Resonances in the $\mathrm{\Lambda }nn$ system

Background: A bound state of the $\mathrm{\Lambda }nn$ system has been reported, but at least three theoretical papers question the existence of such a bound state.

Purpose: We address the alternative question of whether there might exist a resonance in the $\mathrm{\Lambda }nn$ system, using a rank-one separable potential formulation of the Hamiltonian.

Methods: We examine the eigenvalues of the kernel of the Faddeev equation in the complex energy plane using contour rotation to allow us to analytically continue the kernel onto the second energy sheet. The model $\mathrm{\Lambda }n$ interaction is fitted to the $\mathrm{\Lambda }p$ scattering length and effective range.

Results: We follow the largest eigenvalue as the $\mathrm{\Lambda }n$ potentials are scaled and the $\mathrm{\Lambda }nn$ continuum is turned first into a resonance, and then into a bound state of the system.

Conclusions: Because a change in the strength of the $\mathrm{\Lambda }n$ potential of as little as 5% will produce a $\mathrm{\Lambda }nn$ resonance, we infer that an experiment of the ${}^3\text{H}(\text{e},{\text{e}}^{\prime }{\text{K}}^{+}){}_{\mathrm{\Lambda }}{}^3\text{n}$ type at JLAB could be used to constrain the properties of the $\mathrm{\Lambda }n$ interaction.

Figure 1

Trajectory of the resonance pole as one varies the strength of the $\mathrm{\Lambda }N$ interaction. In this case we have used the $NN$ and $\mathrm{\Lambda }N$ interaction for Tables IV and VI of Ref. [11] with a tensor interaction for the $\mathrm{\Lambda }N$ potential.

Figure 2

Trajectory of the resonance pole as one varies the strength of the $\mathrm{\Lambda }n$ interaction. In this case we use the same $nn$ potential given in Eq. (14) for all four curves. The $\mathrm{\Lambda }n$ potentials correspond to Yamaguchi fits as given in Table 2 with Mod D for Nijmegen model D, Chiral for chiral $\left(\mathrm{\Lambda }=600\right)$, NSC97f for Nijmegen NSC97f, and Jülich04 for the Jülich one boson exchange potential. (In scaling the $\mathrm{\Lambda }n$ potential we have insured that no two-body bound state is formed. For example, for Mod D the scaling of the potential by $s=1.35$ moved the two-body pole closest to the real axis from $k=-0.315i$ fm to $k=-0.164i$ fm.)

Figure 3

We plot the variation in the singlet (red) and triplet (blue) effective range parameters for Nijmegen model D with variation in the scaling $s$ between $s=1.0$ and $s=1.275$ in intervals of $\mathrm{\Delta }s=0.025$. Also included are the effective range parameters of the four potentials considered in Table 2.