  A non-perturbative approach to the single-band attractive Hubbard model is
presented in the general context of functional derivative approaches to
many-body theories. As in previous work on the repulsive model, the first step
is based on a local-field type ansatz, on enforcement of the Pauli principle
and a number of crucial sum-rules. The Mermin-Wagner theorem in two dimensions
is automatically satisfied. At this level, two-particle self-consistency has
been achieved. In the second step of the approximation, an improved expression
for the self-energy is obtained by using the results of the first step in an
exact expression for the self-energy where the high- and low-frequency
behaviors appear separately. The result is a cooperon-like formula. The
required vertex corrections are included in this self-energy expression, as
required by the absence of a Migdal theorem for this problem. Other approaches
to the attractive Hubbard model are critically compared. Physical consequences
of the present approach and agreement with Monte Carlo simulations are
demonstrated in the accompanying paper (following this one).
