PACS: 21.45.-v; 25.10.+s; 11.80.Jy

1. Introduction

Strange nuclear physics is a very topical subject. The hyperon-nucleon (*YN*) and hyperon-hyperon (*YY*) interactions constitute the input for microscopic calculations of few- and many-body systems involving strangeness, such as exotic neutron star matter ^{1}^{,}^{2}^{,}^{3}^{,}^{4}^{,}^{5} or hypernuclei ^{6}^{,}^{7}^{,}^{8}. There are theoretical debates ^{9}^{,}^{10}^{,}^{11}^{,}^{12}^{,}^{13}^{,}^{14} on the possible existence of a neutral bound state of two neutrons and a ^{15}. There have been also recent proposals regarding the stability of ^{14}, the existence of ^{6}^{,}^{7}^{,}^{8}, or the existence of a strangeness -2 hypertriton ^{16}^{,}^{17}. Obviously, all these predictions are subject to the uncertainties of our knowledge of the baryon-baryon interaction, in particular in the strangeness -2 sector. Experimentally, it has been recently reported an emulsion event, the so-called KISO event, providing evidence of a possible deeply bound state of ^{18}. Although microscopic calculations are impossible in this case and, consequently, their interpretation will be always affected by uncertainties, the ESC08c Nijmegen potential has been recently updated ^{19}^{,}^{20}^{,}^{21} to give account for the most recent experimental information of the strangeness -2 sector, the KISO ^{18} and the NAGARA^{22} events. A thorough discussion of the present status of the experimental and theoretical progress in hypernuclear physics can be found in Refs. ^{23} and ^{24}.

When a two-baryon interaction is attractive, if the system is merged with nuclear matter and the Pauli principle does not impose severe restrictions, the attraction may be reinforced. Simple examples of the effect of a third or a fourth baryon in two-baryon systems could be given. The deuteron, (I)*J*
^{
P
} = (0) 1^{+}, is bound by 2.225 MeV, while the triton, (I)*J*
^{
P
} = (1/2)1/2^{+}, is bound by 8.480 MeV, and the *J*
^{
P
} = (0)0^{+}, is bound by 28.295 MeV. The binding per nucleon *B*/*A* increases as 1: 3: 7. A similar argument could be employed for strangeness -1 systems. Whereas the existence of dibaryon states is still under discussion^{i}, the hypertriton *J*
^{
P
} = (0)1/2^{+}, is bound with a separation energy of 130 ± 50keV keV, and the *J*
^{
P
} = (0)0^{+}, is bound with a separation energy of 2.12 ± 0.01 (stat) ± 0.09 (syst) MeV ^{26}. This cooperative effect of the attraction in the two-body subsystems when merged in few-baryon states was also made evident in the prediction of a *J*
^{
P
} = (1)1/2^{+} channel very near threshold ^{27}^{,}^{28}. Such ^{29}.

In this paper, we review our recent studies of the three-body systems: ^{19}^{,}^{20}^{,}^{30} accounting for the recent KISO ^{18} and NAGARA ^{22} events in the strangeness -2 sector. As discussed above, the existence of two-body attractive interactions or bound states could give rise to other stable few-body systems when merged with other nucleons or hyperons. For example, the overall attractive character of the ^{31} together with other indications of certain emulsion data ^{20}^{,}^{21}^{,}^{30}. Besides the recent update of ESC08c Nijmegen model, ^{32} and chiral quark models ^{33} found a ^{19}^{,}^{30}^{,}^{34}^{,}^{35}^{,}^{36}^{,}^{37}^{,}^{38}. It is worth to mention that preliminary studies of the ^{39} indicate that lattice QCD calculations of multibaryon systems are now within sight. Analogously, if a second ^{22} and the reinforcement of the ^{14}.

One should bear in mind how delicate is the few-body problem in the regime of weak binding, as demonstrated in Ref. ^{40} for the *YN* interaction, like the hybrid quark-model based analysis of Ref. ^{41}, the effective field theory approach of Ref.^{42}, or even some of the earlier models of the Nijmegen group ^{34} that, in general, predict interactions weakly attractive or repulsive. One does not expect that these models will give rise to stable three- or four-body states. However, it is worth to emphasize that current hypernuclei studies ^{6}^{,}^{7}^{,}^{8}^{,}^{32}^{,}^{40} have been performed by means of interactions derived from the Nijmegen models and, thus, the present review complements such previous work for the simplest systems that can be studied exactly. To advance in the knowledge of the details of the *YN* interaction, high-resolution spectroscopy of ^{12}C targets in (K^{-}, K^{+}) reactions has been awaited ^{43}^{,}^{44} and it is now planned at J-PARC ^{45}. The new hybrid experiment *E*07 recently approved at J-PARC is expected to record of the order of 10^{4}^{46}, one order of magnitude larger than the previous *E*373 experiment, and will hopefully clarify the phenomenology of some of the systems studied in the present work.

The review is organized as follows. In Sec. 2 we describe the technical details to solve the three-body bound state Faddeev equations as well as the generalized Gaussian variational method used to look for bound states of the four-body problem. In Sec. 3 we construct the two-body amplitudes needed for the solution of the bound state three- and four-body problems. The results are presented and discussed in Sec. 4. Finally, in Sec. 5 we summarize our main conclusions.

2. The three- and four-body bound-state problems

In this section we outline the solution of the three- and four-body bound-state problems. We will restrict ourselves to configurations where all particles are in *S*-wave states. The three-body problem has been widely discussed in the literature and we refer the reader to Refs. ^{47}^{,}^{48}^{,}^{49} for a more detailed discussion. The Faddeev equations for a system with total isospin *I* and total spin *J* are,

where *i*
_{
i
} and spin *j*
_{
i
}
*p*
_{
i
} is the momentum of the pair *jk* (with *ijk* an even permutation of 123) and *q*
_{
i
} the momentum of particle *i* with respect to the pair *jk*. *v*
_{
i
} are the corresponding reduced masses, and

Expanding the amplitude

where

with

The four-body problem has been addressed by means of the variational method, specially suited for studying low-lying states. The nonrelativistic hamiltonian is be given by,

where the potential

The variational wave function must include all possible spin-isospin channels contributing to a given configuration. For each channel *s*, the wave function will be the tensor product of a spin (

where

where the eigenvalues are obtained by a minimization procedure.

For the description of the four-body wave function we consider the Jacobi coordinates:

The total wave function should have well-defined permutation properties under the exchange of identical particles. The spin part can be written as,

where the spin of the two *N*’s (*Y*’s) is coupled to *S*
_{12} (*S*
_{34}). Two identical spin-1/2 fermions in a S = 0 state are antisymmetric (*A*) under permutations while those coupled to *S* = 1 are symmetric (*S*). We summarize in Table I the corresponding vectors for each total spin together with their symmetry properties^{ii}

The most general radial wave function with total orbital angular momentum L = 0 may depend on the six scalar quantities that can be constructed with the Jacobi coordinates of the system, they are: *generalized Gaussians*,

where *n* is the number of Gaussians used for each spin-isospin component. *n* × *n*
_{
s
} variational parameters, where *n*
_{
s
} is the number of different channels allowed by the Pauli principle. Eq. (10) should have well-defined permutation symmetry under the exchange of both *N*’s and *Y*’s,

where *P*
_{x} and *P*
_{y} are -1 for antisymmetric states, (*A*), and +1 for symmetric ones, (*S*). Thus, one can build the following radial combinations, (*P*
_{
x
}
*P*
_{
y
} ) = (*SS*), (*SA*), (*AS*), and (*AA*):

The last equations can be expressed in a compact manner by defining the following function,

and the vectors

and

which allows to write Eqs. (13)-(15) as,

The radial wave function includes all possible internal relative orbital angular momenta coupled to *L* = 0. It has also well-defined symmetry properties on the

To evaluate radial matrix elements we use the notation introduced in Eq. (19):

where *F*
^{
ij
} is a matrix whose element

being

where we have shortened the previous notation according to

where the functions *f* (*x*, *y*, *z*) are the potentials. Being all of them radial functions (not depending on angular variables) one can solve the previous integral by noting:

where

One can extract some useful relations for the radial matrix elements using simple symmetry properties. Let us rewrite Eq. (21)

If *f* (*x*, *y*, *z*) depends only in one coordinate, for example

The radial wave function described in this section is adequate to describe not only bound states, but also it is flexible enough to describe states of the continuum within a reasonable accuracy ^{50}^{,}^{51}^{,}^{52}.

3. Two-body amplitudes

We have constructed the two-body amplitudes for all subsystems entering the three- and four-body problems studied by solving the Lippmann-Schwinger equation of each (*i*, *j*) channel,

where

and the two-body potentials consist of an attractive and a repulsive Yukawa term, *i.e.*,

The parameters of the ^{19}, -2 ^{20} and -3 and -4 ^{30} ESC08c Nijmegen potentials. In the case of the *NN* interaction we use the Malfliet-Tjon models ^{53} with the parameters given in Ref. ^{54}. The low-energy data and the parameters of these models are given in Table II. It is worth to note that the scattering length and effective range of the most recent update of the ^{42}) unlike the earlier version used in Ref. ^{14} (see Table IV of Ref. ^{55}) reporting remarkably small effective ranges.

The ^{1}
*S*
_{0} (*I* = 0) potential was fitted to the ^{20} without taking into account the inelasticity, *i.e.*, assuming *NN* and *YN* scattering data, supplied with constraints on the *YN* and *YY* interaction originating from the G-matrix information on hypernuclei ^{19}.

The potentials obtained are shown in Fig. 1. In Fig. 1(a) we show the ^{3}
*S*
_{1} (*I* = 1) ^{*} bound state ^{18} with a binding energy of 1.6 MeV. We also confirm how all the *J* = 1 and *I* = 1 ^{iii}
^{30}. Regarding the ^{1}
*S*
_{0} (*I* = 1) potential, that although having bound states in earlier versions of the ESC08c Nijmegen potential ^{34}, in the most recent update of the strangeness -4 sector it does not present a bound state ^{30}. The existence of bound states in the ^{35}^{,}^{36}^{,}^{37}. It can be definitively stated that all models agree on the fairly important attractive character of this channel, either with or without a bound state ^{38}. Finally, in Fig. 1(d) we show the *NN* and *YN* data, and SU(3) symmetry ^{20}^{,}^{21}. It gives account of the pivotal results of strangeness -2 physics, the NAGARA ^{22} and the KISO ^{18} events. Although other double-^{43}, are not explicitly taken into account, the G-matrix nuclear matter study of ^{12}C and ^{14}N (see section VII of Ref. ^{20}), concludes that the

4. Results and discussion

Let us first of all show the reliability of the input potentials. We compare in Fig. 2 the ^{1}
*S*
_{0} (*I* = 0) potential was fitted to the ^{20}. Once we have described the phase shifts, the ^{iv}. We have also tested the two-body interactions in the three-body problem of systems made of *N*’s and

The reasonable description of the known two- and three-body problems gives confidence to address the study of other three- and four-body systems. We show in Table III the channels of the different two-body subsystems contributing to each (*I*, *J*) three- and four-body state that we will study. For the *I* = 2 channels, because the *I* = 0 and

4.1. Three-body systems

We show in Fig. 3 the Fredholm determinant of all ^{59}^{,}^{60}. As we can see in Fig. 3(b), a bound state is found for the (*I*)*J*
^{
P
} = (3/2)1^{+}/2 *B*
_{2} is the binding energy of the D^{*}^{+}/2 state, which would make it easy to identify experimentally as a sharp resonance lying some 17.2 MeV below the *I*)*J*
^{
P
} = (1/2)3^{+}/2 state in a pure *S*-wave configuration ^{60}. One would need at least the spectator nucleon to be in a *D* -wave or that the *P*-wave channels, with the nucleon spectator also in a *P*-wave. Thus, due to the angular momentum barriers the resulting decay width of the (1/2)3^{+}/2 state is expected to be very small.

For the *i*, *j*) = (1, 1) ^{27},

Thus, due to the negative sign in the r.h.s. the ^{61}. This is the reason why the Fredholm determinant for the (*I*, *J*) = (3/2, 3/2)

Finally, we show in Fig. 4 the Fredholm determinant of all *I*)*J*
^{
P
} = (3/2)3/2^{+} channel is not shown in Fig. 4(b) for the same reason explained above for the *I*)*J*
^{
P
} = (3/2)1^{+}/2, 2.9 MeV below the lowest threshold, *B*
_{2} stands for the binding energy of the D^{*}^{1}
*S*
_{0}(*I* = 1) potential has not bound states, as it is however predicted by several models in the literature. If bound states would exist for the *I* = 1/2 channels are also attractive but they are not bound.

Let us finally mention that our results for three-body systems containing a ^{62}.

4.2. Four-body systems

In the previous section we have seen that all three-body systems made of *N*’s and *i.e*., systems consisting only of neutrons and negative *NN*, *N*’s and *I* = 2. The most favorable configuration to minimize the effect of the Pauli principle is the *N*’s and two *J* = 0 ^{63}.

The binding energy of the ^{64}^{,}^{65}^{,}^{66} and tested against the hyperspherical harmonic formalism with comparable results ^{51}^{,}^{52}. We show in Fig. 5 the binding energy of the (*I*)*J*
^{
P
} = (2)0^{+}*B*
_{2} = 3.2 MeV, where *B*
_{2} is the binding energy of the D^{*}^{*}

One can also study the behavior of the root mean square radius (RMS) of the four-body system, defined in the usual way,

The results are shown in Fig. 6, where besides the RMS radius we have also calculated the root mean square radii of the different Jacobi coordinates. As seen in Table III, only the ^{1}S_{0}(*I* = 1) *NN* and *I*)*J*
^{
P
} = (2)0^{+}^{1}S_{0}(*I* = 1) *NN* and ^{1}S_{0}(*I* = 1) and ^{3}S_{1}(*I* = 1) channels, the last one presenting the *D*
^{*} bound state, which is the responsible of the smallest radius in the

We have finally evaluated the binding energy of the *I*)*J*
^{
P
} = (1)0^{+}^{67}. The system is unbound appearing just above threshold and thus it does not seem to be Borromean, a four-body bound state without two- or three-body stable subsystems. An unbound result was also reported in Ref. ^{68}, although in this case the authors made use of repulsive gaussian-type potentials for any of the two-body subsystems (see the figure on pag. 475) what does not allow for the existence of any bound state.

We have studied the dependence of the binding on the strength of the attractive part of the different two-body interactions entering the four-body problem. For this purpose we have used the following interactions,

with the same parameters given in Table II. The system hardly gets bound for a reasonable increase of the strength of the the ^{20}, however, one needs ^{40} and this modification would produce an almost *NN* interaction. However, such modification would make the ^{1}
*S*
_{0}
*NN* potential as strong as the ^{3}
*S*
_{1}^{53} and thus the singlet *S*-wave would develop a dineutron bound state, ^{1}
*S*
_{0} and ^{3}
*S*
_{1}. We show in Fig. 7 the binding energy of the (*I*)*J*
^{
P
} = (1)0^{+}

Reference ^{14} tackled the same problem by fitting low-energy parameters of older versions of the Nijmegen-RIKEN potential ^{30}^{,}^{69} or chiral effective field theory ^{55}^{,}^{70}, by means of a single Yukawa attractive term or a Morse parametrization. The method used to solve the four-body problem is similar to the one we have used in our calculation, thus the results might be directly comparable. Our improved description of the two- and three-body subsystems and the introduction of the repulsive barrier for the ^{1}
*S*
_{0}
*NN* partial wave, relevant for the study of the triton binding energy (see Table II of Ref. ^{71}), leads to a four-body state just above threshold, that cannot get bound by a reliable modification in the two-body subsystems. As clearly explained in Ref. ^{14}, the window of Borromean binding is more an more reduced for potentials with harder inner cores.

5. Summary

This manuscript intends to summarize our recent work on few-body systems made of *N*’s, *I*)*J*
^{
P
} = (1/2)1/2^{+}, is bound by 144 keV, and the recently discussed *I*)*J*
^{
P
} = (1/2)1/2^{+} system is unbound. We have found that the *I*)*J*
^{
P
} = (3/2)1/2^{+} and (1/2)3/2^{+}, the last one being a deeply bound state lying 15 MeV below the *I*)*J*
^{
P
} = (3/2)1/2^{+}, in spite of having used the most recent update of the ESC08c Nijmegen potential that does not predict *N*’s and *I*)*J*
^{
P
} = (2)0^{+}, lying 7.4 MeV below the *I*)*J*
^{
P
} = (1)0^{+}