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ΛΛ interaction, we adopt several ΛΛ potentials. Most of them are based on the Nijmegen models and are used in double-Λ hypernuclei studies, which are of the three-range Gaussian form 3 

<img src='content_image/368135.jpg'>

The short-range term provides for a strong soft-core repulsion, whereas the medium-range and long-range terms provide for attraction. The parameters v$_{i}$ and β$_{i}$ are taken from Refs. [50–53], and we list them in Table I. The ND1 potential was given in Ref. [50] as an effective soft-core interaction fitted to the Nijmegen model D (ND) hard-core interac- tion. Another simulation of the ND interaction, called ND2 in this article, was obtained in Ref. [51]. The ESC00, NSC97b, NSC97e, and NSC97f potentials given in Ref. [51] were obtained by changing the strength of the medium-range attractive component of the three- range Gaussian potential such that they could reproduce the scattering length and the effective range as close to values by the corresponding Nijmegen models. It is well known that the Nagara event provides unambiguous identification of 6 $_{ΛΛ}$He production with a pre- cise ΛΛ binding energy value B$_{ΛΛ}$ and has had a significant impact on strangeness nuclear physics. The NFs and NSC97s potentials given in Refs. [52, 53] were obtained by adjusting parameters to reproduce the experimental value of B$_{ΛΛ}$ ( 6 $_{ΛΛ}$He) from the Nagara event. We have also chosen an Urbana-type potential that has been successfully used to explain the experimental values of hypernuclei. The Urbana potential could be found in Ref. [54]. 

We plot in Fig. 1 all ΛΛ potentials considered in the present work. The strongest ΛΛ interaction is the ESC00 potential, whereas the weakest ΛΛ interaction is the NSC97f po- tential. We note that the NFs, NSC97s, and Urbana potentials simulate the experimental value of B$_{ΛΛ}$ ( 6 $_{ΛΛ}$He) from the Nagara event. 

## IV. RESULTS AND DISCUSSION 

In this section, we investigate the $^{1}$S$_{0}$ superfluidity of Λ hyperons in neutron star matter and neutron stars. We employ the RMF model with the parameter sets TM1 and NL3 to calculate the properties of neutron star matter, which is known to provide excellent descriptions of the ground states of finite nuclei, including unstable nuclei. The meson- hyperon couplings play an important role in determining the properties of neutron star matter. We use the values constrained by reasonable hyperon potentials that include the 

