$^{6}_{\Lambda\Lambda}$He and $^{10}_{\Lambda\Lambda}$Be hypernuclei calculations with realistic interactions

Abstract

We study the $^{6}_{\Lambda\Lambda}$He and $^{10}_{\Lambda\Lambda}$Be hypernuclei in the three- and four-body models, respectively, employing harmonic oscillator bases and presently most realistic alpha-alpha, alpha-lambda and lambda-lambda interactions. In order to improve convergence we use correlation functions in the case of $^{10}_{\Lambda\Lambda}$Be. Comparison is made with results obtained using similar interactions and other methods, and it is performed an analysis concerning the possibility of a unified description of both hypernuclei.

REGULAR ARTICLES

He and Be hypernuclei calculations with realistic interactions

O. Portilho

Instituto de Física, Universidade de Brasília, 70919-970, Brasília, DF, Brazil

ABSTRACT

We study the He and Be hypernuclei in the three- and four-body models, respectively, employing harmonic oscillator bases and presently most realistic a-a, a-L and L-L interactions. In order to improve convergence we use correlation functions in the case of Be. Comparison is made with results obtained using similar interactions and other methods, and it is performed an analysis concerning the possibility of a unified description of both hypernuclei.

1 Introduction

It was detected recently in the KEK E373 experiment [1] a new candidate for He, with a smaller binding energy than the old data due to Prowse [2] which are now considered under suspicion of misinterpretation. Furthermore, we have recently demonstrated [3] the applicability of Jastrow-type correlation functions to harmonic oscillator (HO) bases in order to accelerate convergence in the case of Be, considered as a four-body aaLL system. This lead us to apply such a technique to the same hypernucleus, now using more realistic interactions, and to review our former calculations on He [4] in the same scope, hoping that a satisfactory and unified description of both nuclei can be made with an adequate choice of potentials. Filikhin and Gal [5] have published a calculation on the same hypernuclei by solving Faddeev-Yakubovsky equations in the s-waves approximation (for Be) and Faddeev equations (for He). They have also introduced ¹S₀LL potentials as a sum of three gaussians simulating LL Nijmegen soft-core [6, 7] (NSC97), extended soft-core [8] (ESC00) and hard-core [9, 10] (ND) interaction models. We make use here of these more realistic potentials, instead of the old one-gaussian attractive Dalitz potential [11, 12], employed in our former paper [3] with the purpose of comparing convergence of the HO basis without correlation functions [13].

In Section II we have a brief presentation of the formalism and in Section III we show our results for the ground-state energies, separation distances, and contributions of the partial waves to the energy, and compare them with results due to other authors. We also discuss the best choice of combination of the several potentials employed in the calculations.

2 Formalism

The method we employ in the study of He and Be hypernuclei in the three- (aaL) (see Ref. [4]) and four-body (aaLL) (see Ref. [3]) models, respectively, is based in a variational calculation where the a-particles are structureless. A convenient coordinate set that splits the internal coordinates from that of the center of mass is given by the Jacobi variables for unequal mass particles. In the case of He the internal (non-dimensionless) coordinates are

where

is the total mass and

is the reduced mass. Coordinate r¢₁ and mass m₁ refer to the a-particle while coordinates r¢₂, r¢₃ and respective masses m₂ and m₃ are related to the L-particles. In an analogous way, we have for Be the coordinates

with

Coordinates r¢₁, r¢₂ and respective masses m₁ and m₂ refer to the a-particles while coordinates r¢₃, r¢₄ and masses m₃ and m₄ are related to the L-particles.

We use as bases for the hypernuclei wave functions products of the HO wave functions in the internal Jacobi variables, coupled to a well defined value for the total angular momentum and its z projection. In the case of He the wave function is expanded in terms of

while in the case of Be the basis is constituted of

In the l.h.s. of above two equations we have omitted the M_J label since the physical quantities we calculate using the wave functions are independent of it. The harmonic oscillators have common quantum energy

which is changed variationally. Moreover, for this hypernucleus we introduce Jastrow-type correlation functions between the a-particles and between the L-particles in order to improve convergence so that the trial function becomes

with

The correlation functions are conveniently taken as 1 + sum of two gaussians, one repulsive and one attractive

Parameters c_a, , c_b, , are changed variationally and satisfy the conditions c_a > 0, c_b > 0, < 0, < 0, 1 + c_a + > 0, 1 + c_b + > 0. Unprimed coordinates mean they are dimensionless and the relation between primed and unprimed coordinates is r¢ = r. On the other hand, the parameters

are also changed variationally, and so are , , defined analogously, satisfying b_a < and b_b < .

For the aa interaction we use the Chien-Brown potential [14], which is given by

with

Here erf is the error function. The potential parameters are V_0N = 287.5 MeV, m¢ = (0.635 fm^–1)² (for l_a = 0), V_0N = 176.5 MeV, m¢ = (0.620 fm^–1)² (for l_a = 2), V_D0 = 85 MeV, g = 1.35 fm^–1, k = 0.514 fm^–2.

For the aL interaction we have two potentials. The first one is the Isle type [15] which is a sum of two gaussians such as

and that was used by Filikhin and Gal [5] as well, with parameters V_rep = 450.4 MeV, V_att = –404.9 MeV, b_rep = 1.25 fm, b_att = 1.41 fm. The second is the one-gaussian attractive potential labeled as Gibson I (r_GI) by Daskaloyannis et al. [16], with parameters V_att = –43.48 MeV and b_att = 1.5764 fm, which produced a reasonable ground-state energy for Be in the aaL model [4].

In our former papers [3, 4, 13] we used as LL interaction the one-gaussian attractive Dalitz potential [11, 12]. However more realistic potentials with repulsive core have been published, especially the ¹S₀ channel simulations to the Nijmegen models due to Filikhin and Gal [5], conveniently expressed as a sum of three gaussians

The parameters are the following: V₁ = -21.49 MeV, V₂ = –379.1g¢ MeV, V₃ = 9324 MeV, b₁ = 1.342 fm, b₂ = 0.777 fm, b₃ = 0.350 fm, with g¢ = 0.4804 (NSC97b model), g¢ = 0.5463 (NSC97e model), g¢ = 1 (ND model) and g¢ = 1.2044 (ESC00 model).

Using the HO bases described in this section we calculated the hamiltonian matrix elements and also, in the case of Be, the norm matrix elements. The bases are truncated by choosing a limit to N, defined as 2(n_a + n_b) (for He) or as 2(n_a + n_b + n_c) (for Be). For each N corresponds a certain amount of possible combinations of n_a, n_b and n_c, together with combinations of l_a, l_b and l_c chosen appropriately, giving the basis dimension (size). After the diagonalisation process we obtained the ground-state energy and the coefficients a_n that appear in Eq. (10). With the wave function so available we proceeded to calculating rms distances between the particles involved in each hypernucleus and expectation values for the kinetic and potential energies. More details on the formalism can be found in the former papers [3, 4].

3 Results and discussion

He

We have revisited our former calculations on He [4] in view of the more recent LL potentials with repulsive core simulating the Nijmegen model, as mentioned in the previous section. We present in Table I the 0⁺ ground-state energies obtained with combinations of Gibson I (r_GI) [16] and Isle [15] aL potentials with four Nijmegen model simulations of LL interactions [5]: NSC97b, NSC97e, ND, ESC00. In the column labeled as ground-state energy we show the results obtained with the (0, 0)+(2, 2)+(4, 4) combination of (l_a, l_b) in the HO basis, what means dimension 408 and N = 2(n_a + n_b) limited to 30. The values of e were chosen in the N = 26 approximation and used in the larger bases. In order to check convergence, we have also shown the extrapolated values of the ground-state energy for N = ¥ by supposing, following Delves [17], that

is obeyed so that, to leading order,

with DE_N = E_N+1 – E_N, and corresponding estimates of the errors involved in the extrapolation. The extrapolation we have made for He in our former paper [4] using the older Bando aL potential [12] was confirmed by Filikhin, Gal and Suslov in their recent Faddeev calculations [18]. Therefore there is no reason to doubt that the results in Table I are also reliable. We notice that results obtained following the sequence NSC97b, NSC97e, ND, ESC00 of LL potentials have convergence improved in the case of the Gibson aL potential since the errors become smaller, while it gets worse in the case of the Isle potential. We also show in the last column the Faddeev equation calculations of Filikhin and Gal [5] who employed the same combinations of Isle aL potential and Nijmegen LL potentials. The experimental ground-state energy is just -B_LL from the recent results of Takahashi et al. [1]. By comparing the several results with the experimental value we find that the best choice of potentials is the combination Isle+NSC97e, whose extrapolated energy coincides with the upper limit of the experimental value. The ESC00 potential was constructed with the purpose of obtaining the old value of experimental ground-state energy of -10.8±0.6 MeV due to Prowse [2], and our calculation with this potential combined with Isle potential shows it is successful in this aim, although these data are now discarded in favor of the results of Takahashi et al. [1]. Some additional results obtained using also the ND interaction, combined with G matrix methodology, are -9.23 MeV [19], -9.34 MeV [20], -9.4 MeV [21], which are a little bit smaller than our result and the one due to Filikhin and Gal, -7.25 MeV [22]; and -7.33 MeV (RPA) [23].

We have also calculated rms distances between the L-particles and between the a-particle and the LL center-of-mass. This is shown in Table II. We notice that the distances obtained with the Isle aL potential are larger than the results that emerge from the Gibson potential. The results obtained by Filikhin and Gal are also a little bit larger than ours. Besides, we notice that the more bound is the nucleus the more compact it is, as it should be. Results from other authors include ^1/2 = 3.31 fm, ^1/2 = 2.14 fm [19] and ^1/2 = 3.20 fm [20].

In Table III we show the contributions of the partial waves (0, 0), (2, 2), (4, 4) to the ground-state energy and to the wave function.

Be

In order to improve convergence of the HO basis in four-body calculations we included the Jastrow-type correlations functions as shown in Eqs. (10)-(13). In all calculations we obtained c_a = c_b = 0, = = -1, = 2.5 fm as best parameters, leaving changes only in . We limited N = 2(n_a + n_b + n_c) to 12 which proved to be enough from the standpoint of convergence in a former calculation of ours [3] with the purely attractive LL Dalitz potential [11, 12]. We show in Table IV our results for the ground-state energy of Be in the N = 12 approximation (dimension 336) considering the combinations (0, 0, 0), (2, 2, 0), (2, 0, 2) and (0, 2, 2) to (l_a, l_b, l_c) and using the aa potential of Chien and Brown [14]. The values of e and were fixed in the N = 8 approximation (dimension 140) and extended to the N = 12 approximation. We consider both Gibson I (r_GI) [16] and Isle [15] aL potentials. We notice again that the combination Isle+NSC97e gives a result close to the experimental ground-state energy [24, 25], as happened with He. We also show the Faddeev-Yakubovsky results of Filikhin and Gal [5]. It is clear that our results present more binding than theirs. As aa interaction they use the older Ali-Bodmer a₀ potential [26] but this cannot account for the discrepancy. They expect an uncertainty of 0.5 MeV in their figures due to limiting the calculations to s-waves. In order to have a precise idea about the convergence quality of our calculations we plot in Fig. 1 the behaviour of the ground-state energy against N for all combinations of Gibson and Isle aL potentials with NSC97b, NSC97e, ND and ESC00 LL potentials. It is also shown the experimental value and the error limits. The results obtained with the combination of the Isle potential with NSC97b and NSC97e potentials are inside the experimental error limits. We see that the convergence is satisfactory. However we should mention that the simulations to the Nijmegen LL potentials due to Filikhin and Gal are supposed to be applied to the ¹S₀ interaction. Since in Table IV the energies were calculated with bases involving l_b > 0 we are assuming that those interactions are still valid for higher relative angular momenta between the L-particles. Other results for the ground-state energy, obtained also with simulations of the Nijmegen D model, combined with the G matrix formalism, are -17.6 MeV [19], -17.15 MeV [20], -17.0 MeV [21], -15.05 MeV [22]; and -16.7 MeV (RPA) [23].

We show in Table V the rms distances between the a-particles, between the L-particles and between the aa and LL centers-of-mass. We notice that ^1/2 is very sensitive to the LL potential: the deeper is its well, the shorter is ^1/2. Other results for ^1/2 are 3.40 fm [20] and 3.44 fm [22], for ^1/2 are 2.81 fm [19], 3.02 fm [20] and 2.8 fm [21], and for ^1/2 are 1.67 fm [19] and 1.90 fm [20].

In Table VI we present the corresponding contributions of the several combinations of (l_a, l_b, l_c) to the ground-state energy in the N = 12 approximation. We notice that the (0, 0, 0) restriction is completely unsatisfactory in our approach, although it can be perfectly acceptable in the Faddeev-Yakubovsky formalism of Filikhin and Gal. We also see from Table VI that the l_b ¹ 0 contributions are not negligible, what reinforces the necessity of detailed studies of the LL interaction beyond the ¹S₀ channel.

In Fig. 2 we plot B_LL, the LL separation energies for He (which are just the extrapolated ground-state energies in Table I with sign changed, -E_g.s.) as a function of the corresponding energies for Be (which in this case are -E_g.s.+ 0.09 MeV - see Table IV), each point calculated with the same set of aL and LL potentials. We notice that there is a correlation between those points through straight lines, what was already observed by Filikhin and Gal [5], Bodmer et al. [27] and Wang et al. [28]. The full line fits the points calculated with the Isle aL potential and the dashed line fits points calculated with the Gibson aL potential. We also indicate through horizontal and vertical lines the experimental values and errors of B_LL for He and Be, respectively. We note that the full line crosses the rectangle defined by the dotted lines that represent the limits of the experimental errors and that the point that corresponds to the combination Isle+NSC97e of aL and LL potentials is just inside the rectangle. This could indicate, as far as the extrapolated ground-state energies of He in Table I are reliable, and the Faddeev calculations of Filikhin, Gal and Suslov [18] seem to confirm that since they reproduce our old result [4] obtained with the Bando aL potential, and the application of LL potentials from Nijmegen model simulations beyond the ¹S₀ channel is acceptable, that a unified description of both He and Be hypernuclei is possible, as represented through the combination Isle+NSC97e. However, for a consistent description, the Isle aL potential should reproduce in our model the ground-state energies of the single hypernuclei He and Be. The results are respectively -3.095 MeV and -8.307 MeV [4], that should be compared with the experimental values -3.12 ±0.02 MeV and -6.62±0.04 MeV. It is clear that while the Isle potential reproduces the ground-state energy of He quite well, it overbinds Be by about 1.7 MeV. For this reason we cannot state that the combination Isle+NSC97e gives a consistent description of the single and double hypernuclei He, Be,He and Be. On the other hand, the Gibson aL potential gives -3.086 MeV and -6.839 MeV [4] for the ground-state energies of He and Be, respectively, being more reasonable than the Isle potential in this respect. However the dashed line in Fig. 2, which represent results for He and Be, is far from the rectangle it should cross. The conclusion is that a consistent description is not possible, agreeing with Filikhin and Gal [5] and Yamamoto et al. [19] and contrary to the conclusion of Albertus et al. [23].

Acknowledgments

We would like to thank the Laboratório de Cálculo Científico of the Institute of Physics of University of Brasília and CENAPAD MG/CO for providing the machines where calculations were performed.

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Received on 22 May, 2003. Revised version received on 23 July, 2003

Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Accepted
  23 July 2003
• Received
  22 May 2003