei A−1 Z are shown in Sec. IV. In Sec. V, results for A Λ Z and the modification of the spin-dependent ΛN interactions are given. The paper is summarized in Sec. VI. In appendixes A and B, validity of the folding potential model approximation and spin rearrangement effects are discussed, respectively.

II. FRAMEWORK
We apply two models to describe structures of core nuclear part. One is the microscopic cluster model with the generator coordinate method (GCM) [56,57], which has been used in the previous works [55,58], and the other is the AMD+VAP. The framework of the AMD model is explained, for example, in Ref. [51]. For details of the frameworks, see those papers and references therein. In the cluster model with the GCM, the dynamical inter-cluster motion is taken into account by means of superposition of cluster wave functions having various inter-cluster distances. However, the cluster model contains only a part of intrinsic-spin configurations because it ignores cluster breaking. To overcome this problem of the cluster model and investigate spin-dependent contributions of the ΛN interactions to energy spectra, we apply the AMD model. A basis AMD wave function is given by a Slater determinant of single-nucleon Gaussian wave functions, in which Gaussian centroids and intrinsic-spin orientations of all nucleons are independently treated as variational parameters. The AMD model does not rely on a priori assumption of clusters and can describe the cluster breaking and cluster formation. Compared with the cluster model, the AMD is a flexible model, in particular, for intrinsic-spin degrees of freedom. However, in description of dynamical inter-cluster motion, the present AMD+VAP calculation is more limited than the cluster model because only a few AMD configurations are superposed.
The ΛN interactions contain spin-independent (V 0 ) and spin-dependent (V 1 ) parts as, The spin-independent part (V 0 ) is dominant and gives leading contributions, whereas the spin-dependent part (V 1 ) is relatively weak. In the present calculation, we first consider the spin-independent ΛN interaction V 0 as the leading part for the mean potential of the 0s-orbit Λ, and then perturbatively take into account the spin-dependent ΛN interaction V 1 . For p-shell A Λ Z, we investigate the leading contributions with cluster and AMD models using only V 0 by ignoring the spin dependence of the ΛN interactions. In order to investigate the spin-dependent contributions from V 1 , we apply the AMD model. For 19 Λ F, we investigate the leading and perturbative contributions with the microscopic cluster model of 16 O+p+n.
A. Calculations of core nuclei with microscopic structure models

Cluster model with GCM for p-shell nuclei
Core nuclei A−1 Z in A Λ Z are calculated with the microscopic cluster model in the same way as the previous calculations for p-shell Λ and double-Λ hypernuclei in Refs. [55,58]. In the model, microscopic A N -nucleon wave functions are expressed by the Brink-Bloch cluster wave functions [59] and superposed by means of the GCM. Here, A N = A − 1 is the mass number of core nuclei. The cluster wave functions of α + d, α + t, 2α, 2α + n, 2α + p, 2α + n 2 , 2α + d, 2α + t, 2α + h, 3α, and 3α + h cluster wave functions are adopted for 6 Li, 7 Li, 8 Be, 9 Be, 9 B, 10 Be, 10 B, 11 B, 11 C, 12 C, and 15 O systems, respectively. d, n 2 , t, h, and α clusters are written by harmonic oscillator (h.o.) 0s configurations. For 10 Be, 11 B, 11 C, and 12 C, additional configurations are included in the model space to take into account cluster breaking components as explained in Ref. [58]. Namely, the 6 He + α wave functions [60] are added to the 2α + n 2 wave functions for 10 Be, and the p 3/2 configurations are added to the 2α + t, 2α + h, and 3α cluster wave functions for 11 B, 11 C and 12 C [55,61]. Those cluster models (with and without additional configurations) are denoted by the label "CL" in the paper. The h.o. width parameter ν of clusters is commonly chosen as ν = 0.235 fm −2 for A N ≤ 12 nuclei. For 15 O, the value ν = 0.16 fm −2 which reproduces the nuclear size of the p-closed 16 O is used.
The Brink-Bloch cluster wave function for a A Nnucleon system consisting of C 1 , . . . , C k clusters is denoted as Φ BB (S 1 , . . . , S k ) with the parameters S j (j = 1, . . . , k) of cluster center positions. k is the number of clusters. To take into account inter-cluster motion, the GCM is applied to the angular-momentum and parity projected Brink-Bloch cluster wave functions with respect to the generator coordinates S j . The wave function Ψ N (I π n ) for the nuclear angular momentum and parity I π n state is given by a linear combination of the Brink-Bloch wave functions with various configurations of {S 1 , . . . , S k } as where P Iπ MK is the angular momentum and parity projection operator. The coefficients c I π n S1,...,S k ,K are determined by solving Griffin-Hill-Wheeler equations [56,57], which is equivalent to the diagonalization of the Hamiltonian and norm matrices, Here H N is the Hamiltonian of the nuclear part described later.

AMD+VAP for p-shell nuclei
The AMD+VAP method is applied for p-shell nuclei to investigate contributions of the spin-dependent ΛN interactions in A Λ Z. In the AMD framework, a basis wave function is given by a Slater determinant where A is the antisymmetrizer, and ϕ i is the ith singleparticle wave function written by a product of spatial, spin, and isospin wave functions, where φ X i and χ i are the spatial and intrinsic-spin functions, respectively, and τ i is the isospin function fixed to be proton or neutron. The width parameter ν is chosen to be the same value as that used in the cluster model. The AMD wave function is specified by a parameter set Z ≡ {X 1 , . . . , X AN , ξ 1 , . . . , ξ AN }. The Gaussian centroids X i and intrinsic-spin orientations ξ i of all nucleons are independently treated as variational parameters in the energy variation. Owing to the flexibility of spatial and intrinsic-spin configurations of single-nucleon Gaussian wave packets, the AMD wave function can express various cluster structures with cluster breaking degrees of freedom as well as various intrinsic spin configurations. Moreover, it can also describe shell-model wave functions because of the antisymmetrization. In the AMD+VAP, the energy variation is done after the angular-momentum and parity projections in the AMD model space as in order to obtain the optimum solution of the parameter set Z for the lowest I π states. For higher I π states, the VAP is done for the component orthogonal to the lower I π states already obtained by the VAP. The method is a version of the AMD and usually called the AMD+VAP. In this paper, it is simply denoted as the AMD.

Microscopic three-body model for 18 F
For 18 F, we use the microscopic 16 O + p + n wave functions adopted in the previous study of 18 F [62]. The wave functions are written in the form of the Brink-Bloch cluster wave functions for C 1 = p, and C 2 = n, and C 3 = 16 O and are superposed with the GCM. The same parametrization of the generator coordinates as Ref. [62] is used, In the present calculation, D = {1, 2, . . . , 7 fm} are chosen. For all D values, the coordinate set (q x , r y ) = (0, 0) is used corresponding to the d cluster. In addition, q x = {0.5, 1, 1.5, 2 fm} and r y = {0, 1 fm} are used for D = 2 fm and q x = {1, 2 fm} and r y = {0, 1 fm} are used for D = 3, 4, 5 fm to take into account the d-cluster breaking at the nuclear surface by the nuclear spin-orbit interactions from 16 O. In the total angular momentum projection P Jπ MK , |K| = 1 components are adopted. Higher |K| states are not included to save computational costs in numerical integration of the Euler angles in the projection. The wave functions are automatically projected onto the isospin T = 0 eigen states because of the |K| = 1 projection.
The inert 16 O cluster is assumed in the 16 O + p + n model. In order to see possible 4α-cluster vibration effect in 16 O, a 4α + p + n model is also applied in the calculation of the leading V 0 contribution. We use the label "CL" for the former model ( 16 O + p + n) with the inert 16 O cluster and CL-4α for the latter one (4α + p + n) with the 16 O vibration. In the CL-4α model, regular tetrahedral 4α configurations with the length of a side r = 0.5, 1.5, 2.5 fm are adopted in the GCM. As shown later, the vibration effect is found to be minor.

Nuclear energy and density
The Hamiltonian of the nuclear part consists of the kinetic terms, effective N N interactions, and Coulomb interactions as follows, where T G is the center of mass (cm) kinetic energy, V N N is the effective N N interactions, and V coulomb is the Coulomb interaction in the A N -nucleon system. The nuclear energy E N = Ψ N (I π n )|H N |Ψ N (I π n ) and nuclear density ρ I π N (r) are calculated for the nuclear wave functions Ψ N (I π n ) (normalized as | Ψ N (I π n )|Ψ N (I π n ) | = 1) obtained with the CL and AMD models. In the calculation of the nuclear energy and density, the cm motion of core nuclei is removed exactly and the radial coordinate r in ρ I π N (r) is defined by the distance from the cm of core nuclei.
B. Λ single-particle state with folding potential model in Λ hypernuclei

Λ wave function
The Λ wave functions in Λ hypernuclei are calculated by assuming a 0s-orbit Λ within a folding potential model as done in the previous work [55]. In the folding potential model, the Λ-nucleus potential is obtained by folding the spin-independent ΛN central interactions (V 0 ) with the nuclear density ρ I π N . The single-particle Hamiltonian h Λ,0 for a 0s-orbit Λ around the core is given as The nuclear density matrix in the exchange potential is approximated with the density matrix expansion in the local density approximation [63] as done in the previous works. For a given nuclear density ρ I π N (r) of the core nucleus I π state, the single-particle energy ε Λ,0 (ρ I π N ) and wave function φ Λ,0 (ρ I π N ; r) are calculated with the Gaussian expansion method [64,65]. In Appendix A, we compare the approximately calculated Λ-potential energies φ Λ,0 |U 0 (ρ)|φ Λ,0 in 7 Λ Li(I π = 1 + , 3 + ) with those of the microscopically calculated values Ψ N (I π )φ Λ,0 |V 0 |Ψ N (I π )φ Λ,0 . It is found that the approximation in the exchange potential gives only minor contribution to energy spectra.
In the cluster model calculation of A Λ Z with A ≤ 13, we take into account the core polarization, i.e., the nuclear size change induced by the Λ through the spinindependent ΛN central interactions (V 0 ) in the same way as done in the previous works. It should be commented that, in the previous works, the core polarization in A > 10 hypernuclei was found to be small and gives only minor effect to energy spectra. The core polarization is omitted in the cluster model calculation of 16 Λ O and 19 Λ F for simplicity. In the AMD calculation, we adopt the frozen core approximation without the core polarization. Namely, we omit the core polarization and use the nuclear wave functions obtained for isolate A−1 Z systems without the Λ.

C. Energy contributions of spin-dependent ΛN interactions
Energy contributions of the spin-dependent part V 1 of the ΛN interactions in A Λ Z(J π ) are perturbatively calculated as ΨA Λ Z (J π )|V 1 |ΨA Λ Z (J π ) by using the unpertur-bative A-body microscopic wave functions obtained with V 0 . Here χ Λ is the Λ-spin function coupling with the nuclear spin I to the total angular momentum J. In the present perturbative treatment, modification of the Λ wave function by V 1 is omitted. Moreover, rearrangement of spin configurations by the V 1 contribution is ignored. We checked the spin rearrangement effect to energy spectra and found that it is mostly minor unless energies of two J π states for different I π are close to each other.

D. Calculation procedure
The procedures of the present calculation are summarized below. The calculation is done in three steps as follows.
1. Calculation of the core nuclear part is performed with two kinds of structure models, cluster and AMD models. (For 19 Λ F, two kinds of cluster model calculations are done.) The nuclear wave function Ψ N (I π ), nuclear energy E N (I π ), and nuclear density ρ I π N (r) for A−1 Z(I π ) are obtained with the Hamiltonian given by Eq. (15).
2. Using the nuclear density ρ I π N obtained in the first step, the 0s-orbit Λ state in A Λ Z is calculated by the folding potential model with the spin-independent part V 0 of the ΛN interactions. The Λ singleparticle energy ε Λ,0 (ρ I π N ) and the Λ wave function φ Λ,0 (ρ I π N ) are obtained with the Hamiltonian given by Eq. (16). The core polarization is taken into account in the cluster model calculations of A Λ Z with A ≤ 13. It is omitted in other calculations.
3. The energy contributions from the spin-dependent part V 1 of the ΛN interactions are perturbatively calculated with the unperturbative wave functions ΨA Λ Z (J π ) given by Ψ N (I π n ) and φ Λ,0 , which are obtained in the first and second steps, respectively. The spin rearrangement and φ Λ,0 change induced by V 1 are ignored. The third step calculation is done by using the nuclear wave functions Ψ N (I π n ) obtained with the AMD (for 19 Λ F, that with the cluster model).

E. Energies
The ΛN interactions (V ΛN ) contains the spinindependent part (V 0 ) and the spin-dependent part (V 1 ). We take into account four spin-dependent terms of V 1 as where V σ , V SΛ , V SN , and V T are the spin-spin (σ Λ · σ N ), the Λ-spin spin-orbit (l · s Λ ), the nucleon-spin spin-orbit (l · s N ), and the tensor (S 12 ) terms, respectively. Following the notation usually used in the shell model (SM) calculations [26][27][28], we denote the contributions from V σ , V SΛ , V SN , and V T terms as the ∆ σ , S Λ , S N , and T contributions, respectively. The ΛΣ contribution from the ΛΣ coupling is ignored in the present calculation. In this section, we explain definitions of energies and respective contributions.

Total energy and excitation energy
The total energy of A Λ Z is given as We regard the sum of the first and second terms as the leading term contributed from V 0 and the third term as the perturbative term from V 1 . In the present paper, we evaluate energies of A Λ Z in two ways as follows. In the first calculation, we evaluate both the leading and the perturbative terms with the AMD model calculation as where are the nuclear energy, Λ single-particle energy, and A Λ Z wave function obtained by the AMD. This is a consistent calculation, in which the V 0 and V 1 contributions are calculated with the AMD. However, inter-cluster motion is not necessarily described sufficiently in the present AMD calculation because of the number of basis wave functions is limited as mentioned previously. The inter-cluster motion may give important contributions, in particular, to the leading term. Therefore, we also adopt an alternative way of energy evaluation by replacing the leading terms with those obtained by the CL calculation as which we call the "CL+AMD" calculation. Here E CL N and ε CL Λ,0 are the nuclear energy and Λ single-particle energy obtained by the CL with V 0 .
In both the AMD and CL+AMD calculations, excitation energies are given by the energy difference between the ground and excited states as

The spin-averaged energies
The spin-averaged energyĒA Λ Z (I π ) is defined by the energy averaged for the spin-doublet partners J π > and J π < in A Λ Z states for the core . A−1 Z(I π ) state. It is given by sum of the leading and perturbative terms corresponding to the V 0 and V 1 contributions as The perturbative term is contributed by the S N term of the spin-dependent ΛN interactions (V SN in V 1 ). Note that, in the present calculation without the spin rearrangement, the S N contribution has no Λ-spin dependence and depends only on the core nuclear spin I π . The spin-averaged binding energyBA Λ Z (I π ) is given by the sum of leading and perturbative terms as whereB Λ,0 andB Λ,SN are the V 0 and S N contributions toB Λ , respectively. The spin-averaged excitation energȳ E x (I π ) is defined as We define the spin-averaged excitation energy shift The energy shift δ Λ (Ē x ) can be separated into two components of the V 0 and S N contributions In the calculation without the core polarization, the first and second components are given by energy differences in ε Λ,0 and S N contributions, respectively, between the ground and excited states as

spin-doublet splitting energy
The splitting energy between spin-doublet J π > and J π < states is given as Here the splitting energy is defined by the energy of the J π > state measured from the J π < state, and a negative splitting energy means the reverse ordering case . In the present perturbative treatment, the splitting energy is sum of ∆ σ , S Λ , and T contributions,

A. Effective N N interactions
As for the effective N N interactions, the finite-range N N interactions of the Volkov central and G3RS spinorbit forces [66,67] are adopted. These interactions are widely used in structure studies of p-shell nuclei.  These effective N N interactions have adjustable parameters, which are usually tuned A N -and modeldependently. The interaction parameters used in the present calculation are summarized in Table I 16 O+p+n cluster model calculations [62]. The CL calculation with these default parameters of the N N interactions globally describes low-lying energy spectra in core nuclei. However, the AMD calculation with the default parameters some-times fails to reproduce the experimental energy spectra in such nuclei as 10 B and 15 O. Instead of the NN-a, we also use an alternative set (NN-a') of modified parameters w = 0.40, m = 0.60, b = h = 0.06, and u 1 = −u 2 = 1300 MeV, which have been used in the AMD calculation of 10 B [54]. For the AMD calculation of 15 O, we try the NN-c. In most cases, the modifications of N N interaction parameters give only minor changes of nuclear structures except for energy spectra. We use the label AMD for the AMD calculation with the default parameters of the N N interactions, and the label AMD' for the AMD calculation with the set NN-a' (for 16 Λ O, the label AMD' for the calculation with the set NN-c).
The parameter sets of effective N N interactions are summarized in Table I.
We use the ΛN G interactions, which are derived with the G-matrix theory from the the Nijmegen extendedsoft-core (ESC) model. The ΛN G interactions used here are even and odd central, triplet-odd spin-orbit, and triplet-odd tensor interactions as For simplicity, the Λ-N relative momentum p in the l = r × p term of the spin-orbit interactions V SΛ and V SN is approximated to be p = (p N − p Λ )/2 corresponding to the equal mass (m N = m Λ ) approximation. We start from the ESC08a version of the ΛN G interactions [45][46][47], and then consider tuning of its spindependent terms as described in later. In Ref. [47], the original ESC08a ΛN G interaction is given by three-range Gaussian local potentials. v e,o 0,σ (r) for the central interac- with the Gaussian range parameters β 1 = 0.5 fm, β 2 = 0.9 fm, and β 3 = 2.0 fm. The density dependence is taken into account by the k F parameter. The values of c 1E n,i , c 3E n,i , c 1O n,i , and c 3E n,i for the ESC08a are listed in Table  II of Ref. [47].  ) and tensor (VT ) terms in the ESC08a ΛN G interaction at kF = 1.0 fm −1 from Ref. [47].
For the spin-orbit and tensor interactions, we use the density-independent interactions fixed at The values of range and strength parameters taken from Ref. [47] are listed in Table II.
For the k F parameter in the central interactions, there are a couple of treatments. One is the density-dependent (DD) k F treatment called "averaged density approximation (ADA)", and another is the density-independent (DI) k F treatment with a fixed k F value. The ΛN G interactions generally have density dependence reflecting nuclear medium effects, which are taken into account in the G-matrix theory. ESC08 versions of the ΛN G interactions was originally designed as density-dependent interactions to globally reproduce the Λ binding energies of A Λ Z in a wide mass number region [42,43,47], whereas the DI k F treatment has been often used in studies of energy spectra of p-shell hypernuclei. In the previous works [55,58], applicability of the DD and DI k F treatments for description of p-shell A Λ Z energy spectra has been tested focusing on excitation energy shifts by Λ particle. The DD k F treatment is found to be not suitable to describe the observed excitation energy shifts in A Λ Z. The DI k F treatment can describe a trend of the excitation energy shifts but tends to overestimate the observed values. It suggests that a moderate density-dependence weaker than the DD k F treatment is favored, and therefore the intermediate version (hybrid k F treatment) between DD and DI has been proposed as an alternative k F treatment in Ref. [58]. In the present calculation, we adopt the hybrid k F treatment described as follows.
In the DD k F treatment (ADA), the k F is taken to be k F = k F Λ , where k F Λ is the averaged Fermi momentum for the Λ distribution as In the hybrid k F treatment, the average of the DD and DI interactions are used. Namely, the k F parameter is chosen to be with a weight factor e = 0.5.
Herek F is the fixed input parameter. In the DD and hybrid k F treatments, k F Λ is self-consistently determined for each state in the Λnucleus potential model. In the hybrid k F treatment, the input parameterk F is chosen for each system and taken to be the mean value of k F Λ determined by the DD k F treatment for the ground and excited states of the system. The used values of the input parameterk F in the hybrid k F treatment are shown in Table I.

IV. PROPERTIES OF CORE NUCLEI
In order to investigate energy spectra in A Λ Z, it is important that the structure models properly reproduce the structure properties such as energy spectra and radii of core nuclear states ( A−1 Z) without a Λ. In particular, nuclear spin properties of core nuclei A−1 Z are essential to discuss spin-dependent contributions of the ΛN interactions in A Λ Z. In this section, we show the calculated result for A−1 Z. In addition to energy spectra of A−1 Z, a particular attention is paid on spin configurations, which are directly reflected in spin-doublet splittings in A Λ Z.
A. Energies and radii of core nuclei A−1 Z The excitation energies calculated with the CL, AMD, AMD' and experimental values are summarized in Table  III. The calculated energy spectra depend on the adopted N N interactions as well as the structure models. The CL result reasonably describes the low-lying energy spectra of p-shell nuclei. The AMD generally gives similar results to the CL, but there are some exceptions. For example, excitation energies of 10 B(1 + ) and 15 O(3/2 − ) states are much overshot by the AMD calculation. The excitation energies of these states strongly depend on the strength of the N N spin-orbit interactions. The overshooting is improved in the AMD' calculation because of the weaker N N spin-orbit interactions in the NN-a' and NN-c than that in the NN-a. Table IV shows the comparison of root-mean-square (rms) radii (R p ) of proton distribution obtained by the CL and AMD calculations as well as the experimental values reduced from rms charge radii. The AMD calculation generally gives smaller R p values than the CL calculation because the dynamical inter-cluster motion is not sufficiently taken into account in the present AMD. On the other hand, the CL model tends to give larger R p than the AMD because the cluster breaking is not taken into account in the model. This is the major reason why we adopt the two typical nuclear models, CL and AMD, in this paper. The experimental values are found to be in between theoretical values of two calculations.
B. Spin configurations and magnetic moments of core nuclei A−1 Z Spin configurations are not so sensitive to choice of N N interactions as the energy spectra except for p 3/2shell closed nuclei such as 10 Be, 11 B, 11 C, and 12 C.
In A Λ Z with odd-odd and even-odd (odd-even) core nuclei, the spin-doublet splitting sensitively reflects the z component S z of the total nuclear intrinsic-spin S through the σ Λ · σ N term in the ∆ σ contribution. It is able to check reliability of structure models for spin properties in comparison of µ moments in A−1 Z between theory and experiment. The AMD and AMD' results for µ moments and nuclear intrinsic-spin and orbital angler momentum in nuclear states ( A−1 Z) are shown in Table  III together with the experimental µ moments. The calculation reasonably reproduces the experimental µ moments indicating that spin configurations of core nuclei are reasonably described with the AMD model.
In Z = N odd-odd nuclei, 6 Li, 10 B, and 18 F, I π = 1 + and I π = 3 + states are dominantly described by pn pairs in L = I − 1 states with the intrinsic spin S = 1, which is approximately aligned to the total nuclear spin (I) as indicated by S z ≈ 1. In particular, in 6 Li and 18 F states, α + pn and 16 O + pn structures are formed, respectively, and µ moments of 1 + , 3 + , and 5 + states are close to the values µ = 0.88, 1.88 and 2.88 µ N for the ideal S = 1 pn pairs in the L = 0, 2, and 4 states, respectively.
In even-odd and odd-even nuclei, a valence nucleon spin s = 1/2 tends to align to the total nuclear spin in I π = 3/2 − states, but the alignment is not necessarily perfect because of significant configuration mixing as seen in deviation from S z = 0.5. In I π = 1/2 − states, S z is roughly equal to s z = −1/6 for the p 1/2 single-particle contribution, but non-negligible configuration mixing is contained as indicated by the µ moments slightly deviating from the Schmidt values (µ Shchmidt = −0.79 µ N for a proton and 0.64 µ N for a neutron) except for 15 O(1/2 − ). In 15 O(1/2 − ), the spin configuration is understood by almost the pure p 1/2 hole configuration.
In Z = N = 2n nuclei, nα-cluster structures are favored. Particularly, the 8 Be(0 + , 2 + ) states have remarkable 2α-cluster structures and almost pure S = 0 components. On the other hand, the 12 C(0 + ) and 12 C(2 + ) states contain significant S = 0 components because of 3α-cluster breaking. The S = 0 mixing is sensitive to the N N spin-orbit interactions especially in 12 C(0 + ). The mixing in 12 C(0 + ) is less in the AMD' than the AMD because of the weaker N N spin-orbit interaction. Compared with 12 C(0 + ), the S = 0 component in 12 C(2 + ) is relatively small and not so sensitive to the N N interactions, but it still gives non-negligible ∆ σ and T contributions to the spin-doublet splitting in 13 Λ C spectra as discussed later.
In this subsection, we show calculated results of averaged structure properties of A Λ Z states obtained with the leading part V 0 (spin-independent) of the ΛN interactions without the perturbative part V 1 . Table V shows the calculated results of the Λ binding energies (B Λ,0 ), rms radii of the Λ and nuclear distributions (r Λ and R N ), and averaged Fermi momentum ( k F Λ ) obtained with the CL and AMD. The experimental data of the the Λ binding energy (B Λ ) and spinaveraged one (B Λ ) are shown for comparison. The systematics of the observed Λ binding energies is reasonably described by the leading part V 0 of ECS08a(Hyb) interaction. The model dependence ofB Λ,0 between the CL and AMD is not so strong. Quantitatively, the Λ binding is slightly deeper in the AMD than the CL except for 12 Λ B(3/2 − gs ), 12 Λ C(3/2 − gs ), and 13 Λ C(0 + ), because the AMD tends to give smaller nuclear radii R N , i.e., the higher nuclear density contributing the deeper Λ-core potential.
The Λ distribution size (r Λ ) is larger than the nuclear matter distribution size (R N ) in light-mass A Λ Z because of small Λ binding energies. With increase of the mass number A, r Λ becomes gradually small as the Λ binding becomes deep. The nuclear matter radii R N increase with the increase of A, and in heavy-mass p-shell A Λ Z it is eventually as large as r Λ . Densities of Λ and nuclear distributions are shown in Fig. 1. The mass number dependence of the Λ distribution is very mild compared with that of nuclear matter distributions.

B. Tuning of spin-dependence of ΛN interactions
The ∆ σ term of the ESC08(a,b) ΛN G interaction is known to be inappropriate to describe the observed spindoublet splitting energies. For example, the ESC08a gives the reverse ordering of the J > and J < states in 12 Λ C and 11 Λ B for the ground state core nuclei inconsistently with the experimental observation as pointed out in Ref. [47]. Other spin-dependent terms of the ESC08a ΛN G interaction have not been well tested yet. In the present work, we phenomenologically tune the spindependent terms of the ESC08a ΛN G interaction by modifying the original strength parameters to describe energy spectra in p-shell Λ hypernuclei as follows.
• For the spin-independent term V 0 , the original parameters are used.
• The ∆ σ term (V σ ) is adjusted so as to globally describe the 3/2 − -1/2 − and 7/2 − -5/2 − splittings in 7 Λ Li and 11 Λ B, which are dominantly contributed by the ∆ σ term because of the total nuclear intrinsicspin S = 1 component. The even and odd parts of V σ are multiplied by factors of 2 and 0.3, respectively.
• For the T term (V T ), the observed value of the 1 − -0 − splitting in 16 Λ O is used as an input parameter for tuning. In the 1 − -0 − splitting, the T contribution is relatively large compared with other systems and cancels the ∆ σ contribution. The strength of V T is multiplied by a factor of 6 to fit the small 1 − -0 − splitting observed in 16 Λ O. • The spin-orbit terms (V SΛ and V SN ) are not modified. These contributions in splitting energies are generally small and it is difficult to definitely determine these terms without ambiguity from existing data, and therefore, these terms are tentatively left as they are. However, it is likely that a larger V SΛ term with a factor of ∼ 2 than the original one is favored to reproduce the 5/2 + -3/2 + splitting in 9 Λ Be.
As a result, above modifications v(r) →ṽ(r) are expressed asṽ with f e σ = 2, f o σ = 0.3, f SΛ = f SN = 1, and f T = 6. We call the ESC08a ΛN G interaction with thus modified spin dependence (msd) "ESC08a-msd". In the following sections, we first show the crucial problem of the original spin-dependent interactions of ESC08a in reproducing the experimental spin-doublet splitting energies and how the results are improved by the modified interactions, ESC08a-msd. Then, we discuss the details of energy spectra in A Λ Z based on the calculations with the ESC08a-msd, which we use as the default ΛN interactions unless otherwise specified.

C. Spin-doublet splitting energies: general features
The spin-doublet splitting energies and respective contributions calculated with the original ESC08a and ESC08a-msd interactions are shown in Table VI together with the experimental data. Significant splittings have been experimentally observed for such core nuclear states as A−1 Z(1 + 1 ) and A−1 Z(3 + 1 ) in Z = N odd-odd nuclei and A−1 Z(3/2 − gs ) and A−1 Z(5/2 − 1 ) in even-odd and oddeven core nuclei. However, the original ESC08a ΛN G interaction gives opposite-sign splitting energies, namely, the reverse ordering of the spin-doublet partners because opposite-sign contributions from the odd term of V σ dominate the ∆ σ contributions. It is a crucial problem of the spin dependence of the original ESC08a as pointed out in Ref. [47]. In contrast, the spin-doublet splitting energies are reasonably reproduced by the ESC08a-msd with the modified spin-dependence of the ΛN interactions. The significant splittings observed for A−1 Z(1 + 1 ) and A−1 Z(3 + 1 ) of Z = N odd-odd nuclei, and A−1 Z(3/2 − gs ) and A−1 Z(5/2 − 1 ) of even-odd and odd-even core nuclei are described by the dominant ∆ σ contributions in the present AMD calculation with the ESC08a-msd. T contributions are usually small compared with dominant ∆ σ contributions except for core A−1 Z(1/2 − ) states in oddeven and even-odd nuclei, in which the T and ∆ σ contributions are comparable order and almost cancel with each other. The S Λ term gives minor contributions in general.
The present result of splitting energies is compared with SM calculations in Table VII. Each contribution is also compared with that of the SM calculation by Millener et al. [28]. In general, the ∆ σ , S Λ , and T contributions obtained in the present calculation are similar to those of the Millener's SM calculation. This is a natural consequence because, in both calculations, spin-dependent contributions are phenomenologically adjusted to fit the observed splitting energies in p-shell A Λ Z. The present AMD calculation with the ESC08a-msd interaction, reasonably describes the global feature of observed data of spin-doublet splittings in p-shell A Λ Z though agreements with the experimental value are not so precise as the Millener's SM calculation [28]. It should be commented that, in the Millener's SM calculation, the spin-dependent ΛN interaction parameters are independently adjusted to the light-and heavy-mass pshell regions, whereas in the present calculation the massnumber independent ΛN interactions are used.
Let us discuss the N N interaction dependence of splitting energies. In Table VIII, we compare the AMD result (the stronger N N spin-orbit interaction) and the AMD' result (the weaker N N spin-orbit interaction). The difference of the N N spin-orbit interaction causes slight difference in the nuclear intrinsic-spin configurations. Generally, weaker N N spin-orbit interaction enhances the LS-coupling component and reduces the jjcoupling component (the cluster breaking). However, in most cases, the N N interaction difference between the AMD' and AMD gives only minor difference in the splittings because nuclear intrinsic-spin configurations are not so sensitive to the N N interactions as shown in Table III. D. Spin-doublet splitting energies: characteristics in cases of odd-odd, even-odd(odd-even), and even-even core nuclei We here discuss characteristics of splitting energies in cases of odd-odd, even-odd(odd-even), and even-even core nuclei based on the AMD results with the ESC08amsd in Table VI  In the case of A Λ Z with Z = N odd-odd core nuclei, one of the characteristics of T = 0 states is remarkably large splitting energies because of the dominant nuclear intrinsic-spin S = 1 component contributed by two valence nucleons, a proton and a neutron. As shown in Table III, the core nuclear states, I π = 1 + , 3 + , and 5 + have S = 1 and T = 0 pn pairs in the L = 0, 2, and 4 waves as dominant components, respectively. The aligned nuclear intrinsic-spin S z ≈ 1 provides the significant ∆ σ contribution in the splitting.
The mass number dependence of ∆ σ comes from the difference in spatial overlap between valence nucleon and Λ orbits. In 19 Λ F, the 0s-orbit Λ has a smaller overlap with the sd-orbit proton and neutron than in 7 Λ Li and 11 Λ B having the valence proton and neutron in the p-shell.
In each A Λ Z system, higher J states tend to have smaller splittings because of negative contributions from the T and S Λ terms. The splittings in 7 Λ Li have been investigated in details by the OCM (semi-microscopic) cluster model calculation with the NSC97f ΛN G interactions [14]. In the OCM calculations with the NSC97f, the ∆ σ contribution in the 3/2 + -1/2 + (7/2 + -5/2 + ) splitting is ∆ σ = 0. 71 The present result is qualitatively consistent with but quantitatively slightly smaller than the OCM calculation and also slightly underestimates the experimental data. For more precise reproduction of the splittings in 7 Λ Li, higher order effects beyond the s-wave Λ approximation should be taken into account.
For 19 Λ F, the AMD calculation gives the 3/2 + -1/2 + splitting of 0.249 MeV, which reasonably agrees with the experimental value 0.315 MeV recently observed by the γ-ray measurement [4]. For the 7/2 + -5/2+ and 11/2 + -9/2+ splittings in 19 Λ F, the present calculation predicts smaller splittings than the 3/2 + -1/2 + splitting because of the cancellation of the dominant ∆ σ contribution by opposite-sign T and S Λ contributions. Our result is consistent with Millener's SM prediction but seems appreciably different from the prediction estimated with NSC97f G-matrix interaction by Umeya et al. [90]. In A Λ Z with even-odd and odd-even core nuclei, the 2 − -1 − and 3 − -2 − splittings for core I π = 3/2 − and 5/2 − states are moderate. They are mainly contributed by the valence nucleon spin S = 1/2 in the p-shell through the ∆ σ term. The exception is the splitting for the core 7 Li(5/2 − ) state, which is described by the t cluster orbiting in the L = 3 wave around the α cluster. In the I π = 5/2 − state, the t-cluster intrinsic-spin S = 1/2 is coupling with the L = 3 with the anti-parallel orientation, and gives a negative contribution to the splitting. As a result, the ordering of the 3 − and 2 − states are reverse in 8 Λ Li. As for the splittings in 10 Λ Be, our result is similar to the SM predictions in Refs. [28,88]. On the other hand, the four-body OCM cluster model calculation [16] gives quite small values in 10 Λ Be as 0.08 MeV and 0.05 MeV for the 2 − -1 − and 3 − -2 − splittings, which are inconsistent with our result and SM predictions.

Case of even-odd and odd-even core nuclei: splittings in
In 8 Λ Li, 10 Λ Be, 12 Λ B, 12 Λ C, and 16 Λ O, the 1 − -0 − splittings for core I π = 1/2 − states are remarkably small because significant negative contributions of the tensor (T ) term cancels the ∆ σ contribution. As pointed out by Millener, this cancellation is essential to describe the small 1 − -0 − splitting observed in 16 Λ O. Indeed, the Millener's SM calculation predicts the small 1 − -0 − splittings in 8 Λ Li, 10 Λ Be, 12 Λ B, and 12 Λ C, and our result is consistent with it. The 1 − -0 − splitting is experimentally known only for 16 Λ O but not for other A Λ Z. In both the present and Millener's SM calculations, the value for 16 Λ O is used as an input data in phenomenological tuning of the spindependent ΛN interactions. In order to check validity of the tensor term of the spin-dependent ΛN interactions, experimental data for other systems are requested.
Let us turn to the 2 − -1 − splitting in 12 Λ B( 12 Λ C) for the excited core, I π = 3/2 − 2 state. The nuclear intrinsicspin and orbital angular momentum coupling in the 11 B(3/2 − 2 )( 11 C(3/2 − 2 )) state is quite different from that in the ground state, I π = 3/2 − 1 (cf. Table III). The I π = 3/2 − 2 state dominantly contains the nuclear orbital angular momentum L = 2 excitation coupled with a porbit valence nucleon as [p 3/2,1/2 ⊗ L = 2] J=3/2 as indicated by the large L 2 in Table III. Because of the significant coupling with L = 2, the nuclear intrinsic-spin of the p 3/2 valence nucleon is not aligned to the I direction, and therefore, gives a relatively small ∆ σ contribution to the 2 − -1 − splitting in 12 Λ B( 12 Λ C). On the other hand, the T and S Λ terms give negative contributions. As a result, the AMD (AMD') calculation give the negative values −0.045(−0.071) and −0.047(−0.073) MeV of the total 2 − -1 − splittings namely, the reverse ordering for the core states 11 B(3/2 − 2 ) and 11 C(3/2 − 2 ), respectively (cf. Tables VI and VIII). The experimental value of this splitting has not been determined yet. The 1 − state at 6.050 MeV in 12 Λ C has been determined by the γ-ray measurement [87], whereas the 1 − and 2 − states in 12 Λ B are not separated but both are included in the peak observed at 5.92 ± 0.13 MeV in the (e, e ′ K + ) reaction experiment [77]. Provided that the Coulomb shift between mirror states in 12 Λ C-12 Λ B is the same as that in 11 C-11 B, the 2 − -1 − splitting for I π = 3/2 − 2 is estimated to be −0.215 MeV. The reverse ordering is consistent with the AMD prediction, but quantitatively the agreement is not so good. The negative splitting (reverse ordering) is also predicted by the SM calculation (theoretical value is −0.122 MeV) [89]. More detailed experimental spectra are demanded to determine the splitting energy.
3. Case of even-even Z = N core nuclei: 5/2 + -3/2 + splitting in 9 Λ Be and 13 Λ C For 9 Λ Be and 13 Λ C, we discuss spin-dependent contributions in the 5/2 + -3/2 + splittings for the excited core states, 8 Be(2 + 1 ) and 12 C(2 + 1 ). The splittings are generally small because nα-cluster structures are favored and the nuclear intrinsic-spin is almost saturated. In 9 Λ Be, since the core state 8 Be(2 + 1 ) has the ideal 2α-cluster structure with the relatively L = 2 wave, the ∆ σ and T contributions almost vanish and only the S Λ term contributes to the splitting. It means that the 5/2 + -3/2 + splitting in 9 Λ Be can be a sensitive probe to test the S Λ term of the ΛN interactions. A tiny splitting −0.023 MeV is obtained in the AMD calculation. In the present calculation, the strength of the S Λ term is not modified. If the strength is tuned to fit the observed value −0.043, a slightly stronger S Λ term by a factor of ∼ 2 is favored. This modification of the S Λ term gives only minor effects to the splittings in other systems.
In 13 Λ C, the core state 12 C(2 + 1 ) contains the slight S = 0 component because of 3α-cluster breaking as can be seen in non-zero expectation value S 2 = 0 in Table III. The S = 0 component from the cluster breaking gives nonnegligible ∆ σ and T contributions, which cancel the negative S Λ contribution in the 5/2 + -3/2 + splitting (see Table VIII). Consequently, the predicted 5/2 + -3/2 + splitting in 13 Λ C is a small positive value, 0.064 MeV in the AMD and 0.026 MeV in the AMD' result. The difference between the AMD and AMD' results originates in the nuclear spin-orbit interaction dependence of the cluster breaking in the core state. If the twice stronger S Λ interaction adjusted to the 5/2 + -3/2 + splitting in 9 Λ Be is adopted, further cancellation occurs in 13 Λ C. The ΛN spin-orbit splittings in 9 Λ Be and 13 Λ C have been investigated by the 2α-and 3α-cluster OCM calculations [12]. In the OCM cluster model calculations, the nuclear intrinsic-spin completely vanishes and only the S Λ term contributes to the 5/2 + -3/2 + splittings because α clusters are assumed. The cluster OCM calculations with the NSC97f predicted the negative 5/2 + -3/2 + splittings as −0.16 MeV in 9 Λ Be and −0.29 MeV in 13 Λ C. Compared with the later observed value −0.043 MeV in 9 Λ Be, the prediction suggests that the S Λ term of the NSC97f interaction may be too strong.
E. SN contributions to energy spectra and BΛ

Excitation energy shifts
In the present perturbative treatment of the spindependent part V 1 of the ΛN interactions, the S N term gives no contribution to the spin-doublet splitting but contributes to the spin-averaged excitation energy shift δ Λ (Ē x (I π )). The spin-averaged excitation energy shift originates in the nuclear structure difference between the ground and excited states, and is given by two contributions, the V 0 contribution (δ Λ,0 (Ē x )) and the S N contribution (δ Λ,SN (Ē x )), as explained in Eq. (32). The V 0 contribution reflects mainly the difference in the core nuclear size, whereas the S N contribution is sensitive to that in the nuclear intrinsic-spin and orbital angular momentum configurations. Table IX shows the V 0 and S N contributions in δ Λ (Ē x ) as well as total values obtained by AMD and CL+AMD calculations together with the experimental data. In both the calculations, the S N contributions are calculated with the AMD except for 19 Λ F as explained previously. The expectation values V SN calculated with the AMD are also shown in the table. In Fig. 2, the V 0 contributions and the total excitation energy shifts are shown compared with observed values of δ Λ (Ē x ). As shown in the table and figure, the CL calculation tends to give larger V 0 contributions than the AMD calculation because it gives larger size difference between the ground and excited states. The S N contributions are relatively minor in light-mass nuclei, whereas they are comparable or even larger than the V 0 contributions in heavy-mass nuclei. Both the AMD and CL+AMD calculations qualitatively describe systematic behavior of the experimental δ Λ (Ē x ) in p-shell A Λ Z. Quantitatively, the CL+AMD (AMD) calculation more or less overestimates (underestimates) the observed data in the A ≥ 11 region.
In Figs. 2(c) and (d), we also show the CL+AMD' result of δ Λ (Ē x ) to see the N N spin-orbit interaction dependence. Note that the difference between the CL+AMD and CL+AMD' calculations is the difference in the S N contribution between the AMD and AMD' calculations. In most states except for 12 Λ B(I π = 3/2 − 2 ) and 13 Λ C(I π = 2 + ), difference between two calculations is rather small indicating that the dependence in the excitation energy shift is minor. However, significant difference is found in 12 Λ B(I π = 3/2 − 2 ) and 13 Λ C(I π = 2 + ), in which nuclear intrinsic-spin configurations are sensitive to the N N spin-orbit interaction as seen in Table IX. In these states, the S N contribution is smaller in the AMD' calculation because of the less cluster breaking than the AMD calculation.
Let us discuss the 16 O core vibration effect to the V 0 contribution in 16 Λ O and 19 Λ F, which are taken into account in the CL calculation of 16 Λ O and the CL-4α calculation of 19 Λ F. For 19 Λ F, the CL-4α result is compared with the CL one, and for 16 Λ O, the AMD result is compared with the CL+AMD one in Table IX. In both systems, the 4α vibration gives only minor effect in the total excitation energy shifts.

SN contributions to binding energiesBΛ
The S N term of the ΛN interactions also contributes to the Λ binding energiesB Λ . In particular, spin-orbit favored states in core nuclei gain much potential energy because the 0s-orbit Λ enhances the single-nucleon (mean) spin-orbit potential through the l · s N term in V SN . Its expectation value in the ground state is nothing but the S N contribution to the Λ binding energies as B Λ,SN = − V SN . As expected, the S N term gives nonnegligible contributions toB Λ for the spin-orbit favored ground states such as 11 Λ Be(I π = 0 + ), 11 Λ B(I π = 3 + ), 12 Λ B(I π = 3/2 − gs ), 12 Λ C(I π = 3/2 − gs ), and 13 Λ C(I π = 0 + ), whereas it gives minor contributions to well clustered states in light-mass A Λ Z (see Table IX). There remains ambiguity in the result because the S N contribution depends on the N N spin-orbit interaction. For 11 Λ Be(I π = 0 + ), 11 Λ B(I π = 3 + ), 12 Λ B(I π = 3/2 − gs ), 12 Λ C(I π = 3/2 − gs ), and 13 Λ C(I π = 0 + ), the AMD' result of V SN is about half of the AMD result as V SN = −0.27, −0.36, −0.38, −0.38 and −0.30 in MeV, respectively. Another ambiguity comes from the V SN term of the ΛN interactions, which has not been checked yet in the present work. In the spin-averaged energy spectra in 7 Λ Li and 11 Λ B, the excitation energy of 6 Li(3 + ) is shifted downward in 7 Λ Li(3 + ), whereas that of 10 B(1 + ) is shifted upward in 11 B(1 + ) because of the V 0 and S N contributions (cf. Table IX). In the energy spectra in 7 Λ Li and 11 Λ B, the significant spin-doublet splitting occurs mainly because of the ∆ σ term contributed by S = 1 pn pairs around the αand 2α-cluster structures, respectively.
The present result of excitation energy shifts and spindoublet splitting energies in 7 Λ Li and 11 Λ B are qualitatively consistent with the experimental spectra. Strictly speaking, however, the agreement with the energy spectra is not perfect. For example, the excitation energies of 7 Λ Li(7/2 + ) and 7 Λ Li(5/2 + ) are underestimated, in particular, by the CL calculation. The reason may be that the s-orbit Λ approximation is not enough for the core nucleus 6 Li having the remarkable α + d cluster structure, and may overestimate the energies of 7 Λ Li(3/2 + ) and 7 Λ Li(1/2 + ) for the core 6 Li(1 + ) state. In order to discuss detailed energy spectra in 7 Λ Li, more precise calculations with microscopic three-body or four-body cluster models are needed needed as has been tried in Refs. [11,14].

Energy spectra of 12 Λ C
Available data of energy spectra in 12 Λ C are reasonably reproduced by the calculations. The excitation energies for 11 C(1/2 − ), 11 C(5/2 − ), and 11 C(3/2 − 2 ) are significantly raised because of the stronger binding energy between Λ and 11 C(3/2 − gs ) (cf. Table V). The spin-doublet splittings for the core 11 C(3/2 − gs ) and 11 C(5/2 − ) are moderate because, as a leading oder, one valence nucleon spin contributes to the splittings. The spin-doublet splittings have not been measured yet except for the 2 − 1 -1 − 1 splitting in 12 Λ C. For the core excited state, 11 C(3/2 − 2 ), the negative splitting energy, i.e., the reverse ordering of the spin-doublet states is predicted. The present prediction is supported by the experimental excitation energy of the mirror 2 − state in 12 Λ B measured by the production cross section analysis [77]. 16 Λ O

Energy spectra of
The experimental spectra in 16 Λ O are reproduced well by the calculations. The spin-averaged excitation energy for the core 15 O(3/2 − ) state is shifted upward by the Λ mainly because of the S N contribution. The V 0 contribution to the excitation energy shift is minor. The AMD' and CL+AMD results are similar indicating that the vibration effect in the 16 O core is minor because the 15 O structure is rather robust differently from fragile cluster structures of light-mass p-shell nuclei. The spin-doublet splittings for the core states, 15 O(1/2 − ) and 15 O(3/2 − ), are tiny and moderate, respectively, reflecting the nuclear spin configurations in the core nucleus.
Let us discuss the detail of the excitation energy E x (5/2 + ) in 19 Λ F. The experimental energy shift −0.042 MeV of 19 Λ F(5/2 + ) is reduced from the observed E x (5/2 + ) = 0.895 MeV in 19 Λ F and E x (3 + )=0.937 MeV in 18 F. The calculated energy shift is −0.03 (−0.07) MeV in the CL (CL 4α +CL) calculations. The result reasonably agrees with the experimental data. In the CL calculation, the S N and V 0 contributions are −0.05 MeV and −0.04 MeV, respectively. In addition, the spin-doublet splitting energy gives positive contribution of +0.07 MeV to the excitation energy shift because it causes larger energy gain in the ground 19 Λ F(1/2 + ) state than the 19 Λ F(5/2 + ) state. For more detailed discussion, the experimental measurement of the excitation energy for the spin-doublet partner 19 Λ F(7/2 + ) is highly requested.

VI. SUMMARY
Energy spectra of 0s-orbit Λ states in p-shell Λ hypernuclei ( A Λ Z) and those in 19 Λ F were studied with the AMD+VAP and microscopic cluster model using the ΛN G interactions. The spin-dependent terms of the ESC08a ΛN G interaction were tested in comparison of the calculated energy spectra with the observed ones. A modification of the spin-dependence of the ESC08a ΛN G interaction was proposed by phenomenological tuning of the spin-spin (σ Λ · σ N ) and tensor (S 12 ) terms to adjust available data of energy spectra in p-shell A Λ Z. The spin-dependent contributions of the ΛN interactions to spin-doublet splittings were discussed. In the case of odd-odd, even-odd, and odd-even core nuclei, the ∆ σ contribution is usually dominant, whereas the T and S Λ contributions are relatively minor in most cases. There are some exceptions such as I π = 1/2 − states in even-odd and odd-even core nuclei, in which the significant tensor contribution cancels the ∆ σ contribution. In 13 Λ C, the cluster breaking component gives non-negligible contributions to the splitting energy for the core 12 C(2 + ) state through the ∆ σ and tensor terms of the ΛN interactions. The V 0 and S N contributions to the excitation energy shifts were also discussed. Calculated energy spectra as well as spin-averaged energy spectra in A Λ Z were compared with experimental data. The calculations reasonably reproduce the observed spectra in p-shell A Λ Z and 19 Λ F. The extensive data of p-shell A Λ Z observed by highresolution γ-ray measurements are useful information to obtain comprehensive understanding of the energy spectra and structures of Λ hypernuclei. Moreover, they are useful to test the effective ΛN interactions in hypernuclei. In particular, the spin-doublet splitting is good probe to check the spin-dependence of the ΛN interactions. In the present systematic investigation of energy spectra in A Λ Z, we proposed the modified spin-dependent ΛN interactions which can reasonably reproduce the observed spin-doublet splitting energies in p-shell A Λ Z. The present work may shed a light on spin-dependence of the effective ΛN interactions in p-shell A Λ Z and enable us to predict spectra for unobserved excited states. In order to explore such systematic investigations of Λ hypernuclei in a wide mass number region, further γ-ray spectroscopic studies of Λ hypernuclei in heavier mass regions are requested.
In the present work, two models, the AMD+VAP and microscopic cluster models, were adopted. The former is useful to treat nuclear intrinsic-spin configurations in detail, whereas the latter is suitable to describe the dynamical inter-cluster motion. In order to investigate energy spectra in A Λ Z precisely, further advanced frameworks that can describe details of intrinsic-spin configurations as well as dynamical structure change are needed. The HAMD method is one of the promising tools. The ambiguity in the effective N N interactions is also a remaining problem to be solved. In the present calculation, the Λ wave functions are obtained by folding the spin-independent central term (V 0 ) of the ΛN interactions. The nuclear density matrix in the exchange potential is approximated with the den-sity matrix expansion in the local density approximation [63]. We discuss here its validity of the approximation for energy spectra in 7 Λ Li. In Table X, the approximated Λ-potential energy φ Λ,0 |U 0 (ρ I π N )|φ Λ,0 is compared with the microscopically calculated energy ΨA Λ Z (J π )|V 0 |ΨA Λ Z (J π ) . Here, ΨA Λ Z (J π ) = [Ψ N (I π n )φ Λ,0 χ Λ ] J is the microscopic A-body wave function for A Λ Z and φ Λ,0 (ρ I π N ; r) is fixed to be that obtained by the folding potential model. It should be noted that, in the folding potential model, the nuclear density ρ I π N is defined for intrinsic wave functions of A−1 Z without the cm motion, and the Λ recoil effect is properly taken into account. However, the microscopic Abody wave function ΨA Λ Z contains the cm motion. For consistency, we also perform the approximated and the microscopic calculations of the Λ-potential energy by using the nuclear density ρ I π N,cm with the cm motion instead of ρ I π N without the cm motion. As shown in the table, errors of the approximation are only < 7% and < 2% in the Λ-potential energy calculated with ρ I π N and ρ I π N,cm , respectively. Moreover, the errors are almost state-independent and give only global shifts meaning that the approximation gives minor effect to energy spectra.
Appendix B: Core rearrangement effect to spin-doublet splitting energy In the present perturbative treatment of the spindependent part (V 1 ) of the ΛN interactions, the nuclear spin rearrangement is ignored. Generally, its effects are expected to be minor because V 1 is relatively weak compared with the N N interactions and also the spinindependent part (V 0 ) of the ΛN interactions. Possible exception is the case that two energy levels with the same J π eventually exist close to each other. In order to see nuclear spin rearrangement effects, we calculate the energy spectra of 8 Λ Li, 10 Λ Be, 12 Λ C, and 16 Λ O with the AMD by diagonalization of the full Hamiltonian including V 0 and V 1 terms of the ΛN interactions. The spin-doublet splittings calculated with and without the rearrangement are compared in Table XI. One can see that the rearrangement effect is minor in most of states.      VII: Spin-doublet splittings in A Λ Z for core A−1 Z(I π ) states obtained by the AMD calculation with the ESC08a-msd compared with the experimental and SM calculations. The ∆σ, SΛ, T contributions, and total splittings are listed together with the Millener's SM calculation [28]. ΛΣ and SN contributions in the Millener's calculation are also shown. Other SM calculations are taken from Refs. [88][89][90]. Information of the experimental values is explained in the caption of    The total energy shift (δΛ(Ēx) = δΛ(Ēx) + δΛ,0(Ēx)) and the V0 contribution (δΛ,0(Ēx)) obtained by the AMD calculation. (c)(d) The total energy shift obtained by the CL+AMD calculation, and the V0 contribution obtained by the CL calculation. In (c) and (d), the total energy shift obtained by the CL+AMD' calculation is also shown by green cross points. For 19 Λ F, the total energy shift and V0 contribution obtained by the CL calculation are plotted by red filled circles and blue filled triangles, respectively. Information of the experimental data is explained in the caption of Table IX.   [2,85,87]. As for the 12 Λ C(J π = 2 − ) state for the core 11 C(3/2 − 2 ) state, the observed energy of the mirror state in 12 Λ B [77] is plotted assuming the Coulomb shift between the mirror states in 12 Λ C-12 Λ B is the same as that in 11 C-11 B. X: Comparison of the Λ potential energy in 7 Λ Li(I π = 1 + , 3 + ) between the folding potential model approximation and the microscopic calculation. The energies calculated using the nuclear densities ρ I π N and ρ I π N,cm (without and with the cm motion) are shown. The difference between approximated and microscopic calculations is also shown. Energies are in MeV.