  The Hubbard model is the simplest model that is believed to exhibit
superconductivity arising from purely repulsive interactions, and has been
extensively applied to explore a variety of unconventional superconducting
systems. Here we study the evolution of the leading superconducting
instabilities of the single-orbital Hubbard model on a two-dimensional square
lattice as a function of onsite Coulomb repulsion $U$ and band filling by
calculating the irreducible particle-particle scattering vertex obtained from
dynamical cluster approximation (DCA) calculations, and compare the results to
both perturbative Kohn-Luttinger (KL) theory as well as the widely used random
phase approximation (RPA) spin-fluctuation pairing scheme. Near half-filling we
find remarkable agreement of the hierarchy of the leading pairing states
between these three methods, implying adiabatic continuity between weak- and
strong-coupling pairing solutions of the Hubbard model. The $d_{x^2-y^2}$-wave
instability is robust to increasing $U$ near half-filling as expected. Away
from half filling, the predictions of KL and RPA at small $U$ for transitions
to other pair states agree with DCA at intermediate $U$ as well as recent
diagrammatic Monte Carlo calculations. RPA results fail only in the very dilute
limit, where it yields a $d_{xy}$ ground state instead of a $p$-wave state
established by diagrammatic Monte Carlo and low-order perturbative methods, as
well as our DCA calculations. We discuss the origins of this discrepancy,
highlighting the crucial role of the vertex corrections neglected in the RPA
approach. Overall, comparison of the various methods over the entire phase
diagram strongly suggests a smooth crossover of the superconducting interaction
generated by local Hubbard interactions between weak and strong coupling.

Superconductivity;Strongly Correlated Electrons