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Extensively reported experimental observations indicate that on varyingpressure ($p$) within the $\textit{T-p}$ phase diagram of most quantum criticalheavy fermion (HF) superconductors, one identifies a cascade of distinctelectronic states which may be magnetic, of Kondo-type, non-conventionalsuperconducting, Fermi Liquid (FL), or non-FL character. Of particular interestto this work is the part of the phase diagram lying below a specific phaseboundary, $T^{*}_\text{{FL}}(p^{*})$, across which the transport andthermodynamic properties switch over from non-FL into FL behavior. Remarkably,this nontrivial manifestation of FL phase is accompanied by (i) thecharacteristic $\rho_o + \textit{AT}^2$ dependence ($\rho_o$ = residualresistivity), (ii) a superconductivity below $T_{c} \leqT^{*}_\text{{FL}}(p^{*})$, and (iii) a universal scaling of $T_c$ and $A$:$ln\frac{T_{c}}{\theta}\propto A^{-\frac{1}{2}}$ ($\theta$ = characteristicenergy scale). We consider that such features are driven by afluctuation-mediated electron-electron scattering channel with the mediatingquasiparticles being either spin fluctuations [Mathur et al., Nature 394,39(1998)] or valence fluctuations [Miyake and Watanabe, Phil. Mag. 97, 3496(2017)] depending on the character of the neighboring instability. On adoptingsuch a scattering channel and applying standard theories of Migdal-Eliashberg(superconductivity) and Boltzmann (transport), we derive analytic expressionsthat satisfactorily reproduce the aforementioned empirical correlations inthese heavy fermion superconductors.

Correlation between normal and superconducting states within the  Fermi-liquid region of the $\textit{T-p}$ phase diagram of quantum-critical  heavy-Fermion superconductors