density region depend on the pairing interaction and the EOS of neutron star matter. The effect of the Nagara event on Λ superfluidity has been discussed in Refs. [23, 24], where the weak attractive ΛΛ interaction suggested by the Nagara event leads to very small pairing gap in dense neutron matter [23] or the disappearance of Λ superfluidity in neutron star matter [24]. All these studies indicate that the ΛΛ pairing gap in dense matter depends both on the ΛΛ interaction and on the EOS of matter. The key role of Λ superfluidity in neutron star cooling motivates us to investigate the possibility of forming a Λ superfluid in neutron stars by carefully considering the pairing interaction and the description of neutron star matter with the updated information from recent developments in hypernuclear physics. This article is arranged as follows. In Sec. II, we briefly describe the RMF theory for the calculation of neutron star matter properties. In Sec. III, we discuss the ΛΛ interaction used in the gap equation. We present the numerical results in Sec. IV. Section V is devoted to a summary. We use the RMF theory to describe the neutron star matter, which is composed of a chemically equilibrated and charge-neutral mixture of nucleons, hyperons, and leptons. In the RMF approach, baryons interact through the exchange of scalar and vector mesons. The baryons considered in this work are nucleons ( p and n ) and hyperons (Λ, Σ, and Ξ). The exchanged mesons include isoscalar scalar and vector mesons ( σ and ω ), an isovector vector meson ( ρ ), and two additional hidden-strangeness mesons ( σ ∗ and φ ). The total Lagrangian density of neutron star matter takes the form II. RELATIVISTIC MEAN FIELD THEORY L RMF = ∑ B ¯ ψ$_{B}$ [ iγ$_{µ}$∂ µ − m$_{B}$ − g$_{σB}$σ − g$_{σ}$ ∗ $_{B}$σ ∗ − g$_{ωB}$γ$_{µ}$ω µ − g$_{φB}$γ$_{µ}$φ µ − g$_{ρB}$γ$_{µ}$τ$_{iB}$ρ µ i ] ψ$_{B}$ + 1 2 ∂$_{µ}$σ∂ $^{µ}$σ − 1 2 m 2 $_{σ}$σ 2 − 1 3 g$_{2}$σ 3 − 1 4 g$_{3}$σ 4 − 1 4 W$_{µν}$W µν + 1 2 m 2 $_{ω}$ω$_{µ}$ ω µ + 1 4 c$_{3}$ ( ω$_{µ}$ω $^{µ}$) 2 − 1 4 R$_{iµν}$R µν i + 1 2 m 2 $_{ρ}$ρ$_{iµ}$ ρ µ i + 1 2 ∂$_{µ}$σ $^{∗}$∂ $^{µ}$σ ∗ − 1 2 m 2 σ ∗ σ ∗ 2 − 1 4 S$_{µν}$S µν + 1 2 m 2 $_{φ}$φ$_{µ}$ φ µ + ∑ l ¯ ψ$_{l}$ [ iγ$_{µ}$∂ µ − m$_{l}$ ] ψ$_{l}$, (1) 5