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We present a systematic study of the interaction, doping, and layer dependence of the $d_{x^2-y^2}$-wave pairing susceptibility of the Hubbard model for a stacked 2D square lattice. We perform a multi-index perturbative expansion up to fourth-order to obtain coefficients in powers of the Hubbard $U$, the inter-layer $V$, and the pair-hopping $J$ interactions. We evaluate the vertex diagrams that contribute to the pairing susceptibility for $\ell= 2,3, 4$ layered models in the $U$-$V$-$J$ interaction space. This provides unprecedented access to the pairing amplitudes, allowing us to identify the processes that enhance or reduce pairing. We distinguish pairing within the diagonal channel, $P^{\parallel}_{d}$, and off-diagonal channel, $P^{\perp}_{d}$, and find that, in the absence of $J$, the qualitative behavior of the layered system is equivalent to the single-layer model. In the presence of $J$, we show that pairing is enhanced sublinearly with increasing $\ell$ and is primarily mediated by the $P^{\perp}_{d}$ component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized $\ell$-dependent equation for $ P^{\perp}_{d}$ to speculate pairing beyond $\ell=4$., Comment: 21 pages, 9 figures in the main text, 10 figures in the supplemental material (appended)

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., Comment: 9 pages - 10 figures