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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: 

<img src='content_image/352007.jpg'>

<img src='content_image/352008.jpg'>

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: 

<img src='content_image/352009.jpg'>

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 

<img src='content_image/352011.jpg'>

<img src='content_image/352010.jpg'>

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 

<img src='content_image/352012.jpg'>

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 

<img src='content_image/352013.jpg'>

