\section{Introduction}

In hypernuclear physics the lambda-nucleon-nucleon ($\Lambda N N$) three-body problem plays a very important role, akin to that of the NN and NNN systems in the non-strange sector of nuclear physics. In particular, the spin-isospin channel $\Lambda n p$ ($J^{\pi}=1/2^+$) is traditionally used for constraining lambda-nucleon forces. After a bound state (or resonance?) was inferred for $\Lambda n n$ ($J^{\pi}=1/2^+$) \cite{rap2013}, these two systems together now serve to provide more constraints on lambda-nucleon forces. The lambda hypertriton ($\Lambda n p$) is important because it can be used to constrain both the $\Lambda p$ and $\Lambda n$ forces, while $\Lambda n n$ is crucial for constraining the $\Lambda n$ force. It is therefore of uppermost importance that the same set of $\Lambda p$ and $\Lambda n$ forces, with charge symmetry breaking, is used to study both three-body problems. Through this approach, the following important \textit{double} question can be answered: what $\Lambda n p$ binding energy is consistent with a $\Lambda n n$ bound state or resonance? 

More than half a century after the first observation of a hypernucleus in nuclear emulsions, there is still no published data for $\Lambda n$ scattering. This is because there are no lambda hyperon targets and, in the case of inverse kinematics, lambda hyperon beams are very difficult to prepare. The lack of $\Lambda n$ data is a serious setback for all methods used in the development of $\Lambda N$ potentials: Meson-Exchange theory, Chiral Effective Field theory, quantum chromodynamics and, recently, inverse scattering theory. As a result, basic structural properties like binding energy, size and lifetime of the simplest hypernucleus cannot be \textit{accurately} computed. 

Since 1958, a number of calculations on $\Lambda n n$ predicted that a bound state was not possible \cite{dal1958a, dal1958b, dow1959, gar1987,bel2008}. After evidence for a possible bound state was inferred from heavy-ion experiments by the HypHI Collaboration at GSI, a number of recent theoretical studies based on improved $\Lambda N$ forces from Meson-Exchange theory, Chiral Effective Field theory and quantum chromodynamics have also argued against the existence of such a bound state \cite{gal2014, gar2014, hiy2014, afn2015, kam2016, fil2016, gib2019}. In order to provide a new perspective on the $\Lambda N$ force, inverse scattering theory has recently been applied on theoretical $\Lambda p$ and $\Lambda n$ scattering data \citep{meo2019}. Spin averages of the recovered $\Lambda p$ and $\Lambda n$ potentials (GLM-YN0) were tested in \cite{meo2020} by computing the binding energy and root-mean-square radius of $\Lambda n p$ ($J^{\pi}=1/2^+$), using Malfliet-Tjon V potential for the NN interaction. The aim of this paper is to use GLM-YN0 potentials with the semi-realistic spin-dependent Malfliet-Tjon I/III NN potential that correctly binds the deuteron, in exploring the $\Lambda n p$ ($J^{\pi}=1/2^+$) and $\Lambda n n$ ($J^{\pi}=1/2^+$) three-body problems.

\section{Lambda-neutron and lambda-proton interactions: GLM-YN0 potentials}

The charge-asymmetric pair of $\Lambda p$ and $\Lambda n$ potentials used in this paper were developed through Gel'fand-Levitan-Marchenko theory \citep{meo2019}. These potentials were restored through the application of this method on sub-threshold theoretical $s$-wave scattering phases. Based on the spin multiplicities of the $^1\text{S}_0$ and $^3\text{S}_1$ $\Lambda N$ states, effective potentials are constructed from one-quarter spin singlet and three-quarters spin triplet potentials from inverse scattering theory \cite{meo2020}:  

\begin{align}
\label{eq:spinlp}
V_{\Lambda p} = \frac{1}{4}V_{\Lambda p}(^1\text{S}_0) + \frac{3}{4}V_{\Lambda p}(^3\text{S}_1)
\end{align} 

\begin{align}
\label{eq:spinln}
V_{\Lambda n} = \frac{1}{4}V_{\Lambda n}(^1\text{S}_0) + \frac{3}{4}V_{\Lambda n}(^3\text{S}_1)
\end{align} 
The data obtained from these spin-averages, which will henceforth be referred to as GLM-YN0 potentials, was fitted with the following three-range Gaussians:
\begin{align}
\label{eq:three_gaussians}
V_{\Lambda N}(r) = \sum_{i=1}^{3} V_{i} \exp \left \{ \frac{-(r - \mu_i)^2}{\sigma^2_i} \right \}.
\end{align}

Figure \ref{fig:both_effective1} illustrates the features of GLM-YN0 potentials, especially a noticeable charge symmetry breaking. The theoretical scattering phases used for inverse scattering theory in \cite{meo2019} carry in them not only this feature of charge asymmetry, but also $\Lambda - \Sigma$ coupling. Therefore, it can be stated that even though GLM-YN0 potentials only explicitly have a central term, the effects of charge symmetry breaking and $\Lambda - \Sigma$ coupling are \textit{built} into this central term, since phase equivalence with NSC1997f was verified in \cite{meo2019}. The parameters $V_{i}, \mu_i$ and $\sigma_i$ in Equation \eqref{eq:three_gaussians} were determined through a nonlinear least squares fit. Estimated values are reported in Table \ref{fit}.

\begin{figure}[!h]
\centering
\includegraphics[width=1.0\linewidth]{both_effective1.eps}
\caption[]{GLM-YN0 effective $\Lambda p$ and $\Lambda n$ potentials: these potentials are constructed from one-quarter of the spin singlet state and three-quarters of the triplet state potential from inverse scattering theory.}
\label{fig:both_effective1} 
\end{figure}

\begin{table}[h!]
\centering
\caption[]{Parameters for GLM-YN0 effective $\Lambda p$  and $\Lambda n$ potentials.}
\begin{tabular}{p{1.0cm}p{3.0cm}p{3.5cm}p{3.6cm}}
           &                  &   $\Lambda p$             &                               \\  \hline
           &  $V_i$ /MeV      &   $\mu_i$/ fm             &  $\sigma_i$/fm                \\ \hline 
$i=1$      & $45.88      $    &   $0.1148 \pm 0.0006601$  &  $-0.3932 \pm 0.0008502$      \\ 
$i=2$      & $8.106e+07  $    &   $-1.193 \pm 0.001948$   &  $0.3575 \pm 0.0005306$       \\ 
$i=3$      & $-47.04     $    &   $0.3748 \pm 0.0001386$  &  $0.1667 \pm 0.0002179$       \\   \hline
           &                  &                           &                               \\
           &                  &                           &                               \\
           &                  &    $\Lambda n$            &                               \\ \hline
           &  $V_i$ /MeV      &   $\mu_i$/ fm             &  $\sigma_i$/fm                \\ \hline 
$i=1$      & $186.9      $    &   $- 0.3476 \pm 0.001364$ &  $-0.5469 \pm 0.001125$       \\ 
$i=2$      & $6.74e+04   $    &   $- 0.383 \pm 0.001433$  &  $0.191 \pm 0.0005638$        \\ 
$i=3$      & $- 52.14    $    &   $0.3243 \pm 0.0001977$  &  $0.2013 \pm 0.0002466$       \\   \hline
\end{tabular}
\label{fit}
\end{table}

\section{Neutron-proton and neutron-neutron interactions: Malfliet-Tjon I/III potential} 

Malfliet-Tjon I/III potential is used to represent the NN interaction. In the spin-triplet states, the $(S,I)=(1,0)$ potential is

\begin{align}
\label{eq:mtiii_np}
V_{np}(r) =  [V^t_A \text{exp}(-\mu_a r ) + V^t_R \text{exp}(-\mu_r r )]/r
\end{align}

In the spin-singlet state, $(S,I)=(0,1)$, the potential is give by 

\begin{align}
\label{eq:mti_nn}
V_{nn}(r) =  [V^s_A \text{exp}(-\mu_a r ) + V^s_R \text{exp}(-\mu_r r )]/r
\end{align}

The parameters used for these NN potentials, shown in Table \ref{mt_iii_i}, are taken from \cite{fri1990}. These parameters were established so that the Malfliet-Tjon I/III potential can be used as a tool for benchmarking few-body computations in nuclear physics. With these parameters, the binding energy of the deuteron was found to be -2.2307 MeV.

\begin{table}[h!]
\centering
\caption[]{Parameters for Malfliet-Tjon I/III NN potentials, from \cite{fri1990}.}
\begin{tabular}{p{3.0cm}p{3.0cm}p{3.0cm}p{3.0cm}}
$V^t_A$    & $V^t_R$          &   $\mu_a$                 &  $\mu_r $                \\ \hline 
-626.885   & 1438.72          &   1.55                    &  3.11                    \\ 
$V^t_A$    & $V^t_R$          &                           &                          \\ \hline 
-513.968   & 1438.72          &                           &                           \\   \hline
\end{tabular}
\label{mt_iii_i}
\end{table}

\section{Hyperspherical method} 

Let $\{\vec{x}_i, \vec{y}_i \}$ be Jacobi coordinates, after elimination of centre-of-mass motion \cite{tho2004}. These coordinates are transformed into six-dimensional hyperspherical coordinates $\{\rho, \Omega_5\}$, where $\rho = \arctan (y_i/x_i)$ is the hyperradius and $\Omega_5$ represents one hyperangle ($\theta_i$) and two sets of polar angles ($\nu_{x_i}, \nu_{y_i}, \omega_{x_i}, \omega_{y_i}$). Faddeev amplitudes are given by
\begin{align}
\label{eq:faddeev}
 \psi_{\alpha_i}^{i,J}(x_i, y_i) =  \sum_{K_i=K_{min}}^{K_{max}} \rho^{-5/2} \chi_{\alpha_i, K_i}^{i,J}(\rho) \phi_{K_i}^{\ell_{xi}, \ell_{yi}}(\theta_i),
\end{align}
where $K_i$ are hyperangular momenta and $\alpha_i = \{ (\ell_{xi}, \ell_{yi})L_i, (s_j,s_k)S_{xi} \}J_i;s_i$ is the coupling order for Faddeev amplitudes. The $\Lambda NN $ three-body problem is simulated through the following system of coupled hyperradial equations \cite{smi1977}:

\begin{align}
\label{eq:hyperradial}
 \left \{ -\frac{\hbar^2}{2m}\frac{d^2}{d \rho^2} + \frac{\hbar^2}{2 m \rho^2}\mathcal{L}_{K_i}(\mathcal{L}_{K_i} +1)  - E  \right \} \chi_{\alpha_i, K_i}^{i,J}(\rho) =- \sum_{ j \alpha_j K_j } V_{\alpha_i K_i, \alpha_j K_j}^{ij} (\rho)       \chi_{\alpha_j, K_j}^{j,J}(\rho),
\end{align}
where $m$ is a unit mass (taken here to be the mass of a nucleon, 939.0 MeV), $\mathcal{L}_{K_i}=K_i+3/2$ and $V_{\alpha_i K_i, \alpha_j K_j}^{ij}$ are coupling potentials. This coupled system of hyperradial equations is converted to the following eigenvalue problem \cite{tho2004}:
\begin{align}
\label{eq:eigenvalue}
H \mathbf{a}=E\mathbf{a}
\end{align}
where $\mathbf{a}$ is the eigenvector and $E$ the eigenvalue.

\section{Results and discussion}

\subsection{$\Lambda n p$ ($J^{\pi}=1/2^+$)}

The eigenvalue problem in Equation \ref{eq:eigenvalue} was solved using the computer code \texttt{FaCE} \cite{tho2004}. For $\Lambda n p$ ($J^{\pi}=1/2^+$), the $np$ potential in Equation \ref{eq:mtiii_np} was used, together with GLM-YN0 $\Lambda p$ and $\Lambda n$ potentials presented in this paper. The binding energy and rms matter radius computed are found to be (-3.0759 MeV, 7.7 fm). This gives a lambda separation energy of $B_{\Lambda}=0.85$ MeV, relative to the deuteron binding energy for the $np$ potential used in the calculation (-2.2307 MeV). In a recent experiment it was found that $B_{\Lambda}=0.41$ MeV \cite{ada2020a}, which is a consequential increase from the widely used value $B_{\Lambda}=0.13$ MeV \cite{jur1973}. A more accurate experiment, that is \textit{expected} to measure an even higher $\Lambda n p$ binding energy, is in the planning phase at JLab \cite{gog2020}. A higher lambda separation energy in $\Lambda n p$ has many consequences in hypernuclear physics, some of which are discussed within chiral effective field theory in \cite{le2020} for $B_{\Lambda}=0.23$ to 0.51 MeV.

The rms radius of $\Lambda n p$ is another property that is dependent on the $\Lambda n p$ wavefunction, even though it has never been measured in any experiment. As expected, the value computed here is smaller than that from models in which $\Lambda n p$ has a lower binding energy. For example, a $\Lambda$-deuteron rms radius of 10.3 fm was recently computed in pionless effective field theory \cite{hil2019} while a value of 10.6 fm is reported in \cite{bra2019}. 

A recent high-precision measurement by the STAR Collaboration at RHIC-BNL \cite{ada2019} found the lifetime of $\Lambda n p$ ($J^{\pi}=1/2^+$) to be about 50\% shorter than that of a free $\Lambda$ hyperon: this clearly implies that $\Lambda p$ and $\Lambda n$ forces are much stronger than currently used forces that compute lambda hypertriton lifetimes nearly equal to that of a free $\Lambda$ hyperon. Our potentials are therefore in accord with the STAR Collaboration lifetime measurements that challenge the notion of $\Lambda n p$ being a loose $\Lambda$ hyperon around a deuteron core. Therefore, an important test for GLM-YN0 $\Lambda p$ and $\Lambda n$ potentials lies in the computation of the lifetime of $\Lambda n p$ ($J^{\pi}=1/2^+$).  


\subsection{$\Lambda n n$ ($J^{\pi}=1/2^+$)}

For $\Lambda n n$ ($J^{\pi}=1/2^+$), the $nn$ potential in Equation \ref{eq:mti_nn} was used, together with GLM-YN0 $\Lambda n$ potential. By examining all the computed eigenvalues of Equation \ref{eq:eigenvalue}, two possible sets of values for the binding energy and rms matter radius are found: (-0.0695 MeV, 12.3 fm) and (-0.6112 MeV, 10.1 fm). The HypHI Collaboration, in their announcement of a possible $\Lambda n n$ bound state, did not state any value for its binding energy. Neither was a spin and parity assigned. There is therefore no experimental results to compare with. Notwithstanding, one can immediately suspect that the second value is too high because there are no $\Lambda n$ or $n n$ bound states, compared to $\Lambda n p$ for which there is a $n p$ bound state. 

The most likely binding energy from our calculation (-0.0695 MeV) is close to that reported in \cite{afn2015, gib2019} as -0.069 MeV. The result from \cite{afn2015, gib2019} is obtained by scaling up meson theory $\Lambda N$ potentials to about 35\% in a few-body calculation carried out through the Complex Energy Method. In \cite{bel2008}, a bound state was found by scaling up their $\Lambda N$ potential by 50\%, though the value of the binding energy computed was not stated. Motivated by the charge asymmetry relation between $\Lambda p$ and $\Lambda n$ forces, an important question arises from the studies in \cite{bel2008, afn2015, gib2019}, and all other studies that computed a $\Lambda n n$ bound state by scaling up the attraction in the $\Lambda n$ force: what is the binding energy of $\Lambda n p$ after both $\Lambda p$ and $\Lambda n$ are scaled up by the same amount? In \cite{hiy2014}, this question was explored using a lambda-nucleon force which is phase-equivalent with the NSC1997f force: by scaling up the tensor component in this force (which acts in the $^3\text{S}_1$ state) by 1.20, a bound state is found for $\Lambda n n$ at -0.054 MeV, and the corresponding lambda separation energy for $\Lambda n p$ computed is $B_{\Lambda}=0.83$ MeV. This value of $B_{\Lambda}$ is nearly the same as that presented in this paper for GLM-YN0 potentials ($B_{\Lambda}=0.85$ MeV). 

Recently, a new experiment to search for a bound state or resonance in $\Lambda n n$ ($J^{\pi}=1/2^+$) was carried out at JLab \cite{tan2018}. At the time of writing this paper, data from this experiment was still being analysed. 



\section{Conclusions}

In this paper, charge asymmetric lambda-nucleon potentials have been proposed. These potentials were obtained by averaging spin-dependent lambda-nucleon potentials recovered through the Gel'fand-Levitan-Marchenko method on theoretical scattering data. The potentials (GLM-YN0) are constructed from one-quarter spin singlet plus three-quarters spin triplet potentials. GLM-YN0 potentials were used in solving the $\Lambda NN$ three-body problem, together with Malfliet-Tjon I/III nucleon-nucleon potential. $\Lambda n p$ ($J^{\pi}=1/2^+$) binding energy and root-mean-square matter radius computed were found to be (-3.0759 MeV, 7.7 fm). These two results reveal that $\Lambda p$ and $\Lambda n$ forces are much stronger, a fact that is strongly evident from the 2018 $\Lambda n p$ ($J^{\pi}=1/2^+$) lifetime measurements by the STAR Collaboration. GLM-YN0 $\Lambda n$ forces gave rise to a bound $\Lambda n n$ ($J^{\pi}=1/2^+$) state, whose binding energy and root-mean-square radius are reported as (-0.0695 MeV, 12.3 fm) or (-0.6112 MeV, 10.1 fm). This $\Lambda n n$ ($J^{\pi}=1/2^+$) bound state arises \textit{naturally} within GLM-YN0 potentials. Computing $\Lambda n p$ ($J^{\pi}=1/2^+$) lifetime and the size of charge symmetry breaking in isospin multiplets are decisive tests on the accuracy of our lambda-nucleon forces from inverse scattering theory.

\section{References}



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