Abstract
Abstract. The hypernuclear systems N N Ξ and ΞΞN are considered as an analogue of nnp ( 3 H) nuclear system (with the notation as AAB system). We use the recently proposed modification for the s-wave Malfliet-Tjon potential. The modification simulates the Extended-Soft-Core model (ESC08c) for baryon-baryon interactions. The ΞN spin/isospin triplet (S, I) = (1, 1) potential generates a bound state with the energy B 2 (AB)=1.56 MeV. Three-body binding energy B 3 for the states with maximal total isospin is calculated employing the configuration-space Faddeev equations. Comparison with the results obtained within the integral representation for the equations is presented. The different types of the relation between B 2 and B 3 (V AA = 0) are discussed.

.
Introduction
The first Ξ-hypernuclear bound state has been reported in Ref. [1] . The lifetime of a Ξ-hypernucleus is long enough to enable the hypernuclear state to be well defined. According the current experimental data, the Ξ-nucleus interactions are attractive [2] . In particular, the hyprnucleus 12 Ξ Be can be interpreted by assuming a nucleus Wood-Saxon potential with a strength parameter of about -14 MeV [3] . Another hypernucleus 15 Ξ C is considered to be the cluster system 14 N(ground state) + Ξ, where Ξ can be in s or p-wave state [4] .
The stable states in the systems ΞN N and ΞΞN were recently predicted in Refs. [5, 6, 7] based on the recent update of the Extended-Soft-Core (ESC08c) model [8, 9, 10] for baryon-baryon interactions. This model has predicted the ΞN bound spin/isospin triplet (S, I) = (1, 1) state with three-body energy B 3 to be equal to 1.56 MeV. This bound state of proton and Ξ 0 or neutron and Ξ − has maximal isospin of the ΞN pair. For the three-body systems when all pairs N N , ΞN , and ΞΞ are in triplet isospin states, the strong decay N Ξ → ΛΛ is forbidden. Such three-body systems can be stable under the strong interaction. The first calculations [6, 7] based on the assumption yield the existence of bound states for the ΞN N and ΞΞN systems.
In the presented work, we use the differential Faddeev equations to mathematically formulate the bound state problems for the ΞN N and ΞΞN systems. The alternative treatment is presented in Refs. [6, 7] where the integral Faddeev equations were applied. Our calculations for the systems are generally in the agreement with the results [6, 7] . However, we found that a small correction for the results is needed. We present our results along with the correction [11] of the results published in Refs. [6, 7] . Additionally, the binding energy for the spin, isospin (0, 1) bound state for the ΞΞα system is calculated. This state was not considered in Refs. [6, 7] .
The models for ΞN N and ΞΞN (ΞΞα) are restricted by the s-wave approach. The coupling to higher-mass channels, ΣΛ and ΣΣ, does not taken into account assuming that their contributions have the second order of smallness to the binding energy of three-body system. The calculations do not also take into account the Coulomb force. 
Formalism

Faddeev equations for AAB system
The differential Faddeev equations [12] can be reduced to a simpler form for the case of two identical particles (like an AAB system). In this case the total wave function of the system is decomposed into the sum of the Faddeev components U and W corresponding to the (AA)B and (AB)B types of rearrangements: Ψ = U + W ± P W , where P is the permutation operator for two identical particles. In the latter expression the sign "+" corresponds to two identical bosons, while the sign "−" corresponds to two identical fermions, respectively. The set of the Faddeev equations is written as following:
Here, H 0 is the operator of kinetic energy of the Hamiltonian taken for corresponding Jacobi coordinates. The functions V AA and V AB describe the pair interactions between the particles. The model space is restricted to the states with the total angular momentum L = 0, the momentum of pair l = 0, and momentum λ = 0 of the third particle respectively to the center of mass of the pair.
s-wave approach
The description of the above mentioned AAB systems is distinguished by the masses of particles and the type of AA and AB interactions. We use s-wave V AA and V AB potentials, which are spin-isospin dependent. This requires to write Eq. (1) with the corresponding spin-isospin configurations.
The separation of spin-isospin variables leads to the Faddeev equations for the considering systems in the following form:
where matrices D and G are defined by the nuclear system under consideration, W is a column matrix with the singlet and triplet parts of the W component of the wave function of a nuclear system, and the exchange operator p acts on the coordinates of identical particles. For the 3 H nucleus, considered as pnn system in the state (S, I)=(1/2, −), we applied the isospin-less approach proposed in Ref. [13] . The inputs into Eq. (2) 
where W s and W t are the spin singlet and spin triplet parts of the W component. Within the isospin formalism when the protons and neutron are identical particles, instead of Eq. (2), which is a set of three equations, one has the set of two equations for the state (S, I)=(1/2, 1/2) of the three nucleon system N N N :
where
and p ± are the operators of cyclical permutations for coordinates of the particles.
Spin-isospin configurations
In Eq. (1), the Faddeev component U (and W ) of the total wave-function is expressed in terms of spin and isospin:
The graphical representation of the spin-isospin configurations in the ΞΞN and N N Ξ systems is given in Fig. 1 . Here, we have taken into account that the spin (isospin) basis of the spin (isospin) 3/2 state for three-body AAB system is formed by a single basis element. Thus, the Faddeev equations for each system considered have the form (2)- (3). The equation for the state ΞΞα (S, I)=(0, 1) has a "scalar form" instead the form (2)- (3):
where the V AB and W are scalars: V AB = v AB . Here, we used what the spin-isospin part of the wave function of the fermion pair ΞΞ is antisymmetric relatively to the permutation P in Eq. (1). Let us assume that V AA = 0, then Eq. (5) is reduced to a single equation:
This equation is similar to one obtained in Ref. [6] for the N N Ξ, (S, I)=(3/2, 3/2) state within the integral Faddeev equations. However, the right hand side of Eq. (6) has opposite sign. The restriction V AA = 0 corresponds to the situation when N N potential can be neglected for the spin/isospin triplet (S, I) = (1, 1) state. The differential Eq. (6) shows that the right hand side term is attractive (including attractive Ξα potential) and can give additional contribution into the binding energy coming from the left hand side term. The corresponding term for the N N Ξ, (S, I)=(3/2, 3/2) state is repulsive due to symmetry of 3/2 spin/isospin basis functions relatively permutation of two identical particles that holds the sign "minus" before the operator P in Eq. (1). The state N N Ξ, (S, I)=(3/2, 3/2) is unbound [6] .
Interactions
In this section we consider the two-body interactions, which are the inputs to our present study. To describe a nucleon-nucleon interaction, we use the semi-realistic Malfliet and Tjon MT I-III [14] potential with the modification from Ref. [15] . where the strength parameters are given in MeV and range parameters are given in fm −1 . The parameters were chosen in Ref. [14] to reproduce the experimental data for np-scattering. It has to be noted that we do not use isospin formalism for the nnp system. Thus, the protons and neutrons are not identical. The details of such treatment are presented in Ref. [13] . To take into account that the nn interaction is not equivalent to np interaction (that is known as the charge dependence of N N interaction), we have made modification of the spin singlet (S, I) = (0, 1) component of the MT I-III potential according Ref. [13] and have defined spin singlet nn potential. The modification was performed by scaling strength parameter. The scaling parameter γ is fixed as γ=0.975 to reproduce experimental nn scattering length for which we used the value of -18.8 fm [16, 17] . By this way, we have obtained three potentials v The parameters of the potentials were fixed to reproduce the scattering lengths and effective radii given by the ESC08c Nijmegen model for the baryon-baryon interaction [8, 9, 10] .
For the Ξα interaction we use the Isle-type potential [18] which has the Gaussian form:
V Ξα (r) = 450.4exp(−(r/1.269
with parameters from Ref. [19] .
.
Numerical results
The ground state binding energies B 3 of the N N N , nnp, N N Ξ, ΞΞN , ΞΞα systems were calculated using the models suggested above. The numerical results are presented in Table 1 . For each system, we show the spin-isospin state (S, I) and two-body energies B 2 (AA) and B 2 (AB) for AA and AB pairs. Additionally, we present the three-body binding energy calculated under the condition V AA = 0. Our results are compared with ones obtained within the integral representation of Refs. [7, 11] . One can see that results of both approaches are in the agreement with high accuracy. The "spin/isospin complication" [20] of the Faddeev equations for the considered systems is appeared by the matrix form of Eq. (3) and leads to the following evaluation for the three-body binding energy of the N N Ξ system in the spin-isospin states (S, I)=(3/2, 1/2), (1/2, 3/2):
The value of B 3 (V AA = 0) is restricted by 3.34 MeV. The similar result we have for the nnp system. In this case, B 3 (V AA = 0) is restricted by 4.46 MeV. In contrast, the scalar form (5) of Eq. (2) for the case ΞΞα (S, I)=(0, 1) leads to the relation:
This relation is known as the mass polarization effect which takes place when m B /m A > 1 [21, 22] . For the spin-isospin state (S, I)=(0, 1) of ΞΞα system, the mass polarization energy can be evaluated [20] . The contribution of this energy (B 3 (V AA = 0) − 2B 2 (AB))/B 3 (V AA = 0) in the three-body bound energy is equal 3.6% that is compatible with the values of 2%-4% [22, 20] for the similar nuclear system ΛΛα. The similarity takes place due to approximate equality of the masses of non-identical particles: Z) [21, 22] . Calculating the incremental energy, one can evaluate the strength of the ΛΛ interaction. For the system ΞΞN (1/2, 3/2), the energy includes significant contribution of the mass polarization energy because m N /m Ξ ∼ 1. Regardless that the relation (7) is not satisfied, the more appropriate value for an evaluation of the strength of the ΞΞ interaction in ΞΞN (1/2, 3/2) is the value of B 3 − B 3 (V ΞΞ = 0). The corresponding evaluation can be obtained from Table 1 . The ΞΞ interaction is attractive in ΞΞN (1/2, 3/2). Analogically, the spin singlet N N interaction is also attractive in the N N Ξ (1/2, 3/2) system. These attractive forces add about 1 Mev to the binding energies of the mirror systems. Thus, the matrix elements < Ψ|V AA |Ψ > have the close values for the systems. It is possible, because the ΞΞN system is more compact (larger B 3 value) and the ΞΞ potential has a minimum closer to the origin as is shown in Fig. 2 . The Faddeev components U , W for the N N Ξ (1/2, 3/2) and ΞΞN (1/2, 3/2) systems are presented in Fig. 3 . From the figure, one can see that the system ΞΞN (1/2, 3/2) is more compact than the N N Ξ (1/2, 3/2) system. For both systems, the rearrangement channel A + (AB) dominates due to existence of the isospin singlet ΞN bound state.
The mirror N N Ξ (1/2, 3/2) and ΞΞN (1/2, 3/2) systems under the condition V AA = 0 can be transformed "one into another" by changing the particle masses. The parameter ξ ≥ 0 sets this transformation by the formula: m (7) is obvious. To show this we have repeated the calculations for more deep spin triplet N Ξ potential. The potential has been scaled by the factor of 1.05. The result is shown in Fig. 4b ). The relation (7) is satisfied for all possible values ξ for this case.
It has to be noted that, as follows from Table 1 for the N N Ξ (3/2, 1/2) state, the three-body system having two bound subsystems has a deep bound state. The value of this N N Ξ (3/2, 1/2) binding energy is related with two-body energies as B 3 >> 2B 2 (AB) + B 2 (AA). Obviously, the V AA potential plays a key role for formation of the bound state. We assume that it is a general property of such three-body systems. 
Conclusions
We studied the hypernuclear system N N Ξ (and ΞΞN ) based on the configurationspace Faddeev equations. The baryon-baryon potential of ESC08c model, which generates the ΞN (S, I) = (1, 1) s-wave bound state, results in the stable states for these three-body systems. The stability relatively N Ξ → ΛΛ conversion is provided by fixing the states with maximal isospin. Our results and ones obtained within the integral Faddeev equation formalism [7, 11] are in agreement with high accuracy. Additionally, we have calculated the binding energy of the ΞΞα (S, I) = (0, 1) state. The relations between B 2 and B 3 (V AA = 0) were proposed for the "spin/isospin complicated" and "scalar" states. The corresponding relations are significantly different. .

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