Investigation of the $nn\mathrm{\Lambda }$ bound state in pionless effective theory

The possibility of an $nn\mathrm{\Lambda }$ bound state is investigated in the framework of pionless effective field theory at leading order. A system of coupled integral equations are constructed in the spin-isospin basis, of which numerical solutions are investigated. In particular, we make use of the limit cycle behavior, i.e., cyclic singularities of coupled integral equations of the system, which would be associated with the formation of a three-body bound state, the so-called Efimov state, in the unitary limit. Furthermore, we find that, when the sharp momentum cutoff introduced in the integral equations is taken to be significantly larger than the hard scale of the effective theory, the coupling of a three-body contact interaction becomes cyclically singular indicating the onset of an Efimov-like bound state formation. However, the paucity of empirical information to determine the parameters of the theory precludes a definitive conclusion on the existence of such a bound state. As a simple test of the feasibility of the $nn\mathrm{\Lambda }$ bound system in nature, we explore the cutoff dependence of the theory, and uncertainties of the present study are discussed as well.

Figure 1

Diagrams for the dressed dibaryon propagator for the spin singlet $nn$ channel. The single and double lines denote the neutron and dibaryon fields, respectively.

Figure 2

Diagrams for the dressed dibaryon propagator for the spin singlet and triplet $n\mathrm{\Lambda }$ channels. The double line denotes the dibaryon field and the thin (thick) line the neutron ($\mathrm{\Lambda }$ hyperon) field.

Figure 3

Diagrammatic representation of the coupled integral equations for the neutron and spin triplet $n\mathrm{\Lambda }$ dibaryon elastic scattering in the spin-doublet channel without three-body contact interaction. Blobs with the horizontal lines denote the elastic amplitudes for the neutron and spin triplet $n\mathrm{\Lambda }$ dibaryon $\left[nt\right(n\mathrm{\Lambda }\left)\right]$ channel, while those with the vertical and crossed lines denote the two inelastic channels, namely, $nt\left(n\mathrm{\Lambda }\right)$ to neutron and spin singlet $n\mathrm{\Lambda }$ dibaryon, and $\mathrm{\Lambda }$ hyperon and the spin singlet $nn$ dibaryon channels. See the captions of Figs. 1 and 2 as well.

Figure 4

Strength of the coupling $g\left({\mathrm{\Lambda }}_c\right)$ of the three-body contact interaction as a function of the cutoff ${\mathrm{\Lambda }}_c$, where zero binding energy $\left(B=0\right)$ of the three-body $nn\mathrm{\Lambda }$ system is maintained.

Figure 5

Binding energy of the $nn\mathrm{\Lambda }$ three-body system as a function of the cutoff ${\mathrm{\Lambda }}_c$ without the three-body contact interaction.