Origin of the Coulomb pseudopotential

We address the outstanding problem of electron pairing in presence strong Coulomb repulsion at small to moderate values parameter, ${r}_{s}\ensuremath{\lesssim}2$, and demonstrate that pseudopotential framework is fundamentally biased uncontrolled. Instead, one has break net result into two distinctively different effects: Fermi liquid renormalization factor change effective low-energy coupling. The latter quantity shown behave nonmonotonically with an extremum ${r}_{s}\ensuremath{\approx}0.75$. Within random-phase approximation, interaction starts enhance coupling ${r}_{s}>2$, suppression critical temperature entirely due renormalized properties. Leading vertex corrections this picture only quantitatively. Our results call for radical reconsideration widely accepted repulsive approach show need precise microscopic treatment interactions superconducting instability.