A Skyrme force for Ξ − hypernuclei

. Experimental data for the cascade hypernuclei 15 Ξ C, 12 Ξ Be, and 13 Ξ B are analyzed within a Skyrme-Hartree-Fock theoretical approach and optimal Skyrme parameters are determined. The important role of deformation for 13 Ξ B is pointed out. Predictions for 7 Ξ H and 10 Ξ Li are made.


Formalism
In the present work, we study Ξ hypernuclei with A = 12, 13, and 15 and focus on the Ξ − 1s and 1p states and their hyperon separation energy, We employ a model based on the self-consistent SHF method [22,23], first extended to the theoretical description of Λ hypernuclei in Ref. [24], and now used for Ξ − hyperons here. The fundamental SHF local energy-density functional of hypernuclear matter is written as and depends on the one-body densities ρ q , kinetic densities τ q , and spin-orbit currents J q , where φ i q (i = 1, N q ) are the self-consistently calculated single-particle (s.p.) wave functions of the N q occupied states for the species q = n, p, Y in a hypernucleus.
The functional ε N is the usual nucleonic part [22,23] and a possible standard parametrization for the hyperonic part is [10,20,[24][25][26] from which one obtains the corresponding hyperonic SHF mean fields and a hyperon effective mass 1 2m * which appear in the SHF Schrödinger equation where V C is the Coulomb field and W N the nucleonic spin-orbit mean-field [23]. The 'threebody' parameter α is kept here to its standard value of 2, but also an alternative value of 7/6, used in several NN Skyrme forces, is evaluated. Contrary to Λ hypernuclei, the Coulomb interaction is very important for Ξ − hypernuclei discussed here and provides a substantial part of the Ξ − binding. An approximate c.m. correction is applied as usual [22,23,27] by replacing the bare masses: 1/m q → 1/m q −1/M, where M = (N n + N p )m N + N Y m Y is the total mass of the (hyper)nucleus. Solving the Schrödinger equation provides the wave functions φ i q (r) and the s.p. energies −e i q for the different s.p. levels i and species q. We use in this work the standard nucleonic Skyrme force SLy4 [28], but the results for hyperonic observables hardly depend on that choice [29].
It should be noted that the Ξ − hypernuclei decay into double-Λ hypernuclei by the ΞN-ΛΛ coupling [30]. Therefore, the ΞN interaction should have an imaginary part to represent  the decay width. However, since we have so far no useful experimental information on this coupling by Refs. [9,11,[13][14][15][16], here the imaginary part is omitted. While the core nuclei 14 N and 11 B are (nearly) spherical, 12 C is an axially-deformed oblate nucleus [20,[31][32][33][34][35][36]. In our approach, the deformed SHF Schrödinger equation is solved in cylindrical coordinates (r, z) within the axially-deformed harmonic-oscillator basis [22,23]. The geometric quadrupole deformation parameter of the nuclear core is expressed as As there are currently not enough data to determine uniquely the five ΞN interaction parameters a i , we proceed as follows. The effective-mass parameter a 1 is kept zero, since recent Brueckner-Hartree-Fock calculations [37] indicate a rather flat Ξ − s.p. spectrum and thus m * Ξ /m Ξ close to unity. Also the spin-orbit parameter is disregarded, a 4 = 0, as the inclusion of Ξ − spin-orbit splitting is clearly premature. For the same reason we do not introduce further parameters for the isospin dependence of the interaction at this stage.
The surface-energy parameter a 2 is essential to determine the shape of the Ξ − mean field V Ξ in the hypernucleus, Eq. (5), which is important for comparison with the WS mean field that was used in the experimental analysis of BNL-E885 for 12 Ξ Be, Refs. [9,15], For fixed a 1 = a 4 = 0 and chosen a 2 , the remaining volume parameters a 0 and a 3 are then determined by fitting the removal energies B Ξ = 8.00 and 1.13 MeV for the Ξ − 1s and 1p states in 15 Ξ C, respectively, as claimed for the KINKA and KISO+IBUKI events. The resulting three-parameter force is termed SLX3.  Table 1. Increasing values of a 2 produce deeper well depths V Ξ (0). The Woods-Saxon mean field deduced in [15] is shown for comparison. Table 1 lists the parameter values a 0,2,3 obtained, together with the Ξ − 1s and 1p removal energies of 13 Ξ B, 12 Ξ Be, 10 Ξ Li, and 7 Ξ H that are predicted. Strongly-bound 1p states with positive removal energies are only found for deformed 13 Ξ B nuclei, but not in spherical approximation (numbers in brackets). 10 Ξ Li and 7 Ξ H are bound for all versions of the ΞN force, although these nuclei are probably too light systems to be calculated reliably within the SHF approach. One obtains reasonable values for the parameters a 0 and a 3 . For comparison, in the SLL4 ΛN Skyrme force [25,26], the equivalent optimal parameters are a ΛN 0 ≈ 320 MeV fm 3 and a ΛN 3 ≈ 700 MeV fm 6 . For increasing surface parameter a 2 , more Ξ − binding is provided by the associated terms in Eqs. (4,5), and therefore both a 0 and a 3 decrease in magnitude.

Results and discussion
In Fig. 1, the different SHF mean field potentials in the 12 Ξ Be hypernucleus are plotted, including the local Coulomb field V C , the strong mean field V Ξ , Eq. (5), and for comparison the WS mean field Eq. (10) with V 0 = 14, 16, 18 MeV, as used in the analysis of BNL-E885 [9,15]. One observes that the SHF results cover the range between the 14 and 18 MeV curves. We consider this as a very good agreement with the analysis of E885. It can clearly be seen that the zero-momentum value V Ξ (0) alone is not a useful unique indicator of the interaction strength, as the shape of the mean field controlled by the parameter a 2 plays an essential role. A value of about a 2 ≈ 20 − 30 MeV fm 5 gives the closest correspondence to a WS shape, whereas a 2 = 0 generates a flat or well shape in the core region.
The hypernuclei 15 Ξ C and 12 Ξ Be and their core nuclei discussed so far are spherical nuclei. The case of 13 Ξ B is more delicate, as the core nucleus 12 C is axially deformed in the SHF approach [20,[31][32][33][34][35][36]. Its deformation derived from the proton quadrupole moment Q p is rather large [38], with R 0 ≡ 1.2 A 1/3 fm. In the SHF approach the size of nuclear deformation can be controlled by adjusting the spin-orbit parameter of the NN Skyrme force [20,[31][32][33][34][35][36], and we follow this procedure to reproduce the proper β value. As discussed in detail in [20], the oblate deformation favors the binding of a Ξ − 1p orbital because of the improved geometrical overlap of wavefunction and embedding potential, such that this state becomes more bound than in spherical approximation. The results of Table 1 for 13 Ξ B are obtained in this way and demonstrate that the proposed interpretation of the KEK-E176 events as 13 Ξp B states might be solely due to the fact that 12 C is a strongly deformed nucleus. Excellent agreement with the experimental value B Ξ ≈ 0.82 MeV is obtained with small values of a 2 and both choices of α.

Summary
We have tried to fit all current data for the cascade hypernuclei 15 Ξ C, 12 Ξ Be, and 13 Ξ B within a global SHF approach for the ΞN interaction. Our main conclusions are: 1) Of all proposed B Ξ ( 15 Ξs C) values, the KINKA (8.00 MeV) interpretation seems to be the one most compatible with both the 12 Ξ Be and 13 Ξ B data, although slightly lower values down to about 7 MeV might also be possible and consistent with 12 Ξ Be, but would not allow interpretation of the KEK-176 events as 13 Ξp B states. 2) Combining KINKA ( 15 Ξs C) and KISO+IBUKI ( 15 Ξp C) data, the predicted Ξ − mean field in 12 Ξ Be is very similar to the best-choice WS mean field for BNL-E885 with a depth of about 14-16 MeV [15], and also compatible with KEK-E224 [16] and BNL-E906 [17].
3) 13 Ξ B is a delicate case, as in the current model this hypernucleus and its core nucleus 12 C are strongly axially deformed, and the claimed B Ξ ≈ 0.8 MeV values of some KEK-E176 emulsion data might be explained as a consequence of this deformation, which energetically favors the extended Ξ − 1p orbit. Treating these nuclei as undeformed does not produce a sufficiently bound Ξ − 1p state (which is actually another possible interpretation of the data). Furthermore, 7 Ξ H is predicted as a bound hypernucleus. The proposed Skyrme force SLX3 with moderate values of the parameter a 2 is thus able to fit satisfactorily all current data, although due to their limited accuracies and ambiguities a firm statement cannot yet be made. Hopefully a large number of emulsion events obtained in KEK-E373 and J-PARC-E07 will be analyzed soon for confrontation. Future improvements of the approach include constraining better the SHF ΞN interaction parameters, going beyond the mean-field treatment, and including the imaginary parts due to the ΞN-ΛΛ decay.