Renormalization Group Study of the Electron-phonon Interaction in High $T_c$ Cuprates

We apply a numerical renormalization group scheme to study the phonon-mediated retarded interactions in the high $T_c$ cuprates. We find that three sets of phonon-mediated retarded quasiparticle scatterings grow under RG flow. These scatterings share the following common features: 1) the initial and final quasiparticle momenta are in the antinodal regions, and 2) the scattering amplitudes have a $x^2-y^2$ symmetry. All three sets of retarded interactions are driven to strong coupling by the magnetic fluctuations around $(\pi,\pi)$. After growing strong, these retarded interactions can trigger density wave orders with d-wave symmetry. However, due to the d-wave form factor they will leave the nodal quasiparticle unaffected. We conclude that the main effect of electron-phonon coupling in the cuprates is to promote these density wave orders.