What determines the magnitude of Tc in HTSC?

Neither Eliashberg equations nor the BCS one are predictive to the critical temperature, Tc, of a superconductor. A certain amount or phenomenological equations have been constructed with the aim of predicting the experimental values found for Tc. They preserve the form of the BCS-Tc equation but introduce different combinations of the electron-phonon interaction parameter, A. and the electron-electron repulsion parameter, mu*, that appear in the Eliashberg-Tc equations. The agreement with experiment is, in general, rather poor. But since these are the only instrument that exists to predict Tc, they are widely used. A criterion that "the higher the lambda, the higher file Tc" has emerged from those equations and the value of this parameter became the accepted criterion to discard or accept the electron-phonon interaction as a possible mechanism in HTSC, In this paper we analyze the theoretical foundations of this criterion and the validity of the criterion itself and compare to the results of Eliashberg-Migdal theory to conclude, first, that they contradict each other whenever we are dealing with HTSC and, second, that a low electron-phonon interaction parameter, lambda, is not an argument solid enough to discard the e-ph interaction as a mechanism in HTSC.