  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 2D Hubbard Model. We study the
static susceptibility and its dependence on the pair momentum $\mathbf{q}$ for
a range of temperature, interaction strength, and chemical potential. We show
that $d_{x^2-y^2}$-wave pairing is expected in the model in the $U/t\to 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 $(\pi,\pi)$ that can be
well fit as separate bosonic modes. We extract amplitudes and correlation
length scales where we find a predominantly local $(\pi,\pi)$ pairing and
non-local $\mathbf{q}=(0,0)$ pairs and present the relative weights of these
modes for variation in temperature, doping, and interaction strength.
