Anomalous BCS equation for a Luttinger superconductor

In the context of the Anderson theory of high Tc cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular Tc predicted by the BCS equation for Luttinger liquids is quite higher than the usual Tc for Fermi liquids. 1 The anomalous BCS equation The equivalent of BCS theory for a Luttinger liquid has not formally worked out, despite the relevance of such theory for the problem of superconductivity in the high-Tc cuprates, see [1] or the discussion in the following section. Such theory should describe superconductors which in their normal state are Luttinger liquids. In this paper we develop such theory using constructive quantum field theory techniques, applied in many other Luttinger liquid problems, see [2], [3]. In particular, we compute in a rigorous way the BCS self-consistence equation for a spinning Luttinger liquid (the Mattis model, see [4]) coupled with a BCS anomalous potential. We stress that it should be difficult to find similar results using different techniques: in fact the Mattis model plus a BCS interaction is not exactly solvable, and bosonization or conformal quantum field theory cannot be used to compute the correlation functions of theories with gap like the present one (see for instance [5]). We will assume that an anomalous self energy has been introduced by some external influence in a Luttinger liquid See [1] pag.209 so that the model is described by the following hamiltonian