The ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ system

TL;DR

This study investigates whether a hypothetical hypernucleus, ${}_{\Lambda\Lambda}^{4}n$, can exist as a bound state, using realistic nuclear interactions, and concludes it is likely unbound, just above the threshold.

Contribution

The paper applies updated Nijmegen potentials and Yukawa-type interactions to assess the existence of the ${}_{\Lambda\Lambda}^{4}n$ hypernucleus, providing new insights into its unbound nature.

Findings

${}_{\Lambda\Lambda}^{4}n$ is unbound, just above threshold.

The $^1S_0$ $nn$ repulsive term influences binding.

Updated potentials do not support bound ${}_{\Lambda\Lambda}^{4}n$.

Abstract

Using local central Yukawa-type Malfliet-Tjon interactions reproducing the low-energy parameters and phase shifs of the $nn$ system and the latest updates of the $n\Lambda$ and $\Lambda\Lambda$ Nijmegen ESC08c potentials we study the possible existence of a ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ bound state. Our results indicate that the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ is unbound, being just above threshold. We discuss the role played by the $^1S_0$ $nn$ repulsive term of the Yukawa-type Malfliet-Tjon interaction.

Tables

Table 1. Table 1: Low-energy parameters and parameters of the local central Yukawa-type potentials given by Eq. ( 3 ) for the N ​ N 𝑁 𝑁 NN   Gib90 , Λ ​ N Λ 𝑁 \Lambda N   Nae15 , and Λ ​ Λ Λ Λ \Lambda\Lambda   Nag15 systems contributing to the ( I ) ​ J P = ( 1 ) ​ 0 + 𝐼 superscript 𝐽 𝑃 1 superscript 0 (I)J^{P}=(1)0^{+} n Λ ​ Λ     4 subscript superscript 𝑛 4 Λ Λ {}_{\Lambda\Lambda}^{\,\,\,\,4}n state. See text for details.

	$(i,j)$	$a\left(\mathrm{fm}\right)$	$r_0\left(\mathrm{fm}\right)$	$A$(MeV fm)	${\mu }_A({\mathrm{fm}}^{-1}$)	$B$(MeV fm)	${\mu }_B\left({\mathrm{fm}}^{-1}\right)$
$NN$	$(1,0)$	$-23.56$	$2.88$	$513.968$	$1.55$	$1438.72$	$3.11$
$\Lambda N$	$(1/2,0)$	$-2.62$	$3.17$	$416$	$1.77$	$1098$	$3.33$
	$(1/2,1)$	$-1.72$	$3.50$	$339$	$1.87$	$968$	$3.73$
$\Lambda \Lambda$	$(0,0)$	$-0.853$	$5.126$	$121$	$1.74$	$926$	$6.04$

Equations

$$t^{ij}(p,p^{\prime };e)=V^{ij}(p,p^{\prime })+{\int }_0^{\infty }{p^{\prime \prime }}^{2}d{p}^{\prime \prime }{V}^{ij}(p,p^{\prime \prime })\frac{1}{e-{p^{\prime \prime }}^2/2\mu }{t}^{ij}(p^{\prime \prime },p^{\prime };e),$$

$$V^{ij}(p,p^{\prime })=\frac{2}{\pi }{\int }_0^{\infty }{r}^{2}dr{j}_0(pr)V^{ij}(r)j_0(p^{\prime }r),$$

$$V^{ij}(r)=-A\frac{e^{-{\mu }_{A}r}}{r}+B\frac{e^{-{\mu }_{B}r}}{r}.$$

$$H=\sum _{i=1}^4\frac{{\vec{p}}_i^2}{2m_i}+\sum _{i<j=1}^{4}V({\vec{r}}_{ij}),$$

$$|{\varphi }_s⟩=|S_{s_1}⟩\otimes |I_{s_2}⟩\otimes |R_{s_3}⟩,$$

$$\sum _{s^{\prime }s}\sum _i{\beta }_{s_3}^{(i)}[⟨R_{s_3^{\prime }}^{(j)}|H|R_{s_3}^{(i)}⟩-E⟨R_{s_3^{\prime }}^{(j)}|R_{s_3}^{(i)}⟩{\delta }_{s,s^{\prime }}]=0\forall j,$$

$${\vec{r}}_{NN}=\vec{x}={\vec{r}}_1-{\vec{r}}_2,$$

$${\vec{r}}_{\Lambda \Lambda }=\vec{y}={\vec{r}}_3-{\vec{r}}_4,$$

$${\vec{r}}_{NN-\Lambda \Lambda }=\vec{z}=\frac{1}{2}\left({\vec{r}}_1+{\vec{r}}_2\right)-\frac{1}{2}\left({\vec{r}}_3+{\vec{r}}_4\right),$$

$${\vec{R}}_{\mathrm{CM}}=\vec{R}=\frac{\sum m_i{\vec{r}}_i}{\sum m_i}.$$

$$|R_{s_3}⟩=\sum _{i=1}^n{\beta }_{s_3}^{(i)}{R}_{s_3}^i(\vec{x},\vec{y},\vec{z})=\sum _{i=1}^n{\beta }_{s_3}^{(i)}{R}_{s_3}^i,$$

$$P_{12}(\vec{x}\to -\vec{x})R_{s_3}^i$$

$$P_{34}(\vec{y}\to -\vec{y})R_{s_3}^i$$

$$g(s_1,s_2,s_3)=\mathrm{Exp}\left(-a_s^i{\vec{x}}^2-b_s^i{\vec{y}}^2-c_s^i{\vec{z}}^2-s_1{d}_s^i\vec{x}\cdot \vec{y}-s_2{e}_s^i\vec{x}\cdot \vec{z}-s_3{f}_s^i\vec{y}\cdot \vec{z}\right),$$

$$\vec{G}_{s}^{i}=\left(\begin{array}[]{l}g(+,+,+)\\ g(-,+,-)\\ g(-,-,+)\\ g(+,-,-)\end{array}\right)\,,$$

$${\vec{\alpha }}_{SS}$$

$${\vec{\alpha }}_{SA}$$

$${\vec{\alpha }}_{AS}$$

$${\vec{\alpha }}_{AA}$$

$$(SS)$$

$$(SA)$$

$$(AS)$$

$$(AA)$$

$$V^{B_1{B}_2}(r)=-g_{B_1{B}_2}A\frac{e^{-{\mu }_{A}r}}{r}+B\frac{e^{-{\mu }_{B}r}}{r}$$

Full text

baryon-baryon interactions, few-body systems

I Introduction

It is well-established the non-existence of two-body bound states made of neutrons and/or $\Lambda$’s, the lightest hyperon. The situation is much more cumbersome for three-, four- and in general few-body systems made of nucleons and hyperons Ric15 . For example, it has been proposed that dineutrons could become bound in the presence of additional nucleons Mig73 . This is the mechanism responsible for the properties of some bound nuclei that have a neutron excess, like 11Li, where a pair of external neutrons form a remote halo around the core of 9Li Ber16 . Such possibility has been recently drawn in a lighter system by the experimental HypHI Collaboration Rap13 , suggesting the existence of a neutral bound state of two neutrons and a $\Lambda$ hyperon, ${}^{3}_{\Lambda}n$. They analyze the experimental data obtained from the reaction 6Li +12C at 2A GeV to study the invariant mass distribution of $d+\pi^{-}$ and $t+\pi^{-}$. The signal observed in the invariant mass distributions of $d+\pi^{-}$ and $t+\pi^{-}$ final states was attributed to a strangeness-changing weak process corresponding to the two- and three-body decays of an unknown bound state of two neutrons associated with a $\Lambda$, ${}^{3}_{\Lambda}n$, via ${}^{3}_{\Lambda}n\to t+\pi^{-}$ and ${}^{3}_{\Lambda}n\to t^{*}+\pi^{-}\to d+n+\pi^{-}$. This is an intriguing conclusion since one would naively expect the $nn\Lambda$ system to be unbound. In the $nn\Lambda$ system the two nucleons interact in the ${}^{1}S_{0}$ partial wave while in the $np\Lambda$ system they interact in the ${}^{3}S_{1}$ partial wave. Thus, since the nucleon-nucleon ($NN$) interaction in the ${}^{1}S_{0}$ channel is weaker than the ${}^{3}S_{1}$ channel, and the $np\Lambda$ system is bound by only 0.13 MeV, one may have anticipated that the $nn\Lambda$ system should be unbound. The absence of binding of the $nn\Lambda$ system was first demonstrated by Dalitz and Downs Dal58 using a variational approach, and later from the solution of the Faddeev equations with separable interactions Gar87 . The theoretical debate on the possible existence of a neutral bound state of two neutrons and a $\Lambda$ hyperon, ${}^{3}_{\Lambda}n$, is still open and has lately deserved an important theoretical effort Gar14 ; Hiy14 ; Gal14 ; And15 ; Afn15 ; Ric15 .

In the four-body case, the analysis of the missing-mass spectrum in the double-charge-exchange reaction 4He(8He,8Be) at 186 MeV/u has unveiled the possible existence of a tetraneutron resonance $0.83\pm 0.65\,{\rm(stat)}\,\pm 1.25\,{\rm(syst)}$ MeV above the threshold of four-neutron decay with a significance level of 4.9$\,\sigma$ Kis16 . In 2002 one collaboration claimed to have found a bound tetraneutron in a 14Be breakup reaction Mar02 . This result remains unconfirmed, and theorists quickly showed that based on the best knowledge of the $NN$ interaction the existence of a bound tetraneutron was nearly impossible, although they could not rule out the existence of a short-lived resonant state on the basis of a dineutron-dineutron structure Ber03 ; Pie03 ; Tim03 ; Laz05 . The stability of a tetraneutron state cannot be established even with potentials made artificially deeper to produce a dineutron bound state (the dineutron is a virtual state 66 keV above the two-neutron threshold), due to the Pauli principle which forbids two identical fermions from occupying the same quantum state. For four-neutrons only one pair can be in the lowest-energy state, forcing the second pair into a state of higher energy, thereby making the tetraneutron unstable. Thus, one could think of the stability of a modified tetraneutron with Bose statistics, where a pair of neutrons is replaced by a pair of neutral light baryons enforcing in this way antisymmetrization with all particles in the lowest-energy state. This is the case of the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ = $(n,n,\Lambda,\Lambda)$ recently discussed in Ref. Ric15 and suggested as a possible Borromean state.

The relevance of the addition of further baryons on an almost bound two-body system has also been discussed recently by some of us looking for stable bound states of $N$’s and $\Xi$’s. In Ref. Gar15 we pointed out that 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 $\alpha$ particle, $(I)J^{P}=(0)0^{+}$, is bound by $28.295$ MeV. The binding per nucleon $B/A$ increases from $1:3:7$. A similar argument could be employed for strangeness $-1$ systems. Whereas there is no evidence for dibaryon states, the hypertriton ${}^{3}_{\Lambda}$H, $(I)J^{P}=(0)1/2^{+}$, is bound with a separation energy of $130\pm 50$ keV, and the ${}^{4}_{\Lambda}$H, $(I)J^{P}=(0)0^{+}$, is bound with a separation energy of $2.12\pm 0.01\,{\rm(stat)}\,\pm 0.09\,{\rm(syst)}$ MeV Ess15 . 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 $\Sigma NN$ quasibound state in the $(I)J^{P}=(1)1/2^{+}$ channel very near threshold Gar07 . Such $\Sigma NN$ quasibound state has been recently suggested in ${}^{3}\rm{He}(K^{-},\pi^{\mp})$ reactions at 600 MeV/c Har14 .

Thus, if a second $\Lambda$ would be added to the uncertain $nn\Lambda$ state, the weakly attractive $\Lambda\Lambda$ interaction Tak01 and the reinforcement of the $N\Lambda$ potential without paying a price for antisymmetry requirements, may give rise to a stable bound state. This would be our goal in this paper, to address the study of the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ state making use of potentials compatible with the low-energy data and phase-shifts of the $nn$, $n\Lambda$, and $\Lambda\Lambda$ systems. A first examination of this problem has been presented in Ref. Ric15 based on potentials with a single Yukawa attractive term or a Morse parametrization.

II Two-body interactions

For the identical pairs, $nn$ and $\Lambda\Lambda$, the S wave interaction is in the ${}^{1}S_{0}$ channel due to the Pauli principle, while for the $N\Lambda$ pair both ${}^{1}S_{0}$ and ${}^{3}S_{1}$ channels contribute. As it is well-known, the $NN$ ${}^{1}S_{0}$ channel is almost bound, the virtual state lying slightly below the $nn$ threshold in the unphysical sheet. In the case of the $NN$ ${}^{1}S_{0}$ channel we use the Malfliet-Tjon I model Mal69 with the parameters given in Ref. Gib90 . For the two-body interactions containing $\Lambda$’s, $N\Lambda$ and $\Lambda\Lambda$, we use the most recent update on the ESC08c Nijmegen potentials Nae15 ; Nag15 ; Rij16 . Regarding the two-body interactions containing a single $\Lambda$, they are constrained by a simultaneous fit to the combined $NN$ and $YN$ scattering data, supplied with constraints on the $YN$ and $YY$ interaction originating from the G-matrix information on hypernuclei Nae15 . The $\Lambda\Lambda$ strangeness $-2$ interaction is mainly determined by the $NN$ and $YN$ data, and SU(3) symmetry Nag15 ; Rij16 . It gives account of the pivotal results of strangeness $-2$ physics, the NAGARA Tak01 and the KISO Naa15 events. Although other double-$\Lambda$ hypernuclei events, like the DEMACHIYANAGI and HIDA events Nak10 , are not explicitly taken into account, the G-matrix nuclear matter study of $\Xi^{-}$ capture both in 12C and 14N (see section VII of Ref. Nag15 ), concludes that the $\Xi N$ attraction in the ESC08c potential is consistent with the $\Xi$-nucleus binding energies given by the emulsion data of the twin $\Lambda$-hypernuclei.

We have constructed the two-body amplitudes for all subsystems entering the four-body problem studied by solving the Lippmann–Schwinger equation of each $(i,j)$ channel,

[TABLE]

where

[TABLE]

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

[TABLE]

The parameters of the $\Lambda N$ and $\Lambda\Lambda$ channels were obtained by fitting the low-energy data and the phase-shifts of each channel as given in the most recent update of the strangeness $-1$ Nae15 and $-2$ Nag15 ESC08c Nijmegen potential. The low-energy data and the parameters of these models, together with those of the $NN$ interaction from Ref. Gib90 , are given in Table 1. It is worth to note that the scattering length and effective range of the most recent updates of the $\Lambda\Lambda$ interaction derived from chiral effective field theories are very much like those of the ESC08c Nijmegen potential (see Table 2 of Ref. Hai16 ) unlike the earlier version used in Ref. Ric15 (see Table 4 of Ref. Pol07 ) reporting remarkably small effective ranges.

If it is assumed that only singlet and triplet S wave contribute in the two-particle channel, the parametrization of the $NN$ interaction used in this work, set III for the triplet partial wave and set I for the singlet partial wave, gives a triton binding energy of 8.3 MeV Mal69 . The effect of the repulsive core on the singlet two-body channel is crucial to get this result, while the repulsion on the triplet two-body channel has almost no effect on the binding. In fact, if the repulsive core in the singlet partial wave is not considered the triton gains around 2 MeV of binding (see Table II of Ref. Mal70 ). Based on predictions for separable potentials, in Ref. Mal69 it is suggested that the inclusion of the tensor force in the triplet interaction changes the binding energy by 0.3 MeV. Indeed, this is the result obtained in Ref. Fuj02 , where as can be seen in Table III a five channel calculation (S and D partial waves) differs from a two channel calculation (only S partial waves) about 0.3 MeV. The influence of local tensor forces in Malfliet-Tjon Yukawa type interactions has also been studied in Ref. Maf69 , showing that the inclusion of tensor forces reduces the binding energy of the three-body problem by 1 to 1.5 MeV, depending on the D wave percentage. Thus, the local Yukawa-type potentials with tensor interaction would lack binding in the three-body problem at difference of separable potentials that would drive to overbinding Phi68 . Note that in the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ the $NN$ ${}^{3}S_{1}$ partial wave does not contribute, thus although this system is free of any uncertainty related to the triplet partial wave, the repulsive core on the singlet $NN$ channel might play some role.

III The four-body problem

The four-body problem has been addressed by means of a generalized variational method. The nonrelativistic hamiltonian will be given by,

[TABLE]

where the potentials $V(\vec{r}_{ij})$ have been discussed in the previous section. For each channel $s$, the variational wave function will be the tensor product of a spin ($\left|S_{s_{1}}\right>$), isospin ($\left|I_{s_{2}}\right>$), and radial ($\left|R_{s_{3}}\right>$) component,

[TABLE]

where $s\equiv\{s_{1},s_{2},s_{3}\}$. Once the spin and isospin parts are integrated out, the coefficients of the radial wave function are obtained by solving the system of linear equations,

[TABLE]

where the eigenvalues are obtained by a minimization procedure.

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

[TABLE]

The total wave function should have well-defined permutation properties under the exchange of identical particles. The most general S wave radial wave function may depend on the six scalar quantities that can be constructed with the Jacobi coordinates of the system, they are: $\vec{x}^{\,2}$, $\vec{y}^{\,2}$, $\vec{z}^{\,2}$, $\vec{x}\cdot\vec{y}$, $\vec{x}\cdot\vec{z}$ and $\vec{y}\cdot\vec{z}$. We define the variational spatial wave function as a linear combination of generalized Gaussians,

[TABLE]

where $n$ is the number of Gaussians used for each spin-isospin component. $R_{s_{3}}^{i}$ depends on six variational parameters: $a^{i}_{s}$, $b^{i}_{s}$, $c^{i}_{s}$, $d^{i}_{s}$, $e^{i}_{s}$, and $f^{i}_{s}$, one for each scalar quantity. Therefore, the four-body system will depend on $6\times n\times n_{s}$ variational parameters, where $n_{s}$ is the number of different channels allowed by the Pauli principle. Eq. (8) should have well defined permutation symmetry under the exchange of both $N$’s and $\Lambda$’s,

[TABLE]

where $P_{x}$ and $P_{y}$ are $-1$ for antisymmetric states, $(A)$, and $+1$ for symmetric ones, $(S)$.

If we now define the function,

[TABLE]

and the vectors,

[TABLE]

and

[TABLE]

we can build any symmetry for the radial wave function, $(P_{x}P_{y})=(SS)$, $(SA)$, $(AS)$ and $(AA)$,

[TABLE]

including all possible relative orbital angular momenta coupled to an S wave. 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 Suz98 ; Vij09 ; Via09 .

The numerical method described in this section has been successfully tested in different few-body calculations in comparison with the hyperspherical harmonic formalism, see for example Refs. Via09 ; Vin09 , or the stochastic variational approach of Ref. Suz98 for some of the results presented in Ref. Vij16 .

IV Results and discussion

Let us first of all show the reliability of the input potentials. We compare in Fig. 1 the $N\Lambda$ and $\Lambda\Lambda$ phase shifts reported by the ESC08c Nijmegen potential and those obtained by our fits with the two-body potential of Eq. (3) and the parameters given in Table 1. As can be seen the agreement is good. Once we have described the phase shifts, the $N\Lambda$ and $\Lambda\Lambda$ potentials include in an effective manner the coupling to other two-body channels as it may be the $N\Sigma$ or $N\Xi$ two-body systems.

We have also tested the two-body interactions in the three-body problem of systems made of $N$’s and $\Lambda$’s. The hypertriton is bound by 144 keV, and the $nn\Lambda$ system is unbound. The reasonable description on the two- and three-body problem gives confidence to address the study of the $nn\Lambda\Lambda$ state.

Using the variational method described in the last section, we have evaluated the binding energy of the $nn\Lambda\Lambda$ system with quantum numbers $(I)J^{P}=(1)0^{+}$. 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. Lek14 , 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,

[TABLE]

with the same parameters given in Table 1. The system hardly gets bound for a reasonable increase of the strength of the attractive part of the $\Lambda\Lambda$ interaction, $g_{\Lambda\Lambda}$. Although one cannot exclude that the genuine $\Lambda\Lambda$ interaction in dilute states as the one studied here could be slightly stronger that the one reported in Ref. Nag15 , however, one needs $g_{\Lambda\Lambda}\geq 1.8$ to get a bound state, what would destroy the agreement with the ESC08c Nijmegen $\Lambda\Lambda$ phase shifts. Note also that this is also a very sensitive parameter for the study of double-$\Lambda$ hypernuclei Nem03 . Taking a factor $1.2$ in the attractive part of the ${}^{1}S_{0}$ $NN$ interaction, that would make the ${}^{1}S_{0}$ $NN$ potential as strong as the ${}^{3}S_{1}$ Mal69 and thus the singlet S wave would develop a dineutron bound state, the four-body system would start to be bound. The situation is slightly different when dealing with the $N\Lambda$ interaction. We have used a common factor $g_{N\Lambda}$ for attractive part of the two $N\Lambda$ partial waves, ${}^{1}S_{0}$ and ${}^{3}S_{1}$. We show in Fig. 2 the binding energy of the $(I)J^{P}=(1)0^{+}$ $nn\Lambda\Lambda$ state as a function of the multiplicative factor $g_{N\Lambda}$, for $g_{NN}=g_{\Lambda\Lambda}=1$. As one can see the four-body system develops a bound state for $g_{N\Lambda}=1.1$.

Ref. Ric15 studied the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ system based on the fit of Nijmegen-RIKEN Rij10 ; Rij13 or chiral effective field theory Pol07 low-energy parameters 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. Mal70 ), 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. Ric15 , the window of Borromean binding is more an more reduced for potentials with harder inner cores.

As already discussed in Ref. Ric15 , many effects are still to be taken into account after arriving to any definitive conclusion. Among the refinements that would eliminate uncertainties, it would be a future challenge to consider three-body forces that may have an attractive component as suggested when studying the triton and 4He Bev77 . Although by fitting the $N\Lambda$ phase shifts, the coupling to the $N\Sigma$ system has been included in an effective manner, it would also be interesting to unfold the effective $\Lambda N$ interaction, separating the contribution from $\Lambda N\leftrightarrow\Sigma N$. As it has been discussed in the literature  Gar14 ; Hiy14 ; Gar07 ; Miy95 ; Gib77 ; Gib79 the hypertriton does not get bound by considering only $NN\Lambda$ channels, but it is necessary to include also $NN\Sigma$ channels. Similar considerations hold from the $\Lambda\Lambda\leftrightarrow N\Xi$ coupling, that is expected to play a minor role in this case, because the nucleon generated in the transition must occupy an excited $p-$shell, the lowest $s-$shell being forbidden by the Pauli principle Nem03 ; Gar13 .

V Summary

In brief, based on a reasonable approach to the interactions of two-body subsystems contributing to the $(I)J^{P}=(1)0^{+}$ $nn\Lambda\Lambda$ state, it does not present a bound state. We have fitted not only the low-energy parameters of the two-body subsystems, but also the phase-shifts. We have considered the repulsive barrier in the two-body interactions, that it is relevant for a correct description of the triton binding energy. We have also studied the strange three-body subsystems involved in the problem, the hypertriton bound by 144 keV, and the $nn\Lambda$ system that it is unbound. Thus, the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ four-body system does not seem to be Borromean. Finally, although our arguments on the unbound nature of the ${}_{\Lambda\Lambda}^{\,\,\,\,4}n$ are strong, one should bear in mind how delicate is the few-body problem in the regime of weak binding, as demonstrated in Ref. Nem03 .

VI Acknowledgments

This work has been partially funded by COFAA-IPN (México), by Ministerio de Economía, Industria y Competitividad and EU FEDER under Contracts No. FPA2013-47443, FPA2015-69714-REDT and FPA2016-77177, by Junta de Castilla y León under Contract No. SA041U16, and by Generalitat Valenciana PrometeoII/2014/066.