the LSR holds if we consider the sign changes the Green’s function undergoes not at ω = 0 but at ω = µ . The residual interactions between the Fermi pockets and the Cooperons will lead to instabilities in the RPA solutions as temperature goes to zero. Provided a finite chemical potential is present the leading instability will be to a superconducting state. While gapless quasi-particles only exist in the A-bands, both A and B bands will go superconducting simultaneously. The general form of the Cooperon-quasiparticle interaction is Here L is the length of the ladders, a is the interladder spacing, and N is the number of ladders in the array. Φ$_{A,B}$ are the Cooperon fields whose bare propagators are defined as We see that g has the dimensionality of energy × length 2 and Γ$_{i}$ has the dimensionality of energy × length 1 / $^{2}$. The different terms in Eqn. (3.11) have different origins. The strongest interactions are presumably Γ$_{i}$ as this term already exists for uncoupled ladders. Inter-ladder interactions, such as interladder Coulomb repulsion, also contribute to Γ$_{i}$. However interladder hoping does not – this contribution is suppressed due to a mismatch between the Fermi momenta of the A$_{i}$ and B$_{i}$ bands. The coupling g is smaller than Γ$_{i}$: it arises only in second or- der perturbation theory from intraladder interactions and from presumed weak inter-ladder Coulomb interactions. The pair susceptibility for the quasiparticles Δ$_{QPA}$ in an RPA approximation is given by 11 χ RPA $_{QPA}$( ω$_{n}$, q ) = 1 LNa ∑ k$_{1}$ , $_{k$_{2}$}$∫ β 0 dτe iω$_{n}$$^{τ}$〈 T Δ$_{QPA}$( k$_{1}$ , q , τ )Δ † $_{QPA}$( k$_{2}$ , q , 0) 〉 D 0 $_{i}$( ω$_{n}$, k ) = 〈 T Φ$_{i}$( k , ω$_{n}$ )Φ † $_{i}$( k , ω$_{n}$ ) 〉 0 = v$_{Fi}$ − ( iω$_{n}$ − 2 µ ) 2 + Δ 2 i + ( v$_{Fi}$k$_{x}$ ) $_{2}$. (3.12) H$_{φQP}$ = ∑ i = A,B ; k , q Γ$_{i}$( k , q ) ( NLa ) 1 / 2 [ Φ$_{i}$( q )Δ † $_{QPA}$( k , q ) + h.c. ] + 1 2 ∑ q , k , k ′ g ( q , k , k $^{′}$) NLa Δ † $_{QPA}$( k , q )Δ$_{QPA}$( k $^{′}$, q ) Δ † $_{QPA}$( k , q ) = ϵ$_{σσ}$ ′ [ A † 1 $_{σ}$( k + q ) A † 1 σ ′ ( − k ) − A † 2 $_{σ}$( k + q ) A † 2 σ ′ ( − k )] . (3.11) 2. Superconducting Instability