  We introduce a new multiscale decomposition of the Fermi propagator based on
its parametric representation. We prove that the corresponding sliced
propagator obeys the same direct space bounds than the previous decomposition
used by the authors. Therefore non perturbative bounds on completely convergent
contributions still hold. In addition the new slicing better preserves momenta,
hence should become an important new technical tool for the rigorous analysis
of condensed matter systems. In particular it should allow to complete the
proof that a three dimensional interacting system of Fermions with spherical
Fermi surface is a Fermi liquid in the sense of Salmhofer's criterion.
