Cooper channel and the singularities in the thermodynamics of a Fermi liquid

We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient gamma(T)-gamma(0)=A(T)T in a two-dimensional Fermi liquid. We show that A(T) is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.