  The problem of weakly correlated electrons on a square lattice is formulated
in terms of one-loop renormalization group. Starting from the action for the
entire Brillouin zone (and not with a low-energy effective action) we reduce
successively the cutoff $\Lambda$ about the Fermi surface and follow the
renormalization of the coupling $U$ as a function of three energy-momenta. We
calculate the intrinsic scale $T_{co}$ where the renormalization group flow
crosses over from the regime ($\Lambda > T_{co}$) where the electron-electron
(e-e) and electron-hole (e-h) terms are equally important to the regime
($\Lambda < T_{co}$) where only the e-e term plays a role. In the low energy
regime only the pairing interaction $V$ is marginally relevant, containing
contributions from all renormalization group steps of the regime $\Lambda >
T_{co}$. After diagonalization of $V_{\Lambda =T_{co}}$, we identify its most
attractive eigenvalue $\lambda _{\min}$. At low filling, $\lambda _{\min}$
corresponds to the $B_2$ representation ($d_{xy}$ symmetry), while near half
filling the strongest attraction occurs in the $B_1$ representation
($d_{x^2-y^2}$ symmetry). In the direction of the van Hove singularities, the
order parameter shows peaks with increasing strength as one approaches half
filling. Using the form of pairing and the structure of the renormalization
group equations in the low energy regime, we give our interpretation of ARPES
experiments trying to determine the symmetry of the order parameter in the
Bi2212 high-$T_{c}$ compound.
