Phonons andd-wave pairing in the two-dimensional Hubbard model

We analyze the influence of phonons on the $d_{x^2-y^2}$-pairing instability in the Hubbard model on the two-dimensional square lattice at weak to moderate interaction $U$, using a functional renormalization group scheme with frequency-dependent interaction vertices. As measured by the pairing scale, the $B_{1g}$ buckling mode enhances the pairing, while other phonon modes decrease the pairing. When various phonon modes are included together, the net effect on the scale is small. However, in situations where $d$-wave superconductivity and other tendencies, e.g., antiferromagnetism, are closely competing, the combined effect of different phonons may be able to tip the balance towards pairing.