E$_{B}$$_{c}$ ( k$_{x}$ ) = √ k 2 x + Δ 2 B − 2 µ, (3.19) χ RPA $_{QPA}$( ω$_{n}$, q ) = 2 C ( ω$_{n}$, q ) 1 + g ( q ) C ( ω$_{n}$, q ) − 2Γ 2 $_{B}$( q ) C ( ω$_{n}$, , q ) D 0 $_{B}$( ω$_{n}$, q ) ; χ RPA B ( ω$_{n}$, q ) = D 0 $_{i}$( ω$_{n}$, k ) + g ( q ) C ( ω, q ) 1 + g ( q ) C ( ω$_{n}$, q ) − 2Γ 2 $_{B}$( q ) C ( ω$_{n}$, , q ) D 0 $_{B}$( ω$_{n}$, q ) . (3.20) C (0 , 0) = 1 aπv$_{FA}$ log( E$_{FA}$$_{1}$E$_{FA}$$_{2}$ T 2 ) . (3.21) D 0 $_{B}$( ω$_{n}$ = 0 , q = 0) ∼ v$_{FB}$ v 2 $_{FA}$a $^{2}$(( x A c ) 2 − x $^{2}$) . (3.22) T$_{c}$ = ( ϵ$_{FA}$$_{1}$ϵ$_{FA}$$_{2}$ ) 1 / $^{2}$exp [$_{−}$ πav$_{FA}$ ( 2Γ 2 $_{B}$(0) v$_{FB}$ v 2 $_{FA}$a $^{2}$(( x $^{A}$) 2 − x $^{2}$) − g (0) ) − $^{1}$] , (3.23) 14 and we see that the critical temperature grows extremely fast with doping, similar to the transition temperature determined in Scenario I. Again we emphasize that the critical doping x A c as defined above does not coincide with optimal doping – in this model we are always in the underdoped regime. For this range of doping we obtain a transition temperature of the form Because the A-quasi-particles are already gapless, a finite µ dopes the A-bands with doping x $^{A}$( µ ). Thus E$_{FA}$$_{i}$ ( µ ) = E$_{FA}$$_{i}$ ( µ = 0) − µv$_{FA}$ . If we denote the critical doping, µ$_{c}$ , as the doping when the B-Cooperon becomes soft (i.e. µ$_{c}$ = Δ$_{B}$ / 2) and x A c = x $^{A}$( µ$_{c}$ ) the corresponding doping of the A-bands, we can rewrite the form of the B-Cooperon propagator, D 0 $_{B}$(0 , 0), as As we no longer have pockets as in Scenario 1, but instead have an anisotropic 2D Fermi liquid whose Fermi surface consists of slightly deformed lines (see Figure 4), the divergent with temperature behaviour of C (0 , 0) now takes the form where C ( ω$_{n}$, q ) is defined as in Eqn. (3.15). The form of the quasi-particle-Cooperon interaction is that of Eqn. (3.13) (though of course, now we have no A-Cooperon and so this coupling is absent). This system, like in Scenario 1, has a pairing instability to superconductivity. The pairing susceptibilities in an RPA approximation take a similar form as for Scenario 1: where E$_{A}$$_{i}$ ( k$_{x}$ ) is given in Eqn. (2.1) and t eff i ( k$_{y}$ ) in Eqn. (3.4). We illustrate the two dimensional Fermi surface of the A-bands in Figure 4.