Possible phononic mechanism for $d_{x^2-y^2}$ superconductivity in the presence of short-range antiferromagnetic 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 arises from a transverse phonon oxygen mode with a finite buckling angle as it appears in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$. 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 ground state. The critical temperature ($T_c$) and the O-isotope effect coefficient (${\alpha }_{\mathrm{O}}$) versus hole density ($x$) are in qualitative agreement with experiments for the cuprates. The minimum (maximum) of ${\alpha }_{\mathrm{O}}\left(T_c\right)$ 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.