content_image/333249.jpg content_image/333248.jpg content_image/333247.jpg
propagator: 

<img src='content_image/333249.jpg'>

where Z is a numerical constant. Thus 

<img src='content_image/333248.jpg'>

Substituting the latter expression into RPA expressions for the pairing susceptibilities (Eqns. 3.20) we obtain an estimate for the critical temperature upon doping as follows: 

<img src='content_image/333247.jpg'>

This dependence on the doping is much weaker than (Eqn. 3.27). It holds in the region where phase fluctuations are already strong. 

Thus we have obtained two regimes with different doping dependence of T$_{c}$ . The first one is the BCS-like with T$_{c}$ given by Eqn. (3.23). It corresponds to lowest doping levels. The other regime, which in our model still describes a situation an anisotropic 1D-like Fermi surface, is the regime with strong phase fluctuations. The mean field transition temperature in this regime is given by Eqn. (3.29). A further increase of doping presumably will lead to a change in the Fermi surface topology and is not considered in this paper. 

## IV. DISCUSSION 

Phenomenological models based on coupled fermions and bosons similar to that derived here, have been proposed much earlier Refs.(32–35) to describe the high temperature su- perconductors. The closest similarity are to the models proposed by Geshkenbein, Ioffe and Larkin 34 and by Chubukov and Tsvelik. 35,36 Both these phenomenological models examined Fermi arcs centered on the nodal directions, coupled in the d-wave channel to Cooperons associated with the antinodal regions. The model studied in Ref. (34) had dispersionless Cooperons which provided BCS-style coupling for the nodal quasiparticles. The result was a superconducting transition with weak fluctuations, similar to our x < x$_{c}$ case. In the model considered in Ref. 35 the Cooperons possessed a one-dimensional dispersion which resulted in strong fluctuations as takes place in our case for x > x$_{c}$ . The authors of Refs. 34–36 con- sidered the fluctuation regime above T$_{c}$ when the Cooperon energy is close to the chemical potential and drew comparisons to experiments in several underdoped cuprates. The key 

