Contrasting Dynamic Spin Susceptibility Models and their Relation to High Temperature Superconductivity

We compare the normal-state resistivities $\rho$ and the critical temperatures $T_c$ for superconducting $d_{x^2-y^2}$ pairing due to antiferromagnetic (AF) spin fluctuation exchange in the context of the two phenomenological dynamical spin susceptibility models, recently proposed by Millis, Monien, and Pines (MMP) and Monthoux and Pines (MP) and, respectively, by Radtke, Ullah, Levin, and Norman (RULN), for the cuprate high-$T_c$ materials. Assuming comparable electronic bandwidths and resistiviies in both models, we show that the RULN model gives a much lower d-wave $T_c$ ($\lsim20$K) than the MMP model (with $T_c\sim100$K). We demonstrate that these profound differences in the $T_c$'s arise from fundamental differences in the spectral weight distributions of the two model susceptibilities and are {\it{not}} primarily caused by differences in the calculational techniques employed by MP and RULN. The MMP model, claimed to fit NMR data in YBCO, exhibits substantial amounts of spin fluctuation spectral weight up to an imposed cut-off of 400meV, whereas, in the RULN model, claimed to fit YBCO neutron scattering data, the weight is narrowly peaked and effectively cut-off by 100meV. Further neutron scattering experiments, to explore the spectral weight distribution at all wavevectors over a sufficiently large excitation energy range, will thus be of crucial importance to resolve the question whether AF spin fluctuation exchange provides a viable mechanism to account for high-$T_c$ superconductivity. The large high-frequency boson spectral weight, needed to generate both a high d-wave $T_c$ and a low normal-state resistivity, also implies large values, of order unity, for the Migdal smallness parameter, thus casting serious doubt on the validity of the very