III. GAP EQUATION AND ΛΛ INTERACTION ρ$_{p}$ + ρ$_{Σ}$ + = ρ$_{e}$ + ρ$_{µ}$ + ρ$_{Σ}$ − + ρ$_{Ξ}$ − , (14) µ$_{B}$ = √ k B F 2 + m ∗ 2 B + g$_{ωB}$ω + g$_{φB}$φ + g$_{ρB}$τ$_{3}$$_{B}$ρ, (12) µ$_{l}$ = k l F 2 + m 2 $_{l}$, (13) Δ ( k ) = − 1 4 π 2 ∫ k ′ $^{2}$dk ′ V ( k, k $^{′}$) Δ ( k $^{′}$) √ [ E ( k ′ ) − E ( k Λ $_{F}$)] 2 + Δ $^{2}$( k ′ ) , (15) E ( k ) = √ k 2 + m ∗ 2 Λ + g$_{ω}$$_{Λ}$ω + g$_{φ}$$_{Λ}$φ. V ( k, k $^{′}$) = 〈 k | V$_{ΛΛ}$ ( $^{1}$S$_{0}$ ) | k $^{′}$〉 = 4 π ∫ r $^{2}$dr j$_{0}$ ( kr ) V$_{ΛΛ}$ ( r ) j$_{0}$ ( k $^{′}$r ) , (16) 8 of baryons and leptons are given by respectively. The electric charge neutrality condition is expressed by where ρ$_{i}$ = ( k $_{F}$) $^{3}$/ (3 π $^{2}$) is the number density of species i . We solve the coupled Eqs. (2)– (6), (8)–(11), and (14) self-consistently at a given baryon density ρ$_{B}$ . Then we can calculate the EOS and the composition of neutron star matter as well as the effective mass and the Fermi momentum of Λ hyperons, which are crucial in the study of ΛΛ pairing. We study the $^{1}$S$_{0}$ superfluidity of Λ hyperons in neutron star matter. The crucial quantity in determining the onset of superfluidity is the energy gap function Δ ( k ), which can be obtained by solving the gap equation where E ( k ) is the single-particle energy of Λ with momentum k . For Λ hyperons in neutron star matter, the single-particle energy in the RMF approach is given by The effective mass m ∗ Λ and the Fermi momentum k Λ F are computed self-consistently at a given baryon density ρ$_{B}$ within the RMF approach. For the ΛΛ pairing interaction in the $^{1}$S$_{0}$ channel, the potential matrix element can be written as where j$_{0}$ ( kr ) = sin( kr ) / ( kr ) is the spherical Bessel function of order zero and V$_{ΛΛ}$ ( r ) is the $^{1}$S$_{0}$ ΛΛ interaction potential in coordinate space. Because of large uncertainties in the