Impact of $e-e$ interactions on the superfluid density of dirty superconductors

Landau's theory of the Fermi liquid is adapted to analyze the impact of electron-electron ($e-e$) interactions on the deficit of the superfluid density $\rho_{s0}=\rho_s(T=0)$ in dirty superconducting electron systems in which the damping $\gamma$ of single-particle excitations exceeds the zero temperature BCS gap $\Delta_0$. In the dirty strong-coupling limit $\gamma/\Delta_0\gg 1,m^*/m_e\gg 1$, the formula derived for $\rho_{s0}$ is shown to coincide with the well-known empirical Uemura relation provided pair-breaking contributions are nonexistent. The roles of the crystal lattice and magnetic pair-breaking effects in the observed decline of the zero-temperature superfluid density $\rho_{s0}$ in overdoped LSCO compounds are also discussed. Our method is also applied to elucidation of results from the pioneering experimental studies performed recently by Bozovi\`c and collaborators in overdoped LSCO compounds.