$(I,J^P)=(1,1/2^+)$ $\Sigma NN$ quasibound state

JLab has recently found indications of the possible existence of a $\Sigma NN$ resonance at $(3.14 \pm 0.84) - i (2.28 \pm 1.2)$ MeV. In the past, using models that exploit symmetries between the two-baryon sector with and without strangeness, hyperon-nucleon interactions have been derived that reproduce the experimental data of the strangeness $-1$ sector. We make use of these interactions to review existing Faddeev studies of the $\Lambda NN$-$\Sigma NN$ system that show theoretical evidences about a $(I,J^P)=(1,1/2^+)$ $\Sigma NN$ quasibound state near threshold. The calculated position of the pole is at 2.92$\,-i\,$2.17 MeV, in reasonable agreement with the experimental findings.

Hall A Collaboration at Jefferson Lab has made use of the (e, e ′ K + ) reaction to study the possible existence of neutral three-body Λ and Σ hypernuclei [1]. They reported an excess of events around the Σ thresholds. The most significant enhancement appears 3.14 ± 0.84 MeV below the Σ 0 nn threshold and has a width of σ ≈ 2.28 ± 1.2 MeV. It possibly hints at a bound Σ 0 nn (I = 1) state.
The existing experimental data and the expected forthcoming optimized data call for theoretical studies that could help with their interpretation. In this letter it is our purpose to emphasize the relevant findings of existing Faddeev studies of the ΛNN − ΣNN system.
The theoretical results obtained are a valuable tool to analyze the Hall A Collaboration data.
We have carried out a detailed study of the ΛNN −ΣNN three-body system at threshold looking for bound states or resonances [2]. The strangeness −1 two-body interactions have been derived from the chiral quark cluster model (CQCM) [3], by exploiting the symmetries with the two-nucleon sector. In the CQCM hadrons are clusters of massive (constituent) quarks. As color carriers, massive quarks are confined through a confining potential. They interact through a one-gluon exchange potential arising from Quantum Chromodynamics (QCD) perturbative effects. The non-perturbative effects generate one-boson exchange potentials between quarks [3,4].
The nucleon-nucleon (NN) and hyperon-nucleon (Y N) interactions describe reasonably well the NN and Y N two-body observables [4]. In particular, the two-nucleon system low-energy parameters, the NN S−wave phase shifts, and the triton binding energy are described correctly [3]. Besides, there is a reasonable agreement with the hyperon-nucleon elastic and inelastic scattering cross sections and the hypertriton binding energy. Finally, the isospin one Λnn system is unbound [2,4,5]. At the two-body level, the NΛ − NΣ coupling as well as the tensor force, responsible for the coupling between S and D waves, have been considered. The Λ ↔ Σ conversion is crucial to have a correct description of the ΛNN system [2]. The NN and Y N interactions contain sizable non-central terms which are responsible, among others, for the deuteron binding energy. The relevance of the Y N tensor force becomes apparent when studying the Σ − p → Λn reaction. Such process is controlled by the ΣN(ℓ = 0) → ΛN(ℓ = 2) transition so that if only the central interaction ΣN(ℓ = 0) → ΛN(ℓ = 0) is considered, the cross section cannot be described correctly [6]. The non-central NΛ − NΣ interaction induces a three-body force through the coupling between Y NN channels with (ℓ, λ) = (0, 0) and (ℓ, λ) = (2, 2), where the Y N relative orbital angular momentum is denoted by ℓ and λ stands for that of the spectator nucleon respect to the Y N system.
For this study different models have been designed by choosing sets of spin-singlet and spin-triplet ΛN scattering lengths describing correctly the available experimental data. In particular, besides a reasonable description of the Y N cross sections, the hypertriton binding energy corresponds to its experimental value within the error bars B 0,1/2 = 0.130 ± 0.050 MeV [7]. The upper limit of the ΛN spin-triplet scattering length, a ΛN 1/2,1 , has been established by requiring that the (I, J P ) = (0, 3/2 + ) ΛNN state does not become bound [2]. The lower limit was set by requiring a correct description of the Y N cross sections, which deteriorates markedly as the ΛN spin-triplet scattering length decreases. Thus, it was found that 1.41 ≤ a ΛN 1/2,1 ≤ 1.58 fm. Once the ΛN spin-triplet scattering length has been defined, the ΛN spinsinglet scattering length, a ΛN 1/2,0 , was constrained by demanding that the hypertriton binding energy is in the experimental interval B = 0.130±0.050 MeV, leading to 2.33 ≤ a ΛN 1/2,0 ≤ 2.48 fm. Without loss of generality we take the model with a ΛN 1/2,1 = 1.41 and a ΛN 1/2,0 = 2.48 as the reference model. All calculations have been performed for several models within the scattering lengths intervals and the conclusions remain unchanged. For the reference model the hypertriton binding energy obtained is 129 keV. Just to illustrate the relevant role played by the D waves of the three-body system, it is worth to note that considering only S wave three-body channels the hypertriton binding energy is 89 keV, out of the experimental range.
The solution of the three-body problem have been described elsewhere [2] and are out of the scope of this letter. We focus on the results concerning the possible existence of a ΣNN (I, J P ) = (1, 1/2 + ) resonance [1].
Let us first discuss the attractive or repulsive character of the different J P = 1/2 + ΣNN channels. Fig. 1  This can be easily understood as follows. We show in Table I   (i N (Σ) , s N (Σ) ), and N N channels with Λ spectator (i N (Λ) , s N (Λ) ) that contribute to a given J P = 1/2 + ΛN N − ΣN N state with total isospin I. the imaginary part has a maximum, which are the typical signals of a quasibound state [8]. Gibson found a near threshold I = 0 resonance while exploring Λd elastic scattering by a continuum Faddeev calculation [11]. Recent preliminary calculations [12] have suggested that the pole for the I = 1 resonance is also located near the ΣNN threshold, but the two resonances are unlikely to be distinguished experimentally.