Superconducting properties of high-T_c cuprates from power-law liquid

Recent photoemission spectroscopy measurements (arXiv:1509.01611) on cuprate superconductors have inferred that over a wide doping range, the imaginary part of the electron self-energy scales as Σ''~(ω² + π²T²)^α, with α = 1 in the overdoped Fermi-liquid state and α ≤ 0.5 in the optimal to underdoped regime. We show that this power-law self-energy leads to an enhancement of superconductivity at α = ½. By evaluating the BCS gap equation, we find that the minimal coupling needed for pairing instability behaves as |α - ½| near α = ½, where the non-analyticity at α = ½ stems from a logarithmic divergence in the gap equation. In addition, we find that the enhancement of superconductivity at α = ½ is amplified by including the effects of impurity scattering. This qualitatively reproduces the superconducting dome in the cuprates, suggesting that a power-law self-energy could provide a natural framework for studying these systems. Within this framework, we find that a pseudogap suppresses superconductivity by lifting the logarithmic divergence at α = ½.