We compute the diagrammatic expansion of the particle-particle susceptibility via algorithmic Matsubara integration and compute the correlated pairing susceptibility in the thermodynamic limit of the two-dimensional Hubbard model. We study the static susceptibility and its dependence on the pair momentum
q for a range of temperature, interaction strength, and chemical potential. We show that ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing is expected in the model in the $U/t\ensuremath{\rightarrow}{0}^{+}$ limit from direct perturbation theory. From this, we identify key second- and third-order diagrams that support pairing processes and note that the diagrams responsible are not a part of charge or spin susceptibility expansions. We find two key components for pairing at momenta (0,0) and $(\ensuremath{\pi},\ensuremath{\pi})$ that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local $(\ensuremath{\pi},\ensuremath{\pi})$ pairing and nonlocal
q=(0,0) pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.