Ξ hypernuclei predicted by the new interaction model ESC 08

The features of the new interaction model ESC08 in ΛN, ΣN andΞN channels are demonstrated by the partial wave contributions to single hyperon potentials UY (Y = Λ,Σ, Ξ) in nuclear matter on the basis of the G-matrix theory.Ξ hypernuclei are studied with the ΞN G-matrix interactions derived from ESC08.


Introduction
The properties of baryon many-body systems, which contains not only nucleons but also hyperons with strangeness, link closely to the underlying hyperon(Y)-nucleon(N) interactions.As demonstrated in our previous works [1] [2], it is very important to test the YN interaction models in analyses of various hypernuclear phenomena.For such a purpose the G-matrix theory is very convenient.The features of free-space YN interactions are related to the properties of hypernuclei through the corresponding G-matrix interactions.These G-matrix interactions are considered as effective interactions used in the model spaces.Thus, the hypernuclear phenomena and the underlying YN interaction models are linked through the models of hypernuclei.SU(3)-invariant interaction models give useful guidance toward an entire picture of strong interactions among octet baryons.Epoch-making development for such an approach has been accomplished by the Extended Soft Core (ESC) models, in which two-meson and meson-pair exchanges are taken into account explicitly and no effective boson is included differently from the usual one-boson exchange models.Recently, Th.A. Rijken, M.M. Nagels and Y.Y. have proposed the latest version the ESC08 [3].In the parameter fitting of ESC08, the G-matrix results were used as an important guidance to impose constraints so that the resulting interaction model was consistent with the hypernuclear properties and especially gave the attractive ΞN sector substantially.The two versions ESC08a and ESC08b have been proposed.In this report, first the features of the ESC08 model in ΛN, ΣN and ΞN channels are demonstrated by the partial wave contributions to single hyperon potentials U Y (Y = Λ, Σ, Ξ) in nuclear matter on the basis of the G-matrix theory.Next, as an application of the attractive ΞN G-matrix interaction, some Ξ hypernuclei are studied with the four-body cluster models.a e-mail: yamamoto@tsuru.ac.jp

Λ, Σ and Ξ in nuclear matter
Here, we demonstrate the properties of the ΛN, ΣN and ΞN G-matrix interactions derived from ESC08a/b.For simplicity, here, the G-matrix equations are solved with the QTQ prescription (Gap choice), where no potential term is taken into account in intermediate propagations of correlated YN pairs.A two-particle state with isospin (T ), spin (S ), orbital and total angular momenta (L and J) is represented as 2T +1,2S +1 L J .An isospin quantum number is often omitted, when it is evident.In Table 1 we show the potential energies U Λ for a zero-momentum Λ and their partial-wave contributions at normal density ρ 0 (k F =1.35 fm −1 ).Hereafter, a statistical factor (2T + 1)(2J + 1) is included in each 2S +1 L J contribution.The 3 P contribution means the sum of 3 P J (J = 0, 1, 2) contributions, and the D contribution does the sum of 1 D 2 and 3 D J contributions.The obtained results for ESC08a/b are compared with those for ESC04a [1] and NSC97f [2].In the Tables, ESC08a/b, ESC04a and NSC97f are denoted as 08a/b, 04a and 97f, respectively.The S -state contributions in ESC08a (ESC08b) are found to be comparable to (slightly less attractive than) those in ESC04a and NSC97f.The P-state contributions in ESC08a/b are very similar to each other, and they are less attractive than those in ESC04a.On the other hand, those in NSC97f are repulsive contrastively to the ESC models.
The spin-dependent features of the ΛN G-matrix interactions are very important, because they are tested by EPJ Web of Conferences indications from hypernuclear data.The contributions to U Λ from S -state spin-spin components can be seen qualitatively in values of U σσ = (U Λ ( 3 S 1 ) − 3U Λ ( 1 S 0 ))/12.Various analyses suggest that the reasonable value of U σσ (ρ 0 ) is between those of NSC97e and NSC97f, which are 1.05 and 1.70 MeV, respectively [2].The obtained values for ESC08a and ESC08b are 1.32 and 1.44 MeV, respectively: It turns out that the S -state spin-spin components of ESC08a/b are of nice strengths.In Table 2 we show the potential energies U Σ (ρ 0 ) and the partial wave contributions.It should be noted here that the strongly repulsive values of U Σ can be obtained for ESC08a/b.Contrastively, the U Σ values for all NSC and ESC models are attractive [1] [2].In the case of the quarkcluster models, the repulsive nature of U Σ is due to the existence of almost Pauli-forbidden states in T = 1/2 1 S 0 and T = 3/2 3 S 1 states, where the latter contribution with the large statistical weight is distinctly important for the repulsive value of U Σ .On the other hand, in the conventional Nijmegen soft-core models (NSC, ESC), the repulsive cores are given by pomeron and ω meson exchanges etc., which are of similar strengths in all channels.In the case of ESC08 modeling, the above excellent feature of the quark-cluster model is taken into account phenomenologically by strengthening the pomeron coupling in these states.Then, it should be noted that the contributions in the T = 3/2 3 S 1 states for ESC08a/b are remarkably repulsive contrastively those for ESC04a and NSC97f, as found in Table 2. Experimentally, the repulsive Σ-nucleus potentials are suggested in the observed (π − , K − )spectra [4].It is quite interesting that the repulsive values of U Σ can be realized under the specific modeling for the core part of the YN interaction.
In Table 3 we show the potential energies U Ξ (ρ 0 ) and the partial wave contributions for ESC08a/b and ESC04d, where the U Ξ values are found to be substantially attractive in the cases of ESC08a/b and ESC04d.However, the partial-wave contributions to U Ξ are distinctly different from each other: In the case of ESC08a/b (ESC04d), the attractive contributions to U Ξ are dominated by those in the T = 1 (T = 0) 3 S 1 state.As discussed in Ref. [1], the strong T = 0 3 S 1 -state attraction in ESC04d is because the contributions of vector and axial-vector meson exchanges are strongly cancelled in this channel.The strong T = 1 3 S 1state attractions in ESC08a/b are caused by the strong ΞN- are also given in Table 3, the contributions of which come dominantly from the ΛΛ-ΞN-ΣΣ coupling interactions in T = 0 1 S 0 states.Here, it is found that the values of Γ Ξ for ESC08a/b are substantially smaller than that for ESC04d.

Some Ξ-bound systems
Though there is almost no experimental information for ΞN interactions, the BNL-E885 experiment [5] indicates that a Ξ single particle potential in 11 B core is given by the attractive Wood-Saxon potential with the depth ∼ −14 MeV (called WS14).It is found that the values of U Ξ (ρ 0 ) for ESC08a/b in Table 3 are rather more attractive than this WS depth, though the latter should not be compared strictly with the former quantities.Here, we modify ESC08a so as to be comparable to WS14 by weakening artificially the ΞN-ΛΣ-ΣΣ coupling interaction in the T = 1 3 S 1 state.This version is denoted as ESC04a' in Table 3.The U Ξ value −14.5 MeV for ESC08a' is noted to be very similar to the depth of WS14.In the present calculations, the results for ESC08a' are compared with those for ESC04d.Then, also ESC04d is modified so as to be consistent with WS14 according to the way given in Ref. [6].
For applications to finite Ξ systems, the obtained complex ΞN G-matrix interactions are represented as k Fdependent local potentials where k F is a Fermi momentum of nuclear matter.The suffices (+) and (−) specify even and odd, respectively.The interaction parameters (a i , b i , c i and β i ) of the G-matrix interaction for ESC08a' are tabulated in Table 4.When this G-matrix interaction is applied to finite Ξ systems, the k F 07007-p.2 parameters should be taken so as to agree with their adequate values in respective systems.It is difficult, however, to perform this procedure strictly in the present case, because of no experimental information on Ξ hypernuclei.It is inevitable here to choose the k F values rather arbitrarily in a reasonable region (0.8 ∼ 1.2 fm −1 in light p-shell systems).In Ref. [6], the 7 Ξ − H and 10 Ξ − Li systems were studied with use of the αnnΞ − and ααnΞ − four-body cluster models, which can be produced by the (K − , K + ) reaction on 7 Li and 10 B targets, respectively.In this previous work, the ΞN G-matrix interactions were derived from ESC04d and the Nijmegen hard-core model D, from which the Ξα interactions were obtained by the folding procedures.In the same framework, the calculations are performed with the ΞN G-matrix interaction for ESC08a', and the results are compared with those for ESC04d.In calculations, it is inadequate to use the G-matrix interaction for Ξn parts in the four-body models, because the degrees of freedom are fully taken into account for n and Ξ: We use a simple single-channel Ξn interaction, as explained in Ref. [6].
Table 5 gives the calculated values of the Ξ − binding energies B Ξ − and the conversion widths Γ Ξ − for an αΞ − ground state, where the same value 0.9 fm −1 of k F is chosen in both cases of ESC08a' and ESC04d.αΞ − Coulomb interactions are taken into account.Here, we obtain the similar values of B Ξ − (αΞ − ) for ESC08a' and ESC04d.This means that the spin-and isospin-averaged αΞ interactions  Table 6 gives the values of B Ξ − and Γ Ξ − for 1 − and 2 − states of 12  Ξ − Be calculated with the ααtΞ − cluster model.Here, the k F values for ESC08a' and ESC04d are chosen so as to reproduce the B Ξ value of 2.2 MeV without the Coulomb interaction.This value is close to the s-state Ξ binding energy obtained from WS14.Thus, both the Gmatrix interactions derived from ESC08a' and ESC04d are found to be adjusted so as to be consistent with the Ξnucleus attractions indicated in the E885 experiment.

Now, let us show the result for the 7
Ξ − H.The core nucleus 6 He is in a bound state composed of α and weakly-07007-p.3EPJ Web of Conferences bound two neutrons.In Ref. [6], the αnnΞ system was demonstrated to be in a bound state when the αΞ folding interaction was attractive enough to make an αΞ bound state.Though the αΞ interactions derived from ESC04d and ESC08a' are similar to each other, their Ξ − n parts are very different: That for ESC08a' (ESC04d) and are substantially (very weakly) attractive.The calculated Ξ − energies in 7  Ξ − H for ESC04d and ESC08a' are given by solid lines in Fig. 1.The dashed lines show the (αΞ − ) cal + n + n threshold energies, (αΞ − ) cal being the calculated value of the αΞ − binding energy.The Ξ − state given by ESC08a' is found to be more bound than that by ESC04d.This is an example that the strong ΞN attraction in the T = 1 3 S 1 state works favorably to Ξ − states produced by (K − , K + ) reactions on available targets.The corresponding result for 10 Ξ − Li is given in Fig. 2. Here, the difference between the results for ESC04d and ESC08a' is found to be reduced in comparison with the 7 Ξ − H case.The reason is understood as follows: The main difference of the two interaction models is in their Ξ − n parts.There are two Ξ − n bonds in 7 Ξ − H, but only one bond in 10 Ξ − Li.It is very important that the T = 1 ΞN attractions are strong in the case of ESC08a/b.In the present experimental situation, the most promising production of Ξ − hypernuclei are by (K − , K + ) reactions.Then, produced Ξ − systems have to be in neutron-excess, because of ∆T z = 1 transfers on available nuclear targets.For such systems, the T = 1 ΞN attractions work favorably.On the basis of ESC08a/b, it is expected that there exist various light Ξ − bound systems other than 7  Ξ − H (αnnΞ − ) and 10 Ξ − Li (ααnΞ − ).For instance, a few-body bound systems such as nnΞ − , pnΞ − , pnnΞ − , αnΞ − and αtnΞ − might be produced by (K − , K + ) reactions on 3 H, 3 He, 4 He, 6 Li and 9 Be targets, respectively.

Fig. 2 .
Fig. 2. Calculated energies of the 2 − and 1 − states in 10 Ξ − Li are shown by solid lines.The dashed lines show the (ααΞ − ) cal + n threshold energies.

Table 4 .
The G-matrix interaction for ESC08a' represented asG(k F ; r) = i (a i + b i k F + c i k 2 F ) exp −(r/β i ) 2 .

Table 5 .
Calculated values of B Ξ − and Γ Ξ − for an αΞ − system.The parameter k F is taken as 0.9 fm −1 .Coulomb interactions are included.

Table 6 .
Calculated values of B Ξ − and Γ Ξ − for 1 − and 2 − states of 12 Ξ − Be (ααtΞ − ) in MeV.The values in parentheses are calculated without Coulomb interactions.