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and s$_{1}$ = sin( π/ 5) and s$_{2}$ = sin(2 π/ 5). We have assumed the hopping is real and that the low energy contribution to G 2D RPA comes from the A -bands as Δ$_{A}$ ≪ Δ$_{B}$. Thus G$_{A}$$_{1}$ ( ω, k$_{x}$ ) /G$_{A}$$_{2}$ ( ω, k$_{x}$ ) are the Green’s functions of the A -band electrons on a given 4- leg ladder. As we have discussed in the previous section G$_{A}$$_{1}$/G$_{A}$$_{2}$ are no more than the bonding/anti-bonding electron Greens functions for a 2-leg ladder. The RPA does not mix G$_{A}$$_{1}$ and G$_{A}$$_{2}$ as the weights of the two are found near differing Fermi wavevectors (i.e. we can take G$_{A}$$_{1}$ ( k ) G$_{A}$$_{2}$ ( k ) ∼ 0 safely for all k ). The presence of Γ$_{1}$( k$_{y}$ ) and Γ$_{2}$( k$_{y}$ ) act as structure factors which cause the quasiparticle weight at various k$_{y}$ to be negligible. While the denominator of G 2D RPA has the periodicity of the reduced Brillouin zone i.e. k$_{y}$ and k$_{y}$ + π/ 2 are identified) these structure functions merely have the periodicity of the original zone i.e. k$_{y}$ and k$_{y}$ + 2 π are identified). 

