Sensitivity of Λ single-particle energies to the ΛN spin-orbit coupling and to nuclear core structure in p-shell and sd-shell hypernuclei
aa r X i v : . [ nu c l - t h ] M a y Sensitivity of Λ single-particle energies to the Λ N spin-orbit coupling and to nuclear core structure inp-shell and sd-shell hypernuclei P. Vesel´y , E. Hiyama , J. Hrt´ankov´a and J. Mareˇs Nuclear Physics Institute, Czech Academy of Sciences, 250 68 ˇReˇz, Czech Republic RIKEN Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan
Abstract
We introduce a mean field model based on realistic 2-body baryon interac-tions and calculate spectra of a set of p -shell and sd -shell Λ hypernuclei - C, O, Ne,
Si and
Ca. The hypernuclear spectra are compared withthe results of a relativistic mean field (RMF) model and available experi-mental data. The sensitivity of Λ single-particle energies to the nuclear corestructure is explored. Special attention is paid to the effect of spin-orbit Λ N interaction on the energy splitting of the Λ single particle levels 0 p / and0 p / . In particular, we analyze the contribution of the symmetric (SLS) andthe anti-symmetric (ALS) spin-orbit terms to the energy splitting. We givequalitative predictions for the calculated hypernuclei. Keywords:
Λ hypernuclei, spin orbit splitting, Λ N interacion, mean fieldmodel
1. Introduction
One of the major goals in hypernuclear physics is to obtain information onbaryon-baryon interactions in a unified way. However, due to the limitation ofhyperon ( Y ) nucleon ( N ) scattering data, the Y N potential models proposedso far, such as the Nijmegen models, have a large degree of ambiguity. There-fore, quantitative analyses of light Λ hypernuclei, where the features of Λ N interactions appear rather clearly in observed level structures are indispens-able. For this purpose, accurate measurements of γ -ray spectra [1, 2, 3, 4, 5]and high resolution ( π + , K + ) reaction experiment [6] have been performedsystematically. These experiments are source of invaluable information about Preprint submitted to Nuclear Physics A October 1, 2018 pin-dependent components of Λ N interaction. Useful constraints on the Y N interaction components have been provided by shell model [7] and few-bodycluster [8, 9] calculations. Among the
Y N spin-dependent components, spin-orbit terms are important, since they are intimately related to modeling theshort-range part of the interaction. For example, it is well known that theantisymmetric spin-orbit (ALS) forces come out qualitatively different in one-boson-exchange (OBE) models [10, 11, 12, 13] and in quark models [14, 15].It was pointed out that the ALS force based on quark model [15] is as strongas to cancel the symmetric spin-orbit (SLS) force, while the ALS force ofNijmegen potentials [10, 11, 12, 13] is of a smaller strength. To extract in-formation on these spin-orbit forces, specific γ -ray experiments [2, 3] and( π + , K + ) reaction experiments [6] have been performed. In the γ -ray experi-ments, spin-orbit dominated energy splittings of 43 ± / +1 and5 / +1 doublet of levels in Be [3] and 152 ± ±
36 keV for the 1 / − − / − doublet in C [2] were measured. Considerably larger energy splittings of1 . ± .
20, 1 . ± .
14 and 1 . ± . ± .
10 MeV were reported from theanalysis of the p -orbit, d -orbit and f -orbit peaks, respectively, observed inthe Y spectrum of the ( π + , K + ) reaction [6]. However, these splittings aremost likely caused by core-excited configurations and have little to do withthe spin-orbit interaction [16].The strength of the spin-orbit forces was studied within few-body [17]and shell model calculations [18] of Be and
C. Reasonable strengths wereintroduced to reproduce the experimental data. It is worth mentioning thatsmall energy splittings for Be and
C were predicted by Hiyama et al.[17] using ALS forces based on a quark model. On the other hand, it waspointed out in the shell-model calculation by Millener [19] that a Λ N tensorcontribution was as important as spin-orbit for these energy splittings.More information about spin-orbit interaction in other Λ hypernuclei iscertainly needed. For this purpose, it is planned to explore the level structureof some medium heavy Λ hypernuclei at JLab. Our aim in the present paperis to study the spin-orbit doublet states p / and p / in Λ hypernuclei builton nα nuclear cores. It is expected that the spin-spin and tensor termsof the Λ N interaction are not effective as a consequence of the α -clusterdominating structure, and we can thus safely assess Y N spin-orbit forces.However, if α clusters in these hypernuclei are broken, the spin-spin and thetensor interactions might contribute to the p / and p / energy splitting aswell. In this paper, we investigate spin-spin and spin-orbit contributions tothis energy splitting and give qualitative predictions of the splittings in C,2 O, Ne,
Si and
Ca. For this aim we apply the ESC08c potential, whichhas been recently proposed by Nijmegen group, using a mean field approachbased on realistic baryon interactions [20]. In the next section, we brieflydescribe our method. Selected results are presented and discussed in thethird section, and a summary is given in the final section.
2. Methodology
We calculate spectra of single Λ hypernuclei using a Hamiltonian of theform H = T + V NN + V Λ N − T CM , (1)where T = P Aa =1 p a / m is the kinetic term, V NN ( V Λ N ) denotes interactionamong nucleons (Λ and nucleons) and T CM = ( P Aa =1 p a + P Aa = b ~p a .~p b ) / mA is a center of mass term. In our calculations we use a realistic Λ N interactionand a realistic N N interaction corrected by the density dependent term whichsimulates 3 N force. If we introduce the creation (annihilation) operators a † i ( a i ) for nucleons and c † i ( c i ) for Λ, we can express the Hamiltonian (1) in theformalism of second quantization H = X ij t Nij a † i a j + X ij t Λ ij c † i c j + 14 X ijkl V NNijkl a † i a † j a l a k + X ijkl V Λ Nijkl a † i c † j c l a k , (2)with the kinetic matrix elements t ij = (cid:18) − A (cid:19) h i | p m | j i , (3)antisymmetrized N N interaction matrix elements V NNijkl = h ij | (cid:18) V NN − ~p .~p mA (cid:19) | kl − lk i , (4)and Λ N interaction matrix elements V Λ Nijkl = h ij | (cid:18) V Λ N − ~p .~p mA (cid:19) | kl i , (5)all expressed in the harmonic oscillator basis. The harmonic oscillator basisdepends on one parameter ~ ω HO which defines the oscillator lengths b N and b Λ for the wave functions of nucleons and Λ, respectively, due to the relations b N = s ~ c m N c ~ ω HO , b Λ = s ~ c m Λ c ~ ω HO . (6)3n basis which is large enough the physical results do not depend on ~ ω HO .In our calculations we use N max = 10 and ~ ω HO = 16 MeV.The mean field is constructed in our model as follows. First we solve thenuclear part of Hamiltonian (2) ( P i t Nij a † i a i + P ijkl V NNijkl a † i a † j a l a k ) withinthe Hartree-Fock approximation. As a result we obtain the wave function | HF i as well as the nuclear density ρ Nlk = h HF | a † k a l | HF i of the nuclear core ofa hypernucleus. Then we calculate the single-particle energies e Λ i by diago-nalizing the matrix ( t Λ ij + U Λ Nij ) where U Λ Nij = X τ = p,n X kl V Λ Nkilj ρ Nlk , (7)assuming that the Λ hyperon interacts with the mean field of the core nucleus.The hypernuclear wave function at the level of mean-field approximation isdefined as | i i = c † i | HF i .In our model, we used the realistic N N interaction NNLO opt [21]. It isa chiral next-to-next-to leading order potential with parameters optimizedto minimize the effect of three-body interactions (their effect, however, stillremains relatively important), which makes this force useful for many-bodycalculations (for more details about the optimization procedure see [21]).Matrix elements of this interaction were generated by the CENS code [22].However, the effect of three-body forces is still not negligible. If we performthe calculations purely with the two-body
N N interactions we do not obtaincorrect distribution of the nuclear density. The nuclear density distributionis more compressed which leads to much smaller nuclear rms radii than arethe experimental values. In general, this has influence on the single particleenergies of Λ, particularly on the splitting between the 0 s and 0 p states ascan be deduced from the Bertlmann-Martin inequalities [23]. For this reasonwe add a corrective density dependent (DD) term of the form V NN,DD = C ρ P σ ) ρ (cid:18) ~r + ~r (cid:19) δ ( ~r − ~r ) , (8)where C ρ is the coupling constant and P σ is the spin exchange operator. Thisdensity dependent interaction term was first introduced in [24]. It was shown[25] that it gives the same contribution to the Hartree-Fock energy as thecontact three-body interaction V NNN = C ρ δ ( ~r − ~r ) δ ( ~r − ~r ) . (9)4he term (8) is necessary for reasonable description of correct nuclear singleparticle spectra within the mean field calculations with realistic N N inter-actions [26] and improves significantly the description of the nuclear densitydistributions and radii [27]. In this paper we fix the values of C ρ for eachhypernucleus independently to get the realistic values of nuclear radii as wellas nuclear densities with respect to the available experimental data [28] andthe calculations within the RMF model [29]. Our future goal is to implementdirectly the chiral N N N interaction instead of the density dependent term(8). In this case we would not need to fit any independent parameter C ρ .The Λ N interaction implemented in our model is the YNG force derivedfrom the Nijmegen model ESC08 [30], namely its version ESC08c [31]. It isrepresented in a three-range Gaussian form: G ( r ; k F ) = X i =1 ( a i + b i k F + c i k F )exp( − r /β i ) . (10)For more details including the values of the parameters a i , b i , c i , β i see [31].We represent the Λ N interaction (10) in the form of the interaction elementsof Eq. (5). It should be noted that there are no tensor terms in the ESC08cversion [31] used in this work.The Λ N interaction depends explicitly on the Fermi momentum k F . Wecan either consider k F as a free parameter of our model and fit its value tothe observed hypernuclear spectra or we can fix the value of k F within theThomas-Fermi approximation through the relation k F = (cid:18) π h ρ i (cid:19) / . (11)The average density h ρ i in Eq. (11) can be expressed within the AverageDensity Approximation (ADA) by the following prescription [32] < ρ > = Z d rρ N ( ~r ) ρ Λ ( ~r ) , (12)where ρ N ( ~r ) is the density of the nuclear core and ρ Λ ( ~r ) is the density of Λin the hypernucleus. Note that we have to perform the hypernuclear calcu-lation to obtain ρ Λ ( ~r ) and determine the value of Fermi momentum k F . Inother words the value of k F has to be evaluated self-consistently because theequation (12) depends on the result of a calculation which itself depends on k F . 5he symmetric (SLS) and antisymmetric (ALS) spin-orbit terms in theΛ N potential are included within the Scheerbaum approximation [33]. Dueto this approximation we include the effect of the SLS and ALS terms directlyat the mean field level. We add the following contribution into the matrix(7): U N Λ , ls ij = h i | K Λ r d ρ d r~l.~s | j i , (13)where K Λ = K SLSΛ + K ALSΛ = − π S SLS + S ALS ) , (14)and S SLS , ALS = 3¯ q Z ∞ r j (¯ qr ) G ( r ; k F )d r. (15)The value of ¯ q in (15) is set to 0.7 fm − [33]. The form of the function G ( r ; k F ) in (15) is identical for the SLS and ALS terms, they only differ bythe values of the input parameters. It should be noted that the Λ N -Σ N coupling term in ESC08c is renormalized into the Λ N -Λ N part of G-matrixinteraction, giving rise to an important part of the density dependence [31].In the RMF approach, the strong interactions among point-like hadronsare mediated by effective mesonic degrees of freedom. The formalism is basedon the Lagrangian density of the form L = L N + L Λ , L Λ = ¯ ψ Λ [i γ µ ∂ µ − g ω Λ γ µ ω µ − ( M Λ + g σ Λ σ )] ψ Λ + L T , (16) L T = f ω Λ M Λ ¯ ψ Λ σ µν ∂ ν ω µ ψ Λ . Here, L N is the standard nuclear Lagrangian [29] and we used the NL-SHparametrization in this work [34]. The L T is the ω ΛΛ anomalous (tensor)coupling term. This term is crucial in order to get negligible Λ spin-orbitsplitting for larger values of the Λ couplings required by constituent quarkmodel [35, 36].The system of coupled field equations for both baryons ( N , Λ) and con-sidered meson fields results from L using standard techniques and approxi-mations [29, 36].For the coupling constants g ω Λ and f ω Λ we used the naive quark modelvalues and g σ Λ was tuned so as to reproduce the binding energy of Λ in the0 s state in O [36]. 6 . Results
In this section, we present selected results of our calculations of the hy-pernuclei C, O, Ne,
Si and
Ca. In the hypernuclei with the doublymagic nuclear core -
O and
Ca - the calculations within Hartree-Fockmethod are straightforward. However, in the case of C, Ne and
Si,the ground states of the corresponding core nuclei have more complex struc-ture [37, 38] and it is necessary to take into account configuration mixing andperform calculations within a deformed basis in order to describe their struc-ture properly. Nevertheless, even calculations within the spherical HO basiscould provide interesting information about these hypernuclei if we considervarious configurations of the corresponding nuclear cores. In this work, weperformed calculations for the following configurations: p / and p / p / in C; d / , s / , and d / in Ne; d / , d / s / and d / s / d / in Si. Wetreated these configurations always symmetrically for both protons and neu-trons. It is to be noted that more configurations could be realized in theground states of the above nuclei. We selected just some of them in orderto illustrate the effect of the wave function of the nuclear core on the en-ergy splitting of the Λ single particle levels 0 p / and 0 p / in the consideredhypernuclei.For each particular nucleus we first fixed the parameter C ρ to obtain rea-sonable density distribution and rms radius of the nuclear core, comparablewith the available experimental data and RMF calculations within the NL-SH parametrization [34]. The values of the charge rms radii are summarizedin Table 1. Table 1: The charge rms radii (in fm) of considered nuclei in selected g.s. configurations,calculated without the DD term (8), C ρ = 0 MeV · fm − (A), for the fitted values of C ρ (B) and within the RMF model NL-SH [34] (see text for details). The experimental values(exp.) are taken from [28]. A B RMF exp. C 2.37 2.50 2.46 2.47 O 2.44 2.72 2.70 2.70 Ne 2.62 2.95 2.88 3.01 Si 2.73 3.14 3.04 3.12 Ca 2.99 3.48 3.45 3.487 ρ (f m - ) RMFC ρ = 0C ρ fitted ρ (f m - ) C Si Ca O Figure 1: The nuclear core density distributions in selected g.s. configurations of C, O, Si, and Ca, calculated without the DD term ( C ρ = 0) and with the DD term ( C ρ fitted). The density distributions calculated within the RMF model NL-SH [34] are shownfor comparison (see text for details). We used the values C ρ = 600 MeV · fm − for C, C ρ = 1600 MeV · fm − for O, C ρ = 1700 MeV · fm − for Ne, C ρ = 1600 MeV · fm − for Si, and C ρ =2100 MeV · fm − for Ca. In case of hypernuclei with the open-shell core wedid not repeat tunning of the parameter C ρ for each configuration separatelybut we fixed it for one case (the configuration p / for C, d / for Neand d / for Si). The corresponding radii for the remaining configurationsdiffer from the values shown in Table 1 but the differences are much smallerthan the differences between the values in the columns A and B for eachgiven nucleus. We chose to fit the radii and radial distributions for the aboveconfigurations because they are the lowest configurations in energy due tothe empirical ordering of the s and d levels.In Fig. 1, we present the radial nuclear density distributions calculatedwithin the mean field model based on realistic 2-body baryon interactionsand the RMF model NL-SH [34]. The figure illustrates the importance ofthe DD term in the N N interaction (8) on selected nuclei – C (in the p / O, Ne (in the d / configuration), Si ( in the d / config-uration), and Ca. Calculations performed without the DD term ( C ρ = 0)yield unrealistically large central densities and, as a consequence, the corre-sponding rms radii are too small. After including the DD term the densitydistributions become more diffused and get closer to the RMF predictionswhich are in agreement with empirical density distributions [29, 39] (comparealso charge rms radii in Table 1).The values of the Fermi momentum k F used in the present calculationswere determined using the ADA approximation (Eqs. (11) and (12)). Weobtained k F = 1 .
20 fm − for C (in the configuration p / ), k F = 1 .
20 fm − for O, k F = 1 .
21 fm − for Ne (in the configuration d / ), k F = 1 .
27 fm − for Si (in the configuration d / ), and k F = 1 .
29 fm − for Ca.Before focusing on the main objective of the present work, the energysplittings between p / and p / , we discuss on the Λ single particle energies inthe considered hypernuclei. Fig. 2 shows our hypernuclear spectra, calculatedfor the configurations for which the parameters C ρ and k F were tuned, and thespectra calculated within the RMF model NL-SH for the same configurations.We see that our results are in good agreement with the experimental datashown for comparison. However, it should be pointed out that the data inthe figure are for O, Si, and
Ca since the data for hypernuclei with thesame Z and closed nuclear cores are not available at present (unlike the Ccase). The most pronounced difference between the HF calculations basedon realistic interactions and the RMF model NL-SH occurs in the spectrumof
Si. Here the RMF model predicts significantly more binding for Λ - thelowest level 0 s / has nearly the same energy as the 0 s / level in Ca. Itwill be demonstrated in Fig. 4 that the discrepancy between the calculated
Si spectra is smaller for other nuclear core configurations and could beattributed to considerably different nuclear spin-orbit splittings in the twoconsidered models. We expect that proper calculations allowing for nuclearcore deformation and configuration mixing of the nuclear core wave functioncould decrease this discrepancy in predicted Λ spectra in
Si. It is to benoted that the Λ energies calculated in both considered models could getcloser to each other if we fine tuned e.g. the k F parameter for each particularconfiguration of the nuclear core in the mean field model based on realisticinteractions and/or the RMF g σ Λ coupling separately for Si. However,being aware of the limitations of our current hypernuclear calculations we donot intend to do so in the present study.9 -20-15-10-50 e Λ ( M e V ) C O Ne Si Ca Λ Λ Λ Λ Λ ps Figure 2: The Λ single particle energies in
C in the nuclear core configuration p / , O, Ne in the configuration d / , Si in the configuration d / , and Ca, calculatedwithin the mean-field model with realistic interactions (black lines) and the RMF modelNL-SH (red lines). The 0 p / levels are denoted by dotted lines. The experimental valuesfor C are taken from [40, 41]. We show also energies of the s and p levels measured for O [40, 42],
Si [42, 43] and
Ca [42, 44].
Since our Λ single particle energies reproduce the data reasonably well,let us discuss on the main subject of our study, the energy splitting ∆ p =[( E (0 p / ) − E (0 p / ))] between the Λ single particle levels 0 p / and 0 p / . InTable 2, we present the calculated values of ∆ p for the following options: thespin-orbit (SLS+ALS) forces completely switched off (A), only the SLS termincluded (B), both the SLS and ALS terms included (C). The results obtainedwithin the RMF model are presented for comparison. The configurations forwhich the parameters C ρ and k F were tuned are given in bold face.It is to be stressed that the energy splittings ∆ p calculated within themean field based on realistic baryon interactions and RMF models are ofdifferent origin. In the former case, the splitting is caused by the 2-bodyΛ N interaction with its various spin dependent terms. On the other hand,10 able 2: The Λ energy splitting ∆ p = ( E (0 p / ) − E (0 p / )) (in MeV) in C, O, Ne,
Si, and
Ca for selected nuclear core configurations, calculated without SLS and ALSterms (A), with SLS term only (B), and with SLS + ALS terms (C). Negative values of ∆ p indicate inverse ordering of the 0 p / and 0 p / levels. In columns (B) and (C) we presentthe SLS and ALS contribution to ∆ p , respectively. The values of ∆ p calculated withinthe RMF model are shown for comparison. Those configurations for which we tuned theparameters C ρ and k F are shown in bold face. core A B C RMFconfiguration ∆ p SLS ∆ p ALS ∆ p ∆ p C p / -0.58 1.57 0 .
99 -1.05 -0.06 0.27 p / p / .
07 0.86 0 .
93 -0.57 0 .
36 0.13 O p / p / -0.07 0.89 0 .
82 -0.59 0 .
23 0.24 d / -0.22 0.69 0 .
47 -0.46 0 .
01 0.26 Ne s / -0.10 1.28 1 .
18 -0.85 0 .
33 0.25 d / .
20 0.56 0 .
76 -0.37 0 .
38 — d / -0.64 0.39 -0.25 -0.25 -0.50 0.29 Si d / s / -0.51 1.14 0 .
64 -0.74 -0.10 0.32 d / s / d / .
01 1.06 1 .
07 -0.69 0 .
37 0.24 Ca d / d / s / -0.01 0.58 0 .
57 -0.37 0 .
20 0.21the splitting of the spin orbit partners in the RMF model results from adelicate balance between strong scalar and vector mean fields in the Diracequation [35]. This explains qualitatively different predictions for ∆ p withinthe two approaches. Clearly the calculations based on realistic baryon in-teractions give considerably larger variations of the ∆ p values in the studiedhypernuclei than the RMF approach.The ESC08c Λ N potential gives a non-zero, negative energy splitting ofthe 0 p / and 0 p / levels in several configurations in C, Ne and
Si evenif the SLS and ALS forces are switched off (see column A in Table 2). Thespin-spin term in the ESC08c Λ N interaction thus contributes significantlyto the ∆ p splitting in these configurations of hypernuclear cores. In the caseof O and
Ca (hypernuclei with the doubly magic nuclear core) and also11 f(r) ( M e V f m - ) f(r) ( M e V f m - ) SLS O -ALS Ne Λ Λ
SLS-ALS Λ -ALSSLS Si Figure 3: The functions f SLS ( r ) (SLS, solid line) and − f ALS ( r ) (-ALS, dashed line)defined by Eq. (17) calculated for O (left), for
Ne in the d / configuration of thenuclear core (middle), and for Si in the d / core configuration (right) (see text fordetails). in C for the configuration p / p / , in Ne for the configuration s / and in Si for the configuration d / s / d / the 0 p / and 0 p / levels are close to bedegenerate. We checked that these states are roughly degenerate also in theother two hypernuclei with doubly magic core - Zr and
Pb. When theSLS term is included (B), the ∆ p splitting becomes positive in all consideredhypernuclei except Si in the d / core configuration. The ALS term acts inan opposite way to the SLS term (thus weakens the effect of SLS) and itsmagnitude is ≈ / p splitting in most ofthe cases except Si in the core configurations d / and d / s / , and Cin the core configuration p / . It is also to be pointed out that in the caseswhen the states p / and p / are close to be degenerate in the column A, theenergy splitting ∆ p comes almost entirely from the SLS and ALS spin-orbitinteraction terms.Finally, there is a missing RMF value for the Ne configuration d / in12able 2 – this configuration could not be calculated since the proton d / level was found to be unbound in the applied RMF parametrization.The competition between the SLS and ALS forces can be illustrated withthe help of the function f ( r ) defined as follows: f ( r ) = 4 πr φ ∗ p / ( r ) K Λ r d ρ d r~l.~sφ p / ( r ) − πr φ ∗ p / ( r ) K Λ r d ρ d r~l.~sφ p / ( r ) , (17)evaluated for K SLSΛ ( K ALSΛ ). The radial integral of f ( r ) determines the con-tribution of the SLS (ALS) force to the energy splitting between the 0 p / and 0 p / levels (see Eq. 13).In Fig. 3, we compare f ( r ) for K SLSΛ with f ( r ) for − K ALSΛ in O (leftpanel),
Ne in the configuration d / (middle panel) and Si in the con-figuration d / (right panel). The difference between the areas delimited bySLS and -ALS curves above and below zero determines the spin-orbit (SLS+ ALS) contribution to the energy splitting ∆ p . This difference is relativelylarge in O while in the case of
Ne, the ”negative” and ”positive” con-tributions compensate more each other. Even larger compensation effect isseen for the
Si calculated within the configuration d / . Consequently, the(SLS + ALS) splitting in Si calculated in this configuration is about twicesmaller than in
O (compare Table 2).In Fig. 4, we show the Λ single particle energies in
Si with three dif-ferent configurations of the nuclear core, calculated within the mean-fieldmodel with realistic interactions (black lines) and the RMF model NL-SH(red lines). The 0 p / levels are denoted by dotted lines. The experimentalvalues for Si [42, 43] are shown for comparison (there are no data for
Si).The figure demonstrates how the Λ single particle spectrum is affected bythe wave function of the nuclear core. We can see that the Λ levels 0 p / and0 p / switch their ordering in the case of mean-field based on realistic baryonforces. This is in contrast to the RMF model for which the energy splittingof both levels remains roughly constant. We found possible explanation forthis effect by analyzing the nucleon single particle energies. While the RMFmodel is able to reproduce well the empirical spin-orbit splitting of the 0 p / and 0 p / states for protons and neutrons, the mean field model based onrealistic baryon forces appears to be quite sensitive to different configura-tions considered for the ground state. We do not obtain realistic splitting ofthe nucleonic 0 p / and 0 p / states in the configurations d / and d / s / –in the former case we even get wrong ordering of these levels. Only for the13 e Λ ( M e V ) Si Λ d d + s d + d + s Figure 4: The Λ single particle energies in
Si with three different configurations ofthe nuclear core, calculated within the mean-field model with realistic interactions (blacklines) and the RMF model NL-SH (red lines). The 0 p / levels are denoted by dottedlines. The energies of the s and p levels measured for Si are taken from [42, 43]. configuration d / s / d / we obtain satisfactory agreement of the nucleonicspin-orbit splitting with the empirical values (and also the RMF values).Consequently, we also get standard ordering of the Λ 0 p / and 0 p / levels(see Table 2).
4. Conclusions
In this work, we performed calculations of selected p - and sd - shell hy-pernuclei, namely C, O, Ne,
Si and
Ca within the mean field modelbased on realistic 2-body baryon interactions and compared the results withthe predictions of the RMF model NL-SH. We introduced the density depen-dent 2-body interaction term which mimics the effect of the 3-body
N N N force in order to get realistic charge radii and density distributions of thenuclear cores of the studied hypernuclei. This appeared important also inthe calculations of hypernuclear spectra since the ESC08c Λ N interactiondepends explicitly on the Fermi momentum k F which was determined usingthe averaged density approximation. Reasonable description of the densitydistributions in the studied (hyper)nuclear systems is thus crucial.14he main objective of the present calculations is to study the influenceof SLS and ALS spin orbit terms on the energy splitting ∆ p of the Λ levels0 p / and 0 p / . The ∆ p splittings in O and
Ca, calculated within themean field model based on realistic baryon interactions and the RMF modelNL-SH are very close to each other. In the case of C, Ne and
Si it is de-sirable to perform calculations within deformed basis and take into accountconfiguration mixing of the nuclear core wave function. Nevertheless, our cal-culations in the spherical HO basis, which considered several configurationsof the nuclear core of these hypernuclei yielded valuable insight into the issueof the spin dependence of the Λ N interaction and the Λ spin-orbit splittingin these open-shell hypernuclei. We found that the energy splittings of theΛ levels 0 p / and 0 p / calculated using realistic N N and Λ N interactionsdepend strongly on the chosen configuration of the nuclear core, unlike theRMF approach. For the configurations which give the energy splitting closeto zero when the Λ N spin-orbit interaction is switched off, the ∆ p splittingis almost entirely due to the ALS and SLS terms and is in rough agreementwith the RMF values. By comparing the results for the ESC08c model shownin columns B and C in Table 2 we conclude that the magnitude of the ALSterm which acts in an opposite way to the SLS term is about 2/3 of the SLSmagnitude.Our results demonstrate that it is highly desirable to explore further theenergy splitting of the Λ 0 p / and 0 p / levels in p - and sd -shell hypernuclei,both experimentally and theoretically, in order to extract important infor-mation about the spin-dependence of the Λ N interaction, as well as the innerstructure of the hypernuclei under study.There are several issues left for further improvements of the present cal-culations. First, we intend to develop the code which will allow to performcalculations within an axially symmetric single particle basis and allow forconfiguration mixing in the nuclear core wave function. In this case we wouldbe able to calculate open-shell hypernuclei (such as Ne or
Si) more pre-cisely. Second, we intend to implement directly the 3-body
N N N forcesinstead of the 2-body density dependent term in our Hamiltonian. Third, itis desirable to incorporate the Λ N tensor terms and explore their contribu-tion to the energy splitting.Another extension is to include the core polarization effects. We intendto develop a scheme which couples Λ single-particle states with one-phononor possibly multi-phonon excitations of the core nucleus within an Equationof Motion Phonon Model (EMPM) [45] treating nuclear excitations in mul-15iphonon basis. In this method the Tamm-Dancoff phonon operators Q † ν areused to build Hilbert space spanned by one-, two- and three-phonon config-urations. In this way we will get rather complex description of hypernucleiwhich includes not only core polarization effects but also beyond mean fieldcorrelations.
5. Acknowledgments
This work was partly supported by the GACR grant P203/15/04301S.Highly appreciated was the access to computing and storage facilities pro-vided by the MetaCentrum under the program LM2010005 and the CERIT-SC under the program Center CERIT Scientific Cloud, part of the Op-erational Program Research and Development for Innovations, Reg. no.CZ.1.05/3.2.00/08.0144. P. V. thanks RIKEN for the kind hospitality duringhis stay. This work was partly supported by RIKEN iTHES Project.
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