  Nonlocal effective interactions are inherent to non-relativistic quantum
many-body systems, but their systematic resummation poses a significant
challenge known as the ``vertex problem" in many-body perturbation theory. We
introduce a renormalization scheme based on a projection-based renormalization
condition that selectively resums the most essential nonlocal contributions to
the effective interaction vertex, avoiding the computational complexity of the
full vertex function. This enables us to derive a renormalized Feynman
diagrammatic series with large parameters canceled by counter-diagrams,
efficiently generated using a perturbative expansion of the parquet equations
and computed using a diagrammatic Monte Carlo algorithm. Applying our approach
to a 3D Yukawa Fermi liquid, we demonstrate that the renormalized perturbation
theory remains predictive even in the strongly correlated regime and uncover
significant sign cancellations between different channels contributing to the
scattering amplitude. Our work establishes a novel framework for investigating
strong correlations in quantum many-body systems, offering a systematic
approach to explore nonlocal theories for challenging systems like the electron
liquid in material science.
