Absence of an isotope effect in the magnetic resonance in high-$T_c$ superconductors

An inelastic neutron scattering experiment has been performed in the high-temperature superconductor $rm YBa_2Cu_3O_{6.89}$ to search for an oxygen-isotope shift of the well-known magnetic resonance mode at 41 meV. Contrary to a recent prediction (I. Eremin, {it et al.}, Phys. Rev. B {bf 69}, 094517 (2004)), a negligible shift (at best $leq$ +0.2 meV) of the resonance energy is observed upon oxygen isotope substitution ($^{16}$O$to^{18}$O). This suggests a negligible spin-phonon interaction in the high-$T_c$ cuprates at optimal doping.