Momentum dependence of the electron-phonon coupling, phonon-induced pairing interaction, and self-energy effects in YBa$_2$Cu$_3$O$_7$ within the local density approximation

Using the local density approximation (LDA) and a realistic phonon spectrum we calculate the momentum and frequency dependence of the electron-phonon coupling in YBa$_2$Cu$_3$O$_7$ and determine its consequences for the phonon-induced pairing interaction and for the electronic self-energy in the normal state. The phonon-induced interaction has a pronounced peak for large momentum transfers and the interband contributions between bonding and antibonding band are of the same magnitude as the intraband ones. The dimensionless coupling constant in the d-wave channel $\lambda^d$, relevant for superconductivity, is only 0.022, i.e., even about ten times smaller than the small value of the s-wave channel. For electronic states at the Fermi energy, the maximum in the real part of the phonon-induced self-energy at low frequencies is about a factor 5 too small compared to the experiment, resulting in a very small and smooth change in the slope of the electronic dispersion [1]. These findings suggest that phonons are not the important low-energy excitations, and cannot produce well-pronounced kinks in YBa$_2$Cu$_3$O$_7$, at least, within LDA. [1] Heid, Bohnen, Zeyher,Manske, PRL {\bf 100}, 137001 (2008).