A Possible Phononic Mechanism for $d_{x^2 - y^2}$-Superconductivity in the Presence of Short-Range AF Correlations

We discuss the high temperature superconductors in a regime where the antiferromagnetic (AF) correlation length is only a couple of lattice spacings. In the model proposed here, these short-range AF fluctuations play an essential role in the dressing of the carriers, but the attraction needed for superconductivity (SC) arises from a transverse phonon oxygen mode with a finite buckling angle as it appears in ${\rm YBa_2Cu_3O_{7-\delta}}$. A simple fermion-phonon model analog to the Holstein model is introduced to account for this effect. We argue that the model has a $d_{x^2 - y^2}$-wave superconducting groundstate. The critical temperature ($T_c$) and the O-isotope effect coefficient ($\alpha_O$) vs hole density ($x$) are in qualitative agreement with experiments for the cuprates. The minimum (maximum) of $\alpha_O$ ($T_c$) at optimal doping is caused by a large peak in the density of states of holes dressed by AF fluctuations, as discussed in previous van Hove scenarios.