= 2 C ( ω$_{n}$, q ) 1 + g ( q ) C ( ω$_{n}$, q ) − 2 ∑ $_{i}$Γ 2 $_{i}$( q ) C ( ω$_{n}$, , q ) D 0 $_{i}$( ω$_{n}$, q ) . (3.13) C ( ω$_{n}$, q ) = 2 ∫ dk$_{x}$dk$_{y}$ 4 π 2 [ f ( ϵ$_{A}$$_{1}$ ( k + q )) − f ( − ϵ$_{A}$$_{1}$ ( − k )) iω$_{n}$ − ϵ$_{A}$$_{1}$ ( k + q ) ϵ$_{A}$$_{1}$ ( k ) + ( ϵ$_{A}$$_{1}$ ↔ ϵ$_{A}$$_{2}$ ) ] . (3.14) χ RPA i ( ω$_{n}$, q ) = 〈 T φ$_{i}$ ( q, τ ) φ † $_{i}$( q, 0) 〉 = D 0 $_{i}$( ω$_{n}$, k ) + ( D 0 $_{i}$( ω$_{n}$, k )) $^{2}$Γ2 $_{i}$( q ) χ RPA $_{QPA}$( ω$_{n}$, q ) = D 0 $_{i}$( ω$_{n}$, k ) + g ( q ) C ( ω, q ) − 2 D 0 ˜ $_{i}$( ω$_{n}$, q ) C ( ω$_{n}$, q )Γ 2 ˜ $_{i}$( q ) 1 + g ( q ) C ( ω$_{n}$, q ) − 2 ∑ $_{i}$Γ 2 $_{i}$( q ) C ( ω$_{n}$, , q ) D 0 $_{i}$( ω$_{n}$, q ) . (3.15) C (0 , 0) ( g (0) − 2 ∑ i Γ 2 $_{i}$(0) v$_{Fi}$ Δ 2 i − 4 µ 2 ) − 1 = 0 . (3.16) T$_{c}$ = max { T$_{c}$$_{1}$, T$_{c}$$_{2}$ } ; T$_{c}$ ≈ ϵ$_{Fi}$ exp 1 √ m$_{|}$$_{|}$$_{i}$m$_{⊥}$$_{i}$ [ 2Γ 2 $_{A}$(0) v$_{FA}$ (Δ 2 A − 4 µ $^{2}$) − g (0) ] − 1 , (3.17) x ( µ ) = κ$_{0}$ 2 7 / $^{2}$π$^{2}$| t eff i (0) | 1 / 2 (2Δ$_{i}$ + | t eff i (0) | + 2 µ )( − 2Δ$_{i}$ + | t eff i (0) | + 2 µ ) √ | t eff i (0) | + 2 µ (3.18) 12 where ϵ$_{Fi}$ = γ 2 i 4 ( t eff i (0)) 2 t eff i (0) − 2 $_{µ}$. If we suppose that µ and t eff i are such that we only have hole pockets, the density of dopants is equal to We note that this vanishing occurs simultaneously in all channels. If g > 0 (though inter- ladder Coulomb repulsion is repulsive, the interactions between quasi-particles on a given ladder is attractive leaving the sign of g indeterminate) the instability occurs only when the chemical potential approaches sufficiently close to Δ$_{A}$ / 2 so that the resulting effective inter- action becomes attractive. This chemical potential corresponds to minimal doping at which the superconductivity appears. Taking Δ$_{A}$ ≪ Δ$_{B}$, the corresponding transition temperature takes the form where ˜ A = B, ˜ B = A . The superconducting instability occurs when the denominator in Eqns. (3.3 and 3.5) vanishes at ω$_{n}$ = 0 , q = 0, that is Here ϵ$_{A}$$_{1}$$_{/}$$_{2}$ ( k ) are the bare dispersions of the A$_{1}$$_{/}$$_{2}$ quasi-particles. As T → 0, C ( ω$_{n}$, q = 0) develops a logarithmic divergence: C ( ω$_{n}$, q = 0) ≈ √ m$_{|}$$_{|}$m$_{⊥}$ log( ϵ$_{F}$ + µ T ). The pair susceptibility for the Cooperons fields has a similar RPA form: We have assumed that the couplings g ( q , k , k $^{′}$) and Γ$_{i}$ ( k , q ) are such that we can ignore their dependence on k and k $^{′}$. Here C ( ω$_{n}$, q ) is the Cooper bubble: